Quinta Essentia - Part 3

QUINTA ESSENTIA A Practical Guide to Space-Time Engineering

PART 3 “METRIC ENGINEERING & THE QUASI-UNIFICATION OF PARTICLE PHYSICS” “To Oliya” RESEARCH NOTES Key Words: Balmer Series, Bohr Radius, Buckingham Π Theory, Casimir Force, ElectroMagnetics, Equivalence Principle, Euler’s Constant, Fourier Series, Fundamental Particles, General Relativity, Gravity, Harmonics, Hydrogen Spectrum, Newtonian Mechanics, Particle Physics, Physical Modelling, Planck Scale, Polarisable Vacuum, Quantum Mechanics, Zero-Point-Field.

2nd Edition Project Initiated: July 1, 1996 Project Completed: October 12, 2005 Revised: Thursday, 24 November 2011 RICCARDO C. STORTI1

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[email protected] © Copyright 2011: Delta Group Engineering (dgE): All rights reserved.

SCIENTIFIC ACHIEVEMENTS The physical characteristics derived herein (from 1st principles), based upon a single paradigm [i.e. the application of Buckingham “Π” Theory (BPT) and Dimensional Analysis Techniques (DAT’s)] may be articulated as follows (many of which are experimentally verified or implied), 1. The spectral quantisation of gravity. 2. The application of the spectral quantisation of gravity to Metric Engineering principles. 3. The experimentally implicit validation of the Polarisable Vacuum (PV) model of gravity. 4. The formulation of the Electro-Gravi-Magnetic (EGM) Spectrum. 5. The experimentally implicit validation of the EGM Spectrum by the calculation of highly precise physically verified fundamental particle properties. 6. The Quasi-Unification of particle physics illustrating that all fundamental particles may be described as harmonic multiples of each other. 7. The Zero-Point-Field (ZPF) equilibrium radius. 8. The experimental Root Mean Square (RMS) charge radius of the Proton. 9. The classical RMS charge radius of the Proton. 10. The experimental Proton Electric Radius. 11. The experimental Proton Magnetic Radius. 12. The experimental Mean Square (MS) charge radius of the Neutron. 13. The conversion of the conventional representation of the experimental Neutron “MS” charge radius to a more intuitively meaningful positive form. 14. The experimental Neutron Magnetic Radius. 15. The precise experimental graphical properties of the Neutron charge distribution. 16. The experimental mass-energies and radii of all Quarks and Bosons consistent with the Particle Data Group (PDG) and ZEUS Collaboration (ZC). 17. The charge radii of all Neutrino’s, consistent with the interpretation of experimental data from the Sudbury Neutrino Observatory (SNO). 18. The experimental mass-energy of the Top Quark as defined by the D-Zero Collaboration (D0C) based upon the observation of Top events. 19. The Photon mass-energy threshold consistent with PDG interpretation of experimental evidence. 20. The Photon and Graviton mass-energies and radii consistent with Quantum Mechanical (QM) expectations. 21. The derivation of the Fine Structure Constant “α” in terms of Electron and Proton radii. 22. The derivation of “α” in terms of Neutron, Muon and Tau radii. 23. The derivation of the Casimir Force based upon the spectral quantisation of gravity. 24. The optimisation of an energy / gravitational experiment associated with the Casimir Force. 25. An experimentally implicit definition of the Planck Scale. 26. An experimentally implicit definition of the Bohr Radius. 27. The experimental Hydrogen atom emission / absorption spectrum (Balmer Series). 28. The prediction of three new Leptons and associated Neutrino's. 29. The prediction of two new Intermediate Vector Bosons (IVB’s). 30. A physically implicit value and limit for “π” at the “QM” level – subject to uncertainty principles. 31. A physically implicit value and limit for the Euler-Mascheroni Constant “γ” at the “QM” level – subject to uncertainty principles. 32. The formulation of a single mathematical algorithm incorporating (1 - 31). Note: where possible, calculated results have been compared to physical measurement. Cognisant of experimental uncertainty, many predictions herein may be considered to be exact.

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RECENT DEVELOPMENTS Particle physics is a rapidly expanding and highly dynamic sphere of knowledge supporting a landscape of constantly changing hues. Experimental boundaries are being shifted with exciting reductions in uncertainty at a staggering pace. This text develops the Electro-Gravi-Magnetic (EGM) construct to define relationships between the distributions of mass-energy over space-time of fundamental particles. The EGM construct was finalized in 2004 and tested against published PDG data of the day (i.e. 2005 values). Particle Data Group (PDG) Mass-Energy Ranges (2006) Annually, the PDG reconciles its published values of particle properties against the latest experimental and theoretical evidence. The 2006 changes in PDG mass-energy range values not impacting EGM are as follows: 1. Strange Quark = “70 < msq(MeV) < 120”. 2. Charm Quark = “1.16 < mcq(GeV) < 1.34”. 3. “W” Boson = “80.374 < mW(GeV) < 80.432”. 4. “Z” Boson = “91.1855 < mZ(GeV) < 91.1897”. Electron Neutrino and Up / Down / Bottom Quark Mass The EGM construct relates “mass to size” in harmonic terms. However, contemporary Physics is currently incapable of specifying the mass and size of most fundamental particles precisely and concurrently. Subsequently, EGM is required to approximate values of either mass or radius to predict one or the other (i.e. mass or size). Hence, the EGM predictions articulated in Particle Summary Matrix (3.2, 3.4) denote values based upon estimates of either mass or radius. Consequently, some of these results are approximations and subject to revision as new experimental evidence regarding particle properties (particularly mass), come to light. The 2006 changes in PDG mass-energy values affecting these results are shown below. In this data set, the EGM radii are displayed as a range relating to its mass-energy influence. Note: the average value of EGM “Up + Down Quark” mass from these tables [i.e. 5.2574(MeV)] remains within the 2006 average mass range specified by the PDG [i.e. 2.5 to 5.5(MeV)]. Particle Electron Neutrino (ν ν e) Up Quark (uq) Down Quark (dq) Bottom Quark (bq)

EGM Radii x10-16(cm)

EGM Mass-Energy (utilized)

ren < 0.0811 0.5469 < ruq < 0.7217 0.7217 < rdq < 1.0128 1.0719 > rbq > 1.0863

PDG Mass-Energy Range (2006 Values)

PDG Mass-Energy Range (2006 Values) men(eV) < 2 1.5 < muq(MeV) < 3 3 < mdq(MeV) < 7 4.13 < mbq(GeV) < 4.27

The predicted radii ranges above demonstrate that no significant deviation from 2005 EGM values exist. This emphasizes that the EGM harmonic representation of fundamental particles is a robust formulation and is insensitive to minor fluctuations in particle mass, particularly in the absence of experimentally determined Root Mean Square (RMS) charge radii. Therefore, we may conclude that the EGM construct continues to predict experimentally verified results within the SM to high computational precision.

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Top Quark Mass Dilemma The Collider Detector at Fermilab (CDF) and “D-ZERO” (D0) Collaborations have recently revised their world average value of “Top Quark” mass (mtq) from “178.0(GeV/c2)” to “172.0(GeV/c2)” in 2005, [80, 81] to “172.5” in early 2006, then to “171.4” in mid 2006. [85] Note: since the precise value of “mtq” is subject to frequent revision, we shall utilize the late 2005 value in the resolution of the dilemma as it sits between the 2006 values, resulting in an EGM construct error of approximately “< 3.640(%)” rather than the “< 0.280(%)” displayed in “Particle Summary Matrix 3.1”. Resolution The EGM method utilizes fundamental particle RMS charge radius to determine mass. Currently, Quark radii are not precisely known and approximations were applied in the formulation of “mtq” displayed herein. However, if one utilizes the revised experimental value of “mtq = 172.0(GeV/c2)” to calculate the RMS charge radius of the Top Quark “rtq”, based upon Proton harmonics, it is immediately evident that a decrease in “rtq” of “< 1.508(%)” produces the new world average value precisely. The revised radius calculation may be performed simply (the denominator of the proceeding equation), producing a result of “0.9156x10-16(cm)”. The decrease in “Top Quark” RMS charge radius (relative to its approximated value in chapter 3.12) based upon the “new world average Top Quark” mass may be determined as follows, r tq

1 = 1.5076 ( % )

5

GeV 172. 2 c 1 . r π. 9 mp 140

2

where, (i) “rπ”, “mp” and “rtq” denote the RMS charge radius and mass of the Proton and the initial approximation of the RMS charge radius of the “Top Quark” respectively (see chapter 3.12). (ii) rπ = 830.5957 x10-16(cm), mp = 1.67262171 x10-27(kg) and rtq = 0.9294 x10-16(cm). Note: the mid 2006 value for revised “mtq” modifies the error defined above to “< 1.65(%)”. Therefore, since the change in “rtq” is so small and its experimental value is not precisely known, we may conclude the EGM construct continues to predict experimentally verified results within the SM to high computational precision. The revised “Top Quark Mass” presented above is not definitive. Other experimental efforts have produced slightly different results in favour of the EGM construct. However, this text utilises the presented measurement as a “quasi-certain” boundary limit. Subsequently, the reader is encouraged to review the latest experimental results utilising the Cornell University Library in [86]. The following “keywords” produce a robust suite of experimentally based scientific papers for the review of “recent developments”: i. ALEPH, ALICE, ANTRES, ATHENA, ATLAS, BABAR, BELLE, BES, CCFR, CDF, CDF II, CKM, CLAS, CLEO, CMD2, CMS, COMPASS, D0 (D-ZERO), DELPHI, DISTO, E143, E787, E949, FOCUS, G-2, H1, HERA-B, HERMES, KLOE, KTev, L3, LEP, NA48, NA50, NA52, NEMO, New Muon, NOMAD, NuTev, OPAL, PHENIX, SELEX, SLD, SNO, STAR, Tevatron, TOTEM, TWIST, UA8, ZEUS. ii. Collaboration, Electroweak, Flavour, Working Group. 4

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ACKNOWLEDGEMENTS Generally speaking, it is difficult for readers of literature to fully appreciate the effort and commitment involved in producing a book of any description or function if the reader has not travelled the publication path. It is a milestone which involves the support of many people, the sum of professional life experience and a society by which to gain these attributes. Writing any scientific or engineering text has its unique set of difficulties based upon the fundamental need for all material contained within it to be factually correct, not simply personal opinion. In the case of novels, one may extensively draw upon one’s own perceptions and views without the level of factual scrutiny associated with the scientific method. In my specific case, this document and the scientific material contained herein could not have been possible without the support network of many people, both directly and indirectly. For instance, an author of scientifically based literature requires at least some degree of formal education. One cannot simply commence writing scientific based literature without knowing and understanding the facts to be presented. Firstly, I must acknowledge the enormous and deciding financial scarifies made by my parents (Alberto and Nives) in providing the academic foundation from which this text is derived. The years of arduous labour involved in precipitating my skills into this text, would not have been possible without their help and support. I would not have acquired the tools necessary for completion of this personal milestone without them. Secondly, the encouragement provided by my sister (Mary) cannot be overstated. Without her boundless optimism, I would not have had the stamina to complete this journey. As I mentioned previously, it is difficult for the non-author to fully appreciate the focused psychology required to complete such a protracted work of completely original content as this text. Finally, I would like to thank the following list of colleagues, friends and organisations: Colleagues A. Prof. P. Jarvis

University of Tasmania For providing positive feedback on this body of work.

Dr. V. Karmanov (Lead Researcher)

Lebedev Physical Institute For persistence in understanding, an open mind, great encouragement, professional guidance, providing challenges and provoking the formulation of appendices 3.G and 3.I.

E. Prof. R. Kiehn

University of Houston For providing great encouragement, recognising the scientific potential of Electro-Gravi-Magnetics (EGM) and whose thoughts I value deeply.

Prof. G. Modanese

University of Bolzano For recognising the scientific potential of the EGM construct.

Dr. H. Puthoff

EarthTech International, Inc. For providing much of the inspiration for the EGM construct.

Prof. C. Rangacharyulu

University of Saskatchewan For providing great encouragement, recognising the scientific potential of EGM and whose thoughts I value deeply.

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Publications Chapters herein have been re-printed with the permission of: Physics Essays Publication: 2012 Woodglen Cres., Ottawa Ontario K1J 6G4, Canada Authors:

Riccardo C. Storti Todd J. Desiato

The International Society for Optical Engineering (SPIE) and are taken from the symposia proceedings of the 50th SPIE conference: The Nature of Light: What is a Photon? Proceedings Volume 5866 (pg. 207 – 217) Authors:

Riccardo C. Storti Todd J. Desiato

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ARTICLE

3.1

OVERVIEW

Albert Einstein: 1879 – 1955

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DOCUMENT PURPOSE AND OBJECTIVES Overall To derive solutions (from a single paradigm) for the design and construction of localised gravity modification experiments for application to “off-the-shelf” ElectroMagnetic (EM) simulation packages. The solutions presented are based upon well established engineering principles and tested against the derivation of fundamental particle property characteristics for validity. Articles Article 3.1:

To present an introduction and summary of all the material contained herein.

Article 3.2:

To derive engineering principles for localised gravity modification experiments.

Article 3.3:

To verify the engineering principles derived by application to fundamental questions in Physics, facilitating the derivation of experimentally verified results from a single paradigm.

Appendices App. 3.A:

To present a summary of key equations derived herein.

App. 3.B:

To present the formulations, derivations, characteristics and proofs utilised in the preceding articles.

App. 3.C:

To present a set of analytical simplifications utilised in the preceding articles.

App. 3.D:

To present a computational sub-routine for the numerical derivation of Lepton radii, complimenting derivations described in preceding articles.

App. 3.E:

To present a computational sub-routine for the numerical derivation of Quark and Boson mass-energies and radii, complimenting derivations described in preceding articles.

App. 3.F:

To visualise the harmonic principles derived and applied to the preceding articles.

App. 3.G:

To present the derivation of certain ElectroMagnetic characteristics of the Neutron and Proton based upon the outputs of the preceding articles.

App. 3.H:

To present an explanation for the “missing” Neutrino’s associated with the Standard Model in particle Physics and the lack of detection of the appropriate number of Solar Neutrino’s.

App. 3.I:

To present a derivation of the emission / absorption spectrum of the Hydrogen atom based upon the methods developed in the preceding articles.

App. 3.K:

To present a complete computational algorithm / simulation capable of generating all the results and claims contained within this document. This also confirms the lack of numerical errors presented in the preceding articles.

App. 3.L:

To present a simplified calculation engine / algorithm based upon the preceding appendix to reduce computational error and improve accuracy. This also confirms the lack of numerical errors presented in the preceding articles.

App. 3.M:

To present a simplified calculation engine / algorithm based upon the preceding appendix to minimise computational error and improve accuracy. This is achieved by the utilisation of a more advance computational / simulation environment and also confirms the lack of numerical errors presented in the preceding articles. 8

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TABLE OF CONTENTS 1

SUMMARY OF CONTENTS

Scientific Achievements Recent Developments

2 3

Article 3.1:

7 8 10 20 21 30 39 40

Chapter 3.0:

Overview Document Purpose and Objectives Errata Errata Glossary of Terms (by chapter) Definition of Terms Spiral Galaxy (Photograph - NASA) History of the Universe (CERN) Introduction 1 General 2 EGM Construct Process Summary 3 Particle Summary Matrices

43 65 68

Article 3.2: Derivation of Engineering Principles Fundamental Engineering – The Pyramids at Giza (Photograph) Chapter 3.1: Dimensional Analysis Chapter 3.2: General Modelling and the Critical Factor Chapter 3.3: The Engineered Metric Chapter 3.4: Amplitude and Frequency Spectra Chapter 3.5: General Similarity Chapter 3.6: Harmonic and Spectral Similarity Chapter 3.7: The Casimir Effect

83 84 85 97 107 115 125 145 159

Article 3.3: Application of Derived Engineering Principles Advanced Engineering – Mankind on the Moon (Photograph) Chapter 3.8: Derivation of the Photon Mass-Energy Threshold Chapter 3.9: Derivation of Fundamental Particle Radii (Electron, Proton and Neutron) Chapter 3.10: Derivation of the Photon and Graviton Mass-Energies and Radii Chapter 3.11: Derivation of Lepton Radii Chapter 3.12: Derivation of Quark and Boson Mass-Energies and Radii Chapter 3.13: The Planck Scale, Photons, Predicting New Particles and Designing an Experiment to Test the Negative Energy Conjecture

167 168 169 175 183 189 195

Appendices App. 3.A: App. 3.B: App. 3.C: App. 3.D: App. 3.E: App. 3.F: App. 3.G:

Key Artefacts Formulations, Derivations, Characteristics and Proofs Simplifications Derivation of Lepton Radii Derivation of Quark and Boson Mass-Energies and Radii Harmonic Representations 1 Conversion of the Neutron Positive Core Radius 2 Derivation of the Neutron Magnetic Radius 3 Derivation of the Proton Electric Radius 4 Derivation of the Proton Magnetic Radius 5 Derivation of the Classical Proton RMS Charge Radius 9

205

219 227 243 245 247 251 255 261 262 262 262

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App. 3.H: App. 3.I: App. 3.J:

Calculation of L2, L3 and L5 Associated Neutrino Radii Derivation of the Hydrogen Atom Spectrum (Balmer Series) and an Experimentally Implicit Definition of the Bohr Radius Glossary of Terms (alphabetical order)

263 265 269

Bibliography 3

276

Numerical EGM Simulations App. 3.K: MathCad 8 Professional (Complete Simulation) App. 3.L: MathCad 8 Professional (Calculation Engine) App. 3.M: MathCad 12 (High Precision Calculation Results)

279 281 365 387

Experimentally Verified or Implied Algorithms, Calculations and Derivations 1 Derivation of Photon and Graviton Mass-Energies and Radii (chapter 3.10) 2 Mathematical Algorithm for the Calculation of All Lepton Radii (App. 3.D) 3 Mathematical Algorithm for the Calculation of All Quark and Boson Mass-Energies and Radii (App. 3.E) 4 Derivation of the Proton Electric and Magnetic Radii and the Neutron Magnetic Radius (App. 3.G) 5 Derivation of the Hydrogen Atom Spectrum (Balmer Series) (App. 3.I) 6 MathCad 8 Professional (Complete Simulation) (App. 3.K) 7 MathCad 8 Professional (Calculation Engine) (App. 3.L) 8 MathCad 12 (High Precision Calculation Results) (App. 3.M)

255 265 281 365 387

Index

395

Periodic Table of the Elements

402

Notes

183 245 247

42, 64, 82, 96, 105, 106, 124, 143, 144, 157, 158, 173, 174, 188, 194, 203, 204, 242, 254, 264, 268, 280, 282, 364, 366, 388, 393, 394, 405-407 ERRATA

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2

DETAIL OF CONTENTS

Scientific Achievements Recent Developments

2 3

Article 3.1:

7 8 10 20 21

Overview Document Purpose and Objectives Errata Errata Glossary of Terms (by chapter) Definition of Terms Alpha Forms Amplitude Spectrum Background Field Bandwidth Ratio Beta Forms Buckingham Π Theory Casimir Force Change in the Number of Modes Compton Frequency Cosmological Constant Critical Boundary Critical Factor Critical Field Strengths Critical Frequency Critical Harmonic Operator Critical Mode Critical Phase Variance Critical Ratio Curl DC-Offsets Dimensional Analysis Techniques Divergence Dominant Bandwidth EGM EGM Spectrum Energy Density Engineered Metric Engineered Refractive Index Engineered Relationship Function Experimental Prototype Experimental Relationship Function Fourier Spectrum Frequency Bandwidth Frequency Spectrum Fundamental Beat Frequency Fundamental Harmonic Frequency General Modelling Equations General Relativity General Similarity Equations Gravitons Graviton Mass-Energy Threshold 11

30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 33 33 33 33 33 33 33 www.deltagroupengineering.com

Group Velocity Harmonic Cut-Off Frequency Harmonic Cut-Off Function Harmonic Cut-Off Mode Harmonic Inflection Mode Harmonic Inflection Frequency Harmonic Inflection Wavelength Harmonic Similarity Equations IFF Impedance Function Kinetic Spectrum Mode Bandwidth Mode Number Number of Permissible Modes Phenomena of Beats Photon Mass-Energy Threshold Polarisable Vacuum Polarisable Vacuum Beat Bandwidth Polarisable Vacuum Spectrum Potential Spectrum Poynting Vector Precipitations Primary Precipitant Radii Calculations by EGM Range Factor Reduced Average Harmonic Similarity Equations Reduced Harmonic Similarity Equations Refractive Index Representation Error RMS Charge Radii (General) RMS Charge Radius of the Neutron Similarity Bandwidth Spectral Energy Density Spectral Similarity Equations Subordinate Bandwidth Unit Amplitude Spectrum ZPE ZPF ZPF Spectrum ZPF Beat Bandwidth ZPF Beat Cut-Off Frequency ZPF Beat Cut-Off Mode 1st Sense Check 2nd Reduction of the Harmonic Similarity Equations 2nd Sense Check 3rd Sense Check 4th Sense Check 5th Sense Check 6th Sense Check Physical Constants Mathematical Constants and Symbols Solar System Statistics 12

33 33 33 33 34 34 34 34 34 34 34 34 34 34 34 34 35 35 35 35 35 35 35 35 35 36 36 36 36 36 36 36 36 36 36 36 37 37 37 37 37 37 37 37 37 37 37 38 38 38 38 38

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Chapter 3.0:

Spiral Galaxy (Photograph - NASA) 39 History of the Universe (CERN) 40 Introduction and Document Statistics 1 General 1.1 Introduction Laying the Foundations 43 The EGM Approach 47 EGM Achievements 49 EGM Formulation Tips 52 Tips for Applying EGM to Particle Physics 53 Accidental Particle Property Predictions by EGM 53 1.1.1 Current Problems 1.1.1.1 Physics 54 1.1.1.2 Mathematics 55 1.1.2 How EGM Works 55 1.2 Key Results and Findings PV and ZPF 61 Photons, Gravitons and Euler's Constant 61 All Other Particles 62 The Casimir Force 63 The Experimentally Implicit Planck Scale 63 The Prediction of New Particles 63 The Experimentally Implicit Bohr Radius 64 1.3 Building an Experiment 64 2 EGM Construct Process Summary Modelling Foundations 65 The Casimir Force 65 Mass-Energy and Radii of Photons and Gravitons 65 Mass-Energy and Radii of all other Standard Model Part. 66 The Planck Scale 66 Theoretical Particles Beyond the Standard Model 67 Designing and Constructing an Experiment 67 The Bohr Radius 67 3 Particle Summary Matrices 3.1 Detailed Summary Matrices 68 3.2 Concise Matrix EGM Harmonic Representation of Particles 79 Refined EGM Ch. Radii and Mass-Energies of Part. 79 Periodic Table of Elementary Particles 80

Article 3.2: Derivation of Engineering Principles Fundamental Engineering – The Pyramids at Giza (Photograph) Chapter 3.1: Dimensional Analysis Abstract Process Flow 3.1 1 Introduction 2 Theoretical Modelling 3 Mathematical Modelling 3.1 Formulation of Π Groupings 3.2 Technical Verification of Π Groupings 4 Domain Specification 4.1 General Characteristics 13

83 84 85 86 87 88 88 89 90

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4.2

Chapter 3.2:

Chapter 3.3:

Chapter 3.4:

Precipitations of the General Form 4.2.1 Frequency Domain Precipitation 4.2.2 Displacement Domain Precipitation 4.2.3 Wavefunction Precipitation 5 Experimental Relationship Functions 6 The Polarisable Vacuum Model 6.1 Refractive Index 6.2 Superposition 6.3 Constant Acceleration 6.4 Complex Fourier Series 7 Conclusions General Modelling and the Critical Factor Abstract Process Flow 3.2 1 Introduction 1.1 Hypothesis to Be Tested 1.2 What Is Derived? 2 Theoretical Modelling 2.1 Primary Precipitant 2.2 Interpretations of the Primary Precipitant 3 Mathematical Modelling 3.1 Separation of Primary Forms 3.2 General Modelling Equations 3.3 Critical Factor 4 Physical Modelling 4.1 Poynting Vector 4.2 Poisson and Lagrange 5 Conclusions The Engineered Metric Abstract Process Flow 3.3 1 Introduction 1.1 Description 1.2 Critical Ratio 1.3 Metric Engineering 2 Theoretical Modelling 2.1 Mathematical Similarity 2.2 Critical Factor 2.3 Critical Ratio 3 Mathematical Modelling 3.1 Engineering the Relationship Functions 3.2 Engineering the Refractive Index 4 Physical Modelling 5 Metric Engineering 5.1 Polarisable Vacuum 5.2 Engineered Metrics 6 Engineered Metric Effects 7 Conclusions Amplitude and Frequency Spectra Abstract Process Flow 3.4 1 Introduction 14

90 90 91 91 92 92 93 94 95 97 98 99 99 100 102 102 103 103 104 104 105 107 108 109 109 109 109 111 111 112 112 113 113 114 114 114 115 116

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Chapter 3.5:

1.1 General 1.2 Harmonics 1.3 Experimentation 2 Theoretical Modelling 2.1 Time Domain 2.2 Displacement Domain 3 Mathematical Modelling 3.1 Constant Acceleration 3.2 Frequency Spectrum 3.3 Energy Density 3.4 Spectral Characteristics 3.4.1 Cut-Off Mode and Frequency 3.4.2 Zero-Point-Field 4 Physical Modelling 4.1 Polarisable Vacuum 4.2 Test Volumes 4.3 Test Object 5 Sample Calculations 5.1 Background Gravitational Field 5.1.1 Fundamental Frequency 5.1.2 Frequency Bandwidth 5.2 Applied Experimental Fields 5.2.1 Mode Bandwidth 5.2.2 Engineering Considerations 6 Conclusions General Similarity Abstract Process Flow 3.5 1 Introduction 1.1 General 1.2 Harmonics 2 Theoretical Modelling 3 Mathematical Modelling 3.1 Introduction 3.2 Phenomena of Beats 3.2.1 Frequency 3.2.2 Wavelength 3.2.3 Group 3.2.3.1 Velocity 3.2.3.2 Error 3.2.4 Beat Bandwidth Characteristics 3.2.4.1 Frequency 3.2.4.2 Modes 3.2.4.3 Critical Ratio 3.3 Critical Boundary 3.3.1 Frequency 3.3.2 Mode 3.4 Bandwidth Ratio 4 Physical Modelling 4.1 General Similarity Equations 4.1.1 Overview 4.1.2 GSEx 15

117 117 117 117 118 118 118 119 120 121 121 121 122

122 122 122 123 123 125 126 127 127 127 127 128 129 129 129 130 130 130 131 131 131

132 132

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Chapter 3.6:

Chapter 3.7:

4.2 Qualitative Limits 5 Metric Engineering 5.1 Polarisable Vacuum 5.2 Design Considerations 5.2.1 Range Factor 5.2.2 Sense Checks and Rules of Thumb 6 Engineering Characteristics 6.1 Beat Spectrum 6.2 Considerations 6.3 EGM Wave Propagation 6.4 Dominant and Subordinate Bandwidths 6.5 Kinetic and Potential 7 Conclusions 7.1 Conceptual 7.2 Physical Modelling Characteristics Harmonic and Spectral Similarity Abstract Process Flow 3.6 1 Introduction 1.1 General 1.2 Practical Methods 1.3 Objectives 1.4 Results 2 Theoretical Modelling 3 Mathematical Modelling 3.1 Design Matrix 3.2 Engineering Considerations 4 Physical Modelling 4.1 Harmonic Similarity Equations 4.2 Visualisation of HSEx Operands 4.3 Reduction of HSEx 4.4 Visualisation of HSEx R 4.5 Spectral Similarity Equations 4.6 Critical Phase Variance 4.7 Critical field Strength 4.8 DC-Offsets 5 Maxwell’s Equations 5.1 General 5.2 Critical Frequency 6 Conclusions The Casimir Effect Abstract Process Flow 3.7 1 Introduction 2 Theoretical Modelling 3 Mathematical Modelling 4 Physical Modelling 4.1 The Casimir Force 4.2 Cosmological Constant 4.3 Refinement of Classical Casimir Equation 5 Conclusions

16

134 135 136 136 138 138 139 139 140 140 141 145 146 147 147 147 147 148 149 150 150 151 153 153 155 155 156 156 156 157 157 159 160 161 161 163 164 165 166 166

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Article 3.3: Application of Derived Engineering Principles Advanced Engineering – Mankind on the Moon (Photograph) Chapter 3.8: Derivation of the Photon Mass-Energy Threshold Abstract Process Flow 3.8 1 Introduction 2 Mathematical Modelling 3 Physical Modelling 4 Conclusions Chapter 3.9: Derivation of Fundamental Particle Radii (Electron, Proton and Neutron) Abstract Process Flow 3.9 1 Introduction 2 Theoretical Modelling 2.1 Sense Checks and Rules of Thumb 2.2 The Proton 3 Mathematical Modelling 3.1 Derivation of Proton and Neutron Radii 3.1.1 Numerical 3.1.2 Analytical 3.2 Derivation of Electron Radius 3.2.1 Numerical 3.2.2 Analytical 3.3 Derivation of the Fine Structure Constant 3.4 Electron Cut-Off Frequency 3.5 Refinement of Electron Radius 3.6 Derivation of Electron Scattering Mass 3.7 Harmonic Cut-Off Frequencies 4 Physical Modelling 4.1 Electron 4.2 Proton 4.3 Neutron 5 Experimentation 6 Conclusions Chapter 3.10: Derivation of the Photon and Graviton Mass-Energies and Radii Abstract Process Flow 3.10 1 Introduction 2 Theoretical Modelling 3 Mathematical Modelling 4 Physical Modelling 5 Conclusions Chapter 3.11: Derivation of Lepton Radii Abstract Process Flow 3.11 1 Introduction 2 Theoretical Modelling 3 Mathematical Modelling 3.1 Electron Radius 3.2 Muon - Tau Radii and the Fine Structure Constant 3.3 Neutrino Radii 4 Physical Modelling 17

167 168 169 170 171 171 172 173 175 176 177 177 177

178 178 179 180 180 180 180 180 181 181 181 182 182 182 183 184 185 185 186 186 187 189 190 191 191 191 192 192 193

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5 Conclusions Chapter 3.12: Derivation of Quark and Boson Mass-Energies and Radii Abstract Process Flow 3.12 1 Introduction 2 Theoretical Modelling 2.1 Statistical Considerations 2.2 Generalised Similarity 2.3 Relative Similarity 3 Mathematical Modelling 4 Physical Modelling 4.1 Quark Radii 4.2 Quark Mass 4.3 Refinement of Top Quark Radius 4.4 Boson Radii 5 Conclusions Chapter 3.13: The Planck Scale, Photons, Predicting New Particles and Designing an Experiment to Test the Negative Energy Conjecture Abstract Process Flow 3.12 1 Introduction 2 The Planck Scale 2.1 Convergent Bandwidth 2.2 Planck Characteristics 2.3 Experimental Relationship Functions 2.4 Experimentally Implicit Values of Planck Char. 2.5 Impact of Experimentally Implicit Values 3 Theoretical Modelling 3.1 Background 3.2 Leptons 3.3 Quark / Bosons 4 Mathematical Modelling 4.1 Background 4.2 Bandwidth Ratio 4.3 Optimal Separation 5 Physical Modelling 5.1 Inflection Wavelength 5.2 Critical Field Strengths 5.3 Critical Phase Variance 6 Conclusions Appendices App. 3.A:

Key Artefacts Refractive Index and Experimental Relationship Function Summation of sinusoids producing a constant function Critical Factor General Modelling Equation1 General Modelling Equation2 Critical Ratio Engineered Relationship Function Engineered Refractive Index Gravitational amplitude spectrum 18

193 195 196 197 197 198 198 199 199 200 201 201 202

205 206 207 207 207 209 210 210 211 211 212 212 214 214 214 215 215 216

219 219 219 219 220 220 220 220 220

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App. 3.B:

Gravitational frequency spectrum Harmonic cut-off mode Harmonic cut-off function Harmonic cut-off frequency Critical Boundary EGM Wave Propagation EGM Spectrum Critical Phase Variance Critical Field Strengths Spectral Similarity Equations4,5 DC-Offsets Critical Frequency Harmonic Inflection Mode Critical Mode Harmonic Inflection Frequency EGM Casimir Force Photon mass-energy threshold The Fine Structure Constant Harmonic cut-off frequency Proton and Neutron radii The mass-energy of a Graviton The mass-energy of a Photon The radius of a Photon The radius of a Graviton Harmonic cut-off frequency ratio Electron, Muon and Tau radii Electron, Muon and Tau Neutrino radii Quark and Boson harmonic representations Quarks and Bosons as harmonic multiples of the Electron Planck Scale Experimental Relationship Functions Approximation of the radius of a free Photon, relating physical properties of the Lepton family Theoretical particles (Lepton / Quark / Boson) The optimal configuration of a Classical Casimir Experiment to test the negative energy conjecture Neutron Charge Distribution Neutron Charge Density Radius Intercept “To” Neutron Mean Square Charge Radius Conversion Equation “From” Neutron Mean Square Ch. Radius Conversion Equation Neutron Magnetic Radius Proton Electric Radius Proton Magnetic Radius Classical Proton Root Mean Square Charge Radius The 1st Term of the Balmer Series Formulations, Derivations, Characteristics and Proofs Chapter 3.2 Chapter 3.3 Chapter 3.4 Chapter 3.5 Chapter 3.6 Chapter 3.7 - 3.9

19

220 220 220 220 221 221 221 221 221 222 222 222 222 222 222 222 222 223, 224 223 223 223 223 223 223 223 224 224 224 224 224 225 225 225 225 225 226 226 226 226 226 226 226 227 230 230 234 235 240

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App. 3.C:

App. 3.D: App. 3.E: App. 3.F: App. 3.G:

App. 3.H: App. 3.I: App. 3.J:

Simplifications Harmonic cut-off function Harmonic cut-off mode Harmonic cut-off frequency Derivation of Lepton radii Derivation of Quark and Boson mass-energies and radii Harmonic Representations 1 Conversion of the Neutron Positive Core Radius 2 Derivation of the Neutron Magnetic Radius 3 Derivation of the Proton Electric Radius 4 Derivation of the Proton Magnetic Radius 5 Derivation of the Classical Proton RMS Charge Radius Calculation of L2, L3 and L5 Associated Neutrino radii Derivation of the Hydrogen Atom Spectrum (Balmer Series) and an Experimentally Implicit Definition of the Bohr Radius Glossary of Terms (alphabetical order)

243 243 244 245 247 251 255 261 262 262 262 263 265 269

Bibliography 3

276

Numerical EGM Simulations App. 3.K: MathCad 8 Professional (Complete Simulation) App. 3.L: MathCad 8 Professional (Calculation Engine) App. 3.M: MathCad 12 (High Precision Calculation Results)

279 281 365 387

Index

395

Periodic Table of the Elements

402

Notes

42, 64, 82, 96, 105, 106, 124, 143, 144, 157, 158, 173, 174, 188, 194, 203, 204, 242, 254, 264, 268, 280, 282, 364, 366, 388, 393, 394, 405-407 ERRATA

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GLOSSARY OF TERMS •

Acronyms

BNL BPT CCFR CERN CHARM-II D0C DAT DELPHI DONUT E734 EGM EM EP ERF FNAL FS GME1 GME2 GMEx GPE GR GSE1 GSE2 GSE3 GSE4 GSE5 GSEx HERA HSE1 HSE2 HSE3 HSE4 HSE5 HSEx IFF IHEP INFN LANL LEP LHS MCYT MEXT MS

Brookhaven National Laboratory Buckingham Π Theory Chicago Columbia Fermi-Lab Rochester European Organisation for Nuclear Research Experiment: study of Neutrino-Electron scattering at CERN D-Zero Collaboration: an international research effort of leading scientists utilising facilities at FNAL in Illinois, USA Dimensional Analysis Techniques Detector with Lepton, Photon and Hadron Identification Experiment: a search for direct evidence of the Tau Neutrino at Fermi-Lab Experiment: a measurement of the elastic scattering of Neutrino's from Electrons and Protons (at BNL) Electro-Gravi-Magnetics: a mathematical method based upon the modification of the vacuum polarisability by the superposition of EM fields ElectroMagnetic Experimental Prototype Experimental Relationship Function Fermi National Accelerator Laboratory (FERMI-LAB) Fourier Series General Modelling Equation One General Modelling Equation Two Generalised reference to GME1 and GME2 Gravitational Potential Energy General Relativity General Similarity Equation One General Similarity Equation Two General Similarity Equation Three General Similarity Equation Four General Similarity Equation Five Generalised reference to GSE1, GSE2, GSE3, GSE4 or GSE5 Hadron Electron Ring Accelerator in Hamburg, Germany Harmonic Similarity Equation One Harmonic Similarity Equation Two Harmonic Similarity Equation Three Harmonic Similarity Equation Four Harmonic Similarity Equation Five Generalised reference to HSE1, HSE2, HSE3, HSE4 or HSE5 If and only if Institute of High Energy Physics National Institute of Nuclear Physics (Italy) Los Alamos National Laboratories Large Electron-Positron storage ring Left hand side Ministry of Science and Technology (Spain) Ministry of Science (Japan) Mean Square 21

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NIST NuTeV PDG PV RFBR RHS RMS SK SLAC SM SNO SSE1 SSE2 SSE3 SSE4 SSE5 SSEx TRISTAN US NSF USDoE ZC ZPF •

General Symbols

Symbol B

c E

G H h h-bar i J k L M M0 ME mh

National Institute of Standards & Technology Neutrino's at the Tevatron Particle Data Group: an international research effort of leading scientists Polarisable Vacuum Russian Foundation for Basic Research Right hand side Root Mean Square Super-Kamiokande Collaboration Stanford Linear Accelerator Centre Standard Model in particle physics Sudbury Neutrino Observatory Spectral Similarity Equation One Spectral Similarity Equation Two Spectral Similarity Equation Three Spectral Similarity Equation Four Spectral Similarity Equation Five Generalised reference to SSE1, SSE2, SSE3, SSE4 or SSE5 Particle collider at the Japanese High Energy Physics Laboratory (KEK) United States National Science Foundation United States Department of Energy ZEUS Collaboration: an international research effort of leading scientists utilising facilities at HERA Zero-Point-Field

Description Units Magnitude of Magnetic field vector T Magnitude of Magnetic field vector (at infinity) in the PV model of gravity: Ch. 3.2 Velocity of light in a vacuum m/s Velocity of light in a vacuum (at infinity) in the PV model of gravity: Ch. 3.1 Energy: Ch. 3.3 J Magnitude of Electric field vector V/m Magnitude of Electric field vector (at infinity) in the PV model of gravity: Ch. 3.2 Universal Gravitation Constant m3kg-1s-2 Hydrogen Magnetic field strength Oe Height: Ch. 3.4 m Planck's Constant (plain h form) Js Planck's Constant (2π form) Complex number Initial condition Vector current density A/m2 Wave vector 1/m Length m Mass kg or eV Zero mass (energy) condition of free space Mass of the Earth Planck Mass 22

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MJ MM MS Q r

RE RJ RM RS S t ∆r α ε0 λ λCe λCN λCP λh µ0 ρ ω ωCe ωCN ωCP ωh •

Mass of Jupiter Mass of the Moon Mass of the Sun Magnitude of Electric charge Arbitrary radius with homogeneous mass (energy) distribution Generalised notation for length (e.g. r → λ/2π): Ch. 3.1 Generalised notation for length (locally) in the PV model of gravity: Ch. 3.1 Magnitude of position vector from centre of spherical object with homogeneous mass (energy) distribution Reciprocal of the wave number: Ch. 3.1 Mean radius of the Earth Mean radius of Jupiter Mean radius of the Moon Mean radius of the Sun Poynting Vector Time Plate separation of a Classical Casimir Experiment Practical changes in benchtop displacement values An inversely proportional description of how energy density may result in an acceleration: Ch. 3.2 Fine Structure Constant Permittivity of a vacuum Wavelength Electron Compton Wavelength Neutron Compton Wavelength Proton Compton Wavelength Planck Length Permeability of a vacuum Charge density Field frequency Field frequency (at infinity) in the PV model of gravity: Ch. 3.2 Electron Compton Frequency Neutron Compton Frequency Proton Compton Frequency Planck Frequency

kg or eV

C m

W/m2 s m m/s2

F/m m

N/A2 C/m3 Hz

Specific Symbols by Chapter EGM Construct - Ch. 3.1: Dimensional Analysis

Symbol a a∞ F(k,n,t) f(t) F0(k) In,P K0(r,X) K0(X)

Description Magnitude of acceleration vector Mean magnitude of acceleration over the fundamental period in a FS representation in EGM Complex FS representation of EGM Magnitude of the ambient gravitational acceleration represented in the time domain Amplitude spectrum / distribution of F(k,n,t) Macroscopic intensity of Photons within a test volume ERF by displacement domain precipitation Generalised ERF 23

Units m/s2

W/m2

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K0(ω ω,r,E,B,X) ERF by wavefunction precipitation K0(ω ω,X) ERF by frequency domain precipitation The intensity of the background PV field at specific frequency modes Kn,P Refractive Index of PV KPV Field harmonic (harmonic frequency mode) n, N Polarisation vector P Transformed value of generalised length (locally) in the PV model of gravity rc Local value of the velocity of light in a vacuum vc All variables within the experimental environment that influence results and X behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis Dimensional grouping derived by application of BPT Π Symbol a1 a2 ax(t) B0 c0 D

E0

K1 K2 KC r0 α1 αx β β1 βx ρ0 ω0

Symbol BA BPV EA EPV g00 g11

EGM Construct - Ch. 3.2: General Modelling and the Critical Factor Description Acceleration with respect to General Modelling Equation One Acceleration with respect to General Modelling Equation Two Arbitrary acceleration in the time domain Amplitude of applied Magnetic field: Ch. 3.6 Magnitude of Magnetic field vector (locally) in the PV model of gravity Velocity of light (locally) in the PV model of gravity Experimental configuration factor: a specific value relating all design criteria; this includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy Amplitude of applied Electric field: Ch. 3.6 Energy (locally) in the PV model of gravity Magnitude of Electric field vector (locally) in the PV model of gravity ERF formed by re-interpretation of the primary precipitant ERF formed by re-interpretation of the primary precipitant Critical Factor Length (locally) in the PV model of gravity The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to α1 and α2 A directly proportional description of how energy density may result in an acceleration The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to β1 and β2 Spectral energy density Field frequency (locally) in the PV model of gravity Field frequency (locally) in the PV model of gravity by EGM EGM Construct - Ch. 3.3: The Engineered Metric Description Magnitude of applied Magnetic field vector Magnitude of PV Magnetic field vector Magnitude of applied Electric field vector Magnitude of PV Electric field vector Tensor element Tensor element 24

W/m2 C/m2 m m/s

Units m/s2

T m/s

V/m J V/m (V/m)2 T-2 PaΩ m m/s2

Pa/Hz Hz

Units T V/m

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g22 g33 kA KEGM kPV KR L0 m0 nA nPV Ug Z ∆aPV ∆g ∆K0(ω ω,X) ∆K1 ∆K2 ∆ KC ∆t ∆t0 ∆ Ug ∆UPV

Tensor element Tensor element Harmonic wave vector of applied field Engineered Refractive Index Harmonic wave vector of PV Critical Ratio Length (locally) in the PV model of gravity by EGM Mass (locally) in the PV model of gravity by EGM Harmonic frequency modes of applied field Harmonic frequency modes of PV Initial state GPE per unit mass described by any appropriate method Impedance function Change in the magnitude of the local PV acceleration vector Change in magnitude of the local gravitational acceleration vector Engineered Relationship Function by EGM Change in K1 by EGM Change in K2 by EGM Change in Critical Factor by EGM Change in time (at infinity) in the PV model of gravity by EGM Change in time (locally) in the PV model of gravity by EGM Change in Gravitational Potential Energy (GPE) per unit mass induced by any suitable source Change in energy density of gravitational field

EGM Construct - Ch. 3.4: Amplitude and Frequency Spectra Symbol Description Magnitude of PV Magnetic field vector BPV Amplitude of fundamental frequency of PV (nPV = 1) CPV(1,r,M) CPV(nPV,r,M) Amplitude spectrum of PV Magnitude of PV Electric field vector EPV K0(ω ωPV,r,EPV,BPV,X) ERF equivalent to K0(ω,r,E,B,X) Permissible mode bandwidth of applied experimental fields N∆r Harmonic cut-off mode of PV nΩ Poynting Vector of PV Sω Rest mass-energy density Um Field energy density of PV Uω Frequency bandwidth of PV ∆ωPV Harmonic cut-off function of PV Ω Generalised reference to ωPV(nPV,r,M) ωPV Fundamental frequency of PV (nPV = 1) ωPV(1,r,M) Frequency spectrum of PV ωPV(nPV,r,M) Harmonic cut-off frequency of PV ωΩ

1/m 1/m m kg (m/s)2 Ω m/s2

(V/m)2 T-2 PaΩ s (m/s)2 Pa

Units T m/s2 V/m

W/m2 Pa Hz Hz

EGM Construct - Ch. 3.5: General Similarity Symbol nΩ ZPF nβ RError Stα

Description ZPF beat cut-off mode Mode Number (Critical Boundary Mode) of ωβ Representation error Range factor 25

Units

% PaΩ www.deltagroupengineering.com

Stβ Stδ Stε Stγ ∆GME1 ∆GME2 ∆GMEx ∆nS ∆UPV ∆vΩ ∆vδr ∆λΩ ∆λδr ∆ωR ∆ωS ∆ωZPF ∆ωΩ ∆ωδr λPV ωΩ ZPF ωβ

1st Sense check 3rd Sense check 4th Sense check 2nd Sense check Change in GME1 Change in GME2 Generalised reference to changes in GME1 and GME2 Change in the number of ZPF modes Change in rest mass-energy density Terminating group velocity of PV Group velocity of PV Change in harmonic cut-off wavelength of PV Change in harmonic wavelength of PV Bandwidth ratio Similarity bandwidth ZPF beat bandwidth Beat bandwidth of PV Beat frequency of PV Wavelength of PV ZPF beat cut-off frequency Critical boundary

Symbol BC Brms DC EC Erms HSE4A R HSE5A R HSEx R KEGM H KPV H KR H nB nE Ug H ∆K0 H φ φC ωB ωC ωE

EGM Construct - Ch. 3.6: Harmonic and Spectral Similarity Description Critical Magnetic field strength Root Mean Square of BA Offset function Critical Electric field strength Root Mean Square of EA Time average form of HSE4 R Time average form of HSE5 R Generalised reference to the reduced form of HSEx Harmonic form of KEGM Harmonic form of KPV Critical harmonic operator (based upon the unit amplitude spectrum) Harmonic Mode Number of the ZPF with respect to BA Harmonic Mode Number of the ZPF with respect to EA Harmonic form of Ug Harmonic form of ∆K0 Relative phase variance between EA and BA Critical phase variance Harmonic frequency of the ZPF with respect to BA Critical frequency Harmonic frequency of the ZPF with respect to EA

m/s2

Pa m/s m

Hz

m Hz

Units T % V/m

(m/s)2 θc Hz

EGM Construct - Ch. 3.7: The Casimir Effect Symbol A APP

Description 1st Harmonic term Parallel plate area of a Classical Casimir Experiment 26

Units m2 www.deltagroupengineering.com

D FPP FPV KP NC NT NTR NX StN ∆ΛPV ΣH ΣHR ωX Symbol bq cq dq e, e-

g H Km KS KX Kλ Kω L2 L3 L5 mbq mcq mdq me men mgg mH mL(2) mL(3) mL(5) mn mp mQB(5) mQB(6) msq mtq muq

Common difference The Casimir Force by classical representation The Casimir Force by EGM A refinement of a constant in FPP Critical mode Number of terms The ratio of the number of terms Harmonic inflection mode nth Harmonic term Change in the local value of the Cosmological Constant by EGM The sum of terms The ratio of the sum of terms Harmonic inflection frequency Particles Physics: Ch. 3.8 - 3.13 Description Bottom Quark: elementary particle in the SM Charm Quark: elementary particle in the SM Down Quark: elementary particle in the SM Charge Electron: subatomic / elementary particle in the SM Exponential function: mathematics Gluon: theoretical elementary particle in the SM Magnitude of gravitational acceleration vector Higgs Boson: theoretical elementary particle in the SM Experimentally implicit Planck Mass scaling factor Neutron MS charge radius by EGM Neutron MS charge radius (determined experimentally) in the SM Experimentally implicit Planck Length scaling factor Experimentally implicit Planck Frequency scaling factor Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Bottom Quark rest mass (energy) by EGM Charm Quark rest mass (energy) by EGM Down Quark rest mass (energy) by EGM Electron rest mass (energy) according to NIST Electron Neutrino rest mass (energy) according to PDG Graviton rest mass (energy) by EGM Higgs Boson rest mass (energy) according to PDG Rest mass (energy) of the L2 particle by EGM Rest mass (energy) of the L3 particle by EGM Rest mass (energy) of the L5 particle by EGM Neutron rest mass (energy) according to NIST Proton rest mass (energy) according to NIST Rest mass (energy) of the QB5 particle by EGM Rest mass (energy) of the QB6 particle by EGM Strange Quark rest mass (energy) by EGM Top Quark rest mass (energy) according (energy) to PDG Up Quark rest mass (energy) by EGM 27

N

Hz2

Hz

Units

C

m/s2 m2

kg or eV

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mW mZ mε mγ mγg mγγ mµ mµn mτ mτn n p QB5 QB6 rBoson rbq rcq rdq re ren rgg rH rL rp rQB rsq rtq ru ruq rW rxq rZ rε rγγ rµ rµn rν rν2 rν3 rν5 rνM rνx rπ rπE rπM rτ rτn sq

W Boson rest mass according (energy) to PDG Z Boson rest mass according (energy) to PDG Electron rest mass (energy) in high energy scattering experiments Photon rest mass (energy) threshold according to PDG Graviton rest mass (energy) threshold according to PDG Photon rest mass (energy) by EGM Muon rest mass (energy) according to NIST Muon Neutrino rest mass (energy) according to PDG Tau rest mass (energy) according to NIST Tau Neutrino rest mass (energy) according to PDG Neutron: subatomic particle in the SM Proton: subatomic particle in the SM Theoretical elementary particle (Quark or Boson) by EGM Theoretical elementary particle (Quark or Boson) by EGM Generalised RMS charge radius of a Boson by EGM RMS charge radius of the Bottom Quark by EGM RMS charge radius of the Charm Quark by EGM RMS charge radius of the Down Quark by EGM Classical Electron radius in the SM RMS charge radius of the Electron Neutrino by EGM RMS charge radius of the Graviton by EGM RMS charge radius of the Higgs Boson utilising ru Average RMS charge radius of the rε, rµ and rτ particles Classical RMS charge radius of the Proton in the SM Average RMS charge radius of the QB5 / QB6 particles by EGM utilising ru RMS charge radius of the Strange Quark by EGM RMS charge radius of the Top Quark by EGM Heisenberg uncertainty range RMS charge radius of the Up Quark by EGM RMS charge radius of the W Boson utilising ru Generalised RMS charge radius of all Quarks as determined by the ZC within the SM RMS charge radius of the Z Boson by utilising ru RMS charge radius of the Electron by EGM RMS charge radius of the Photon by EGM RMS charge radius of the Muon by EGM RMS charge radius of the Muon Neutrino by EGM Neutron RMS charge radius (by analogy to KS) RMS charge radius of the ν2 particle by EGM RMS charge radius of the ν3 particle by EGM RMS charge radius of the ν5 particle by EGM Neutron Magnetic radius by EGM Generalised reference to rν2, rν3 and rν5 RMS charge radius of the Proton by EGM Proton Electric radius by EGM Proton Magnetic radius by EGM RMS charge radius of the Tau by EGM RMS charge radius of the Tau Neutrino by EGM Strange Quark: elementary particle in the SM 28

kg or eV

m

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〈rBoson〉 〈rQuark〉 〈r〉〉

5th Sense check 6th Sense check A positive integer value representing the harmonic cut-off frequency ratio between two proportionally similar mass (energy) systems Top Quark: elementary particle in the SM Up Quark: elementary particle in the SM W Boson: elementary particle in the SM Z Boson: elementary particle in the SM RMS charge diameter of the Graviton by EGM RMS charge diameter of the Photon by EGM Photon: elementary particle in the SM Mathematical Constant: Euler-Mascheroni (Euler's) Constant Graviton: theoretical elementary particle in the SM Muon: elementary particle in the SM Theoretical elementary Neutrino of the L2 particle by EGM Theoretical elementary Neutrino of the L3 particle by EGM Theoretical elementary Neutrino of the L5 particle by EGM Electron Neutrino: elementary particle in the SM Muon Neutrino: elementary particle in the SM Tau Neutrino: elementary particle in the SM Tau: elementary particle in the SM Average mass (energy) of all Quarks by EGM Average mass (energy) of all Quarks according to PDG Average RMS charge radius of all Bosons in the SM utilising ru Average RMS charge radius of all Quarks by EGM Average RMS charge radius of all Quarks and Bosons by EGM utilising ru

Symbol E mAMC mx nq Qe rBohr rx R∞ ∆E λA λB µ

Appendices Description Electronic energy level Atomic Mass Constant Imaginary particle mass Quantum number Magnitude of Electric charge Classical Bohr radius Bohr radius by EGM Rydberg Constant Change in electronic energy level 1st term of the Balmer Series by EGM Classical Balmer Series wavelength Reduced mass of Hydrogen

Stη Stθ Stω tq uq W Z φgg φγγ γ γ γg µ, µν2 ν3 ν5 νe νµ ντ τ, τ〈 mQuark〉

29

m

kg or eV m

Units J kg or eV

C m J m kg or eV

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DEFINITION OF TERMS Alpha Forms “αx” • An inversely proportional description of how energy density may result in acceleration. Amplitude Spectrum • A family of wavefunction amplitudes. • The amplitudes associated with a frequency spectrum. • See: Frequency Spectrum. Background Field • Reference to the background (ambient) gravitational field. • Reference to the local gravitational field at the surface of the Earth. Bandwidth Ratio “∆ωR” • The ratio of the bandwidth of the ZPF spectrum to the Fourier spectrum of the PV. Beta Forms “βx” • A directly proportional description of how energy density may result in acceleration. Buckingham Π Theory (BPT) • Arrangement of variables determined by DAT's into Π groupings. These groupings represent sub-systems of dimensional similarity for scale relationships. • Minimises the number of experiments required to investigate phenomena. • See: DAT's. Casimir Force “FPP” • Attractive (non-gravitational) force between two parallel and neutrally charged mirrored plates of equal area. Change in the Number of Modes “∆nS” • The difference between the ZPF beat cut-off mode and the Mode Number at the Critical Boundary as a function of the Critical Ratio. • See: Mode Number “nβ”. • See: Critical Ratio “KR”. Compton Frequency “ωCx” • The generalised definition of Compton frequency applied globally herein is: ωCx = mxc2 / h-bar = 2πm 2π xc2/ h = 2πc 2π 2/ λCx. • This is the only equation in which the “h-bar” form of Planck's Constant is used. Cosmological Constant • A constant introduced into the equations of GR to facilitate a steady state cosmological solution. • See: General Relativity. Critical Boundary “ωβ” • Represents the lower boundary (commencing at the ZPF beat cut-off frequency) of the ZPF spectrum yielding a specific proportional similarity value. • See: Zero-Point-Field Beat Cut-Off Frequency “ωΩ ZPF”. 30

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•

See: Critical Ratio “KR”.

Critical Factor “KC” • A proportional measure of the applied EM field intensity (or magnitude of Poynting Vectors) within an experimental test volume. • The ratio of two experimentally determined relationship functions. Critical Field Strengths “EC and BC” • RMS strength values of applied Electric and Magnetic fields for complete dynamic, kinematic and geometric similarity with the background gravitational field. • See: Background Field. Critical Frequency “ωC” • The minimum frequency for the application of Maxwell's Equations within an experimental context. Critical Harmonic Operator “KR H” • A representation of the Critical Ratio at ideal dynamic, kinematic and geometric similarity utilising a unit amplitude spectrum. Critical Mode “NC” • The ratio of the critical frequency to the fundamental harmonic frequency of the PV. • See: Critical Frequency “ωC”. • See: Fundamental Harmonic Frequency “ωPV(1,r,M)” Critical Phase Variance “φC” • The difference in phase between applied Electric and Magnetic fields for complete dynamic, kinematic and geometric similarity with the background gravitational field. • See: Background Field. Critical Ratio “KR” • A proportional indication of anticipated experimental configurations by any suitable measure. Typically, this is the magnitude of the ratio of the applied EM experimental fields to the ambient background gravitational field. Curl •

The limiting value of circulation per unit area.

DC-Offsets • A proportional value of applied RMS Electric and / or Magnetic fields acting to offset the applied function/s. Dimensional Analysis Techniques (DAT's) • Formal experimentally based research methods facilitating the derivation, from first principles, of any number or combination of parameters considered important by an experimentalist. • See: BPT. Divergence • The rate at which “density” exits a given region of space.

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Dominant Bandwidth • The bandwidth of the EGM spectrum which dominates gravitational effects. • See: Electro-Gravi-Magnetics (EGM) Spectrum. Electro-Gravi-Magnetics (EGM) • A method of calculation (not a theory) based upon energy density. • Being a calculation method, it does not favour or bias any particular theory in the Standard Model of particle physics. • Developed as a tool for engineers to modify gravity. • The modification of vacuum polarisability based upon the superposition of EM fields. Electro-Gravi-Magnetics (EGM) Spectrum • A simple but extreme extension of the EM spectrum (including gravitational effects) based upon a Fourier distribution. Energy Density (General) • Energy per unit volume. Engineered Metric • A metric tensor line element utilising the Engineered Refractive Index. Engineered Refractive Index “KEGM” • An EM based engineered representation of the Refractive Index. Engineered Relationship Function “∆K0(ω,X)” • A change in the Experimental Relationship Function resulting from a modification in the local value of the magnitude of acceleration by similarity of applied EM fields to the background gravitational field. Experimental Prototype (EP) • Reference to the gravitational acceleration through a practical benchtop volume of space-time in a laboratory at the surface of the Earth. Experimental Relationship Function “K0(ω,X)” • A proportional scaling factor relating an experimental prototype (typically herein, it is the local gravitational field or ambient physical conditions) to a mathematical model. Fourier Spectrum • Two spectra combined into one (an amplitude spectrum and a frequency spectrum) obeying a Fourier Series. • See: Amplitude Spectrum. • See: Frequency Spectrum. Frequency Bandwidth “∆ωPV” • The bandwidth of the Fourier spectrum describing the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). Frequency Spectrum • A family of wavefunction frequencies. • The frequencies associated with an amplitude spectrum. 32

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•

See: Amplitude Spectrum.

Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)” • The change in fundamental harmonic frequency of the PV across an elemental displacement. • See: Fundamental Harmonic Frequency “ωPV(1,r,M)”. • See: Polarisable Vacuum (PV). Fundamental Harmonic Frequency “ωPV(1,r,M)” • The lowest frequency in the PV spectrum utilising Fourier harmonics. General Modelling Equations (GMEx) • Proportional solutions to the Poisson and Lagrange equations resulting in acceleration. General Relativity (GR) • The representation of space-time as a geometric manifold of events where gravitation manifests itself as a curvature of space-time and is described by a metric tensor. General Similarity Equations (GSEx) • Combines General Modelling Equations with the Critical Ratio by utilisation of the Engineered Relationship Function. • See: Critical Ratio “KR”. Gravitons “γg” • Conjugate Photon pairs responsible for gravitation. This is an inherent mathematical conclusion arising from similarity modelling utilising a Fourier distribution in Complex form and the PV model of gravity considerate of ZPF Theory (due to harmonic symmetry about the “0th” mode). Graviton Mass-Energy Threshold “mγg” • The upper boundary value of the mass-energy of a Graviton as defined by the Particle Data Group. Group Velocity • The velocity with which energy propagates. Harmonic Cut-Off Frequency “ωΩ” • The terminating frequency of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). Harmonic Cut-Off Function “Ω” • A mathematical function associated with the harmonic cut-off mode and frequency. • See: Harmonic Cut-Off Mode “nΩ”. • See: Harmonic Cut-Off Frequency “ωΩ”. Harmonic Cut-Off Mode “nΩ” • The terminating mode of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). 33

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Harmonic Inflection Mode “NX” • The mode at which the phase variance between the Electric and Magnetic wavefunctions describing the PV in a classical Casimir experiment begins to alter dramatically. • A conjectured resonant mode of the PV in a classical Casimir experiment. • See: Casimir Force “FPP”. • See: Polarisable Vacuum (PV). Harmonic Inflection Frequency “ωX” • The frequency associated with the harmonic inflection mode. • See: Harmonic Inflection Mode “NX”. Harmonic Inflection Wavelength “λX” • The wavelength associated with the harmonic inflection frequency. Harmonic Similarity Equations (HSEx) • A harmonic representation of General Similarity Equations utilising the Critical Harmonic Operator. • A family of equations defined by relating the experimental prototype to a mathematical model (General Similarity Equations). • See: Critical Harmonic Operator “KR H”. • See: General Similarity Equations (GSEx). IFF •

If and only if.

Impedance Function • A measure of the ratio of the permeability to the permittivity of a vacuum. • Resistance to energy transfer through a vacuum. Kinetic Spectrum • Another term for the ZPF spectrum. • See: ZPF Spectrum. Mode Bandwidth • The modes associated with a frequency bandwidth. Mode Number (Critical Boundary Mode) “nβ” • The ratio of the Critical Boundary frequency to the fundamental frequency of the PV. • The harmonic mode associated with the Critical Boundary frequency. Number of Permissible Modes “N∆r” • The number of modes permitted for the application of Maxwell's Equations within an experimental context, based upon the harmonic cut-off frequency. • See: Harmonic Cut-Off Frequency “ωΩ”. Phenomena of Beats • The interference between two waves of slightly different frequencies. Photon Mass-Energy Threshold “mγ” • The upper boundary value of the mass-energy of a Photon as defined by the Particle Data Group. 34

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Polarisable Vacuum (PV) • The polarised state of the Zero-Point-Field due to mass influence. • Characterised by a Refractive Index. • Obeys a Fourier distribution. • A bandwidth of the EGM Spectrum. • See: Electro-Gravi-Magnetics (EGM). • See: Electro-Gravi-Magnetics (EGM) Spectrum. Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ” • The change in harmonic cut-off frequency across an elemental displacement. • See: Harmonic Cut-Off Frequency “ωΩ”. • See: Phenomena of Beats. • See: Polarisable Vacuum (PV). Polarisable Vacuum (PV) Spectrum • Another term for the Fourier spectrum applied by EGM to describe the PV harmonically. • A bandwidth of the EGM Spectrum. • See: Electro-Gravi-Magnetics (EGM). • See. Fourier Spectrum. • See: Polarisable Vacuum (PV). Potential Spectrum • Another term for the PV spectrum. • See: Polarisable Vacuum (PV) Spectrum. Poynting Vector • Describes the direction and magnitude of EM energy flow. • The cross product of the Electric and Magnetic field. Precipitations • Results driven by the application of limits. Primary Precipitant • The frequency domain precipitation. • See: Precipitations. Radii Calculations by Electro-Gravi-Magnetics (EGM) • Radii calculations by EGM represent the radial position of energetic equilibrium between the energy density of a homogeneous spherical mass with the ZPF. • The radii predictions calculated by EGM coincide with the RMS charge radii of all charged fundamental particles. • See: Electro-Gravi-Magnetics (EGM). • See: Zero-Point-Field (ZPF). Range Factor “Stα” • The product of the change in energy density and the Impedance Function. • An “at-a-glance” tool indicating the boundaries of the applied energy requirements for complete dynamic, kinematic and geometric similarity with the background field. • See: Energy Density. • See: Background Field. 35

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•

See: Impedance Function.

Reduced Average Harmonic Similarity Equations (HSExA R) • See: 2nd Reduction of the Harmonic Similarity Equations. Reduced Harmonic Similarity Equations (HSEx R) • A simplification of the Harmonic Similarity Equations by substitution of RMS expressions for the time varying representations of applied Electric and Magnetic field harmonics. • A simplification of the Harmonic Similarity Equations facilitating the investigation of the effects of phase variance [on a modal (per mode) basis]. Refractive Index “KPV” • Characterisation value of the PV. Representation Error “RError” • Error associated with the mathematical representation of a physical system. RMS Charge Radii (General) • The RMS charge radius refers to the RMS value of the charge distribution curve. RMS Charge Radius of the Neutron “rν” • The RMS charge radius of a Neutron “rν” is so termed by analogy to the Neutron Mean Square charge radius “KX” which is typically represented as a squared length quantity “fm2”. Therefore, the dimensional square root of “KX” represents “rν” by analogy. • “rν” represents the cross-over radius (the node) on the Neutron charge distribution curve. Similarity Bandwidth “∆ωS” • The difference between the ZPF beat cut-off frequency and the critical boundary frequency. • A measure of similarity between the background gravitational field spectrum and the applied field frequencies (commencing at the ZPF beat cut-off frequency). • See: Background Field. • See: Critical Boundary “ωβ”. • See: Zero-Point-Field Beat Cut-Off Frequency “ωΩ ZPF”. Spectral Energy Density “ρ0(ω)” • Energy density per frequency mode. Spectral Similarity Equations (SSEx) • A representation of the complete spectrum of the PV utilising the 2nd Reduction of the Harmonic Similarity Equations by application of similarity principles. Subordinate Bandwidth • The EM spectrum. • See: Dominant Bandwidth. • See: Electro-Gravi-Magnetics (EGM) Spectrum. Unit Amplitude Spectrum • A harmonic representation of unity (the number one) utilising the amplitude spectrum of a Fourier distribution. 36

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Zero-Point-Energy (ZPE) • The lowest possible energy of the space-time manifold described in quantum terms. Zero-Point-Field (ZPF) • The field associated with ZPE. Zero-Point-Field (ZPF) Spectrum • The spectrum of amplitudes and frequencies associated with the ZPF. Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF” • The difference between the ZPF beat cut-off frequency and the fundamental beat frequency. • See: Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)”. • See: Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF”. Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF” • The terminating frequency of the ZPF spectrum across an elemental displacement. Zero-Point-Field (ZPF) Beat Cut-Off Mode “nΩ ZPF” • The terminating mode of the ZPF spectrum across an elemental displacement. 1st Sense Check “Stβ” • A common sense test relating the ZPF beat bandwidth to the Compton frequency of an Electron. • See: Compton Frequency “ωCx”. • See: Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF”. 2nd Reduction of the Harmonic Similarity Equations (HSExA R) • A time averaged simplification of the Reduced Harmonic Similarity Equations. 2nd Sense Check “Stγ” • A common sense test relating the PV beat bandwidth to the Compton frequency of an Electron. • See: Compton Frequency “ωCx”. • See: Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ”. 3rd Sense Check “Stδ” • A common sense test relating the harmonic cut-off mode across an elemental displacement. • See: Harmonic Cut-Off Mode “nΩ”. 4th Sense Check “Stε” • A common sense test relating the representation error across an elemental displacement. • See: Representation Error “RError”. 5th Sense Check “Stη” • A common sense test relating the harmonic cut-off frequency of a Proton to the Compton frequency of a Proton. • See: Compton Frequency “ωCx”.

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6th Sense Check “Stθ” • A common sense test relating the harmonic cut-off frequency of a Neutron to the Compton frequency of a Neutron. • See: Compton Frequency “ωCx”. Physical Constants [1] Symbol α c G ε0 µ0 h h-bar λCe λCP λCN λCµ λCτ me mp mn mµ mτ re rp λh mh th ωh eV

Description Fine Structure Constant Velocity of light in a vacuum Universal Gravitation Constant Permittivity of a vacuum Permeability of a vacuum Planck's Constant Planck's Constant (2π form) Electron Compton Wavelength Proton Compton Wavelength Neutron Compton Wavelength Muon Compton Wavelength Tau Compton Wavelength Electron rest mass Proton rest mass Neutron rest mass Muon rest mass Tau rest mass Classical Electron radius Classical Proton RMS charge radius Planck Length Planck Mass Planck Time Planck Frequency Electron Volt

NIST value utilised by EGM 7.297352568 x10-3 299792458 6.6742 x10-11 8.854187817 x10-12 4π x10-7 6.6260693 x10-34 1.05457168 x10-34

Units None m/s m3kg-1s-2 F/m N/A2 Js

= h / (me,p,n,µ,τ c)

m

9.1093826 x10-31 1.67262171 x10-27 1.67492728 x10-27 1.88353140 x10-28 3.16777 x10-27 2.817940325 x10-15 0.8750 x10-15 = √(Gh/c3) = √(hc/G) = √(Gh/c5) = 1/th 1.60217653 x10-19

kg

m

kg s Hz J

Mathematical Constants and Symbols • Euler-Mascheroni Constant (Euler's Constant) [2] “γ” = 0.5772156649015328 • “∩” Refers to an intersection. • “∪” Refers to a union. • “→” Or “↓” refers to a process: “leads to”. Solar System Statistics [3] Symbol MM ME MJ MS RM RE RJ RS

Description Mass of the Moon Mass of the Earth Mass of Jupiter Mass of the Sun Mean Radius of the Moon Mean Radius of the Earth Mean Radius of Jupiter Mean Radius of the Sun 38

Value utilised by EGM Units 7.35 x1022 kg 24 5.977 x10 1898.8 x1024 1.989 x1030 1.738 x106 m 6.37718 x106 7.1492 x107 6.96 x108 www.deltagroupengineering.com

SPIRAL GALAXY

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 CERN (http://doc.cern.ch//archive/electronic/cern/others/PHO/photo-di/9108002.jpeg)

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CHAPTER

3.0

Introduction and Document Statistics Statistic Appendices Significant Equations Significant Figures Images Pages Tables Words (approx.)

41

Value 13 458 47 27 407 67 64000

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NOTES

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1

GENERAL

1.1

INTRODUCTION

Laying the Foundations Since the early 20th century, gravitational physics has been dominated by Albert Einstein’s concept of General Relativity (GR) where space-time is defined as a manifold described by geometric events. Gravitational influence on the manifold is represented by space-time curvature and is typically visualised by analogy to a bowling ball on a trampoline. The mat deflects under the weight of the bowling ball and denotes the curvature of the space-time manifold actioned by the object. It is also visualised by the path of light around a massive gravitational object as being “bent”. GR has proven to be a useful and reliable tool by which to map and predict the interaction of large scale mass systems. GR has two main issues associated with its application as a practical engineering tool. Firstly and most importantly, at least planetary sized masses are required to affect the state of the manifold in any practical and meaningful way to engineers. A non-practical / non-viable approach is to bombard a region of space-time with a sufficient level of ElectroMagnetic (EM) energy, in the appropriate manner, to alter its geometric state.

The second main issue is complexity. [4] As can be clearly seen from the analytical representation of only a portion of Einstein’s field equations (left), it is unwieldy and can only be solved in practical application terms by numerical methods involving powerful computers. Einstein’s equations do have a shorthand representation utilising Tensors (e.g. Gµν = 8πTµν), but this is for scientific communication purposes and still requires numerical evaluation 43

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by computers in order to visualise the interaction of gravitational systems. Complexity becomes even more problematic if less idealised systems are considered. That is, the absence of symmetries in the system under consideration causes the solution to swell into thousands of terms in each equation. Hence, it is discouragingly apparent that GR is not an engineering tool of choice by which to modify the space-time manifold, in any practical sense, on a laboratory test bench. So what can we do about it? What can we do to engineer the space-time manifold, or at the very least, develop experiments to assess if the space-time manifold can indeed be modified utilising existing equipment and engineering methodologies? To even begin to answer these questions, we must first establish what investigative tools to use. A thorough search of the scientific literature will find volumes of information stating scientific opinions of what gravity is based upon personal interpretation by the individual author, but absolutely no specific literature exists defining what gravity is factually and unmistakably known to be. This is self evident in the fact that humanity does not currently know how to engineer gravity beyond physically arranging masses. Commonsense compels us to establish a way forward based upon the notion that any practical benchtop experimental configuration would involve applied sinusoidal EM fields. Unfortunately, there are no other practical and effective methods of delivering energy to a region of space-time, so our first decision is actually made for us by the world that we currently live in. By recognising this boundary, we accept by necessity, the existence of unification between Electric, Magnetic and gravitational forces. We shall term this unification, Electro-Gravi-Magnetics (EGM). Given that we have, by necessity, established the existence of EGM, we must also establish an engineering definition. In order to facilitate this, we should determine if any physical models exist, other than GR, which are wavefunction based and bias engineering methodologies as our intention is to investigate gravitational acceleration based upon the superposition of applied sinusoidal EM fields. An effortless search of the scientific literature reveals that a flat region of space-time (a region of zero gravitational strength) may be described by a Zero-Point-Field (ZPF). The ZPF may be described as an endless sea of randomly orientated Photons / wavefunctions at the Zero-Point-Energy (ZPE) ground state (incrementally above zero) in accordance with Quantum Mechanical (QM) models. The ZPF may be well visualised (below) by the ripples on a pond during light rain. The ripples are analogous to the Photons / wavefunctions of the ZPF and the randomly co-ordinated raindrops are analogous to their orientation. So, how we can give a ZPF description to gravity at the surface of the Earth? To answer this question and facilitate a method by which we can assign a ZPF description to the surface of the Earth (which cannot be at the ZPF ground state of free space because the gravitational acceleration is non-zero), we require an additional model, other than GR and ZPF Theory. Again, an effortless search reveals the Polarisable Vacuum (PV) model of gravity to be a practical alternative to GR, at minimum, isomorphic in the weak field. Image depicting ZPF analogy to pond ripples,

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The PV model of gravity, as the name suggests, acts to polarise the ZPF analogous to the field lines produced by a magnet (right). Because the vacuum is polarised at the surface of the Earth, the orientation of the gravitational vector is straight down and gravity may be usefully described as a one dimensional (1-D) phenomenon in human experience. In other words, gravity in human experience is not three dimensional (3-D). The importance of this realisation is of paramount significance as it greatly simplifies the mathematics involved within EGM. So, between what we understand of ZPF Theory, the PV model of gravity and the EGM method thus far, we have partially deduced a suite of tools for investigative engineering. Image depicting PV analogy to field lines, The magnitude of gravitational acceleration may be usefully approximated to being a constant mathematical function at all points vertically across a room. This may be decomposed into a spectral family of constituent harmonics utilising a simple Fourier distribution. The lower spectral limit is termed the fundamental frequency “ωPV(1,r,M)”, whilst the upper spectral limit is termed the harmonic cut-off frequency “ωΩ”. Image depicting Fourier harmonics, The image above depicts a Fourier representation of a square wave for “ωPV(1,r,M)” and the first 15 harmonics. A usefully approximate Fourier description of any function involves the near infinite summation of harmonics. The physical problem that is intended to be described by application of this method may be over a displacement domain, or simply in the time domain without any intention to be representative of wave propagation. In the case of gravity, we may apply a Fourier description to a specific mathematical point above the surface of the Earth, by considering the wavefunctions at that point as being “pseudopropagating”. That is, we can mathematically regard them as propagating wavefunctions, without an actual physical requirement to do so. This concept is reinforced by the fact that the collective behaviour of the entire spectrum of wavefunctions at the mathematical point under consideration will have a group velocity of zero due to the mathematical summation at that point to a constant value. That is, the wavefunction mathematically cannot propagate. This agrees with physical observation since gravitational waves are not observed to propagate from planetary bodies. Hence, a “pseudo-propagating” representation in the Fourier domain is a useful engineering tool. Therefore, the magnitude of gravitational acceleration at the surface of the Earth may be mathematically described in the Fourier domain by the magnitude of the square wave depicted in the preceding image. Subsequently, by applying a Fourier series to practical human experience, we can determine the spectral composition of the polarised state of the ZPF at the surface of the Earth to great precision. We shall define the resulting Fourier representation as “the PV spectrum”. The next step in assessing if the space-time manifold can indeed be modified is to study the best example of space-time available to us: the practical volume of space-time on a laboratory test bench! To achieve this, we shall reverse engineer the volume utilising Dimensional Analysis Techniques (DAT’s) and Buckingham Π Theory (BPT). 45

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The following statement is a verbatim quotation from [5, 6] Dimensional Analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantities may be added, subtracted, compared, or equated. When like or unlike dimensioned quantities are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantities are raised to a power or a power root, the same is done to the dimensions attached to those quantities. The dimensions of a physical quantity is associated with symbols such as “M, L, T” which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable speed is “distance / time (L/T)” and the dimension of force is “mass × distance / time² (ML/T²)”. In mechanics, every dimension can be expressed in terms of distance (which physicists often call “length”), time and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another set of dimensions. In ElectroMagnetism, for example, it may be useful to use dimensions of “M, L, T, and Q”, where “Q” represents the quantity of electric charge. The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but length always has dimension “L” whether it is measured in meters, feet, inches, miles or micrometres. In the most primitive form, dimensional analysis may be used to check the “correctness” of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of “3(kg)” is undefined, but the logarithm of “3” is “0.477”. It should be noted that very different physical quantities may have the same dimensions: work and torque, for example, have the same dimensions, “M L2T-2”. An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. The value of a dimensional physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. When like dimensioned quantities are added, subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, conceptually, there is no problem adding quantities of the same dimension expressed in different units. BPT forms the basis of the central tool of Dimensional Analysis. This theorem describes how every physically meaningful equation involving “n” variables can be equivalently rewritten as an equation of “n-m” dimensionless parameters, where “m” is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. BPT is a systematic method of Dimensional Analysis, whereby variables that are relevant to a particular situation are formed into dimensionless Π groups. The number of dimensionless groups equals the original number of variables minus the number of fundamental dimensions present in all the variables. This analysis reduces the degrees of freedom for a physical situation and can be used to guide experimental design programs. Proofs of BPT often begin by considering the space of fundamental and derived physical units as a vector space, with the fundamental units as basis vectors and with multiplication of physical units as the “vector addition” operation and raising to powers as the “scalar multiplication” 46

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operation. Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical unit vector space. Two systems for which these parameters coincide are called similar; they are equivalent for the purposes of the equation and the experimentalist whom wishes to determine the form of the equation can choose the most convenient one. BPT uses linear algebra: the space of all possible physical units can be seen as a vector space over rational numbers if we represent a unit as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of BPT is essentially a Gauss-Jordan elimination carried out in this vector space. In 1941, “Sir Geoffrey I. Taylor” used Dimensional Analysis to estimate the energy released in an atomic bomb explosion. The first atomic bomb was detonated in New Mexico on July 16, 1945. In 1947, movies of the explosion were declassified, allowing “Sir Taylor” to complete the analysis and estimate the energy released in the explosion, even though the energy release was still classified! The actual energy released was later declassified and its value was remarkably close to Taylor's estimate. Taylor supposed that the explosive process was adequately described by five physical quantities, the time “t” since the detonation, the energy “E” which is released at a single point in space at detonation, the radius “R” of the shock wave at time “t”, the ambient atmospheric pressure “p” and density “ρ”. There are only three fundamental physical units in this combination (MLT) which yield Taylor's equation. Once the radius of the explosion as a function of the time was known, the energy of the explosion was calculated. End of verbatim quotation. The EGM Approach We shall utilise BPT to relate gravitational acceleration, EM acceleration by the superposition of applied fields, ZPF Theory and the PV model of gravity via Einstein’s equivalence principle. Dimensionally, there is no difference between gravitational and EM acceleration. The equivalence principle provides a well accepted vehicle for the logical application of BPT and DAT’s to gravity. Much of Thermodynamics and Fluid Mechanics is built form the application of BPT and DAT’s. BPT facilitates the ability to string together any number of variables in a way that permits one to test one’s own idea. So, it is really a mix between science and art. There is nothing wrong with any grouping formed utilising BPT, it is simply a question of how “well” a grouping tends to fit physical observation. The BPT formalism affords an engineer the ability to phrase the dynamics of an Experimental Prototype (EP) in multiple ways resulting in an equation describing the system mathematically. BPT provides the mathematical syntax upon which an equation may be constructed. An engineer designs one yielding a robust depiction of the EP. Parameters may be included or removed from the construct until an appropriate mathematical model is formulated. To derive the PV spectrum, we take the standard ZPF spectral energy density equation that describes the energy density in a region of space as a smooth cubic distribution and combine it with a Fourier distribution. This yields the beginning and endpoint of the spectrum. In other words, objects with mass polarise the ZPF which may be described as a Fourier distribution at the surface of the object. The surface is the equilibrium boundary between the energy contained within the 47

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object and the polarised state of the ZPF surrounding it. The difference between the EGM, PV and ZPF spectra is that the EGM spectrum commences incrementally above “0(Hz)” and approaches the Planck Frequency. The PV spectrum is mass specific and represents a bandwidth of the EGM spectrum commencing at a non-zero fundamental frequency. The EGM and PV spectra follow a Fourier distribution. The ZPF spectrum has the same frequency bandwidth of the EGM spectrum, but does not follow a Fourier distribution. So, the EGM spectrum is the polarised form of the ZPF spectrum, whilst the PV spectrum is an object specific subset of the EGM spectrum following a Fourier distribution. Note: the EGM spectrum is a simple, but extreme, extension of the EM spectrum.

ElectroMagnetic spectrum, DAT’s and BPT bring to the research and design table, the following key elements: [7] • It helps to assess the reasonableness of a model and which variables it should contain. • It reduces the number of variables and parameters to a minimum. • It reduces the number of needed experiments, on computers as well as in the laboratory. • It provides the fundamental theory behind experiments on scale models. • It is a systematic method for the analysis of problems. • It forces you to make estimates and to understand the problem. • It helps you understand what is important and what is not. • It produces dimensionless equations with small (or large) parameters. • It facilitates a “reverse engineering” approach to gravity if a region of space-time on a laboratory test bench is considered to be the Experimental Prototype (EP) for the mathematical model produced by the application of DAT’s and BPT. Subsequently, the mathematical model can be applied to the EP for scaling purposes, leading to gravity control experiments. Note: DAT’s and BPT should be applied before numerical computations are done. We develop a dynamic, kinematic and geometric equivalent of the ZPF, expressed in Fourier terms, which describes gravity at the surface of the Earth as a PV. The EGM spectrum is a simple, but extreme, extension of the EM spectrum. In the same way that radio waves, visible light, ultra violet, x-rays and gamma rays exist, gravitational waves exist as a spectrum of frequencies. The EGM spectrum is in fact the EM spectrum (subject to a Fourier distribution) but with an “end point” approaching the Planck Frequency at conditions of maximum permissible energy density. 48

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Typically, for the surface of the Earth for example, the vast majority of gravitational waves exist well above the Terahertz (THz) range. It is extremely important to note that gravity does not exist as a single wave; it exists as a spectrum of frequencies with a group propagation velocity of zero. EGM does not differentiate between EM and gravitational spectra but does predict the endpoint as being far above what we currently measure the EM spectrum to be. EGM Achievements We may indirectly test the validity of the EGM model with respect to gravity by utilising EGM to determine fundamental particle properties such as mass and radii. If we are able to make mathematical predictions for these characteristics which can be experimentally verified (as a litmus test), it follows that the EGM method is qualitatively and quantitatively validated. As it turns out, much in terms of mass and radii that is currently known in particle physics can be derived from first principles utilising EGM. In other words, the mathematical predictions made by the EGM method with respect to particle mass and radii have been experimentally verified, or at the very least (if not yet experimentally verified), satisfy Particle Data Group (PDG) mass-energy ranges. We are able to show utilising the EGM method that all particles (relative to an arbitrary selected base / reference particle) can be described as harmonic multiples of each other and indeed, all matter may be described in terms of Photons. For example, all flavours of Quarks may be described as exact harmonic multiples of the “Up or Down Quark”. Alternatively, they may also be described as exact harmonic multiples of the Electron. None of the particle predictions made herein, or EGM for that matter, contradict the Standard Model (SM) of particle physics in any way. Most importantly, EGM is the simple recognition of a mathematical pattern in nature. The amplitude spectrum within a Fourier series is comprised of an inverted harmonic series. The frequency spectrum within a Fourier distribution is a typical arithmetic sequence. If we assume that nature is truly quantum, we are able to find the fundamental spectral frequency and possess a method by which to describe the entire spectrum. This results in a very neat and complete harmonic description of the Universe. Effectively, the harmonic representation is “the power of one” (the number “1” represented harmonically with a Fourier distribution). It is possible to characterise objects with mass, say planets, by their spectral signature. This could be either the 49

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terminating or average spectral frequencies. Either way, it is characteristic of a spectral signature. The EGM model looks at mass and fields as harmonic wavefunction radiators. That is not to say that the Earth (for example) physically radiates waves because they are not detected radiating from planetary bodies. There is a mathematical argument for this as well as a common sense argument. The common sense argument being that the group velocity of the spectrum must be zero and that is why we do not detect waves being radiated. The mathematical argument is that because each mathematical point away from the object / planet is physically known to be in a constant gravitational state, the summation of a near infinite number of waves under a Fourier distribution results in a group propagation velocity of zero, which concurs with physical observation. Consequently, in the fullness of time, humanity may be able to develop gravitational telescopes that may be tuned to specific frequencies relating to masses etc. Alternatively, if the principle could be proven and the technology developed, it would be possible to differentiate between a genuine incoming ballistic missile threat and a highly reflective radar decoy with “100(%)” accuracy. EGM, in effect, means that Electricity, Magnetism and gravity have been unified from the sub-atomic level up to objects “about” the size of Neutron Stars, with approximately a “5(%)” error based upon the change in the Cosmological Constant in terms of energy density over small practical laboratory benchtop displacement values. We apply a calculated value of ZPF beat cutoff frequency “ωΩ ZPF” to derive the Casimir Force to high computational precision at the surface of the Earth (physically verified) and illustrate it to be different in other gravitational fields such as on the surface of other planets. Currently, its classical representation depicts its value as being constant throughout the Universe, independent of gravitational field strength. We show that the modes excluded from the plates (typically more modes are excluded on planets with lower gravitational field strengths) create a pressure imbalance that is responsible for the Casimir Force. This means that planets with greater gravitational field strengths have a compressed bandwidth which is shifted toward the upper end of the EGM spectrum. Hence, there are fewer low frequency modes and a greater number of higher frequency modes that simply pass through the plates, for gravitational fields of higher strength (its terminating spectral frequency will be greater).

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Note: EGM can also be utilised to produce repulsive Casimir Forces in accordance with current Finite Element Analysis (FEA) models. As mass increases, the PV spectrum is compressed (like a spring) such that the number of modes within that spectrum approaches unity and “ωPV(1,r,M) = ωΩ” in the case of a Planck particle. This was not a design goal in EGM development, it is a consequence of good mathematical formulation and has not been pre-empted in any way. By contrast, in free space where the spacetime manifold is completely flat (by analogy: the spring is decompressed due to the absence of mass), the ZPF is comprised of an infinite number of modes with “ωΩ” tending towards zero. That is, the ZPF spectrum is infinitely broad but bounded by a low frequency end-point. This arises from the notion that the fundamental harmonic frequency of a completely (or nearly completely) flat space-time manifold is extremely low [incrementally above “0(Hz)”]. For the sake of argument, the ZPF “may” have a fundamental frequency of “10-N(Hz)” (where “N” denotes a very large number), in which case if the harmonic cut-off mode “nΩ” was “near infinite”, say the “10Nth” mode, then “ωΩ” is still only “1(Hz)”! The same method is used to describe particle properties and to calculate the Casimir Force. Currently, no other methods are known to exist that can derive the Casimir Force from particle properties or vice versa based upon ZPF Theory or the PV model of gravity. One particular mathematical constant used in EGM is called Euler’s constant “γ” (Leonhard Euler (right): 1707 – 1783). This is a purely mathematical construct and currently has no physical meaning at the quantum level. We apply “γ” to calculate “nΩ” and “ωΩ” which is utilised to produce experimentally verified fundamental particle properties. Consequently, if the mass-energy of a Photon “mγγ” can be physically verified, the relevant equation may be transposed and solved for Euler’s Constant. Therefore, it may be possible to determine the natural physical limit of Euler’s Constant at the quantum level, implying that mathematics itself has a natural physical limit! EGM is also able to determine an experimentally implicit calculation of the Planck Scale [Max Planck (left): 1858 – 1947]. By EGM estimation, the Planck Scale is “about 16(%)” too small and the Bohr radius [Niels Bohr (right): 1885 – 1962] is “about 0.35(%)” too large. The experimentally implicit calculation of the Bohr radius is based upon the ZPF equilibrium state of the Hydrogen atom. This also leads to the first term of the Balmer series for the emission / absorption spectrum of the Hydrogen atom and by inference, the entire series may be derived. 51

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EGM Formulation Tips Note: apply financial investment logic, i.e. you will only find high return investments if you look where nobody else is looking to invest. Therefore, one should keep the following in mind: 1. GR is correct, but it is an ineffective engineering tool. 2. Nature follows basic mathematical patterns. 3. Planetary geometry may be usefully approximated to spherical. 4. Consider a volume of space-time on a practical laboratory benchtop and reverse engineer the gravitational contents of that volume. 5. Don’t try to describe what things actually are, only how we perceive them to be. 6. Gravity is a “1-D” phenomenon in human experience. No human being has ever directly experienced gravity as being “3-D”, even though astronomical observation shows it to be. 7. Be practical: IFF gravity can be controlled artificially, then by necessity, it must be EM in nature because ElectroMagnetism is the only practical tool available to us. 8. The EGM formulation process is an engineering approach where useful approximations are better than unworkable exact statements. “It is better to be approximately correct, than very precisely wrong.” 9. Imagine that all mass radiates a spectrum of Photons existing as conjugate pairs in accordance with Fourier mathematics in complex form. 10. Relate gravitational and EM acceleration via the equivalence principle. 11. The equivalence principle is a relativistic manifestation of DAT's and BPT. 12. Gravity arises from a change in energy density and physically manifests in terms of an EM Poynting Vector. 13. Humanity does not factually know the true nature of EM waves. 14. Recognise that the classical representation of an EM wave (i.e. a sinusoid) is merely a human representation of observed effects described mathematically. 15. If Electricity, Magnetism and gravity can be unified utilising sinusoidal wavefunctions, then the relationship between the volume of space-time to be reverse engineered and the mathematical model applied to describe it (i.e. sinusoidal wavefunctions), is ideal because it would actually be relating the same system twice. 16. Assume that unification manifests at the most fundamental level possible in nature such that all systems may be usefully described as linear at this level. 17. Recognise that, under ideal conditions, the relationship function typically determined experimentally when applying DAT’s and BPT (i.e. relating the experimental prototype to the mathematical model) has a value of unity. 18. DAT's and BPT are geometrically based. Since GR is also geometrically based, there is an obvious connection between GR, DAT's and BPT. 19. Assume that a ZPF equilibrium point exists (the point where the energy inside the object is in balance with the field energy outside the object). 20. Recognise that everything in nature behaves as a system. 21. Controls systems engineering principles should be considered at all times. Controls systems engineering is all about, as the name suggests, controlling the behaviour of machines, systems or situations. It is an extremely important function in the sphere of engineering. Without such a field of study, we wouldn’t have much of what we currently know. Image depicting feedback control loop, 52

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To exist in steady state is extremely important to nature, engineers and society. Nature itself, always wants to achieve steady state. Typically, this takes the form of the lowest possible energy level and the way it achieves this may be represented by negative feedback control loops in many situations. So, it is really quite important to look at things as systems and the information exchange between the system elements, feeding back error to achieve steady state. Tips for Applying EGM to Particle Physics Image depicting particles in a bubble chamber, One should keep the following in mind: 1. Recognise that there is no energy in fundamental particles (at rest) beyond their mass as is clearly illustrated by Einstein (E = mc2). There is no charge term in the equation; therefore, charge must be a physical manifestation of its “mass” in some unknown direct or indirect manner. 2. Spherical particle geometry is the natural shape of the lowest energy state. 3. If an observer was “on the surface” of a fundamental particle, it would appear spherical. 4. Special Relativity (SR) effects may be usefully neglected. If an observer is sufficiently far away, ellipsoidal distinction in not possible. Considering how small the sub-atomic scale is relative to the laboratory test bench and the human observer, supports this contention. 5. The equilibrium point of the ZPF will always be in the same frame of reference as the particle itself; hence the particle will always be approximately spherical relative to the ZPF equilibrium radius. Accidental Particle Property Predictions by EGM The possibility of the highly precise, experimentally verified particle mass and radii predictions made by EGM to be “luck or accidental” should be considered. We may apply common sense digestion of this possibility as well as develop mathematical arguments to determine the true likelihood of EGM predictions being a “fluke”. For simplicity and brevity, we shall consider the Proton RMS charge radius “rπ” and the Neutron Mean Square charge radius “KX” as both are regarded, by the particle physics community, to have been “precisely measured” [rπ = 0.8307(fm) and KX = –0.113(fm2)]. It is shown by EGM that “KX” may be converted to the RMS charge radius [rν = 0.8269(fm)]. Both particle radii predictions by EGM are within experimental uncertainty, so we shall consider them to be exactly correct physical values. If we consider the radii predictions to be a string of dimensionless digits based upon conversion of the “fm” scale, probability boundaries may be conjectured and represented as follows, • “rπ” becomes “0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-8-3-0-7” • “rν” becomes “0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-8-2-6-9”

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If each digit in the “rπ” string has a “1 in 10” chance of coming up, the probability of getting a string of “19” numbers correct is “1/10, 19 times”. Hence, the probability of “rπ” being a “fluke match” with the experimentally verified result is: “Pπ = 10-19”. Moreover, if we consider the Neutron as well, the total probability of both particles being numerically correct in relation to experimentally verified results, is equal to the probability of both particles being “fluked” and may be written as: “Pπ+ν = Pπ2 = 10-38”. In addition, if we apply the same rationale to the predictions of Electric and Magnetic radii of the Proton and the Magnetic radius of the Neutron by EGM, the total probability “PT” of error becomes even smaller. Furthermore, if we also consider the mass-energy predictions of the “Top Quark” by EGM, then “PT << 10-38”. Note: the total probability of the EGM method being in error and achieving experimentally verified results by “fluke” is trivial and may be usefully approximated to zero (PT → 0). 1.1.1 CURRENT PROBLEMS 1.1.1.1 PHYSICS There are several major stops currently facing ZPF Theory and the SM in particle physics that have been addressed by the development of the EGM construct herein. Some of these may be articulated as follows: 1. Dilemma: The precise spectral composition of the ZPF is unknown. •

Resolution: EGM resolves this problem by relating Fourier Harmonics to energy density via the PV model of gravity. This precisely defines the spectral composition of the ZPF at the surface of a solid spherical object of homogeneous mass-energy distribution.

2. Dilemma: The SM in particle physics does not allow for the existence of any new fundamental particles beyond current predictions. •

Resolution: EGM predicts the existence of three new Leptons (and associated Neutrino's) and two new Quarks or Bosons. However, it is likely that these are Intermediate Vector Bosons (IVB’s - force carriers).

3. Dilemma: Particle properties such as mass-energy and radii are completely unknown for many particles. •

Resolution: EGM facilitates the calculation of mass-energy and radii for all fundamental particles.

4. Dilemma: Particle properties such as mass-energy and radii are calculated in different ways, depending on the particle. That is, there is no uniformity of approach. •

Resolution: EGM facilitates the calculation of mass-energy and radii from a common footing. That is, a common method of solution is presented for all fundamental particles.

5. Dilemma: The Solar Neutrino mass detected at laboratories on Earth only accounts for “about half” of what should be ejected from the Sun according to the SM in particle physics. •

Resolution: The prediction of additional Leptons and associated Neutrino's by EGM may account for the absence of Solar Neutrino mass. 54

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6. Dilemma: The Planck Scale is a purely theoretical construct: •

Resolution: EGM produces an experimentally implicit definition of the Planck Scale.

7. Dilemma: The Bohr radius is a purely theoretical construct: •

Resolution: EGM produces an experimentally implicit definition of the Bohr radius based upon the ZPF equilibrium state of the Hydrogen atom. This also leads to the first term of the Balmer series for the emission / absorption spectrum of the Hydrogen atom and by inference, the entire series may be derived.

1.1.1.2 MATHEMATICS The following statement is a verbatim quotation from [2] The Euler-Mascheroni Constant (Euler's Constant “γ”) was first defined by Euler in 1735 (using the letter “C”) and stated that it was “worthy of serious consideration” and represents the limit of a harmonic sequence. The symbol “γ” was first used by Mascheroni in 1790. It is not known if the constant is irrational, let alone transcendental. It is rumoured that the famous English mathematician “G.H. Hardy” allegedly offered his chair at Oxford to anyone who can prove “γ” to be irrational, although no written reference to this quote seems to be known. “Hilbert” mentioned the irrationality of “γ” to be an unsolved problem that is “unapproachable”. End of verbatim quotation. Euler's Constant represents an extremely important characteristic in mathematics and cuts across many areas including Merten's Theorem and the Reimann Zeta Function. Currently, “γ” is only known to exist as a purely mathematical construct, therefore: 8. Dilemma: “γ” has no physical meaning: •

Resolution: EGM facilitates an experimentally implicit calculation of “γ” based upon the determination that the diameter of a Photon at rest is precisely the Planck Length.

1.1.2 HOW EGM WORKS To understand the way in which EGM works, one requires a basic knowledge of engineering principles. Primarily, EGM is a method of calculation (not a theory) based upon fundamental engineering principles and techniques. It does not compete with or contradict the SM in particle physics in any manner. The creation and development of EGM was driven by necessity. A scan of contemporary approaches in gravitational physics illustrates an obvious lack of mathematical tools facilitating engineering of the space-time manifold. Or rather, engineering possibilities are obvious, but require massive objects on a planetary, stellar or galactic scale. Therefore, to facilitate gravity control, a new tool is required permitting engineering of the space-time manifold. To begin the process, we must first make some basic assumptions based upon the availability and practicality of existing tools by which we may construct further tools. We shall use one tool to build another. EGM is nothing more than an engineering tool constructed from other engineering tools and should be always regarded as such. Engineering is fundamentally a practical discipline that does not search for highly precise numerical or exact results. Instead, it aims to achieve physically meaningful quantitative solutions. Again, practicality and common sense must prevail and, by necessity, must commence with the assumption that any realistic attempt at gravity control must physically fit on a laboratory test bench. There is no benefit in developing a tool requiring non-practical scales of reality.

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Einstein brought forth the concept that mass and energy are interchangeable. This is trivially obvious by virtue of his now famous equation (E = mc2). This, combined with practical thinking, clearly suggests that EM radiation is the mechanism of choice. Hence, we have established the basic requirements going forward. That is, we are necessarily bounded in research and design terms by practical benchtop EM fields. The next step is to find a tool that facilitates the construction of relationships tying EM fields to acceleration. For an experienced engineer, the answer is obvious. In situations where little has been established previously, Dimensional Analysis Techniques (DAT's) and Buckingham's Π Theory (BPT) are solid first steps. In addition to being able to connect seemingly unrelated parameters, it also serves to minimise the number of experiments required to investigate physical behaviour. BPT is a similarity method that has been tried and proven experimentally for many years. In fact, much of present day Thermodynamics and Fluid Mechanics knowledge may be attributed to DAT's and BPT. Mainstream understanding of gravity is based upon GR which is a geometric approach. It describes space-time curvature as a set of geometric artefacts resulting in what we experience as gravity. Assuming Einstein was correct and the enormous collective scientific effort since 1905 has not been a poor investment, it follows that any geometrically based engineering tool is an excellent starting point. Deeper understanding of BPT reveals that it is a method based upon dynamic, kinematic and geometric similarity. Being geometric in nature makes it ideally suited to gravitational problems in keeping with GR. However, a strict GR approach is unwieldy and a simpler description would be highly advantageous. Subsequently, we utilise the PV model of gravity as a substitute to GR, which is isomorphic in the weak field, is conducive to engineering approaches and facilitates the development of the EGM construct. Thus far, we have established several of the baseline elements forming a skeletal EGM structure. To add flesh, we require a way to relate the geometric output of BPT to the PV model of gravity. The relationship between the two may be bridged by assuming the equivalence principle applies cross discipline. Considering the need for an EM mechanism, we shall assume that the PV model of gravity denotes a polarised state of the ZPF representing a sinusoidal manifestation of the space-time manifold by virtual particles, Photons or wavefunctions. Consequently, it follows that the representation of gravity at a mathematical point by Fourier Harmonics is a useful tool by which to represent the ZPF. Therefore, we may relate the logic of the preceding arguments in a solution algorithm constituting the EGM construct by five simple steps as follows, i. Apply DAT's, BPT and similarity principles to combine Electricity, Magnetism and resultant EM acceleration in the form of Π groupings. ii. Apply the equivalence principle to the Π groupings formed in (i). iii. Apply Fourier Harmonics to the equivalence principle. iv. Apply ZPF Theory to Fourier Harmonics. v. Apply the PV model of gravity to the ZPF. These steps may be logically granulated according to the application of basic engineering principles producing an iterative cascade design approach as follows, Chapter 3.1:

Application of Dimensional Analysis Techniques and Buckingham Π Theory The relationship between EM fields and acceleration is demonstrated by the application of BPT. It is illustrated that, for physical modelling applications, manipulating the full spectrum of the PV is not required and optimal PV coupling may exist at specific frequency modes. This dramatically simplifies the design of experimental prototypes and suggests that the PV may be usefully approximated to a discrete wave spectrum by applying an intense 56

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superposition of fields within a single frequency mode. Chapter 3.2:

↓ The development of general modelling equations and the critical factor

Chapter 3.3:

A critical factor, proportional to the Poynting Vector, is identified by application of similarity principles. This may be determined by direct measurement of the intensity of the EM field strength at each harmonic frequency mode. ↓ The development of an engineered metric

Chapter 3.4:

Engineering expressions are developed for experimental investigations involving coupling between EM fields and gravity that may be characterised by the magnitude of the superposition of Poynting Vectors. Based upon dimensional similarity and the equivalence principle, it is concluded that an engineered acceleration may be used to modify the gravitational acceleration at the surface of the Earth by an engineered change in the value of the Refractive Index. ↓ The derivation of gravitational amplitude and frequency spectra

Chapter 3.5:

It is concluded that the delivery of EM radiation to a test object may be used to alter the weight of the object. If the test object is bombarded by EM radiation, at high energy density and frequency, the gravitational spectral signature of the test object may undergo constructive or destructive interference. ↓ The development of general similarity relationships It is concluded that the frequency dependent conditions for gravitational similarity at the surface of the Earth are enormous. Summarising yields: i. The ZPF spectrum of free space is composed of an infinite number of modes, with frequencies tending to “0(Hz)”. ii. The group velocity produced by the PV at a mathematical point and across practical values of “∆r” at the surface of the Earth is “0(m/s)”. Consequently, gravitational wavefunctions are not observed to propagate from the centre of a planetary body. iii. Planetary mass-energy density is proportional to the spectral energy density at its surface. iv. Gravitational acceleration exists (at practical benchtop experimental conditions / dimensions) as a relatively narrow band of beat frequencies in the “PHz” [x1015(Hz)] range. Spectral frequency compositions below this range [approx. less than 42(THz)] are negligible [similarity ≈ 0(%)]. v. General Similarity Equations facilitate the construction of computational models to assist in designing optimal experiments. Moreover, they can readily be coded into “off-the-shelf-3D-EM” simulation tools to facilitate the experimental investigation process. vi. A solution for optimal experimental similarity utilising EM configurations exists when Maxwell's Equations at steady state conditions are observed such that: (a) The divergence of the applied Electric field and curl of the applied Magnetic field is maximised. 57

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(b) The magnitude and curl of the applied Electric field is minimised. Chapter 3.6:

↓ The development of harmonic and spectral similarity relationships

Chapter 3.7:

A number of tools that facilitate the experimental design process are presented. These include the development of a design matrix based upon the unit amplitude spectrum, the derivation of Harmonic and Spectral Similarity Equations, Critical Phase Variance, Critical Field Strengths and Critical Frequency. ↓ The derivation of the Casimir effect An experimental prediction is formulated hypothesising the existence of a resonant modal condition for application to classical parallel plate Casimir experiments. The resonant condition is subsequently utilised to derive the Casimir Force to high precision. The results obtained suggest Casimir Forces arise due to PV pressure imbalance between the plates induced by the presence of a physical boundary excluding low energy harmonic modes.

Chapter 3.8:

↓ The derivation of the Photon mass-energy threshold It is illustrated that the PV model of gravity based upon the existence of a spectrum of frequencies makes the following predictions, i. The Photon mass-energy threshold for a mode normalised population of Photons is believed to be “< 5.75 x10-17(eV)”, based upon the physical properties of an Electron. ii. Experimental validation of the Photon mass-energy boundary predicted herein may be natural evidence of Euler’s Constant at a quantum level.

Chapter 3.9:

↓ The derivation of fundamental particle radii (Electron, Proton and Neutron) It is illustrated that the EGM model of gravity predicts experimentally supported RMS charge radii values of a free Electron, Proton and Neutron from an almost entirely mathematical foundation. Experimental predictions have been derived from first principles for the RMS charge radii of a free Electron, Proton and Neutron to high computational precision. This places the derived value of Proton radius to within “0.38(%)” of the average “Simon” and “Hand” predictions, arguably the two most precise and widely cited references since the 1960's. Most importantly, the SELEX Collaboration has experimentally verified the Proton radius prediction derived herein to extremely high precision {√[0.69(fm2)] = 0.8307(fm)}. The derived value of Electron radius compares favourably to results obtained in High-Energy scattering experiments conducted at “LANL”. It has also been illustrated that a change in Electron mass of “≈ +0.04(%)” accompanies the High-Energy scattering measurements. This suggests that the Electron radius depends on the manner in which it is measured and the energy absorbed by the Electron during the measuring process. The Fine Structure Constant is also derived, to within “0.026(%)” of its NIST value, utilising the Electron and Proton radii construct herein. In addition, it is predicted that the terminating gravitational spectral frequency for each 58

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particle may be expressed simply in terms of Compton frequencies. ↓ Chapter 3.10: The derivation of the Photon and Graviton mass-energies and radii The mass-energies and RMS charge diameters of a Photon and Graviton are derived. The results agree with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of Planck length. ↓ Chapter 3.11: The derivation of Lepton radii The RMS charge radii of all Leptons are derived to high computational precision; the Fine Structure Constant “α” is also derived to within “7.6 x10-3 (%)” of its NIST 2002 value. ↓ Chapter 3.12: The derivation of Quark and Boson mass-energies and radii The mass-energies and RMS charge radii of all Quarks are derived in agreement with PDG estimates, experimental observations and generalisations made by the ZEUS Collaboration (ZC). The “Top” Quark mass-energy derived is shown to be within “0.35(%)” of the value concluded by the D-Zero Collaboration (D0C). The RMS charge radii of the “W”, “Z” and Higgs Boson are also derived and it is illustrated that all flavours of Quarks and Bosons exist as exact harmonic multiples of the Electron. The derived harmonic relationships between the Lepton, Quark and Boson groups, suggests that all fundamental particles radiate populations of Photons at specific frequencies. ↓ Chapter 3.13: The derivation of an experimentally implicit definition of the Planck Scale, prediction of new particles and the design of an experiment to test the negative energy conjecture This chapter derives: i. An experimentally implicit increase of the Planck Scale. ii. An approximation of the RMS charge radius of a free Photon, utilising physical properties of the Lepton family, specifically all Electron-Like particles. iii. The existence of three (3) new particles in the Lepton family. iv. The existence of two (2) new particles in the Quark / Boson families. v. The optimal practical benchtop configuration of a Classical Casimir Experiment to test the negative energy conjecture. ↓ App. 3.D:

Derivation of Lepton radii A precise numerical result for all Lepton radii is achieved utilising the analytical representations in chapter 3.1 - 3.12 as boundary conditions.

App. 3.E:

↓ Derivation of Quark and Boson mass-energies and radii A precise numerical result for all Quark and Boson mass-energies and radii is achieved utilising the analytical representations in chapter 3.1 - 3.12 as boundary conditions. 59

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App. 3.G:

1

↓ Conversion of the Neutron Positive Core Radius

2

↓ Derivation of the Neutron Magnetic Radius

3

↓ Derivation of the Proton Electric Radius

4

↓ Derivation of the Proton Magnetic Radius

5

↓ Derivation of the Classical Proton RMS Charge Radius

App. 3.H:

↓ Calculation of L2, L3 and L5 Neutrino radii

App. 3.I:

↓ Derivation of the Hydrogen Atom Spectrum

An essential mathematical subroutine facilitating this process is the derivation of similarity equations. This is the most complicated procedure undertaken herein, is extremely important and may be articulated as follows, Dimensional Analysis Techniques (DAT's) ↓ Buckingham Π Theory (BPT) ↓ General Modelling Equations (GMEx) ↓

Amplitude and Frequency Spectra (CPV & ωPV) ↓ General Similarity Equations (GSEx) ↓ Harmonic Similarity Equations (HSEx) ↓ Reduced Harmonic Similarity Equations (HSEx R) ↓ 2nd Reduction of Harmonic Similarity Equations (Reduced Average Harmonic Similarity Equations) (HSExA R) ↓ Spectral Similarity Equations (SSEx) ↓ Fundamental Particle Properties, the Hydrogen Atom Spectrum and the Casimir Force 60

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1.2

KEY RESULTS AND FINDINGS

The most important results determined by the EGM construct may be categorised into five main areas as follows: i. Polarisable Vacuum and Zero-Point-Field. ii. Photons, Gravitons and Euler's Constant. iii. All other particles. iv. The Casimir Force. v. The Planck Scale and the Bohr radius. Hence (equation numbers appear on the RHS of the page): PV and ZPF •

Gravitational amplitude spectrum “CPV” G.M .

C PV n PV, r , M

2

r

•

(3.64)

Gravitational frequency spectrum “ωPV” n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

•

2 . π n PV

(3.67)

Harmonic cut-off frequency “ωΩ” ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73)

Photons, Gravitons and Euler's Constant Note: Euler's Constant “γ” may be calculated by: (i) physical measurement of “mγγ” and (ii), the assumption that “2 x rγγ” is precisely equal to the experimentally implicit value of the Planck Length characterised by “Kλ x λh”. •

The mass-energy of a Graviton “mgg” mgg = 2mγγ

•

The mass-energy of a Photon “mγγ” 3

h .

m γγ

re

•

(3.216)

3

π .r e 2 .c .G.m e

.

512.G.m e

2

.

c .π

n Ω r e, m e ln 2 .n Ω r e , m e

2

γ

2

(3.220)

The radius of a Photon “rγγ” 5

2

m γγ

r γγ r e .

m e .c

r γγ K ω .

2

(3.225)

G.h . r µ c

61

3

rτ

(3.274)

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•

The radius of a Graviton “rgg” 5

r gg

4 .r γγ

(3.227)

All Other Particles •

The Fine Structure Constant “α” α

rε

2

.e

3

rπ

(3.204) rµ

α

rε

rτ

.e

rν

•

(3.236)

Harmonic cut-off frequency ratio (the ratio of two particle spectra) “Stω” 2

•

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

9

r2

St ω

r1

(3.230)

Neutron Magnetic Radius “rνM” r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

•

(3.420)

Proton Electric Radius “rπE” r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

•

(3.423)

Proton Magnetic Radius “rπM” ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

•

Classical Proton Root Mean Square Charge Radius “rp” r P r πE

•

(3.426)

1. 2

r νM

rν

(3.429)

The first term of the Hydrogen Spectrum (Balmer Series) “λA” [by EGM] utilising the Bohr radius “rBohr” and the fundamental PV wavelength “λPV” λA

λ PV 1 , K ω .r Bohr , m p

2 .n Ω K ω .r Bohr , m p

62

(3.457) www.deltagroupengineering.com

•

EGM Prediction versus Experimental Measurement Particle / Atom EGM Prediction Experimental Measurement Proton (p) rπ = 830.5957 x10-16(cm) rπ = 830.6624 x10-16(cm) rπE = 848.5274 x10-16(cm) rπE = 848 x10-16(cm) rπM = 849.9334 x10-16(cm) rπM = 857 x10-16(cm) rp = 875.0 x10-16(cm) rp = 874.5944 x10-16(cm) Neutron (n) rν = 826.8379 x10-16(cm) rX ≈ 825.6174 x10-16(cm) -26 2 KS = -0.1133 x10 (cm ) KX = -0.113 x10-26(cm2) rνM = 878.9719 x10-16(cm) rνM = 879 x10-16(cm) Top Quark (tq) mtq(GeV) ≈ 178.4979 mtq(GeV) ≈ 178.0 Hydrogen (H) λA = 657.3290(nm) λB = 656.4696(nm) rx = 0.0527(nm) rBohr = 0.0529(nm) Particle Summary Matrix 3.1,

(%) Error < 0.008 < 0.062 < 0.825 < 0.046 < 0.148 < 0.296 < 0.003 < 0.280 < 0.131 < 0.353

Note: “rp = 875.0 x10-16(cm)” [i.e. the classical RMS charge radius of the Proton] and “rBohr = 0.0529(nm)” [i.e. the Bohr radius] are not experimental values, they denote the official values listed by the National Institute of Standards and Technology (NIST). [1] The Casimir Force The Casimir Force by EGM “FPV” is derived to within “0.01(%)” of its historically predicted value, whilst experimental evidence confirming the existence of the force has a “5(%)” measure of uncertainty. [8] F PV A PP , r , ∆r , M

A PP .∆U PV( r , ∆r , M ) .

N C( r , ∆r , M ) N X( r , ∆r , M )

2

.ln

N X( r , ∆r , M ) N C( r , ∆r , M )

4

(3.179)

The Experimentally Implicit Planck Scale Experimentally implicit modification factors for the Planck Scale are derived based upon experimentally verified particle radii predictions and may be articulated as follows: 3

Kω

Kω

Kω

Such that: i. Planck Frequency becomes: ii. Planck Length becomes: iii. Planck Mass becomes:

2 π

(3.270)

1 Kλ

(3.264)

1 Km

(3.265)

Kω x ωh Kλ x λh Km x mh

The Prediction of New Particles EGM predicts the existence of new particles beyond the Standard Model (SM). This includes, but is not limited to, i. 3 Leptons with mass-energies of “9(MeV), 57(MeV) and 566(MeV)”. ii. 3 Neutrino's with mass-energies approximating the Electron, Muon and Tau Neutrino’s. 63

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iii. 2 Quarks or Bosons with mass-energies of “10(GeV) and 22(GeV)”. However, it is likely that these are Intermediate Vector Bosons (IVB's → force carriers). The Experimentally Implicit Bohr Radius An experimentally implicit definition of the Bohr radius is presented in “Appendix 3.I” based upon exact correlation between the first term of the Hydrogen atom spectrum (Balmer series) predicted by EGM, to the experimentally verified value. 1.3

BUILDING AN EXPERIMENT

An experimental configuration is presented in chapter 3.13 based upon a Classical Casimir Experiment. It is suggested that a physical experiment, in accordance with the characteristics presented in the proceeding table, may reveal new and exciting phenomena for further investigation and may take several manifestations. The primary focus of the proposed experiment is to the negative energy conjecture argued to exist in ZPF Theory. It is currently unknown if energy can be efficiently and usefully extracted from the ZPF, however, the experiment suggested represents a point of mathematical interest based upon the derivation of the Casimir Force presented in chapter 3.7. Alternatively and probably more likely, a carefully configured experiment based upon XRay Laser wavelengths may produce a gravitational effect on a test object. If this can be experimentally observed and verified, then the EGM construct and the notion that all masses are wavefunction radiators may be transposed from calculation methodology to physical theory. Characteristic ∆r λX Erms Brms φ4,5

Description Value Plate separation ≈ 16.5 x10-3 Inflection wavelength ≈ 1.8 x10-8 Critical Electric field strength ≈ 550 Critical Magnetic field strength ≈ 1.8 x10-6 Critical phase variance = 0, ±π or ±π/2 Design Specification Matrix 3.1,

Units m V/m T θc

NOTES

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2 •

EGM CONSTRUCT PROCESS SUMMARY Modelling Foundations: Chapter 3.1 - 3.5

1. Assume a relationship exists between Electricity, Magnetism and resultant ElectroMagnetic (EM) acceleration. 2. Apply Dimensional Analysis Techniques (DAT's) and Buckingham's Π Theory (BPT) to combine Electricity, Magnetism and resultant EM acceleration. 3. Apply the equivalence principle to the resultant EM acceleration. 4. Assume a Zero-Point-Field (ZPF) exists in the space-time manifold. 5. Combine ZPF Theory with the equivalence principle to conclude that gravity is a spectrum of frequencies that can be investigated at specific conditions satisfying BPT. 6. Assume that the PV model of gravity denotes the polarisation of the ZPF by the presence of mass. 7. Identify that the Polarisable Vacuum (PV) model of gravity is a concise isomorphic representation of General Relativity (GR) in terms of a Refractive Index. 8. Assume dimensional similarity in accordance with BPT and apply a Fourier representation of a constant function of a mathematical point to the PV model of gravity (i.e. a gravitational field is equivalent to a spectrum of frequencies radiating into space). 9. Assume that the radiated spectrum is equivalent to the space-time manifold described by GR. 10. Assume that the radiant energy is equal to the ZPF spectral energy (the ZPF spectrum is considered continuous). 11. Consequently, the spectral composition of the PV is derived via an operation analogous to a gauge transformation (spectral compression) and is determined to be finite and discrete. 12. Assume that the PV spectrum across an elemental displacement forms beats due to the difference in spectra across the element. 13. Amalgamate the two spectra (PV and ZPF) across the elemental displacement. •

The Casimir Force: Chapter 3.6, 3.7

14. Formulate expressions for harmonic similarity between an applied EM field and the fundamental beat frequency of the PV across an elemental displacement. Subsequently, this leads to the formulation of expressions for spectral similarity between applied EM fields and the complete PV spectrum. 15. Identify that the sum of all modes of a double-sided reciprocal harmonic spectrum, about the “0th” mode, approaches the sum of all modes of a one-sided reciprocal harmonic spectrum with vanishing error. 16. Consequently, the Casimir Force is derived coinciding with experimental measurement. •

Mass-Energy and Radii of Photons and Gravitons: Chapter 3.8, 3.10

17. Identify that the PV is a double-sided frequency spectrum, extending from negative infinity to positive infinity. 18. Identify that, in the Complex Frequency domain, either side of the spectrum is a conjugate representation of the alternate side.

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19. Identify that the Real Component of gravitational acceleration in the PV model on the Complex Plane is always positive. 20. Assume the PV to consist of conjugate Photon pair populations. 21. Identify that Electrons are natural Photon emitters. 22. Assume that Electrons at rest radiate populations of Photons continuously. 23. Assume that an Electron at rest has spherical geometry. 24. Assume that the amplitude spectrum of the Fourier distribution applied to the PV model of gravity is proportional to the conjugate Photon pair population. 25. Assume that one conjugate Photon pair defines a Graviton. 26. Derive the quantity of gravitational energy being radiated as Gravitons (conjugate Photon pairs) per fundamental harmonic spectral period. 27. Identify that, due to the mathematical nature of Fourier harmonics for constant functions, Gravitons only exist at odd frequency modes. The sum of all even modes equals zero. 28. Identify that there are half as many odd modes as there are “odd + even” modes in a Fourier distribution. 29. Identify that the Graviton to Photon mass-energy ratio equates to half the sum of a one-sided reciprocal harmonic spectrum. 30. Consequently, the Photon mass-energy threshold is derived coinciding with experimental observation. 31. Assume the Photon mass-energy threshold is accurately calculated. 32. Assume that the terminating spectral frequency of the PV for an Electron is equal to the frequency of a single Photon. 33. Consequently, the Photon and Graviton mass-energies and radii are derived. •

Mass-Energy and Radii of all other Standard Model Particles: Chapter 3.9, 3.11 - 3.12 Appendix 3.D, 3.E, 3.G

34. Assuming the spectral distribution derived for the PV model of gravity is correct, it follows that the ratio of two spectra of two solid spherical masses must be proportionally related by similarity in accordance with BPT. 35. Identify that, at a fundamental particle level in nature, mass-energy is a unifying property. 36. Assume that the terminating spectral frequency of the PV is a proportional measure of the massenergy of a fundamental particle. 37. Formulate a generalised relationship for the ratio of two terminating spectral frequencies. 38. Identify the formation of mathematical patterns. 39. Consequently, all fundamental mass-energies and radii may be derived coinciding with experimental measurement (where applicable). •

The Planck Scale: Chapter 3.13

40. Identify the constants used to define Planck Frequency, Length and Mass. 41. Apply standard Dimensional Analysis Techniques (DAT's) and BPT. 42. Assume the derived spectrum describing the PV is correct. 43. Solve for experimental relationship functions. 44. Consequently, an experimentally implicit Planck Scale is derived.

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•

Theoretical Particles beyond the Standard Model: Chapter 3.13

45. Assume the ratio of two spectra of two solid spherical masses must be proportionally related by similarity in accordance with BPT. 46. Determine the average Electron-like Lepton radii based upon previous calculations. 47. Determine the average Boson / Quark radii based upon previous calculations and available experimentally implied or verified data. 48. Solve the spectra ratio equations for mass-energy at the appropriate harmonic conditions. 49. Consequently, multiple new particles are theorised beyond the Standard Model. •

Designing and Conducting an Experiment: Chapter 3.13

50. Assume an experimental configuration analogous to a classic Casimir experiment. 51. Determine the optimal separation distance. 52. Determine the inflection wavelength and frequency. 53. Determine Critical Field Strengths. 54. Determine Critical Phase Variance. 55. Trap EM energy by reflection at the inflection frequency and Critical Phase Variance inside the cavity. 56. Permit the Root Mean Square (RMS) intensities of the Electric and Magnetic fields inside the cavity to attain Critical Field Strength. 57. Ensure that the Electric and Magnetic field vectors are orthogonal inside the cavity. 58. Ensure that a standing wave forms in three dimensions (3-D) within the cavity. 59. Ensure any unexpected effects / events are observed. •

The Bohr Radius:

Appendix: 3.I 60. Assume the Bohr radius defines a usefully approximate position of the ZPF equilibrium radius. 61. Assume that the fundamental wavelength of the PV spectrum of the Hydrogen atom coincides with the longest wavelength in the Balmer series. 62. Assume that the Hydrogen atom may be usefully represented by an “imaginary particle” (spherical) of Bohr radius with approximately the mass of the Atomic mass constant. 63. Assume that the ZPF mass-energy within this “imaginary particle” (at approximately “rBohr”) is in equilibrium with the “imaginary field” surrounding the particle. That is, an “imaginary field exists” at approximately the Bohr radius. 64. Derive the appropriate mathematical relationship. 65. Substitute the experimental value for the first term of the Hydrogen atom spectrum (Balmer series) into the relationship derived (considering the Planck re-scaling factor derived in chapter 3.13). 66. Consequently, an experimentally implicit definition of the Bohr radius is derived.

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3

PARTICLE SUMMARY MATRICES

3.1

DETAILED MATRIX - UTILISING 2005 PDG DATA

Existing Particle Proton (p) Derived in Ch. 3.9

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.212) 5

2

rε

. c .e r e ω Ce

3

Mass-Energy Harmonic Cut-Off Freq. National Institute of Standards & Technology ωΩ(rπ,mp) = 2.6174 x1027(Hz) (NIST) [1]: mp(MeV) = 938.272029

rπ

2 4 c . ω Ce 27. ω h ω Ce . . 3 4 ω 4 . ω CP 32. π CP

rπ ≈ 830.5952 ± 0.0004 Experimental Measurement: [9] rp = 830.6624 ± 12

Neutron (n)

Interpretation: Satisfactory result → in agreement with experimental measurement. EGM Prediction: Equation (3.215) NIST: mn(MeV) = 939.565360

Derived in Ch. 3.9

ωΩ(rν,mn) ≈ ωΩ(rπ,mp)

5

rπ

rε π

2 4 c . ω Ce 27. ω h ω Ce . . rν 3 4 ω 4 . ω CN 32. π CN

rν ≈ 826.8898 ± 0.0519 Experimental Measurement: [10] rν ≈ 825.4152 ± 18.3 (see Appendix 3.G) Interpretation: Satisfactory result → in agreement with experimental measurement. 68

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Existing Particle Electron (e)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.231)

Derived in Ch. 3.11 Stω = 2 utilising “rπ” from 5 Ch. 3.9 rε rπ.

1

.

St ω

Muon (µ µ)

me

≈ 11.8024

Interpretation: Satisfactory result → in agreement with scattering experiments conducted by Los Alamos National Laboratory: [11] EGM Prediction: Equation (3.234) NIST: mµ(MeV) = 105.6583692

5

1 St ω

Tau (ττ)

.

mµ

2

≈ 8.2122

me

9

Interpretation: Insufficient scientific opinion available. EGM Prediction: Equation (3.234)

NIST: mτ(MeV) = 1776.99

ωΩ(rτ,mτ) = 6 ωΩ(rε,me) ωΩ(rτ,mτ) = 12 ωΩ(rπ,mp)

Derived in Ch. 3.11 Stω = 6 utilising “rε” therein 5

rτ rε.

ωΩ(rµ,mµ) = 4 ωΩ(rε,me) ωΩ(rµ,mµ) = 8 ωΩ(rπ,mp)

Derived in Ch. 3.11 Stω = 4 utilising “rε” therein rµ rε.

Harmonic Cut-Off Freq. ωΩ(rε,me) = 2 ωΩ(rπ,mp)

2

mp

9

Mass-Energy NIST: me(MeV) = 0.510998918

1 St ω

. 9

mτ me

2

≈ 12.2407

Interpretation: Insufficient scientific opinion available. 69

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Existing Particle RMS Charge Radius x10-16(cm) Electron Neutrino (ν νe) EGM Prediction: Equation (3.238) 5 Derived in Ch. 3.11 m en utilising “rε” therein r en r ε .

≈ 0.0954

ren2 ≈ 9.0971 x10-3 [x10-32(cm2)] Experimental Measurement: -5.5 ≤ 〈 rA2(νe)[x10-32(cm2)] 〉 ≤ 9.8

Where: δm = 10-100 Particle Data Group (PDG) Expectation: [12] men(eV) < 3 Interpretation: Satisfactory assumption → in agreement with PDG expectation.

Interpretation: Satisfactory result → in agreement with an extensive review of experimental data by “Hirsch et. Al.”. [13] EGM Prediction: Equation (3.238) Mass-Energy value utilised for radius calculation: mµn(MeV) ≈ 0.19 - δm

5 Derived in Ch. 3.11 m µn utilising “rε” therein r µn r µ . m

Harmonic Cut-Off Freq. ωΩ(ren,men) = ωΩ(rε,me) ωΩ(ren,men) = 2 ωΩ(rπ,mp)

2

me

Muon Neutrino (ν νµ)

Mass-Energy Mass-Energy value utilised for radius calculation: men(eV) ≈ 3 - δm

ωΩ(rµn,mµn) = ωΩ(rµ,mµ) ωΩ(rµn,mµn) = 4 ωΩ(rε,me)

2

≈ 0.6552

µ

rµn2 ≈ 4.2933 [x10-33(cm2)] Experimental Measurement: -5.2 ≤ 〈 rA2(νµ)[x10-33(cm2)] 〉 ≤ 6.8

PDG Expectation: mµn(MeV) < 0.19

ωΩ(rµn,mµn) = 8 ωΩ(rπ,mp)

Interpretation: Satisfactory assumption → in agreement with PDG expectation.

Interpretation: Satisfactory result → in agreement with an extensive review of experimental data by “Hirsch et. Al.”.

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Existing Particle Tau Neutrino (ν ν τ)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.238)

5 Derived in Ch. 3.11 m τn utilising “rε” therein r τn r τ .

mτ

Harmonic Cut-Off Freq. ωΩ(rτn,mτn) = ωΩ(rτ,mτ) ωΩ(rτn,mτn) = 6 ωΩ(rε,me)

2

≈ 1.9587

rτn2 ≈ 3.8364 [x10-32(cm2)] Experimental Measurement: -8.2 ≤ 〈 rA2(ντ)[x10-32(cm2)] 〉 ≤ 9.9

Up Quark (uq)

Mass-Energy Mass-Energy value utilised for radius calculation: mτn(MeV) ≈ 18.2 - δm PDG Expectation: mτn(MeV) < 18.2

ωΩ(rτn,mτn) = 12 ωΩ(rπ,mp)

Interpretation: Satisfactory assumption → in agreement with PDG expectation.

Interpretation: Satisfactory result → in agreement with an extensive review of experimental data by “Hirsch et. Al.”. EGM Prediction: Equation (3.242) EGM Prediction: Equation (3.246)

1 Derived in Ch. 3.12 5 2 m dq utilising “rε” from Ch. r 3 .r xq. 2 ≈ 0.7682 m uq 3.11 and “rπ” from uq Ch. 3.9 Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data by the ZEUS Collaboration. [14]

ωΩ(ruq,muq) = 7 ωΩ(rε,me) ωΩ(ruq,muq) = 14 ωΩ(rπ,mp)

Stω = 7

9 m uq m e . St ω .

r uq rε

5

≈ 3.5083(MeV)

PDG Expectation: 1.5 < muq(MeV) < 4 Interpretation: Satisfactory result → in agreement with PDG expectation.

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Existing Particle Down Quark (dq)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.241)

Derived in Ch. 3.12 Stω = 1 utilising “rε” from Ch. 5 3.11 and “rπ” from 1 . m dq Ch. 3.9 r dq r uq . St ω

≈ 1.0136

Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data by the ZEUS Collaboration. Strange Quark (sq)

EGM Prediction: Equation (3.244)

Derived in Ch. 3.12 Stω = 2 utilising “rε” from Ch. 5 3.11 and “rπ” from Ch. 3.9 r sq r uq .

Harmonic Cut-Off Freq. ωΩ(rdq,mdq) = ωΩ(ruq,muq)

Stω = 7

ωΩ(rdq,mdq) = 7 ωΩ(rε,me)

2

m uq

9

Mass-Energy EGM Prediction: Equation (3.246)

9 m dq m e . St ω .

r dq

ωΩ(rdq,mdq) = 14 ωΩ(rπ,mp)

5

≈ 7.0166(MeV)

rε

PDG Expectation: 4 < mdq(MeV) < 8 Interpretation: Satisfactory result → in agreement with PDG expectation. EGM Prediction: Equation (3.246) ωΩ(rsq,msq) = 2 ωΩ(ruq,muq) ωΩ(rsq,msq) = 14 ωΩ(rε,me)

Stω = 14 1 St ω

. 9

m sq m uq

2

≈ 0.8879

Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data gathered by the ZEUS Collaboration.

m sq m e . St ω

9.

r sq

ωΩ(rsq,msq) = 28 ωΩ(rπ,mp)

5

rε

≈ 114.0201(MeV)

PDG Expectation: 80 < msq(MeV) < 130 Interpretation: Satisfactory result → in agreement with PDG expectation.

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Existing Particle Charmed Quark (cq)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.244)

Derived in Ch. 3.12 Stω = 3 utilising “rε” from Ch. 5 3.11 and “rπ” from Ch. 3.9 r cq r uq .

1 St ω

. 9

m cq

≈ 1.0913

m uq

EGM Prediction: Equation (3.244)

Derived in Ch. 3.12 Stω = 4 utilising “rε” from Ch. 5 3.11 and “rπ” from 1 . m bq Ch. 3.9 r bq r uq . St ω

9

m uq

Harmonic Cut-Off Freq. ωΩ(rcq,mcq) = 3 ωΩ(ruq,muq)

Stω = 21

ωΩ(rcq,mcq) = 21 ωΩ(rε,me)

2

Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data gathered by the ZEUS Collaboration. Bottom Quark (bq)

Mass-Energy EGM Prediction: Equation (3.246)

9 m cq m e . St ω .

r cq

ωΩ(rcq,mcq) = 42 ωΩ(rπ,mp)

5

≈ 1.1841(GeV)

rε

PDG Expectation: 1.15 < mcq(GeV) < 1.35 Interpretation: Satisfactory result → in agreement with PDG expectation. EGM Prediction: Equation (3.246) ωΩ(rbq,mbq) = 4 ωΩ(ruq,muq) ωΩ(rbq,mbq) = 28 ωΩ(rε,me)

Stω = 28 2

≈ 1.071

Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data gathered by the ZEUS Collaboration.

m bq m e . St ω

9.

r bq

ωΩ(rbq,mbq) = 56 ωΩ(rπ,mp)

5

rε

≈ 4.1223(GeV)

PDG Expectation: 4.1 < mbq(GeV) < 4.4 Interpretation: Satisfactory result → in agreement with PDG expectation.

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Existing Particle Top Quark (tq)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.247)

Derived in Ch. 3.12 Stω = 10 utilising “rε” from Ch. 5 3.11 and “rπ” from 1 . m tq Ch. 3.9 r tq r uq . 9 St ω m uq

≈ 0.9294

W Boson

EGM Prediction: Equation (3.251)

Derived in Ch. 3.12

Stω = 7

r W r uq

.

1 St ω

. 9

mW m uq

Harmonic Cut-Off Freq. ωΩ(rtq,mtq) = 10 ωΩ(ruq,muq)

Stω = 70

ωΩ(rtq,mtq) = 70 ωΩ(rε,me)

2

Interpretation: Satisfactory result → in agreement with generalised conclusions based upon experimental data gathered by the ZEUS Collaboration.

5

Mass-Energy EGM Prediction: Equation (3.246)

2

9 m tq m e . St ω .

r tq rε

ωΩ(rtq,mtq) = 140 ωΩ(rπ,mp)

5

≈ 178.6141(GeV)

D-ZERO Collaboration: [15] mtq(GeV) = 178.0 ± 4.3 PDG Expectation: 169.2 < mtq(GeV) < 179.4 Interpretation: Satisfactory result → in agreement with D-ZERO Collaboration and PDG expectation. Mass-Energy value utilised for radius calculation: ωΩ(rW,mW) = 7 ωΩ(ruq,muq) mW(GeV) ≈ (80.387 + 80.463) / 2 ≈ 80.425 ωΩ(rW,mW) = 49 ωΩ(rε,me) PDG Expectation: 80.387 < mW(GeV) < 80.463 ωΩ(rW,mW) = 98 ωΩ(rπ,mp)

≈ 1.2835

Interpretation: Satisfactory result → in agreement with Heisenberg Uncertainty Range. [16]

Interpretation: Satisfactory assumption → in agreement with PDG expectation.

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Existing Particle Z Boson

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.251)

Derived in Ch. 3.12

Stω = 8 r Z r uq .

1 St ω

(Theoretical) Derived in Ch. 3.12

Harmonic Cut-Off Freq. ωΩ(rZ,mZ) = 8 ωΩ(ruq,muq) ωΩ(rZ,mZ) = 56 ωΩ(rε,me)

5

Higgs Boson (H)

Mass-Energy Mass-Energy value utilised for radius calculation: mZ(GeV) ≈ (91.1855 + 91.1897) / 2 ≈ 91.1876

. 9

mZ m uq

PDG Expectation: 91.1855 < mZ(GeV) < 91.1897

2

ωΩ(rZ,mZ) = 112 ωΩ(rπ,mp)

≈ 1.0613 Interpretation: Satisfactory assumption → in agreement with PDG expectation.

Interpretation: Satisfactory result → in agreement with Heisenberg Uncertainty Range. EGM Prediction: Equation (3.251) Mass-Energy value utilised for radius calculation: ωΩ(rH,mH) = 9 ωΩ(ruq,muq) mH(GeV) ≈ 114.4 + δm Stω = 9 ωΩ(rH,mH) = 63 ωΩ(rε,me) PDG Expectation: 5 mH(GeV) > 114.4 2 ωΩ(rH,mH) = 126 ωΩ(rπ,mp) 1 . mH . ≈ 0.9401 r H r uq 9 Interpretation: St ω m uq Satisfactory assumption → in agreement with PDG expectation. Interpretation: Satisfactory result → in agreement with Heisenberg Uncertainty Range.

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Existing Particle Photon (γγ)

RMS Charge Diam. (Planck Lengths) EGM Prediction: Equation (3.225)

5 Derived in Ch. 3.8, m γγ 3.10 2 .r γγ 2 .r e . 2

m e .c

2

≈ Planck Length

Planck Length ≈ 4.0513 x10-35(m) (Plain “h” form)

Mass-Energy EGM Mass-Energy Threshold: Equation (3.193) mγ <

512.h .G.m e c . π .r e

2

.

Harmonic Cut-Off Freq. Not Applicable

n Ω r e, m e ln 2 .n Ω r e , m e

γ

mγ < 5.75 x10-17(eV) PDG Mass-Energy Threshold: mγ < 6 x10-17(eV)

Interpretation: Satisfactory result → implicitly supports Interpretation: theories of Quantum Mechanics. Satisfactory result → in agreement with PDG Mass-Energy Threshold expectation. EGM Mass-Energy: Equation (3.220) 3

m γγ

h . re

3

π .r e 2 .c .G.m e

.

512.G.m e c .π

2

2

.

n Ω r e, m e ln 2 .n Ω r e , m e

γ

2

mγγ ≈ 3.2 x10-45(eV)

Graviton (γγg)

EGM Prediction: Equation (3.227)

Interpretation: Satisfactory result → in agreement with PDG Mass-Energy Threshold expectation. Not Applicable EGM Prediction: Equation (3.216)

(Theoretical)

5 2 .r gg 2 . 4 .r γγ

m gg 2 .m γγ

Derived in Ch. 3.10

≈ 1.5 x Planck Length

≈ 6.4 x10-45(eV)

Interpretation: Interpretation: Satisfactory result → implicitly supports Insufficient scientific opinion available. theories of Quantum Mechanics. 76

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Theoretical Particle L2 (Lepton)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.276, 3.278)

rµ rε rτ Derived in Ch. 3.13 ≈ 10.7518 utilising results from r L 3 Ch. 3.11

Mass-Energy EGM Prediction: Equation (3.279, 3.280)

Harmonic Cut-Off Freq. ωΩ(rL,mL(2)) = 2 ωΩ(rε,me)

Stω = 2

ωΩ(rL,mL(2)) = 4 ωΩ(rπ,mp) 9 m e . St ω .

rL

5

Interpretation: m L St ω ≈ 9(MeV) rε Insufficient scientific opinion available. The Standard Model (SM) in particle physics does not predict this average Interpretation: Insufficient scientific opinion available. The SM value. in particle physics does not predict the existence of the L2 particle. Note: L3 (Lepton) It is possible that associated Neutrino's EGM Prediction: Equation (3.281) ωΩ(rL,mL(3)) = 3 ωΩ(rε,me) exist for the L2, L3 and L5 particles Derived in Ch. 3.13 predicted herein. Stω = 3 ωΩ(rL,mL(3)) = 6 ωΩ(rπ,mp) utilising results from Ch. 3.11 In the proceeding Periodic Table of mL(3) ≈ 57(MeV) Elementary Particles, L2, L3 and L5 Neutrino mass-energy values have been Interpretation: assumed based upon radii calculations Insufficient scientific opinion available. The SM contained in Appendix 3.H in particle physics does not predict the existence of the L3 particle. EGM Prediction: Equation (3.282) L5 (Lepton) ωΩ(rL,mL(5)) = 5 ωΩ(rε,me) Derived in Ch. 3.13 utilising results from Ch. 3.11

ωΩ(rL,mL(5)) = 10 ωΩ(rπ,mp)

Stω = 5 mL(5) ≈ 566(MeV) Interpretation: Insufficient scientific opinion available. The SM in particle physics does not predict the existence of the L5 particle. 77

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Theoretical Particle QB5 (Quark or Boson)

RMS Charge Radius x10-16(cm) EGM Prediction: Equation (3.284)

Mass-Energy EGM Prediction: Equation (3.283, 3.285)

Harmonic Cut-Off Freq. ωΩ(rQB,mQB(5)) = 5 ωΩ(ruq,muq)

rQB = 〈r〉 ≈ 1.0052

Stω = 5

ωΩ(rQB,mQB(5)) = 35 ωΩ(rε,me)

Derived in Ch. 3.12

QB6 (Quark or Boson)

Interpretation: ωΩ(rQB,mQB(5)) = 70 ωΩ(rπ,mp) 5 r 9 . QB Satisfactory result → in agreement with m . ≈ 10(GeV) QB St ω m uq St ω r uq Heisenberg Uncertainty Range, scientific expectation and experimental evidence to date. [16] Interpretation: Insufficient scientific opinion available. The SM in particle physics does not predict the existence of the QB5 particle. EGM Prediction: Equation (3.286) ωΩ(rQB,mQB(6)) = 6 ωΩ(ruq,muq) Stω = 6

ωΩ(rQB,mQB(6)) = 42 ωΩ(rε,me)

mQB(6) ≈ 22(GeV)

ωΩ(rQB,mQB(6)) = 84 ωΩ(rπ,mp)

Derived in Ch. 3.12

Interpretation: Insufficient scientific opinion available. The SM in particle physics does not predict the existence of the QB6 particle.

Particle Summary Matrix 3.2, 78

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3.2 •

CONCISE MATRIX EGM Harmonic Representation of Particles

Existing and Theoretical Particles Proton Harmonics Electron Harmonics x ωΩ(rπ,mp) x ωΩ(rε,me) Proton (p), Neutron (n) Stω = 1 Stω = 1/2 2 1 Electron (e), Electron Neutrino (ν νe ) L2 (Theoretical Lepton) 4 2 L3 (Theoretical Lepton) 6 3 8 4 Muon (µ µ), Muon Neutrino (ν νµ) L5 (Theoretical Lepton) 10 5 12 6 Tau (ττ), Tau Neutrino (ν ντ ) Up Quark (uq), Down Quark (dq) 14 7 Strange Quark (sq) 28 14 Charm Quark (cq) 42 21 Bottom Quark (bq) 56 28 QB5 (Theoretical Quark or Boson) 70 35 QB6 (Theoretical Quark or Boson) 84 42 W Boson 98 49 Z Boson 112 56 Higgs Boson (H) (Theoretical) 126 63 Top Quark (tq) 140 70 Particle Summary Matrix 3.3, •

Quark Harmonics x ωΩ(ruq,muq) Stω = 1/14 1/7 2/7 3/7 4/7 5/7 6/7 1 2 3 4 5 6 7 8 9 10

Refined EGM Charge Radii and Mass-Energies of Particles (see Appendix 3.D, 3.E)

Existing SM Particle Proton (p) Neutron (n) Electron (e) Muon (µ µ) Tau (ττ) Electron Neutrino (ν ν e) Muon Neutrino (ν νµ) Tau Neutrino (ν ν τ) Up Quark (uq) Down Quark (dq) Strange Quark (sq) Charm Quark (cq) Bottom Quark (bq) Top Quark (tq) W Boson Z Boson Higgs Boson (H) Photon (γγ) Graviton (γγg)

EGM Radii x10-16(cm) rπ = 830.5957 rν = 826.8379 rε = 11.8055 rµ = 8.2165 rτ = 12.2415 ren ≈ 0.0954 rµn ≈ 0.6556 rτn ≈ 1.9588 ruq ≈ 0.7682 rdq ≈ 1.0136 rsq ≈ 0.8879 rcq ≈ 1.0913 rbq ≈ 1.071 rtq ≈ 0.9294 rW ≈ 1.2839 rZ ≈ 1.0616 rH ≈ 0.9403 rγγ = ½Kλλh rgg = 2(2/5)rγγ

EGM Mass-Energy (computed or utilized)

PDG Mass-Energy Range (2005 Data)

Mass-Energy is precisely known by physical measurement See: National Institute of Standards & Technology (NIST) Note: δm = 10-100 men(eV) ≈ 3 - δm mµn(MeV) ≈ 0.19 - δm mτn(MeV) ≈ 18.2 - δm muq(MeV) ≈ 3.5060 mdq(MeV) ≈ 7.0121 msq(MeV) ≈ 113.9460 mcq(GeV) ≈ 1.1833 mbq(GeV) ≈ 4.1196 mtq(GeV) ≈ 178.4979 mW(GeV) ≈ 80.425 mZ(GeV) ≈ 91.1876 mH(GeV) ≈ 114.4 + δm mγγ ≈ 3.2 x10-45(eV) mgg = 2mγγ

79

men(eV) < 3 mµn(MeV) < 0.19 mτn(MeV) < 18.2 1.5 < muq(MeV) < 4 4 < mdq(MeV) < 8 80 < msq(MeV) < 130 1.15 < mcq(GeV) < 1.35 4.1 < mbq(GeV) < 4.4 169.2 < mtq(GeV) < 179.4 80.387 < mW(GeV) < 80.463 91.1855 < mZ(GeV) < 91.1897 mH(GeV) > 114.4 mγ < 6 x10-17(eV) No definitive commitment

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New Particles (Theoretical) L2 (Lepton) L3 (Lepton) L5 (Lepton) ν2 (L2 Neutrino) ν3 (L3 Neutrino) ν5 (L5 Neutrino) QB5 (Quark or Boson) QB6 (Quark or Boson)

EGM Radii x10-16(cm)

EGM Mass-Energy mL(2) ≈ 9(MeV) rL ≈ 10.7518 mL(3) ≈ 57(MeV) mL(5) ≈ 566(MeV) rν2,ν3,ν5 mν2 ≈ men ≈ mν3 ≈ mµn ren,µn,τn mν5 ≈ mτn rQB ≈ 1.0052 mQB(5) ≈ 10(GeV) mQB(6) ≈ 22(GeV) Particle Summary Matrix 3.4,

PDG Mass-Energy Range or Threshold

Not predicted or considered

where, (i) “Kλ” denotes a Planck scaling factor, determined to be “(π/2)1/3” in Ch. 3.13. (ii) “λh” denotes Planck length [4.05131993288926 x10-35(m)]. (iii) “rL” and “rQB” denote the average radii of SM Leptons and Quark / Bosons (respectively) utilized to calculate the mass-energy of the proposed “new particles”. Note: (a) A formalism for the approximation of ν2, ν3 and ν5 mass-energy is shown in “Appendix 3.H”. (b) It is shown in Ch. (3.8, 3.10, 3.13) that the RMS charge diameters of a Photon and Graviton are “λh” and “1.5λh” respectively, in agreement with Quantum Mechanical (QM) models. (c) The “new theoretical particles” are believed to be extremely short lived (unstable). Please refer to Ch. 3.13 for “the answer to some important questions” in this matter. This includes: (i) What causes harmonic patterns to form? (ii) Why haven’t the “new” particles been experimentally detected? (iii) Why is EGM a method and not a theory? (iv) What would one need to do, in order to disprove the EGM method? •

Periodic Table of Elementary Particles (utilising 2006 PDG data)

Assuming “QB5,6” to be Intermediate Vector Bosons (IVB's), we shall conjecture that the Periodic Table of Elementary Particles may be constructed as follows, Group I

Standard Model Leptons

Quarks

Up

14 +2/3,1/2,[R,G,B] uq 1.5 < muq(MeV) < 3 Down 14 -1/3,1/2,[R,G,B] dq 3 < mdq(MeV) < 7 Electron 2 -1,1/2 e = 0.5110(MeV) Electron Neutrino 2 0,1/2 νe < 2(eV)

Types of Matter Group II Charm 42 +2/3,1/2,[R,G,B] cq ≈ 1.1833(GeV) Strange 28 -1/3,1/2,[R,G,B] sq ≈ 113.9460(MeV) Muon 8 -1,1/2 µ = 105.7(MeV) Muon Neutrino 8 0,1/2 νµ < 0.19(MeV) 80

Group III Top

140 +2/3,1/2,[R,G,B] tq ≈ 171.4(GeV) Bottom 56 -1/3,1/2,[R,G,B] bq 4.13 < mbq(GeV) < 4.27 Tau 12 -1,1/2 τ = 1.777(GeV) Tau Neutrino 12 0,1/2 ντ < 18.2(MeV)

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EGM Leptons

L2

4 L3 -1,1/2

L2 ≈ 9(MeV) L2 Neutrino

4 0,1/2

L3 ≈ 57(MeV) L3 Neutrino

6 L5 -1,1/2

6 0,1/2

ν3 ≈ mµn Standard Model and EGM Bosons Photon N/A Gluon ? QB6 84 1,Colour,1 1,Weak Charge,10-6 1,Charge,α gl QB6 γ -45 < 10(MeV) ≈ 22(GeV) ≈ 3.2 x10 (eV) Graviton N/A QB5 70 W Boson 98 -39 -6 2,Energy,10 1,Weak Charge,10 1,Weak Charge,10-6 QB5 W γg ≈ 10(GeV) ≈ 80.27(GeV) = 2mγγ Particle Summary Matrix 3.5,

L5 ≈ 566(MeV) L5 Neutrino

ν2

≈ men

Quarks

Legend Leptons

10 -1,1/2

10 0,1/2

ν5 ≈ mτn Z Boson 112 1,Weak Charge,10-6 Z ≈ 91.1875(GeV) Higgs Boson 126 0,Higgs Field,? H ≈ 114.4(GeV)

Bosons

Name

Stω Name Stω Name Stω Charge(e),Spin,Colour Charge(e),Spin Spin,Source,SC Symbol Symbol Symbol Mass-Energy Mass-Energy Mass-Energy Particle Summary Matrix 3.6, where: (i) “SC” denotes coupling strength at “1(GeV)”. [17] (ii) The values of “Stω” utilise the Proton as the reference particle. This is due to its RMS charge radius and mass-energy being precisely known by physical measurement.

Note: the theoretical particles predicted may also be interpreted as transient states of Standard Model particles. Please refer to Ch. 3.13 for a detailed discussion.

****** IMPORTANT ****** The EGM Harmonic Representation of Fundamental Particles (i.e. Particle Summary Matrix 3.3) is applicable to the size relationship between the Proton and Neutron (i.e. to calculate “rπ” from “rν” and vice-versa utilising “Stω = 1”) as an approximation only. For precise calculations based upon similar forms, the reader should refer to Ch. 3.9 [Eq. (3.212, 3.215)]. EGM is a method and not a theory because: (i) it is an engineering approximation and (ii), the mass and size of most subatomic particles are not precisely known. It harmonizes all fundamental particles relative to an arbitrarily chosen reference particle by parameterising ZPF equilibrium in terms of harmonic cut-off frequency “ωΩ”. The formulation of Particle Summary Matrix 3.3 is a robust approximation based upon PDG data. Other interpretations are possible, depending on the values utilized. For example, if one reapplies the method presented in Ch. 3.12 based upon other data, the values of “Stω” in Particle Summary Matrix 3.3 might differ. However, in the absence of exact experimentally measured mass and size information, there is little motivation to postulate alternative harmonic sequences, particularly since the current formulation fits the available experimental evidence extremely well. 81

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If all mass and size values were exactly known by experimental measurement, the main sequence formulated in Ch. 3.12 (or a suitable variation thereof) will produce a precise harmonic representation of fundamental particles, invariant to interpretation. Particle Summary Matrix 3.3 values cannot be dismissed due to potential multiplicity before reconciling how: i. “ωΩ”, which is the basis of the Particle Summary Matrix 3.3 construct, produces the experimentally verified formulation of Eq. (3.212, 3.215) as derived in Ch. 3.9. These generate radii values substantially more accurate than any other contemporary method. Infact, it is a noteworthy result that EGM is capable of producing the Neutron Mean Square (MS) charge radius as a positive quantity. Conventional techniques favour the non-intuitive form of a negative squared quantity. ii. “ωΩ” is capable of producing “a Top Quark” mass value – the SM cannot. iii. EGM produces the results defined in Particle Summary Matrix 3.1. iv. Extremely short-lived Leptons (i.e. with lifetimes of “TΩ”) cannot exist, or do not exist for a plausible harmonic interpretation. v. Any other harmonic interpretation, in the absence of exact mass and size values determined experimentally, denote a superior formulation. Therefore, EGM is a method facilitating the harmonic representation of fundamental particles. NOTES

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ARTICLE

3.2

DERIVATION OF ENGINEERING PRINCIPLES

Edgar Buckingham: 1867 – 1940 [18]

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FUNDAMENTAL ENGINEERING – THE PYRAMIDS AT GIZA

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CHAPTER

3.1

Dimensional Analysis [63] Abstract It is hypothesised that coupling exists between ElectroMagnetic (EM) fields and the magnitude of the local value of gravitational acceleration “g”. Buckingham’s Π Theory (BPT) is applied to establish a mathematical relationship that precipitates a set of modelling equations termed Π groupings. The Π groupings are reduced to a single expression in terms of the speed of light and an experimental relationship function. This function is interpreted to represent the Refractive Index and is demonstrated to be equivalent to the Polarisable Vacuum (PV) Model representation of General Relativity (GR). Assuming dynamic, kinematic and geometric similarity between the PV and the BPT derivation, it is implied that the PV may also be represented as a superposition of ElectroMagnetic (EM) fields. It is conjectured that by applying an intense superposition of fields within a single frequency mode, it may be possible to modify the Refractive Index within the test volume of an experiment. This may significantly reduce the experimental complexity and energy requirements necessary to locally affect “g”.

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Process Flow 3.1,

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1

INTRODUCTION

To date, great strides have been made by General Relativity (GR) to our understanding of gravity. GR is an excellent tool that represents space-time as a geometric manifold of events, where gravitation manifests itself as a curvature of space-time and is described by a metric tensor. [19] However, GR does not easily facilitate engineering solutions that may allow us to design electromechanical devices with which to affect the space-time metric. If mankind wishes to engineer the space-time metric, alternative tools must be developed to compliment those already available. Subsequently, the Electro-Gravi-Magnetics (EGM) methodology was derived to achieve this goal. EGM is defined as the modification of vacuum polarisability by applied ElectroMagnetic fields. It provides a theoretical description of space-time as a Polarisable Vacuum (PV) derived from the superposition of ElectroMagnetic (EM) fields. The PV representation of GR is a heuristic tool and is isomorphic to GR by weak field approximation. utilising EGM, EM fields may be applied to affect the state of the PV and thereby facilitate interactions with the local gravitational field. To demonstrate practical modelling methods of the PV, we apply Buckingham's Π Theory (BPT). BPT is a powerful tool that has been in existence, tried and experimentally proven for many years. BPT is an excellent tool that may be applied to the task of determining a practical relationship between gravitational acceleration and applied EM fields. The underlying principle of BPT is the preservation of dynamic, kinematic and geometric similarity between a mathematical model and an Experimental Prototype (EP). [20] Historically, BPT has been used extensively in the engineering field to model, predict and optimise fluid flow and heat transfer. However, in principle, it may be applied to any system that is dynamically, kinematically and geometrically founded – such as the geometric space-time manifold. Typical examples of experimentally verified Π groupings in fluid mechanics are Froude, Mach, Reynolds and Weber numbers. [20] Thermodynamic examples are Eckert, Grashof, Prandtl and Nusselt numbers. [21] Moreover, the Planck Length commonly used in theories of Quantum Gravity shares its origins with the Dimensional Analysis Technique (the foundation of BPT). [22] The application of BPT is not an attempt to answer fundamental physical questions but to apply universally accepted engineering design methodologies to real world problems. It is primarily an experimental process. It is not possible to derive system representations without involving experimental relationship functions. We represent these functions as “K0(X)”, where “X” denotes all variables within the experimental environment that influence results and behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis. Once the Π groupings have been formed, they may be manipulated or simplified as required to test ideas and determine experimental relationship functions. Ultimately, the relationship functions validate the system equations developed. For the proceeding BPT construct, we shall hypothesise that: Coupling exists between a superposition of EM fields and the local value of gravitational acceleration. Ideally, experimental relationship functions possess values of unity relative to the distant observer. This indicates a loss-less relationship between the EP and the mathematical model utilised to describe the EP. Typically, due to viscous forces and energy loss / transformation effects, experimental relationship functions take extreme values of magnitude (i.e. large or small). If we consider the EP to be the ambient gravitational environment (local space-time manifold) and the mathematical model to be the PV model of gravity, then we expect all experimental relationship functions to approach unity, as shall be demonstrated in the proceeding construct.

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The reasons for this are: i. The true nature of gravity is currently unknown to physics. ii. The mathematical descriptions used to predict gravitational behaviour are constructed from observation of effects, not cause. iii. A mathematical description is nothing more than just that. It is a non-physical manifestation of human understanding. For example, GR is a Tensor based mathematical formulation only - there is no physical evidence to validate the contention that the true nature of space-time is physically geometric with Planck Scale grid lines radiating from cosmological objects. iv. There can be no physical losses between two mathematical representations of the same thing. 2

THEORETICAL MODELLING

BPT commences with the selection of significant parameters. There are no right or wrong choices with respect to the selection of these parameters. Often, the experience of the researcher exerts the greatest influence to the beginning of the process and the choice of significant parameters are validated or invalidated by experimentation. [22] When applying BPT, it is important to avoid repetition of dimensions. Subsequently, it is often desirable to select variables that may be formulated by the manipulation of simpler variables already chosen. The selected variables used in EGM are shown in table (3.1) of the following section. These parameters have been selected to facilitate experiments utilising EM fields and assume that there is a physical device to be tested, located on a laboratory test bench. The objective of the experiment is to utilise a superposition of EM fields to reduce the weight of a test-mass when placed in the volume of space located directly above the device. Therefore, the significant parameters are those factors that may affect the acceleration of the test-mass within this volume. Our selection of significant parameters involves the magnitude of vector quantities and scalars. This avoids unnecessary repetition of fundamental units in accordance with the application of BPT methodology. [22] The significant vector magnitude parameters are acceleration, Magnetic field, Electric field and position. The scalar quantities are Electric charge and frequency. Since static charge on the device or the test-mass may also exert strong Lorentz forces and therefore accelerations, the scalar value of static charge is included to determine its contribution. If the device is small then the distance between the surface of the device and the test-mass suspended in the volume above it is trivial and that the magnitude of the position vector is usefully constant. Mechanical height adjustments and conventional Radio Frequency (RF) test and measurement equipment may be used to sweep the values of position and frequency in a controlled manner, throughout a range of practical values. 3

MATHEMATICAL MODELLING

3.1

FORMULATION OF Π GROUPINGS

The formulation of Π groupings begins with the determination of the number of groups to be formed. The difference between the number of significant parameters and the number of dimensions represents the number of Π groups required (two). where, Variable a B E r

Description Magnitude of acceleration vector Magnitude of Magnetic field vector Magnitude of Electric field vector Magnitude of position vector 88

Units m/s2 T V/m m

Composition kg0 m1 s-2 C0 kg1 m0 s-1 C-1 kg1 m1 s-2 C-1 kg0 m1 s0 C0

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Q ω

Magnitude of Electric charge Field frequency Table 3.1,

C Hz

kg0 m0 s0 C1 kg0 m0 s-1 C0

Note: the traditional representation of mass (M), length (L) and time (T), in BPT methodology has been replaced by dimensional representations familiar to most readers (kg, m and s). “C” denotes Coulombs, the MKSA units representing charge. We may write the general formulation of significant parameters as, x1 x2 x3 x4 x5 . . . .

a K 0( X ) .B

ω

E

r

Q

(3.1)

where, “K0(X)” represents an experimentally determined dimensionless relationship function. Subsequently, the general formulation may be expressed in terms of its dimensional composition as follows, 0 1 2 0 kg .m .s .C

1 0 1 K 0( X ) . kg .m .s .C

1

x1

. kg1 .m1 .s 2 .C

x2

1

. kg0 .m0 .s 1 .C0

x3

. kg0 .m1 .s 0 .C0

x4

. kg0 .m0 .s 0 .C1

x5

(Eq. 3.2) Applying the indicial method [22] yields, x1

x2 0

x2

x4 1

x1 x1

2 .x 2 x2

x3

2

solve , x 2 , x 3 , x 4 , x 5

x1 x1

2 x1

1 0

x5 0

(3.3)

Substituting the expressions for “xn” into the general formulation and grouping terms yields, a r .ω

2

K 0( X ) .

B.ω .r

x1

E

(3.4)

Note: “Q” has evaporated from the general formulation indicating that the acceleration derived is not to be associated with the Lorentz force. 3.2

TECHNICAL VERIFICATION OF Π GROUPINGS

The formulation of Π groupings may be verified by a simple check of dimensionless homogeneity as follows, 1

a r .ω

1 2

2

(3.5)

B .ω .r E

x1

(3.6)

By inspection - both Π groupings are dimensionless: no technical error has been made in their formulation. [21]

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4

DOMAIN SPECIFICATION

4.1

GENERAL CHARACTERISTICS

The application of basic assumptions regarding the practical nature of experimental configurations enables precipitations of the general formulation. Precipitations are defined as results derived by the application of limits, whereby the value of “x1” may be calculated. To achieve this, we shall assume that all significant parameters have been selected correctly and that the relationship between experimental observation and the general formulation is a single valued function “-∞ < K0(X) < ∞”. This assumption does not remove the necessity for experimental determination, as the form of the experimental relationship function is a consequence of the precipitation process. Due to the practical nature of experimental investigation, “ω” and “r” are dominating factors because; (i) they are repeated in both Π groupings; (ii) numerically, they have the largest relative contribution to the behaviour of both Π groupings and (iii) experimentally, they are practical parameters to sweep and modify. 4.2

PRECIPTITATIONS OF THE GENERAL FORM

4.2.1 FREQUENCY DOMAIN PRECIPITATION For investigations where “0 < ω < ∞”, solving equation (3.4) for “x1” and applying limits yields, Low frequency solution, a

lim + . 2 ω 0 rω

B.ω .r K 0( X ) . E

solve , x 1

x1

ln( a )

expand

ln K 0( X )

2

ln( ω )

( ln( B)

2 .ln( ω )

ln( r ) ln( r )

ln( E) )

factor

(3.7)

High frequency solution, lim ω ∞

a r .ω

2

B .ω .r K 0( X ) . E

solve , x 1

x1

ln( a )

expand

ln K 0( X )

( ln( B)

ln( ω )

ln( r )

2 .ln( ω ) 2

ln( r )

ln( E) )

factor

(3.8)

Hence, the precipitated relationship may be expressed in Π form as, a r .ω

2

K 0( ω , X ) .

E . B ω .r

2

(3.9)

Alternatively, in a general form in terms of the acceleration as, a K 0( ω , X ) .

1 . E r B

2

(3.10)

4.2.2 DISPLACEMENT DOMAIN PRECIPITATION For investigations where “0 < r < ∞”, repeating the procedure defined above yields, Solution for small “r”,

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lim r

B.ω .r K 0( X ) . E

a 0 + r .ω2

x1

solve , x 1

ln( a )

expand

ln K 0( X ) ln( ω )

( ln( B)

2 .ln( ω )

ln( r )

1 ln( r )

ln( E) )

factor

(3.11)

Solution for large “r”, lim ∞

r

B .ω .r K 0( X ) . E

a -

r .ω

2

x1

solve , x 1

ln( a )

expand

ln K 0( X ) ln( ω )

( ln( B)

ln( r )

2 .ln( ω ) 1

ln( r )

ln( E) )

factor

(3.12)

The precipitated relationship may be expressed in Π form as, a r .ω

2

K 0( r , X ) .

E . B ω .r

(3.13)

Alternatively, in a general form as, a K 0( r , X ) .ω .

E B

(3.14)

4.2.3 WAVEFUNCTION PRECIPITATION For PV model investigations involving transverse plane wave solutions in a vacuum, Maxwell’s equations require “E/B = c” when “r → λ/2π” in the frequency range “0 < ω < ∞”, Low frequency solution, lim + ω 0

lim r

lim c ω

E

a ω .λ . r .ω2 B 2 .π

B.ω .r K 0( X ) . E

x1

solve , x 1

ln( a )

expand

ln K 0( X ) ln( ω )

( ln( B)

ln( r )

2 .ln( ω ) 1

ln( r )

ln( E) )

factor

(Eq. 3.15) High frequency solution, lim ω ∞

lim r

lim c ω

E

a ω .λ . r .ω2 B 2 .π

B.ω .r K 0( X ) . E

x1

solve , x 1

ln( a )

expand

ln K 0( X )

( ln( B)

ln( ω )

ln( r )

2 .ln( ω ) 1

ln( r )

ln( E) )

factor

(Eq. 3.16) The precipitated relationship may be expressed in Π form as, a r .ω

2

K 0( ω , r , E, B, X ) .

ω .r c

(3.17)

Alternatively, in general form as, a K 0( ω , r , E, B , X ) .

5

3 2 ω .r

c

(3.18)

EXPERIMENTAL RELATIONSHIP FUNCTIONS

By application of the forms obtained in the frequency, displacement and wavefunction domains, we may determine an ideal solution for the experimental relationship functions. Applying limits corresponding to wavefunction solutions “ω → c/r” and “E → cB” to equation (3.10) and (3.14) yields, 91

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lim ω

K 0( ω , X ) 2 c . r

2

1 . E lim K 0( ω , X ) . r B E c .B

(3.19)

2

E lim K 0( r , X ) .ω . B c E c .B

c . K 0( r , X ) r

(3.20)

r

Thus, K 0( ω , X ) K 0( r , X )

(3.21)

Substituting “ωr = c” into (3.18) yields, a K 0( ω , r , E, B, X ) .

c

2

r

(3.22)

Therefore, when wavefunction solutions are applied to each precipitation, the relationship functions are equal “K0(ω,X) = K0(r,X) = K0(ω,r,E,B,X) = K0(X)”. The wavefunction precipitation we require for investigations involving a superposition of waves may then be represented by, a K 0( X) .

c

2

r

(3.23)

where, “X” represents all other physical variables not specified in the equation. 6

THE POLARISABLE VACUUM MODEL

6.1

REFRACTIVE INDEX

It is known that for complete dynamic, kinematic and geometric similarity between Π groupings according to BPT, “K0(X) = 1” representing ideal experimental behaviour. Since BPT is based upon the dynamic, kinematic and geometric similarity between a mathematical model and an EP, we may usefully represent the PV by the general form of equation (3.23). In the PV model, [23, 24] the vacuum is characterised by the value of the Refractive Index “KPV”. Subsequently, if we consider “a”, “c” and “r” in the preceding equation to be at infinity, then “K0(X)” may be expressed locally by “vc” and “rc” such that “a = vc2 / rc”, “c → vc * KPV” and “rc → r * √KPV”. Hence, substituting these relationships into equation (3.23) yields an expression for “K0(X)” explicitly in terms of the Refractive Index in the PV model of gravity as follows, 2

c a K 0( X ) . substitute , c v c . K PV, r r c . K PV , a r

vc

2

rc

, solve , K 0( X )

1 3

K PV

2

(3.24)

The equivalence principle indicates that an accelerated reference frame is equivalent to a uniform gravitational field. Therefore, assuming “a” is equivalent to the magnitude of the gravitational acceleration vector “g” as in the PV model, we may determine the value of “K0(X)” at the surface of the Earth by using the weak field approximation to the gravitational potential [23, 24] as follows, 2

K PV K 0( X )

6.2

3

(3.25)

SUPERPOSITION

BPT relates the scale of two similar systems by Π groupings [22] and the PV background field is assumed derivable from a superposition of applied EM fields. The Π groupings are 92

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compared directly and scaled to determine the required applied fields. The ratio “B1/E1 = 1/c” represents the velocity of light at ambient background PV conditions within the test volume and the ratio “B2/E2 = 1/vc” represents the modified velocity of light “vc” within the test volume as determined by the applied EM fields. Scaling of the Π groupings may be experimentally applied according to equation (3.26), B 1 .ω 1 .r 1 E1

x1

B 2 .ω 2 .r 2 E2

x1

substitute , E 1 c .B 1 , E 2 v c .B 2 , r 1 r 2 , solve , v c

ω 2.

c ω1

(3.26)

“KPV” may then be determined by the ratio of frequency modes between the EM fields of the PV model and the local gravitational field. Additional notation is required to indicate the discrete spectrum of the superposition of waves within the test volume. The subscript “n” and “P” denote the applied spectral frequency modes and polarisation vectors respectively. Substituting a superposition of wavefunctions, “K0(X)” may be constructed by design according to, Kn , P

2

K 0( X )

3

c

In , P . ω n , P

( n , P) Kn , P .ω n , P

vc ( n , P)

(3.27)

where, “In,P” represents the macroscopic intensity of Photons within the test volume and “Kn,P” is an undetermined relationship function representing the intensity of the PV background field at each frequency mode. For the Zero-Point-Field (ZPF) ground state of the vacuum, predicted by Quantum Electrodynamics, “Kn,P = ½” and “In,P = 0”. Equation (3.27) implies that, when in a gravitational field, the vacuum field is not in the ZPF ground state. Therefore, within the test volume at ambient background conditions, we would generally expect “Kn,P ≠ ½”. Equation (3.27) describes the relative change in the spectral energy density and thereby represents a modification of polarisability of the vacuum within the test volume. 6.3

CONSTANT ACCELERATION

Fourier Series (FS), representing the summation of trigonometric functions, may be applied to define a constant vector field “a” over the period “0 ≤ t ≤ 1/ω”. A constant function is termed “even” due to symmetry about the “Y-Axis”, subsequently; the Fourier representation contains only certain terms and may be expressed in complex form. We may relate the principles of Complex FS to EGM superposition by the application of equation (3.10). Let an arbitrary transverse EM plane wave be defined by, F( k , n , t ) F 0( k ) . e

( π .n .ω .t ) .i

(3.28)

Where: (i) “k” and “n” denote the wave vector and field harmonic respectively, (ii) “ω” denotes the fundamental field frequency such that, B( k, n , t ) Re( F( k, n , t ) )

(3.29)

E( k, n , t ) Im( F( k, n , t ) )

(3.30)

Substituting equation (3.29) and (3.30) into equation (3.10) yields,

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N E( k , n , t ) a( t )

K 0( ω , X ) r

2

. n= N N B( k , n , t )

2

n= N

(3.31)

Acceleration

It has been numerically simulated that the effect of phase variance between superimposed field wavefunctions may be usefully approximated to zero, when applied to equation (3.31), for field harmonic values “N ≥ 20” (approx.) [i.e. as N → ∞, a(t) → constant]. This may be graphically illustrated by (N = 20), 1

1

2 .ω

ω

a( t ) a∞

t Time

Figure 3.1, The mean value “a∞” of equation (3.31) over the fundamental period “1/ω” also represents the magnitude of the acceleration vector “a” as “N → ∞”. Hence, 1 ω a∞ ω.

6.4

a( t ) d t

0.( s )

(3.32)

COMPLEX FOURIER SERIES

Equation (3.28) is analogous to a Fourier representation by the term “F0(k)”, which represents the EM amplitude distribution within the experimental environment. Subsequently, we may write the direct equivalence of equation (3.28) to Complex FS representation by the following expression, 1 F 0( k )

ω. 2

ω

f( t )

0. ( s )

e

( π .n .ω .t ) .i

dt

(3.33)

Hence, “F0(k)” may take the form of the complex Fourier coefficient typically denoted as “Cn”. [25] This correlation may enable the experimentalist to design and control the geometry of forcing configurations to exact analytical targets. Therefore, it has been illustrated that we may relate Fourier approximations of a constant vector field to EGM by the summation of EM wavefunctions representing the superposition of waves at each frequency mode. This may be accomplished by the determination of the experimental relationship function “K0(ω,X)”. 94

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Moreover, equation (3.31) represents a useful and practical relationship between experimental observation and engineering research. By sequentially scanning field harmonics between “-N” and “N”, values of “K0(ω,X)” may be calculated when the localised value of ambient acceleration has been reduced. Experimental determination of the function “K0(ω,X)” will permit engineering applications to be developed by direct scaling. An illustrational Complex FS representation of constant acceleration magnitude “a”, by the summation of harmonics over the interval “0 ≤ t ≤ 1/ω” may be written as follows, [25] 1 N a( t ) n= N

ω. 2

ω 0. ( s )

f( t ) e

( π .n .ω .t ) .i

(π d t .e

.n .ω .t ) .i

(3.34)

Acceleration

And may be graphically illustrated by,

Re( a( t ) ) Im( a( t ) ) f( t )

t Time

Real Terms (Non-Zero Sum) Imaginary Terms (Zero Sum) Constant Function (eg. "g")

Figure 3.2, where, Units Variable Description th n “n ” harmonic of integer value None th “N ” Fourier polynomial corresponding to the spectral frequency mode such N that “-∞ < Ν < ∞” - Figure (3.2) displays an illustrational value of “N = 10” Period over which “a” is constant - Figure (3.2) displays an illustrational s 1/ω value of “1/ω = 1(s)” f(t) Constant function being represented by the summation of Fourier m/s2 polynomials Table 3.2, Important features: i. The Real Terms are odd numbered harmonics producing a Non-Zero Sum. ii. The Imaginary Terms are even numbered harmonics producing a Zero Sum. 7

CONCLUSIONS

The relationship between EM fields and acceleration has been demonstrated by the application of BPT. Equation (3.26) and (3.27) indicate that, for physical modelling applications, manipulating the full spectrum of the PV is not required and optimal PV coupling may exist at specific frequency modes. This dramatically simplifies the design of experimental prototypes and suggests that the PV may be usefully approximated to a discrete wave spectrum by applying an intense superposition of fields within a single frequency mode. 95

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NOTES

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CHAPTER

3.2

General Modelling and the Critical Factor [64] Abstract In chapter 3.1, by application of Buckingham’s Π Theory, it was demonstrated how constant acceleration may be derived from a superposition of ElectroMagnetic (EM) fields. An experimentally determined relationship function “K0(ω,X)” was predicted which couples gravitational acceleration to the intensity of an applied EM field, in agreement with the equivalence principle and the Polarisable Vacuum (PV) Model. The EM field was then decomposed into its constituent frequency modes and their respective intensities to show that their summation results in constant acceleration as the number of harmonic frequencies in the field tends to infinity. This chapter is an extension of previous work, intended to present a hypothesis to be tested and to demonstrate how “K0(ω,X)” may be expressed, by decomposition, as two relationship functions that may be directly measured by experimentation. This results in two representations that are proportional to solutions of the Poisson and Lagrange equations. It is demonstrated that the ratio of the resulting relationship functions is proportional to the square of the magnitude of the resultant Poynting Vector. This property, in conjunction with the orientation of the resultant Poynting Vector, may be utilised as a practical design tool for engineering the PV by the application of “off-theshelf” EM modelling software.

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Process Flow 3.2,

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1

INTRODUCTION

1.1

HYPOTHESIS TO BE TESTED

It was illustrated in chapter 3.1 that an experimentally determined relationship function “K0(ω,X)”, may be used to characterise the relationship between the magnitude of the acceleration vector “a” and the energy densities of discrete frequency modes “N” of an applied ElectroMagnetic (EM) field. This relationship was shown to be equivalent to the Polarisable Vacuum (PV) Model approach to General Relativity (GR) [23, 26 - 30] such that the Newtonian gravitational potential is approximated by an exponential function [30] and that a weak gravitational field may be represented by a superposition of EM wavefunctions. Historically, variations in the energy density are known to result in gravitation from the solutions of Poisson’s equation in Newtonian gravity. [31] An equivalent result is presented in terms of the energy densities at the discrete frequency modes of the applied EM field. The relationship function “K0(ω,X)” may also be derived from the results of experiments that determine other relationship functions. It is demonstrated how these experimentally determined relationship functions may be found and used to address experimental design issues. To achieve this, experiments must be designed that test the following hypothesis, There are three key factors in achieving a local modification of the magnitude of the acceleration vector “a” of a gravitational field. These are; (i) an increase in the energy density of the EM field at specific frequency modes, (ii) the superposition of time varying EM fields at specific frequency modes and (iii), the equivalence principle, which indicates that an accelerated reference frame is equivalent to a uniform gravitational field. It has been conjectured that if the frequency and phase of all modes in the PV where known, then by application of the appropriate EM fields, the interaction at those modes may facilitate destructive interference resulting in a complete cancellation of the local value of gravitational acceleration. However, in the hypothesis to be tested, it is assumed that it is impossible to decrease the energy density of a gravitational field by applying an EM field to a region of space-time. Therefore, the hypothesis requires that in any practical experiment it is only possible to increase the energy density at specific frequency modes. 1.2

WHAT IS DERIVED?

This chapter derives three key design considerations. These are, i. The “α” forms, which are an inversely proportional description of how energy density may result in an acceleration “ax(t)”. ii. The “β” forms, which are a directly proportional description of how energy density may result in an acceleration “ax(t)”. iii. The Critical Factor “KC”, which is the ratio of the experimentally determined relationship functions “K1” and “K2” presented in section 3. The key design considerations are derived from the hypothesis to be tested, which seeks to couple gravitational acceleration to ElectroMagnetism. Dimensional Analysis was utilised in chapter 3.1 to demonstrate that coupling may exist between ElectroMagnetism and gravity by the application of Buckingham’s Π Theory. Analytical results herein suggest that the square of the magnitude of the resultant Poynting Vector may be a useful design tool. Calculation and visualisation of the orientation and intensity of the Poynting Vector is standard functionality in many “off-the-shelf-EM” simulation products. This provides a convenient platform from which to design practical laboratory benchtop experiments.

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The key design considerations are derived by identifying possible interpretations of equation (3.31) that impact the hypothesis to be tested, as illustrated in section 2. Equation (3.31) is then separated into subordinate elements based upon these interpretations, as illustrated in section 3. The subordinate elements are then used to determine “KC” by solving for the ratio of the experimental relationship functions, “K1” and “K2” defined in table (3.5). 2

THEORETICAL MODELLING

2.1

PRIMARY PRECIPITANT The frequency domain precipitation derived in chapter 3.1 may be written as, N E( k , n , t ) a( t )

K 0( ω , X ) r

2

. n= N N B( k , n , t )

2

n= N

(3.31)

where, Variable a(t) E(k,n,t) B(k,n,t) r ω n, N k K0(ω,X)

Description Magnitude of acceleration vector Magnitude of Electric field vector Magnitude of Magnetic field vector Magnitude of position vector Field frequency Harmonic frequency modes Magnitude of the harmonic wave vector Experimental relationship function Table 3.3,

Units m/s2 V/m T m Hz None 1/m None

Equation (3.31) is termed the primary precipitant and may be manipulated to alternate forms by incorporation. Incorporation is the redefinition of an as yet undetermined relationship function to include a variable contained within the equation under consideration {e.g. equation (3.10) may be written as “a = K0(r,ω,X)(E/B)2”}. In our case, incorporation is utilised to visualise constant acceleration by the superposition of EM waves to promote changes in energy density in the local space-time manifold. Fourier Series (FS), representing the summation of trigonometric functions, may be applied to facilitate this change by defining a constant vector field “a” over the period “0 ≤ t ≤ 1/ω”. A constant function is termed even due to symmetry about the “Y-Axis”, subsequently; the Fourier representation contains only certain terms and may be expressed in complex form and graphically illustrated for “N → ∞”. Firstly, visualising the applied EM forcing function yields,

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EM Function

1

2

ω

ω

Re( F ( k , 1 , t ) ) Re( F ( k , 2 , t ) ) Re( F ( k , 3 , t ) )

t Time

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

EM Function

Figure 3.3, 1

2

ω

ω

Im( F ( k , 1 , t ) ) Im( F ( k , 2 , t ) ) Im( F ( k , 3 , t ) )

t Time

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

Figure 3.4,

EM Wave-Function Superposition

Producing the representation where, “E(t) = ΣE(k,n,t)2” and “ΣB(t)= ΣB(k,n,t)2”. 1

1

2 .ω

ω

ΣE( t ) ΣB( t )

t Time

Electric Field Magnitude Magnetic Field Magnitude

Figure 3.5,

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In the PV representation, the value of the EM field at infinity replaces the fields in equation (3.31) according to: “E(k,n,t) → E0(k,n,t)”, “B(k,n,t) → KPVΒ0(k,n,t)”, “r → r0/√KPV” and “ω → ω0/√KPV” hence, N E 0( k , n , t )

a r0

2.2

2

K 0 ω 0, X . n= N N 3 r 0 . K PV 2 B 0( k , n , t ) n= N

(3.35)

INTERPRETATIONS OF THE PRIMARY PRECIPITANT

The primary precipitant is subject to two interpretations. (i) It may be increased or decreased as a function of the magnitude of “E0(k,n,t)”, or (ii), it may be increased or decreased as a function of the magnitude of “B0(k,n,t)”. Subsequently, if “E0(k,n,t)” is constant in an experiment we may incorporate it into the experimental relationship function as follows, α1

K 1 ω 0, r 0, E 0, D , X

(3.36)

N 3 r 0 . K PV .

B 0( k , n , t )

2

n= N

Incorporating “B0(k,n,t)” represents the second interpretation of the primary precipitant. β1

K 2 ω 0, r 0, B 0, D , X . r 0 . K PV

3

N E 0( k , n , t ) n= N

2

(3.37)

where, Variable α1 β1 K1(ω0,r0,E0,D,X) K2(ω0,r0,B0,D,X) D

c = c0 / KPV

Description Units 2 The subset formed, as “N → ∞”, by the method of incorporation m/s applied to equation (3.35). The subset formed, as “N → ∞”, by the method of incorporation applied to equation (3.35). Experimental relationship function (V/m)2 Experimental relationship function T-2 Experimental configuration factor: a specific value relating all design None criteria. This includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy. Velocity of light in the PV m/s Table 3.4,

3

MATHEMATICAL MODELLING

3.1

SEPARATION OF PRIMARY FORMS

The “α” and “β” forms, equation (3.36) and (3.37) respectively, may be used to generate subset expressions with respect to the hypothesis to be tested. The subsets have been termed the first and second alpha subsets (α1,α2) and the first and second beta subsets (β1,β2) to better characterise anticipated results. 102

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The “αx” and “βx” subsets and experimental relationship functions “K1(ω0,r0,E0,D,X)” and “K2(ω0,r0,B0,D,X)” may be formulated as follows, Var. / Eq. Description K 1 ω 0 , r 0, E 0, D, X Eq.(3.36) αx α1

Formulation Formed by incorporation applied to equation (3.35) as N → ∞.

N 3 r 0. K PV .

B 0( k, n , t )

2

Units m/s2

n= N

Eq.(3.38)

Formed by substitution assuming a transverse EM wave relationship, c0 = E0/B0 into α1.

2

K ω , r , E , D, X . 1 0 0 0 N 3 r 0 . K PV 2 E 0( k , n , t ) n= N c0

α2

Eq. (3.39)

2 Formed by relating α1 to (V/m) equation (3.35) and solving. Formed by incorporation m/s2 applied to equation (3.35) as N → ∞.

N K 1 ω 0, r 0, E 0, D, X

K 0 ω 0, X .

E 0( k , n , t )

2

n= N

βx

Eq.(3.37) β1

N

K 2 ω 0, r 0, B 0, D, X .

E 0( k , n , t )

r 0 . K PV

3

Εq.(3.40)

n= N . 3

K 2 ω 0, r 0 , B 0 , D , X

Formed by substitution assuming a transverse EM wave relationship, c0 = E0/B0 into β1. Formed by relating β1 to equation (3.35) and solving.

N

2

c0

β 2 K 2 ω 0 , r 0 , B 0, D, X .

r 0. K PV

Eq. (3.41)

2

B 0( k, n , t )

2

n= N

K 0 ω 0, X N B 0( k , n , t )

2

T-2

n= N

Table 3.5, 3.2

GENERAL MODELLING EQUATIONS

The subsets in table (3.5) suggest two experimental avenues with respect to the hypothesis to be tested. These have been termed General Modelling Equations: GME1 and GME2 as follows, Description Form GME1 a1(r0) = ±½(β1 + β2) GME2 a2(r0) = ±½(β1 - β2) Table 3.6, 3.3

Equation (3.42) (3.43)

CRITICAL FACTOR

The resulting relationship functions may be characterised by the Critical Factor “KC”. It takes the form of a squared term and is a measure of the applied EM field intensity within the experimental test volume. The Left Hand Side of equation (3.44) “KC2” is an arbitrary definition as a consequence of its units of measure “(PaΩ)2”. “KC2” may be derived from the ratio of “K1(ω0,r0,E0,D,X)” to “K2(ω0,r0,B0,D,X)” by taking the limit as “N → ∞”, K C K 1, K 2

2

K 1 ω 0, r 0, E 0, D , X K 2 ω 0, r 0, B 0, D, X

N

N E 0( k , n , t ) . 2

n= N

103

B 0( k , n , t ) n= N

2

(3.44)

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The permittivity and permeability of free space, “ε0” and “µ0” respectively, may be included to express equation (3.44) in units of energy density squared. 4

PHYSICAL MODELLING

4.1

POYNTING VECTOR

Equation (3.44) illustrates that “KC” may be usefully approximated by proportionality to the magnitude of the resultant Poynting Vector. Assuming a transverse EM wave relationship, the magnitude of the Poynting Vector at a particular frequency mode is “S = EH” where, “µ0H = B”. [32] The intensity of each field mode is the power per unit area and are summed to yield the total intensity. This indicates the intensity of the field passing through a surface of an experimental test volume. The resultant Poynting Vector from a benchtop experimental device may be oriented parallel to the position vector “r” such that the acceleration acts to alter the magnitude of the local value of gravitational acceleration “g”. 4.2

POISSON AND LAGRANGE

Table (3.6) defines two expressions that may be applied to experimental investigations illustrating modelling significance, N

a r0

±

β1

β2

±

2

K 2 ω 0, r 0, B 0, D, X . 2 .r 0 . K PV

3

N

E 0( k , n , t )

N E 0( k , n , t )

2

2 c0 .

n= N

B 0( k , n , t )

2

±

n= N

2

K 0 ω 0, X . n= N N 3 2 .r 0 . K PV 2 B 0( k , n , t )

2

c0

n= N

(Eq. 3.45) N

a r0

±

β1

β2 2

±

K 2 ω 0, r 0, B 0, D, X . 2 .r 0 . K PV

3

N

E 0( k , n , t )

N E 0( k , n , t )

2

2 c0 .

n= N

B 0( k , n , t ) n= N

2

±

2

K 0 ω 0, X . n= N N 3 2 .r 0 . K PV 2 B 0( k , n , t ) n= N

(Eq. 3.46) Equation (3.45) is proportional to a solution of the Poisson equation [31] applied to Newtonian gravity where the resulting acceleration is a function of the geometry of the energy densities. Equation (3.46) is proportional to a solution of the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities of the EM field harmonics in a vacuum. [32] These demonstrate that “K2(ω0,r0,B0,D,X)” is the same in both instances. This becomes significant when considering that the EM Zero-Point-Field (ZPF) of the quantum vacuum is described in terms of the energy density per frequency mode (Spectral Energy Density) by, ρ 0( ω )

2 .h .ω c

3

3

(3.47)

where, “h” denotes Planck’s Constant [6.6260693 x10-34(Js)] and “ω” is in “(Hz)”. Therefore, the functions “E0(k,n,t)2” and “B0(k,n,t)2” are proportional to the applied energy density at each frequency mode with respect to specific experimental configurations.

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2

c0

5

CONCLUSIONS

In an experiment, the value of “KC2” may be determined by direct measurement of the intensity of the EM field strength at each harmonic frequency mode. If a resulting acceleration vector is also measured, the hypothesis to be tested as presented in section 1 is validated. NOTES

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NOTES

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CHAPTER

3.3

The Engineered Metric [65] Abstract Engineering expressions are developed for experimental investigations involving coupling between ElectroMagnetism and gravity. It is illustrated that an accelerated reference frame may be derived from a superposition of applied ElectroMagnetic (EM) fields and may be characterised by the magnitude of the Poynting Vectors. Based upon dimensional similarity and the equivalence principle, the engineered acceleration may be used to modify the gravitational acceleration at the surface of the Earth. An engineered change in the value of the Refractive Index corresponds to an incremental change in the gravitational potential energy. The magnitude of this change and the similarity between an experimental test volume and the local gravitational environment, may then be characterised by a Critical Ratio “KR” such that the gravitational acceleration is reduced to “zero” when “KR = 1”. An Engineered Refractive Index equation is derived that may be used for EM metric engineering purposes. An engineered Polarisable Vacuum (PV) metric line element is then presented as an example.

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1

INTRODUCTION

1.1

DESCRIPTION

To demonstrate practical modelling methods of the Polarisable Vacuum (PV), utilising Buckingham Π Theory (BPT), experiments must be designed that test the hypothesis stated in chapter 3.2. Subsequently, three key design considerations were derived and may be characterised by two system regimes, i. The Critical Factor “KC”, which is a proportional measure of the magnitude of the applied Poynting Vectors. ii. The General Modelling Equations (GMEx). (a) GME1 is proportional to a solution of the Poisson equation applied to Newtonian gravity where the resulting acceleration is a function of the geometry of the energy densities. [31] (b) GME2 is proportional to a solution of the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities of the ElectroMagnetic (EM) field harmonics in a vacuum. [32] 1.2

CRITICAL RATIO

Based upon principles of similarity, as defined by BPT, [20 - 22] an engineering parameter termed the Critical Ratio “KR” has been formulated to indicate proportional experimental conditions (section 2.3). It is defined as the ratio of the applied EM fields to the change in the gravitational field in terms of the change in energy densities. In addition, it is shown that “KR” may be used to enhance the representation of the changing experimental relationship function “∆K0(ω,X)” and leads to interactions with the PV as illustrated in section 3 and 4. “KR” is a dimensionless parameter of the hypothesis to be tested as presented in chapter 3.2. 1.3

METRIC ENGINEERING

An engineered metric tensor line element is developed in section 5 utilising an Engineered Refractive Index term constructed in section 3.2. The exponential metric tensor line element as stated in the PV representation of GR [30] is also presented and a table of metric effects articulated in section 6. 2

THEORETICAL MODELLING

2.1

MATHEMATICAL SIMILARITY

It has been illustrated in chapter 3.1 that the magnitude of an acceleration vector field “a”, formed utilising BPT methodology is equivalent to the magnitude of the gravitational acceleration vector field “g” by dimensional similarity and utilisation of the equivalence principle. In the PV representation, this may be expressed as “|n| → ∞” by the more generalised form as follows,

a r0

K 0 ω 0, X ( n , k) . r 0 . K PV

E 0( k , n , t )

2

B 0( k , n , t )

3

( n , k)

2

(3.35)

where, “ω0”, “E0”, “B0” and “r0” denote physical properties as “KPV → 1” asymptotically at infinity such that, 109

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Variable a(r0) E0(k,n,t) B0(k,n,t) r0 ω0 n k KPV K0(ω0,X)

Description Magnitude of PV acceleration vector Magnitude of PV Electric field vector Magnitude of PV Magnetic field vector Magnitude of position vector Field frequency Harmonic frequency modes of PV Harmonic wave vector of PV Refractive Index Experimental relationship function Table 3.7,

Units m/s2 V/m T m Hz None 1/m None

By replacing “E0 → EPV”, “B0 → BPV / KPV” and “r0 → r * √KPV”, an engineered change in “g” may be expressed in local form that may be used to solve for the applied “E” and “B” fields as “|nPV| → ∞” according to,

∆g ≡ ∆a PV

∆K 0( ω , X )

.

E PV k PV, n PV, t

2

B PV k PV, n PV, t

2

n PV, k PV

r n PV, k PV

(3.48)

i

such that, by the application of dimensional similarity and the equivalence principle, the acceleration may be affected by an applied EM field as follows, N E PV k PV, n PV, t

2

n PV, k PV

2

B A k A,n A,t

2

nA= N B PV k PV, n PV, t

n PV, k PV

EA k A,n A,t N

2

i

(3.49)

nA= N

where, Variable ∆g ∆aPV EPV(kPV,nPV,t) BPV(kPV,nPV,t) EA(kA,nA,t) BA(kA,nA,t) nPV kPV nA kA i ∆K0(ω,X)

Description Change of gravitational acceleration vector Change in PV acceleration vector Magnitude of PV Electric field vector Magnitude of PV Magnetic field vector Magnitude of applied Electric field vector Magnitude of applied Magnetic field vector Harmonic frequency modes of PV Harmonic wave vector of PV Harmonic frequency modes of applied field Harmonic wave vector of applied field Denotes initial conditions of PV Engineered relationship function Table 3.8,

Units m/s2 V/m T V/m T None 1/m None 1/m None

It shall be illustrated in section 3 that equation (3.48) may be utilised to develop an engineering solution.

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2.2

CRITICAL FACTOR

An engineering solution can be further advanced by application of the Critical Factor “KC”, which is a measure of the applied EM field intensity within an experimental test volume. Hence, the Change in Critical Factor “∆KC” (specifically from zero) represents a proportional measure of the magnitude of the applied Poynting Vectors as “|nA| → ∞” for the local observer as follows, ∆K C ∆K 1 , ∆K 2

2

∆K 1( ω , r , E, D , X )

N 1

N

.

∆K 2( ω , r , B, D , X ) K PV

EA k A,n A,t

2

2

nA= N

.

B A k A,n A,t nA= N

2

(3.50)

N ∆K 1( ω , r , E, D , X ) ∆K 0( ω , X ) .

E A k A,n A,t

2

nA= N

(3.51)

∆K 0( ω , X ) .K PV

2

∆K 2( ω , r , B, D , X )

N B A k A,n A,t

2

(3.52)

nA= N

where, Variable ∆K1(ω,r,E,D,X) ∆K2(ω,r,B,D,X) D

2.3

Description Change in experimental relationship function Change in experimental relationship function Experimental configuration factor Table 3.9,

Units (V/m)2 T-2 None

CRITICAL RATIO

Practical engineering of the hypothesis to be tested may be possible by calculation of the Critical Ratio “KR”. This may be defined by consideration of the equivalence principle applied to equation (3.48) as “|nPV| → ∞”. Complete similarity occurs when “|KR| = 1” and proportional similarity at “|KR| ≠ 1”, therefore it follows that “KR” may be used to represent the proportional relationships in terms of potential, acceleration, energy densities or any suitable measure as follows, KR

∆U g ∆a PV ∆K C( ∆r ) Ug

g

∆U PV( ∆r )

.

ε0 µ0

(3.53)

where, the permittivity and permeability of free space, “ε0” and “µ0” respectively, act as the Impedance Function “Z = √(µ0/ε0)” and is independent of “KPV” in the PV representation. Variable Ug

Description Units Initial state of Gravitational Potential Energy (GPE) per unit mass described (m/s)2 by any appropriate method Change in “GPE” per unit mass induced by any suitable source ∆Ug ∆KC(∆r) Change in Critical Factor with respect to “∆r” PaΩ Pa ∆UPV(∆r) Change in energy density of the gravitational field with respect to “∆r” Table 3.10,

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3

MATHEMATICAL MODELLING

3.1

ENGINEERING THE RELATIONSHIP FUNCTIONS

For experimental investigations, we require a model from which to design and predict behaviour in accordance with the hypothesis to be tested in chapter 3.2. In figure (3.6): (i) the arrows pointing downwards represent a uniform gravitational field, (ii) the arrows pointing upwards represent a uniformly applied system field, (iii) the sphere represents the experimental test volume residing at co-ordinates (0,0,r) and (iv) the square section represents an EM flux area.

Figure 3.6, The hypothesis to be tested assumes coupling exists between propagating transverse EM plane waves and gravity such that the local value of “g” is reduced to zero and complete similarity is achieved as “[|nPV|,|nA|] → ∞”. Substituting “c2 → ΣEPV2/ΣBPV2” into equation (3.48), solving for “∆K0(ω,X)” and recognising that “∆aPV → g KR” yields, ∆K 0( ω , X )

G.M . r .g . KR KR 2 2 c r .c

(3.54)

Hence, expressions for all experimental and Engineered Relationship Functions have been obtained in terms of a scalar multiple of the magnitude of the resultant Poynting Vector and the magnitudes of the superimposed EM fields. 3.2

ENGINEERING THE REFRACTIVE INDEX

The hypothesis to be tested suggests that “KPV” [30] may be engineered in the same manner as “∆K0(ω,X)”. Equation (3.54) indicates that “|KR| = 1” at complete similarity between the applied EM fields and the local gravitational field. At this condition, the magnitude of “∆KC/Z” is proportional to the magnitude of “∆UPV” at the surface of the Earth within the test volume. Utilising the classical weak field exponential approximation of “KPV”, [30] a useful approximation for practical laboratory benchtop experiments at the surface of the Earth may be derived as follows, 2.

K PV e

G .M 2 r .c

(3.55)

Therefore, an EM affected representation of “KPV” may be expressed by the Engineered Refractive Index “KEGM” as follows, K EGM K PV. e

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2 . ∆K 0( ω , X )

(3.56) www.deltagroupengineering.com

4

PHYSICAL MODELLING

The mathematical construct herein is based upon the modification of “g” by similarity of applied EM fields to the local gravitational environment. Subsequently, we may characterise physical modelling design criteria by the following engineering functions, Initial Value 0

Key Characteristics Range: -∞ < KR < ∞

Engineered Solution KR

0(T )

.

∆U PV( ∆r )

g

ε0 µ0

N

1. Configuration specific 2. Determined experimentally

-2

Ug

∆K C( ∆r )

G.M . ∆K 0( ω , X) KR 2 r .c

Range: -∞ < ∆K0(ω,X) < ∞ 0(V/m)2

∆U g ∆a PV

∆K 1( ω , r , E, D , X ) ∆K 0( ω , X ) .

EA k A,n A,t

2

nA= N

1. Configuration specific 2. Determined experimentally

∆K 0( ω , X ) .K PV

2

∆K 2( ω , r , B , D , X )

N B A k A,n A,t

2

nA= N

0(PaΩ)2

1. 2. 3. 1. 2. 3.

0(m/s2)

Change in Critical Factor Configuration specific Key design consideration Change in GME1 Configuration specific Key design consideration

∆K C ∆K 1 , ∆K 2

∆K 1( ω , r , E, D , X )

2

∆K 2( ω , r , B, D , X ) N E A k A,n A,t

∆a 1( r )

∆K 0( ω , X ) 2 .r

.

2

nA= N

c

2

c

2

N B A k A,n A,t

2

E A k A,n A,t

2

nA= N N

1. Change in GME2 2. Configuration specific 3. Key design consideration

∆a 2( r )

∆K 0( ω , X ) 2 .r

.

nA= N N B A k A,n A,t

2

nA= N

e

g .r 2. 2 c

|KR| = 1 Range: KPV ≥ 1

K PV e

|KR| > 0 Range: KEGM > KPV

G .M 2. 2 r .c

e

K EGM K PV. e

2 .∆K 0 ω , X , K R

2 . ∆K 0( ω , X )

Normal Matter Form

Range: 0 < KEGM < KPV

K PV

K EGM e

2 . ∆K 0( ω , X )

Exotic Matter Form Table 3.11, 5

METRIC ENGINEERING

5.1

POLARISABLE VACUUM

The exponential metric tensor line element in the PV representation of GR (in the weak field limit) may be defined in Spherical Coordinates as follows, [30] ds

2

µ υ c .dt g µυ .dx .dx K PV 2

2

2 K PV. dr

113

2 2 r .dθ

2 2 2 r .sin ( θ ) .dψ

(3.57)

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where, g 00

1 K PV

(3.58)

g 11 g 22 g 33 K PV

5.2

(3.59)

ENGINEERED METRICS

The engineered metric tensor line element for weak field approximations using exponential components may be expressed as, ds

2

µ υ g µυ .dx .dx

2 2 c .dt

K EGM

2 K EGM. dr

2 2 r .dθ

2 2 2 r .sin ( θ ) .dψ

(3.60)

where, g 00

1 K EGM

(3.61)

g 11 g 22 g 33 K EGM

6

(3.62)

ENGINEERED METRIC EFFECTS

Engineered metric effects may be represented for the “normal matter form” as follows, Variable Velocity of Light: vc(KEGM) Mass: m(KEGM) Frequency: ω(KEGM) Time Interval: ∆t(KEGM) Energy: E(KEGM) Length Dim.: L(KEGM)

7

Determining Eq. vc = c / KEGM m = m0 * KEGM3/2 ω = ω0 / √KEGM ∆t = ∆t0 * √KEGM E = E0 / √KEGM L = L0 / √KEGM Table 3.12,

KEGM > KPV (Engineered Metric) Velocity of light < c Effective mass increases Redshift toward lower frequencies Clocks run slower Lower energy states Objects contract

CONCLUSIONS

Engineering expressions are developed for experimental investigations involving coupling between EM fields and gravity that may be characterised by the magnitude of the superposition of Poynting Vectors. Based upon dimensional similarity and the equivalence principle, the engineered acceleration may be used to modify the gravitational acceleration at the surface of the Earth. An engineered change in the value of the Refractive Index corresponds to a change in the gravitational potential energy. The magnitude of this change may be characterised by “KR” such that the gravitational acceleration is reduced to “zero” within a practical benchtop test volume when “KR = 1”. This leads to “KEGM” which may be used for metric engineering purposes.

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CHAPTER

3.4

Amplitude and Frequency Spectra [66] Abstract An experimental prediction is developed considering gravitational acceleration as a harmonic function. This expands potential experimental avenues in relation to the hypothesis to be tested presented in chapter 3.2. Subsequently, this chapter presents: (i) a harmonic representation of gravitational fields at a mathematical point arising from geometrically spherical objects of uniform mass-energy distribution using modified Complex Fourier Series (FS): (ii) characteristics of the amplitude spectrum based upon (i): (iii) derivation of the fundamental harmonic frequency based upon (i): (iv) characteristics of the Frequency spectrum of an implied Zero-Point-Field (ZPF) based upon (i) and the assumption that an ElectroMagnetic (EM) relationship exists over a change in displacement within a practical benchtop test volume.

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Process Flow 3.4, 116

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1

INTRODUCTION

1.1

GENERAL

A metric engineering description was presented in chapter 3.3 based upon the principles of similarity. An engineering parameter, termed the Critical Ratio “KR”, has been formulated to indicate proportional experimental conditions, which may be stated as the ratio of the applied ElectroMagnetic (EM) fields to the induced change of gravitational field strength. 1.2

HARMONICS

To further articulate the applicability of “KR”, a harmonic description of gravitational fields is developed. This acts to expand potential experimental avenues in relation to the hypothesis to be tested as stated in chapter 3.2. Subsequently, this chapter presents: i. A harmonic representation of gravitational fields at a mathematical point arising from geometrically spherical objects of uniform mass-energy distribution using modified Complex Fourier Series (FS). ii. Characteristics of the amplitude spectrum based upon (i). iii. Derivation of the Fundamental harmonic frequency based upon (i). iv. Characteristics of the frequency spectrum of an implied Zero-Point-Field (ZPF) based upon (i) and the assumption that an EM relationship exists over a change in displacement across a practical benchtop test volume. The proceeding construct obeys the following hierarchy, v. A harmonic representation of the magnitude of gravitational acceleration “g” is developed in section 3.1. vi. The frequency spectrum of (v) is derived in section 3.2 by application of Buckingham Π Theory (BPT) and dimensional similarity developed in chapter 3.1. vii. The ZPF energy density is related to (vi) in section 3.3 based upon the assumption that engineered EM changes in “g” may be produced across the dimensions of a practical benchtop test volume. viii. Spectral characteristics of the Polarisable Vacuum (PV) are derived in section 3.4 based upon (vii). ix. A description of physical modelling criteria is presented in section 4. x. A set of sample calculations and illustrational plots are presented in section 5. 1.3

EXPERIMENTATION

The method of solution contained herein facilitates the determination of the following PV / ZPF experimental design boundaries at practical benchtop conditions, i. Amplitude and frequency spectra. ii. Poynting Vectors. 2

THEORETICAL MODELLING

2.1

TIME DOMAIN

FS may be applied to represent a periodic function as a trigonometric summation of sine and cosine terms. It may also be applied to represent a constant function over an arbitrary period by the same method. Since the classical PV model is a weak field isomorphic approximation of General Relativity (GR) and the frequency spectrum is postulated to range from “-∞ < ω < ∞”, it follows that FS represent a useful tool by which to describe gravity. 117

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2.2

DISPLACMENT DOMAIN

The time domain modelling in the proceeding section may be applied over the displacement domain of a practical benchtop test volume by considering the relevant changes over the dimensions of that volume. This is illustrated by sample calculations presented in section 5. 3

MATHEMATICAL MODELLING

3.1

CONSTANT ACCELERATION

Constant functions may be expressed as a summation of trigonometric terms. Subsequently, it is convenient to model a gravitational field utilising modified Complex FS, according to the harmonic distribution “nPV = -N, 2 - N ... N”, where “N” is an odd number harmonic. Hence, “g” may be usefully represented by the magnitude of a periodic square wave solution as “|nPV| → ∞”, g( r, M )

G. M . 2

r

n PV

2 . i . π .n PV .ω e π . n PV

.. PV ( 1 , r , M ) t i

(3.63)

where, the wavefunction amplitude spectrum “CPV” is calculated to be, C PV n PV, r , M

G.M . 2

r

2 π .n PV

(3.64)

such that, Variable ωPV(nPV,r,M) ωPV(1,r,M) nPV r M G

3.2

Description Units Hz Frequency spectrum of PV Fundamental frequency of PV Harmonic frequency modes of PV None Magnitude of position vector from the centre of mass m Mass kg Universal Gravitation Constant m3kg-1s-2 Table 3.13,

FREQUENCY SPECTRUM

It was illustrated in chapter 3.1 that dimensional similarity and the equivalence principle could be applied to represent the magnitude of an acceleration vector “aPV” as follows, a PV K 0 ω PV, r , E PV, B PV, X .

3 2 ω PV .r

c

(3.65)

where, Variable K0(ωPV,r,EPV,BPV,X) ωPV EPV BPV c

Description Experimental relationship function Harmonic frequency modes of PV Magnitude of PV Electric field vector Magnitude of PV Magnetic field vector Velocity of light in a vacuum Table 3.14,

Units None Hz V/m T m/s

In accordance with the harmonic representation of “g” illustrated by equation (3.63), “K0(ωPV,r,EPV,BPV,X)” is a frequency dependent experimental function. It was illustrated in chapter 3.1 that “K0(ωPV,r,EPV,BPV,X) = K0(X) = KPV-3/2”. Hence, an expression for the frequency spectrum 118

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may be derived in terms of harmonic mode. This may be achieved by assuming the acceleration described by equation (3.65) is dynamically, kinematically and geometrically similar to the amplitude of the 1st harmonic (|nPV| = 1) described by equation (3.64) as follows, aPV ≡ CPV(1,r,M)

(3.66)

The assumption associated with the preceding equation manifests by recognising that a FS is the hybridisation of the “CPV” and “ωPV” distributions where, “CPV” decreases as “nPV” increases and “ωPV” increases as “nPV” increases, intersecting at “|nPV| = 1”. Therefore, utilising equation (3.65) and (3.66), it follows that all frequency modes may be represented by, n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

(3.67)

Hence, the fundamental frequency (|nPV| = 1) as a function of planetary radial displacement may be graphically represented as follows,

Fundamental Frequency

RE

ω PV 1 , r , M E ω PV 1 , R E , M E

r Radial Distance

Figure 3.7, where, Variable RE ME

3.3

Description Radius of the Earth Mass of the Earth Table 3.15,

Units m kg

ENERGY DENSITY

The gravitational field surrounding a homogeneous solid spherical mass may be characterised by its energy density. If the magnitude of this field is directly proportional to the mass-energy density of the object, then the field energy density of the PV “Uω” may be evaluated over the difference between successive odd frequency modes. The reason for evaluation over odd frequency modes is due to the mathematical properties of FS for constant functions. For such cases – as appears in standard texts, the summed contribution of all even modes equals zero. [33] Subsequently, only odd mode contributions need be considered when modelling a constant function as follows [refer to Appendix 3.B for derivation], U ω n PV, r , M

U ω( r , M ) .

n PV

2

4

4

n PV

(3.68)

where, U ω( r , M )

h . 4 ω PV( 1, r , M ) 3 2.c

119

(3.69)

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Variable Description Uω(nPV,r,M) Energy density per change in odd harmonic mode h Planck’s Constant [6.6260693 x10-34] Table 3.16, 3.4

Units Pa Js

SPECTRAL CHARACTERISTICS

3.4.1 CUT-OFF MODE AND FREQUENCY Utilising the approximate rest mass-energy density “Um” of a solid spherical object, as described by equation (3.70), an expression relating the terminating harmonic frequency mode to “r” and “M” may be derived as follows, U m( r , M )

3 .M .c

2

4 .π .r

3

(3.70)

Assuming that “|Um(r,M)| = |Uω(nPV,r,M)|”, equation (3.70) may be related to equation (3.68) and solved for “|nPV|”. Hence, we may form the harmonic cut-off mode “nΩ” as follows, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

where, “Ω(r,M)” is termed the harmonic cut-off function, 3

108.

Ω ( r, M )

U m( r , M ) U ω( r , M )

12. 768 81.

U m( r , M )

2

U ω( r , M )

(3.72)

Subsequently, the upper boundary of the ZPF frequency spectrum “ωΩ” termed the harmonic cut-off frequency may be calculated as follows, ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73)

Therefore, “nΩ” and “ωΩ” may be graphically represented for the Earth as follows, RE n Ω R E, M E n Ω r, M E ω Ω r, M E ω Ω R E, M E

r Radial Distance

Cutoff Mode Cutoff Frequency

Figure 3.8, The derivation of equation (3.71 - 3.73) is based upon the compression of energy density to one change in odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordance with BPT. The preservation of similarity across one change in odd mode is due to the 120

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mathematical properties of constant functions in FS as discussed section 3.3. The subsequent application of these results to equation (3.63) acts to decompress the energy density over the Fourier domain yielding a highly precise reciprocal harmonic representation of “g” whilst preserving dynamic, kinematic and geometric similarity to the Newtonian, PV and GR representations. 3.4.2 ZERO-POINT-FIELD The cross-fertilisation of the amplitude and frequency characteristics of the Fourier spectrum with the ZPF spectral energy density distribution is a useful tool by which to analyse expected characteristics. This may be achieved by graphing the ZPF Poynting Vector in PV form “Sω” as follows,

ZPF Poynting Vector

S ω n PV, r , M

c .U ω n PV, r , M

(3.74)

S ω n PV , R E , M E

n PV Harmonic

Figure 3.9, 4

PHYSICAL MODELLING

4.1

POLARISABLE VACUUM

The spectral characteristics of the PV at the surface of the Earth may be articulated by assuming, i. The PV physically exists as a spectrum of frequencies and wave vectors. ii. The sum of all PV wave vectors at the surface of the Earth is coplanar with the gravitational acceleration vector. This represents the only vector of practical experimental consequence. iii. A modified Complex FS representation of “g” is representative of the magnitude of the resultant PV wave vector. iv. A physical relationship exists between Electricity, Magnetism and gravity such that the local value of gravitational acceleration may be investigated and modified utilising the equations defined in the preceding section. 4.2

TEST VOLUMES

The application of modified FS to define the modes of oscillation of physical systems has been experimentally verified since its development by Joseph Fourier (1768 - 1830). [34] The representation developed in the preceding section is defined in the time domain but may also be applied over an arbitrary displacement domain “∆r” as appears in standard engineering texts for beams, membranes, strings, control systems and wave equations. [25, 33] If we consider a small (experimentally practical) cubic test volume of length “∆r” to be filled with a large number of incremental displacement elements, frequency characteristics of the test volume may be hypothesised. Assuming each element within the test volume may be described by sinusoids of appropriate amplitude and frequency, it may be conjectured that the system 121

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interaction of the elements produces an amplitude and frequency spectrum consistent with a modified FS representation of “g” over “∆r”. The resultant wave vector at each frequency mode of the test volume is required to be coplanar with the gravitational acceleration vector for it to be representative of physical reality. Hence, only a line of action vertically downward through the cubic element is required for experimental consideration. Moreover, the mathematical representation of forces acting through the test volume is further simplified by approximating “g” as constant over the vertical dimension of the test volume. 4.3

TEST OBJECT

In accordance with PV and ZPF theories, test objects are assumed to produce a gravitational spectral signature in the same manner as the signature produced by planetary masses. Gravitational spectral signature is defined as the spectrum of amplitudes and frequencies unique to “r” and “M” by the application of modified FS. 5

SAMPLE CALCULATIONS

5.1

BACKGROUND GRAVITATIONAL FIELD

5.1.1 FUNDAMENTAL FREQUENCY The fundamental frequency mode of the PV at the surface of the Earth may be usefully approximated as follows, ωPV(1,RE,ME) ≈ 0.04(Hz) (3.75) 5.1.2 FREQUENCY BANDWIDTH An expression may be defined representing the frequency bandwidth of the local gravitational field as follows, ∆ω PV( r , M ) ω Ω ( r , M )

ω PV( 1 , r , M )

(3.76)

Assuming an ideal relationship between the mathematical model and the background gravitational field yields, ∆ωPV(RE,ME) ≈ 520(YHz) (3.77) where, “YHz = 1024(Hz)”. 5.2

APPLIED EXPERIMENTAL FIELDS

5.2.1 MODE BANDWIDTH Assuming a practical benchtop cubic element of length “∆r” possesses spectral attributes over the displacement domain, the number of permissible modes “Ν∆r” starting from “ωΩ” at “r” over “∆r” as “|nPV| → nΩ” may be approximated by, N ∆r( r , M ) ω Ω ( r , M ) .

∆r c

(3.78)

In figure (3.10), 1. The arrows pointing downwards represent a uniform gravitational field. 2. The arrows pointing upwards represent a uniformly applied system field. 3. The cube represents the experimental test volume of length “∆r”, with base residing at coordinates (0,0,r). 122

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4. The square section represents an EM flux area. 5. “h” represents the vertical displacement above the EM flux area.

∆r

h -X

Y

X

-Y Figure 3.10, 5.2.2 ENGINEERING CONSIDERATIONS

The factors to be considered in experimental design configurations are as follows: i. Where feasible, the experiment should attempt to maximise the applied energy density with preference towards the highest frequency bombardment possible. ii. Optimal energy delivery conditions occur at the highest achievable frequencies tending towards the harmonic cut-off mode “nΩ”. iii. Optimal experimental conditions occur when the ratio of the applied Poynting Vector to the Impedance Function approaches unity. iv. EM modes within the test volume are subject to normal physical influence. For example, the fundamental frequency mode cannot exist within a typical Casimir experiment; hence, the equivalent gravitational acceleration harmonic cannot exist. Hence, the relative contribution of low harmonic mode numbers to “g” is trivial. 6

CONCLUSIONS

The delivery of EM radiation to a test object may be used to alter the weight of the object. If the test object is bombarded by EM radiation, at high energy density and frequency, the gravitational spectral signature of the test object may undergo constructive or destructive interference.

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NOTES

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CHAPTER

3.5

General Similarity [67] Abstract An experimental prediction is developed considering gravitational acceleration “g” as a harmonic function across an elemental displacement utilising modified Complex Fourier Series (FS). This is evaluated to illustrate that the contribution of low frequency harmonics is trivial relative to high frequency harmonics when considering “g”. Moreover, the formulation and development of the Critical Boundary “ωβ” leading to the proposition that the dominant bandwidth arising from the formation of beat spectra is several orders of magnitude above the Tera-Hertz (THz) range, terminating at the ZPF beat cut-off frequency is presented. In addition, it is proposed that the modification of “g” is dominated by the magnitude of the applied Magnetic field vector “BA” and that the Electro-Gravi-Magnetic (EGM) spectrum is an extension of the classical ElectroMagnetic (EM) spectrum.

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Process Flow 3.5,

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1

INTRODUCTION

1.1

GENERAL

The Polarisable Vacuum (PV) model provides a theoretical description of space-time that may be derived from the superposition of ElectroMagnetic (EM) fields. The space-time metric may be engineered utilising Electro-Gravi-Magnetics (EGM), where EM fields may be applied to affect the state of the PV and thereby facilitate interactions with the local gravitational environment. This chapter continues previous work leading to practical modelling methods of the PV based upon the assumption that dimensional similarity exists between the space-time geometric manifold and applied EM fields. In accordance with Buckingham's Π Theory (BPT), experiments must be designed that tests the hypothesis stated in chapter 3.2. 1.2

HARMONICS

This chapter facilitates the following additions to the global EGM construct, i. Derivation of the fundamental harmonic beat frequency across an elemental displacement “∆r”, IFF “∆r << r”. This is evaluated to illustrate that the contribution of low frequency harmonics is trivial relative to high frequency harmonics when considering gravitational acceleration “g” across “∆r”. ii. Group velocities across “∆r”. iii. Formulation and development of the Critical Boundary “ωβ” leading to the proposition that the dominant bandwidth arising from the formation of beat spectra is several orders of magnitude above the Tera-Hertz (THz) range, terminating at the ZPF beat cut-off frequency “ωΩ ZPF”. iv. The development of General Similarity Equations (GSEx) applicable to experimental investigations. v. The proposition that the modification of “g” is dominated by the magnitude of the applied Magnetic field vector “BA”. vi. The proposition that the EGM spectrum is an extension of the classical EM spectrum. 2

THEORETICAL MODELLING

Fourier Series (FS) may be applied to represent a constant function over an arbitrary period by the infinite summation of sinusoids. Since the PV model of gravitation is an isomorphic approximation of General Relativity (GR) in the weak field, it follows that FS may present a useful tool by which to describe gravity as the number of harmonic frequency modes tends to infinity. The frequency spectrum of the PV is postulated to range from “-∞ < ωPV < ∞”. 3

MATHEMATICAL MODELLING

3.1

INTRODUCTION

The spectral composition of the PV / Zero-Point-Field (ZPF) is an important design consideration for experimental investigations. It was illustrated that the harmonic cut-off mode “nΩ” may be quantified by a system of equations. Taking limits of “nΩ” as described in chapter 3.4 yields the free space harmonic cut-off mode. Free space refers to a flat space-time manifold where the magnitude of the acceleration vector is “0(m/s2)”: hence, lim r

∞

-

n Ω ( r, M )

127

→∞

(3.79)

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Therefore, the spectral modelling characteristics of the PV / ZPF may be articulated as follows, i. The free space harmonic mode bandwidth is “-∞ < nPV < +∞”. ii. The magnitude of the free space harmonic cut-off mode tends to infinity. iii. The fundamental harmonic frequency of free space tends to zero. iv. The presence of a planetary mass superimposed on the ZPF alters the free space harmonic mode spectrum to “-nΩ(r,M) ≤ nPV ≤ +nΩ(r,M)”. v. The fundamental and cut-off harmonic frequencies of the PV / ZPF for a planetary mass increases as “r” decreases according to: “ωPV(1,r-∆r,M) > ω PV(1,r,M)”, “ωΩ(r-∆r,M) > ωΩ(r,M)” and “nΩ(r-∆r,M) < nΩ(r,M)” ωΩ(r,M) YHz ωΩ(RE,M0) → 0 ωΩ(RE,MM) ≈ 196 ωΩ(RE,ME) ≈ 520 ωΩ(RE,MJ) ≈ 2x103 ωΩ(RE,MS) ≈ 9x103 Table 3.17,

ωPV(1,r,M) Hz ωPV(1,RE,M0) → 0 ωPV(1,RE,MM) ≈ 0.008 ωPV(1,RE,ME) ≈ 0.0358 ωPV(1,RE,MJ) ≈ 0.2445 ωPV(1,RE,MS) ≈ 2.4841

nΩ(r,M) nΩ(RE,M0) → ∞ nΩ(RE,MM) ≈ 2.4x1028 nΩ(RE,ME) ≈ 1.5x1028 nΩ(RE,MJ) ≈ 7.6x1027 nΩ(RE,MS) ≈ 3.5x1027

where, “YHz = 1024 (Hz)”. Variable nΩ(r,M) ω PV(nPV,r,M) ωΩ(r,M) nPV r ∆r M RE M0 MM ME MJ MS

3.2

Description Units Harmonic cut-off mode of PV None Frequency spectrum of PV Hz Harmonic cut-off frequency of PV Harmonic frequency modes of PV None Magnitude of position vector relative to m the centre of mass of a planetary body Magnitude of change of position vector Mass of the planetary body kg Radius of the Earth m Zero mass condition of free space kg Mass of the Moon Mass of the Earth Mass of Jupiter Mass of the Sun Table 3.18,

PHENOMENA OF BEATS [35]

3.2.1 FREQUENCY It was illustrated in chapter 3.4 that it is convenient to model a gravitational field at a mathematical point utilising Complex FS obeying an odd number harmonic distribution. Subsequently, it follows that a beat frequency “∆ωδr” spectrum forms across “∆r” since “nΩ(r,M) ≠ nΩ(r ± ∆r,M)”. Hence, the change in frequency (also termed a beat) across “∆r” may be usefully approximated by, ∆ω δr n PV, r , ∆r , M

ω PV n PV, r

∆r , M

ω PV n PV, r , M

(3.80)

The fundamental beat frequency occurs when “|nPV| = 1” and may be expressed as “∆ωδr(1,r,∆r,M)” and the change in harmonic cut-off frequency “∆ωΩ” (also termed the PV beat bandwidth across “∆r”) becomes, 128

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∆ω Ω ( r , ∆r , M )

ω Ω(r

∆r , M )

ω Ω ( r, M )

(3.81)

where, “ωΩ” represents the harmonic cut-off frequency of the PV. 3.2.2 WAVELENGTH The change in harmonic wavelength “∆λδr” across “∆r” may be determined in a similar manner as follows, ∆λ δr n PV, r , ∆r , M

λ PV n PV, r

∆r , M

λ PV n PV, r , M

(3.82)

where, λ PV n PV, r , M

c ω PV n PV, r , M

(3.83)

Therefore, the change in harmonic cut-off wavelength “∆λΩ” may be given by, ∆λ Ω ( r , ∆r , M ) c .

1 ω Ω( r

1 ∆r , M )

ω Ω ( r, M )

(3.84)

3.2.3 GROUP 3.2.3.1 VELOCITY Group velocity is a term used to describe the resultant velocity of propagation of a set or family of interacting wavefunctions. Within the bounds of this book, we consider two distinct scenarios by which to construct the mathematical model. The first scenario concerns itself with engineering representations at a mathematical point “r”. At “r”, a spectrum of harmonic modes exists according to “-nΩ ≤ nPV ≤ +nΩ”. Superposition of these modes produces the constant function “g”. Therefore, it follows that the group velocity at a mathematical point is zero. Consequently, gravitational wavefunctions are not observed to radiate from a planetary body. The second scenario considers group velocities over a differential element “∆r”. Recognising that the change in modal amplitude across practical values of “∆r” at the surface of the Earth tends to zero, the group velocity “∆vδr” at each harmonic frequency mode may be defined as follows, ∆v δr n PV, r , ∆r , M

∆ω δr n PV, r , ∆r , M .∆λ δr n PV, r , ∆r , M

(3.85)

The terminating group velocity “∆vΩ” is the group velocity induced by the change in frequency at the highest harmonic mode “nΩ”. Since the number of modes varies significantly with “r”, the group velocity terminates with respect to the induced beat across “∆r” at the highest common mode number “nΩ(r,M)” (recalling that “nΩ” increases with “r”). Subsequently, “∆vΩ” occurs at the lower harmonic cut-off mode and may be defined as follows, ∆v Ω ( r , ∆r , M ) ∆v δr n Ω ( r , M ) , r , ∆r , M

(3.86)

3.2.3.2 ERROR Evaluating equation (3.85, 3.86) reveals incremental non-zero magnitudes at low harmonics tending to zero “([∆vδr],[∆vΩ]) → 0(m/s)” as “|nPV| → nΩ”. However, the expected result is that the group velocity is exactly zero at all modes “([∆vδr],[∆vΩ]) = 0(m/s)”. However, if “∆r → ∞”, then “∆vδr” is non-trivial and a mathematical statement has been made predicting the radiation of gravitational waves from the centre of mass of a planetary body. 129

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Therefore, we may consider the calculation of “∆vδr” and “∆vΩ” as being proportional measures of the mathematical representation error “RError” across “∆r”. It should be noted that the error revealed by equation (3.85, 3.86) is introduced by the simplification that the magnitude of the amplitude of “nPV” is constant across “∆r”. Typically, for practical values of “∆r” at the surface of the Earth, “RError → (∆vδr ≈ ∆vΩ) → 0(%)”. 3.2.4 BEAT BANDWIDTH CHARACTERISTICS 3.2.4.1 FREQUENCY Thus far, it has been illustrated that an amplitude and frequency spectrum exists at each mathematical point over the domain “0 < |r| < ∞”. The preceding body of work has defined certain characteristics, including change over the domain “∆r”. However, the variation in spectral bandwidth from “r” to “r+∆r” requires further consideration. Assuming the ZPF energy across “∆r” is equal to the change in the magnitude of the rest mass-energy density influence “|∆UPV(r,∆r,M)|” yields, h . ω Ω ( r , ∆r , M ) 3 ZPF 2.c

∆U PV( r , ∆r , M )

4

∆ω δr( 1, r , ∆r , M )

4

(3.87)

where, the ZPF beat cut-off frequency “ωΩ ZPF” becomes, 4

ω Ω ( r , ∆r , M )

ZPF

2 .c . ∆U PV( r , ∆r , M ) h 3

∆ω δr( 1 , r , ∆r , M )

4

(3.88)

Therefore, the ZPF beat bandwidth “∆ωZPF” may be defined as, ∆ω ZPF( r , ∆r , M ) ω Ω ( r , ∆r , M ) ZPF

∆ω δr( 1, r , ∆r , M )

(3.89)

3.2.4.2 MODES The ZPF beat cut-off mode “nΩ ZPF” corresponding to “ωΩ ZPF” may be determined utilising equation (3.90) developed in chapter 3.4 as follows, 1 . 2 .c .G.M . K PV( r , M ) r π .r 3

ω PV( 1 , r , M )

(3.90)

where, “ωPV(nΩ ZPF,r,M) = ωΩ(r,∆r,M)ZPF” and “|nPV| = nΩ(r,∆r,M)ZPF”. n Ω ( r , ∆r , M )

ω Ω ( r , ∆r , M ) ZPF

ZPF

ω PV( 1 , r , M )

(3.91)

3.2.4.3 CRITICAL RATIO “KR” is defined as the ratio of the applied fields to the ambient background field by any suitable measure. Consequently, “KR” in terms of the ratio of energy densities may be defined by, ω Ω ( r , ∆r , M ) KR

ω Ω ( r , ∆r , M )

4 ZPF

4 ZPF

ωβ

4

∆ω δr( 1 , r , ∆r , M )

130

4

(3.92)

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3.3

CRITICAL BOUNDARY

3.3.1 FREQUENCY The Critical Boundary “ωβ” represents the lower boundary of the ZPF spectrum yielding a specific proportional similarity value as follows, ω β r , ∆r , M , K R

4

ω Ω ( r , ∆r , M )

K R . ω Ω ( r , ∆r , M ) ZPF

4 ZPF

4

∆ω δr( 1 , r , ∆r , M )

4

(3.93)

Therefore, the similarity bandwidth “∆ωS” is given by, ∆ω S r , ∆r , M , K R

ω Ω ( r , ∆r , M ) ZPF

ω β r , ∆r , M , K R

(3.94)

3.3.2 MODE The Mode Number (Critical Boundary Mode) of “ωβ” may be calculated by re-use of equation (3.90) as follows, n β r , ∆r , M , K R

ω β r , ∆r , M , K R ω PV( 1, r , M )

(3.95)

Consequently, the change in the number of modes as a function of “KR” may be given by, ∆n S r , ∆r , M , K R

3.4

n Ω ( r , ∆r , M )

ZPF

n β r , ∆r , M , K R

(3.96)

BANDWIDTH RATIO

A bandwidth ratio “∆ωR” may be defined relating “∆ωZPF” to “∆ωΩ”. This represents the ratio of the bandwidth of the ZPF spectrum to the Fourier spectrum of the PV. “∆ωR” provides a useful conversion relationship between forms over practical benchtop values of “∆r” and may be defined as follows,

Bandwidth Ratio

∆ω R( r , ∆r , M )

∆ω ZPF( r , ∆r , M ) ∆ω Ω ( r , ∆r , M )

(3.97)

∆ω R R E , ∆r , M E

∆r Change in Radial Displacement

Figure 3.11,

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4

PHYSICAL MODELLING

4.1

GENERAL SIMILARITY EQUATIONS

4.1.1 OVERVIEW It was illustrated in chapter 3.2 that acceleration may be represented by the superposition of wavefunctions. The Primary Precipitant was decomposed to form General Modelling Equations GMEx. Therefore, for applied experimental fields (commencing from zero strength), the change in GMEx is equal to the required change of the magnitude of the gravitational acceleration vector “g”. ∆GME1 is proportional to a solution of the Poisson equation applied to Newtonian gravity, where the resulting acceleration is a function of the geometry of the energy densities. “∆GME2” is proportional to a solution of the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities of the EM field harmonics in a vacuum. Assuming proportional similarity (|KR| ≠ 1) between the ambient gravitational field across “∆r” and the mathematical model, a family of General Similarity Equations (GSEx) may be defined where “∆GME1 ≠ ∆GME2” for all “∆r” as “|nPV| → nΩ ZPF” and “+nΩ ZPF < +∞”. 4.1.2 GSEx GSE1,2 may be formed utilising the following energy balancing equations, ∆GME x

g 0

(3.98)

∆GME x

g 2 .g

(3.99)

such that, N EA k A,n A,t ∆GME x

∆K 0( ω , X ) 2 .r

.

2

nA= N

±c

2

(3.100)

N B A k A,n A,t

2

nA= N

∆K 0( ω , X )

G.M . KR 2 r .c

(3.54)

where, “∆K0(ω,X)” is the Engineered Relationship Function as derived in chapter 3.3, “kA” denotes the applied wave vector and the permittivity and permeability of free space, “ε0” and “µ0” respectively, act as the Impedance Function. Substituting equation (3.54, 3.100) into (3.98, 3.99) and solving for “KR” yields the Critical Ratio explicitly in terms of applied fields as “|nA| → nΩ ZPF” such that “|KR| → 1” as follows, N 2 2 .c .

KR

B A k A,n A,t

2

nA= N N

(3.101)

N EA k A,n A,t

2

2 ±c .

nA= N

B A k A,n A,t

2

nA= N

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Subsequently, proportional representations of similarity over the domain “1< |nA|
GSE E A , B A , k A , n A , t

B A k A,n A,t

2

nA= N 1, 2

N

N E A k A,n A,t

2 ±c .

2

nA= N

2

B A k A,n A,t nA= N

(3.102)

Similarly, it follows that GSE3 may be written utilising the following equation, KR

ε ∆K C ∆K 1 , ∆K 2 . 0 ∆U PV( r , ∆r , M ) µ0

(3.103)

where, ∆K C ∆K 1 , ∆K 2

2

∆K 1( ω , r , E, D , X )

N

N

1

. EA k A,n A,t ∆K 2( ω , r , B, D , X ) K 2 PV n A = N

2

.

B A k A,n A,t nA= N

2

(3.50)

Substituting equation (3.50) into (3.103) when “|KR| = 1” yields GSE3 as follows, GSE E A , B A , k A , n A , t , r , ∆r , M

1 3

K PV( r , M ) .∆U PV( r , ∆r , M )

.

ε0 µ0

N

N

.

EA k A,n A,t

2

nA= N

.

B A k A,n A,t nA= N

(Eq. 3.104) GSE4,5 may be formed by combining GSE1,2 with GSE3 as follows, GSE E A , B A , k A , n A , t , r , ∆r , M

GSE E A , B A , k A , n A , t , r , ∆r , M 4, 5

GSE E A , B A , k A , n A , t

3

1,2

(3.105)

where, Variable ∆GMEx g EA(kA,nA,t) BA(kA,nA,t) ∆KC(∆K1,∆K2) ∆UPV(r,∆r,M) c G

Description Change in applied acceleration vector Magnitude of gravitational acceleration vector Magnitude of applied Electric field vector Magnitude of applied Magnetic field vector Change in Critical Factor with respect to changes in experimental relationship functions Change in energy density of the gravitational field with respect to “r, ∆r and M” Velocity of light in a vacuum Universal Gravitational Constant Table 3.19,

133

Units m/s2 V/m T PaΩ Pa m/s m kg-1s-2 3

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2

4.2

QUALITATIVE LIMITS

Theoretical qualitative behaviour may be obtained for GSE1,2 by taking the limits of the Right Hand Side (RHS) of equation (3.102) with respect to applied EM fields. By performing the appropriate substitutions (|KR| → 1 as [|nPV|,|nA|] → nΩ ZPF) the following results are obtained, lim lim GSE E A , B A , k A , n A , t - B 0+ EA ∞ A

1, 2

lim lim GSE E A , B A , k A , n A , t - E + 0 BA ∞ A

1, 2

→0

(3.106)

→2

(3.107)

GSE1,2(EA,BA,kA,nA,t) qualitatively imply that achieving complete dynamic, kinematic and geometric similarity between the applied EM fields and “g” is facilitated by maximising “BA” whilst minimising “EA”. This suggests the proposition that “BA” dominates the local modification of “g”. The result, “|lim GSE1,2(EA,BA,kA,nA,t)| → 2” as “EA → 0+” and “BA → ∞-”, arises from the final energy density state of the PV after successful experimentation being twice the initial state. This results in a net magnitude of acceleration of “2g” and may be represented by the following equations, where “f” denotes the final state of the PV for complete similarity: N E f k PV, n PV, t

2

2.

n PV, k PV

E A k A,n A,t

2

nA= N

(3.108)

N B f k PV, n PV, t

2

n PV, k PV

2.

B A k A,n A,t

2

nA= N

(3.109)

As “|nA| → nΩ ZPF”, the superposition of applied wavefunctions describes the magnitudes of the Electric and Magnetic field vectors as constant (steady state) functions. Therefore, Maxwell's Equations (in MKS units) may define the system characteristics as follows (where: “ρ” is the charge density and “J” is the vector current density), [36] ∇ .E A

ρ ε0

,∇

E A 0 , ∇ .B A 0 , ∇

B A µ 0 .J

(3.110)

Consequently as “|nA| → nΩ ZPF”, optimal similarity occurs when: i. The divergence of “EA” is maximised. ii. The magnitude and curl of “EA” is minimised. iii. The magnitude and curl of “BA” is maximised. As the square root of the ratio of the sum of the applied field’s approach “c”, GSE1 approaches unity as follows, GSE E A , B A , k A , n A , t

lim N

c

E A k A,n A,t

2

B A k A,n A,t

2

1

nA= N N nA= N

→1 134

(3.111)

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Similarly, the applied field’s influence on GSE2 may be expressed as follows, GSE E A , B A , k A , n A , t

lim N

c

E A k A,n A,t

2

B A k A,n A,t

2

2

nA= N N nA= N

→ |Undefined| (3.112)

Consequently, characteristics of equation (3.106 - 3.112) are such that: iv. GSE1(EA,BA,kA,nA,t) qualitatively implies: “|KR| = 1” when “[ΣEA(kA,nA,t)2/ΣBA(kA,nA,t)2] → c2” as “|nA| → nΩZPF”. v. GSE1(EA,BA,kA,nA,t) qualitatively implies use over the range: “0 ≤ |GSE1(EA,BA,kA,nA,t)| < 2”. vi. GSE2(EA,BA,kA,nA,t) qualitatively implies use over the range: “0 ≤ |GSE2(EA,BA,kA,nA,t)| < 1 ∪ 1 < |GSE2(EA,BA,kA,nA,t)| < 2”. The results presented above should not be taken as definitive mathematical solutions or experimental predictions. However, deeper consideration may suggest that GSE1 represents an expression biasing constructive EGM interference, whilst GSE2 biases destructive EGM interference with “g”. The “undefined” result, indicated by equation (3.112), suggests that the local space-time manifold cannot be totally flattened in the presence of applied EM fields. The applied fields represent energy contributions that inherently modify the geometry of the local space-time manifold. 5

METRIC ENGINEERING

5.1

POLARISABLE VACUUM

Utilising GSE3, we may write (in terms of the applied Poynting Vector) the exponential metric tensor line element for the PV model representation of GR in the weak field limit analogous to the form specified in chapter 3.3 as follows, ds

2

µ υ g µυ .dx .dx

2 2 c .dt

K EGM

2 K EGM. dr

2 2 r .dθ

2 2 2 r .sin ( θ ) .dψ

(3.113)

g 00

1 K EGM

(3.114)

g 11 g 22 g 33 K EGM

where, 2.

K EGM e

G .M . 1 2 r .c

1. 2

GSE 3 3 K PV . e

(3.115) ∆K 0( ω , X )

(3.116) Note: i. “KEGM” is a function of the applied fields and constituent characteristics “(EA,BA,kA,nA,t)”. ii. “|nA| >> 1”.

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5.2

DESIGN CONSIDERATIONS

5.2.1 RANGE FACTOR The range factor “Stα(r,∆r,M)” is the product of “∆UPV(r,∆r,M)” and the Impedance Function “Z”. It is a useful “at-a-glance” design tool that indicates the boundaries of the applied energy requirements for experiments. The greater the magnitude of the range factor, the greater the magnitude of applied energy required for complete dynamic, kinematic and geometric similarity with the ambient background field and may be represented as follows, µ0

St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) .

ε0

(3.117)

We may determine specific limiting characteristics of the range factor for an ideal experimental solution where the upper limiting value is defined by, 3.M .c . 4.π 2

lim St ( r , 0, M ) + α ∆r 0

1 (r

∆r )

3

1 . µ0 3 ε0 r

St α ( r , 0, M ) 0

(3.118)

The lower limiting value is defined by, 1

µ0 3 .M .c . 4 .π 2

lim St α ( r , ∞ , M ) ∆r ∞

1 (r

∆r )

3

1 . µ0 3 ε0 r

St α ( r , ∞ , M )

3 . . 2. ε 0 Mc 3 4 π .r

2

(3.119)

The range of “|Stα(r,∆r,M)|” over the domain “0 < |∆r| < ∞” is given by, 0

St α ( r , ∆r , M ) <

2 3 .M .c . µ 0 3 ε0 4.π .r

(3.120)

5.2.2 SENSE CHECKS AND RULES OF THUMB For non-experimentally validated engineering undertakings, it is common practice to sense check predicted behaviour before proceeding. We may develop simple sense checks and rules of thumb by further considering the predicted mathematical results herein, in relation to other physical phenomena. For example, it is widely believed by proponents of the PV and ZPF models (of gravity and inertia respectively) that the Compton frequency of an Electron “ωCe” represents some sort of boundary condition. Subsequently, we may define the ratio of “∆ωZPF” to “ωCe” as the 1st Sense Check “Stβ” as defined by equation (3.121). This acts as an indicator regarding order-of-magnitude relationships and results. The Electron represents a fundamental particle in nature and it would seem inappropriate that “Stβ >> 1” (∆ωZPF >> ωCe) as it would imply that the beat bandwidth of ZPF frequencies, over practical benchtop values of “∆r” is much larger that the Compton frequency of an Electron, contradicting contemporary belief. Similarly, if “Stβ → 0”, then “ωCe >> ∆ωZPF” and would seem to imply that, assuming “ωCe” is representative of a natural gravitational boundary condition, proportional similarity (|KR| ≈ 1) by artificial means is not experimentally practical and the mathematical model derived to achieve similarity is inappropriate. Therefore, we expect that “0 << Stβ < 1”. Hence, 136

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∆ω ZPF( r , ∆r , M )

St β ( r , ∆r , M )

ω Ce

(3.121)

The 2nd Sense Check “Stγ” may be defined as the ratio of the magnitude of “∆ωΩ” to “ωCe”, therefore it follows that “Stβ ≥ Stγ” (ZPF bandwidth > the Fourier cut-off change). St γ ( r , ∆r , M )

∆ω Ω ( r , ∆r , M )

(3.122)

ω Ce

The 3rd Sense Check “Stδ” may be defined as the ratio of the harmonic cut-off modes across “∆r” (expected to be: ≈ 1). St δ ( r , ∆r , M )

n Ω( r

∆r , M )

n Ω ( r, M )

(3.123)

Therefore, it follows that, ∆ω R( r , ∆r , M )

St β ( r , ∆r , M ) St γ ( r , ∆r , M )

(3.124)

th

The 4 Sense Check “Stε” may be defined in terms of “RError” across “∆r” as follows (expected to be: ≈ 1), St ε n PV, r , ∆r , M

∆v δr n PV, r , ∆r , M ∆v Ω ( r , ∆r , M )

(3.125)

Hence,

Sense Check

RE

St β R E , ∆r , M E

St β r , ∆r , M E St γ r , ∆r , M E

St γ R E , ∆r , M E

r Radial Distance

Figure 3.12 [above, Y-Axis is logarithmic scale]: Figure 3.13 [below],

Sense Check

N

N

St ε n PV , R E , ∆r , M E

n PV Harmonic

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6

ENGINEERING CHARACTERISTICS

6.1

BEAT SPECTRUM

Characteristics of the beat PV / ZPF spectrum, over “∆r = 1(mm)”, at the surface of the Earth may be approximated according to the following table [PHz = 1015(Hz)], Characteristic Evaluated Approximation Wavelength λPV(1,RE,ME) ≈ 8.4x106(km) Change in Wavelength ∆λδr(1,RE,∆r,ME) ≈ 1.8(m) Change in Cut-Off Wavelength ∆λΩ(RE,∆r,ME) ≈ 0(m) Group Velocity ∆vδr(1,RE,∆r,ME) ≈ 1.3x10-11(m/s) Terminating Group Velocity ∆vΩ(RE,∆r,ME) ≈ 1.3x10-11(m/s) Representation Error RError ≈ 1.3x10-9(%) Fundamental Beat Frequency ∆ωδr(1,RE,∆r,ME) ≈ 7.5x10-12(Hz) Change in Cut-Off Frequency ∆ωΩ(RE,∆r,ME) ≈ 45(PHz) Beat Cut-Off Frequency ωΩ(RE,∆r,ME)ZPF ≈ 371(PHz) Beat Cut-Off Mode nΩ(RE,∆r,ME)ZPF ≈ 1x1019 Beat Bandwidth ∆ωZPF(RE,∆r,ME) ≈ 371(PHz) Critical Boundary Frequency ωβ (RE,∆r,ME,50%) ≈ 312(PHz) Critical Boundary Mode nβ(RE,∆r,ME,50%) ≈ 8.7x1018 Similarity Bandwidth ∆ωS(RE,∆r,ME,50%) ≈ 59(PHz) Similarity Modes ∆nS(RE,∆r,ME,50%) ≈ 1.7x1018 Bandwidth Ratio ∆ωR(RE,∆r,ME) ≈ 8.2 Bandwidth Ratio (∆r = 17mm) ∆ωR(RE,∆r,ME) ≈ 1 Range Factor |Stα(RE,∆r,ME)| ≈ 88(MPa MΩ) Range Factor Upper Limit |Stα(RE,∞,ME)| ≈ 2x105(GPa GΩ) 1st Sense Check Stβ(RE,∆r,ME) ≈ 4.8x10-4 nd 2 Sense Check Stγ(RE,∆r,ME) ≈ 5.8x10-5 3rd Sense Check Stδ(RE,∆r,ME) ≈ 1 th 4 Sense Check Stε(nPV,RE,∆r,ME) ≈ 1 Table 3.20, 6.2

CONSIDERATIONS

Some of the factors to be considered in experimental design configurations may be articulated as follows: i. The experimental design should attempt to maximise the applied energy density with the highest frequency conditions possible. ii. Optimal conditions occur approaching the ZPF beat cut-off mode “nΩ ZPF”. iii. EM modes within an experimental volume are subject to normal physical influence. The fundamental frequency mode will not exist within a Casimir experiment. Hence, the equivalent gravitational acceleration harmonic cannot exist. iv. Numerical solutions to equation (3.93) indicate that greater than “99.99(%)” of the EGM beat spectrum occurs in the “PHz” range. “KR ≈ 1” when “ωβ(RE,∆r,ME,99.999999999999 %) ≈ 312(PHz)” and “∆ωZPF(RE,∆r,ME) ≈ 371(PHz)”.

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6.3

EGM WAVE PROPAGATION

The gravitational effect generated by a specifically applied EM field harmonic may be conceptualised as a modified EM wave. Figure (3.14) depicts the manner in which pseudo-wave propagation occurs. This has been termed EGM Wave Propagation and has 5 components as follows, i. The Electric Field Wave. ii. The Magnetic Field Wave. iii. The Electro-Gravitic Coupling Wave (co-planar with the Electric Field Wave). iv. The Magneto-Gravitic Coupling Wave (co-planar with the Magnetic Field Wave). v. The Poynting Vector indicated in Figure (3.14) as the wave propagation arrow.

Figure 3.14, not to scale: 6.4

DOMINANT AND SUBORDINATE BANDWIDTHS

The EGM spectrum is fictitious and is derived from the concept of similarity. However, practical benefits to facilitate understanding of the concepts presented herein may be realised by the articulation, in terms of applied experimental fields, of the conventional representation of the EM spectrum. [37, 38] The EGM spectrum represents all frequencies within the EM spectrum but may be simplified into two regimes. These have been termed the dominant and subordinate gravitational bandwidths (“∆ωEGM δ” and “∆ωEGM σ” respectively) as indicated in Figure (3.15).

Figure 3.15, not to scale:

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At the surface of the Earth, over practical benchtop values of “∆r”, “∆ωEGM δ” is responsible for significantly more than “99.99(%)” of the spectral composition of “g”. Therefore, utilising table (3.17) we may approximate the classical EM spectral representation for frequencies of Gamma Rays “ωγ” at a mathematical point with displacement “r” as follows [YHz = 1024(Hz)], i. “105(PHz) > ωγ > 1(YHz)”. ii. “ωg > 1(YHz)”. where, “ωg” represents the gravitational frequency of the applied experimental fields for complete dynamic, kinematic and geometric similarity with the background gravitational field at the surface of the Earth. 6.5

KINETIC AND POTENTIAL

The EGM spectrum may be considered a hybrid function of an amplitude and frequency distribution. The harmonic behaviour across an element “∆r” has been described in terms of, i. The Fourier spectrum – termed the potential spectrum and is non-physical. ii. The ZPF spectrum – termed the kinetic spectrum and is physical. Properties of the Fourier spectrum are such that wavefunction amplitude decreases as frequency increases, whereas properties of the ZPF spectrum dictate constant amplitude with increasing frequency. Consequently, merging the two distributions as defined by equation (3.92) produces engineering properties and boundaries seemingly consistent with common-sense expectations. The potential spectrum has the advantage of being able to fictitiously represent ZPF behaviour at a mathematical point in addition to “∆r”. This is otherwise not possible due to the ZPF being a physical manifestation of “g” and the constituent wavefunctions possess finite wavelengths. 7

CONCLUSIONS

7.1

CONCEPTUAL

The construct herein suggests that the delivery of EM radiation to a test object may be used to modify its weight. Specifically, at high energy density and frequency, the gravitational spectral signature of the test object may undergo constructive or destructive interference. However, the frequency dependent conditions for gravitational similarity at the surface of the Earth are enormous: [ωβ ≈ 312(PHz) and ∆ωZPF ≈ 371(PHz)]. Summarising yields: i. The ZPF spectrum of free space is composed of an infinite number of modes “nPV”, with frequencies tending to “0(Hz)”, as illustrated in table (3.17). ii. The group velocity produced by the PV at a mathematical point and across practical values of “∆r” at the surface of the Earth is “0(m/s)”. Consequently, gravitational wavefunctions are not observed to propagate from the centre of a planetary body. iii. “|∆UPV(r,∆r,M)|” is proportional to “∆ωZPF(r,∆r,M)”. iv. “g” exists (at practical benchtop experimental conditions / dimensions) as a relatively narrow band of beat frequencies in the “PHz” range. Spectral frequency compositions below this range [approximately less than 42(THz)] are negligible [similarity ≈ 0(%)]. v. General Similarity Equation (GSEx) facilitates the construction of computational models to assist in designing optimal experiments. Moreover, they can readily be coded into “off-theshelf-3D-EM” simulation tools to facilitate the experimental investigation process. vi. A solution for optimal experimental similarity utilising EM configurations exists when Maxwell's Equations at steady state conditions are observed such that: (a) The divergence of “EA” and curl of “BA” is maximised. (b) The magnitude and curl of “EA” is minimised. 140

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7.2

PHYSICAL MODELLING CHARACTERISTICS

For “∆r << r” yields: 50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

Re ω β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.16, 50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

Im ω β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.17, 50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

ω β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.18, 50 .% Re ∆ω S R E , ∆r , M E , K R Im ∆ω S R E , ∆r , M E , K R

0

0.5

100 .% ∆ω S R E , ∆r , M E , 50 .% 1

1.5

2

KR Critical Ratio

Figure 3.19,

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50 .%

100 .%

∆ω S R E , ∆r , M E , K R ∆ω S R E , ∆r , M E , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.20, n β R E , ∆r , M E , 50 .% 100 .%

50 .%

Re n β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.21, n β R E , ∆r , M E , 50 .% 100 .%

50 .%

Im n β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.22, n β R E , ∆r , M E , 50 .% 100 .%

50 .%

n β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.23,

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50 .%

100 .% ∆n S R E , ∆r , M E , 50 .%

Re ∆n S R E , ∆r , M E , K R Im ∆n S R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.24, 50 .%

100 .%

∆n S R E , ∆r , M E , K R ∆n S R E , ∆r , M E , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

Figure 3.25, NOTES

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NOTES

144

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CHAPTER

3.6

Harmonic and Spectral Similarity [68] Abstract A number of tools to facilitate the experimental design process are presented. These include the development of a design matrix based upon: (i) the Critical Harmonic Operator “KR H” based upon a unit amplitude spectrum: (ii) the derivation of Harmonic and Spectral Similarity Equations (HSEx and SSEx): (iii) Critical Phase Variance “φC”: (iv) Critical Field Strengths (EC and BC) and (v), Critical Frequency “ωC”.

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Process Flow 3.6,

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1

INTRODUCTION

1.1

GENERAL

In previous chapters, a number of practical engineering tools for application to the Polarisable Vacuum (PV) model of gravity were derived by application of Buckingham Π Theory (BPT) and dimensional analysis techniques. BPT is a well-tested and experimentally verified method that relates a mathematical model to an Experimental Prototype (EP). The EP represents the PV at the surface of the Earth by which all similarity conditions are referenced. The power of BPT to facilitate and articulate the derivation of mathematical constructs has advanced theoretical boundaries to a higher level. The tools derived have volunteered precise calculations leading to the Critical Ratio “KR” and General Similarity Equations (GSE's). “KR” is defined as the ratio of the sum of the magnitudes of the applied ElectroMagnetic (EM) fields, to the magnitude of the background gravitational field. The General Modelling Equation’s (GME’s) derived in chapter 3.2 exploit these definitions to produce GSEx. 1.2

PRACTICAL METHODS

Practical engineering of the hypothesis to be tested may be realised by application of the equivalence principle with respect to “KR”. Complete similarity occurs when “|KR| = 1” and proportional similarity at “|KR| ≠ 1”, therefore it follows that “KR” may be used to represent relationships in terms of potential, acceleration, energy densities or any suitable measure in harmonic form. The harmonic representation of “KR” in the Fourier domain leads to a useful engineering tool facilitating the experimental design process. 1.3

OBJECTVES

This chapter assists in the qualitative and quantitative experimental design process as follows, i. Harmonic representation of “|KR| = 1” in the Fourier domain over an elemental displacement “∆r” termed the Critical Harmonic Operator “KR H” based upon a unit amplitude spectrum. ii. Utilisation of “KR H” to formulate harmonic representations of various other physical variables for consideration in the experimental design process. iii. Utilisation of “KR H” to simplify GSEx, on a modal basis, to Harmonic Similarity Equations (HSEx). iv. Graphical visualisation of HSEx based upon Complex Phasor Forms of the magnitude of the applied Electric and Magnetic fields (“EA” and “BA” respectively). v. The Reduction of HSEx into simplified ElectroMagnetic (EM) design consideration forms HSEx R. vi. Spectral Similarity Equations (SSE): these qualify and quantify the similarity of a singularly applied experimental EM source to the frequencies that inhabit the ambient Electro-GraviMagnetic (EGM) spectrum. vii. Determination of the applied EM phase requirements with respect to the background gravitational field utilising SSEx. viii. Assess the suitability of Maxwell's Equations to experimental investigations utilising SSEx. 1.4

RESULTS

The results obtained may be articulated by the development of a design matrix based upon, i. The derivation of “KR H”. ii. The derivation of HSEx R and SSEx. iii. Critical Phase Variance “φC”. 147

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iv. Critical Field Strengths “EC” and “BC” (Electric and Magnetic field strengths respectively). v. Critical Frequency “ωC”. 2

THEORETICAL MODELLING

Assuming complete dynamic, kinematic and geometric similarity between the EP and the mathematical model (|KR| =1) where the harmonic mode of the PV “nPV” approaches the harmonic cut-off mode “nΩ” [|nPV| → nΩ and nΩ < ∞], “KR” has many representations. One such representation incorporating the change in harmonic frequency modes across “∆r” shall be derived. The spectral characteristics of the EP may be articulated at the surface of the Earth assuming spherical geometry with uniform mass distribution, i. The Zero-Point-Field (ZPF) physically exists as a spectrum of frequencies and wave vectors. ii. The summed effect of all ZPF wave vectors at the surface of the Earth is coplanar with the gravitational acceleration vector. iii. A modified Complex FS representation of “g” is physically real and is representative of the magnitude of the resultant ZPF wave vector. iv. A physical relationship exists between gravity, Electricity and Magnetism such that the physical interaction of applied EM fields with the PV, in accordance with the hypotheses to be tested as defined in chapter 3.2, may be investigated and potentially modified. It was illustrated in chapter 3.3 that, for an engineered change in “g” by application of BPT and the equivalence principle, a change in the PV may be described [as |nPV| → nΩ] by,

∆g ≡ ∆a PV

∆K 0( ω , X )

E PV k PV, n PV, t

2

B PV k PV, n PV, t

2

n PV, k PV

.

r n PV, k PV

(3.48)

i

where, Variable ∆g ∆aPV EPV(kPV,nPV,t) BPV(kPV,nPV,t) ω kPV i ∆K0(ω,X) r

Description Change of gravitational acceleration vector Change in PV acceleration vector Magnitude of PV Electric field vector Magnitude of PV Magnetic field vector Field frequency Harmonic wave vector of PV Denotes initial conditions of PV Engineered relationship function Magnitude of position vector from centre of mass Table 3.21,

Units m/s2 V/m T Hz 1/m None m

Subsequently, considering only the resultant ZPF wave vector relating to “g” in a practical laboratory experiment, equation (3.48) may be usefully simplified by removing “kPV” notation and relating it to a generalised Fourier representation of constant “g” over “∆r” as “|nPV| → nΩ”, analogous to the form utilised in chapter 3.4, E PV n PV, t ∆K 0( ω , X ) n PV . r B n PV

2

G.M . PV n PV, t

2

i .

2

r

n PV

2 π .n PV

.e

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

(3.126)

i

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where, “∆ωδr” denotes the beat frequency across “∆r” as defined in chapter 3.5 and “∆K0(ω,X)” is the Engineered Relationship Function. E PV n PV, t c

2

2

n PV B PV n PV, t

2

n PV

(3.127)

G.M . ∆K 0( ω , X ) KR 2 r .c

(3.54)

Substituting equation (3.127) and (3.54) into (3.126) yields the PV - EM harmonic representation of the ideal value of the magnitude of “KR” for the complete reduction of “g” over “∆r” in a laboratory at the surface of the Earth as “|nPV| → nΩ”, K R( r , ∆r , M )

2. π

i . n PV

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

1 . e n PV

(3.128)

where, “i” on the Right Hand Side (RHS) of equation (3.126, 3.128) represents complex number notation and the maximum amplitude occurs at time index, t n PV, r , ∆r , M

1 2 . n PV .∆ω δr( 1 , r , ∆r , M )

(3.129)

Yields the unit amplitude spectrum analogous to the result previously found in chapter 3.4 as “|nPV| → nΩ(r+∆r)”, K R n PV

3

MATHEMATICAL MODELLING

3.1

DESIGN MATRIX

2

(3.130)

π . n PV

H

Utilising equation (3.130) a table of expressions for the magnitude of the amplitude spectrum of various experimental design considerate relationships may be formulated for complete dynamic, kinematic and geometric similarity between the EP and the mathematical model (|KR| =1) as “|nPV| → nΩ”, Description Eng. Rel. Func.

Primitive Form

Harmonic Form

∆K 0( ω , X)

∆K 0 n PV, r , M , X

G.M . KR 2 r .c

Refractive Index

2.

K EGM K PV. e U g( r, M )

Critical Factor KR

K PV n PV, r , M

2 r .c

K PV( r , M ) e

Engineered Refractive Index GPE / kg

G .M

2 . ∆K 0( ω , X )

G.M .m . 1 r m

∆K C( r , ∆r , M ) ∆U PV( r , ∆r , M )

.

ε0 µ0

H

G.M . K R n PV 2 H H r .c K PV( r , M ) .K R n PV

K EGM n PV, r , M , K R

U g n PV, r , M

H

H

H

ΣKPV H → KPV

H

K EGM r , M , K R

U g ( r , M ) .K R n PV

∆K C n PV, r , ∆r , M

Result Σ∆K0 H → ∆K0

.K n R PV

H

ΣKEGM H → KEGM ΣUg H → Ug

H

∆U PV( r , ∆r , M ) .K R n PV . H

µ0

Σ∆KC H → ∆KC

ε0

Table 3.22, where, the permittivity and permeability of free space (“ε0” and “µ0” respectively) act as the Impedance Function such that, 149

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Variable ∆UPV(r,∆r,M) ∆KC(r,∆r,M) G

3.2

Description Change in energy density of PV Change in Critical Factor Universal Gravitation Constant Table 3.23,

Units Pa PaΩ m3kg-1s-2

ENGINEERING CONSIDERATIONS

Factors to be considered in experimental design configurations when applying equations defined in table (3.22) are as follows: i. The actual EM modes over “∆r” are subject to normal physical influence. The fundamental frequency mode will not exist within a Casimir experiment; hence, the equivalent gravitational acceleration harmonic cannot exist. ii. The relative contribution of the fundamental frequency mode to the gravitational acceleration vector “g” is trivial. 4

PHYSICAL MODELLING

4.1

HARMONIC SIMILARITY EQUATIONS

A family of HSEx may be defined by relating the EP to the mathematical model on a modal basis, termed discrete similarity for “|∆r| << ∞”. Utilising GSE1,2 derived in chapter 3.5 yields HSE1,2; formed from the ratio of “KR(r,∆r,M)” to “GSE1,2” as follows, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

i . E A k A, n A, t

2

1, 2

2 ± c .B A k A , n A , t

2

.e

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2 π . n PV. c . B A k A , n A , t

2

(Eq. 3.131) Similarly, HSE3 may be formed utilising the ratio of “KR(r,∆r,M)” to “GSE3” as follows, π .n

HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

3

.∆ω

( 1 , r , ∆r , M ) .t .i

PV δr 2 .i .K PV( r , M ) .St α ( r , ∆r , M ) .e π .n PV.E A k A , n A , t .B A k A , n A , t

(Eq. 3.132) Hence, HSE4,5 may be formed utilising the ratio of “KR(r,∆r,M)” to “GSE4,5” as follows, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

2 4 .i .K PV( r , M ) .St α ( r , ∆r , M ) .c .B A k A , n A , t .e 4, 5

π .n PV.E A k A , n A , t . E A k A , n A , t

2

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2 ± c .B A k A , n A , t

2

(Eq. 3.133) Recognising that, i .e

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

1

(3.134)

Yields, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

1. 1,2

2

E A k A, n A, t

2

2 ± c .B A k A , n A , t

2 c .B A k A , n A , t

2

2

.K

R n PV

H

(Eq. 3.135)

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HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

K PV( r , M ) .St α ( r , ∆r , M ) 3

.K n R PV

E A k A , n A , t .B A k A , n A , t

H

(Eq. 3.136) HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

4, 5

2.

2 K PV( r , M ) .St α ( r , ∆r , M ) .c .B A k A , n A , t

E A k A,n A,t . EA k A,n A,t

2

2.

± c B A k A,n A,t

2

.K n R PV

(Eq. 3.137) where, St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) . 3 .M .c . 4 .π 2

∆U PV( r , ∆r , M )

Variable EA(kA,nA,t) BA(kA,nA,t) c M

4.2

µ0 ε0

1 (r

∆r )

(3.117) 1

3

3

r

(3.118)

Description Units Magnitude of applied Electric field vector V/m Magnitude of applied Magnetic field vector T Velocity of light in a vacuum m/s Mass kg Table 3.24,

VISUALISATION OF HSEx OPERANDS

Visualisation of HSE operands - the expression inside the magnitude notation on the Right Hand Side (RHS) of equation (3.131 - 3.133) - provides valuable information regarding the differences between forms. For example, it shall be demonstrated that HSE4,5 suggest constructive and destructive EM interference considerations. To achieve this, we shall utilise the following definitions for the applied EM fields in Complex Phasor Form, E A E 0 , n E, r , ∆ r , M , t

E 0.e

2 .π .ω E n E , r , ∆r , M .t

π . i 2

(3.138) B A B 0, n B, φ , r , ∆ r , M , t

B 0.e

2 .π .ω B n B , r , ∆r , M .t

π 2

φ .i

(3.139) Note: since “g” on a laboratory test bench at the surface of the Earth is usefully approximated to a one-dimensional (1D) situation and complete dynamic, kinematic and geometric similarity between the EP and the mathematical model (|KR| =1) is assumed, the harmonic wave vector “kA” has been omitted for simplicity. where, E rms

E0 2

B rms

B0 2

151

(3.140)

(3.141)

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H

Variable EA(E0,nE,r,∆r,M,t) BA(B0,nB,φ,r,∆r,M,t) E0 B0 nE nB φ ωE ωB Erms Brms

Description Applied Electric field vector Applied Magnetic field vector Amplitude of Electric field vector Amplitude of Magnetic field vector Harmonic mode number of the ZPF with respect to EA Harmonic mode number of the ZPF with respect to BA Relative phase variance between EA and BA Harmonic frequency of the ZPF with respect to EA Harmonic frequency of the ZPF with respect to BA Root Mean Square of EA Root Mean Square of BA Table 3.25,

Units V/m T V/m T None θc Hz V/m T

Equations (3.138, 3.139) are functions in Complex Form and contain Real and Imaginary components. For visualisation purposes, only the Real component is required. Figure (3.26) includes a graphical representation of “EA” and “BA” for arbitrary illustrational values. The representations for “Re(HSE1,2)” have been accentuated for illustrational purposes by a large value of “φ” (180°). Typically, values of “0°” would be expected in accordance with classical EM propagation, or “90°” in accordance with Maxwell’s Equations.

Re E A 1 .

V m

, 1 , R E , ∆r , M E , t

Re B A 1 .( T ) , 1 , 180 .( deg ) , R E , ∆r , M E , t V Re HSE 1 1 . , 1 .( T ) , 1 , 1 , 180 .( deg ) , 3 , R E , ∆r , M E , t m V Re HSE 2 1 . , 1 .( T ) , 1 , 1 , 180 .( deg ) , 3 , R E , ∆r , M E , t m

t

Electric Forcing Function Magnetic Forcing Function HSE 1 HSE 2

Figure 3.26, Figure (3.27) includes arbitrary illustrational values but also contains important information regarding “φ”. Exploratory graphical analysis demonstrates that “Re(HSE3)” is in-phase with “Re(HSE4)” and out-of-phase with “Re(HSE5)” for key values (0° and 90°) of “φ”. The significance of this being that “Re(HSE3)” is analogous to the Poynting Vector and implies that “Re(HSE4)” is representative of constructive EGM interference and “Re(HSE5)” is representative of destructive EGM interference. HSE4,5 were formed from General Modelling Equation “1” and “2” (GME1,2) as described in chapter 3.5. GME1 is proportional to a solution of the Poisson equation applied to Newtonian gravity where the resulting acceleration is a function of the geometry of the energy densities. GME2 is proportional to a solution of the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities of the EM field harmonics in a vacuum. 152

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Therefore, experimental investigations with the objective of reducing the local gravitational acceleration on a test bench, by means of EGM interference, should bias engineering designs governed by HSE5. However, designs favouring HSE4 should not be discounted and should form part of any complete design process.

Re HSE 3 1 .

V

Re HSE 4 1 .

V

Re HSE 5 1 .

V

m

m

m

, 1 .( T ) , 1 , 1 , 0 .( deg ) , 3 , R E , ∆r , M E , t , 1 .( T ) , 1 , 1 , 0 .( deg ) , 3 , R E , ∆r , M E , t , 1 .( T ) , 1 , 1 , 0 .( deg ) , 3 , R E , ∆r , M E , t

t

HSE 3 HSE 4 HSE 5

Figure 3.27, 4.3

REDUCTION OF HSEx

HSEx may be simplified by performing the appropriate substitution of equation (3.138 3.141). The simplified equations carry the subscript “R” (of the form HSEx R) and facilitate the investigation of the influence of “φ” on a modal basis. This becomes important in a practical sense because experimental investigations will involve “1” (or very few) applied forcing function frequencies. The reproduction of the entire background EGM spectrum would be technically difficult to achieve. Subsequently, experimental configurations will need to consider “φ” influence very carefully. Assuming the forcing function frequency of “EA” is equal to that of “BA” yields HSEx R as follows, HSE 1 φ, n PV

HSE 2 φ, n PV

2 .( cos ( 2 .φ) π .n PV

R

2 .( cos ( 2 .φ) π .n PV

R

HSE 3 E rms, B rms, n PV, r , ∆r , M

(3.142) 1)

(3.143)

K PV( r , M ) .St α ( r , ∆r , M ) π .n PV.E rms.B rms

R

HSE 4 E rms , B rms , φ , n PV, r , ∆r , M

HSE 5 E rms , B rms , φ , n PV, r , ∆r , M

4.4

1)

1 R

cos ( φ ) 1

R

sin ( φ )

.HSE E 3 rms , B rms , n PV, r , ∆r , M

.HSE E 3 rms , B rms , n PV, r , ∆r , M

(3.144) R

R

(3.145)

(3.146)

VISUALISATION OF HSEx R

Significant design information leading to complete dynamic, kinematic and geometric similarity with the background field (|KR| =1) may be obtained by visualisation of HSEx R. Assigning arbitrary values where required to analyse modelling behaviour facilitating the 153

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experimental design process yields, Figure (3.28) analysis: i. Harmonic similarity is maximised at “φ = 0°” and “φ = 90°”. Intuitively, this appears to agree with expectation; these phase angles are observed in classical vacuum EM wave propagation and Maxwell's Equations respectively. ii. Maximum harmonic amplitude occurs at “|nPV| = 1”: This implies that a low frequency carrier wave encasing a high frequency Poynting Vector maximises similarity of the applied fields with the background gravitational field. Intuitively, this appears to agree with expectation as the population of Photons in the ZPF is maximised at the fundamental harmonic. iii. “HSE1 = HSE2” at “φ = 45°” and “φ = 135°”. π

π

Harmonic Similarity

2

HSE 1_R ( φ , 1 ) HSE 1_R ( φ , 2 ) HSE 2_R ( φ , 1 ) HSE 2_R ( φ , 2 )

φ Phase Variance

Figure 3.28, Note: alteration of notation is required for graphing purposes. It is a limitation of the graphing software used herein that axial arguments may not be written precisely in the form HSE1 R(φ,1) etc. Figure (3.29) analysis [Y-Axis is logarithmic]: iv. “|HSE3 R| → 1” as “|nPV| → nΩ ZPF”: This is consistent with Poynting Vector characteristics described in chapter (3.2, 3.3). where,

Harmonic Similarity

Variable nΩ(r,∆r,M)ZPF RE ME

Description ZPF beat cut-off mode across “∆r” at “r” Radius of the Earth Mass of the Earth Table 3.26,

Units None m kg

HSE 3_R E rms , B rms , n PV , R E , ∆r , M E

n PV Harmonic Mode

Figure 3.29,

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4.5

SPECTRAL SIMILARITY EQUATIONS

The preceding sections define the requirements for complete dynamic, kinematic and geometric similarity with any specific mode in the background EGM field. However, reproduction of only one specific mode for experimental investigations is extremely limiting. Alternatively, it is highly advantageous to consider the reproduction of a harmonically averaged distribution for each HSE, termed Spectral Similarity Equations (SSE's). SSE's are defined as a family of equations that quantify and qualify the similarity of a single field source defined by HSE with respect to the spectrum of frequencies that inhabit the background EGM field. SSE's differs from GSE's in that GSE's represents similarity of multiple EM sources with respect to the background EGM field. Therefore, utilising the HSE's above, the magnitude of the average spectral similarity per frequency mode with respect to the applied forcing function may be generalised as follows, 1

SSE

.

n Ω ( r , ∆r , M )

1 ZPF

HSE

n PV

(3.149)

where, “nPV” has the odd harmonic distribution: “-nΩ ZPF, 2 - nΩ ZPF …. nΩ ZPF”. Recognising that (with error “< 6.7x10-6(%)” at “nΩ ZPF > 106”), 1 n Ω ( r , ∆r , M )

1 ZPF

ln 2 .n Ω ( r , ∆r , M ) ZPF

1

.

n PV

n PV

n Ω ( r , ∆r , M )

γ 1

ZPF

(3.150)

As “nPV → nΩ ZPF” and “nΩ ZPF >>1”: Substituting HSE's into equation (3.149) yields, SSE 1( φ, r , ∆r , M )

SSE 2( φ, r , ∆r , M )

SSE 3 E rms , B rms , r , ∆r , M

2 .( cos ( 2 .φ)

ln 2 .n Ω ( r , ∆r , M )

π

n Ω ( r , ∆r , M )

2 .( cos ( 2 .φ)

1) .

γ ZPF

1

(3.151)

ZPF

ln 2 .n Ω ( r , ∆r , M ) ZPF

π

n Ω ( r , ∆r , M )

γ 1

ZPF

. γ K PV( r , M ) . St α ( r , ∆r , M ) ln 2 n Ω ( r , ∆r , M ) ZPF . π .E rms .B rms n Ω ( r , ∆r , M ) 1 ZPF

SSE 4 E rms , B rms , φ, r , ∆r , M SSE 5 E rms , B rms , φ, r , ∆r , M

4.6

1) .

1 cos ( φ) 1 sin ( φ )

.SSE E 3 rms , B rms , r , ∆r , M .SSE E 3 rms , B rms , r , ∆r , M

(3.152)

(3.153) (3.154) (3.155)

CRITICAL PHASE VARIANCE

“φC” is defined as the phase difference between “EA” and “BA” for complete dynamic, kinematic and geometric similarity with the background EGM field (|SSEx| = 1). Therefore, by analyses of the preceding figures and the appropriate transformation of the preceding equations, “φC” may be easily determined. For proportional solutions to the Poisson equation applied to Newtonian gravity where the resulting acceleration is a function of the geometry of the energy densities, “φC = 0°”. For proportional solutions to the Lagrange equation where the resulting acceleration is a function of the 155

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Lagrangian densities, “φC = 90°”. 4.7

CRITICAL FIELD STRENGTH

“EC” and “BC” are derived utilising the reciprocal harmonic distribution describing the EGM amplitude spectrum. Solutions to “|SSE4,5| = 1” represent conditions of complete dynamic, kinematic and geometric similarity with the amplitude of the background EGM spectrum. “EC” and “BC” denote Root Mean Square (RMS) values satisfying the proceeding equation, π SSE 5 E rms , B rms , , r , ∆r , M 2

SSE 4 E rms , B rms , 0 , r , ∆r , M

1

(3.156)

where, E rms E C( r , ∆r , M ) B rms

4.8

(3.157)

E C( r , ∆r , M ) c

(3.158)

DC-OFFESTS

The value of “EC” and “BC” may be decreased by the application of an offset function “DC”. This denotes a percentage offset of the forcing function and may be applied to facilitate a specific experimental configuration. For example, if “DC = 100(%)” the value of “EC” and “BC” computed above yield, SSE 4 ( 1 DC) .E rms, B rms, 0, r , ∆r , M SSE 4 ( 1 DC) .E rms, ( 1 DC) .B rms, 0, r , ∆r , M

π SSE 5 E rms, ( 1 DC) .B rms, , r , ∆r, M 2

1 2

π SSE 5 ( 1 DC) .E rms, ( 1 DC) .B rms, , r , ∆r, M 2

(3.159) 1 4

(3.160)

Therefore, by re-computing the value of “EC” and “BC” at “|SSE4,5| = 1” a decrease in Critical Field Strength shall be observed. 5

MAXWELL’S EQUATIONS

5.1

GENERAL

By considering Maxwell's Equations in relation to the applied EM fields and the requirements of similarity, it is possible to deduce important design characteristics for further consideration. Maxwell's Equations (in MKS units) for time-varying fields are as follows (where, “ρ” is the charge density and “J” is the vector current density), [39] ∇ .E

ρ ε0

,∇

E

∂B ∂t

, ∇ .B 0 , ∇

B µ 0 .J

ε 0 .µ 0 .

∂E ∂t

(3.161)

Consequently as “|SSE5| → 1”, optimal similarity occurs when: i. The divergence of “EA” is maximised. ii. The curl of “BA” is maximised.

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5.2

CRITICAL FREQUENCY

“ωC” is defined as a half wavelength over “∆r” by applied fields and represents a minimum frequency for the application of Maxwell's Equations within this experimental context, ω C( ∆r )

6

c 2 .∆r

(3.162)

CONCLUSIONS

A number of tools that facilitate the experimental design process are presented. These include the development of a design matrix based upon the unit amplitude spectrum, the derivation of Harmonic and Spectral Similarity Equations (HSEx and SSEx), Critical Phase Variance “φC”, Critical Field Strengths (EC and BC) and Critical Frequency “ωC”. Note: equations (3.147, 3.148) were deleted from this section due to redundancy. NOTES

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NOTES

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CHAPTER

3.7

The Casimir Effect [69] Abstract An experimental prediction is formulated hypothesising the existence of a resonant modal condition for application to classical parallel plate Casimir experiments. The resonant condition is subsequently utilised to derive the Casimir Force “FPP” to high precision for a specific plate separation distance of “∆r = 1(mm)”; ignoring finite conductivity + temperature effects and evading the requirement for Casimir Force corrections due to surface roughness. The results obtained suggest Casimir Forces arise due to Polarisable Vacuum (PV) pressure imbalance between the plates induced by the presence of a physical boundary excluding low energy harmonic modes.

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Process Flow 3.7,

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1

INTRODUCTION

A Dutch Physicist named “Hendrick Casimir” predicted that quantum ElectroMagnetic field fluctuations would produce an attractive force between two neutrally charged reflective parallel plates, inexplicable by gravitational attraction (nowadays known as the Casimir Effect). This effect has been experimentally verified and its derivation states it to be cosmologically homogeneous. This chapter challenges that assertion and demonstrates that it depends upon environmental conditions (i.e. the magnitude of the ambient gravitational field strength). Chapter 3.6 established two Reduced Average Harmonic Similarity Equations (HSE4A,5A R). It shall be demonstrated that HSE4A,5A R may be utilised to describe the characteristics of Relative Phase Variance “φ” over the range of the Polarisable Vacuum (PV) harmonic “nPV”. Subsequently, deriving the Casimir Force and hypothesising a calculation of the PV inflection mode and frequency of a classical Casimir plate experiment. HSE4A,5A R presented in chapter 3.6 are as follows, HSE 4A E rms , B rms , φ, n PV, r , ∆r , M HSE 5A E rms , B rms , φ, n PV, r , ∆r , M

1 R

cos ( φ)

R

sin ( φ)

1

.HSE E 3 rms , B rms , n PV, r , ∆r , M

.HSE E 3 rms , B rms , n PV, r , ∆r , M

HSE 3 E rms, B rms, n PV, r , ∆r , M

R

(3.147) (3.148)

K PV( r , M ) .St α ( r , ∆r , M ) π .n PV.E rms.B rms

R

St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) . 3 .M .c . 4 .π 2

∆U PV( r , ∆r , M )

R

µ0

(r

(3.117)

ε0

1 ∆r )

(3.144)

1 3

3

r

(3.118)

where, the permittivity and permeability of free space (“ε0” and “µ0” respectively) act as the Impedance Function. Variable HSE3 R r ∆r c M Stα ∆UPV KPV Erms Brms

2

Description Units Reduced Harmonic Similarity Equation proportional None to the Poynting Vector of the PV Magnitude of position vector from centre of the Earth m Separation distance between parallel Casimir Plates Velocity of light in a vacuum m/s Planetary mass kg Range Factor PaΩ Change in energy density of PV Pa Refractive Index of PV None Root Mean Square of EA (applied Electric Field) V/m Root Mean Square of BA (applied Magnetic field) T Table 3.27,

THEORETICAL MODELLING

Spectral Similarity Equations (SSEx) were developed from HSEx. SSEx represent the average magnitude per harmonic mode, analogous to a solution of field pressure equilibrium with respect to the intensity of the amplitude spectrum. Of particular importance, SSE3 denotes a proportional formulation of the ambient (i.e. required applied) Poynting Vector as follows, 161

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SSE 3 E rms, B rms, r , ∆r , M

N X( r , ∆r , M )

K PV( r , M ) . St α ( r , ∆r , M ) π .E rms.B rms.N X( r , ∆r , M ) n Ω ( r , ∆r , M )

(3.163)

1 ZPF

ln 2 .n Ω ( r , ∆r , M ) ZPF

γ

(3.164)

where, “γ” denotes Euler’s Constant and “NX” is termed the harmonic inflection mode. Utilising equation (3.163) and assuming complete similarity between the PV and SSE3 yields the Critical Field Strengths “EC” and “BC” as follows, SSE 3 E rms, B rms, r , ∆r , M

1

(3.165)

E rms E C( r , ∆r , M ) B rms

(3.157)

E C( r , ∆r , M ) c

(3.158)

Substituting equation (3.157, 3.158) into equation (3.163) and solving for “EC” yields, E C( r , ∆r , M )

c .K PV( r , M ) . St α ( r , ∆r , M ) π .N X( r , ∆r , M )

(3.166)

Therefore, utilising equation (3.147, 3.148) and assuming complete similarity between the PV and HSE4A,5A R, an expression for “φ4,5” in terms of “nPV” for each harmonic form may be articulated as follows, φ 4 E C( r , ∆r , M ) , B C( r , ∆r , M ) , n PV, r , ∆r , M

Re acos HSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , n PV, r , ∆r , M

R

(Eq. 3.167) φ 5 E C( r , ∆r , M ) , B C( r , ∆r , M ) , n PV, r , ∆r , M

Re asin HSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , n PV, r , ∆r , M

R

(Eq. 3.168) Hence, π N C R E , ∆r , M E

N X R E , ∆r , M E

π

φ 4 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E

2

φ 5 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E

n PV

Figure 3.30, where, “RE” and “ME” denote radius and mass of the Earth and “NC” indicates the Critical Mode representing the condition of minimum permissible wavelength between the parallel plates over “∆r = 1(mm)”. “ωC” and “ωPV(1,r,M)” denote the Critical Frequency and fundamental harmonic frequency respectively, 162

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ω C( ∆r )

N C( r , ∆r , M )

ω PV( 1, r , M )

(3.169)

c ω C( ∆r ) . 2 ∆r

(3.162)

Analysis of figure (3.30) illustrates that “NX” represents a point of graphical inflection where the rate of change of “φ4,5” with respect to “nPV” is non-trivial. Notably, “φ4 = π” and “φ5 = π/2” over the range “1 ≤ nPV ≤ NX” and are influenced by the manner in which the applied forcing functions (“EA” and “BA” – representing the PV by similarity) have been initially defined. Since the PV cannot be uniquely described by a single mode, the arbitrary value of “φ” initially utilised in the mathematical construct is unimportant. Hence, the phase similarity on a modal basis may be disregarded. The Critical Phase Variance “φC” defined in chapter 3.6 considers the entire PV spectrum when defining the required value of “φ” for complete similarity. Subsequently, we may conjecture that the corresponding frequency at “NX” relates to a resonant condition where the Harmonic Inflection Frequency “ωX” may be defined as follows, ω X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

3

(3.170)

MATHEMATICAL MODELLING

To derive a relationship incorporating harmonic PV characteristics with the Casimir Force for parallel plates, we shall bring to the fore a suite of mathematical approximations resulting in a highly precise representation of the Casimir Force. Recognising that the sum of odd modes of a double sided reciprocal harmonic spectrum, symmetrical about the “0th” mode, approaches the sum of all modes of a one-sided reciprocal harmonic spectrum, with vanishing error, as “|nPV| → nΩ ZPF” and “nΩ ZPF >> 1” according to [refer to Appendix 3.B for derivation], n Ω ( r , ∆r , M ) 1 n PV

ZPF

1

ln( 2 )

n PV

n PV

n PV = 1

ln 2 .n Ω ( r , ∆r , M )

γ ZPF

(3.171) where, i. The Left Hand Side (LHS) of the preceding equation denotes the summation of all odd modes across the entire spectrum, symmetrical about the “0th” mode, following the sequence: “nPV = -nΩ ZPF, 2 - nΩ ZPF ... nΩ ZPF”. ii. The middle expression of the preceding equation represents the summation of all odd and even modes on the Right Hand Side (RHS) side of the spectrum following the sequence “nPV = 1, 2 … nΩ ZPF”. iii. “γ” on the RHS of the preceding equation denotes Euler’s Constant. Subsequently, the difference in sum between “NX” and “NC” may be usefully approximated as follows, ln 2 .N X( r , ∆r , M )

γ

ln 2 .N C( r , ∆r , M )

γ

ln

N X( r , ∆r , M ) N C( r , ∆r , M )

(3.172)

By contrast to the preceding equations, we shall apply classical arithmetic progression to facilitate the derivation of the Casimir Force. A Fourier distribution describing a constant function is composed of a reciprocal harmonic series governing amplitude characteristics and an arithmetic progression governing frequency characteristics. The interaction of these two spectral distributions intersects at the fundamental harmonic (|nPV| = 1).

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Subsequently, we expect that an expression relating the Casimir Force to harmonic distributions of the PV should consider aspects of a classical arithmetic sequence and a reciprocal harmonic series. Hence, Let: “A”, “D” and “StN” denote the values of the 1st harmonic term, common difference and “NTth” harmonic term respectively in a classical arithmetic sequence [40] as follows, St N A

NT

1 .D

(3.173)

where, the number of terms “NT” is, N T A , D , St N

St N

A

D

D

(3.174)

Hence, the ratio of the number of terms “NTR” relating “NX” to “NC” is, N T A , D , N X( r , ∆r , M )

N TR( A , D , r , ∆r , M )

N T A , D , N C( r , ∆r , M )

(3.175)

Considering the sum of terms yields, Σ H A , D, N T

NT

. 2 .A

D. N T

2

1

(3.176)

where, the ratio of the sum of terms “ΣHR” relating “NX” to “NC” is, Σ HR( A , D, r , ∆r , M )

Σ H A , D, N X( r , ∆r , M ) Σ H A , D, N C( r , ∆r , M )

(3.177)

Therefore, when “A = 1” and “D = 1,2”: N X( r , ∆r , M ) N C( r , ∆r , M )

4

PHYSICAL MODELLING

4.1

THE CASIMIR FORCE

N TR( 1 , 1 , r , ∆r , M )

Σ HR( 1 , 2 , r , ∆r , M )

(3.178)

Analysis and consideration of the mathematical characteristics of equation (3.171 - 3.178) facilitates the formulation of the Casimir Force “FPV” in terms of “NX” and “NC” [for a specific configuration of “∆r = 1(mm)”] as follows, F PV A PP , r , ∆r , M

A PP .∆U PV( r , ∆r , M ) .

N C( r , ∆r , M ) N X( r , ∆r , M )

2

.ln

N X( r , ∆r , M ) N C( r , ∆r , M )

4

(3.179)

where, “APP” denotes the projected area of a parallel plate in a classical Casimir experiment. We shall now compare the classical representation of the Casimir Force for parallel plates “FPP” to the preceding equation by performing a sample calculation, [8] F PP

π .h .c .A PP 4 480.∆r

(3.180)

Considering a Casimir plate area equal to planetary surface area in equation (3.179), yields a result to within “10-2(%)” of the classical representation of the Casimir Force described by equation (3.180).

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Analysis of equation (3.179) indicates that “FPP” decreases with increasing ambient gravitational environment. This concurs with chapter 3.5 and suggests the exclusion of fewer available low frequency modes. The mathematical construct defined in chapter 3.5 states that, as gravitational acceleration at the surface of a planetary body increases, “ωPV(1,r,M)” also increases. Therefore, an Earth based equivalent Casimir experiment conducted on Jupiter will exclude fewer low frequency modes – preserving higher frequency modes that simply pass through the plates, resulting in a smaller Casimir Force. By contrast, the same experiment conducted on the Moon will produce a larger Casimir Force. Notably, a Casimir Experiment conducted in free space will produce an extremely small force (tending to zero) due to the lack of initial background field pressure. Since the Casimir Force arises from a pressure imbalance, the lack of significant ambient field pressure between the plates prevents the formation of large Casimir Forces. 4.2

COSMOLOGICAL CONSTANT

The Cosmological Constant “Λ” is a function of the vacuum energy density, typically symbolised by “ρvac” or “UZPF”. Since the vacuum energy density is modified by the presence of gravitational fields (i.e. it becomes polarised in accordance with the “PV” model of gravity), one must differentiate between the cosmological average and local value of “Λ”. The proportional change in the local value of the Cosmological Constant “∆Λ” across small values of “∆r” may also be approximated harmonically commencing with the definition, 8 .π .G . ∆U PV( r , ∆r , M ) 2 3 .c

∆Λ PV( r , ∆r , M )

(3.181)

Exploratory factor analysis of “ωPV(1,r,M)”, “∆ωδr(1,r,∆r,M)”, “Um(r,M)” and “∆UPV(r,∆r,M)” produces the following approximate relationship for practical experimental values of “∆r”, ω PV( 1 , r , M )

2 U m( r , M ) 3 . 2 ∆U PV( r , ∆r , M )

∆ω δr( 1 , r , ∆r , M )

(3.182)

where, U m( r , M )

3 .M .c

2

4 .π .r

3

(3.70)

Hence, ∆Λ ( r , ∆r , M )

9 .G.M . ∆ω δr( 1, r , ∆r , M ) 2.r

3

ω PV( 1 , r , M )

(3.183)

Therefore, equation (3.183) is a useful weak field approximation as illustrated by the proceeding table defining errors with practical experimental values of “∆r” at “r”, Object The Moon The Earth Jupiter The Sun White Dwarf: “r ≈ 4200 (km) @ 3 x105 Earth Masses” [41] Red Giant: “r ≈ 200 Solar Radii @ 4 Solar Masses” [42] Neutron Star: “r ≈ 20 (km) @1 Solar Mass” [43] Table 3.28, 165

Mag. of error (%) ≈ 2.4545 x10-7 ≈ 6.5632 x10-5 ≈ 4.0931 x10-4 ≈ 3.6992 x10-3 ≈ 0.0238 ≈ 0.1952 ≈ 5.2482

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Typically in engineering research, predictions and experimental expectations with less than approximately “5(%)” error are generally considered to be “acceptable and useful” approximations for “back of the envelope” calculations. Subsequently, we may state that the Electro-GraviMagnetic (EGM) method of practical experimental modelling is useful over the range of sub-atomic particles to Neutron Stars. 4.3

REFINEMENT OF CLASSICAL CASIMIR EQUATION

Historically, integrating from infinity to the surface of a planet derives “FPP”. This approach assumes that the fundamental frequency of the ZPF at the surface of a planetary body is the same as free space [0(Hz)]. However, there is no physical evidence to support this contention and it shall be illustrated in proceeding chapters that non-zero fundamental frequencies lead to precise calculations of fundamental particle mass-energy and radii. “∆Λ” May be utilised to refine “FPP” to a solution precisely satisfying equation (3.179, 3.183). By appropriately relating equation (3.179, 3.180, 3.183), a Planetary Casimir Factor “KP” may be defined. “KP” represents a refinement of the value of “480” residing in the denominator of “FPP” and takes the generalised form, 2

K P( r , ∆r , M )

2 3 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 4 N C( r , ∆r , M ) 27.c .M .∆r N C( r , ∆r , M )

4

.

ω PV( 1 , r , M ) ∆ω δr( 1 , r , ∆r , M )

(3.184)

Assuming “KP” to be a representation of greater precision than the value of “480” in “FPP”, we may re-formulate “FPP” to be, F PP

π .h .c .A PP

. 4 480.0436∆r

(3.185)

-34

where, “h” denotes Planck’s Constant [6.6260693 x10 (Js)]. 5

CONCLUSIONS

An experimental prediction has been formulated hypothesising the existence of a resonant modal condition for application to classical parallel plate Casimir experiments. The resonant condition was subsequently utilised to derive the Casimir Force “FPP” to high precision for a specific configuration of “∆r = 1(mm)”; ignoring finite conductivity + temperature effects and evading the requirement for Casimir Force corrections due to surface roughness. The results obtained suggest Casimir Forces arise due to PV pressure imbalance between the plates induced by the presence of a physical boundary excluding low energy harmonic modes.

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ARTICLE

3.3

APPLICATION OF DERIVED ENGINEERING PRINCIPLES

Jean–Baptiste Joseph Fourier: 1768 – 1830

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ADVANCED ENGINEERING – MANKIND ON THE MOON

“ONE SMALL STEP FOR MAN, ONE GIANT LEAP FOR MAKIND” Neil Armstrong

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CHAPTER

3.8

Derivation of the Photon Mass-Energy Threshold [70] Abstract An analytical representation of the mass-energy threshold of a Photon is derived utilising finite reciprocal harmonics. The derived value is “< 5.75 x10-17(eV)” and is within “4.3(%)” of the Eidelman et. Al. value endorsed by the Particle Data Group (PDG) “< 6 x10-17(eV)”. The PDG value is an adjustment of theoretical predictions to fit physical observation. The derivation presented herein is without adjustment and may represent physical evidence of the existence of Euler’s Constant in nature at the quantum level.

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Process Flow 3.8,

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1

INTRODUCTION

It shall be demonstrated that the Polarisable Vacuum (PV) model of gravitation, complimenting General Relativity (GR) in the weak field, is capable of predicting the Photon massenergy threshold to within “4.3(%)” of the Particle Data Group (PDG) prediction presented by “Eidelman et. Al.” of “< 6 x10-17(eV)”. [12] The PDG is a collaboration of leading Nuclear and Theoretical Particle physicists funded by the USDoE, CERN, INFN (Italy), US NSF, MEXT (Japan), MCYT (Spain), IHEP and RFBR (Russia). The derived Photon mass-energy threshold “mγ”, based upon the physical properties of the Electron, may be usefully described by a finite reciprocal harmonic series representation as the number of harmonic modes approaches infinity, producing the result “mγ < 5.75 x10-17(eV)”. The proceeding section sets the foundation from which a complete construct may be formed based upon practical modelling methods. The use of physical modelling techniques will be shown to be highly advantageous in the development of “mγ”. 2

MATHEMATICAL MODELLING

The PV spectrum is conjectured to be composed of mathematical wavefunctions, over the symmetrical frequency domain “-ωΩ < ωPV < ωΩ”, which physically manifest as conjugate Photon pair populations. It shall be illustrated that the pending definition leads to a solution for the massenergy threshold of a population of Photons based upon the physical properties of an Electron as defined by, i. The geometry of a free Electron at rest is usefully approximated to spherical. ii. Electrons radiate a spectrum of conjugate Photon pairs through their spherical geometric boundary. iii. The term “conjugate Photon pair” denotes a theoretical particle population involving energy transfer resulting in the magnitude of the local acceleration vector “g”. The existence of conjugate Photon pair populations requires experimental validation and is conjectured to be equivalent to the Polarisation Electric field, “4πP” of a polarised dielectric medium coupled to the source field. iv. The modes of the PV spectrum contributing to gravitational effects exist as odd harmonics over the domain “nPV = -nΩ, 2 - nΩ ... nΩ”, symmetrical about the “0th” mode. The even modes (Imaginary component) of the complex Fourier function are disregarded due to null summation for all “|nPV|”. v. The amplitude spectrum of the Fourier distribution is proportional to the conjugate Photon pair population. We shall continue the construct by establishing a useful mathematical operator for subsequent use. It takes the form of the average value at each harmonic mode utilising the summation operand defined by equation (3.63) and may be generated as follows, 1 n PV.ω PV( 1 , r , M ) n PV.ω PV( 1 , r , M ) . 0 .( s )

2 .i . π .n PV .ω e π .n PV

.. PV( 1 , r , M ) t i

dt

4 2 n PV.π

(3.186)

where, “ωPV(1,r,M)” is the fundamental harmonic frequency derived in chapter 3.4. By considering an Electron at rest as a solid spherical particle with uniform surface and homogeneous mass-energy distribution, we may determine the magnitude of the average power at each odd harmonic mode. An important aspect to this, assuming an Electron radiates a spectrum of conjugate Photon 171

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pairs through its geometric boundary, is the proportional rest mass-energy power flow “c*Um(re,me)” through the surface “4πre2”. Hence, the mass-energy power flow at each mode “Ste” may be formed as follows, St e n PV

2 4 .π .r e . c .U m r e , m e

4

. 2

π . n PV

(3.187)

where, “re” and “me” denote the classical Electron radius and rest mass (kg) respectively. Subsequently, the magnitude of the average energy per odd harmonic period on either side of the PV spectrum “Stg” is defined by, St g n PV

St e n PV n PV .ω PV 1 , r e , m e

(3.188) th

Recognising that the PV spectrum is symmetrical about the “0 ” mode, we may formulate an expression for the mass-energy of the odd harmonic conjugate Photon pair population “mg”. Assuming that “|nPV| = nΩ” at the spherical boundary of an Electron, an upper limit for “mg” may be defined as follows, 2 . St g n Ω r e , m e Ng

mg N g

(3.189)

where, “Ng” denotes the Photon pair population. Evaluating equation (3.189) assuming that the population of conjugate Photon pairs is mode normalised to unity (Ng = 1) yields, mg ≈ 1.2 x10-15(eV) 3

(3.190)

PHYSICAL MODELLING

To predict the mass-energy threshold of a Photon “mγ”, we shall utilise the conjugate Photon pair population principles defined above. Firstly, we shall establish some useful mathematical relationships that facilitate the concise representation of “mγ”. It has been illustrated that the summation of the odd harmonic modes are representative of the magnitude of the acceleration vector “g”. Therefore, summing the odd modes across both sides of the spectrum leads to the following representation with vanishing error. This is proportional to the sum of all modes on the positive side of the spectrum as “|nPV| → nΩ” and “nΩ >> 1” as stated in chapter 3.7 [refer to Appendix 3.B for derivation], n Ω ( r, M ) 1 n PV

1

ln( 2 )

n PV

n PV = 1

ln 2 .n Ω ( r , M )

n PV

γ

(3.191)

There are half as many odd modes as there are “odd + even” modes when “|nPV| → nΩ”. Hence, we may deduce “mγ” by the following ratio, mg 1 > . ln 2 .n Ω r e , m e mγ 2

γ

(3.192)

Performing the appropriate substitutions and recognising that the preceding equation may be further reduced by usefully approximating the Refractive Index “KPV” to unity, yields the Photon massenergy threshold to be, mγ<

512.h .G.m e c . π .r e

2

.

n Ω r e, m e ln 2 .n Ω r e , m e

172

γ

(3.193) www.deltagroupengineering.com

Evaluating yields, mγ < 5.75 x10-17(eV)

(3.194)

By comparing the value of “mγ” derived to the value for the Photon mass-energy threshold endorsed by the PDG “< 6 x10-17(eV)”, [12] it is apparent that “mγ” compares favourably. 4

CONCLUSIONS

It has been illustrated that the PV model of gravity based upon the existence of a spectrum of frequencies makes the following predictions, i. The Photon mass-energy threshold for a mode normalised population of Photons is believed to be “< 5.75 x10-17(eV)”, based upon the physical properties of an Electron. ii. Experimental validation of the Photon mass-energy boundary predicted herein may be natural evidence of Euler’s Constant at a quantum level. NOTES

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NOTES

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CHAPTER

3.9

Derivation of Fundamental Particle Radii [71] Electron, Proton and Neutron Abstract Experimental predictions are derived from first principles for the Root Mean Square (RMS) charge radius of a free Electron and Proton, and Mean Square (MS) charge radius of a free Neutron to high computational precision [0.0118(fm), 0.8305 ± 0.0001(fm) and 0.8269(fm) respectively]. This places the derived value of Proton radius to within “0.38(%)” of the average “Simon” and “Hand” predictions [0.8335(fm)], arguably the two most precise and widely cited references since the 1960's. Most importantly, the SELEX Collaboration has experimentally verified the Proton radius prediction derived herein to extremely high precision as being {√[0.69(fm2)] = 0.8307(fm)}. The derived value of Electron radius compares favourably to results obtained in High-Energy scattering experiments [0.01(fm)] as reported by “Milonni et. Al.” It is also illustrated that a change in Electron mass of “≈ +0.04(%)” accompanies the High-Energy scattering measurements. This suggests that the Electron radius depends on the manner in which it is measured and the energy absorbed by the Electron during the measuring process. The Fine Structure Constant “α” is also derived, to within “0.026(%)” of its “National Institute of Standards and Technology” (NIST) value, utilising the Electron and Proton radii construct herein. In addition, it is also illustrated that the terminating gravitational spectral frequency for each particle may be expressed simply in terms of Compton frequencies. Precise2 calculations for: (i) the Neutron Mean Square (MS) charge and Magnetic radii, (ii) the Proton Electric, Magnetic and Classical RMS charge radii are derived in “Appendix 3.G”. Note: within this chapter, “ħ” (i.e. Dirac’s Constant) is applied to Compton Frequencies, whilst “h” (i.e. Planck’s Constant) is utilised in Compton Wavelengths.

2

Shown to be in agreement with experimental observations. 175

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Process Flow 3.9,

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1

INTRODUCTION

It is widely hypothesised, by proponents of the Polarisable Vacuum (PV) and Zero-PointField (ZPF) models of gravity and inertia respectively, that the Compton frequency of an Electron “ωCe” represents some sort of boundary condition. We may expand this hypothesis by recognising that the Compton frequency of a Proton “ωCP” and Neutron “ωCN” are multiples of “ωCe”. Hence, it follows that “ωCP” and “ωCN” may represent natural boundary conditions. The construct herein utilises Electro-Gravi-Magnetics (EGM) principles to facilitate the derivation of the Root Mean Square (RMS) charge radius of a free Electron and Proton, and Mean Square (MS) charge radius of a free Neutron to high computational precision [rε = 0.0118(fm), rπ = 0.8305 ± 0.0001(fm) and rν = 0.8269(fm) respectively]3. The Fine Structure Constant “α” may be formulated in terms of “rε” and “rπ” to within “0.026(%)” of its NIST value4 and utilised to numerically refine the derived value of “rε”. Subsequently, it is conjectured that High-Energy scattering measurements of “rε” result in a change in Electron mass “∆me” of “≈ +0.04(%)”. Thus, the harmonic cut-off frequency “ωΩ” for the Electron, Proton and Neutron are simplified to “ωΩ(rε,me) = 2ωΩ(rπ,mp)”, “ωΩ(rπ,mp) = ωCP2/ωCe” and “ωΩ(rν,mn) = ωCN2/ωCe” respectively. The subscripts “e, ε” denote classical and scattered Electron parameters respectively derived herein. 2

THEORETICAL MODELLING

2.1

SENSE CHECKS AND RULES OF THUMB

A series of sense checks and rules of thumb were defined in chapter 3.5 acting as indicators for order-of-magnitude relationships and results. Considering “ωCP” and “ωCN” as hypothetical boundaries, it follows that the Sense Checks (“Stη” and “Stθ”: 5th and 6th respectively) may be formulated utilising the ratio of “ωΩ” (as defined in chapter 3.4) of the Proton and Neutron to their respective Compton frequencies. Since “|ωΩ| → 0” as “r → ∞”, we might also expect that “[Stη, Stθ] → 0” as “r → ∞” according to the “1x2” matrix block as follows, St η r π , m p

St θ r ν , m n

ω Ω r π, m p

ω Ω r ν ,mn

ω CP

ω CN

(3.195)

where, “mp” and “mn” denote the rest mass of a Proton and Neutron respectively. 2.2

THE PROTON

When “Stη” is forced to consider RMS charge radii predictions for free Protons as reported by “Stein”5, tempting assumptions may be inferred. Table (3.29), illustrates the value of “Stη” in relation to four possible radii configurations. Based upon the computed values of “Stη” and “Stθ” as stated in table (3.29), we may hypothesise that the accuracy of the RMS charge radius of a free Proton may be numerically and analytically derived. By equating the value of “Stη” to the Proton to Electron mass ratio “mp/me”, highly precise radii predictions may be articulated.

3

The National Institute of Standards and Technology (NIST) states that the RMS charge radius of a free Proton to be [2002]: “rp = 0.8750 ± 0.0068(fm)” [1] where, “fm” represents femtometre [1(fm) = 10-15(m)]. 4 α = 7.297352568 x10-3 [2002] [1]. 5 “0.805 ± 0.011(fm)” and “0.862 ± 0.012(fm)”. [44] 177

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Stη(rp,mp) Stη(0.875(fm),mp) Stη(0.862(fm),mp) Stη(0.845(fm),mp) Stη(0.805(fm),mp) Stη(0.834(fm),mp)

Value 1783.8 1798.7 1818.7 1868.4 1832.6

Description Utilising the NIST 2002 value of rp [1] Utilising the value of rp as reported by Simon et. Al [44 - 46] Utilising the value of rp as reported by Andrews et. Al [47] Utilising the value of rp as reported by Stein [44 - 46] Utilising the average value of rp above as reported by [44 - 46] Table 3.29,

3

MATHEMATICAL MODELLING

3.1

DERIVATION OF PROTON AND NEUTRON RADII

3.1.1 NUMERICAL Utilising the results defined in table (3.29), we shall hypothesise that a numerically exact relationship exists between the ratio of the Compton wavelength of an Electron “λCe” to the Compton wavelength of a Proton “λCP” and the Proton to Electron mass ratio. Similarly, we shall hypothesise that a numerically exact relationship exists between the ratio of “λCe” to the Compton wavelength of a Neutron “λCN” and the Neutron to Electron mass ratio according to the “2x2” matrix block as follows, λ Ce m p St η r π , m p

St η r π , m p

λ CP m e

St θ r ν , m n

St θ r ν , m n

λ Ce m n λ CN m e

(3.196)

where, “rπ” and “rν” denote values satisfying equation (3.196) utilising the “Given” function within the “MathCad 8 Professional” environment. Hence, [rπ rν] = [0.8306 0.8269] (fm)

(3.197)

Comparing the results for “rπ” to the values illustrated in table (3.29), it is apparent that “rπ” compares favourably [within 1.8(%)] to the prediction [0.845(fm)] determined by “Andrews et. Al”. [47] Moreover, considering “rπ” in relation to the predictions derived by “Simon et. Al” [44] and “Hand et Al”, [48] arguably the two most precise and cited relevant works referenced by science since the 1960’s, [49] we find that “rπ” is within “0.38(%)” of the average “Simon et. Al” and “Hand et. Al” predictions [0.8335(fm)]. 3.1.2 ANALYTICAL Performing the appropriate substitutions from “Appendix 3.C” into the mass ratio relationships for “Stη” and “Stθ” in equation (3.196), useful analytical representations for “rπ” and “rν” may be formed in terms of Compton wavelengths and particle mass as follows, λ CP rπ

c .m e

rν

8 .π

2

5

.

4 27.m e 3 128.G.π .h

.

K PV r π , m p .m p λ CN K PV r ν , m n .m n

178

5

.

4

λ CP

4 2 K PV r π , m p .m p 5

.

4

λ CN

4 2 K PV r ν , m n .m n

(3.198)

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where, “c” represents the speed of light in a vacuum. “G” and “h” denote the Gravitational and Planck Constants respectively; the Refractive Index “KPV” may be usefully approximated to unity. Utilising the approximations and exact expressions described in “Appendix 3.C”, equation (3.198) may be simplified in terms of Compton, Planck (λh, ωh, mh) and particle mass characteristics. Hence, three highly precise analytical approximation forms of “rπ” and “rν” may be written as follows, 3

rπ

λ CP

.

λ CN

2 16.π .λ Ce

λ CP

27

. 2 4 . . 4 π λ h λ Ce

2 16.π .λ Ce 3

rν

5

5

.

27

. 2

λ CN 4

4 .π .λ h λ Ce

c .ω Ce

5

5

2 4 2 4 27.ω h ω Ce h .m e 27.m h m e . . . . 3 4 2 3 mp 4 .π 4 .ω CP 32.π ω CP 16.c .π .m p

c .ω Ce

5

(3.199)

5

2 4 2 4 27.ω h ω Ce h .m e 27.m h m e . . . . 3 4 2 3 mn 4 .π 4 .ω CN 32.π ω CN 16.c .π .m n

(3.200)

Subsequently, the analytical approximation error relative to the numerically precise result for “rπ” and “rν” returned may be shown to be trivial [< 10-6(%)]. 3.2

DERIVATION OF ELECTRON RADIUS

3.2.1 NUMERICAL It was illustrated in chapter 3.6 that the mass-energy threshold of a Photon “mγ” based upon the classical Electron radius “re” may be deduced by the summation of a finite reciprocal harmonic series. Since there are half as many odd harmonics as there are “odd + even” harmonics in a broad Fourier distribution6, the following relationship was derived [refer to Appendix 3.B for derivation], mg 1 > . ln 2 .n Ω r e , m e mγ 2

γ

(3.192)

where, “mg” represents the odd harmonic spectral mass-energy contribution and “γ” denotes Euler’s Constant. Applying Buckingham Π Theory (BPT) in terms of dynamic, kinematic and geometric similarity to the preceding equation and recognising that the mass-energy terms may be replaced by “ωΩ”, leads to an expression where “rε” may be numerically determined as follows, ω Ω r ε, m e

1.

ω Ω r e, m e

2

ln 2 .n Ω r e , m e

γ

(3.201)

It was illustrated in chapter (3.4 - 3.6) that a gravitational spectrum may be characterised by a frequency distribution terminating at “ωΩ”. Subsequently, it follows that IFF “re” represents a conditional experimental observation parameter; we may conjecture that the radius of an Electron occupies a range of values dependent upon how it is measured as suggested by recent scattering experiments. [11] Therefore, the preceding equation represents a robust mathematical condition defining the lower boundary of the Electron radius that preserves the gravitational nature of the works covered in chapter (3.1 - 3.6). Utilising the “Given” function within the “MathCad 8 Professional” environment, a highly precise numerical approximation for “rε” is determined to be, r ε 0.0118.( fm)

6

(3.202)

As the harmonic cut-off mode “nΩ” tends to infinity. 179

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3.2.2 ANALYTICAL An analytical representation of equation (3.201) may be formulated by performing the appropriate substitutions for “ωΩ” as defined in chapter 3.4 leading to the following relationship with trivial error, 9

r ε r e.

3.3

1. 2

ln 2 .n Ω r e , m e

5

γ

(3.203)

DERIVATION OF THE FINE STRUCTURE CONSTANT

An analytical approximation of “α” incorporating “rε” and “rπ” may be obtained utilising exploratory factor analysis. Applying the radii approximations above, a useful exponential relationship between the Proton and Electron may be defined to within “0.026(%)” of its NIST value based upon approximations of “rε” and “rπ” derived herein as follows, rε

α

2

.e

3

rπ

(3.204)

We shall conjecture that the “2/3” index is a qualitative indicator by considering the derivations in chapter 3.1 where it was illustrated that “2/3” relates the experimental relationship function “K0” to “KPV”. This assumption shall be further developed in the proceeding section. 3.4

ELECTRON CUT-OFF FREQUENCY

The calculated results imply that EGM may be a useful tool by which to enhance Nuclear understanding in the fields of Quantum-Electro-Dynamics (QED) and Quantum-Chromo-Dynamics (QCD). Subsequently, exploratory factor analysis in conjunction with the preceding formulations suggest that “ωΩ” for a free Electron may be usefully approximated to within “0.018(%)” as follows, ωΩ(rε,me) = 2ωΩ(rπ,mp) (3.205) 3.5

REFINEMENT OF ELECTRON RADIUS

Assuming equation (3.204, 3.205) to be exact representations may provide an opportunity for greater computational precision of “rε”. This may be achieved by utilising the “Given” function satisfying the following “1x3” matrix block within the “MathCad 8 Professional” environment, α

r ε ω Ω r ε, m e

rε

r e ω Ω r π, m p

rπ

9

2

.e

3

1. 2

ln 2 .n Ω r e , m e

5

γ

2

(3.206)

returns the result [ωΩ(0.011802(fm),me) = 2ωΩ(rπ,mp) to within 10-6(%)], rε = 0.011802 (fm) 3.6

(3.207)

DERIVATION OF ELECTRON SCATTERING MASS

The mass of the Electron based upon its classical radius has been established and scientifically accepted for many years. However, considering that scattering experiments have cast doubt on its radius, we may conjecture that the introduction of energy to the state of the Electron during radius measurements by scattering techniques affects its mass. 180

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It is also well accepted that the mass of a particle increases as its energy state increases leading to a reduction in its physical dimensions. Subsequently, we may conjecture that “rε” derived herein is accompanied by a “∆me” when “rε” is measured utilising High-Energy techniques as conducted by (for example) “Los Alamos National Laboratories” (LANL) [11] and the Stanford Linear Accelerator (SLAC). [50] Therefore, the Electron scattering mass “mε” may be determined utilising the “Given” function satisfying the following “1x2” matrix block within the “MathCad 8 Professional” environment, ω Ω r ε, m ε

ω Ω r ε, m ε

1.

ω Ω r e, m e

ω Ω r π, m p

2

ln 2 .n Ω r e , m e

γ

2

(3.208)

yields, ∆me ≈ +0.04 (%) 3.7

(3.209)

HARMONIC CUT-OFF FREQUENCIES

Utilising the preceding construct in conjunction with exploratory factor analysis, a simple family of equations may be formulated expressing the terminating gravitational spectral frequency for a free Electron, Proton and Neutron explicitly in terms of Compton frequencies in the form of a “1x3” matrix as follows, ω Ω r ε, m e

ω Ω r π, m p

4

PHYSICAL MODELLING

4.1

ELECTRON

2 .ω Ω r π , m p

ω Ω r ν ,mn

ω CP

2

ω CN

2

ω Ce

ω Ce

(3.210)

Equation (3.202, 3.207) agrees favourably with the results of High-Energy scattering experiments reported by “Milonni et. Al.”. [11] It states that, if the Electron is not a point particle, its physical dimensions are approximately no larger than “0.01(fm)” and it seems improbable that the Electron has any “structure”. These results strongly support EGM because “nΩ” is a function of radius and mass. Hence, it may be stated that EGM also implies that “structure” does not exist within “rε”. Therefore, we may conjecture that the free Electron radius and mass varies according to its energy level and may be physically modelled over the following set, {(r,M): rε ≤ r ≤ re ∩ me ≤ M ≤ mε} 4.2

(3.211)

PROTON

Relating equation (3.204) to the standard calculation form “α = 2πre / λCe” yields a set of highly precise physical modelling boundaries for “rπ” in terms of Compton, Planck and exponential characteristics as follows, rε

. c .e r e ω Ce

2 3

c .ω Ce

5

2 4 27.ω h ω Ce . . rπ 3 4 4 .ω CP 32.π ω CP

(3.212)

Therefore, “rπ” may be approximately written as: “rπ = 0.8305 ± 0.0001(fm)”.

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4.3

NEUTRON

In addition to “rν” predicted above, a set of useful physical modelling approximations [within “0.3(%)”] to assist with experimental design considerations may be defined based upon the proceeding equations, r ν λ CN ω CP m p r π λ CP ω CN m n

(3.213)

Notably, a convenient shorthand physical-modelling tool is the ratio of “rε” to the difference in radii between a free Proton and Neutron as follows, rε rπ

rν

≈π (3.214)

If we assume that equation (3.214) represents an exact analytical boundary solution where “rε” from equation (3.204) is utilised in conjunction with “rπ” from equation (3.199), the result returned for “rν” may be expressed in terms of Compton, Planck and trigonometric characteristics. The error on the Left Hand Side (LHS), with respect to the Right Hand Side (RHS) is less than “0.013(%)” as follows, rπ

rε π

rν

c .ω Ce 3

4 .ω CN

5

.

27.ω h ω Ce . 4 32.π ω CN 2

4

(3.215)

Therefore, “rν” may be approximately written as: “rν = 0.8269(fm)”. 5

EXPERIMENTATION

The SELEX Collaboration is an international effort pursuing experimental verification of particle properties such as the radius of a Proton. The Proton radius prediction derived herein has been experimentally verified to extremely high precision. [9] 6

CONCLUSIONS

It has been demonstrated that the EGM model of gravity predicts experimentally supported radii values of a free Electron, Proton and Neutron from an almost entirely mathematical foundation. Experimental predictions have been derived from first principles for the radii of a free Electron, Proton and Neutron to high computational precision. This places the derived value of Proton radius to within “0.38(%)” of the average “Simon et. Al” and “Hand et. Al” predictions, arguably the two most precise and widely cited references since the 1960's. Most importantly, the SELEX Collaboration has experimentally verified the Proton radius prediction derived herein to extremely high precision {√[0.69(fm2)] = 0.8307(fm)}. The derived value of Electron radius compares favourably to results obtained in HighEnergy scattering experiments conducted at “LANL”. It has also been illustrated that a change in Electron mass of “≈ +0.04(%)” accompanies the High-Energy scattering measurements. This suggests that the Electron radius depends upon the manner in which it is measured and the energy absorbed by the Electron during the measuring process. The Fine Structure Constant has also been derived, to within “0.026(%)” of its NIST value, utilising the Electron and Proton radii construct herein. In addition, it is predicted that the terminating gravitational spectral frequency for each particle may be expressed simply in terms of Compton characteristics. 182

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CHAPTER

3.10

Derivation of the Photon and Graviton Mass-Energies and Radii [72] Abstract The construct herein utilises the Photon mass-energy threshold “mγ” to facilitate the precise derivation of the mass-energies of a Photon and Graviton [mγγ = 3.2 x10-45(eV) and mgg = 6.4 x1045 (eV) respectively]. Moreover, recognising the wave-particle duality of the Photon, the Root Mean Square (RMS) charge radii of a free Photon and Graviton [rγγ = 2.3 x10-35(m) and rgg = 3.1 x1035 (m) respectively] is derived to high computational precision. In addition, the RMS charge diameters of a Photon and Graviton (“φγγ” and “φgg” respectively) are shown to be in agreement with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of the Planck length “λh”. The value of “φγγ” is illustrated to be “≈λh”, whilst the value of “φgg” is demonstrated to be “≈1.5λh”.

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Process Flow 3.10,

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1

INTRODUCTION

It has been demonstrated, based upon the physical properties of an Electron, that the Polarisable Vacuum (PV) model of gravitation, complimenting General Relativity (GR) in the weak field, is capable of predicting the Photon mass-energy threshold “mγ” to within “4.3(%)” of the Particle Data Group (PDG) prediction. [12] This chapter articulates the precise derivation of the mass-energies of a Photon and Graviton [mγγ = 3.2 x10-45(eV) and mgg = 6.4 x10-45(eV) respectively]. Moreover, recognising the waveparticle duality of the Photon, the Root Mean Square (RMS) charge radii of a free Photon and Graviton [rγγ = 2.3 x10-35(m) and rgg = 3.1 x10-35(m) respectively] is derived to high computational precision. In addition, the RMS charge diameters of a Photon and Graviton (“φγγ” and “φgg” respectively) are derived and shown to be in agreement with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of the Planck length “λh”. [51] Utilising the “Plain h” form where “λh = 4.05131993288926 x10-35(m)” [calculated from National Institute of Standards and Technology (NIST) 2002 [1]], the value of “φγγ” is illustrated to be “≈λh”, whilst the value of “φgg” is demonstrated to be “≈1.5λh”. 2

THEORETICAL MODELLING

Assuming that “mγ” as illustrated in chapter 3.8 represents an exact boundary value, a precise expression for “mγγ” may be derived utilising equation (3.193), mγ

512.h .G.m e c . π .r e

2

.

n Ω r e, m e ln 2 .n Ω r e , m e

γ

(3.193)

where, Variable h G c me re nΩ γ

Description Planck's Constant Universal Gravitation Constant Velocity of light in a vacuum Electron rest mass Classical Electron radius Harmonic cut-off mode of PV Euler's Constant Table 3.30,

Units Js m3kg-1s-2 m/s kg m None

To initiate the derivation process, we require a definition of “mgg” from which to apply dynamic, kinematic and geometric similarity with respect to “mγγ”. It has been illustrated that only the odd modes of a finite reciprocal harmonic distribution contribute to the magnitude of gravitational acceleration “g” according to “nPV = -nΩ, 2 - nΩ ... nΩ” being symmetrical about the “0th” mode where, “nPV” represents the modes of space-time manifold in the PV model of gravitation terminating at “nΩ”. The PV spectrum is conjectured to be composed of mathematical wavefunctions, over the symmetrical frequency domain “-ωΩ < ωPV < ωΩ”, which physically manifest as conjugate Photon pair populations. Subsequently, we shall define the odd frequency modes to be representative of conjugate Photon pair populations constituting a population of Gravitons. Therefore, “1” Graviton shall be defined as “1” conjugate Photon pair according to the following relationship, mgg = 2mγγ

(3.216)

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3

MATHEMATICAL MODELLING

Recognising that the Photon energy “EΩ” [52] at the harmonic cut-off frequency “ωΩ” is proportional to the conjugate Photon pair population, we may determine the Photon population “Nγ” at the mass-energy threshold as follows, E Ω h .ω Ω r e , m e

(3.217)

EΩ

Nγ

mγ

(3.218)

Performing the appropriate substitutions utilising equations defined in “Appendix 3.C” yields, 2 3 . . . c .π .r e 2 c Gme . . ln 2 .n Nγ Ω r e, m e 512.G.m e π .r e

γ

(3.219)

Hence, 3

m γγ

h . re

3

π .r e 2 .c .G.m e

.

512.G.m e

2

.

c .π

n Ω r e, m e ln 2 .n Ω r e , m e

2

γ

2

(3.220)

Evaluating yields, Nγ = 1.8 x1028

(3.221)

[mγγ mgg] = [3.2 6.4] x10-45(eV) 4

(3.222)

PHYSICAL MODELLING

In accordance with the preceding definition of Photon and Graviton mass-energy, we may apply Buckingham Π Theory (BPT) in terms of dynamic, kinematic and geometric similarity between two mass-energy systems defined at “ωΩ”. Subsequently, it follows that any two dimensionally similar systems may be represented by, ω Ω r 1, M 1

ω Ω r 2, M 2

(3.223)

where, “r1,2” and “M1,2” denote arbitrary radii and mass values. Subsequently, utilising equations defined in “Appendix 3.C” and performing the appropriate substitutions, the preceding equation may be simplified as follows, M1

2

r1

M2

5

r2

(3.224)

Let “M1 = mγγ/c2”, “M2 = me”, “r1 = rγγ” and “r2 = re”: solving for “rγγ” yields, 5

r γγ r e .

2

m γγ m e .c

2

(3.225)

where, “rγγ” may be expressed in terms of Compton and Planck characteristics as follows [to within 5 x10-3(%) of the precise numerical result], r γγ γ . λ h .

λ CN

c . ω CP h .m p λ CP ω h ω CN c .m h m n

186

(3.226)

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where, Variable λh λCN λCP ωCN ωCP ωh mn mp mh

Description Planck Length Neutron Compton Wavelength Proton Compton Wavelength Neutron Compton Frequency Proton Compton Frequency Planck Frequency Neutron rest mass Proton rest mass Planck Mass Table 3.31,

Units m

Hz

kg

Hence, r gg

5

4 .r γγ

(3.227)

Therefore, the Photon and Graviton RMS charge diameters may be expressed as multiples of the Planck length as follows, r 2 . γγ λ h r gg

1.1529 1.5213

[φγγ φgg] ≈ [1 1.5] λh 5

(3.228) (3.229)

CONCLUSIONS

The construct herein derives the mass-energies and RMS charge diameters of a Photon and Graviton. The results agree with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of Planck length “λh” according to “φγγ ≈ λh” and “φgg ≈ 1.5λh”.

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NOTES

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CHAPTER

3.11

Derivation of Lepton Radii [73] Abstract This chapter predicts the Root Mean Square (RMS) charge radii of free Electron “e-”, Muon “µ ” and Tau “τ-” particles. The Fine Structure Constant “α” is also derived as a function of Muon and Tau radii (“rµ” and “rτ” respectively) to within “7.6 x10-3(%)” of its 2002 National Institute of Standards and Technology (NIST) value. In addition, the Mean Square (MS) charge radii of free Electron, Muon and Tau Neutrino’s (“ren”, “rµn” and “rτn” respectively) is derived to high computational precision. These are shown to be in favourable agreement to experimental observations made by “the Sudbury Neutrino Observatory (SNO), Super-Kamiokande, Tristan, LEP, LEP-1.5, LEP-2, NuTeV, CHARM-II, CCFR, BNL E734 and DONUT” as analysed by “Hirsch et. Al.”. -

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Process Flow 3.11,

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1

INTRODUCTION

Electro-Gravi-Magnetics (EGM) principles may be utilised to facilitate the precise derivation of the mass-energies of a Photon and Graviton. Recognising the wave-particle duality of the Photon, the Root Mean Square (RMS) charge diameters of a free Photon and Graviton were derived in chapter 3.10 to high computational precision by methods of dimensional similarity. They were shown to be in agreement with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of the Planck length. Similarly, this chapter predicts the RMS charge radii of free Electron “e-”, Muon “µ-” and Tau “τ-” particles. Subsequently, the Fine Structure Constant “α” is derived as a function of Muon and Tau radii (“rµ” and “rτ” respectively) to within “7.6 x10-3(%)” of its 2002 National Institute of Standards and Technology (NIST) value. In addition, the Mean Square (MS) charge radii of free Electron, Muon and Tau Neutrino’s (“ren”, “rµn” and “rτn” respectively) are derived to high computational precision. These are shown to be in favourable agreement to experimental observations made by “the Sudbury Neutrino Observatory (SNO), Super-Kamiokande, Tristan, LEP, LEP-1.5, LEP-2, NuTeV, CHARM-II, CCFR, BNL E734 and DONUT” as analysed by “Hirsch et. Al.”. 2

THEORETICAL MODELLING

The application of Buckingham Π Theory (BPT) in terms of dynamic, kinematic and geometric similarity between two mass-energy systems defined at the harmonic cut-off frequency “ωΩ”, leads to a solution for the mass-energies and radii of the Photon and Graviton. Subsequently, it follows that any two completely similar systems may be represented by “ωΩ(r1,M1) = ωΩ(r2,M2)” where, “r1,2” and “M1,2” denote arbitrary radii and mass values. Therefore, utilising the equations defined in “Appendix 3.C” and performing the appropriate substitutions, an expression for proportional similarity may be stated as follows, 2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

r2 r1

9

St ω

(3.230)

where, “Stω” represents the harmonic cut-off frequency ratio between two proportionally similar mass-energy systems. 3

MATHEMATICAL MODELLING

3.1

ELECTRON RADIUS

It was illustrated in chapter 3.9 that “ωΩ(rε,me) / ωΩ(rπ,mp) = 2” to high computational precision. Hence, substituting “Stω = 2” into equation (3.230) yields an approximation for “rε”, 5

1 . me r ε r π. 9 2 mp

2

(3.231)

where, Variable rε, r1 rπ, r2 me, M1 mp, M2

Description RMS charge radius of a free Electron RMS charge radius of a free Proton Electron rest mass Proton rest mass Table 3.32, 191

Units m kg

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3.2

MUON - TAU RADII AND THE FINE STRUCTURE CONSTANT

Consideration of equation (3.230) provides initial guidance as to a method of determining the RMS charge radii of free “µ-” and “τ-” particles. Since there is little apparent difference in the physical behaviour of these particles with respect to “e-”, we may surmise that “rµ” and “rτ” are proportional to “rε”. Moreover, because the Muon rest mass “mµ” is less than the Tau rest mass “mτ” and are members of the same family, we expect that “rµ < rτ”. Subsequently, the application of BPT and dimensional similarity principles imply that “rµ” and “rτ” may be proportionally approximated as follows, 2

[rµ rτ] ≈

me

r ε. 1

2

9

me

1

mµ

9

mτ

(3.232)

It is important to note that equation (3.232) is not a definitive mathematical statement and requires further development. To facilitate this, we shall consider the effect of these radii approximations on “ωΩ” and “Stω” as follows, 1 ω Ω r ε, m e

. ω Ω r µ,mµ

≈ [4 6]

ω Ω r τ,m τ

(3.233)

Assuming the values of “Stω” determined in equation (3.233) represent exact analytical boundary conditions, highly precise representations for “rµ” and “rτ” may be formulated as follows, 5

rµ rτ

r ε.

1 . mµ 9 4 me

2 5

1 . mτ 9 6 me

2

(3.234)

Evaluating yields, [rµ rτ] ≈ [8.2122x10-3 0.0122] (fm)

(3.235)

Consequently, “α” may be expressed in exponential form utilising equation (3.234) [to within 7.6 x10-3(%) of the NIST 2002 value: α = 7.297352568 x10-3 [1]] as follows, rµ

α

rε

.e

rτ

rν

(3.236)

where, “rν” denotes the MS charge radius of a free Neutron as derived chapter 3.9 and evaluated to be “≈ 0.8269(fm)” [see also: Eq. (3.418)]. 3.3

NEUTRINO RADII

Lepton Neutrino’s are categorised within the family group into types. [53] Assuming that each Neutrino type (Electron Neutrino “νe”, Muon Neutrino “νµ” and Tau Neutrino “ντ”) shares a common value of “ωΩ” with its parent particle (“e-”, “µ-” or “τ-”), a highly precise representation for “ren”, “rµn” and “rτn” is possible and may be formulated as follows, Let: 2

ω Ω rε , µ , τ , me , µ , τ

me , µ , τ

ω Ω ren , µn , τn , men , µn , τn

men , µn , τn

192

5

9

.

rεn , µn , τn rε , µ , τ

9

1

(3.237)

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such that, 5

r en r µn r τn

r ε.

m en me

2

5

r µ.

m µn

2

mµ

5

r τ.

m τn

2

mτ

(3.238)

where, Variable men mµn mτn

Description [12] Electron Neutrino rest mass [< 3(eV/c2)] Muon Neutrino rest mass [< 0.19(MeV/c2)] Tau Neutrino rest mass [< 18.2(MeV/c2)] Table 3.33,

Units kg

Evaluating the preceding equation yields, [ren rµn rτn] < [9.5379x10-5 6.5524x10-4 1.9587x10-3] (fm)

(3.239)

Note: these results are only as accurate as the values of the Neutrino rest masses utilised. 4

PHYSICAL MODELLING

“Hirsch et. Al.” thoroughly revisited available observations from “the Sudbury Neutrino Observatory (SNO), Super-Kamiokande, Tristan, LEP, LEP-1.5, LEP-2, NuTeV, CHARM-II, CCFR, BNL E734 and DONUT” [54] as summarised in table (3.34). Hence, the radii predictions returned in equation (3.239) satisfy “Hirsch” conclusions. In addition, the Neutrino radii boundary value of “〈rε,µ,τ2〉 < 10-31(cm2)” as derived by “Joshipura et. Al.” [55] is also satisfied by equation (3.239). The authors conducted a worthwhile and thorough scientific analysis, but the “Hirsch et. Al.” study has greater scope and is the main focus for observational comparisons to the preceding construct. Hirsch Radii Range -5.5 ≤ 〈 rA2(νe) 〉 ≤ 9.8 -5.2 ≤ 〈 rA2(νµ) 〉 ≤ 6.8 -8.2 ≤ 〈 rA2(ντ) 〉 ≤ 9.9

EGM Derived Radii ren2 ≈ 9.0971x10-3 rµn2 ≈ 4.2933 rτn2 ≈ 3.8364 Table 3.34,

Scale x10-32(cm2) x10-33(cm2) x10-32(cm2)

where, “rA” denotes the axial vector charge radius. 5

CONCLUSIONS

The preceding construct derives the charge radii of the “µ-”, “τ-”, “νe”, “νµ” and “ντ” to high computational precision; “α” is also derived to within “7.6 x10-3(%)” of its NIST 2002 value. This result, in conjunction with experimental observations, implicitly validates all radii predictions derived herein. Note: a “MathCad 8 Professional” calculation algorithm utilising the analytical representations derived in chapter (3.9, 3.11) as exact boundary conditions, is defined in “Appendix 3.D” and evaluated.

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CHAPTER

3.12

Derivation of Quark and Boson Mass-Energies and Radii [74] Abstract This chapter assumes classical form factors to derive mass-energies [〈mallQuarks〉 = 30.6742(GeV)], in agreement with Particle Data Group (PDG) estimates [〈mallQuarks〉 = 29.9856(GeV)]. The Root Mean Square (RMS) charge radii of all flavours of Quarks is also derived [〈rallQuarks〉 = 0.9602 x10-16(cm)], in agreement with experimental observations and generalisations made by the ZEUS Collaboration (ZC) [〈rupQuark+downQuark〉 < 0.85 x10-16(cm)]. The “Top” Quark mass-energy derived [178.6(GeV)] is shown to be within “0.35(%)” of the value concluded by the D-ZERO Collaboration (D0C) [178.0(GeV)], in agreement with the Standard Model (SM) electroweak fit. The RMS charge radii of the “W”, “Z” and Higgs Boson is also derived and it is illustrated that all flavours of Quarks and Bosons exist as exact multiples of the harmonic cut-off frequency “ωΩ” of an Electron.

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Process Flow 3.12,

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1

INTRODUCTION

Quarks are fundamental natural constituent particles of prime importance to our understanding of the Universe. It was not until our recent scientific past that Quarks were known to exist. Prior to this knowledge, it was believed that Protons and Neutrons represented a natural particulate boundary state. Since the discovery of the Quark, an arsenal of considerable scientific expertise has been applied and further research is being pursued vigorously. The ZEUS Collaboration (ZC) has performed a major contribution to the body of knowledge regarding the physical dimensions of Quarks. The ZC is an international effort utilising its detector at HERA to determine physical characteristics of Quarks. [14] The experiments demonstrated no significant deviations from the predictions of the Standard Model (SM) and a generalised Root Mean Square (RMS) charge radius for the Quark “rxq”, based upon classical form factor, was concluded to be “< 0.85 x10-16(cm)” April 2004. Electro-Gravi-Magnetics (EGM) principles have been applied to facilitate the precise derivation of mass-energies and radii of a variety of particles to high computational precision. This chapter utilises a similar procedure, based upon methods of dimensional similarity, to predict the RMS charge radii of free Quarks. The term “free” indicates particles with classical form factor. This is achieved by the application of Buckingham’s Π Theory (BPT) in terms of dynamic, kinematic and geometric similarity between two mass-energy systems defined at a harmonic cut-off frequency “ωΩ”. Subsequently, it follows that any two completely similar systems may be represented by “ωΩ(r1,M1) = ωΩ(r2,M2)”, with proportional similarity when “ωΩ(r1,M1) ≠ ωΩ(r2,M2)” where, “r1,2” and “M1,2” denote arbitrary radii and mass values. This chapter assumes classical form factors to derive mass-energies [〈mallQuarks〉 = 30.6742(GeV)] in agreement with Particle Data Group (PDG) estimates [〈mallQuarks〉 = 29.9856(GeV)]. [12] The RMS charge radii of all flavours of Quarks are also derived, in agreement with experimental observations and generalisations made by the ZC [〈rupQuark+downQuark〉 < rxq]. The “Top” Quark mass-energy “mtq” derived [178.6(GeV)] is shown to be within “0.35(%)” of the value concluded by the D-ZERO Collaboration7 (D0C) [mtq = 178.0(GeV) - June 2004], [15] in agreement with the Standard Model (SM) electroweak fit. [12] The RMS charge radii of the “W”, “Z” and Higgs Boson is also derived and it is illustrated that all flavours of Quarks and Bosons exist as exact harmonic multiples of “ωΩ” for an Electron. Consequently, this suggests that the Photon may be the fundamental particle in nature from which all others may be described. The derived harmonic relationships between the Lepton, Quark and Boson groups, suggests that all fundamental particles radiate populations of Photons at specific frequencies. 2

THEORETICAL MODELLING

2.1

STATISTICAL CONSIDERATIONS

This chapter scrutinises important elements of the ZC and extends theoretical boundaries by application of the experimental data gathered. We shall commence by noting that the collisions studied released the constituent Quarks of the Proton. [14] Subsequently, the translation of data gathered into RMS charge radii does not differentiate between “Up” Quark radius “ruq” and “Down” Quark radius “rdq”. Protons are composed of two “Up” Quarks and one “Down” Quark; therefore, we shall assume that the ratio of collision data translating to Quark radii predictions obeys a “2:1” ratio. This leads to an equation relating radii to mean values of a large population sample as follows, 7

An international collaboration of leading scientists. 197

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r xq

2.2

2.r uq

r dq

3

(3.240)

GENERALISED SIMILARITY

Utilising the equations defined in “Appendix 3.C” and performing the appropriate substitutions, a generalised expression for similarity may be stated as follows, 2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

9

r2

St ω

r1

(3.230)

where, “Stω” represents the harmonic cut-off frequency ratio between two similar mass-energy systems. Therefore, assuming “ωΩ(rdq,mdq) = ωΩ(ruq,muq)” due to confinement within the Proton, an expression for “rdq” is possible in terms of “ruq” from equation (3.230) as follows, 5

r dq r uq .

m dq

2

m uq

(3.241)

Substituting equation (3.241) into (3.240) and solving for “ruq” yields, 1

5

r uq 3 .r xq. 2

m dq

2

m uq

(3.242)

where, “muq” and “mdq” represent the rest mass of the “Up” and “Down” Quark respectively. 2.3

RELATIVE SIMILARITY

At present, the mass-energy of Quarks has not been precisely measured. Consequently, our ability to mathematically predict Quark radii is restricted and mass-energy approximations must be utilised to apply EGM principles. However, the impact of the experimental gap may be minimised by assessing characteristics relative to an acceptable datum. This may be achieved by utilising “ωΩ” of the “Up” Quark to describe “ωΩ” of all other Quarks. Equation (3.243) represents a matrix of all Quark flavours, which acts to normalise “Stω” relative to the lightest particle as follows, St dq

ω Ω r dq , m dq

St sq

ω Ω r xq, m sq

St cq < St bq

1 ω Ω r uq , m uq

. ω Ω r xq, m cq ω Ω r xq, m bq ω Ω r xq, m tq

St tq

(3.243) Where: i. The subscripts “dq, sq, cq, bq and tq” denote “Down, Strange, Charm, Bottom and Top” Quarks respectively. ii. The subscript “ω” in “Stω” is replaced by “dq, sq, cq, bq and tq” as appropriate. This assists in distinguishing between harmonic cut-off frequency ratios of different Quarks relative to the “Up” Quark. iii. “m” denotes rest mass. 198

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Note: “rxq” is used in equation (3.243) to compensate for the lack of definitive experimental data for the radii of specific Quark flavours. In addition, the “ZC's” experimental generalisation of “rxq” validates its qualitative use in this forum. The proceeding construct utilises “rxq” as an initialisation value for subsequent development. 3

MATHEMATICAL MODELLING

Current scientific community assessments of experimental data pertaining to Quark rest mass vary significantly. Perhaps the most reliable estimates reside with the PDG. The PDG specify a range of Quark mass for each flavour, bounded by upper and lower limits based upon experimental observation and a variety of theoretical approaches. The “Top” Quark has a relatively narrow banded estimate based upon observation of top events. [12] By deductive reasoning, upper PDG and SLAC boundary estimates provide the most sensible values to be employed by equation (3.243) in a preliminary capacity. Subsequently, “Stω” values for all flavours of Quarks may be evaluated according to, Upper Limit [12] Units muq < 4 MeV mdq < 8 msq < 130 GeV mcq < 1.35 mbq < 4.7 [50] mtq < 179.4 Table 3.35, Hence, the “Stω” threshold values, in accordance with equation (3.243) utilising table (3.35) PDG estimates, are calculated in table (3.36). In addition, “Stω” threshold values are rounded down to the nearest integer to produce “Stω” harmonic values. This is an extremely important reduction as shall be illustrated in the proceeding section. Stω Threshold Stω Harmonic Stdq = 1 (normalised) Stdq = 1 Stsq < 2.0491 Stsq = 2Stdq Stcq < 3.4468 Stcq = 3Stdq Stbq < 4.5479 Stbq = 4Stdq Sttq < 10.2166 Sttq = 10Stdq Table 3.36, 4

PHYSICAL MODELLING

4.1

QUARK RADII

It shall be demonstrated that the physical properties (radii and mass-energy) of all flavours of Quarks may be described as integer multiples of the “Up” Quark in the form “ωΩ(rQuark,mQuark) / ωΩ(ruq,muq) = Stω”. This acts to unify the physical description of Quarks in terms of “ωΩ”. Transformation of equation (3.230) followed by the appropriate substitutions utilising “Stω” harmonic values defined in the preceding table, facilitate the precise determination of Quark radii as exact harmonics of the “Up” Quark as follows, 5

rsq , cq , bq , tq r uq .

1 Stsq , cq , bq , tq

. 9

msq , cq , bq , tq m uq

2

(3.244)

Yields, 199

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EGM Radii x10-16(cm) ruq = 0.7682 rdq = 1.0136 rsq = 0.8879 rcq = 1.0913 rbq = 1.071 rtq ≈ 0.8834 Table 3.37, Note: “rtq” has been listed as approximate, whereas all other flavours are listed as highly precise. This is an important statement and shall be explored in detail in the proceeding section. 4.2

QUARK MASS

It was illustrated in chapter 3.9 that an exact harmonic relationship exists between the Electron and Proton according to “ωΩ(rε,me) / ωΩ(rπ,mp) = 2”. Subsequently, it follows that exact harmonic relationships should exist between Electrons and Quarks. It shall be demonstrated that exact harmonic solutions satisfy all currently known boundaries regarding the radii and mass-energy of Quarks in accordance with ZC, PDG and D0C estimates as follows, Assuming, St uq

ω Ω r uq , m uq

St dq

ω Ω r dq , m dq

St sq St cq

<

1

.

ω Ω r ε, m e

ω Ω r sq , m sq ω Ω r cq , m cq

St bq

ω Ω r bq , m bq

St tq

ω Ω r tq , m tq

(3.245)

Yields, Stω Threshold Stω Harmonic Stuq < 7.207 Stuq = 7 Stdq < 7.207 Stdq = Stuq Stsq < 14.4141 Stsq = 2Stuq Stcq < 21.6211 Stcq = 3Stuq Stbq < 28.8281 Stbq = 4Stuq Sttq < 72.0703 Sttq ≈ 72 Table 3.38, where, “rε” denotes the Electron radius defined in chapter 3.11 and “me” represents Electron rest mass. Subsequently, transformation of equation (3.230) yields, muq , dq , sq , cq , bq , tq

m e.

Stuq , dq , sq , cq , bq , tq

9.

ruq , dq , sq , cq , bq , tq

5

rε

(3.246)

Followed by the appropriate substitution produces excellent results as follows, EGM Mass-Energy muq = 3.5083 mdq = 7.0166 msq = 114.0201

PDG Mass-Energy Range 1.5 < muq < 4 4 < mdq < 8 80 < msq < 130 200

Units MeV

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1.15 < mcq < 1.35 4.1 < mbq < 4.4 169.2 < mtq < 179.4 Table 3.39,

mcq = 1.1841 mbq = 4.1223 mtq = 178.6141

GeV

Therefore, by satisfaction of ZC experimental observations and PDG boundary conditions, it has been demonstrated that the radii and mass-energy characteristics of Quarks may be represented as exact harmonic multiples of “ωΩ(ruq,muq)”. Notably, D0C has suggested (June 2004) a new “world average” for the value of the “Top” Quark mass-energy to be “mtq = 178.0 ± 5.1(GeV)”. [15] Hence, the EGM mass-energy prediction is within “0.35(%)” of the D0C result. 4.3

REFINEMENT OF TOP QUARK RADIUS

Utilising “mtq” from table (3.39) and the exact “Stω” harmonic value in table (3.36), a refined prediction for “rtq” may be formulated satisfying the ZC, PDG, D0C and EGM as follows, 5

r tq r uq

1 . m tq 9 10 m uq

.

2

(3.247)

Evaluating yields, rtq = 0.9294x10-16(cm) 4.4

(3.248)

BOSON RADII

Utilising PDG mass-energy estimates defined in table (3.40), it shall be demonstrated that the “W”, “Z” and Higgs Boson “H” may also be described in terms of harmonic multiples of the “Up” Quark as follows, PDG Mass-Energy [12] Units GeV mW = 80.425 (range average) mZ = 91.1876 (range average) mH ≈ 114.4 (boundary value) Table 3.40, Bosons are exchange particles, therefore we may approximate their radii utilising the Heisenberg Uncertainty Range “ru” relationship, [16] where “rBoson ≈ ru” as follows, ru(M) ≈

h 4 .π .c .M

(3.249)

Hence, Heisenberg Radii x10-16(cm) ru(mW) ≈ 1.2268 ru(mZ) ≈ 1.082 ru(mH) ≈ 0.8624 Table 3.41, Assuming, ω Ω r u mW ,mW

St W St Z St H

≈

1 ω Ω r uq , m uq

. ω Ω r u mZ ,mZ ω Ω r u mH ,mH

(3.250)

Calculating and rounding to the nearest integer harmonic representation yields, 201

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Stω Threshold Stω Harmonic StW = 7 StW ≈ 7.1781 StZ = 8 StZ ≈ 7.9147 StH = 9 StH ≈ 9.4414 Table 3.42, Hence applying, 5

rW , Z , H r uq

.

1 StW , Z , H

.

mW , Z , H

2

m uq

9

(3.251)

Yields, EGM Radii x10-16(cm) rW ≈ 1.2835 rZ ≈ 1.0613 rH ≈ 0.9401 Table 3.43, We may perform a sense check of the EGM results by considering the Heisenberg Uncertainty approximation illustrated by equation (3.249). If the ratio of the predicted radii is approximately equal to “ru” [rBoson ≈ ru], then the predicted results appear feasible as follows,

5

rW

rZ

rH

ru mW

r u mZ

ru mH

( 1.0463 0.9809 1.09 )

(3.252)

CONCLUSIONS

The construct herein assumes classical form factors to derive mass-energies and RMS charge radii in agreement with PDG estimates, experimental observations and generalisations made by the ZC. The “Top” Quark mass-energy derived was shown to be within “0.35(%)” of the value concluded by D0C as illustrated in table (3.44). The RMS charge radii of the “W”, “Z” and Higgs Boson were also derived and it was illustrated that all flavours of Quarks and Bosons exist as exact harmonic multiples of the Electron as described by equation (3.253, 3.254). The derived harmonic relationships between the Lepton, Quark and Boson groups, suggests that all fundamental particles radiate populations of Photons at specific frequencies. Key results, EGM Radii x10-16(cm) ruq = 0.7682 rdq = 1.0136 rsq = 0.8879 rcq = 1.0913 rbq = 1.071 rtq = 0.9294 〈rQuark〉 = 0.9602 ≈ rxq rW ≈ 1.2835 rZ ≈ 1.0613 rH ≈ 0.9401 〈rBoson〉 ≈ 1.095 〈r〉 ≈ 1.0052

EGM Mass-Energy muq = 3.5083 mdq = 7.0166 msq = 114.0201 mcq = 1.1841 mbq = 4.1223 mtq = 178.6141 〈mQuark〉 = 30.6742

PDG Mass-Energy Range 1.5 < muq < 4 4 < mdq < 8 80 < msq < 130 1.15 < mcq < 1.35 4.1 < mbq < 4.4 169.2 < mtq < 179.4 〈mQuark〉 = 29.9856 80.387 < mW < 80.463 91.1855 < mZ < 91.1897 mH > 114.4

Units MeV

GeV

Table 3.44, 202

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1

.

ω Ω r uq , m uq 1 ω Ω r ε,m e

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

1 2 3 4

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

7 8 9 10

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

7 14 21 28

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

49 56 63 70

(3.253)

(3.254)

Note: the “MathCad 8 Professional” calculation algorithm utilised to derive all computational results presented in this chapter is contained in “Appendix 3.E”. NOTES

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NOTES

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CHAPTER

3.13

The Planck Scale, Photons, Predicting New Particles and Designing an Experiment to Test the Negative Energy Conjecture [75] Abstract This chapter utilises previous works to predict a “16(%)” experimentally implicit increase of the Planck Scale. An approximation of the Root Mean Square (RMS) charge radius of a free Photon “rγγ” utilising physical properties of Lepton family particles is derived. Moreover, the existence of three (3) new particles in the Lepton family is also predicted at the 2nd, 3rd and 5th Electron harmonics with mass-energies of approximately “9(MeV), 57(MeV) and 566(MeV)” respectively. The existence of two (2) new particles in the Quark / Boson families is also predicted at the 5th and 6th “Up” Quark harmonics with mass-energies of approximately “10(GeV) and 22(GeV)” respectively. In addition, the optimal configuration of a Classical Casimir Experiment to test the negative energy conjecture is also presented. It is concluded that the optimal practical benchtop physical conditions to test the conjecture exist in the “X-Ray” Laser range.

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Process Flow 3.13, 206

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1

INTRODUCTION

This chapter utilises previous works to derive: i. A “16(%)” experimentally implicit increase of the Planck Scale. ii. An approximation of the Root Mean Square (RMS) charge radius of a free Photon “rγγ”, utilising physical properties of the Lepton family. iii. The existence of three (3) new particles in the Lepton family at the 2nd, 3rd and 5th Electron harmonics with mass-energies of approximately “9(MeV), 57(MeV) and 566(MeV)” respectively. iv. The existence of two (2) new particles in the Quark / Boson families at the 5th and 6th “Up” Quark harmonics with mass-energies of approximately “10(GeV) and 22(GeV)” respectively. v. The optimal configuration of a Classical Casimir Experiment to test the negative energy conjecture. It is concluded that the optimal practical benchtop physical conditions to test the conjecture exist at: 1. A plate separation distance of “≈ 16.5(mm)”. 2. An applied ElectroMagnetic (EM) beam wavelength of “≈ 18(nm)”. This is indicative of the “X-Ray” Laser range. 3. An RMS Electric Field Intensity of “≈ 550(V/m)”. 4. An RMS Magnetic Flux Density of “≈ 18(milli-gauss)”. 5. A phase variance between the applied Electric and Magnetic fields of “0, ±π or ±π/2”. 2

THE PLANCK SCALE

2.1

CONVERGENT BANDWIDTH

The Planck Scale is considered a limiting natural condition by which to formulate mathematical constructs and to qualitatively validate derivations. Subsequently, we may utilise Planck properties of length and mass (“λh” and “mh” respectively) to qualitatively validate the Electro-Gravi-Magnetic (EGM) construct. By considering all works covered in chapter 3.1 - 3.12, we expect that the Polarisable Vacuum (PV) spectral frequency bandwidth to converge to a single mode at conditions of maximum energy density. Planck Mass denotes such a condition and represents a useful qualitative measure. Computing the value of the harmonic cut-off mode “nΩ” and the harmonic cut-off frequency “ωΩ” to fundamental harmonic frequency “ωPV” ratio, a value of unity for equation (3.255) [to within 0.2(%)] indicates that the PV spectral frequency bandwidth converges to a single mode at conditions of maximum energy density. n Ω λ h,mh

2.2

ω Ω λ h,mh ω PV 1, λ h , m h

1

(3.255)

PLANCK CHARACTERISTICS

It is important to note that we would not expect “ωΩ” to be equal to the Planck Frequency “ωh”. The reason for this is due to the manner in which “ωh” is derived. Historically, it involved dimensionally combining the Universal Gravitational Constant “G” with Planck's Constant “h” [6.6260693 x10-34(Js)] and the velocity of light in a vacuum “c”. Simply combining variables does not take into account the contribution of Experimental Relationship Functions (ERF's) in accordance with accepted Dimensional Analysis Techniques (DAT's) or Buckingham Π Theory (BPT) as utilised in chapter 3.1. Since there is no direct method facilitating the determination of these ERF's, a value of unity has historically been assumed. 207

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This assumption implicitly places Planck characteristics in the domain of the nonphysical. However, we may utilise the properties of the preceding equation to determine a value of “ωh” that is physically meaningful. To proceed, we shall apply DAT's as follows, Let a “3x1” matrix represent Planck characteristics in the following form: 5

c G.h

K ω.

ωh λh

G.h

K λ.

c

mh

3

h .c K m. G

(3.256)

Where: “Kω”, “Kλ” and “Km” denote ERF's governing Planck Frequency, Length and Mass respectively. Determining the “mh” to “λh” ratio yields, mh λh

h .c K m. G G.h

K λ.

c

K m c2 . Kλ G

3

(3.257)

Hence, Kλ mh . G Km λh

c

2

(3.258)

Substituting classical definitions of “mh” and “λh” produces the result, G.h 3 Kλ c . c K m G h .c 2

1

G

(3.259)

Hence, Kλ Km

(3.260)

Recognising the classical “λh” to “ωh” relationship, λh

c ωh

(3.261)

Performing the appropriate substitutions by relating equation (3.256, 3.261) yields, K λ.

G.h c

3

c . G.h K ω c5

(3.262)

Simplifying, K ω .K λ c .

3 G.h . c 1 . 5 c Gh

(3.263)

Hence, 208

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Kω

1 Kλ

(3.264)

Therefore, Kω

2.3

1 Km

(3.265)

EXPERIMENTAL RELATIONSHIP FUNCTIONS

Utilising properties of equation (3.255) and the relationships of equation (3.264, 3.265), we may formulate an experimentally based solution for “Kω”, “Kλ” and “Km”. The solution is experimentally based because “ωΩ” and “ωPV” represent elements of the PV spectral frequency bandwidth. These were used to produce an experimentally verified result for the RMS charge radius of a free Proton with classical form factor, as determined by the SELEX Collaboration [9] and illustrated in chapter 3.9. Recognising that the classical representation of the Refractive Index “KPV” described by equation (3.55) is a weak field exponential approximation, we shall remove its contribution to “ωPV” in determining an experimentally based solution for “Kω”, “Kλ” and “Km”. Secondary justification for the removal of “KPV” from “ωPV” stems from the recognition of the mathematical properties of equation (3.255). “KPV” does not contribute numerically to the modification of the “ωΩ / ωPV” ratio. If “nΩ = 1” at Planck conditions in accordance with equation (3.255) [the PV spectral bandwidth is convergent]: “Kω”, “Kλ” and “Km” may be determined. This may be accomplished by considering the ratio of the fundamental PV spectral frequency of a Planck Particle “ωPV(1,λh,mh)” [with removal of “KPV”] to the classical representation of “ωh” as follows, 2.

K PV( r , M ) e ω PV n PV, r , M

G .M 2 r .c

(3.55)

n PV 3 2 .c .G.M . . K ( r, M ) PV r π .r

(3.67)

Substituting Planck characteristic notation at the fundamental harmonic “nPV = 1” produces, ω PV 1 , λ h , m h

3 . . . 1 . 2 c G mh. K PV λ h , m h λh π .λ h

(3.266)

Hence, “Kω” may be represented by, Kω

ω PV 1, λ h , m h K PV λ h , m h

. 1 ωh

(3.267)

Simplifying yields, 3 . . . 1 . 2 c Gmh Kω λ h .ω h π .λ h

(3.268)

Substituting classical Planck definitions produces, 3 3 1 2 .c .G . h .c . c Kω . c G G.h π

(3.269)

Therefore, 209

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3

Kω

2.4

2 π

(3.270)

EXPERIMENTALLY IMPLICIT VALUES OF PLANCK CHARACTERISTICS

Utilising equation (3.255, 3.270) and classical Planck definitions, it may be demonstrated that “ωPV → ωΩ → the Planck Frequency”, independent of “KPV” in the PV model of gravitation by the following relationship, ω PV 1 , λ h , m h K PV λ h , m h

.

ω Ω λ h,m h

1 K ω .ω h

K PV λ h , m h

.

1 K ω .ω h

1

(3.271)

Therefore, an experimentally based determination of Planck Frequency, Length and Mass may be implicitly defined as follows, K ω. ωh

1 . G.h K ω c3

λh mh

2.5

5

c G.h

1 . h .c Kω G

(3.272)

IMPACT OF EXPERIMENTALLY IMPLICIT VALUES

The impact to the classical definition of the Planck Scale is to raise its value by approximately “16(%)” as illustrated by the following equation, 1

1 16.2447.( % )

Kω

(3.273)

Consequently, we may express the RMS charge radius of a free Photon “rγγ” [which was derived in chapter 3.10 from the physical properties of an Electron], in terms of free Muon and Tau particle RMS charge radii (“rµ” and “rτ” respectively) according to, r γγ K ω .

G.h . r µ c

3

rτ

(3.274)

such that, the error in relation to equation (3.225) in chapter 3.10 is less than “0.12(%)” and may be expressed as follows, 1

r γγ Kω

.

3 c .r τ 0.1192.( % ) G.h r µ

(3.275)

Therefore, it is clear from equation (3.274) that the determination of “Kω” leads to a useful approximation relating physical properties of the Lepton family, specifically all Electron-Like particles, to Photons. Note: the value of “Kλ” reaffirms the conclusion [to within 0.83(%)] stated in chapter 3.10, that the diameter of a Photon coincides with the Planck Length as defined by equation (3.272). This implies that the diameter of a Photon is the natural limit of the Quantum scale.

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3 3.1

THEORETICAL MODELLING BACKGROUND

In chapter 3.11, equation (3.230) was derived by the application of BPT which relates the mass and radius of two particles by similarity where, “r1,2” and “M1,2” denote arbitrary radii and mass values respectively. The result produced by equation (3.230) [chapter 3.11, 3.12] agrees with physical experiment and contemporary expectation. Hence, we may conclude that the EGM construct is well formulated and equation (3.230) is fit for further theoretical particle predictions. 2

3.2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

r2

9

St ω

r1

(3.230)

LEPTONS

As stated in the proceeding section, we shall utilise equation (3.230) to predict the existence of additional Lepton family particles that are not currently known or predicted by the Standard Model in particle physics. To proceed, we shall assume that any as yet undiscovered particles exist as harmonic multiples of the Electron in terms of “ωΩ” where, “Stω” represents the harmonic cut-off frequency ratio between two proportionally similar mass-energy systems. We shall also assume that the RMS charge radii of a free Electron as implied by scattering experiments “rε” (see chapter 3.11), “rµ” and “rτ” produces a usefully approximate Electron-Like Lepton average RMS charge radii “rL” as follows, rL 5

rL

rπ

m . 1. e 3 29 m p

rµ

rε

rτ

3

(3.276)

5

2

. 1

1 . mµ 9 4 me

2

5

1 . mτ 9 6 me

2

(3.277)

rL ≈ 10.7518 x10-16(cm) ≈ 0.0108(fm)

(3.278)

Hence, a generalised mass-energy relationship may be stated as, m L St ω

9 m e . St ω .

rL

5

rε

(3.279) nd

rd

th

Therefore, the mass-energies of three (3) new theoretical particles at the 2 , 3 and 5 Electron harmonics are, mL(2) ≈ 9(MeV) (3.280) mL(3) ≈ 57(MeV)

(3.281)

mL(5) ≈ 566(MeV)

(3.282)

Note: the 1st, 4th and 6th Electron harmonics denote the mass-energies of the Electron “me”, Muon “mµ” and Tau “mτ” particles respectively (see chapter 3.11).

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3.3

QUARKS / BOSONS

Similarly, new particles may be predicted in the Quark / Boson families utilising the same method. Equation (3.253) describes Quarks and Bosons as harmonic multiples of the “Up” Quark. The integer pattern is obvious and suggests the existence of two (2) new theoretical particles at the 5th and 6th harmonics. Utilising equation (3.230) and the average Quark / Boson radii defined in table (3.44), the mass-energies of the two (2) new theoretical particles [at “Up” Quark harmonic multiples] may be predicted as follows, m QB St ω

9 m uq . St ω .

r QB

5

r uq

(3.283)

where, table (3.44) indicates: rQB = 〈r〉 ≈ 1.0052 x10-16(cm)

(3.284) th

th

Therefore, the mass-energies of two (2) new theoretical particles at the 5 and 6 “Up” Quark harmonics are, (3.285) mQB(5) ≈ 10(GeV) mQB(6) ≈ 22(GeV) st

nd

rd

th

th

th

th

(3.286)

th

Note: the 1 , 2 , 3 , 4 , 7 , 8 , 9 and 10 harmonics denote known or currently theoretical particles (Higgs Boson) as articulated in chapter 3.12. 4

MATHEMATICAL MODELLING

4.1

BACKGROUND

We shall now methodically design an experiment based upon a Classical Casimir Configuration (parallel plates) with the ambition of achieving PV resonance. The existence of a resonant condition may reveal the possibility of negative vacuum energy exploitation as conjectured by “Puthoff et. Al.”. [23, 24, 30] The design approach involves the determination of an optimal physical configuration by dynamic, kinematic and geometric similarity with the experimentally verified force predictions of equation (3.179) and graphical analysis of figure (3.30), F PV A PP , r , ∆r , M

A PP .∆U PV( r , ∆r , M ) .

N C( r , ∆r , M ) N X( r , ∆r , M )

2

.ln

N X( r , ∆r , M )

4

N C( r , ∆r , M )

(3.179)

where, Variable r ∆r M FPV App ∆UPV NC NX RE ME

Description Magnitude of position vector from the centre of mass Change in magnitude of position vector Mass Casimir force predicted by the PV model Projected area of a parallel plate Change in energy density of PV Critical mode Harmonic inflection mode Radius of the Earth Mass of the Earth Table 3.45, 212

Units m kg N m2 Pa None m kg

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π N C R E , ∆r , M E

N X R E , ∆r , M E

π

φ 4 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E

2

φ 5 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E

n PV

Figure 3.30, Figure (3.30) illustrates the behaviour of the Phase Variance “φ4,5” between the Electric and Magnetic field of the PV, with respect to harmonic mode “nPV”, derived from Reduced Harmonic Similarity Equations (see chapter 3.6, 3.7) for a classical Casimir configuration. Analysis of figure (3.30) indicates that the rate of change of “φ4,5” is approximately constant until “nPV” approaches “NX”. Rapid changes in system states are typically associated with conditions of spectral sympathy. Therefore, the rapid rate of change in “φ4,5” commencing at “NX” may represent a state of natural resonance at the associated frequency “ωX”. It is unclear as to how the PV might respond to forced EM oscillations at “ωX”, but it is obvious that “NX” denotes a point of mathematical interest in relation to the negative energy conjecture. Subsequently, the first step in the design of a resonant Casimir cavity is to determine the optimal plate separation for complete similarity between the Casimir Force predicted by ZPF Theory and the gravitational force associated with the PV model, in this case “∆r” in accordance with figure (3.31),

D

B C A Figure 3.31, where, Variable A B C D

Description EM beam generator EM beam Mating face material for resonant cavity Resonant cavity of height “∆r” Table 3.46,

Note: figure (3.31) is for illustrational purposes only. Materials of construction should be reflective inside the cavity and neutrally charged. 213

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The basic principles of experimental operation should be to: i. Trap EM energy by reflection at frequency “ωX” and Phase Variance “φ4,5” inside the cavity. ii. Permit the RMS intensities of the Electric and Magnetic fields inside the cavity to attain the values specified in section 5.2 (as a minimum). iii. Ensure that the Electric and Magnetic field vectors are orthogonal inside the cavity. iv. Ensure that a standing wave forms in three dimensions (3D) within the cavity. v. Ensure any unexpected effects / events are observed. 4.2

BANDWIDTH RATIO

A bandwidth ratio “∆ωR” was defined in chapter 3.5 relating the Zero-Point-Field (ZPF) beat bandwidth “∆ωZPF” to a change in harmonic cut-off frequency “∆ωΩ”. This represents the ratio of the bandwidth of the ZPF spectrum to the Fourier spectrum of the PV. “∆ωR” provides a useful conversion relationship between forms over practical benchtop values of “∆r” and was defined as follows, ∆ω R( r , ∆r , M )

∆ω ZPF( r , ∆r , M ) ∆ω Ω ( r , ∆r , M )

(3.97)

It shall be demonstrated that equation (3.97) may be applied to determine the optimal value of “∆r” for practical benchtop experimental investigation of the negative energy conjecture. 4.3

OPTIMAL SEPARATION

We may utilise “∆ωR” to facilitate the calculation of the optimal value of “∆r” by determining the physical properties that satisfy the solution “|∆ωR| = 1”. Once achieved, the ZPF spectrum used to derive the Casimir Force (see chapter 3.4 - 3.7) is completely similar to the Fourier spectrum of the PV used to derive fundamental particle properties (see chapter 3.4, 3.8 3.12). This technique is easily applied utilising the “Given” and “Find” commands in the “MathCad 8 Professional” environment by the following algorithm, Given |∆ωR(RE,∆r,ME)| = 1 ∆r = Find(∆r) Therefore, the optimal practical benchtop value of “∆r” is, ∆r ≈ 16.5(mm) 5

PHYSICAL MODELLING

5.1

INFLECTION WAVELENGTH

(3.287)

Utilising equation (3.287), the harmonic inflection wavelength is applied to determine the type of energy delivery system to be used in experimentation and may be calculated as follows, λ X( r , ∆r , M )

c ω X( r , ∆r , M )

(3.288)

λX(RE,∆r,ME) ≈ 18(nm)

(3.289)

Evaluating yields,

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Therefore, based upon the wavelength specified by equation (3.289), an “X-Ray” laser system should be utilised in experimental investigations for a plate separation distance of “∆r ≈ 16.5(mm)”. 5.2

CRITICAL FIELD STRENGTHS

In addition to wavelength, the critical field strengths in terms of the applied Electric and Magnetic RMS values (“Erms” and “Brms” respectively) must also be achieved for complete similarity with the background gravitational field. This may be determined by calculating the value of Erms which satisfies the condition “|SSE3| = 1” (see chapter 3.6) for a plate separation distance of “∆r ≈ 16.5(mm)”. Similarly, this is easily computed utilising the “Given” and “Find” commands in the “MathCad 8 Professional” environment by the following algorithm, Given |SSE3(Erms, Erms/c,RE,∆r,ME)| = 1 Erms = Find(Erms) where, Brms = Erms/c: Therefore,

5.3

Erms ≈ 550(V/m)

(3.290)

Brms ≈ 18(milli-gauss)

(3.291)

CRITICAL PHASE VARIANCE

The final criteria required to achieve complete similarity with the background gravitational field is “φ4,5”. This may be determined by calculating the value of “φ4,5” which satisfies the condition “|SSE4,5| = 1” utilising equation (3.290, 3.291). As before, this is easily computed utilising the “Given” and “Find” commands in the “MathCad 8 Professional” environment by the following algorithm, Given |SSE4(Erms, Brms,φ4,RE,∆r,ME)| = 1 |SSE5(Erms, Brms,φ5,RE,∆r,ME)| = 1 [φ4 φ5] = Find(φ4,φ5) Yields, [φ4 φ5] = [0 ±π/2]

(3.292)

Performing a graphical analysis of “|SSE4,5(φ)| = 1” produces, π 2

π

SSE 4 E rms , B rms , φ , R E , ∆r , M E SSE 5 E rms , B rms , φ , R E , ∆r , M E 1

φ

Figure 3.32, Therefore, utilising equation (3.292) and figure (3.32), optimal phase variance between the applied Electric and Magnetic fields occurs at “0, ±π or ±π/2”. 215

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6

CONCLUSIONS

This chapter derives: i. An experimentally implicit increase of the Planck Scale. ii. An approximation of the RMS charge radius of a free Photon “rγγ”, utilising physical properties of the Lepton family, specifically all Electron-Like particles. iii. The existence of three (3) new particles in the Lepton family. iv. The existence of two (2) new particles in the Quark / Boson families. v. The optimal practical benchtop configuration of a Classical Casimir Experiment to test the negative energy conjecture. THE ANSWERS TO SOME IMPORTANT QUESTIONS • (a)

WHAT CAUSES HARMONIC PATTERNS TO FORM? ZPF Equilibrium

A free fundamental particle is regarded by EGM as a “bubble” of energy equivalent mass. Nature always seeks the lowest energy state: so surely, the lowest state for a free fundamental particle “should be” to diffuse itself to “non-existence” in the absence of “something” acting to keep it contained? This provokes the suggestion that a free fundamental particle is kept contained by the surrounding space-time manifold. In other words, free fundamental particles are analogous to “neutrally buoyant bubbles” floating in a locally static fluid (the space-time manifold). EGM is an approximation method, developed by the application of standard engineering tools, which finds the Zero-Point-Field (ZPF) equilibrium point between the mass-energy equivalence of the particle and the space-time manifold (the polarized ZPF) surrounding it - as depicted below,

Figure 3.43, (b)

Inherent Quantum Characteristics

If one assumes that the basic nature of the Universe is built upon quantum states of existence, it follows that ZPF equilibrium is a common and convenient feature amongst free fundamental particles by which to test this assumption. Relativity tells us that no absolute frames of reference exist, so a logical course of action is to define a datum as EGM is derived from a gravitational base. In our case, it is an arbitrary choice of fundamental particle. To be representative of the quantum realm, it follows that ZPF equilibrium between free fundamental particles should also be analogous to quantum and fractional quantum numbers – as one finds with the “Quantum Hall Effect”. Subsequently, the harmonic patterns of Particle Summary Matrix 3.3 form because the determination of ZPF equilibrium is applied to inherently quantum characteristic objects – i.e. fundamental particles. 216

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Hence, it should be no surprise to the reader that comparing a set of inherently quantum characterized objects to each other, each of which may be described by a single wavefunction at its harmonic cut-off frequency, results in a globally harmonic description. That is, the EGM harmonic representation of fundamental particles is a quantum statement of ZPF equilibrium – as one would expect. In-fact, it would be alarming if Particle Summary Matrix 3.3, or a suitable variation thereof, could not be formulated. Therefore, harmonic patterns form due to inherent quantum characteristics and ZPF equilibrium. •

WHY HAVEN’T THE “NEW” PARTICLES BEEN EXPERIMENTALLY DETECTED?

EGM approaches the question of particle existence, not just by mass as in the Standard Model (SM), but by harmonic cut-off frequency “ωΩ” (i.e. by mass and ZPF equilibrium). It was shown in Ch. 3.5 that the bulk of the PV spectral energy [i.e. “>> 99.99(%)”] at the surface of the Earth exists well above the “THz” range. Hence, generalizing this result to any mass implies that the harmonic cut-off period “TΩ” (i.e. the inverse of “ωΩ”) defines the minimum detection interval to confirm (or refute) the existence of the proposed “L2, L3, L5” Leptons and associated “ν2, ν3, ν5” Neutrino’s. In other words, a particle exists for at least the period specified by “TΩ”. Quantum Field Theory (QFT) approaches this question from a highly useful, but extremely limited perspective compared to the EGM construct. QFT utilizes particle mass to determine the minimum detection period (in terms of eV) to be designed into experiments. To date, this approach has been highly successful, but results in the conclusion that no new Leptons exist beyond the SM in the mass-energy range specified by the proposed Leptons. Whilst QFT is a highly useful yardstick, it is by no means a definitive benchmark to warrant termination of exploratory investigations for additional particles. Typically in the SM, short lived particles are seen as resonances in cross sections of data sets and many Hadrons in the data tables are revealed in this manner. Hence, the SM asserts that the more unstable particles are, the stronger the interaction and the greater the likelihood of detection. The EGM construct regards the existing Leptons of the SM as long-lived particles. It also asserts that the SM does not adequately address the existence or stability of the extremely shortlived Leptons proposed. This assertion is supported by the fact that detection of these particles is substantially beyond current capabilities due to: i. The minimum detection interval (with negligible experimental error) being “< 10-29(s)”. ii. The possibility that the proposed Leptons are transient (intermediate) states of particle production processes which decay before detection. For example, perhaps an Electron passes through an “L2” phase prior to stabilization to Electronic form (for an appropriate production process). Subsequently, this would be not be detected if the transition process is very rapid and the accelerator energies are too low. iii. The possibility of statistically low production events. Hence: i. The proposed Leptons are too short-lived to appear as resonances in cross-sections. ii. The SM assertion that the more unstable particles are, the stronger the interaction and the greater the likelihood of detection is invalid for the proposed Leptons. Therefore, contemporary particle experiments are incapable of detecting the proposed Leptons at the minimum accelerator energy levels required to refute the EGM construct. •

WHY SHOULD ONE BELIEVE THAT ALL FUNDAMENTAL PARTICLES MAY BE DESCRIBED AS HARMONIC MULTIPLES OF EACH OTHER?

Because of the precise experimental and mathematical evidence presented in Particle Summary Matrix (3.1, 3.2, 3.4). These results were achieved by construction of a model based upon a single gravitational paradigm. Moreover, the Casmir force was also derived in Ch. 3.7. 217

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•

WHY IS EGM A METHOD AND NOT A THEORY?

EGM is a method and not a theory because: (i) it is an engineering approximation and (ii), the mass and size of most subatomic particles are not precisely known. It harmonizes all fundamental particles relative to an arbitrarily chosen reference particle by parameterising ZPF equilibrium in terms of harmonic cut-off frequency “ωΩ”. The formulation of Particle Summary Matrix 3.3 is a robust approximation based upon PDG data. Other interpretations are possible, depending on the values utilized. For example, if one reapplies the method presented in Ch. 3.12 based upon other data, the values of “Stω” in Particle Summary Matrix 3.3 might differ. However, in the absence of exact experimentally measured mass and size information, there is little motivation to postulate alternative harmonic sequences, particularly since the current formulation fits the available experimental evidence extremely well. If all mass and size values were exactly known by experimental measurement, the main sequence formulated in Ch. 3.12 (or a suitable variation thereof) will produce a precise harmonic representation of fundamental particles, invariant to interpretation. Particle Summary Matrix 3.3 values cannot be dismissed due to potential multiplicity before reconciling how: i. “ωΩ”, which is the basis of the Particle Summary Matrix 3.3 construct, produces the experimentally verified formulation of Eq. (3.212, 3.215) as derived in Ch. 3.9. These generate radii values substantially more accurate than any other contemporary method. In-fact, it is a noteworthy result that EGM is capable of producing the Neutron Mean Square (MS) charge radius as a positive quantity. Conventional techniques favour the non-intuitive form of a negative squared quantity. ii. “ωΩ” is capable of producing “a Top Quark” mass value – the SM cannot. iii. EGM produces the results defined in Particle Summary Matrix 3.1. iv. Extremely short-lived Leptons (i.e. with lifetimes of “TΩ”) cannot exist, or do not exist for a plausible harmonic interpretation. v. Any other harmonic interpretation, in the absence of exact mass and size values determined experimentally, denote a superior formulation. Therefore, EGM is a method facilitating the harmonic representation of fundamental particles. •

WHAT WOULD ONE NEED TO DO, IN ORDER TO DISPROVE THE EGM METHOD?

Explain how measurements of charge radii and mass-energy by collaborations such as CDF, D0, L3, SELEX and ZEUS in [9, 14, 15, 80, 82-85]; do not correlate to EGM calculations. •

WHY DOES THE EGM METHOD PRODUCE CURRENT QUARK MASSES AND NOT CONSTITUENT MASSES?

The EGM method is capable of producing current and constituent Quark masses, only current Quark masses are presented herein. This text is limited to current Quark masses because it is the simplest example of ZPF equilibrium applicable whereby a particle is treated as “a system” and the equilibrium radius is calculated. Determination of the constituent Quark mass is more complicated, but it is possible. For example, “Appendix 3.I” calculates an experimentally implicit value of the Bohr radius by treating the atom as “a system” in equilibrium with the polarized ZPF. •

WHY DOES THE EGM METHOD YIELD ONLY THE THREE OBSERVED FAMILIES?

This occurs because it treats all objects with mass as a system (e.g. the Bohr atom) in equilibrium with the Polarized ZPF (its own gravitational field). Therefore, since fundamental particles with classical form factor denote states (or systems: Quarks in the Proton and Neutron) of polarized ZPF equilibrium, it follows that only the three families will be predicted. 218

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APPENDIX 3.A KEY ARTEFACTS Chapter •

3.1

Refractive Index and Experimental Relationship Function 2

K PV K 0( X )

(3.25)

Summation of sinusoids produces a constant function

Re( a( t ) )

Acceleration

•

3

Im( a( t ) ) f( t )

t Time

Real Terms (Non-Zero Sum) Imaginary Terms (Zero Sum) Constant Function (eg. "g")

Figure 3.2, Chapter •

3.2

Critical Factor “KC” K C K 1, K 2

•

2

K 1 ω 0, r 0, E 0, D , X K 2 ω 0, r 0, B 0, D, X

N

N 2 E 0( k , n , t ) .

n= N

B 0( k , n , t ) n= N

2

(3.44)

General Modelling Equation1 (GME1) N 2

a r0

±

β1

β2 2

K 2 ω 0, r 0, B 0, D, X . ±

N

2 .r 0 . K PV

n= N

3

N E 0( k , n , t )

2

2 c0 .

B 0( k , n , t ) n= N

2

E 0( k , n , t ) K 0 ω 0, X n = N . ± N 3 2 .r 0 . K PV 2 B 0( k , n , t ) n= N

(Eq. 3.45)

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2

c0

General Modelling Equation2 (GME2) N

a r0

±

β1

β2

±

K 2 ω 0, r 0, B 0, D, X

2

N

N

.

2 .r 0 . K PV

3

E 0( k , n , t )

2 c0 .

2

n= N

B 0( k , n , t ) n= N

2

±

K 0 ω 0, X

E 0( k , n , t )

2

. n= N N 3 2 .r 0 . K PV 2 B 0( k , n , t ) n= N

(Eq. 3.46) Chapter •

3.3

Critical Ratio “KR” KR

•

ε ∆U g ∆a PV ∆K C( ∆r ) . 0 g Ug ∆U PV( ∆r ) µ 0

Engineered Relationship Function “∆K0(ω,X)” ∆K 0( ω , X )

•

r .g . G.M . KR KR 2 2 c r .c

Chapter

2 . ∆K 0( ω , X )

(3.56)

3.4

Gravitational amplitude spectrum “CPV” G.M .

C PV n PV, r , M

2

r

•

(3.64)

n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

(3.67)

Harmonic cut-off mode “nΩ” n Ω ( r, M )

•

2 π .n PV

Gravitational frequency spectrum “ωPV” ω PV n PV, r , M

•

(3.54)

Engineered Refractive Index “KEGM” (normal matter form) K EGM K PV. e

•

(3.53)

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

Harmonic cut-off function “Ω” 3

Ω ( r, M )

•

108.

U m( r , M ) U ω( r , M )

12. 768 81.

U m( r , M ) U ω( r , M )

2

(3.72)

Harmonic cut-off frequency “ωΩ” ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

220

(3.73) www.deltagroupengineering.com

2

c0

Chapter •

3.5

Critical Boundary “ωβ” ω β r , ∆r , M , K R

•

4

ω Ω ( r , ∆r , M )

4 ZPF

K R . ω Ω ( r , ∆r , M )

4 ZPF

∆ω δr( 1 , r , ∆r , M )

4

(3.93)

EGM Wave Propagation

Figure 3.14, •

EGM Spectrum

Figure 3.15, Chapter

3.6

•

Critical Phase Variance “φC = 0°, 90°”

•

Critical Field Strengths (“EC and BC”)

“EC” and “BC” are derived utilising the reciprocal harmonic distribution describing the EGM amplitude spectrum. Solutions to “|SSE4,5| = 1” represent conditions of complete dynamic, kinematic and geometric similarity with the amplitude of the background EGM spectrum.

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•

Spectral Similarity Equations4,5 (SSE4,5) π SSE 5 E rms , B rms , , r , ∆r , M 2

SSE 4 E rms , B rms , 0 , r , ∆r , M

•

1

(3.156)

DC-Offsets SSE 4 ( 1

SSE 4 ( 1

DC) .E rms , B rms , 0 , r , ∆r , M

DC) . E rms, ( 1

SSE 5 E rms , ( 1

DC) .B rms, 0, r , ∆ r , M

π DC) .B rms , , r , ∆r , M 2

SSE 5 ( 1 DC) . E rms, ( 1

1 2

DC) .B rms,

π 2

(3.159) , r, ∆ r, M

1 4

(Eq. 3.160) •

Critical Frequency “ωC” c . 2 ∆r

ω C( ∆r )

Chapter •

3.7

Harmonic Inflection Mode “NX” N X( r , ∆r , M )

•

n Ω ( r , ∆r , M )

1 ZPF

ln 2 .n Ω ( r , ∆r , M ) ZPF

γ

(3.164)

Critical Mode “NC” N C( r , ∆r , M )

•

(3.162)

ω C( ∆r ) ω PV( 1 , r , M )

(3.169)

Harmonic Inflection Frequency “ωX” ω X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

•

EGM Casimir Force “FPV” F PV A PP , r , ∆r , M

Chapter •

(3.170)

A PP .∆U PV( r , ∆r , M ) .

N C( r , ∆r , M ) N X( r , ∆r , M )

.ln

2

N X( r , ∆r , M ) N C( r , ∆r , M )

4

(3.179)

3.8

Photon mass-energy threshold “mγ” mγ<

512.h .G.m e c . π .r e

2

.

n Ω r e, m e ln 2 .n Ω r e , m e

222

γ

(3.193)

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Chapter •

3.9

The Fine Structure Constant “α” 2

rε

α

.e

3

rπ

•

Harmonic cut-off frequency “ωΩ” ω Ω r ε, m e

•

ω Ω r π, m p

2 .ω Ω r π , m p

ω Ω r ν ,mn

ω CP

2

ω CN

2

. c .e r e ω Ce

rπ

Chapter

ω Ce

ω Ce

rε

2 4 27.ω h ω Ce . . rπ 3 4 4 .ω CP 32.π ω CP

c .ω Ce

rν

π

5

.

3

4 .ω CN

(3.212)

27.ω h ω Ce . 4 ω . 32 π CN 2

4

(3.215)

3.10

The mass-energy of a Graviton “mgg” (3.216)

The mass-energy of a Photon “mγγ” 3

h .

m γγ

re

•

3

π .r e 2 .c .G.m e

.

512.G.m e

2

c .π

ln 2 .n Ω r e , m e

2

2

(3.220)

2

m γγ m e .c

2

(3.225)

The radius of a Graviton “rgg” r gg

Chapter

5

4 .r γγ

(3.227)

3.11

Harmonic cut-off frequency ratio (the ratio of two particle spectra) “Stω” 2

•

γ

The radius of a Photon “rγγ” r γγ r e .

•

n Ω r e, m e

.

5

•

(3.210)

5

c .ω Ce

3

mgg = 2mγγ •

2

Proton and Neutron radii (rπ, rν) rε

•

(3.204)

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

r2 r1

9

St ω

(3.230)

Electron, Muon and Tau radii (rε, rµ, rτ) 223

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5

2

1 . me r ε r π. 9 2 mp 5

•

5

2

1 . mµ 9 4 me

rε .

rµ rτ

(3.231) 1 . mτ 9 6 me

2

(3.234)

The Fine Structure Constant “α” rµ

rε

α

.e

rτ

rν

•

(3.236)

Electron, Muon and Tau Neutrino radii (ren, rµn, rτn) 5

r en r µn r τn

Chapter •

r µ.

me

m µn mµ

2

5

r τ.

m τn

2

mτ

(3.238)

Quark and Boson harmonic representations .

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

1 2 3 4

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

7 8 9 10

ω Ω r uq , m uq

(3.253)

Quarks and Bosons as harmonic multiples of the Electron 1 ω Ω r ε,m e

Chapter •

r ε.

5

2

3.12

1

•

m en

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

7 14 21 28

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

49 56 63 70

(3.254)

3.13

Planck Scale Experimental Relationship Functions 3

Kω

Kω

Kω

224

2 π

(3.270)

1 Kλ

(3.264)

1 Km

(3.265)

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•

Approximation of the radius of a free Photon “rγγ”, relating physical properties of the Lepton family, specifically all Electron-Like particles G.h . r µ

r γγ K ω .

rτ

(3.274)

mL(2) ≈ 9(MeV)

(3.280)

mL(3) ≈ 57(MeV)

(3.281)

mL(5) ≈ 566(MeV)

(3.282)

mQB(5) ≈ 10(GeV)

(3.285)

mQB(6) ≈ 22(GeV)

(3.286)

c

• • • • • •

•

3

Theoretical particle (Lepton)

Theoretical particle (Lepton)

Theoretical particle (Lepton)

Theoretical particle (Quark / Boson)

Theoretical particle (Quark / Boson)

The optimal configuration of a Classical Casimir Experiment to test the negative energy conjecture exists at: ∆r ≈ 16.5(mm) (3.287) λX(RE,∆r,ME) ≈ 18(nm)

(3.289)

Erms ≈ 550(V/m)

(3.290)

Brms ≈ 18(milli-gauss)

(3.291)

The optimal phase variance between the applied Electric and Magnetic fields occurs at “0, ±π or ±π/2”

Appendix 3.G •

Neutron Charge Distribution “ρch” r

ρ ch ( r )

•

KS

2. 3

3.

5 2 π rν . x

. e

rν

2

1.

e

r x .r ν

2

3

1

x

(3.406)

Neutron Charge Density Gradient Radius Intercept “rdr” r dr

5. 3

rν

225

(3.391)

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•

“To” Neutron Mean Square Charge Radius Conversion Equation “KS” KS

•

3. π .r ν

2

. (1

x) . x

1

x x

8

3

2

“From” Neutron Mean Square Charge Radius Conversion Equation “b1 and rX” b1

2 . KS 3.r ν

2

2

x

1

(3.394)

6 .b 1 .K X . x

2

rX

•

(3.396)

3 .b 1 . x

2

1

rν KS

1

. K .K S X

(3.418)

Neutron Magnetic Radius “rνM” r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

•

(3.420)

Proton Electric Radius “rπE” r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

•

(3.423)

Proton Magnetic Radius “rπM” ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

•

(3.426)

Classical Proton Root Mean Square Charge Radius “rp” r P r πE

1. 2

r νM

rν

(3.429)

Appendix 3.I •

The first term of the Hydrogen Spectrum (Balmer series) “λA” [by EGM] utilising the Bohr radius “rBohr” and the fundamental PV wavelength “λPV” λA

λ PV 1 , K ω .r Bohr , m p 2 .n Ω K ω .r Bohr , m p

226

(3.457)

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APPENDIX 3.B FORMULATIONS, DERIVATIONS, CHARACTERISTICS AND PROOFS CHAPTER

3.2 substitute , E( k , n , t ) E 0( k , n , t ) E( k , n , t )

a( t )

K 0( ω , X )

2

substitute ,

. n

r

B( k , n , t )

2 K PV .

2

n B( k , n , t )

2

n

2

a r0

r0

substitute , r

n

B 0( k , n , t )

K 0 ω 0, X .n 3

r 0 .K PV

K PV

E 0( k , n , t )

2

B 0( k , n , t )

2

2

n

substitute , ω ω 0 substitute , a( t ) a r 0

a r0

K 0 ω 0, X n . 3

r 0 .K PV

E 0( k , n , t )

2

α1

substitute , K 0 ω 0 , X B 0( k , n , t )

2

substitute , a r 0

(3.35)

K 1 ω 0, r 0, E 0, D , X

2

substitute ,

n

E 0( k , n , t )

2

K 1 ω 0, r 0 , E 0, D , X

α1

1

3

r 0 .K PV

n

2

.

B 0( k , n , t )

2

n

(3.36)

substitute , α 1 α 2 α1

K 1 ω 0, r 0, E 0, D, X

E 0( k , n , t )

3

r 0 .K PV

2

.

B 0( k , n , t ) n

2

B 0( k , n , t )

substitute , n

2

n 2

c0

2

α2

K 1 ω 0, r 0 , E 0, D , X

.c 2 0

3

r 0 .K PV

2

.

E 0( k , n , t ) n

227

2

(3.38)

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a r0

K 0 ω 0, X

substitute , a r 0

2

.n

3

r 0 .K PV

β1

E 0( k , n , t )

substitute , K 0 ω 0 , X B 0( k , n , t )

2

substitute ,

E 0( k , n , t )

2

.

E 0( k , n , t )

3

1

r 0 .K PV

2

n

2

(3.37)

2

substitute ,

n

E 0( k , n , t )

2

2.

K 0 ω 0, X n . 3

E 0( k , n , t )

β2

K 2 ω 0, r 0, B 0 , D , X .c 2 . 0

B 0( k , n , t )

1

r 0 . K PV

3

2

2

n

(3.40)

2

2

K 0 ω 0, X .

3 2

2

n

K 1 ω 0, r 0, E 0, D, X r 0 .K PV

n

B 0( k , n , t )

2

B 0( k , n , t )

c0

n

r 0 .K PV substitute , α 1

B 0( k , n , t )

K 2 ω 0, r 0, B 0, D , X

substitute , β 1 β 2

substitute , a r 0

α1

β1

n

K 2 ω 0, r 0, B 0, D , X . 3

K 2 ω 0, r 0, B 0 , D , X

2

n

r 0 . K PV

a r0

β1

.

B 0( k , n , t )

2

E 0( k , n , t )

2

n

n solve , K 1 ω 0 , r 0 , E 0 , D , X

(3.39)

228

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substitute , a r 0

K 0 ω 0, X

β1

substitute , β 1

2

.n

3

r 0 .K PV a r0

E 0( k , n , t )

B 0( k , n , t )

2

2

n

K 0 ω 0, X

K 2 ω 0, r 0 , B 0, D , X .

E 0( k , n , t )

3

r 0 .K PV

2

B 0( k , n , t )

n

2

2

n

solve , K 2 ω 0 , r 0 , B 0 , D , X

(3.41) substitute , K 1 ω 0 , r 0 , E 0 , D , X

K C ω 0, r 0, E 0, B 0, D, X

2

E 0( k , n , t )

2

n

K 1 ω 0 , r 0, E 0, D , X K 2 ω 0 , r 0, B 0, D, X

K 0 ω 0, X .

substitute , K 2 ω 0 , r 0 , B 0 , D , X

K C ω 0, r 0, E 0, B 0, D , X

K 0 ω 0, X B 0( k , n , t )

2

2 E 0( k , n , t ) .

n

2

B 0( k , n , t )

2

n

n

(Eq. 3.44) substitute , β 1

K 2 ω 0, r 0, B 0, D , X . r 0 .K PV

a r0

β1

β2 2

E 0( k , n , t )

3

n

2

E 0( k , n , t )

K 2 ω 0, r 0, B 0, D , X .c 2 . substitute , β 2 0 3 r 0 .K PV

B 0( k , n , t ) n

2

substitute , K 2 ω 0 , r 0 , B 0 , D , X

2

2

a r0

2

1 . K 0 ω 0, X . n 3 2 2 B 0( k , n , t ) 2 r 0 .K PV n

1 . K 0 ω 0, X . 2 c0 3 2 2 r 0 .K PV

K 0 ω 0, X B 0( k , n , t )

2

n expand

(3.45) 229

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substitute , β 1

K 2 ω 0, r 0, B 0, D , X . r 0 .K PV

a r0

β1

β2

substitute , β 2

2

E 0( k , n , t )

3

n

2

E 0( k , n , t )

K 2 ω 0, r 0, B 0, D , X .c 2 . 0

B 0( k , n , t )

3

r 0 .K PV

2

1 . K 0 ω 0, X . n 3 2 2 B 0( k , n , t ) 2 . r 0 K PV n

2

a r0

n

2

substitute , K 2 ω 0 , r 0 , B 0 , D , X

2

1 . K 0 ω 0, X . 2 c0 3 2 2 r 0 .K PV

K 0 ω 0, X B 0( k , n , t )

2

n expand

CHAPTER

(3.46)

3.3 E PV k PV, n PV, t

∆a PV

∆K 0 ω 0 , X n PV . r B PV k PV, n PV, t n PV

2

substitute ,

E PV k PV, n PV, t

2

2 c .

n PV 2

B PV k PV, n PV, t

n PV

substitute , ∆a PV g .K R

2

g .K R.

r c

solve , ∆K 0 ω 0 , X

2

(3.54)

The change in amplitude spectrum for “∆K0(ω,X)” is proportional to the Fourier amplitude at each mode within the spectrum. Subsequently, the change in amplitude spectrum over “∆r” is trivial. CHAPTER

3.4

Integrating equation (3.47), ρ 0( ω )

2 .h .ω c

3

3

(3.47)

Yields, 230

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2 .h . c

1. h . 4 ω 2 c3

3

ω dω

3

(3.293)

where, “ω ≡ ωPV”: utilising equation (3.67), ω PV

n PV 3 2 .c .G.M . . K PV r π .r

(3.67)

Yields a generalised frequency change representation according to, Uω

h . ωPV 3 2.c

4 2

ωPV

4

(3.294)

Substituting equation (3.67) into (3.293) yields the generalised change in odd mode representation according to, U ω( r , M ) .

U ω n PV, r , M

n PV

2

4

4

n PV

(3.68)

where, U ω( r , M )

3 h .G.M . 2 .c .G.M . 2 K PV 2. 5 .r π . πc r

(3.295)

Note: equation (3.295) is a modified representation of equation (3.69). Subsequently, if: U m( r , M )

3 .M .c 4 .π .r

3

And assuming: U m( r , M )

2

(3.70) U m( r , M )

U ω( r , M ) .

U ω n PV, r , M

n PV

2

4

then, 4

n PV

(3.296)

231

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Next, let: D

U m( r , M ) U ω( r , M )

(3.297)

Hence, D

n PV

2

4

4

n PV

(3.298)

Solving for “nPV” yields, 2 1

1 . 108.D 12

2 12. 768 81.D

1

3

1

2

48

12. 108.D

108.D

2 12. 768 81.D

2

D

n PV 2

4

4

n PV

solve , n PV, factor

1. 24

108.D

2 12. 768 81.D

3

2

1

48 24. 108.D

12. 768 81.D

2

2

3

2

1

i . 3 . 108.D

12. 768 81.D

2

3

2

48.i . 3

1 1

108.D 1

1 . 108.D

2

1

3

2

2

1 1

1

12. 768 81.D

3

12. 768 81.D

2

12. 768 81.D

2

2

1

3

3

1

2

48

24. 108.D

12. 768 81.D

2

3

2

2

2 1

i . 3 . 108.D

2 12. 768 81.D

2

3

48.i . 3

1

24 1

108.D

12. 768 81.D

2

3

2

(Eq. 3.299)

232

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Analysing the structure of the preceding equation leads to simplification by assigning temporary definitions of “F” and “L” for use with equation (3.299). This approach is required to fully exploit the “MathCad 8 Professional” symbolic calculation environment and may be articulated as follows, Let: “F = 108D+12√(768+81D2)” and “F = L3”. Hence, an expression for “nPV” as a function of “L” may be defined by, 1. L 1 12

solve , n PV, factor D

n PV 2

4

substitute , 108.D

4

n PV

2 12. 768 81.D F 2

1

3

substitute , F L, 3

1

,F

3

2

L

F L

4 L

1. . i 3 24

1 . L 1 24

2 .i . 3

1. . i 3 24

1 . L 24

2 .i . 3

1

2

L 2

L

(3.300)

collect , L

Equation (3.300) is a simplifying intermediary step leading to the harmonic cut-off function “Ω(r,Μ)” subject to the redefinition of “L” as follows, Let: “L = Ω(r,M)” and “nΩ(r,M) = nPV + 2” hence, n Ω ( r , M ) n PV 2

Ω ( r, M )

4

12

Ω ( r, M )

1

2

(3.301)

Therefore, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

Performing the appropriate substitutions of “D” into “L3 = 108D+12√(768+81D2)” for application to equation (3.71) yields, 3

Ω ( r, M )

108.

U m( r , M ) U ω( r , M )

12. 768 81.

U m( r , M )

2

U ω( r , M )

(3.72)

Hence, ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73) 233

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CHAPTER

3.5 / 3.6

The HSE1,2 operand may be formed utilising the ratio of “KR(r,∆r,M)” to “GSE1,2”, 2 .i . e . π n PV

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2.

2 .c B A k A , n A , t EA k A,n A,t

2

exp i .π .n PV.∆ω δr( 1 , r , ∆r , M ) .t . E k ,n ,t A A A 2 2 π .n PV c .B A k A , n A , t i

simplify, factor 2

2 c .B A k A , n A , t

.

2

2 c .B A k A , n A , t

2

2

(3.302)

The HSE3 operand may be formed utilising the ratio of “KR(r,∆r,M)” to “GSE3”, 2 .i . e π .n PV

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

simplify

E A k A , n A , t .B A k A , n A , t K PV( r , M ) .St α ( r , ∆r , M )

2.

exp i .π .n PV.∆ω δr( 1 , r , ∆r , M ) .t .K ( r , M ) .St ( r , ∆r , M ) PV α π .n PV E A k A , n A , t .B A k A , n A , t i

.

(3.303)

The HSE4,5 operand may be formed utilising the ratio of “KR(r,∆r,M)” to “GSE4,5”, 2 .i . e π .n PV EA k A,n A,t

2

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2 c .B A k A , n A , t

2

.E k , n , t .B k , n , t A A A A A A

2 2 .c .K PV( r , M ) .St α ( r , ∆r , M ) .B A k A , n A , t

simplify

2 4 .i .St α ( r , ∆r , M ) .K PV( r , M ) .c .B A k A , n A , t .

exp i .π .n PV.∆ω δr( 1 , r , ∆r , M ) .t π .n PV.E A k A , n A , t

3

2 π .n PV.E A k A , n A , t .c .B A k A , n A , t

2

(Eq. 3.304)

234

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2

CHAPTER

3.6

HSE1 R may be formed as follows, substitute , E A

2 i . EA

2 2 c .B A

2 2 π .n PV.c .B A

substitute , B A

E 0 .e B 0 .e

substitute , ω E n E

2 .π .ω E n E .t

π . i 2

2 .π .ω B n B .t

π 2

φ .i

1

i

ωB nB

substitute , B 0

2 .B rms

substitute , E 0

2 .E rms

π .n PV

.( exp( 2 .i .φ )

1) →

i π .n PV

.( exp( 2 .i .φ )

1 ) simplify

1. π

2.

( cos ( 2 .φ )

1)

2

2

n PV

substitute , E rms c .B rms simplify

(Eq. 3.305) HSE2 R may be formed as follows, substitute , E A

2 i . EA

2 2 c .B A

2 2 π .n PV.c .B A

substitute , B A

E 0 .e B 0 .e

substitute , ω E n E

2 .π .ω E n E .t

π . i 2

2 .π .ω B n B .t

π

ωB nB

substitute , B 0

2 .B rms

substitute , E 0

2 .E rms

2

φ .i

1

i π .n PV

.( exp( 2 .i .φ )

1) →

( exp( 2 .i .φ ) i . π .n PV

1)

simplify, factor

1. π

2.

( cos ( 2 .φ )

1)

2

2

n PV

substitute , E rms c .B rms simplify

(Eq. 3.306) 235

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HSE3 R may be formed as follows, E 0 .e

substitute , E A

2 .i .K PV( r , M ) .St α ( r , ∆r , M ) π .n PV.E A .B A

B 0 .e

substitute , B A

2 .π .ω E n E .t

2 .π .ω B n B .t

substitute , ω E n E

ωB nB

substitute , ω B n B

ω EM n EM

substitute , B 0

2 .B rms

substitute , E 0

2 .E rms

π . i 2 π 2

φ .i

i .K PV( r , M ) .

St α ( r , ∆r , M ) π .n PV.E rms .B rms

.exp i . 4 .π .ω . EM n EM t

π

φ

simplify

i .K PV( r , M ) .

St α ( r , ∆r , M ) π .n PV.E rms.B rms

(3.307)

.exp i . 4.π .ω . EM n EM t

π

φ

1.

simplify

π

K PV( r , M ) .

St α ( r , ∆r , M ) n PV.E rms.B rms

(3.308)

HSE4 R may be formed as follows (Eq. 3.309),

2 4 .i .St α ( r , ∆r , M ) .K PV( r , M ) .c .B A 2 π .n PV.E A . E A

2.

c BA

2

substitute , E A

E 0 .e

substitute , B A

B 0 .e

2 .π .ω E n E .t

π . i 2

2 .π .ω B n B .t

π

substitute , ω E n E

ωB nB

substitute , ω B n B

ω EM n EM

substitute , B 0

2 .B rms

substitute , E 0

2 .E rms

substitute , c

2

φ .i

2 .i .St α ( r , ∆r , M ) .

K PV( r , M ) E rms .B rms .π .n PV. exp i .π . 4 .ω EM n EM .t 1

exp i . 4 .π .ω EM n EM .t

π

2 .φ

.exp( i .φ )

E rms B rms

simplify, factor

236

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1

2 .i .St α ( r , ∆r , M ) .

K PV( r , M ) E rms .B rms .π .n PV. exp i .π . 4 .ω EM n EM .t

1

exp i . 4 .π .ω EM n EM .t

π

2 .φ

1.

.exp( i .φ ) simplify, expand , simplify

π

K PV( r , M )

2.

St α ( r , ∆r , M )

2

2

2 2 2 2 B rms .E rms .n PV .cos ( φ )

(Eq. 3.310) HSE5 R may be formed as follows,

2 4 .i .St α ( r , ∆r , M ) .K PV( r , M ) .c .B A 2 π .n PV.E A . E A

2 2 c .B A

substitute , E A

E 0 .e

substitute , B A

B 0 .e

2 .π .ω E n E .t

2 .π .ω B n B .t

substitute , ω E n E

ωB nB

substitute , ω B n B

ω EM n EM

substitute , B 0

2 .B rms

substitute , E 0

2 .E rms

substitute , c

π . i 2 π

φ .i

2

2.i .exp( i .φ ) .K PV( r , M ) .

St α ( r , ∆r , M ) E rms .B rms .π .n PV. exp i .π . 4 .ω EM n EM .t

1

exp i . 4 .π .ω EM n EM .t

π

2 .φ

E rms B rms

simplify, factor

(Eq. 3.311) 1

2 .i .St α ( r , ∆r , M ) .

K PV( r , M ) E rms.B rms.π .n PV. exp i .π . 4 .ω EM n EM .t

1

exp i . 4 .π .ω EM n EM .t

π

2 .φ

.exp( i .φ ) simplify, expand , simplify

1. π

St α ( r , ∆r , M )

2.

K PV( r , M )

2

2

2 2 2 2 B rms .E rms .n PV .sin ( φ )

(Eq. 3.312)

237

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Thus, from equations (3.310, 3.312), HSE4,5 R may be formed utilising HSE3 R as follows, HSE 4 E rms , B rms , φ , n PV, r , ∆r , M HSE 5 E rms , B rms , φ , n PV, r , ∆r , M

1 R

cos ( φ ) 1

R

sin ( φ )

.HSE E 3 rms , B rms , n PV, r , ∆r , M

.HSE E 3 rms , B rms , n PV, r , ∆r , M

R

(3.313)

R

(3.314)

Note: equations (3.315 – 3.320) were deleted from this section due to redundancy. In addition to graphical methods illustrated in chapter 3.6, “φC” may be determined as follows, 1

d

d φ C cos φ C

1

d

d φ C sin φ C

.HSE E 3 rms , B rms , n PV, r , ∆r , M

.HSE E 3 rms , B rms , n PV, r , ∆r , M

R

R

0 solve , φ C

0 solve , φ C

0 OR

“φC = 2π”

(3.321)

1. π 2 1. π 2

(3.322)

A useful approximation of the average amplitude per harmonic mode utilised in SSEx may be numerically proven as follows, Let: N = 106 + 1 Considering a double sided odd number distribution: “nPV = -N, 2 – N … N” The approximation error may be numerically evaluated utilising “MathCad 8 Professional” to be, 1 N

. 1

1 N

n PV

1 n PV

.( ln( 2 .N )

1 = 6.6287.10

6

(%)

γ)

1

(3.323)

Subsequently, vanishing error is implied as “N → [|nΩ|,|nΩ ZPF|]” and SSEx may be formed according to,

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1 N

1

ln( 2 .N )

n PV

N

. 1

n PV

γ

1

(3.324)

CHAPTER

3.7 – 3.9

Mathematical summation characteristics presented in chapter (3.7 – 3.9) may be numerically proven utilising “MathCad 8 Professional” as follows, Let: N = 106 + 1 where, equation (3.285) represents matrix “M” such that: i. The matrix element “M0,0” follows the integer one-sided distribution: “nPV = 1, 2 … N” ii. The matrix element “M2,0” follows the double sided odd number distribution: “nPV = -N, 2 – N … N” N 1

ln( 2 ) n PV = 1 ln( 2 .N )

n PV

γ

15.0859

1 n PV

15.0859 = 15.0859

n PV

(3.325)

Subsequently, vanishing error is implied as “N → [|nΩ|,|nΩ ZPF|]” by, 1 n PV

n PV . 1 = 3.314410

6

( %)

N 1

ln( 2 ) n PV = 1

n PV

(3.326)

Therefore, a highly precise useful approximation may be formulated as follows,

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N 1 n PV

1

ln( 2 )

n PV

n PV = 1

ln( 2 .N )

γ

n PV

(3.327)

Next, considering the error for a one-sided odd spectrum following the distribution “nPV = 1, 3 … N” yields, 1 n PV

n PV

1. ( ln( 2 .N ) 2

1 = 6.6287.10

6

(%)

γ)

(3.328)

Therefore, the relationship between odd and “odd + even” harmonic modes, to high computational precision, is usefully represented as “N → |nΩ(re,me)|” by, mg 1 > . ln 2 .n Ω r e , m e mγ 2

γ

(3.329) NOTES

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NOTES

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NOTES

242

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APPENDIX 3.C SIMPLIFICATIONS

me

h .ω Ce 2 .π .c

2

λ Ce .m e

λ Ce .m e

λ CP

λ CN

ω CP.m e

,mp

ω Ce

,mn

ω CN.m e ω Ce

h .ω CP

h .ω CN

2 2 .π .c

2 2 .π .c

G.h λh , ωh mh

c

3 5

c G.h h .c G

(3.330)

U m( r , M ) 3 .r2 .c4 3 π .r . U ω( r , M ) 4 .h .G 2 .c .G.M 3

Ω ( r , M ) 3 .c .

(3.331)

6 .r .c . π .r . . h G 2 c .G.M 2

3

(3.332)

3

2 3 Ω ( r , M ) c . 6 .r .c . π .r n Ω ( r, M ) . . 12 4 h G 2 c .G.M

ω Ω r ε, m e ω Ω r e, m e

n Ω r ε , m e .ω PV 1 , r ε , m e n Ω r e , m e .ω PV 1 , r e , m e

(3.333) 3 . . . n Ω r ε,m e 2 c G me . 3 rε π .r ε n Ω r ε, m e r e r e . . 3 . . . n Ω r e,m e 2 c G m e n Ω r e, m e r ε r ε . re π .r e

243

(3.334)

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3

. 2 . 3 π .r c. 6 rε c. ε 3 3 . . . . Ω r ε, m e r e r e 4 hG 2 c G me r e r e . . . . 3 rε rε Ω r e, m e r ε r ε . 2 . 3 π .r c. 6 re c. e 4 h .G 2 .c .G.m e

5 3

2 3

rε

3

3

rε re re . . re rε rε

.

re

re rε

.

rε

3

.

re

rε

re

re

rε

9

(3.335)

G .M

e

2 r .c

1

(3.336) 1 3

2.

π .r c . 6 .r c . 3 3 2 3 3 2 . . π .r . 2 .c .G.M c . 6 .r .c . π .r 4 h G 2 c .G.M . 2 .c .G.M c . 6 .r .c . ω Ω ( r, M ) r 4 .r h .G 2 .c .G.M 4 .r h .G 2 .c .G.M π .r π .r 3

1 2.

c . 6 .r c ω Ω ( r, M ) 4 .r h .G

1 3 ω Ω ( r , M ) 3 .h

1

3

. . . . 2 c GM π .r

2

13

3.

2

9

14

5

1

1

3

9

π .r . . 2 c .G.M

2

.π 9 .c 9 .r 9 .M 9 .G

3 h

3

3

2.

c . 12.r .c M 4 .r π .h

1

π .r . . 2 c .G.M

1

1

1 9

1

.

c

14

13 2 2 .π .G

9

1

9

3 ω Ω ( r, M ) c . . 2

ωh 4 .π .h

2

.M

14

5

3

1

. . . . 2 c GM π .r

3

2

1. 3 . 9 . 9 . 9 . 9 . 12 2 c r M h 4

(3.337) 2

1

1

3.

9.

G

3

π

9

(3.338)

1

9

. M

2

9

5

r

(3.339)

2 Utilising c5 ω h .G.h yields,

3

1

1

(3.340)

2

5

r

(3.341) 244

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APPENDIX 3.D DERIVATION OF LEPTON RADII Assuming the analytical representations derived in chapter (3.9, 3.11) denote exact boundary conditions, particle radii may be calculated utilising the following “MathCad 8 Professional” algorithm [satisfying all criteria between the “Given” and “Find” commands], Given 5

1 rπ

c .ω Ce

rν

4

5

.

2.

. 3

27.ω h ω Ce ω CP . 5 4 1 . 32.π 4

α

(3.199, 3.200) 1 ω CN

3

ω CN

rε

1 ω CP

2

.e

3

rπ

(3.204)

rε rπ

π rν

(3.214)

5

1 . me r ε r π. 9 2 mp

2

(3.231)

5

rε .

rµ rτ

1 . mµ 9 4 me

2

5

1 . mτ 9 6 me

2

(3.234)

rµ

α

rε

.e

rτ

rν

(3.236) 5

r en r µn r τn

r ε.

m en me

2

5

r µ.

m µn mµ

2

5

r τ.

m τn

2

mτ

(3.238)

rε rπ rν rµ rτ

Find r ε , r π , r ν , r µ , r τ , r en , r µn , r τn

r en r µn r τn

(3.342) 245

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rε

0.0118

rπ

0.8306

rν

0.8268

rµ

. 8.216210

rτ

=

3

( fm)

0.0122

r en r µn r τn

. 9.540410

5

. 6.555610

4

. 1.958710

3

(3.343)

The radii results may be tested against the calculation accuracy of “α” and “π” as follows, 1 .r ε . e α rπ

2 3

100

rµ

1 .r ε . e α rν 1. π rπ

rτ

= 100 ( % ) 100

rε rν

(3.344)

where, “α” and “π” accuracy is displayed to high precision. The change in Electron mass, as discussed in chapter 3.9 may be re-computed subject to the preceding equation set as follows, Given ω Ω r ε, m ε

ω Ω r ε,m ε

1.

ω Ω r e, m e

ω Ω r π, m p

2

ln 2.n Ω r e , m e

γ

2

(3.208)

me = Find(mε)

(3.345)

where, the Electron scattering mass-energy becomes, mεc2 = 0.511533744627484(MeV)

(3.346)

The Electron mass-energy increase becomes, (mε/me) – 1 = 0.105(%)

(3.347)

Considering the mass-energy increase defined in chapter 3.9 and equation (3.306) yields, ∆me < 0.11(%)

(3.348)

where, “+0.04(%) ≈ +0.11(%)” due to physical measurement limitations.

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APPENDIX 3.E DERIVATION OF QUARK AND BOSON MASS-ENERGIES AND RADII Assuming the analytical representations derived in chapter (3.4, 3.9, 3.11, 3.12) denote exact boundary conditions, particle properties may be calculated utilising the following “MathCad 8 Professional” algorithm, n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

(3.67)

√KPV(r,M) ≈ 1

(3.349)

h . 4 ω PV( 1, r , M ) 3 2.c

U ω( r , M )

3 .M .c

U m( r , M )

(3.69)

2

4 .π .r

3

n Ω ( r, M )

(3.70)

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

3

U m( r , M )

108.

Ω ( r, M )

12. 768 81.

U ω( r , M )

U m( r , M )

2

U ω( r , M )

(3.72)

ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73) 5

1 rπ

c .ω Ce

rν

4

5

.

1

2 4 ω CP 27.ω h .ω Ce ω CP . 5 4 1 . 1 32.π 3 ω CN ω CN 3

5

1 . me r ε r π. 9 2 mp

r uq 3 .r xq. 2

5

m dq

(3.199, 3.200)

2

(3.231) 1

5

r dq r uq .

.

m dq

2

m uq

(3.350)

2

m uq

(3.351)

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St dq

ω Ω r dq , m dq

St sq

ω Ω r xq, m sq

St cq St bq

1 ω Ω r uq , m uq

. ω Ω r xq, m cq ω Ω r xq, m bq ω Ω r xq, m tq

St tq St dq

floor St dq

St sq

floor St sq

St cq

floor St cq

St bq

floor St bq

St tq

floor St tq

(3.353) 5

m sq

2

St sq

9

m cq

2

5

r sq r cq r bq

5

r uq .

1 m uq

St cq

. 2

5

m bq

r tq

9

2

St bq 5

m tq

9

2

St tq

9

St uq

ω Ω r uq , m uq

St dq

ω Ω r dq , m dq

St sq

1

St cq

ω Ω r ε ,me

.

ω Ω r cq , m cq ω Ω r bq , m bq

St tq

ω Ω r tq , m tq floor St uq

St dq

floor St dq

St sq

floor St sq

St cq

floor St cq

St bq

floor St bq

St tq

floor St tq

(3.354)

ω Ω r sq , m sq

St bq

St uq

(3.352)

(3.355)

(3.356)

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9 5 St uq .r uq

m uq

9 5 St dq .r dq

m dq m sq

me

m cq

rε

m bq

9 5 St sq .r sq

.

9 5 St cq .r cq

5

9 5 St bq .r bq

m tq

9 5 St tq .r tq

5

r tq r uq .

2

1 . m tq 9 10 m uq

r u( M )

h . 4 π .c .M

rW

r u mW

rZ

r u mZ

rH

r u mH

(3.247)

(3.358)

(3.359) ω Ω r u mW ,mW

St W St Z St H

1

. ω Ω r u mZ ,mZ

ω Ω r uq , m uq

ω Ω r u mH ,mH

St W

round St W , 0

St Z

round St Z , 0

St H

round St H , 0 5

rW rZ rH

(3.357)

(3.361) 1

St W 5

r uq .

1 m uq

9

.m 2 W

5

1 . 2 mZ 9 St Z

5

1 . 2 mH

. 2

(3.360)

9

St H

(3.362)

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m uq

3.506.10

m dq

7.0121.10

m sq

3

1.1833

m bq

4.1196

m tq

178.4979

r uq

=

r cq

0.8879

10

1.0913

16 .

cm

0.9294

r tq

(3.364)

rW

1.2839

rZ

= 1.0616

rH

0.9403

6

(3.363)

1.071

r bq

1.

2

1.0136

r sq

6

c

0.7682

r dq

1.

GeV

0.1139

=

m cq

3

r uq

r dq

m uq

10

16 .

cm

(3.365)

r sq

m dq

m sq

r cq

m cq

rZ

rH

ru mW

ru mZ

r u mH

3

16 .

cm

(3.366)

m tq = 30.6542

m bq

GeV c

rW

1.

r tq = 0.9602 10

r bq

rW

r H = 1.0953 10

rZ

1

.

ω Ω r uq , m uq

2

(3.367)

= ( 1.0465 0.9811 1.0903 )

(3.252) 16 .

cm

(3.368)

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

=

1 2 3 4 7 8 9 10

(Eq. 3.253) 1 ω Ω r ε ,me

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

=

7 14 21 28 49 56 63 70

(Eq. 3.254)

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APPENDIX 3.F HARMONIC REPRESENTATIONS Commencing with the classical representation of gravitational acceleration as a function of planetary radial distance, we shall illustrate the harmonic modes of gravity in the EGM model as follows, Gravitational Acceleration

Acceleration

R E

G .M E r

g

2

r Radial Distance

Figure 3.33, Assuming the fundamental harmonic period “TPV” to be, T PV n PV, r , M

1 ω PV n PV, r , M

(3.369)

The harmonic modes of acceleration “aPV” in the EGM representation of the PV model of gravity in the weak field approximation (√KPV(r,M) ≈ 1) at “r” may be stated as, a PV n PV, r , M , t

i .C PV n PV, r , M .e

π .n PV .ω PV ( 1 , r , M ) .t .i

(3.370)

where, the gravitational representation of the preceding equation may be written as, g PV n PV, r , M , t

a PV n PV, r , M , t n PV

(3.371)

Representing equation (3.370) graphically yields,

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T PV 1 , R E , M E

T PV 1 , R E , M E

6

2

Acceleration

C PV 1 , R E , M E

Re a PV 1 , R E , M E , t Re a PV 3 , R E , M E , t

C PV 3 , R E , M E

Re a PV 5 , R E , M E , t

t Time

1st +ve Harmonic 3rd +ve Harmonic 5th +ve Harmonic

Figure 3.34,

Acceleration

where, the Real Component of “aPV” is equal to the amplitude spectrum of “gPV” as follows,

Re a PV n PV , R E , M E ,

T PV n PV , R E , M E 2

C PV n PV , R E , M E

n PV Harmonic

Real Component Harmonic Amplitude Spectrum

Figure 3.35, Representing equation (3.371) graphically yields,

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T PV 1 , R E , M E 2

Acceleration

g

g PV n PV , R E , M E , t

t Time

Figure 3.36, A useful graphical representation for “KR H” presented in chapter 3.6 is termed the Critical Harmonic Operator with composition as follows, 1

T δr n PV, r , ∆r , M

K R( r , ∆r , M , t )

∆ω δr n PV, r , ∆r , M 2. π

H

i . n PV

1 . e n PV

(3.372)

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

(3.373) T δr( 1 , r , ∆r , M )

K R_av ( r , ∆r , M )

H

∆ω δr n PV, r , ∆r , M .

K R( r , ∆r , M , t ) d t H

0 .( s )

(3.374)

Note that the average value of “KR H” over the fundamental period “Τδr” [“KR av H”], is approximately “98.2(%)” when “N = 21”. This indicates rapid convergence with vanishing error as “|nPV| → nΩ”. Representing equation (3.373) graphically yields,

Unit Harmonic Operator

K R_av_H R E , ∆r , M E 1

K R_H R E , ∆r , M E , t 0.5

t Time

Figure 3.37,

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NOTES

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APPENDIX 3.G [76] Results,

1

CONVERSION OF NEUTRON POSITIVE CORE RADIUS

The Mean Square (MS) charge radius of a free Neutron “rν”, as derived in chapter 3.9, may be converted to the conventional MS charge radius “KX” representation of “-fm2”. This may be achieved by utilising the Neutron Charge Distribution “ρch” curve as follows, bi

1 .

ρ ch ( r )

π

2

r

2

r

1 . b1. e 3 3 a1 π

ai

.e

3

i= 1 a i

3

a1

2

r

b2

.e

2

a2

3

a2

(3.375)

where, “a1, a2, b1 and b2” are mathematical constants physically satisfying the preceding equation, “r” denotes the magnitude of the radial position vector and “fm” denotes “femtometre” [x10-15(m)]. Recognising that, ∞ ρ ch ( r ) d r d r d r 4 . π .

Q

2 r . ρ ch ( r ) d r b 1

b2 0

0 ∞

3 4 2 r .ρ ch ( r ) d r . a 1 2

4. π . 0

(3.376)

2 a 2 .b 1

(3.377) -3

where, “Q” denotes the charge density per unit Coulomb and takes the units “fm ”. Subsequently, equation (3.376) yields the relationship “b2 = -b1” such that, 2

r

b1

ρ ch ( r )

π

1 . e

.

a1

3

2

a2

1 . e 3

a1

3

r

a2

(3.378)

“rν” represents the Zero-Point-Field (ZPF) equilibrium radius and intersects the radial axis at “r = rν” in accordance with equation (3.375). Hence, an expression for “rν” may be defined in terms of “a1, a2 and b1” as follows, rν

b1 π

rν

.

3

2

1 . e

2

rν

a1

1 . e

3

a2 a1

0

3

a1

3 . ln

a2

2

a2

.

a 1.a 2 2

a2

(3.379) 2 2

a1

(3.380)

The maximum value of “ρch” occurs at “r = 0” and may be determined (assuming spherical Neutron geometry) according to, V( r )

4. . 3 πr 3

(3.381)

Hence, the Charge Density per unit Coulomb “Q(r)” is expressed by [Q(r) → C/m3 * 1/C = 1/m3]: Q( r )

1 V( r )

(3.382) 255

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Therefore, the Charge Density per unit Coulomb per unit Quark “Qch(r)” may be written as, Q( r )

Q ch ( r )

3

(3.383)

Evaluating yields, 1

Q ch r ν = 0.1408

3

fm

(3.384)

This result may be expressed analytically by relating equation (3.378) to (3.381 – 3.383) when “r = 0” as follows, b1

1 4.π . r ν

3

π

.

1

1 3

3

a1

3

a2

(3.385)

Hence, 3 . π . a1a2 4. π .b 1 a 3 a 3 2 1 3

rν

3

(3.386)

The radial position “rdr” (as a function of “rν”) for which the gradient of the Charge Density “dρch(r)/dr” is zero may be determined as follows, 2

r

d b1 . 1 . e dr 3 a 3 1 π

r

a1

1 . e

2

a2

3

a2

r

2 .b 1.r π

3.

a 1.a 2

. a 5 .e 1

a2

2

r 5 a 2 .e

a1

2

0

5

(3.387)

Simplifying yields, r dr 5 a 1 .e

2

a2

r dr

2

a1

5 a 2 .e

(3.388)

Therefore, r dr

rν

2

r dr

2

5 . ln

a1

a 1.a 2

.

a2

2

a2

ln a 2 2 3 r dr . . 5 ln a 1

5. 3

2 2

a1

(3.389)

ln a 1 ln a 2

(3.390)

rν

(3.391)

Evaluating yields, r dr = 1.0674 ( fm)

(3.392)

Exploratory factor analysis, with respect to equation (3.378), indicates that an infinite family of solutions for “a1, a2 and b1” exists to satisfy “ρch”. Therefore, we shall assume that “a2 = xa1” and “a1 = rν”. Subsequently, the values of “a2, b1 and x” may be determined as follows, 256

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substitute , a 2 x. a 1 3.

2 a 2 .b 1 K S

2

a1

2

KS

2.

substitute , a 1 r ν

3

solve , b 1

2 rν . 1

2

x

(3.393)

where, “KS” denotes the MS charge radius of a Neutron as derived utilising EGM methodology. Hence, 2 . KS

b1

3 .r ν

2

2

x

1

(3.394)

Substituting “b1” into equation (3.385) yields an expression for “KS” in terms of “rν and x” as follows, substitute , b 1 b1

1 4.π .r ν

3

π

1

.

a1

3

KS

3 a 2 2 . substitute , a 2 x a 1

1 3

2.

3

a2

2

a1

1

3. 8

2.

rν π

2

. x3 .

( x 1) 2

x

substitute , a 1 r ν

x 1

solve , K S

(3.395)

Hence, 3. π .r ν

KS

2

. (1

x) . x

1

x x

8

3

2

(3.396)

A solution for “x” may be found by performing the appropriate substitutions into equation (3.380) and solving numerically utilising the “Given” and “Find” commands within the “MathCad 8 Professional” environment as follows, rν

a2

3 . ln

2

.

a1

a 1 .a 2 2

a2

2

substitute , a 2 x. a 1

2

substitute , a 1 r ν

a1

rν

2

2 3 . ln( x) . r ν .

factor

2

x ( ( x 1) .( x 1) )

(3.397)

Given 2

x

ln( x) . 2

x

x

1 1 3

(3.398)

Find( x)

(3.399)

Evaluating yields, x = 0.6829 a1 a2

=

0.8268 0.5647

(3.400) ( fm)

(3.401)

b 1 = 0.2071

(3.402) 2

K S = 0.1133 fm

(3.403)

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The error produced by “KS” in relation to its experimental value “KX” [10] may be calculated according to “1 – KS / KX” as follows, KX 1

2 0.113. fm

KX

(3.404)

= 0.295 ( % )

KS

(3.405)

Note: the experimental uncertainty of “KX” is “±0.005(fm2)” (as defined by [10]). Consequently, “KS” matches experimental measurement precisely, with zero error. The error described by equation (3.405) assumes an exact experimental value as defined by equation (3.404). We may graphically reinforce the preceding derivation by substituting the results for “KS, rν and x” into equation (3.378) and working in dimensionless form as follows, r

ρ ch ( r )

KS

2. 3

3 5 2 π .r ν . x

. e

2

rν

1.

e

r x .r

2 ν

3

x

1

(3.406) Neutron Charge Distribution

Charge Density

rν

r dr

ρ ch( r ) ρ ch r 0 ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

Figure 3.38, Evaluating “ρch” at specific conditions yields the appropriate results, ρ ch r 0

0.1408

ρ ch r ν

= 5.768.10

12 ρ ch 10 .( fm)

0

9

1 3

fm

(3.407)

Utilising the “Given” and “Find” commands, we may determine graphical inflections at “r1” and “r2” according to, Given

258

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r1

d

2

KS

2.

d r 12 3

3

π .r ν

5.

rν

. e 2

x

3

d r 23 3

r1

3.

π rν

5.

1

x

e

x .r ν

0

(3.408) 2

rν

. e 2

1.

2

3

KS

2.

r1

x

r2

d

2

r2

1.

e

2

x .r ν

0

3

x

1

(3.409)

Find r 1, r 2

r2

(3.410)

Evaluating yields [r1 = 0.3766(fm), r2 = 0.6624(fm)], r1

=

r2

0.3766 0.6624

(3.411) Neutron Charge Distribution r1

r2

ρ ch( r ) ρ ch r 0 d

Neutron Charge Characteristic

dr

ρ ch( r )

d dr 1 d

ρ ch r 1

2

d r2 d

ρ ch( r )

2

d r 22 d

2

d r 02

ρ ch r 2

ρ ch r 0

r Radius

Figure 3.39, Evaluating specific conditions yields the appropriate results, d ρ ch r 1 dr 1 d

2

d r 22 d

2

d r 02

0.2539 ρ ch r 2

= 0.5447 1.1032

ρ ch r 0

(3.412) 259

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rν 4 r . ρ ch ( r ) d r

0 ∞ 4 r . ρ ch ( r ) d r

4.π .

0.0166

rν

0.13

=

rν

0.0705

2.

r ρ ch ( r ) d r

0.0705

0 ∞ 2 r . ρ ch ( r ) d r

rν

(3.413)

We shall perform an additional test to ensure that no obvious algebraic errors have been inadvertently performed. To achieve this, we shall employ the exact analytical representation of the integrand, in this case “ρch”, as defined by standard mathematics tables as follows, [34] ∞ 2

r

b1 π

.

3

1 . e

2

r

a1

a2

1 . e

3

dr

3

a1

a2

b1

. 1 2.π a 2 1

1 2

a2

0

(3.414)

Substituting appropriately produces, 2 . KS b1

.

1

3.r ν

1

2.π a 2 1

2

2

x

1

2.π

2

a2

.

1 rν

KS

1 2

x. r ν

2

4 2 3.π .r ν .x

(3.415)

Evaluating yields, KS 4 2 3 .π .r ν .x

= 0.0552

(3.416)

Whereas the result computed by numerical approximation is, ∞ ρ ch ( r ) d r = 0.0552 0

(3.417)

Since the results of the two preceding equations are identical, no obvious algebraic or numerical errors have been performed. Assuming “KX” has zero uncertainty, equation (3.394) may be transposed and utilised to convert “KX” to an equivalent RMS charge radius form “rX” as follows, 6 .b 1 .K X . x

2

rX

3 .b 1 . x

2

1

1

rν KS

. K .K S X

(3.418)

Evaluating and converting dimensionally produces: 0.8071 ≤ rX(fm) ≤ 0.8437

260

(3.419)

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2

NEUTRON MAGNETIC RADIUS

Continuing in dimensionless form, the Neutron Magnetic Radius “rνM” may be usefully approximated to high computational precision {to within 3.2 x10-3(%) of the experimental result [0.879(fm)]} [56] utilising “ρch”, “rν” and “rdr”. Firstly, recognising that “d2ρch(rdr/rν)/dr2 = 0” and graphing over the domain “rν ≤ r ≤ 1.8(fm)”, r dr

r dr

d2 d r2

rν

ρ ch( r )

r Radius

Figure 3.40, Provokes the solution, Given r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

r νM

(3.420)

Find r νM

(3.421)

Evaluating and converting dimensionally produces, rνM = 0.87897(fm)

(3.422)

Visualising graphically over the domain “rν ≤ r ≤ 1.8(fm)” yields, Neutron Charge Distribution r νM

r dr

ρ ch r νM ρ ch( r )

r Radius

Figure 3.41,

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3

PROTON ELECTRIC RADIUS

Similarly, in dimensionless form, the Proton Electric Radius “rπE” may be usefully approximated to high computational precision {to within 6.2 x10-2(%) of the experimental result [0.848(fm)]} [56] utilising “ρch”, “rν” and “rdr” as follows, Given r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

r πE

(3.423)

Find r πE

(3.424)

Evaluating and converting dimensionally produces, rπE = 0.84853(fm)

4

(3.425)

PROTON MAGNETIC RADIUS

Again, in dimensionless form, the Proton Magnetic Radius “rπM” may be usefully approximated to high computational precision {to within 0.82(%) of the experimental result [0.857(fm)]} [56] utilising “ρch”, “rν” and “rdr” as follows, Given ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

r πM

(3.426)

Find r πM

(3.427)

Evaluating and converting dimensionally produces, rπM = 0.84993(fm)

5

(3.428)

CLASSICAL PROTON RMS CHARGE RADIUS

Finally, in dimensionless form, the Proton RMS charge radius “rp” may be usefully approximated to high computational precision {to within 0.05(%) of the National Institute of Standards and Technology (NIST) result [0.8750(fm)]} [1] as follows, r P r πE

1. 2

r νM

rν

(3.429)

Evaluating and converting dimensionally produces, rp = 0.87459(fm)

(3.430)

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APPENDIX 3.H [76] Results,

CALCULATION OF L2, L3 AND L5 ASSOCIATED NEUTRINO RADII We may deduce the approximate masses of the “L2, L3 and L5” Neutrino particles by inference. This may be achieved by initially assuming their masses to be approximately equal to the Neutrino masses articulated in chapter 3.11 and solving for their radii in accordance with equation (3.230, 3.238). Utilising the masses of the Electron, Muon and Tau Neutrino’s (men, mµn and mτn respectively), the radii of the “Lx” Neutrino particles “rνx”, may be determined relative to the Electron mass as follows, 5 5

r ν2 r ν3 r ν5

m en 1 . r ε. 2 9 me 2

5

2

r µ.

m µn

5

2

9

r τ.

3

m τn

2

9

5

(3.431)

Evaluating produces Neutrino radii of, r ν2 r ν3 r ν5 = ( 0.0274 0.76557 2.82054) 10

16 .

cm

(3.432)

Determining the average “rνx” radius value yields, 1. 3

r ν2

r ν3

r ν5 = 1.2045 10

16 .

cm

(3.433)

Determining the average Electron, Muon and Tau Neutrino radii produces (chapter 3.11), 1. 3

r en

r µn

r τn = 0.90323 10

16 .

cm

(3.434)

Comparing equation (3.433, 3.434) yields, r ν2

r ν3

r ν5

r en

r µn

r τn

= 1.33356

≈

4 3

(3.435)

Therefore, since the average value of both radii groupings approximate unity (4/3), the initial assumption that their masses (by matter type) are approximately equal appears qualitatively validated.

263

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NOTES

264

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APPENDIX 3.I DERIVATION OF THE HYDROGEN ATOM SPECTRUM (BALMER SERIES) AND AN EXPERIMENTALLY IMPLICIT DEFINITION OF THE BOHR RADIUS [76] Results, It is possible to utilise Electro-Gravi-Magnetics (EGM) to derive the first term in the Balmer series of the Hydrogen atom spectrum. Subsequently by inference, the remaining terms may also be produced. Moreover, an experimentally implicit definition of the Bohr radius “rBhor” may also be derived. Classical Derivation of the Atomic Emission / Absorption Spectrum [57] 1. Calculate the reduced mass of Hydrogen “µ”, m e .m p

µ

me

mp

(3.436)

2. Calculate the Rydberg Constant “R∞” [Joules] (Qe denotes Electric charge), 2 4 2.π .µ .Q e

R∞

h

2

(3.437)

3. Calculate the Electronic energy level “E” at an arbitrary quantum number “nq”, R∞

E nq

nq

2

(3.438)

4. Calculate the transition energy “∆E”, ∆E n q

E nq

E( 2 )

(3.439)

5. Calculate the Balmer series wavelength “λB”, λB nq

h .c ∆E n q

(3.440)

6. Specify the quantum range variable “nq = 3, 4…12” and plot the spectrum, The Hydrogen Spectrum (Balmer Series)

1

350

400

450

500

550

600

650

700

λB nq nm Wavelength

Figure 3.42, 7. Evaluate the first term: λB(3) = 656.46962(nm) 265

(3.441) www.deltagroupengineering.com

General Formulation of Atomic Emission / Absorption Spectra by EGM Assumptions: 1. “rBohr” defines a usefully approximate position of the Zero-Point-Field (ZPF) equilibrium radius. 2. The fundamental wavelength of the Polarisable Vacuum (PV) spectrum of the Hydrogen atom coincides with the longest wavelength in the Balmer series. 3. The Hydrogen atom may be usefully represented by an “imaginary particle” (spherical) of radius “rBohr” with approximately the mass of a Proton. 4. The ZPF is in equilibrium with an “imaginary field” surrounding the atom at approximately “rBohr”. EGM has utilised Fourier series to develop a spectral representation of the PV model of gravity. EGM describes the field energy induced by mass as a spectrum of frequencies. The EGM spectrum is defined by a discrete set of frequencies commencing from incrementally above “0(Hz)” to the Planck frequency. Or in other words, the EGM spectrum is a discrete version of the continuous ZPF spectrum based upon a Fourier distribution. The PV spectrum is a subset of the EGM spectrum with a non-zero fundamental frequency. It occupies a bandwidth of the EGM spectrum and is system or particle specific, based upon the distribution of energy density. It assumes that, in the case of a spherical particle for example, the ZPF energy outside a region with certain radius is in equilibrium with the field energy of the particle or system inside that region. Since EGM is based upon a Fourier distribution, the amplitude spectrum within the Fourier distribution is usefully represented by a decay function (asymptotically tending to zero). Subsequently, we would expect that the ratio of the fundamental PV wavelength to the longest wavelength in the Balmer Series might relate to the total number of modes by an index value. Since the PV spectrum as described by EGM is double sided and symmetrical about the th “0 ” mode, the wavelength of the PV spectrum “λPV” for a spherical mass may be applied to determine the first term in the Balmer series of the Hydrogen atom spectrum as follows, c

λ PV n PV, r , M

ω PV n PV, r , M

(3.83)

where, “λPV(1,r,M)” denotes the fundamental (starting) wavelength of the PV spectrum of arbitrary mass and radius. If “λA” approximates the first term of the Balmer series [i.e. the longest wavelength such that λA ≅ λB(3)] in the Hydrogen atom emission / absorption spectrum, then a relationship to the EGM method may be assumed and tested as follows, Let: λ PV( 1 , r , M ) λ A( r, M )

ψ

2 .n Ω ( r , M )

(3.442)

where, “2nΩ” denotes the total number of modes (odd + even) on both sides of the EGM spectrum, symmetrical about the “0th” mode defined by, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.7)

Hence, testing an obvious value of “ψ = 2” yields the general formulation, λ A( r, M )

λ PV( 1, r , M )

266

2.n Ω ( r , M )

(3.443) www.deltagroupengineering.com

Application of the General Formulation by EGM Method 1: The Bohr radius is a non-physical quantum average property incorporating Planck’s Constant “h” and may be defined as follows, ε 0 .h

r Bohr

2

π .m e .Q e

2

(3.444)

Evaluating yields, [58] rBohr = 5.291772108 x10-11(m)

(3.445)

It was illustrated in chapter 3.13 that the Planck Scale was approximately “16(%)” too small. Since “h” is a function of “rBohr” and represents a non-physical quantum average property, it follows that “rBohr” is approximately “16(%)” too large and must also be re-scaled for application to equation (3.443) under the EGM method by a factor of “Kω”. Hence, 3

Kω

2 π

(3.270)

λ A K ω .r Bohr , m p = 657.32901 ( nm)

(3.446)

Evaluating “λA” and comparing to “λB” yields the EGM error associated with the first term in the Balmer series as follows, λ A K ω .r Bohr , m p λB

1 = 0.13091 ( % )

(3.447)

Method 2: If we assume “rBohr” to be correct and constrain the EGM predicted Balmer Series wavelength to be exactly equal to the classical representation, then we may calculate the required “imaginary particle mass” (mx) utilising the “Given” and “Find” commands within the “MathCad 8 Professional” environment as follows, Given λ A K ω .r Bohr , m x

1

λB mx

(3.448)

Find m x

m x = 1.68052 10

(3.449) 27 .

kg

(3.450)

Notably, “mx” is very close to the Proton mass and the Atomic Mass Constant “mAMC”. Determining EGM mass errors yields, mx

1 = 0.47208 ( % )

mp mx

(3.451) 1 = 1.20316 ( % )

m AMC

(3.452)

Method 3: 267

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If we assume “mAMC” to be correct and apply similar logic as previously (Method 2), we may determine the correct value of ZPF equilibrium radius based upon the experimentally implicit definition of the Planck Scale derived in chapter 3.13 as follows, Given λ A K ω .r x, m AMC λB rx

1

(3.453)

Find r x

r x = 5.27319.10

(3.454) 11

( m)

(3.455)

Comparing “rx” to “rBohr” yields the difference between them, r Bohr

1 = 0.35238 ( % )

rx

(3.456)

Hence, the ZPF equilibrium radius coincides with the Bohr Radius to within “0.353(%)” and suggests an experimentally implicit8 definition of “rBohr”. Therefore, a useful approximation to the first term in the Hydrogen atom spectrum (Balmer series) may be given by, λA

λ PV 1 , K ω .r Bohr , m p 2 .n Ω K ω .r Bohr , m p

(3.457)

NOTES

APPENDIX 3.J 8

Refer to chapter 3.13 for factors of experimental implication. 268

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GLOSSARY OF TERMS •

General and Specific Symbols (alphabetical order)

Symbol A a a1 a2 APP ax(t) a∞

Description 1st Harmonic term Magnitude of acceleration vector Acceleration with respect to General Modelling Equation One Acceleration with respect to General Modelling Equation Two Parallel plate area of a Classical Casimir Experiment Arbitrary acceleration in the time domain Mean magnitude of acceleration over the fundamental period in a FS representation in EGM Magnitude of Magnetic field vector B Magnitude of Magnetic field vector (at infinity) in the PV model of gravity: Ch. 3.2 Amplitude of applied Magnetic field: Ch. 3.6 B0 Magnitude of Magnetic field vector (locally) in the PV model of gravity Magnitude of applied Magnetic field vector BA Critical Magnetic field strength BC Magnitude of PV Magnetic field vector BPV Bottom Quark: elementary particle in the SM bq Root Mean Square of BA Brms Velocity of light in a vacuum c Velocity of light in a vacuum (at infinity) in the PV model of gravity: Ch. 3.1 Velocity of light (locally) in the PV model of gravity c0 Amplitude of fundamental frequency of PV (nPV = 1) CPV(1,r,M) Amplitude spectrum of PV CPV(nPV,r,M) Charm Quark: elementary particle in the SM cq Common difference D Experimental configuration factor: a specific value relating all design criteria; this includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy Offset function DC Down Quark: elementary particle in the SM dq Energy: Ch. 3.3 E Magnitude of Electric field vector Magnitude of Electric field vector (at infinity) in the PV model of gravity: Ch. 3.2 Electronic energy level Charge e, eElectron: subatomic / elementary particle in the SM Exponential function: mathematics Amplitude of applied Electric field: Ch. 3.6 E0 Energy (locally) in the PV model of gravity Magnitude of Electric field vector (locally) in the PV model of gravity Magnitude of applied Electric field vector EA Critical Electric field strength EC Magnitude of PV Electric field vector EPV 269

Units m/s2 m2 m/s2

T

T m/s m/s2

% J V/m

J C

V/m J V/m V/m

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Root Mean Square of EA m/s2 Complex FS representation of EGM Magnitude of the ambient gravitational acceleration represented in the time domain Amplitude spectrum / distribution of F(k,n,t) F0(k) The Casimir Force by classical representation N FPP The Casimir Force by EGM FPV Gluon: theoretical elementary particle in the SM g Magnitude of gravitational acceleration vector m/s2 3 -1 -2 Universal Gravitation Constant m kg s G Tensor element g00 Tensor element g11 Tensor element g22 Tensor element g33 Height: Ch. 3.4 m h Higgs Boson: theoretical elementary particle in the SM H Hydrogen Magnetic field strength Oe Js Planck’s Constant (plain h form) h h-bar Planck’s Constant (2π form) HSE4A R Time average form of HSE4 R HSE5A R Time average form of HSE5 R Generalised reference to the reduced form of HSEx HSEx R Complex number i Initial condition Macroscopic intensity of Photons within a test volume W/m2 In,P Vector current density A/m2 J Wave vector 1/m k K0(r,X) ERF by displacement domain precipitation Generalised ERF K0(X) K0(ω ω,r,E,B,X) ERF by wavefunction precipitation K0(ω ω,X) ERF by frequency domain precipitation K0(ω ωPV,r,EPV,BPV,X) ERF equivalent to K0(ω,r,E,B,X) ERF formed by re-interpretation of the primary precipitant (V/m)2 K1 ERF formed by re-interpretation of the primary precipitant T-2 K2 Harmonic wave vector of applied field 1/m kA Critical Factor KC PaΩ Engineered Refractive Index KEGM Harmonic form of KEGM KEGM H Experimentally implicit Planck Mass scaling factor Km The intensity of the background PV field at specific frequency modes W/m2 Kn,P A refinement of a constant in FPP KP Harmonic wave vector of PV 1/m kPV Refractive Index of PV KPV Harmonic form of KPV KPV H Critical Ratio KR Critical harmonic operator (based upon the unit amplitude spectrum) KR H Neutron MS charge radius by EGM m2 KS m2 Neutron MS charge radius (determined experimentally) in the SM KX Experimentally implicit Planck Length scaling factor Kλ Erms F(k,n,t) f(t)

270

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Kω L L0 L2 L3 L5 M m0 M0 mAMC mbq mcq mdq me ME men mgg mH mh MJ mL(2) mL(3) mL(5) MM mn mp mQB(5) mQB(6) MS msq mtq muq mW mx mZ mε mγ mγg mγγ mµ mµn mτ mτn n n, N nA nB NC nE nPV

Experimentally implicit Planck Frequency scaling factor Length Length (locally) in the PV model of gravity by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Mass Mass (locally) in the PV model of gravity by EGM Zero mass (energy) condition of free space Atomic Mass Constant Bottom Quark rest mass (energy) by EGM Charm Quark rest mass (energy) by EGM Down Quark rest mass (energy) by EGM Electron rest mass (energy) according to NIST Mass of the Earth Electron Neutrino rest mass (energy) according to PDG Graviton rest mass (energy) by EGM Higgs Boson rest mass (energy) according to PDG Planck Mass Mass of Jupiter Rest mass (energy) of the L2 particle by EGM Rest mass (energy) of the L3 particle by EGM Rest mass (energy) of the L5 particle by EGM Mass of the Moon Neutron rest mass (energy) according to NIST Proton rest mass (energy) according to NIST Rest mass (energy) of the QB5 particle by EGM Rest mass (energy) of the QB6 particle by EGM Mass of the Sun Strange Quark rest mass (energy) by EGM Top Quark rest mass (energy) according (energy) to PDG Up Quark rest mass (energy) by EGM W Boson rest mass according (energy) to PDG Imaginary particle mass Z Boson rest mass according (energy) to PDG Electron rest mass (energy) in high energy scattering experiments Photon rest mass (energy) threshold according to PDG Graviton rest mass (energy) threshold according to PDG Photon rest mass (energy) by EGM Muon rest mass (energy) according to NIST Muon Neutrino rest mass (energy) according to PDG Tau rest mass (energy) according to NIST Tau Neutrino rest mass (energy) according to PDG Neutron: subatomic particle in the SM Field harmonic (harmonic frequency mode) Harmonic frequency modes of applied field Harmonic mode number of the ZPF with respect to BA Critical mode Harmonic mode number of the ZPF with respect to EA Harmonic frequency modes of PV 271

m

kg or eV

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nq NT NTR NX N∆r nΩ nΩ ZPF nβ P p Q, Qe QB5 QB6 r

r0 rBohr rBoson rbq rc rcq rdq re RE ren RError rgg rH RJ rL RM rp rQB RS rsq rtq ru ruq rW rx rxq rZ rε rγγ rµ

Quantum number Number of terms The ratio of the number of terms Harmonic inflection mode Permissible mode bandwidth of applied experimental fields Harmonic cut-off mode of PV ZPF beat cut-off mode Mode Number (Critical Boundary Mode) of ωβ Polarisation vector Proton: subatomic particle in the SM Magnitude of Electric charge Theoretical elementary particle (Quark or Boson) by EGM Theoretical elementary particle (Quark or Boson) by EGM Arbitrary radius with homogeneous mass (energy) distribution Generalised notation for length (e.g. r → λ/2π): Ch. 3.1 Generalised notation for length (locally) in the PV model of gravity: Ch. 3.1 Magnitude of position vector from centre of spherical object with homogeneous mass (energy) distribution Reciprocal of the wave number: Ch. 3.1 Length (locally) in the PV model of gravity Classical Bohr radius Generalised RMS charge radius of a Boson by EGM RMS charge radius of the Bottom Quark by EGM Transformed value of generalised length (locally) in the PV model of gravity RMS charge radius of the Charm Quark by EGM RMS charge radius of the Down Quark by EGM Classical Electron radius in the SM Mean radius of the Earth RMS charge radius of the Electron Neutrino by EGM Representation Error RMS charge radius of the Graviton by EGM RMS charge radius of the Higgs Boson utilising ru Mean radius of Jupiter Average RMS charge radii of the rε, rµ and rτ particles Mean radius of the Moon Classical RMS charge radius of the Proton in the SM Average RMS charge radius of the QB5 / QB6 particles by EGM utilising ru Mean radius of the Sun RMS charge radius of the Strange Quark by EGM RMS charge radius of the Top Quark by EGM Heisenberg uncertainty range RMS charge radius of the Up Quark by EGM RMS charge radius of the W Boson utilising ru Bohr radius by EGM Generalised RMS charge radius of all Quarks as determined by the ZC within the SM RMS charge radius of the Z Boson by utilising ru RMS charge radius of the Electron by EGM RMS charge radius of the Photon by EGM RMS charge radius of the Muon by EGM 272

C/m2 C

m

% m

m

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RMS charge radius of the Muon Neutrino by EGM Neutron RMS charge radius (by analogy to KS) RMS charge radius of the ν2 particle by EGM RMS charge radius of the ν3 particle by EGM RMS charge radius of the ν5 particle by EGM Neutron Magnetic radius by EGM Generalised reference to rν2, rν3 and rν5 RMS charge radius of the Proton by EGM Proton Electric radius by EGM Proton Magnetic radius by EGM RMS charge radius of the Tau by EGM RMS charge radius of the Tau Neutrino by EGM Rydberg Constant Poynting Vector Strange Quark: elementary particle in the SM nth Harmonic term Range factor 1st Sense check 3rd Sense check 4th Sense check 2nd Sense check 5th Sense check 6th Sense check A positive integer value representing the harmonic cut-off frequency ratio between two proportionally similar mass (energy) systems Poynting Vector of PV Sω Time t Top Quark: elementary particle in the SM tq Initial state GPE per unit mass described by any appropriate method Ug Harmonic form of Ug Ug H Rest mass-energy density Um Up Quark: elementary particle in the SM uq Field energy density of PV Uω Local value of the velocity of light in a vacuum vc W Boson: elementary particle in the SM W All variables within the experimental environment that influence results and X behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis Impedance function Z Z Boson: elementary particle in the SM Change in electronic energy level ∆Ε Change in the magnitude of the local PV acceleration vector ∆aPV Change in magnitude of the local gravitational acceleration vector ∆g ∆GME1 Change in GME1 ∆GME2 Change in GME2 ∆GMEx Generalised reference to changes in GME1 and GME2 Harmonic form of ∆K0 ∆K0 H ∆K0(ω ω,X) Engineered Relationship Function by EGM Change in K1 by EGM ∆K1 rµn rν rν2 rν3 rν5 rνM rνx rπ rπE rπM rτ rτn R∞ S sq StN Stα Stβ Stδ Stε Stγ Stη Stθ Stω

273

J W/m2

PaΩ

W/m2 s (m/s)2 Pa Pa m/s

Ω J m/s2

(V/m)2

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∆K2 ∆ KC ∆nS ∆r ∆t ∆t0 ∆ Ug ∆UPV ∆vΩ ∆vδr ∆ΛPV ∆λΩ ∆λδr ∆ωPV ∆ωR ∆ωS ∆ωZPF ∆ωΩ ∆ωδr Π ΣH ΣHR Ω α α1 αx β β1 βx ε0 φ φC φgg φγγ γ γg λ λΑ λΒ λCe λCN

Change in K2 by EGM Change in Critical Factor by EGM Change in the number of ZPF modes Plate separation of a Classical Casimir Experiment Practical changes in benchtop displacement values Change in time (at infinity) in the PV model of gravity by EGM Change in time (locally) in the PV model of gravity by EGM Change in Gravitational Potential Energy (GPE) per unit mass induced by any suitable source Change in energy density of gravitational field Change in rest mass-energy density Terminating group velocity of PV Group velocity of PV Change in the local value of the Cosmological Constant by EGM Change in harmonic cut-off wavelength of PV Change in harmonic wavelength of PV Frequency bandwidth of PV Bandwidth ratio Similarity bandwidth ZPF beat bandwidth Beat bandwidth of PV Beat frequency of PV Dimensional grouping derived by application of BPT The sum of terms The ratio of the sum of terms Harmonic cut-off function of PV An inversely proportional description of how energy density may result in an acceleration: Ch. 3.2 Fine Structure Constant The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to α1 and α2 A directly proportional description of how energy density may result in an acceleration The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to β1 and β2 Permittivity of a vacuum Relative phase variance between EA and BA Critical phase variance RMS charge diameter of the Graviton by EGM RMS charge diameter of the Photon by EGM Mathematical Constant: Euler-Mascheroni (Euler’s) Constant Photon: elementary particle in the SM Graviton: theoretical elementary particle in the SM Wavelength 1st term of the Balmer Series by EGM Classical Balmer Series wavelength Electron Compton Wavelength Neutron Compton Wavelength 274

T-2 PaΩ m s (m/s)2 Pa m/s Hz2 m Hz Hz

m/s2 m/s2

F/m θc m

m

m

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λCP λh λPV µ, µµ µ0 ν2 ν3 ν5 νe νµ ντ ρ ρ0 τ, τω

Proton Compton Wavelength Planck Length Wavelength of PV Muon: elementary particle in the SM Reduced mass of Hydrogen Permeability of a vacuum Theoretical elementary Neutrino of the L2 particle by EGM Theoretical elementary Neutrino of the L3 particle by EGM Theoretical elementary Neutrino of the L5 particle by EGM Electron Neutrino: elementary particle in the SM Muon Neutrino: elementary particle in the SM Tau Neutrino: elementary particle in the SM Charge density Spectral energy density Tau: elementary particle in the SM Field frequency Field frequency (at infinity) in the PV model of gravity: Ch. 3.2 Field frequency (locally) in the PV model of gravity ω0 Field frequency (locally) in the PV model of gravity by EGM Harmonic frequency of the ZPF with respect to BA ωB Critical frequency ωC Harmonic frequency of the ZPF with respect to EA ωE Electron Compton Frequency ωCe Neutron Compton Frequency ωCN Proton Compton Frequency ωCP Planck Frequency ωh Generalised reference to ωPV(nPV,r,M) ωPV Fundamental frequency of PV (nPV = 1) ωPV(1,r,M) ωPV(nPV,r,M) Frequency spectrum of PV Harmonic inflection frequency ωX Harmonic cut-off frequency of PV ωΩ ZPF beat cut-off frequency ωΩ ZPF Critical boundary ωβ 〈 mQuark〉 Average mass (energy) of all Quarks according to PDG Average mass (energy) of all Quarks by EGM Average RMS charge radius of all Bosons in the SM utilising ru 〈rBoson〉 Average RMS charge radius of all Quarks by EGM 〈rQuark〉 Average RMS charge radius of all Quarks and Bosons by EGM utilising ru 〈r〉〉

275

kg or eV N/A2

C/m3 Pa/Hz Hz

kg or eV m

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Bibliography 3 [1] National Institute of Standards and Technology (NIST), http://physics.nist.gov/cuu/ [2] Mathworld, http://mathworld.wolfram.com/Euler-MascheroniConstant.html [3] Software: MathCad 8 Professional, http://www.mathsoft.com/ [4] University of Illinois, http://archive.ncsa.uiuc.edu/Cyberia/NumRel/mathmine1.html [5] http://en.wikipedia.org/wiki/Dimensional_analysis [6] http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem [7] Norwegian University of Science and Technology, http://www.math.ntnu.no/~hanche/kurs/matmod/1998h/ [8] University of California, Riverside, http://math.ucr.edu/home/baez/physics/Quantum/casimir.html [9] The SELEX Collaboration, http://arxiv.org/abs/hep-ex/0106053v2 [10] “Karmanov et. Al.”, http://arxiv.org/abs/hep-ph/0106349v1 [11] “P. W. Milonni”, The Quantum Vacuum – An Introduction to Quantum Electrodynamics, Academic Press, Inc. 1994. Page 403. [12] Particle Data Group, http://pdg.lbl.gov/, “S. Eidelman et Al.” Phys. Lett. B 592, 1 (2004). [13] “Hirsch et. Al.”, Bounds on the tau and muon neutrino vector and axial vector charge radius, http://arxiv.org/abs/hep-ph/0210137v2 [14] The ZEUS Collaboration, http://arxiv.org/abs/hep-ex/0401009v2 [15] The D-ZERO Collaboration, http://arxiv.org/abs/hep-ex/0406031v1 [16] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/forces/exchg.html [17] “James William Rohlf”, Modern Physics from α to Z, John Wiley & Sons, Inc. 1994. [18] Norwegian University of Science and Technology, A micro-biography of Edgar Buckingham, http://www.math.ntnu.no/~hanche/notes/buckingham/ [19] “W. Misner, K. S. Thorne, J. A. Wheeler”, Gravitation, W. H. Freeman & Co, 1973. Ch. 1, Box 1.5, Ch. 12, Box 12.4, sec. 12.4, 12.5. [20] “B.S. Massey”, Mechanics of Fluids sixth edition, Van Nostrand Reinhold (International), 1989, Ch. 9. [21] “Rogers & Mayhew”, Engineering Thermodynamics Work & Heat Transfer third edition, Longman Scientific & Technical, 1980, Part IV, Ch. 22. [22] “Douglas, Gasiorek, Swaffield”, Fluid Mechanics second edition, Longman Scientific & Technical, 1987, Part VII, Ch. 25. [23] “Puthoff et. Al.”, Polarizable-Vacuum (PV) representation of general relativity, v2, Sept., 1999 http://xxx.lanl.gov/abs/gr-qc/9909037 [24] “Puthoff et. Al.”, Polarizable-vacuum (PV) approach to general relativity, Found. Phys. 32, 927 – 943 (2002). [25] “Erwin Kreyszig”, Advanced Engineering Mathematics Seventh Edition, John Wiley & Sons, 1993, Ch. 10. [26] “H.A. Wilson”, An electromagnetic theory of gravitation, Phys. Rev. 17, 54 – 59 (1921). [27] “R.H. Dicke”, Gravitation without a principle of equivalence. Rev. Mod. Phys. 29, 363 – 376, 1957. [28] “R.H. Dicke”, Mach’s principle and equivalence, in Proc. Of the International School of Physics “Enrico Fermi” Course XX, Evidence for Gravitational Theories, ed. C. Møller, Academic Press, New York, 1961, pp. 1 – 49. [29] “A.M. Volkov, A.A. Izmest’ev, and G.V. Skrotskii”, The propagation of electromagnetic waves in a Riemannian space, Sov. Phys. JETP 32, 686 – 689 1971. [30] “Puthoff et. Al.”, Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight, JBIS, Vol. 55, pp.137, 2002 , http://xxx.lanl.gov/abs/astro-ph/0107316v2. [31] “G. Arfken”, Mathematical Methods for Physicists – Third Edition, Academic Press, Inc. 1985 ISBN 0-12-059820-5. Ch. 1, pp. 77. [32] “J.D. Jackson”, Classical Electrodynamics, Third Edition, 1998, ISBN 0-471-30932-x, Ch. 6, 276

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Secs. 6.7 – 6.9, Ch. 12, Sec. 12.7. [33] “K.A. Stroud”, “Further Engineering Mathematics”, MacMillan Education LTD, Camelot Press LTD, 1986, Programme 17. [34] “Lennart Rade, Bertil Westergren”, “Beta Mathematics Handbook Second Edition”, ChartwellBratt Ltd, 1990, Page 470. [35] Scienceworld, http://scienceworld.wolfram.com/physics/BeatFrequency.html [36] Scienceworld, http://scienceworld.wolfram.com/physics/MaxwellEquationsSteadyState.html [37] Scienceworld, http://scienceworld.wolfram.com/physics/ElectromagneticRadiation.html [38] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html [39] Scienceworld, http://scienceworld.wolfram.com/physics/MaxwellEquations.html [40] http://www.mathcentre.ac.uk/students.php/all_subjects/series [41] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/whdwar.html [42] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/redgia.html [43] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/pulsar.html [44] “Stein, B. P”. Physics Update, Physics Today 48, 9, Oct. 1995. [45] “Simon et Al.”, Nucl. Phys. A333, 381 (1980). [46] Scienceworld, http://scienceworld.wolfram.com/physics/Proton.html [47] “Andrews et Al.”, 1977 J. Phys. G: Nucl. Phys. 3 L91 – L92. [48] “L.N. Hand, D.G. Miller, and R. Wilson”, Rev. Mod. Phys. 35, 335 (1963). [49] A Proposal to the MIT-Bates PAC. Precise Determination of the Proton Charge Radius, August 19 (2003) – Spokespersons: H. Gao, J.R. Calarco [e-mail: [email protected], phone: (617) 258-0256, fax: (617) 258-5440]. [50] Stanford Linear Accelerator, http://www.slac.stanford.edu/ http://www2.slac.stanford.edu/vvc/theory/quarks.html [51] Scienceworld, http://scienceworld.wolfram.com/physics/PlanckLength.html [52] Scienceworld, http://scienceworld.wolfram.com/physics/Photon.html [53] Stanford Linear Accelerator, http://www2.slac.stanford.edu/vvc/theory/fundamental.html [54] See: [13] [55] “Joshipura et. Al.”, Bounds on the tau neutrino magnetic moment and charge radius from Super-K and SNO observations, http://arxiv.org/abs/hep-ph/0108018v1 [56] “Hammer and Meißner et. Al.”, http://arxiv.org/abs/hep-ph/0312081v3 [57] University of Tel Aviv, http://www.tau.ac.il/~phchlab/experiments/hydrogen/balmer.htm [58] Scienceworld, http://scienceworld.wolfram.com/physics/BohrRadius.html [59] “Albert Einstein”, http://nobelprize.org/physics/laureates/1921/index.html [60] “Jean Baptiste Joseph Fourier”, http ://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html [61] http://stores.lulu.com/dge [62] http://www.veoh.com/users/DeltaGroupEngineering [63] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – I, Physics Essays: Vol. 19, No. 1: March 2006. [64] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – II, Physics Essays: Vol. 19, No. 2: June 2006. [65] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – III, Physics Essays: Vol. 19, No. 3: September 2006. [66] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – IV, Physics Essays: Vol. 19, No. 4: December 2006. [67] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – V, Physics Essays: Vol. 20, No. 1: March 2007. [68] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – VI, Physics Essays: Vol. 20, No. 2: June 2007. [69] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarisable vacuum – VII, Physics Essays: Vol. 20, No. 3: September 2007. 277

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[70] “Riccardo C. Storti, Todd J. Desiato”, Derivation of the Photon mass-energy threshold, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005) [pg. 207 – 213]. [71] “Riccardo C. Storti, Todd J. Desiato”, Derivation of fundamental particle radii (Electron, Proton & Neutron), Physics Essays: Vol. 22, No. 1: March 2009. [72] “Riccardo C. Storti, Todd J. Desiato”, Derivation of the Photon & Graviton mass-energies & radii, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005) [pg. 214 – 217]. [73, 74, 75] See [76]. [76] “Riccardo C. Storti”, The Natural Philosophy of Fundamental Particles, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 6664 (SPIE, Bellingham, WA, 2007). [77] http://www.deltagroupengineering.com/Docs/QE3_-_Summary.pdf [78] http://www.deltagroupengineering.com/Docs/QE3_-_Calculation_Engine.pdf [79] http://www.deltagroupengineering.com/Docs/QE3_-_High_Precision_(MCAD12).pdf [80] Progress in Top Quark Physics (Evelyn Thomson): Conference proceedings for PANIC05, Particles & Nuclei International Conference, Santa Fe, New Mexico (USA), October 24 – 28, 2005: http://arxiv.org/abs/hep-ex/0602024v1 [81] http://www-cdf.fnal.gov/physics/new/top/top.html#PAIR [82] W Mass & Properties (the CDF & D0 Collaborations): http://arxiv.org/abs/hep-ex/0511039v1 [83] Measurement of the Mass and the Width of the W Boson at LEP (the L3 Collaboration): http://arxiv.org/abs/hep-ex/0511049v1 [84] Precision Electroweak Measurements on the Z Resonance (the ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the SLD Electroweak & Heavy Flavour Groups): http://arxiv.org/abs/hep-ex/0509008v3 [85] Combination of CDF and D0 Results on the Mass of the Top Quark, Fermilab-TM-2347-E, TEVEWWG/top 2006/01, CDF-8162, D0-5064: http://arxiv.org/abs/hep-ex/0603039v1 [86] Cornell University Library: http://www.arxiv.org/

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NUMERICAL EGM SIMULATIONS

279

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NOTES

280

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MATHCAD 8 PROFESSIONAL COMPLETE SIMULATION [77]

281

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NOTES

282

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APPENDIX 3.K Computational Environment NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated.

Units of Measure (Definitions) Scale 1

10

Scale 2

10

3

3

10 6

10

6

10

9

9

10

10

10

12

12

10

10 15

15

10

18

10

18

10

21

10

24

10

Scale 1 .( Hz)

( mHz µHz nHz pHz fHz aHz zHz yHz ) Scale 1 .( J )

Scale 1 .( W )

( mW µW nW pW fW aW zW yW )

Scale 1 .( ohm )

( mΩ µΩ nΩ pΩ fΩ aΩ zΩ yΩ )

Scale 1 .( V)

( mV µV nV pV fV aV zV yV )

Scale 1 .( Pa )

( mPa µPa nPa pPa fPa aPa zPa yPa )

Scale 1 .( T )

( mT µT nT pT fT aT zT yT )

Scale 1 .( Ns )

( mNs µNs nNs pNs fNs aNs zNs yNs ) ( mN µN nN pN fN aN zN yN )

Scale 1 .( newton )

( mgs µgs ngs pgs fgs ags zgs ygs )

Scale 1 .( gauss ) Scale 1 .( gm)

( mgm µgm ngm pgm fgm agm zgm ygm ) ( mSt µSt nSt pSt fSt aSt zSt ySt ) ( kSt MSt GSt TSt PSt ESt ZSt YSt )

Scale 1 Scale 2

( kHz MHz GHz THz PHz EHz ZHz YHz ) ( kN MN GN TN PN EN ZN YN ) ( kJ MJ GJ TJ PJ EJ ZJ YJ )

24

10

Scale 1 .( m)

( mm µm nm pm fm am zm ym )

( mJ µJ nJ pJ fJ aJ zJ yJ )

21

Scale 2 .( Hz)

Scale 2 .( newton )

Scale 2 .( J )

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Scale 2 .( W )

( kW MW GW TW PW EW ZW YW )

Scale 2 .( ohm )

( kΩ MΩ GΩ TΩ PΩ EΩ ZΩ YΩ )

Scale 2 .( V)

( kV MV GV TV PV EV ZV YV )

Scale 2 .( Pa )

( kPa MPa GPa TPa PPa EPa ZPa YPa )

Scale 2 .( T )

( kT MT GT TT PT ET ZT YT )

Scale 2 .( eV)

( keV MeV GeV TeV PeV EeV ZeV YeV )

Ns newton .s

Constants (Definitions) G

ε0 α

6.6742.10

3

m

11 .

kg .s

. 8.85418781710

c

m 299792458. s

h

6.6260693.10

2

12 .

F

µ0

34 .

7 newton 4.π .10 . 2 A

( J .s )

. eV 1.6021765310

19 .

γ

19 .

( J)

m . 7.29735256810

3

. 1.6021765310

Qe

( C)

0.5772156649015328

Fundamental Particle Characteristics (Definitions or Initialisation Values) m e m p m n m µ m τ m AMC

. 9.109382610

λ Ce λ CP λ CN λ Cµ λ Cτ

ω Ce ω CP ω CN ω Cµ ω Cτ

31

h. 1

. 1.6726217110

27

1

1

1

. 1.6749272810

27

. 1.883531410

28

. 3.1677710

27

. 1.6605388610

27

.( kg )

1

c me mp mn mµ mτ 2 2.π .c .

h

me mp mn mµ mτ

eV 6 6 3 0.19.10 18.2.10 . 2 c

m en m µn m τn

Note: for the Bottom Quark, the “SLAC” estimate is utilised initially. m uq m dq m sq m cq m bq m tq

4.10

( 80.425 91.1876 114.4) .

mW mZ mH

r xq

0.85.10

3

GeV 0.13 1.35 4.7 179.4 . 2 c

GeV c

re rp rn

8.10

3

2

( 2.817940325 0.875 0.85 ) .( fm) 16 .

( cm )

r Bohr

. 0.529177210810

10

( m)

284

λB

.( nm ) 656.469624182052

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Planck Characteristics (Definitions) G.h

λh

c

h .c

mh

3

G.h

th

G

c

1

ωh

5

th

Astronomical Statistics 24 24 24 30 0.0735.10 5.977.10 1898.8.10 1.989.10 .( kg )

MM ME MJ MS

5 1738 6377.18 71492 6.96.10 .( km)

RM RE RJ RS 2 c .R E 2 .G

M BH

R BH

c

200.R S

R RG

2 .G.M BH

M RG

2

4 .M S

M NS

1 .M S

R NS

R WD

4200.( km)

M WD

20 .( km)

3 300.10 .M E

Other . M BH = 4.2937906795847110

33

( kg )

mx

mp

rx

r Bohr

Arbitrary Values for Illustrational Purposes ω

KR

1 .( Hz)

k

1

X

R max

1

4 10 .( km)

1

r

∆R max

RE

F 0( k )

1

K 0( ω , X )

1

R max 250

Chapter 3.1 Specifying arbitrary values for illustrational purposes facilitates the representation of constant acceleration by the superposition of wavefunctions as follows: N

10

n

B( k, n , t )

N, 1 N.. N

Re( F( k, n , t ) ) .( T )

( π .n .ω .t ) .i

F( k , n , t )

F 0( k ) .e

E( k , n , t )

Im( F( k , n , t ) ) .

V m

N E( k , n , t ) a( t )

K 0( ω , X ) r

1

2

ω

. n= N N

a∞ B( k , n , t )

2

ω.

0 .( s )

a( t ) d t

n= N

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Harmonic Representation of Acc. 1

1

2 .ω

ω

Acceleration

a( t ) a∞

t Time

The contribution (to a constant function) of the sine and cosine terms may be represented for illustrational purposes as follows: 1 N

f( t )

g

ω.

a( t ) n= N

ω

f( t )

2

e

0 .( s )

( π .n .ω .t ) .i

(π d t .e

.n .ω .t ) .i

Real & Complex Harmonic Contributions

Acceleration

Re( a( t ) ) Im( a( t ) ) f( t )

t Time

Real Terms (Non-Zero Sum) Imaginary Terms (Zero Sum) Constant Function (eg. "g")

Chapter 3.2 Additional harmonic characteristics may be usefully represented for illustrational purposes as follows: N

N

5

ΣE( t )

N E( k , n , t )

2

ΣB( t )

n= N

B( k , n , t )

2

n= N

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EM Function

1

2

ω

ω

Re( F ( k , 1 , t ) ) Re( F ( k , 2 , t ) ) Re( F ( k , 3 , t ) )

t Time

EM Function

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

1

2

ω

ω

Im( F ( k , 1 , t ) ) Im( F ( k , 2 , t ) ) Im( F ( k , 3 , t ) )

t Time

EM Wave-Function Superposition

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

1

1

2 .ω

ω

ΣE( t ) ΣB ( t )

t Time

Electric Field Magnitude Magnetic Field Magnitude

Chapter 3.3 Assuming an experiment may be conducted such that the magnitude of the local value of gravitation is either reduced to zero or doubled, the behaviour of the Engineered Refractive Index may be illustrated by the following equation set: 287

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Note: “ ∆K 0 ” does not refer to a change in “ K 0 ”. 2.

K PV( r , M )

e

G .M

3

2 r .c

K 0( r , M , ω , X )

∆K 0( r , M , ω , X )

G.M . KR 2 r .c

K EGM_N( r , M )

K PV( r , M ) .e

e

2

Engineered Relationship Function, 2 . ∆K 0( r , M , ω , X )

K PV( r , M )

K EGM_E( r , M )

K PV( r , M )

Engineered Refractive Index (normal matter form),

Engineered Refractive Index (exotic matter form),

2 . ∆K 0( r , M , ω , X )

By considering astronomical objects as point gravitational masses, we may compare characteristics (to six decimal places) as follows: K PV R E, M M K PV R E, 2 .M M

K PV R E, M E K PV R E, 2 .M E

K PV R E, M J K PV R E, 2 .M J

K 0 R E, M M , ω , X

K 0 R E, M E, ω , X

K 0 R E, M J , ω , X

∆K 0 R E, M M , ω , X

∆K 0 R E, M E, ω , X

∆K 0 R E, M J , ω , X

K EGM_N R E, M M

K EGM_N R E, M E

K EGM_N R E, M J

1

1

1.000001

K EGM_E R E, M M

K EGM_E R E, M E

K EGM_E R E, M J

1

1

1

3 K PV R E, M E .e

∆K 0 R E , M E , ω , X

K PV R E, M S K PV R E, 2 .M S K 0 R E, M S , ω , X ∆K 0 R E, M S , ω , X

1

1

1

1

1.000001

1

1 . 8.55887110

12

. 6.96005110

0.999999 10

2.2111.10

7

∆K 0 R E , M E , ω , X

K 0 R E, M E, ω , X

=1

1.000463 1.000927 =

0.999305 . 2.31613510

K EGM_N R E, M S

1.000927

K EGM_E R E, M S

1

3 K PV R S , M S .e

e

=1

=

1

∆K 0 R S , M S , ω , X

4

∆K 0 R S , M S , ω , X

= 1.000008

e

∆K 0_min

∆K 0_divs

K 0 R S, M S, ω , X

= 1.000008

Hence: ∆K 0_min

1 .10

K EGM r , M , ∆K 0

7

∆K 0_max

K PV( r , M ) e

2 .∆K 0

∆K 0

∆K 0_min 100

∆K 0_min, ∆K 0_divs .. ∆K 0_max

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Engineering Refractive Characteristics

Engineered Refractive Index

0

K PV R E , M E K EGM R E , M E , ∆K 0

∆K 0 Engineered Relationship Function

3 K PV( r , M ) .e

Hence:

∆K 0( r , M , ω , X )

e

∆K 0( r , M , ω , X )

K 0( r , M , X )

Chapter 3.4 Amplitude Spectrum of “g” The time dependent amplitude spectrum of a Fourier representation of “g” at a mathematical point (in Complex form over the time domain) may be represented as follows: Note: “negative amplitude” harmonics are equivalent to “positive amplitude” harmonics as illustrated in the graphs. N, 2

N .. N

t

0 .( s ) ,

1 2 .. . . 25 N ω ω

i .

a PV n PV, t

2 .g . e . π n PV

π .n PV .ω .t .i

Harmonic Amplitudes of Acceleration

Acceleration

n PV

Re a PV n PV , t

t Time

Real Component

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Harmonic Amplitudes of Acceleration

Acceleration

Re a PV( 1 , t ) Re a PV( 1 , t ) Re a PV( 3 , t ) Re a PV( 5 , t )

t Time

1st Negative Harmonic 1st Positive Harmonic 3rd Positive Harmonic 5th Positive Harmonic

Hence, the time-independent amplitude spectrum “ C PV” may be determined by substitution of t

1 2 .n PV.ω

a PV n PV, t

into

C PV n PV, r , M

which produces:

G.M . 2

r

2 . π n PV

Fundamental Frequency of “g” It was illustrated that the frequency spectrum may be given by: r

R max, R max ∆R max.. R max

Hence, it follows that:

ω PV n PV, r , M

T PV n PV, r , M

n PV 3 2 .c .G.M . . K ( r, M ) PV r π .r 1

λ PV n PV, r , M

ω PV n PV, r , M

c ω PV n PV, r , M

Fundamental characteristics occur when “ n PV 1 ” such that: ω PV 1 , R E, M M ω PV 1 , R E, M E ω PV 1 , R E, M J ω PV 1 , R E, M S

. 8.27226110 =

0.035839

T PV 1 , R E, M M

3

( Hz)

T PV 1 , R E, M E

0.244543

T PV 1 , R E, M J

2.484128

T PV 1 , R E, M S

290

120.885935 =

27.902544 4.089263

(s)

0.402556

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λ PV 1 , R E, M M

. 7 3.62406910

λ PV 1 , R E, M E

=

λ PV 1 , R E, M J λ PV 1 , R E, M S

. 6 8.36497210 . 6 1.2259310

( km)

. 5 1.20683210

Fundamental Gravitational Frequency

Fundamental Frequency

RE

ω PV 1 , r , M M ω PV 1 , r , M E ω PV 1 , r , M J ω PV 1 , r , M S ω PV 1 , R E , M E

r Radial Distance

The Moon The Earth Jupiter The Sun

Harmonic Representation of “g” Since “g” is physically constant and never negative in the time domain, we may model “the real world” by taking the magnitude of the appropriate Fourier function. This solution is provoked by the preceding graphs where negative harmonics produce the same results as positive harmonics. Therefore, a generalised representation of the magnitude of “g” at a mathematical point over a fundamental period may be given by: N

21

a PV( r , M , t )

n PV

N, 2

N .. N

i .

C PV n PV, r , M .e

t

0.( s ) ,

T PV 1, R E, M E 25.N

.. T PV 1, R E, M E

π .n PV .ω PV ( 1 , r , M ) .t .i

n PV

291

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Harmonic Representation of Gravity T pv

Acceleration

g

Re a PV R E , M E , t

t Time

Harmonic Cut-off Function, Mode and Frequency It has been illustrated that the mass-energy density for a sold spherical object of homogeneous distribution may be given by: 3 .M .c

U m( r , M )

2

4 .π .r

3

Subsequently, the energy stored in the gravitational field surrounding this object may be given by: 4 h . ω PV( 1, r , M ) 3 2.c

U ω( r , M )

The mass-energy stored in the gravitational field denotes the Polarized Vacuum form of the ZeroPoint-Field. Where, the harmonic cut-off function, mode and frequency are given by “Ω”, “ n Ω ” and “ ω Ω ” respectively – as follows: 3

Ω ( r, M )

n Ω ( r, M ) ω Ω ( r, M ) ∆ω PV( r , M )

108.

U m( r , M )

12. 768 81.

U ω( r , M )

Ω ( r, M )

4

12

Ω ( r, M )

U m( r , M )

2

U ω( r , M )

1

n Ω ( r , M ) .ω PV( 1 , r , M ) ω Ω ( r, M )

ω PV( 1 , r , M )

The gravitational Poynting Vector, according to the PV model of gravity, is characterised by: S m( r , M )

c .U m( r , M )

292

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Evaluating and graphing the preceding system of equations yields: U m R E, M M U m R E, M E U m R E, M J

Ω R E, M M

6.080707

Ω R E, M E

494.481475 =

( EPa)

. 5 1.57089110

Ω R E, M J

. 29 2.83606210 . 29 1.73968910

=

. 28 9.17216810

U m R E, M S

. 1.64551410

Ω R E, M S

n Ω R E, M M

. 28 2.36338510

ω Ω R E, M M

n Ω R E, M E

. 1.44974110

ω Ω R E, M E

. 27 7.64347410

ω Ω R E, M J

. 27 3.5284510

ω Ω R E, M S

. 3 8.76512110

S m R E, M M

0.182295

n Ω R E, M J n Ω R E, M S

8

28

=

∆ω PV R E, M M ∆ω PV R E, M E ∆ω PV R E, M J

195.505363 . 3 1.86915710

( YHz)

519.573099 =

. 8.76512110

. 3 1.86915710

14.824182 =

S m R E, M J

3

∆ω PV R E, M S

195.505363

S m R E, M E

519.573099 =

. 28 4.2341410

( YHz)

YW

. 3 4.70941210

2

cm

. 4.93312710

6

S m R E, M S

Cutoff Function vs Radial Distance RM

RE

Ω R E, M E

Cutoff Function

Ω r, M M Ω r, M E Ω r, M J

Ω R E, M J

Ω r, M S

r Radial Distance

The Moon The Earth Jupiter The Sun

293

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Cutoff Mode vs Radial Distance RM

RE n Ω R E, M E

Cutoff Mode

n Ω r, M M n Ω r, M E n Ω r, M J

n Ω R E, M J

n Ω r, M S

r Radial Distance

The Moon The Earth Jupiter The Sun

Cutoff Frequency vs Radial Distance RM

RE

Cutoff Frequency

ω Ω r, M M ω Ω r, M E ω Ω R E, M J

ω Ω r, M J ω Ω r, M S

ω Ω R E, M E

r Radial Distance

The Moon The Earth Jupiter The Sun

294

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Poynting Vector vs Radial Distance

Magnitude of PV Poynting Vector

RM

RE

S m r, M M S m r, M E S m r, M J S m r, M S S m R E, M J

S m R E, M E

r Radial Distance

The Moon The Earth Jupiter The Sun

295

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Cutoff Mode & Frequency vs Radial Dist. RM

RE

n Ω r, M M

n Ω R E, M E

n Ω r, M E n Ω r, M J n Ω r, M S ω Ω r, M M ω Ω r, M E ω Ω r, M J

ω Ω R E, M E

ω Ω r, M S

r Radial Distance

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

Gravitational Poynting Vector per Change in Odd Harmonic Mode The harmonic contribution of the gravitational Poynting Vector, according to the PV model of gravity, per change in odd mode may be represented as follows. U ω n PV, r , M

h . 4 ω PV( 1, r , M ) . 3 . 2c

S ω n PV, r , M

c .U ω n PV, r , M

n PV

2

4

4

n PV

The following graphs show that the change in energy density per odd frequency mode is trivial, but the cumulative effect is “g”. It also shows that the energy density per mode increases with frequency.

296

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S ω n PV , R E , M M S ω n PV , R E , M E S ω n PV , R E , M J S ω n PV , R E , M S

n PV Harmonic

The Moon The Earth Jupiter The Sun Poyn. Vec. vs Change in Harm. Freq. Mode

Magnitude of PV Poynting Vector

Magnitude of PV Poynting Vector

Poyn. Vec. vs Change in Harm. Freq. Mode

S ω n PV , R E , M E

n PV Harmonic

297

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Applied Experimental Field – Mode Bandwidth The number of permissible modes fitting within practical benchtop displacement geometries “ N ∆r ” may be defined as follows:

∆r

1 .( mm)

∆r ω Ω ( r, M ) . c

N ∆r( r , M )

N ∆r R E, M M

. 14 6.52135710

N ∆r R E, M E

. 15 1.73310910

N ∆r R E, M J

=

N ∆r R E, M S

. 15 6.23483610 . 16 2.9237310

Chapter 3.5 The behaviour of the EGM construct over a practical laboratory benchtop elemental displacement “ ∆r ” in terms of the PV and ZPF, may be characterised by the following system of equations: ∆ω δr n PV, r , ∆r , M

ω PV n PV, r

∆λ δr n PV, r , ∆r , M ∆λ Ω ( r , ∆r , M )

λ PV n PV, r

c.

ω PV n PV, r , M λ PV n PV, r , M

1 ω Ω ( r, M )

∆r , M )

∆ω δr n PV, r , ∆r , M .∆λ δr n PV, r , ∆r , M

∆v δr n Ω ( r , M ) , r , ∆r , M 3 .M .c . 4 .π 2

∆U PV( r , ∆r , M )

∆K C( r , ∆r , M )

4

∆ω ZPF( r , ∆r , M )

n Ω_ZPF( r , ∆r , M )

1 (r

∆r )

1 3

∆U PV( r , ∆r , M ) .

ω Ω_ZPF ( r , ∆r , M )

KR

∆r , M

1 ω Ω(r

∆v δr n PV, r , ∆r , M

∆v Ω ( r , ∆r , M )

∆r , M

3

r

µ0 ε0

2 .c . ∆U PV( r , ∆r , M ) h 3

ω Ω_ZPF( r , ∆r , M )

∆ω δr( 1 , r , ∆r , M )

4

∆ω δr( 1 , r , ∆r , M )

ω Ω_ZPF( r , ∆r , M ) ω PV( 1, r , M )

0 , 0.0025.. 2

ω β r , ∆r , M , K R n β r , ∆r , M , K R

4

ω Ω_ZPF( r , ∆r , M )

4

4 K R . ω Ω_ZPF( r , ∆r , M )

∆ω δr( 1 , r , ∆r , M )

4

ω β r , ∆r , M , K R ω PV( 1 , r , M )

298

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∆n S r , ∆r , M , K R ∆ω Ω ( r , ∆r , M )

n Ω_ZPF( r , ∆r , M ) ω Ω(r

∆ω Ω ( r , ∆r , M )

∆ω S r , ∆r , M , K R

St β ( r , ∆r , M )

St γ ( r , ∆r , M )

St δ( r , ∆r , M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M ) .

201

ω β r , ∆r , M , K R

µ0 ε0

∆ω ZPF( r , ∆r , M ) ω Ce ∆ω Ω ( r , ∆r , M ) ω Ce n Ω(r

∆r , M )

n Ω ( r, M )

St ε n PV, r , ∆r , M

N

ω Ω ( r, M )

∆ω ZPF( r , ∆r , M )

∆ω R( r , ∆r , M )

St α ( r , ∆r , M )

∆r , M )

n β r , ∆r , M , K R

∆v δr n PV, r , ∆r , M ∆v Ω ( r , ∆r , M ) N, 2

n PV

∆ω δr 1 , R E, ∆r , M M ∆ω δr 1 , R E, ∆r , M E

=

∆ω δr 1 , R E, ∆r , M J

N .. N

1.729554

∆λ δr 1 , R E, ∆r , M M

7.493187

∆λ δr 1 , R E, ∆r , M E

( pHz )

51.128768 519.469801

∆ω δr 1 , R E, ∆r , M S

∆λ δr 1 , R E, ∆r , M J

. 1.33585910

4

∆v δr 1 , R E, ∆r , M M

∆λ Ω R E, ∆r , M E

. 5.02660110

5

∆v δr 1 , R E, ∆r , M E

. 1.39724710

5

∆λ Ω R E, ∆r , M S

. 2.97920610

6

∆v Ω R E, ∆r , M M

13.105112

∆λ Ω R E, ∆r , M J

∆v Ω R E, ∆r , M E ∆v Ω R E, ∆r , M J ∆v Ω R E, ∆r , M S

=

( ym )

∆v δr 1 , R E, ∆r , M J

∆U PV R E, ∆r , M E

13.105115

s

∆U PV R E, ∆r , M J ∆U PV R E, ∆r , M S

299

0.256316

( m)

13.105101 =

13.10513

pm

13.105131

s

13.109717

∆U PV R E, ∆r , M M pm

1.74894 0.025237

∆v δr 1 , R E, ∆r , M S

13.105121 13.109693

=

∆λ δr 1 , R E, ∆r , M S

∆λ Ω R E, ∆r , M M =

7.577156

2.860531 232.617621 =

4 7.3899.10

( GPa)

. 7 7.74094810

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∆K C R E, ∆r , M M

ω Ω_ZPF R E, ∆r , M M

1.077649

∆K C R E, ∆r , M E

87.634109 =

∆K C R E, ∆r , M J

. 4 2.78399910

ω Ω_ZPF R E, ∆r , M E

( MPa .MΩ )

370.868276 =

ω Ω_ZPF R E, ∆r , M J

. 7 2.9162510

∆K C R E, ∆r , M S

123.501066 . 3 1.56573710

ω Ω_ZPF R E, ∆r , M S

. 3 8.90753610

∆ω ZPF R E, ∆r , M M

123.501066

n Ω_ZPF R E, ∆r , M M

. 19 1.49295410

∆ω ZPF R E, ∆r , M E

370.868276

n Ω_ZPF R E, ∆r , M E

. 19 1.03481710

=

∆ω ZPF R E, ∆r , M J

( PHz)

. 3 1.56573710 . 3 8.90753610

∆ω ZPF R E, ∆r , M S

=

n Ω_ZPF R E, ∆r , M J

( PHz)

. 18 6.40270810

n Ω_ZPF R E, ∆r , M S

. 18 3.5857810

n β R E, ∆r , M M , K R2

. 15 1.78829110

KR2 = 99.99999999999999(%) ω β R E, ∆r , M M , K R2

14.793206

ω β R E, ∆r , M E, K R2

=

ω β R E, ∆r , M J , K R2

41.841506

n β R E, ∆r , M E, K R2

( THz)

167.366022

n β R E, ∆r , M J , K R2

946.765196

ω β R E, ∆r , M S , K R2

. 19 1.49277510

∆ω Ω R E, ∆r , M M

∆n S R E, ∆r , M E, K R2

19 1.0347.10

∆ω Ω R E, ∆r , M E

∆n S R E, ∆r , M J , K R2 ∆n S R E, ∆r , M S , K R2 ∆ω R R E, ∆r , M M ∆ω R R E, ∆r , M E

=

∆ω R R E, ∆r , M J ∆ω R R E, ∆r , M S St α R E, ∆r , M M St α R E, ∆r , M E St α R E, ∆r , M J

18

. 6.40202410

∆ω Ω R E, ∆r , M J

. 18 3.58539910

∆ω Ω R E, ∆r , M S

9.615565

∆ω S R E, ∆r , M J , K R2

11.66707

∆ω S R E, ∆r , M S , K R2 St β R E, ∆r , M M

. 2.78399910

4

St β R E, ∆r , M E

( MPa .MΩ )

St β R E, ∆r , M J

7

St β R E, ∆r , M S

. 2.19383110

5

St δ R E, ∆r , M M

St γ R E, ∆r , M E

. 5.83032610

5

St δ R E, ∆r , M E

St γ R E, ∆r , M S

. 2.0974410 . 9.83425710

St δ R E, ∆r , M J

4

St δ R E, ∆r , M S

4

300

45.263389 162.833549

( PHz)

763.476685

∆ω S R E, ∆r , M E, K R2

St γ R E, ∆r , M M

St γ R E, ∆r , M J

=

8.19356

. 2.9162510

=

17.031676

∆ω S R E, ∆r , M M , K R2

87.634109

St α R E, ∆r , M S

. 14 3.81125810

7.251258

1.077649 =

. 14 6.84403710

n β R E, ∆r , M S , K R2

∆n S R E, ∆r , M M , K R2 =

. 15 1.16748410

=

=

123.486273 370.826434 =

. 3 1.56556910

( PHz)

. 3 8.90658910

. 1.59080310

4

. 4.77711210

4

. 2.01680710

3

0.011474

1 =

1 1 1

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St ε 1 , R E, ∆r , M M St ε 1 , R E, ∆r , M E

=

St ε 1 , R E, ∆r , M J St ε 1 , R E, ∆r , M S 2.

e

G .M M . 1 2 R E .c

1.

0.999999

St ε n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

1.000001

St ε n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E

1.000001

St ε n Ω_ZPF R E, ∆r , M J , R E, ∆r , M J

1.000002

St ε n Ω_ZPF R E, ∆r , M S , R E, ∆r , M S 2.

2

2

=1

e

G .M E . 1 2 R E .c

1.

1.000001 =

1 1.000003 1

2

2

=1

Hence:

K EGM e

2.

e

G .M J . 1 2 R E .c

G .M . 1 2 r .c

1. 2

1. 2

GSE 3 3.

K PV( r , M ) e

2.

2

= 1.000001

e

∆K 0( r , M , ω , X )

e

∆K 0( r , M , ω , X )

K 0( r , M , X )

G .M S . 1 2 R E .c

1. 2

2

= 1.000927

Critical Boundary 50 .%

100 .%

ω β R E , ∆r , M J , 50 .%

Re ω β R E , ∆r , M M , K R Critical Boundary

2.

ω β R E , ∆r , M E , 50 .%

Re ω β R E , ∆r , M E , K R Re ω β R E , ∆r , M J , K R Re ω β R E , ∆r , M S , K R

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

301

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Critical Boundary 100 .%

150 .%

ω β R E , ∆r , M J , 150 .%

Critical Boundary

Im ω β R E , ∆r , M M , K R Im ω β R E , ∆r , M E , K R ω β R E , ∆r , M E , 150 .%

Im ω β R E , ∆r , M J , K R Im ω β R E , ∆r , M S , K R

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

302

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Critical Boundary 50 .%

100 .%

ω β R E , ∆r , M J , 50 .%

Critical Boundary

ω β R E , ∆r , M M , K R ω β R E , ∆r , M E , 50 .%

ω β R E , ∆r , M E , K R ω β R E , ∆r , M J , K R ω β R E , ∆r , M S , K R

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

303

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Similarity Bandwidth 50 .%

100 .% ∆ω S R E , ∆r , M J , 50 .%

Re ∆ω S R E , ∆r , M M , K R Re ∆ω S R E , ∆r , M E , K R

∆ω S R E , ∆r , M E , 50 .%

Similarity Bandwidth

Re ∆ω S R E , ∆r , M J , K R Re ∆ω S R E , ∆r , M S , K R Im ∆ω S R E , ∆r , M M , K R

0

0.5

1

1.5

2

Im ∆ω S R E , ∆r , M E , K R Im ∆ω S R E , ∆r , M J , K R Im ∆ω S R E , ∆r , M S , K R

KR Critical Ratio

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

304

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Similarity Bandwidth 50 .%

100 .%

∆ω S R E , ∆r , M J , 50 .%

Similarity Bandwidth

∆ω S R E , ∆r , M M , K R ∆ω S R E , ∆r , M E , K R ∆ω S R E , ∆r , M J , K R ∆ω S R E , ∆r , M S , K R

∆ω S R E , ∆r , M E , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

305

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Mode Number 50 .%

100 .%

n β R E , ∆r , M E , 50 .%

Mode Number

Re n β R E , ∆r , M M , K R Re n β R E , ∆r , M E , K R Re n β R E , ∆r , M J , K R Re n β R E , ∆r , M S , K R

n β R E , ∆r , M J , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

306

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Mode Number 100 .%

Im n β R E , ∆r , M E , 150 .%

Im n β R E , ∆r , M M , K R Mode Number

150 .%

Im n β R E , ∆r , M E , K R Im n β R E , ∆r , M J , K R Im n β R E , ∆r , M S , K R

Im n β R E , ∆r , M J , 150 .%

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

307

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Mode Number 50 .%

100 .%

n β R E , ∆r , M E , 50 .%

Mode Number

n β R E , ∆r , M M , K R n β R E , ∆r , M E , K R n β R E , ∆r , M J , K R n β R E , ∆r , M S , K R

n β R E , ∆r , M J , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

308

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Mode Number 50 .%

100 .%

Re ∆n S R E , ∆r , M M , K R Re ∆n S R E , ∆r , M E , K R ∆n S R E , ∆r , M E , 50 .% ∆n S R E , ∆r , M J , 50 .%

Mode Number

Re ∆n S R E , ∆r , M J , K R Re ∆n S R E , ∆r , M S , K R Im ∆n S R E , ∆r , M M , K R

0

0.5

1

1.5

2

Im ∆n S R E , ∆r , M E , K R Im ∆n S R E , ∆r , M J , K R Im ∆n S R E , ∆r , M S , K R

KR Critical Ratio

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

309

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Change in Mode Number

Change in Mode Number

50 .%

100 .%

∆n S R E , ∆r , M M , K R ∆n S R E , ∆r , M E , K R ∆n S R E , ∆r , M J , K R ∆n S R E , ∆r , M S , K R

∆n S R E , ∆r , M E , 50 .% ∆n S R E , ∆r , M J , 50 .%

0

0.5

1

1.5

2

KR Critical Ratio

The Moon The Earth Jupiter The Sun

310

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Sense Checks RE

St β r , ∆r , M M St β r , ∆r , M E

Sense Check

St β r , ∆r , M J St β r , ∆r , M S St γ r , ∆r , M M

St β R E , ∆r , M E

St γ r , ∆r , M E St γ r , ∆r , M J St γ r , ∆r , M S

St γ R E , ∆r , M E

r Radial Distance

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

311

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Range Factor RM

RE

Range Factor

St α r , ∆r , M M St α r , ∆r , M E St α r , ∆r , M J St α r , ∆r , M S

St α R E , ∆r , M E

St α R E , ∆r , M M

r Radial Distance

The Moon The Earth Jupiter The Sun

312

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Sense Check

Sense Check

N

N

St ε n PV , R E , ∆r , M M

n PV Harmonic

Sense Check

Sense Check

N

N

St ε n PV , R E , ∆r , M E

n PV Harmonic

Sense Check

Sense Check

N

N

St ε n PV , R E , ∆r , M J

n PV Harmonic

313

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Sense Check

Sense Check

N

N

St ε n PV , R E , ∆r , M S

n PV Harmonic

∆r min

0 10 .( mm)

∆r max

2 10 .( m)

∆r

∆r min,

∆r max .. ∆r max 100

Bandwidth Ratio

Bandwidth Ratio

∆ω R R E , ∆r , M M ∆ω R R E , ∆r , M E ∆ω R R E , ∆r , M J ∆ω R R E , ∆r , M S

∆r Change in Radial Displacement

The Moon The Earth Jupiter The Sun

Chapter 3.6 Representation of “g” (in harmonic form “ g ∆r ”) over a practical laboratory benchtop elemental displacement “ ∆r ” in terms of the PV and ZPF, may be characterised by the following system of approximations: 314

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N

t

21

0 .( s ) ,

N, 2

n PV

T δr 1 , R E, ∆r , M E 200

K R( r , ∆r , M , t )

2.

i .

π

n PV

1 .( mm)

∆r

N .. N

T δr n PV, r , ∆r , M

1 ∆ω δr n PV, r , ∆r , M

.. 4 .T δr 1 , R E, ∆r , M E 1 . e n PV

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

g ∆r( r , ∆r , M , t )

G.M . K R( r , ∆r , M , t ) 2 r

Unit Fourier Spectrum 1.5

T δ_r

1 Ideal Critical Ratio

1

K R R E , ∆r , M E , t

0.5

t Time

Harmonic Representation of "g" T δ_r

g

Acceleration

10

g ∆r R E , ∆r , M E , t

5

t Time

Harmonic Similarity Equations Utilising unit magnitude values where appropriate for illustrational purposes, the Harmonic Similarity Equations may be visualised as follows: E0 EE B0 BB n E n B φ

1.

V m

ω E n E, r , ∆r , M

n E.∆ω δr( 1 , r , ∆r , M )

ω B n B, r , ∆r , M

n B.∆ω δr( 1 , r , ∆r , M )

0.

V

1 .( T ) 0 .( T ) 1 1 0 .( deg )

m

315

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Let the EM forcing functions “EA” and “BA” be represented as Complex Numbers in Phasor Form: (http://mathworld.wolfram.com/ComplexNumber.html) – Where “E0” and “B0” are magnitudes of the Electric Field Intensity and Magnetic Flux Density amplitudes respectively. E A E 0 , n E, r , ∆r , M , t

B A B 0 , n B , φ, r , ∆r , M , t

E 0 .e

2 .π .ω E n E , r , ∆r , M .t

B 0 .e

π . i 2

2 .π .ω B n B , r , ∆r , M .t

π

E rms

1 . E0 2

B rms

1 . B0 2

φ .i

2

The Phasor form has Real and Complex components, as graphically illustrated below:

Electric Field

Re E A 1 .

V m

, 1 , R E , ∆r , M E , t

V Im E A 1 . , 1 , R E , ∆r , M E , t m E rms

t Time

Magnetic Field

Electric Field

Re B A 1 .( T ) , 1 , 90 .( deg ) , R E , ∆r , M E , t Im B A 1 .( T ) , 1 , 90 .( deg ) , R E , ∆r , M E , t B rms

t Time

Magnetic Field

Visualisation of Harmonic Similarity Equation Operands HSE 1 E 0 , B 0 , n E, n B , φ, n PV, r , ∆r , M , t

HSE 2 E 0 , B 0 , n E, n B, φ, n PV, r , ∆r , M , t

i . E A E 0 , n E, r , ∆r , M , t

2

2 c .B A B 0 , n B , φ, r , ∆r , M , t

2

2 π .n PV.c .B A B 0 , n B , φ, r , ∆r , M , t

i . E A E 0 , n E, r , ∆r , M , t

2

2 c .B A B 0 , n B, φ, r , ∆r , M , t

2

2 π .n PV.c .B A B 0 , n B, φ, r , ∆r , M , t

316

.e

.e

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2

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π .n

.∆ω

( 1 , r , ∆r , M ) .t .i

PV δr 2 .i .K PV( r , M ) .St α ( r , ∆r , M ) .e π .n PV.E A E 0 , n E, r , ∆r , M , t .B A B 0 , n B, φ, r , ∆r , M , t

HSE 3 E 0 , B 0 , n E, n B, φ, n PV, r , ∆r , M , t

2 4 .i .St α ( r , ∆r , M ) .K PV( r , M ) .c .B A B 0 , n B, φ, r , ∆r , M , t .e

HSE 4 E 0 , B 0, n E, n B, φ, n PV, r , ∆r , M , t

π .n PV.E A E 0 , n E, r , ∆r , M , t . E A E 0 , n E, r , ∆r , M , t

2

2 c .B A B 0 , n B, φ, r , ∆r , M , t

2 4 .i .St α ( r , ∆r , M ) .K PV( r , M ) .c .B A B 0 , n B, φ, r , ∆r , M , t .e

HSE 5 E 0 , B 0, n E, n B, φ, n PV, r , ∆r , M , t

π .n PV.E A E 0 , n E, r , ∆r , M , t . E A E 0 , n E, r , ∆r , M , t

2

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i 2

π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i

2 c .B A B 0 , n B, φ, r , ∆r , M , t

2

T δr 1 , R E , ∆r , M E 2

Re E A 1 .

V m

, 1 , R E , ∆r , M E , t

Re B A 1 .( T ) , 1 , 180 .( deg ) , R E , ∆r , M E , t Re HSE 1 1 .

V m

, 1 .( T ) , 1 , 1 , 180 .( deg ) , 3 , R E , ∆r , M E , t

V Re HSE 2 1 . , 1 .( T ) , 1 , 1 , 180 .( deg ) , 3 , R E , ∆r , M E , t m

t

Electric Forcing Function Magnetic Forcing Function HSE 1 HSE 2

T δr 1 , R E , ∆r , M E

2 .T δr 1 , R E , ∆r , M E

V Re HSE 3 1 . , 1 .( T ) , 1 , 1 , φ , 3 , R E , ∆r , M E , t m V Re HSE 4 1 . , 1 .( T ) , 1 , 1 , φ , 3 , R E , ∆r , M E , t m V Re HSE 5 1 . , 1 .( T ) , 1 , 1 , φ , 3 , R E , ∆r , M E , t m

t

HSE 3 HSE 4 HSE 5

The preceding graph indicates that “HSE4” may be considered to be the “Constructive Form” whilst “HSE5” may be considered to be the “Destructive Form”. Reduction of the Harmonic Similarity Equations N

201

HSE 1_R φ, n PV

n PV

N, 2

N .. N

2 .( cos ( 2 .φ) π .n PV

1)

φ

0 .( deg ) , 2 .( deg ) .. 360 ( deg )

HSE 2_R φ, n PV

317

∆r x

2 .( cos ( 2 .φ) π .n PV

1)

10.( cm) 0.01.( mm) , .. 10.( cm) 2 10

n EM

nE

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E rms c

Harmonic Similarity

B rms

ω EM n EM

n EM.( Hz)

HSE 3_R E rms , B rms , n PV, r , ∆r , M

K PV( r , M ) .St α ( r , ∆r , M ) π .n PV.E rms .B rms

HSE 3_R E rms , B rms , n PV , R E , ∆r , M M HSE 3_R E rms , B rms , n PV , R E , ∆r , M E HSE 3_R E rms , B rms , n PV , R E , ∆r , M J HSE 3_R E rms , B rms , n PV , R E , ∆r , M S

n PV Harmonic Mode

The Moon The Earth Jupiter The Sun HSE 4 E rms , B rms , φ , n PV, r , ∆r , M

HSE 5 E rms , B rms , φ , n PV, r , ∆r , M

1 R

cos ( φ ) 1

R

sin ( φ )

.HSE E 3 rms , B rms , n PV, r , ∆r , M

.HSE E 3 rms , B rms , n PV, r , ∆r , M

318

R

R

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π

π

Harmonic Similarity

2

HSE 1_R ( φ , 1 ) HSE 1_R ( φ , 2 ) HSE 2_R ( φ , 1 ) HSE 2_R ( φ , 2 )

φ Phase Variance

π 2

Harmonic Similarity π

HSE 4_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M M , R E , ∆r , M M HSE 4_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M E , R E , ∆r , M E

Harmonic Similarity

HSE 4_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M J , R E , ∆r , M J HSE 4_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M S , R E , ∆r , M S HSE 5_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M M , R E , ∆r , M M HSE 5_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M E , R E , ∆r , M E HSE 5_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M J , R E , ∆r , M J HSE 5_R E rms , B rms , n EM , φ , n Ω _ZPF R E , ∆r , M S , R E , ∆r , M S

φ Phase Variance

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

319

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Harmonic Similarity

HSE 4_R E rms , B rms , n EM , 0 , n PV , R E , ∆r , M M HSE 4_R E rms , B rms , n EM , 0 , n PV , R E , ∆r , M E

Harmonic Similarity

HSE 4_R E rms , B rms , n EM , 0 , n PV , R E , ∆r , M J HSE 4_R E rms , B rms , n EM , 0 , n PV , R E , ∆r , M S HSE 5_R E rms , B rms , n EM , HSE 5_R E rms , B rms , n EM ,

π 4 π 4

, n PV , R E , ∆r , M M , n PV , R E , ∆r , M E

π HSE 5_R E rms , B rms , n EM , , n PV , R E , ∆r , M J 4 HSE 5_R E rms , B rms , n EM ,

π 4

, n PV , R E , ∆r , M S

n PV Harmonic

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

Note: the Phase Variance has been set to enable graphical distinction between curves and at ideal conditions, graphical overlap occurs.

320

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Spectral Similarity Equations SSE 1( φ, r , ∆r , M )

. 1 ) . ln 2 n Ω_ZPF( r , ∆r , M )

2 .( cos ( 2 .φ) π

n Ω_ZPF( r , ∆r , M )

π

SSE 3 E rms, B rms, r , ∆r , M

SSE 4 E rms , B rms , φ, r , ∆r , M

SSE 5 E rms , B rms , φ, r , ∆r , M

1

. 1 ) . ln 2 n Ω_ZPF( r , ∆r , M )

2 .( cos ( 2 .φ)

SSE 2( φ, r , ∆r , M )

γ

n Ω_ZPF( r , ∆r , M )

γ 1

K PV( r , M ) . St α ( r , ∆r , M ) ln 2.n Ω_ZPF( r , ∆r , M ) γ . π .E rms.B rms n Ω_ZPF( r , ∆r , M ) 1 1 cos ( φ) 1 sin ( φ)

.SSE E 3 rms , B rms , r , ∆r , M

.SSE E 3 rms , B rms , r , ∆r , M

SSE 4 E rms , B rms , 0 , R E, ∆r , M M

SSE 4 E rms , B rms , 0 , R E, ∆r , M J

π SSE 4 E rms , B rms , , R E, ∆r , M M 4

π SSE 4 E rms , B rms , , R E, ∆r , M J 4

π SSE 5 E rms , B rms , , R E, ∆r , M M 4

π SSE 5 E rms , B rms , , R E, ∆r , M J 4

π

625.721384

. 7 884.903667 5.23117610 . 7 884.903667 5.23117610

π

SSE 5 E rms , B rms , , R E, ∆r , M M 2

SSE 5 E rms , B rms , , R E, ∆r , M J 2

SSE 4 E rms , B rms , 0 , R E, ∆r , M E

SSE 4 E rms , B rms , 0 , R E, ∆r , M S

π SSE 4 E rms , B rms , , R E, ∆r , M E 4

π SSE 4 E rms , B rms , , R E, ∆r , M S 4

π SSE 5 E rms , B rms , , R E, ∆r , M E 4

π SSE 5 E rms , B rms , , R E, ∆r , M S 4

π SSE 5 E rms , B rms , , R E, ∆r , M E 2

π SSE 5 E rms , B rms , , R E, ∆r , M S 2

321

7 3.699.10

=

625.721384

7 3.699.10

. 4 6.83180210 . 10 7.28183810 . 5 9.66162710 . 10 1.02980710 . 5 9.66162710 . 10 1.02980710 . 4 6.83180210 . 10 7.28183810

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π

π

4

2

Spectral Similarity

SSE 1 φ , R E , ∆r , M M SSE 1 φ , R E , ∆r , M E

Spectral Similarity

SSE 1 φ , R E , ∆r , M J SSE 1 φ , R E , ∆r , M S SSE 2 φ , R E , ∆r , M M SSE 2 φ , R E , ∆r , M E SSE 2 φ , R E , ∆r , M J SSE 2 φ , R E , ∆r , M S

φ Phase Variance

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun Spectral Similarity

Spectral Similarity

SSE 3 E rms , B rms , R E , ∆r x , M M SSE 3 E rms , B rms , R E , ∆r x , M E SSE 3 E rms , B rms , R E , ∆r x , M J SSE 3 E rms , B rms , R E , ∆r x , M S

∆r x Change in Radial Displacement

The Moon The Earth Jupiter The Sun

322

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Spectral Similarity

SSE 1 0 , R E , ∆r x , M M SSE 1 0 , R E , ∆r x , M E SSE 1 0 , R E , ∆r x , M J

Spectral Similarity

SSE 1 0 , R E , ∆r x , M S π SSE 2 , R E , ∆r x , M M 16 π SSE 2 , R E , ∆r x , M E 16 π SSE 2 , R E , ∆r x , M J 16 SSE 2

π 16

, R E , ∆r x , M S

∆r x Change in Radial Displacement

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

Note: the Phase Variance has been set to enable graphical distinction between curves and at ideal conditions, graphical overlap occurs. Spectral Similarity .

π

3π

4

4

SSE 4 E rms , B rms , φ , R E , ∆r , M M SSE 4 E rms , B rms , φ , R E , ∆r , M E

Spectral Similarity

SSE 4 E rms , B rms , φ , R E , ∆r , M J SSE 4 E rms , B rms , φ , R E , ∆r , M S SSE 5 E rms , B rms , φ , R E , ∆r , M M SSE 5 E rms , B rms , φ , R E , ∆r , M E SSE 5 E rms , B rms , φ , R E , ∆r , M J SSE 5 E rms , B rms , φ , R E , ∆r , M S

φ Phase Variance

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

323

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Spectral Similarity

SSE 4 E rms , B rms , 0 , R E , ∆r x , M M SSE 4 E rms , B rms , 0 , R E , ∆r x , M E SSE 4 E rms , B rms , 0 , R E , ∆r x , M J

Spectral Similarity

SSE 4 E rms , B rms , 0 , R E , ∆r x , M S SSE 5 E rms , B rms ,

π 4

, R E , ∆r x , M M

π SSE 5 E rms , B rms , , R E , ∆r x , M E 4 π SSE 5 E rms , B rms , , R E , ∆r x , M J 4 π SSE 5 E rms , B rms , , R E , ∆r x , M S 4

∆r x Change in Radial Displacement

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

Note: the Phase Variance has been set to enable graphical distinction between curves and at ideal conditions, graphical overlap occurs. Phase Variance Required for Optimal Similarity Conditions φ 4C_H E rms , B rms, n PV, r , ∆r , M

Re acos HSE 3_R E rms, B rms , n PV, r , ∆r , M

φ 5C_H E rms, B rms, n PV, r , ∆r , M

Re asin HSE 3_R E rms , B rms, n PV, r , ∆r , M

φ 1C_S( r , ∆r , M )

. 1 1 π n Ω_ZPF ( r , ∆r , M ) 1 Re .acos . 2 2 ln 2 .n Ω_ZPF ( r , ∆r , M ) γ

φ 2C_S( r , ∆r , M )

1 Re .acos 2

. 1 . π n Ω_ZPF ( r , ∆r , M ) 1 2 ln 2 .n Ω_ZPF ( r , ∆r , M ) γ

324

2

1

2

1

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Phase Variance π

Phase Variance

2 φ 1C_S R E , ∆r x , M E

0

φ 2C_S R E , ∆r x , M E

∆r x Change in Radial Displacement

φ 4C_S E rms, B rms, r , ∆r , M

K PV( r , M ) . St α ( r , ∆r , M ) . ln 2.n Ω_ZPF( r , ∆r , M ) π .E rms.B rms. n Ω_ZPF( r , ∆r , M ) 1

γ

Re acos

φ 5C_S E rms, B rms, r , ∆r , M

K PV( r , M ) . St α ( r , ∆r , M ) . ln 2 .n Ω_ZPF( r , ∆r , M ) π .E rms.B rms. n Ω_ZPF( r , ∆r , M ) 1

γ

Re asin

Phase Variance

Phase Variance

φ 4C_H E rms , B rms , n PV , R E , ∆r , M E

π

φ 5C_H E rms , B rms , n PV , R E , ∆r , M E π 2

n PV Harmonic

Phase Variance π

Phase Variance

2 φ 4C_S E rms , B rms , R E , ∆r x , M E

0

φ 5C_S E rms , B rms , R E , ∆r x , M E

∆r x Change in Radial Displacement

325

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φ 4C_H E rms , B rms , 1 , R E, ∆r , M M

φ 4C_H E rms , B rms , 1 , R E, ∆r , M J

φ 4C_H E rms , B rms , n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

φ 4C_H E rms , B rms , n Ω_ZPF R E, ∆r , M J , R E, ∆r , M J

φ 5C_H E rms , B rms , 1 , R E, ∆r , M M

φ 5C_H E rms , B rms , 1 , R E, ∆r , M J

φ 5C_H E rms , B rms , n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

φ 5C_H E rms , B rms , n Ω_ZPF R E, ∆r , M J , R E, ∆r , M J

90

90

φ 1C_S R E, ∆r , M M

φ 1C_S R E, ∆r , M J

90

90

φ 2C_S R E, ∆r , M M

φ 2C_S R E, ∆r , M J

φ 4C_S E rms , B rms , R E, ∆r , M M

φ 4C_S E rms , B rms , R E, ∆r , M J

φ 5C_S E rms , B rms , R E, ∆r , M M

φ 5C_S E rms , B rms , R E, ∆r , M J

φ 4C_H E rms , B rms , 1 , R E, ∆r , M E

φ 4C_H E rms , B rms , 1 , R E, ∆r , M S

φ 4C_H E rms , B rms , n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E

φ 4C_H E rms , B rms , n Ω_ZPF R E, ∆r , M S , R E, ∆r , M S

φ 5C_H E rms , B rms , 1 , R E, ∆r , M E

φ 5C_H E rms , B rms , 1 , R E, ∆r , M S

φ 5C_H E rms , B rms , n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E φ 1C_S R E, ∆r , M E

180 180 180 180

=

0

0

90

90

0

0

90

90

180 180

( deg )

180 180 90

90

90

90

φ 5C_H E rms , B rms , n Ω_ZPF R E, ∆r , M S , R E, ∆r , M S

0

0

φ 1C_S R E, ∆r , M S

90

90

φ 2C_S R E, ∆r , M E

φ 2C_S R E, ∆r , M S

0

0

90

90

φ 4C_S E rms , B rms , R E, ∆r , M E

φ 4C_S E rms , B rms , R E, ∆r , M S

φ 5C_S E rms , B rms , R E, ∆r , M E

φ 5C_S E rms , B rms , R E, ∆r , M S

Critical Field Strengths Given SSE 4 E rms,

SSE 5 E rms,

E rms

E rms c E rms c

, 0.( deg ) , R E, ∆r , M E

, 90.( deg ) , R E, ∆r , M E E rms = 190.811924

Find E rms

E rms

B rms

1

1 V m

B rms = 6.364801 ( mgs )

c

DC-Offsets SSE 4 ( 1

DC

100.( % )

DC) .E rms , B rms , 0 .( deg ) , R E, ∆r , M E

SSE 5 E rms , ( 1

0.5

DC) .B rms , 90.( deg ) , R E, ∆r , M E

SSE 4 ( 1

DC) .E rms , ( 1

DC) .B rms , 0 .( deg ) , R E, ∆r , M E

SSE 5 ( 1

DC) .E rms , ( 1

DC) .B rms , 90.( deg ) , R E, ∆r , M E

=

0.5 0.25 0.25

Critical Frequency ω C( ∆r )

c . 2 ∆r

ω C( ∆r ) = 149.896229( GHz)

λ C( ∆r )

326

c ω C( ∆r )

λ C( ∆r ) = 2 ( mm)

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Chapter 3.7 Graphical Representation, Analysis and Optimal Conditions of Similarity φ 4 E rms , B rms, n PV, r , ∆r , M

φ 4C_H E rms, B rms , n PV, r , ∆r , M

φ 5 E rms, B rms, n PV, r , ∆r , M

φ 5C_H E rms, B rms, n PV, r , ∆r , M

N max

n Ω_ZPF R M , ∆r , M M

N X( r , ∆r , M )

B C( r , ∆r , M )

λ X( r , ∆r , M )

n PV

n Ω_ZPF( r , ∆r , M )

1

ln 2.n Ω_ZPF( r , ∆r , M ) E C( r , ∆r , M ) c c ω X( r , ∆r , M )

γ

ω X( r , ∆r , M )

N C( r , ∆r , M )

1,

1 . N max 1 .. N max 500

E C( r , ∆r , M )

c .K PV( r , M ) . St α ( r , ∆r , M ) π .N X( r , ∆r , M )

N X( r , ∆r , M ) .ω PV( 1 , r , M ) ω C( ∆r ) ω PV( 1 , r , M )

ω C( ∆r ) = 149.896229( GHz)

N X R M , ∆r , M M

. 17 2.15162910

E C R M , ∆r , M M

N X R E, ∆r , M E

. 17 2.29685210

E C R E, ∆r , M E

17

. 3.15778710

E C R J , ∆r , M J

. 17 3.76223110

E C R S , ∆r , M S

N X R J , ∆r , M J

=

N X R S , ∆r , M S B C R M , ∆r , M M B C R E, ∆r , M E B C R J , ∆r , M J

=

λ X R M , ∆r , M M

λ X R J , ∆r , M J λ X R S , ∆r , M S

ω X R M , ∆r , M M

6.364801

ω X R E, ∆r , M E

0.76984

( mgs )

0.240852

B C R S , ∆r , M S

λ X R E, ∆r , M E

9.8181

36.419294 97.406507 167.343325

=

N C R E, ∆r , M E N C R J , ∆r , M J N C R S , ∆r , M S

327

V

23.079214

m

10.073108 =

8.231693 3.077746

( PHz)

1.791481

N C R M , ∆r , M M ( nm )

190.811924 7.220558

ω X R S , ∆r , M S

29.761666 =

ω X R J , ∆r , M J

294.339224

. 12 3.20180310 =

. 12 4.18248610 . 13 1.53794510 . 13 3.14792110

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π

Phase Variance N C R E , ∆r , M E

N X R E , ∆r , M E

φ 4 E C R M , ∆r , M M , B C R M , ∆r , M M , n PV , R M , ∆r , M M φ 4 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E

Phase Variance

φ 4 E C R J , ∆r , M J , B C R J , ∆r , M J , n PV , R J , ∆r , M J π

φ 4 E C R S , ∆r , M S , B C R S , ∆r , M S , n PV , R S , ∆r , M S

2

φ 5 E C R M , ∆r , M M , B C R M , ∆r , M M , n PV , R M , ∆r , M M φ 5 E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E φ 5 E C R J , ∆r , M J , B C R J , ∆r , M J , n PV , R J , ∆r , M J φ 5 E C R S , ∆r , M S , B C R S , ∆r , M S , n PV , R S , ∆r , M S

n PV Harmonic

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun Harmonic Similarity N X R E , ∆r , M E Im acos HSE 3_R E C R M , ∆r , M M , B C R M , ∆r , M M , n PV , R M , ∆r , M M Im acos HSE 3_R E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E Im acos HSE 3_R E C R J , ∆r , M J , B C R J , ∆r , M J , n PV , R J , ∆r , M J π

Im acos HSE 3_R E C R S , ∆r , M S , B C R S , ∆r , M S , n PV , R S , ∆r , M S

Harmonic Similarity

Re acos HSE 3_R E C R M , ∆r , M M , B C R M , ∆r , M M , n PV , R M , ∆r , M M Re acos HSE 3_R E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E Re acos HSE 3_R E C R J , ∆r , M J , B C R J , ∆r , M J , n PV , R J , ∆r , M J Re acos HSE 3_R E C R S , ∆r , M S , B C R S , ∆r , M S , n PV , R S , ∆r , M S

π

acos HSE 3_R E C R M , ∆r , M M , B C R M , ∆r , M M , n PV , R M , ∆r , M M

2

acos HSE 3_R E C R E , ∆r , M E , B C R E , ∆r , M E , n PV , R E , ∆r , M E acos HSE 3_R E C R J , ∆r , M J , B C R J , ∆r , M J , n PV , R J , ∆r , M J acos HSE 3_R E C R S , ∆r , M S , B C R S , ∆r , M S , n PV , R S , ∆r , M S

n PV Harmonic

The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun The Moon The Earth Jupiter The Sun

π

Spectral Similarity

2

Spectral Similarity π

SSE 4 E C R E , ∆r , M E , B C R E , ∆r , M E , φ , R E , ∆r , M E SSE 5 E C R E , ∆r , M E , B C R E , ∆r , M E , φ , R E , ∆r , M E

1

φ Phase Variance

328

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φ 4 E C R M , ∆r , M M , B C R M , ∆r , M M , N X R M , ∆r , M M , R M , ∆r , M M 180

φ 4 E C R E, ∆r , M E , B C R E, ∆r , M E , N X R E, ∆r , M E , R E, ∆r , M E

180

φ 4 E C R J , ∆r , M J , B C R J , ∆r , M J , N X R J , ∆r , M J , R J , ∆r , M J

180

φ 4 E C R S , ∆r , M S , B C R S , ∆r , M S , N X R S , ∆r , M S , R S , ∆r , M S φ 5 E C R M , ∆r , M M , B C R M , ∆r , M M , N X R M , ∆r , M M , R M , ∆r , M M

=

180 90

( deg )

90

φ 5 E C R E, ∆r , M E , B C R E, ∆r , M E , N X R E, ∆r , M E , R E, ∆r , M E

90

φ 5 E C R J , ∆r , M J , B C R J , ∆r , M J , N X R J , ∆r , M J , R J , ∆r , M J

90

φ 5 E C R S , ∆r , M S , B C R S , ∆r , M S , N X R S , ∆r , M S , R S , ∆r , M S SSE 4 E C R M , ∆r , M M , B C R M , ∆r , M M , 0 , R M , ∆r , M M

π SSE 5 E C R M , ∆r , M M , B C R M , ∆r , M M , , R M , ∆r , M M 2

SSE 4 E C R E, ∆r , M E , B C R E, ∆r , M E , 0 , R E, ∆r , M E

π SSE 5 E C R E, ∆r , M E , B C R E, ∆r , M E , , R E, ∆r , M E 2 π

SSE 4 E C R J , ∆r , M J , B C R J , ∆r , M J , 0 , R J , ∆r , M J

SSE 5 E C R J , ∆r , M J , B C R J , ∆r , M J , , R J , ∆r , M J 2

1 1 =

1 1 1 1 1 1

π

SSE 4 E C R S , ∆r , M S , B C R S , ∆r , M S , 0 , R S , ∆r , M S

SSE 5 E C R S , ∆r , M S , B C R S , ∆r , M S , , R S , ∆r , M S 2

Mathematical Approximations, Limits, Patterns and Series N N

6

10

Σ1

1

1

ln( 2 ) n PV = 1

N, 2

n PV

1

Σ2

N .. N

n PV

Σ Error

Σ2

n PV

. Σ Error = 3.31435710

1

Σ1

n PV

6

Σ 1 = 15.085875

ln( 2 .N )

Σ 2 = 15.085875

Σ2

γ = 15.085874

. Σ 1 = 4.99999810

7

(%)

N 1

ln( 2 ) n PV = 1 ln( 2 .N )

1

n PV

15.085875 = 15.085874

γ

1 n PV = 1

n Ω ( r, M ) 1 n PV

. 6 ( %) 1 = 3.31435710

ln( 2 )

n PV

Hence:

n PV N

15.085875

1 n PV

n PV

n PV

n PV

ZPF

1

ln( 2 ) n PV = 1

n PV

ln 2 .n Ω ( r , M ) ZPF

γ

Note: for large mode numbers, the error is trivial.

329

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N 1

1

ln( 2 )

n PV

n PV

n PV = 1

1 , 3 .. N

n PV

ln( 2 .N )

γ

n PV

Considering only one side of the spectrum,

1 n PV

n PV

. 1 = 6.62871210

1. ( ln( 2 .N ) 2

1 1

n PV 1

N 1

1

N

γ

n PV

= 1.508584.10

1

n PV 6

1

(%) 1

γ)

1 1

ln( 2 .N )

n PV

N

n PV

n PV

n PV

.( ln( 2 .N )

N

Average,

1

.

N . = 6.62870610

5

1

1

.

N

ln( 2 .N )

5

n PV

.( ln( 2 .N )

N

Error,

Considering both sides of the spectrum,

. = 1.50858510

. 1

1

1

N .. N

1

.

N

(%)

γ)

N, 2

n PV

6

. 1 = 6.62870610

6

(%)

γ)

1

γ

1

It can be numerically proven that the average function is: 1 n Ω ( r, M )

•

. 1 ZPF

n PV

1

ln 2 .n Ω ( r , M ) ZPF n Ω ( r, M )

n PV

γ 1

ZPF

The LHS of the equation includes the odd modes over the entire spectrum from left to right. The RHS of the equation includes all modes (odd and even).

•

Hence: 1. 2

ln 2 .N X R M , ∆r , M M

γ

ln 2 .N C R M , ∆r , M M

ln 2 .N X R E, ∆r , M E

γ

ln 2 .N C R E, ∆r , M E

1. 2 1. 2 1. 2

ln 2 .N X R J , ∆r , M J

γ

ln 2 .N C R J , ∆r , M J

ln 2 .N X R S , ∆r , M S

γ

ln 2 .N C R S , ∆r , M S

330

γ

γ

γ

γ

1 . N X R M , ∆r , M M ln 2 N C R M , ∆r , M M 1 . N X R E, ∆r , M E ln 2 N C R E, ∆r , M E 1 . N X R J , ∆r , M J ln 2 N C R J , ∆r , M J

5.557718 5.557718 =

5.45678 5.45678 4.964882 4.964882 4.694305 4.694305

1 . N X R S , ∆r , M S ln 2 N C R S , ∆r , M S

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A D St N

N T A , D, St N

(1 1 1 )

N TR( A , D , r , ∆r , M )

Σ HR( A , D , r , ∆r , M )

N T A , D, N X( r , ∆r , M ) N T A , D, N C( r , ∆r , M ) Σ H A , D , N X( r , ∆r , M ) Σ H A , D , N C( r , ∆r , M )

Form Check_1( r , ∆r , M )

Form Check_2( r , ∆r , M )

1

.

N R( r , ∆r , M )

St N

A

D

D

Σ H A , D, N T

N R( r , ∆r , M )

NT

. 2.A

2

D. N T

1

N X( r , ∆r , M ) N C( r , ∆r , M )

N TR( 1 , 1 , r , ∆r , M ) Σ HR( 1 , 2 , r , ∆r , M )

N R( r , ∆r , M ) 4

Σ HR( 1 , 2 , r , ∆r , M )

1

Form Check_3( r , ∆r , M )

N R( r , ∆r , M )

ln

. 1 . ln 2 .N ( r , ∆r , M ) X 2

γ

ln 2 .N C( r , ∆r , M )

γ

N T 1 , 2 , N C R M , ∆r , M M

N T 1 , 2 , N C R J , ∆r , M J

. 12 7.68972610 . 12 1.60090210

N T 1 , 2 , N X R M , ∆r , M M

N T 1 , 2 , N X R J , ∆r , M J

. 17 1.57889410 . 17 1.07581410

N T 1 , 2 , n Ω_ZPF R M , ∆r , M M

N T 1 , 2 , n Ω_ZPF R J , ∆r , M J

N T 1 , 2 , N C R E, ∆r , M E

N T 1 , 2 , N C R S , ∆r , M S

N T 1 , 2 , N X R E, ∆r , M E

N T 1 , 2 , N X R S , ∆r , M S

. 17 1.88111510 . 17 1.14842610

N T 1 , 2 , n Ω_ZPF R E, ∆r , M E

N T 1 , 2 , n Ω_ZPF R S , ∆r , M S

. 18 8.57004510 . 18 5.17408410

=

. 18 7.16489910 . 18 4.83975610 . 12 1.57396110 . 13 2.09124310

N TR 1 , 1 , R M , ∆r , M M

. 4 6.72005410

Σ H 1 , 2 , n Ω_ZPF R M , ∆r , M M

. 37 9.36929710

N TR 1 , 1 , R E, ∆r , M E

. 4 5.49159510

Σ H 1 , 2 , n Ω_ZPF R E, ∆r , M E

. 38 1.07084610

. 4 2.05325110

Σ H 1 , 2 , n Ω_ZPF R J , ∆r , M J

N TR 1 , 1 , R S , ∆r , M S

. 4 1.19514810

Σ H 1 , 2 , n Ω_ZPF R S , ∆r , M S

Σ HR 1 , 2 , R M , ∆r , M M

. 9 4.51591310

N TR 1 , 1 , R J , ∆r , M J

Σ HR 1 , 2 , R E, ∆r , M E Σ HR 1 , 2 , R J , ∆r , M J Σ HR 1 , 2 , R S , ∆r , M S

=

=

=

. 38 2.05343110 . 38 2.93782710

. 9 3.01576110 . 8 4.21583910 . 8 1.42837810

1 Form Check_1 R M , ∆r , M M = 1

1 Form Check_1 R J , ∆r , M J = 1

331

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Form Check_2 R M , ∆r , M M Form Check_2 R E, ∆r , M E

=

Form Check_2 R J , ∆r , M J Form Check_2 R S , ∆r , M S

1

Form Check_3 R M , ∆r , M M

1

Form Check_3 R E, ∆r , M E

1

Form Check_3 R J , ∆r , M J

1

Form Check_3 R S , ∆r , M S

1

Form Check_1 R E, ∆r , M E =

Form Check_1 R S , ∆r , M S =

1

1 =

1 1 1

1 1

Casimir Force A PP( r )

4 .π .r

2

F PV( r , ∆r , M )

F PP( r , ∆r )

π .h .c .A PP( r )

A PP( r ) .∆U PV( r , ∆r , M ) .

4 480.∆r

N C( r , ∆r , M ) N X( r , ∆r , M )

Casimir force per unit area,

.ln

2

N X( r , ∆r , M ) N C( r , ∆r , M )

Casimir force per unit area,

F PP R M , ∆r

F PV R M , ∆r , M M

A PP R M

A PP R M

F PP R E, ∆r A PP R E F PP R J , ∆r

=

1.300126

F PV R E, ∆r , M E

1.300126

A PP R E

1.300126

( fPa )

F PV R J , ∆r , M J

1.300126

A PP R J

4

A PP R J

F PP R S , ∆r

F PV R S , ∆r , M S

A PP R S

A PP R S

2.349179 =

1.300007 0.074224

( fPa )

0.015617

Discrepancy between the classical representation and EGM, F PP R M , ∆r

1

F PV R M , ∆r , M M F PP R E, ∆r F PV R E, ∆r , M E F PP R J , ∆r F PV R J , ∆r , M J F PP R S , ∆r F PV R S , ∆r , M S KM KE KJ

44.65616

1 = 1

. 9.15864310

3

. 3 1.65163110

( %)

. 3 8.22480110

1

(1 1 1 )

332

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Given

F PV R M , ∆r , M M

2 2 π .h .c .R M

K M .∆r

4

2 2 π .h .c .R E

F PV R E, ∆r , M E

4 K E.∆r 2 2 π .h .c .R J

F PV R J , ∆r , M J

K J .∆r

4

KM KE

KM Find K M , K E, K J

KJ

KE

=

66.412608

KM

265.650432

120.01099

4. K E

= 480.043961

. 2.10195710

KJ

. 3 8.40782810

3

KJ

The Proportional Change in the Value of the Cosmological Constant By Casimir Force ∆Λ ( r , ∆r , M )

8 .π .G . ∆U PV( r , ∆r , M ) 2 3 .c 2

8 .π .G . F PV( r , ∆r , M ) . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 2 A PP( r ) N C( r , ∆r , M ) N C( r , ∆r , M ) 3 .c

St ∆Λ ( r , ∆r , M )

Λ R( r , ∆r , M )

4

∆Λ ( r , ∆r , M ) St ∆Λ ( r , ∆r , M )

∆Λ R M , ∆r , M M ∆Λ R E, ∆r , M E ∆Λ R J , ∆r , M J

Λ R R S , ∆r , M S

10

0.029107 . 3.39437710

Λ R R M , ∆r , M M

Λ R R J , ∆r , M J

1.447168

=

∆Λ R S , ∆r , M S

Λ R R E, ∆r , M E

St ∆Λ R M , ∆r , M M

3.225809 15 .

St ∆Λ R E, ∆r , M E

2

Hz

St ∆Λ R J , ∆r , M J

3

St ∆Λ R S , ∆r , M S

3.225809 =

1.447168 10

0.029107 . 3.39437710

15 .

2

Hz

3

1 =

1 1 1

333

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By Fundamental Harmonics ω PV 1 , R M , M M

2 U m R M,M M 3 . 2 ∆U PV R M , ∆r , M M

∆ω δr 1 , R M , ∆r , M M ω PV 1 , R E, M E

. 9 1.303510

∆ω δr 1 , R E, ∆r , M E

=

ω PV 1 , R J , M J ∆ω δr 1 , R J , ∆r , M J

. 9 4.78288210

9 1.3035.10

. 11 5.22005110

. 9 4.78288510

=

2 U m R J, M J 3 . 2 ∆U PV R J , ∆r , M J

. 10 5.36192210

10 5.3619.10

. 11 5.21985810

2 U m R S, M S 3 . 2 ∆U PV R S , ∆r , M S

ω PV 1 , R S , M S ∆ω δr 1 , R S , ∆r , M S

∆ω δr_Error( r , ∆r , M )

2 U m R E, M E 3 . 2 ∆U PV R E, ∆r , M E

ω PV( 1 , r , M )

2 U m( r , M ) . 3 . ∆ω δr( 1 , r , ∆r , M ) 2 ∆U PV( r , ∆r , M )

1

∆ω δr_Error R M , ∆r , M M

∆ω δr_Error R E, ∆r , M E

∆ω δr_Error R J , ∆r , M J

∆ω δr_Error R S , ∆r , M S

∆ω δr_Error R WD , ∆r , M WD

∆ω δr_Error R RG, ∆r , M RG

∆ω δr_Error R NS , ∆r , M NS

∆ω δr_Error R BH, ∆r , M BH

=

1

. 2.45448210

7

. 4.09314210

4

. 3.69917510

0.023754

0.195216

5.248215

27.272806

∆Λ EGM( r , ∆r , M )

Making the appropriate substitutions yields:

. 6.56319310

5 3

(%)

9 .G.M . ∆ω δr( 1 , r , ∆r , M ) ω PV( 1 , r , M )

2 .r

3

where, any suitable harmonic mode may be utilised to produce an equivalent result. The first harmonic has been represented here for convenience. Note: additional notation is required [“EGM”] to distinguish between the harmonic and classical representations. ∆Λ Error( r , ∆r , M )

1

∆Λ ( r , ∆r , M ) ∆Λ EGM( r , ∆r , M )

∆Λ EGM R M , ∆r , M M

∆Λ EGM R E, ∆r , M E

∆Λ EGM R J , ∆r , M J

∆Λ EGM R S , ∆r , M S

∆Λ EGM R WD , ∆r , M WD

∆Λ EGM R RG, ∆r , M RG

∆Λ EGM R NS , ∆r , M NS

∆Λ EGM R BH, ∆r , M BH

∆Λ Error R M , ∆r , M M

∆Λ Error R E, ∆r , M E

∆Λ Error R J , ∆r , M J

∆Λ Error R S , ∆r , M S

∆Λ Error R WD , ∆r , M WD

∆Λ Error R RG, ∆r , M RG

∆Λ Error R NS , ∆r , M NS

∆Λ Error R BH, ∆r , M BH

=

3.225809

1.447169

0.029107

. 3.39425210

3

. 6 2.30813410

. 8.47616310

12

. 15 5.25385210

=

334

. 2.45448210

7

. 4.09314210

4

10

15 .

2

Hz

. 9 1.42948610 . 6.56319310 . 3.69917510

0.023754

0.195216

5.248215

27.272806

5 3

(%)

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Casimir Force Correction 2

8 .π .G . F PV( r , ∆r , M ) . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 2 A PP( r ) N C( r , ∆r , M ) N C( r , ∆r , M ) 3 .c

4

9 .G.M . ∆ω δr( 1 , r , ∆r , M ) ω PV( 1 , r , M )

2 .r

3

N ( r , ∆r , M ) ∆ω δr( 1 , r , ∆r , M ) 8 .π .G 2 .r3 π .h .c .A PP 1 N X( r , ∆r , M ) . . . . .ln X . . 2 4 App N C( r , ∆r , M ) N C( r , ∆r , M ) ω PV( 1 , r , M ) 3 .c 9 G M 480.∆r 2

∆ω δr( 1 , r , ∆r , M )

Let:

ω PV( 1 , r , M )

2

2 3 1 . 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln K P 27.c .M .∆r4 N C( r , ∆r , M ) N C( r , ∆r , M )

4

4

where, KP is a planetary factor. 2

2 3 1 . 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln K P 27.c .M .∆r4 N C( r , ∆r , M ) N C( r , ∆r , M )

St PP K P , r , ∆r , M

St PP 4 .K E, R E, ∆r , M E

St PP 4 .K E, R E, ∆r , M E

St PP 480, R E, ∆r , M E

St PP 480, R E, ∆r , M E

1

4

.

99.999934

=

ω PV( 1 , r , M ) ∆ω δr( 1 , r , ∆r , M )

. 5 6.56319310

. 3 100.009093 9.09300510

1

( %)

Hence: ∆ω δr( 1 , r , ∆r , M ) ω PV( 1 , r , M )

2

2 3 1 . 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln K P 27.c .M .∆r4 N C( r , ∆r , M ) N C( r , ∆r , M ) 2

K P( r , ∆r , M )

2 3 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 4 N C( r , ∆r , M ) 27.c .M .∆r N C( r , ∆r , M )

4 .K M K P R M , ∆r , M M 4 .K E K P R E, ∆r , M E 4 .K J K P R J , ∆r , M J

4

.

1

7

= 6.56319710 .

5

. 4.09312510

ω PV( 1 , r , M ) ∆ω δr( 1 , r , ∆r , M )

1 . 2.45448210

4

K P R M , ∆r , M M K P R E, ∆r , M E

( %)

K P R J , ∆r , M J

4

K P R S , ∆r , M S

1

265.650431 480.043646 =

. 3 8.40786210 . 4 3.99605210

Additionally: ∆Λ EGM( r , ∆r , M )

2.G.M . ∆U PV( r , ∆r , M ) 3

r

U m( r , M )

1

2.G.M . (r

∆r )

335

1 3

3

r

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Checking yields: 2 .G.M M ∆U PV R M , ∆r , M M . 3 U m R M,M M RM 2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE

3.225809 =

2 .G.M J ∆U PV R J , ∆r , M J . 3 U m R J, M J RJ

1.447168 0.029107 . 3.39437710

10

15 .

2

10

15 .

2

Hz

3

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS 1

2 .G.M M .

1 ∆r

RM

3

1

2 .G.M E.

3

1

2 .G.M J . RJ

RS

3.225809 3

RE 1

∆r

3

∆r

3

1

2 .G.M S .

3

1 ∆r

RE

RM

3

RJ

=

1.447168 0.029107 . 3.39437710

Hz

3

1 3

RS

2 .G.M M ∆U PV R M , ∆r , M M 1 . . 2 .G.M . M 3 U R , M RM m M M R M ∆r 2 .G.M E ∆U PV R E, ∆r , M E 1 . . 2 .G.M . E 3 U m R E, M E RE R E ∆r 2 .G.M J ∆U PV R J , ∆r , M J 1 . . 2 .G.M . J 3 U R , M RJ m J J R J ∆r 2 .G.M S ∆U PV R S , ∆r , M S 1 . . 2 .G.M . S 3 U m R S, M S RS R S ∆r

1

1 3

RM

1 3 1

1 3

1 = 1

1 3

1 3

RJ

0 0

(%)

0

1

1 3

0

3

RE

1 3

RS

336

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∆Λ EGM R M , ∆r , M M .

.

∆Λ EGM R E, ∆r , M E

2 .G.M M ∆U PV R M , ∆r , M M . 3 U m R M, M M RM

1

1

2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE

1

1 1

2 .G.M J ∆U PV R J , ∆r , M J . ∆Λ EGM R J , ∆r , M J . 3 U m R J, M J RJ .

∆Λ EGM R S , ∆r , M S

1 ∆r

3

∆r

RE

3

∆Λ EGM R J , ∆r , M J . 2 .G.M J .

3

3

1 3

∆r

RS

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

(%)

1

1 3

=

1

RJ

1

∆Λ EGM R S , ∆r , M S . 2 .G.M S .

1 3

1

1 ∆r

RJ

1

RE

1

(%)

. 3 3.69903810

1

1

5

1

RM

1

∆Λ EGM R E, ∆r , M E . 2 .G.M E.

. 6.56319710

1

1

RM

7

. 4 4.09312510

1

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

∆Λ EGM R M , ∆r , M M . 2 .G.M M .

=

. 2.45448210

1 3

RS

Chapter 3.8 and 3.10 512.h .G.m e

mγ

c . π .r e

n Ω r e, m e ln 2 .n Ω r e , m e

.

2

EΩ

γ 5

m γγ

mγ Nγ

m γ = 5.746734 10

m γγ m gg

1

=

2 .m γγ

m gg

17 .

3.195095 6.39019

r e.

2

m γγ m e .c

5

r gg

2

4 .r γγ

Nγ

EΩ mγ

φ γγ φ gg

2.

r γγ r gg

. 28 N γ = 1.79861110

eV

10

r γγ

h .ω Ω r e , m e

45 .

eV

λ CN c ω CP γ . h .m p . λ h. r γγ λ CP ω h ω CN c .m h m n

r γγ r gg

=

2.335379 3.081551

10

35 .

m

φ 1 . γγ λ h φ gg

=

1.152898 1.521258

. 3 4.80847710 . 3 4.80847710 . 3 ( %) = 4.80847710

337

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Chapter 3.9 Representation 1 ω Ω ( r, M )

St ζ( r , M )

St η ( r , M )

ω Ce

ω Ω ( r, M )

St θ ( r , M )

ω CP

ω Ω ( r, M ) ω CN

Given λ Ce

St η r p , m p

λ CP mp

St η r p , m p

me

λ Ce

St θ r n , m n

λ CN mn

St θ r n , m n

rπ

me

rπ

Find r p , r n

rν

=

rν

830.594743 826.941624

( am)

r ν = 3.653119 ( am )

rπ

Representation 2 5

λ CP c .m e 8 .π

2

5

4 27.m e

.

.

K PV r p , m p .m p

3 128.G.π .h

3

λ CN

5

27

.

2 16.π .λ Ce

.

h .m e

4 2 K PV r p , m p .m p 4

λ CN

.

2

830.594743 826.941624

3

( am)

5

λ CN

4

830.594743

c .ω Ce

= 830.594743 ( am)

3

4 .ω CN

830.594743

5

2 4 27.m h m e . . 2 3 mp 4 .π 16.c .π .m p

5

.

2

4

4 .π .λ h λ Ce 2 4 27.ω h ω Ce . 4 32.π ω CN

h .m e 2 3 16.c .π .m n

338

λ . CN

27

.

2 16.π .λ Ce

4 .π .λ h λ Ce

5

=

4 2 K PV r n , m n .m n

λ CP

2 4 27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

c .ω Ce

.

5

K PV r n , m n .m n λ CP

4

λ CP

5

.

826.941624 = 826.941624 ( am) 826.941624

2 4 27.m h m e . mn 4 .π

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Representation 3 5

1 rπ

c .ω Ce

rν

4

1

.

9

2 4 ω CP 27.ω h .ω Ce ω CP . . 5 4 1 1 . 32.π 3 ω CN ω CN 5

3

rε

r e.

1. 2

ln 2 .n Ω r e , m e

5

γ

Refinement of radii predictions, rε

2

.e

3

. = 7.29429710

3

rπ

1 .r ε . e α rπ

2 3

= 99.958131 ( % )

Accuracy in relation to “α” utilising the NIST 2002 value. ω Ω r ε,m e

Recognising that:

= 2.000178

ω Ω r π, m p

and that the ratio of “re” to “rp - –n” is approximately

“π”, we shall conjecture a set of physical rules as follows: Given

α

r ε ω Ω r ε, m e r e ω Ω r π, m p

rν

rε rπ

rε rν

9

2

.e

3

rπ

1. 2

ln 2 .n Ω r e , m e

5

γ

2 π

Find r ν , r ε

rε

Any changes in radii predictions due to refinement methods can be shown to be negligible as follows, rπ rν

830.594743 = 826.837911 ( am) 11.802437

rε

rε π. r π

1 .r ε . e α rπ

= 100 ( % ) rν

2 3

= 99.974102( % )

A possible change in Electron mass may be calculated according to, let:

mε

me

Given ω Ω r ε, m ε

ω Ω r ε,m ε

1.

ω Ω r e, m e

ω Ω r π, m p

2

mε

m ε = 9.112989kg 10

Find m ε

. m ε .c = 5.11201210 ( eV) 2

ln 2.n Ω r e , m e

5

γ

2

mε

31

me

1 = 0.039588 ( % )

mε

= 1.000396

me

. m e .c = 5.10998910 ( eV) 2

5

339

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Particle Characteristics λ Ce m p λ Ce m n λ CP m e λ CN m e r ν λ CN ω CP m p r π λ CP ω CN m n rν

1

. 3 1.83615310 . 3 1.83868410 . 3 1.83868410 . 3 = 1.83615310

= ( 0.995477 0.998623 0.998623 0.998623)

λ CP ω CN m n

.

= ( 0.315088 0.315088 0.315088) ( % )

r π λ CN ω CP m p

St ζ r e , m e

St η r π , m p

. 5 1.83615310 . 3 1.83881210 . 3 = 3.21927910

St θ r ν , m n

ω PV 1 , r e , m e

ω PV 1 , r π , m p

ω PV 1 , r ν , m n

ω Ω r e, m e

ω Ω r π, m p

ω Ω r ν,mn

ω PV 1 , r π , m p

ω Ω r π, m p

ω PV 1 , r e , m e

ω Ω r e,m e

ω PV 1 , r ν , m p

ω Ω r ν,mn

ω PV 1 , r e , m e

ω Ω r e,m e

ω Ω r ν ,mn

ω PV 1 , r e , m e

ω PV 1 , r ν , m p

ω PV 1 , r ν , m n

2

λ Ce

2 .π .c .

ω Ce

λ CP 2

ω Ω r ν ,mn

ω CN

ω Ce

ω Ω r ε, m e 2 .ω Ω r π , m p

ω CP.

2

2 .π .c .

λ Ce 2

λ CN

mp

. 2.6174110

18

35.738651 ( GHz)

. 18 2.62481410

2

. 17 7.32711610 . 16 7.34446910 . 16 = 4.39398910

3 3 3 3 = 2.61741.10 2.61741.10 2.61741.10 2.61741.10

( YHz)

me

ω CN.

ω . CP ω Ω r π , m p ω Ce 1

35.506976

62.803639 10.50233

ω Ω r π, m p

ω CP

. 2.49926810

17

62.425172 10.472707

=

ω Ω r e, m e

ω Ω r π, m p

0.568793

=

mn

. 3 2.62463110 . 3 2.62463110 . 3 2.62463110 . 3 = 2.62481410

( YHz)

me 2

ω . CN ω Ω r ν , m n ω Ce 1

= ( 100 100 99.993032) ( % )

Chapter 3.11 2

5

rε

r π.

1 . me 9 2 mp

2

r µ0 r τ0

r ε. 1

me

2

9

1

mµ

me

9

mτ 5

St ω

1 ω Ω r ε,m e

. ω Ω r µ0 , m µ

ω Ω r τ0 , m τ

rµ rτ

340

r ε.

1 . mµ 9 4 me

2 5

1 . mτ 9 6 me

2

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5

r ε.

r en r µn r τn

rπ

m en

5

2

r µ.

me

830.594743

r τ0

11.802436

rε St ω St ω

0,0

=

0,1

=

4.005149

rµ

5.629206

rτ

=

mτ

8.193164

( am)

13.730068

8.212157 12.240673

2

m τn

r τ.

mµ

r µ0

= 826.837911 ( am)

rν

5

2

m µn

( am)

rµ

1 .r ε . e α rν

1

r en

rτ

. = 4.99870410

3

( %)

r en

0.095379

r µn = 0.655235 ( am)

1

.

ω Ω r ε,m e

2

. 9.09712910

r µn

1.958664

r τn

r ν mp . = 0.589336 ( % ) r π mn

1

r τn

ω Ω r µ,mµ

=

ω Ω r τ,m τ

2

=

3

0.429333

10

32 .

2

cm

3.836365

2

4

ω Ω r µ,m µ

6

ω Ω r τ, m τ

=

. 28 2.09392810 . 28 3.14089210

( Hz)

Chapter 3.12

1

5

r uq

m dq

3 .r xq. 2

m uq

5

2

r dq

r uq .

m dq

2

St dq

ω Ω r dq , m dq

St sq

ω Ω r xq, m sq

St cq

m uq

St bq

1 ω Ω r uq , m uq

1

St dqn

floor St dq

St dqn

floor St sq

St sq

2 = 3

St sq

2.049066

St sq

St cq

= 3.446836

St cq

floor St cq

St cq

4.547918

St bq

floor St bq

St bq

St bq St tq

10.216613

St tq

floor St tq

341

ω Ω r xq, m bq ω Ω r xq, m tq

St tq St dq

. ω Ω r xq, m cq

St tq

1

4 10

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5

2

m sq

9

St sq 5

m cq

r sq r cq r bq

5

r uq

1

.

.

m uq

2

St cq

9

m bq

5

r tq

m tq

St cq St bq St tq

=

St dq

ω Ω r dq , m dq ω Ω r sq , m sq

1

2

St cq

ω Ω r ε, m e

9

St bq

ω Ω r bq , m bq

St tq

ω Ω r tq , m tq

.

ω Ω r cq , m cq

9

St uq

floor St uq

St uq

7.207028

St dq

floor St dq

St dq

14.414056

St sq

floor St sq

St sq

21.621085

St cq

floor St cq

St cq

28.828113

St bq

floor St bq

St bq

St tq

floor St tq

St tq

7.207028

St dq

ω Ω r uq , m uq

2

St tq St uq

St uq

St sq

St bq 5

St sq

2

72.070282

7 7 =

14 21 28 72

9 5 St uq .r uq

m uq

9 5 St dq .r dq

m dq m sq m cq m bq

me rε

5

9 5 St sq .r sq

.

r tq

r uq .

9 5 St cq .r cq

5

1 . m tq 9 10 m uq

2

r u( M )

h 4 .π .c .M

9 5 St bq .r bq

m tq

9 5 St tq .r tq

ω Ω r u mW ,mW

rW

r u mW

St W

rZ

r u mZ

St Z

rH

r u mH

St H

St W

7.178111

St W

round St W , 0

St W

7

St Z

= 7.914688

St Z

round St Z , 0

St Z

= 8

St H

9.44142

St H

round St H , 0

St H

9

1 ω Ω r uq , m uq

342

. ω Ω r u mZ ,mZ ω Ω r u mH ,mH

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5

1 St W

rW

5

1

r uq .

rZ

5

m uq

rH

9

m uq

.m 2 W

m dq m sq

1 . 2 mZ 9 St Z

. 2 5

9

St H

r uq

1.013628

r sq

0.887904

=

r cq

1.091334

6 1. 9 1. 3

3

0.11402

GeV

1.184055

m bq

4.122266

m tq

178.61407

c

2

= 1.081984 ( am)

r u mZ

( am)

rW

1.226776

rZ

0.862443

r u mH

1.283533 = 1.061303 ( am) 0.940072

rH

0.92938

r tq

1.

r u mW

1.070961

r bq

6

. 7.01662310

0.768186

r dq

1.

3

=

m cq

1 . 2 mH

. 3.50831210

r uq

r dq

m uq

r sq

m dq

r cq

m sq

r tq = 0.960232 ( am)

r bq

m cq

m tq = 30.674156

m bq

GeV c

r uq

r dq

rW

rZ

r sq

r cq

r bq

rZ

rH

ru mW

ru mZ

r u mH

1

.

ω Ω r uq , m uq

ω Ω r ε,m e

rW

r H = 1.005145 ( am)

rZ

r H = 1.09497 ( am)

rW

1

r tq

2

.

= ( 1.046265 0.980887 1.09001 )

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

=

=

1 2 3 4 7 8 9 10

7 14 21 28 49 56 63 70

Chapter 3.13 The Planck Scale n Ω λ h , m h = 1.001996

ω PV 1, λ h , m h ωh

ω Ω λ h,m h ω PV 1 , λ h , m h 3

= 2.338413

Kω

2 π

1 = 0.199602 ( % )

Kλ

1 Kω

343

Km

ω Ω λ h,mh ωh

= 2.34308

Kλ

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K ω.

K ω K λ K m = ( 0.860254 1.162447 1.162447 )

1 . G.h . 11 ( ym) = 4.70944610 K ω c3

1

5

c . 18 ( YHz) = 6.36576910 . Gh

1 . h .c . 8 ( kg ) = 6.34179210 Kω G r γγ

1 = 16.244735( % )

Kω

= 0.998808

G.h . r µ K ω. 3 c rτ r γγ

1 K ω.

K λ .λ h

K λ .λ h

r γγ .K λ .

3 c .r τ = 0.119179 ( % ) G.h r µ

rτ

3

= 0.991785

2 .r gg

1

G.h . r µ c

2 .r γγ

= 0.119179 ( % )

1

2 .r γγ K λ .λ h

ω PV 1 , λ h , m h

1 = 30.866795 ( % )

2 .r gg

= 0.821515 ( % )

K PV λ h , m h

.

K λ .λ h

= 1.308668

1 = 100 ( % ) K ω .ω h

Note: these results indicate that the fundamental frequency for a Planck particle (at the experimentally implicit scale derived by EGM) is the harmonic cut-off frequency. That is, only one mode exists. φ γγ

K λ .λ h

φ gg

φ γγ = 4.709446 10

5

35 .

4 .φ γγ

5

4 = 131.950791 ( % )

φ gg = 6.214151 10

m

35 .

m

Theoretical Particles Leptons

rL

rµ

rε

rτ

3

m L 2, r L

m L 3, r L

m L St ω , r L

m L 5, r L

9 m e . St ω .

rL

5

r L = 10.751756 ( am )

rε

= ( 9.158498 56.785167 565.658456)

MeV c

2

m L 1, r ε

m L 2, r L

m L 3, r L

m L 4, r µ

m L 5, r L

m L 6, r τ

m L 7, r L

m L 8, r L

. 3 2.57116810 . 3 4.68915110 . 3 565.658456 1.77698910

m L 9, r L

m L 10, r L

m L 11, r L

m L 12, r L

. 3 1.27993910 . 4 1.96542410 . 4 2.90740110 . 4 7.96673810

MeV

m L 13, r L

m L 14, r L

m L 15, r L

m L 16, r L

. . . . 4.16806410 5.81788910 7.93596810 1.06103410

c

m L 17, r L

m L 18, r L

m L 19, r L

m L 20, r L

. 5 1.80266710 . 5 2.2992210 . 5 2.89617110 . 5 1.39382910

m L 21, r L

m L 22, r L

m L 23, r L

m L 24, r L

. 5 4.44724810 . 5 5.4320610 . 5 6.57869710 . 5 3.60724910

0.510999

=

9.158498

4

344

56.785167

4

4

105.65837

5

2

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Quarks and Bosons r QB

1. 9

r uq

r dq

r cq

9 m uq . St ω .

m QB St ω , r QB

m QB 5, r QB

r sq

m QB 6, r QB

r bq

r QB

r tq

rW

rZ

rH

5

r QB = 1.005145 ( am )

r uq

= ( 9.602148 21.811422)

GeV c

2

m QB 1 , r dq

m QB 2 , r sq

m QB 3, r cq

m QB 4, r bq

. 7.01662310

0.11402

1.184055

4.122266

m QB 5 , r QB

m QB 6 , r QB

m QB 7, r W

m QB 8 , r Z

9.602148

21.811422

80.425

91.1876

m QB 9 , r H

m QB 10, r tq

m QB 11, r QB

m QB 12, r QB

114.4

178.61407

333.634108

493.536148

m QB 13, r QB

m QB 14, r QB

m QB 15, r QB

m QB 16, r QB

=

3

707.535843

. 3 1.80112310 . 3 987.596451 1.34714410

m QB 17, r QB

m QB 18, r QB

m QB 19, r QB

m QB 20, r QB

. 3 3.06005810 . 3 3.90296410 . 3 2.36604810

m QB 21, r QB

m QB 22, r QB

m QB 23, r QB

m QB 24, r QB

. 3 7.54927810 . 3 9.22101310 . 3 1.11674510 . 4 6.12336610

GeV c

2

. 3 4.916310

Experimental Design Specifications for a Resonant Casimir Cavity Optimal Displacement ∆r .( 1 1 1 1 )

∆r M ∆r E ∆r J ∆r S

Given ∆ω R R M , ∆r M , M M

∆ω R R E, ∆r E, M E ∆ω R R J , ∆r J , M J

1

1 1

∆ω R R S , ∆r S , M S

1

∆r M

∆r M

∆r E

∆r E

∆r J

Find ∆r M , ∆r E, ∆r J , ∆r S

∆r J

∆r S

∆r S

5.358102 =

16.518308 122.49972

( mm)

855.41628

Harmonic Inflection Wavelength and Frequency λ X R M , ∆r M , M M λ X R E, ∆r E, M E λ X R J , ∆r J , M J λ X R S , ∆r S , M S

=

19.744081

ω X R M , ∆r M , M M

18.346216

ω X R E, ∆r E, M E

30.054415 32.089744

( nm )

ω X R J , ∆r J , M J ω X R S , ∆r S , M S

345

15.183915 =

16.340834 9.974989

( PHz)

9.342314

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Critical Frequency ω C ∆r M

27.97562

ω C ∆r E

9.074551

=

ω C ∆r J

( GHz)

1.223645 0.175232

ω C ∆r S

Critical Field Strengths E C R M , ∆r M , M M

554.936781

B C R M , ∆r M , M M

E C R E, ∆r E, M E

550.421992

V

B C R E, ∆r E, M E

141.888993

m

B C R J , ∆r J , M J

=

E C R J , ∆r J , M J

92.476743

E C R S , ∆r S , M S

B C R S , ∆r S , M S

18.510699 =

18.360101 4.732907

( mgs )

3.084692

Target Resonant Field Pressure at Similarity Characteristics Specified Above ∆U PV R E, ∆r , M M

2.860531

∆U PV R E, ∆r , M E

232.617621 =

∆U PV R E, ∆r , M J

( GPa)

4 7.3899.10

. 7 7.74094810

∆U PV R E, ∆r , M S

Appendix 3.D Derivation of Lepton Radii Given 5

1 rπ

c .ω Ce

rν

4

5

.

2.

.

ω CP

3

ω CN

ω CN 5

1 . me r ε r π. 9 2 mp

rµ rτ

1 . mµ 9 4 me

2

5

5

r en r µn r τn

α

rε

1

2

5

r ε.

1

3

27.ω h ω Ce ω CP . 5 4 1 . 32.π 4

r ε.

m en me

1 . mτ 9 6 me 2

2

5

r µ.

m µn mµ

2

5

r τ.

m τn

2

mτ

2

.e

3

rπ

346

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rµ

rε

α

.e

rτ

rν rε rπ

π rν

rε

rε

0.011806

rπ

rπ

0.830596

rν

rν

0.826838

rµ

rµ

. 8.21650110

Find r ε , r π , r ν , r µ , r τ , r en , r µn , r τn

rτ

=

rτ

r en

r en

r µn

r µn

r τn

r τn

1 .r ε . e α rπ 3

3

100

rµ

( fm)

0.012241

2

1 .r ε . e α rν

. 9.54036910

5

. 6.55581610

4

1.

. 1.95879510

3

π rπ

rτ

= 100 ( % ) 100

rε rν

Given ω Ω r ε, m ε

ω Ω r ε,m ε

1.

ω Ω r e, m e

ω Ω r π, m p

2

mε

ln 2.n Ω r e , m e

γ

2

Find m ε

m ε .c = 0.511534 ( MeV )

mε

m e .c = 0.510999 ( MeV )

2

2

1 = 0.104669 ( % )

me

Appendix 3.E Derivation of Quark and Boson Mass-Energies and Radii

1

5

r uq

3 .r xq. 2

m dq m uq

5

2

r dq

r uq .

m dq

2

St dq

ω Ω r dq , m dq

St sq

ω Ω r xq, m sq

St cq

m uq

St bq St tq

347

1 ω Ω r uq , m uq

. ω Ω r xq, m cq ω Ω r xq, m bq ω Ω r xq, m tq

www.deltagroupengineering.com

5

2

m sq

9

St sq St dq

floor St dq

St sq

floor St sq

St cq

floor St cq

St bq

floor St bq

St tq

floor St tq

5

m cq

r sq r cq r bq

5

r uq

.

1

.

m uq

2

St cq

9

m bq

2

5

2

r tq

St bq 5

m tq

9

2

St tq

9

St uq

ω Ω r uq , m uq

St uq

floor St uq

St dq

ω Ω r dq , m dq

St dq

floor St dq

ω Ω r sq , m sq

St sq

floor St sq

ω Ω r cq , m cq

St cq

floor St cq

St bq

ω Ω r bq , m bq

St bq

floor St bq

St tq

ω Ω r tq , m tq

St tq

floor St tq

St sq

1

St cq

ω Ω r ε, m e

.

9 5 St uq .r uq

m uq

9 5 St dq .r dq

m dq m sq m cq m bq

me rε

.

5

5

9 5 St sq .r sq

1 . m tq 9 10 m uq

r uq .

r tq

9 5 St cq .r cq

2

r u( M )

h . . 4 π c .M

9 5 St bq .r bq

m tq

9 5 St tq .r tq

rW

r u mW

St W

rZ

r u mZ

St Z

rH

r u mH

St H

ω Ω r u mW ,mW 1 ω Ω r uq , m uq

5

St W

round St W , 0

rW

St Z

round St Z , 0

rZ

St H

round St H , 0

rH

. ω Ω r u mZ ,mZ ω Ω r u mH ,mH

1 St W

5

r uq .

1 m uq

9

.m 2 W

5

1 . 2 mZ 9 St Z

5

1 . 2 mH

. 2

9

St H

348

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m uq m dq m sq

. 3.50603110

3

r uq

. 7.01206110

3

r dq

1.013628

r sq

0.887904

=

m cq

GeV

0.113946 1.183285

c

r cq

2

m bq

4.119586

r bq

m tq

178.49794

r tq

1. 6

1. 6

r uq

r dq

m uq

r sq

m dq

r cq

m sq

1.091334

rH

ru mW

ru mZ

r u mH

1.

rW

1.283867 = 1.06158 ( am) 0.940317

rH

0.92938

GeV c

rZ

rZ

( am)

1.070961

m tq = 30.654213

m bq

rW

3

=

rW

r tq = 0.960232 ( am)

r bq

m cq

0.768186

2

= ( 1.046537 0.981142 1.090294)

r H = 1.095254 ( am)

rZ

The following two result sets are accurate to “4” decimal places: 1

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r uq , m uq 1

.

ω Ω r ε,m e

mW

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

m en

80.425 = 91.1876

mZ

GeV c

114.4

mH 1.

r QB

9

r uq

r dq

2

r sq

r cq

3 .10

0.19

m τn

18.2

r tq

=

7 8 9 10

7 14 21 28 49 56 63 70

6

m µn =

r bq

1 2 3 4

=

rW

MeV c

rZ

2

rH

r QB = 1.00524 ( am )

Appendix 3.G 4. . 3 πr 3

V( r )

Q( r )

1 V( r )

Q ch ( r )

Q( r )

r dr

3

5. rν 3

Let:

x

1 2

Given 2

x

ln( x) . 2

x

x

1 1 3

Find( x)

349

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Therefore: a1 a2

r0

3 . π .r ν ( 1 x) .x3 . 2 8 1 x x 2

1 r ν. x

KS

0 .( fm)

r max

1.8.( fm)

δr

ρ ch ( r )

KS

2. 3

2

1. e 3 x

1

a1

x = 0.682943

2

K S = 0.113334 fm

KX

ρ ch r 0 ρ ch r ν

rν

. e

3 5 2 π .r ν . x

r dr = 1.067443 ( fm)

r max r 0

r

100 r

a2

2 0.113. fm

r x .r

12 ρ ch 10 .( fm)

0

3 .r ν

KS

. 2

2

x

1

r 0 , δr .. r max

2 ν

Q ch r ν = 0.140776

1 3

fm

=

0.826838 0.564683

1

KX

b 1 = 0.20712

( fm)

= 0.294995 ( % )

KS

0.140776 = 5.76803210 . 9

2

b1

1 3

fm

1 . rν

rν ρ ch ( r ) d r = 0.071089

1 3

fm

0

1 1. ρ ch r 0 = 0.070388 3 2 fm Neutron Charge Distribution

Charge Density

rν

r dr

ρ ch( r ) ρ ch r 0 ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

350

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rν

rν.

r0

0

1

r dr

fm

r max

1

r dr .

fm

δr

1.8

KS

2. 3

3.

π rν

5.

K S.

r max r 0

2

x

2

rν

. e

1. e 3 x

1

1 2

fm

r

100 r

ρ ch ( r )

KS

r x .r

r 0 , δr .. r max

2 ν

Let:

r1 r2

0.38

r dr

0.38 0.38 2

Given 2

r1

d

2

KS

2.

d r 12 3

3.

π rν

5.

. e 2

x

rν

1. e 3 x

1 2

r2

3

d r 23 3

r1 r2

KS

2. 3

π .r ν

5.

. e 2

x

rν

r1 r2

=

2

x .r ν

r2

1. e 3 x

1

Find r 1 , r 2

0

2

x .r ν

0

0.376649 0.662409 Neutron Charge Distribution r1

r2

ρ ch( r ) ρ ch r 0 d dr Neutron Charge Characteristic

d

r1

ρ ch( r )

d dr 1 d

ρ ch r 1

2

d r2 d

ρ ch( r )

2

d r 22 d2 d r 02

ρ ch r 2

ρ ch r 0

r Radius

351

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rν 4 r .ρ ch ( r ) d r

0

d ρ ch r 1 dr 1 d

2

d r 22 d

2

d r 02

∞ 4 r .ρ ch ( r ) d r

0.253851 ρ ch r 2

4 .π .

= 0.544657 1.103201

rν

0.016626 =

rν 2.

r ρ ch ( r ) d r

ρ ch r 0

0.129961 0.070507 0.070506

0 ∞ 2 r .ρ ch ( r ) d r

rν

Hence:

4 2 3 .π .r ν .x

6 .b 1 .K X . x

2

rX

∞

KS

3 .b 1 . x

2

6 .b 1 . K X

0

1

r X = 0.825617 ( fm)

2 2 0.005. fm . x 2

1

r ν .( fm)

= 0.147606 ( % )

6 .b 1 . K X

= 0.807145 ( fm)

2 2 0.005. fm . x

3 .b 1 . x

2

2 0.005. fm

3 .b 1 . x

2

2 0.005. fm

2 3 .b 1 . x

1

rX

1

6 .b 1 . K X

6 .b 1 . K X

ρ ch ( r ) d r = 0.055162

matches

1

3 .b 1 . x

r ν .( fm)

= 0.055162

. x2

. x2

1

1

= 0.843686 ( fm)

1

= 0.019693 ( fm)

1

1

r ν .( fm) = 0.016848 ( fm)

1

The Neutron Magnetic Radius may be determined as follows: r νM

rν

Given r dr rν r ν .ρ ch r νM

ρ ch ( r ) d r rν

r νM

r νM = 0.878972

Find r νM

Hence:

d

2

d r2

ρ ch

r dr

. ρ ch r νM = 2.93889110

3

0

rν

352

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Neutron Charge Distribution r νM

r dr

ρ ch r νM ρ ch( r )

r Radius

r dr

d2 d r2

r dr rν

ρ ch( r )

r Radius

The Proton Electric Radius may be determined as follows: r πE

rν

Given r dr r ν .ρ ch r πE

ρ ch ( r ) d r rν

r πE

r πE = 0.848527

Find r πE

. ρ ch r πE = 1.35418110

3

r πE

1 = 0.062194 ( % )

0.848

The Proton Magnetic Radius may be determined as follows: r πM

rν

∞

10.r ν

Given ∞ r ν .ρ ch r πM

ρ ch ( r ) d r r dr rν

353

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r πM

. ρ ch r πM = 1.43530110

r πM = 0.849933

Find r πM

3

The Classical Proton RMS Charge Radius may be determined as follows: r πE

1. 2

r ν = 0.874594

r νM

Performing dimensional conversions yields: r νM

r νM .( fm)

r πE.( fm)

r πE

r νM = 878.971907 ( am )

r πM

r πM .( fm)

r πE = 848.527406 ( am )

rν

r ν .( fm)

KS

K S . fm

2

r πM = 849.933378 ( am )

Appendix 3.H 5 5

r ν2 r ν3 r ν5

m en 1 . r ε. 2 9 me 2

2

5

r µ.

m µn

5

2

9

r τ.

3

m τn 9

5

1.

r ν2 r ν3 r ν5 = ( 0.027398 0.7656 2.820647 ) ( am )

r en r µn r τn

r ν2

r ν3

r ν5

r en

r µn

r τn

2

3 1.

= ( 0.095404 0.655582 1.958795 ) ( am )

3

r ν2

r ν3

r ν5 = 1.204548 ( am)

r en

r µn

r τn = 0.90326 ( am)

= 1.333557

Appendix 3.I λ A( r, M )

λ PV( 1 , r , M ) 2 .n Ω ( r , M )

Method 1 λ A K ω .r Bohr , m p

λ A K ω .r Bohr , m p = 657.329013( nm)

λB

1 = 0.130911( % )

Method 2 Given λ A K ω .r Bohr , m x λB

mx

Find m x

1

m x = 1.680518 10

mx

27 .

kg

mp

354

1 = 0.472081 ( % )

mx

1 = 1.203162 ( % )

m AMC

www.deltagroupengineering.com

Method 3 Given λ A K ω .r x, m AMC

1

λB

rx

. r x = 5.27319110

Find r x

11

r Bohr

( m)

1 = 0.352379 ( % )

rx

Particle Summary Matrix 3.1

r πE

2 0.69. fm 0.848.( fm)

r πM

0.857.( fm)

rπ

KX KS

=

0.113

2

fm

0.113334

1.

r πE

2

830.595686 830.662386

= r νM

rν

rp

848.527406

848

849.933378

857

874.594421

875

( am)

826.837876 825.617412 rν

rX

r νM

0.879.( fm)

878.971907

879

2

rε

.e

3

rπ

m tq = 178.49794

GeV c

rε

2

.e

rµ

. 7.29735310

3

rτ

= 7.29735310 .

3

rν

λ A K ω .r Bohr , m p λB

3.141593

=

657.329013 656.469624

( nm)

rε rπ

1 .r ε . e α rπ

rν rµ

2

1 .r ε . e α rν

3

rπ M Error

2 0.69. fm

1 . 1. r νM rp 2

M Error =

rν

rτ

π rπ r πM

0.848.( fm)

0.857.( fm) 1

r πE

rν

KS

rX

KX

m tq .c

0.879.( fm)

178.( GeV)

2

λ A K ω .r Bohr , m p

14

. 4.44089210

. 8.02976710

3

0.062194

0.824577

0.147824

0.295867

0.279741

0.130911

3

14

. 8.88178410

λB

. 1.11022310

. 3.19605810

rν

r πE

r νM

0.046352

rε

1.

14

(%)

355

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1. M Error 0,0 12 + M Error 2, 0

Error Av

M Error

M Error

0, 1

M Error

0, 2

M Error

M Error

2, 1

M Error

1, 0

M Error

2, 2

M Error

1,1

1, 2

M Error

3,0

...

M Error

3, 1

3, 2

Error Av = 0.149891 ( % )

Particle Summary Matrix 3.2 2

rε

. c .e r e ω Ce

r π_1

rπ

∆r π

r π_av

c .ω Ce

r ν_2

( 0.69 0.02) . fm

r ν_av

2

r π_1

r π_2

r ν_1

r ν_2

5

1.

r ν_av

2

1 . r π_av

r π_Error

r ν_1

∆r π

r π_2

2 0.69. fm = 830.662386 ( am )

∆r ν

( 0.69 0.02) . fm

2

1.

r π_av

2 4 27.ω h ω Ce . . 3 4 4 .ω CN 32.π ω CN

∆r ν

r π_1

5

2 4 27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

r ν_2

π

1 . r ν_av

r ν_Error

c .ω Ce

r π_2

rε

r ν_1

1.

3

2

= 12.03985 ( am)

2

r π_1

r π_2

r ν_1

r ν_2

r π_av r ν_av ∆r π

=

830.59568

830.594743

826.837876

826.941624

830.595212 . 4.68527810

∆r ν

826.88975

4

1. 2

∆K X

rX KX

rX KX

∆K X

rX KX

∆K X

r ν_Error

2 6 .b 1 .K X . x

rX KX

2 3 .b 1 . x

∆r X_av

∆K X

rX KX

0

( %)

1

2 0.005. fm

∆K X

1

r X_av

∆K X

rX KX

843.685579 807.144886

=

r X_av

825.415232

∆r X_av

18.270346

. r X_Error = 1.11022310

m gg = 6.39019 10

0

1=

0.051874

. 3 ( YHz) ω Ω r π , m p = 2.61740910

r X_av

r π_Error

( am)

14

( %)

45 .

eV

( am)

r X_Error

. m γ = 5.74673410

28

2 .r e λh

5

.

45 .

eV

∆K X

∆r X_av

= 1.152898 2

1

r X_av

m γγ = 3.195095 10 5

2

m γγ m e .c

10

rX KX

5 2 . 4 .r e .

λh

eV

2

m γγ m e .c

45 .

= 1.521258 2

Also, see Chapter 3.11 - 3.13. 356

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Particle Summary Matrix 3.3 The following is accurate to “1 or 2” decimal places (as implied by the results): ω Ω r π, m p ω Ω r ν ,mn

ω Ω r ν,mn

ω Ω r ε,m e

ω Ω r ε, m e

0.5

ω Ω r en , m en

1

ω Ω r en , m en

0.5

ω Ω r L, m L 2 , r L

2

ω Ω r L, m L 2 , r L

1

ω Ω r L, m L 3 , r L

2

ω Ω r L, m L 3 , r L

1

ω Ω r µ,mµ

ω Ω r π, m p

.

6

2

ω Ω r µ,mµ

3

ω Ω r µn , m µn

8

ω Ω r µn , m µn

4

ω Ω r L, m L 5 , r L

8

ω Ω r L, m L 5 , r L

4

ω Ω r τ,m τ

10

ω Ω r τ, m τ

ω Ω r τn , m τn 1

4

12

1

12

ω Ω r ε,m e

.

ω Ω r τn , m τn ω Ω r uq , m uq

ω Ω r uq , m uq

= 14

ω Ω r dq , m dq

14

ω Ω r dq , m dq

28

ω Ω r sq , m sq

ω Ω r sq , m sq

42

5 6 =

6 7 7 14 21

ω Ω r cq , m cq

56

ω Ω r cq , m cq

28

ω Ω r bq , m bq

70

ω Ω r bq , m bq

35

ω Ω r QB, m QB 5 , r QB

84

ω Ω r QB, m QB 5 , r QB

42

98

49

112

ω Ω r QB, m QB 6 , r QB

56

ω Ω r W,mW

126

ω Ω r W,mW

63

ω Ω r Z, m Z

140

ω Ω r Z, m Z

70

ω Ω r QB, m QB 6 , r QB

ω Ω r H, m H

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r tq , m tq

357

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ω Ω r π, m p ω Ω r ν,mn ω Ω r ε, m e

0.07

ω Ω r en , m en

0.07

ω Ω r L, m L 2 , r L

0.14

ω Ω r L, m L 3 , r L

0.14 0.29

1

ω Ω r µ,mµ

0.43

14

ω Ω r µn , m µn

0.57

1

ω Ω r L, m L 5 , r L

0.57

14

0.71

1

ω Ω r τ, m τ 1 ω Ω r uq , m uq

.

ω Ω r τn , m τn ω Ω r uq , m uq ω Ω r dq , m dq ω Ω r sq , m sq

0.86 =

0.86

7

2

3

3

ω Ω r bq , m bq

5

ω Ω r QB, m QB 5 , r QB

6

0.07

2

1

4

0.07

1 7

1

ω Ω r cq , m cq

7

7 4 7 4

0.14 0.14 0.29 = 0.43 0.57 0.57 0.71

7

7

0.86

ω Ω r QB, m QB 6 , r QB

8

5

0.86

ω Ω r W,mW

9

7

ω Ω r Z, m Z

10

6 7

ω Ω r H, m H

6

ω Ω r tq , m tq

7

358

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Particle Summary Matrix 3.4 φ γγ φ gg

=

4.709446 6.214151

10

35 .

m

rε

φ 1 . γγ K λ .λ h φ gg

=

1

m γγ

1.319508

m gg

11.805507

mp

. 4 5.10998910

rν

830.595686

mn

0.938272

rµ

826.837876

mµ

0.939565

rτ

8.216501

mτ

r en

12.241488

m en

0.095404

r µn

=

10

45 .

eV

3 .10

0.768186

m uq =

( am)

1.091334

9

4 1.9.10

1.958795

0.887904

r sq

1.776989

m τn

1.013628

r dq

0.0182 . 3 3.50603110

m dq

. 3 7.01206110

m sq

0.113946

r cq

1.070961

m cq

1.183285

r bq

0.92938

m bq

4.119586

r tq

1.283867

m tq

rW

1.06158

mW

91.1876

mZ

114.4

0.940317

rZ rH

GeV c

2

178.49794 80.425

mH m L 2, r L

r QB

6.39019

0.105658

m µn

0.655582

r τn

rL

3.195095

me

rπ

r uq

=

. 9.15849810

m L 3, r L

=

10.751756 1.00524

( am)

0.056785

m L 5, r L

=

0.565658

m QB 5 , r QB

9.604416

m QB 6 , r QB

21.816575

3

GeV c

2

3 λ Ce λ CP λ CN λ Cµ λ Cτ = 2.42631.10 1.32141 1.319591 11.734441 0.697721 ( fm)

. ω Ce ω CP ω CN ω Cµ ω Cτ = 7.76344110

4

1.425486 1.427451 0.160523 2.699721 ( YHz)

359

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Comparison Tables ω Ω r ε, m e

. 3 5.23406410

ω Ω r ε,m e

. 3 2.61740910

ω Ω r π, m p

ω Ω r ν,mn

. 3 2.62481410

ω Ω r ν ,mn

ω Ω r µ,mµ

. 4 2.09331310

ω Ω r µ,mµ

ω Ω r τ, m τ

. 4 3.14077610

ω Ω r τ,m τ

52.002066

ω Ω r en , m en

. 3 5.23406410

ω Ω r en , m en

78.023133

ω Ω r µn , m µn

. 4 2.09331310

ω Ω r µn , m µn

13.002457

ω Ω r τn , m τn

. 4 3.14077610

ω Ω r τn , m τn

ω Ω r π, m p

ω Ω r uq , m uq

= 3.66384510 .

4

( YHz)

1 ω Ω R M,M M

13.002457 6.502164 6.52056

52.002066 78.023133

. ω Ω r uq , m uq

= 91.017198

ω Ω r dq , m dq

91.017198

ω Ω r dq , m dq

. 4 3.66384510

ω Ω r sq , m sq

. 4 7.3276910

ω Ω r sq , m sq

ω Ω r cq , m cq

. 5 1.09915310

ω Ω r cq , m cq

364.068792

ω Ω r bq , m bq

. 5 1.46553810

ω Ω r bq , m bq

910.17198

ω Ω r tq , m tq

5

. 3.66384510

ω Ω r tq , m tq

637.120386

ω Ω r W,mW

. 5 2.56469110

ω Ω r W,mW

ω Ω r Z, m Z

. 5 2.93107610

ω Ω r Z, m Z

ω Ω r H, m H

. 5 3.2974610

ω Ω r H, m H

ω Ω r ε,m e

ω Ω r ε, m e

ω Ω r π, m p

ω Ω r π, m p

ω Ω r ν ,mn ω Ω r µ,mµ

ω Ω R E, M E

273.051594

728.137584 819.154782

ω Ω r ν,mn

5.037614

ω Ω r µ,m µ

5.051867

10.723101 5.362322 5.377493

40.2891

ω Ω r τ, m τ

42.886002

ω Ω r en , m en

60.449171

ω Ω r en , m en

64.345524

ω Ω r µn , m µn

10.073778

ω Ω r µn , m µn

10.723101

ω Ω r τ,m τ

ω Ω r τn , m τn 1

10.073778

182.034396

40.2891 60.449171

. ω Ω r uq , m uq

= 70.516448

ω Ω r dq , m dq

70.516448

ω Ω r sq , m sq ω Ω r cq , m cq ω Ω r bq , m bq ω Ω r tq , m tq ω Ω r W,mW

ω Ω r τn , m τn 1 ω Ω R J, M J

141.032895

64.345524

. ω Ω r uq , m uq

= 75.061704

ω Ω r dq , m dq

75.061704

ω Ω r sq , m sq

211.549343

42.886002

150.123408 225.185113

282.065791

ω Ω r cq , m cq

300.246817

705.164477

ω Ω r bq , m bq

750.617042

493.615134

ω Ω r tq , m tq

525.431929

564.131581

ω Ω r W,mW

634.648029

ω Ω r Z, m Z

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r H, m H

360

600.493634 675.555338

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ω Ω r ε, m e ω Ω r π, m p ω Ω r ν,mn ω Ω r µ,mµ

4.059933 32.378335

ω Ω r en , m en

48.579976

ω Ω r τn , m τn ω Ω R S, M S

4.048478

ω Ω r τ, m τ

ω Ω r µn , m µn

1

8.095792

8.095792 32.378335 48.579976

. ω Ω r uq , m uq

= 56.670543

ω Ω r dq , m dq

56.670543

ω Ω r sq , m sq

113.341086 170.011629

ω Ω r cq , m cq

226.682172

ω Ω r bq , m bq

566.70543

ω Ω r tq , m tq

396.693801

ω Ω r W,mW

453.364344 510.034887

ω Ω r Z, m Z ω Ω r H, m H

NOTES

361

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NOTES

362

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NOTES

363

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NOTES

364

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MATHCAD 8 PROFESSIONAL CALCULATION ENGINE [78]

365

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NOTES

366

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APPENDIX 3.L Computational Environment NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated.

PV / ZPF Equations 2.

K PV( r , M )

e

G .M

3

2 r .c

K 0( r , M )

K PV( r , M ) .e

K EGM_N( r , M )

K PV( r , M )

2

2 . ∆K 0( r , M )

G.M . KR 2 r .c

∆K 0( r , M )

K PV( r , M )

K EGM_E( r , M )

e G.M .

C PV n PV, r , M

2

r

2

1

T PV n PV, r , M

n PV 3 2 .c .G.M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

π .n PV

c

λ PV n PV, r , M

ω PV n PV, r , M

2 . ∆K 0( r , M )

U m( r , M )

ω PV n PV, r , M

3 .M .c

2

4 .π .r

3

3

U ω( r , M )

n Ω ( r, M )

h . 4 ω PV( 1, r , M ) 3 2.c Ω ( r, M )

4

12

Ω ( r, M )

∆ω PV( r , M )

ω Ω ( r, M )

∆ω δr n PV, r , ∆r , M ∆λ δr n PV, r , ∆r , M ∆λ Ω ( r , ∆r , M )

c.

∆v δr n PV, r , ∆r , M

1

λ PV n PV, r

ω Ω ( r, M )

n Ω ( r , M ) .ω PV( 1 , r , M )

U ω( r , M )

S m( r , M )

∆r , M ∆r , M

1 ω Ω(r

108.

ω PV( 1 , r , M )

ω PV n PV, r

U m( r , M )

Ω ( r, M )

12. 768 81.

c .U m( r , M )

2

U ω( r , M )

N ∆r( r , M )

ω Ω ( r, M ) .

∆r c

ω PV n PV, r , M λ PV n PV, r , M

1 ∆r , M )

ω Ω ( r, M )

∆ω δr n PV, r , ∆r , M .∆λ δr n PV, r , ∆r , M 3 .M .c . 4 .π 2

∆v Ω ( r , ∆r , M )

U m( r , M )

∆v δr n Ω ( r , M ) , r , ∆r , M

∆U PV( r , ∆r , M )

367

1 (r

∆r )

1 3

3

r

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∆ω ZPF( r , ∆r , M )

4

∆ω δr( 1 , r , ∆r , M )

ω Ω(r

∆ω S r , ∆r , M , K R

∆r , M )

ω Ω ( r, M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M ) .

∆ω Ω ( r , ∆r , M ) ω Ce

µ0

4

ω PV( 1 , r , M )

∆ω δr( 1 , r , ∆r , M )

n Ω_ZPF( r , ∆r , M )

∆ω R( r , ∆r , M )

4

n β r , ∆r , M , K R

∆ω ZPF( r , ∆r , M ) ∆ω Ω ( r , ∆r , M )

ω β r , ∆r , M , K R

St β ( r , ∆r , M )

ε0

∆ω δr( 1 , r , ∆r , M )

ω Ω_ZPF( r , ∆r , M )

n Ω_ZPF( r , ∆r , M )

∆n S r , ∆r , M , K R

ω PV( 1 , r , M )

∆ω Ω ( r , ∆r , M )

3

4 K R . ω Ω_ZPF( r , ∆r , M )

4

ω β r , ∆r , M , K R

n β r , ∆r , M , K R

St γ ( r , ∆r , M )

ε0

ω Ω_ZPF( r , ∆r , M )

2 .c . ∆U PV( r , ∆r , M ) h

ω Ω_ZPF ( r , ∆r , M )

ω Ω_ZPF( r , ∆r , M )

ω β r , ∆r , M , K R

St α ( r , ∆r , M )

4

µ0

∆U PV( r , ∆r , M ) .

∆K C( r , ∆r , M )

St δ( r , ∆r , M )

n Ω(r

∆ω ZPF( r , ∆r , M ) ω Ce ∆r , M )

n Ω ( r, M )

∆v δr n PV, r , ∆r , M

St ε n PV, r , ∆r , M

∆v Ω ( r , ∆r , M )

Casimir Equations ω C( ∆r )

c . 2 ∆r

E C( r , ∆r , M )

ω X( r , ∆r , M )

N C( r , ∆r , M )

λ C( ∆r )

c ω C( ∆r )

c .K PV( r , M ) . St α ( r , ∆r , M ) π .N X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

ω C( ∆r ) ω PV( 1 , r , M )

N TR( A , D , r , ∆r , M )

Σ HR( A , D , r , ∆r , M )

A D St N

N T A , D, N X( r , ∆r , M ) N T A , D, N C( r , ∆r , M ) Σ H A , D , N X( r , ∆r , M ) Σ H A , D , N C( r , ∆r , M )

N X( r , ∆r , M )

B C( r , ∆r , M )

λ X( r , ∆r , M )

(1 1 1 )

Σ H A , D, N T

N R( r , ∆r , M )

368

n Ω_ZPF( r , ∆r , M )

1

ln 2.n Ω_ZPF( r , ∆r , M )

γ

E C( r , ∆r , M ) c c ω X( r , ∆r , M )

St N

N T A , D, St N

NT 2

. 2.A

N C( r , ∆r , M )

D

D

D. N T

N X( r , ∆r , M )

A

1

A PP( r )

4 .π .r

2

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π .h .c .A PP( r )

F PP( r , ∆r )

A PP( r ) .∆U PV( r , ∆r , M ) .

F PV( r , ∆r , M )

4 480.∆r

N C( r , ∆r , M ) N X( r , ∆r , M )

N X( r , ∆r , M )

.ln

4

N C( r , ∆r , M )

2

8 .π .G . ∆U PV( r , ∆r , M ) 2 3 .c

∆Λ ( r , ∆r , M )

2

4

8 .π .G . F PV( r , ∆r , M ) . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 2 A PP( r ) N C( r , ∆r , M ) N C( r , ∆r , M ) 3 .c

St ∆Λ ( r , ∆r , M )

∆Λ ( r , ∆r , M )

Λ R( r , ∆r , M )

∆ω δr_Error( r , ∆r , M )

St ∆Λ ( r , ∆r , M )

9 .G.M . ∆ω δr( 1 , r , ∆r , M )

∆Λ EGM( r , ∆r , M )

2 U m( r , M ) . 3 . ∆ω δr( 1 , r , ∆r , M ) 2 ∆U PV( r , ∆r , M )

∆Λ Error( r , ∆r , M )

ω PV( 1 , r , M )

2 .r

3

ω PV( 1 , r , M )

1

2

2 3 1 . 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln K P 27.c .M .∆r4 N C( r , ∆r , M ) N C( r , ∆r , M )

St PP K P , r , ∆r , M

2

4

2 3 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 4 N C( r , ∆r , M ) 27.c .M .∆r N C( r , ∆r , M )

K P( r , ∆r , M )

∆Λ ( r , ∆r , M )

1

4

1

∆Λ EGM( r , ∆r , M ) ω PV( 1 , r , M )

.

∆ω δr( 1 , r , ∆r , M )

ω PV( 1 , r , M )

.

∆ω δr( 1 , r , ∆r , M )

Fundamental Particle Equations mγ

512.h .G.m e c . π .r e

2

.

n Ω r e, m e ln 2 .n Ω r e , m e 5

m gg

φ γγ φ gg

2 .m γγ

2.

r γγ

r e.

r γγ

γ

2

m γγ m e .c

r gg

2

ω Ω ( r, M )

St ζ( r , M )

r gg

h .ω Ω r e , m e

EΩ

ω Ce

5

3

4 .r γγ

Kω

St η ( r , M )

2

Nγ

Kλ

π

ω Ω ( r, M ) ω CP

EΩ mγ

1 Kω

St θ ( r , M )

m γγ

Km

mγ Nγ

Kλ

ω Ω ( r, M ) ω CN

Note: the highlighted equation is not included as a constraint. This is the most significant difference between the calculation engine and the “complete algorithm” of Appendix 3.K. 5

1 rπ

c .ω Ce

rν

4

5

2.

. 3

27.ω h ω Ce ω CP . . 5 4 1 . 32.π 4

3

ω CN

1 ω CP 1

5

rε

1 . me r π. 9 2 mp

2

ω CN

369

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5

r ε.

rµ rτ

1 . mµ 9 4 me

2 5

1 . mτ 9 6 me

5

2

r ε.

r en r µn r τn

m en me

2

5

r µ.

5

2

m µn

m τn

r τ.

mµ

2

mτ

Given 5

1 . me r ε r π. 9 2 mp rε

α

2

2

.e

3

rπ rµ

rε

α

.e

rτ

rν rε rπ

π rν

rε rπ rν rµ rτ

1

5

Find r ε , r π , r ν , r µ , r τ , r en , r µn , r τn

3 .r xq. 2

r uq

m dq

5

2

r dq

m uq

r uq

m dq

.

2

m uq

r en r µn r τn 5

St dq

ω Ω r dq , m dq

St dq

floor St dq

St sq

ω Ω r xq, m sq

St sq

floor St sq

St cq St bq St tq

1 ω Ω r uq , m uq

. ω Ω r xq, m cq

St cq

floor St cq

ω Ω r xq, m bq

St bq

floor St bq

ω Ω r xq, m tq

St tq

floor St tq

m sq

2

St sq

9

m cq

2

5

r sq r cq r bq

5

r uq

.

1 m uq

St cq

. 2

5

m bq

r tq

2

St bq 5

m tq

9

2

St tq

370

9

9

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St uq

ω Ω r uq , m uq

St uq

floor St uq

St dq

ω Ω r dq , m dq

St dq

floor St dq

ω Ω r sq , m sq

St sq

floor St sq

ω Ω r cq , m cq

St cq

floor St cq

St bq

ω Ω r bq , m bq

St bq

floor St bq

St tq

ω Ω r tq , m tq

St tq

floor St tq

St sq

1

St cq

ω Ω r ε, m e

.

9 5 St uq .r uq

m uq

9 5 St dq .r dq

m dq m sq

me

m cq

rε

m bq

5

9 5 St sq .r sq

.

1 . m tq r uq . 9 10 m uq

r tq

9 5 St cq .r cq

5

2

r u( M )

h 4 .π .c .M

rW

r u mW

rZ

r u mZ

rH

r u mH

9 5 St bq .r bq

m tq

9 5 St tq .r tq

ω Ω r u mW ,mW

St W

round St W , 0

. ω Ω r u mZ ,mZ

St Z

round St Z , 0

ω Ω r u mH ,mH

St H

round St H , 0

St W 1

St Z

ω Ω r uq , m uq

St H

5

1 St W

rW

5

m uq

rH

5

1

r uq .

rZ

. 2

9

.m 2 W

1 . 2 mZ 9 St Z

5

rµ

rL

rε

rτ

3

1 . 2 mH 9

St H

r QB

1. 9

r uq

m QB St ω , r QB

Let:

x

r dq

r sq

r cq

9 m uq . St ω .

r bq

r QB r uq

r tq

rW

rZ

rH

m L St ω , r L

9 m e . St ω .

rL

5

rε

5

V( r )

4. . 3 πr 3

Q( r )

1 V( r )

Q ch ( r )

Q( r ) 3

r dr

5. rν 3

1 2

371

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Given 2

x

ln( x) .

1 1 3

2

x

x

Find( x)

KS

rX

2 3 . π .r ν ( 1 x) .x3 . 2 8 1 x x

2 6 .b 1 .K X . x

3 .b 1 . x

2

1

r νM

KS

2. 3

rν

rν.

rν

. e

3 5 2 π .r ν . x

r πE

2

2

x

1 fm

r

2 0.113. fm

1

r dr .

1 fm

1

K S.

KS

2

fm

2

x .r ν

1. e 3 x

r πM

KX

r dr

2

rν

1

rν

KS

.

3 .r ν

1

r

ρ ch ( r )

2

b1

10.r ν

∞

rν

V

volt

Given r dr rν r ν .ρ ch r νM

ρ ch ( r ) d r rν

r dr r ν .ρ ch r πE

ρ ch ( r ) d r rν ∞

r ν .ρ ch r πM

ρ ch ( r ) d r r dr rν

r νM r πE

Find r νM , r πE, r πM

r πM

r νM

r νM

r πE

r πE .( fm)

r πM

r πM

5 5

r ν2 r ν3 r ν5

m en 1 . r ε. 2 9 me 2

2

5

r µ.

m µn 9

3

2

rν

5

r τ.

m τn

r ν .( fm)

KS

K S . fm

2

2

9

5

372

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λ PV( 1 , r , M )

λ A( r, M )

2 .n Ω ( r , M )

Given λ A K ω .r x, m AMC

1

λB rx

Find r x

Particle Summary Matrix 3.1 2 0.69. fm 0.848.( fm)

rπ r πE

KX KS

=

0.113

2

0.113364

0.857.( fm)

r πM

fm

1.

r πE

2

830.702612 830.662386

= r νM

rν

rp

848.636631

848

850.059022

857

874.696943

875

( am)

826.944318 825.617412 rν

rX

r νM

0.879.( fm)

879.064943

879

2

rε

.e

3

rπ

m tq = 178.440506

GeV c

rε

2

.e

rµ

. 7.29735310

3

rτ

= 7.29735310 .

3

rν

λ A K ω .r Bohr , m p λB

3.141593

=

657.329013 656.469624

( nm)

rε rπ

1 .r ε . e α rπ

rµ

2

1 .r ε . e α rν

3

rπ M Error

2 0.69. fm

1 . 1. r νM rp 2

rν

rν

rτ

rε

1. π rπ

rν

r πE

r πM

0.848.( fm)

0.857.( fm) 1

r πE

rν

KS

rX

KX

r νM

m tq .c

0.879.( fm)

178.( GeV)

2

λ A K ω .r Bohr , m p λB

373

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0 . 4.8425510

M Error =

. 1.11022310

0 3

0.034635 . 7.38826910

3

0.075074

0.809916

0.160717

0.321692

0.247475

0.130911

1. M Error 0,0

Error Av

12

13

M Error

M Error

0, 1

+ M Error 2, 0

(%)

M Error

0, 2

M Error

M Error

2, 1

M Error

1, 0

1,1

M Error

2, 2

3,0

M Error

3, 1

M Error

1, 2

...

M Error

3, 2

Error Av = 0.149388 ( % )

Particle Summary Matrix 3.2 2

rε

. c .e r e ω Ce

r π_1 r π_2

c .ω Ce

5

3

rπ r ν_1

2

r π_av

r π_Error r ν_Error

∆r π

r π_av

r π_1

2 r ν_1

r ν_2

∆r ν

r ν_av

r ν_1

∆r π

1 . r ν_av r ν_2

∆r ν

2 6 .b 1 .K X . x

rX KX

3

r π_2

1 . r π_av r π_2

3 .b 1 . x

2

1

r π_Error r ν_Error

r π_1

r π_2

r ν_1

r ν_2

r π_av r ν_av

1

∆r π

=

∆K X

2

( 0.69 0.02) . fm

2

π

27.ω h ω Ce . 4 ω . 32 π CN 2

. 14 2.22044610

4

( %)

0

830.702606

830.594743

826.944318

826.941624

830.648674

826.942971 . 1.34683810

( am)

3

2 0.005. fm

2 0.69. fm

( 0.69 0.02) . fm

1=

0.053931

∆r ν

. 3 ( YHz) ω Ω r π , m p = 2.61722210

1.

.

4 .ω CN

r π_1

1.

r ν_av

c .ω Ce

r ν_2

4

27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

5

rε

=

830.662386 12.03985

( am)

2

r X_av

r X_Error

1. 2

rX KX rX KX

∆K X ∆K X

rX KX ∆r X_av

∆K X

∆r X_av

r X_av

rX KX

∆K X

1

r X_av

374

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rX KX

∆K X

rX KX

∆K X

843.685579 807.144886

=

r X_av

825.415232

∆r X_av

18.270346

m γ = 5.746734 10

φ 1 . γγ λ h φ gg

=

m γγ

17 .

eV

1.152898 1.521258

r X_Error = 0 ( % )

( am)

m gg

=

3.195095 6.39019

φ 1 . γγ K λ .λ h φ gg

=

10

45 .

eV

0.991785 1.308668

Particle Summary Matrix 3.3 The following is accurate to “1 or 2” decimal places (as implied by the results): ω Ω r π, m p

ω Ω r ν ,mn

ω Ω r ν,mn

ω Ω r ε,m e

ω Ω r π, m p

0.5

2

ω Ω r L, m L 2 , r L

1

4

ω Ω r L, m L 3 , r L

1

6

ω Ω r µ,mµ

3

ω Ω r µn , m µn

4 4

ω Ω r L, m L 2 , r L

2

ω Ω r L, m L 3 , r L

ω Ω r µn , m µn

8

ω Ω r L, m L 5 , r L

8

ω Ω r τ,m τ

10

ω Ω r L, m L 5 , r L

12

ω Ω r τ, m τ

12

ω Ω r τn , m τn

ω Ω r τn , m τn .

0.5

ω Ω r en , m en

1

ω Ω r µ,mµ

1

ω Ω r ε, m e

ω Ω r en , m en

ω Ω r uq , m uq

= 14

ω Ω r dq , m dq

14

1 ω Ω r ε,m e

.

ω Ω r uq , m uq

2

5 6 =

6 7 7

28

ω Ω r dq , m dq

42

ω Ω r cq , m cq

ω Ω r sq , m sq

56

ω Ω r bq , m bq

70

ω Ω r cq , m cq

28

84

ω Ω r bq , m bq

35

ω Ω r QB, m QB 5 , r QB

ω Ω r QB, m QB 5 , r QB

42

ω Ω r QB, m QB 6 , r QB

98 112

ω Ω r W,mW

ω Ω r QB, m QB 6 , r QB

126

56

ω Ω r Z, m Z

140

ω Ω r W,mW

63

ω Ω r Z, m Z

70

ω Ω r sq , m sq

ω Ω r H, m H

14 21

49

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r tq , m tq

375

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ω Ω r π, m p ω Ω r ν,mn ω Ω r ε, m e

0.07

ω Ω r en , m en

0.07

ω Ω r L, m L 2 , r L

0.14

ω Ω r L, m L 3 , r L

0.14

14

0.29

ω Ω r µ,mµ

1

0.43

7

ω Ω r µn , m µn

0.57

0.07

1

0.57

0.07

ω Ω r L, m L 5 , r L

7

0.71

2

0.86

7

0.86

3

1

7

1

4

2

7

3

4

ω Ω r cq , m cq

4

7

0.86

ω Ω r bq , m bq

5

5

0.86

ω Ω r QB, m QB 5 , r QB

6

ω Ω r τ, m τ 1

ω Ω r τn , m τn

.

=

ω Ω r uq , m uq

ω Ω r uq , m uq

ω Ω r dq , m dq ω Ω r sq , m sq

1 14 1

8

ω Ω r W,mW

9

ω Ω r Z, m Z

10

0.14 0.29 = 0.43 0.57 0.57 0.71

7 6

7

ω Ω r QB, m QB 6 , r QB

0.14

7 6 7

ω Ω r H, m H ω Ω r tq , m tq

Particle Summary Matrix 3.4 φ γγ φ gg r Bohr rx

=

4.670757 6.163101

10

35 .

1 = 0.352379 ( % )

m

. r x = 5.27319110

m γγ m gg

=

11

( m)

3.195095 6.39019

10

376

φ 1 . γγ K λ .λ h φ gg

=

0.991785 1.308668

45 .

eV

www.deltagroupengineering.com

rε

me

rπ

11.807027

mp

. 4 5.10998910

rν

830.702612

mn

0.938272

rµ

826.944318

mµ

0.939565

rτ

8.215954

mτ

r en

12.240673

m en

0.095379

r µn

r uq

=

0.768186

m uq =

( am)

1.091334

9

4 1.9.10

1.958664

0.887904

r sq

3 .10

m τn

1.013628

r dq

1.776989

m µn

0.655235

r τn

0.105658

0.0182 . 3 3.50490310

m dq

. 3 7.00980510

m sq

0.113909

r cq

1.070961

m cq

1.182905

r bq

0.92938

m bq

4.11826

r tq

1.284033

m tq

rW

1.061716

mW

91.1876

mZ

114.4

0.940438

rZ rH

r QB

c

2

178.440506 80.425

mH m L 2, r L

rL

GeV

. 9.15554710

m L 3, r L

=

10.754551 1.005287

0.056767

m L 5, r L

( am)

3

=

0.565476

m QB 5 , r QB

9.596205

m QB 6 , r QB

21.797922

GeV c

2

3 λ Ce λ CP λ CN λ Cµ λ Cτ = 2.42631.10 1.32141 1.319591 11.734441 0.697721 ( fm)

. ω Ce ω CP ω CN ω Cµ ω Cτ = 7.76344110

1. 6

1. 6

r uq

m uq

r dq

r sq

m dq

r cq

m sq

r bq

m cq

4

1.425486 1.427451 0.160523 2.699721 ( YHz)

r tq = 0.960232 ( am)

m bq

m tq = 30.644349

GeV c

2

Similarity Equations SSE 3 E rms, B rms, r , ∆r , M

φ 4C_S( r , ∆r , M )

K PV( r , M ) . St α ( r , ∆r , M ) ln 2.n Ω_ZPF( r , ∆r , M ) γ . π .E rms.B rms n Ω_ZPF( r , ∆r , M ) 1

Re acos SSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , r , ∆r , M

377

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φ 5C_S( r , ∆r , M )

Re asin SSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , r , ∆r , M 1

SSE 4 φ, DC_E, DC_B, E rms , B rms , r , ∆r , M

.SSE ( 1 3

DC_E) .E rms , ( 1

DC_B ) .B rms , r , ∆r , M

.SSE ( 1 3

DC_E) .E rms , ( 1

DC_B ) .B rms , r , ∆r , M

cos ( φ) 1

SSE 5 φ, DC_E, DC_B , E rms , B rms , r , ∆r , M

sin ( φ)

Calculation Results K PV R E, M M

K PV R E, M E

K PV R E, M J

K PV R E, 2 .M M

K PV R E, 2 .M E

K PV R E, 2 .M J

K 0 R E, M M

K 0 R E, M E

K 0 R E, M J

∆K 0 R E, M M

∆K 0 R E, M E

∆K 0 R E, M J

K EGM_N R E, M M

K EGM_N R E, M E

K EGM_N R E, M J

1

1

1.000001

K EGM_E R E, M M

K EGM_E R E, M E

K EGM_E R E, M J

1

1

1

=

1

1

1

1

1

1.000001

1 . 8.55887110

1 12

. 6.96005110

K PV R E, M S K PV R E, 2 .M S 3 K PV R E, M E .e

K PV R S , M S

3.

e

∆K 0 R E , M E

∆K 0 R S , M S

ω PV 1 , R E, M M ω PV 1 , R E, M E

K 0 R E, M E

= 1.000008

. 8.27226110 =

e

=1

0.035839

∆K 0 R E , M E

e

( Hz)

K 0 R S, M S

2.484128

T PV 1 , R E, M S

λ PV 1 , R E, M M

. 7 3.62406910

λ PV 1 , R E, M J λ PV 1 , R E, M S Ω R E, M M Ω R E, M E Ω R E, M J Ω R E, M S

=

. 6 8.36497210 . 6 1.2259310

1.000927

K EGM_E R E, M S

1

=

U m R E, M E U m R E, M J

27.902544 4.089263

6.080707 494.481475 =

. 5 1.57089110 . 8 1.64551410

. 29 2.83606210

n Ω R E, M M

. 28 2.36338510

. 29 1.73968910

n Ω R E, M E

. 28 9.17216810

n Ω R E, M J

. 28 4.2341410

n Ω R E, M S

378

(s)

0.402556

U m R E, M S

. 5 1.20683210

4

120.885935

U m R E, M M ( km)

. 2.31613510

K EGM_N R E, M S

T PV 1 , R E, M E

ω PV 1 , R E, M S

=

0.999305

= 1.000008

T PV 1 , R E, M J

λ PV 1 , R E, M E

=

∆K 0 R S , M S

0.244543

ω PV 1 , R E, M J

1.000927

∆K 0 R E, M S

T PV 1 , R E, M M

3

7 2.2111.10

1.000463

K 0 R E, M S

=1

0.999999 10

=

( EPa)

. 28 1.44974110 . 27 7.64347410 . 27 3.5284510

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ω Ω R E, M M ω Ω R E, M E

. 3 1.86915710

ω Ω R E, M S

. 3 8.76512110

S m R E, M M

0.182295

S m R E, M J

( YHz)

S m R E, M S

∆ω PV R E, M J

. 4.70941210

N ∆r R E, M E

YW

N ∆r R E, M J

2

cm

. 6 4.93312710

∆ω δr 1 , R E, ∆r , M E

=

∆ω δr 1 , R E, ∆r , M J

=

1.729554

∆λ δr 1 , R E, ∆r , M M

7.493187

∆λ δr 1 , R E, ∆r , M E

( pHz )

519.469801

∆ω δr 1 , R E, ∆r , M S

. 14 6.52135710

N ∆r R E, M S

51.128768

∆λ δr 1 , R E, ∆r , M J

. 1.33585910

4

∆v δr 1 , R E, ∆r , M M

∆λ Ω R E, ∆r , M E

. 5.02660110

5

∆v δr 1 , R E, ∆r , M E

. 1.39724710

5

∆λ Ω R E, ∆r , M J ∆λ Ω R E, ∆r , M S

. 2.97920610

∆v Ω R E, ∆r , M M

13.105112

∆v Ω R E, ∆r , M E ∆v Ω R E, ∆r , M J

=

∆K C R E, ∆r , M J ∆K C R E, ∆r , M S

∆v δr 1 , R E, ∆r , M J

. 16 2.9237310

7.577156 =

pm

∆U PV R E, ∆r , M E

13.105115

s

∆U PV R E, ∆r , M J ∆U PV R E, ∆r , M S

1.74894 0.256316

13.105101 =

13.10513

pm

13.105131

s

13.109717

2.860531 232.617621 =

. 2.78399910

4

. 7 7.74094810

ω Ω_ZPF R E, ∆r , M E

( MPa .MΩ )

ω Ω_ZPF R E, ∆r , M J

. 7 2.9162510

( GPa)

. 4 7.389910

ω Ω_ZPF R E, ∆r , M M

87.634109

( m)

0.025237

∆U PV R E, ∆r , M M

1.077649 =

. 15 6.23483610

∆v δr 1 , R E, ∆r , M S

6

13.105121

∆K C R E, ∆r , M M ∆K C R E, ∆r , M E

( ym )

13.109693

∆v Ω R E, ∆r , M S

. 15 1.73310910

∆λ δr 1 , R E, ∆r , M S

∆λ Ω R E, ∆r , M M =

( YHz)

. 3 1.86915710 . 3 8.76512110

∆ω PV R E, M S

3

∆ω δr 1 , R E, ∆r , M M

519.573099 =

N ∆r R E, M M

14.824182 =

195.505363

∆ω PV R E, M E

519.573099 =

ω Ω R E, M J

S m R E, M E

∆ω PV R E, M M

195.505363

123.501066 370.868276 =

. 3 1.56573710

ω Ω_ZPF R E, ∆r , M S

. 3 8.90753610

n Ω_ZPF R E, ∆r , M M

. 19 1.49295410

( PHz)

KR2 = 99.99999999999999(%) ∆ω ZPF R E, ∆r , M M ∆ω ZPF R E, ∆r , M E ∆ω ZPF R E, ∆r , M J ∆ω ZPF R E, ∆r , M S

123.501066

n Ω_ZPF R E, ∆r , M E

370.868276 =

. 3 1.56573710

( PHz)

n Ω_ZPF R E, ∆r , M J

. 3 8.90753610

n Ω_ZPF R E, ∆r , M S

379

=

. 19 1.03481710 . 18 6.40270810 . 18 3.5857810

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ω β R E, ∆r , M M , K R2 ω β R E, ∆r , M E, K R2

=

ω β R E, ∆r , M J , K R2

41.841506 167.366022

∆n S R E, ∆r , M M , K R2 ∆n S R E, ∆r , M E, K R2 ∆n S R E, ∆r , M J , K R2

∆ω R R E, ∆r , M M =

∆ω R R E, ∆r , M J ∆ω R R E, ∆r , M S St α R E, ∆r , M M

St α R E, ∆r , M J

∆ω Ω R E, ∆r , M J

. 18 3.58539910

∆ω Ω R E, ∆r , M S

8.19356

∆ω S R E, ∆r , M E, K R2

9.615565

∆ω S R E, ∆r , M J , K R2

11.66707

∆ω S R E, ∆r , M S , K R2 St β R E, ∆r , M M St β R E, ∆r , M E

( MPa .MΩ )

St β R E, ∆r , M J

5

St δ R E, ∆r , M M

St γ R E, ∆r , M E

. 5.83032610

5

St δ R E, ∆r , M E

. 2.0974410

St γ R E, ∆r , M S

. 9.83425710

St ε 1 , R E, ∆r , M M St ε 1 , R E, ∆r , M E St ε 1 , R E, ∆r , M J

=

St ε 1 , R E, ∆r , M S 2.

e 2.

G .M M . 1 2 . REc

G .M J . 1 2 R E .c

1.

St δ R E, ∆r , M J

4

3

0.011474

1 =

1 1 1

1.000001

St ε n Ω_ZPF R E, ∆r , M J , R E, ∆r , M J

1.000002

St ε n Ω_ZPF R E, ∆r , M S , R E, ∆r , M S

=1

e 2.

2

2

= 1.000001

e

G .M S . 1 2 R E .c

1.

( PHz)

. 3 8.90658910

. 2.01680710

1.000001

G .M E . 1 2 . REc

. 3 1.56556910

4

St ε n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E

2.

370.826434 =

. 4.77711210

St ε n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

2

( PHz)

123.486273

4

0.999999

2

1.

=

St δ R E, ∆r , M S

4

162.833549

. 1.59080310

St β R E, ∆r , M S

. 2.19383110

45.263389 763.476685

∆ω S R E, ∆r , M M , K R2

St γ R E, ∆r , M M =

=

7.251258

. 7 2.9162510

St γ R E, ∆r , M J

e

. 18 6.40202410

. 4 2.78399910

St α R E, ∆r , M S

17.031676

∆ω Ω R E, ∆r , M E

1.0347.10

87.634109 =

. 14 3.81125810

∆ω Ω R E, ∆r , M M

1.077649

St α R E, ∆r , M E

. 14 6.84403710

n β R E, ∆r , M S , K R2

19

∆n S R E, ∆r , M S , K R2

. 15 1.16748410

=

n β R E, ∆r , M J , K R2

. 19 1.49277510 =

. 15 1.78829110

n β R E, ∆r , M E, K R2

( THz)

946.765196

ω β R E, ∆r , M S , K R2

∆ω R R E, ∆r , M E

n β R E, ∆r , M M , K R2

14.793206

1.000001 =

1 1.000003 1

2

2

=1 1.

2

2

= 1.000927

380

www.deltagroupengineering.com

N X R M , ∆r , M M N X R E, ∆r , M E N X R J , ∆r , M J

=

B C R M , ∆r , M M

B C R J , ∆r , M J

=

λ X R M , ∆r , M M

λ X R J , ∆r , M J

2

9.8181

ω X R M , ∆r , M M

6.364801

ω X R E, ∆r , M E

0.76984

( mgs )

=

36.419294 97.406507

294.339224 =

=

=

N C R J , ∆r , M J N C R S , ∆r , M S

ln 2 .N X R E, ∆r , M E

γ

ln 2 .N C R E, ∆r , M E

ln 2 .N X R J , ∆r , M J

γ

ln 2 .N C R J , ∆r , M J

ln 2 .N X R S , ∆r , M S

γ

ln 2 .N C R S , ∆r , M S

8.231693 3.077746

( PHz)

. 12 3.20180310

N C R E, ∆r , M E

γ

m

1.791481

N C R M , ∆r , M M

167.343325

volt

23.079214

10.073108

ω X R S , ∆r , M S

( nm )

190.811924 7.220558

ω X R J , ∆r , M J

ln 2 .N C R M , ∆r , M M

1.

2

E C R S , ∆r , M S

γ

2

1.

. 17 3.76223110

ln 2 .N X R M , ∆r , M M

1.

2

E C R J , ∆r , M J

29.761666

λ X R S , ∆r , M S 1.

. 17 3.15778710

0.240852

B C R S , ∆r , M S

λ X R E, ∆r , M E

. 2.29685210

E C R E, ∆r , M E

17

N X R S , ∆r , M S

B C R E, ∆r , M E

E C R M , ∆r , M M

. 17 2.15162910

. 12 4.18248610 . 13 1.53794510 . 13 3.14792110

1 . N X R M , ∆r , M M ln 2 N C R M , ∆r , M M

γ

γ

γ

1 . N X R E, ∆r , M E ln 2 N C R E, ∆r , M E

5.557718 5.557718 =

N X R J , ∆r , M J

1. ln 2 N C R J , ∆r , M J

5.45678 5.45678 4.964882 4.964882 4.694305 4.694305

1 . N X R S , ∆r , M S ln 2 N C R S , ∆r , M S

N T 1 , 2 , N C R M , ∆r , M M

N T 1 , 2 , N C R J , ∆r , M J

. 12 7.68972610 . 12 1.60090210

N T 1 , 2 , N X R M , ∆r , M M

N T 1 , 2 , N X R J , ∆r , M J

. 17 1.57889410 . 17 1.07581410

N T 1 , 2 , n Ω_ZPF R M , ∆r , M M

N T 1 , 2 , n Ω_ZPF R J , ∆r , M J

N T 1 , 2 , N C R E, ∆r , M E

N T 1 , 2 , N C R S , ∆r , M S

N T 1 , 2 , N X R E, ∆r , M E

N T 1 , 2 , N X R S , ∆r , M S

. 17 1.88111510 . 17 1.14842610

N T 1 , 2 , n Ω_ZPF R E, ∆r , M E

N T 1 , 2 , n Ω_ZPF R S , ∆r , M S

. 18 8.57004510 . 18 5.17408410

N TR 1 , 1 , R M , ∆r , M M N TR 1 , 1 , R E, ∆r , M E N TR 1 , 1 , R J , ∆r , M J N TR 1 , 1 , R S , ∆r , M S

=

=

. 18 7.16489910 . 18 4.83975610 . 12 1.57396110 . 13 2.09124310

. 4 6.72005410

Σ H 1 , 2 , n Ω_ZPF R M , ∆r , M M

. 4 5.49159510

Σ H 1 , 2 , n Ω_ZPF R E, ∆r , M E

. 4 2.05325110

Σ H 1 , 2 , n Ω_ZPF R J , ∆r , M J

. 4 1.19514810

Σ H 1 , 2 , n Ω_ZPF R S , ∆r , M S

381

. 37 9.36929710 =

. 38 1.07084610 . 38 2.05343110 . 38 2.93782710

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F PP R M , ∆r A PP R M

Σ HR 1 , 2 , R M , ∆r , M M

. 9 4.51591310

F PP R E, ∆r

Σ HR 1 , 2 , R E, ∆r , M E

. 9 3.01576110

A PP R E

. 4.21583910

F PP R J , ∆r

. 8 1.42837810

A PP R J

=

Σ HR 1 , 2 , R J , ∆r , M J Σ HR 1 , 2 , R S , ∆r , M S

8

1.300126 =

1.300126

( fPa )

1.300126 1.300126

F PP R S , ∆r A PP R S F PV R M , ∆r , M M

F PP R M , ∆r

A PP R M

F PV R M , ∆r , M M

F PV R E, ∆r , M E

2.349179

F PP R E, ∆r

A PP R E

1.300007

F PV R E, ∆r , M E

=

F PV R J , ∆r , M J

0.074224

( fPa )

F PP R J , ∆r

0.015617

A PP R J

F PV R J , ∆r , M J

F PV R S , ∆r , M S

F PP R S , ∆r

A PP R S

F PV R S , ∆r , M S

∆Λ R M , ∆r , M M ∆Λ R E, ∆r , M E ∆Λ R J , ∆r , M J

=

44.65616

1 = 1

1.447168

St ∆Λ R E, ∆r , M E

0.029107

15 .

2

Hz

. 3 1.65163110

St ∆Λ R J , ∆r , M J

3

3

( %)

1

St ∆Λ R M , ∆r , M M 10

. 9.15864310

. 3 8.22480110

3.225809

. 3.39437710

∆Λ R S , ∆r , M S

1

3.225809 =

St ∆Λ R S , ∆r , M S

1.447168 10

0.029107 . 3.39437710

15 .

2

Hz

3

ω PV 1 , R M , M M ∆ω δr 1 , R M , ∆r , M M Λ R R M , ∆r , M M Λ R R E, ∆r , M E Λ R R J , ∆r , M J Λ R R S , ∆r , M S

=

1

ω PV 1 , R E, M E

1

∆ω δr 1 , R E, ∆r , M E

1

ω PV 1 , R J , M J

1

∆ω δr 1 , R J , ∆r , M J

. 9 1.303510 =

. 9 4.78288210 . 10 5.36192210 . 11 5.22005110

ω PV 1 , R S , M S ∆ω δr 1 , R S , ∆r , M S

382

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2 U m R M,M M 3 . 2 ∆U PV R M , ∆r , M M 2 U m R E, M E 3 . 2 ∆U PV R E, ∆r , M E

9 1.3035.10

=

2 U m R J, M J 3 . 2 ∆U PV R J , ∆r , M J

. 9 4.78288510 10 5.3619.10

. 11 5.21985810

2 U m R S, M S 3 . 2 ∆U PV R S , ∆r , M S

∆ω δr_Error R M , ∆r , M M

∆ω δr_Error R E, ∆r , M E

∆ω δr_Error R J , ∆r , M J

∆ω δr_Error R S , ∆r , M S

∆ω δr_Error R WD , ∆r , M WD

∆ω δr_Error R RG, ∆r , M RG

∆ω δr_Error R NS , ∆r , M NS

∆ω δr_Error R BH, ∆r , M BH

=

. 2.45448210

7

. 6.56319310

. 4.09314210

4

. 3.69917510

0.023754

0.195216

5.248215

27.272806

∆Λ EGM R M , ∆r , M M

∆Λ EGM R E, ∆r , M E

3.225809

1.447169

∆Λ EGM R J , ∆r , M J

∆Λ EGM R S , ∆r , M S

0.029107

. 3.39425210

3

∆Λ EGM R WD , ∆r , M WD

∆Λ EGM R RG, ∆r , M RG

. 8.47616310

12

∆Λ EGM R NS , ∆r , M NS

∆Λ EGM R BH, ∆r , M BH

∆Λ Error R M , ∆r , M M

∆Λ Error R E, ∆r , M E

∆Λ Error R J , ∆r , M J

∆Λ Error R S , ∆r , M S

∆Λ Error R WD , ∆r , M WD

∆Λ Error R RG, ∆r , M RG

∆Λ Error R NS , ∆r , M NS

∆Λ Error R BH, ∆r , M BH

K P R M , ∆r , M M K P R E, ∆r , M E K P R J , ∆r , M J K P R S , ∆r , M S

=

. 2.30813410

6

. 15 5.25385210

=

. 2.45448210

7

. 4.09314210

4

5 3

(%)

10

15 .

2

Hz

. 9 1.42948610

. 6.56319310 . 3.69917510

0.023754

0.195216

5.248215

27.272806

5 3

(%)

265.650431 480.043646 =

. 3 8.40786210 . 4 3.99605210

2 .G.M M ∆U PV R M , ∆r , M M . 3 U m R M,M M RM 2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE 2 .G.M J ∆U PV R J , ∆r , M J . 3 U m R J, M J RJ

3.225809 =

1.447168 10

0.029107 . 3.39437710

15 .

2

Hz

3

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

383

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1

2 .G.M M .

1 ∆r

RM

3

RM

1

2 .G.M E.

1 ∆r

RE

3

RJ

1 ∆r

3

∆r

3

RS

1.447168

=

3

RJ

1

2 .G.M S .

3.225809 3

RE

1

2 .G.M J .

3

. 3.39437710

15 .

10

0.029107

2

Hz

3

1 3

RS

2 .G.M M ∆U PV R M , ∆r , M M 1 . . 2 .G.M . M 3 U R , M RM m M M R M ∆r 2 .G.M E ∆U PV R E, ∆r , M E 1 . . 2 .G.M . E 3 U m R E, M E RE R E ∆r 2 .G.M J ∆U PV R J , ∆r , M J 1 . . 2 .G.M . J 3 U R , M RJ m J J R J ∆r

3

3

3

3

∆r

RE

3

1

∆Λ EGM R J , ∆r , M J . 2 .G.M J .

∆r

RJ

3

1

∆Λ EGM R S , ∆r , M S . 2 .G.M S . RS

3

1

1

1 1

1 =

. 2.45448210

7

. 6.56319710

5

. 4 4.09312510

(%)

. 3 3.69903810

1

1

1 3 1

1 3

RE

=

1

1 3

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

( %)

1

1 ∆r

0

1

RM

RJ

(%)

3

1

1

0

1

1

1

0

RS

1

1

∆Λ EGM R E, ∆r , M E . 2 .G.M E.

1 3

1

1

0 =

1

RJ

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

3

1

1

2 .G.M J ∆U PV R J , ∆r , M J . ∆Λ EGM R J , ∆r , M J . 3 U m R J, M J RJ

∆r

1 3

2 .G.M E ∆U PV R E, ∆r , M E . ∆Λ EGM R E, ∆r , M E . 3 U m R E, M E RE

RM

3

RE

2 .G.M M ∆U PV R M , ∆r , M M . ∆Λ EGM R M , ∆r , M M . 3 U m R M, M M RM

∆Λ EGM R M , ∆r , M M . 2 .G.M M .

1

RM 1

2 .G.M S ∆U PV R S , ∆r , M S 1 . . 2 .G.M . S 3 U m R S, M S RS R S ∆r

∆Λ EGM R S , ∆r , M S .

1

1

1 3

RS

384

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5

λ CP c .m e

5

4 27.m e

.

.

K PV r p , m p .m p

3 128.G.π .h

8 .π

2

3

λ CN

5

2 16.π .λ Ce

c .ω Ce

2

5

2 3 16.c .π .m p

4

λ CP m e λ CN m e r ν λ CN ω CP m p r π λ CP ω CN m n rν

.

830.594743 = 830.594743 ( am)

4 .ω CN

3

830.594743

4

2 4 27.ω h ω Ce . 4 32.π ω CN

2 3 16.c .π .m n

5

.

826.941624 = 826.941624 ( am) 826.941624

2 4 27.m h m e . mn 4 .π

= ( 0.315205 0.315205 0.315205) ( % )

St θ r ν , m n

. 5 1.83602.103 1.83868.103 = 3.21927910

ω PV 1 , r e , m e

ω PV 1 , r π , m p

ω PV 1 , r ν , m n

ω Ω r e, m e

ω Ω r π, m p

ω Ω r ν ,mn

ω PV 1 , r π , m p

ω Ω r π, m p

ω PV 1 , r e , m e

ω Ω r e,m e

ω PV 1 , r ν , m p

ω Ω r ν,mn

ω PV 1 , r e , m e

ω Ω r e,m e

=

ω Ω r ν ,mn

ω PV 1 , r e , m e

ω PV 1 , r ν , m p

ω PV 1 , r ν , m n

2 .π .c .

λ Ce 2

λ CP

0.568793

35.500829

35.73252

. 17 2.61722210 . 18 2.62462610 . 18 2.49926810

( GHz)

62.792864 10.50158

ω Ω r π, m p

2

=

62.414364 10.471952

ω Ω r e, m e

ω Ce

.

h .m e

St η r π , m p

ω CP

5

2

4 .π .λ h λ Ce

= ( 0.995476 0.998623 0.998623 0.998623)

r π λ CN ω CP m p

ω Ω r π, m p

c .ω Ce

λ . CN

27

. 3 1.83615310 . 3 1.83868410 . 3 1.83868410 . 3 = 1.83615310

λ CP ω CN m n

St ζ r e , m e

5

.

2 16.π .λ Ce

2 4 27.m h m e . mp 4 .π

λ Ce m p λ Ce m n

( am)

4 2 K PV r n , m n .m n 3

2 4 27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

1

λ CN

.

4 .π .λ h λ Ce

.

826.941624

λ CN

5

h .m e

830.594743

=

4

λ . CP

27

.

4 2 K PV r p , m p .m p 5

K PV r n , m n .m n λ CP

4

λ CP

.

ω CP.

mp

. 17 7.32784510 . 16 7.34520410 . 16 = 4.39398910

. 3 2.61741.103 2.61741.103 2.61741.103 = 2.61722210

( YHz)

me

385

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2

ω CN

ω Ω r ν ,mn

ω Ce

λ Ce

2 .π .c .

ω CN.

2

λ CN

2

ω Ω r ε,m e 2 .ω Ω r π , m p

mn

. 3 2.62463110 . 3 2.62463110 . 3 2.62463110 . 3 = 2.62462610

( YHz)

me 2

ω . CP ω Ω r π , m p ω Ce

ω . CN ω Ω r ν , m n ω Ce

1

1

= ( 99.985611 100.007215 100.000181) ( % )

m L 1, r ε

m L 2, r L

m L 3, r L

m L 4, r µ

m L 5, r L

m L 6, r τ

m L 7, r L

m L 8, r L

m L 9, r L

m L 10, r L

m L 11, r L

m L 12, r L

m L 13, r L

m L 14, r L

m L 15, r L

m L 16, r L

m L 17, r L

m L 18, r L

m L 19, r L

m L 20, r L

. 5 1.80208610 . 5 2.29847910 . 5 2.89523810 . 5 1.3933810

m L 21, r L

m L 22, r L

m L 23, r L

m L 24, r L

. 5 4.44581510 . 5 5.4303110 . 5 6.57657710 . 5 3.60608710

9.155547

0.510999

. 565.476231 1.77526210

3

=

56.766874

105.677748

. 2.5703410

. 3 4.6876410

3

. 3 1.27952710 . 4 1.96479110 . 4 2.90646410 . 4 7.96417210

MeV

. 4 5.81601510 . 4 7.93341210 . 4 1.06069210 . 5 4.16672110

c

m QB 1 , r dq

m QB 2 , r sq

m QB 3 , r cq

m QB 4 , r bq

. 7.00980510

m QB 5 , r QB

m QB 6 , r QB

m QB 7 , r W

m QB 8 , r Z

9.596205

21.797922

80.425

91.1876

m QB 9 , r H

m QB 10, r tq

m QB 11, r QB

m QB 12, r QB

114.4

178.440506

333.427609

493.23068

m QB 13, r QB

m QB 14, r QB

m QB 15, r QB

m QB 16, r QB

707.097922

986.98519

. 3 1.80000810 . 3 1.3463110

=

3

0.113909

1.182905

4.11826

m QB 17, r QB

m QB 18, r QB

m QB 19, r QB

m QB 20, r QB

. 2.36458310

m QB 21, r QB

m QB 22, r QB

m QB 23, r QB

m QB 24, r QB

. 3 7.54460610 . 3 9.21530610 . 3 1.11605410 . 4 6.11957610

∆U PV R E, ∆r , M M ∆U PV R E, ∆r , M E ∆U PV R E, ∆r , M J ∆U PV R E, ∆r , M S

3

. . 3.05816410 3.90054810 3

2

3

GeV c

2

. 4.91325710

3

2.860531 232.617621 =

4 7.3899.10

( GPa)

. 7 7.74094810

The following two result sets are accurate to “13” decimal places: 1

.

ω Ω r uq , m uq 1 ω Ω r ε,m e

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

=

=

1 2 3 4 7 8 9 10

7 14 21 28 49 56 63 70

Resonant Casimir Cavity Design Specifications (Experimental) Given ∆ω R R E, ∆r , M E ∆r

Find ( ∆r )

1

∆r = 16.518377 ( mm)

E C R E, ∆r , M E = 550.422869

V m

ω X R E, ∆r , M E = 16.340851 ( PHz)

B C R E, ∆r , M E = 18.360131( mgs )

SSE 4 0 .( deg ) , 0 .( % ) , 0 .( % ) , E C R E, ∆r , M E , B C R E, ∆r , M E , R E, ∆r , M E SSE 5 90.( deg ) , 0 .( % ) , 0 .( % ) , E C R E, ∆r , M E , B C R E, ∆r , M E , R E, ∆r , M E

386

=

1 1

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MATHCAD 12 HIGH PRECISION CALCULATION RESULTS [79]

387

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NOTES

388

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APPENDIX 3.M Computational Environment NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED This appendix denotes high precision calculation results obtained from the “MathCad 12” computational environment utilising the calculation engine defined in “Appendix 3.L”. • • •

Convergence Tolerance (TOL): 10-14. Constraint Tolerance (CTOL): 10-14. Calculation Display Tolerance: 6 figures – unless otherwise indicated.

Particle Summary Matrix 3.1

      rπE +    

( 2)

rπ

0.69⋅ fm

rπE

0.848⋅ ( fm)

rπM 1 2

⋅ ( rνM − rν ) rν rνM

  0.857⋅ ( fm)   rp    rX  0.879⋅ ( fm) 

( )

 KX   −0.113  2  =  fm  KS   −0.113348 2  − rε  ⋅e 3  rπ  rµ  −  rε ⋅ e rτ  rν   rε  rπ − rν

         

 830.647087 830.662386  848.579832 848    849.993668 857   ( am) =  874.643564 875   826.889045 825.617615    879.016508 879 

mtq = 178.470327

GeV 

 2   c 

 λA( Kω ⋅ rBohr , mp )   657.329013  =  ( nm) λB    656.469624

 7.297353× 10− 3    = −3 7.297353× 10    3.141593 

rµ   2 − − r r r r 1 ε 1 ε 1 ε  3 τ  ⋅ ⋅e ⋅ ⋅e ⋅   α rπ α rν π rπ − rν   rπ rπE rπM     0.848⋅ ( fm) 0.857⋅ ( fm) 2 0.69⋅ fm M Error :=  −1 1 1  rν KS  ⋅  ⋅ ( rνM − rν ) + rπE  rX KX   rp  2    2 rνM mtq ⋅ c λA ( Kω ⋅ rBohr , mp )     0.879⋅ ( fm) 178⋅ ( GeV) λB  

( )

389

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− 14 − 13   2.220446× 10 0 1.110223× 10    −1.841834× 10− 3  0.068376 −0.817542 M Error =   ( %) −0.040736 0.153997 0.308232     −3 0.264229 0.130911  1.87806× 10 

1

ErrorAv :=

12

⋅  M Error

+ M Error

+ M Error

+ M Error

+ MError

+ MError

...

 + M 0, 0 + M 0, 1 + M 0, 2 + M 1, 0 + M 1, 1 + M 1, 2 Error2 , 0 Error2 , 1 Error2 , 2 Error3 , 0 Error3 , 1 Error3 , 2 

  

ErrorAv = 0.148979(%) Particle Summary Matrix 3.2 2  −  rε c 3  ⋅ ⋅e r ω e Ce  rπ_1     :=  5 2 4  rπ_2   c⋅ ωCe 27⋅ ωh ωCe  ⋅ ⋅  4⋅ ω 3 32⋅ π4 ωCP CP 

  rν_2  rν_av   ∆rν  rπ_2

rε   rπ − π    rν_1    5 :=    2 4 r c⋅ ωCe 27⋅ ωh ωCe  ν_2   ⋅ ⋅   4⋅ ωCN3 32⋅ π4 ωCN   

 1 ⋅ (r   π_av + ∆rπ )   rπ_Error   rπ_2    :=   r 1  ν_Error   rν_2 ⋅ ( rν_av + ∆rν )   

 ∆rπ   rπ_av − rπ_1    :=    ∆rν   rν_av − rν_1 

 rπ_1 r  ν_1  rπ_av   ∆rπ

        

 830.647081 830.594743 826.889045 826.941624 = ( am)  830.620912 826.915335  −0.026169 0.02629   

rX ( KX ) :=

 rπ_av  1  rπ_1 + rπ_2    := ⋅    rν_av  2  rν_1 + rν_2 

 rπ_Error  0   − 1 =   ( %) 0  rν_Error 

(2 ) 2 3⋅ b 1⋅ ( x − 1)

−6⋅ b 1⋅ KX ⋅ x − 1

( 2)

ωΩ ( rπ , mp ) = 2.617319× 10 ( YHz) 3

∆KX := 0.005⋅ fm

( )

2   0.69⋅ fm    830.662386 1  =  ( am) 2 2  ⋅  ( 0.69 + 0.02) ⋅ fm − ( 0.69 − 0.02) ⋅ fm    12.03985  2 

( )

rX_av :=

1 2

( (

( )

)

(

⋅ rX KX − ∆KX + rX KX + ∆KX

rX_Error :=

(

)

rX KX − ∆KX − ∆rX_av rX_av

−1

))

(

∆rX_av := rX_av − rX KX + ∆KX

 rX( KX − ∆KX)   rX( KX + ∆KX)    rX_av     ∆rX_av  

390

)

 843.685786   807.145085  = ( am)  825.415435  18.270351   

www.deltagroupengineering.com

− 14

rX_Error = 2.220446× 10

 φγγ   1.152898  =  λh  φgg   1.521258  1

⋅

(

− 17

mγ = 5.746734 10

( %)

)

⋅ eV

(

)

 mγγ   3.195095 − 45   =  10 ⋅ eV  mgg   6.39019 

 φγγ   0.991785 =  Kλ ⋅ λh  φgg   1.308668  1

⋅

Particle Summary Matrix 3.3 The following is accurate to “1 or 2” decimal places (as implied by the results): ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2, rL) )     ωΩ ( rL , mL( 3, rL) )    ωΩ ( rµ , mµ )   ωΩ ( rµn , mµn)    ω (r , m (5, r ) )  Ω L L L     ωΩ ( rτ , mτ )   ωΩ ( rτn , mτn)   1   ωΩ ( ruq , muq) ⋅  ωΩ ( rπ , mp )  ωΩ ( rdq , mdq)     ωΩ ( rsq , msq )     ωΩ ( rcq , mcq)   ωΩ ( rbq , mbq)    ω (r , m (5 , r ) )   Ω QB QB QB   ωΩ ( rQB , mQB( 6 , rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH)     ωΩ ( rtq , mtq )  

ωΩ ( rπ , mp )     ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2, rL) )   ω (r , m ( 3, r ))  Ω L L L    ωΩ ( rµ , mµ )    ωΩ ( rµn , mµn)    ωΩ ( rL , mL( 5, rL) )    ωΩ ( rτ , mτ )     ωΩ ( rτn , mτn) 1   ⋅  ωΩ ( rε , me)  ωΩ ( ruq , muq)   ωΩ ( rdq , mdq)     ωΩ ( rsq , msq)   ωΩ ( rcq , mcq)     ωΩ ( rbq , mbq)    ωΩ ( rQB , mQB( 5, rQB) )     ωΩ ( rQB , mQB( 6, rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH)     ωΩ ( rtq , mtq )  

 1   2     2   4     6   8   8     10   12   12    =  14   14     28   42   56     70   84   98     112   126     140 

391

 0.5   0.5     1   1   2     3   4     4   5   6    6  =  7   7     14   21     28   35   42     49   56   63     70 

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ωΩ ( rπ , mp )     ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2, rL) )   ω ( r , m ( 3, r ))  Ω L L L   ωΩ ( rµ , mµ )     ωΩ ( rµn , mµn)    ωΩ ( rL , mL( 5, rL) )    ωΩ ( rτ , mτ )     ωΩ ( rτn , mτn) 1   ⋅  ωΩ ( ruq , muq)  ωΩ ( ruq , muq)   ωΩ ( rdq , mdq)     ωΩ ( rsq , msq)   ωΩ ( rcq , mcq)     ωΩ ( rbq , mbq)    ωΩ ( rQB , mQB( 5, rQB) )     ωΩ ( rQB , mQB( 6, rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH)     ωΩ ( rtq , mtq )  

 0.07   0.07     0.14   0.14   0.29     0.43   0.57     0.57   0.71   0.86    0.86   =  1   1     2   3     4   5   6     7   8   9     10 

Particle Summary Matrix 3.4  rε     rπ   11.806238  r  830.647087  ν    826.889045  rµ      8.214055    rτ   12.237844   ren       0.095379   rµn  0.655235   rτn   1.958664      ruq  =  0.768186  ( am)    r   1.013628  dq      rsq   0.887904    1.091334  rcq   1.070961    rbq      0.92938     rtq   1.283947   rW   1.061645       rZ   0.940375  r   H

 me     mp  m   n   mµ     mτ   men     mµn   mτn     muq  m   dq   msq     mcq   mbq     mtq   mW     mZ  m   H

 5.109989× 10− 4     0.938272   0.939565     0.105658   1.776989     3 × 10− 9    −4  1.9 × 10    0.0182   GeV = −3   3.505488× 10  2    c  −3  7.010977× 10   0.113928     1.183102   4.118949     178.470327  80.425     91.1876   114.4  

392

(

)

 φγγ   4.670757 − 35   =  10 ⋅ m  φgg   6.163101

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− 11

rx = 5.273191× 10

 φγγ   0.991785 =  Kλ ⋅ λh  φgg   1.308668 1

( m)

(

⋅

)

 mγγ   3.195095 − 45  rL   10.752712   =  10 ⋅ eV  =  ( am)  mgg   6.39019   rQB   1.005262 

1 6 1 6

rBohr rx

− 1 = 0.352379( %)

 mL( 2, rL)     mL( 3, rL)   m ( 5, r )   L L   mQB( 5, rQB)   m 6, r   QB( QB) 

 9.153163× 10− 3   0.056752    GeV =  0.565329   2   9.597226   c   21.800242   

⋅ ( ruq + rdq + rsq + rcq + rbq + rtq ) = 0.960232( am) ⋅ ( muq + mdq + msq + mcq + mbq + mtq ) = 30.649471

GeV   2   c 

The following two result sets are accurate to “13” decimal places: 1 ωΩ ( ruq , muq)

 ωΩ ( rdq , mdq) ωΩ ( rsq , msq ) ωΩ ( rcq , mcq) ωΩ ( rbq , mbq)   1 2 3 4  =   ωΩ ( rW , mW) ωΩ ( rZ , mZ ) ωΩ ( rH , mH ) ωΩ ( rtq , mtq )   7 8 9 10 

⋅

 ωΩ ( rdq , mdq) ωΩ ( rsq , msq ) ωΩ ( rcq , mcq) ωΩ ( rbq , mbq)   7 14 21 28  =  ωΩ ( rε , me)  ωΩ ( rW , mW) ωΩ ( rZ , mZ ) ωΩ ( rH , mH ) ωΩ ( rtq , mtq )   49 56 63 70  1

⋅

NOTES

393

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NOTES

394

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INDEX 3 •

Instances of significant appearance by mathematical symbol

Symbol A a a1 a2 APP ax(t) a∞

Description 1st Harmonic term Magnitude of acceleration vector Acceleration with respect to General Modelling Equation One Acceleration with respect to General Modelling Equation Two Parallel plate area of a Classical Casimir Experiment Arbitrary acceleration in the time domain Mean magnitude of acceleration over the fundamental period in a FS representation in EGM Magnitude of Magnetic field vector B Magnitude of Magnetic field vector (at infinity) in the PV model of gravity: Ch. 3.2 Amplitude of applied Magnetic field: Ch. 3.6 B0 Magnitude of Magnetic field vector (locally) in the PV model of gravity Magnitude of applied Magnetic field vector BA Critical Magnetic field strength BC Magnitude of PV Magnetic field vector BPV Bottom Quark: elementary particle in the SM bq Root Mean Square of BA Brms Velocity of light in a vacuum c Velocity of light in a vacuum (at infinity) in the PV model of gravity: Ch. 3.1 Velocity of light (locally) in the PV model of gravity c0 Amplitude of fundamental frequency of PV (nPV = 1) CPV(1,r,M) CPV(nPV,r,M) Amplitude spectrum of PV Charm Quark: elementary particle in the SM cq Common difference D Experimental configuration factor: a specific value relating all design criteria; this includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy Offset function DC Down Quark: elementary particle in the SM dq Energy: Ch. 3.3 E Magnitude of Electric field vector Magnitude of Electric field vector (at infinity) in the PV model of gravity: Ch. 3.2 Electronic energy level Electron: subatomic / elementary particle in the SM e, eAmplitude of applied Electric field: Ch. 3.6 E0 Energy (locally) in the PV model of gravity Magnitude of Electric field vector (locally) in the PV model of gravity Magnitude of applied Electric field vector EA Critical Electric field strength EC Magnitude of PV Electric field vector EPV Root Mean Square of EA Erms F(k,n,t) Complex FS representation of EGM 395

Page 164 88 103 164 99 94 88 102 152 102 110 31 110 73 151 38 92 102 119 118 73 164 102

31 72 114 88 100 265 69, 191 151 114 102 110 31 110 151 93

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Magnitude of the ambient gravitational acceleration represented in the time domain Amplitude spectrum / distribution of F(k,n,t) F0(k) The Casimir Force by classical representation FPP The Casimir Force by EGM FPV Gluon: theoretical elementary particle in the SM g Magnitude of gravitational acceleration vector Universal gravitation constant G Tensor element g00 Tensor element g11 Tensor element g22 Tensor element g33 Height: Ch. 3.4 h Higgs Boson: theoretical elementary particle in the SM H Hydrogen Magnetic field strength Planck's Constant (plain h form) h h-bar Planck's Constant (2π form) HSE4A R Time average form of HSE4 R HSE5A R Time average form of HSE5 R Generalised reference to the reduced form of HSEx HSEx R Complex number i Initial condition Macroscopic intensity of Photons within a test volume In,P Vector current density J Wave vector k K0(r,X) ERF by displacement domain precipitation Generalised ERF K0(X) K0(ω ω,r,E,B,X) ERF by wavefunction precipitation K0(ω ω,X) ERF by frequency domain precipitation K0(ω ωPV,r,EPV,BPV,X) ERF equivalent to K0(ω,r,E,B,X) ERF formed by re-interpretation of the primary precipitant K1 ERF formed by re-interpretation of the primary precipitant K2 Harmonic wave vector of applied field kA Critical Factor KC Engineered Refractive Index KEGM Harmonic form of KEGM KEGM H Experimentally implicit Planck Mass scaling factor Km The intensity of the background PV field at specific frequency modes Kn,P A refinement of a constant in FPP KP Harmonic wave vector of PV kPV Refractive Index of PV KPV Harmonic form of KPV KPV H Critical Ratio KR Critical harmonic operator (based upon the unit amplitude spectrum) KR H Neutron MS charge radius by EGM KS Neutron MS charge radius (determined experimentally) in the SM KX Experimentally implicit Planck Length scaling factor Kλ Experimentally implicit Planck Frequency scaling factor Kω Length L f(t)

396

95 94 30 63 81 85 38 114

123 75 63 104 38 238 36 149 110 93 134 93 92 87 92 32 118 99 110 31 32 149 63 93 166 110 36 149 31 226 36 63 46

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L0 L2 L3 L5 M m0 M0 mAMC mbq mcq mdq me ME men mgg mH mh MJ mL(2) mL(3) mL(5) MM mn mp mQB(5) mQB(6) MS msq mtq muq mW mx mZ mε mγ mγg mγγ mµ mµn mτ mτn n n, N nA nB NC nE nPV nq NT

Length (locally) in the PV model of gravity by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Mass Mass (locally) in the PV model of gravity by EGM Zero mass (energy) condition of free space Atomic Mass Constant Bottom Quark rest mass (energy) by EGM Charm Quark rest mass (energy) by EGM Down Quark rest mass (energy) by EGM Electron rest mass (energy) according to NIST Mass of the Earth Electron Neutrino rest mass (energy) according to PDG Graviton rest mass (energy) by EGM Higgs Boson rest mass (energy) according to PDG Planck Mass Mass of Jupiter Rest mass (energy) of the L2 particle by EGM Rest mass (energy) of the L3 particle by EGM Rest mass (energy) of the L5 particle by EGM Mass of the Moon Neutron rest mass (energy) according to NIST Proton rest mass (energy) according to NIST Rest mass (energy) of the QB5 particle by EGM Rest mass (energy) of the QB6 particle by EGM Mass of the Sun Strange Quark rest mass (energy) by EGM Top Quark rest mass (energy) according (energy) to PDG Up Quark rest mass (energy) by EGM W Boson rest mass according (energy) to PDG Imaginary particle mass Z Boson rest mass according (energy) to PDG Electron rest mass (energy) in high energy scattering experiments Photon rest mass (energy) threshold according to PDG Graviton rest mass (energy) threshold according to PDG Photon rest mass (energy) by EGM Muon rest mass (energy) according to NIST Muon Neutrino rest mass (energy) according to PDG Tau rest mass (energy) according to NIST Tau Neutrino rest mass (energy) according to PDG Neutron: subatomic particle in the SM Field harmonic (harmonic frequency mode) Harmonic frequency modes of applied field Harmonic mode number of the ZPF with respect to BA Critical mode Harmonic mode number of the ZPF with respect to EA Harmonic frequency modes of PV Quantum number Number of terms 397

114 77

46 114 128 267 73 72 38 38 70 61 75 38 38 77

38 38 78 38 72 63 71 74 267 75 181 34 33 51 38 70 38 71 63 93, 94 110 151 31 151 110 265 164

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NTR NX N∆r nΩ nΩ ZPF nβ P p Q, Qe QB5 QB6 r

r0 rBohr rBoson rbq rc rcq rdq re RE ren RError rgg rH RJ rL RM rp rQB RS rsq rtq ru ruq rW rx rxq rZ rε rγγ rµ rµn rν rν2

The ratio of the number of terms Harmonic inflection mode Permissible mode bandwidth of applied experimental fields Harmonic cut-off mode of PV ZPF beat cut-off mode Mode number of ωβ Polarisation vector Proton: subatomic particle in the SM Magnitude of Electric charge Theoretical elementary particle (Quark or Boson) by EGM Theoretical elementary particle (Quark or Boson) by EGM Arbitrary radius with homogeneous mass (energy) distribution Generalised notation for length (e.g. r → λ/2π): Ch. 3.1 Generalised notation for length (locally) in the PV model of gravity: Ch. 3.1 Magnitude of position vector from centre of spherical object with homogeneous mass (energy) distribution Length (locally) in the PV model of gravity Classical Bohr radius Generalised RMS charge radius of a Boson by EGM RMS charge radius of the Bottom Quark by EGM Transformed value of generalised length (locally) in the PV model of gravity RMS charge radius of the Charm Quark by EGM RMS charge radius of the Down Quark by EGM Classical Electron radius in the SM Mean radius of the Earth RMS charge radius of the Electron Neutrino by EGM Representation error RMS charge radius of the Graviton by EGM RMS charge radius of the Higgs Boson utilising ru Mean radius of Jupiter Average RMS charge radius of the rε, rµ and rτ particles Mean radius of the Moon Classical RMS charge radius of the Proton in the SM Average RMS charge radius of the QB5 / QB6 particles by EGM utilising ru Mean radius of the Sun RMS charge radius of the Strange Quark by EGM RMS charge radius of the Top Quark by EGM Heisenberg uncertainty range RMS charge radius of the Up Quark by EGM RMS charge radius of the W Boson utilising ru Bohr radius by EGM Generalised RMS charge radius of all Quarks as determined by the ZC within the SM RMS charge radius of the Z Boson by utilising ru RMS charge radius of the Electron by EGM RMS charge radius of the Photon by EGM RMS charge radius of the Muon by EGM RMS charge radius of the Muon Neutrino by EGM Neutron RMS charge radius (by analogy to KS) RMS charge radius of the ν2 particle by EGM 398

164 34 33 37 34 93 63 46, 265 78 88 91 92 88 102 267 201 73 92 73 72 38 38 70 36 62 75 38 77 38 38 78 38 72 74 201 71 74 268 197 75 69 61 69 70 36 263

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RMS charge radius of the ν3 particle by EGM RMS charge radius of the ν5 particle by EGM Neutron Magnetic radius by EGM Generalised reference to rν2, rν3 and rν5 RMS charge radius of the Proton by EGM Proton Electric radius by EGM Proton Magnetic radius by EGM RMS charge radius of the Tau by EGM RMS charge radius of the Tau Neutrino by EGM Rydberg Constant Poynting Vector Strange Quark: elementary particle in the SM nth Harmonic term Range factor 1st Sense check 3rd Sense check 4th Sense check 2nd Sense check 5th Sense check 6th Sense check A positive integer value representing the harmonic cut-off frequency ratio between two proportionally similar mass (energy) systems Poynting Vector of PV Sω Time t Top Quark: elementary particle in the SM tq Change in Gravitational Potential Energy (GPE) per unit mass induced by Ug any suitable source Harmonic form of Ug Ug H Rest mass-energy density Um Up Quark: elementary particle in the SM uq Field energy density of PV Uω Local value of the velocity of light in a vacuum vc W Boson: elementary particle in the SM W All variables within the experimental environment that influence results and X behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis Impedance function Z Z Boson: elementary particle in the SM Change in electronic energy level ∆E Change in the magnitude of the local PV acceleration vector ∆aPV Change in magnitude of the local gravitational acceleration vector ∆g ∆GME1 Change in GME1 ∆GME2 Change in GME2 ∆GMEx Generalised reference to changes in GME1 and GME2 Harmonic form of ∆K0 ∆K0 H ∆K0(ω ω,X) Engineered Relationship Function by EGM Change in K1 by EGM ∆K1 Change in K2 by EGM ∆K2 Change in Critical Factor by EGM ∆ KC rν3 rν5 rνM rνx rπ rπE rπM rτ rτn R∞ S sq StN Stα Stβ Stδ Stε Stγ Stη Stθ Stω

399

263 62 263 53 62 69 71 265 104 72 164 35 37

38 62 121 47 63 111 149 120 71 119 92 59 87 111 59 265 110 132

149 32 111

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∆nS ∆r ∆t ∆t0 ∆ Ug ∆UPV ∆vΩ ∆vδr ∆ΛPV ∆λΩ ∆λδr ∆ωPV ∆ωR ∆ωS ∆ωZPF ∆ωΩ ∆ωδr Π ΣH ΣHR Ω α α1 αx β β1 βx ε0 φ φC φgg φγγ γ γg λ λA λB λCe λCN λCP λh λPV

Change in the number of ZPF modes Plate separation of a Classical Casimir Experiment Practical changes in benchtop displacement values Change in time (at infinity) in the PV model of gravity by EGM Change in time (locally) in the PV model of gravity by EGM Initial state GPE per unit mass described by any appropriate method Change in energy density of gravitational field Change in rest mass-energy density Terminating group velocity of PV Group velocity of PV Change in the local value of the Cosmological Constant by EGM Change in harmonic cut-off wavelength of PV Change in harmonic wavelength of PV Frequency bandwidth of PV Bandwidth ratio Similarity bandwidth ZPF beat bandwidth Beat bandwidth of PV Beat frequency of PV Dimensional grouping derived by application of BPT The sum of terms The ratio of the sum of terms Harmonic cut-off function of PV An inversely proportional description of how energy density may result in an acceleration: Ch. 3.2 Fine Structure Constant The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to α1 and α2 A directly proportional description of how energy density may result in an acceleration The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to β1 and β2 Permittivity of a vacuum Relative phase variance between EA and BA Critical phase variance RMS charge diameter of the Graviton by EGM RMS charge diameter of the Photon by EGM Mathematical Constant: Euler-Mascheroni (Euler's’ Constant Photon: elementary particle in the SM Graviton: theoretical elementary particle in the SM Wavelength 1st term of the Balmer Series by EGM Classical Balmer Series wavelength Electron Compton Wavelength Neutron Compton Wavelength Proton Compton Wavelength Planck Length Wavelength of PV 400

30 64 57 114 111 111 130 129 165 129 32 30 36 37 35 128 30 164 33 99 38 102 103 99 102 30 38 152 31 183 38 76 33 91 266 265 38

62

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µ, µµ µ0 ν2 ν3 ν5 νe νµ ντ ρ ρ0 τ, τω

Muon: elementary particle in the SM Reduced mass of Hydrogen Permeability of a vacuum Theoretical elementary Neutrino of the L2 particle by EGM Theoretical elementary Neutrino of the L3 particle by EGM Theoretical elementary Neutrino of the L5 particle by EGM Electron Neutrino: elementary particle in the SM Muon Neutrino: elementary particle in the SM Tau Neutrino: elementary particle in the SM Charge density Spectral energy density Tau: elementary particle in the SM Field frequency Field frequency (at infinity) in the PV model of gravity: Ch. 3.2 Field frequency (locally) in the PV model of gravity ω0 Field frequency (locally) in the PV model of gravity by EGM Harmonic frequency of the ZPF with respect to BA ωB Critical frequency ωC Harmonic frequency of the ZPF with respect to EA ωE Electron Compton Frequency ωCe Neutron Compton Frequency ωCN Proton Compton Frequency ωCP Planck Frequency ωh Generalised reference to ωPV(nPV,r,M) ωPV Fundamental frequency of PV (nPV = 1) ωPV(1,r,M) ωPV(nPV,r,M) Frequency spectrum of PV Harmonic inflection frequency ωX Harmonic cut-off frequency of PV ωΩ ZPF beat cut-off frequency ωΩ ZPF Critical boundary ωβ 〈 mQuark〉 Average mass (energy) of all Quarks according to PDG Average mass (energy) of all Quarks by EGM Average RMS charge radius of all Bosons in the SM utilising ru 〈rBoson〉 Average RMS charge radius of all Quarks by EGM 〈rQuark〉 Average RMS charge radius of all Quarks and Bosons by EGM utilising ru 〈r〉〉

401

69, 189 265 38 80

70 71 134 36 69, 189 89 102 114 152 31 152 136 177 38 61 33 118 34 33 37 30 202

78

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Periodic Table of the Elements

Group** Period

1 IA 1A 1

1

H 1.008

2

3

4

5

6

7

18 VIIIA 8A 2 IIA 2A

13 IIIA 3A

14 IVA 4A

15 VA 5A

16 VIA 6A

17 VIIA 7A

2

He 4.003

3

4

5

6

7

8

9

10

Li

Be

B

C

N

O

F

Ne

6.941

9.012

10.81

12.01

14.01

16.00

19.00

20.18

11

12

Na

Mg

22.99

24.31

3 IIIB 3B

4 IVB 4B

5 VB 5B

6 VIB 6B

7 VIIB 7B

8 9 10 ------- VIII ------------- 8 -------

11 IB 1B

12 IIB 2B

13

14

15

16

17

18

Al

Si

P

S

Cl

Ar

26.98

28.09

30.97

32.07

35.45

39.95

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

K

Ca

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

39.10

40.08

44.96

47.88

50.94

52.00

54.94

55.85

58.47

58.69

63.55

65.39

69.72

72.59

74.92

78.96

79.90

83.80

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

Xe

85.47

87.62

88.91

91.22

92.91

95.94

(98)

101.1

102.9

106.4

107.9

112.4

114.8

118.7

121.8

127.6

126.9

131.3

55

56

57

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

Cs

Ba

La*

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

132.9

137.3

138.9

178.5

180.9

183.9

186.2

190.2

190.2

195.1

197.0

200.5

204.4

207.2

209.0

(210)

(210)

(222)

89

104

87

88

Fr

Ra

(223)

(226)

Lanthanide Series*

Actinide Series~

Ac~ Rf

105

106

107

108

109

110

111

112

114

116

118

Db

Sg

Bh

Hs

Mt

---

---

---

---

---

---

()

()

()

(227)

(257)

(260)

(263)

(262)

(265)

(266)

()

()

()

58

59

60

61

62

63

64

65

66

67

68

69

70

71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

140.1

140.9

144.2

(147)

150.4

152.0

157.3

158.9

162.5

164.9

167.3

168.9

173.0

175.0

95

96

100

101

90

91

92

93

94

97

98

99

102

103

Th

Pa

U

Np

Pu

Am Cm

Bk

Cf

Es

Fm Md

No

Lr

232.0

(231)

(238)

(237)

(242)

(243)

(247)

(249)

(254)

(253)

(254)

(257)

(247)

402

(256)

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HARMONIC REPRESENTATION OF FUNDAMENTAL PARTICLES Illustrational only Wavefunction “ψ” based upon Proton harmonics, sin St ω .2 .π .ω Ω r π , m p .t

ψ St ω , t

(3.458)

1. T Ω r π ,m p 2

ψ( 1, t ) ψ( 2, t ) ψ( 4, t )

5 .10

0

29

1 .10

28

1.5 .10

28

2 .10

28

2.5 .10

28

3 .10

28

3.5 .10

28

ψ( 6, t )

t

Proton, Neutron Electron, Electron Neutrino L2, v2 L3, v3

Figure 3.44, 1 . T Ω r π ,m p

16

ψ( 8, t) ψ ( 10 , t ) ψ ( 12 , t )

0

5 .10

30

1 .10

29

1.5 .10

29

2 .10

29

2.5 .10

29

3 .10

29

3.5 .10

29

4 .10

29

4.5 .10

29

ψ ( 14 , t )

t

Muon, Muon Neutrino L5, v5 Tau, Tau Neutrino Up and Down Quark

Figure 3.45,

403

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1 . T Ω r π ,m p

56

ψ ( 28 , t ) ψ ( 42 , t ) ψ ( 56 , t )

0

1 .10

30

2 .10

30

3 .10

30

4 .10

30

5 .10

30

6 .10

30

30

7 .10

8 .10

30

9 .10

30

3 .10

30

1 .10

29

1.1 .10

29

1.2 .10

29

1.3 .10

29

ψ ( 70 , t )

t

Strange Quark Charm Quark Bottom Quark QB5

Figure 3.46, 1 . T Ω r π ,m p

168

ψ ( 84 , t ) ψ ( 98 , t ) ψ ( 112 , t ) ψ ( 126 , t ) 0

5 .10

31

1 .10

30

1.5 .10

30

2 .10

30

2.5 .10

30

3.5 .10

30

4 .10

30

4.5 .10

30

ψ ( 140 , t )

t

QB6 W Boson Z Boson Higgs Boson Top Quark

Figure 3.47,

404

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NOTES

405

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NOTES

406

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NOTES

407

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90000 ID: 471178 www.lulu.com

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Quinta Essentia: A Practical Guide to Space-Time Engineering - Part 3

ISBN 978-1-84753-353-1