Quinta Essentia - Part 2

QUINTA ESSENTIA A Practical Guide to Space-Time Engineering

PART 2 “ELECTRO-GRAVI-MAGNETICS” (EGM) “To my sister”: from Ricc “To Mike”: from Geoff RESEARCH NOTES Particle-Physics 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, ParticlePhysics, Physical Modelling, Planck Scale, Polarisable Vacuum, Quantum Mechanics, Zero-PointField. Cosmology Key Words: Big-Bang, CMBR, Cosmological Evolution / Expansion / History / Inflation, Dark Energy / Matter, Gravitation, Hubble constant.

2nd Edition Project Initiated: April 15, 2007 Project Completed: December 3, 2007 Revised: Thursday, 24 November 2011 RICCARDO C. STORTI1 & GEOFFREY S. DIEMER

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

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Synopsis “One does not find gold prospecting in a field filled with miners. One must break new ground, not perpetually overturn familiar soil.” • Riccardo C. Storti Quinta Essentia: A Practical Guide to Space-Time Engineering (the series), describes the development of a mathematical method termed “Electro-Gravi-Magnetics” (EGM); so named because it facilitates the representation of gravitational fields in ElectroMagnetic (EM) terms. EGM examines, based upon standard engineering principles, whether it might be possible to modify the gravitational force acting on a test object explicitly utilising EM energy. The EGM method is rooted within the foundations of General Relativity (GR) and Quantum Mechanics (QM). GR states that matter generates “curvature” within the fabric of space-time surrounding it. As objects pass through regions of curved space, the spatial curvature determines their motion, resulting in what we perceive to be “gravity”. However, it must be noted that spacetime “curvature” is a physically meaningless term. It is a mathematical contrivance acting to describe (not explain) the physical phenomenon we call gravity. EGM presumes mass-energy must do “work” on the space-time manifold in order to generate “curvature”, such that instead of being “curved”, space-time becomes “refractive” in the presence of matter. It describes gravity as a by-product of EM exchange2 between matter and the space-time manifold surrounding it, resulting in the formation of radial energy density gradients. Objects passing through these gradients behave in precisely the same manner as predicted by GR. The key difference between GR and EGM, however, is that EGM explicitly describes why spacetime physically becomes refractive in the presence of matter; GR does not. The EGM construct is an engineering tool producing astonishing results, revealing the EM architecture of gravitational fields and unveiling a universal principle applicable from the subatomic scale to the Cosmological. It is the authors’ sincere hope that the reader will learn and utilise the EGM method in their own research. Author contribution, • Geoffrey S. Diemer and Riccardo C. Storti: Synopsis, Preface, Ch. 1-2.6. • Riccardo C. Storti: Ch. 2.7-9 + Appendices.

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i.e. in accordance with the principles of QM. 3

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Preface “My advice to those who wish to learn the art of scientific prophesy is not to rely on abstract reason, but to decipher the secret language of Nature from Nature’s documents: the facts of experience.” • Max Born Humanity teeters on the brink of profound scientific developments. We approach a level of technological advancement exposing unexplored opportunities to solve pressing global crises. Our breadth of knowledge continues to expand; we are explorers evolving beyond our bounds. Our destiny is amongst the stars and our future depends upon overcoming two formidable technological hurdles: mastering the force of gravity and surpassing the speed of light. In 1968, a Russian Physicist and winner of the 1975 Nobel Peace Prize, Andrei Sakharov, proposed a hypothesis for the Quantum-ElectroMagnetic origin of gravity and inertia3. Similar ideas introduced by Harold Wilson and Robert Dicke in the 1920’s and 1950’s respectively, have spawned a contemporary movement offering an alternative to the apparently insurmountable limits of interstellar travel. Their ideas rest upon the most fundamental yet ineffable entity in the Universe (i.e. light). Light is energy – ElectroMagnetic (EM) radiation, which may be partitioned into units called “Photons” simultaneously possessing particulate and sinusoidal characteristics. General Relativity (GR)4 is derived from the manner in which light propagates through space. QuantumElectroDynamics (QED) describes the elegant and indissoluble dance of light and matter defining the material world. The Photon also establishes the basis for an optical model of gravity, first described by Sir Isaac Newton centuries ago. This optical model has a contemporary representative; the “Polarisable Vacuum (PV) Representation of GR” (i.e. the PV model). The vacuum of space is an effervescent matrix of energy based upon the principles of Quantum Mechanics (QM) such that light may be conceptualised as a fluctuation in the fabric of space-time. When we consider space to be composed of light and not simply a void through which it transits, it becomes possible to pinpoint the origin of gravitational and inertial forces on matter, unifying the ostensibly disparate forces of gravity and ElectroMagnetism. QM comes into sharp focus and the baffling consequences of GR become intuitive and comprehensible. This new perspective also suggests that the seemingly immutable forces of gravity and inertia might eventually be subject to manipulation such that supraluminal propulsion systems (i.e. “warp-drive”) might be feasible! Herein describes the development of a mathematical method; Electro-Gravi-Magnetics (EGM). It provides a framework for representing the PV model in measurable, quantifiable terms, allowing researchers to perform highly accurate calculations and predictions. It has been applied to determine experimentally verified properties of matter to a level of accuracy substantially beyond the capabilities of the Standard Model (SM) of Particle-Physics and Cosmology, yielding astonishing conclusions. EGM characterises how energy is distributed throughout the Cosmos in terms of a matterspace-time system. From this perspective, matter is not merely floating inertly in the vacuum of empty space; rather it exists as an inextricable part of the space it inhabits. When EGM methods are applied to represent the PV model, it is not only possible to validate GR in a manner fully incorporating QM and ElectroMagnetism, but also gain a heuristic understanding of what Relativistic and Quantum Mechanics represent physically. The Scientist’s natural ally is reduction; “a generalised phenomenon” is decomposed into its basic constituents. An engineer’s natural ally introduces the opposing perspective by integrating 3

Vacuum quantum fluctuations in curved space and the theory of gravitation. A.D. Sakharov. 1967. {Reprinted in Sov. Phys. Usp. 34 (1991) 394 [Usp. Fiz. Nauk 161 (1991) No. 5 64-66]}. 4 A geometric interpretation of gravity and space-time. 5

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basic principles and formulating a systemic solution congruent with experimental observation. Utilising engineering logic, matter and space are dynamic, interdependent and function in a feedback system of interactions such that “the phenomenon” may be “reverse-engineered”. In our era, professional science has been corrupted by conceit and wizardry. We have developed cumbersome and un-testable theories, conjuring upwards of twenty dimensions and an infinite array of simultaneous realities in an attempt to explain basic universal principles. Our conceit grows from the belief that the more baroque and complicated a theory is, the more accurate it must be. The hallmark of genius is “the ability to make your thoughts reality”. This is achieved by the articulation of unique vision and the implementation of unconventional reasoning in an elegant manner. Genius cannot be obtained by regurgitation of established doctrine. Sir Isaac Newton and Albert Einstein laboured great works, not because they recited the toil of those whom preceded them, but because they presented novel concepts unveiling great truths of Nature. The famed statistician George Box once wrote: “essentially, all models are wrong, but some are useful”5. The guiding principle underlying the work presented herein is a desire to represent gravity, not only philosophically, intuitively and elegantly, but in useful terms. EGM does not intend to imply that GR (or any other theory) is wrong, only that there may be an alternative way to interpret Nature. In order to facilitate technological progress in gravity control or supraluminal travel, we must reconstruct GR in a format promoting invention. EGM provides a tool to model the dynamics of matter-space-time interactions solely in the language of EM radiation, offering engineering potential. This text, Part Two of the “Quinta Essentia” series, is a summarised presentation of the key results and findings of Parts Three and Four. The EGM method generates new predictions and confirms well-established experimental observations, particularly in the fields of Particle-Physics and Cosmology. It is our explicit hope that the material presented in the “Quinta Essentia” series, will inspire new ideas and experiments dealing directly with matter-space-time modification, by either applying EGM methods or through the development of one’s own approach. Note: references to “Quinta Essentia – Part 3 and 4” are denoted by QE3 and QE4 respectively.

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Empirical Model-Building and Response Surfaces. Wiley, New York (1987) pp. 424. George E.P. Box and Norman R. Draper. 6

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Scientific Achievements The physical characteristics reported herein (derived from 1st principles based upon a single paradigm) may be articulated as follows (many of which are experimentally verified or implied), “Quinta Essentia – Part 3” (QE3) 1. 2. 3. 4. 5.

The spectral quantisation of gravity. The application of the spectral quantisation of gravity to Metric Engineering principles. The experimentally implicit validation of the Polarisable Vacuum (PV) model of gravity. The formulation of the Electro-Gravi-Magnetic (EGM) Spectrum. 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).

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“Quinta Essentia – Part 4” (QE4) •

Astro-Physics 33. Derivation of the minimum physical “Schwarzschild-Black-Hole” (SBH) mass and radius. 34. Derivation of maximum permissible energy density. 35. Derivation of the harmonic mode and frequency characteristics and profiles of a SBH. 36. Derivation of the SBH singularity radius.

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Cosmology • General 37. Derivation of the minimum gravitational lifetime of matter. 38. Derivation of the present Cosmological age. 39. Derivation of the present Cosmological size. 40. Derivation of the total Cosmological mass. 41. Derivation of the present Cosmological mass-density. • Hubble constant 42. Derivation of the Hubble constant at the instant of the “Big-Bang”. 43. Derivation of the maximum Hubble constant since the “Big-Bang”. 44. Derivation of the present Hubble constant within experimental tolerance. 45. Derivation of the Hubble constant in the time domain. 46. Derivation of the rates of change of the Hubble constant in the time domain. • Cosmic Microwave Background Radiation (CMBR) temperature 47. Derivation of the CMBR temperature at the instant of the “Big-Bang”. 48. Derivation of the maximum Cosmological temperature since the “Big-Bang”. 49. Derivation of the present CMBR temperature within experimental tolerance. 50. Derivation of the CMBR temperature in the time domain. 51. Derivation of the rates of change of the CMBR temperature in the time domain. • Evolutionary processes 52. Categorisation of the Cosmological evolution process into two regimes: comprised of four distinct periods. 53. Determination of the impact of “Dark Matter / Energy” on the Hubble constant and CMBR temperature. 54. Articulation of the precise history of the Universe. • Cosmological constant 55. Experimentally implicit derivation of the Zero-Point-Field (ZPF) energy density threshold, yielding an insight into the Cosmological constant.

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Particle-Physics 56. Derivation of the Photon and Graviton mass-energies lower limit. 57. Derivation of the Photon and Graviton Root-Mean-Square (RMS) charge radii lower limit. 58. Derivation of the Photon charge threshold. 59. Derivation of the Photon charge upper and lower limits.

“Quinta Essentia – Part 2” (QE2) 60. Derivation of experimentally implicit values of the deceleration parameter and Cosmological constant. Note: where possible, calculated results have been compared to physical measurement. Cognisant of experimental uncertainty, key predictions herein may be considered to be exact. 8

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Table of Contents Synopsis ......................................................................................................................................... 3 Preface ........................................................................................................................................... 5 Scientific Achievements ................................................................................................................ 7 1

Quinta Essentia ................................................................................................................... 23

1.1 The aether ......................................................................................................................... 23 1.1.1 The void ................................................................................................................... 23 1.1.2 The platonic solids.................................................................................................... 24 1.1.3 The laws of motion................................................................................................... 26 1.1.4 The luminiferous aether............................................................................................ 27 1.1.5 Michelson and Morely.............................................................................................. 29 1.1.6 Space-Time .............................................................................................................. 30 1.1.7 The Casimir Effect ................................................................................................... 30 1.2

Inertia................................................................................................................................ 32

1.3

Material waves .................................................................................................................. 33

1.4

Equilibration and virtual reality ......................................................................................... 33

1.5

QVIH ................................................................................................................................ 34

1.6

Bridging the gaps .............................................................................................................. 35

1.7 The Polarisable Vacuum.................................................................................................... 36 1.7.1 Blind-sighted ............................................................................................................ 36 1.7.2 Optical gravity.......................................................................................................... 36 1.7.3 Shaping the lens ....................................................................................................... 37 1.7.4 Asymmetry, equilibrium and “KPV”.......................................................................... 38 1.7.5 Conflux .................................................................................................................... 38 2

Electro-Gravi-Magnetics (EGM) ........................................................................................ 41

2.1

Introduction....................................................................................................................... 41

2.2

Similitude.......................................................................................................................... 41

2.3

Precepts and principles ...................................................................................................... 42

2.4

Gravity .............................................................................................................................. 43

2.5

Elementary particles .......................................................................................................... 47

2.6

Cosmology ........................................................................................................................ 48

2.7 Technical summary ........................................................................................................... 51 2.7.1 Synopsis ................................................................................................................... 51 2.7.2 The QV spectrum ..................................................................................................... 53 9

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2.7.3 2.7.4 2.7.5 2.7.6 2.7.7 2.7.8 2.7.9 2.7.10 2.7.11 2.7.12 2.7.13 2.7.14 2.8 3

The EGM spectrum .................................................................................................. 53 The ZPF spectrum .................................................................................................... 53 The PV spectrum ...................................................................................................... 54 The EGM, PV and ZPF spectra................................................................................. 56 The Casimir Effect ................................................................................................... 56 Comparative spectra ................................................................................................. 57 Characterisation of the gravitational spectrum .......................................................... 61 Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” characteristics ..... 62 Fundamental Cosmology .......................................................................................... 62 Advanced Cosmology............................................................................................... 62 Gravitational Cosmology.......................................................................................... 63 Particle Cosmology .................................................................................................. 63

Key point summary ........................................................................................................... 63 The PV Model of Gravity.................................................................................................... 65

3.1

Synopsis ............................................................................................................................ 65

3.2

Introduction....................................................................................................................... 65

3.3

Precepts and principles ...................................................................................................... 65

3.4

ZPF transformations .......................................................................................................... 66

3.5

PV transformations............................................................................................................ 67

3.6 The Schwarzschild solution ............................................................................................... 69 3.6.1 Special note .............................................................................................................. 69 3.6.2 Abstract.................................................................................................................... 69 3.6.3 Introduction.............................................................................................................. 69 3.6.4 “KPV” ....................................................................................................................... 69 3.6.5 “F(KPV)”................................................................................................................... 70 3.6.6 “LD(KPV)”................................................................................................................. 70 3.6.7 Conclusion ............................................................................................................... 71 3.7 The Reissner-Nordstrom solution ...................................................................................... 71 3.7.1 Special note .............................................................................................................. 71 3.7.2 Abstract.................................................................................................................... 71 3.7.3 “LD(KPV)”................................................................................................................. 71 3.7.4 “ψ1,2”........................................................................................................................ 71 3.7.5 Conclusion ............................................................................................................... 72 3.8 The generalised PV equations of motion............................................................................ 73 3.8.1 Special note .............................................................................................................. 73 3.8.2 Abstract.................................................................................................................... 73 3.8.3 Time-independent solutions...................................................................................... 73 3.8.4 Co-ordinate systems ................................................................................................. 73 3.8.4.1 Cartesian................................................................................................................ 73 3.8.4.2 Spherical................................................................................................................ 73 3.8.4.3 Cylindrical ............................................................................................................. 74 3.8.5 “KL” ......................................................................................................................... 74 3.8.6 Conclusion ............................................................................................................... 74 10

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The Natural Philosophy of Fundamental Particles ............................................................ 75

Abstract ........................................................................................................................................ 75 4.1

Harmonic representation of gravitational acceleration........................................................ 76

4.2

Poynting Vector “Sω” ........................................................................................................ 78

4.3

The size of the Proton, Neutron and Electron (radii: “rπ”, “rν”, “rε”) .................................. 79

4.4 The harmonic representation of fundamental particles ....................................................... 81 4.4.1 Establishing the foundations ..................................................................................... 81 4.4.2 Improving accuracy .................................................................................................. 81 4.4.3 Formulating an hypothesis........................................................................................ 82 4.4.4 Identifying a mathematical pattern............................................................................ 82 4.4.5 Results...................................................................................................................... 83 4.4.5.1 Harmonic evidence of unification .......................................................................... 83 4.4.5.2 Recent developments ............................................................................................. 84 4.4.5.2.1 PDG mass-energy ranges ................................................................................. 84 4.4.5.2.2 Electron Neutrino and Up / Down / Bottom Quark mass................................... 85 4.4.5.2.3 Top Quark mass ............................................................................................... 85 4.4.5.2.3.1 The dilemma ............................................................................................. 85 4.4.5.2.3.2 The resolution ........................................................................................... 85 4.4.6 Discussion ................................................................................................................ 86 4.4.6.1 Experimental evidence of unification ..................................................................... 86 4.4.6.2 The answers to some important questions............................................................... 87 4.4.6.2.1 What causes harmonic patterns to form? .......................................................... 87 4.4.6.2.1.1 ZPF equilibrium ........................................................................................ 87 4.4.6.2.1.2 Inherent quantum characteristics................................................................ 87 4.4.6.2.2 Why haven’t the “new” particles been experimentally detected? ...................... 88 4.4.6.2.3 Why can all fundamental particles be described in harmonic terms?................. 88 4.4.6.2.4 Why is EGM a method and not a theory? ......................................................... 89 4.4.6.2.5 What would one need to do, in order to disprove EGM?................................... 89 4.4.6.2.6 Why does EGM produce current and not constituent Quark masses? ................ 89 4.4.6.2.7 Why does EGM yield only the three observed families?................................... 89 4.5

What may the periodic table of elementary particles look like under EGM?....................... 90

4.6

Graphical representation of fundamental particles under EGM .......................................... 91

4.7

Concluding remarks........................................................................................................... 92

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The Natural Philosophy of the Cosmos .............................................................................. 93

Abstract ........................................................................................................................................ 93 5.1

Introduction....................................................................................................................... 94

5.2 Objectives and scope ......................................................................................................... 95 5.2.1 What is derived?....................................................................................................... 95 5.2.2 How is it achieved? .................................................................................................. 95 5.3

Derivation process............................................................................................................. 95 11

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5.3.1 5.3.2 5.3.3 5.3.4

Hubble constant “HU”............................................................................................... 95 CMBR temperature “TU”.......................................................................................... 96 “HU → HU2, TU → TU2 → TU3”................................................................................. 97 Rate of change “dHdt”............................................................................................... 99

5.4 Sample results ................................................................................................................. 100 5.4.1 Numerical evaluation and analysis.......................................................................... 100 5.4.1.1 Cosmological properties....................................................................................... 100 5.4.1.2 Significant temporal ordinates.............................................................................. 102 5.4.2 Graphical evaluation and analysis........................................................................... 103 5.4.2.1 Average Cosmological temperature vs. age .......................................................... 103 5.4.2.2 Magnitude of the Hubble constant vs. Cosmological age...................................... 104 5.4.3 Cosmological evolution process.............................................................................. 105 5.4.4 History of the Universe according to EGM ............................................................. 106 5.5 Discussion ....................................................................................................................... 107 5.5.1 Conceptualization................................................................................................... 107 5.5.1.1 “λx” ..................................................................................................................... 107 5.5.1.2 “TL” ..................................................................................................................... 108 5.5.1.3 “CΩ_J” .................................................................................................................. 108 5.5.1.4 “Stω”.................................................................................................................... 109 5.5.2 Dynamic, kinematic and geometric similarity ......................................................... 109 5.5.2.1 “HU” .................................................................................................................... 109 5.5.2.2 “TU”..................................................................................................................... 110 5.6

Concluding remarks......................................................................................................... 110

5.7 Graphical summary ......................................................................................................... 112 5.7.1 “TU3 vs. Hβ”: Figure 4.22........................................................................................ 112 5.7.2 “TU3 vs. t = (HβHα)-1” (i): Figure 4.23..................................................................... 113 5.7.3 “TU3 vs. t = (HβHα)-1” (ii): Figure 4.24.................................................................... 114 5.7.4 “TU3 vs. t = (HβHα)-1” (iii): Figure 4.25................................................................... 115 5.7.5 “TU3 vs. H = (HβHα)” (i): Figure 4.26 ..................................................................... 116 5.7.6 “TU3 vs. H = (HβHα)” (ii): Figure 4.27 .................................................................... 117 5.7.7 “TU3 vs. r = (HβHα)-1c” (i): Figure 4.28 ................................................................... 118 5.7.8 “TU3 vs. r = (HβHα)-1c” (ii): Figure 4.29.................................................................. 119 5.7.9 “dTU4/dt vs. t = (HβHα)-1” (i): Figure 4.30............................................................... 120 5.7.10 “dTU4/dt vs. t = (HβHα)-1” (ii): Figure 4.31.............................................................. 121 5.7.11 “d2TU4/dt2 vs. t = (HβHα)-1” (i): Figure 4.32 ............................................................ 122 5.7.12 “d2TU4/dt2 vs. t = (HβHα)-1” (ii): Figure 4.33 ........................................................... 123 5.7.13 “|d3TU4/dt3| vs. t = (HβHα)-1” (i): Figure 4.34........................................................... 124 5.7.14 “|d3TU4/dt3| vs. t = (HβHα)-1” (ii): Figure 4.35.......................................................... 125 5.7.15 “dH/dt vs. (HβηHα)-1” (i): Figure 4.36 ..................................................................... 126 5.7.16 “dH/dt vs. (HβηHα)-1” (ii): Figure 4.37 .................................................................... 127 5.7.17 “dH/dt vs. (HβηHα)-1” (iii): Figure 4.38 ................................................................... 128 5.7.18 “dH/dt vs. (HβηHα)-1” (iv): Figure 4.39.................................................................... 129 5.7.19 “d2H/dt2 vs. (HβηHα)-1” (i): Figure 4.40................................................................... 130 5.7.20 “d2H/dt2 vs. (HβηHα)-1” (ii): Figure 4.41.................................................................. 131 5.7.21 “d2H/dt2 vs. (HβηHα)-1” (iii): Figure 4.42................................................................. 132 5.7.22 “d2H/dt2 vs. (HβηHα)-1” (iv): Figure 4.43................................................................. 133 5.7.23 “d2H/dt2 vs. (HβηHα)-1” (v): Figure 4.44 .................................................................. 134 12

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5.7.24 5.7.25 5.7.26 5.7.27 5.7.28 6 6.1

“d2H/dt2 vs. (HβηHα)-1” (vi): Figure 4.45 ................................................................. 135 “|H| vs. (HβηHα)-1” (i): Figure 4.46.......................................................................... 136 “|H| vs. (HβηHα)-1” (ii): Figure 4.47......................................................................... 137 “TU2,3 vs. |H|”: Figure 4.48...................................................................................... 138 “TU2 vs. |H|”: Figure 4.49........................................................................................ 139

“q0, Λ0”............................................................................................................................... 141 SMoC.............................................................................................................................. 141

6.2 EGM ............................................................................................................................... 141 6.2.1 “Ro, MG, ΩEGM, Ω ZPF”............................................................................................. 141 6.2.2 “ΩZPF → –q0 → qSM_1” ........................................................................................... 142 6.2.3 “Λ0”........................................................................................................................ 143 6.2.4 “UΛ, UZPF, Uλ”........................................................................................................ 143 6.2.5 “T0”........................................................................................................................ 144 6.2.6 “ΛZPF → qSM_2”....................................................................................................... 145 6.2.7 “qSM_2 ≈ ±½” .......................................................................................................... 145 6.2.7.1 Construct ............................................................................................................. 145 6.2.7.2 Sample calculations ............................................................................................. 146 6.2.7.3 Analysis............................................................................................................... 147 6.3

EGM vs. SMoC ............................................................................................................... 147

6.4

Conclusion ...................................................................................................................... 148

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Definition of Terms ........................................................................................................... 149 Numbering conventions................................................................................................... 149

7.2 Quinta Essentia – Part 3................................................................................................... 149 7.2.1 Alpha Forms “αx”................................................................................................... 149 7.2.2 Amplitude Spectrum............................................................................................... 149 7.2.3 Background Field ................................................................................................... 149 7.2.4 Bandwidth Ratio “∆ωR”.......................................................................................... 149 7.2.5 Beta Forms “βx” ..................................................................................................... 149 7.2.6 Buckingham Π Theory (BPT)................................................................................. 149 7.2.7 Casimir Force “FPP”................................................................................................ 149 7.2.8 Change in the Number of Modes “∆nS” .................................................................. 149 7.2.9 Compton Frequency “ωCx” ..................................................................................... 149 7.2.10 Cosmological Constant........................................................................................... 150 7.2.11 Critical Boundary “ωβ”........................................................................................... 150 7.2.12 Critical Factor “KC”................................................................................................ 150 7.2.13 Critical Field Strengths “EC and BC”....................................................................... 150 7.2.14 Critical Frequency “ωC” ......................................................................................... 150 7.2.15 Critical Harmonic Operator “KR H” ......................................................................... 150 7.2.16 Critical Mode “NC”................................................................................................. 150 7.2.17 Critical Phase Variance “φC” .................................................................................. 150 7.2.18 Critical Ratio “KR” ................................................................................................. 150 7.2.19 Curl ........................................................................................................................ 150 7.2.20 DC-Offsets ............................................................................................................. 150 13

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7.2.21 7.2.22 7.2.23 7.2.24 7.2.25 7.2.26 7.2.27 7.2.28 7.2.29 7.2.30 7.2.31 7.2.32 7.2.33 7.2.34 7.2.35 7.2.36 7.2.37 7.2.38 7.2.39 7.2.40 7.2.41 7.2.42 7.2.43 7.2.44 7.2.45 7.2.46 7.2.47 7.2.48 7.2.49 7.2.50 7.2.51 7.2.52 7.2.53 7.2.54 7.2.55 7.2.56 7.2.57 7.2.58 7.2.59 7.2.60 7.2.61 7.2.62 7.2.63 7.2.64 7.2.65 7.2.66 7.2.67 7.2.68 7.2.69 7.2.70 7.2.71

Dimensional Analysis Techniques (DAT's) ............................................................ 151 Divergence ............................................................................................................. 151 Dominant Bandwidth.............................................................................................. 151 Electro-Gravi-Magnetics (EGM) ............................................................................ 151 Electro-Gravi-Magnetics (EGM) Spectrum............................................................. 151 Energy Density (General) ....................................................................................... 151 Engineered Metric .................................................................................................. 151 Engineered Refractive Index “KEGM”...................................................................... 151 Engineered Relationship Function “∆K0(ω,X)”....................................................... 151 Experimental Prototype (EP) .................................................................................. 151 Experimental Relationship Function “K0(ω,X)”...................................................... 151 Fourier Spectrum.................................................................................................... 151 Frequency Bandwidth “∆ωPV” ................................................................................ 152 Frequency Spectrum ............................................................................................... 152 Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)” ...................................................... 152 Fundamental Harmonic Frequency “ωPV(1,r,M)”.................................................... 152 General Modelling Equations (GMEx) .................................................................... 152 General Relativity (GR).......................................................................................... 152 General Similarity Equations (GSEx) ...................................................................... 152 Gravitons “γg” ........................................................................................................ 152 Graviton Mass-Energy Threshold “mγg” ................................................................. 152 Group Velocity....................................................................................................... 152 Harmonic Cut-Off Frequency “ωΩ” ........................................................................ 152 Harmonic Cut-Off Function “Ω” ............................................................................ 153 Harmonic Cut-Off Mode “nΩ” ................................................................................ 153 Harmonic Inflection Mode “NX”............................................................................. 153 Harmonic Inflection Frequency “ωX”...................................................................... 153 Harmonic Inflection Wavelength “λX”.................................................................... 153 Harmonic Similarity Equations (HSEx)................................................................... 153 IFF ......................................................................................................................... 153 Impedance Function ............................................................................................... 153 Kinetic Spectrum.................................................................................................... 153 Mode Bandwidth .................................................................................................... 153 Mode Number (Critical Boundary Mode) “nβ”........................................................ 153 Number of Permissible Modes “N∆r” ...................................................................... 154 Phenomena of Beats ............................................................................................... 154 Photon Mass-Energy Threshold “mγ”...................................................................... 154 Polarisable Vacuum (PV) ....................................................................................... 154 Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ” .................................................. 154 Polarisable Vacuum (PV) Spectrum........................................................................ 154 Potential Spectrum.................................................................................................. 154 Poynting Vector...................................................................................................... 154 Precipitations.......................................................................................................... 154 Primary Precipitant................................................................................................. 154 Radii Calculations by Electro-Gravi-Magnetics (EGM) .......................................... 155 Range Factor “Stα” ................................................................................................. 155 Reduced Average Harmonic Similarity Equations (HSExA R) .................................. 155 Reduced Harmonic Similarity Equations (HSEx R) .................................................. 155 Refractive Index “KPV”........................................................................................... 155 Representation Error “RError” .................................................................................. 155 RMS Charge Radii (General).................................................................................. 155 14

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7.2.72 7.2.73 7.2.74 7.2.75 7.2.76 7.2.77 7.2.78 7.2.79 7.2.80 7.2.81 7.2.82 7.2.83 7.2.84 7.2.85 7.2.86 7.2.87 7.2.88 7.2.89 7.2.90 7.2.91 7.2.92 7.2.93

RMS Charge Radius of the Neutron “rν”................................................................. 155 Similarity Bandwidth “∆ωS” ................................................................................... 155 Spectral Energy Density “ρ0(ω)” ............................................................................ 156 Spectral Similarity Equations (SSEx) ...................................................................... 156 Subordinate Bandwidth .......................................................................................... 156 Unit Amplitude Spectrum ....................................................................................... 156 Zero-Point-Energy (ZPE)........................................................................................ 156 Zero-Point-Field (ZPF) ........................................................................................... 156 Zero-Point-Field (ZPF) Spectrum ........................................................................... 156 Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF” ................................................... 156 Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF”....................................... 156 Zero-Point-Field (ZPF) Beat Cut-Off Mode “nΩ ZPF” .............................................. 156 1st Sense Check “Stβ”.............................................................................................. 156 2nd Reduction of the Harmonic Similarity Equations (HSExA R)............................... 156 2nd Sense Check “Stγ” ............................................................................................. 156 3rd Sense Check “Stδ” ............................................................................................. 157 4th Sense Check “Stε”.............................................................................................. 157 5th Sense Check “Stη” ............................................................................................. 157 6th Sense Check “Stθ” ............................................................................................. 157 Physical Constants.................................................................................................. 157 Mathematical Constants and Symbols..................................................................... 158 Solar System Statistics............................................................................................ 158

7.3 Quinta Essentia – Part 4................................................................................................... 159 7.3.1 “Big-Bang”............................................................................................................. 159 7.3.2 Black-Hole “BH” ................................................................................................... 159 7.3.3 Broadband Propagation .......................................................................................... 159 7.3.4 Buoyancy Point ...................................................................................................... 159 7.3.5 CMBR Temperature “T0” ....................................................................................... 159 7.3.6 EGM-CMBR Temperature “TU”............................................................................. 159 7.3.7 EGM Flux Intensity “CΩ_J”..................................................................................... 159 7.3.8 EGM Hubble constant “HU” ................................................................................... 159 7.3.9 Event Horizon “RBH”.............................................................................................. 159 7.3.10 Galactic Reference Particle “GRP” ......................................................................... 159 7.3.11 Gravitational Interference ....................................................................................... 159 7.3.12 Gravitational Propagation ....................................................................................... 159 7.3.13 Hubble Constant “H0”............................................................................................. 160 7.3.14 Narrowband Propagation ........................................................................................ 160 7.3.15 Non-Physical .......................................................................................................... 160 7.3.16 Physical.................................................................................................................. 160 7.3.17 Primordial Universe................................................................................................ 160 7.3.18 Schwarzschild-Black-Hole “SBH”.......................................................................... 160 7.3.19 Schwarzschild-Planck-Black-Hole “SPBH”............................................................ 160 7.3.20 Schwarzschild-Planck-Particle................................................................................ 160 7.3.21 Singularity.............................................................................................................. 160 7.3.22 Singularity Radius “rS” ........................................................................................... 160 7.3.23 Solar Mass.............................................................................................................. 160 7.3.24 Super-Massive-Black-Hole “SMBH”...................................................................... 161 7.3.25 Total Mass-Energy ................................................................................................. 161 7.3.26 Astronomical / Cosmological statistics ................................................................... 161

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8

Glossary of Terms ............................................................................................................. 163

8.1 Quinta Essentia – Part 3................................................................................................... 163 8.1.1 Acronyms ............................................................................................................... 163 8.1.2 Symbols in alphabetical order................................................................................. 165 8.2 Quinta Essentia – Part 4................................................................................................... 172 8.2.1 Acronyms ............................................................................................................... 172 8.2.2 Symbols by chapter ................................................................................................ 173 8.2.3 Symbols in alphabetical order................................................................................. 177 9

Key Artefact and Equation Summary .............................................................................. 181

9.1 Quinta Essentia – Part 3................................................................................................... 181 9.1.1 Dimensional Analysis............................................................................................. 181 9.1.1.1 “KPV, K0(X)” ....................................................................................................... 181 9.1.1.2 “a(t)”.................................................................................................................... 181 9.1.2 General modelling and the critical factor ................................................................ 181 9.1.2.1 “KC” .................................................................................................................... 181 9.1.2.2 “GME1” ............................................................................................................... 181 9.1.2.3 “GME2” ............................................................................................................... 182 9.1.3 The engineered metric ............................................................................................ 182 9.1.3.1 “KR” .................................................................................................................... 182 9.1.3.2 “∆K0(ω,X)”.......................................................................................................... 182 9.1.3.3 “KEGM” (normal matter form)............................................................................... 182 9.1.4 Amplitude and frequency spectra............................................................................ 182 9.1.4.1 “CPV” ................................................................................................................... 182 9.1.4.2 “ωPV” ................................................................................................................... 182 9.1.4.3 “nΩ” ..................................................................................................................... 182 9.1.4.4 “Ω”...................................................................................................................... 182 9.1.4.5 “ωΩ” .................................................................................................................... 182 9.1.5 General similarity ................................................................................................... 183 9.1.5.1 “ωβ”..................................................................................................................... 183 9.1.5.2 EGM Wave Propagation ...................................................................................... 183 9.1.5.3 EGM Spectrum .................................................................................................... 183 9.1.6 Harmonic and spectral similarity ............................................................................ 183 9.1.6.1 “φC = 0°, 90°”, “EC, BC”....................................................................................... 183 9.1.6.2 “SSE4,5” ............................................................................................................... 184 9.1.6.3 DC-Offsets........................................................................................................... 184 9.1.6.4 “ωC”..................................................................................................................... 184 9.1.7 The Casimir Effect ................................................................................................. 184 9.1.7.1 “NX” .................................................................................................................... 184 9.1.7.2 “NC” .................................................................................................................... 184 9.1.7.3 “ωX” .................................................................................................................... 184 9.1.7.4 “FPV” ................................................................................................................... 184 9.1.7.5 “∆r, λx, Erms, Brms, 0c, ±π, ±π/2”........................................................................... 184 9.1.8 Physical characteristics........................................................................................... 185 9.1.8.1 Photon / Graviton................................................................................................. 185 9.1.8.1.1 “mγ” ............................................................................................................... 185 9.1.8.1.2 “mgg”.............................................................................................................. 185 9.1.8.1.3 “mγγ” .............................................................................................................. 185 9.1.8.1.4 “rγγ”................................................................................................................ 185 16

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9.1.8.1.4.1 Primary ................................................................................................... 185 9.1.8.1.4.2 Secondary................................................................................................ 185 9.1.8.1.5 “rgg” ............................................................................................................... 185 9.1.8.2 “α” ...................................................................................................................... 185 9.1.8.2.1 Primary .......................................................................................................... 185 9.1.8.2.2 Secondary ...................................................................................................... 185 9.1.8.3 “Stω”.................................................................................................................... 186 9.1.8.4 Electron, Muon, Tau ............................................................................................ 186 9.1.8.4.1 “rε, rµ, rτ”........................................................................................................ 186 9.1.8.4.2 “ren, rµn, rτn”.................................................................................................... 186 9.1.8.5 Proton, Neutron.................................................................................................... 186 9.1.8.5.1 “ωΩ” .............................................................................................................. 186 9.1.8.5.2 “rπ, rν”............................................................................................................ 186 9.1.8.5.3 “ρch”............................................................................................................... 186 9.1.8.5.4 “rdr”................................................................................................................ 186 9.1.8.5.5 “KS”............................................................................................................... 187 9.1.8.5.6 “b1, rX”........................................................................................................... 187 9.1.8.5.7 “rνM” .............................................................................................................. 187 9.1.8.5.8 “rπE”............................................................................................................... 187 9.1.8.5.9 “rπM” .............................................................................................................. 187 9.1.8.5.10 “rp”............................................................................................................... 187 9.1.8.6 Quark / Boson harmonics..................................................................................... 187 9.1.8.6.1 “Up Quark”.................................................................................................... 187 9.1.8.6.2 Electron ......................................................................................................... 187 9.1.8.7 Hydrogen Spectrum: “λA”.................................................................................... 188 9.1.9 Theoretical propositions ......................................................................................... 188 9.1.9.1 The Planck scale: “Kω, Kλ, Km” ........................................................................... 188 9.1.9.2 Particles ............................................................................................................... 188 9.1.9.2.1 Leptons: “mL(2), mL(3), mL(5)”...................................................................... 188 9.1.9.2.2 Quarks / Bosons: “mQB(5), mQB(6)”................................................................ 188 9.2 Quinta Essentia – Part 4................................................................................................... 189 9.2.1 Gravitation ............................................................................................................. 189 9.2.1.1 “Stg” .................................................................................................................... 189 9.2.1.2 “ωΩ_2”.................................................................................................................. 189 9.2.1.3 “aEGM_ωΩ” ............................................................................................................ 189 9.2.1.4 “StG”.................................................................................................................... 189 9.2.1.5 “ωΩ_3”.................................................................................................................. 189 9.2.1.6 “λΩ_3” .................................................................................................................. 189 9.2.1.7 “G” ...................................................................................................................... 189 9.2.1.8 “ωPV(nPV,r,M)3” ................................................................................................... 189 9.2.1.9 “StJ”..................................................................................................................... 189 9.2.1.10 “CΩ_J1, CΩ_Jω” .................................................................................................. 189 9.2.1.11 “nΩ_2”............................................................................................................... 190 9.2.1.12 “KDepp”............................................................................................................. 190 9.2.1.13 “KPV” ............................................................................................................... 190 9.2.1.14 “TL” ................................................................................................................. 190 9.2.1.15 “ωg” ................................................................................................................. 190 9.2.1.16 “ngg”................................................................................................................. 190 9.2.1.17 “rω” .................................................................................................................. 190 9.2.1.18 “aPV” ................................................................................................................ 190 17

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9.2.1.19 “ag” .................................................................................................................. 190 9.2.1.20 “gav”................................................................................................................. 191 9.2.2 Planck-Particles...................................................................................................... 191 9.2.2.1 “mx”..................................................................................................................... 191 9.2.2.2 “λx” ..................................................................................................................... 191 9.2.2.3 “ρm, ρS” ............................................................................................................... 191 9.2.2.4 “r3, M3”................................................................................................................ 191 9.2.3 SBH’s..................................................................................................................... 191 9.2.3.1 “StBH” .................................................................................................................. 191 9.2.3.2 “ωΩ_4”.................................................................................................................. 191 9.2.3.3 “rS” ...................................................................................................................... 191 9.2.3.4 “nΩ_4” .................................................................................................................. 192 9.2.3.5 “nΩ_5” .................................................................................................................. 192 9.2.3.6 “nBH” ................................................................................................................... 192 9.2.3.7 “ωΩ_5”.................................................................................................................. 192 9.2.3.8 “ωBH”................................................................................................................... 192 9.2.3.9 “ωΩ_6”.................................................................................................................. 192 9.2.3.10 “ωΩ_7”.............................................................................................................. 192 9.2.3.11 “ωPV_1”............................................................................................................. 192 9.2.3.12 “ng”.................................................................................................................. 192 9.2.4 Cosmology ............................................................................................................. 192 9.2.4.1 “r2, M2”................................................................................................................ 192 9.2.4.2 “λy” ..................................................................................................................... 192 9.2.4.3 “KU” .................................................................................................................... 193 9.2.4.4 “AU” .................................................................................................................... 193 9.2.4.5 “RU” .................................................................................................................... 193 9.2.4.6 “HU, HU2, HU5, |H|” .............................................................................................. 193 9.2.4.7 “Hα” .................................................................................................................... 193 9.2.4.8 “ρU, ρU2”.............................................................................................................. 193 9.2.4.9 “MU”.................................................................................................................... 194 9.2.4.10 “KT”................................................................................................................. 194 9.2.4.11 “TW” ................................................................................................................ 194 9.2.4.12 “StT” ................................................................................................................ 194 9.2.4.13 “TU, TU2, TU3, TU4, TU5” ................................................................................... 194 9.2.4.14 “dTdt, dT2dt2, dT3dt3” ........................................................................................ 194 9.2.4.15 “dHdt, dH2dt2”................................................................................................... 195 9.2.4.16 “t1, t2, t3, t4, t5”.................................................................................................. 195 9.2.5 ZPF ........................................................................................................................ 195 9.2.5.1 “ΩEGM” ................................................................................................................ 195 9.2.5.2 “ΩZPF”.................................................................................................................. 196 9.2.5.3 “UZPF”.................................................................................................................. 196 9.2.6 EGM Construct limits............................................................................................. 196 9.2.6.1 “ML” .................................................................................................................... 196 9.2.6.2 “rL” ...................................................................................................................... 196 9.2.6.3 “tL” ...................................................................................................................... 196 9.2.6.4 “ML / rL = MEGM / REGM = tL / tEGM” ..................................................................... 196 9.2.7 Particle-Physics ...................................................................................................... 196 9.2.7.1 “mγγ2” .................................................................................................................. 196 9.2.7.2 “mgg2” .................................................................................................................. 196 9.2.7.3 “rγγ2” .................................................................................................................... 196 9.2.7.4 “rgg2”.................................................................................................................... 197 18

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9.2.7.5 “Nγ” ..................................................................................................................... 197 9.2.7.6 “Qγ” ..................................................................................................................... 197 9.2.7.7 “Qγγ” .................................................................................................................... 197 9.2.7.8 “Qγγ2”................................................................................................................... 197 9.2.7.9 “tL / TL = mγγ / mγγ2 = Qγγ / Qγγ2”........................................................................... 197 9.2.8 Other useful relationships ....................................................................................... 197 APPENDIX 2.A......................................................................................................................... 199 APPENDIX 2.B ......................................................................................................................... 215 APPENDIX 2.C......................................................................................................................... 217 APPENDIX 2.D......................................................................................................................... 223 Quinta Essentia – Part 3 .............................................................................................................. 223 • MathCad 8 Professional: calculation engine............................................................ 223 a. Computational environment ........................................................................................ 223 b. Units of measure (definitions) ..................................................................................... 223 c. Constants (definitions) ................................................................................................ 224 e. Planck characteristics (definitions) .............................................................................. 225 f. Astronomical statistics ................................................................................................ 225 g. Other........................................................................................................................... 225 h. Arbitrary values for illustrational purposes.................................................................. 225 i. PV / ZPF equations ..................................................................................................... 226 j. Casimir equations........................................................................................................ 227 k. Fundamental particle equations ................................................................................... 228 l. Particle summary matrix 3.1........................................................................................ 232 m. Particle summary matrix 3.2........................................................................................ 233 n. Particle summary matrix 3.3........................................................................................ 234 o. Particle summary matrix 3.4........................................................................................ 235 p. Similarity equations .................................................................................................... 236 q. Calculation results....................................................................................................... 237 r. Resonant Casimir cavity design specifications (experimental)..................................... 245 • MathCad 12: High precision calculation results ...................................................... 247 a. Computational environment ........................................................................................ 247 b. Particle summary matrix 3.1........................................................................................ 247 c. Particle summary matrix 3.2........................................................................................ 248 d. Particle summary matrix 3.3........................................................................................ 249 e. Particle summary matrix 3.4........................................................................................ 250 Quinta Essentia – Part 4 .............................................................................................................. 253 • MathCad 8 Professional.......................................................................................... 253 a. Complete simulation ................................................................................................... 253 i. Computational environment..................................................................................... 253 ii. Units of measure (definitions).................................................................................. 253 iii. Constants (definitions) ......................................................................................... 253 iv. Astronomical statistics ......................................................................................... 253 v. Characterisation of the gravitational spectrum.......................................................... 253 1. “Ω → Ω1, nΩ → nΩ_1, ωΩ → ωΩ_1”....................................................................... 253 2. “g → ωΩ”............................................................................................................. 255 19

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“ωΩ_1 → ωΩ_2” ................................................................................................. 255 “ωΩ_1 → ωΩ_3” ................................................................................................. 257 3. “g → ωPV3” .......................................................................................................... 257 4. “SωΩ → c⋅Um”...................................................................................................... 258 5. “CΩ_J” .................................................................................................................. 258 vi. Derivation of “Planck-Particle” and SBH characteristics...................................... 259 1. “λx, mx” ............................................................................................................... 259 2. “ρm(λxλh,mxmh), Um(λxλh,mxmh)” ........................................................................ 261 3. Physicality of “Kλ” .............................................................................................. 261 4. “KPV @ λxλh”....................................................................................................... 261 i. “KPV = Undefined”........................................................................................... 261 ii. “KDepp = KPV”................................................................................................... 263 5. “ωΩ_3”.................................................................................................................. 264 6. “ωΩ_4”.................................................................................................................. 265 7. “rS” ...................................................................................................................... 266 i. “rS(λxλh)” ......................................................................................................... 266 ii. “rS(ΜΒΗ), rS(RΒΗ)” ........................................................................................... 266 iii. “MBH(rS)” ........................................................................................................ 267 8. “r → RBH”............................................................................................................ 268 i. “nΩ → nΩ_4, nΩ_5, nBH” ..................................................................................... 268 ii. “ωΩ → ωΩ_5, ωBH” ........................................................................................... 269 iii. “ωΩ_6, ωΩ_7, ωPV_1” .......................................................................................... 270 9. “TL” ..................................................................................................................... 271 10. “ωg, ngg”........................................................................................................... 272 11. BH’s ................................................................................................................ 273 vii. Fundamental Cosmology...................................................................................... 275 1. “Hα, HU” .............................................................................................................. 275 i. “AU, RU, HU”.................................................................................................... 275 ii. “Hα”................................................................................................................. 276 iii. “ρU”................................................................................................................. 276 iv. “MU”................................................................................................................ 277 2. “TU”..................................................................................................................... 277 3. “TU → TU2” ......................................................................................................... 278 4. “TU2 → Ro, MG, HU2, ρU2”.................................................................................... 279 5. “UZPF”.................................................................................................................. 281 viii. Advanced Cosmology .......................................................................................... 281 1. “nΩ_2 → nΩ_6” ...................................................................................................... 281 2. “KU2 → KU3” ....................................................................................................... 282 3. “HU2 → HU3, TU2 → TU3”..................................................................................... 282 4. “HU3 → HU4, TU3 → TU4”..................................................................................... 282 5. “HU4 → HU5, TU4 → TU5”..................................................................................... 282 6. “HU3, HU4, HU5, TU3, TU4, TU5” ............................................................................. 283 7. Time dependent characteristics ............................................................................ 283 8. History of the Universe ........................................................................................ 292 9. “ML, rL, tL, tEGM”.................................................................................................. 293 10. Radio astronomy .............................................................................................. 294 ix. Gravitational Cosmology ..................................................................................... 295 x. Particle Cosmology.................................................................................................. 297 b. Calculation engine ...................................................................................................... 299 i. Computational environment..................................................................................... 299 i. ii.

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ii. Standard relationships.............................................................................................. 299 iii. Derived constants................................................................................................. 299 iv. Base approximations / simplifications .................................................................. 300 v. SBH mass and radius ............................................................................................... 301 vi. “nΩ” ..................................................................................................................... 302 vii. “ωΩ, TΩ, λΩ” ........................................................................................................ 303 viii. Gravitation........................................................................................................... 305 ix. Flux intensity ....................................................................................................... 306 x. Photon and Graviton populations ............................................................................. 308 xi. Hubble constant and CMBR temperature ............................................................. 309 xii. SBH temperature ................................................................................................. 316 xiii. ZPF...................................................................................................................... 317 xiv. Cosmological limits ............................................................................................. 318 xv. Particle Cosmology.............................................................................................. 318 • MathCad 12............................................................................................................ 321 c. High precision calculation engine................................................................................ 321 i. Computational environment..................................................................................... 321 ii. Astronomical statistics ............................................................................................. 321 iii. Derived constants................................................................................................. 321 iv. Algorithm ............................................................................................................ 321 d. Various forms of the derived constants........................................................................ 322 Bibliography 2........................................................................................................................... 323 Periodic Table of the Elements ................................................................................................. 324 Cosmological Evolution Process ............................................................................................... 325 Notes

22, 40, 50, 64, 72, 140, 158, 161-162, 164, 171-172, 176, 180, 198, 214, 216, 221222, 246, 251-252, 298, 320, 322, 327, 328 ERRATA

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NOTES

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1 Quinta Essentia 1.1

The aether

“Among the great things which are found among us, the existence of nothing is the greatest.” • Leonardo da Vinci6 1.1.1 The void The Sun, the Earth and all the planets of our solar system float in the vast expanse of space along with every other material object in the Universe. Our world is effortlessly, almost magically, suspended in an equally mystical, indefinable void. It may easily be assumed that few people today ever give a moment of thought to the question of what space actually is. To others, this question is an obsession. The nature of space has been the source of philosophical and scientific debate for thousands of years. This debate began as a rational argument to substantiate the existence of “nothing”. Before humanity had any experiential knowledge of space, a debate raged about whether a threedimensional volume could be completely devoid of all substance. If there was in-fact a true void, could it even be thought to exist? Over the centuries, “the void” eventually gained acceptance as a truism, but then the debate shifted to questions concerning the physical nature of nothingness. Was the void truly nothing, or is it composed of an aethereal substance? The question posed by philosophers throughout the ages is: how can “nothing” exist as part of our reality, that is, since “nothing” represents a state of non-existence, it is a paradox and a contradiction in terms. Some ancient Greek philosophers expressly opposed the existence of the void for this reason. But the precise definition of the void at that time was considered to be a true and complete nothingness. One interpretation of the vacuum was related to the idea of “zero”, which is in many ways just as unfathomable as the concept of “infinity”. The Roman poet Lucretius is well known for the phrase: “ex nihilo nihil fit”, meaning, “nothing comes from nothing” – an idea originally expressed by the Greek philosopher Empedocles (495-435 BC). Empedocles’ view was that everything in our material Universe had to be born of something else, something tangible. Something cannot be created from nothing, nor could anything simply disappear into nothingness. To the Greek philosophers in this particular camp, everything that is, is and forever will be, so there was no rational way to include the idea of nothing or the state of non-existence into arguments regarding the nature of matter. It is this overlying concept that marks the birth of what is referred to today as “conservation of energy” in contemporary Physics. This means that energy can neither be created nor destroyed, but only transformed or exchanged. It’s like accounting, or balancing your bank account. Although we all may wish that money could magically appear in our bank account, or that we could just “addon” an extra zero to the end of our balance, we can’t. The money has to come from somewhere. In the same way that currency is exchanged for goods and services, paid to us for the work we do, the same is true of energy – the currency of the Universe. Leucippus (5th century BC) and his student Democritus (460-370 BC) are both referred to as being “atomists”. This is because they introduced the notion that matter is composed of eternal, indivisible, fundamental units. A pure substance, the atomists would say, could be divided and subdivided again and again until at some point it could be divided no further. The end-point of matter was called “atomos”, meaning “without parts”. But the philosophical and logical invention of the atom required something special – namely, a void. All of those unseen atoms that make up matter would need some free space to move around in – to rearrange themselves and form structures within. If there were no space, then there would be no movement and no transformation of matter that we witness in our commonplace experience. There would likewise be no cause-and-effect and the ever-dynamic motions of the Universe would cease. The Cosmos would be frozen without time. 6

The Notebook, translated and edited by E. Macurdy, London (I954), p.6I. Leonardo da Vinci. 23

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The Pythagoreans, as Aristotle wrote, believed that: “It is the void which keeps things distinct, being a separation and division of things”7. Aristotle (384-322 BC) didn’t completely agree with the atomists. In-fact, it was Aristotle himself who maintained that “Nature abhors a vacuum”. However, he didn’t necessarily disagree with them either, because his argument wasn’t actually rooted in a denial of the void itself. The argument now became an issue of defining terms. When we speak of a vacuum, what do we mean? Aristotle would contend that if one tried to create new space where there wasn’t space before; matter would always immediately rush in to fill that space. To use an example based in our own time, if one were to take a zip-sealed plastic sandwich bag, flatten it completely to remove all the air and then zip it shut, one will find that it isn’t possible to pull the sides apart in any way that could create a new space inside. Indeed, if one were to construct a similar experiment utilising something more rigid, like glass or metal, we know that it is in-fact possible to create a vacuum largely free of air and matter, but the creation of that vacuum doesn’t entail that a zone of “non-existence” has been substituted in its place. For example, the new vacuum between the walls of glass would still transmit light. This point is indicative of the direction the void would take in philosophical terms. The void was a necessity in the atomists’ view, but it was still impossible because a true void could never actually be created by any natural process. Something, whatever it may be, still had to occupy the new space that was created. But what was it, exactly, that was rushing in to fill the space if it wasn’t some form of matter? The spaces that both permeate objects and separate them from each other must be composed of something for this line of reasoning to be compatible with experience and observation. When a vacuum is created, it may be devoid of all matter, but according to Aristotle, it must still be something. It was the Hellenistic philosopher, Zeno of Citium (333-264 BC), who’s teachings mark the beginnings of “Stoicism”8, so named because of the Painted Porch from which he taught. Like Aristotle, the Stoics also believed in a continuum of matter, or at least an absence of a true void in the presence of matter. They believed that there must be some kind of substance that occupied the space around objects, yet also completely permeated them, as if to say that all matter was imbibed with a spirit which imparts purpose for being. They called this substance “pneuma”, which was thought to be a mixture of fire and air – an energising fluid. But unlike Aristotle, who’s void-substance was somewhat static and eternal, the Stoics’ pneuma was dynamic and protected matter from simply dissolving away into the true void of nothingness which they believed existed, but only surrounded or encapsulated the pneuma. It is this concept of nothing which has its roots in what Empedocles called the “aether” – a mysterious and ubiquitous medium which surrounded and permeated matter. This so-called aether, supremely rarefied and quintessential, became the so-named aetherial substance that gives form to the void. The debate over nothingness (i.e. non-existence), became a futile endeavour beyond the realm of empirical study or solution. However, the nature and composition of the vacuum as a real substance called the aether would take the focus of debate evermore. 1.1.2 The platonic solids It was Plato (427-347 BC) who derived a mathematical interpretation of the aether. Plato, in a similar manner to the atomists, boiled down matter into its quintessential elemental constituents. Study of Pythagorean and Euclidian mathematics quite possibly provided the inspiration for his development of a rather poetic model of the Universe based on geometric symmetry. In his treatise called “Timaeus”, Plato describes a complete theory of matter based on what he called the “five perfect solids”.

7 8

Aristotle quoted in, An Introduction to Greek Philosophy, Boston, (1968), p. 75. J. Robinson. “Stoa” is Greek for “porch”. 24

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These solids represent the only perfectly symmetrical polyhedrons9 whose outer surfaces are entirely composed of a single type of regular polygon such as an equilateral triangle, a square or a pentagon. Other shapes, such as the hexagon, cannot form a polyhedron with a surface comprised only of hexagons. The first of these perfect polyhedrons is known as the “tetrahedron”; a three-dimensional shape consisting of four (4) equilateral triangles connected along their edges to form a three-legged pyramid structure. The next order of polyhedron is the “octahedron”; composed of two standard four-sided pyramids sandwiched together by sharing the square base of each pyramid, forming a diamond shape from eight (8) triangles. Even though the center of the diamond shape is a square on the inside, the surface is entirely composed of triangles. The third solid is the hexahedron (i.e. a simple cube). The fourth perfect solid, composed of twenty (20) equilateral triangles, is the “icosahedron”. The fifth and most unique solid, the “dodecahedron”, is composed of twelve (12) identical pentagons, forming a shape approximating a soccer ball. Each of these five solids formed Plato’s version of what we might think of today as a periodic table of elements which were thought to make up all material objects in the Universe10. Infact, the dodecahedron was considered so important by ancient philosophers that its existence was kept secret from the general population11. These five elements would go on to form the basis of Alchemy, which was practiced over the next few thousand years. Empedocles, before Plato, held the belief that there were only four elements, not including the aether, forming the basic atomic constituents of all matter; various concoctions of these four elements could create all objects and substances. The four elements themselves were Earth, Air, Fire and Water. The existence of the five Platonic solids implied that there must be an additional “fifth element” of matter, called “Quinta Essentia”. The Quinta Essentia was the aether itself; the substance that the heavens were made of. It was considered to be eternal, immutable and the source of all things. This marked a rather different way of viewing the aether because it transformed the void into the quintessential origin of all things. The Quinta Essentia didn’t simply infuse matter with spirit in the way that the pneuma was believed to do, rather, it was considered to be both the fabric of the void and the basis of matter itself. Plato surmised that these five elements, unlike Empedocles’ four elements, could split and merge into entirely new and larger atoms and thus form different substances, whereas the four fundamental elements of Empedocles were combined in various recipes to form substances with unique characteristics. In Plato’s model, the five elements correspond to each of the five perfect solids, Element Earth Air Fire Water Aether

Geometry Hexahedron Octahedron Tetrahedron Icosahedron Dodecahedron

Plato describes how the first four elements could recombine to form new elements; however, the dodecahedron was unique. The aether could not be broken up into more fundamental subunits or recombined with other elements like the others could. This is due to the fact that the surfaces of the other four solids may be further subdivided into two types of right triangles. One of these is formed by slicing a square diagonally through its centre. The other is produced by dividing an equilateral triangle by drawing a line from one tip through to the centre of the base, thus dividing it in half. What makes the dodecahedron unique in this case is that it is not possible to build a pentagon from 9

Three-dimensional shapes. Each solid represented one of “the five” atoms. 11 Carl Sagan, “Cosmos” television series. 10

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just these two types of right triangles, as it is for the other shapes. The other elements were malleable, whereas the aether was eternal. The Quinta Essentia thus became the fabric of the Cosmos upon which all matter was thought to be embroidered. Oddly enough, this kind of triangular symmetry is mirrored in the subatomic particles and Quarks comprising the atoms as we have come to understand them today. Our contemporary atomic model is composed of three subatomic units; Protons, Neutrons and Electrons. Moreover, the Proton is composed of two “Up” and one “Down” Quark; whereas the Neutron is composed of one “Up” and two “Down” Quarks. These Quark triplicates (even the triplicate subatomic components of the atom) can be likened to Plato’s sub-elemental triangles! Even though we now know that Plato’s conjectures were nothing more than philosophical representations of reality, it is quite surprising that the basic tenets of his theory display such prescience. One must wonder whether it was forethought and logic in his arguments holding true, or whether this connection hints at a deeper order in Nature which Plato was able to illuminate through his careful study of mathematical symmetry. 1.1.3 The laws of motion From Roman times to the dark ages, through the middle ages and the Renaissance, the Platonic solids formed the basis of Physics and Alchemy. The Quinta Essentia contributed to “Sir Isaac Newton’s” philosophical and scientific stance regarding the aether and remained a key ingredient in the many concoctions of alchemical practice, greatly contributing to the development of the modern Scientific Method. In his writings12, Newton was very careful to remind his readers that he would “feign no hypotheses” for what the aether could physically be, it remained the basis for his reasoning throughout. Newton felt that the aether should remain in the realm of the occult and metaphysics, but in Newton’s time, very little distinction was made between Alchemy and science. These were completely overlapping methods at a time when formalised science was beginning to burgeon. In Newton’s laws of motion, gravity was thought to be a “force” which attracted bodies to each other in the heavens, as it does on Earth. Objects invariably fall to the Earth and it was thought that an actual, physical force existed pulling everything to the surface. Even to this day, a colloquial notion of what gravity is persists in our language. We continue to call gravity a “force” and erroneously refer to it as though it has the ability to reach out and pull in objects from afar. Gravity in Newton’s time was thought to be transmitted instantaneously through space via the aether, imparting a force on objects. Even though Newton implicated the aether as the medium transmitting force, he could not logically reconcile how the nature and behaviour of solids moving in a fluid could allow a “fluidlike” description of the aether, demonstrating how fluids act to impede the movements of objects. The planets moved eternally and without resistance through the aether, so how could objects move in a fluid without any resistance to slow their motion? If the aether was some kind of substance, it should cause resistance to the orbital motion of planets and cause them to spiral into the Sun. Newton writes in his work, “Opticks”: p.528, Qu.28. “A dense fluid can be of no use for explaining the phenomena of Nature, the motions of the planets and comets being better explained without it. It serves only to disturb and retard the motions of those great bodies, and make the frame of Nature languish; . . . so there is no evidence for its existence; and, therefore, it ought to be rejected. . . . the main business of natural philosophy is to argue from phenomena without feigning hypotheses, and to deduce causes from effects, till we come to the very first cause, which certainly is not mechanical and not only to unfold the 12

“Principia” (1687) and “Opticks” (1704). 26

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mechanism of the world, but chiefly to resolve these and such like questions. What is there in places almost empty of matter, and whence is it that the Sun and planets gravitate towards one another, without dense matter between them? Whence is it that Nature doth nothing in vain; and whence arises all that order and beauty which we see in the world?” Thus, Newton recognised that “something” occupying the spaces between objects accounted for transmission of heat and gravitational force. Otherwise, how could one object like the Earth affect the motions of the moon so far away with just empty space between them? In Newton’s time, this strange, disconnected cause-and-effect relationship was referred to as “action-at-a-distance”, sparking another great debate in Physics which raged until Albert Einstein’s development of General Relativity (GR) approximately two hundred years later. Newton’s work however, was immune from this argument, even though it remained a point of great contention, pestering and haunting him ceaselessly. Newton eloquently demonstrated that simply understanding the regular, predictable behaviour of Nature can often suffice, i.e., it is sometimes adequate to formulate a mathematical description of Nature’s laws “without feigning hypotheses”. He formulated a mathematical structure describing the motions of the planets without espousing a mechanical, physical manifestation of its behaviour. If it works, so be it; thus, it became possible to discuss the aether in purely philosophical terms without invoking it as a necessity for the laws of gravitation and mechanics. Newton’s equations have allowed us to design rockets to the moon, enabling rover missions to Mars and made other planetary explorations of our solar system possible. It was also Newton’s principles of optics which enabled us to design the photographic equipment producing the images of these adventures, igniting our imaginations. All of this has been possible without having to understand the mechanics of the aether. The need for the aether evaporated, even though its existence could be debated; the precise nature of it remained as mysterious and indefinable as ever. 1.1.4 The luminiferous aether One of the most triumphant and influential discoveries in scientific history was James Clerck Maxwell’s development of the four equations for ElectroMagnetism in 1864. Based upon earlier work by Michael Faraday, the introduction of the laws of ElectroMagnetism transformed the world forever. In much the same way that Newton derived the laws of motion and gravitation from first principles, by feigning no hypotheses and an uncorrupted observation of Nature, Maxwell was able to successfully merge the forces of Electricity and Magnetism into a system of interactions he termed “ElectroMagnetism”. Maxwell’s equations describe the behaviour and interaction of Electric and Magnetic fields with matter. He was the first to demonstrate that light is an oscillating wave of intertwined Electric and Magnetic fields. This hailed the development of Relativity and Quantum Mechanics (QM). His equations were a monumental achievement, not only because of their elegance, or because of their immense usefulness for technological applications, but because they verified a fundamental connection between Electricity and Magnetism, as these were once considered to be completely disparate phenomena. During the latter part of the 19th century, British Physicists13 continually returned to the notion of the “luminiferous” aether to assess emergent theories. This embodiment of the aether was so-named because it was believed to represent the medium which carried light, i.e., ElectroMagnetic (EM) waves; thus, termed “luminiferous”. During the Victorian period, technological advancements spawning the industrial revolution contributed to the rapid development of a mechanistic world-view. British society was bearing witness to the triumph of the machine; rapidly and drastically transforming the social and cultural landscape. The technological developments of the age shaped the spectacles through which British 13

Those following Maxwell’s lead to further describe ElectroMagnetism. 27

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Scientists would view the Universe, skewing the perception of theorists, causing them to conceptualise the fabric of space in terms of cogs and wheels14. This emergent perspective gave Physicists cause to explain ElectroMagnetism in terms of the mechanical language of the day. Light was discovered by Maxwell to be an EM wave propagating through space. But what did this actually mean? What were these waves of light propagating in? What were they made of? We can easily imagine waves of light propagating through space like waves on the surface of the ocean rolling towards the shore. However, the concept of waves implies movement through a medium. So the question was; what substance carried these waves of light? The aether was an attempt to provide a physical explanation for an abstract mathematical representation of Maxwell’s equations for ElectroMagnetism. The form of this physical substance was modelled after the most prevalent mechanistic imagery of the time. In our own era, some theorists argue for an “information” philosophy as the basis for conceptualising elementary particle phenomenon, so that matter itself may be conceptualised in terms of binary bits of information. The fundamental constituent of our known Universe, according to information theory, is not really bits of “stuff”, rather bytes of information15. In the “information and computing age”, we are naturally tempted to create philosophical models reflecting our own zeitgeist16; in precisely the same manner as Maxwellian Physicists in their quest for a mechanical interpretation of physical reality. The aether provided Scientists at the time with a convenient pedagogical tool for describing the manner in which Electric and Magnetic forces interacted. A driving desire existed to validate the aether, unifying physical phenomena as purely mechanical movements, in-line with the late 19th century’s view of the Universe. If the problem of the aether could be resolved, it would be a scientific triumph rivalling all others, providing instant fame and glory for those whom resolved it. Any mechanical proof of the aether’s structure eliminates the nagging problem of “action-at-adistance”, which had been a source of debate since Newton’s time. The Maxwellians wanted a complete theory quieting metaphysical questions regarding precisely how EM signals propagated and planets separated by vast distances in space could interact with one another. Following the lead of Descartes, theorists even developed models demonstrating how matter might be understood as a manifestation of the aether in the form of “vortex rings”. In this model, atoms are conceptualised as tiny stable vortices within the “fluidic” aether. But why hold on to these purely hypothetical models of space if one could successfully predict and harness the phenomenon of ElectroMagnetism via mathematical reasoning alone? By modelling the luminiferous aether, the classical Physicist could present a working hypothesis for EM waves demonstrating “how” they mechanically propagate through the aether. Physicists at the time wrestled with the concepts of “theory” and “method”. If the method works, is there any real need to provide a concrete theory to explain why the method works? Is it not enough to simply provide an elegant set of formulae which may be used to describe how Nature works, even if we still don’t understand why it works that way? The fervour with which the Maxwellians sought to solve the structure of the aether was largely motivated by a desire to unify Physics in its entirety. This desire remains just as strong today as it did in Maxwell’s time. Born of a want to provide an all-encompassing theory utilising the mechanical language of the Victorian era, the aether had such a manageable and useful quality that it seemed its discovery was not only attainable, but almost within reach. The aether became the Holy Grail for the British Maxwellians, not only to make their work on ElectroMagnetism credible, but to also render it immune to doubt and criticism. There remained a desperate need to hold onto the idea of the aether, whether it stemmed from a desire for personal triumph, or a sense of obligation to uphold the essence of Maxwell’s theory; even more persuasive was the tantalizing hint 14

An invisible, intricate and seamlessly connected clockwork of interactions via which the movements of objects and light travelled through space. 15 Fire in the Mind: Science, Faith and the Search for Order. Alfred A Knopf Publishers, New York (1995). George Johnson. 16 Spirit of the time. 28

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that there was a golden opportunity to finally unveil the mysterious inner structure of the Cosmos. The concept of the aether had great philosophical value, but only in its use as a tool. Eventually, with a greater reliance on purely mathematical approaches to the problems of ElectroMagnetism, emphasised by the results of the Michelson-Morely experiment, the aether was eventually abandoned. As the 19th century began to ebb away into the 20th, Einstein later explained that; “mechanics as the basis of Physics was being abandoned, almost unnoticeably, because its adaptability to the facts presented itself finally as hopeless”17. 1.1.5 Michelson and Morely The final nail in the coffin for the luminiferous aether came in the form of an experiment performed by Albert Michelson and Edward Morley in 188718; however, the basis for this came decades earlier. In 1803, Thomas Young demonstrated that when light was directed at an opaque screen with two slits, the light coming through each slit interfered with one another, forming a pattern on the wall behind the screen. Known as the “two-slit” experiment, Young discovered light to be “wave-like” creating peaks and troughs of interference. The Michelson and Morely experiment was formulated on the premise that if the Earth was moving through a fluid-like medium, we should be able to detect our movement through it. If the aether existed in the form envisioned by the Maxwellians, the speed of light through the aether should be relative to some “ground speed” of the aether itself. Michelson and Morley tested for this by emitting two perpendicular beams of light from a single point source, reflected by mirrors back to a single detector. Because of Young’s pioneering two-slit experiment, Michelson and Morley knew that the two beams would interfere indicating whether they travelled at different speeds. Taking into account the rotation of the Earth and relative motions around the Sun, they demonstrated that no matter what relative direction the beams of light were travelling, no interference pattern is observed indicative of a preferred direction of the aether. This experiment silenced the debate over the notion that a mechanical fluid-like aether filled space, acting as the medium through which light propagated. But Michelson and Morley more accurately proved that no preferred reference frame exists from which we can measure the propagation of light signals. This idea became the spring-board for Einstein’s Relativity theory, where light speed is constant and everything else, including time, is observed relative to the speed of light.

17 18

The Maxwellians, Cornell University Press, Ithaca and London, (1991). Bruce J. Hunt. Philos. Mag. S.5, 24 (151), p.449-463 (1887). A. A. Michelson and E.W. Morley. 29

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1.1.6 Space-Time Michelson and Morley disproved the existence of the mechanical luminiferous aether conceptualised in Maxwell’s era, but it did little to arrest the emergence of a contemporary version. Einstein is, at least partially, responsible for destroying the mechanical aether of old and replacing it with a new aether all his own. Although this time, unlike the Maxwellians, Einstein feigned no hypotheses for what physical manifestation the aether might take. Einstein’s development of Relativity and the notion of a new aether termed “curved space-time” evaporated the concept that gravity was a force mediated by the ill-defined aether of Newton’s time. Einstein’s equations demonstrate that an object’s motion in a gravitational field is determined by its “geodesic” path19. Einstein introduced this concept to describe gravitational interactions between mass-objects, eliminating the necessity for “action-at-a-distance”. Curved space-time is a geometric contrivance, but exactly what is being curved? And if the vacuum of space is indeed a formless void, then how may “nothing” have shape? GR not only invokes, but requires the existence of a medium (i.e. manifold) capable of conveying information indicating whether the space-time a mass-object transits is curved. On May 5th, 1920 at the University of Leiden in the Netherlands, Einstein gave an address on the issue of the aether stating, “According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense.” To Einstein, the physicality of the space-time fabric was as undeniable as it was indefinable. He desired an accurate and complete explanation for the physical Universe, in much the same way that the Maxwellians needed to interpret the physical meaning of ElectroMagnetism through an understanding of the luminiferous aether. Newton, like Maxwell, feigned no hypotheses in his explanation for why his equations were true, he only demonstrated that they were; Einstein did the same with Relativity. However, we must not take the notion of curvature too literally, but we are still left wrestling with the “imponderable” demon that is the aether. 1.1.7 The Casimir Effect A French nautical handbook from the 1830’s20 tells of the strange attraction between two large ships in the open ocean. The optimal condition for this attraction occurs when they are positioned side by side in moderate swell with little or no wind. If the ships drift within a distance of “30-40” meters from each other, the handbook states that they are gradually pulled together and their riggings entangled. To avert disaster, a small boat manned by “20-30” rowers would tow the ships apart. So what are we to make of this strange observation? Why is it that the two ships are drawn together in such a way? The effect, as it so happens, has to do with “boundary conditions”. The hulls of the two ships may be represented as two parallel lines in a uniform sea. The two ships delineate the otherwise uniform surface of the ocean, which in turn divides the ocean’s surface into distinct zones. Each ship establishes a physical boundary separating the region between them from the greater environment21. The Dutch Physicist, Sipko Boersma, noticed Causee’s nautical description of this strange effect during a visit to the Amsterdam Shipping Museum in the mid 1990’s and published a paper

19

i.e. the shortest temporal path between two points in a curved space-time manifold. L’Album du Marin, Charpentier, Nantes (1836). P.C. Causee. 21 The importance of boundary conditions cannot be overstated; they are the essential hallmark of dynamic systems. 20

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about it in 199622. Boersma sagaciously recognised that this nautical phenomenon observed in the early 1800’s was a real-world analogy for what is known as the “Casimir Effect” in QM. Hendrik Casimir, also a Dutch Physicist, predicted a similar effect based on the rules of QM nearly “50” years earlier. Casimir’s discovery marked a key turning point in the philosophical debate which has raged since ancient times on the true nature of the vacuum. The discovery of the Casimir Effect provided strong evidence, in the form of a physical, measurable force that the socalled empty vacuum of space is in-fact something more resembling a plenum of energy. The Casimir Effect is a QM manifestation of the nautical observations described in the 19th century French naval handbook, except that in the Casimir case, the environmental conditions affected by parallel boundaries are EM waves, propagating in an ocean of energy known as the Quantum Vacuum (QV). Quantum Field Theory (QFT) models the vacuum of space as being something quite different than what most of us might think. People generally assume that the vacuum of deep space is a true and complete void. Objects occupy a three-dimensional volume in the void and we conceptualise space as nothing more than a matrix containing matter; once the matter is removed, we are left with an empty space of the same volume as the matter which has been removed. However, QFT states that if we were to take a volume of space at the surface of the Earth for example, evacuate every last molecule of air, shield all thermal radiation so that the vacuum was at absolute zero temperature, a vacuum of space filled with energy remains; it can never be completely “evacuated” from a given volume like air. Energy will propagate throughout the volume because the QV is composed of energy propagating in sinusoidal form and can never fully come to rest; within QM, energy must cycle about its ground state. The Casimir Effect occurs when two neutrally charged conducting plates are placed in close proximity and parallel to one another, establishing boundary conditions in the QV. In such a configuration, an attractive force is observed between the plates, beyond that which may be attributed to gravitational attraction. The QV is comprised of EM wavefunctions which may only exist between the plates if their lengths are equal to or less than the plate separation distance “∆r”; any wave of longer length cannot exist within the gap! For example, if “∆r = 1(µm)”, only the QV modes of wavelength less than “1(µm)” may physically exist within that space. As the plates are drawn closer together, an increasing number of QV modes are excluded from existence between the plates. This implies that more Quantum-Vacuum-Energy (QVE) exists outside the plates than in between them. This energy difference pushes the plates together with a force inversely proportional to the plate separation distance. That is to say, as the plates get closer together, the force pushing them together becomes greater. It was not until 1997-8 that the Casimir Force was physically confirmed by two independent experiments. The first measurement was performed in 1997 by Steve Lamoreaux at the University of Washington, Seattle23. The second measurement was conducted in 1998 by Umar Mohideen and Anushree Roy at the University of California, Riverside24. In an experiment similar to that of Lamoreaux, Mohideen and Roy used an atomic Force Microscope capable of measuring values as low as “10-18(N)”. Although the Casimir Force is miniscule at the separation distances measured, confirmation of the Casimir Effect provided physical evidence for the existence of QVE, suggesting that aether-like energy forms the physical basis of space-time. 22

A maritime analogy of the Casimir Effect, American J Physics, 64, p. 539 (1996). S.L. Boersma. Demonstration of the Casimir Force in the 0.6 to 6 µm Range, Phys. Rev. Lett. 78, 5–8 (1997). S. K. Lamoreaux. 24 A Precision Measurement of the Casimir Force between 0.1 to 0.9 mm, Physical Review Letters, vol.81, (no.21), APS, (1998). U. Mohideen and Anushree Roy. 23

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1.2

Inertia “Give me a firm place to stand and I will move the earth.” • Archimedes (287-212 BC)

For thousands of years the aether has been invoked as a means of explaining various physical phenomena. However, all attempts have been relegated to the realm of speculation and philosophical exercise. Human psychology seems to require a medium to account for the balance of forces we experience in our lives. Birds fly by manipulating the balance of forces beneath their wings, fish swim by manipulating the balance of forces encasing their bodies and humans walk by manipulating the balance of forces beneath their feet. We observe force balancing everywhere; for an object to change velocity, a “push” is required to overcome the balance of uniform motion and in doing so, encounters an acceleration reaction force termed “inertia”. Objects in uniform motion will remain in that state unless otherwise acted upon by an outside force25. Newtonian gravity is analogous to inertia because objects experience the force of gravity in free space and respond accordingly. Why do objects not experience a force when in a uniform state of motion, but suddenly do when it changes from one uniform state to another? And what peculiar attribute of space provides us with this ability to register the difference between them? Why does matter resist acceleration if there is nothing in the way to impede it? The similarity between the manner in which matter experiences gravitational and inertial forces is primarily responsible for the development of modern Physics. The Czech-Austrian Physicist, Ernst Mach, proposed a possible mechanism for inertial forces and their connection to gravitation in the late 1800’s, while Einstein was only in his teens and just beginning to explore the frame of thought which would later lead to the development of Relativity. In-fact it was Einstein himself who, in describing Mach’s ideas on the subject of inertia, coined the term “Mach’s Principle”26. Mach’s original proposal was based upon the notion that all matter in the Universe is homogeneously distributed, connected by a web of gravitational interactions. Mach reasoned that all co-ordinates in space should experience the averaged effect of gravitational fields from all matter in the Universe. Thus, regardless of location, objects experience gravitational resistance opposing changes in motion. Mach studied the work of Johan Christian Andreas Doppler27 in great detail; consequently, Mach’s principle for the origin of inertia is analogous to a gravitational Doppler Effect28. Objects are attracted uniformly in all directions such that motion induces an immediate opposing force; the averaged gravitational field is compressed in the direction of motion29. Mach’s view implies that an object’s motion relative to the fabric of space30 induces inertial forces. Thus, energy input is required to counter the opposing force as it moves uniformly31. This is problematic because objects travelling with uniform motion do not experience inertial forces. If Mach’s principle were true, objects should experience inertial force at all times. However, inertia is only experienced upon acceleration. Although Mach’s principle was never formally developed into a quantitative, physical theory, a compelling aspect exists. Despite its inadequacies, Mach’s conceptualisation was correct in its premise that a relationship exists between gravitational and inertial forces.

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Newton’s first law of motion. http://en.wikipedia.org/wiki/Mach's_principle 27 http://en.wikipedia.org/wiki/Christian_Doppler 28 http://en.wikipedia.org/wiki/Doppler_effect 29 i.e. motion induces a decompression wake within space. 30 i.e. a pan-universal gravitational matrix. 31 A natural conclusion considering the human experience and knowledge of the era. 26

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The relationship between gravitational and inertial forces provides the basis for the “equivalence principle”; a key premise of GR asserting that geometric interpretations of curved space-time demonstrate why gravity is the same force as inertia. GR states that any diversion from an object’s geodesic path of motion in a curved space-time manifold induces an inertial reaction force. The most far-reaching of Einstein’s theories is the concept of mass-energy equivalence, described by “E = mc2”. Einstein realised this connection whilst considering inertia; energy is required to accelerate an object or alter its geodesic path through curved space-time, implying that the energy required to change the motion of mass is directly proportional to the mass itself. Thus, it follows that mass and energy are equivalent – mass is energy and energy is mass! Einstein’s epiphany leading to the derivation of “E = mc2” was that the more energy provided to accelerate an object, the more massive it becomes. Understanding the nature of inertia will allow us to understand the true nature of space and matter. The equivalence principle necessitates that if we can modify inertial force, we would also be able to manipulate gravity, giving ourselves a “firm place to stand” in Archimedes’ challenge. 1.3

Material waves

When Louis De Broglie32 was a student at the University of Paris in the early 1920’s, he learned that mass-less Photons possess momentum and a Photon’s frequency was a measure of its energy. Thus, if Photons of light possess the wave characteristic of frequency, could the Electron also have characteristics of waves? Or greater still, could all forms of matter possess wave-like characteristics as well? The answer, De Broglie discovered, was yes! Three years after De Broglie derived his hypothesis it was verified experimentally by Clinton Davisson and Lester Germer at Bell Labs. Experimental confirmation of De Broglie’s hypothesis earned him the Nobel Prize in 1929. Electrons not only appeared to move in waves, they may be considered to be waves. Subsequently, matter may be characterised by what is now termed its “De Broglie wavelength”. Note: the “Quinta Essentia” series extends this hypothesis by describing matter33 as a spectrum of wavelengths, rather than a single De Broglie wavelength. The physical attributes of wavelength and frequency are the characteristics by which we describe energy, providing the basis for the concept of mass-energy equivalence. Thus, by considering the fabric of space as being composed of energy, i.e. Plato’s fifth element “Quinta Essentia”, space is the basis for all matter. 1.4

Equilibration and virtual reality

We may consider the space-time curvature induced by the presence of matter to be a marriage of the manifold to the mass-energy introduced, constituting “a system”. All natural systems energetically equilibrate with their environment; hence, the QVE of space surrounding matter is as important as the matter it circumscribes. The reverse concept implies that the QV adopts different states dependent upon the local matter content. Thus, it follows that the presence of matter invokes a local boundary condition in the vacuum such that the mass-energy of the matter present must equilibrate with the local QVE34. QVE is also referred to as “vacuum energy”35 32

http://nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-bio.html In terms of mass-energy density. 34 This partitioning of space becomes particularly important at the level of subatomic particles. 35 This is a casual reference commonly utilised for generalised conversation. Please refer to the “Electro-Gravi-Magnetic (EGM): Technical summary” section for further information. 33

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because the energy considered is associated with the vacuum of space and not with matter. QM permits systems to briefly “borrow” vacuum energy fluctuations for the creation of “virtual particles”36, followed by re-absorption of the energy back into the QV. However, the process occurs so rapidly that we may never be able to detect them directly. One might wonder why particles are considered virtual instead of “real”; since the particles cannot be detected directly, we consider them to be theoretical37 – however, they induce real and measurable effects. The most dramatic insight to be gained by understanding the QV is that space affects matter and matter affects space; a relationship exists between matter, space and energy which cannot be severed. Although QVE is predicted and obliged to exist by the rules of QM, our psychological acceptance of the QV appears to be a suspension of disbelief rather than a sincere conviction. The broader scientific community is required to modify its collective perspective, emphasising future research pathways in terms of systems of interactive wholes rather than disconnected entities. 1.5

QVIH

QM states that space is replete with QV fluctuations. In the early 1990’s, Astrophysicist Bernard Haisch and Physicist Alfonso Rueda applied the concept of radiation pressure to the QV. A Poynting Vector38 is associated with each of the Photons39 within the QV; however, their chaotic and random distribution requires that the QV does not possess net direction. Thus, in a flat spacetime manifold the QV is said to be isometric40. Haisch and Rueda wondered what the QV might become from an accelerated reference frame. By applying standard ElectroDynamic principles, they determined that transformation of the QV from a stationary to an accelerated reference frame induces asymmetry; the field was no longer random and isometric. The QV in the accelerated frame appeared to have a net Poynting Vector associated with it. They discovered that the net Poynting Vector generated by the QV was proportional to the magnitude of the applied acceleration; thus, the greater the applied acceleration, the greater the QV resistance. Haisch and Rueda surmised that upon acceleration, matter experiences an EM drag-force against the QV, akin to radiation pressure. Object’s will only be affected by the QV when it appears to have a net direction, i.e., when it is asymmetric. The fact that the magnitude of the QV asymmetry appeared to be acceleration-dependent implies a physical basis for the existence of inertia. The model proposed by Haisch and Rueda is called the “QV Inertia Hypothesis” (QVIH). Why is it that an asymmetry in the QV arises only during acceleration and not during uniform motion? The QV is predicted to posses a “cubic frequency” distribution in a flat space-time manifold. Thus, at low frequencies, the QVE density is minimal; however, at high frequencies the QVE density is maximal (paralleling the cube of the frequency along the EM spectrum). This means that the highest frequency ranges of the EM spectrum contain the most QVE, making the 36

Virtual particles are invoked to explain conservation of energy and momentum during particle decay processes. They are also utilised to explain the Electro-Weak and Strong-Nuclear force inside the atom by virtual particle exchange between subatomic elements. The Electro-Weak force is generally described as resulting from a subatomic transfer of “virtual Photons”. Moreover, virtual particles are utilised to plot the creation and annihilation of intermediate particles formed at accelerator laboratories. Particles are collided at near light-speed and the resulting high-energy subatomic debris is explained through a mapping process often requiring the use of adjunct virtual particles. In addition, virtual particles are also utilised to describe the interaction and partitioning of energy in these exceedingly short-lived events. 37 See: Hawking and Davies-Unruh radiation. 38 The Poynting Vector is a physical measure of the direction and power associated with EM radiation. 39 Virtual and / or real. 40 i.e. equal in all directions. 34

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energy density of free-space inconceivably energetic. In-fact, it has been estimated that the amount of QVE contained in a coffee cup volume of empty space would be enough to boil away the Earth’s oceans41,42. Haisch and Rueda assert that the cubic frequency distribution of the QV explains why an object does not experience inertia during uniform motion. An object moving uniformly does not experience inertia because, regardless of velocity, ambient high-frequency QV Photons compensate for the Doppler Effect in the trailing direction of motion43. Hence, the distribution of the QV ensures that space-time appears flat and isometric for any object travelling in uniform motion. Therefore, the QVIH model remains consistent with GR and the inertial-frame view of space-time. GR states that an observer travelling in uniform motion through space-time perceives the Universe as being flat; however, in an accelerated reference frame space-time appears to be curved. Haisch and Rueda’s classical ElectroDynamics model of inertia asserts a congruent position; during uniform motion the QV appears symmetric. However, in an accelerated reference frame, asymmetry44 manifests in the QV; proportional to the magnitude of the applied acceleration. Thus, rather than the metaphysical non-intuitive terminology of GR, i.e., “flat” or “curved” spacetime, the QVIH facilitates the substitution of intuitive terminology, i.e., physically meaningful reference to “symmetrical” or “asymmetrical” QVE distributions. 1.6

Bridging the gaps

Einstein relied upon the equivalence principle to demonstrate that the space-time geometry of an accelerated reference frame is equivalent to a gravitational field; the same may be said for the QVIH. Haisch and Rueda have utilised the equivalence principle to demonstrate the manner in which QV asymmetry appears in an accelerated reference frame and a reference frame held fixed in a gravitational field. In the Earth’s gravitational field, an object experiences force due to local QVE asymmetry, producing a net energy flux downwards. QM was largely formulated several years after Einstein developed GR and he considered the whole field to be rather unpalatable. Einstein lacked the tools to offer any physical basis for why inertia existed, or why matter curved space-time. The QV had not yet been conceived at the time he was developing GR; thus, he had no foundation from which to derive a potential physical basis for gravity and inertia, aside from the “luminiferous aether”, which he believed did not exist. QM, through its prediction of the QV, states that the energy density of the Universe is enormous. However, when viewed through GR, the energy density of the QV should cause a catastrophic gravitational collapse of the Universe. This creates a major dilemma for Physicists and Astronomers; either QM or GR is somehow fundamentally flawed or incomplete, yet they have proven themselves to be highly accurate means of representing physical systems.

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Emerging Possibilities for Space Propulsion Breakthroughs, Interstellar Propulsion Society Newsletter, Vol. I, No. 1, (July 1, 1995). Marc G. Millis. 42 This is the mainstream view, not the view of the EGM construct in the “Quinta Essentia” series (i.e. QE3,4) where the opposite conclusion is mathematically derived. That is, QE3,4 mathematically demonstrate that “free space” does not contain a near infinite amount of energy in a vanishing volume. 43 Analogous to the negation of pressure-drag associated with motion through a Newtonian fluid. 44 i.e. anisotropy. 35

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1.7

The Polarisable Vacuum

1.7.1 Blind-sighted GR is a geometric model of gravity representing space-time as a four-dimensional manifold of events. It has profoundly enhanced our understanding of the Universe, yielding highly accurate predictions, yet does not explain why matter produces a gravitational field, or the specific mechanism by which matter experiences inertial forces upon acceleration. The GR space-time manifold is a vacuum – a void. If this is indeed the case, then an obvious question arises: how can nothing have curved four-dimensional geometry? 20th century Physics has replaced Newtonian “action-at-a-distance” with the equally abstract concept of “curved space-time”. Physicists have largely ignored the question of why objects “feel” force upon acceleration, surrendering without protest to the notion that it comes from nowhere, content to nod vacantly in peer agreement and accept the logical contradictions of “geometric nothingness” in lieu of predicting Nature. Many are so comfortable with this that they insist the relativistic tensor mathematics is space-time – substituting the abstraction for the phenomenon45. Physicist John Wheeler is noted for his concise description of GR: “matter tells space how to curve and curved space tells matter how to move.” However, does “matter curving space” cause light to bend? Ask an Engineer to bend light and they won’t attempt it by curving space-time; if you want to bend light, put it through a lens! 1.7.2 Optical gravity Bernard Haisch and Alfonso Rueda introduced a model describing matter as being immersed-in and wholly dependent upon the QV for its existence. This fed an intuitively appealing interpretation of space-time curvature termed the “Polarisable Vacuum (PV) Approach to GR”46. Harold Puthoff first introduced the PV model in 2002, having drawn upon earlier work by Harold Wilson, Robert Dicke and Andrei Sakharov. The PV model is an optical interpretation of gravity because it applies optical principles to define the topological features of space-time, otherwise represented geometrically within GR. It attributes space-time with a variable Refractive Index “KPV”, not “curvature”. The value of “KPV” is proportional to the energy density associated with a gravitational field. As light passes a massobject, it transits through regions of variable “KPV” and refracts (i.e. bends), in accordance with the experimentally verified results within the GR construct. Light is refracted as it passes from one gravitational field to another of differing strength (i.e. energy density). When light transits from a region of low “KPV” to high, it slows down; causing it to bend47. The degree to which it refracts also depends upon the “angle of incidence”. Sir Isaac Newton worked extensively in this field and we owe the science of Optics to him. He studied how lenses of differing shape and density bend light. However, its core principles are based upon Photon exchange in the governing theory; termed, Quantum-ElectroDynamics (QED). The PV model ascribes a value of “KPV” to the QV48 such that all matter generates a gradient in the energy density of the QV surrounding it. The gradient relates to a change in “KPV” acting as a space-time lens causing light to bend49. Hence, the PV model demonstrates that substituting the metaphysical conceptualisation of space-time curvature with a physically meaningful optical construct yields a congruent interpretation of gravity to that of GR. 45

GR is a descriptive tool, not a literal explanation of Nature. H. E. Puthoff, “Polarizable-Vacuum (PV) approach to general relativity”, Found. Phys. 32, 927943 (2002). 47 A change in propagation rate induces refraction. 48 The existence of the QV was derived from QM and is essential to QED. 49 Referred to as “gravitational lensing”. 46

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The key difference between interpretations is that the PV model describes the physical manner by which space-time is “curved”, GR does not. However, neither GR nor the PV model specifically addresses the precise mechanism by which matter physically polarises space-time. Fortunately, the PV model is not required to do so because QED explains this mechanism based upon the premise that within a volume of space-time devoid of matter, a chaotic and equally distributed mix of virtual Electron-Positron particle pairs is said to “pop” into and out of existence. 1.7.3 Shaping the lens The PV model asserts that matter polarises50 the QV into variable regions of energy density which, in turn, generates regions of variable “KPV”. A well-developed precedent for the existence of vacuum polarisation exists, based upon the generally accepted model of the Electron. The contemporary model of the Electron stems from QED51, modelling it as a negatively charged point core surrounded by a cloud of virtual particle pairs52, constantly emerging from and disappearing into the QV. The presence of the negatively charged core attracts the virtual positive charges and repels the virtual negative charges present in the vacuum, biasing the QV, resulting in a vacuum gradient as it segregates clustered regions of virtual charges. In this state the vacuum is no longer uniform – it has been polarised. The effect of an Electron on the QV is termed “vacuum polarisation” and the property of charge emerges due to a change in the QVE distribution of the surrounding space-time. Thus, if the QV is effervescent with virtual particle pairs, we must consider its effect on all elementary particles, not just the Electron. From the perspective of the PV model; matter polarises the QV, forming gravitational fields because its atomic constituents are composed of countless numbers of elementary particles, all generating their own localised polarisations of the vacuum such that the cumulative effect results in a large-scale, synergistic polarisation. Conceptualising the space-time manifold in terms of vacuum polarisation yields an isomorphic representation of GR. Moreover, the PV interpretation combined with the QVIH advances a meaningful physical explanation for why matter experiences gravitational and inertial force, whereas GR offers very little in this regard53. In the PV model, matter generates a polarised gradient manifesting as a change in “KPV” such that gravitational and inertial force is experienced whenever the QV appears asymmetrical. Thus, if matter is capable of polarising the QVE distribution asymmetrically, then we immediately understand why matter generates a gravitational field. 50

i.e. enforces direction and order. http://Physics.nist.gov/cuu/Constants/alpha.html: according to QED and the relativistic QFT of the interaction of charged particles and Photons, an Electron may emit virtual Photons which, in turn, may become virtual Electron-Positron pairs. The virtual Positrons are attracted to the “bare” Electron whilst the virtual Electrons are repelled from it. The bare Electron is therefore screened due to polarisation. 52 Hawking and Davies-Unruh radiation are also derived from the principle of virtual particle pair formation. 53 Within GR, gravitational weight and inertial force are generated through a geometric manifold defined by the geodesic paths of light in curved space-time. When an object deviates from the geodesic topology of space-time, it experiences a force and energy input is required to affect the deviation from the geodesic path. When an object accelerates or is held fixed in a gravitational field, space-time appears curved and the object experiences a force. When the object moves with uniform motion in flat space-time, or falls along the geodesic path in a curved manifold, space-time will appear to be flat and no force will be experienced. 51

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When the optical effects of asymmetrical energy densities are considered, it is clear how the PV state affects the propagation of light. As light enters an area of vacuum polarisation, it is affected by the changing “KPV” and arcs towards the mass-object, as if it were passing through a lens. The curved path resulting from the changing “KPV” is precisely congruent to the behaviour predicted geometrically within GR. 1.7.4 Asymmetry, equilibrium and “KPV” Three important features of the PV model incorporating the QVIH are, i. Gravitational and inertial forces arise due to QV asymmetry. ii. Mass-energy equilibrates to the local QVE density. iii. “KPV”. •

QV asymmetry

The QV surrounding a mass-object exists in a state of asymmetrical energy polarisation such that material objects equilibrate by falling with gravity. QV asymmetry is the reason why constant acceleration requires energy input and why inertia is experienced during acceleration. Energy input is required to maintain QV asymmetry; hence, once energy input ceases, the object’s motion becomes uniform. •

Mass-energy equilibration

As an object accelerates away from an observer to near light speed, it perceives an increase in the local QVE density. From the perspective of the object, its mass-energy is invariant; however, to the distant observer, the object’s mass-energy appears to increase as it absorbs mass-energy from the local QV environment (i.e. the object equilibrates). •

“KPV”

“KPV” may be conceptualised as an observed change in energy density; the greater the QV asymmetry, the greater the observed change. “KPV” has a minimum permissible value of unity, applicable to a mass-less observer performing measurements on an object from infinity. 1.7.5 Conflux For the PV model to be isomorphic to GR54, “KPV” must consider the speed of light, length contraction, mass scaling, frequency-shifts and time dilation. “KPV” is demonstrated to consider these metrics for the distant stationary observer according to, • •

The speed of light Light passing from a region of lower to higher “KPV” slows down and is refracted to the region of higher “KPV”. Length contraction Objects equilibrate to the local QVE density, akin to the manner in which a balloon is affected by atmospheric pressure. Thus, an object appears to contract as it approaches light speed because the value of “KPV” appears to increase as the object55 equilibrates.

54

GR is based upon the trajectories of light through curved space-time relative to an observer; thus, light (i.e. energy) is the basis for articulating gravitational effects and all metrics descriptive of material objects (e.g. mass, size and time) are also subject to the motion of light. 55 Its length will not seem to change from its own perspective; however, the Universe will appear to increase in energy. 38

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•

•

•

Mass scaling Accelerating mass encounters inertial resistance56 due to disequilibrium between itself and the local QVE density. Thus, it absorbs QVE57 and the value of “KPV” appears to increase as the object equilibrates. Frequency shifts Light emitted by an object accelerating away at near light speed is measured as being refracted by the space-time it moves through. The frequency of the light appears to red-shift because the observer perceives the object as moving from a lower58 to higher “KPV” value as the object equilibrates. The spectral shift is solely dependent upon the relative difference between the “KPV” values of the observer and the object. Time dilation Time is a function of “KPV” because its value affects the propagation of light59. Thus, an object moving from a lower to higher “KPV” value equilibrates and time is dilated.

We can now see how the “KPV” value of the QV is synonymous with the notion of “curved space-time” in explaining the behaviour of objects accelerating or moving through gravitational fields. Utilising an optical model of gravity, it is possible to understand the physical basis for the conclusions derived within the metaphysical GR construct. Note: a table of physical relationships involving “KPV” is articulated in a proceeding chapter.

56

i.e. QV asymmetry. Hence, the object’s mass appears to increase. 58 i.e. the observer’s local value. 59 Within Relativity, the speed of light is constant; thus, time, mass and length simultaneously change to compensate. 57

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NOTES

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2 Electro-Gravi-Magnetics (EGM) 2.1

Introduction

Electricity and Magnetism, once thought to be entirely disparate entities, have been unified into a single set of interactions termed ElectroMagnetism. Physics has long considered the creation of an all-encompassing “Theory of Everything” (ToE) to be its greatest and final purpose – to unify gravity (i.e. GR) and ElectroMagnetism into a single relationship. Within GR, matter generates “space-time curvature” directly affecting the propagation of EM energy and the motion of material objects. As a beam of light transits through a gravitational field, its trajectory arcs in direct response. The dynamic behaviour of EM energy in proximity to matter defines “space-time curvature” which, in turn, defines the manner in which material objects interact with gravitational fields. The PV model of gravity asserts that the metaphysical concept of “space-time curvature” may be replaced by an optical representation of QV polarisation. Thus, it follows that the formation of gravitational fields are a result of QVE displacement due to the presence of matter. Recognising that QVE is EM in composition, a fundamental relationship between matter, EM energy and gravity is implied. This is described utilising a mathematical method termed Electro-Gravi-Magnetics (EGM)60, developed from the application of standard engineering principles, modelling the manner in which matter equilibrates with, and is constrained by, the local QV as a system. 2.2

Similitude

The initial premise in the development of the EGM method is the assumption that gravity and ElectroMagnetism may be unified via QM in terms of the QV, utilising Buckingham “Π” Theory (BPT). BPT is a well established and widely used engineering principle developed by Edgar Buckingham in the early 1900’s. BPT is applied to simplify complex systems and determine which parameters are necessary (or unnecessary) to adequately represent it. The Greek letter “Π” denotes the formulation of dimensionless groups describing the system. BPT is analogous to grammar and sentence structure. In this regard, the dimensionless “Π” groups represent the words of the sentence and the grammatical structure and choice of words are analogous to the equation best describing the system being analysed. For example, a single event may be articulated many ways, utilising different words or combinations of words, placed in various order, and yet still yield an adequate description of the event. There is no “right-or-wrong” sentence, only one that best describes the event being observed. One may elect to recount a single event quite differently from another person, or rephrase the details of one event in various ways. However, the desired result remains unchanged; that the information is communicated adequately. 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. Syntax provides the structural framework upon which concepts are communicated. The basic rules of syntax permit a limited number of words to be arranged into a large number of expressions. Syntax provides structure and meaning to language so that ideas are conveyed. BPT provides the mathematical syntax upon which an equation may be constructed. No “right-or-wrong” equation exists because an engineer designs one yielding a robust depiction of the EP. Parameters may be included or removed from the construct until a mathematical model is formulated predicting the outcome of a specific or generalised simulation. BPT is utilised to model the behaviour of a whole system61 without requiring precise interactional knowledge of all components simultaneously. For example, it is unnecessary to 60 61

A method derived from a single paradigm, cross-fertilising core physical concepts. Particularly when scaling physical relationships to the size of benchtop EP’s. 41

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determine the motion of every water molecule in the ocean to adequately model or predict the movement of a wave passing through it. BPT formulations are executed within the structural framework of Dimensional Analysis Techniques (DAT’s), indicating that similar systems may be described in like terms. An important consideration involving DAT’s and BPT is the rule of “similitude”. In order to compare a mathematical model to a physical system, certain criteria must be satisfied. The model must have dynamic, kinematic or geometric similarity to the real-world system (any of, or all of these if applicable). “Dynamic similarity” relates forces, “Kinematic similarity” relates motion62 and “Geometric similarity” relates shape63. Once the design principles of similitude are satisfied, the mathematical model is considered applicable to the real-world system64. DAT’s and BPT bring to the research and design table, the following key elements65: • 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. The famed English Physicist, “Sir Geoffrey I. Taylor”, masterfully demonstrated how DAT’s and BPT may be applied to develop mathematical predictions. Taylor accurately predicted the energy released by the first atomic bomb detonated outside Alamogordo, New Mexico in 1945, utilising declassified high-speed camera images of the explosion. Taylor surmised that the five physical factors involved in the explosion were; its energy, the radius of the shockwave, the atmospheric pressure and density acting to containing it, and the time associated with the shockwave’s expansion. These five physical terms only have three fundamental units (i.e. mass, length and time). Therefore, only two dimensionless “Π” groupings are required to determine the energy released. 2.3

Precepts and principles

Quinta Essentia: A Practical Guide to Space-Time Engineering (the series) specifies the development of a modelling approach termed EGM, articulating how gravitational fields may be described in terms of ElectroMagnetism. EGM is an engineering approach66, not a theory; it is a tool for mathematically simulating real-world systems in order to model physical problems in GR and QM utilising standard engineering techniques. Gravity is the result of an interaction between matter and the space-time manifold; leading to the following precepts, i. An object at rest polarises the QV surrounding it. ii. An object at rest is in equilibrium with the QV surrounding it. iii. The QVE67 surrounding an object at rest is equivalent to “E = mc2”.

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Synonymous with the time domain. i.e. the topology of space-time curvature within the context of GR. 64 Refer to a standard Engineering text for worked examples of DAT’s and BPT. 65 Norwegian University of Science and Technology, http://www.math.ntnu.no/~hanche/kurs/matmod/1998h/ 66 It is not “new Physics”. 67 i.e. gravitational field energy. 63

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iv.

The frequency distribution of the spectral energy density of the QV surrounding an object at rest is cubic.

EGM is a mathematical modelling method considerate of standard control-systems engineering philosophy. Control systems engineering is commonly applied to design cruise control devices in cars and in general, any automated technology utilising feedback to maintain a steadystate. Subsequently, EGM regards material objects as being part of a matter-space-time system, modelling the manner in which matter behaves in terms of energy exchange with its space-time environment. EGM precepts state that the mass of any object at rest may be expressed as an equivalent energy via “E = mc2” which, in turn, may be represented as a spectrum of EM energy. The EGM method commences by mathematically representing mass as an equivalent localised density of wavefunction energy, contained by the QV surrounding it. Properties of Fourier harmonics are utilised to mathematically decompile the mass-energy into a spectrum of EM frequencies. Mass-energy may be represented through principles of similitude as demonstrated by “Sir Geoffrey I. Taylor” when he modelled the energy of an atomic bomb blast in the atmosphere. He modelled the physical parameters of the atomic blast as an action-reaction dynamic between the energy released and the surrounding environment acting to contain it. The QV is predicted and required by QM and QED, both dictating that virtual energy must exist within the fabric of space-time. The term “virtual” is applied because it refers the nebulous boundary between existence and non-existence68. The EGM construct represents matter as a precisely defined spectrum of EM energy utilising Fourier techniques and models its interaction with the QV as a dynamic system. Subsequently, the unique spectral “signatures” of matter are superimposed upon the QV demonstrating that a change in Poynting Vector69 (∆P) results in a gravitational effect. 2.4

Gravity

All natural systems seek to find equilibrium; this implies that the energy condensed as matter exists in a state of equilibrium within the Universe surrounding it. Consequently, EGM asserts that mass is relativistic because it equilibrates to the ambient energy conditions of its local environment. The methodology developed in QE3 determines the mass-energy equilibrium point between an object and the space-time manifold such that the metaphysical conception of “curvature” is reinterpreted as being a local polarisation of the QV, explicable by the superposition of EM fields, yielding a change in the Poynting Vector “∆P”. The gradient in “P” is analogous to variations in the “KPV”70 value of the space-time manifold in an optical model of gravity. An optical interpretation of gravity was first suggested approximately three hundred years ago by Sir Isaac Newton in his treatise entitled Opticks. Newton theorised that the aether should be most dense far away from an object like the Earth, and conversely, more subtle and rarefied nearby or within it. Two passages from Newton’s Opticks illustrate the optical model of gravity exceedingly well: [sic] Qu. 20. “Doth not this aetherial medium in passing out of water, glass, crystal, and other compact and dense bodies into empty spaces, grow denser and denser, by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curved lines? And doth not the gradual condensation of this medium extend to some distance from the bodies, and thereby cause the inflexions of the rays of light, which pass by the edges of dense bodies, at some distance from the bodies?” 68

The origin and physical constitution of virtual energy is too complex to be describe herein and has been omitted for brevity. 69 i.e. in the displacement domain. 70 Refer to preceding chapters. 43

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Qu. 21. “Is not this medium much rarer within the dense bodies of the Sun, stars, planets and comets, than in the empty celestial spaces between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great bodies towards one another, and of their parts towards the bodies; every body endeavouring to go from the denser parts of the medium towards the rarer? For if this medium be rarer within the Sun’s body than at its surface . . . and rarer there than at the orb of Saturn, I see no reason why the increase of density should stop anywhere, and not rather be continued through all distances from the Sun to Saturn, and beyond. And if the elastic force of this medium be exceedingly great, it may suffice to impel bodies from the denser parts of the medium towards the rarer, with all that power which we call gravity.”71 Newton’s optical model of gravity has a contemporary equivalent known as the PV Representation of GR – a title originally coined by Physicist H.E. Puthoff in 1994, based upon an earlier body of work introduced by Harold Wilson and Robert Dicke in the 1950’s. The PV model replaces the concept of “space-time curvature” with a variable “KPV” value induced by the polarisation of the QV surrounding an object. Newton wrote that a gradual change in the density of the aether curves paths of light. Regional changes in “KPV” result in the refraction of light as though passing through a lens. The EGM construct models vacuum polarisation by the superposition of mass-energy and QV spectra. A key difference distinguishing mass-energy from QVE is that the energy contained within matter is highly localised, whereas QVE is distributed homogeneously throughout the vast regions of freespace72. Haisch, Rueda and Puthoff (HRP) determined that the QV spectrum obeys a cubic frequency distribution73. However, this presents a rather formidable dilemma. This type of distribution implies that the energy density of empty space is staggering. Calculating the total energy represented by the HRP interpretation suggests that every cubic centimetre of empty space is so packed full of energy that it should cause the Universe to collapse in on itself. Because of this theoretical result, many Physicists discount the existence of the QV in cubic frequency form, believing that something must be fundamentally wrong with the derivation, despite the fact that it is derived utilising standard QM. However, the EGM construct does not suffer from this ailment and emphatically rejects the assertion that an infinite quantity of energy is contained within the vanishing volume associated with the QM derivation of the QV74. The physical justification for this emphatic rejection spawns from the derivation of the present Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0” utilising the harmonic representation of fundamental particles75. “H0” and “T0” are measures of Cosmological expansion since the instant of the “Big-Bang”. The Standard Model (SM) of Cosmology (SMoC) determines “H0” by measuring the red-shifted light from receding galaxies as they are pulled apart by the expanding fabric of space; “T0” is measured from the microwave frequencies permeating space, representative of the residual high-energy radiation from the “BigBang” being stretched to the microwave frequency range by cosmic expansion.

71

Opticks. Sir Isaac Newton. Chicago, Encyclopaedia Britannica [1955, c1952] Book III, Part I p.520-521. 72 i.e. flat space-time geometry. 73 i.e. the energy density of QV spectral modes increases to the cube of the frequency. 74 Please refer to the “Technical summary” section of this chapter for further information, or QE3 for rigorous mathematical verification. 75 See: the chapters titled “The Natural Philosophy of Fundamental Particles” and “The Natural Philosophy of the Cosmos” herein or QE3,4 for further information. 44

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The derivation of “H0” and “T0” within the EGM construct yields experimentally impressive results, substantially beyond the abilities of the SMoC, without the “vanishing volume” implications of QM; hence, the “emphatic rejection” asserted herein is substantiated. Applying EGM to the energy dynamics of Hubble expansion spectrally, “H0” is derived by modelling the QV spectrum of the “Primordial-Universe”76 as a single high-frequency wavefunction representing the energy of the entire Universe. Instantaneously after the “Big-Bang”, the single wavefunction rapidly decomposed into a broad spectrum of lower-frequency wavefunctions, forming localised gradients through the condensation of mass77. Summing the energy associated with all lower-frequency wavefunctions in the present QV yields the total energy of the Universe, equalling the total energy at an instant prior to the “Big-Bang”; hence, energy is conserved and the cubic frequency distribution of the QV spectrum predicted by HRP is preserved78. The QV spectrum of a geometrically flat Universe is described within the EGM construct by a cubic frequency distribution, comprised of a large number of modes such that the terminating spectral frequency approaches zero. Hence, the “infinite energy in a vanishing volume” problem associated with the QM derivation of the QV spectrum does not exist under EGM, representing a significant correction over contemporary assertions. Setting the QV spectrum temporarily aside, we shall now define and describe the energy spectrum associated with matter; termed “the EGM spectrum”. This is a wavefunction representation of mass-energy obeying a Fourier distribution such that the number of modes decreases as energy density increases79 (see: QE3), implying that the energy density of free-space approaches zero, avoiding the “infinite energy in a vanishing volume” problem. Consider the action of adding a point mass to an empty Universe. This action superimposes the EGM spectrum of the point mass onto the QV spectrum of the Universe; doing so forms the PV spectrum80 surrounding the point mass, inducing a mode population gradient in space-time between the point mass and the edge of the Universe. The mode population gradient modifies the “KPV” value of the vacuum such that it changes at the same rate as gravitational acceleration “g” from the point mass. Thus, the gradient is “curved” in an analogous manner to space-time within GR. The obvious question arising from the formation of the PV spectrum is; what induces the modal population gradient?81 …. The nature of the Universe is to expand such that the energy within it is “stretched out”. The process of expansion results in the bifurcation of the relatively few high-frequency modes in proximity to the point mass, into a larger number of low-frequency modes at the edge of the Universe. The Universe continually strives to reach its lowest energy state and greatest stability, yet it never can; and in so doing, undergoes modal bifurcation. A mass-object pushes the vacuum around it “uphill”, against the natural flux of expansion. Mass may be modelled as doing work82 on the surrounding vacuum by “curving” it. This occurs because the nature of the Universe is to expand and upon encountering resistance to its normal flux 76

i.e. instantaneously prior to the “Big-Bang”. This is a mathematical representation of the spectral energy dynamic of the expansion model, not a literal interpretation. 78 Although the cubic frequency distribution of the QV spectrum derived by HRP is physically meaningful, their derivation implies that the energy density of the present Universe is greater than at the time of the “Big-Bang”; when the spectral distribution could be defined by a relatively small number of high-frequency (i.e. high energy) modes. Summing the many high-energy modes comprising the HRP specification of the QV spectrum, yields an energy density value predicting Cosmological collapse. 79 i.e. the number of modes is inversely proportional to the energy density of the space-time manifold. 80 i.e. a quantised representation of the gravitational field. 81 i.e. why does the vacuum become polarised? 82 i.e. expending energy. 77

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from high to low energy, the Universe “pushes back” as it strives to find balance (i.e. equilibrium). Thus, the matter-Universe interaction is a dynamic mass-energy-vacuum exchange system rather than material inertly suspended in a vast expanse of nothingness. EGM mathematically represents matter as radiating a spectrum of conjugate EM frequencies. However, if we consider matter to radiate a spectrum of “Gravitons”83, the EGM construct may be represented in quasi-physical form84 in equilibrium with its environment as a system, such that Gravitons emerge as a vehicle for the feedback of information between the EGM spectrum of matter and the QV spectrum of the local space-time manifold. EGM considers the spectral energy of a gravitational field to be equivalent to the massenergy of the object generating the field, expressible in terms of a PV spectrum analogous to spacetime curvature within GR. It models each of the conjugate EM frequencies as two populations of “conjugate Photon pairs”, i.e., each population is “180°” out of phase with its conjugate, consistent with a Fourier harmonics representation of a constant function in complex form (see: QE3). A conjugate Photon pair constitutes the definition of a Graviton within the EGM construct. The density of Gravitons surrounding a mass-object is maximal nearby, gradually decreasing with radial distance; thus, the greater the population density of Gravitons, the stronger the gravitational field. These factors are consistent with the manner in which the PV spectrum is defined via Fourier harmonics, resulting in a spectrum which increases in mode number with radial distance from the mass-object85. The tendency of the space-time manifold is to expand; however, the presence of matter interrupts this movement, polarising the QV. Energy is required to alter its state to fewer modes of higher frequency, counteracting the thermodynamic tendency of any system to move towards a state of lowest energy and greatest stability. Subsequently, an observer held fixed within a QV gradient senses that the mode energy is asymmetrical86 and based upon the QVIH, vacuum asymmetry results in an apparent acceleration force on the observer, perceived as gravity. Rather than a geometric curvature of nothingness, the manifestation of “g” is better represented as back-pressure from the vacuum as mass-energy exerts its influence upon it. Anything caught in the inward flow of space-time, so to speak, is pulled along with the current. EGM represents this process as the superposition of two spectra, resulting in a mathematical description of “g”, utilising Fourier harmonics. Thus, it may be stated that the EGM construct yields a quantised description of gravity. The EGM interpretation of gravity is akin to Newton’s conceptualisation of optical gravity. According to Newton, the aether was presumed to be “denser” farther away from a mass-object and “less dense” nearby. The change (i.e. gradient) in the density of the aether causes light and the movements of objects through it, to follow trajectories characteristic of gravitational attraction. The increasing density of Newton’s aether may be substituted with the analogous concept of increasing mode population in the QV, proportional to the distance from a mass-object. The denser the mass, the fewer modes it has in its PV spectrum because each mode within it possesses higher energy (i.e. frequency). The modal bandwidth of the PV spectrum for a very dense object is narrower than that of a less dense object. Hence, the more massive the object, the “steeper” the gradient (i.e. change) in mode number between its centre of mass and the edge of the Universe, resulting in gravitational acceleration proportional to mass87.

83

i.e. the elementary particles presumed to mediate gravitational force. Science has yet to detect or rigorously define Gravitons; consequently, sufficient latitude exists to interpret the Graviton in a manner suitable to the EGM construct. 85 i.e. QV mode number decreases with “Graviton” density. 86 i.e. higher in the direction of the centre of mass of an object and lower out in space. 87 EGM derives the Casimir Force from first principles, demonstrating that it differs depending on the gravitational field strength of where it is measured. For example, EGM asserts that the strength of the Casimir Force on Jupiter will be smaller than on the surface of the Moon (see: QE3). 84

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2.5

Elementary particles

Particle-Physicists research often involves the act of smashing subatomic particles together at near light-speed velocities and analysing the bewildering array of debris formed in the collision. This process is commonly described as being similar to smashing two cars together and attempting to determine how they worked by analysing the pieces. The discipline of Particle-Physics is also referred to as High-Energy Physics (HEP). This term is also applied because the particles resulting from such collisions are only able to exist in extremely high energy environments. Subatomic particles often only exist as interlinked components of another greater particle system and not as free entities in and of themselves. They often only exist when we cause them to exist. At the instant of a particle collision, each particle’s total kinetic energy and thus its mass, has been greatly amplified. However, the subatomic particle products generated in the collision may only be measured (or even be generated in the first place) by having increased the energy of the environment in which the parent particles are smashed together. A Proton is composed of three Quark subunits; however, Quarks themselves are not known to exist as free Quarks. The configuration of the Proton system acts as a boundary condition, containing the Quarks in a composite form called the “Proton”. Extremely high energies are thus required to smash Protons into their individual Quark constituents and the Quarks released in the collision can only exist freely for an extremely brief period of time88. The energy of any object, whether particle or otherwise, is equilibrated by the ambient energy in its local environment. However, only when equilibrium is artificially shifted, as occurs in a high energy collision, is the energy balance destabilised sufficiently to allow high-energy Quarks to exist autonomously for a brief moment. Shortly after the “Big-Bang”, the Universe was a soup of free Quarks in a hot and dense environment. In the first moments after the “Big-Bang”, the total energy of the early Universe was much more densely packed than it is today. These Quarks could exist freely in the early Universe because the ambient energy density allowed them to exist in this more energetic form. When particles are accelerated to extremely high energies in a collider, we are re-creating the dense energy conditions of the early Universe and free particles exist for a brief period89. EGM models the energy-density environmental equilibrium dynamics of systems, where matter is affected by ambient conditions, by mathematically decomposing its mass-energy into an “EGM spectrum” of frequencies utilising Fourier harmonics. This is a mathematical construct only, modelling the system as a whole, not specifically intended to be taken as physically representative. Nevertheless, the process of mathematically translating units of mass into spectral information articulates the equilibrium between matter and the QV, yielding a natural harmonic relationship between all subatomic particles. Note: where appropriate, due to the principle of mass-energy equivalence and the law of conservation of energy90, the EGM spectrum may also be referred to as the PV spectrum. The PV spectrum is derived as a representation of mass-energy density equilibration such that spectral characteristics differ from object to object; e.g., the Proton possesses a lower harmonic mode population and higher harmonic cut-off91 frequency than a star [see: Fig. (2.1, 2.2)]. However, the PV spectra of fundamental particles may be characterised by their harmonic cut-off frequency such that they may be represented as harmonic multiples of an arbitrarily selected reference particle. 88

i.e. until energy conditions return to normal. As the Universe continued to expand and cool, its energy density decreased, subsequently permitting the condensation of composite particles such as the Proton and Neutron, followed by even more complex, low-energy composites (i.e. the atom) as the energy density decreased further. 90 In terms of equilibration. 91 Spectral limit. 89

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Based upon this harmonic principle of order, a periodic table of subatomic particles may be formulated [see: Tab. (4.5)] mirroring the hierarchical basis upon which the chemical elements are arranged. Note: the harmonic pattern is derived by considering all matter to be radiators of populations of conjugate Photon pairs92, suggesting that the quintessential building-block of all atoms, chemical elements, molecules and material forms in the Cosmos is the Photon – energy itself! 2.6

Cosmology

EGM represents a single paradigm which may be applied to precisely derive Cosmological measurements such as “H0” and93 “T0”. The EGM harmonic representation of fundamental particles serves to validate and substantiate the evolutionary epochs of our Universe, as science has come to understand them, since the time of the “Big-Bang”. The Planck blackbody radiation phenomenon demonstrates that matter radiates a spectrum of EM radiation based upon its temperature. This principle of spectral information is mirrored by the EGM construct because the PV spectra of mass-objects are generated by Fourier harmonics. That is to say, each PV spectrum is a mathematical decomposition of the gravitational energy of a mass-object into its cognate spectrum of harmonic frequencies. “Wien’s displacement law” describes the relationship between the temperature of an object and its blackbody radiation spectrum. Comparing hot and cold objects, we see that the blackbody spectrum for each object type possesses identical shape; depicting peak Photon prevalence at a specific frequency range, trailing off at the high and low spectral limits. Differences in peak emission frequencies obey a scaling factor relationship defined by Wien’s displacement law. (Right) Blackbody radiation curves depicting spectra from objects of varying temperature. Wien’s displacement law describes the relationship between peak emissions (in wavelength) vs. prevalence of Photons (Y-axis) according to temperature (T). When we directly measure the temperature of empty space, we are in-fact measuring the residual energy from the “Big-Bang”. The temperature of empty space is approximately “2.7(K)” such that the Planck blackbody radiation wavelength is about “1(mm)” (i.e. within the microwave frequency range of the EM spectrum). At Bell Laboratories in 1964, while working with a large horn antenna designed for Radio Astronomy and satellite communications, Arno Penzias and Robert Woodrow Wilson discovered a ubiquitous white-noise that could not be eliminated. It was audible day and night in all directions, falling within the microwave frequency range. What Penzias and Wilson heard with their antenna was the radiation left over from the birth of our Universe! The discovery of “T0” earned Penzias and Wilson the Nobel Prize in 1978. The physical detection and measurement of “T0” was momentous because at the time of its discovery, the “Big-Bang” model of cosmic history was merely conjecture. The idea emerged from Hubble’s observation that the Universe was apparently expanding in all directions. It was presumed that it should be possible to trace this expansion back in time when all the matter and energy in the Universe was packed together in a much denser form. However, in the intervening decades between 92

The majority of energy contained within a PV spectrum occurs at the spectral limit; hence, the spectrum may be usefully approximated by a single conjugate wavefunction pair at the harmonic cut-off frequency. See: the chapter herein titled “The Natural Philosophy of Fundamental Particles” or QE3 for further information. 93 It is demonstrated that “T0” may be derived from “H0”. 48

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the time Hubble expansion was discovered and “T0” was measured94, the “Big-Bang” model was by no means on solid ground. The favourable pairing of prediction and observation meant that, as strange as it may seem, our Universe must have suddenly burst into being as if from nowhere. The Universe could no longer be considered as a timeless, “steady-state” Universe, but was instead finite – having a beginning (and perhaps an end). EGM models mass-objects as being in equilibrium with the QV such that the energy state of matter describes the energy state of the vacuum. Hence, “H0” and “T0” represent observational evidence of Cosmological mass-energy equilibration. Invoking principles of similitude, “H0” is derived by relating the PV spectrum of a “Planck-Particle”95 to the present-day utilising the “MilkyWay” Galaxy as a basis for comparison. Within the EGM construct, a “Planck-Particle” denotes the condition of maximum permissible energy density, representing the Universe compacted to a point. As mass-energy density increases, the PV modal bandwidth compresses such that for a “PlanckParticle”, the PV spectrum converges into a single mode approaching the Planck Frequency. Galaxies are homogeneously distributed throughout the Universe and are “approximately” in the same stage of evolution. Hence, it follows that we may utilise our own “Milky-Way” Galaxy as a universal reference to yield an average value of Cosmological gravitational intensity. Utilising astronomical estimates of total galactic mass and radius, we may represent the “Milky-Way” as a “particle” at the centre of the galaxy, termed the “Galactic Reference Particle” (GRP). The radiant gravitational intensity of the GRP may be calculated from its PV spectral limit. The GRP is representative of the total mass-energy density and vacuum equilibrium state of the Universe at the present time96; as viewed by instrumentation within our solar system. Thus, “H0” is derived by comparing the “Planck-Particle Universe” at the instant of creation to the GRP; facilitated by utilisation of the harmonic representation of fundamental particles. “T0” is derived from “H0”; relating the Cosmological expansion of the primordial “PlanckParticle Universe” to the GRP yields an expansive scaling factor “KT”. Subsequently, Wien’s displacement law is applied to determine a thermodynamic scaling factor “TW” quantifying the manner in which Photons radiated at the instant of the “Big-Bang” have red-shifted to the microwave range after Hubble time. The microwave frequency is converted to temperature by relating “KT” and “TW”, producing a value of “T0” precisely matching measurement! The EGM harmonic representation of fundamental particles facilitates the articulation of the Hubble constant since the instant of the “Big-Bang” to the present day. The resulting history of the CMBR temperature corroborates with all epochs of cosmic evolution as predicted by the SMoC. The theory of early “cosmic inflation” is reinforced and the recently measured “accelerated expansion” is derived. Cosmic inflation is an epoch thought to have occurred within the first fractions of an instant after the “Big-Bang”. Note: the Cosmological inflation and accelerated expansion phenomena emerge naturally within the EGM construct and are not presumed “a priori” as part of the modelling process. Cosmic inflation was originally introduced to the SMoC as a requisite so that the “BigBang” theory “fits” observational evidence. Without this inflationary epoch, the Universe would not exist as observed; it would be flat and featureless, with no clumps of matter and would be so small today that the entire Universe, after billions of years, would fit on the head of a pin. However, the EGM construct generates the inflationary epoch from first principles, derived from Particle-Physics!

94

Representing the red-shifted (i.e. stretched-out) Photons of the early Universe; the EM waves we observe today, were once extremely high-frequency Photons moments after Cosmological creation. Billions of years later, those Photons have become stretched by cosmic expansion to such a degree that now they are approximately “1(mm)” in wavelength. 95 Representing the Universe at the instant of the “Big-Bang”. 96 As one homogeneous “piece” of the Cosmos. 49

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Scientists wondered whether there might be enough matter in the Universe to halt cosmic expansion, causing the Universe to meet its end in a “Big Crunch”. However, Astronomers were vexed to discover that the Universe is actually accelerating at a rate exceeding predictions. The discrepancy between prediction and observation within the SMoC is so vast, that Cosmologists invented the concepts of “dark energy” and “dark matter” to make sense of the findings. Our best measurements of expansion are so far from the predicted value that theorists estimate “72(%)” of the Universe must be composed of dark energy and “23(%)” must be dark matter, meaning that “95(%)” of our Universe exists in an unknown and unobservable form! According to observation, it is thought that a substantial portion of matter is “missing” because of the peculiar manner in which galaxies rotate. Instead of rotating fastest in the centre and slower at the periphery, the spiral arms of galaxies rotate about the central axis at the same rate as the stars near the centre. This suggests that “something” must be present in undetectable halos surrounding galaxies. Thus, dark matter has been contrived to explain why galaxies rotate uniformly. Similarly, dark energy has been contrived to explain why the Universe continues to expand at an accelerated rate, despite the addition of dark matter to the SMoC. Not withstanding “dark + visible” matter, the remainder of the Universe is thought to be in the form of an energy field inducing negative pressure in the space-time manifold, counteracting gravity on a very large scale, causing intergalactic voids of space-time to expand like giant balloons. Even though the cosmic inflation epoch is also a contrivance introduced to fit a theory, EGM substantiates its existence because it emerges as a natural consequence of the derivation of “H0” and “T0”. However, the existence of dark energy / matter must be questioned due to the fact that the EGM method requires no contrivances in order to predict “H0”, “T0” and Cosmological inflation / accelerated expansion, without invoking dark matter or energy; producing results substantially more precise than the SMoC97. We have come full circle, from alpha to omega and back, uncovering physical proof for what the Pythagoreans and ancient Babylonians hypothesised millennia ago. We appreciate the stark beauty of the “Laws of Cosmic Harmony” which Pythagoras exhorted with such passion. We hold substantive evidence that authenticates the philosophical beliefs of our ancient scientific predecessors, who contemplated and understood the Cosmos to be more than a “void” in which matter resides. Their Cosmos encompassed all forms in the Universe, from the miniscule to immense, both living and inert; it was an expression of the “Laws of Musical Proportion” and “sympathetic affinity” connecting all things. NOTES

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Within the EGM construct, the contribution of dark matter / energy to the Cosmological model is shown to be “< 1(%)”. 50

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2.7

Technical summary

2.7.1 Synopsis • • • • • •

Applicable definitions, Quantum Vacuum (QV): a quantum representation of the space-time manifold within GR. Quantum-Vacuum-Energy (QVE): the spectral energy associated with the QV. Zero-Point-Field (ZPF): the QV field associated with globally flat space-time geometry. However, such a configuration cannot physically exist; thus, the ZPF takes the form of a generalised reference to the QV field throughout the “Quinta Essentia” series (i.e. QE2-4). Zero-Point-Energy (ZPE): the spectral energy associated with the ZPF. Polarisable Vacuum (PV): a polarised representation of the ZPF. Electro-Gravi-Magnetics (EGM): a theoretical relationship between EM fields and “g”.

We shall outline the method developed in QE2-4 to describe “g” in harmonised terms, yielding new predictions and highly precise experimentally verified results beyond the Standard Models (SM’s) of Particle-Physics and Cosmology. The EGM construct derives (see: QE3): 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. 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 ZPF based upon (i) and the assumption that an EM relationship exists over a change in displacement across a practical benchtop test volume. The derivational procedure obeys the following hierarchy, v. A harmonic representation of “g” is developed. vi. The frequency spectrum of (v) is derived by application of Buckingham “Π” Theory (BPT) and dimensional similarity. vii. The ZPF energy density is related to (vi) 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 PV98 are derived based upon (vii). ix. A description of physical modelling criteria is presented. x. A set of sample calculations and illustrational plots are presented. Fourier series99 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 PV model is (historically) a weak field isomorphic approximation of GR and the frequency spectrum is postulated to range from negative to positive infinity, it follows that Fourier series represent a useful tool by which to describe gravity. Time domain modelling may be applied over the displacement domain of a practical benchtop test volume by considering the relevant changes over the dimensions of that volume. Constant functions may be expressed as a summation of trigonometric terms; subsequently, it is convenient to model a gravitational field utilising modified Complex Fourier series according to an odd number harmonic distribution. Hence, “g” may be usefully represented by the magnitude of a periodic square wave solution as the number of waves utilised to describe it, approaches infinity, It is demonstrated in QE3 that dimensional similarity and the equivalence principle may be applied to represent the magnitude of an acceleration vector such that an expression for the 98

Refer to the section titled “The PV Model of Gravity” for a complete description. A Fourier series representation of a constant function involves the hybridisation of amplitude and frequency spectra (i.e. a Fourier distribution contains two embedded spectra). 99

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frequency spectrum is derived in terms of harmonic mode. This is achieved by assuming that ElectroMagnetically induced acceleration is dynamically, kinematically and geometrically similar to “g” as constructed by Fourier series wave summation. 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 may be evaluated over the difference between successive odd frequency modes. The reason for this is due to the mathematical properties of Fourier series for constant functions. For such cases – as appears in standard texts, the summed contribution of all even modes equals zero. Subsequently, only odd mode contributions need be considered when modelling a constant function. Utilising the approximate rest mass-energy density of a solid spherical object, an expression relating the terminating harmonic cut-off mode may be derived by assuming that the equivalent quantity of mass-energy within an object is also stored in the gravitational field surrounding it. Subsequently, the upper boundary of the frequency spectrum, termed the harmonic cut-off frequency, may be calculated; the derivation is based upon the compression of energy density of the “random ZPF form” to one change in odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordance with BPT. The compressed “random ZPF form” is subsequently decompressed over the Fourier domain (assigning structure), yielding a highly precise reciprocal harmonic representation of “g”; preserving dynamic, kinematic and geometric similarity to the Newtonian, PV and GR representations. The cross-fertilisation of the amplitude and frequency characteristics of a constant function described by Fourier series with the ZPF spectral energy density distribution derived by Haisch and Rueda, is a useful tool by which to determine the spectral characteristics of the PV representation of GR (proposed by Puthoff) at the surface of the Earth (for example) by assuming, xi. The PV physically exists as a spectrum of frequencies and wave vectors. xii. 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. xiii. A modified Complex Fourier series representation of “g” is representative of the magnitude of the resultant PV wave vector. xiv. A physical relationship exists between Electricity, Magnetism and Gravity such that “g” may be investigated and modified. Therefore, we may summarise the solution algorithm constituting the harmonically based EGM construct by five simple steps as follows, xv. Apply Dimensional Analysis Techniques (DAT's), BPT and similarity principles to combine Electricity, Magnetism and resultant EM acceleration in the form of “Π” groupings. xvi. Apply the equivalence principle to the “Π” groupings formed in (xv). xvii. Apply Fourier Harmonics to the equivalence principle. xviii. Apply ZPF Theory100 to Fourier Harmonics. xix. Apply the PV model of gravity to the ZPF. Within the EGM construct, the Poynting Vector “P” represents the propagation of energy (i.e. conjugate Photon pairs, see: QE3), radially outwards from the centre of mass; however, “g” is the result of the change in “P” (i.e. “∆P”) between two points in the displacement domain. This may appear counter-intuitive since “P” propagates away from the centre of mass, but “g” is a consequence of “∆P” not “P”. A “∆P” arises due to the superposition of the “P” field upon the ZPF. The ZPF acts to constrain the “P” field, yielding “g” as predicted by Newtonian mechanics and GR.

100

See also: ZPF equilibrium as described in the chapter titled “The Natural Philosophy of Fundamental Particles”. 52

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This principle may be demonstrated by a simple example; let the value of “P” at positive radial displacements from a mass-object “r1” and “r2” be given by the positive values “P1” and “P2” respectively. Hence, if “r2 > r1” then “P2 < P1” because “P2 → 0” as “r2 → ∞” such that the difference between “P1” and “P2” is negative [i.e. “(∆P = P2 – P1) < 0”], indicating that “g” acts towards the centre of mass and opposite to the direction of propagation of “P”. “P” represents the propagation of spectral mass-energy equivalence in the form of populations of conjugate Photon pairs. An equilibrium gradient in the displacement domain arises due to the mathematical interaction between the mass-energy and ZPF spectra, equivalent to spacetime curvature under GR because the intensity of “P” varies congruently with “g”. Hence, the radial gradient in “P” is analogous to variations in the Refractive Index of the space-time manifold in an optical model of gravity. 2.7.2 The QV spectrum Historically, the QV has been considered to be composed of a near infinite spectrum of randomly orientated wave functions, each of specific frequency and amplitude, analogous to the static one observes on a dead television channel. However, the EGM construct disagrees with this historical conception as it implies the existence of a near infinite quantity of energy in a vanishing volume (i.e. free space contains a near infinite amount of energy). EGM asserts that the QV is more appropriately described as a finite spectrum whose wave function population is determined by the quantity of mass-energy occupying a specific volume (i.e. free space contains a near zero amount of energy). Subsequently, the QV spectrum may be characterised by the following statements, xx. It is a generalised reference to a quantum description of the space-time manifold. xxi. In flat space-time geometries, it transforms to the ZPF spectrum. xxii. In curved geometries (i.e. gravitational fields), it transforms to the PV spectrum. Note: a vanishing volume containing infinite energy does not exist within the EGM construct. 2.7.3 The EGM spectrum The EGM spectrum is a harmonic description of mass-energy represented as conjugate EM wavefunction pairs; incrementally above “ω = 0(Hz)”, tending to the Planck Frequency “ωh” and obeying a Fourier distribution. Key generalised spectral features are, xxiii. It is discrete and harmonically continuous “–ωh ← ω → +ωh”. xxiv. The terminating frequency is a harmonic multiple of the fundamental (i.e. lowest freq.). xxv. Each wavefunction represents a population of Photons such that each conjugate Photon pair constitutes a Graviton. xxvi. Where appropriate, due to the principle of mass-energy equivalence and the law of conservation of energy101, it may also be referred to as the PV spectrum. 2.7.4 The ZPF spectrum The ZPF spectrum may be partially described by its contrast to the EGM spectrum. The EGM spectrum relates the mass of an object to the gravitational field surrounding it utilising Fourier harmonics; hence, it is “somewhat localised”. However, the energy of the ZPF is dispersed homogenously throughout the Universe. The historical conception of the ZPF implies the existence of a near infinite quantity of energy in a vanishing volume (i.e. free space contains a near infinite amount of energy). 101

i.e. the mass-energy within an object is energetically equivalent to the gravitational field surrounding the object. 53

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Fortunately, EGM resolves this conflict such that a vanishingly small volume of flat space-time does not contain an infinite amount of energy. This is achieved by merging the continuous cubic frequency characteristic of the ZPF with a discrete and finite Fourier distribution such that, xxvii. The number of harmonic modes approaches infinity. xxviii. The highest frequency tends to zero. A determination of available ZPF energy is demonstrated in the chapter titled “The Natural Philosophy of the Cosmos” where it is shown that (i), the average value of ZPF energy density “UZPF” throughout the Universe may be stated as “UZPF < –2.52 x10-13(Pa)” and (ii), the gradient of the Hubble constant in the time domain is presently positive102. Hence, On a human scale, this translates to levels of ZPF energy according to, xxix. “< –252(yJ/mm3)”: where, “yJ = 10-24(Joule)” and “mm = millimetre”. On an astronomical scale, this becomes, xxx. “< –0.252(mJ/km3)”: where, “mJ = 10-3(Joule)” and “km = kilometre”. xxxi. “< –7.4 x1012(YJ/pc3)”: where, “YJ = 1024(Joule)” and “pc = parsec”. On a Cosmological scale, this becomes, xxxii. “< –6.6 x1041(YJ/RU3)”: where, “RU” denotes the Hubble size of the visible Universe by the EGM method (i.e. “RU ≈ 14.58 Billion Light Years”). Note: although on the human scale the quantity of ZPF energy is trivial, on the astronomical or Cosmological scale, it becomes extremely large when approaching the dimensions of the visible Universe. 2.7.5 The PV spectrum The PV spectrum may be formulated by merging the EGM and ZPF spectral distributions. Energy condensed as mass is finite; representing a small fraction of the total energy in the Universe. The finite parameters of matter dictate the form that the mass-energy spectrum will take. The resulting harmonic description is termed the PV spectrum. PV spectral formation may be conceptualised by considering a Universe populated by a singular spherical object of homogeneous mass-energy density. When such an object is added to an empty Universe, the EGM spectrum of the object is superimposed upon the background ZPF spectrum. Merging the EGM and ZPF spectra results in the cross-fertilisation of characteristics; the complete mathematical derivation is contained in QE3. Descriptions of the specific mathematical events required are as follows, xxxiii. Integrate the HRP spectral energy density equation over the frequency domain “ω”: [see: Eq. (3.47, 3.293)]. xxxiv. Recognise that, for any Fourier summation resulting in a constant function, only odd harmonic modes are required due to the null summation of even modes. This is a fundamental property of Fourier mathematics and should not be dismissed103. xxxv. Formulate an expression for the change in energy density with respect to odd harmonics, in terms of “ω”, utilising the integrated HRP spectral energy density equation [see: Eq. (3.294)]. xxxvi. Substitute the harmonic frequency “ωPV” relationship [see: Eq. (3.67)] into Eq. (3.294). xxxvii. Solving appropriately, one obtains the harmonic cut-off mode and frequency (i.e. “nΩ” and “ωΩ” respectively). “nΩ” denotes the highest harmonic mode contained in the merged spectra (i.e. the PV spectrum) and “ωΩ” represents the terminating spectral frequency relative to a fundamental value (i.e. its lowest permissible magnitude).

102 103

Facilitating an explanation of the “accelerated Cosmological expansion” phenomenon. Refer to any standard text for further information regarding Fourier techniques. 54

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Hence, all required attributes have been derived to completely describe “g” in harmonic terms. The next step is to understand how the EGM method produces a PV spectrum such that the “infinite energy” dilemma of ZPF Theory (derived by contemporary QM methods), is averted. The deductive reasoning may be articulated as follows, xxxviii. The HRP derivation implies that the majority of ZPE exists at the spectral limit104. xxxix. Assume that the ZPE at an arbitrary mathematical point in the space-time manifold is constant such that the associated spectrum may be described harmonically relative to the magnitude of “some” fundamental frequency at the point under consideration. xl. The Fourier characteristics of a constant function demonstrate that only odd harmonic modes are required for summation. xli. Principles of equivalence and similitude imply that the highest spectral transition of odd harmonic mode may be utilised in the representation of the total localised ZPE. xlii. Assume that the mass-energy density of an object is equal to the spectral energy density of the gravitational field surrounding it. xliii. Integrating the HRP spectral energy density relationship yields the total ZPE, which may be expressed locally as a narrow high-frequency bandwidth of equivalent energy. Equating this result to the mass-energy density of an object yields the PV spectrum surrounding it, preserving similitude. Therefore, when the EGM and ZPF spectra are merged, the continuous ZPF spectrum is compressed and equated to the Fourier distribution of the EGM spectrum such that the resulting PV spectrum is a decompressed form of the merged spectra and the properties of its spectral limits may be determined. This process mathematically transforms the continuous ZPF spectrum to a discrete and finite Fourier distribution of equivalent energy. Thus, as radial displacement “r” at a mathematical point from a mass-object increases, xliv. Gravitational field strength decreases. xlv. Spectral energy density decreases. xlvi. The number of harmonic modes increases (i.e. bifurcation). xlvii. Greater numbers of modes are required to be summed for energetic equivalence. The EGM interpretation of Gravity is similar to Newton’s thoughts of an optical model such that the aether was presumed to be “denser” farther away. The gradient in aether density causes light and objects to follow trajectories characteristic of GR. EGM demonstrates that the increasing density of Newton’s aether is analogous to increases in mode population in the PV spectrum. Hence, the PV is an EM frequency spectrum obeying a Fourier distribution at displacement “r” describing a mass “M” induced gravitational field such that, xlviii. It denotes a polarised form of the ZPF spectrum105. xlix. The population of spectral modes “nΩ(r,M)” decreases as mass increases. l. Maximum spectral frequency “ωΩ(r,M)” increases as mass increases. li. The fundamental spectral frequency “ωPV(1,r,M)” increases as mass increases. lii. Spectral frequency bandwidth106 increases as mass increases. Sample calculations107, ωPV(1,r,M) Hz ωPV(1,RE,M0) → 0 ωPV(1,RE,MM) ≈ 0.008

ωΩ(r,M) YHz ωΩ(RE,M0) → 0 ωΩ(RE,MM) ≈ 196

nΩ(r,M) nΩ(RE,M0) → ∞ nΩ(RE,MM) ≈ 2.4x1028

104

i.e. low frequency energy contribution is comparatively trivial. Mass pushes the ZPF surrounding it “uphill”, against the natural flux of space-time manifold expansion. 106 i.e. the difference in magnitude between the highest and lowest frequencies “∆ωPV”. 107 The radius of the Earth “RE” has been applied as a standard unit of measure for demonstration purposes only. 105

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ωPV(1,RE,ME) ≈ 0.0358 ωΩ(RE,ME) ≈ 520 ωPV(1,RE,MJ) ≈ 0.2445 ωΩ(RE,MJ) ≈ 2x103 ωPV(1,RE,MS) ≈ 2.4841 ωΩ(RE,MS) ≈ 9x103 Table 3.17, where, “YHz = 1024 (Hz)”. Variable r RE M0 MM, ME, MJ, MS

nΩ(RE,ME) ≈ 1.5x1028 nΩ(RE,MJ) ≈ 7.6x1027 nΩ(RE,MS) ≈ 3.5x1027

Description Magnitude of position vector relative to the centre of mass Radius of the Earth Zero mass condition of free space Mass of the Moon, Earth, Jupiter and the Sun respectively Table 2.1,

Units m kg

2.7.6 The EGM, PV and ZPF spectra 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, whereas the ZPF spectrum possesses the same frequency bandwidth of the EGM spectrum, but does not follow a Fourier distribution. Thus, 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. 2.7.7 The Casimir Effect The Casimir Effect108 demonstrates that when small distances separate two flat neutral metal plates, Photons in the PV field with wavelengths larger than the plate separation distance are excluded from the spatial cavity, resulting in an attractive force between the plates due to the bias in vacuum energy across the system109. Gravity, in this regard, is analogous to a long-range Casimir effect because EGM asserts that mass induced gravitational effects may be described by changes in mode population across a region of space. The EGM construct was applied in QE3 to derive the Casimir Force from first principles, demonstrating that it differs depending upon ambient gravitational field strength! For example, the Casimir Force will be slightly different on Earth than Jupiter or the Moon. QE3 states that, liii. “…. 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.” liv. “…. 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 plates110 prevents the formation of large Casimir Forces.”

108

Presently, it is only experimentally confirmed to exist in gravitational fields (i.e. PV fields). “The Effect” has not been physically verified in flat space-time geometries (i.e. the free-space “0g”condition). 109 i.e. the vacuum energy density is lower between the plates. 110 i.e. in and around the experimental zone. 56

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2.7.8 Comparative spectra EGM bandwidth comparisons of PV spectra associated with physical categories of objects may be formulated and represented graphically based upon ZPF equilibria. Determination of the ZPF equilibrium radius of subatomic particles is a sophisticated process, summarised herein by the chapter titled “The Natural Philosophy of Fundamental Particles”. A complete and rigorous derivation is presented in QE3. Note: the ZPF equilibrium radius of astronomical bodies coincides with the mean radius (see: QE3), representing the mathematical boundary (within EGM) delineating mass composition and the gravitational field surrounding it. Utilising the EGM construct, the HRP spectral energy density equation with cubic frequency distribution [see: Eq. (3.47)] may be graphically categorised into four regions (i.e. zones), these are; “Planck-Scale” energy densities, “Particle-Physics”, “Astro-Physics” and “Cosmology”, subject to the following generalised characteristics [see: Fig. (2.1, 2.2)], lv. Planck Scale energy densities111 [see: QE4] • Narrowband high-frequency spectrum. • Narrowband modal spectrum. lvi. Particle-Physics112 [see: Tab. (2.4)] • Broadband high-frequency spectrum. • Narrowband modal spectrum. lvii. Astro-Physics [see: Tab. (2.2)] • Moderateband113 high-frequency spectrum. • Moderateband modal spectrum. lviii. Cosmology [see: Tab. (2.6, 2.7)] • Narrowband low-frequency spectrum. • Broadband modal spectrum. To demonstrate these characteristics in tabular form, the PV frequency and modal bandwidths (“∆ωPV” and “∆nPV” respectively) are given by, ∆ωPV(r,M) = ωΩ(r,M) – ωPV(1,r,M)

(2.1)

where, “ωΩ(r,M) >> ωPV(1,r,M)” hence “∆ωPV(r,M) ≈ ωΩ(r,M)” and: ∆nPV(r,M) = nΩ(r,M) – 1

(2.2)

where, “nΩ(r,M) >> 1” hence “∆nPV(r,M) ≈ nΩ(r,M)”. Sample results, Astronomical Object The Moon The Earth Jupiter The Sun

∆ωPV(r,M) YHz ∆ωPV(RM,MM) ≈ 403 ∆ωPV(RE,ME) ≈ 520 ∆ωPV(RJ,MJ) ≈ 488 ∆ωPV(RS,MS) ≈ 646 Table 2.2,

∆nPV(r,M) ∆nPV(RM,MM) ≈ 8.6x1027 ∆nPV(RE,ME) ≈ 1.5x1028 ∆nPV(RJ,MJ) ≈ 5.0x1028 ∆nPV(RS,MS) ≈ 1.4x1029

111

Refers to particulate representations of maximum permissible energy densities (i.e. “BlackHole” singularities). 112 The radial dimension “r” denotes the position of ZPF equilibrium (refer to “The Natural Philosophy of Fundamental Particles” or QE3). 113 A generalised reference to spectral bandwidth relative to “narrow” and “broad” descriptors. 57

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ωPV(1,r,M) Hz ωPV(1,RM,MM) ≈ 0.05 ωPV(1,RE,ME) ≈ 0.04 ωPV(1,RJ,MJ) ≈ 9.8x10-3 ωPV(1,RS,MS) ≈ 4.8x10-3 Particle Proton Neutron Electron Electron Neutrino Muon Muon Neutrino Tau Tau Neutrino Up Quark Down Quark Strange Quark Charmed Quark Bottom Quark W Boson Z Boson Higgs Boson Top Quark

ωΩ(r,M) YHz ωΩ(RM,MM) ≈ 403 ωΩ(RE,ME) ≈ 520 ωΩ(RJ,MJ) ≈ 488 ωΩ(RS,MS) ≈ 646 Table 2.3,

∆ωPV(r,M) YHz ∆ωPV(rπ,mp) ≈ 2.6172x103 ∆ωPV(rν,mn) ≈ 2.6246x103 ∆ωPV(rε,me) ≈ 5.2337x103 ∆ωPV(ren,men) ≈ 5.2348x103 ∆ωPV(rµ,mµ) ≈ 2.0934x104 ∆ωPV(rµn,mµn) ≈ 2.0939x104 ∆ωPV(rτ,mτ) ≈ 3.1409x104 ∆ωPV(rτn,mτn) ≈ 3.1409x104 ∆ωPV(ruq,muq) ≈ 3.6636x104 ∆ωPV(rdq,mdq) ≈ 3.6636x104 ∆ωPV(rsq,msq) ≈ 7.3272x104 ∆ωPV(rcq,mcq) ≈ 1.0991x105 ∆ωPV(rbq,mbq) ≈ 1.4654x105 ∆ωPV(rW,mW) ≈ 2.5645x105 ∆ωPV(rZ,mZ) ≈ 2.9309x105 ∆ωPV(rH,mH) ≈ 3.2972x105 ∆ωPV(rtq,mtq) ≈ 3.6636x105 Table 2.4,

ωPV(1,r,M) THz ωPV(1,rπ,mp) ≈ 0.0355 ωPV(1,rν,mn) ≈ 0.0357 ωPV(1,rε,me) ≈ 0.8421 ωPV(1,ren,men) ≈ 9.3719 ωPV(1,rµ,mµ) ≈ 8.0751 ωPV(1,rµn,mµn) ≈ 28.6052 ωPV(1,rτ,mτ) ≈ 12.1584 ωPV(1,rτn,mτn) ≈ 30.3949 ωPV(1,ruq,muq) ≈ 61.1401 ωPV(1,rdq,mdq) ≈ 53.2256 ωPV(1,rsq,msq) ≈ 160.8502 ωPV(1,rcq,mcq) ≈ 266.5401 ωPV(1,rbq,mbq) ≈ 414.2502 ωPV(1,rW,mW) ≈ 875.8276 ωPV(1,rZ,mZ) ≈ 1.1768x103 ωPV(1,rH,mH) ≈ 1.4920x103 ωPV(1,rtq,mtq) ≈ 1.7578x103

nΩ(r,M) nΩ(RM,MM) ≈ 8.6x1027 nΩ(RE,ME) ≈ 1.5x1028 nΩ(RJ,MJ) ≈ 5.0x1028 nΩ(RS,MS) ≈ 1.4x1029 ∆nPV(r,M) ∆nPV(rπ,mp) ≈ 7.3723x1016 ∆nPV(rν,mn) ≈ 7.3452x1016 ∆nPV(rε,me) ≈ 6.2154x1015 ∆nPV(ren,men) ≈ 5.5857x1014 ∆nPV(rµ,mµ) ≈ 2.5924x1015 ∆nPV(rµn,mµn) ≈ 7.3201x1014 ∆nPV(rτ,mτ) ≈ 2.5833x1015 ∆nPV(rτn,mτn) ≈ 1.0334x1015 ∆nPV(ruq,muq) ≈ 5.9921x1014 ∆nPV(rdq,mdq) ≈ 6.8831x1014 ∆nPV(rsq,msq) ≈ 4.5553x1014 ∆nPV(rcq,mcq) ≈ 4.1235x1014 ∆nPV(rbq,mbq) ≈ 3.5376x1014 ∆nPV(rW,mW) ≈ 2.9281x1014 ∆nPV(rZ,mZ) ≈ 2.4906x1014 ∆nPV(rH,mH) ≈ 2.2100x1014 ∆nPV(rtq,mtq) ≈ 2.0842x1014

ωΩ(r,M) YHz ωΩ(rπ,mp) ≈ 2.6172x103 ωΩ(rν,mn) ≈ 2.6246x103 ωΩ(rε,me) ≈ 5.2337x103 ωΩ(ren,men) ≈ 5.2348x103 ωΩ(rµ,mµ) ≈ 2.0934x104 ωΩ(rµn,mµn) ≈ 2.0939x104 ωΩ(rτ,mτ) ≈ 3.1409x104 ωΩ(rτn,mτn) ≈ 3.1409x104 ωΩ(ruq,muq) ≈ 3.6636x104 ωΩ(rdq,mdq) ≈ 3.6636x104 ωΩ(rsq,msq) ≈ 7.3272x104 ωΩ(rcq,mcq) ≈ 1.0991x105 ωΩ(rbq,mbq) ≈ 1.4654x105 ωΩ(rW,mW) ≈ 2.5645x105 ωΩ(rZ,mZ) ≈ 2.9309x105 ωΩ(rH,mH) ≈ 3.2972x105 ωΩ(rtq,mtq) ≈ 3.6636x105 Table 2.5,

58

nΩ(r,M) nΩ(rπ,mp) ≈ 7.3723x1016 nΩ(rν,mn) ≈ 7.3452x1016 nΩ(rε,me) ≈ 6.2154x1015 nΩ(ren,men) ≈ 5.5857x1014 nΩ(rµ,mµ) ≈ 2.5924x1015 nΩ(rµn,mµn) ≈ 7.3201x1014 nΩ(rτ,mτ) ≈ 2.5833x1015 nΩ(rτn,mτn) ≈ 1.0334x1015 nΩ(ruq,muq) ≈ 5.9921x1014 nΩ(rdq,mdq) ≈ 6.8831x1014 nΩ(rsq,msq) ≈ 4.5553x1014 nΩ(rcq,mcq) ≈ 4.1235x1014 nΩ(rbq,mbq) ≈ 3.5376x1014 nΩ(rW,mW) ≈ 2.9281x1014 nΩ(rZ,mZ) ≈ 2.4906x1014 nΩ(rH,mH) ≈ 2.2100x1014 nΩ(rtq,mtq) ≈ 2.0842x1014

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Sample plots,

Figure 2.1 (illustrational only - not to scale), where, Region / Zone Applicable Category Gravitational Model Space-Time Geom.

A Cosmology ZPF Flat

B Astro-Physics PV Curved Table 2.6,

C Particle-Physics PV Flat

D Planck Scale PV Curved

On a Cosmological scale114, the ZPF upper spectral limit is influenced by the average energy density of the present Universe. The spectral density of the ZPF remains cubic; however, the upper spectral frequency limit is lower than it was in the early Universe. Hence, the majority of ZPE is presently in the form of low-frequency modes, each containing a relatively small amount of energy. The few high-frequency modes characterising the early Universe have bifurcated into a very large bandwidth of lower-frequency modes as the Universe expanded to its present form. The total energy of the Universe remains constant, but is spread out over a much greater volume as Cosmological expansion continues115. It is demonstrated by derivation in QE3 and confirmed in 114

i.e. on average, with a flat space-time manifold as determined by “WMAP”. The information in this paragraph should not be confused with the PV spectrum of a specific body such as a planet, in which case, the bulk of the gravitational energy [i.e. >> 99.99(%)] occurs at the harmonic cut-off frequency. The low frequency modes do not contribute significantly and may be usefully neglected from most calculations. This phenomenon has been thoroughly and rigorously explored in QE3. 115

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QE4, that the majority proportion of the gravitational effect in a field occurs at the harmonic cut-off frequency “ωΩ” such that all other frequencies may be usefully neglected. Subsequently, utilising this proportional spectral frequency characteristic in the harmonic representation of gravitational fields by the EGM method, the bifurcation phenomenon may be mathematically articulated by a simple derivation as follows, Utilising “ωΩ” from QE3,

ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73)

Note: “Eq. 3.xx, 4.xx” denotes reference to QE3,4 respectively. An expression for the Spectral Energy Density, as derived by HRP [see: Eq. (3.47)], per cubic mode population “ρ0(ω) / nΩ3” may be defined according to, ρ 0 ω Ω ( r, M ) n Ω ( r, M )

2 .h . ω Ω ( r , M ) n Ω( r, M )

c

3

3

2 .h . 3 ω PV( 1 , r , M ) c

(2.3)

Hence, the Spectral Energy Density per cubic mode population may be written in terms of the fundamental harmonic PV spectral frequency “ωPV(1,r,M)” as, 3 2 .h . ω PV( 1 , r , M ) c

ρ 0( r , M )

(2.4)

Associating the preceding expression with the energy density of a PV field “Uω” derived in QE3, U ω ( r, M )

h . 4 ω PV( 1 , r , M ) 3 2 .c

(3.69)

yields, 3 2 .c .U ω ( r , M )

ω PV( 1 , r , M )

4

c .ρ 0( r , M ) 2 .ω PV( 1 , r , M )

3

(2.5) such that: the representation of Spectral Energy Density as a function of harmonic frequency may be written as, ρ 0 n PV, r , M

2 4 .c .U ω ( r , M )

ω PV n PV, r , M

(2.6)

where, the frequency spectrum of the harmonised gravitational field “ωPV” is given by Eq. (3.67), ω PV n PV, r , M

n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

(3.67)

Therefore: by substitution, ρ0 ∝ 1 / nPV

(2.7)

Graphing yields,

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Figure 2.2 (illustrational only - not to scale), where, Region / Zone Applicable Category Gravitational Model Space-Time Geom.

E F Planck Scale Particle-Physics PV PV Curved Flat Table 2.7,

G Astro-Physics PV Curved

H Cosmology ZPF Flat

2.7.9 Characterisation of the gravitational spectrum Note: refer to the glossary of terms if required. The EGM equations, utilised to describe fundamental particles in harmonic terms “Stω”, are simplified for values of Refractive Index approaching unity “KPV → 1”. This facilitates the representation of “g” utilising the PV harmonic cut-off frequency “ωΩ”, leading to the formulation of a generalised cubic frequency expression “g → ωPV3”. It is demonstrated that the PV spectrum is dominated by “ωΩ” such that the magnitude of the associated gravitational Poynting Vector “SωΩ” is usefully approximated by the total energy density “SωΩ → c⋅Um”, resulting in an expression for EGM Flux Intensity “CΩ_J”. The derivation sequence proceeds as follows, lix. Simplification of the EGM equations. lx. Derivation of “g” in terms of “ωΩ”. lxi. Formulation of a generalised cubic frequency expression in terms of “g”: “g → ωPV3”. lxii. Determination of the gravitationally dominant EGM frequency: “SωΩ → c⋅Um”. lxiii. Derivation of EGM Flux Intensity “CΩ_J”. 61

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2.7.10 Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” characteristics The minimum physical dimensions of “Schwarzschild-Planck-Particle” mass and radius is derived, leading to the determination of the value of “KPV” at the event horizon of a “Schwarzschild-Planck-Black-Hole” (SPBH). Consequently, the magnitude of the harmonic cut-off frequency “ωΩ” at the event horizon “RBH” of a “Schwarzschild-Black-Hole” (SBH) is presented, yielding the singularity radius “rS” and harmonic cut-off profiles (“nΩ” and “ωΩ” as “r → RBH”). The minimum gravitational lifetime of matter “TL” is also advanced such that the value of generalised average emission frequency per Graviton “ωg” may be calculated. These determinations assist in the supplemental EGM interpretation with respect to the visibility of “Black-Holes” (BH’s). The derivation sequence proceeds as follows, lxiv. Derivation of the minimum physical “Schwarzschild-Planck-Particle” mass and radius. lxv. Derivation of the value of the “KPV” at the event horizon of a “Schwarzschild-PlanckBlack-Hole” (SPBH). lxvi. Derivation of “ωΩ” at the event horizon of a SPBH. lxvii. Derivation of “ωΩ” at the event horizon of a SBH. lxviii. Derivation of “rS”. lxix. “nΩ” and “ωΩ” profiles (as “r → RBH”) of SBH’s. lxx. Derivation of the minimum gravitational lifetime of matter “TL”. lxxi. Derivation of the average emission frequency per Graviton “ωg”. lxxii. Why can't we observe BH’s? 2.7.11 Fundamental Cosmology The primordial and present values of the Hubble constant are derived (“Hα” and “HU” respectively), leading to the determination of the Cosmic Microwave Background Radiation (CMBR) temperature “TU”. This facilitates the determination of the impact of “Dark Matter / Energy” on “HU” and “TU” such that a generalised expression for “TU” in terms of “HU” is formulated. An experimentally implicit derivation of the ZPF energy density threshold “UZPF” is also presented. The derivation sequence proceeds as follows, lxxiii. Derivation of the primordial and present Hubble constants “Hα, HU”. lxxiv. Derivation of the Cosmic Microwave Background Radiation (CMBR) temperature “TU”. lxxv. Numerical solutions for “Hα, AU, RU, ρU, MU, HU” and “TU”. lxxvi. Determination of the impact of “Dark Matter / Energy” on “HU” and “TU”. lxxvii. “TU” as a function of a generalised Hubble constant “TU → TU2”. lxxviii. Derivation of “Ro”, “MG”, “HU2” and “ρU2” from “TU2”. lxxix. Experimentally implicit derivation of the ZPF energy density threshold “UZPF”. 2.7.12 Advanced Cosmology A time dependent derivation of “TU” is performed, including its rate of change and relationship to “HU”. This facilitates the articulation of the Cosmological evolution process into four distinct periods dealing with the inflationary and early expansive phases. Subsequently, the history of the Universe116 is developed and compared to the Standard Model (SM) of Cosmology (SMoC). This assists in determining the Cosmological limitations of the EGM construct. The question of the practicality of utilising conventional radio telescopes for gravitational astronomy is also addressed. The derivation sequence proceeds as follows, lxxx. Time dependent CMBR temperature “TU2 → TU3”. lxxxi. Rates of change of CMBR temperature “TU3 → TU4 → d1,2,3TU4/dt1,2,3”. 116

As defined by the EGM construct. 62

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lxxxii. lxxxiii. lxxxiv. lxxxv. lxxxvi.

Rates of change of the Hubble constant “d1,2H/dt1,2”. Cosmological evolution process. History of the Universe. EGM Cosmological construct limitations. Are conventional radio telescopes, practical tools for gravitational astronomy?

2.7.13 Gravitational Cosmology An engineering model is developed to explain how gravitational effects are transmitted through space-time in terms of EGM wavefunction propagation and interference. The derivation sequence proceeds as follows, lxxxvii. Gravitational propagation: the mechanism for interaction. lxxxviii. Gravitational interference: the mechanism of interaction. 2.7.14 Particle Cosmology The following characteristics are derived utilising EGM principles: lxxxix. The Photon and Graviton mass-energies lower limit. xc. The Photon and Graviton Root-Mean-Square (RMS) charge radii lower limit. xci. The Photon charge threshold. xcii. The Photon charge upper limit. xciii. The Photon charge lower limit. 2.8

Key point summary

Under the EGM construct, the following assertions were derived: xciv. The EGM spectrum is a harmonic description of mass-energy represented as conjugate EM wavefunction pairs; incrementally above “ω = 0(Hz)”, tending to the Planck Frequency “ωh” and obeying a Fourier distribution. Key generalised spectral features are, • It is discrete and harmonically continuous “–ωh ← ω → +ωh”. • The highest frequency is a harmonic multiple of the fundamental (i.e. lowest freq.). • Each wavefunction represents a population of Photons such that each conjugate Photon pair constitutes a Graviton. • Where appropriate, due to the principle of mass-energy equivalence and the law of conservation of energy, it may also be referred to as the PV spectrum. xcv. The ZPF is an EM frequency spectrum referring to the QV spectrum of globally flat space-time geometry. However, such a configuration cannot physically exist; thus, the ZPF takes the form of a generalised reference to the QV field throughout the “Quinta Essentia” series (i.e. QE2-4) such that, • The number of harmonic modes approaches infinity. • The highest frequency tends to zero. xcvi. The PV is an EM frequency spectrum obeying a Fourier distribution at displacement “r” describing a mass “M” induced gravitational field such that, • It denotes a polarised form of the ZPF spectrum. • The population of spectral modes “nΩ(r,M)” decreases as mass increases. • Maximum spectral frequency “ωΩ(r,M)” increases as mass increases. • The fundamental spectral frequency “ωPV(1,r,M)” increases as mass increases. • Spectral frequency bandwidth increases as mass increases. xcvii. A vanishing volume containing infinite energy does not exist under the EGM construct. xcviii. Although on the human scale the quantity of ZPF energy is trivial, on the astronomical or Cosmological scale, it becomes extremely large when approaching the dimensions of the visible Universe. 63

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xcix. The EGM spectrum is a simple, but extreme, extension of the EM spectrum. c. The ZPF equilibrium radius of astronomical bodies coincides with the mean radius (see: QE3), representing the mathematical boundary (within EGM) delineating mass composition and the gravitational field surrounding it. NOTES

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3 The PV Model of Gravity 3.1

Synopsis

The Polarisable Vacuum (PV) representation of General Relativity (GR) is applied to derive mathematical tools facilitating the formulation of “metric engineering” principles utilising ElectroMagnetic (EM) fields. An Engineering interpretation of the PV model is articulated, termed Electro-Gravi-Magnetics (EGM), demonstrating how transformations may be applied to the ZeroPoint-Field (ZPF) to describe variations in energy density as a superposition of EM waves. 3.2

Introduction

The Polarisable Vacuum (PV) model of Gravity is an optical representation of General Relativity (GR) such that the curvature of the space-time manifold is expressed in terms of a Refractive Index “KPV” yielding, i. The potential to supplement GR in terms of the propagation of light through an optical medium, with a scalar theory of gravitation featuring formal analogies with Maxwell's theory of ElectroMagnetism. ii. The potential to unify gravitation and ElectroMagnetism in a theory of Electro-Gravity by quantising the field with populations of conjugate Photon pairs (i.e. Gravitons as defined within the EGM construct). iii. The potential to provide a physical mechanism for how space-time “acquires curvature” in GR, suggesting the possibility of “metric engineering” for spacecraft propulsion etc. The PV model has been shown to pass five crucial tests of GR: (i) predictions of gravitational red shift, (ii) bending of light by a star, (iii) the advance of the perihelion of Mercury (iv), a metric representation leading to the Schwarzschild solution of GR and (v), the inclusion of charge leads to a representation of the Reissner-Nordstrom metric. The success of the PV representation of GR is remarkable given that it is not a geometric model. It is based upon the polarisability of the ZPF and described by variations in permeability “µ0” and permittivity “ε0” as a function of system co-ordinates, in accordance with “THµε” methodology117 and classical macroscopic theory of polarisable media. 3.3

Precepts and principles

Let us consider light bending around an influential gravitational body in a vacuum. One expects that the velocity of light at infinity118 “c0” should be modified by space-time curvature to “c”; hence, the value of “KPV” of space-time may be described such that “KPV = c0 / c”. Since “c0” is a definition (i.e. not a measurement), we conclude that “KPV > 1” for any gravitational field. However, when measurements of “c” are performed utilising instrumentation in the locally inertial reference frame, the instruments (i.e. “rulers and clocks”) are also modified by the gravitational field such that the measured value of Refractive Index is always unity. Hence, one may intelligently question; what is the value of such an approach if the determination of the Refractive Index is always unity (i.e. “KPV = 1”)? The answer may be derived by considering a distant observer119 of an appropriate Gravity well. In such a case, the observer conceptually introduces “KPV” by postulating that “c0” is different in the reference frame of the Gravity well. Thus, relative to the distant observer utilising a globally 117

In the Gaussian system of units, the values of inductance and capacitance are historically derived from geometrical units of length and time. 118 i.e. in a globally flat space-time manifold. 119 i.e. approaching infinity. 65

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flat coordinate system in the local rest frame, variations in the gravitational field strength manifest as changes in the value of Refractive Index120 (i.e. “KPV > 1”). The PV model theorizes that controlling the value of “KPV” affects aspects of Special Relativity (SR) such as time dilation, length contraction and relativistic mass, thereby permitting supraluminal travel as determined by the distant observer. Subsequently, the impact of this upon space-time provokes the Electro-Gravi-Magnetic (EGM) construct, formulated utilising standard engineering design methodologies and principles rather than GR such that, i. The PV model is a tool for understanding gravitation in terms of “KPV”, which determines the intensity of space-time curvature. ii. The ZPF provides the refractive medium (i.e. “KPV”) required by the PV model in a globally flat space-time manifold121. iii. EGM is a tool allowing “KPV” to be understood in terms of spectral energy122. The physical definition of EGM may be stated as: the modification of vacuum polarisability by application of ElectroMagnetic (EM) fields.123 iv. EGM relates gravitational acceleration “g” to locally applied EM fields via the equivalence principle124, utilising Dimensional Analysis Techniques (DAT’s) and Buckingham “Π” Theory (BPT). v. EGM is a discrete interpretation of classical EM field theory where the ZPF is considered to be an EM field with a continuous spectrum of frequency modes. vi. EGM describes the PV as an EM field of densely packed discrete frequencies obeying a Fourier distribution. vii. EGM demonstrates that the PV may be usefully represented by a superposition of EM fields, interpreted theoretically as the local Spectral Energy Density of the vacuum state as derived in Quantum ElectroDynamics (QED). viii. EGM theorizes “metric engineering” of local vacuum polarisability by the manipulation of energy density via by the superposition of applied EM fields. 3.4

ZPF transformations

If the ZPF is an EM field with Lorentz invariant Spectral Energy Density125 “ρ0”, existing as a continuous cubic frequency distribution126 “ω3”, the relationship between them may be written as follows, ρ 0( ω )

2 .h .ω 3

c0

3

(3.47)

where, “h” denotes Planck’s Constant [6.6260693 x10-34(Js)] and “ω” is in “(Hz)”. Relating the preceding equation to an elemental cubic volume (i.e. a box) of homogeneous127 Photon population, the cubic wavelength transforms analogously such that “ρ0” is Lorentz invariant. Therefore, in all locally inertial reference frames, one expects to observe constant “ρ0” at each harmonic frequency mode “nPV”. To articulate variations in the energy density of the

120

Its value may be determined by measurements of gravitational red shift by the distant observer. The presence of gravitational fields significantly modifies the Spectral Energy Density of the ZPF. This is rigorously explored in QE3. 122 Resulting in a well defined “local” representation of the PV. 123 i.e. affecting the state of the PV medium. 124 By dynamic, kinematic and geometric similarity. 125 i.e. each “nPV” in the field possesses a minimum energy of “h⋅ω/2”. 126 Distinct from the discrete PV spectrum derived in QE3 utilising Eq. (3.47). 127 Inferring a constant “g” environment. 121

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ZPF, “nPV” transforms128 by wavefunction superposition such that “ρ0” remains Lorentz invariant129. Therefore, given that the PV model passes several crucial tests of GR utilising “KPV”, it follows that the EGM construct also passes these tests because “rulers and clocks” depend upon the local value of “ρ0” (by wavefunction superposition within the EGM construct), just as they depend upon “KPV” in the PV model. Hence, it is conjectured that the presence of matter or energy results in discrete variations in the spectrum, thereby breaking the symmetry of the ZPF ground state resulting in gravitational effects via the equivalence principle described by similitude techniques. 3.5

PV transformations

The historical derivation of the PV model exhibits isomorphism to GR in weak field approximation. By comparison, the similarities between EGM and the PV model demonstrate that EGM is also isomorphic to GR in the weak field. However, differences exist between the two representations, primarily due to the introduction of a superposition of fields, facilitating the formulation of engineering tools which may be utilised in practical applications. Within the EGM construct, “KPV” is a function of “ρ0” by wavefunction superposition at each point in a gravitational field. EGM supports the conjecture of the PV model such that measurements by “rulers and clocks” depend upon “KPV” of the medium, by applying transformations to the ZPF. Hence, utilising the weak field approximation “Eq. (3.55)” or the assigned form of the “Depp” solution130 “Eq. (4.106)”, a PV transformation table for application to “metric engineering” effects may be formulated utilising, 2.

K PV( r , M ) e

G .M r .c 0

2

(3.55) 1

K Depp ( r , M ) 1

2 .G.M r .c 0

2

(4.106)

For weak field applications, it may be proven by numerical evaluation131 that Eq. (4.106) is usefully approximated by Eq. (3.55) such that, K PV( r , M ) K PV( r , M )

K Depp ( r , M )

2

K Depp ( r , M )

(4.110)

where, “r” and “M” denote the magnitude of the position vector from the centre of mass of the object to the observer and object mass respectively. Note: • For strong fields, “KPV << KDepp2” and “KDepp2” [i.e. Eq. (4.106)] must be applied. Whilst “KDepp2” is valid for all gravitational field strengths compliant with the Schwarzschild limit, the exponential form of Refractive Index “KPV” [i.e. Eq. (3.55)] is only valid in the weak field. • It is the preference of the primary author, for no reason other than scientific respect, to apply the historically original weak field approximation of Refractive Index “KPV” derived by Puthoff et. Al., throughout the “Quinta Essentia” series (i.e. QE2-4); where appropriate. 128

It is demonstrated in QE3 that “nPV” is also a function of coordinates. Quantised within the applied boundary conditions. 130 The assigned form of the Schwarzschild solution specified in QE4 for purposes of convenience with application to Eq (3.67), from the original derivation by Depp et. Al. 131 Refer to QE4. 129

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A transformation table articulating the weak field exponential approximation132 of the “PV Representation of GR” may be generated as follows133, Physical Constant Velocity of light Planck Dirac134 Gravitation Permeability Permittivity Impedance Unit of Measure135 Mass (m) Length (r) Time (t) Energy (E) Planck Measure136 Mass (mh) Length (λh) Time (th) Energy (Eh) Relative Measure137 Mass Length Time Energy

PV Representation of GR c(KPV) = KPV-1⋅c0 h(KPV) = h0 = h ħ(KPV) = ħ0 = ħ G(KPV) = G0 = G µ(KPV) = KPV⋅µ0 ε(KPV) = KPV⋅ε0 Z(KPV) = Z0 = (µ0/ε0)½ m(KPV) = KPV3/2⋅m0 r(KPV) = KPV-½⋅r0 t(KPV) = KPV½⋅t0 E(KPV) = KPV-½⋅E0 mh(KPV) = KPV-½⋅mh_0 λh(KPV) = KPV3/2⋅λh_0 th(KPV) = KPV5/2⋅th_0 Eh(KPV) = KPV-5/2⋅Eh_0 m(KPV)/mh(KPV) = KPV2⋅(m0/mh_0) r(KPV)/λh(KPV) = KPV-2⋅(r0/λh_0) t(KPV)/th(KPV) = KPV-2⋅(t0/th_0) E(KPV)/Eh(KPV) = KPV2⋅(E0/Eh_0) Table138 2.8,

where, “Eh = mh⋅c02” and, th

G.h 5

c0

mh

(4.16)

h .c 0 G

(4.18)

λh

G.h 3

c0

(4.19)

Hence, the preceding table permits Engineers (in principle) to design and develop new technologies within the EGM construct, to affect the PV medium controlling relative polarisability (i.e. “KPV, KDepp2”), at any point in a gravitational field by the superposition of applied EM wavefunctions. Note: to apply the preceding table to strong fields, “KPV” must be replaced by the assigned form of the “Depp” Schwarzschild solution (i.e. “KDepp2”). 132

i.e. the historically original form derived by Puthoff et. Al. The subscript “0” relates to values in a globally flat space-time manifold. The non-subscripted parameters of “µ”, “ε” and “Z” do not refer to the classical representation of relative permeability, permittivity and impedance (i.e. they are generalised references to the constants only). 134 Where, “ħ” denotes Dirac’s Constant (i.e. “ħ ≡ h / 2π”). 135 PV transformations at the physical scale (i.e. “rulers and clocks”). 136 PV transformations at the Planck scale. 137 This demonstrates the consistency of relative measures of the PV model (i.e. the relationship between the physical and Planck scales). 138 By Desiato et. Al. 133

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3.6

The Schwarzschild solution

3.6.1 Special note i. ii.

This section is based upon formalisms by “Joseph G. Depp”. All mathematical statements herein were formulated utilising “MathCad 8 Professional” and appear in standard product notation. It is a MathCad computational environmental requirement to specify all derivatives utilising full differential form. However, the notation utilised within this section refers to partial derivatives where appropriate (i.e. “dKPV ≡ δKPV”).

3.6.2 Abstract The PV model is scrutinised for mathematical equivalence to the Schwarzschild GR solution. An alternative is proposed such that the resulting Lagrangian Density and equation of motion are presented for further investigation. 3.6.3 Introduction In 1957, Dicke et. Al. published an oft-cited paper139 investigating the possibility that GR could be derived from field theory. The form of the Lagrangian Density “LD” for the Refractive Index was taken as, L D K PV

K L.F K PV . ∇ K PV

2

K PV d . K PV c 0 dt

2

(2.8)

where, “KPV” is a generalised reference140 to Refractive Index, “KL” and “F(KPV)” represent a constant and scalar function respectively. Dicke et. Al. formalised a solution such that “F(KPV) → 1/KPV”, whilst Puthoff et. Al. investigated “F(KPV) → 1/KPV2”. Herein, it is concluded that the form “F(KPV) → 1/KPV4” satisfies the Schwarzschild solution for congruence with GR. 3.6.4 “KPV” The development of the Schwarzschild metric is found in many introductory texts to GR141. The generalised spherically symmetric, time-independent metric line element may be written according to, ds

2

A ( r ) . c 0 .dt

B( r ) .dr

2

2

2 2 r . dθ

( sin ( θ ) .dφ)

2

(2.9)

where, A(r) and B(r) are intrinsically positive functions such that the metric element is typically solved obtaining the solution by means of “c0 = G = 1” as follows, A( r)

1 B( r )

1

2 .G.M r .c 0

2

1

2 .M r

(2.10)

139

R. H. Dicke, “Gravitation without a principle of equivalence”, Rev. Mod. Phys. 29, 363–376, (1957). 140 i.e. not specifically relating to the weak field exponential approximation shown by Eq. (3.55). 141 R. Adler, M. Bazin, M. Schiffer, “Introduction to General Relativity”, McGraw-Hill 1994 Ch 6, 164-173. 69

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where, Desiato et. Al. assert142 that the Schwarzschild solution of the Refractive Index takes the form “KPV(r,M) → A(r,M)-1 = B(r,M)” hence, 1 2 .G.M

KPV(r,M) → 1

1 1

r .c 0

2 .M r

2

(2.11i)

3.6.5 “F(KPV)” The PV equation of motion with scalar function “F(KPV)” may be written as, 2

∇ K PV

K PV c0

2

2 .d K PV d t2

1 . f K PV . ∇ K PV 2 F K PV

2

2

K PV

K PV f K PV . . 1 . d K PV 2 F K PV c 2 d t 0

2

(2.12)

where, “f(KPV) = F`(KPV)” such that for a spherically symmetric time-independent solution the preceding equation reduces to, d

2

d r2

K PV

2 .d r dr

1 . f K PV . d K PV 2 F K PV d r

K PV

2

(2.13)

As stated in the introduction, Dicke et. Al. formalised a solution where “F(KPV) → 1/KPV”, whilst Puthoff et. Al. investigated “F(KPV) → 1/KPV2”. Hence, based upon the Dicke / Puthoff et. Al. formalisms, an obvious evolutionary proposition to investigate is “F(KPV) → 1/KPV4”. To test the stated Desiato et. Al. assertion, we hypothesise that the result “F(KPV) = 1/ KPV4” may be obtained by substitution of Eq. (2.11i) into Eq. (2.13); executing the procedure yields, f K PV

4

F K PV

K PV

(2.14) 143

Solving for “F(KPV)” with a constant of integration set to zero

produces,

1

F K PV

4

K PV

(2.15)

Therefore, the Desiato et. Al. assertion is validated and the assigned form of the Refractive Index articulated by Eq. (4.106) (i.e. “KDepp2”) is verified as a satisfactory Schwarzschild GR solution. 3.6.6 “LD(KPV)” Substituting “F(KPV) = 1/ KPV4” into Eq. (2.8, 2.12) yields, L D K PV

2

∇ K PV

142 143

KL

. ∇K PV 4 K PV

K PV c0

2

2

K PV d . K PV c 0 dt

2 2 . ∇ K PV .d K PV K PV d t2

2

2

K PV d . K PV 2 c 0 dt

(2.16) 2

(2.17)

T. J. Desiato, R. C. Storti, “Event horizons in the PV model”, E-print 2003. Due to the inclusion of the arbitrary constant “KL” in “LD”. 70

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3.6.7 Conclusion Eq. (2.16) differs from the Lagrangian Densities utilised by Dicke / Puthoff et. Al. However, it produces an expression for the Refractive Index in exact agreement with the Schwarzschild GR144 solution. Moreover, the Desiato et. Al. assertion implies that co-ordinates “(r,θ,φ,t)” in Eq. (2.16, 2.17) are equivalent to those in the metric element defined in Eq. (2.8), implicitly satisfying dynamic, kinematic and geometric similarity associated with the utilisation of DAT’s and BPT within the EGM construct. 3.7

The Reissner-Nordstrom solution

3.7.1 Special note i. ii.

This section is based upon formalisms by “Joseph G. Depp”. All mathematical statements herein were formulated utilising “MathCad 8 Professional” and appear in standard product notation. It is a MathCad computational environmental requirement to specify all derivatives utilising full differential form. However, the notation utilised within this section refers to partial derivatives where appropriate (i.e. “dKPV ≡ δKPV”).

3.7.2 Abstract The PV model is scrutinised for mathematical equivalence to the Reissner-Nordstrom GR solution. An alternative is proposed such that the resulting Lagrangian Density and equation of motion are presented for further investigation. 3.7.3 “LD(KPV)” Utilising Eq. (2.8), “LD” for the Refractive Index of an EM field may be written as, L D K PV

KL

. ∇K PV 4 K PV

2

K PV d . K PV c 0 dt

2

1.

2

B( r ) 2 K PV.µ 0

2 K PV.ε 0 .E( r )

(2.18)

For a static point charge (i.e. “B = 0”), the field is given in “MKS” units by, E( r )

Q 2 4 .π .K PV.ε 0 .r

(2.19)

where, only non-source terms such as “c”, “µ” and “ε” transform (i.e. incorporate “KPV”). 3.7.4 “ψ1,2” Utilising Eq. (2.18), the PV equation of motion may be written according to, 2 .K L

2 2 d . ∇2 K K PV . K PV PV 4 d ( c .t ) 2 K PV

144

2 . ∇ K PV K PV

2

d K PV. K PV d ( c .t )

2

1

2

.1 0 4 2 .K PV ( 4 .π ) .ε 0 r (2.20) .

Q

2

2

The Dicke / Puthoff et. Al. formalisms do not achieve Schwarzschild compliance. 71

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Simplifying yields, 2 .K L

2 . d K PV 4 K PV d r 2

d

2

d r2

K PV

2 .d K PV r dr

2 . d K PV K PV d r

2 .d K PV r dr

2 . d K PV K PV d r

2

1

2

.1 0 4 2 .K PV ( 4 .π ) ε 0 r .

2

2

Q

2.

2

Q

K . PV

2 64.π .ε 0 .K L

r

2

(2.21)

4

(2.22)

Let, 2 .ψ 2

2

Q

2 64.π .ε 0 .K L

(2.23)

Hence, d

2

d r2

K PV

2 .d r dr

K PV

2 . d K PV K PV d r

2

2 2 .K PV .ψ 2 4

r

(2.24)

Pursuant to the Desiato et. Al. Schwarzschild assertion in the preceding section, the “Reissner-Nordstrom” metric line element takes the following Refractive Index form, ds

1

2

K PV( r )

. c .dt 0

2

K PV( r ) .dr

2

2 2 r . dθ

( sin ( θ ) .dφ)

2

(2.25)

such that: KPV(r) = [1 ± (ψ1/r) ± (ψ2/r2)]-1 where, by means of “c0 = G = 1”, ψ1

2 .G.M

(2.26)

2 .M

2

c0

(2.27)

3.7.5 Conclusion It has been demonstrated that the modified PV Lagrangian Density presented in the preceding section yields the Reissner-Nordstrom metric line element. NOTES

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3.8

The generalised PV equations of motion

3.8.1 Special note i. ii.

iii.

This section is based upon formalisms by “Todd J. Desiato”. The Schwarzschild solution for the Refractive Index is utilised in this section. Desiato et. Al. derived (i.e. in addition to Depp et. Al.) the Schwarzschild solution “KDesiato” by unique means. The derivation of “KDesiato” has been omitted for brevity which is intended to foster an appreciation for the identical PV representations of GR formulated by two independent sources, reinforcing the EGM construct. All mathematical statements herein were formulated utilising “MathCad 8 Professional” and appear in standard product notation. It is a MathCad computational environmental requirement to specify all derivatives utilising full differential form. However, the notation utilised within this section refers to partial derivatives where appropriate (i.e. “dKPV ≡ δKPV”).

3.8.2 Abstract Time-independent solutions of the equations of motion for the Refractive Index in the PV model are derived by Desiato et. Al. It is demonstrated that these equations may be applied to obtain solutions and equations of motion for the metric component functions, identical to GR. The equations of motion in the PV model are easier to solve than the equations of GR as they do not require Tensor mathematics or geometric interpretations to be understood. 3.8.3 Time-independent solutions The time-independent equations of motion for the Refractive Index in the PV model are found from the Laplace equation, “∇2φ(r) = 0” as follows, 2

∇ φ( r ) ∇

2

c0 K PV( r )

∇

2

1

0

K PV

(2.28)

where, “φ(r) = c0 / KPV(r)” is abbreviated to “1/KPV” by means of “c0 = G = 1”. 3.8.4 Co-ordinate systems 3.8.4.1 Cartesian In Cartesian co-ordinates “(x1,x2,x3)”, solutions for “1/KPV” are a family of straight lines of the form145 “1/KPV = a⋅x1,2,3 + b”. 3.8.4.2 Spherical In spherical co-ordinates “(r,θ,φ)” for a symmetric potential, Eq. (2.28) becomes, d

2

d r2

K PV

2 .d K PV r dr

2 . d K PV K PV d r

2

0

(2.29)

where, the solution for “KPV” is identical to the Schwarzschild GR solution as follows146, 145 146

From the generalised classical representation of “y = m⋅x + c”. i.e. derived by Depp / Desiato et. Al. independently. 73

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1 2 .G.M

KPV(r,M) → KDepp(r,M)2 → KDesiato(r,M) → 1

1 1

r .c 0

2 .M r

2

(2.11ii)

The Schwarzschild metric component is then interpreted as the Refractive Index of the ZPF surrounding a spherical homogeneous mass “M”, centred at the origin of the co-ordinate system. Moreover, if we include an ElectroMagnetic (EM) “source” term such as charge “Q” located at the origin of the coordinate system, Eq. (2.29) becomes, d

2

d r2

2 .d

K PV

r dr

K PV

2 . d K PV K PV d r

2

2 2 .K PV .ψ 2 4

r

(2.24)

where, any value of “KPV” of the form “KPV(r) = [1 ± (ψ1/r) ± (ψ2/r2)]-1” is a solution such that147, ψ2

2 G.Q

4 .π .ε 0 .c 0

4

(2.30)

3.8.4.3 Cylindrical In cylindrical co-ordinates “(ρ,θ,z)”, the equation of motion for an “infinite wire” is derived from Eq. (2.28) as follows, d

2

dρ2

K PV

1 .d ρ dρ

K PV

2 . d K PV K PV d ρ

2

(2.31)

Eq. (2.31) is identical to that derived under GR; thus, possessing identical Refractive Index solutions. 3.8.5 “KL” Equating Eq. (2.23, 2.30), a solution for “KL” may be determined according to, 2 G.Q

2

Q

2 4 128.π .ε 0 .K L 4 .π .ε 0 .c 0

(2.32)

Hence, 4

KL

c0

32.π .G

(2.33)

Therefore, “KL” is in agreement with the value suggested by Dicke et. Al. based on heuristic arguments, demonstrating that the PV model satisfies the Schwarzschild and Reissner-Nordstrom GR solutions. 3.8.6 Conclusion Three different coordinate systems were utilised to determine the equations of motion of the Refractive Index. In all cases, the solutions were found to be identical to those under GR. Therefore, the PV model may provide insight into alternative models of Quantum Gravity (QG), without the need for cumbersome geometric interpretations of the space-time manifold, simply the consistent application of the Refractive Index to typical scalar field theories. 147

Including the Reissner-Nordstrom solution, identical to GR. 74

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4 The Natural Philosophy of Fundamental Particles “The alpha males of academia [on occasion], project the convenience that Nature is beyond simplicity and its secrets are only privy to themselves.” • Riccardo C. Storti Taken from “Quinta Essentia – Part 3, 4” (QE3,4) Abstract Electro-Gravi-Magnetics (EGM) is a term describing a hypothetical harmonic relationship between Electricity, Gravity and Magnetism. The hypothesis may be mathematically articulated by the application of Dimensional Analysis Techniques (DAT’s) and Buckingham “Π” Theory (BPT), both being well established and thoroughly tested geometric engineering principles, via Fourier harmonics. The hypothesis may be tested by the correct derivation of experimentally verified fundamental properties not predicted within the Standard Model (SM) of Particle-Physics. Theoretical estimates and correlations, based upon the EGM method, are presented for the RootMean-Square (RMS) charge radius and mass-energy of many well established subatomic particles. The estimates and correlations coincide to astonishing precision with experimental data presented by the Particle Data Group (PDG), CDF, D0, L3, SELEX and ZEUS Collaborations. Our tabulated results clearly demonstrate a possible natural harmonic pattern representing all fundamental subatomic particles. In addition, our method predicts the possible existence of several other subatomic particles not contained within the Standard Model (SM). The accuracy and simplicity of our computational estimates demonstrate that EGM is a useful tool to gain insight into the domain of subatomic particles.

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4.1

Harmonic representation of gravitational acceleration

It is demonstrated in “Quinta Essentia – Part 3” (QE3) that a theoretical representation of constant acceleration at a mathematical point in a gravitational field may be defined by a summation of trigonometric terms utilising modified complex Fourier series in exponential form, according to the harmonic distribution “nPV = -N, 2 - N ... N”, where “N” is an odd number harmonic. Hence, the magnitude of the gravitational acceleration vector “g” (via the equivalence principle) may be usefully represented by Eq. (3.63) 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)

such that, the frequency spectrum of the harmonic gravitational field “ωPV” is given by Eq. (3.67), ω PV n PV, r , M

n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

(3.67)

where: “c0 → c”, Variable ωPV(1,r,M) KPV

nPV r M G

Description Units Fundamental spectral frequency Hz Refractive Index of a gravitational field in the Polarisable Vacuum (PV) model of Gravity, only contributing significantly when a large gravitational mass (i.e. a strong gravitational field) is considered. For None all applications herein, the effect is approximated to KPV(r,M) = 1. Harmonic modes of the gravitational field Magnitude of position vector from centre of mass m Mass kg Gravitational constant m3kg-1s-2 Table 4.3,

Subsequently, the harmonic (Fourier) representation of the magnitude of the gravitational acceleration vector (in the time domain) at the surface of the Earth up to “N = 21” is graphically shown to be,

Gravitational Acceleration

g

Time

Figure 4.3: harmonic representation of gravitational acceleration, As “N → ∞”, the magnitude of the gravitational acceleration vector becomes measurably constant. Hence, Eq. (3.63, 3.67) illustrate that the Newtonian representation of “g” is easily harmonised over the Fourier domain, from geometrically based methods (i.e. DAT’s and BPT). Therefore, unifying (in principle) Newtonian, geometric (relativistic) and quantum (harmonised) models of Gravity. QE3 demonstrates that the spectrum defined by Eq. (3.67) is discrete and finite. The lower boundary value is given by “ωPV(1,r,M)”, whilst the upper boundary value “ωΩ” (also termed the harmonic cut-off frequency) is given by Eq. (3.73), 76

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ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73)

supported by the following equation set, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

3

Ω ( r, M )

108.

U m( r , M )

12. 768 81.

U ω( r , M )

U m( r , M )

U ω( r , M )

3 .M .c

U m( r , M )

2

U ω( r , M )

(3.72)

2

4 .π .r

(3.70)

h . 4 ω PV( 1 , r , M ) 3 2 .c

(3.69)

3

where, Variable nΩ Ω Um Uω h

Description Units None Harmonic cut-off mode [mode number at ωΩ] Harmonic cut-off function Mass-energy density of a solid spherical gravitational object Pa Energy density of mass induced gravitational field scaled to the fundamental spectral frequency Planck’s Constant [6.6260693 x10-34] Js Table 4.4,

Since the relationship between trigonometric terms, at each amplitude and corresponding frequency, is mathematically defined by the nature of Fourier series, the derivation of Eq. (3.71, 3.72) is based on 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 mathematical properties of constant functions utilising Fourier series as discussed in QE3. The subsequent application of these results to Eq. (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 Newtonian Gravity, identified by the “compression technique” stated above. Key gravitational characteristics for the Earth148 in the displacement domain 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

Figure149 3.7, 148 149

“RE” and “ME” denote the radius and mass of the Earth respectively. Fundamental frequency (|nPV| = 1) as a function of planetary radial displacement. 77

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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

Figure150 3.8, 4.2

Poynting Vector “Sω”

“Haisch, Puthoff and Rueda” conjectured that “Inertia” may have ElectroMagnetic (EM) origins due to the Zero-Point-Field (ZPF) of Quantum-ElectroDynamics (QED), manifested by the Poynting Vector, via the equivalence principle. Hence, it follows that gravitational acceleration may also be EM in nature and the Polarisable Vacuum (PV) model of Gravity is an EM polarised state of the ZPF with a Fourier distribution, assigning physical meaning to Eq. (3.63). Subsequently, it follows that the energy density of a mass induced gravitational field may be scaled to changes in odd harmonic mode numbers satisfying the mathematical properties of any constant function described in terms of Fourier series utilising Eq. (3.69) - such that, U ω n PV, r , M

U ω( r , M ) .

n PV

2

4

4

n PV

(3.68)

151

Therefore, the Poynting Vector of the polarised Zero-Point (ZP) gravitational field “Sω” surrounding a solid spherical object with homogeneous mass-energy distribution is given by, S ω n PV, r , M

c .U ω n PV, r , M

(3.74)

ZPF Poynting Vector

and may be graphically represented as follows,

S ω n PV , R E , M E

n PV Harmonic

Figure 3.9, Fig. (3.9) illustrates that the Poynting Vector of the ZP gravitational field increases with “nPV”. Further work in QE3 showed that “>>99.99(%)” of the effect in a gravitational field exists 150 151

Harmonic cut-off mode “nΩ” and frequency “ωΩ” as a function of planetary radial displacement. Per change in odd harmonic mode number. 78

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well above the “THz” range. Hence, it becomes apparent that “nΩ” and “ωΩ” are important characteristics of gravitational fields and were used to “quasi-unify” Particle-Physics in harmonic form. 4.3

The size of the Proton, Neutron and Electron (radii: “rπ”, “rν”, “rε”)

QE3 derives the mass-energy threshold of the Photon utilising “nΩ” and the classical Electron radius, to within “4.3(%)” of the Particle Data Group (PDG) value152 stated in [1], then proceeds to derive the mass-energies and radii of the Photon and Graviton by the consistent utilisation of “nΩ”. The method developed was utilised to derive the sizes153 of the Electron, Proton and Neutron. The motivation for this was to test the hypothesis presented by direct comparison of the computed size values to experimentally measured fact. One may argue that highly precise computational predictions’, agreeing with experimental evidence beyond the abilities of the SM to do so, is conclusive evidence of the validity of the harmonic method developed in QE3. To date, highly precise measurements have been made of the Root-Mean-Square (RMS) charge radius of the Proton by [2] and the Mean-Square (MS) charge radius of the Neutron as demonstrated in [3]. However, the calculations presented in QE3 are considerably more accurate than the physical measurements articulated in [2,3], lending support for the harmonic representation of the magnitude of the gravitational acceleration vector stated in Eq. (3.63). The basic approach utilised in QE3 was to determine the equilibrium position between the polarised state of the ZPF and the mass-energy of the fundamental particle inducing space-time curvature as would appear in General Relativity (GR). In other words, one may consider the curvature of the space-time manifold surrounding an object to be a “virtual fluid” in equilibrium with the object itself154. This concept is graphically represented in Fig. (4.4). A free fundamental particle with classical form factor is depicted in equilibrium with the surrounding space-time manifold. The ZPF is polarised by the presence of the particle in accordance with the PV model of Gravity, which is (at least) isomorphic to GR in the weak field.

Figure 4.4: free fundamental particle with classical form factor, In the case of the Proton, the ZPF equilibrium radius coincides with the RMS charge radius “rπ” [Eq. (3.199)] producing the experimentally verified result “rp” by the SELEX Collaboration as stated in [2]155, 152

Consistent with experimental evidence and interpretation of data. From first principles and from a single paradigm. 154 The intention is not to suggest that the space-time manifold is actually a fluid, it is merely to present a method by which to solve a problem. 155 rπ = 0.8306(fm), rp = 0.8307 ± 0.012(fm). 153

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rπ

h .m e

5

4

. . m . 27 h c . e 2 3 4 .π .G m p 16.c .π .m p

(3.199)

where, “me” and “mp” denote Electron and Proton rest-mass respectively. In the case of the Neutron, the ZPF equilibrium radius coincides with the radial position of zero charge density “rν” [Eq. (3.200)] with respect to the Neutron charge distribution as illustrated in Fig. (4.5). It is shown in QE3 that “rν” relates to the MS charge radius “KS” by a simple formula [Eq. (3.396)] producing the experimentally verified result “KX” as presented in [3]156, h .m e

5

4

. . m . 27 h c . e rν 2 3 4 .π .G m n 16.c .π .m n

(3.200)

where, “mn” denotes Neutron rest-mass. Neutron Charge Distribution

Charge Density

rν

r dr

ρ ch( r ) ρ ch r 0

r dr

5. 3

rν

ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

Figure 4.5: Neutron charge distribution, KS 157

where, “x” is solved numerically

3. π .r ν 8

2

. (1

x) . x

1

x x

3

2

(3.396)

within the “MathCad” environment by the following algorithm,

Given 2

x

ln( x) . 2

x x

1

(3.398)

1 3

(3.399)

Find ( x)

Utilising “KS”, “KX” may be converted to determine an experimental zero charge density radial position value “rX” according to Eq. (3.418),

156 157

rν = 0.8269(fm), KS = -0.1133(fm2), KX = -0.113 ± 0.005(fm2). x = 0.6829, rX = 0.8256 ± 0.018(fm). 80

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rν

rX

KS

. K .K S X

(3.418)

In the case of the Electron (as with the Proton), the ZPF equilibrium radius coincides with the RMS charge radius “rε” [Eq. (3.203)] producing an experimentally implied result158 as stated in [4], 9

r ε r e.

1. 2

ln 2 .n Ω r e , m e

5

γ

(3.203)

where, “re” and “γ” [5] denote the classical Electron radius and Euler-Mascheroni constant respectively. 4.4

The harmonic representation of fundamental particles

4.4.1 Establishing the foundations Motivated by the physical validation of Eq. (3.199, 3.200), Storti et. Al. conducted thought experiments in QE3 to investigate harmonic and trigonometric relationships by analysing various forms of radii combinations for the Electron, Proton and Neutron consistent with the DAT’s and BPT derivations – yielding the following useful approximations, ω Ω r ε, m e

ω Ω r ε, m e

ω Ω r π, m p

ω Ω r ν ,mn

rε rπ

α

2

(4.1)

π rν

rε

(3.214) 2

.e

3

rπ

(3.204)

where, i. ii.

iii. iv.

“α” and “e” denote the fine structure constant and exponential function respectively. Eq. (4.1) error: (a) Associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “8.876 x10-3(%)” (b) Associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.266(%)”. Eq. (3.214) error is “2.823(%)”. Eq. (3.204) error is “0.042(%)”.

4.4.2 Improving accuracy Since the experimental value of the RMS charge radius of the Proton is considered by the scientific community to be precisely known159, the accuracy of Eq. (3.214, 3.204) may be improved by re-computing the value of “rν” and “rε”. This action further strengthens the validity of Eq. (4.1) by verifying trivial deviation utilising the re-computed values. Hence, it follows that numerical solutions for “rν” and “rε”, constrained by exact mathematical statements [Eq. (3.203, 3.204, 3.214, 4.1)], suggests that the gravitational relationship between the Electron and Proton, as inferred by the result “ωΩ(rε,me)/ωΩ(rπ,mp) = 2”, is harmonic. 158 159

rε ≥ 0.0118(fm), γ = 0.577215664901533. To a degree of accuracy significantly greater than the Electron or Neutron. 81

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The computational algorithm supporting this contention may be stated as follows, Given α

r ε ω Ω r ε, m e r e ω Ω r π, m p

rν rε

rε rπ

rε rν

rπ

9

2

.e

3

1. 2

ln 2 .n Ω r e , m e

5

γ

2 π

Find r ν , r ε

(4.2)

(4.3)

yields, rν

0.826838

rε

0.011802

.( fm)

(4.4)

where, i.

ii. iii.

Eq. (4.1) error: (a) Associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “4.493 x10-7(%)”. (b) Associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.282(%)”. Eq. (3.214) error is “1.11 x10-13(%)”. Eq. (3.204) error is “0.026(%)”.

4.4.3 Formulating an hypothesis In the preceding calculations utilising known particle mass and radii as a reference, it was found that the harmonic cut-off frequency ratio of an Electron to a Proton was precisely “2”. This provokes the hypothesis that a simple harmonic pattern may exist describing the relationship of all fundamental particles relative to an arbitrarily chosen base particle according to, ω Ω r 1, M 1 ω Ω r 2, M 2

St ω

(3.230i)

Performing the appropriate substitutions utilising Eq.(3.69 – 3.73), Eq. (3.230i) may be simplified to, M1 M2

2

.

r2

5

r1

St ω

9

(3.230ii)

where, “Stω” represents the ratio of two particle spectra. Subsequently, “rε” may be simply calculated according to, 5

1 . me r ε r π. 9 2 mp

2

(3.231)

4.4.4 Identifying a mathematical pattern Utilising Eq. (3.230ii), Storti et. Al. identify mathematical patterns in QE3 showing that “Stω” may be represented in terms of the Proton, Electron and Quark harmonic cut-off frequencies derived from the respective particle. Potentially, three new Leptons (L2, L3, L5 and associated Neutrino’s: ν2, ν3, ν5) and two new Quark / Boson’s (QB5 and QB6) are predicted, beyond the SM as shown in table (4.5).

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The EGM Harmonic Representation of Fundamental Particles (i.e. table (4.5)) is applicable to the size relationship between the Proton and Neutron (i.e. to calculate “rπ” from “rν” and viceversa utilising “Stω = 1”) as an approximation only. For precise calculations based upon similar forms, the reader should refer to Eq. (3.199, 3.200). Note: although the newly predicted Leptons are within the kinetic range160 and therefore “should have been experimentally detected”, there are substantial explanations discussed in the proceeding sections. Proton Electron Quark Harmonics Harmonics Harmonics Proton (p), Neutron (n) Stω = 1 Stω = 1/2 Stω = 1/14 2 1 1/7 Electron (e), Electron Neutrino (ν νe) 4 2 2/7 L2, ν2 (Theoretical Lepton, Neutrino) 6 3 3/7 L3, ν3 (Theoretical Lepton, Neutrino) 8 4 4/7 Muon (µ µ), Muon Neutrino (ν νµ) 10 5 5/7 L5, ν5 (Theoretical Lepton, Neutrino) 12 6 6/7 Tau (ττ), Tau Neutrino (ν ντ ) Up Quark (uq), Down Quark (dq) 14 7 1 Strange Quark (sq) 28 14 2 Charm Quark (cq) 42 21 3 Bottom Quark (bq) 56 28 4 QB5 (Theoretical Quark or Boson) 70 35 5 QB6 (Theoretical Quark or Boson) 84 42 6 W Boson 98 49 7 Z Boson 112 56 8 Higgs Boson (H) (Theoretical) 126 63 9 Top Quark (tq) 140 70 10 Table 4.5: harmonic representation of fundamental particles,

Existing and Theoretical Particles

4.4.5 Results 4.4.5.1 Harmonic evidence of unification Exploiting the mathematical pattern articulated in table (4.5), EGM predicts the RMS charge radius and mass-energy of less accurately known particles, comparing them to expert opinion. The values of “Stω” shown in table (4.5), predict possible particle mass and radii for all Leptons, Neutrinos, Quarks and Intermediate Vector Bosons (IVB’s), in complete agreement with the SM, PDG estimates and studies by Hirsch et. Al in [6] as shown in table (4.6), Particle Proton (p) Neutron (n) Electron (e) Muon (µ µ) Tau (ττ) Electron Neutrino (ν ν e) Muon Neutrino (ν νµ) 160

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

EGM Mass-Energy (computed or utilised)

PDG Mass-Energy Range (2005 Values)

Mass-Energy precisely known, See: National Institute of Standards and Technology (NIST) [7] Note: δm = 10-100 men(eV) ≈ 3 - δm mµn(MeV) ≈ 0.19 - δm

men(eV) < 3 mµn(MeV) < 0.19

A region extensively explored in Particle-Physics experiments. 83

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Tau Neutrino (ν ν τ) rτn ≈ 1.9588 mτn(MeV) ≈ 18.2 - δm mτn(MeV) < 18.2 Up Quark (uq) 1.5 < muq(MeV) < 4 ruq ≈ 0.7682 muq(MeV) ≈ 3.5060 Down Quark (dq) 3 < mdq(MeV) < 8 rdq ≈ 1.0136 mdq(MeV) ≈ 7.0121 Strange Quark (sq) 80 < msq(MeV) < 130 rsq ≈ 0.8879 msq(MeV) ≈ 113.9460 1.15 < mcq(GeV) < 1.35 Charm Quark (cq) mcq(GeV) ≈ 1.1833 rcq ≈ 1.0913 Bottom Quark (bq) 4.1 < mbq(GeV) < 4.4 rbq ≈ 1.071 mbq(GeV) ≈ 4.1196 Top Quark (tq) 169.2 < mtq(GeV) < 179.4 rtq ≈ 0.9294 mtq(GeV) ≈ 178.4979 W Boson 80.387 < mW(GeV) < 80.463 rW ≈ 1.2839 mW(GeV) ≈ 80.425 Z Boson 91.1855 < mZ(GeV) < 91.1897 rZ ≈ 1.0616 mZ(GeV) ≈ 91.1876 Higgs Boson (H) mH(GeV) ≈ 114.4 + δm mH(GeV) > 114.4 rH ≈ 0.9403 Photon (γγ) rγγ = ½Kλλh mγγ ≈ 3.2 x10-45(eV) mγ < 6 x10-17(eV) No definitive commitment Graviton (γγg) rgg = 2(2/5)rγγ mgg = 2mγγ L2 (Lepton) mL(2) ≈ 9(MeV) rL ≈ 10.7518 mL(3) ≈ 57(MeV) L3 (Lepton) L5 (Lepton) mL(5) ≈ 566(MeV) ν2 (L2 Neutrino) rν2,ν3,ν5 mν2 ≈ men Not predicted or considered ≈ ν3 (L3 Neutrino) mν3 ≈ mµn ren,µn,τn ν5 (L5 Neutrino) mν5 ≈ mτn QB5 (Quark or Boson) rQB ≈ 1.0052 mQB(5) ≈ 10(GeV) QB6 (Quark or Boson) mQB(6) ≈ 22(GeV) Table 4.6: RMS charge radii and mass-energies of fundamental particles, where, i. “Kλ” denotes a Planck scaling factor, determined to be “(π/2)1/3” in QE3. 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) utilised to calculate the mass-energy of the proposed “new particles”. Note: iv. A formalism for the approximation of ν2, ν3 and ν5 mass-energy is shown in QE3. v. It is shown in QE3 that the RMS charge diameters of a Photon and Graviton are “λh” and “1.5λh” respectively, in agreement with Quantum Mechanical (QM) models. 4.4.5.2 Recent developments 4.4.5.2.1 PDG mass-energy ranges The EGM construct was finalized by Storti et. Al. in 2004 and tested against published PDG data of the day [i.e. the 2005 values shown in table (4.6)]. Annually, as part of their “continuous improvement cycle”, 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: i. Strange Quark = “70 < msq(MeV) < 120”. ii. Charm Quark = “1.16 < mcq(GeV) < 1.34”. iii. “W” Boson = “80.374 < mW(GeV) < 80.432”. iv. “Z” Boson = “91.1855 < mZ(GeV) < 91.1897”. 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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4.4.5.2.2 Electron Neutrino and Up / Down / Bottom Quark mass Particle-Physics research is a highly dynamic field supporting a landscape of constantly changing hues. The EGM construct relates “mass to size” in harmonic terms. If one applies Eq. (3.230ii) and utilizes the Proton as the reference particle in accordance with table (4.5), one obtains a single expression with two unknowns, as implied by Eq. (3.231). Since contemporary Physics is currently incapable of specifying the mass and size of most fundamental particles precisely and concurrently, EGM is required to approximate values of either mass or radius to predict one or the other (i.e. mass or size). Subsequently, the EGM predictions articulated in table (4.6) denote values based upon estimates of either mass or radius. Hence, some of the results in table (4.6) 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 table (4.6) 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 table (4.6) [i.e. 5.2574(MeV)] remains within the 2006 average mass range specified by the PDG [i.e. 2.5 to 5.5(MeV)]. EGM Radii x10-16(cm)

PDG Mass-Energy Range (2006 Values) men(eV) < 2 Electron Neutrino (ν νe) ren < 0.0811 PDG Mass-Energy Up Quark (uq) 1.5 < muq(MeV) < 3 0.5469 < ruq < 0.7217 Range (2006 Values) Down Quark (dq) 3 < mdq(MeV) < 7 0.7217 < rdq < 1.0128 Bottom Quark (bq) 1.0719 > rbq > 1.0863 4.13 < mbq(GeV) < 4.27 Table 4.7: RMS charge radii and mass-energies of fundamental particles, Particle

EGM Mass-Energy (utilised)

The predicted radii ranges above demonstrate that no significant deviation from table (4.6) values exists. 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 RMS charge radii. Therefore, we may conclude that the EGM construct continues to predict experimentally verified results within the SM to high computational precision. 4.4.5.2.3 Top Quark mass 4.4.5.2.3.1 The dilemma The Collider Detector at Fermilab (CDF) and “D-ZERO” (D0) Collaborations have recently revised their world average value of “Top Quark” mass from “178.0(GeV/c2)” in 2004 [8] to, “172.0” in 2005 [9], “172.5” in early 2006, then “171.4” in July 2006. [10] Note: since the precise value of “mtq” is subject to frequent revision, we shall utilise the 2005 value in the resolution of the dilemma as it sits between the 2006 values. 4.4.5.2.3.2 The 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 in table (4.6). 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 on Proton 85

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harmonics, it is immediately evident that a decrease in “rtq” of “< 1.508(%)” produces the new world average value precisely. The relevant calculations may be performed simply as follows, The revised “Top Quark” radius based upon the “new world average Top Quark” mass, 5

GeV 172. 2 1 . c r π. 9 mp 140

2

= 0.9156 10

16 .

cm

(4.5)

The decrease in “Top Quark” RMS charge radius [relative to the table (4.6) value] based upon the “new world average Top Quark” mass becomes, r tq

1 = 1.5076 ( % )

5

GeV 172. 2 1 . c r π. 9 mp 140

2

(4.6)

where, “rtq” denotes the RMS charge radius of the “Top Quark” from table (4.6). 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. Note: the 2006 value for revised “mtq” modifies the error defined by Eq. (4.6) to “< 1.65(%)”. 4.4.6 Discussion 4.4.6.1 Experimental evidence of unification Table (4.5, 4.6, 4.7) display mathematical facts demonstrating that all fundamental particles may be represented as harmonics of an arbitrarily selected reference particle, in complete agreement with the SM. Considering that the EGM method is so radically different and quantifies the physical world beyond contemporary solutions, one becomes tempted to disregard table (4.5, 4.6, 4.7) in favour of concluding these to be “coincidental”. However, it is inconceivable that such precision from a single paradigm spanning the entire family of fundamental particles could be “coincidental”. The derivation of the “Top Quark” massenergy is in itself, an astonishing result that the SM is currently incapable of producing. Moreover, the derivation of (a), EM radii characteristics of the Proton and Neutron (rπE, rπM and rνM) (b), the classical RMS charge radius of the Proton (c), the 1st term of the Hydrogen atom spectrum “λA” and (d), the Bohr radius “rx”: all from the same paradigm, strengthens the harmonic case. Additionally, QE3 demonstrates that the probability of coincidence is “<< 10-38” based upon the results shown in table (4.8), Particle / Atom EGM Prediction (QE3) Proton (p) rπ = 830.5957 x10-16(cm) rπE = 848.5274 x10-16(cm) rπM = 849.9334 x10-16(cm) rp = 874.5944 x10-16(cm) Neutron (n) rν = 826.8379 x10-16(cm) KS = -0.1133 x10-26(cm2)

Experimental Measurement rπ = 830.6624 x10-16(cm) [2] rπE = 848 x10-16(cm) [11,12] rπM = 857 x10-16(cm) [11,12] rp = 875.0 x10-16(cm) [7] rX ≈ 825.6174 x10-16(cm) KX = -0.113 x10-26(cm2) [3] 86

(%) Error < 0.008 < 0.062 < 0.825 < 0.046 < 0.148 < 0.296

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Neutron (n) Top Quark (tq) Hydrogen (H)

where, i. ii. iii. iv. v.

rνM = 878.9719 x10-16(cm) rνM = 879 x10-16(cm) [11,12] mtq(GeV) ≈ 178.4979 mtq(GeV) ≈ 172.0 [9] λA = 657.3290(nm) λB = 656.4696(nm) [13] rx = 0.0527(nm) rBohr = 0.0529(nm) [7] Table 4.8: experimentally verified EGM predictions,

< 0.003 < 3.64 < 0.131 < 0.353

“rπE” and “rπM” denote the Electric and Magnetic radii of the Proton respectively. “rX” denotes the conversion of the experimentally determined implicit conventional form “KX” to the explicit EGM form. “rνM” denotes the Magnetic radius of the Neutron. “λA” and “λB” denote the first term of the Hydrogen atom spectrum (Balmer series). “rp = 875.0 x10-16(cm)” and “rBohr = 0.0529(nm)” are not experimental values, they denote the classical RMS charge radius of the Proton and the Bohr radius, i.e. the official values listed by NIST.

Note: numerical simulations generating all values in table (4.5, 4.6, 4.8) can be found in QE3. 4.4.6.2 The answers to some important questions 4.4.6.2.1 What causes harmonic patterns to form? 4.4.6.2.1.1 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 ZPF equilibrium point between the mass-energy equivalence of the particle and the space-time manifold (the ZPF) surrounding it - as depicted by Fig. (4.4). 4.4.6.2.1.2 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 table (4.5) form because the determination of ZPF equilibrium is applied to inherently quantum characteristic objects – i.e. fundamental particles. Hence, it should be no surprise to the reader that comparing a set of inherently quantum characterised 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 table (4.5), or a suitable variation thereof, could not be formulated. 87

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Therefore, harmonic patterns form due to inherent quantum characteristics and ZPF equilibrium. 4.4.6.2.2 Why haven’t the “new” particles been experimentally detected? EGM approaches the question of particle existence, not just by mass as in the SM, but also harmonic cut-off frequency “ωΩ” (i.e. by mass and ZPF equilibrium). Storti et. Al. showed in QE3 that the bulk of the PV spectral energy161 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 period162 “TΩ” defines the minimum detection interval to confirm (or refute) the existence of the proposed “L2, L3, L5” Leptons and associated “ν2, ν3, ν5” Neutrinos. In other words, a particle exists for at least the period specified by “TΩ” – i.e. its minimum lifetime. 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: iv. The proposed Leptons are too short-lived to appear as resonances in cross-sections. v. 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. 4.4.6.2.3 Why can all fundamental particles be described in harmonic terms? Because of the precise experimental and mathematical evidence presented in table (4.5, 4.6, 4.8). These results were achieved by construction of a model based upon a single gravitational paradigm. Moreover, Storti et. Al. also derives the Casmir force in QE3 utilising Eq. (3.63, 3.67, 3.73).

161 162

“>> 99.99(%)”. The inverse of “ωΩ”. 88

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4.4.6.2.4 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 table (4.5) is a robust approximation based upon PDG data. Other interpretations are possible, depending on the values utilised. For example, if one re-applies the method presented in QE3 based upon other data; the values of “Stω” in table (4.5) 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 QE3 (or a suitable variation thereof) will produce a precise harmonic representation of fundamental particles, invariant to interpretation. Table (4.5) values cannot be dismissed due to potential multiplicity before reconciling how: i. “ωΩ”, which is the basis of the table (4.5) construct, produces Eq. (3.199, 3.200) as derived in QE3. 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 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 table (4.8). iv. Extremely short-lived Leptons [i.e. with lifetimes of “< 10-29(s)”] 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. 4.4.6.2.5 What would one need to do, in order to disprove EGM? Explain how experimental measurements of charge radii and mass-energy by international collaborations such as CDF, D0, L3, SELEX and ZEUS in [2,8-10,14-17], do not correlate to EGM calculations. 4.4.6.2.6 Why does EGM produce current and not constituent Quark masses? The EGM method is capable of producing current and constituent Quark masses; only current Quark masses are presented herein. This manuscript 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 a more complicated process, but the method of solution remains basically the same. For example, Storti et. Al. calculate an experimentally implicit value of the Bohr radius in QE3 by treating the atom as “a system” in equilibrium with the polarised ZPF. 4.4.6.2.7 Why does EGM 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 polarised ZPF (the objects own gravitational field). Therefore, since 89

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fundamental particles with classical form factor denote fundamental states (or systems: Quarks in the Proton and Neutron) of polarised ZPF equilibrium, it follows that only the three families will be predicted. 4.5

What may the periodic table of elementary particles look like under EGM?

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, Types of Matter Group II Group III Up 14 Charm 42 Top 140 +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] uq cq tq 1.5 < muq(MeV) < 3 ≈ 1.1833(GeV) ≈ 172.0(GeV) 56 Down 14 Strange 28 Bottom -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] dq sq bq 3 < mdq(MeV) < 7 4.13 < mbq(GeV) < 4.27 ≈ 113.9460(MeV) Electron 2 Muon 8 Tau 12 -1,1/2 -1,1/2 -1,1/2 e µ τ = 0.5110(MeV) = 105.7(MeV) = 1.777(GeV) Electron Neutrino 2 Muon Neutrino 8 Tau Neutrino 12 0,1/2 0,1/2 0,1/2 νe νµ ντ < 2(eV) < 0.19(MeV) < 18.2(MeV) L2 4 L3 6 L5 10 -1,1/2 -1,1/2 -1,1/2 L2 L3 L5 ≈ 9(MeV) ≈ 57(MeV) ≈ 566(MeV) L2 Neutrino 4 L3 Neutrino 6 L5 Neutrino 10 0,1/2 0,1/2 0,1/2 ν2 ν3 ν5 ≈ men ≈ mµn ≈ mτn Standard Model and EGM Bosons Photon N/A Gluon ? QB6 84 Z Boson 112 -6 1,Colour,1 1,Weak Charge,10 1,Weak Charge,10-6 1,Charge,α gl Q B6 Z γ -45 < 10(MeV) ≈ 22(GeV) ≈ 91.1875(GeV) ≈ 3.2 x10 (eV) Graviton N/A QB5 70 W Boson 98 Higgs Boson 126 2,Energy,10-39 1,Weak Charge,10-6 1,Weak Charge,10-6 0,Higgs Field,? QB 5 W H γg ≈ 10(GeV) ≈ 80.27(GeV) > 114.4(GeV) = 2mγγ Table 4.9: predicted periodic table of elementary particles, E GM Leptons

Standard Model Leptons

Quarks

Group I

Quarks

Legend Leptons

Bosons Stω Name Stω Name Stω Charge(e),Spin,Colour Charge(e),Spin Spin,Source,*SC Symbol Symbol Symbol Mass-Energy Mass-Energy Mass-Energy

Name

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(i) *Where, “SC” denotes coupling strength at “1(GeV)”. [18] (ii) The values of “Stω” in table (4.9) utilise the Proton as the reference particle. This is due to its RMS charge radius and mass-energy being precisely known by physical measurement. Table 4.9: particle legend, 4.6

Graphical representation of fundamental particles under EGM

Illustrational (only) wavefunction “ψ” [Eq. (3.458)] based on 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, 91

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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, 4.7

Concluding remarks

A concise mathematical relationship has been used to combine gravitational acceleration and ElectroMagnetism into a method producing fundamental particle properties to extraordinary precision. This also results in the representation of fundamental particles as harmonic forms of each other, beyond the Standard Model – suggesting the following: i. The potential for new Physics at higher accelerator energies. ii. Physical limitations on the value of two extremely important mathematical constants [i.e. “π” and “γ”] at the QM level – subject to uncertainty principles. 92

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5 The Natural Philosophy of the Cosmos “Neither fame, nor station, nor reputation is a scientific compass.” • Riccardo C. Storti Taken from “Quinta Essentia – Part 4” (QE4) Abstract We utilise principles of mass-energy distribution and similitude by ZPF equilibria to derive the values of the present Hubble constant “H0” and CMBR temperature “T0”. It is demonstrated that a mathematical relationship exists between the Hubble constant and CMBR temperature such that “T0” is derived from “H0”. The values derived are “67.0843(km/s/Mpc)” and “2.7248(K)” respectively. The derivations are possible by assuming that, instantaneously prior to the “BigBang”, the “Primordial Universe” was analogous to a homogeneous Planck scale particle of maximum permissible energy density, characterised by a single wavefunction. Simultaneously, we represent the “Milky-Way” as a Planck scale object of equivalent total Galactic mass “MG”, acting as a Galactic Reference Particle (GRP) characterised by a large number of wavefunctions with respect to the solar distance from the Galactic centre “Ro”. This facilitates a comparative analysis between the Primordial and Galactic particle representations by application of a harmonic relationship, yielding “H0” in terms of “Ro” and “MG”. Consequently, utilising the experimental value of “T0”, we derive improved estimates for “Ro” and “MG” as being “8.1072(kpc)” and “6.3142 x1011(solar-masses)” respectively. The construct herein implies that the observed “accelerated expansion” of the Universe is attributable to the determination of the ZPF energy density threshold “UZPF” being “< -2.52 x10-13(Pa)”. Moreover, it is graphically illustrated that the gradient of the Hubble constant in the time domain is presently positive.

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5.1

Introduction

We extend the principles of Electro-Gravi-Magnetics (EGM) to two important aspects of Cosmology [i.e. the present value of the Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0”]. Subsequently, the reader is actively encouraged to review QE3,4, to obtain a full appreciation of the EGM method. QE3 develops an equation facilitating the harmonic representation of all fundamental particles relative to an arbitrarily chosen base particle. It is demonstrated, for example, that the EGM wavefunction frequency of an Electron “ωΩ(rε,me)” is twice that of the Proton “ωΩ(rπ,mp)”, and the harmonic relationship between them “Stω” has a value of “2”. Hence, a table of fundamental particle harmonics was formulated. This resulted in a relationship between the mass-energy and size of fundamental particles based upon Zero-Point-Field (ZPF) equilibria. Although the EGM harmonic representation is an approximation derived from basic engineering principles, it produces experimentally verified results substantially beyond the current abilities of the Standard Model (SM) of Particle-Physics to do so, to at least four orders of magnitude. We utilise the principles of mass-energy distribution and similitude by ZPF equilibria developed in QE3, to derive “H0” and “T0”. It is demonstrated that a mathematical relationship exists between the Hubble constant and CMBR temperature such that “T0” is derived from “H0”. Consequently, this enables the complete and precise specification of the thermodynamic, inflationary and expansive history of the Universe from the “Big-Bang” to the present day. Astonishingly, the application of the EGM construct to Cosmology produces “Black-BodyRadiation” curve characteristics, without the application of the “Black-Body-Law”, further reinforcing the validity of the “H0” and “T0” formulations of approximately “67.0843(km/s/Mpc)” and “2.7248(K)” respectively. Considering that the experimental tolerance of the CMBR temperature is presently “2.725 ± 0.001(K)”, it is obvious that any determination within such a narrow band should be given serious consideration. The derivation of “H0” and “T0” is possible assuming that, instantaneously prior to the “BigBang”, the “Primordial Universe” was analogous to a homogeneous Planck scale particle of maximum permissible energy density, characterised by a single EGM wavefunction. Simultaneously, we represent the “Milky-Way” as a Planck scale object of equivalent total Galactic mass “MG”, acting as a Galactic Reference Particle (GRP) characterised by a large number of EGM wavefunctions with respect to the solar distance from the Galactic centre “Ro”. This facilitates a comparative analysis between the Primordial and Galactic particle representations utilising the harmonic equation derived in QE3, yielding “H0” in terms of “Ro” and “MG”. Moreover, we extend the analysis by determining the theoretical frequency shift of a fictitious EGM wavefunction being radiated from the Primordial particle, yielding “T0” in terms of “H0”. Consequently, by utilising the measured value of “T0”, we derive improved estimates for “Ro” and “MG” as being approximately “8.1072(kpc)” and “6.3142 x1011(solar-masses)” respectively. Because the value of “H0” is still widely debated and the associated experimental tolerance is much broader than “T0”, the EGM construct implies that the observed “accelerated expansion” of the Universe is attributable to the determination of the ZPF energy density threshold “UZPF” being “< -2.52 x10-13(Pa)”. Moreover, it is graphically illustrated that the gradient of the Hubble constant in the time domain is presently positive. Subsequently, it is demonstrated that the majority of what is currently conjectured to constitute “Dark Matter / Energy” by the scientific community, is nothing more than Photons due to the definition of a Graviton under the EGM construct. In addition, it is mathematically shown that the magnitude of the impact of “Dark Matter / Energy” on the value of the Hubble constant and CMBR temperature is “< 1(%)” such that the Universe is composed of: • “> 94.4(%) Photons”. • “< 1(%) Dark Matter / Energy”. • “4.6(%) Atoms”. 94

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5.2

Objectives and scope

Note: all chapter citations refer to QE4. 5.2.1 What is derived? The present Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0” denote two of the most important Cosmological phenomenon to have been identified in the last hundred years and may hold significant insight into the natural philosophy of the Cosmos. Experimental measurements of “H0” and “T0” are advancing dramatically and have raised some important aspects regarding the nature of the Cosmological evolution process. QE4 is a companion to QE3, applying a method termed Electro-Gravi-Magnetics (EGM). Storti et. Al. derived the EGM construct, utilising Dimensional Analysis Techniques (DAT’s) and Buckingham “Π” Theory (BPT)163, to represent fundamental particles in harmonic form to high computational precision in favourable agreement with the Standard Model (SM) of Particle-Physics and experimental measurement. One of the key findings was that, at a fundamental physical level, mass-energy is distributed over space-time in only one manner164. The EGM construct has been re-applied to Cosmology with the following derivational objectives (within experimental tolerance where applicable): i. The Hubble constant (see: Ch. 7.1, 7.3, 7.6, 8.3). ii. The CMBR temperature (see: Ch. 7.2, 7.3, 7.5, 8.1, 8.2). iii. The ZPF energy density threshold (see: Ch. 7.7). iv. The Cosmological evolution process (see: Ch. 8.4). v. The history of the Universe (see: Ch. 8.5). 5.2.2 How is it achieved? The primary tool employed to achieve our objectives is similitude165, subject to the following simplified constraints (see: Ch. 6.1, 7.1 – 7.3), i. The Cosmos at an instant prior to the “Big-Bang” is termed the “Primordial Universe”. It was characterised by a single wavefunction with maximum permissible energy density distributed homogeneously, analogous to a Planck scale particle of radius “λxλh” and mass “mxmh” such that it was dynamically, kinematically and geometrically similar to a “Schwarzschild-Black-Hole” (SBH). ii. The relationship between the “Primordial Universe” and its present visible size obeys the EGM harmonic representation of fundamental particles. iii. The “Milky-Way” (MW) Galaxy may be represented as a Planck scale particle of homogeneous energy density and equivalent total mass. This configuration has been termed the Galactic Reference Particle (GRP), such that dynamic, kinematic and geometric similarity exists between the “Primordial Universe” and the GRP. 5.3

Derivation process

5.3.1 Hubble constant “HU” i. Utilising harmonic cut-off frequency in “ωΩ_3” form (see: Ch. 5.2.2), derive an expression for EGM Flux Intensity “CΩ_J1”: (see: Ch. 5.5), Output: 163

Refer to the many standard texts relating to DAT’s and BPT. In accordance with Zero-Point-Field equilibria. 165 A reference to DAT’s and BPT. 164

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St J

C Ω_J1( r , M )

9

. M

2

5

8

r

r

(4.52)

where, St J

9 .c . St G 4 .π

St G

3.

4

• • •

2 9

(4.51) 2

3 .ω h

. c 2

4 .π .h

9

(4.35)

“c = 299792458(m/s)”. “h = 6.6260693 x10-34(Js)”. “ωh = 1 / th = 1 / √(Gh/c5), G = 6.6742 x10-11(m3kg-1s-2)”.

ii. Derive an expression for the minimum gravitational lifetime of matter “TL”: (see: Ch. 6.7.2.2), Output: TL

h m γγ

(4.196)

where, • “mγγ” denotes the mass-energy of a Photon defined in QE3. • “mγγ = 3.195095 x10-45(eV)”. iii. Derive an expression for the EGM Hubble constant “HU” utilising the EGM harmonic representation of fundamental particles: (see: Ch. 7.1), Output: λ y r 2, M 2

K U r 2, r 3, M 2, M 3

ln

1 ln n Ω_2 r 2 , M 2

(4.229)

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2

A U r 2, r 3, M 2, M 3

H U r 2, r 3, M 2, M 3

(4.231)

TL K U r 2, r 3, M 2, M 3

5

(4.233)

1 A U r 2, r 3, M 2, M 3

(4.235)

where, “nΩ_2” denotes the non-refractive form of “nΩ” defined in QE3. 5.3.2 CMBR temperature “TU” iv. Derive an expression for the average number of Gravitons “ng” radiated by a SBH at frequency “ω”: (see: Ch. 6.7.1.1), Output: n g ω , M BH

E M BH E g( ω )

(4.177)

where, “MBH” denotes SBH mass. 96

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v. Derive an expression for the value of the EGM Hubble constant at the instant of the “BigBang”, termed the primordial Hubble constant “Hα”: (see: Ch. 7.1.3.2), Output: H α r 3, M 3

2.

2. . . π G ρ m r 3, M 3 3

(4.237)

vi. Derive an expansive scaling factor “KT” incorporating “ng”, “Hα” and “HU”: (see: Ch. 7.2.3), Output: K T r 2, r 3, M 2, M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

H α r 3, M 3 H U r 2, r 3 , M 2, M 3

(4.240)

where, “ωΩ_3” has a generalised definition according to, 9

ω Ω_3( r , M )

2

M St G. 5 r

(4.36)

vii. Derive a thermodynamic scaling factor “TW” incorporating Wien’s displacement constant “KW” and EGM wavelength of the form “λΩ_3”: (see: Ch. 7.2.3), Output: T W r 2, r 3, M 2 , M 3

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3

(4.241)

where, “ωΩ_3(r,M) → ωΩ_3(RU(r2,M2,r3,M3),M3)” “λΩ_3(RU(r2,M2,r3,M3),M3) = c / ωΩ_3(RU(r2,M2,r3,M3),M3)” R U r 2, r 3 , M 2, M 3

•

c .A U r 2 , r 3 , M 2 , M 3

(4.234)

“KW = 2.8977685 x10-3(mK)”.

viii. Derive an expression for EGM Cosmological temperature “TU” utilising “KT” and “TW”: (see: Ch. 7.2.3), Output: T U r 2, r 3, M 2, M 3

K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3

(4.242)

5.3.3 “HU → HU2, TU → TU2 → TU3” ix. Derive the minimum physical dimensions of mass and radius for a SBH with maximum permissible energy density at the Planck scale: (see: Ch. 6.1.3), Output: mx

λx

λx 2

(4.71)

4 . 2 6 π 3

(4.72)

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Hence, the dimensions of a SBH at maximum permissible energy density at the Planck scale is given by, • “Mass = mxmh” when “mh = √(hc/G)”. • “Radius = λxλh” when “λh = √(Gh/c3)”. x. Assume that the “Primordial Universe” (i.e. the Universe instantaneously prior to the “Big-Bang”) is analogous to a SBH of Planck scale dimensions at a condition of maximum permissible energy density, with radius “r3 = λxλh” and mass “M3 = mxmh = λxmh / 2”: (see: Ch. 7.3.1), xi. Formulate generalised expressions for “r2” and “M2” incorporating the EGM adjusted Planck Length and mass: (see: Ch. 7.3.1), Output: r2(r) = Kλ⋅r

(4.247)

M2(M) = Km⋅M = Kλ⋅M

(4.248)

where, “Kλ = Km = [π / 2](1 / 3) ≈ 1.162447” as defined in QE3. xii. Simplify “ng”: (see: Ch. 7.3.1, 7.6), Output: For “r3 = λxλh” and “M3 = mxmh = λxmh / 2”: “ng[ω,MBH] = ng[ωΩ_3(r3,M3),M3] = 8 / 3”. xiii. Simplify “Hα”: (see: Ch. 7.3.2), Output: H α λ x.λ h , m x.m h

ωh λx

(4.249)

For brevity in future applications, let: “Hα = ωh / λx”. xiv. Transform “HU” to “HU2”: (see: Ch. 7.6.1), Output:

H U K λ .r , λ x.λ h , K m.M , m x.m h

H U2( r , M )

(4.276)

xv. Transform “TU” to “TU2”: (see: Ch. 7.5), Output: T U2( H )

K W .St T .ln

ωh λ x.H

9

. H5

(4.275)

where, “H” denotes a generalised reference to Hubble constant and “StT” is a constant according to. 9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

98

2

(4.274)

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xvi. Transform “TU2” to “TU3”: (see: Ch. 8.1.3), Output: T U3 H β

K W .St T .ln

1 Hβ

. H .H β α

5 .µ

2

(4.318)

where, “µ = 1 / 3” and “Hβ” denotes a dimensionless range variable such that “1 ≥ Hβ > 0”. xvii. Select values of “r” and “M” for application to “r2(r), M2(M)” utilising the following measures: (see: Ch. 7.3.2): Input: i. ii. iii. iv. v. vi.

“r = Ro” denotes the mean distance from the Sun to the MW Galactic centre. “Ro = 8(kpc)” as defined by the PDG [19] (“kpc” = kilo-parsec). “M = MG” denotes the total mass (i.e. visible + dark) of the MW Galaxy. “MG ≈ 6 x1011” solar masses as defined by [20]. “H0 = 71(km/s/Mpc)” as defined by the PDG [21] (“Mpc” = Mega-parsec). “T0 = 2.725(K)” as defined by the PDG. [19]

5.3.4 Rate of change “dHdt” xviii. Derive a generalised expression for the rate of change of the EGM Hubble constant in the time domain “dHdt” as a function of the dimensionless range variable “Hγ” such that: “1 ≥ Hγ > 0” and “Hγ ∝ Hβη”: (see: Ch. 8.3.3), Output: dH dt H γ

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

1

(4.361)

Note: “dHdt” is alternative notation introduced to replace the typical differential form “dH/dt”, for application in the “MathCad 8 Professional” computational environment. xix. For solutions where the deceleration parameter is zero, derive an expression for the magnitude of the EGM Hubble constant “|H|” in the time domain166: (see: Ch. 8.3.3), Output: d H dt

H

(4.378)

xx. Devise a numerical approximation method facilitating the graphical representation of “|H|” in terms of an indicial power “η” (see: Ch. 8.3.3) such that, Input: t

1 H γ .H α

Hγ Hβ

(4.359)

η

(4.376)

166

This terminology is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”, as indicated by the equation. 99

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xxi. For non-zero deceleration parameter solutions, derive the ZPF energy density threshold “UZPF” (see: Ch. 7.7), Output: 3 .c . H U2 R o , M G Ω ZPF . 8 .π .G 2

U ZPF

2

(4.315)

where, Ω ZPF

1

Ω EGM

(4.313)

ρ U2 r x5.R o , m g5 .M G

Ω EGM

ρ U2 R o , M G

(4.308)

xxii. Reduce the expression for the EGM Hubble constant and Cosmological temperature to their simplest functionally dependent forms: “HU5” and “TU5” respectively (see: App. 4.B: “MathCad 8 Professional – b. Calculation engine – xi”, “MathCad 12 – c. High precision calculation engine – iv”). Output: H U5( r , M )

1 . ln TL

T U5( r , M )

( 3 .π )

µ

2

32

256

KW c

5.4

7 .µ .

µ

µ

. µ m .ln ( 3 π ) . h 4 M

. .ln . 4µ H U5( r , M ) λ h Hα

2 .µ

7 .µ

2

. r λh

.

1 π .H α

2 7 .µ

5

.

mh

5 .µ

2

. r λh

M 2 .µ

2

2 26 .µ

(4.529)

. 2

.H ( r , M ) 5 µ U5

(4.530)

Sample results

5.4.1 Numerical evaluation and analysis 5.4.1.1 Cosmological properties Evaluating “AU”, “RU”, “HU” and “TU” yields, 9 A U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .yr

(4.250)

9 R U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .Lyr

(4.251)

H U K λ .R o , λ x.λ h , K m.M G, m x.m h = 67.084304

km s .Mpc

(4.254)

T U K λ .R o , λ x.λ h , K m.M G, m x.m h = 2.724752 ( K )

(4.255)

The EGM construct error associated with “HU” and “TU” with respect to expert opinion and physical measurement is given by, 1 . H U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 5.515064 ( % )

H0

(4.256)

1 . T U K λ .R o , λ x.λ h , K m.M G, m x.m h

T0

. 1 = 9.08391310

3

(%)

(4.257) 100

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It is possible to calculate the value of “HU” and “TU” based upon the “visible mass only” of the MW Galaxy by a simple substitution of values (i.e. “M2 / 3 = KmMG / 3”) as follows, 1 km H U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 67.753267 . 3 s Mpc

(4.262)

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 2.739618 ( K ) 3

(4.263)

Hence, the magnitude of the impact of “Dark Matter / Energy” on the value of “HU” and “TU” is demonstrated to be “< 1(%)” when compared to the previously derived value according to, H U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 0.987352 ( % )

1 H U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3 T U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.264) 1 = 0.542607 ( % )

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

(4.265)

A simple test verifying “TU2” is demonstrated below. Since the computed value of “TU2[HU2(Ro,MG/3)]” based upon visible MW Galactic mass “MG/3” is exactly compliant with “TU” (i.e. “TU = TU2”), no technical error exists. Moreover, the result “TU2(H0) ≈ T0” agrees precisely with historical expectation (i.e. prior to measurement by satellite) of “T0”. 1 T U2 H U2 R o , .M G 3 T U2 H 0

=

2.739618 2.810842

( K)

(4.277)

Note: the validation of “TU = TU2” above, also verifies that “HU = HU2”. In addition, it is also demonstrated and numerically verified in “App. 4.B” that “HU2 = HU5”. The preceding results demonstrate that the impact of “Dark Matter / Energy” on “HU” and “TU” is very small. This implies that the constitution of the Universe under the EGM construct is quite different from current thinking. The contemporary view167 is that the constitution of the Universe is, i. “72(%) Dark Energy”. ii. “23(%) Dark Matter”. iii. “4.6(%) Atoms”. However, the EGM construct generalises the constitution of the Universe as being, iv. “> 94.4(%) Photons”. v. “< 1(%) Dark Matter / Energy”. vi. “4.6(%) Atoms”. For solutions where the deceleration parameter is zero, “η” may be numerically approximated utilising the “Given” and “Find” commands within the “MathCad 8 Professional” computational environment, subject to the constraint that “dHdt” as a function of the present value of “Hβ” [i.e. “≈ HU2(Ro,MG) / Hα”] raised to an indicial power, is equal to the square of the present Hubble constant as determined by the EGM construct “HU2(Ro,MG)2” according to the following algorithm,

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Given dH dt

H U2 R o , M G

η

Hα

H U2 R o , M G

η

1

(4.379)

Find( η )

(4.380)

Hence, “η = 4.595349”. 5.4.1.2 Significant temporal ordinates (See: Ch. 8.3.4) Significant temporal ordinates of Cosmological evolutionary events (marked on the proceeding graphs) are given in matrix form as follows, 1

t1

e

2 5 .µ .

Hα 10 .µ

t2

1

e

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

2

. 1 Hα

2.206287 2.206287 4.196153 4.196153

2

2 2 2 5 .µ . 5 .µ . 5 .µ

3

2

. 1 Hα

e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

t5 e

10

42 .

s

20.932666 20.932666 8.385263 8.385263

1

t4

= 6.205726 6.205726

1

2

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα

1

2

. 1 Hα

(4.384)

where, “t5” denotes the temporal ordinate of the local minima of the “2nd” time derivative of the Hubble constant (see: Ch. 8.3.3, 8.3.6.10).

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5.4.2 5.4.2.1

Graphical evaluation and analysis Average Cosmological temperature vs. age

(See: Ch. 8.2.5.1, 8.2.5.2) Av. Cosmological Temperature vs. Age 1 Hα

Av. Cosmological Temperature (K)

T U3 H β

t1

3 .1031

1 T U3 e

T U3 e

T U3 e

5 .µ

2

2 .1031

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

1 .1031

1 .10

43

1 .10

42

1 .10

41

1 .10

40

1 .10 1

39

1 .10

38

1 .10

37

1 .10

36

H β .H α Cosmological Age (s)

Figure 4.24, Av. Cosmological Temperature vs. Age t2t3

3 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2

2 .1031

2 1 10 .µ 2 2 5 .µ . 5 .µ 1 2 2 2 2 15 .µ . 5 .µ 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

1 .1031

1 .10

43

1 .10

42

1 .10

41

1 .10

40

1 .10 1

39

1 .10

38

1 .10

37

1 .10

36

H β .H α Cosmological Age (s)

Figure 4.25,

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5.4.2.2

Magnitude of the Hubble constant vs. Cosmological age

(See: Ch. 8.3.6.11, 8.3.6.12) Mag. of Hubble Cons. vs. Cosm. Age 2.5 .10 dH dt H β

dH dt e

η

1 2 5 .µ

2 .10

(Hz) dH dt e

1 Hα

t1

42

1 1.5 .1042

1 dH dt e

42

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

1 .1042

5 .10

41

0 43 42 41 40 39 38 37 36 35 34 33 32 31 30 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 η H β .H α Cosmological Age (s)

Figure 4.46, Mag. of Hubble Cons. vs. Cosm. Age 2.5 .10 dH dt H β

dH dt e

η

1 2 5 .µ

2 .10

(Hz) dH dt e

1 Hα

t4

42

1 1.5 .1042

1 dH dt e

42

2 2 5 .µ . 5 .µ

1

2 2 2 4 5 .µ . 5 .µ 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

1 .1042

5 .10

41

0 43 42 41 40 39 38 37 36 35 34 33 32 31 30 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 η H β .H α Cosmological Age (s)

Figure 4.47,

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5.4.3 Cosmological evolution process Based upon the preceding graphical output, the Cosmological evolution process may be categorised into two regimes, comprised of four distinct periods (i.e. three inflationary and one expansive) as follows, Time Temperature Hubble Constant -1 -∞ < TU2 < 0 +∞ > |H| > Hα 0 < t < Hα 0 → Hα-1 -∞ → 0 +∞ → Hα -1 -1 0 ≤ TU2 < TU2(t1 ) Hα ≥ |H| > 0 Hα ≤ t < t1 Hα-1 → t1 0 → TU2(t1-1) Hα → 0 -1 -1 t1 ≤ t < t4 TU2(t1 ) ≥ TU2 > TU2(t4 ) 0 ≤ |H| < √|dHdt[(t4Hα)-1]| -1 -1 t1 → t4 TU2(t1 ) → TU2(t4 ) 0 → √|dHdt[(t4Hα)-1]| t4 ≤ t < AU TU2(t4-1) ≥ TU2 ≥ TU2(HU2) √|dHdt[(t4Hα)-1]| ≥ |H| ≥ HU2 -1 t4 → AU TU2(t4 ) → TU2(HU2) √|dHdt[(t4Hα)-1]| → HU2 Description Primordial Inflation (prior to the “Big-Bang”): the Universe may be described as “inverted and non-physical” such that the interior of the Cosmos existed outside the exterior boundary “RBH” in accordance with the “Primordial Universe” model described in Ch. (7, 8) such that: 1. “TU2” increases from negative infinity to zero. 2. “dHdt” increases from negative infinity to “-Hα2”. 3. “|H|” decreases from positive infinity to “Hα”. Thermal Inflation: the period from the instant of the “Big-Bang” to the instant of maximum Cosmological temperature such that: 4. “TU2” increases from zero to its maximum value “TU2(t1-1)”. 5. “dHdt” increases from “-Hα2” to zero. 6. “|H|” decreases from “Hα” to zero. Hubble Inflation: the period from the instant of maximum Cosmological temperature to the instant of maximum post-primordial “|H|” such that: 7. “TU2” decreases from its maximum value to “TU2(t4-1)”. 8. “dHdt” increases from zero to its maximum physical value “dHdt[(t4Hα)-1]”. 9. “|H|” increases from zero to its maximum physical value “√|dHdt[(t4Hα)-1]|”. Hubble Expansion: the period from the maximum post-primordial “|H|” to the present day such that: 10. “TU2” decreases from “TU2(t4-1)” to “TU2(HU2)”. 11. “dHdt” decreases from its maximum physical value to “HU22”. 12. “|H|” decreases from its maximum physical value to “HU2”. Symbol Definition / Value The EGM Hubble constant at the instant of the “Big-Bang”: Hα ≈ 2.742004 x1042(Hz) ≈ 8.460941 x1061(km/s/Mpc) -Hα2 ≈ -7.518587 x1084(Hz2) ≈ -7.158752 x10123(km/s/Mpc)2 HU2 The present value of the EGM Hubble constant: = HU2(Ro,MG) ≈ 67.084304(km/s/Mpc) HU22 ≈ 4.500304 x103(km/s/Mpc)2 H0 The PDG Hubble constant: ≈ 71(km/s/Mpc) 2 H0 ≈ 5.041 x103(km/s/Mpc)2 Hα-1 The instant of the “Big-Bang”: ≈ 3.646967 x10-43(s) t1 The instant of max. Cosmological temperature: ≈ 2.206287 x10-42(s) t4 The instant of maximum physical “|H|”: ≈ 2.093267 x10-41(s)

Physical @ {RBH ≥ rS}

Non-Physical @ {RBH < rS}

Period Primordial Inflation Thermal Inflation Hubble Inflation Hubble Expansion Regime

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AU TU2(Hα) TU2(t1-1) TU2(t4-1) TU2(HU2) T0 dHdt[(t4Hα)-1] √|dHdt[(t4Hα)-1]| RU 2 Hα ⋅(dHdt[(t4Hα)-1])-1 Hα⋅(√|dHdt[(t4Hα)-1]|)-1

The EGM Cosmological age: = HU2-1 ≈ 14.575885 x109(yr) The EGM Cosmological temperature at the instant of the “Big-Bang”: = 0(K) The Maximum EGM Cosmological temperature: ≈ 3.195518 x1031(K) The EGM Cosmological temperature at the instant of maximum physical “|H|”: ≈ 2.059945 x1031(K) The present EGM Cosmological temperature: = TU3(HU2Hα-1) ≈ 2.724752(K) The present experimentally measured CMBR temperature: ≈ 2.725(K) The approximated maximum rate of change of the physical EGM Hubble constant: ≈ 1.553518 x1084(Hz2) ≈ 1.479167 x10123(km/s/Mpc)2 The approximated maximum physical “|H|”: ≈ 1.246402 x1042(Hz) ≈ 3.845994 x1061(km/s/Mpc) The EGM Cosmological size: = c⋅HU2-1 ≈ 14.575885 x109(Lyr) ≈ 4.839718 ≈ 2.199936 Table 4.10,

Time 0 Hα-1 ≈ 3.646967 x10-43(s) t1 ≈ 2.206287 x10-42(s) t4 ≈ 2.093267 x10-41(s) AU ≈ 14.575885 x109(yr)

TU2 (K) dHdt (km/s/Mpc)2 -∞ -∞ 0 ≈ -7.158752 x10123 0 ≈ 3.195518 x1031 31 ≈ 2.059945 x10 ≈ 1.479167 x10123 ≈ 2.724752 ≈ 4.500304 x103 Table 4.11,

|H|| (km/s/Mpc) +∞ ≈ 8.460941 x1061 0 ≈ 3.845994 x1061 ≈ 67.084304

5.4.4 History of the Universe according to EGM Utilising “TU2”, the history of the Universe may be articulated as follows, Epoch or Event

Time Domain t

Primordial epoch

Grand unification epoch

Electroweak / Quark Epoch

Lepton Epoch

Boundary Temperature Value

1

T U2 H α = 0 ( K )

Hα 1

< t 10

34 .

(s)

Hα

10-34 < t(s) ≤ 10-10

10-10 < t(s) ≤ 102

1

T U2 10

10

10 .

T U2

( K)

(s)

1

T U2

. = 1.92400510

28

34 .

. 15 ( K ) = 3.43308810

(s)

1 2.

. 9 ( K) = 1.01325410

10 ( s )

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Photon Epoch

Universe becomes transparent

102 < t(s) ≤ 1013 1013(s) ≈ 3 x105(yr)

T U2

3 x105 < t(yr) ≤ 109

T U2

1 13 .

10

= 978.724031 ( K )

(s)

1 9.

= 11.838588 ( K )

10 ( yr )

109 < t(yr) ≤ 5 x109

First Supernovae

T U2

5 x109 < t(yr) ≤ 14.58 x109

Present Epoch

1 9.

5 .10 ( yr )

= 4.898955 ( K )

T U2 H U2 R o , M G

= 2.724752 ( K )

Table 4.12, T U2

1

T U2 T U2

1 .( day ) 1 . 31 ( day )

T U2 T U2 T U2

1 1 .( s )

1 1 .( yr ) 1 2 10 .( yr )

1 3.

1 4.

10 ( yr )

5.5

5.

10 ( yr ) 1

T U2

6.

10 ( yr ) 1

T U2

7.

10 ( yr ) T U2 T U2 T U2

10 ( yr ) T U2

1

T U2

1

. 7 2.52413210

521.528169

. 3.86401510

147.71262

6

= 1.00307810 . 6

41.823796

1

. 4 8.07751510

11.838588

9 10 .( yr )

. 2.29089210

3.35005

1

. 6.49496110

0.947724

8 10 .( yr )

4 3

( K)

10 .

10 T U2

. 10 1.84076810 . 3 1.2497710

( yr )

1 11 .

10

( yr )

(4.405)

Discussion

5.5.1 Conceptualization 5.5.1.1 “λx” A physical interpretation of “λx” is possible utilising the Stefan-Boltzmann Law by considering the energy flux emitted from a “Black-Body” and equating it to the peak average Cosmological temperature. “λx” is shown to be proportional to the “4th power-root” of the energy flux of the Universe at the peak average Cosmological temperature (see: App. 4.A).

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5.5.1.2 “TL” The minimum gravitational lifetime of matter “TL” is a simple concept to embrace by considering all matter to represent a vast store of Gravitons within, being ejected at a uniform rate with an emission frequency of “ωg” (see: Ch. 6.7.2.2, 6.8). 5.5.1.3 “CΩ_J” The initial step in conceptualizing the method of solution for the derivation of the Hubble constant and CMBR temperature presented herein is to understand the nature of EGM Flux Intensity “CΩ_J”. The EGM construct represents gravitational fields as a spectrum of conjugate wavefunction pairs, each comprising of a population of Photons. The spectrum is gravitationally dominated by the energy of the population of conjugate Photon pairs at the harmonic cut-off frequency168 “ωΩ” (see: Ch. 5.4). Subsequently, all gravitational objects may be usefully represented by approximation as wavefunction radiators of a single population of conjugate Photon pairs (see: Ch. 9.2.2.2, 9.2.3.2). The EGM spectrum is derived from the application of Fourier series Harmonics, involving the hybridization of “2” spectra (i.e. an amplitude spectrum and a frequency spectrum). The relationship between “CΩ_J” and harmonic cut-off mode “nΩ” (which also denotes the total number of modes in the PV spectrum169) is analogous to the relationship between the amplitude and frequency spectra inherent in Fourier series Harmonics. Thus, i. “CΩ_J” decreases with Cosmological expansion and is analogous to the decrease in PV spectral amplitude as the distance to the subject increases (i.e. the gravitational influence decreases). ii. Instantaneously after the “Big-Bang”, there were no Galaxies and as the Universe expanded, energy condensed into matter and the EGM spectrum developed into its current form such that matter radiates a spectrum of conjugate wavefunction pairs, each comprising of a population of Photons. Therefore, a single frequency mode describing the “Primordial Universe” becomes “many modes” when describing matter in the present state of the Universe. Hence, “nΩ” increases with Cosmological expansion as the distance to the subject increases. iii. EGM finds the convergent solution relating “2” spectra of opposing gradient. That is, “CΩ_J” decreases and “nΩ” increases as the Universe expands. iv. For solutions to “ωΩ” where the Refractive Index “KPV” approaches unity170, it is demonstrated that “ωΩ → ωΩ_3” (see: Ch. 5.1, 5.2), consequently “CΩ_J” may be simplified to “CΩ_J1” (see: Ch. 5.5.1) and a definition stated as follows: EGM Flux Intensity is a representation of gravitational field strength (i.e. the gradient in the energy density of the space-time manifold) expressed in “Jansky’s” (Jy). v. The gravitational forces governing the formation of the “Milky-Way” Galaxy are equivalent to the gravitational forces responsible for the current state of the Universe as a whole. Subsequently, the average EGM Flux Intensity of the “Milky-Way” Galaxy is proportional to the average value of the present Universe and the peak value of the “Primordial Universe” instantaneously prior to the “Big-Bang”. This means that the EGM Flux Intensity of the “Milky-Way” Galaxy acts a baseline reference.

168

i.e. the high-end terminal spectral frequency. The PV spectrum is a bandwidth of the EGM spectrum. 170 The typical representation of “KPV” is an isomorphic weak field approximation to General Relativity (GR). 169

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5.5.1.4 “Stω” The EGM harmonic representation of fundamental particles “Stω” demonstrates that the mass-energy distribution over the space-time manifold at the elementary level, utilising the condition of ZPF equilibria, occurs in only one manner. The significance of this is that it provokes an obvious question with respect to Cosmology. That is: “perhaps it applies on a Cosmological scale?” Simply described, the representation works by expressing the values of “ωΩ” of two fundamental particles171, as an integer ratio (i.e. a harmonic of the reference particle). Subsequently, it follows that “CΩ_J” may be expressed in a similar manner as it is derived utilising “ωΩ”. Thus, if the EGM harmonic representation of fundamental particles with respect to mass-energy distribution over the space-time manifold were universally valid, we would expect that in order to apply it cosmologically: i. The ratio of the presently observable Cosmological size “rf”, to the initial size “ri” of the “Primordial Universe” instantaneously prior to the “Big-Bang”, is proportional to the corresponding EGM Flux Intensity {i.e. “(rf / ri) ∝ [CΩ_J1(rf) / CΩ_J1(ri)]”}. ii. The value of “CΩ_J” at the periphery of the “Primordial Universe” (i.e. instantaneously prior to the “Big-Bang”) is substantially greater than the value at the edge of the presently observable Universe. That is, the gradient of the energy density of the “Primordial Universe”, instantaneously prior to the “Big-Bang”, was substantially greater than the gradient of the energy density at the periphery of the presently observable Universe. iii. Since the values of wavefunction amplitude in the EGM spectrum decrease inversely with “nΩ”, and “nΩ” increases with radial displacement, it follows that “some sort” of naturally logarithmic or exponential relationship should exist between the ratio of the sizes described above and the associated EGM Flux Intensities. iv. “Stω9” represents the harmonic relationship between the values of “ωΩ” of two dimensionally similar particles. Hence, recognising that the frequency and time domains are interchangeable, we may apply “Stω9” as the ratio of “TL” to the present “Hubble age” of the Universe by the EGM method “AU”. Hence, it follows that the ratio of the sizes described above is proportional to the ratio “TL : AU” (see: Ch. 6.7.2.2). 5.5.2 Dynamic, kinematic and geometric similarity 5.5.2.1 “HU” The “Primordial Universe” was analogous to a spherical particle on the Planck scale with radius “r1” and homogeneous mass distribution “M1”, described by a single wavefunction whereas the presently observable Universe is described by a spectrum of wavefunctions. The maximum EGM Flux Intensity measured by an observer at the edge of the “Primordial Universe” is given by “CΩ_J1(r1,M1)”. Matter radiates Gravitons172 at a spectrum of frequencies such that the Cosmological majority of it exists in Photonic form, resulting in an approximately homogeneous mass-energy distribution throughout the Universe whereby any Galactic formation is dynamically, kinematically and geometrically equivalent to a spherical particle of homogeneous mass distribution and may be represented as a Planck scale object to be utilised as a Galactic Reference Particle (GRP). The associated EGM Flux Intensity of the GRP is given by “CΩ_J1(r2,M2)” where, “r2” denotes the mean “H0” measurement distance173 to the Galactic centre and “M2” represents total 171

One of them being an arbitrarily selected reference particle from which to compare all others. Coherent populations of conjugate Photon pairs for a minimum period of “TL”. 173 i.e. the distance relative to the Galactic centre from where a physical measurement of “H0” is performed. 172

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Galactic mass (i.e. visible + dark). The definition of “r2” comes from the scientific requirement to compare calculation or prediction to measurement. Subsequently, one should also utilise parameters within the same frame of reference as the measurement, against which the construct is being tested. It is not known by physical validation that “H0” is measured as being the same from all locations in the Universe. It is believed to be the case by contemporary theory; however it is not factually known to be true. To verify it physically, one would be required to perform the “H0” measurement from a significantly different location in space. Thus, to minimise potential modelling errors, we shall confine “r2” to the same frame of reference174 as the measurement of “H0” (see: Ch. 7.1). 5.5.2.2 “TU” EGM defines the “Primordial Universe” as a single mode wavefunction, therefore any temperature calculation must be scaled accordingly for application to black-body radiation (i.e. black-bodies emit a spectrum of thermal frequencies, not just one). Hence, we would expect that the peak CMBR temperature since the “Big-Bang” is proportional to the average number of Gravitons being radiated per harmonic period by the “Primordial Universe” instantaneously prior to the “BigBang” (see: Ch. 7.2). 5.6

Concluding remarks ⇒ The CBMR temperature is a function of the Hubble constant. ⇒ The Hubble constant and CBMR temperature instantaneously prior to the “Big-Bang” is calculated to be: • Hα = ωh / λx ≈ 8.460941 x1061(km/s/Mpc). • TU2[Hα] = 0(K). ⇒ Physical Laws become real instantaneously after the “Big-Bang”. For example, the “2nd Law of Thermodynamics” is not violated at “TU2[H > Hα]” because “TU2 > 0(K)”. ⇒ The magnitude of the EGM Hubble constant175 at the instant of maximum EGM Cosmological temperature is graphically illustrated to be: • |H(t1)| = 0(km/s/Mpc). ⇒ The maximum EGM Cosmological temperature is calculated to be: • TU2(t1-1) ≈ 3.195518 x1031(K). ⇒ The magnitude of the maximum physical (i.e. post “Big-Bang”) EGM Hubble constant (abbreviated reference) is calculated to be: • |H(t4)| = √|dHdt[(t4Hα)-1]| ≈ 3.845994 x1061(km/s/Mpc). ⇒ The EGM Cosmological temperature at the instant of maximum physical EGM Hubble constant (abbreviated reference) is calculated to be: • TU2(t4-1) ≈ 2.059945 x1031(K).

174

The solar system. This terminology is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”.

175

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⇒ The present EGM Hubble constant and average EGM Cosmological temperature is calculated to be: • HU2(Ro,MG) ≈ 67.084304(km/s/Mpc). • TU2[HU2(Ro,MG)] ≈ 2.724752(K). ⇒ The present CMBR temperature is measured to be: • T0 ≈ 2.725 ± 0.001(K). ⇒ The present Hubble constant is stated by the PDG176 to be: • H0 = 71, +1/-2(km/s/Mpc). ⇒ The EGM Cosmological temperature based upon the PDG Hubble constant is calculated to be: • TU2[H0] ≈ 2.810842(K). ⇒ The Universe is composed of: • “> 94.4(%) Photons”. • “< 1(%) Dark Matter / Energy”. • “4.6(%) Atoms”. ⇒ The magnitude of the impact of “Dark Matter / Energy” on the value of the Hubble constant and CMBR temperature is “< 1(%)”. ⇒ The EGM construct exhibits characteristics satisfying the observed phenomena of “accelerated Cosmological expansion” due to: • The ZPF energy density threshold value “UZPF < -2.52 x10-13(Pa)”. • The gradient of the Hubble constant in the time domain is presently positive. On a human scale, this translates to levels of ZPF energy according to, i. “< -252(yJ/mm3)”. On an astronomical scale, this becomes, ii. “< -0.252(mJ/km3)”. iii. “< -7.4 x1012(YJ/pc3)”. On a Cosmological scale, this becomes, iv. “< -6.6 x1041(YJ/RU3)”. The deceleration parameter, v. “ΩEGM” may be utilised to obtain non-zero deceleration parameter solutions. Note: although on the human scale the quantities of ZPF energy are extremely small, on the astronomical or Cosmological scales, they become extremely large when approaching the dimensions of the visible Universe according to “RU → RU(KλRo,λxλh,KmMG,mxmh)”.

176

http://pdg.lbl.gov/2006/reviews/hubblerpp.pdf (pg. 20 - “WMAP + All”).

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5.7

Graphical summary

5.7.1

“TU3 vs. Hβ”: Figure 4.22 Av. Cosmological Temperature

1

31 3.5 .10 e

5 .µ

2

3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2 1.5 .1031

31 1 .10

5 .1030

1

0.1

0.01

1 .10 3 Hβ Dimensionless Range Variable

1 .10 4

1 .10 5

1 .10 6

Average Cosmological Temperature Maximum Av. Cosmological Temperature

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Maximum Av. Cosrature

5.7.2

“TU3 vs. t = (HβHα)-1” (i): Figure 4.23 Av. Cosmological Temperature vs. Age

1

31 3.5 .10

2

1 Hα

e

5 .µ . 1 Hα

3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2

1.5 .1031

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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5.7.3

“TU3 vs. t = (HβHα)-1” (ii): Figure 4.24 Av. Cosmological Temperature vs. Age 1 Hα

31 3.5 .10

t1

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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5.7.4

“TU3 vs. t = (HβHα)-1” (iii): Figure 4.25 Av. Cosmological Temperature vs. Age t2

31 3.5 .10

t3

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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5.7.5

“TU3 vs. H = (HβHα)” (i): Figure 4.26 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

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5.7.6

“TU3 vs. H = (HβHα)” (ii): Figure 4.27 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 1 t2 t3

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

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5.7.7

“TU3 vs. r = (HβHα)-1c” (i): Figure 4.28 Av. Cosmological Temperature vs. Size c Hα

31 3.5 .10

t 1 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

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5.7.8

“TU3 vs. r = (HβHα)-1c” (ii): Figure 4.29 Av. Cosmological Temperature vs. Size

31 3.5 .10

t 2 .c t 3 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

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5.7.9

“dTU4/dt vs. t = (HβHα)-1” (i): Figure 4.30 1st Derivative of Av. Cosmological Temp.

1 .1072 t1

t2

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

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5.7.10

“dTU4/dt vs. t = (HβHα)-1” (ii): Figure 4.31 1st Derivative of Av. Cosmological Temp.

1 .1072 t2

t3

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

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5.7.11

“d2TU4/dt2 vs. t = (HβHα)-1” (i): Figure 4.32 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t1

t2

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

122

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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5.7.12

“d2TU4/dt2 vs. t = (HβHα)-1” (ii): Figure 4.33 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t2

t3

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

123

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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5.7.13

“|d3TU4/dt3| vs. t = (HβHα)-1” (i): Figure 4.34 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t1

t2

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

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5.7.14

“|d3TU4/dt3| vs. t = (HβHα)-1” (ii): Figure 4.35 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t2

t3

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

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5.7.15

“dH/dt vs. (HβηHα)-1” (i): Figure 4.36 1st Derivative of the Hubble Constant 1.6 .1084

t1

t4

84 1.4 .10

84 1.2 .10 dH dt H β

(Hz^2)

dH dt e

dH dt e

dH dt e

η

5 .µ

1 2

1 .1084 1

1 2 2 5 .µ . 5 .µ

1

8 .1083

2 2 5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

83 6 .10

83 4 .10

2 .1083

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

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5.7.16

“dH/dt vs. (HβηHα)-1” (ii): Figure 4.37 1st Derivative of the Hubble Constant 2 .1084 1 Hα

t1

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

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5.7.17

“dH/dt vs. (HβηHα)-1” (iii): Figure 4.38 1st Derivative of the Hubble Constant 2 .1084 t2

t3

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

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5.7.18

“dH/dt vs. (HβηHα)-1” (iv): Figure 4.39 1st Derivative of the Hubble Constant 2 .1084 t5

t4

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

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5.7.19

“d2H/dt2 vs. (HβηHα)-1” (i): Figure 4.40 2nd Derivative of the Hubble Constant 1 Hα

4 .10127

t1

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

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5.7.20

“d2H/dt2 vs. (HβηHα)-1” (ii): Figure 4.41 2nd Derivative of the Hubble Constant t2

4 .10127

t3

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

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5.7.21

“d2H/dt2 vs. (HβηHα)-1” (iii): Figure 4.42 2nd Derivative of the Hubble Constant t5

4 .10127

t4

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

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5.7.22

“d2H/dt2 vs. (HβηHα)-1” (iv): Figure 4.43 2nd Derivative of the Hubble Constant 125 8 .10

t1

t2

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

133

1 .10 40

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5.7.23

“d2H/dt2 vs. (HβηHα)-1” (v): Figure 4.44 2nd Derivative of the Hubble Constant 125 8 .10

t3

t4

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

134

1 .10 40

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5.7.24

“d2H/dt2 vs. (HβηHα)-1” (vi): Figure 4.45 2nd Derivative of the Hubble Constant 125 8 .10

t5

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

135

1 .10 40

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5.7.25

“|H| vs. (HβηHα)-1” (i): Figure 4.46 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t1

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

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5.7.26

“|H| vs. (HβηHα)-1” (ii): Figure 4.47 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t4

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

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5.7.27

“TU2,3 vs. |H|”: Figure 4.48 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

T U3 H β 1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β , H β .H α Hubble Constant (Hz)

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5.7.28

“TU2 vs. |H|”: Figure 4.49 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β Hubble Constant (Hz)

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NOTES

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6 “q0, Λ0” 6.1

SMoC

Two of the most important aspects of Cosmology are the present values of the deceleration parameter “q0” and the Cosmological constant “Λ0”. In-fact, both these values are so important that this chapter discusses them exclusively. “q0”177 is a dimensionless measure of the cosmic acceleration associated with the expansion of the Universe. Recent measurements imply that the expansion of the Universe is accelerating; “q0” was initially thought to be positive, now it is believed to be negative. “Λ0”178 was proposed by Albert Einstein as a modification of his original theory of General Relativity (GR) to achieve a stationary Universe. The contemporary view [i.e. the Standard Model (SM) of Cosmology] is that a problem exists because most Quantum Field Theories (QFT’s) predict a huge value of “Λ0” from the energy of the Zero-Point-Field (ZPF). This would need to be cancelled almost, but not exactly, by an equally large term of opposite sign. Some SuperSymmetric theories require a value of “Λ0” to be exactly zero. This is known as the “Λ0” problem179 and no known natural manner exists in which to derive the miniscule “Λ0” used in Cosmology from Particle-Physics. The SM of Cosmology (SMoC) defines “q0” as follows180, q0

Ω0

Λ0

2

3 .H 0

2

(2.34)

where, “Ω0” and “H0” denote the present values of the density parameter and Hubble constant respectively. Note: astronomical observations imply that “Λ0” cannot exceed “10-46(km-2)” 181,182. 6.2

EGM

6.2.1 “Ro, MG, ΩEGM, Ω ZPF” Fortunately, the Electro-Gravi-Magnetic (EGM) construct does not suffer from the same afflictions as the SMoC regarding “q0” and “Λ0”. That is, precise and meaningful values of “q0” and “Λ0” may be derived in agreement with physical observation, from the EGM Particle-Physics model derived in QE3. Therefore, EGM demonstrates that the SMoC assertion of “no known natural manner exists in which to derive the miniscule “Λ0” used in Cosmology from Particle-Physics” is incorrect. An algorithm is presented in QE4 such that the value of the ZPF density parameter “Ω ZPF” is derived utilising the difference between the predicted Cosmic Microwave Background Radiation (CMBR) temperature by the EGM method “TU2” and its presently measured value “T0”, facilitated by improved determinations of Milky-Way Galactic radius “Ro” and total mass “MG” from the 177

http://en.wikipedia.org/wiki/Deceleration_parameter Einstein abandoned the concept after the observation of the Hubble red-shift indicated that the Universe might not be stationary. However, the discovery of cosmic acceleration in the “1990’s” has renewed interest in “Λ0” (http://en.wikipedia.org/wiki/Cosmological_constant). 179 Believed to be the worst problem of fine-tuning in Physics. 180 http://radio.astro.gla.ac.uk/page184.gif 181 Michael, E., University of Colorado, Department of Astrophysical and Planetary Sciences, “The Cosmological Constant” (http://super.colorado.edu/~michaele/Lambda/lambda.html). 182 i.e. approximately “4.815526 x10-10(Pa)”. 178

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application of EGM principles. Utilising the algorithm defined in QE4 (i.e. considerate of the experimental tolerance of “T0”)183, a table of corresponding “Ro”, “MG”, “Ω EGM” and “ΩZPF = 1 – ΩEGM” values may be formulated according to, Ro (kpc) M G / MS 7.996943 6.331133 x1011 8.107221 6.314167 x1011 8.218926 6.296982 x1011

ΩEGM ΩZPF 0.998993 1.006904 x10-3 1.000331 -3.314007 x10-4 1.001671 -1.671006 x10-3 Table 2.9,

CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726

where, “MS” and “ΩEGM” denote solar mass (i.e. a standard value) and net Cosmological density parameter as defined by the EGM method respectively. 6.2.2 “ΩZPF → –q0 → qSM_1” “T0” is higher than the EGM determination184 of “2.724752(K)”; this implies that, because the Universe is “hotter” than the EGM prediction, it is slightly smaller than expected. This expectation is clearly set by the “TU2(H)” curve demonstrating that the Universe cools down as it expands (i.e. as it gets bigger) over time. Subsequently, “something” is acting to retard the expansion of the Universe. QE4 attributes this effect to the ZPF energy density threshold value “UZPF” derived from “ΩZPF”. Hence, assuming null experimental error associated with “T0”, the retardation energy is negative (i.e. “UZPF < 0”). Subsequently, by precise measurement of “T0”, the EGM construct facilitates the calculation of Cosmological size. Thus; i. If the Universe is smaller (i.e. hotter) than predicted by the EGM construct then, “q0 > 0” and “UZPF < 0”. ii. If the Universe is larger (i.e. cooler) than predicted by the EGM construct then, “q0 < 0” and “UZPF > 0”. Assuming “|ΩZPF| = |q0|”, a generalised representation of “q0” may be formulated in accordance with the EGM construct as follows, q0 = –ΩZPF (2.35) Note: the EGM construct mathematically derives the experimentally observed “accelerated Cosmological expansion phenomenon” hence, “q0” is positive from the EGM perspective (i.e. presently “dH/dt > 0”: refer to QE4). However, within the framework of the SMoC, the present observational situation should be “dH/dt < 0”, but this has been experimentally determined to be incorrect in favour of the EGM prediction of “dH/dt > 0” (i.e. from the perspective of the SMoC, “q0” is negative – opposite to the EGM perspective). Hence, it is necessary to differentiate between the EGM and SMoC perspectives of the deceleration parameter. Thus, let the SMoC deceleration parameter “qSM_1” be written as, qSM_1 = –q0

(2.36)

Tabulating an EGM and SMoC deceleration parameter comparison (i.e. “q0” and “qSM_1” respectively) yields,

183

Refer to “Appendix 2.A”. Produced by utilising a deceleration parameter of zero (i.e. “q0 = 0”), representing the “vanilla” solution which may be utilised as a baseline reference for many other calculations – as demonstrated in QE4.

184

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q0185 -1.006904 x10-3 3.314007 x10-4 1.671006 x10-3

qSM_1186 1.006904 x10-3 -3.314007 x10-4 -1.671006 x10-3 Table 2.10,

CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726

6.2.3 “Λ0” “Λ0” may be easily calculated utilising the SMoC relationship187 [i.e. Eq. (2.34)] such that: “Ω0 → ΩEGM”, “q0 → (qSM_1 = ΩZPF)” and “ΩZPF = 1 – ΩEGM” according to, Λ0

Ω EGM

3 .H

2

2

Ω ZPF

Ω EGM 2 .Ω ZPF Ω EGM 2 . 1 2

Ω EGM

2

3. Ω EGM 1 2

(2.37)

where, “H” denotes a generalised reference to the Hubble constant. If “H = HU2(Ro,MG) = 67.084304(km/s/Mpc)”, “Λ0” may be written according to, Λ 0 Ω EGM

Λ0 (km/s/Mpc)2 6.730065 x103 6.757167 x103 6.784296 x103 √Λ0 (km/s/Mpc) 82.036971 82.20199 82.366838

2 3 3 .H U2 R o , M G . .Ω EGM 1 2

Λ0 / c2 (km-2) 7.864602 x10-47 7.896273 x10-47 7.927975 x10-47 Table 2.11,

(2.38)

CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726

1/HU2 – 1/√ √Λ0 (yr) 1/√ √Λ0 (yr) 11.919176 x109 2.656709 x109 11.895249 x109 2.680636 x109 11.871442 x109 2.704443 x109 Table 2.12,

CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726

6.2.4 “UΛ, UZPF, Uλ” The “Big-Bang” was an explosion of space-time such that the Cosmological expansion of the Universe is expected, by the SMoC, to be slowing down (i.e. “dH/dt < 0”) whilst the EGM construct predicts188 that “dH/dt > 0”. If we view the “Big-Bang” through a force pairing Newtonian lens, expansive Cosmological pressure “Uλ” must be counter represented by contractive Cosmological pressure189 – interpreted by EGM to be approximately equal to the ZPF energy density threshold “UZPF” derived in QE4. 185

The experimentally observed “accelerated expansion phenomenon” was correctly predicted by the EGM construct (i.e. “dH/dt > 0”). Hence, the deceleration parameter is positive because (at a CMBR temperature of “T0”) the Universe is smaller (i.e. hotter) than the CMBR temperature solution derived utilising the EGM method, for “q0 = 0”. 186 The SMoC predicts that the deceleration parameter should be positive, but it has been experimentally observed to be negative. Hence, the SMoC “perceives” the deceleration parameter by the EGM construct to be negative. 187 Where, the deceleration parameter is considered to be negative. 188 (i) confirmed by experimental observation; (ii) no requirement for the existence of “Dark Matter / Energy”. 189 i.e. the resistance of the space-time manifold to being “stretched” by Cosmological expansion. 143

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“Λ0” may be utilised to determine the net expansive Cosmological pressure “UΛ” as follows, 2

U Λ Ω EGM

c . Λ 0 Ω EGM . 8 π .G

(2.39)

The contractive Cosmological pressure “UZPF” is given by, 3 .c . Ω ZPF . H U2 R o , M G 8 .π .G 2

U ZPF Ω ZPF

2

(4.315)

Note: “UZPF” is derived from the critical Cosmological mass density by the EGM method “ρU2(Ro,MG) = 3⋅HU2(Ro,MG) / (8πG)”: refer to QE4, Eq. (4.304). The expansive Cosmological pressure “Uλ” is given by, Uλ(ΩEGM,ΩZPF) = UΛ(ΩEGM) – UZPF(ΩZPF) UΛ (Pa) 3.787219 x10-10 3.802471 x10-10 3.817737 x10-10

UZPF (Pa) 7.649839 x10-13 -2.51778 x10-13 -12.695284 x10-13

Uλ (Pa) 3.77957 x10-10 3.804989 x10-10 3.830432 x10-10 Table 2.13,

|Uλ / UZPF| 494.071804 1.511247 x103 301.720886

(2.40) CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726

6.2.5 “T0” At this juncture, it is important to review the results presented in the preceding table. One should note that the results corresponding to “T0 – ∆T0” demonstrate positive values of ZPF pressure. This implies that (@ “T0 – ∆T0”), the effects of the ZPF are expansive, not contractive. Consequently, if EGM insists upon the ZPF applying a Cosmological contractive influence, we may interpret this contradiction as evidence of “Dark Matter / Energy”. Thus, two distinct physical possibilities exist dependent upon the precise measurement of the CMBR temperature as follows, i. If a precise measurement of the CMBR temperature190 demonstrates that “T0 ≥ TU2”, then “Dark Matter / Energy” does not exist as described by the SMoC because the EGM construct naturally derives the experimentally confirmed “accelerated expansion phenomenon” (i.e. “dH/dt > 0”). However, the SMoC requires the existence of “Dark Matter / Energy” to account for accelerated Cosmological expansion191. ii. If a precise measurement of the CMBR temperature demonstrates that “T0 < TU2”, then “Dark Matter / Energy” does exist, but in substantially less quantity than required by the SMoC. In-fact, for a worse case scenario192 of “T0 – ∆T0”, the influence of “Dark Matter / Energy” upon the CMBR temperature by the EGM method is “< 1(%)”193. Thus, insisting upon the ZPF possessing negative energy density facilitates the estimation of an improved “T0” tolerance as follows, TU2 < T0 ≤ (T0 + ∆T0) (2.41) Therefore, iii. We may advance the CMBR temperature prediction by approximately one-order-ofmagnitude to be: “2.7254 ± 0.0006(K)”. iv. “1 – {[TU2 – (T0 – ∆T0)] / [2⋅∆T0]}” demonstrates that the probability of the ZPF acting as a Cosmological contractive force is “> 62(%)”. 190

i.e. beyond the present resolution of “T0 ± ∆T0”. This is a major failing of the SMoC. 192 In relation to “TU2”. 193 As demonstrated in QE4. 191

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6.2.6 “ΛZPF → qSM_2” Many contemporary Cosmologists believe that ZPF energy is a positive quantity pseudonym for “Λ0”, considering it to be responsible for the experimentally observed “accelerated expansion phenomenon”. Utilising this notion (i.e. from this perspective), we may determine the value of deceleration parameter “qSM_2” in SMoC terms as follows, Λ0 → Λ ZPF Ω ZPF

8 .π .G . U ZPF Ω ZPF 2 c

(2.42)

Substituting into Eq. (2.34) yields, ΛZPF (km/s/Mpc)2 -13.594118 4.474212 22.560109 √ΛZPF (km/s/Mpc) 3.68702i 2.115233 4.749748

qSM_2 ΛZPF / c2 (km-2) -49 -1.588578 x10 0.500503 0.522846 x10-49 0.499834 2.636323 x10-49 0.499164 Table 2.14,

CMBR Temp. (K) Д Д Д

1/HU2 – 1/√ √ΛZPF (yr) (14.575885+265.20416i) x109 -447.696095 x109 -191.290414 x109 Table 2.15,

1/√ √ΛZPF (yr) -265.20416i x109 462.27198 x109 205.866299 x109

CMBR Temp. (K) Д Д Д

“Д”: calculation deferred until proceeding section. Note: “UZPF” is multiplied by a negative operator; as required by the SMoC due to the “accelerated Cosmological expansion phenomenon” (i.e. “dH/dt > 0”) suggesting that “Λ0 > 0”. Analysis of the preceding tables demonstrates that the EGM construct, by application of “UZPF”, may be easily transformed into SMoC terms. However, in doing so, one directly attributes ZPF energy to being the cause of the experimentally observed “accelerated Cosmological expansion phenomenon”, rather than being a force pairing consequence of “dH/dt > 0”194 as presented in the “qSM_1” solution. Hence, i. “√ΛZPF” becomes complex for solutions where “T0 < TU2” suggesting that, for SMoC transformations of EGM construct predictions, “qSM_2 ≤ ½”. ii. It is known by physical measurement (i.e. from measurements by the WMAP satellite) that the Universe is flat; hence, “qSM_2 ≤ ½” and the preceding tables of results are supported by Kellermann et. al.195 6.2.7 “qSM_2 ≈ ±½” 6.2.7.1 Construct Utilising the EGM construct, it is possible to qualitatively analyse the applicability of a range of deceleration parameters within the SMoC by applying the relationship196, q

1 . d H 2 H dt

2

H

(W.3)

194

Which the EGM construct naturally derives. http://www.nature.com/nature/journal/v361/n6408/abs/361134a0.html 196 http://en.wikipedia.org/wiki/Hubble_parameter; see also: QE4. 195

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However, we shall commence the analysis by validating Eq. (W.3) against expected results derived in QE4. The solution algorithm articulated in “Appendix 2.A” to facilitate the validation process is a generalisation of the rigorously tested numerical approximation technique (i.e. a secondary derivation of “HU2”) also presented in QE4. Hence, assuming “q → qSM_1 → ΩZPF” yields “H(ΩZPF)” as follows, Ω ZPF

d H dt 2

1 . d H 2 H dt

2

H

(2.43)

1 Ω ZPF

H

d H dt 2

(2.44)

Ω ZPF

1

H 2

H

(2.45) 1 Ω ZPF

. d H 1 dt

(2.46)

Hence, let: H

1 Ω ZPF

1

. d H dt

(2.47)

6.2.7.2 Sample calculations Evaluating the preceding equation in accordance with the solution algorithm articulated in “Appendix 2.A” yields the following table of results, Note: the proceeding table is formulated utilising the derivation of “TU2” in QE4 (i.e. for “q0 = 0”) as a point of reference and transforming the SMoC range of “qSM_2 ≈ ±½” relative to the “TU2” solution.

√dHdt (km/s/Mpc) 1/√ √dHdt (yr) 67.050522 14.583229 x109 67.095419 14.57347 x109 67.14033 14.563722 x109 √dHdt (km/s/Mpc) 1/√ √dHdt (yr) 47.411877 20.623801 x109 47.443626 20.609999 x109 47.475383 20.596213 x109 √dHdt (km/s/Mpc) 1/√ √dHdt (yr) 82.174949 11.899163 x109 82.15662 11.901818 x109 82.138273 11.904476 x109

q → qSM_1 → ΩZPF qSM_1 √dHdt/HU2 – 1 (%) -0.050358 1.006904 x10-3 0.016569 -3.314007 x10-4 0.083515 -1.671006 x10-3 q → qSM_2 ≈ +½ qSM_2 √dHdt/HU2 – 1 (%) -29.324934 0.500503 -29.277606 0.499834 -29.230268 0.499164 q → qSM_2 ≈ -½ qSM_2 √dHdt/HU2 – 1 (%) 22.495045 -0.500503 22.467722 -0.499834 22.440373 -0.499164 Table 2.16,

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CMBR Temp. (K) T0 – ∆T0 = 2.724 T0 = 2.725 T0 + ∆T0 = 2.726 CMBR Temp. (K) 2.252547 2.253374 2.254201 CMBR Temp. (K) 3.045402 3.045029 3.044656

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where, i. “q → qSM_1 → ΩZPF” denotes the reference calculation. ii. “qSM_2 ≈ ±½” denotes the SMoC deceleration range under consideration for comparison to the reference calculation. iii. “dHdt” is a convenient notational representation of “dH/dt” for application within the “MathCad 8 Professional” computational environment. iv. “√dHdt ≈ HU2”: the Hubble constant by the EGM method. v. “1/√dHdt”: the Hubble age by the EGM method. vi. “√dHdt/HU2 – 1”: the error between two EGM methods of Hubble constant determination. 6.2.7.3 Analysis i. The values of “√dHdt/HU2 – 1” associated with “q → qSM_1 → ΩZPF” (i.e. the reference calculation) demonstrate that the SMoC interpretation of the deceleration parameter, as derived by the EGM method, is consistent with the physical determination of the CMBR temperature. Hence, the approach utilised to produce the tabulated results is valid. ii. The values of “√dHdt/HU2 – 1” associated with “qSM_2 ≈ ±½” demonstrate that the solutions are approximately symmetrical. iii. The boundary conditions “qSM_2 ≈ ±½” exhibit over / under estimation of the Hubble age. iv. The values of CMBR temperature associated with “qSM_2 ≈ ±½” span a thermal bandwidth of “≈ 0.8(K)” and possess an average value of “≈ 2.649202(K)”197. v. As a generalisation, the tabulated results imply that the SMoC values of “qSM_2 ≈ ±½” are subjectively reasonable198 for Cosmological application within contemporary frameworks. However, from the perspective of the highly precise correlation to experimental observations produced by the EGM construct, these mathematically “convenient limits” are “rough approximations” and should not be utilised beyond qualitative analysis. vi. Proposed SMoC values of deceleration parameter should be tested against the solution algorithm defined in “Appendix 2.A” for validity in theoretical Cosmological constructs. 6.3

EGM vs. SMoC Key Mathematical Fact Dark matter / energy required Max Cosmological Temp ≈ 1031(K) Big-Bang Temperature = 0(K) Unification with Particle-Physics Relationship between “H0” and “T0” “H0” and “T0” are calculable to high precision “H0” and “T0” were derived from Particle-Physics Precise determination of distinct Cosmological evolutionary phases Sign of the deceleration parameter is in agreement with expectation Prediction of “accelerated Cosmological expansion” (i.e. “dH/dt > 0”) Experimentally implicit determination of “H0, q0 , Λ0” from “T0” Table 2.17,

SMoC Yes Yes No No No No No No No No No

EGM No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

197

To within approximately “2.781592(%)” and “2.77276(%)” of the physically measured and EGM derived CMBR temperature values respectively. 198 i.e. the impact upon the Hubble constant is significant, but less pronounced with respect to the CMBR temperature. 147

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6.4

Conclusion Four interpretations of deceleration parameter are presented (i.e. “q0”, “qSM_1” and “±qSM_2”), • “q0 = –ΩZPF”: the EGM interpretation of Cosmological expansion. • “qSM_1 = –q0”: the SMoC interpretation of “q0 = –ΩZPF”. • “±qSM_2”: a SMoC interpretation of ZPF energy. Two interpretations of Cosmological constant “Λ0” are presented, • EGM decomposes “Λ0” into expansive and contractive components. • ZPF energy as a positive quantity pseudonym for “Λ0”. Several key determinations of “qSM_2”, with respect to the SMoC, are presented, • SMoC values of “qSM_2 ≈ ±½” are subjectively reasonable for Cosmological application within contemporary frameworks. • “qSM_2 ≈ +½” → an under-estimation of “H0” and “T0”. • “qSM_2 ≈ –½” → an over-estimation of “H0” and “T0”. • The values of “T0” associated with “qSM_2 ≈ ±½” span a thermal bandwidth of “≈ 0.8(K)” and possess an average value of “≈ 2.649202(K)”. • From the perspective of the EGM construct, “qSM_2 ≈ ±½” are “rough approximations” and should not be utilised beyond qualitative analysis (i.e. the EGM construct is substantially more precise than the SMoC). • Proposed SMoC values of deceleration parameter should be tested against the solution algorithm defined in “Appendix 2.A” for validity in theoretical Cosmological constructs.

From the perspective of the SMoC, Cosmological deceleration was expected: however, Cosmological acceleration has been experimentally confirmed, in agreement with the EGM derived prediction. Subsequently, an under-estimation of “H0” and “T0” implies that the Universe is larger (i.e. cooler) than expected. In which case, the SMoC perceives “q0” to be negative. In terms of the preceding terminology, this may be represented as “q0 → –qSM_2” (alt. “qSM_2 = –q0”). Moreover, the SMoC interpretation of the sign “±” associated with ZPF energy is opposite to the EGM construct. That is, the SMoC interprets ZPF energy as a positive quantity; EGM interprets it as a negative quantity. Consequently, one is required to exercise caution when interpreting the appropriate sign of “q0”, due to observationally defiant physical formulations within the SMoC. The common feature across the various interpretations of “q0” and “Λ0” is ZPF energy. “Dark Matter / Energy” has not been considered because, although the SMoC is completely reliant upon its existence to explain the experimentally observed “accelerated Cosmological expansion phenomenon”, it is not required by the EGM construct as “dH/dt > 0” is naturally derived. Thus, in one manner or another, ZPF energy is required to derive precise and meaningful values of “q0” and “Λ0”. Moreover, because the value of “H0” is still widely debated and the associated experimental tolerance is much broader than “T0”, the EGM construct implies that the observed “accelerated expansion” of the Universe is attributable to the determination of the ZPF energy density threshold “UZPF” being “< -2.52 x10-13(Pa)”.

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7 Definition of Terms 7.1

Numbering conventions • •

7.2

References of the form “3.*” refer to “Quinta Essentia – Part 3”. References of the form “4.*” refer to “Quinta Essentia – Part 4”. Quinta Essentia – Part 3

7.2.1 Alpha Forms “αx” • An inversely proportional description of how energy density may result in acceleration. 7.2.2 Amplitude Spectrum • A family of wavefunction amplitudes. • The amplitudes associated with a frequency spectrum. • See: Frequency Spectrum. 7.2.3 Background Field • Reference to the background (ambient) gravitational field. • Reference to the local gravitational field at the surface of the Earth. 7.2.4 Bandwidth Ratio “∆ωR” • The ratio of the bandwidth of the ZPF spectrum to the Fourier spectrum of the PV. 7.2.5 Beta Forms “βx” • A directly proportional description of how energy density may result in acceleration. 7.2.6 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. 7.2.7 Casimir Force “FPP” • Attractive (non-gravitational) force between two parallel and neutrally charged mirrored plates of equal area. 7.2.8 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”. 7.2.9 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.

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7.2.10 Cosmological Constant • A constant introduced into the equations of GR to facilitate a steady state Cosmological solution. • See: General Relativity. 7.2.11 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”. • See: Critical Ratio “KR”. 7.2.12 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. 7.2.13 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. 7.2.14 Critical Frequency “ωC” • The minimum frequency for the application of Maxwell's Equations within an experimental context. 7.2.15 Critical Harmonic Operator “KR H” • A representation of the Critical Ratio at ideal dynamic, kinematic and geometric similarity utilising a unit amplitude spectrum. 7.2.16 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)” 7.2.17 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. 7.2.18 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. 7.2.19 Curl • The limiting value of circulation per unit area. 7.2.20 DC-Offsets • A proportional value of applied RMS Electric and / or Magnetic fields acting to offset the applied function/s. 150

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7.2.21 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. 7.2.22 Divergence • The rate at which “density” exits a given region of space. 7.2.23 Dominant Bandwidth • The bandwidth of the EGM spectrum which dominates gravitational effects. • See: Electro-Gravi-Magnetics (EGM) Spectrum. 7.2.24 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. 7.2.25 Electro-Gravi-Magnetics (EGM) Spectrum • A simple but extreme extension of the EM spectrum (including gravitational effects) based upon a Fourier distribution. 7.2.26 Energy Density (General) • Energy per unit volume. 7.2.27 Engineered Metric • A metric tensor line element utilising the Engineered Refractive Index. 7.2.28 Engineered Refractive Index “KEGM” • An EM based engineered representation of the Refractive Index. 7.2.29 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. 7.2.30 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. 7.2.31 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. 7.2.32 Fourier Spectrum • Two spectra combined into one (an amplitude spectrum and a frequency spectrum) obeying a Fourier series. • See: Amplitude Spectrum. • See: Frequency Spectrum. 151

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7.2.33 Frequency Bandwidth “∆ωPV” • The bandwidth of the Fourier spectrum describing the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). 7.2.34 Frequency Spectrum • A family of wavefunction frequencies. • The frequencies associated with an amplitude spectrum. • See: Amplitude Spectrum. 7.2.35 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). 7.2.36 Fundamental Harmonic Frequency “ωPV(1,r,M)” • The lowest frequency in the PV spectrum utilising Fourier harmonics. 7.2.37 General Modelling Equations (GMEx) • Proportional solutions to the Poisson and Lagrange equations resulting in acceleration. 7.2.38 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. 7.2.39 General Similarity Equations (GSEx) • Combines General Modelling Equations with the Critical Ratio by utilisation of the Engineered Relationship Function. • See: Critical Ratio “KR”. 7.2.40 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). 7.2.41 Graviton Mass-Energy Threshold “mγg” • The upper boundary value of the mass-energy of a Graviton as defined by the Particle Data Group. 7.2.42 Group Velocity • The velocity with which energy propagates. 7.2.43 Harmonic Cut-Off Frequency “ωΩ” • The terminating frequency of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV).

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7.2.44 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”. 7.2.45 Harmonic Cut-Off Mode “nΩ” • The terminating mode of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). 7.2.46 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). 7.2.47 Harmonic Inflection Frequency “ωX” • The frequency associated with the harmonic inflection mode. • See: Harmonic Inflection Mode “NX”. 7.2.48 Harmonic Inflection Wavelength “λX” • The wavelength associated with the harmonic inflection frequency. 7.2.49 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). 7.2.50 IFF • If and only if. 7.2.51 Impedance Function • A measure of the ratio of the permeability to the permittivity of a vacuum. • Resistance to energy transfer through a vacuum. 7.2.52 Kinetic Spectrum • Another term for the ZPF spectrum. • See: ZPF Spectrum. 7.2.53 Mode Bandwidth • The modes associated with a frequency bandwidth. 7.2.54 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.

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7.2.55 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”. 7.2.56 Phenomena of Beats • The interference between two waves of slightly different frequencies. 7.2.57 Photon Mass-Energy Threshold “mγ” • The upper boundary value of the mass-energy of a Photon as defined by the Particle Data Group. 7.2.58 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. 7.2.59 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). 7.2.60 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). 7.2.61 Potential Spectrum • Another term for the PV spectrum. • See: Polarisable Vacuum (PV) Spectrum. 7.2.62 Poynting Vector • Describes the direction and magnitude of EM energy flow. • The cross product of the Electric and Magnetic field. 7.2.63 Precipitations • Results driven by the application of limits. 7.2.64 Primary Precipitant • The frequency domain precipitation. • See: Precipitations.

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7.2.65 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). 7.2.66 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. • See: Impedance Function. 7.2.67 Reduced Average Harmonic Similarity Equations (HSExA R) • See: 2nd Reduction of the Harmonic Similarity Equations. 7.2.68 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]. 7.2.69 Refractive Index “KPV” • Characterisation value of the PV. 7.2.70 Representation Error “RError” • Error associated with the mathematical representation of a physical system. 7.2.71 RMS Charge Radii (General) • The RMS charge radius refers to the RMS value of the charge distribution curve. 7.2.72 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. 7.2.73 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”.

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7.2.74 Spectral Energy Density “ρ0(ω)” • Energy density per frequency mode. 7.2.75 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. 7.2.76 Subordinate Bandwidth • The EM spectrum. • See: Dominant Bandwidth. • See: Electro-Gravi-Magnetics (EGM) Spectrum. 7.2.77 Unit Amplitude Spectrum • A harmonic representation of unity (the number one) utilising the amplitude spectrum of a Fourier distribution. 7.2.78 Zero-Point-Energy (ZPE) • The lowest possible energy of the space-time manifold described in quantum terms. 7.2.79 Zero-Point-Field (ZPF) • The field associated with ZPE. 7.2.80 Zero-Point-Field (ZPF) Spectrum • The spectrum of amplitudes and frequencies associated with the ZPF. 7.2.81 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”. 7.2.82 Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF” • The terminating frequency of the ZPF spectrum across an elemental displacement. 7.2.83 Zero-Point-Field (ZPF) Beat Cut-Off Mode “nΩ ZPF” • The terminating mode of the ZPF spectrum across an elemental displacement. 7.2.84 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”. 7.2.85 2nd Reduction of the Harmonic Similarity Equations (HSExA R) • A time averaged simplification of the Reduced Harmonic Similarity Equations. 7.2.86 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”. 156

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7.2.87 3rd Sense Check “Stδ” • A common sense test relating the harmonic cut-off mode across an elemental displacement. • See: Harmonic Cut-Off Mode “nΩ”. 7.2.88 4th Sense Check “Stε” • A common sense test relating the representation error across an elemental displacement. • See: Representation Error “RError”. 7.2.89 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”. 7.2.90 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”. 7.2.91 Physical Constants 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

157

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

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7.2.92 Mathematical Constants and Symbols • Euler-Mascheroni Constant (Euler's Constant) [33] “γ” = 0.5772156649015328 • “∩” Refers to an intersection. • “∪” Refers to a union. • “→” Or “↓” refers to a process: “leads to”. 7.2.93 Solar System Statistics 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

Value utilised by EGM Units kg 7.35 x1022 5.977 x1024 1898.8 x1024 1.989 x1030 m 1.738 x106 6.37718 x106 7.1492 x107 6.96 x108

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7.3

Quinta Essentia – Part 4

7.3.1 “Big-Bang” • The moment of Cosmological creation. • The explosion of space-time at the moment of Cosmological creation. 7.3.2 Black-Hole “BH” • A massive gravitational body such that light cannot escape. • See: Super-Massive-Black-Hole “SMBH”. 7.3.3 Broadband Propagation • The propagation of the EGM wavefunctions in the PV spectrum, at a group velocity of zero. 7.3.4 Buoyancy Point • The point of gravitational acceleration balance (neutrality) between the Earth and the Moon. 7.3.5 CMBR Temperature “T0” • Cosmic Microwave Background Radiation (CMBR) temperature. • See: EGM-CMBR Temperature “TU”. 7.3.6 EGM-CMBR Temperature “TU” • Derivation of the CMBR temperature by the EGM method. • See: CMBR Temperature “T0”. 7.3.7 EGM Flux Intensity “CΩ_J” • The flux intensity of gravitational energy expressed in Jansky's. • Formulated by considering celestial objects as point radiation sources of a high-frequency EGM wavefunction. 7.3.8 EGM Hubble constant “HU” • Derivation of the Hubble constant by the EGM method. • See: Hubble Constant “H0”. 7.3.9 • • •

Event Horizon “RBH” Refers to “RBH” of a SBH. The radial displacement from the centre of a SBH from which light cannot escape. See: Schwarzschild-Black-Hole “SBH”.

7.3.10 Galactic Reference Particle “GRP” • The total mass / energy of any Galactic formation represented as a particle with dimensions approaching the Planck scale. 7.3.11 Gravitational Interference • The formation of interference patterns from either broadband or narrowband EGM wavefunction propagation between two or more gravitational fields. 7.3.12 Gravitational Propagation • See: Broadband Propagation. • See: Narrowband Propagation.

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7.3.13 Hubble Constant “H0” • A measure of the rate of Cosmological expansion. • See: EGM Hubble constant “HU”. 7.3.14 Narrowband Propagation • An approximation of Broadband characteristics. • See: Broadband Propagation. 7.3.15 • • •

Non-Physical Situations where “nΩ” is less than unity. See: “nΩ”, “Quinta Essentia – Part 3”. See: Physical.

7.3.16 • • •

Physical Situations where “nΩ” is greater than or equal to unity. See: “nΩ”, “Quinta Essentia – Part 3”. See: Non-Physical.

7.3.17 Primordial Universe • The Universe prior to the “Big-Bang”. • The Universe at the instant prior to the “Big-Bang”. 7.3.18 Schwarzschild-Black-Hole “SBH” • A static Black-Hole. • The simplest form of Black-Hole. 7.3.19 Schwarzschild-Planck-Black-Hole “SPBH” • A SBH of maximum permissible energy density existing at the Planck scale such that the singularity and event horizon radii coincide. • The value of harmonic cut-off mode “nΩ” at the periphery is unity. • The minimum physical radius is “λxλh”. • The minimum physical mass is “mxmh”. • See: Schwarzschild-Black-Hole “SBH”. • See: “nΩ”, “Quinta Essentia – Part 3”. 7.3.20 Schwarzschild-Planck-Particle • Generalised reference to a SPBH. 7.3.21 Singularity • The maximum permissible energy density at the centre of a SBH, represented as a particle. • The particle representation (or mathematical point) at the centre of a SBH for which physical laws are unknown to apply. • See: Schwarzschild-Black-Hole “SBH”. 7.3.22 Singularity Radius “rS” • The radius of the singularity at the centre of a SBH. • See: Schwarzschild-Black-Hole “SBH”. 7.3.23 Solar Mass • The mass of the Sun. 160

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7.3.24 Super-Massive-Black-Hole “SMBH” • A BH of greater than “109” solar masses. • See: Black-Hole “BH”. 7.3.25 Total Mass-Energy • Refers to “visible + dark”. 7.3.26 Astronomical / Cosmological statistics Symbol DE2M H0 MG MG/3 MNS RNS Ro T0

Description Mean Earth-Moon distance Hubble constant (present value) Total Galactic mass Visible Galactic mass Mass of Neutron Star Mean Radius of a Neutron Star Mean distance to Galactic centre CMBR temperature (present value)

Value utilised by EGM 3.844 x108 71 6 x1011MS 2 x1011MS ≥ MS 20 8 2.725

Units m km/s/Mpc kg

km kpc K

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8 Glossary of Terms 8.1

Quinta Essentia – Part 3

8.1.1 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

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) 163

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

Ministry of Science (Japan) Mean Square 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 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 NOTES

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8.1.2 Symbols in 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) 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 Charge e, e Electron: 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 Root Mean Square of EA Erms F(k,n,t) Complex FS representation of EGM 165

Units m/s2 m2 m/s2

T

T m/s m/s2

% J V/m

J C

V/m J V/m

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Magnitude of the ambient gravitational acceleration represented in the time m/s2 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 Universal Gravitation Constant m3kg-1s-2 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 Planck’s Constant (plain h form) Js 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 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 m L f(t)

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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 167

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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

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 RMS charge radius of the Muon Neutrino by EGM Neutron RMS charge radius (by analogy to KS) 168

C/m2 C

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% m

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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 Change in K2 by EGM ∆K2 Change in Critical Factor by EGM ∆ KC rν2 rν3 rν5 rνM rνx rπ rπE rπM rτ rτn R∞ S sq StN Stα Stβ Stδ Stε Stγ Stη Stθ Stω

169

m

J W/m2

PaΩ

W/m2 s (m/s)2 Pa Pa m/s

Ω J m/s2

(V/m)2 T-2 PaΩ

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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 λΑ λΒ λCe λCN λCP λh

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 Proton Compton Wavelength Planck Length 170

m s (m/s)2 Pa m/s Hz2 m Hz Hz

m/s2 m/s2

F/m θc m

m

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λPV µ, µµ µ0 ν2 ν3 ν5 νe νµ ντ ρ ρ0 τ, τω

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) Frequency spectrum of PV ωPV(nPV,r,M) 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〉〉

m kg or eV N/A2

C/m3 Pa/Hz Hz

kg or eV m

NOTES

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8.2

Quinta Essentia – Part 4

8.2.1 Acronyms •

Refer to “Quinta Essentia – Part 3: Acronyms” where appropriate.

BH CMBR DAT ED EFT EP GA GRP GUT HEP LFT MW NASA QED QFT QM RF SBH SED SM SMBH SPBH ST ToE VP ZP ZPE

Black-Hole Cosmic Microwave Background Radiation Dimensional Analysis Technique ElectroDynamics Effective Field Theory Experimental Prototype Gravitational Astronomy Galactic Reference Particle Grand Unified Theory High Energy Physics Lattice Field Theory Milky-Way National Aeronautics and Space Administration Quantum Electro Dynamics Quantum Field Theory Quantum Mechanics Radio Frequency Schwarzschild-Black-Hole Spectral Energy Density Standard Model of Particle-Physics or Cosmology Super-Massive-Black-Hole Schwarzschild-Planck-Black-Hole String Theory Theory of Everything Virtual Photon Zero-Point Zero-Point-Energy NOTES

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8.2.2 Symbols by chapter •

Refer to “Quinta Essentia – Part 3: Symbols in alphabetical order” where appropriate.

Symbol aEGM_ωΩ ωΩ CΩ_J CΩ_J1 nΩ_1 Stg StG StJ SωΩ Ω1 ωΩ_1 ωΩ_2 ωΩ_3

Chapter 5: Characterisation of the Gravitational Spectrum Description Gravitational acceleration utilising ωΩ_2 EGM Flux Intensity Non-refractive form of CΩ_J Non-refractive form of nΩ 1st EGM gravitational constant: Stg = 1.828935 x10245 2nd EGM gravitational constant: StG = 8.146982 x10224 3rd EGM gravitational constant: StJ9 = 1.093567 x10-146(kg4m26/s18) Approximated / simplified Poynting Vector Non-refractive form of Ω Non-refractive form of ωΩ Gravitational acceleration form of ωΩ_1 Transformed representation of ωΩ_1

Units m/s2 Jy (Jansky)

m-1s-5 m5kg-2s-9 (kg4m26/s18)(1/9) W/m2 Hz

Chapter 6: Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” Characteristics Symbol Description Units Propagation energy of a Graviton J Eg Ex Proportional relationship between Eg and Eγ Propagation energy of a Photon J Eγ Hubble constant (present value) Hz H0 Refractive Index of PV by Depp KDepp Planck scale experimental relationship function Kh Generalised mass kg M1 Generalised mass M2 Mass of a SBH MBH 2nd SPBH constant mx nBH Harmonic cut-off mode ratio (nΩ_5 : nΩ_4) ng Average number of Gravitons radiated by a SBH per TΩ_4 period Population of Gravitons within starving matter ngg Transformed representation of nΩ_1 nΩ_2 The form nΩ_2 takes as a function of λx nΩ_3 nΩ_1 at the periphery of a SBH singularity nΩ_4 nΩ_1 at the event horizon of a SBH nΩ_5 Average number of Photons radiated by a SBH per TΩ_4 period nγ Population of Photons within starving matter nγγ Hubble size (present value) m r0 Generalised radial displacement r1 Generalised radial displacement r2 Radius of the event horizon of a SBH RBH Range variable for SBH’s Rbh Singularity radius rS ZPF equilibrium radius rZPF Distance from the centre of mass of a celestial object to the Earth rω Average emission period per Graviton s Tg 173

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TL TΩ_3 TΩ_4 V ∆KPV ∆r λVL λx λX-RAY ρm ρS ωBH ωg ωPV_1 ωX-RAY ωΩ_4 ωΩ_5 ωΩ_6 ωΩ_7

Minimum gravitational lifetime of matter 1 / ωΩ_3 1 / ωΩ_4 Volume Change in Refractive Index of PV Change in displacement within the event horizon of a SBH Wavelength of visible light 1st SPBH constant Wavelength of X-Rays Mass density Mass density of a SPBH Harmonic cut-off frequency ratio (ωΩ_5 : ωΩ_4) Average Graviton emission frequency (1 / Tg) Fundamental harmonic frequency ratio (ωΩ_6 : ωΩ_7) Frequency of X-Rays ωΩ_3 at the event horizon of a SBH ωΩ_3 at the periphery of a SBH singularity ωPV(1,r,MBH) at the periphery of a SBH singularity: r = rS(MBH) ωPV(1,r,MBH) at the event horizon of a SBH: r = RBH(MBH)

s

m3 m

m kg/m3

Hz Hz

Chapter 7: Fundamental Cosmology Symbol AU H HU HU2 Hα KT KU KW M3 Mf MG mg1 mg2 mg3 mg4 mg5 Mi MU r3 rf ri Ro RU rx1 rx2 rx3 rx4

Description EGM Cosmological age (present value) Generalised reference to the Hubble constant EGM Hubble constant Transformed representation of HU Primordial Hubble constant Expansive scaling factor rf / ri Wien displacement constant: 2.8977685 x10-3 [35] Generalised mass or mxmh Total Cosmological mass (present value) Total mass of the Milky-Way Galaxy Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Total Cosmological mass (initial value) Total EGM Cosmological mass Generalised radial displacement or λxλh Cosmological size (present value) Cosmological size (initial value) Distance from the Sun to the Galactic centre EGM Cosmological size (present value) Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor 174

Units s Hz

mK kg

kg m

pc (parsec) m

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rx5 T0 TU TU2 TW UZPF ∆ Ro ∆T0 Ω ΩEGM Ωm ΩPDG ΩZPF ΩΛ Ωγ Ων λy λΩ_3 ρ ρc ρU ρU2

Computational pre-factor CMBR temperature (present value) CMBR temperature by the EGM method Transformed representation of TU Thermodynamic scaling factor ZPF energy density threshold Experimental tolerance of Ro Experimental tolerance of T0 Community reference to the net Cosmological density parameter Net Cosmological density parameter as defined by the EGM method Visible mass contribution to the net Cosmological density parameter Net Cosmological density parameter as defined by the PDG ZPE contribution to the net Cosmological density parameter Dark energy contribution to the net Cosmological density parameter Photon contribution to the net Cosmological density parameter Neutrino contribution to the net Cosmological density parameter Generalised representation of λx c / ωΩ_3 Community reference to Cosmological mass-density Critical Cosmological mass-density EGM Cosmological mass-density Transformed representation of ρU

K

Pa pc K

m kg/m3

Chapter 8: Advanced Cosmology Symbol CΩ_Jωω dH2dt2 dHdt dT2dt2 dT3dt3 dTdt Hβ Hβ2 Hγ MEGM ML REGM rL t1 t2 t3 t4 t5 t7 tEGM

Description Equal to CΩ_J 2nd time derivative of H 1st time derivative of H 2nd time derivative of TU4 3rd time derivative of TU4 1st time derivative of TU4 Dimensionless range variable Computational pre-factor Dimensionless range variable Convenient form of MU EGM Cosmological mass limit Convenient form of RU EGM Cosmological size limit • Temporal ordinate (local maxima) of the CMBR temperature • The instant of maximum Cosmological temperature Temporal ordinate (local minima) of the 1st time derivative of the CMBR temperature Temporal ordinate (local maxima) of the 2nd time derivative of the CMBR temperature • Temporal ordinate (local maxima) of the 1st time derivative of H • The instant of maximum physical EGM Hubble constant Temporal ordinate (local minima) of the 2nd time derivative of H Equal to t1 Convenient form of AU 175

Units Jy Hz3 Hz2 K/s2 K/s3 K/s

kg m s

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tL TU3 TU4 η µ

EGM Cosmological age limit Transformed representation of TU2 Transformed representation of TU3 Computed index Indicial constant (µ = 1 / 3)

s K

Symbol ag aPV DE2M gav r4 r5

Chapter 9: Gravitational Cosmology Description High frequency harmonic acceleration Gravitational acceleration harmonic Mean distance from the Earth to the Moon Average high-frequency harmonic acceleration Distance from the centre of mass of the Earth to the buoyancy point Distance from the centre of mass of the Moon to the buoyancy point

Units m/s2 m m/s2 m

Chapter 10: Particle Cosmology Symbol mgg2 mγγ2 γγ Nγ Qγ Qγ_PDG Qγγ Qγγ2 γγ rgg2 rγγ2 γγ

Description Graviton mass-energy lower limit Photon mass-energy lower limit Photon population at Qγ Photon RMS charge threshold by EGM Photon RMS charge threshold by PDG Photon RMS charge upper limit by EGM Photon RMS charge lower limit by EGM Graviton RMS charge radius lower limit Photon RMS charge radius lower limit

Units eV

C

m

Appendix 4.A Symbol TBH Th TSPBH Φ κ σ

Description BH temperature Planck temperature SPBH temperature Energy flux emitted from a “Black-Body” at temperature “T” Boltzmann’s constant: 1.3806505 x10-23 [35] Stefan-Boltzmann constant: 5.670400 x10-8 [35]

Units K W/m2 J/K Wm-2K-4

NOTES

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8.2.3 Symbols in alphabetical order •

Refer to “Quinta Essentia – Part 3: Symbols in alphabetical order” where appropriate.

Symbol aEGM_ωΩ ωΩ ag aPV AU CΩ_J CΩ_J1 CΩ_Jωω DE2M dH2dt2 dHdt dT2dt2 dT3dt3 dTdt Eg Ex Eγ gav H H0 HU HU2 Hα Hβ Hβ2 Hγ KDepp Kh KT KU KW M1 M2 M3 MBH MEGM Mf MG mg1 mg2 mg3 mg4 mg5 mgg2 Mi ML

Description Gravitational acceleration utilising ωΩ_2 High frequency harmonic acceleration Gravitational acceleration harmonic EGM Cosmological age (present value) EGM Flux Intensity Non-refractive form of CΩ_J Equal to CΩ_J Mean distance from the Earth to the Moon 2nd time derivative of H 1st time derivative of H 2nd time derivative of TU4 3rd time derivative of TU4 1st time derivative of TU4 Propagation energy of a Graviton Proportional relationship between Eg and Eγ Propagation energy of a Photon Average high-frequency harmonic acceleration Generalised reference to the Hubble constant Hubble constant (present value) EGM Hubble constant Transformed representation of HU Primordial Hubble constant Dimensionless range variable Computational pre-factor Dimensionless range variable Refractive Index of PV by Depp Planck scale experimental relationship function Expansive scaling factor rf / ri Wien displacement constant: 2.8977685 x10-3 [35] Generalised mass Generalised mass Generalised mass or mxmh Mass of a SBH Convenient form of MU Total Cosmological mass (present value) Total mass of the Milky-Way Galaxy Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Graviton mass-energy lower limit Total Cosmological mass (initial value) EGM Cosmological mass limit 177

Units m/s2

s Jy (Jansky)

m Hz3 Hz2 K/s2 K/s3 K/s J J m/s2 Hz

mK kg

eV kg www.deltagroupengineering.com

MU mx mγγ2 γγ nBH ng ngg nΩ_1 nΩ_2 nΩ_3 nΩ_4 nΩ_5 nγ Nγ nγγ Qγ Qγ_PDG Qγγ Qγγ2 γγ r0 r1 r2 r3 r4 r5 RBH Rbh REGM rf rgg2 ri rL Ro rS RU rx1 rx2 rx3 rx4 rx5 rZPF rγγ2 γγ rω Stg StG StJ SωΩ T0 t1

Total EGM Cosmological mass 2nd SPBH constant Photon mass-energy lower limit Harmonic cut-off mode ratio (nΩ_5 : nΩ_4) Average number of Gravitons radiated by a SBH per TΩ_4 period Population of Gravitons within starving matter Non-refractive form of nΩ Transformed representation of nΩ_1 The form nΩ_2 takes as a function of λx nΩ_1 at the periphery of a SBH singularity nΩ_1 at the event horizon of a SBH Average number of Photons radiated by a SBH per TΩ_4 period Photon population at Qγ Population of Photons within starving matter Photon RMS charge threshold by EGM Photon RMS charge threshold by PDG Photon RMS charge upper limit by EGM Photon RMS charge lower limit by EGM Hubble size (present value) Generalised radial displacement Generalised radial displacement Generalised radial displacement or λxλh Distance from the centre of mass of the Earth to the buoyancy point Distance from the centre of mass of the Moon to the buoyancy point Radius of the event horizon of a SBH Range variable for SBH’s Convenient form of RU Cosmological size (present value) Graviton RMS charge radius lower limit Cosmological size (initial value) EGM Cosmological size limit Distance from the Sun to the Galactic centre Singularity radius EGM Cosmological size (present value) Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor ZPF equilibrium radius Photon RMS charge radius lower limit Distance from the centre of mass of a celestial object to the Earth 1st EGM gravitational constant: Stg = 1.828935 x10245 2nd EGM gravitational constant: StG = 8.146982 x10224 3rd EGM gravitational constant: StJ9 = 1.093567 x10-146(kg4m26/s18) Approximated / simplified Poynting Vector CMBR temperature (present value) • Temporal ordinate (local maxima) of the CMBR temperature • The instant of maximum Cosmological temperature 178

kg eV

C

m

pc (parsec) m

m

m-1s-5 m5kg-2s-9 (kg4m26/s18)(1/9) W/m2 K s

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t2 t3 t4 t5 t7 TBH tEGM Tg Th TL tL TSPBH TU TU2 TU3 TU4 TW TΩ_3 TΩ_4 UZPF V ∆KPV ∆r ∆ Ro ∆T0 Φ Ω Ω1 ΩEGM Ωm ΩPDG ΩZPF ΩΛ Ωγ Ων η κ λVL λx λX-RAY λy λΩ_3 µ ρ ρc ρm

Temporal ordinate (local minima) of the 1st time derivative of the CMBR temperature Temporal ordinate (local maxima) of the 2nd time derivative of the CMBR temperature • Temporal ordinate (local maxima) of the 1st time derivative of H • The instant of maximum physical EGM Hubble constant Temporal ordinate (local minima) of the 2nd time derivative of H Equal to t1 BH temperature Convenient form of AU Average emission period per Graviton Planck temperature Minimum gravitational lifetime of matter EGM Cosmological age limit SPBH temperature CMBR temperature by the EGM method Transformed representation of TU Transformed representation of TU2 Transformed representation of TU3 Thermodynamic scaling factor 1 / ωΩ_3 1 / ωΩ_4 ZPF energy density threshold Volume Change in Refractive Index of PV Change in displacement within the event horizon of a SBH Experimental tolerance of Ro Experimental tolerance of T0 Energy flux emitted from a “Black-Body” at temperature “T” Community reference to the net Cosmological density parameter Non-refractive form of Ω Net Cosmological density parameter as defined by the EGM method Visible mass contribution to the net Cosmological density parameter Net Cosmological density parameter as defined by the PDG ZPE contribution to the net Cosmological density parameter Dark energy contribution to the net Cosmological density parameter Photon contribution to the net Cosmological density parameter Neutrino contribution to the net Cosmological density parameter Computed index Boltzmann’s constant: 1.3806505 x10-23 [35] Wavelength of visible light 1st SPBH constant Wavelength of X-Rays Generalised representation of λx c / ωΩ_3 Indicial constant (µ = 1 / 3) Community reference to Cosmological mass-density Critical Cosmological mass-density Mass density 179

s

K s K s K

s Pa m3 m pc K W/m2

J/K m m m kg/m3

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ρS ρU ρU2 σ ωBH ωg ωPV_1 ωX-RAY ωΩ_1 ωΩ_2 ωΩ_3 ωΩ_4 ωΩ_5 ωΩ_6 ωΩ_7

Mass density of a SPBH EGM Cosmological mass-density Transformed representation of ρU Stefan-Boltzmann constant: 5.670400 x10-8 [35] Harmonic cut-off frequency ratio (ωΩ_5 : ωΩ_4) Average Graviton emission frequency (1 / Tg) Fundamental harmonic frequency ratio (ωΩ_6 : ωΩ_7) Frequency of X-Rays Non-refractive form of ωΩ Gravitational acceleration form of ωΩ_1 Transformed representation of ωΩ_1 ωΩ_3 at the event horizon of a SBH ωΩ_3 at the periphery of a SBH singularity ωPV(1,r,MBH) at the periphery of a SBH singularity: r = rS(MBH) ωPV(1,r,MBH) at the event horizon of a SBH: r = RBH(MBH)

kg/m3

Wm-2K-4 Hz Hz

NOTES

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9 Key Artefact and Equation Summary The following is an itemised account of the key relationships derived in QE3 and QE4: 9.1

Quinta Essentia – Part 3

9.1.1 Dimensional Analysis 9.1.1.1 “KPV, K0(X)” 2

K PV K 0( X )

3

(3.25)

9.1.1.2 “a(t)”

Acceleration

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, 9.1.2 General modelling and the critical factor 9.1.2.1 “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)

9.1.2.2 “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

9.1.2.3 “GME2” N

a r0

±

β1

β2

±

2

K 2 ω 0, r 0, B 0, D, X

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) 9.1.3 The engineered metric 9.1.3.1 “KR” KR

∆U g ∆a PV ∆K C( ∆r ) Ug

.

∆U PV( ∆r )

g

ε0 µ0

(3.53)

9.1.3.2 “∆K0(ω,X)” ∆K 0( ω , X )

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

(3.54)

9.1.3.3 “KEGM” (normal matter form) K EGM K PV. e

2 . ∆K 0( ω , X )

(3.56)

9.1.4 Amplitude and frequency spectra 9.1.4.1 “CPV” G.M .

C PV n PV, r , M

2

r

2 . π n PV

(3.64)

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

ω PV n PV, r , M

(3.67)

9.1.4.3 “nΩ” n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

9.1.4.4 “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)

9.1.4.5 “ωΩ” ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

182

(3.73)

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2

c0

9.1.5 General similarity 9.1.5.1 “ωβ” ω β r , ∆r , M , K R

4

ω Ω ( r , ∆r , M )

4 ZPF

K R . ω Ω ( r , ∆r , M ) ZPF

4

∆ω δr( 1 , r , ∆r , M )

4

(3.93)

9.1.5.2 EGM Wave Propagation

Figure 3.14, 9.1.5.3 EGM Spectrum

Figure 3.15 (illustrational only), 9.1.6 Harmonic and spectral similarity 9.1.6.1 “φC = 0°, 90°”, “EC, BC” •

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. 183

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9.1.6.2 “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)

9.1.6.3 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

SSE 5 ( 1

π DC) .B rms , , r , ∆r , M 2

DC) . E rms, ( 1

DC) . B rms,

1 2

π

, r, ∆ r, M

2

(3.159) 1 4

(Eq. 3.160) 9.1.6.4 “ωC” ω C( ∆r )

c 2 .∆r

(3.162)

9.1.7 The Casimir Effect 9.1.7.1 “NX” N X( r , ∆r , M )

n Ω ( r , ∆r , M )

1 ZPF

ln 2 .n Ω ( r , ∆r , M )

γ

(3.164)

ZPF

9.1.7.2 “NC” N C( r , ∆r , M )

ω C( ∆r ) ω PV( 1 , r , M )

(3.169)

9.1.7.3 “ωX” ω X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

(3.170)

9.1.7.4 “FPV” 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)

9.1.7.5 “∆r, λx, Erms, Brms, 0c, ±π, ±π/2” •

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”

184

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9.1.8 Physical characteristics 9.1.8.1 Photon / Graviton 9.1.8.1.1 “mγ” mγ<

512.h .G.m e

.

2

c . π .r e

n Ω r e, m e ln 2 .n Ω r e , m e

γ

(3.193)

9.1.8.1.2 “mgg” mgg = 2mγγ

(3.216)

9.1.8.1.3 “mγγ” 3

m γγ

h . re

3

π .r e

512.G.m e

.

2 .c .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)

9.1.8.1.4 “rγγ” 9.1.8.1.4.1 Primary 5

2

m γγ

r γγ r e .

m e .c

2

(3.225)

9.1.8.1.4.2 Secondary •

Approximation of the radius of a free Photon “rγγ”, relating physical properties of the Lepton family, specifically all Electron-Like particles r γγ K ω .

G.h . r µ c

3

rτ

(3.274)

9.1.8.1.5 “rgg” r gg

5

4 .r γγ

(3.227)

9.1.8.2 “α” 9.1.8.2.1 Primary α

rε

2

.e

3

rπ

(3.204)

9.1.8.2.2 Secondary rµ

α

rε

.e

rτ

rν

(3.236)

185

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9.1.8.3 “Stω” 2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

r2

9

St ω

r1

(3.230)

9.1.8.4 Electron, Muon, Tau 9.1.8.4.1 “rε, rµ, rτ” 5

2

1 . me r ε r π. 9 2 mp

(3.231)

5

rε .

rµ rτ

5

2

1 . mµ 9 4 me

1 . mτ 9 6 me

2

(3.234)

9.1.8.4.2 “ren, rµn, rτn” 5

m en

r ε.

r en r µn r τn

5

2

r µ.

me

5

2

m µn

r τ.

mµ

m τn

2

mτ

(3.238)

9.1.8.5 Proton, Neutron 9.1.8.5.1 “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

ω Ce

ω Ce

(3.210)

9.1.8.5.2 “rπ, rν” 2

rε

. c .e r e ω Ce

rπ

rε π

3

rν

rπ

5

c .ω Ce

.

3

4 .ω CP

c .ω Ce

5

.

3

4 .ω CN

2 4 27.ω h ω Ce . 4 32.π ω CP

(3.212)

2 4 27.ω h ω Ce . 4 32.π ω CN

(3.215)

9.1.8.5.3 “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)

9.1.8.5.4 “rdr” r dr

5. 3

rν

186

(3.391)

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9.1.8.5.5 “KS” KS

2 3 . π . r ν ( 1 x) . x3 . 2 8 1 x x

(3.396)

9.1.8.5.6 “b1, rX” b1

2 . KS 3.r ν

2

2

1

x

(3.394)

2 6 .b 1 .K X . x

rX

2 3 .b 1 . x

1

rν KS

1

. K .K S X

(3.418)

9.1.8.5.7 “rνM” r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

(3.420)

9.1.8.5.8 “rπE” r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

(3.423)

9.1.8.5.9 “rπM” ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

(3.426)

9.1.8.5.10 “rp” r P r πE

1. 2

r νM

rν

(3.429)

9.1.8.6 Quark / Boson harmonics 9.1.8.6.1 “Up Quark” 1

.

ω Ω r uq , m uq

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

(3.253)

9.1.8.6.2 Electron 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

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

187

(3.254)

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9.1.8.7 Hydrogen Spectrum: “λA” •

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

(3.457)

9.1.9 Theoretical propositions 9.1.9.1 The Planck scale: “Kω, Kλ, Km” 3

Kω

Kω

Kω

2 π

(3.270)

1 Kλ

(3.264)

1 Km

(3.265)

9.1.9.2 Particles 9.1.9.2.1 Leptons: “mL(2), mL(3), mL(5)” mL(2) ≈ 9(MeV)

(3.280)

mL(3) ≈ 57(MeV)

(3.281)

mL(5) ≈ 566(MeV)

(3.282)

9.1.9.2.2 Quarks / Bosons: “mQB(5), mQB(6)” mQB(5) ≈ 10(GeV)

(3.285)

mQB(6) ≈ 22(GeV)

(3.286)

188

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9.2

Quinta Essentia – Part 4

9.2.1 Gravitation 9.2.1.1 “Stg” 6 3 3 .ω h

St g

13 2 2 .π .c

(4.23)

9.2.1.2 “ωΩ_2” ω Ω_2( r , M )

9

St g

.g ( r , M ) 2

r

(4.25)

9.2.1.3 “aEGM_ωΩ” r . 9 ω Ω_2( r , M )

a EGM_ωΩ( r , M )

St g

(4.26)

9.2.1.4 “StG” 3.

St G

2

3 .ω h

9

. c 2

4 .π .h

(4.35)

9.2.1.5 “ωΩ_3” 9

2

M St G. 5 r

ω Ω_3( r , M )

(4.36)

9.2.1.6 “λΩ_3” λΩ_3 = c / ωΩ_3 9.2.1.7 “G” St G

G

St g

(4.37)

9.2.1.8 “ωPV(nPV,r,M)3” 2 .c .n PV

3

ω PV n PV, r , M

3

π .r

.g ( r , M )

2

(4.41)

9.2.1.9 “StJ” 9 .c . St G 4 .π 4

St J

2 9

(4.51)

9.2.1.10 “CΩ_J1, CΩ_Jω” C Ω_J1( r , M )

St J 2

r

189

9

. M

5

8

r

(4.52) www.deltagroupengineering.com

5.2

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

C Ω_Jω ω Ω_3 , M

(4.427)

9.2.1.11 “nΩ_2” 9

n Ω_2( r , M )

. 3 3 .π m h . r 16 M λh 2

7

(4.60)

9.2.1.12 “KDepp” 1

K Depp ( r , M )

2 .G.M

1

r .c

2

(4.106)

9.2.1.13 “KPV” K PV( r , M )

K Depp ( r , M )

2

K Depp ( r , M )

K PV( r , M )

(4.110)

9.2.1.14 “TL” TL

h m γγ

(4.196)

9.2.1.15 “ωg” M .c 2 .h

ω g( M )

2

(4.207)

9.2.1.16 “ngg” n gg ( M ) T L.ω g ( M )

(4.208)

9.2.1.17 “rω” 5

r ω ω Ω_3 , M

St G.

M

2

ω Ω_3

9

(4.212)

9.2.1.18 “aPV” a PV( r , M , t )

i .

C PV n PV, r , M .e

π .n PV .ω PV( 1 , r , M ) .t .i

n PV

(4.436)

9.2.1.19 “ag” a g ( r , M , φ, t )

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

190

φ

(4.439)

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9.2.1.20 “gav”

g av ( r , M )

2

1. T Ω ( r, M ) 2 .

T Ω ( r, M )

a g( r, M , 0, t ) d t

0 .( s )

(4.440)

9.2.2 Planck-Particles 9.2.2.1 “mx” mx

λx 2

(4.71)

4 . 2 6 π 3

(4.72)

9.2.2.2 “λx” λx

9.2.2.3 “ρm, ρS” . 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

(4.78)

ρS = ρm(λxλh,mxmh) 9.2.2.4 “r3, M3” r3 = λxλh

(4.245)

M3 = mxmh = λxmh / 2

(4.246)

9.2.3 SBH’s 9.2.3.1 “StBH” 9

St BH

c.

c .St G ( 2 .G)

5

(4.138)

9.2.3.2 “ωΩ_4” 3

St BH.

ω Ω_4 M BH

1 M BH

(4.139)

9.2.3.3 “rS” 3

r S R BH

2 λ x.λ h .R BH

3

r S M BH 3

r S R BH

191

(4.146)

3 .M BH 4 .π .ρ S

(4.148)

2 3 .c .R BH 8 .π .G.ρ S

(4.150) www.deltagroupengineering.com

9.2.3.4 “nΩ_4” n Ω_4 M BH

n Ω_2 r S M BH , M BH

(4.157)

n Ω_5 M BH

n Ω_2 R BH M BH , M BH

(4.158)

9.2.3.5 “nΩ_5”

9.2.3.6 “nBH” n BH M BH

n Ω_5 M BH n Ω_4 M BH

(4.159)

9.2.3.7 “ωΩ_5” ω Ω_5 M BH

ω Ω_3 r S M BH , M BH

(4.161)

9.2.3.8 “ωBH” ω BH M BH

ω Ω_5 M BH ω Ω_4 M BH

(4.162)

9.2.3.9 “ωΩ_6” ω Ω_5 M BH

ω Ω_6 M BH

n Ω_4 M BH

(4.166)

9.2.3.10 “ωΩ_7” ω Ω_4 M BH

ω Ω_7 M BH

n Ω_5 M BH

(4.167)

9.2.3.11 “ωPV_1” ω Ω_6 M BH

ω PV_1 M BH

ω Ω_7 M BH

(4.168)

9.2.3.12 “ng” n g ω , M BH

E M BH E g( ω )

(4.177)

9.2.4 Cosmology 9.2.4.1 “r2, M2” r2(r) = Kλ⋅r

(4.247)

M2(M) = Km⋅M = Kλ⋅M

(4.248)

9.2.4.2 “λy” λ y r 2, M 2

1 ln n Ω_2 r 2 , M 2

192

(4.229)

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9.2.4.3 “KU” K U r 2, r 3, M 2, M 3

ln

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2 5

5

K U r 2, r 3, M 2, M 3

1

ln

2

9

(4.231)

7

.ln n Ω_2 r 2 , M 2

3.

M3

26

9

.

M2

r2

9

r3

(4.232)

9.2.4.4 “AU” A U r 2, r 3, M 2, M 3

TL K U r 2, r 3, M 2, M 3

5

(4.233)

9.2.4.5 “RU” R U r 2, r 3 , M 2, M 3

c .A U r 2 , r 3 , M 2 , M 3

(4.234)

9.2.4.6 “HU, HU2, HU5, |H|” H U r 2, r 3, M 2, M 3

H U5( r , M )

1 . ln TL

( 3 .π )

µ

(4.235)

H U K λ .r , λ x.λ h , K m.M , m x.m h

H U2( r , M ) 7 .µ .

1 A U r 2, r 3, M 2, M 3

2

32

256

. µ m .ln ( 3 π ) . h 4 M

H

µ

(4.276)

7 .µ

2

. r λh

2 7 .µ

5

.

mh M

d H dt

5 .µ

2

. r λh

2 26 .µ

(4.529)

(4.378)

9.2.4.7 “Hα” H α r 3, M 3

2.

2. . . π G ρ m r 3, M 3 3

(4.237)

Hα(r3,M3) = ωh / λx H α λ x.λ h , m x.m h

ωh λx

(4.249)

9.2.4.8 “ρU, ρU2” ρ U r 2, r 3, M 2, M 3

ρ U2( r , M )

193

3 .H U r 2 , r 3 , M 2 , M 3 8 .π .G 3 .H U2( r , M ) 8 .π .G

2

(4.238)

2

(4.304)

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9.2.4.9 “MU” M U r 2, r 3, M 2, M 3

V R U r 2 , r 3 , M 2 , M 3 .ρ U r 2 , r 3 , M 2 , M 3

(4.239)

9.2.4.10 “KT” K T r 2, r 3, M 2, M 3

H α r 3, M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

H U r 2, r 3 , M 2, M 3

8 . H α r 3, M 3 ln 3 H

K T( H )

(4.240)

(4.268)

9.2.4.11 “TW” T W r 2, r 3, M 2 , M 3

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3

(4.241)

KW

T W( H) λ Ω_3

c λ x. , mh H 2

(4.269)

9.2.4.12 “StT” 9

2

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

(4.274)

9.2.4.13 “TU, TU2, TU3, TU4, TU5” T U r 2, r 3, M 2, M 3

K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3 ωh

K W .St T .ln

T U2( H )

T U3 H β

9

. H5

λ x.H

K W .St T .ln

(4.275)

1

T U5( r , M )

c

µ

.ln

. . 4µ H U5( r , M ) λ h Hα

Hβ

2 .µ

.

5 .µ

. H .H β α

2

(4.318)

1 T U4( t ) K W .St T .ln H α .t . t

KW

(4.242)

5 .µ

2

(4.331) 2 .µ

1 π .H α

2

. 2

.H ( r , M ) 5 µ U5

(4.530)

9.2.4.14 “dTdt, dT2dt2, dT3dt3” dT dt ( t )

K W .St T .

2 5 .ln H α .t .µ

t

dT2 dt2 ( t )

K W .St T .

5 .µ

2

.t

2 2 5 .µ . ln H α .t . 5 .µ

t

194

5 .µ

1

2

.t2

(4.335) 1

2

1

(4.339) www.deltagroupengineering.com

dT3 dt3 ( t )

K W .St T .

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ

t

3

5 .µ

2

2 2 15.µ . 5 .µ

2 .t3

2

2

(4.343)

9.2.4.15 “dHdt, dH2dt2” 1 H γ .H α

t

Hγ Hβ

η

(4.376)

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ . 5µ Hγ

dH dt H γ

dH2 dt2 H γ

(4.359)

1

(4.361)

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ 5 .µ Hγ

1

2

1

(4.371)

9.2.4.16 “t1, t2, t3, t4, t5” 1

t1

e

5 .µ

2

. 1 Hα

10 .µ

t2

e

2

(4.334) 1

2 2 5 .µ . 5 .µ

1

. 1 Hα

2 2 15 .µ . 5 .µ

t3

e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

(4.338) 2

3

2

. 1 Hα

(4.342)

1

t4

e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ

t5

e

. 1 Hα 4

2 2 2 5 .µ . 5 .µ . 5 .µ

(4.366)

2 1

2

. 1 Hα

(4.375)

9.2.5 ZPF 9.2.5.1 “Ω EGM” Ω EGM

ρ U2 r x5.R o , m g5 .M G ρ U2 R o , M G r x5 m g5

=

(4.308)

1.013403 1.052361

195

(4.298)

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9.2.5.2 “Ω ZPF” Ω ZPF

Ω EGM

1

(4.313)

9.2.5.3 “UZPF” 3 .c . H U2 R o , M G Ω ZPF . 8 .π .G 2

U ZPF

2

(4.315)

9.2.6 EGM Construct limits 9.2.6.1 “ML” ML

R EGM

K m.M G.

R EGM

5 5

.

K λ .R o

R EGM K λ .R o

(4.409)

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.410)

9.2.6.2 “rL” rL

R BH M L

(4.411)

9.2.6.3 “tL” tL

rL c

(4.412)

9.2.6.4 “ML / rL = MEGM / REGM = tL / tEGM” M L M EGM rL M EGM t EGM

tL

R EGM t EGM

(4.415)

M U K λ .R o , λ x.λ h , K m.M G, m x.m h A U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.413) (4.414)

9.2.7 Particle-Physics 9.2.7.1 “mγγ2” m γγ2

h tL

(4.446)

2 .m γγ2

(4.447)

9.2.7.2 “mgg2” m gg2

9.2.7.3 “rγγ2” 5

r γγ2

r e.

m γγ2 m e .c

196

2

2

(4.451)

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9.2.7.4 “rgg2” r gg2

4 .r γγ2

5

(4.452)

9.2.7.5 “Nγ” E Ω ( r, M )

N γ( r, M )

mγ

(4.457)

9.2.7.6 “Qγ” Qe

Q γ( r, M )

N γ( r, M )

(4.458)

9.2.7.7 “Qγγ” Q γ( r, M )

Q γγ ( r , M )

N γ( r, M ) Q γ( r, M )

Q γγ ( r , M )

Q γγ

9.2.7.8

Qγ

(4.462)

2

Qe

(4.463)

2

Qe

(4.464)

“Qγγ2” Q γγ2

Q γγ m γγ

.m γγ2

(4.470)

9.2.7.9 “tL / TL = mγγ / mγγ2 = Qγγ / Qγγ2” tL

m γγ

Q γγ

T L m γγ2 Q γγ2

(4.468)

9.2.8 Other useful relationships mγ

2

E Ω r ε,me

ω Ω r e, m e .m γγ ω Ω r ε, m e

E Ω r e, m e

ω Ω r e, m e

197

mγ

(4.474)

2

m γγ mγ

(4.476) 2

h .m γγ

(4.478)

www.deltagroupengineering.com

NOTES

198

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APPENDIX 2.A Note: “Quinta Essentia – Part 2” (QE2) is a summary of “Quinta Essentia – Part 3, 4” (QE3,4). Subsequently, the calculation engine in QE2 utilises QE3,4 as its foundational construct. Please consult QE3,4 as required. Quinta Essentia – Part 2 •

MathCad 8 Professional: calculation engine

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED •

Computational environment • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated.

The following denotes output and data from QE4, serving as inputs (i.e. where required) for the QE2 calculation engine, T0

T U2 H U2 R o , M G

= 2.724752 ( K )

m g1

•

=

0.989364

r x5

1.057292

m g5

=

2.724 = 2.725 ( K )

T0 T0

r x1

∆T 0

∆T 0

2.726

1.013403 1.052361

QE2 calculation engine

Given T U2 H U2 r x1.R o , m g1 .M G r x6 m g6

T0

∆T 0

T0

∆T 0

Find r x1, m g1

Given T U2 H U2 r x1.R o , m g1 .M G r x7 m g7

Find r x1, m g1

Ω EGM r x0, m g0

ρ U2 r x0.R o , m g0 .M G ρ U2 R o , M G

Ω ZPF r x0, m g0

199

1

Ω EGM r x0, m g0

www.deltagroupengineering.com

r x6

7.996943

MG

R o . r x5 = 8.107221 ( kpc ) 8.218926 r x7

MS

m g6

. 11 6.33113310

. m g5 = 6.31416710 . 11 m g7

. 11 6.29698210

Ω EGM r x6, m g6

0.998993

Ω ZPF r x6, m g6

Ω EGM r x5, m g5

= 1.000331

Ω ZPF r x5, m g5

Ω EGM r x7, m g7

1.001671

. 1.00690410

3

. 3.31400710

4

Ω ZPF r x7, m g7

. 1.67100610

3

q SM_1 r x0, m g0

q 0 r x0, m g0

=

For “ΩZPF → –q0 → qSM_1”, q 0 r x0, m g0

Ω ZPF r x0, m g0

q 0 r x6, m g6

. 1.00690410

3

q SM_1 r x6, m g6

q 0 r x5, m g5

= 3.31400710 .

4

q SM_1 r x5, m g5

q 0 r x7, m g7

. 1.67100610

3

q SM_1 r x7, m g7

. 1.00690410 =

3

. 3.31400710

4

. 1.67100610

3

For “Λ0”, Λ 0 r x0, m g0 Λ 0 r x6, m g6

2 3 3 .H U2 R o , M G . .Ω EGM r x0, m g0 2

Λ 0 r x6, m g6

. 3 6.73006510

Λ 0 r x5, m g5

= 6.75716710 . 3

Λ 0 r x7, m g7

. 3 6.78429610

1

km . s Mpc

2

1. Λ 0 2 c Λ0

7.864602

r x5, m g5

= 7.896273

r x7, m g7

7.927975

10

47 2

km

1

Λ 0 r x6, m g6 Λ 0 r x5, m g5

Λ 0 r x6, m g6

82.036971 = 82.20199 82.366838

11.919176 1

km s .Mpc

Λ 0 r x5, m g5 1

Λ 0 r x7, m g7

= 11.895249

9 10 .yr

11.871442

Λ 0 r x7, m g7 1

Λ 0 r x6, m g6

2.656709 1

1 H U2 R o , M G

Λ 0 r x5, m g5 1

= 2.680636

9 10 .yr

2.704443

Λ 0 r x7, m g7 1

Λ 0 r x6, m g6 1 H0

1.852839 1

Λ 0 r x5, m g5 1

= 1.876767

9 10 .yr

1.900574

Λ 0 r x7, m g7

200

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1

1

Λ 0 r x6, m g6

Λ 0 r x6, m g6 1

1 H0

Λ 0 r x5, m g5

0.803869 1

1 H U2 R o , M G

=

Λ 0 r x5, m g5

1

9 10 .yr

0.803869

1

Λ 0 r x7, m g7

0.803869

Λ 0 r x7, m g7 1

Λ 0 r x6, m g6

0.817733 1

H U2 R o , M G .

Λ 0 r x5, m g5 1

= 0.816091 0.814458

Λ 0 r x7, m g7

For “UΛ, UZPF, Uλ”, 2

U Λ r x0, m g0

c . Λ 0 r x0, m g0 8 .π .G

U Λ r x6, m g6

3.787219

U Λ r x5, m g5

= 3.802471

U Λ r x7, m g7

3.817737

3 .c . Ω ZPF r x0, m g0 . H U2 R o , M G 8 .π .G 2

U ZPF r x0, m g0

U λ r x0, m g0

U Λ r x0, m g0

U ZPF r x0, m g0

10

10 .

Pa

U ZPF r x6, m g6 2

7.649839

U ZPF r x5, m g5

=

2.51778

10

13 .

Pa

12.695284

U ZPF r x7, m g7

U λ r x6, m g6

3.77957

U λ r x5, m g5

= 3.804989

U λ r x7, m g7

3.830432

10

10 .

Pa

U λ r x6, m g6 U ZPF r x6, m g6 U λ r x5, m g5 U ZPF r x5, m g5

494.071804 = 1.51124710 . 3

1

301.720886

U λ r x7, m g7

T U2 H U2 R o , M G 2 .∆T

T0

∆T 0

= 62.376832 ( % )

0

U ZPF r x7, m g7

For “q → qSM_2 ≈ ½”,

Λ ZPF r x0, m g0

8 .π .G . U ZPF r x0, m g0 2 c

Λ ZPF r x6, m g6 Λ ZPF r x5, m g5 Λ ZPF r x7, m g7

201

13.594118 =

4.474212 22.560109

km . s Mpc

2

www.deltagroupengineering.com

Λ ZPF r x6, m g6 1. Λ ZPF 2 c Λ ZPF

Λ ZPF r x6, m g6

1.588578

r x5, m g5

= 0.522846

r x7, m g7

2.636323

10

49

Λ ZPF r x5, m g5

2

km

3.68702i = 2.115233 4.749748

km s .Mpc

Λ ZPF r x7, m g7 1

Λ ZPF r x6, m g6

265.20416i 1

Λ ZPF r x5, m g5 1

9 10 .yr

= 462.27198 205.866299

Λ ZPF r x7, m g7 1

Λ ZPF r x6, m g6

14.575885 265.20416i 1

1

=

Λ ZPF r x5, m g5

H U2 R o , M G

447.696095

9 10 .yr

191.290414

1

Λ ZPF r x7, m g7

q SM_2 r x0, m g0

Ω EGM r x0, m g0

Λ ZPF r x0, m g0

2

3 .H U2 R o , M G

2

q SM_2 r x6, m g6

0.500503

q SM_2 r x5, m g5

= 0.499834

q SM_2 r x7, m g7

0.499164

For “q → qSM_1 → ΩZPF”, Let:

η q1 η q2 η q3

η .( 1 1 1 )

Given

1 q SM_1 r x5, m g5

1

.dH dt

H U2 R o , M G

η q1

Hα

1

H U2 R o , M G

1 q SM_1 r x6, m g6

1

.dH dt

H U2 R o , M G

η q2

Hα

1

H U2 R o , M G

1 q SM_1 r x7, m g7

1

.dH dt

H U2 R o , M G

H U2 R o , M G

Hα

η q3

1

202

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η q1

η q2

η q2

η q1 = 4.595344 4.595322 η q3

Find η q1 , η q2 , η q3

η q3 η . η q2

1

dH dt

dH dt

dH dt

η q1

1

η q3

1

H U2 R o , M G

1=

. 3.57192910

4

4.595365

. 1.18155910

4

. 5.9333510

4

(%)

η q2

Hα H U2 R o , M G

η q1

67.050522 = 67.095419

Hα H U2 R o , M G

67.14033

km . s Mpc

η q3

Hα 1

dH dt

H U2 R o , M G

η q2

Hα 1

dH dt

H U2 R o , M G

η q1

14.583229 = 14.57347

Hα

9 10 .yr

14.563722 1

dH dt

H U2 R o , M G Hα

dH dt

1 H U2 R o , M G

η q3

.

dH dt

dH dt

H U2 R o , M G

η q2

Hα H U2 R o , M G

η q1

Hα H U2 R o , M G

0.050358 1 = 0.016569 ( % ) 0.083515

η q3

Hα

203

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T U2

T U2

T U2

dH dt

dH dt

dH dt

H U2 R o , M G

η q2

Hα H U2 R o , M G

η q1

2.724 = 2.725 ( K )

Hα

2.726

H U2 R o , M G

η q3

Hα

For “q → qSM_2 ≈ ½”, Let:

η q4 η q5 η q6

η .( 1 1 1 )

Given

1 q SM_2 r x5, m g5

1

.dH dt

H U2 R o , M G

η q4

Hα

1

H U2 R o , M G

1 q SM_2 r x6, m g6

1

.dH dt

H U2 R o , M G

η q5

Hα

1

H U2 R o , M G

1 q SM_2 r x7, m g7

1

.dH dt

H U2 R o , M G

η q6

Hα

1

H U2 R o , M G η q4

η q5

η q5

η q4 = 4.606653 4.606632 η q6

Find η q4 , η q5 , η q6

η q6

dH dt

dH dt

dH dt

H U2 R o , M G

4.606675

η q5

Hα H U2 R o , M G

η q4

Hα H U2 R o , M G

47.411877 = 47.443626 47.475383

km . s Mpc

η q6

Hα

204

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1

dH dt

H U2 R o , M G

η q5

Hα 1

dH dt

H U2 R o , M G

η q4

20.623801 9 10 .yr

= 20.609999

Hα

20.596213 1

dH dt

H U2 R o , M G Hα

1

.

H U2 R o , M G

Hα

T U2

T U2

η q4

H U2 R o , M G

dH dt

29.324934 1=

Hα

29.277606 ( % ) 29.230268

η q6

H U2 R o , M G

dH dt

Hα

H U2 R o , M G

dH dt

η q5

H U2 R o , M G

dH dt

T U2

η q6

η q5

Hα H U2 R o , M G

dH dt

η q4

2.252547 = 2.253374 ( K )

Hα

2.254201

H U2 R o , M G

dH dt

η q6

Hα

For “q → qSM_2 ≈ -½”, Let:

η q7 η q8 η q9

η .( 1 1 1 )

Given

1 q SM_2 r x5, m g5

1

.dH dt

H U2 R o , M G

H U2 R o , M G Hα

η q7

1

205

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1 q SM_2 r x6, m g6

1

H U2 R o , M G

.dH dt

η q8

Hα

1

H U2 R o , M G

1 q SM_2 r x7, m g7

1

H U2 R o , M G

.dH dt

η q9

Hα

1

H U2 R o , M G

η q7

η q8

η q8

η q7 = 4.588735 4.588742 η q9

Find η q7 , η q8 , η q9

η q9

dH dt

dH dt

dH dt

H U2 R o , M G

4.588728

η q8

Hα H U2 R o , M G

η q7

82.174949 = 82.15662

Hα H U2 R o , M G

82.138273

km . s Mpc

η q9

Hα 1

dH dt

H U2 R o , M G

η q8

Hα 1

dH dt

H U2 R o , M G

η q7

11.899163 = 11.901818

Hα

9 10 .yr

11.904476 1

dH dt

H U2 R o , M G Hα

dH dt

1 H U2 R o , M G

η q9

.

dH dt

dH dt

H U2 R o , M G

η q8

Hα H U2 R o , M G

η q7

Hα H U2 R o , M G

22.495045 1 = 22.467722 ( % ) 22.440373

η q9

Hα

206

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T U2

T U2

T U2

T U2

T U2

T U2

dH dt

dH dt

dH dt

dH dt

dH dt

dH dt

H U2 R o , M G

η q8

Hα H U2 R o , M G

3.045402

η q7

= 3.045029 ( K )

Hα

3.044656

H U2 R o , M G

η q9

Hα

H U2 R o , M G

η q8

T U2

Hα H U2 R o , M G

dH dt

T U2

H U2 R o , M G

dH dt

T U2

0.792854

η q4

= 0.791655 ( K )

Hα

η q9

Hα

η q5

Hα

η q7

Hα H U2 R o , M G

H U2 R o , M G

0.790455

H U2 R o , M G

dH dt

η q6

Hα

Checking calculations, 1

H q

1

. d H dt

q

Ω

Λ

2

3 .H

H

2

1 Ω

Λ

2

3 .H

1

. d H dt

2

For “q → qSM_1 → ΩZPF”, Let:

η q10 η q11 η q12

η .( 1 1 1 )

Given

1 Ω EGM r x5, m g5

Λ 0 r x5, m g5

2

3 .H U2 R o , M G

.dH dt

H U2 R o , M G

η q10

Hα

1 2

1

H U2 R o , M G

1 Ω EGM r x6, m g6

Λ 0 r x6, m g7

2

3 .H U2 R o , M G

.dH dt

H U2 R o , M G

η q11

Hα

1 2

1

H U2 R o , M G

207

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1 Ω EGM r x7, m g7

Λ 0 r x7, m g7

2

3 .H U2 R o , M G

.dH dt

H U2 R o , M G

η q12

Hα

1 2

1

H U2 R o , M G

η q10

η q10

η q11

η q11 = 4.595363 4.595322 η q12

Find η q10 , η q11 , η q12

η q12

dH dt

dH dt

dH dt

H U2 R o , M G

4.595344

η q11

Hα H U2 R o , M G

67.055441

η q10

= 67.095419

Hα H U2 R o , M G

67.14033

km . s Mpc

η q12

Hα 1

dH dt

H U2 R o , M G

η q11

Hα 1

dH dt

H U2 R o , M G

η q10

14.582159 = 14.57347

Hα

9 10 .yr

14.563722 1

dH dt

H U2 R o , M G Hα

dH dt

1 H U2 R o , M G

η q12

.

dH dt

dH dt

H U2 R o , M G

η q11

Hα H U2 R o , M G

η q10

Hα H U2 R o , M G

0.043026 1 = 0.016569 ( % ) 0.083515

η q12

Hα

208

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T U2

T U2

T U2

dH dt

dH dt

dH dt

H U2 R o , M G

η q11

Hα H U2 R o , M G

η q10

2.72411 =

2.725

Hα

( K)

2.726

H U2 R o , M G

η q12

Hα

For “q → qSM_2 ≈ ½”, Let:

η q13 η q14 η q15

η .( 1 1 1 )

1 Ω EGM r x5, m g5 2

.dH dt

Λ ZPF r x5, m g5 3 .H U2 R o , M G

H U2 R o , M G

η q13

Hα

1 2

1

H U2 R o , M G

1 Ω EGM r x6, m g6 2

.dH dt

Λ ZPF r x6, m g7 3 .H U2 R o , M G

H U2 R o , M G

η q14

Hα

1 2

1

H U2 R o , M G

1 Ω EGM r x7, m g7 2

.dH dt

Λ ZPF r x7, m g7 3 .H U2 R o , M G

H U2 R o , M G Hα

1 2

1

H U2 R o , M G η q13

η q14

η q14

η q13 = 4.606653 4.606632 η q15

η q15

Find η q13 , η q14 , η q15

η q15

4.606675

209

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H U2 R o , M G

dH dt

η q14

Hα H U2 R o , M G

dH dt

η q13

47.411877 = 47.443626

Hα

47.475383

H U2 R o , M G

dH dt

km s .Mpc

η q15

Hα 1

dH dt

H U2 R o , M G

η q14

Hα 1

dH dt

H U2 R o , M G

η q13

20.623801 9 10 .yr

= 20.609999

Hα

20.596213 1

dH dt

H U2 R o , M G Hα

dH dt

1 H U2 R o , M G

.

dH dt

dH dt

T U2

T U2

T U2

dH dt

dH dt

dH dt

η q15

H U2 R o , M G

η q14

Hα H U2 R o , M G

η q13

29.324934 1=

Hα H U2 R o , M G

H U2 R o , M G

29.277606 ( % ) 29.230268

η q15

Hα η q14

Hα H U2 R o , M G

η q13

Hα H U2 R o , M G

2.252547 = 2.253374 ( K ) 2.254201

η q15

Hα

210

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For “q → qSM_2 ≈ -½”, Let:

η q16 η q17 η q18

η .( 1 1 1 )

1 Ω EGM r x5, m g5

Λ ZPF r x5, m g5

2

3 .H U2 R o , M G

H U2 R o , M G

.dH dt

η q16

Hα

1 2

1

H U2 R o , M G

1 Ω EGM r x6, m g6

Λ ZPF r x6, m g7

2

3 .H U2 R o , M G

H U2 R o , M G

.dH dt

η q17

Hα

1 2

1

H U2 R o , M G

1 Ω EGM r x7, m g7

Λ ZPF r x7, m g7

2

3 .H U2 R o , M G

H U2 R o , M G

.dH dt

Hα

1 2

1

H U2 R o , M G

η q16 η q17

η q16 Find η q16 , η q17 , η q18

η q18

dH dt

dH dt

dH dt

H U2 R o , M G

η q18

4.588735

η q17 = 4.588728 4.588742 η q18

η q17

Hα H U2 R o , M G

η q16

Hα H U2 R o , M G

82.174949 = 82.15662 82.138273

km s .Mpc

η q18

Hα

211

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1

dH dt

H U2 R o , M G

η q17

Hα 1

dH dt

H U2 R o , M G

η q16

11.899163 9 10 .yr

= 11.901818

Hα

11.904476 1

dH dt

H U2 R o , M G Hα

dH dt

1

.

H U2 R o , M G

dH dt

dH dt

T U2

T U2

T U2

η q18

dH dt

dH dt

dH dt

H U2 R o , M G

η q17

Hα H U2 R o , M G

η q16

22.495045 1 = 22.467722 ( % )

Hα

22.440373

H U2 R o , M G

η q18

Hα

H U2 R o , M G

η q17

Hα H U2 R o , M G

η q16

3.045402 = 3.045029 ( K )

Hα

3.044656

H U2 R o , M G

η q18

Hα

For “Λ0 = ΛZPF = 0”, Let:

η q19 η q20 η q21

1 Ω EGM r x5, m g5

.dH dt

η .( 1 1 1 )

H U2 R o , M G

1

2 H U2 R o , M G

η q19

Hα 1

212

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1 Ω EGM r x6, m g6

.dH dt

η q20

H U2 R o , M G Hα

1

2

1

H U2 R o , M G

1 Ω EGM r x7, m g7

.dH dt

η q21

H U2 R o , M G Hα

1

2

1

H U2 R o , M G

η q19

η q19

η q20

η q20 = 4.606642 4.606686 η q21

Find η q19 , η q20 , η q21

η q21

dH dt

dH dt

dH dt

H U2 R o , M G

4.606664

η q20

Hα H U2 R o , M G

η q19

47.459642 = 47.427906

Hα H U2 R o , M G

47.396115

km s .Mpc

η q21

Hα 1

dH dt

H U2 R o , M G

η q20

Hα 1

dH dt

H U2 R o , M G

η q19

20.603044 = 20.616831

Hα

9 10 .yr

20.630659 1

dH dt

H U2 R o , M G

η q21

Hα

213

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dH dt

1 H U2 R o , M G

.

dH dt

dH dt

T U2

T U2

T U2

dH dt

dH dt

dH dt

H U2 R o , M G

η q20

Hα H U2 R o , M G

η q19

29.253731 1=

Hα H U2 R o , M G

H U2 R o , M G

29.30104 ( % ) 29.348429

η q21

Hα η q20

Hα H U2 R o , M G

η q19

Hα H U2 R o , M G

2.253791 = 2.252965 ( K ) 2.252137

η q21

Hα

NOTES

214

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APPENDIX 2.B The Electro-Gravi-Magnetic (EGM) harmonic representation of fundamental particles [i.e. Eq. (3.230)] may be generalised to incorporate any meaningful physical quantity199 in relation to a reference particle; represented by “r2, M2, S2, Q2” as follows200, M1 M2

2

.

r2

ψ

5

St ω

r1

9

(3.230)

→

KΠ

.

St ι ( ψ )

Π

Π

ψ

St ω

(ψ

1)

(2.48)

ψ=1

Let “ψ = 4”; substituting appropriately yields Eq. (2.49) [below] according to, substitute , St ι ( 1 ) ψ KΠ.

St ι ( ψ ) ψ=1

Π

ψ

Π

St ω

(ψ

1)

substitute , St ι ( 2 ) substitute , St ι ( 3 ) substitute , St ι ( 4 )

M1 M2 r1 r2 S1

Π

KΠ.

M1 M2

Π

1

.

r1 r2

Π

2

.

S1 S2

Π

3

.

Q1 Q2

4

Π

St ω

5

S2 Q1 Q2

where, • “KΠ → KΠ(r1,2,M1,2,S1,2,Q1,2,Π1,2,3,4,5)” denotes the appropriate Experimental Relationship Function (ERF). • “M1,2” denotes particle rest-mass. • “r1,2” denotes Zero-Point-Field (ZPF) equilibrium radius [e.g. Root-Mean-Square (RMS) charge radius]. • “S1,2” denotes Spin Angular Momentum (SAM). • “Q1,2” denotes Electric charge. • Possible solution → “Π1 = 2, Π2 = -5, Π5 = 9”. • Requiring solution or specification → “Π3,4”. Therefore, Eq. (2.49) may be utilised to articulate a single representation of any fundamental particle incorporating mass, size, spin and charge, in relation to an arbitrarily selected reference (i.e. base) particle. Note: a worked example of Eq. (2.49)201 has been deliberately omitted and is proposed as a simple exercise for the reader (hint: a family of representations exist).

199

Such as spin angular momentum or Electric charge in order to completely describe it. Shown on the front cover of QE3. 201 Solving for “KΠ, Π3,4”. 200

215

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NOTES

216

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APPENDIX 2.C Charge If we were compelled to associate the production of Photons with one specific particle, what would it be? The answer to this question is obvious (i.e. the Electron) as we see evidence of Photonic production by Electronic means in our everyday life in the form of television, radio and the internet. However, the Photon itself possesses charge according to the PDG202. Thus, it seems “reasonable” to expect that the particle best associated with the production of Photons, may have its own charge properties expressed in terms of Photon populations. The EGM construct treats all matter as radiators of populations of conjugate Photon pairs. Subsequently, quantifying this characteristic in terms of Electron charge is a natural conclusion. Moreover, the Electron itself is a practical measure of charge and the successful quantification of it implicitly produces all charge multiples (i.e. the Proton etc.). In other words, if one can derive the magnitude of Electron charge, one has derived the magnitudes of all multiples of charge utilising a single paradigm. The Photon charge threshold “Qγ” and upper limit “Qγγ” is derived in QE4. The derived “Qγ” 203 result compares favourably to the PDG value (i.e. to within a factor of “1.884”), leading to the derivation of “Qγγ”. It is possible to derive “Qγγ” again herein (or alternatively, the Electric Charge “Qe”) in a slightly different manner; demonstrating that Electronic charge may be represented as the sum of a population of Photons, each Photon carrying a charge of “Qγγ”. To facilitate this, let the Photon charge per unit mass-energy204 (based upon the Electron) be given by “Qmγγ” according to, Q mγγ

Q γγ m γγ

(2.50)

Q mγγ = 3.534774 10

34 .

C eV

(2.51)

Hence, the Photon charge205 at the Electron harmonic cut-off frequency “QΩ” may be written as follows, QΩ

Q mγγ .E Ω r ε , m e

Q Ω = 7.650948 10

(2.52)

21 .

C

(2.53)

where, “EΩ(rε,me)” denotes the propagation energy of a Photon at the Electron harmonic cut-off frequency. Moreover, recognizing that: Qe

1.

= 20.94089

QΩ

(2.54)

2

ln 2 .n Ω r e , m e

γ

= 20.947273

(2.55)

Provokes the solution, Q e Σ C.Q Ω

(2.56)

ΣC

1. 2

ln 2 .n Ω r e , m e

γ

(2.57)

202

“QPDG < 5 x10-30Qe”: http://pdg.lbl.gov/, W.-M. Yao et al., Journal of Physics G 33, 1 (2006) and 2007 partial update for edition 2008. 203 “Qγ < 2.7 x10-30Qe”. 204 “mγγ” denotes the rest mass-energy of a Photon (see: QE3). 205 The “Pauli Exclusion Principle” does not apply to Photons. Therefore, no distinction may be drawn between a single Photon and a population of coherent Photons propagating at a specific frequency. 217

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Evaluating yields, Σ C .Q Ω = 1.602665 10

19 .

C

(2.58)

Comparing the preceding result to “Qe” produces a derived Electronic charge to within “0.031(%)” of the NIST value as follows, Σ C.Q Ω

1 = 0.030484 ( % )

Qe

(2.59)

Hence; utilising “Qe”, “Qγγ” may be written in agreement with the derivation in QE4206 as follows, Q γγ

Q e .m γγ Σ C.E Ω r ε , m e

Q e .m γγ Σ C.E Ω r ε , m e

= 1.12905 10

(2.60) 78 .

C

(2.61)

The preceding equations demonstrate that Electronic charge may be characterised by the proportional sum of Photonic charges within its radiated wavefunction spectrum propagating at the harmonic cut-off frequency “ωΩ(rε,me)”. The error associated with “Qe” utilising the QE4 approximation of “Qγγ” is quasi-negligible, if the definition of “Qγγ” above is utilised [i.e. Eq. (2.60)], the error is zero. Momentum •

Construct

The radiated wavefunction spectrum is composed of populations of conjugate Photon pairs described by a Fourier distribution producing a constant magnitude function as the number of harmonic modes approaches infinity. A Fourier distribution involves the hybridisation of two spectra [i.e. amplitude207 and frequency208 (see: QE3)] which combine to produce an energetically efficient equivalent representation of gravitational acceleration “g” (see: QE3 for the complete derivation). The reason EGM utilises a Fourier distribution in its formulation, as opposed to any other distribution, is because various other combinations of amplitude and frequency spectra do not harmonically describe “g” at a mathematical point as energetically efficiently as a Fourier distribution (i.e. if the composite wavefunctions are considered to represent Photons). Naturally, one expects that the momentum of the spectrum of radiated conjugate Photon pairs propagating at “c” be conserved. To confirm this expectation, we shall determine the average spectral momentum for comparison to an analogous Newtonian system. If the two results coincide, we have confirmed that the EGM construct is appropriately and consistently formulated. Firstly, considering the momentum of a Photon at the harmonic cut-off frequency “Pγγ” yields, P γγ ( r , M )

h. ω Ω ( r, M ) c

(2.62)

Since “ωΩ” is formulated in accordance with an arithmetic sequence, the average spectral frequency and period is given by “ωav” and “Tav” respectively according to, 206

“Qγγ ≤ 7.05 x10-60Qe”. Spectral amplitude decreases as “nPV” increases. 208 Spectral frequency increases as “nPV” increases. 207

218

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ω av ( r , M )

ω Ω ( r, M ) 2

T av ( r , M )

(2.63)

1 ω av ( r , M )

(2.64)

Hence, the momentum of a Photon at the average spectral frequency is given by “Pγγ_ω_av” such that, h. ω av ( r , M ) c

P γγ _ω_av ( r , M )

(2.65)

The EGM construct describes gravitational fields as a double sided harmonic spectrum in accordance with a Fourier distribution, asserting that all starving matter contains a population of “nγγ” Photons and exists for a minimum gravitational lifetime of “TL” as derived in QE4. Subsequently, Photon emission rates (i.e. per second) per side of a double sided modal spectrum is given by “nγγ_av_s” according to, n γγ _av_s ( M )

n γγ ( M ) 2 .T L

(2.66)

Hence, the average Photon emission per average period “nγγ_av_T” becomes, n γγ _av_T ( r , M )

n γγ _av_s ( M ) .T av ( r , M )

(2.67)

Consequently, the average momentum of each Photon over the average period “Pγγ_av_T” is given by, P γγ _av_T ( r , M )

n γγ _av_T ( r , M ) .P γγ _ω_av ( r , M )

(2.68)

Subsequently, it follows that the analogous orbital angular momentum of a Photon “PO_γγ_av_T” utilising the preceding equation is, P O_γγ _av_T ( r , M )

r .P γγ _av_T ( r , M )

(2.69)

Applying the definition of a Graviton under the EGM construct, the analogous orbital angular momentum of a Graviton “PO_gg_av_T” may be stated as, P O_gg_av_T ( r , M )

2 .P O_γγ _av_T ( r , M )

(2.70)

Thus, by Newtonian analogy, “PO_gg_av_T” may be pictorially illustrated by two counter rotating half masses, representing the conjugate EGM wavefunction construct as follows,

0.5M

0.5M

r

Figure 2.3, The mechanical representation of the orbital angular momentum of a “½M” particle about a radial centre is given by “IO”, I O( v , r , M )

219

v .r .

M 2

(2.71) www.deltagroupengineering.com

where, the magnitude of the orbital angular momentum of the system depicted is “IS”, 2 . I O( v , r , M )

I S( v , r , M )

•

(2.72)

Sample calculation Performing sample calculations utilising “Up Quark” properties yields, P γγ r uq , m uq

P γγ _ω_av r uq , m uq

P gg r uq , m uq

P gg_ ω_av r uq , m uq

n γγ _av_T r uq , m uq

=

n gg_av_T r uq , m uq P γγ _av_T r uq , m uq

=

P gg_av_T r uq , m uq

P O_γγ _av_T r uq , m uq P O_gg_av_T r uq , m uq

=

. 8.0973210

14

. 14 4.0486610

. 1.61946410

13

. 14 8.0973210

. 2.31325910

8

. 4.62651910

8

0.93656

10

1.87312

=

( Ns )

(2.73)

(2.74) 21 .

Ns

(2.75)

0.719452

10

1.438905

39 .

Ns .m

(2.76)

Since the preceding pictorial representation is a non-physical Newtonian analogy, we may ignore relativistic effects and assume that “v → c” at “r” such that for the “Up Quark” and the Electron, I O c , r uq , m uq

=

I O c, r ε , m e

0.719452 1.612205

10

39 .

Ns .m

(2.77)

Hence; “PO_gg_av_T = IS” according to, 1

.

I S c , r uq , m uq 1

.

I S c, r ε , m e

•

P O_γγ _av_T r uq , m uq

=

P O_gg_av_T r uq , m uq

P O_γγ _av_T r ε , m e P O_gg_av_T r ε , m e

=

0.5 1

(2.78)

0.5 1

(2.79)

Computing errors Determining the Newtonian analogy error with respect to other fundamental particles yields, P O_gg_av_T r ε , m e

P O_gg_av_T r π , m p

I S c, r ε , m e

I S c, r π , m p

P O_gg_av_T r ν , m n

P O_gg_av_T r µ , m µ

I S c, r ν, m n

I S c, r µ, m µ

P O_gg_av_T r τ , m τ

P O_gg_av_T r en , m en

I S c, r τ , m τ

I S c , r en , m en

P O_gg_av_T r µn , m µn

P O_gg_av_T r τn , m τn

I S c , r µn , m µn

I S c , r τn , m τn

220

. 14 2.22044610 1=

. 2.22044610

14

. 2.22044610

14

0

. 14 1.11022310

. 1.11022310

14

. 14 1.11022310

. 4.44089210

14

(%)

(2.80)

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P O_gg_av_T r uq , m uq

P O_gg_av_T r dq , m dq

I S c , r uq , m uq

I S c , r dq , m dq

P O_gg_av_T r sq , m sq

P O_gg_av_T r cq , m cq

I S c , r sq , m sq

I S c , r cq , m cq

P O_gg_av_T r bq , m bq

P O_gg_av_T r tq , m tq

I S c , r bq , m bq

I S c , r tq , m tq

P O_gg_av_T r W , m W

P O_gg_av_T r Z , m Z

I S c, r W , m W

I S c, r Z, m Z

P O_gg_av_T r H , m H I S c , r H, m H

. 2.22044610 1=

14

0 . 2.22044610 . 2.22044610

14 14

0 . 2.22044610

14

. 4.44089210

14

. 2.22044610

14

(%)

(2.81) . 1 = 2.22044610

14

( %)

(2.82)

Therefore, based upon the preceding results; the radiated wavefunction spectrum may be represented analogously in terms of classical orbital mechanics at the ZPF equilibrium radius rotating at the speed of light. NOTES

221

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NOTES

222

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APPENDIX 2.D Note: “Quinta Essentia – Part 2” is a summary of “Quinta Essentia – Part 3, 4”. Subsequently, the simulation and calculation engine in “Part 4” is a natural extension of “Part 3”, utilising it as a foundational construct. Hence, the calculation engine developed in “Part 3, 4” has been included (verbatim) herein for reference. Please consult “Part 3, 4” if required. Quinta Essentia – Part 3 •

MathCad 8 Professional: calculation engine a. 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. b. Units of measure (definitions)

Scale 1

10

Scale 2

10

3

3

10 6

10

6

10

9

9

10

10

12

10 12

10 15

10

15

10

18

10

( mm µm nm pm fm am zm ym )

18

10

21

10

10

10

24

24

Scale 1 .( m)

( mHz µHz nHz pHz fHz aHz zHz yHz ) ( mJ µJ nJ pJ fJ aJ zJ yJ )

21

Scale 1 .( Hz)

Scale 1 .( J ) Scale 1 .( W )

( mW µW nW pW fW aW zW yW ) ( mΩ µΩ nΩ pΩ fΩ aΩ zΩ yΩ ) ( mV µV nV pV fV aV zV yV )

Scale 1 .( ohm )

Scale 1 .( V)

( mPa µPa nPa pPa fPa aPa zPa yPa ) ( mT µT nT pT fT aT zT yT )

Scale 1 .( T )

( mNs µNs nNs pNs fNs aNs zNs yNs ) ( mN µN nN pN fN aN zN yN )

Scale 1 .( Pa )

Scale 1 .( Ns )

Scale 1 .( newton )

( mgs µgs ngs pgs fgs ags zgs ygs )

Scale 1 .( gauss )

223

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Scale 1 .( gm )

( mgm µgm ngm pgm fgm agm zgm ygm ) ( mSt µSt nSt pSt fSt aSt zSt ySt )

Scale 1

( kSt MSt GSt TSt PSt ESt ZSt YSt )

Scale 2

( kHz MHz GHz THz PHz EHz ZHz YHz)

Scale 2 .( Hz)

Scale 2 .( newton )

( kN MN GN TN PN EN ZN YN )

Scale 2 .( J )

( kJ MJ GJ TJ PJ EJ ZJ YJ )

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 )

( keV MeV GeV TeV PeV EeV ZeV YeV)

Scale 2 .( eV)

Ns newton .s

c. Constants (definitions)

G

ε0

α

. 6.674210

3

m

11 .

kg .s

. 8.85418781710

2

12 .

F

c

m 299792458. s

h

. 6.626069310

µ0

34 .

( J .s )

7 newton 4 .π .10 . 2 A

. eV 1.6021765310

19 .

( J)

m . 7.29735256810

3

. 1.6021765310

Qe

19 .

( C)

γ

0.5772156649015328

d. Fundamental particle characteristics (definitions or initialisation values) m e m p m n m µ m τ m AMC

. 31 1.6726217110 . 27 1.6749272810 . 27 1.883531410 . 28 3.1677710 . 27 1.6605388610 . 27 .( kg ) 9.109382610

λ Ce λ CP λ CN λ Cµ λ Cτ

ω Ce ω CP ω CN ω Cµ ω Cτ

h. 1 c

1

1

1

1

me mp mn mµ mτ 2 2 .π .c .

h

me mp mn mµ mτ

224

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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. 4 .10

m uq m dq m sq m cq m bq m tq

( 80.425 91.1876 114.4 ) .

mW mZ mH

GeV 0.13 1.35 4.7 179.4 . 2 c

2

( 2.817940325 0.875 0.85 ) .( fm)

re rp rn 0.85.10

3

GeV c

r xq

8 .10

3

16 .

( cm)

. 0.529177210810

r Bohr

10

.( nm ) 656.469624182052

λB

( m)

e. Planck characteristics (definitions) G.h

λh

c

3

h .c

mh

G.h

th

G

c

1

ωh

5

th

f. Astronomical statistics 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

200.R S

R RG

24 24 24 30 0.0735.10 5.977.10 1898.8.10 1.989.10 .( kg )

R BH

2 .G . M BH 2 c

M NS

1 .M S

R NS

M RG

4 .M S

R WD

4200.( km)

M WD

20.( km)

3 300.10 .M E

g. Other .10 M BH = 4.29379067958471

33

( kg )

mx

mp

rx

r Bohr

h. Arbitrary values for illustrational purposes ω

KR

1 .( Hz)

1

k

1

R max

X

4 10 .( km)

r

1

∆R max

RE

F 0( k )

1

K 0( ω , X )

1

R max 250

225

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i. 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

c.

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

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

1

U m( r , M )

Ω ( r, M )

12. 768 81.

c .U m( r , M )

∆ω ZPF( r , ∆r , M )

ω β r , ∆r , M , K R

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 )

3 .M .c . 4 .π 2

∆K C( r , ∆r , M )

2

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

∆v δr n PV, r , ∆r , M

∆v Ω ( r , ∆r , M )

U m( r , M )

∆v δr n Ω ( r , M ) , r , ∆r , M

∆U PV( r , ∆r , M ) .

4

4

µ0

ω Ω_ZPF( r , ∆r , M )

ε0

ω Ω_ZPF( r , ∆r , M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M )

∆ω δr( 1 , r , ∆r , M )

4

(r

1

∆r )

3

2 .c . ∆U PV( r , ∆r , M ) h

3

r

3

n Ω_ZPF( r , ∆r , M )

4 K R . ω Ω_ZPF( r , ∆r , M )

226

1

∆ω δr( 1 , r , ∆r , M )

4

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

∆ω δr( 1 , r , ∆r , M )

4

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ω β r , ∆r , M , K R

n β r , ∆r , M , K R

∆ω Ω ( r , ∆r , M )

ω Ω(r

∆ω S r , ∆r , M , K R

∆r , M )

∆ω Ω ( r , ∆r , M )

St γ ( r , ∆r , M )

ω Ω ( r, M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M ) .

St α ( r , ∆r , M )

∆n S r , ∆r , M , K R

ω PV( 1 , r , M )

ω Ce

µ0

n β r , ∆r , M , K R

∆ω ZPF( r , ∆r , M )

∆ω R( r , ∆r , M )

∆ω Ω ( r , ∆r , M )

ω β r , ∆r , M , K R

St β ( r , ∆r , M )

ε0

n Ω_ZPF( 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 )

j. 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 )

π .N X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

ω C( ∆r ) ω PV( 1 , r , M )

Σ HR( A , D , r , ∆r , M )

∆Λ ( r , ∆r , M )

ω C( ∆r )

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

N TR( A , D , r , ∆r , M )

F PP( r , ∆r )

c

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 )

π .h .c .A PP( r ) 4 480.∆r

F PV( 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 )

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 .A

2

N C( r , ∆r , M )

A PP( r ) .∆U PV( r , ∆r , M ) .

D

D

D. N T

N X( r , ∆r , M )

A

1

A PP( r )

N C( r , ∆r , M ) N X( r , ∆r , M )

2

.ln

4 .π .r

2

N X( r , ∆r , M )

4

N C( r , ∆r , M )

8 .π .G . ∆U PV( r , ∆r , M ) 2 3 .c

227

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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 )

3

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

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 ( r , ∆r , M ) N C( r , ∆r , M ) . . . 27 c M ∆r C

K P( r , ∆r , M )

1

ω PV( 1 , r , M )

1

∆Λ ( r , ∆r , M )

1

4

∆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 )

k. Fundamental particle equations

mγ

512.h .G.m e

.

2

c . π .r e

n Ω r e, m e ln 2 .n Ω r e , m e 5

2 .m γγ

m gg

φ γγ

r e.

r γγ

2.

φ gg

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 )

Nγ

2

Kλ

π

ω Ω ( r, M ) ω CP

EΩ

m γγ

mγ

1

Km

Kω

mγ Nγ

Kλ

ω Ω ( r, M )

St θ ( 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 4 27.ω h .ω Ce 4 32.π

.

ω CP

3

.

ω CP

5

1

.

rµ rτ

r ε.

1 . mµ 9 4 me

2 5

1 . mτ 9 6 me

1

5

rε

1 . me r π. 9 2 mp

2

ω CN

3

ω CN 5

1

5

2

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

Given 5

r ε r π.

1 . me 9 2 mp

2

228

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rε

α

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

2

m sq

9

St sq

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

1 ω Ω r uq , m uq

St tq

. 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

5

m cq

r sq r cq r bq

5

r uq .

1 m uq

. 2

St cq

9

m bq

2

5

r tq

St bq 5

m tq

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

.

229

9

2

St tq St uq

2

9

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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

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

5

1

r uq .

rZ

.

m uq

rH

2

9

.m 2 W

1 . 2 mZ 9 St Z

5

rµ

rL

rε

rτ

3

1 . 2 mH 9

St H

1.

r QB

9

r uq

m QB St ω , r QB

Let:

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

4. . 3 πr 3

V( r )

Q( r )

1 V( r )

Q ch ( r )

Q( r ) 3

r dr

5. rν 3

1

x

2

Given 2

x

ln( x) . 2

x x

KS

1 1 3

Find( x) 2 3 . π .r ν ( 1 x) .x3 . 2 8 1 x x

b1

2 3 .r ν

. 2

KS 2

x

KX

2 0.113. fm

1

230

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6 .b 1 .K X . x

2

rX

3 .b 1 . x

2

1

1

r νM

KS

2. 3

. e

3 5 2 π .r ν . x

rν

r πE

rν

r πM

r dr .

r dr

fm

r x .r

1. e 3 x

1

rν

1

2

r

ρ ch ( r )

rν.

rν

fm

1

K S.

KS

2

fm

2 ν

10.r ν

∞

rν

1

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 Find r νM , r πE, r πM

r πE r πM

r νM

r νM

r πE

r πE .( fm)

r πM

r πM

5 5

r ν2 r ν3 r ν5

λ A( r, M )

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

λ PV( 1 , r , M ) 2 .n Ω ( r , M )

Given

λ A K ω .r x, m AMC λB

rx

1

Find r x

231

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l. 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

rν rµ

2

1 .r ε . e α rν

3

rπ 2 0.69. fm

M Error

1 . 1. r νM rp 2

rν

0.848.( fm)

0.857.( fm) 1

r πE

rν

KS

rX

KX

178.( GeV) . 1.11022310

0

0.034635 . 7.38826910

Error Av

r πM

0.879.( fm)

3

3

0.809916

0.160717

0.321692

0.247475

0.130911

M Error

0, 1

M Error

2, 1

2

λ A K ω .r Bohr , m p λB

13

0.075074

1. M Error 0,0 12 + M Error 2, 0

rν

r πE

m tq .c

. 4.8425510

rε

1. π rπ

r νM

0 M Error =

rτ

(%)

M Error

0, 2

M Error

2, 2

M Error

1, 0

M Error

3,0

M Error

1,1

M Error

3, 1

M Error

1, 2

...

M Error

3, 2

Error Av = 0.149388 ( % )

232

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m. Particle summary matrix 3.2 2

rε

. c .e r e ω Ce

r π_1 r π_2

3

rπ r ν_1

5

r π_av

3

r π_2

∆r π

r π_av

r π_1

2 r ν_1

r ν_2

∆r ν

r ν_av

r ν_1

1 . r π_av

r π_Error

r π_2

r ν_Error

1 . r ν_av r ν_2

∆r π

rX KX

3 .b 1 . x

2

r π_Error

1

r π_1

r π_2

r ν_1

r ν_2

r π_av r ν_av

1

∆r π

∆K X

=

( 0.69 0.02) . fm

2

(%)

830.594743

826.944318

826.941624

830.648674

826.942971 . 1.34683810

( am) 3

2 0.005. fm

=

( 0.69 0.02) . fm

14

830.702606

0.053931

2 0.69. fm

1.

2 4 27.ω h ω Ce . 4 32.π ω CN

0

∆r ν

. 3 ( YHz) ω Ω r π , m p = 2.61722210

π

. 2.22044610

1=

r ν_Error

∆r ν

2 6 .b 1 .K X . x

.

4 .ω CN

r π_1

1.

r ν_av

c .ω Ce

r ν_2

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

c .ω Ce

5

rε

830.662386 12.03985

2

( am)

2 r X_av

r X_Error

1.

∆K X

rX KX

2

∆K X

rX KX

rX KX

∆r X_av

∆K X

∆r X_av

r X_av

rX KX

∆K X

1

r X_av

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

17 .

eV

r X_Error = 0 ( % )

( am)

m γγ m gg

=

3.195095 6.39019

10

233

45 .

eV

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φ 1 . γγ λ h φ gg

=

1.152898 1.521258

φ 1 . γγ K λ .λ h φ gg

=

0.991785 1.308668

n. 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

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 µn , m µn

ω Ω r µ,mµ

8

3

ω Ω r L, m L 5 , r L

8

ω Ω r µn , m µn

4

10

ω Ω r L, m L 5 , r L

4

ω Ω r τ, m τ

12

ω Ω r τ, m τ

12

ω Ω r τn , m τn

ω Ω r en , m en

1

ω Ω r L, m L 2 , r L

2

ω Ω r L, m L 3 , r L ω Ω r µ,mµ

ω Ω r τn , m τn 1 ω Ω r π, m p

.

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

234

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ω Ω r π, m p ω Ω r ν ,mn ω Ω r ε,me

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

1 14

0.29

1

ω Ω r µ,mµ

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,mτ 1

1 14

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

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

o. 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

235

φ 1 . γγ K λ .λ h φ gg

=

0.991785 1.308668

45 .

eV

www.deltagroupengineering.com

rε

me

rπ

11.807027

mp

. 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

0.105658 1.776989

0.655235

m µn

1.958664

m τn

r uq =

0.768186

m uq

r dq

1.013628

r τn

( am)

0.887904

r sq

1.091334

3 .10

9

1.9.10

4

0.0182 =

. 3.50490310

3

m dq

. 7.00980510

3

m sq

0.113909 1.182905

r cq

1.070961

m cq

r bq

0.92938

m bq m tq

rW

1.061716

mW

91.1876

mZ

114.4

rH

c

2

4.11826

1.284033

rZ

GeV

178.440506

r tq

0.940438

4

80.425

mH m L 2, r L

. 9.15554710

m L 3, r L

rL

=

r QB

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 1.32141 1.319591 11.734441 0.697721 ( fm) λ Ce λ CP λ CN λ Cµ λ Cτ = 2.4263110 . 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

p. 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

236

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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)

q. Calculation results 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

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

1

0.999999

. 8.55887110

12

. 6.96005110

K PV R E, M S K PV R E, 2 .M S 3 K PV R E, M E .e

3 K PV R S , M S .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

∆K 0 R E , M E

0.035839

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 E

. 8.36497210

λ PV 1 , R E, M J λ PV 1 , R E, M S

. 6 1.2259310 . 5 1.20683210

1.000927

K EGM_E R E, M S

1

4

120.885935 =

U m R E, M E U m R E, M J U m R E, M S

237

27.902544 4.089263

(s)

0.402556

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

6

=

∆K 0 R S , M S

0.244543

ω PV 1 , R E, M J

7

1.000927

∆K 0 R E, M S

T PV 1 , R E, M M

3

. 2.211110

1.000463

K 0 R E, M S

=1

10

6.080707 494.481475 =

. 5 1.57089110

( EPa)

. 8 1.64551410

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Ω R E, M M Ω R E, M E Ω R E, M J

=

. 29 2.83606210

n Ω R E, M M

. 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

Ω R E, M S ω Ω R E, M M

519.573099 =

ω Ω R E, M J

. 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 E S m R E, M J

( YHz)

14.824182 =

S m R E, M S

. 3 4.70941210

∆ω δr 1 , R E, ∆r , M E

=

∆ω δr 1 , R E, ∆r , M J

. 27 3.5284510 195.505363

∆ω PV R E, M E

519.573099 =

∆ω PV R E, M J

N ∆r R E, M E

YW

N ∆r R E, M J

2

cm

. 14 6.52135710 =

N ∆r R E, M S

1.729554

∆λ δr 1 , R E, ∆r , M M

7.493187

∆λ δr 1 , R E, ∆r , M E

( pHz)

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 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

=

∆K C R E, ∆r , M M ∆K C R E, ∆r , M E ∆K C R E, ∆r , M J ∆K C R E, ∆r , M S

( ym )

∆v δr 1 , R E, ∆r , M J

13.105121

pm

∆U PV R E, ∆r , M E

13.105115

s

∆U PV R E, ∆r , M J

87.634109 . 4 2.78399910

. 16 2.9237310

7.577156 =

∆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 =

. 7 7.74094810

ω Ω_ZPF R E, ∆r , M E ω Ω_ZPF R E, ∆r , M J

. 7 2.9162510

( GPa)

4 7.3899.10

ω Ω_ZPF R E, ∆r , M M ( MPa .MΩ )

( m)

0.025237

∆U PV R E, ∆r , M M

1.077649 =

. 15 6.23483610

∆v δr 1 , R E, ∆r , M S

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

N ∆r R E, M M

519.469801

∆ω δr 1 , R E, ∆r , M S

. 27 7.64347410

∆ω PV R E, M S

. 6 4.93312710

∆ω δr 1 , R E, ∆r , M M

. 28 1.44974110

=

∆ω PV R E, M M

195.505363

ω Ω R E, M E

. 28 2.36338510

ω Ω_ZPF R E, ∆r , M S

123.501066 370.868276 =

. 3 1.56573710

( PHz)

. 3 8.90753610

KR2 = 99.99999999999999(%)

238

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∆ω ZPF R E, ∆r , M M ∆ω ZPF R E, ∆r , M E

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

∆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 ∆n S R E, ∆r , M S , 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

. 18 6.40202410

∆ω Ω R E, ∆r , M J

. 18 3.58539910

∆ω Ω R E, ∆r , M S

∆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

. 2.78399910

St β R E, ∆r , M E St β R E, ∆r , M J

. 2.9162510

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

. 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.

G .M M . 1 2 R E .c

1. 2

St δ R E, ∆r , M J

4

St δ R E, ∆r , M S

4

=1

=

( PHz)

. 3 8.90658910

3

0.011474

1 =

1 1 1

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

e

. 3 1.56556910

. 2.01680710

1.000001

2

370.826434 =

4

1.000001

1.

( PHz)

123.486273

. 4.77711210

St ε n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E

G .M E . 1 2 R E .c

162.833549

4

St ε n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

2.

45.263389

. 1.59080310

0.999999

2

. 14 6.84403710

763.476685

8.19356

St γ R E, ∆r , M M

St γ R E, ∆r , M J

=

∆ω S R E, ∆r , M M , K R2

( MPa .MΩ )

. 15 1.16748410

. 14 3.81125810

7.251258

4

=

=

17.031676

∆ω Ω R E, ∆r , M E

. 1.034710

87.634109 =

. 15 1.78829110

∆ω Ω R E, ∆r , M M

19

=

n β R E, ∆r , M M , K R2

n β R E, ∆r , M S , K R2

. 19 1.49277510

. 18 6.40270810 . 18 3.5857810

n β R E, ∆r , M J , K R2

1.077649

St α R E, ∆r , M S

e

167.366022

. 19 1.03481710

n Ω_ZPF R E, ∆r , M S

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

41.841506

=

n Ω_ZPF R E, ∆r , M J

. 3 8.90753610

∆ω ZPF R E, ∆r , M S

St α R E, ∆r , M J

( PHz)

. 3 1.56573710

. 19 1.49295410

n Ω_ZPF R E, ∆r , M E

370.868276 =

∆ω ZPF R E, ∆r , M J

St α R E, ∆r , M E

n Ω_ZPF R E, ∆r , M M

123.501066

1.000001 =

1 1.000003 1

2

=1

239

www.deltagroupengineering.com

G .M J . 1 2 R E .c

2.

e

1. 2

2.

2

= 1.000001

e

G .M S . 1 2 R E .c

1. 2

2

= 1.000927

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

. 3.15778710

E C R J , ∆r , M J

N X R J , ∆r , M J

=

N X R S , ∆r , M S

. 17 3.76223110

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

2

=

36.419294 97.406507

=

=

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

( mgs )

γ

2

1.

6.364801

ω X R E, ∆r , M E

ln 2 .N X R M , ∆r , M M

1.

2

ω X R M , ∆r , M M

29.761666

λ X R S , ∆r , M S 1.

9.8181 0.76984

=

E C R S , ∆r , M S

0.240852

B C R S , ∆r , M S

λ X R E, ∆r , M E

17

294.339224

. 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 =

1 . N X R J , ∆r , M J 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

240

. 37 9.36929710 =

. 38 1.07084610 . 38 2.05343110 . 38 2.93782710

www.deltagroupengineering.com

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

(%)

1

St ∆Λ R M , ∆r , M M 10

. 3 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 1.3035.10

=

. 9 4.78288210 . 10 5.36192210 . 11 5.22005110

ω PV 1 , R S , M S ∆ω δr 1 , R S , ∆r , M S

241

www.deltagroupengineering.com

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 5.361910 . 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

. 4.09314210

4

. 6.56319310

5

. 3.69917510

3

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 3.39425210

∆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

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

=

. 6 2.30813410 . 15 5.25385210

=

. 8.47616310

12

(%)

10

15 .

2

Hz

. 9 1.42948610

. 2.45448210

7

. 6.56319310

5

. 4.09314210

4

. 3.69917510

3

0.023754

0.195216

5.248215

27.272806

(%)

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

242

www.deltagroupengineering.com

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

RJ

1

2 .G.M S . RS

3.225809 3

RE

1

2 .G.M J .

3

1.447168

=

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

RJ

3

1

1

∆Λ EGM R E, ∆r , M E . 2 .G.M E.

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

0

1

1

1 1

1 = 1

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

(%)

1

1

1

1 3

1 3

RE

=

1

1 3

RJ

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

(%)

1

1 ∆r

(%)

3

1

1

0

1

1

RM 1

∆r

RE

1 3

0

RS

1 3

0 =

1

1

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

∆r

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

RM

1 3

2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE

∆Λ EGM R M , ∆r , M M . 2 .G.M M .

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 S , ∆r , M S .

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 E, ∆r , M E .

1

1

1 3

RS

243

www.deltagroupengineering.com

5

λ CP c .m e

5

27.m e

.

4

.

K PV r p , m p .m p

3 128.G.π .h

2

8 .π

3

λ CN

5

2 16.π .λ Ce

c .ω Ce

5

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

.

830.594743

.

3

2 4 27.ω h ω Ce . 4 32.π ω CN 5

.

2 3 16.c .π .m n

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 1.8360210 . 3 1.8386810 . 3 = 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

. 2.49926810

. 2.61722210

17

18

35.73252 . 18 2.62462610

( GHz)

62.792864 10.50158

ω Ω r π, m p

2

=

62.414364 10.471952

ω Ω r e,m e

ω Ce

5

h .m e

St η r π , m p

ω CP

2 4 4 .π .λ h λ Ce

4 .ω CN

830.594743

λ . CN

= ( 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

= 830.594743 ( am)

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

2 3 16.c .π .m p

( 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

.

2 4 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 2.6174110 . 3 2.6174110 . 3 2.6174110 . 3 ( YHz) = 2.61722210

me

244

www.deltagroupengineering.com

2

ω CN

ω Ω r ν,mn

ω Ce

2 .π .c .

λ Ce

mn ω CN. me

2

λ CN

2

ω Ω r ε, m e 2 .ω Ω r π , m p

. 3 2.62463110 . 3 2.62463110 . 3 2.62463110 . 3 ( YHz) = 2.62462610

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

0.510999

=

9.155547

56.766874

105.677748

. 3 2.5703410 . 3 565.476231 1.77526210

. 3 4.6876410

. 3 1.27952710 . 4 1.96479110 . 4 2.90646410 . 4 7.96417210

MeV

. . . . 4.16672110 5.81601510 7.93341210 1.06069210

c

4

4

4

5

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

r. Resonant Casimir cavity design specifications (experimental) Given ∆ω R R E, ∆r , M E

∆r

Find( ∆r )

1

∆r = 16.518377( mm)

ω X R E, ∆r , M E = 16.340851 ( PHz)

245

www.deltagroupengineering.com

E C R E, ∆r , M E = 550.422869

V m

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

=

1 1

NOTES

246

www.deltagroupengineering.com

•

MathCad 12: High precision calculation results a. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED The high precision calculation results are obtained via the “MathCad 12” computational environment utilising the calculation engine defined in the preceding section. • • •

Convergence Tolerance (TOL): 10-14. Constraint Tolerance (CTOL): 10-14. Calculation Display Tolerance: 6 figures – unless otherwise indicated. b. Particle summary matrix 3.1

      rπE +    

( 2)

0.69⋅ fm

rπ

rπM 1 2

⋅ ( rνM − rν ) rν rνM

830.647087 830.662386    848.579832 848    0.857⋅ ( fm)   849.993668 857   ( am) =   874.643564 875 rp   826.889045 825.617615  rX   879.016508 879   0.879⋅ ( fm)  0.848⋅ ( fm)

rπE

( )

 KX   −0.113  2   =  fm  KS   −0.113348 2  − rε  ⋅e 3  rπ  rµ  −  rε ⋅ e rτ  rν   rε  rπ − rν

         

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  r K ν S  ⋅  ⋅ ( rνM − rν ) + rπE  rX KX   rp  2    2 rνM mtq ⋅ c λA ( Kω ⋅ rBohr , mp )     0.879⋅ ( fm) 178⋅ ( GeV) λB  

( )

247

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− 14 − 13   0 2.220446× 10 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

⋅  MError

0, 0

+ M Error

0,1

+ MError

0, 2

+ M Error

1,0

+ MError

+ M Error

1, 1

1,2

...

+ M Error2 , 0 + MError2 , 1 + M Error2 , 2 + MError3 , 0 + M Error3 , 1 + MError3 , 2 

  

ErrorAv = 0.148979(%) c. Particle summary matrix 3.2 2  −  rε c 3  ⋅ ⋅e re ωCe  rπ_1     :=  5 2 4  rπ_2   c⋅ ωCe 27⋅ ωh ωCe  ⋅ ⋅  4⋅ ω 3 32⋅ π4 ωCP  CP

 ∆rπ

 rπ_Error 

 rπ_av − rπ_1

  rν_2  rν_av   ∆rν  rπ_2

rε   rπ − π   rν_1   5   := 2 4  r ν_2 ⋅ 27⋅ ωh ωCe c ω   Ce  ⋅ ⋅  4⋅ ωCN3 32⋅ π4 ωCN 

 1 ⋅ (r  r π_av + ∆rπ )  π_2    :=    1  rν_Error   rν_2 ⋅ ( rν_av + ∆rν )   

    :=    ∆rν   rν_av − rν_1 

 rπ_1   rν_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)   843.685786  rX( KX + ∆KX)   807.145085   ( am)  =   825.415435 rX_av    18.270351      ∆rX_av  

248

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− 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

⋅

d. 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   ω r , m ⋅ Ω ( uq uq)  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 

249

 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 

e. Particle summary matrix 3.4  rε     rπ   11.806238   r   830.647087  ν     rµ   826.889045    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   r   1.091334   cq   1.070961   rbq       0.92938  r tq    1.283947   rW     1.061645   rZ   0.940375  r   H

 me      5.109989× 10− 4   mp   0.938272  m     n   0.939565   mµ   0.105658      1.776989  mτ     men  −9  3 × 10     mµn   1.9 × 10− 4    mτn     0.0182     GeV  muq  =  −3    m   3.505488× 10   c2    dq    7.010977× 10− 3   msq     m   0.113928  cq    1.183102   mbq       4.118949  m tq    178.470327   mW  80.425       91.1876 m Z     m   114.4   H

250

(

)

 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

− 1 = 0.352379( %)

rx

 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)    ωΩ ( rW , mW) ωΩ ( rZ , mZ ) ωΩ ( rH , mH) ωΩ ( rtq , mtq ) 

⋅

1 2 3 4    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

251

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NOTES

252

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Quinta Essentia – Part 4 •

MathCad 8 Professional a. Complete simulation i. 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. ii. Units of measure (definitions)

Jy

10

W

26 .

pc

2.

. 16 .( m) 3.085677580710

m Hz

( mJy µJy nJy pJy fJy aJy zJy yJy )

Scale 1 .( Jy )

( mpc µpc npc ppc fpc apc zpc ypc )

Scale 1 .( pc )

( kJy MJy GJy TJy PJy EJy ZJy YJy )

Scale 2 .( Jy )

( kpc Mpc Gpc Tpc Ppc Epc Zpc Ypc )

Scale 2 .( pc )

iii. Constants (definitions) σ

. 8. 5.67040010

W

κ

2. 4

. 1.380650510

Th

J K

m K

m h .c

23 .

KW

. 3 .( m.K ) 2.897768510

2

κ

iv. Astronomical statistics Lyr

∆T 0

c .yr

D E2M

0.001.( K )

8 3.844.10 .( m)

Ro

8 .( kpc )

AU

∆R o

149597870660.( m) H 0

0.5.( kpc )

MG

71.

km . s Mpc

T0

2.725.( K )

11 6 .10 .M S

v. Characterisation of the gravitational spectrum 1. “Ω → Ω1, nΩ → nΩ_1, ωΩ → ωΩ_1” Note: “the complete simulation” is the computational algorithm developed for this text and is predominantly without comment. 253

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Commencing with the following relationship set, significant simplifications to primary EGM equations may be derived as follows, 4 h . ω PV( 1 , r , M ) 3 2 .c

U ω( r , M )

c .U ω n PV, r , M

S ω n PV, r , M

U ω( r , M ) .

U ω n PV, r , M

U m( r , M )

3 .M .c

n PV

2

4

4

n PV

2

4 .π .r

3

3

Ω ( r, M )

108.

U m( r , M ) U ω( r , M )

2

U m( r , M )

12. 768 81.

n Ω ( r, M )

U ω( r , M )

Ω ( r, M )

4

12

Ω ( r, M )

1

Hence, 3 .M .c U m( r , M )

2

4 .π .r

3 .M .c

3

U ω( r , M )

5

4 3 h . 4 ω PV( 1 , r , M ) 2 .π .r .h .ω PV( 1 , r , M ) 3 2 .c

U m( r , M )

108.

3

Ω 1( r , M ) 6 .

>> 768”, hence simplifying / approximating forms yields,

U ω( r , M )

3

Ω 1( r , M )

2

U m( r , M )

Typically: “ 81.

108.

U ω( r , M )

U m( r , M )

3 .M .c

6.

U ω( r , M )

216.

U ω( r , M )

3

U m( r , M )

3

U m( r , M ) U ω( r , M )

Ω 1( r , M )

4 3 2 .π .r .h .ω PV( 1 , r , M )

3

6 .c

5

r .ω PV( 1 , r , M )

.

3 .M .c

2

2 .π .h .ω PV( 1 , r , M )

Typically: “ Ω ( r , M ) >> 1” hence, 3

n Ω_1( r , M )

Ω 1( r , M ) 1 U m( r , M ) .

C Ω_1( r , M )

12

G.M . 2

r

ω Ω_1( r , M )

3

c

U ω( r , M ) 2 .r .ω PV( 1 , r , M )

2

.

3 .M .c

2

2 .π .h .ω PV( 1 , r , M )

n Ω_1( r , M )

Ω 1( r , M ) 12

3

2

ω Ω_1( r , M ) n Ω_1( r , M ) .ω PV( 1 , r , M )

π .n Ω_1( r , M )

n Ω_1( r , M ) .ω PV( 1 , r , M )

T Ω_1( r , M )

1 ω Ω_1( r , M )

2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

λ Ω_1( r , M )

c ω Ω_1( r , M )

Checking errors yields, Ω 1 R M,M M

Ω 1 R E, M E

Ω R M, M M

Ω R E, M E

Ω 1 R J, M J

Ω 1 R S, M S

Ω R J, M J

Ω R S, M S

1=

. 14 4.44089210 . 6.66133810

14

. 14 6.66133810 . 4.44089210

14

254

(%)

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Ω 1 R NS , M NS

1 = 0 (%)

Ω R NS , M NS

n Ω_1 R M , M M

n Ω_1 R E, M E

n Ω R M,M M

n Ω R E, M E

n Ω_1 R J , M J

n Ω_1 R S , M S

n Ω R J, M J

n Ω R S, M S

n Ω_1 R NS , M NS n Ω R NS , M NS

ω Ω_1 R E, M E

ω Ω R M,M M

ω Ω R E, M E

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω R J, M J

ω Ω R S, M S

ω Ω R NS , M NS

T Ω_1 R E, M E

T Ω R M,M M

T Ω R E, M E

T Ω_1 R J , M J

T Ω_1 R S , M S

T Ω R J, M J

T Ω R S, M S

T Ω R NS, M NS

λ Ω_1 R E, M E

λ Ω R M,M M

λ Ω R E, M E

λ Ω_1 R J , M J

λ Ω_1 R S , M S

λ Ω R J, M J

λ Ω R S, M S

λ Ω R NS , M NS

. 2.22044610

14

. 4.44089210

14

. 8.88178410

14

(%)

1=

. 6.66133810

14

. 2.22044610

14

. 6.66133810

14

. 8.88178410

14

(%)

1=

. 7.77156110

14

. 2.22044610

14

. 5.55111510

14

. 7.77156110

14

. 7.77156110

14

. 2.22044610

14

. 6.66133810

14

. 7.77156110

14

(%)

1 = 0 ( %)

λ Ω_1 R M , M M

λ Ω_1 R NS , M NS

14

1 = 0 (%)

T Ω_1 R M , M M

T Ω_1 R NS, M NS

. 6.66133810

1 = 0 (%)

ω Ω_1 R M , M M

ω Ω_1 R NS , M NS

1=

1=

(%)

1 = 0 (%)

2. “g → ωΩ” i. “ωΩ_1 → ωΩ_2” 3

ω Ω_1( r , M )

2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

ω Ω_1( r , M )

255

3

3 2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

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ω PV( 1 , r , M )

. 3 . . . 2r . 2πh 3 2 c 3 .M .c ω Ω_1( r , M ) 1

16.π .h . r 5 3 .M .c ω Ω_1( r , M )

3

ω Ω_2( r , M )

th

9

9 2 c . 3 .M .c . 1 . 3 2r 2 .π .h ω PV( 1 , r , M )

G.h c

ωh

5

th

2

c G.h

3 14 2 3 .c .M

4 3 2 2 3 .G .M .ω h .c

13 5 2 3 2 .r .π .h .G

2 13 5 2 2 .r .π .λ h

ω Ω_2( r , M )

g( r , M )

9

6 3 3 .ω h . . GM 2 13 . 2 . . 2 π rc r

G.M

mh

2

St g

9

r

λh

G

5 3 3 .ω h . . GM 13 5 π 2 .λ h .r

ω Ω_2( r , M )

2

h .c

5

1

2

9 2 c . 3 .M .c 2 .r 2 .π .h

2

3

3

G.h c

λh

2

r

6 3 3 .ω h

13 5 2 3 2 .r .π .h .G

3

1

3

5 3 3 .ω h .G.M G.M . 2 3 13 2 .λ h .π .r

3 14 2 3 .c .M

1 . 1 . 2 c .G.M 3 π .r r

.

2

c G.h

G λh

6 3 3 .ω h . . GM 2 13 2 2 .π .r .c r

c 2

3

h

2

245

10

St g = 1.828935

13 2 2 .π .c

1. 2 St g .g ( r , M ) r

ω Ω_2( r , M )

9

5 m.s

St g

.g ( r , M ) 2

r

Checking errors yields, ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω_2 R NS , M NS

1=

. 1.04678510

9

. 2.32001510

8

. 6.57443310

7

. 7.07196310

5

(%)

1 = 2.491576 ( % )

ω Ω_1 R NS , M NS

Therefore, a EGM_ωΩ( r , M )

r . 9 ω Ω_2( r , M )

St g

a EGM_ωΩ R E, M E = 9.809009

m s

2

Checking errors yields, a EGM_ωΩ R M , M M

a EGM_ωΩ R E, M E

g R M,M M

g R E, M E

a EGM_ωΩ R J , M J

a EGM_ωΩ R S , M S

g R J, M J

g R S, M S

1=

. 1.49880110

12

. 1.49880110

12

. 1.5432110

12

. 1.57651710

12

256

(%)

www.deltagroupengineering.com

a EGM_ωΩ R NS , M NS

. 1 = 1.65423210

g R NS, M NS

12

(%)

ii. “ωΩ_1 → ωΩ_3” 3

3

1 U m( r , M ) . 1 ω Ω_1( r , M ) . ω PV( 1 , r , M ) . 2 U ω( r , M ) 2

ω Ω_1( r , M )

U m( r , M ) 4 h . ω PV( 1 , r , M ) 3 2 .c

3 1 . 2 .c . U m( r , M ) 8 h ω PV( 1 , r , M )

3

3 .M .c ω Ω_3( r , M )

.ω ( 1 , r , M ) PV

3 c . U m( r , M ) 4 .h ω PV( 1 , r , M )

9

3

3 3

c 4 .h

3

2

. . . 4πr . 2 c .G.M 3

14

2

27 . c . M 8192 h 3 π2 .r5 .G

5

9

2

27 . c . c . M 8192 G.h h 2 π2 .r5

3

9

2

3 . 2 .c . M ωh 13 2 2 2 h π .r5

π .r

4

9

ω Ω_3( r , M )

c.

3.

2

9

ω Ω_3( r , M )

3 .ω h

2

. M

4 .π .h

2

St G

5

r

3.

3 .ω h

2

. c 2

4 .π .h

M St G. 5 r

ω Ω_3( r , M )

224 .

St G = 8.146982 10

5

m

2 9 kg .s

2

1

2

9

9 M St G .

St G

9

G

St g

5

r

9

Checking errors yields, ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_3 R NS , M NS ω Ω_2 R NS , M NS

. 1 = 6.66133810

1=

14

(%)

. 14 8.88178410

. 1.11022310

13

. 13 1.11022310

. 1.11022310

13

1 . St G

. 1 = 3.33066910

G

(%)

14

(%)

St g

3. “g → ωPV3” 2 .c .n PV

3

ω PV n PV, r , M

3

2 π .r

.g ( r , M )

257

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4. “SωΩ → c⋅Um” Reducing / simplifying / approximating utilising computational features of the environment yields, nΩ

8 .n Ω

3

2

24.n Ω

8 .n Ω . n Ω

4

2

2

nΩ

2

nΩ

2

8 .n Ω

simplify

32.n Ω factor

3 .n Ω factor

2

substitute , n Ω

4

2

8 .n Ω . n Ω

2 8 .n Ω . n Ω

3 .n Ω

2

3

24.n Ω

2

32.n Ω

16

4

3

Hence, nΩ 8 .n Ω

2 3

4

2

2 24.n Ω

S ωΩ ( r , M )

nΩ

2

4

nΩ

4

32.n Ω 8 .n Ω . n Ω

h . 4 ω PV( 1 , r , M ) . 2 . 2c

nΩ 2

2

8 .n Ω

4

3 .n Ω

n Ω ( r, M )

4

24.n Ω

2

32.n Ω

8 .n Ω . n Ω

2

3 .n Ω

4

2

3

n Ω ( r, M )

4

16 2 8 .n Ω . n Ω

3

8 .n Ω

3

h . 4 3 ω PV( 1 , r , M ) .8 .n Ω ( r , M ) 2 . 2c 3

2 c 4 .h . 4 4 3 .M .c 3 4 .h . . S ωΩ ( r , M ) ω PV( 1 , r , M ) .n Ω ( r , M ) ω PV( 1 , r , M ) . 2 2 2 .r .ω PV( 1 , r , M ) 2 .π .h .ω PV( 1 , r , M ) c c

3

4

3 2 3 4 .h .c . ω PV( 1 , r , M ) . 3 .M .c 3 .M .c S ωΩ ( r , M ) 2 3 3 . . . 3 8 .c .r ω PV( 1 , r , M ) 2 π h ω PV( 1 , r , M ) 4 .π .r

Hence, S ωΩ ( r , M ) c .U m( r , M )

5. “CΩ_J” C Ω_J ( r , M )

2 d λ Ω ( r , M ) . U m( r , M ) dr

2

C Ω_J1( r , M )

2

c 9

2

M St G. 5 r 9 .c . St G 4 .π 4

St J

. . . d 3Mc d r 4 .π .r3

2 9

C Ω_J1( r , M )

2

c

2

. r

9

2

St G

St J 2

r

9

5

9

. M

M

2

1

. . .9 M c 4 4 .π .r

2

2

5 9

9 .c . 9 M St G . 26 4 .π r 4

5

8

r

258

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Checking errors yields, C Ω_J1 100.( km) , M M

C Ω_J1 R S , M M

C Ω_J 100.( km) , M M C Ω_J1 100.( km) , M E C Ω_J 100.( km) , M E C Ω_J1 100.( km) , M J C Ω_J 100.( km) , M J C Ω_J1 100.( km) , M S C Ω_J 100.( km) , M S C Ω_J1 100.( km) , M NS C Ω_J 100.( km) , M NS

C Ω_J R S , M M

1=

. 3.63875410

8

. 2.95903310

6

. 9.40034410

4

C Ω_J1 R S , M E C Ω_J R S , M E C Ω_J1 R S , M J

(%)

1=

C Ω_J R S , M J

0.979587

C Ω_J1 R S , M S

0.979587

C Ω_J R S , M S

. 3.86357610

12

. 4.23450210

10

. 1.3506210

7

. 1.41439110

4

. 1.41439110

4

(%)

C Ω_J1 R S , M NS C Ω_J R S , M NS

vi. Derivation of “Planck-Particle” and SBH characteristics 1. “λx, mx” n Ω_1( r , M )

2 .c 1 . U m( r , M ) 1 . 3 .M .c . 8 U ω( r , M ) 8 4 .π .r3 h .ω ( 1 , r , M ) 4 PV 2

3

2 1 . 3 .M .c .

2 .c

2 1 . 3 .M .c .

3

8 4 .π .r3 h .ω ( 1 , r , M ) 4 8 4 .π .r3 PV

1 . 3 .M .c . 8 4 .π .r3

2 .c

2

3

2 .c

3 1 2 .c .G.M h. . r π .r

1 . 3 .M .c . 8 4 .π .r3

3

3

2 .c

2

3 1 2 .c .G.M h. . r π .r

4

4

3

3 1 2 .c .G.M . 2 .c .G.M h. . 4 π .r π .r r

3 .c . r

3

2 1 . 3 .M .c .

8 4 .π .r3

2 .c

3. c . . 2 cr 1 . 4 h .G

3

3 1 2 .c .G.M . 2 .c .G.M 8 h. . 4 π .r π .r r

3 .c . r n Ω_1( r , M )

1. 4

9

8

3

3

n Ω_1( r , M )

2 .c .G.M π .r

3

n Ω_2( r , M )

3

λh

r . π .c . 3 . 16 2 2 GM λh 7

2 .c .G.M π .r

3 . 16

2

π .m h

8

3

λh

2 .c .G.M π .r

2 3 2

3

7 r . π .m h . 3 16 2 2 M λh λh

3

1

1

9

1. 4

2

. r M λh

3 9

7

n Ω_2( r , M )

1. 3 2

259

7

2

.

π .m h M

7

9

. r λh

9

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9

λ x.λ h

n Ω_3 λ x.λ h , m x.m h

7

16

2

mx

3 3 .π . 7 λx 16 2

2 c .R BH 2 .G

R BH

9

m . π . h. 3 m x.m h λ h λ 2 h

3

2

2 .G . M BH 2 c

9

7

3 3 3 .π . λ x 1 . 3 .π . λ x 16 m 2 x 2 mx 2

M BH

.

5

R BH

7

2

2 c .R BH 2 .G

λ x.λ h

5

m x.m h

2

λ x.λ h

5

m x.m h

2

.

5

R BH

2

.

5

R BH 5

2 1. λ x . c 4 R 3 G BH

2

2 λ x.λ h .c

2

λ x.λ h

5

m x.m h

2

λh

.

5

2 1. λ x . c 4 R 3 G BH

5

5

1.

m x.m h

St ω λ x.λ h , m x.m h

2 .G.m x.m h

6

λ x 33 .π .λ 7 x 2 216

9

3

2

λ x 4.

3 3 .π

λh

.

λx

5

m x.m h 2 2

λx

4 λ .λ x h

2

2

. c 3 G

.

λh

2

5

m x.m h

c

2

5

1.

. c 3 G

4 λ .λ x h

2

2 λ x.λ h .c

mh

λx

mx

2

λh

.

1.λ x

1

2

mx

2 .G.m x.m h

λx 2

9

n Ω_3 λ x

2

2 λ x.λ h .c

5

m x.m h

2 .G.m x.m h 2 m x

G λh

4 . 2 6 π 3

2 2

λx

3 1 . 3 .π . λ x 2 λx 2

9

7

1. 2

3.

3 π.

λx

6

2

2 3 9

n Ω_3 λ x

π.

3.

2

λx

2

2

λ x.λ h = 1.093333 10

10 .

ym

n Ω_3 λ x

λx

1 = 0 ( %)

. m x.m h = 7.36147410

8

mx

( kg )

mx

=

2.698709 1.349354

1 = 0.14278 ( % )

2

Km n Ω_3 n Ω_3

1 3

0.248017

1

0.324994

2

n Ω_3( 1 )

=

0.515897 0.818935

n Ω_3( 2 )

1

n Ω_3 λ x

1.073108

n Ω_3( 3 )

260

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2. “ρm(λxλh,mxmh), Um(λxλh,mxmh)” V( r )

4. . 3 πr 3

ρ m( r , M )

. 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

M V( r )

ρ m λ x.λ h , m x.m h

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

ρ m R S, M S

. = 9.55041510

90

3. Physicality of “Kλ” . 42 ( Hz) K ω .ω h = 6.36576910 K λ .λ h

K λ .λ h = 4.709446 10

35 .

m

. K m.m h = 6.34179210

8

( kg )

1 = 0.82832 ( % )

2 .r γγ

4. “KPV @ λxλh” i. “KPV = Undefined” Recognising, U ω λ x.λ h , m x.m h

h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

4

h . ω Ω λ x.λ h , m x.m h 3 . 2c

4

m h c2 λh

G

It follows that, ω PV 1 , λ x.λ h , m x.m h

3 . . . . 1 . 2 c G mx mh . K PV λ x.λ h , m x.m h λ x.λ h π .λ x.λ h

3

3

1 . λ x.λ h

2 .c .G.m x.m h π .λ x.λ h

3

2 .c .G. 1 . λ x.λ h

. K . . PV λ x λ h , m x m h

λx 2 .c .G. .m h 1 . 2 . K . . PV λ x λ h , m x m h . . λxλh π λ x.λ h

λx

.m h 2 . K . . PV λ x λ h , m x m h π .λ x.λ h

3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π ω PV 1 , λ x.λ h , m x.m h

ωh

ωh

3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π

3

. 1. K . . PV λ x λ h , m x m h λx π

3

. 1. K . . PV λ x λ h , m x m h λx π

261

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Performing substitutions yields, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

U m λ x.λ h , m x.m h 4

U m λ x.λ h , m x.m h 4

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 2 .c λ x π

4

3.

4

2 .π . π .c λ x

.K . . PV λ x λ h , m x m h

3 4 3 2 .π . π .c .λ x U m λ x.λ h , m x.m h . 4 . h ωh K PV λ x.λ h , m x.m h

4

.K . . PV λ x λ h , m x m h

h .ω h 3

U m λ x.λ h , m x.m h

3 4 3 2 .π . π .c .λ x

4

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 λ π . 2c x

U m λ x.λ h , m x.m h

h .ω h

4

2

2

2

Checking errors yields, 3 h . ωh. 1 3 2 .c λ x π

h .ω h

4

. 1 = 6.66133810

14

(%)

4

3 4 3 2 .π . π .c .λ x ω 3 . h . h. 1 4 3 2 .c λ x π h .ω h

4

. 1 = 6.66133810

14

(%)

3 4 3 2 .π . π .c .λ x

Evaluating, 3 4 3 2 .π . π .c .λ x .U λ .λ , m .m = 8 m x h x h 4 . h ωh

Checking errors yields, 3 4 3 2 .π . π .c .λ x .U λ .λ , m .m m x h x h 4 . h ωh

. 8 = 8.88178410

13

(%)

3 4 3 2 .π . π .c .λ x

h .ω h

4

= 6.619576

10

87

YPa

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

262

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Simplifying, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

8 K PV λ x.λ h , m x.m h

2

K PV λ x.λ h , m x.m h

2. 2.

U ω λ x.λ h , m x.m h U m λ x.λ h , m x.m h

Recognising that the EGM spectrum converges to a single mode for a SPBH yields, Ω λ x.λ h , m x.m h

n Ω λ x.λ h , m x.m h

12

Ω λ x.λ h , m x.m h

4. 3

4 . Ω λ x λ h , m x.m h

1 1

4 . 3 = 6.928203

3

Ω λ x.λ h , m x.m h

108.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

12. 768 81.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

2

4. 3

By inspection, the only solution which satisfies this equation is, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

0

Checking yields, 3

108.0

2 12. 768 81.0 = 6.928203

Therefore, 2. 2

K PV λ x.λ h , m x.m h

K PV R BH, M BH

Undefined

0

K PV λ x.λ h , m x.m h

ii. “KDepp = KPV”

K Depp ( r , M )

1 2 .G.M

2

1

r .c

1

r .c

2

2

K PV( r , M )

2 .G.M

2 .G.M r .c

K Depp ( r , M )

2

K Depp ( r , M )

K PV( r , M )

2

1

K Depp ( r , M )

r .c

K Depp R E, M E = 1.00000000069601

2

K PV R E, M E = 1.00000000069601

K PV( r , M )

1

263

2 .G.M r .c

2

r .c

1

r .c

2

2

2 .G.M

R BH

2 .G . M BH 2 c

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K PV R BH, M BH

2 .G.M BH

1

1

1

2 .G . 2 M BH.c 2 c

1

K Depp λ x.λ h , m x.m h

λh

0

1 . 2 G.m x.m h

2

K Depp λ x.λ h , m x.m h

m h c2

Undefined

1 . 2 G.m x c2 . 1 2 G . λxc

1

2 λ x.λ h .c

K Depp R BH, M BH

λx

mx

G

1

1

2 .m x λx

2

1

2.

λx 0

Undefined

2

1

λx

Undefined

5. “ωΩ_3”

2

M St G. 5 r

St G.

m x.m h

2

λ x.λ h

5

2

St G m h . 3 5 . 4λx λh

3 4 .λ

3

.

x

3 4 .λ

3

π .h

x 2

ω h .m h π .h

2

ωh

.

St G.

9 St G .

m x.m h

9

9

15 . 2

2

π

9

ω Ω_3 λ x.λ h , m x.m h

9

3 λx

3

.

ωh

9

St G.

9

1.

15 2 2 .π 2

3

2

.

ωh

2

λ x.λ h 3

ωh

.

m h .c

7

2

π .h

x

St G.

5

2

3 4 .λ

2

St G.

ωh

2

.m h

λ x.λ h

2

9 m . c . h 2 λ 5 h

h

15 2 2 .π

λx

.m h

3

.

2

5

ω h .m h

x

π .h

m x.m h

2

λ x.λ h

5

St G m h . 3 5 4 .λ x λ h 2

.

5 4 c .ω h 9

2

3

3

λx

.

ωh

9

15 2 2 .π

1

9

1

.

λx

5

λ x.λ h

3

3 4 .λ

3

1

3

5

2 c .m h . h λx

3

2

.

λ x.λ h

2

St G.

2

5 4 c .ω h .

9

2

9 m . c . h 2 λ 5 h

2 1

m x.m h

λx

.ω h

. 1 = 1.11022310

m x.m h

2

λ x.λ h

5

9

St G.

9

3

. 1 .ω h λ x 26 .π2

1. 2

13

(%)

m x.m h

2

λ x.λ h

5

3 λx

3

.

ωh

9

15 2 2 .π

9

3

3

9

3

1. 1 . 3 . 1 .ω .ω h h 2 π2 4 .λ x λ x 26 .π2 2

9

3

1. 1 . 3 .ω h 2 π2 4 .λ x

3

1 9

. 2

2. π

1

3 . ωh . 4λx

9 St G .

m x.m h

9

. 5

λ x.λ h

9

264

3

1 9

2. π

. 2

1

3 . ωh . 4λx

. 1 = 1.11022310

13

(%)

www.deltagroupengineering.com

9

1. 1 . 2 π2

6

4. 4.

9

3

3

9

4

1. 3 . 6 4 25 π3

3

2

m x.m h 9 St G . λ x.λ h

5 9

1

9

9

4

. 1 . 3 . 6 .ω h 4 25 π3

ω Ω_3 λ x.λ h , m x.m h 9

4

1. 3 . 6 . ωh 4 25 π3

3 3 .π 2

1

ω Ω_3 λ x.λ h , m x.m h

. 1 = 1.11022310

. 1 = 1.11022310

13

13

(%)

. 18 ( YHz) ω Ω_3 λ x.λ h , m x.m h = 1.87219710

(%)

4

1. 3 . 6 . ωh 4 25 π3

1. . 18 ( YHz) ω h = 1.84996810 4

λx

e

e 1

α

1

α

ωh

1.

1 . e λx 1 α

= 2.698589

. 1 = 4.43474910

ω Ω_3 λ x.λ h , m x.m h

ω PV 1 , λ x.λ h , m x.m h

ω PV 1 , λ x.λ h , m x.m h

3

(%)

n Ω_2 λ x.λ h , m x.m h = 1

n Ω_3 λ x

n Ω_3 λ x = 1

1 = 1.18731904721517( % )

4 ω Ω_3 λ x.λ h , m x.m h

ω Ω_3 λ x.λ h , m x.m h

6. “ωΩ_4” 9

ω Ω_4 M BH

2

9

M BH

St G.

2 .G.M BH c

c. 5

c .St G

9

5 3 ( 2 .G) .M BH

ω Ω_4 M BH

St BH.

c .St G 5 ( 2 .G)

2

ω Ω_4 m x.m h 3

St BH

c.

1 M BH

ω Ω_4 M S 10 ω Ω_4 10 .M S

265

. 18 1.87219710 = 6.23977510 . 5

( YHz)

289.624693

www.deltagroupengineering.com

7. “rS” i. “rS(λxλh)”

ρ m( r , M )

3 .M

ρ m λ x.λ h , m x.m h

4 .π .r

3

ρ m λ x.λ h , m x.m h 3 .m h

. 1 = 2.22044610

14

3.

3 .m x.m h

λx 2

4 .π . λ x.λ h

3

.m h

3 .m h

4 .π . λ x.λ h

3 .M BH

ρ m r S , M BH

(%)

2 3 8 .π .λ x .λ h

3

4 .π .r S

3

2 3 8 .π .λ x .λ h

M BH

ρ m λ x.λ h , m x.m h

ρ m r S , M BH

3

ωh

2

R BH( M )

2 .G.λ x

3

2

rS

3

r S M BH

λ x.λ h

2

ωh

ωh

2

2 .G.λ x

2

3

.M . . 2. BH λ h 2 λ x

= 1.195378 10

32 .

G

2

λh

2

ωh

2

kg

. = 5.63257510

94

2

3

m

M BH

3

r S R BH

mh

r S λ x.λ h

2

am

λh

c

2

2 .G.M c

2 2 .G.λ x

m h c2

2 3 2 .λ x .λ h

rS

M BH

mh

3

2 λ x.λ h .R BH

2 λ x.λ h . λ x.λ h

λ x.λ h

ii. “rS(ΜΒΗ), rS(RΒΗ)” 3

ρS

ρ m λ x.λ h , m x.m h

r S M BH

r S R BH

3

2 3 .c .R BH

8 .π .G.ρ S

2 λ x.λ h .R BH M S 3

2.

3 .c R BH M S 8 .π .G.ρ S

1 rS MS

3 .M BH 4 .π .ρ S

3 3

3.

.

4 .π .r S

3

2 λ x.λ h .R BH M S 3

2.

3 .c R BH M S 8 .π .G.ρ S

rS MS 1 = 0 (%)

2 c .R BH 2 .G

1=

. 3.28046310

5.

r S 10 M S

=

10 r S 10 .M S

266

0.015227

ρS

2 3 .c .R BH

8 .π .G.r S

3

. 1.11022310

14

. 1.11022310

14

(%)

4

( am)

0.706754

www.deltagroupengineering.com

ρ m r S m x.m h , m x.m h ρ m r S M S ,M S 1 . 5 5 ρ S ρ m r S 10 .M S , 10 .M S

1=

. 1.29896110

12

. 8.32667310

13

. 7.66053910

13

. 6.7723610

10 10 ρ m r S 10 .M S , 10 .M S

(%)

13

U m r S M S ,M S

. 8.10462810

1 . U m r S 105 .M S , 105 .M S . . U m λ x λ h,m x mh 10 10 U m r S 10 .M S , 10 .M S ρ m r S m x.m h , m x.m h

. 7.2164510

13

. 6.7723610

13

(%)

U m λ x.λ h , m x.m h

ρ m r S M BH , M BH 10 r S 10 .M S

r uq = 0.768186 ( am)

1=

13

U m r S M BH , M BH

1 = 7.996993 ( % )

r uq

iii. “MBH(rS)” 4. . 3 π ρ S .r S 3

M BH r S

M BH r tq

. 10 = 2.27391910

. 40 ( kg ) M BH r tq = 4.52155110 M BH r uq

MS

. 10 = 1.28408510

MS

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 13 1.62379510 . 19 1.60185510 . 19 1.57097210 . 13 4.66247210

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 13 2.45782610 . 7 5.19529810

1 . M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 1.28408510 . 10 2.95005410 . 10 1.9828610 . 10 3.68186410 . 10

M BH r bq

M BH r tq

M BH r W

. 10 2.27391910 . 10 5.99684310 . 10 3.39015710 . 10 3.47948910

M BH r H

M BH r γγ

M BH r gg

M BH r Z 1 .( kg )

MS

. 10 2.3560510

. 9 2.12850410 . 11 7.96867110

0

0

14.554628

. 6 5.06892810

R BH M BH r τn

16.217926

. 7.67248410

R BH M BH r sq

R BH M BH r cq

= 4.00847210 .

R BH M BH r tq

R BH M BH r W

R BH M BH r Z

0.010862

R BH M BH r γγ

R BH M BH r gg

R BH( 1.( kg ) )

. 7.35477510

R BH M BH r ε

R BH M BH r π

R BH M BH r ν

R BH M BH r µ

R BH M BH r τ

R BH M BH r en

R BH M BH r µn

R BH M BH r uq

R BH M BH r dq

R BH M BH r bq R BH M BH r H

ω Ω_4 m x.m h

5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

=

. 5 6.23977510

. 9.2090510

3

. 7.0983910

3

3

0

U m λ x.λ h , m x.m h

. 18 1.87219710

ω Ω_4 M S

3

. 4 1.34431910

5 5 U m R BH 10 .M S , 10 .M S

289.624693

10 10 U m R BH 10 .M S , 10 .M S

267

. 6 5.0004410

4.904034

. 2.48754410

3

0.066445

. 6.18980410

3

0.011494 ( Lyr)

0.01872

0.010583

0

0

. 87 1.20853710

U m R BH M S , M S ( YHz)

6

0

=

. 12 1.65639710

( YPa)

165.639685 . 1.65639710

8

www.deltagroupengineering.com

r S mh 1 . r m .m S x h λh λx

r S m x.m h R BH m x.m h

144.219703

1=

. 4.21884710

(%)

13

. 1 = 4.44089210

M BH r π

1 = 22.109851 ( % )

=

M BH r e

R BH m h

(%)

. 43 9.27104510

M BH r ε r S mh

13

M BH r Bohr

. 49 3.22881910 . 51 1.26038310

( kg )

. 63 8.34661610

8. “r → RBH” i. “nΩ → nΩ_4, nΩ_5, nBH” n Ω_2 r S M BH , M BH

n Ω_4 M BH

n BH M BH

n Ω_5 M BH

n Ω_5 M BH n Ω_4 M BH

n Ω_4 m x.m h

n Ω_5 m x.m h

n BH m x.m h

n Ω_4 M S

n Ω_5 M S

n BH M S

5 n Ω_4 10 .M S

5 n Ω_5 10 .M S

5 n BH 10 .M S

10 n Ω_4 10 .M S

10 n Ω_5 10 .M S

10 n BH 10 .M S

R BH M S

∆R bh

n Ω_2 R BH M BH , M BH

rS MS

200

R bh

1 =

1

1

. 5 9.00254210 . 24 2.56419310 . 19 3.51086810 . 6 1.93953910 . 28 1.0035610 . 22 1.93265910 . 7 4.1786110 . 31 3.92767810 . 24 1.06388810

r S M S , ∆R bh .. R BH M S Harmonic Cut-Off Mode vs Radial Disp.

Harmonic Cut-Off Mode

rS MS

R BH M S

n Ω _2 R bh , M S 5 n Ω _2 R bh , 10 .M S 10 n Ω _2 R bh , 10 .M S n Ω _4 M S

R bh Radial Displacement

Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

268

www.deltagroupengineering.com

ii. “ωΩ → ωΩ_5, ωBH” ω Ω_5 M BH

ω Ω_3 r S M BH , M BH

ω Ω_4 m x.m h

. 5 6.23977510

=

5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

. 18 1.87219710

ω Ω_5 M S ( YHz)

5 ω Ω_5 10 .M S

289.624693

10 ω Ω_5 10 .M S

1

ω BH M S

. 13 7.30358710

10 ω BH 10 .M S

ω Ω_4 M BH

. 4 1.34431910

ω BH m x.m h =

ω Ω_5 M BH

ω Ω_5 m x.m h

. 18 1.87219710

ω Ω_4 M S

5 ω BH 10 .M S

ω BH M BH

=

. 19 4.55727410 . 19 6.9805610

( YHz)

. 20 1.06924110

. 15 5.19263810 . 17 3.69181510

ω Ω_5 m x.m h ω Ω_5 M S 1 . 5 ω h ω Ω_5 10 .M S 10 ω Ω_5 10 .M S

ω Ω_4 m x.m h

0.253004

ω Ω_4 M S 5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

=

0.253004

. 6.158585 8.43227510

14

. 9.433354 1.81667910

15

14.44945

0

Harmonic Cut-Off Freq. vs Radial Disp.

Harmonic Cut-Off Frequency

rS MS

R BH M S

ω Ω _3 R bh , m x .m h ω Ω _3 R bh , M S 5 ω Ω _3 R bh , 10 .M S 10 ω Ω _3 R bh , 10 .M S

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

269

www.deltagroupengineering.com

iii. “ωΩ_6, ωΩ_7, ωPV_1” ω Ω_6 M BH

ω Ω_5 M BH n Ω_4 M BH

ω PV_1 M BH

ω Ω_4 M BH

ω Ω_7 M BH

n Ω_5 M BH

ω Ω_6 M BH ω Ω_7 M BH

ω Ω_6 m x.m h

ω Ω_7 m x.m h

. 42 1.87219710 . 42 1.87219710

ω Ω_6 M S

ω Ω_7 M S

. 38 6.93112610 . 4 1.29804810

5 ω Ω_6 10 .M S

5 ω Ω_7 10 .M S

10 ω Ω_6 10 .M S

10 ω Ω_7 10 .M S

ω PV_1 m x.m h

=

. 37 3.61189510

( Hz)

0.693113

. 37 6.93112610 . 1.00503110

6

1

ω PV_1 M S =

5.

ω PV_1 10 M S

. 33 1.8727810 . 5.21112310

37

1

.

ωh

10 ω PV_1 10 .M S ( Hz)

= 5.103269

. 42 1.45002610

10 ω PV_1 10 .M S

Fundamental Freq. vs Radial Disp. rS MS

R BH M S

Fundamental Frequency

ω Ω _3 R bh , m x .m h n Ω _2 R bh , m x .m h ω Ω _3 R bh , M S n Ω _2 R bh , M S 5 ω Ω _3 R bh , 10 .M S 5 n Ω _2 R bh , 10 .M S ω Ω _3 R bh , 10 n Ω _2 R bh , 10

10 . MS

10 . MS

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

270

www.deltagroupengineering.com

9. “TL” M .c

E( M )

2

n γ ω , M BH

1. n γ ω , M BH 2 E g( ω )

h .ω

E γ( ω )

E g ( ω ) E x.E γ ( ω )

E M BH

E M BH

n g ω , M BH

E γ( ω ) E M BH

E M BH E x.E γ ( ω )

E m x.m h = 6.616163 ( GJ)

2 .E γ ( ω )

n γ ω Ω_4 m x.m h , m x.m h n g ω Ω_4 m x.m h , m x.m h

=

Ex

= 6.616163 ( GJ)

P g ω Ω_4 m x.m h

n γγ( M )

E g ω Ω_4 m x.m h

= 1.240531 ( GJ)

P γ ω Ω_4 m x.m h

c

= 8.275929 ( Ns )

2 .n gg ( M )

T Ω _3( r , M )

T L r S λ x.λ h , m x.m h

10 10 T L r S 10 .M S , 10 .M S

=

m γγ m gg

3.195095

=

6.39019

1

P g( ω )

10

=2

E g( ω ) c

45 .

T L( r , M )

ω Ω _3( r , M )

n gg ( M )

eV

E( M ) m gg

n gg ( M ) .T Ω _3( r , M ) n g ω Ω _3( r , M ) , M

9 10 .yr

. 13 4.10173110 . 13 4.10173110

T L r uq , m uq

. 13 4.10173110

T L R BH M S , M S

10 10 T L R BH 10 .M S , 10 .M S

s

E g ω Ω_4 m x.m h E γ ω Ω_4 m x.m h

= 2.481061 ( GJ)

= 4.137964 ( Ns )

. 13 4.10173110

T L R BH λ x.λ h , m x.m h

5 5 T L R BH 10 .M S , 10 .M S

43 .

. 13 4.10173110

T L r S M S ,M S 5 5 T L r S 10 .M S , 10 .M S

2 .E M BH 2 n γ ω , M BH .E γ ( ω )

2.666667

n g ω Ω_4 m x.m h , m x.m h .E g ω Ω_4 m x.m h

E γ( ω )

1. n γ ω , M BH 2

5.333333

= 6.616163 ( GJ)

P γ( ω )

ω Ω_4 M BH

T Ω_4 m x.m h = 5.341319 10

n γ ω Ω_4 m x.m h , m x.m h .E γ ω Ω_4 m x.m h

E γ ω Ω_4 m x.m h

1

n g ω , M BH

E g( ω )

1. n γ ω , M BH 2

E g( ω )

T Ω_4 M BH

=

. 13 4.10173110 . 4.10173110

13

9 10 .yr

T L r ε, m e T L r π, m p T L r ν,mn

. 13 4.10173110

271

. 13 4.10173110 =

. 13 4.10173110 . 4.10173110

13

9 10 .yr

. 13 4.10173110

www.deltagroupengineering.com

1 m γγ

h.

=

2

. 13 4.10173110 . 4.10173110

13

m γγ T L λ x.λ h , m x.m h . m gg h 2

9.

10 yr

m gg H0

71.

km . s Mpc

=

1

TL

1

h m γγ

. T L.H 0 = 2.97830810

12

10. “ωg, ngg” T PV n PV, r , M

T g n PV, r , M

n g ω PV n PV, r , M , M

n g ω PV n PV, r , M , M T PV n PV, r , M

T PV n PV, r , M

ω PV n PV, r , M .n g ω PV n PV, r , M , M

ω PV n PV, r , M .n g ω PV n PV, r , M , M

ω g n PV, r , M

n g ω PV n PV, r , M , M

ω g n PV, r , M

1 ω PV n PV, r , M . .n γ ω PV n PV, r , M , M 2

E( M ) 1 ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M

1 E( M ) ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M 1 E( M ) ω PV n PV, r , M . . 2 h .ω PV n PV, r , M ω g m x.m h

E( M ) 2 .h

=

10 ω g 10 .M S

. 56 1.34855310 . 61 1.34855310

E MS 1 . 5 m gg E 10 .M S 10 E 10 .M S

M .c 2 .h

2

n gg ( M )

5 n gg 10 .M S 10 n gg 10 .M S

T L.ω g ( M )

. 72 6.46222510

n gg M S

( YHz)

. 66 1.34855310

E m x.m h

ω g( M )

n gg m x.m h

. 18 4.99252510

ωg MS 5 ω g 10 .M S

1 E( M ) ω PV n PV, r , M . . . 2 h ω PV n PV, r , M

=

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

. 72 6.46222510 =

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

272

www.deltagroupengineering.com

11. BH’s r0

c

1

9 r 0 = 13.772016 10 .Lyr

H0

5

St G.

r ω ω Ω_3 , M

M

ω VL λ VL

c

ω VL( 750 ( nm ) )

9

λ VL

ω VL( 400 ( nm ) )

ω Ω_3 r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

5 ω Ω_3 r 0 , 10 .M S

5 ω Ω_3 r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

10 .

218.810356

410.269418

. 4 6.84370610 . 4 3.64997710

ω Ω_3 r 0 , 10 M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

4 ω Ω_3 1.63.10 .r 0 , M S

4 ω Ω_3 5.052.10 .r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

6 5 ω Ω_3 1.63.10 .r 0 , 10 .M S

6 5 ω Ω_3 5.052.10 .r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

ω Ω_3 1.63.10 r 0 , 10 M S

10 .

8.

ω Ω_3 5.052.10 r 0 , 10

ω VL( 400 ( nm ) )

10 C Ω_J1 r 0 , 10 .M S

( THz)

. 3 5.29883310 . 3 = 2.82604410

10 .

MS

5 C Ω_J1 r 0 , 10 .M S

749.481145

27.355887

ω Ω_3 r 0 , M S

C Ω_J1 r 0 , M S

399.723277

= 2.118067 ( EHz)

10 ω Ω_3 r 0 , 10 .M S

8.

=

0.163994

5 ω Ω_3 r 0 , 10 .M S

ω Ω_3 r 0 , 10

H0

2

ω Ω_3

ω Ω_3 r 0 , M S

9 = 13.772016 10 .yr

0.999916 1.000078 = 0.999916 1.000078 0.999916 1.000078

10 .

MS

ω VL( 750 ( nm ) ) . 1.48429110

5

= 8.89809310 .

3

10

20 .

yJy

5.334267

1 10 C Ω_J1 r 0 , 10 .M S .

C Ω_J1 r 0 , M S 1

=

. 5 3.59381410 599.48425

5 C Ω_J1 r 0 , 10 .M S

273

www.deltagroupengineering.com

r ω ω VL( 400 ( nm ) ) , M S

r ω ω VL( 750 ( nm ) ) , M S

5 r ω ω VL( 400 ( nm ) ) , 10 .M S

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

10 r ω ω VL( 400 ( nm ) ) , 10 .M S

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω ω VL( 400 ( nm ) ) , M S

5 r ω 30.( EHz) , 10 .M S

10 r ω 30.( PHz) , 10 .M S

10 r ω 30.( EHz) , 10 .M S

C Ω_J1 r ω 30.( PHz) , 10 M S , 10 M S 10 10 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

C Ω_J1 r ω 30.( EHz) , 10 M S , 10 M S 10 10 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

=

. 6 2.95234410

0.741144

( Lyr)

16 .

yJy

= 2.12751776034345 .103 8.46980075872643 .10

3

.105 2.12751776034345

= 2.93002110 .

7

0.846980075872643

1.166462 116.646228

6 10 .Lyr

. 9 1.16646210 . 4 2.93002110

2.164916 . = 2.16491610

3

. 2.16491610

6

C Ω_J1 r ω 30.( EHz) , M S , M S 5.

0.239057

5

. 5 2.93002110

C Ω_J1 r ω 30.( PHz) , M S , M S 5.

10

=

.10 21.2751776034345 8.46980075872643

10 r ω 30.( EHz) , 10 .M S

5 r ω 30.( PHz) , 10 .M S

r ω 30.( EHz) , m x.m h

1.102778

5 r ω 30.( EHz) , 10 .M S

r ω 30.( EHz) , M S

r ω 30.( PHz) , m x.m h

28.979765

=

r ω 30.( PHz) , M S

5.

. 8 5.05271110 . 8 1.62975410

r ω ω VL( 750 ( nm ) ) , m x.m h

r ω 30.( EHz) , M S

5.

= 1.62975410 . 6 5.05271110 . 6

r ω ω VL( 400 ( nm ) ) , m x.m h

C Ω_J1 r ω ω VL( 750 ( nm ) ) , m x.m h , m x.m h

1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

. 4 5.05271110 . 4 1.62975410

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

C Ω_J1 r ω ω VL( 400 ( nm ) ) , m x.m h , m x.m h

9 10 .Lyr

9 . 9 2.2445.10 6.95860210

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

1. 10 ( Lyr) , 10 .M S = 1.031709 10

r ω 30.( PHz) , M S

= 2.2445.107 6.95860210 . 7

r ω ω VL( 750 ( nm ) ) , M S

1 . r ω ( 400 ( nm ) ) , 105 .M ω VL S r0 10 r ω ω VL( 400 ( nm) ) , 10 .M S

K PV

5 . 5 2.2445.10 6.95860210

10

29 .

10

14 .

yJy

8.618686 . = 8.61868610

3

. 8.61868610

6

yJy

3 10 .km

11.753495

7 C Ω_J1 r ω 30.( PHz) , m x.m h , m x.m h = 6.228302 10 .yJy

C Ω_J1 r ω 30.( EHz) , m x.m h , m x.m h = 2.479532 ( fJy )

274

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vii. Fundamental Cosmology 1. “Hα, HU” i. “AU, RU, HU” 5

C Ω_J1 r 1 , M 1

9

C Ω_J1( r , M )

M St J . 26 r

ln

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

λ y r 2, M 2

ln

9

5

M1

ln

.

M2

r2

r1 M1

r1

K U r 2, r 3, M 2, M 3

9

ln

ln

A U r 2, r 3, M 2, M 3

H U r 2, r 3, M 2, M 3

ln

.

r2

26

5

ln

ri

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

λ y r 2 , M 2 .r 3 λ y r 2, M 2 .M 3 2 5

λy M3 r . . 1 . 2 2 M2 λy r3

C Ω_J1 r 2 , M 2

λy M3 r . . 1 . 2 2 M2 λy r3

rf

r1

C Ω_J1 r 3 , M 3

26

26

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2 5

5

K U r 2, r 3, M 2, M 3

5

M2

26

ln n Ω_2 r 2 , M 2

C Ω_J1 r 2 , M 2

M1

C Ω_J1 r 2 , M 2

9

1

C Ω_J1 r 3 , M 3

9

1 2

9

7

.ln n Ω_2 r 2 , M 2

TL K U r 2, r 3, M 2, M 3

5

1

3.

M3

26

9

.

M2

r2

9

r3

R U r 2, r 3 , M 2, M 3

c .A U r 2 , r 3 , M 2 , M 3

9 A U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .yr

A U r 2, r 3 , M 2, M 3

9 R U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .Lyr

H U K λ .R o , λ x.λ h , K m.M G, m x.m h = 67.084304 1 . H U K λ .R o , λ x.λ h , K m.M G, m x.m h H0

km s .Mpc

1 = 5.515064 ( % )

275

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H U R o , λ x.λ h , M G, m x.m h H U R o , λ h , M G, m h

66.700842

H U K λ .R o , λ h , K m.M G, m h

70.06923

H U R o , λ x.λ h , M G, m x.m h 1 . H0

km . s Mpc

= 69.672169

6.055152

H U R o , λ h , M G, m h

1=

1.870184 ( % )

H U K λ .R o , λ h , K m.M G, m h

1.310944

1 km H U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 67.753267 . 3 s Mpc

H U K λ .R o , λ h , K m.M G, m h 1 H U K λ .R o , λ h , .K m.M G, m h 3

1=

H U K λ .R o , λ x.λ h , K m.M G, m x.m h

0.978843 0.987352

(%)

1 H U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

ii. “Hα” 3 .H ρm 8 .π .G 2

H α r 3, M 3

2.

. 61 H α λ x.λ h , m x.m h = 8.46094110

λx H α λ x.λ h , m x.m h . ωh

2. . . π G ρ m r 3, M 3 3

H α λ x.λ h , m x.m h

ωh

km s .Mpc

. 1 = 4.44089210

λx

14

. = 8.46094110

61

ωh λx km s .Mpc

(%)

iii. “ρU” 3 .H U r 2 , r 3 , M 2 , M 3 8 .π .G

ρ U r 2, r 3, M 2, M 3

2

ρ U K λ .R o , λ x.λ h , K m.M G, m x.m h ρ U K λ .R o , λ h , K m.M G, m h 3 .H 0

2

8.453235 = 9.222226

10

33 .

kg 3

cm

9.468862

8 .π .G

Hence, 8.45 ρ U . 10

33 .

kg

9.23

3

cm

276

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iv. “MU” M U r 2, r 3, M 2, M 3

V R U r 2 , r 3 , M 2 , M 3 .ρ U r 2 , r 3 , M 2 , M 3

. 52 ( kg ) M U K λ .R o , λ x.λ h , K m.M G, m x.m h = 9.28458610

2. “TU” K T r 2, r 3 , M 2, M 3

λ Ω_3 r 3 , M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

c

T0

ω Ω_3 r 3 , M 3

T W r 2, r 3, M 2 , M 3

T U r 2, r 3, M 2, M 3

H α r 3, M 3 H U r 2, r 3, M 2, M 3

2.725.( K )

. K W = 2.89776910

3

( m.K )

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3 K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3

T U K λ .R o , λ x.λ h , K m.M G, m x.m h = 2.724752 ( K )

1 . T U K λ .R o , λ x.λ h , K m.M G, m x.m h

. 1 = 9.08391310

3

(%)

T0

T U R o , λ x.λ h , M G, m x.m h T U R o , λ h , M G, m h

2.716201 = 1.199134 ( K )

T U K λ .R o , λ h , K m.M G, m h

1.202877

T U R o , λ x.λ h , M G, m x.m h 1 . T0

T U R o , λ h , M G, m h T U K λ .R o , λ h , K m.M G, m h

0.322893 1=

55.995089 ( % ) 55.857737

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 2.739618 ( K ) 3

T U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 0.542607 ( % )

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

277

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3. “TU → TU2” 9

c

c

λ Ω_3( r , M )

ω Ω_3( r , M )

9

2

M St G. 5 r

9

5

c c λ x. λ Ω_3 , mh H 2

1 . St G

c.

λx

8 . H α r 3, M 3 ln 3 H

K T( H ) .T W ( H )

ωh 8. . ln 3 λ x.H

ωh 8. . ln 3 λ x.H

2

.m h

1 . 2 St G λ x.m h

2

. c H

λ Ω_3

KW

T U2( H ) K T( H ) .T W ( H )

c λ x. mh λ Ω_3 , H 2

c λ x. , mh H 2

ωh 8. . ln 3 λ x.H

c λ x. , mh H 2

KW 9

c.

1 . 2 St G λ x.m h

2

. c H

9

c.

1 . 2 St G λ x.m h

2

. c H

5

ωh λ .m 8 .K W . . St . x h ln G 3 c 2 λ x.H

. H α = 8.46094110

km . s Mpc

. H c

5

ωh λ .m 8 KW. . St . x h T U2( H ) . ln G 3 c 2 λ x.H 9

. 8 . St G . λ x m h St T 5 3 .c 2 c

8 . 3 .c

2

9

61

9

3.

5

9

KW

λx

9

5

KW

KW

ωh

Hα

c.

T W( H)

ωh 8. . ln 3 λ x.H

λ Ω_3

9

H

2

K T( H )

5

1 . r St G M 2

c.

3 .ω h 4 .π .h c

5

2

. c 2

9

2

3.

2

. 8 . St G . λ x m h 5 3 .c 2 c

8 . 3 .c

3 .ω h 4 .π .h c

5

2

. c 2

2

. H c

5

9

.

λ x.m h

2

2

9

.

λ x.m h 2

2

9

. 8 .c . 3 . 3 ω h 3 .c 2 c5 4 .π .h

2

.

λ x.m h

2

2

278

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9

. 8 .c . 3 . 3 ω h 3 .c 2 c5 4 .π .h 9

4. 3

3

3 4 .c

.

9

3

λ x.m h

9

2

2

2

4.

3

3

6 5 2 .c

3

3 4 .c

3

3

4.

9

2

m λ . h. x π .h λ h

4. 3. . λ x c 3 4 π .h .G

T U2( H )

2

2 λ . x .c π .h G

9

3 λx 4. 3. . c 2 3 3 4 π .c .λ h

K W .St T .ln

ωh λ x.H

.

λ x.m h .ω h π .h

3

4.

3

3

6 5 2 .c

.

4. 3

3 4 .c

4.

π .h

3

9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h

9

2

4. 3

3

.

mh λ x . π .h λ h

3 3 . λx c. 4 π .h .G

2

2

2 9 . 95 St T = 6.35557910

s

5 9

m

T U2 H U K λ .R o , λ x.λ h , K m.M G, m x.m h

. H5

3 4 .c

3

2 λ . x .c π .h G

3

St T

9

2

λ x.m h .ω h

9

2

2

9

9

2

= 2.72475246336977( K )

4. “TU2 → Ro, MG, HU2, ρU2” ∆R o

0.5.( kpc )

T U2 H U2 R o , M G

T U2 H 0

T U2 H U2 R o

1 . T U2 H U2 R o , M G T0

= 2.724752 ( K )

1 T U2 H U2 R o , .M G 3

T U2 H U2 R o

H U K λ .r , λ x.λ h , K m.M , m x.m h

H U2( r , M )

=

2.739618

( K)

2.810842

1 ∆R o , .K m.M G 3 1 ∆R o , .K m.M G 3

=

2.733025 2.741859

. 1 = 9.08391310

T U2 H U2 R o

∆R o , M G

T U2 H U2 R o

∆R o , M G

∆T 0

( K)

Computational environment initialisation values →

=

2.720213 2.729021

3

(%)

( K)

0.001.( K )

r x1

1

m g1

1

r x2

1

m g2

1

Given T U2 H U2 r x1.R o , M G T U2 H U2 R o , m g1 .M G

T0

∆T 0

T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g2 .M G

r x1

r x1

r x2

r x2

m g1 m g2

Find r x1, r x2, m g1 , m g2

m g1

T0

∆T 0

0.989364 =

m g2

279

1.017883 1.057292 0.911791

www.deltagroupengineering.com

T U2 H U2 r x1.R o , M G T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g1 .M G T U2 H U2 R o , m g2 .M G

2.724 =

2.726

R o.

( K)

2.724

r x1

=

r x2

7.914908 8.143063

( kpc )

2.726

. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910

r x1 m g1

1.063645 5.729219

1=

r x2 m g2

1.788292

8.820858

(%)

Given T U2 H U2 r x1.R o , m g1 .M G T U2 H U2 r x1.R o , m g2 .M G

∆T 0

T0

T U2 H U2 r x2.R o , m g1 .M G T U2 H U2 r x2.R o , m g2 .M G

r x3

r x3

r x4

r x4

m g3

Find r x1, r x2, m g1 , m g2

=

2.724

=

2.726

1.013348 0.977007 0.977007

m g4

T U2 H U2 r x3.R o , m g3 .M G T U2 H U2 r x4.R o , m g4 .M G

∆T 0

0.984956

m g3

m g4

T0

R o.

( K)

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410

r x3

=

r x4

7.879647 8.106786

r x3 m g3 r x4 m g4

1=

( kpc )

1.50441 2.29934 1.334822 2.29934

(%)

Hence, if “T0” is exactly correct (i.e. zero experimental uncertainty); “Ro”, “MG” and “HU2” may be approximated as follows, Given T U2 H U2 r x1.R o , m g1 .M G r x5 m g5

Find r x1, m g1

r x5.R o = 8.107221 ( kpc )

T0 r x5 m g5

m g5 .

H U2 r x5.R o , m g5 .M G = 67.095419

=

MG

1.013403

T U2 H U2 r x5.R o , m g5 .M G

1.052361

. = 6.31416710

11

MS

r x5 m g5

1=

1.340256 5.236123

= 2.725 ( K )

( %)

km . s Mpc

ρ m R U K λ .R o , λ x.λ h , K m.M G, m x.m h , M U K λ .R o , λ x.λ h , K m.M G, m x.m h

= 8.453235 10

33 .

kg 3

cm

280

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5. “UZPF”

Ω

ρ ρc

Ω EGM

Ω PDG

Ω EGM Ω PDG

3 .H U2( r , M )

ρ U2( r , M )

ρ U2 R o , M G = 8.453235 10

8 .π .G

Ω EGM = 1.000331

ρ U2 R o , M G

. Ω ZPF = 3.31400710

1

U ZPF

3 .c . H U2 R o , M G Ω ZPF . 8 .π .G

13 .

U ZPF = 2.51778 10

Ω EGM

U ZPF = 251.778016

Pa

fJ

U ZPF = 251.778016

U ZPF = 842.934914

3

kg 3

Ω PDG Ω m Ω γ .. Ω ν

Ω ZPF

2

= 0.997339

33 .

cm

ρ U2 r x5.R o , m g5 .M G

1.003

2

m

2

ΩΛ

4

U ZPF = 251.778016( fPa )

yJ

U ZPF = 0.251778

3

mJ 3

mm

km

EJ

. 12 U ZPF = 7.39723510

AU

3

YJ pc

3

YJ

. 41 U ZPF = 6.60189810

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

3

viii. Advanced Cosmology 1. “nΩ_2 → nΩ_6” 9

n Ω_6( r , M )

. . 3 3 . π mh . K λ r . 16 2 KmM λh

9

7

9 9

9

1 6.

. 3 3 .K λ π m h . r 16 M λh 2 2 . 3 3 . π .π m h . r 16 2 M λh 2

7

7

9

n Ω_6( r , M )

3 .π . m h 4

M

9

6

1

.

π .m h

. r M λh

3 ( 3 .π ) . m h . r 18 M λh 2 1

3

3

3

3 . π 16 2 2

. . 3 3 . π mh . K λ r . 16 2 KmM λh

7

3

3 . π 16 2 2 9

7

3

7 9

6. . 3 3 .K λ π m h . r 16 M λh 2

7

6

.

π .m h

. r M λh

3 ( 3 .π ) . m h . r 18 M λh 2

7

9

1. 4

7

9

2 . 3 3 . π .π m h . r 16 2 M λh 2

3 mh . r ( 3 .π ) . M λh

7

7

7

9

. r λh

9

281

www.deltagroupengineering.com

2. “KU2 → KU3”

K U2( r , M )

ln

Kλ λx

7

5

3

9

.ln n Ω_6( r , M )

.

mh

26 9

. r λh

4 .M

K U3( r , M ) ln ( 3 .π )

7

5

2 . ln n Ω_6( r , M ) 256

3.

3

Kλ

( 3 .π )

λx

7 18 6.

5

2

256

26 9

. r λh

M

.

4

9

mh

9

1

5 7 18 6.

7

5

3. “HU2 → HU3, TU2 → TU3” K U2( r , M )

H U3( r , M )

5

TL 7 5

7 18 5 6 2

ln ( 3 .π ) .

256

7

.ln n Ω_6( r , M )

3.

9

mh M

7 18 5 6 2

9

. r λh

3

1

26

ln ( 3 .π ) .

.ln

3

9

3 .π . m h

256

4

9

. r λh

M

5

7

.

mh M

26

9

. r λh

9

7 7

5

1 . 18 . . 6 . 2 ( 3 π ) ln 256

K U3( r , M )

ln

T U3( r , M )

T U2 H U3( r , M )

1

1

3 ( 3 .π ) . m h

9

4

3

. r λh

M

5

7 9

.

mh M

26

9

9

. r λh

4. “HU3 → HU4, TU3 → TU4” K U3( r , M )

H U4( r , M )

5

T U4( r , M )

TL

T U2 H U4( r , M )

5. “HU4 → HU5, TU4 → TU5”

µ

1

H U5( r , M )

3

9 3

m γγ

( 3 .π )

.ln

h

7 .µ .

µ

2

32

256

µ

m . .ln ( 3 π ) . h 4 M

µ

7 .µ

2

. r λh

9

λx

4 3 1. St T . . 3 4 c5 π .λ 2 h

2

St T

9

9

4 . 3 3 4

3

2 7 .µ

5

.

mh M

5 .µ

2

. r λh

2 26 .µ

9

.1 . c

5

2

λx π .λ h

282

2

9

4 . 3 3 4

3

.1 . c

5

2

λx π .λ h

2

4 .1 . λx 3 c5 π .λ 2 h 6

2

www.deltagroupengineering.com

ωh λx

6

H α r 3, M 3

Hα

2

6 ( 4 .µ ) .

ωh

5

π .H α .λ h

c

St T

1 c

St T

4 .1 . λx 3 c5 π .λ 2 h

3

1 c

µ

2

c

1

2

9

9

.

1 π .H α

. . 4µ λh

6 ( 4 .µ ) .

2 .µ

.

π .H α .λ h

3

. . 4µ λh

5

π .H α .λ h

c

c

π .H α .λ h

1

2

9

9

9

1

2 .µ

3

.

1 π .H α

2

T U5( r , M )

π .H α

. . 4µ λh

KW c

µ

.ln

2

2

1

3

6

c

1

ωh

6

3

2

( 4 .µ ) .

6 ( 4 .µ ) .

2

1

2

1.

3

c

2

1 π .H α

3

6

1

2

9

3

9

1 c

.

1 π .H α

. . 4µ H U5( r , M ) λ h Hα

. . 4µ λh

6

2

. . 4µ λh 2 .µ

.

3

2 .µ

1 π .H α

2

. 2

.H ( r , M ) 5 µ U5

6. “HU3, HU4, HU5, TU3, TU4, TU5” H U3 R o , M G H U4 R o , M G H U5 R o , M G T U3 R o , M G T U4 R o , M G T U5 R o , M G

H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G

= 67.084304 67.095419

T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G

= 2.724752 2.725 ( K )

67.084304 67.095419 67.084304 67.095419

2.724752 2.725 2.724752 2.725

1 . H U4 R o , M G H0 H U5 R o , M G

H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G

T U3 R o , M G

T U3 r x5.R o , m g5 .M G

1 . T U4 R o , M G T0 T U5 R o , M G

T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G

H U3 R o , M G

km . s Mpc

5.515064 5.499409 1=

5.515064 5.499409 ( % ) 5.515064 5.499409

1=

. 9.08391310

3

. 8.37394610

9

. 9.08391310

3

. 8.37394610

9

. 9.08391310

3

. 8.37394610

9

(%)

7. Time dependent characteristics T U3( H ) K W .St T .ln

Hα

9

H H β .H α

. H5

H

T U3 H β

K W .St T .ln

1 Hβ

. H .H β α

5 .µ

2

1

1 . d K W .St T .ln H β .H α dH β Hβ

Hβ

H β_min ,

2 5 .µ

H β_max H β_min 1 .10

5

0

Hβ e

2 5 .µ

.. H β_max

T U3 H β

283

H β_min

10

H β_max

1

K W .St T .ln

1 Hβ

6

. H .H β α

5 .µ

2

www.deltagroupengineering.com

1

T U3 e

1

2 5 .µ

. 31 ( K ) = 3.19551810

2 5 .µ .

. 61 H α = 1.39858410

e

km s .Mpc

1

e

2 5 .µ .

1

= 2.206287 10

H U2 R o , M G

42 .

s

= 7.928705 10

T U2 H α

61

T U3( 1 )

Hα

Hα

=

0

( K)

0

Computational environment initialisation value → H β2 56.4503086205567 Given T U2 10 H β2

10

H β2

273.( K )

.H α

H β2 = 56.450309

Find H β2

H β2

1

.H α

. = 1.02858610

14

10

(s)

10

H β2

H β2

1

.H α

km . s Mpc

.H = 2.99992310 . 5 α

6 = 3.259461 10 .yr

T U2 10

H β2

.H α = 273 ( K )

See Fig. 4.22, 4.23. 1 H β .H α

t

T U3 H β

1 . 1 K W .St T . t t5

d T U4( t ) dt

1

K W .St T .ln

µ

Hβ

t1

e

1

. 5 .ln H .t .µ 2 α

2

1 . 1 K W .St T . 2 5 t t

dT2 dt2 ( t )

µ

µ

3

d t3

T U4( t )

1

2 5 .ln H α .t .µ 5 .µ

2

µ

2

K W .St T .

. 5 .µ 2 . ln H .t . 5 .µ 2 α

ln H α .t . 5 .µ

2

2 2 5 .µ . ln H α .t . 5 .µ

1 . 1 K W .St T . 3 5 t t

. 5 .ln H .t .µ 2 α

1

0

.t

1

2

1

10 .µ

. 5 .µ

2

1

2 2.

5 .µ

2

t

d

1 . 1 K W .St T . 5 t t

t

1 . 1 T U4( t ) K W .St T . 2 5 d t2 t t d

K W .St T .

dT dt ( t )

Hα

1 T U4( t ) K W .St T .ln H α .t . t

2

1 2 5 .µ .

2

5 .µ

. H .H β α

µ

5 .µ

2

1

2

1

1

2

0

t2

e

2

2 2 5 .µ . 5 .µ

1 1

. 1 Hα

1

.t2

2

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

284

3

2

2 2 15.µ . 5 .µ

2

2

www.deltagroupengineering.com

K W .St T .

1 . 1 t

3

t

µ

5

2 2 15 .µ . 5 .µ

t3

e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

3

dT2 dt2 t 2 = 0

s

4.196153

10

2

s

6.205726

1

=

dT2 dt2 t 1

=

dT2 dt2 t 2

. 114 2.02615310

K

0

2

s

. 112 8.77595210

dT2 dt2 t 3

2

K 3

0

K

. 72 1.05719310

s

. 71 9.25283810

dT3 dt3

. 116 7.65967810

2

. 74 1.32321810

Hα

dT dt t 3

Hα

0

1

dT dt t 2

t3

dT2 dt2

2

s

dT dt t 1

42 .

2 2 15.µ . 5 .µ

dT3 dt3 t 3 = 0

dT dt

0.364697

t2

K s

2.206287

2

. 1 Hα

2

1

=

2 2 15.µ . 5 .µ

2

2

K

dT dt t 1 = 0

t1

2

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ 3 2 K W .St T . 2 5 .µ . 3 t t

dT3 dt3 ( t )

Hα

3

1 . 159 6.22716710

Hα

dT3 dt3 t 1

=

. 156 3.77545710

K

. 1.45285710

s

155

dT3 dt3 t 2

3

0

dT3 dt3 t 3

T U2 H α T U2 T U2 T U2

1

0

t1 =

1 t2

. 31 3.19551810 ( K)

. 31 3.03432210

4 . 34 ( K ) T U2 10 .H α = 7.41414610

. 31 2.83254210

1 t3

4 10 .H α

1

= 0.364697 10

46 .

s

T U2 H U2 R o , M G

= 2.724752 ( K )

See Fig. 4.24 – 4.35.

285

www.deltagroupengineering.com

Hα

T U3( H ) K W .St T .ln

Hα . 2 d d .H5 µ K W .St T .ln T U3( H ) dH dH H

9

. H5

H

5 .µ Hα . 2 d .H5 µ K .St . H K W .St T .ln W T dH H H 5 .µ

H K W .St T .

d d T U3( H ) . t dH d T U4( t )

5 .µ

5 .µ

2

H

µ

( H .t )

.

2 5 .µ .

t

µ

2

. 5 .ln H .t .µ 2 α

2

H

1

Hα

1

.µ 2

1

Hα

. 5 .ln

.µ 2

1

H µ

2

. 5 .ln H .t .µ 2 α

2 5 .ln H α .t .µ

2 5 .ln H α .t .µ

5 .ln

1 . 1 t t5

1

2

H

1

t . . 5 .µ 2 . . H α . 2 (H t) 5 ln µ H H

1

. 5 .ln H .t .µ 2 α

.µ 2

H

1

. 5 .ln H .t .µ 2 α

.µ 2

1

H

H

1 . 1 t t5

H

2

Hα

. 5 .ln

H

d H dt

µ

Hα

. 5 .ln

5 .µ

.µ 2

.µ 2

H

H

H

1 . 1 t t5

K W .St T .

Hα

. 5 .ln

Hα

. 5 .ln

2

1 . 1 K W .St T . t t5

2

H

K W .St T .

2

d H dt

1

1

1

2

t

H

1

1 H γ .H α

H

5 .ln H α .

Hα

d H dt H α.

dH dt H γ

1 H γ .H α

5 .µ

. 2

.

5 .ln

1 H γ .H α

Hα

.µ 2

1

1

Hα

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ . 5µ Hγ

2 H α .H γ d . 5 .ln 1 .µ 2 2 dH γ Hγ 5 .µ Hγ

1 .µ 2 . Hγ Hα

1

Hα

1

Hγ

d dH dt H γ dH γ

2 H α .H γ d . 5 .ln 1 .µ 2 2 dH γ Hγ . 5µ Hγ

2

2 5 .µ

. 5 .ln 1 .µ 2 . 1 Hγ

5 .µ

2

1

1

Hα Hγ

2

2 5 .µ

. 5 .ln 1 .µ 2 . 1 Hγ

5 .µ

2

1

0

1

Hγ e

2 2 5 .µ . 5 .µ

286

1 1

t4

e

2 2 5 .µ . 5 .µ

1

. 1 Hα

www.deltagroupengineering.com

dH2 dt2

d

2

d

H

d t2

2

H

d dt

H

d t2

2 5 .ln H α .t .µ

. . 2

5µ . ( H .t ) t

5 .ln

Hα

.µ 2

1 1

H

H

d dt

. . 2

5µ . ( H .t ) t

2 5 .ln H α .t .µ

Hα

5 .ln

1

H

2 2 5 .µ . ln H α .t . 5 .µ

.

. 2

.µ 2

5µ .2 ( H .t ) t

1

Hα

5 .ln

H

d

2 H H α.

1 H γ .H α

2 5 .µ . ln H α .

5 .µ

. 2

.

1 H γ .H α

H α .H γ d . 5.µ 2 . ln 1 . 5 .µ 2 2 dH γ Hγ 5 .µ Hγ 3

2 2 5 .µ . 5 .µ

4

2 2 2 5 .µ . 5 .µ . 5 .µ

Let:

Hγ Hβ

1

1

1

Hα

2

ln

1

1

1

2

2

1

1

1

3 H α .H γ . 5 .µ 2 . 5.µ 2 . 5.µ 2 .ln 1 2 Hγ 5 .µ Hγ

1

Hγ

1

2 .ln

t5

4

2

ln

1 Hγ

1

2 .ln

1

4

2

Hγ

0

Hγ

2 2 5 .µ . 5 .µ 2

.µ 2

1

Hα

2 1

1 . 5 .µ 2 . Hγ Hα

5 .ln

2

3 H α .H γ . 5 .µ 2 . 5 .µ 2 . 5 .µ 2 .ln 1 2 Hγ 5 .µ Hγ

Hγ e

2

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ . 5µ Hγ

dH2 dt2 H γ

.µ 2

2

H

Hα

d t2

1

e

4

2 2 2 5 .µ . 5 .µ . 5 .µ

2 1

2

. 1 Hα

η

Computational environment initialisation value → η

4.595349

Given

dH dt

H U2 R o , M G Hα

H U2 R o , M G

η

η

1

Find( η )

287

www.deltagroupengineering.com

1

t1

e

2 5 .µ .

Hα 10 .µ

t2

1

e

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα

2.206287 2.206287 4.196153 4.196153

2 2

3

= 6.205726 6.205726

. 1 Hα

e

t5 e

s

8.385263 8.385263

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

42 .

20.932666 20.932666

1

t4

10

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα 2

1

2

. 1 Hα 1

dH dt t 1 .H α

1

dH dt e

5 .µ

10 .µ

dH dt t 2 .H α

1

dH dt e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH dt t 3 .H α

1

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 3

2

1

dH dt t 4 .H α

1

dH dt e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH dt t 5 .H α

η = 4.595349

1

dH dt e

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

. 68 7.50137510

. 68 7.50137510

. 83 9.06689310

. 83 9.06689310

= 1.22575310 . 84

. 84 1.22575310

. 84 1.55351810

. 84 1.55351810

. 84 1.38436210

. 84 1.38436210

2

Hz

2 1

H U2 R o , M G

2

η

= 4.726505 10

36 .

2

Hz

Hα

288

www.deltagroupengineering.com

1

dH2 dt2 t 1 .H α

1

dH2 dt2 e

2 5 .µ

10 .µ

dH2 dt2 t 2 .H α

1

2

1

2 2 5 .µ . 5 .µ

dH2 dt2 e

1

2 2 15 .µ . 5 .µ

dH2 dt2 t 3 .H α

1

dH2 dt2 e

. 125 8.50679910

0

0

2

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 125 8.50679910

=

2

3

1

dH2 dt2 t 4 .H α

1

dH2 dt2 e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH2 dt2 t 5 .H α

dH2 dt2

1

dH2 dt2 e

1

. 125 1.16257810

. 124 8.2461110

. 124 8.2461110

. 125 1.33162810

. 125 1.33162810

3

Hz

2

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1.16257810

125

2

1

η

H U2 R o , M G

3

= 0 Hz

Hα

See Fig. 4.36 – 4.45. H

d d 1 H dt dt t

1 t

1 t

2

H

2

Hα

d H dt

H

=1 η

dH dt 1

η

dH dt 1

η

H U2 R o , M G

dH dt

=

. 61 8.46094110 67.084257

km . s Mpc

Hα

See Fig. 4.46, 4.47. Checking errors yields,

H

d H dt

( H .t )

5 .µ

. 2

2 5 .ln H α .t .µ

.t

5 .ln

Hα

.µ

2

5 .µ

1

H

2

. 5 .ln

H

Hα

.µ 2

2 5 .ln H α .t .µ

1 dH

H

t

1

5 .µ

2

1 dt

.t

H 5 .µ

H

2

. 5 .ln

H

5 .µ

H

2

.ln

Hα

.µ 2

H

Hα H

5 .µ

1 dH H

2

.ln

2 5 .ln H α .t .µ

Hα H

ln H α .t t

5 .µ

2

t

has the solution:

H

5 .µ

2

.t

1 dt

ln H α .t t

5 .µ

2

1 t

289

www.deltagroupengineering.com

T U2

η

T U2

dH dt 1

dH dt

T U3( 1 )

T U3

dH dt

e

dH dt e

dH dt

T U2

dH dt

dH dt

1

T U3 e η

2

2 2 5 .µ . 5 .µ

1

T U3 e

0 2

2 2 5 .µ . 5 .µ

0

. 31 3.19551810 . 31 2.97174510

1

2 2 15 .µ . 5 .µ

2 3

2

= 3.18632310 . 31 3.03432210 . 31 ( K )

1

. 31 2.83254210 . 31 3.18071410

η 2

5 .µ

10 .µ

1

2

T U3 e

T U3

Hα

2

2 2 2 5 .µ . 5 .µ . 5 .µ

η

H U2 R o , M G

1

T U2

Hα

η

2 2 2 5 .µ . 5 .µ . 5 .µ

e

( K)

T U3( 1 )

2 2 15 .µ . 5 .µ

T U2

0 2.724752

H U2 R o , M G

2 5 .µ

10 .µ

0 2.724751

=

dH dt 1

1

T U2

Hα

η

T U2

T U2

η

H U2 R o , M G

2 3

2.724751

2.724752

2

H U2 R o , M G Hα

η

2 5 .µ

e

1 = 7.002696 ( % )

1

T U3 e

5 .µ

2

See Fig. 4.48, 4.49.

dH dt H γ

2 H α .H γ . 5 .ln

0

Hγ t7

t 7 = 2.206287 10

=1

t1

1

.µ 2

1

Hγ e

0

Hγ

2 5 .µ

1

1

42 .

s

Hγ Hβ

η

η

2 5 .µ

t7

e

2 5 .µ .

1 Hα

ln H γ

ln t 7 .H α

1

ln H β

ln t 1 .H α

1

1

1

ln t 7 .H α

1

ln t 1 .H α

1

=1

2 H α .e

e

2

1

. 5 .ln 2 5 .µ

1 5 .µ

5 .µ

2

.µ 2

. 68 Hz2 1 = 7.50137510

1

e

5 .µ

2

290

www.deltagroupengineering.com

4

2.

dH dt H γ

Hα Hγ

4 .µ

2

. 5 .ln 1 .µ 2 Hγ

2.

1

5.

Hα e

1

5 .ln

5 .µ

2

1

4

Hα e

. 68 Hz2 1 = 7.50137510

1

e

2.

.µ 2

5.

5 .ln e

5 .µ

2

.µ 2

1

5 .ln

2

1 = 0 Hz

. 1 3

1 1

5.

2

1 =0

2

3

e

1st Derivative of the Hubble Constant

10 42 t 1 .10 8

Scaled Derivative (Hz^2)

6

4 dH dt H β 10

η

79 2

0 0

2

4

2.20624

2.20625

2.20626

2.20627

2.20628

2.20629

2.2063

2.20631

2.20632

η H β .H α

1

2.20633

2.20634

2.20635

2.20636

2.20637

2.20638

.1042

Scaled Cosmological Age (s)

Hα =

dH dt t 4 .H α

1

Hα

H U2 R o , M G H0

=

1

=

. 123 1.47916710

71

2

km s .Mpc

km s .Mpc

1

2

. 1.55351810

84

Hα dH dt

H U2 R o , M G 2

H0

291

. 3.84599410

61

1

. 84 7.51858710

=

. 61 8.46094110

=

dH dt t 4 .H α

dH dt t 4 .H α

. 123 7.15875210

67.084304

Hα ( Hz)

Hα

1

2

dH dt t 4 .H α

. 1.24640210

42

= 2.199936

dH dt t 4 .H α Hα

. 42 2.74200410

km . s Mpc

2

Hz

2

t 4 .H α 2

=

= 4.839718 1

. 3 4.50030410 3 5.041.10

km s .Mpc

2

www.deltagroupengineering.com

1 Hα t1

3.646967 = 22.062867

10

1

43 .

s

9 = 14.575885 10 .yr

H U2 R o , M G

209.326658

t4

A U K λ .R o , λ x.λ h , K m.M G, m x.m h .H U2 R o , M G = 1 T U2 H α T U2 t 1

1

T U2 t 4

1

0 . 31 3.19551810 . 31 = 2.05994510 2.724752

T U2 H U2 R o , M G T U3 H U2 R o , M G .H α

c.

( K)

t1 t4

=

6.614281

10

62.754553

34 .

m

2.724752

1

2.725

T0

t 16.326238 c . 1 = 154.899031 λh t4

c H U2 R o , M G

9 = 14.575885 10 .Lyr

R U K λ .R o , λ x.λ h , K m.M G, m x.m h .H U2 R o , M G

=1

c

8. History of the Universe T U2 H α = 0 ( K )

1

T U2 10 T U2

10 .

T U2

1

. 15 ( K ) = 3.43308810

31

1 13 .

1 9.

5 .10 ( yr )

( K)

1

T U2 10

T U2

(s)

1

. = 1.92400510

28

34 .

( K)

(s)

. 9 ( K) = 1.01325410

2.

10 ( s ) = 978.724031 ( K )

10 ( s ) T U2

. = 3.19551810

t1

T U2

1 9.

= 11.838588 ( K )

10 ( yr ) = 4.898955 ( K )

T U2 H U2 R o , M G

292

= 2.724752 ( K )

www.deltagroupengineering.com

T U2

1 . 1 (s) 1

1 .( day )

T U2

T U2

1 31.( day )

T U2

T U2

1 .( yr ) 1

T U2

2 10 .( yr )

1

T U2

1

1

. 7 521.528169 2.52413210

41.823796

. 4 8.07751510

11.838588

9 10 .( yr )

. 4 2.29089210

3.35005

1

. 3 6.49496110

0.947724

( K)

10 .

( yr )

1 11 .

10

1

= 1.00307810 .

1

10

=

147.71262

6

8 10 .( yr )

T U2

1 . 116 ( day )

. 6 3.86401510

1

T U2

10 ( yr )

T U2

. 10 1.84076810 . 3 1.2497710

7.

T U2

4.

6 10 .( yr )

10 ( yr )

10 ( yr )

T U2

1

T U2

3.

T U2

5.

10 ( yr )

T U2

1

1

T U2

( yr )

. 6 1.87808710 . 3.98831410

( K)

7

TL

9. “ML, rL, tL, tEGM” 5

C Ω_J1 r 1 , M 1

M1

C Ω_J1 r 2 , M 2

26

r1

R EGM

M 2 M 1.

26

r2

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

. 71 ( kg ) M L = 4.86482110

tL

5

M2

rL

rL

. t L = 7.6372910

19

c

M EGM 2 R EGM.c

=1

t EGM

ML

.

r1

K m.M G.

M EGM

r2 r1

R EGM K λ .R o

5 5

.

R EGM K λ .R o

M U K λ .R o , λ x.λ h , K m.M G, m x.m h

A U K λ .R o , λ x.λ h , K m.M G, m x.m h

2 .G

M EGM

5 5

. 19 109 .Lyr r L = 7.6372910

R BH M L

9 10 .yr

r2

t EGM

=1

R EGM c

2 R EGM.c

2 .G

t EGM

R EGM c

M L M EGM rL

tL

R EGM t EGM

293

www.deltagroupengineering.com

ML M EGM

. 18 5.23967510

rL

tL

= 5.23967510 . 18

R EGM

. 18 5.23967510

tL

. 6 = 1.86196810

TL

t EGM

10. Radio astronomy 9 9

9

5

M St J . St J . 26 r

M

M

9

M

St J .

26

St G.

St J .

9

5

5 26

5

2

St G.

9

ω Ω_3

M

M

St J .

5

2

26 9

5

.

ω Ω_3

26

St G

M

5

5

2

9

ω Ω_3

26

M

5

ω Ω_3

.

26

St G

M

5

9

2

26 9

M M

St J .ω Ω_3

5

St J .St G

45

26

5

52

26

5

.St 5 G

M

1

.

27 5

26

5

26

.St 5 G

9

1

5 St J .ω Ω_3 .

27

M

5

26

.St 5 G 4

5 4 9 .c .ω Ω 5. 5. .ω Ω_3 St G M . 4π

3 5

.St 5 G 2

9 .c . 9 St G .St G . 4π 4

5

52

1

9

M 26

26 9 5 St J .ω Ω_3 .

M

5 St J .ω Ω_3 .

26

5

26 45

9 .c . St G 4 .π 4

4 5

5

9 .c . 4 .π 4

C Ω_Jω ω Ω_3 , M

ω Ω_3 4.

St G M

3

5 .ω Ω_3

5.2

C Ω_Jω ω Ω_3 , M

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S = 180.283336( nJy )

Checking errors yields,

294

www.deltagroupengineering.com

5

M

St J .

Test 1 ω Ω_3 , M

9

St G.

M

ω Ω_3

27

.M

45

9

Test 2 3 .( EHz) , M S = 5.438023 10

43 .

Jy

Test 3 3 .( EHz) , M S

Test 2 ω Ω_3 , M Test 1 ω Ω_3 , M

C Ω_Jω 3 .( EHz) , M S = 5.438023 10

. 1 = 5.70654610

43 .

12

12

(%)

C Ω_Jω ω Ω_3 , M

. 1 = 3.66373610

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S

(%)

Jy

. 1 = 2.0428110

Test 4 3 .( EHz) , M S

Jy

43 .

Test 2 ω Ω_3 , M

Test 5 ω Ω_3 , M

C Ω_Jω ω Ω_3 , M

Test 5 3 .( EHz) , M S

5

45

2

Test 1 ω Ω_3 , M

Test 4 ω Ω_3 , M

26

45 .

ω Ω_3

Test 1 3 .( EHz) , M S = 5.438023 10

Test 3 ω Ω_3 , M

St J .St G

Test 2 ω Ω_3 , M

26

26

10 10 C Ω_J1 R BH 10 .M S , 10 .M S

12

(%)

= 0.999999999999968

ix. Gravitational Cosmology G.M E G.M M 2

2

r4 r4 r5

r4

r4

r5

=

. 5 3.46028110 . 3.83719110

4

( km)

a EGM_ωΩ r 5 , M M 0 .( s ) ,

=

T PV 1 , r 4 , M E 500

a PV( r , M , t )

D E2M. M M .M E

i .

g r 4, M E g r 5, M M

. 3.33165310

3

. 3.33165310

3

.. T PV 1 , r 4 , M E

C PV n PV, r , M .e

=

. 3 3.33165310 . 3.33165310

m s

r5

M M .M E

MM

a EGM_ωΩ r 4 , M E

t

r 5 D E2M

3

m s

2

g r 4, M E

r4

g r 5, M M = 0

m s

a EGM_ωΩ r 4 , M E

2

D E2M

a EGM_ωΩ r 5 , M M = 0

2

m s

N

21

n PV

N, 2

2

N .. N

π .n PV .ω PV( 1 , r , M ) .t .i

n PV

295

www.deltagroupengineering.com

Harmonic Acc. & Grav. Interference T PV 1 , r 5 , M M

Acceleration

a PV r 4 , M E , t a PV r 5 , M M , t a PV r 4 , M E , t

a PV r 5 , M M , t

t Time

Gravitational Acceleration due to The Earth Gravitational Acceleration due to The Moon Resultant Acceleration (Interference)

0

ξ

9

a g ( r , M , φ, t )

g av ( r , M )

t

0 .( s ) ,

ξ .T Ω r 5 , M M 200

.. ξ .T Ω r 5 , M M

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

2 T Ω ( r, M )

φ

1. T Ω ( r, M ) 2 . 0 .( s )

g av R E, M E = 9.809009

m s

2

a g( r, M , 0 , t ) d t

ω Ω r 4 , M E = 56.499573 ( YHz) ω Ω r 5 , M M = 72.138509( YHz)

Conjugate WaveFunction Acc. Pairs

a g r 4, M E, 0, t Acceleration

φ

a g r 4, M E, 0, t a g r 5, M M, π , t a g r 5, M M, π , t

t Time

+ve WaveFunction From The Earth -ve WaveFunction From The Earth +ve WaveFunction From The Moon -ve WaveFunction From The Moon

296

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Conjugate WaveFunction Acc. Beats

Acceleration

a g r 4, M E, φ , t

a g r 5, M M, π , t

a g r 4, M E, φ , t

a g r 5, M M, π , t

t Time

+ve WaveFunction Interference Beat -ve WaveFunction Interference Beat (Conjugate)

x. Particle Cosmology h

tL

m γγ 5

m γγ2

r e.

r γγ2

h

m γγ2

m gg2

tL

m γγ2

2 .m γγ2

m gg2

2

r gg2

2 m e .c

5

r γγ2

4 .r γγ2

r gg2

r γγ2

λh

λh

r γγ2

2 .r γγ2

K λ .λ h

K λ .λ h 2 .r γγ2 λh r gg2

2 .r γγ2

0.178967

=

0.357933 0.236148

2 .r γγ2

(%)

λh

7.250508 9.567103

246.127068

2 .r gg2

0.472296

K λ .λ h

211.731798

λh

λh

2 .r gg2

E Ω ( r, M )

Qγ

h .ω Ω ( r , M )

Q γ r ε, m e

N γ( r, M )

Qγ

E Ω ( r, M ) mγ

= 2.655018 10

eV

38 .

m

423.463597

r gg2

2 .r gg2

10

51 .

279.381783

=

0.406294

2 .r gg2

10

3.431956

324.766614

λh K λ .λ h

1.715978

558.763566

λh

0.307913

=

=

30

Qe

297

Q γ( r, M )

Q γ_PDG

Qe N γ( r, M )

5 .10

30 .

Qe

www.deltagroupengineering.com

Q γ_PDG

= 1.883226

Q γγ( r , M )

Qγ

Q γγ

tL

Qγ

Q γ( r, M ) N γ( r, M )

2

Q γγ = 1.129394 10

Qe

m γγ

Q γγ

m γγ

T L m γγ2 Q γγ2

m γγ2

Q γγ2 = 6.065593 10

mγ

85 .

C

Q γγ2

78 .

C

Q γ( r, M )

Q γγ( r , M )

Q γγ

2

Qe

= 7.049122 10

60

Qe

. = 1.86196810

6

= 3.785846 10

Q γγ2

mγ

66

Q γγ m γγ

.m γγ2

2

ω Ω r e, m e .m γγ ω Ω r ε, m e

E Ω r ε,me

Qe

2

E Ω r ε, m e ω Ω r e, m e .m γγ ω Ω r ε, m e

ω Ω r e, m e

mγ

2

h .m γγ

=

1.525768 1.525768

10

46 .

eV

E Ω r e, m e

mγ

2

mγ

h .m γγ

mγ

m γγ

2

=

0.165603 0.165603

( µJ )

m γγ

Qe

ω Ω r e,m e 2

E Ω r e, m e

=

249.926816 249.926816

me ( YHz)

2.

c Q γγ

=

. 11 1.7588210

C

198.286288

kg

m γγ

NOTES

298

www.deltagroupengineering.com

b. Calculation engine i. 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. ii. Standard relationships 1

A0

c

r0

H0

H0

M

ρ m( r , M )

ω VL λ VL

c

9 A 0 = 13.772016 10 .yr

V( r )

2

r

M .c

E( M )

2

G.M

g( r , M )

λ VL

2 .G.M

R BH( M )

V( r )

c

4. . 3 πr 3

E γ ( ω ) h .ω

2

9 r 0 = 13.772016 10 .Lyr

iii. Derived constants 4 . 2 6 π 3

λx

3.

St G

λx

mx

3 .ω h 4 .π .h

2

. c 2

Hα

2

4

St J

10 .µ

t1

e

1

t2

Hα

e

2

t4

e

1

µ

3 1

r3

. 1 Hα

λ x.λ h

= 3.646967 10

2 2 5 .µ . 5 .µ

245

10

5 m.s

t5

M3

43 .

Hα

St g = 1.828935

2

9

9

s

m x.m h

c.

St BH

e

6 3 3 .ω h 13 2 2 .π .c 9

c .St G ( 2 .G)

1

. 1 Hα

t3

4

2 2 2 5 .µ . 5 .µ . 5 .µ

ng

St T

5

1

. 1 Hα

2

mx

3 9 10 .yr

224 .

St G = 8.146982 10

2

2 3

2

. 1 Hα

1

λx

13

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2

8

. T L = 4.10173110

e

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

2 2 15 .µ . 5 .µ

2 2 5 .µ . 5 .µ 1

St g

m γγ

1

1 2 2 5 .µ . 5 .µ

h

TL

λx

9 .c . St G 4 .π

9

1 2 5 .µ .

ωh

5

m

2. 9

kg s

299

=

t7

2.698709 1.349354

e

5 .µ

2

. 1 Hα

. H α = 8.46094110

61

km . s Mpc

. T L.H 0 = 2.97830810

12

1 . St G G

. 1 = 3.33066910

14

(%)

St g

www.deltagroupengineering.com

146 . kg

9

St J = 1.093567 10

4 . 26

s

m

3

18

119

s

10 .

r 3 = 1.093333 10

kg

. St BH = 4.83080210

. M 3 = 7.36147410

ym

8

. St T = 6.35557910 9

3

95

s

5 9

m

( kg )

1

t1

e

5 .µ

10 .µ

t2

e

2

. 1 Hα

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

. 1 Hα

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2.206287 2.206287

2 3

2

. 1 Hα

4.196153 4.196153 =

1

t4

e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

t5 e

1

4

. 1 Hα

6.205726 6.205726

10

20.932666 20.932666

t7

42 .

s

=1

t1

8.385263 8.385263 2.206287 2.206287

2

2 2 2 5 .µ . 5 .µ . 5 .µ

1

2

. 1 Hα

1

t7

e

2 5 .µ .

1 Hα

iv. Base approximations / simplifications

Ω 1( r , M )

6 .c r .ω PV( 1 , r , M )

C Ω_1( r , M )

G.M . 2

r

T Ω_1( r , M )

3

.

3 .M .c

2 . π n Ω_1( r , M )

ω Ω_1( r , M )

Ω 1 R M,M M

Ω 1 R E, M E

Ω R M, M M

Ω R E, M E

Ω 1 R J, M J

Ω 1 R S, M S

Ω R J, M J

Ω R S, M S

Ω R NS , M NS

n Ω_1( r , M )

2 .π .h .ω PV( 1 , r , M )

1

Ω 1 R NS , M NS

2

ω Ω_1( r , M )

λ Ω_1( r , M )

1=

Ω 1( r , M ) 12

n Ω_1( r , M ) .ω PV( 1 , r , M )

c ω Ω_1( r , M )

1

K Depp ( r , M ) 1

. 14 4.44089210 . 6.66133810

14

. 14 6.66133810 . 4.44089210

14

2 .G.M r .c

2

(%)

1 = 0 (%)

300

www.deltagroupengineering.com

n Ω_1 R M , M M

n Ω_1 R E, M E

n Ω R M,M M

n Ω R E, M E

n Ω_1 R J , M J

n Ω_1 R S , M S

n Ω R J, M J

n Ω R S, M S

n Ω_1 R NS , M NS n Ω R NS , M NS

ω Ω_1 R E, M E

ω Ω R M,M M

ω Ω R E, M E

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω R J, M J

ω Ω R S, M S

ω Ω R NS , M NS

T Ω_1 R E, M E

T Ω R M,M M

T Ω R E, M E

T Ω_1 R J , M J

T Ω_1 R S , M S

T Ω R J, M J

T Ω R S, M S

T Ω R NS , M NS

λ Ω_1 R E, M E

λ Ω R M,M M

λ Ω R E, M E

λ Ω_1 R J , M J

λ Ω_1 R S , M S

λ Ω R J, M J

λ Ω R S, M S

λ Ω R NS , M NS

. 2.22044610

14

. 4.44089210

14

. 8.88178410

14

(%)

1=

. 6.66133810

14

. 2.22044610

14

. 6.66133810

14

. 8.88178410

14

(%)

1=

. 7.77156110

14

. 2.22044610

14

. 5.55111510

14

. 7.77156110

14

. 7.77156110

14

. 2.22044610

14

. 6.66133810

14

. 7.77156110

14

(%)

1 = 0 (%)

λ Ω_1 R M , M M

λ Ω_1 R NS , M NS

14

1 = 0 (%)

T Ω_1 R M , M M

T Ω_1 R NS , M NS

. 6.66133810

1 = 0 (%)

ω Ω_1 R M , M M

ω Ω_1 R NS , M NS

1=

1=

K Depp R E, M E

1 = 0 (%)

K PV R E, M E

=

(%)

1.00000000069601 1.00000000069601

v. SBH mass and radius 3

ρS

ρ m λ x.λ h , m x.m h

r S M BH

. 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

3 .M BH 4 .π .ρ S

M BH r S

4. . 3 π ρ S .r S 3

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

301

www.deltagroupengineering.com

r S m x.m h

ρ m λ x.λ h , m x.m h

. 90 = 9.55041510

ρ m R S, M S

0

rS MS

. 3.28046310

=

5.

r S 10 M S

1=

144.219703 . 4.21884710

(%)

13

r S mh

( am)

0.015227 0.706754

10 r S 10 .M S

r S mh 1 . r m .m S x h λh λx

4

1 = 22.109851 ( % )

R BH m h

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 43 3.22881910 . 49 3.18519310 . 49 3.12378410 . 43 9.27104510

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 44 4.88723910 . 37 1.58452310 . 40 4.23240210 . 41 1.03305410

M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 2.55332710 . 40 5.86600510 . 40 3.94279810 . 40 7.32116510 . 40

M BH r bq

M BH r tq

M BH r W

M BH r Z

. 40 4.52155110 . 40 1.19243610 . 41 6.74112410 . 40 6.91875410

M BH r H

M BH r e

M BH r Bohr

M BH r gg

. 40 1.26038310 . 51 8.34661610 . 63 1.64821910 . 9 4.68486410

( kg )

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 13 1.62379510 . 19 1.60185510 . 19 1.57097210 . 13 4.66247210

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 13 2.45782610 . 7 5.19529810

1 . M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 1.28408510 . 10 2.95005410 . 10 1.9828610 . 10 3.68186410 . 10

M BH r bq

M BH r tq

M BH r W

M BH r Z

. 10 2.27391910 . 10 5.99684310 . 10 3.39015710 . 10 3.47948910

M BH r H

M BH r e

M BH r Bohr

M BH r gg

. 10 2.3560510

MS

. 20 6.3385510

. 9 2.12850410 . 11 7.96867110

. 33 4.1975710

14.554628

. 6 5.06892810

R BH M BH r τn

16.217926

. 7.67248410

R BH M BH r sq

R BH M BH r cq

= 4.00847210 .

R BH M BH r tq

R BH M BH r W

R BH M BH r Z

0.010862

R BH M BH r e

R BH M BH r Bohr

R BH M BH r gg

. 7.35477510

R BH M BH r ε

R BH M BH r π

R BH M BH r ν

R BH M BH r µ

R BH M BH r τ

R BH M BH r en

R BH M BH r µn

R BH M BH r uq

R BH M BH r dq

R BH M BH r bq R BH M BH r H

3

3

6

. 9.2090510

3

. 7.0983910

3

. 1.97867710

8

0

. 6 5.0004410

4.904034

. 2.48754410

3

0.066445

. 6.18980410

3

0.011494 ( Lyr )

0.01872

0.010583

. 1.31033610

21

0

vi. “nΩ” 1

1 3 9

n Ω_2( r , M )

n Ω_4 M BH

n BH M BH

1. 3 2

7

2

.

π .m h M

7

9

. r λh

n Ω_2 r S M BH , M BH

n Ω_5 M BH n Ω_4 M BH

3 9

9

n Ω_3 λ x

n Ω_5 M BH

π.

3.

2

2

2

n Ω_2 R BH M BH , M BH

n Ω_2 λ x.λ h , m x.m h n Ω_2 r S m x.m h , m x.m h n Ω_2 λ x.λ h , m x.m h n Ω_2 R BH m x.m h , m x.m h

302

λx

1=

. 3.33066910

13

. 14 4.44089210

(%)

www.deltagroupengineering.com

n Ω_2 r S m x.m h , m x.m h

n Ω_2 λ x.λ h , m x.m h

n Ω_2 r S M S , M S

n Ω_2 R BH M S , M S

5 5 n Ω_2 r S 10 .M S , 10 .M S

5 5 n Ω_2 R BH 10 .M S , 10 .M S

10 10 n Ω_2 r S 10 .M S , 10 .M S

10 10 n Ω_2 R BH 10 .M S , 10 .M S

n Ω_3 n Ω_3

1

1

. . 24 9.00254210 3.51086810 5

=

. 6 1.93953910 . 28 1.93265910 . 7 4.1786110 . 31 1.06388810

1 3

0.248017

1

0.324994

2 =

n Ω_3( 1 )

0.515897 0.818935

n Ω_3( 2 )

1

n Ω_3 λ x

1.073108

n Ω_3( 3 ) n Ω_4 m x.m h

n Ω_5 m x.m h

n BH m x.m h

n Ω_4 M S

n Ω_5 M S

n BH M S

5 n Ω_4 10 .M S

5 n Ω_5 10 .M S

5 n BH 10 .M S

10 n Ω_4 10 .M S

10 n Ω_5 10 .M S

10 n BH 10 .M S

1 =

1

1

. 5 9.00254210 . 24 2.56419310 . 19 3.51086810 . 6 1.93953910 . 28 1.0035610 . 22 1.93265910 . 7 4.1786110 . 31 3.92767810 . 24 1.06388810

vii. “ωΩ, TΩ, λΩ” 2 .c . n PV. g( r, M ) 2 π .r 3

ω PV2 n PV, r , M

9 M St G .

ω Ω_7 M BH

ω PV_1 M BH

ω Ω_4 M BH

St BH.

9

5

r

ω Ω_5 M BH

1. 2 St g .g ( r , M ) r

2

1

ω Ω_3( r , M )

9

ω Ω_2( r , M )

3

9

ω Ω_3 r S M BH , M BH

ω Ω_4 M BH n Ω_5 M BH

ω Ω_6 M BH

ω BH M BH

1 M BH

ω Ω_5 M BH n Ω_4 M BH ω Ω_5 M BH ω Ω_4 M BH

ω Ω_6 M BH ω Ω_7 M BH

303

www.deltagroupengineering.com

ω PV2 1 , R M , M M

ω PV2 1 , R E, M E

ω PV 1 , R M , M M

ω PV 1 , R E, M E

ω PV2 1 , R J , M J

ω PV2 1 , R S , M S

ω PV 1 , R J , M J

ω PV 1 , R S , M S

ω PV2 1 , R NS , M NS ω PV 1 , R NS , M NS

1=

. 3.14037710

9

. 6.96004310

8

. 1.9723310

6

. 2.12158610

4

1 = 7.117159 ( % )

ω PV2 n Ω R M , M M , R M , M M

ω PV2 n Ω R E, M E , R E, M E

ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω PV2 n Ω R J , M J , R J , M J

ω PV2 n Ω R S , M S , R S , M S

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω PV2 n Ω R NS , M NS , R NS , M NS

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω_1 R NS , M NS

1=

. 4.1871410

9

. 9.2800510

8

. 6 2.62977310

. 2.8287810

4

1=

. 1.04678510

9

. 2.32001510

8

. 6.57443310

7

. 7.07196310

5

(%)

1 = 2.491576 ( % )

ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω Ω_2 R J , M J

ω Ω_2 R S , M S

1=

. 14 8.88178410

. 1.11022310

13

. 13 1.11022310

. 1.11022310

13

(%)

ω Ω_4 m x.m h ω Ω_3 R NS , M NS ω Ω_2 R NS , M NS

. 1 = 6.66133810

14

(%)

1 ω Ω_3 λ x.λ h , m x.m h

.

ω Ω_5 m x.m h ω Ω_6 m x.m h ω Ω_7 m x.m h

ω Ω_3 r 0 , m x.m h

. 7.88327910

ω Ω_3 r 0 , M S 5 ω Ω_3 r 0 , 10 .M S 10 ω Ω_3 r 0 , 10 .M S

(%)

1 = 9.375146 ( % )

ω Ω_3 R NS , M NS

ω Ω_2 R NS , M NS

(%)

=

1 =

1 1 1

10

0.163994

( EHz)

2.118067 27.355887

304

www.deltagroupengineering.com

ω Ω_3 r 0 , m x.m h

ω Ω_3 r 0 , m x.m h

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

ω Ω_3 r 0 , M S

ω Ω_3 r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

5.

. 1.05183110

ω Ω_3 r 0 , 10 M S

ω Ω_3 r 0 , 10 M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

10 ω Ω_3 r 0 , 10 .M S

10 ω Ω_3 r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

. 3 5.29883310

. 4 3.64997710

. 4 6.84370610

ω Ω_6 m x.m h

ω Ω_7 m x.m h

ω Ω_4 M S

ω Ω_5 M S

ω Ω_6 M S

ω Ω_7 M S

5.

410.269418

. 2.82604410

ω Ω_5 m x.m h

5.

. 42 1.87219710 . 42 1.87219710 . 42 1.87219710 . 42 1.87219710 =

5.

ω Ω_4 10 M S

ω Ω_5 10 M S

ω Ω_6 10 M S

ω Ω_7 10 M S

10 ω Ω_4 10 .M S

10 ω Ω_5 10 .M S

10 ω Ω_6 10 .M S

10 ω Ω_7 10 .M S

. 29 4.55727410 . 43 1.29804810 . 38 6.93112610 . 4 6.23977510 . 28 6.9805610 . 43 3.61189510 . 37 1.34431910

ω Ω_5 m x.m h

ω Ω_6 m x.m h

ω Ω_7 m x.m h

ω Ω_4 M S

ω Ω_5 M S

ω Ω_6 M S

ω Ω_7 M S

5 ω Ω_5 10 .M S

5 ω Ω_6 10 .M S

5 ω Ω_7 10 .M S

10 .

10 .

ω Ω_5 10 M S

ω BH m x.m h

ω PV_1 m x.m h

ω BH M S

ω PV_1 M S

5 ω BH 10 .M S

5 ω PV_1 10 .M S

10 ω BH 10 .M S

10 ω PV_1 10 .M S

ω Ω_4 10 M S

10 .

0.253004

14

. 6.158585 1.75414910

5

0

. 1.81667910

15

. 9.433354 4.88102410

6

0

. 14.44945 1.35817410

6

0

ω Ω_7 10 M S

1

0

1

. 7.30358710

. 33 1.8727810

13

=

=

0.253004

6

. 8.43227510

10 .

ω Ω_6 10 M S

0.253004

0.253004

( Hz)

0.693113

. 26 1.06924110 . 44 1.00503110 . 37 6.93112610 . 2.89624710

ω Ω_4 m x.m h 1 . 5 ω h ω Ω_4 10 .M S

6

3

ω Ω_4 m x.m h

5.

. 1.97218410

218.810356

=

5.

6

. 15 5.21112310 . 37 5.19263810 . 17 1.45002610 . 42 3.69181510

viii. Gravitation r . 9 ω Ω_2( r , M ) St g

a EGM_ωΩ( r , M )

a g ( r , M , φ, t )

g av ( r , M )

MM

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

2 T Ω ( r, M )

D E2M. M M .M E

r4

r5

M M .M E

D E2M r 4

φ

1. T Ω ( r, M ) 2 . 0 .( s )

a g( r, M , 0 , t ) d t

a EGM_ωΩ R M , M M

a EGM_ωΩ R E, M E

g R M,M M

g R E, M E

a EGM_ωΩ R J , M J

a EGM_ωΩ R S , M S

g R J, M J

g R S, M S

1=

. 1.49880110

12

. 1.49880110

12

. 1.5432110

12

. 1.57651710

12

305

(%)

www.deltagroupengineering.com

a EGM_ωΩ R NS, M NS

. 1 = 1.65423210

g R NS, M NS g r 4, M E g r 5, M M

=

. 3 3.33165310

m

. 3 3.33165310

a EGM_ωΩ r 4 , M E

=

a EGM_ωΩ r 5 , M M

g av R E, M E = 9.809009

s

r5

=

. 5 3.46028110 . 4 3.83719110

g r 5, M M = 0

. 3.33165310

3

m s

a EGM_ωΩ r 4 , M E

2

( km)

m s

3

s

r4

( %)

g r 4, M E

2

. 3.33165310

m

12

2

m

a EGM_ωΩ r 5 , M M = 0

s

2

ω Ω r 4 , M E = 56.499573 ( YHz) ω Ω r 5 , M M = 72.138509 ( YHz)

2

ix. Flux intensity 5

r ω ω Ω_3 , M

St G.

M

2

ω Ω_3

C Ω_J ( r , M )

9

2 d λ Ω ( r , M ) . U m( r , M ) dr

5

C Ω_J1( r , M )

St J .

M

5.2

9

C Ω_Jω ω Ω_3 , M

26

r

9

r ω ω VL( 400 ( nm ) ) , m x.m h

=

r ω ω VL( 750 ( nm ) ) , m x.m h

r ω ( 400 ( nm ) ) , m x.m h 1 . ω VL r 0 r ω ω VL( 750 ( nm ) ) , m x.m h

0.239057 0.741144

=

( Lyr)

. 1.73581410

11

. 5.38152510

11

r ω ω VL( 400 ( nm ) ) , M S

r ω ω VL( 750 ( nm ) ) , M S

5 r ω ω VL( 400 ( nm ) ) , 10 .M S

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

10 r ω ω VL( 400 ( nm ) ) , 10 .M S

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω ω VL( 400 ( nm ) ) , M S

1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

5 . 5 2.2445.10 6.95860210

= 2.2445.107 6.95860210 . 7

r ω ω VL( 750 ( nm ) ) , M S

1 . r ω ( 400 ( nm ) ) , 105 .M ω VL S r0 10 r ω ω VL( 400 ( nm) ) , 10 .M S r ω 30.( PHz) , M S

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

5 r ω ω VL( 750 ( nm ) ) , 10 .M S 10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω 30.( EHz) , M S 5 r ω 30.( EHz) , 10 .M S 10 r ω 30.( EHz) , 10 .M S

9 10 .Lyr

9 . 9 2.2445.10 6.95860210

. 4 5.05271110 . 4 1.62975410 = 1.62975410 . 6 5.05271110 . 6 . 8 5.05271110 . 8 1.62975410

21.2751776034345 8.46980075872643.10

5

= 2.12751776034345 .103 8.46980075872643.10

3

.105 2.12751776034345

306

0.846980075872643

www.deltagroupengineering.com

r ω 30.( PHz) , m x.m h

=

r ω 30.( EHz) , m x.m h

. 6 2.95234410

r 30.( PHz) , m x.m h 1 . ω r 0 r ω 30.( EHz) , m x.m h

3 10 .km

11.753495

r ω 30.( PHz) , M S

r ω 30.( EHz) , M S

5 r ω 30.( PHz) , 10 .M S

5 r ω 30.( EHz) , 10 .M S

10 r ω 30.( PHz) , 10 .M S

10 r ω 30.( EHz) , 10 .M S

r ω 30.( PHz) , M S 1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

. 5 2.93002110 = 2.93002110 .

7

5 r ω 30.( EHz) , 10 .M S 10 r ω 30.( EHz) , 10 .M S

. 21.275178 8.46980110

5

= 2.12751810 . 3 8.46980110 .

3

. 5 2.12751810

0.84698

C Ω_J 100.( km) , M M

C Ω_J R S , M M

C Ω_J 100.( km) , M E C Ω_J1 100.( km) , M J

. 2.95903310

6

. 9.40034410

4

1=

C Ω_J 100.( km) , M J

C Ω_J1 R S , M E C Ω_J R S , M E C Ω_J1 R S , M J

(%)

C Ω_J1 R S , M S

0.979587

C Ω_J 100.( km) , M S

C Ω_J R S , M S

C Ω_J1 100.( km) , M NS

C Ω_J1 R S , M NS

C Ω_J 100.( km) , M NS

C Ω_J R S , M NS

C Ω_Jω ω VL( 400 ( nm) ) , m x.m h C Ω_J1 r ω ω VL( 400 ( nm ) ) , m x.m h , m x.m h C Ω_Jω 30.( PHz) , m x.m h C Ω_J1 r ω 30.( PHz) , m x.m h , m x.m h C Ω_Jω 30.( PHz) , M S C Ω_J1 r ω 30.( PHz) , M S , M S

C Ω_Jω ω VL( 750 ( nm) ) , m x.m h C Ω_J1 r ω ω VL( 750 ( nm ) ) , m x.m h , m x.m h C Ω_Jω 30.( EHz) , m x.m h C Ω_J1 r ω 30.( EHz) , m x.m h , m x.m h C Ω_Jω 30.( EHz) , M S C Ω_J1 r ω 30.( EHz) , M S , M S

5 C Ω_Jω 30.( PHz) , 10 .M S

5 C Ω_Jω 30.( EHz) , 10 .M S

5 5 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

5 5 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

10 C Ω_Jω 30.( PHz) , 10 .M S

10 C Ω_Jω 30.( EHz) , 10 .M S

10 10 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

10 10 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

C Ω_J1 r 0 , m x.m h

10 C Ω_J1 r 0 , 10 .M S

1=

. 3.57491810

12

. 4.23150410

10

. 1.35061710

7

. 1.41439110

4

. 1.41439110

4

(%)

. 1.90958410

12

. 1.92068610

12

. 1.9428910

12

. 1.93178810

12

. 1.58761910

12

. 1.59872110

12

. 1.50990310

12

. 1.50990310

12

. 1.48769910

12

. 1.50990310

12

(%)

0

C Ω_J1 r 0 , M S 5 C Ω_J1 r 0 , 10 .M S

1=

C Ω_J R S , M J

0.979587

C Ω_J1 100.( km) , M S

0

6 10 .Lyr

116.646228

C Ω_J1 R S , M M

8

14

1.166462

C Ω_J1 100.( km) , M M

. 3.63872410

. 2.2659710

. 9 1.16646210 . 4 2.93002110

r ω 30.( EHz) , M S

C Ω_J1 100.( km) , M E

=

=

. 1.48429110

5

. 8.89809310

3

10

20 .

yJy

5.334267

307

www.deltagroupengineering.com

1 C Ω_J1 r 0 , m x.m h

. 26 2.24315810

1

10 C Ω_J1 r 0 , 10 .M S .

C Ω_J1 r 0 , M S

= 3.59381410 . 5 599.48425

1 5 C Ω_J1 r 0 , 10 .M S

C Ω_Jω ω VL( 400 ( nm ) ) , m x.m h

=

C Ω_Jω ω VL( 750 ( nm ) ) , m x.m h

28.979765 1.102778

10

16 .

yJy

C Ω_Jω 30.( EHz) , m x.m h = 2.479532 ( fJy )

7 C Ω_Jω 30.( PHz) , m x.m h = 6.228302 10 .yJy

C Ω_Jω 30.( PHz) , M S 5.

2.164916

C Ω_Jω 30.( PHz) , 10 M S

. = 2.16491610

3

10 C Ω_Jω 30.( PHz) , 10 .M S

. 2.16491610

6

C Ω_Jω 30.( EHz) , M S 5.

10

29 .

10

14 .

yJy

8.618686

C Ω_Jω 30.( EHz) , 10 M S

. = 8.61868610

3

10 C Ω_Jω 30.( EHz) , 10 .M S

. 8.61868610

6

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S 10 10 C Ω_J1 R BH 10 .M S , 10 .M S

yJy

= 0.999999999999968

x. Photon and Graviton populations ω g( M )

M .c 2 .h

2

n gg ( M )

ω g m x.m h

. 18 4.99252510

ωg MS

. 56 1.34855310

5 ω g 10 .M S 10 ω g 10 .M S

=

. 61 1.34855310

T L.ω g ( M )

n γγ( M )

2 .n gg ( M )

( YHz)

. 66 1.34855310

308

www.deltagroupengineering.com

ω g m x.m h ω Ω_4 m x.m h

ω g m x.m h ω Ω_5 m x.m h

ωg MS

ωg MS

2.666667

50

ω Ω_5 M S

. 36 2.95912210

. 57 1.0031510

5 ω g 10 .M S

. 63 4.65620810

5 ω Ω_5 10 .M S

2.666667

ω Ω_4 M S

. 2.1612210

=

5 ω g 10 .M S 5 ω Ω_4 10 .M S

=

. 46 1.26122510

10 ω g 10 .M S

10 ω g 10 .M S

10 ω Ω_4 10 .M S

10 ω Ω_5 10 .M S

n gg m x.m h

. 72 6.46222510

n γγ m x.m h

n gg M S

. 110 1.7455410

n γγ M S

. 115 1.7455410

5 n γγ 10 .M S

. 120 1.7455410

10 n γγ 10 .M S

=

5 n gg 10 .M S 10 n gg 10 .M S

. 41 1.93186910

. 73 1.29244510 . 110 3.4910810

=

. 115 3.4910810 . 120 3.4910810

xi. Hubble constant and CMBR temperature r 2( r )

K λ .r

K m.M

M 2( M )

5

5

K U( r , M )

A U( r , M )

1

ln

9

2

7

.ln n Ω_2 r 2( r ) , M 2( M )

TL K U( r , M )

R U( r , M )

5

Hα

K T( r , M )

n g .ln

T U( r , M )

K T( r , M ) .T W ( r , M )

M U( r , M )

H U5( r , M )

T U5( r , M )

KW c

µ

( 3 .π )

7 .µ .

256

µ

32

2

. .ln . 4µ H U5( r , M ) λ h Hα

2 .µ

.

r3

1 π .H α

H U( r , M )

1 A U( r , M )

KW λ Ω_1 R U( r , M ) , M 3 3 .H U( r , M )

2

8 .π .G

K W .St T .ln

T U2( H )

. µ m .ln ( 3 π ) . h 4 M

9

r 2( r )

c .A U( r , M )

ρ U( r , M )

V R U( r , M ) .ρ U( r , M )

.

M 2( M )

T W( r, M )

H U( r , M )

1 . ln TL

M3

3.

26

9

µ

. 2

.H5 µ

H 7 .µ

2

. r λh 2 .µ

Hα

2

2 7 .µ

5

.

mh M

5 .µ

2

. r λh

2 26 .µ

. 2

.H ( r , M ) 5 µ U5

309

www.deltagroupengineering.com

K W .St T .

dT dt ( t )

2 5 .ln H α .t .µ

t

5 .µ

2

1

K W .St T .

dT2 dt2 ( t )

2 2 5 .µ . ln H α .t . 5 .µ

.t

t

dT3 dt3 ( t )

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ 3 2 K W .St T . 2 5 .µ . 3 t t

dH dt H γ

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

2 2 15.µ . 5 .µ

dH2 dt2 H γ

1

5 .µ

2

1

2

1

.t2

2

2

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ 5 .µ Hγ

1

2

1

Given T U2 H U r x1.R o , M G T U2 H U R o , m g1 .M G

T U2 H U r x2.R o , M G T U2 H U R o , m g2 .M G

∆T 0

T0

T0

∆T 0

r x1 r x2 m g1

Find r x1, r x2, m g1 , m g2

m g2

Given T U2 H U r x1.R o , m g1 .M G T U2 H U r x1.R o , m g2 .M G

∆T 0

T0

T U2 H U r x2.R o , m g1 .M G T U2 H U r x2.R o , m g2 .M G

T0

∆T 0

r x3 r x4 m g3

Find r x1, r x2, m g1 , m g2

m g4

Given T U2 H U r x1.R o , m g1 .M G r x5 m g5

T0

Find r x1, m g1

T U2 H U R o T U2 H U R o

1 ∆R o , .K m.M G 3 1 ∆R o , .K m.M G 3

r x1 =

2.733025 2.741859

r x2

( K)

m g1 m g2

310

0.989364 =

1.017883 1.057292 0.911791

www.deltagroupengineering.com

T U2 H U r x1.R o , M G T U2 H U r x2.R o , M G T U2 H U R o , m g1 .M G T U2 H U R o , m g2 .M G

2.724 =

2.726 2.724

m g3

7.914908 8.143063

( kpc )

1.063645 5.729219 1.788292

=

(%)

8.820858

2.724 2.726

( K)

0.977007

r x3

=

r x4

r x5

7.879647 8.106786

T U2 H U r x5.R o , m g5 .M G

1.052361

MG

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410

( kpc )

1.013403

=

m g5

m g5 .

=

T U2 H U r x3.R o , m g3 .M G T U2 H U r x4.R o , m g4 .M G

0.977007

m g4

R o.

1=

r x2 m g2

1.013348

=

r x2

r x1 m g1

0.984956

r x4

r x1

2.726

. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910 r x3

R o.

( K)

r x5

. 11 = 6.31416710

m g5

MS

1=

r x3 m g3 r x4 m g4

5.236123

( %)

1.50441 2.29934 1.334822 2.29934

( %)

r x5.R o = 8.107221 ( kpc )

= 2.725 ( K )

1.340256

1=

H U r x5.R o , m g5 .M G = 67.095419

km . s Mpc

Given

dH dt

H U R o,M G

η

Hα

1

H U R o,M G

η

Find( η )

9 A U R o , M G = 14.575885 10 .yr

ρ U R o , M G = 8.453235 10

33 .

9 R U R o , M G = 14.575885 10 .Lyr

kg 3

. M U R o , M G = 9.28458610

52

( kg )

cm H U R o,M G 1 H U R o , .M G 3

=

67.084304 67.753267

km . s Mpc

T U R o,M G 1 T U R o , .M G 3

311

=

2.724752 2.739618

( K)

www.deltagroupengineering.com

H U R o,M G

T U R o,M G

H0

T0

H U R o,M G

T U R o,M G

1 H U R o , .M G 3

1 T U R o , .M G 3

T U2 H U R o

. 5.515064 9.08391310 0.987352

3

(%)

0.542607

∆R o , M G 2.720213

T U2 H U R o , M G T U2 H U R o

1=

2.724752

∆R o , M G

H U5 R o , M G H U5 r x5.R o , m g5 .M G

= 2.729021 ( K )

1 T U2 H U R o , .M G 3

2.739618

=

67.084304

km

67.095419

s .Mpc

2.810842

T U2 H 0

H U5 r x5.R o , m g5 .M G T U5 R o , M G T U5 r x5.R o , m g5 .M G

=

2.724752 2.725

H0

( K)

1=

T U5 r x5.R o , m g5 .M G

5.499409 . 8.3739910

9

(%)

T0

T U2 H α T U2 t 1

dT dt

0

1

. 3.19551810

31

T U2 t 2

1

T U2 t 3

1

T U2 t 4

1

. 31 2.05994510

T U2 t 5

1

. 31 2.65086510

dT2 dt2

. 3.03432210 . 2.83254210

dT2 dt2 t 2

. 114 2.02615310

dT2 dt2 t 3 dT2 dt2 t 4 dT2 dt2 t 5

0 . 8.77595210

112

s

dT3 dt3 t 3

. 112 1.612210

dT3 dt3 t 4

. 112 7.1945910

dT3 dt3 t 5

312

s

1 . 159 6.22716710

Hα

dT3 dt3 t 2

2

. 9.25283810

. 71 7.47950610

. 156 3.77545710

dT3 dt3 t 1 K

K

. 71 3.03728910

dT dt t 5

dT3 dt3

. 72 1.05719310 71

dT dt t 4

. 116 7.65967810

=

=

dT dt t 3

1

dT2 dt2 t 1

0

dT dt t 2

( K)

31

Hα

. 74 1.32321810

dT dt t 1

31

=

1 Hα

=

. 155 1.45285710

K

0

s

3

. 153 1.48902210 . 153 9.53337910

www.deltagroupengineering.com

dH dt ( 1 )

dH dt e

0 1

dH dt t 1 .H α

1

dH dt e

5 .µ

10 .µ

dH dt t 2 .H α

1

dH dt e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH dt t 3 .H α

1

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

=

2 3

2

1

dH dt t 4 .H α

1

dH dt e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH dt t 5 .H α

1

dH dt e

1

dH2 dt2 e

. 123 7.15875210

. 107 7.14236410

. 107 7.14236410

. 122 8.63295710

. 122 8.63295710

. 123 1.16708910

. 123 1.16708910

. 123 1.47916710

. 123 1.47916710

. 123 1.31810810

. 123 1.31810810

km s .Mpc

2

2

4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH2 dt2 ( 1 )

. 123 7.15875210

1

2

0 1

dH2 dt2 t 1 .H α

1

dH2 dt2 e

5 .µ

10 .µ

dH2 dt2 t 2 .H α

1

dH2 dt2 e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH2 dt2 t 3 .H α

1

dH2 dt2 e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

=

2 3

2

1

dH2 dt2 t 4 .H α

1

dH2 dt2 e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH2 dt2 t 5 .H α

dH dt

1

H U R o,M G Hα

Hα

dH2 dt2 e

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

η

. 3 = 4.50029710

km . s Mpc

. 186 1.27869510

. 186 1.27869510

. 184 2.49929710

. 184 2.49929710

0

0

. 3.41565310

. 3.41565310

. 183 2.42270610

. 183 2.42270610

. 183 3.91232210

. 183 3.91232210

183

183

km s .Mpc

3

2 1

2

2

dH2 dt2

H U R o,M G Hα

η

=0

km . s Mpc

3

=1 η

dH dt 1

313

www.deltagroupengineering.com

η

dH dt 1

η

1

dH dt e

5 .µ

10 .µ

2

η

2

1

2 2 5 .µ . 5 .µ

dH dt e

. 61 2.55267410 η

2 2 15 .µ . 5 .µ

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

km . s Mpc

. 60 4.13447210 . 60 9.11289510

1

67.084257 η 4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

= 1.16926910 . 61

2

3

η

2 2 5 .µ . 5 .µ

dH dt e

. 61 1.59787310

2

1

dH dt e

. 61 8.46094110

1

2 1

H U R o,M G

2

η

Hα

Hα

Hα

. 61 8.46094110

dH dt t 4 .H α

1

. 61 = 3.84599410 67.084304

H U R o,M G

dH dt t 4 .H α

km s .Mpc

H U R o,M G

71

H0

2

2

H0

. 123 7.15875210 1 2

=

. 123 1.47916710 . 3 4.50030410

km s .Mpc

2

3 5.041.10

1 H α.

dH dt t 4 .H α

1

=

Hα dH dt t 4 .H α

2.199936 4.839718

1

314

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η

T U2

dH dt 1

η

1

T U2

5 .µ

dH dt e

10 .µ

T U2

2

η

2

1

2 2 5 .µ . 5 .µ

dH dt e

. 31 2.97174510 η

2 2 15 .µ . 5 .µ

T U2

2

dH dt e

2 2 5 .µ . 5 .µ

T U2

2.724751 η 2

2 3

2

η

H U R o,M G

dH dt

. 31 3.18071410

1

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt e

. 31 2.72300610

η

2 2 15 .µ . 5 .µ

T U2

= 3.18071410 . 31 ( K )

2

3

1

T U2

. 31 3.18632310

2

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt e

0

1

Hα η

1

1 T U2 t 1

1

.T U2

dH dt e

5 .µ

10 .µ

1 T U2 t 2

1

.T U2

dH dt e

2

η

2

1

2 2 5 .µ . 5 .µ

η

2 2 15 .µ . 5 .µ

1 T U2 t 3

1

.T U2

dH dt e

1

2

2 2 2 5 .µ . 5 .µ . 5 .µ

3

1 T U2 t 4

η

1

.T U2

dH dt e

2 2 5 .µ . 5 .µ

T U2 t 5

.T U2

1 T U2 H U R o , M G

dH dt e

.T U2

(%)

32.18827 . 3.90264410

5

η 2

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

12.291857 19.987768

1

2 2 15 .µ . 5 .µ

1

5.00939

2

1= 1

1

7.002696

2

2 3

2

H U R o,M G

η

Hα

315

www.deltagroupengineering.com

T U2 H α

1

T U2

13 .

10 ( s ) 1

T U2 10

10

10 .

(s)

1

T U2

1

T U2

1 . 31 ( day )

T U2 T U2

1 .( yr ) 1

T U2

2 10 .( yr )

1

T U2

3.

1

. 9 1.01325410

2.724752

( K)

1

1 6.

10 ( yr )

. 10 1.84076810 . 3 1.2497710

1

. 7 521.528169 2.52413210

7.

. 6 3.86401510

147.71262

= 1.00307810 . 6

41.823796

. 4 8.07751510

11.838588

9 10 .( yr )

. 2.29089210

3.35005

1

. 6.49496110

0.947724

1 8 10 .( yr )

1

=

1

3

( yr )

11 .

10

1 116.( day )

4

( K)

1

T U2

4.

T U2

4.898955

5.

10

10 ( yr ) T U2

. 15 3.43308810

10 .

10 ( yr ) T U2

9 5 .10 .( yr )

10 ( yr )

1

T U2

11.838588

10 ( yr )

1 .( day )

T U2

1

T U2

1 .( s ) 1

T U2

=

T U2 H U R o , M G

2 10 .( s )

T U2

T U2

T U2

. 1.92400510

28

9 10 .( yr )

(s)

1

T U2

T U2

34 .

978.724031

0

1

T U2

( yr )

. 6 1.87808710 . 3.98831410

7

( K)

TL

xii. SBH temperature Th

T BH( M )

Th

( 4 .π )

. 2

mh M

. 30 ( K ) = 1.66667410

2 8 .π .λ x

4

T SPBH

. 1. mh c κ h .ω h

2

c. U m λ x.λ h , m x.m h σ

=

. 32 3.55137410 . 32 3.55137410

T BH m x.m h

. 30 ( K ) T BH m x.m h = 1.66667410

h .c

( K)

=1

3

2 16.π .κ .G.m x.m h

10 T BH 10 .M S

h .c

3

T U2 =1

1 t1

T BH m x.m h

T U2 = 19.173025

1 t1

6 .π .T BH m x.m h

1 = 1.716054 ( % )

2 10 16.π .κ .G.10 .M S

316

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T U2

1 t1

3 .T h

T SPBH

. 31 ( K ) T SPBH = 5.02766910

1 = 1.716054 ( % )

T BH m x.m h

= 30.165887

4 .π .λ x

T SPBH T U2

Hα

. 31 ( K ) K ω T SPBH = 3.20071410

= 1.57335

1

T U2

t1 4

.

π

T U2

3 K ω .T SPBH

3.

1

6 .c . 31 ( K ) = 3.20071410 π .σ .G

t1

2 K W .c . 5 G.κ

1 . 15 . h .c λ x 4 .π κ .K W

15 . h .c = 2.659782 4 .π κ .K W

1 = 0.248248 ( % )

1 = 1.442436 ( % )

T BH m e

T BH m p

T BH m n

T BH m µ

T BH m τ

T BH m en

T BH m µn

T BH m τn

T BH m uq

T BH m dq

T BH m sq

T BH m cq

T BH m bq

T BH m tq

T BH m W

T BH m Z

T BH m H T SPBH

=

t1

2 c . KW . 31 ( K ) = 3.18758510 5 G.κ

3

1

1 = 0.162602 ( % )

1

. 47 6.01617410 . 31 5.02766910

. 53 7.33529610 . 49 7.32519910 . 49 6.51392110 . 50 1.34687210 =

. 49 2.29416810 . 58 3.6223710 . 53 3.78159510 . 51 3.87312710 . 52 9.81839510 . 51 6.04208910 . 50 5.81830810 . 49 1.96367910

( K)

. 49 3.8570310 . 47 8.55766610 . 47 7.54763110 . 47 1.67121610

( K)

xiii. ZPF Ω EGM

ρ U r x5.R o , m g5 .M G

3 .c . Ω ZPF . H U R o,M G 8 .π .G 2

U ZPF Ω EGM Ω PDG

Ω PDG

ρ U R o,M G

= 0.997339

U ZPF = 251.778016

U ZPF = 842.934914

yJ 3

2

Ω ZPF

1.003

. Ω ZPF = 3.31400710

4

U ZPF = 251.778016( fPa )

U ZPF = 2.51778 10

13 .

U ZPF = 0.251778

mJ

U ZPF = 251.778016

3

km

EJ

. 12 U ZPF = 7.39723510

AU

Ω EGM

Ω EGM = 1.000331

mm

3

1

fJ 3

m

YJ pc

317

Pa

3

www.deltagroupengineering.com

. 41 U ZPF = 6.60189810

YJ R U R o,M G

3

xiv. Cosmological limits M U R o,M G

M EGM

ML

R EGM

K m.M G.

K λ .R o

R U R o,M G

R EGM 5 5

.

R EGM

rL

K λ .R o

. 71 ( kg ) M L = 4.86482110

t EGM

R BH M L

. 19 109 .Lyr r L = 7.6372910

A U R o,M G

rL

tL

c

. 19 109 .yr t L = 7.6372910

ML M EGM

M EGM 2 R EGM.c

t EGM

=1

2 .G

. 18 5.23967510

rL

=1

R EGM

R EGM

c

tL

tL

= 5.23967510 . 18

. 6 = 1.86196810

TL

. 18 5.23967510

t EGM

xv. Particle Cosmology 5

m γγ2

h tL

E Ω ( r, M )

Qγ

m gg2

h .ω Ω ( r , M )

Q γ r ε, m e

Q γ_PDG

5 .10

2 .m γγ2

r γγ2

N γ( r, M )

Q γγ ( r , M )

30 .

Qe

r e.

m γγ2 m e .c

mγ

m gg2

1.715978 3.431956

4 .r γγ2

5

Q γ( r, M )

Q γγ

N γ( r, M )

=

r gg2

2

E Ω ( r, M )

Q γ( r, M )

m γγ2

2

10

318

51 .

eV

Qγ

Qe N γ( r, M )

2

Qe

Q γγ2

r γγ2 r gg2

Q γγ m γγ

=

.m γγ2

7.250508 9.567103

10

38 .

m

www.deltagroupengineering.com

r γγ2

λh

λh

r γγ2

2 .r γγ2

K λ .λ h

K λ .λ h

2 .r γγ2

0.178967

2 .r γγ2 λh

=

r gg2

0.357933 0.236148

558.763566

λh

0.307913

324.766614

2 .r γγ2

(%)

279.381783

=

λh

423.463597

λh

0.406294

r gg2

246.127068

2 .r gg2

0.472296

K λ .λ h

211.731798

K λ .λ h

2 .r gg2

2 .r gg2

λh

λh

2 .r gg2

Qγ

= 2.655018 10

Q γ_PDG

30

Qe

Q γγ

Q γγ = 1.129394 10

= 1.883226

Qγ

= 7.049122 10

m γγ

60

Qe

. = 1.86196810

6

Q γγ2

Q γγ2 = 6.065593 10

m γγ2 mγ

= 3.785846 10

ω Ω r e, m e .m γγ ω Ω r ε, m e

Qe

E Ω r e, m e mγ

2

C

85 .

C

2

E Ω r ε, m e

66

78 .

=

1.525768 1.525768

10

46 .

eV

ω Ω r e,m e =

0.165603 0.165603

( µJ )

mγ

2

=

h .m γγ

m γγ

249.926816 249.926816

( YHz)

Qe me 2.

c Q γγ

=

. 11 1.7588210

C

198.286288

kg

m γγ

319

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NOTES

320

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•

MathCad 12 c. High precision calculation engine i. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 10-14. Constraint Tolerance (CTOL): 10-14. Calculation Display Tolerance: 6 figures – unless otherwise indicated. ii. Astronomical statistics

H0 := 71⋅ 

   s ⋅ Mpc 

∆H0 := 2⋅ 

   s ⋅ Mpc 

km

km

∆T0 := 0.001( K)

T0 := 2.725⋅ ( K)

H0 − ∆H0 = 69 

   s ⋅ Mpc  km

T0 + ∆T0 = 2.726( K)

iii. Derived constants

µ :=

λx := 4⋅

1

2⋅ µ µ

π

3

Stt := 2⋅ ωh ⋅  4

  µ  3⋅ π  2

7

⋅ 

2

Hα :=

ωh

TL :=

λx

h mγγ

9

 c

iv. Algorithm 7⋅ µ   2 2 2 2 2  5⋅ µ 7⋅ µ µ µ m µ 7⋅ µ  26⋅ µ  m   ( ) ( ) 1  h  3⋅ π ⋅ 32 ⋅ ln 3⋅ π ⋅  h  ⋅  r    r  HU5( r , M) := ⋅ ln  4  M   λ   ⋅  M  ⋅  λ   TL  256   h   h



  H ( r , M )  U5 

KT ( r , M) := 8⋅ µ ⋅ ln

Hα

TU( r , M ) := KT ( r , M ) ⋅ TW( r , M )

TW( r , M) :=

KW



c



HU5 ( r , M)

λΩ 

,

λx 2



⋅ mh 



 Hα  9 5  ⋅ Stt ⋅ H  H 

TU2( H) := KW⋅ ln

 HU5( Ro , MG)   67.084134  km   =    HU5( Ro , µ ⋅ M G)   67.753095  s ⋅ Mpc 

 TU( Ro , MG)   2.724749  =  ( K)  TU( Ro , µ ⋅ MG)   2.739614

 HU5( Ro , M G)   −2.776618 ( %)  −1=  H0 − ∆H0  HU5( Ro , µ ⋅ MG)   −1.807108 1

5

 TU( Ro , M G)   −0.045904 ( %)  −1=  T0 + ∆T0  TU( Ro , µ ⋅ MG)   0.499413  1

⋅

321

⋅

www.deltagroupengineering.com

 TU2( H0 − ∆H0)    TU2( H0)    ∆H    TU2 H0 + 0   2   

 2.767146 =  2.810842 ( K)    2.832481

 H0 − ∆H0   H0  ∆H  H0 + 0 2 

     

 69  km  =  71      s ⋅ Mpc   72 

d. Various forms of the derived constants

 4 ⋅   6π   4⋅ µ ⋅  

  π   2.698709 2.698709  =  6 2⋅ µ   2.698709 2.698709 4⋅ µ µ  π π  2

4

3

3

⋅

6

µ

6 2 1  ⋅  1  ⋅  4⋅ µ    c3  π⋅ Hα   λh     7 9  4  2   2  2⋅ ωh ⋅  µ  ⋅  c      3⋅ π   5 7   1 ⋅ 4  ⋅ 2    4  c   µ    3⋅ π    λh

 6.355579× 1095    5  s   = 6.355579× 1095     9  6.355579× 1095   m   

NOTES

322

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Bibliography 2 [1] Particle Data Group, Photon Mass-Energy Threshold: “S. Eidelman et Al.” Phys. Lett. B 592, 1 (2004): http://pdg.lbl.gov/2006/listings/s000.pdf [2] The SELEX Collaboration, Measurement of the Σ - Charge Radius by Σ- - Electron Elastic Scattering, Phys.Lett. B522 (2001) 233-239: http://arxiv.org/abs/hep-ex/0106053v2 [3] “Karmanov et. Al.”, On Calculation of the Neutron Charge Radius, Contribution to the Third International Conference on Perspectives in Hadronic Physics, Trieste, Italy, 7-11 May 2001, Nucl. Phys. A699 (2002) 148-151: http://arxiv.org/abs/hep-ph/0106349v1 [4] “P. W. Milonni”, The Quantum Vacuum – An Introduction to Quantum ElectroDynamics, Academic Press, Inc. 1994. Page 403. [5] Mathworld, http://mathworld.wolfram.com/Euler-MascheroniConstant.html [6] “Hirsch et. Al.”, Bounds on the tau and muon neutrino vector and axial vector charge radius, Phys. Rev. D67: http://arxiv.org/abs/hep-ph/0210137v2 [7] National Institute of Standards and Technology (NIST): http://Physics.nist.gov/cuu/ [8] The D-ZERO Collaboration, A Precision Measurement of the Mass of the Top Quark, Nature 429 (2004) 638-642: http://arxiv.org/abs/hep-ex/0406031v1 [9] 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 [10] 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 [11] “Hammer and Meißner et. Al”., Updated dispersion-theoretical analysis of the nucleon ElectroMagnetic form factors, Eur. Phys.J. A20 (2004) 469-473: http://arxiv.org/abs/hep-ph/0312081v3 [12] “Hammer et. Al”, Nucleon Form Factors in Dispersion Theory, invited talk at the Symposium "20 Years of Physics at the Mainz Microtron MAMI", October 20-22, 2005, Mainz, Germany, HISKP-TH-05/25: http://arxiv.org/abs/hep-ph/0602121v1 [13] Spectrum of the Hydrogen Atom, University of Tel Aviv. http://www.tau.ac.il/~phchlab/experiments/hydrogen/balmer.htm [14] The CDF & D0 Collaborations, W Mass & Properties, FERMILAB-CONF-05-507-E. http://arxiv.org/abs/hep-ex/0511039v1 [15] The L3 Collaboration, Measurement of the Mass and the Width of the W Boson at LEP, Eur. Phys.J. C45 (2006) 569-587: http://arxiv.org/abs/hep-ex/0511049v1 [16] The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the SLD Electroweak & Heavy Flavor Groups, Precision Electroweak Measurements on the Z Resonance, CERN-PH-EP/2005-041, SLAC-R-774: http://arxiv.org/abs/hep-ex/0509008v3 [17] The ZEUS Collaboration, Search for contact interactions, large extra dimensions and finite Quark radius in ep collisions at HERA, Phys. Lett. B591 (2004) 23-41: http://arxiv.org/abs/hep-ex/0401009v2 [18] “James William Rohlf”, Modern Physics from α to Z, John Wiley & Sons, Inc. 1994. [19] http://pdg.lbl.gov/2006/reviews/astrorpp.pdf [20] http://zebu.uoregon.edu/~imamura/123/lecture-2/mass.html [21] http://pdg.lbl.gov/2006/reviews/hubblerpp.pdf (pg. 20 - “WMAP + All”).

323

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Periodic Table of the Elements

324

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Cosmological Evolution Process

Figure 2.4,

325

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Figure 2.5,

326

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NOTES

327

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NOTES

328

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90000 ID: 1540406 www.lulu.com

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Quinta Essentia: A Practical Guide to Space-Time Engineering - Part 2

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