Electro-gravi-magnetics (egm); Practical Modelling Methods Of The Polarizable Vacuum - Vii

Electro-Gravi-Magnetics (EGM) Practical modelling methods of the polarizable vacuum - VII Riccardo C. Storti1, Todd J. Desiato

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

1

[email protected], [email protected].

1

1

ITRODUCTIO2 [1-11]

Storti et. al. established two Reduced Average Harmonic Similarity Equations (HSE4A,5A R) in [6]. It shall be demonstrated that “HSE4A,5A R” may be utilised to describe the characteristics of Relative Phase Variance “φ” over the range of the Polarizable Vacuum (PV) harmonic “nPV”. Subsequently, deriving the Casimir Force and hypothesising a calculation of the PV resonant mode and frequency (“NX” and “ωX” respectively) of a classical Casimir Plate experiment. “HSE4A,5A R” presented in [6] are as follows, 1 . HSE 4A E rms , B rms , φ , n PV, r , ∆ r , M HSE 3 E rms , B rms , n PV, r , ∆ r , M R cos ( φ ) R (1) HSE 5A E rms , B rms , φ , n PV, r , ∆ r , M

1 R

. HSE

sin ( φ )

3 E rms , B rms , n PV, r , ∆ r , M

R

(2)

Where, the permittivity and permeability of free space (“ε0” and “µ0” respectively) act as the impedance function3, K PV( r , M ) . St α ( r , ∆ r , M ) HSE 3 E rms , B rms , n PV, r , ∆ r , M R π . n PV. E rms . B rms (3) St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) . 2

∆ U PV( r , ∆ r , M )

Variable HSE3 R r ∆r c M Stα ∆UPV KPV Erms Brms

2

3.M .c . 4.π

µ0 ε0

1 (r

∆ r)

(4) 1

3

3

r

Description Reduced Harmonic Similarity Equation proportional to the Poynting Vector of the PV Magnitude of position vector from centre of the Earth 1. Magnitude of elemental displacement “∆r” 2. Separation distance between parallel Casimir Plates Velocity of light in a vacuum Planetary mass Range Factor Change in energy density of PV Refractive Index of PV [1] Root-mean-square of “EA” (applied Electric Field) Root-mean-square of “BA” (applied magnetic field) Table 1,

(5) Units None m m m/s kg PaΩ Pa None V/m T

THEORETICAL MODELLIG

Spectral Similarity Equations “SSEx” were developed in [6] from “HSEx”. “SSEx” represent the average magnitude per harmonic mode, analogous to a solution of field pressure equilibrium with respect to the intensity of amplitude spectrum. Of particular importance, SSE3 denotes a proportional formulation of the ambient (i.e. required applied) Poynting Vector as follows, SSE 3 E rms , B rms , r , ∆ r , M

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

2

(6)

All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 3 The impedance function “ Z = µ 0 /ε 0 ” is independent of KPV in the PV representation.

2

Where4, “γ” denotes Euler’s Constant and “NX” is termed the harmonic inflection mode, n Ω ( r, ∆ r, M ) N X( r , ∆ r , M )

1 ZPF

ln 2 . n Ω ( r , ∆ r , M )

γ ZPF

(7)

Utilising equation (6) and assuming complete similarity between the PV and “SSE3” yields the Critical field Strengths “EC” and “BC” as follows, SSE 3 E rms , B rms , r , ∆ r , M

1

(8)

E rms E C( r , ∆ r , M ) B rms B C( r , ∆ r , M )

(9) E C( r , ∆ r , M ) c

(10)

Substituting equation (8-10) into equation (6) and solving for “EC” yields, c . K PV( r , M ) . St α ( r , ∆ r , M ) π . N X( r , ∆ r , M )

E C( r , ∆ r , M )

(11)

Therefore, utilising equation (1,2) and assuming complete similarity between the PV and “HSE4A,5A R”, an expression for “φ4,5” in terms of “nPV” for each harmonic form may be articulated as follows, φ 4 E C( r , ∆ r , M ) , B C( r , ∆ r , M ) , n PV, r , ∆ r , M

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

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

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

R

(12)

R

(13)

5

Hence , π N C R E , ∆r , M E

N X R E , ∆r , M E

π

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

2

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

12 1 .10

1 .10

13

14 1 .10

1 .10

15

16 1 .10 n PV

1 .10

17

1 .10

18

1 .10

19

1 .10

20

Figure 1, Where, “NC” indicates the Critical Mode representing the condition of minimum permissible wavelength between the parallel plates over a separation distance of “∆r = 1(mm)”. “ωC” and “ωPV(1,r,M)” denote the Critical Frequency [6] and fundamental harmonic frequency [4] respectively, N C( r , ∆ r , M )

ω C( ∆ r ) ω PV( 1 , r , M )

ω C( ∆ r )

4 5

See Appendix A. “RE” and “ME” denote radius and mass of the Earth.

3

c . 2∆r

(14)

(15)

Analysis of figure (1) illustrates that “NX” represents a point of graphical inflection where the rate of change of “φ4,5” with respect to “nPV” is non-trivial. Notably, “φ4 = π” and “φ5 = π/2” over the range “1 ≤ nPV ≤ NX” and are influenced by the manner in which the applied forcing functions (“EA” and “BA” – representing the PV by similarity) have been initially defined. Since the PV cannot be uniquely described by a single mode, the arbitrary value of “φ” initially utilised in the mathematical construct is unimportant. Hence, the phase similarity on a modal basis may be disregarded. The Critical Phase Variance “φC” defined in [6] considers the entire PV spectrum when defining the required value of “φ” for complete similarity. Subsequently, we may conjecture that the corresponding frequency at “NX” relates to a resonant condition where the Resonant Casimir Frequency “ωX” may be defined as follows, ω X( r , ∆ r , M ) N X( r , ∆ r , M ) . ω PV( 1 , r , M )

3

(16)

MATHEMATICAL MODELLIG

To derive a relationship incorporating harmonic PV characteristics with the Casimir Force for parallel plates, we shall bring to the fore a suite of mathematical approximations resulting in a highly precise representation of the Casimir Force. Storti et. al. illustrated in [7] that the sum of odd modes of a double sided reciprocal harmonic spectrum, symmetrical about the 0th mode, approaches the sum of all modes of a one-sided reciprocal harmonic spectrum as “|nPV| → nΩ ZPF and nΩ ZPF >> 1” according to, n Ω ( r , ∆r , M ) 1 n PV

ZPF

1

ln( 2 )

n PV

n PV

n PV = 1

ln 2 .n Ω ( r , ∆r , M ) ZPF

γ

(17)

Subsequently, the difference in sum between “NX” and “NC” may be usefully approximated as follows, ln 2 . N X( r , ∆ r , M )

ln 2 . N C( r , ∆ r , M )

γ

γ

ln

N X( r , ∆ r , M ) N C( r , ∆ r , M )

(18)

By contrast to the preceding equations, we shall apply classical arithmetic progression to facilitate the derivation of the Casimir Force. A Fourier distribution describing a constant function is composed of a reciprocal harmonic series governing amplitude characteristics and an arithmetic progression governing frequency characteristics. The interaction of these two spectral distributions intersects at the fundamental harmonic (|nPV| = 1). Subsequently, we expect that an expression relating the Casimir Force to harmonic distributions of the PV should consider aspects of a classical arithmetic sequence and a reciprocal harmonic series. Hence, Let: “A”, “D” and “StN” denote the values of the 1st harmonic term, common difference and NTth harmonic term respectively in a classical arithmetic sequence [12] as follows, St N A

NT

1 .D

(19)

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

St N

A

D

D

(20)

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

N T A , D , N X( r , ∆ r , M ) N T A , D , N C( r , ∆ r , M )

(21)

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

NT

. 2 .A

2

D. N T

Where, ratio of the sum of terms “ΣHR” relating “NX” to “NC” is,

4

1

(22)

Σ HR( A , D , r , ∆ r , M )

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

(23)

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

4

N TR( 1 , 1 , r , ∆ r , M )

Σ HR( 1 , 2 , r , ∆ r , M )

(24)

PHYSICAL MODELLIG

Analysis and consideration of the mathematical characteristics of equation (17-24) facilitates the formulation of the Casimir Force “FPV” in terms of “NX” and “NC” [for a specific configuration of “∆r = 1(mm)”] as follows, F PV A PP , r , ∆ r , M

A PP . ∆ U PV( r , ∆ r , M ) .

N C( r , ∆ r , M ) N X( r , ∆ r , M )

2

. ln

N X( r , ∆ r , M ) N C( r , ∆ r , M )

4

(25)

Where, “APP” denotes the projected area of a parallel plate in a classical Casimir experiment. We shall now compare the classical representation of the Casimir Force for parallel plates “FPP” to the preceding equation by performing a sample calculation, [13] π . h . c . A PP

F PP

4 480. ∆ r

(26)

Considering a Casimir plate area equal to planetary surface area in equation (25), yields a result to within “10-2 (%)” of the classical representation of the Casimir Force described by equation (26). Analysis of equation (25) indicates that “FPP” decreases with increasing ambient gravitational environment. This concurs with [5] and suggests the exclusion of fewer available low frequency modes. The mathematical construct defined in [5] states that, as gravitational acceleration at the surface of a planetary body increases, “ωPV(1,r,M)” also increases. Therefore, an Earth based equivalent Casimir experiment conducted on Jupiter will exclude fewer low frequency modes – preserving higher frequency modes that simply pass through the plates, resulting in a smaller Casimir Force. %otably, a Casimir Experiment conducted in free space will produce an extremely small force (tending to zero) due to the lack of initial background field pressure. Since the Casimir Force arises from a pressure imbalance, the lack of significant ambient field pressure between the plates prevents the formation of large Casimir Forces. 5

COCLUSIOS

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

5

References [1] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – I”, http://www.deltagroupengineering.com/Docs/EGM_1.pdf [2] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – II”, http://www.deltagroupengineering.com/Docs/EGM_2.pdf [3] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – III”, http://www.deltagroupengineering.com/Docs/EGM_3.pdf [4] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – IV”, http://www.deltagroupengineering.com/Docs/EGM_4.pdf [5] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – V”, http://www.deltagroupengineering.com/Docs/EGM_5.pdf [6] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – VI”, http://www.deltagroupengineering.com/Docs/EGM_6.pdf [7] R. C. Storti, T. J. Desiato, “Derivation of the Photon mass-energy threshold” http://www.deltagroupengineering.com/Docs/Photon_Mass-Energy_Threshold.pdf [8] R. C. Storti, T. J. Desiato, “Derivation of fundamental particle radii (Electron, Proton & Neutron)” http://www.deltagroupengineering.com/Docs/Fundamental_Particle_Radii.pdf [9] R. C. Storti, T. J. Desiato, “Derivation of the Photon & Graviton mass-energies & radii”, http://www.deltagroupengineering.com/Docs/Photon-Graviton_Mass-Energy.pdf [10] R. C. Storti, T. J. Desiato, “Derivation of Lepton radii”, http://www.deltagroupengineering.com/Docs/Lepton_Radii.pdf [11] R. C. Storti, T. J. Desiato, “Derivation of Quark & Boson mass-energies & radii”, http://www.deltagroupengineering.com/Docs/Quark_&_Boson_Mass-Energies_&_Radii.pdf [12] BBC Education: http://www.bbc.co.uk/education/asguru/maths/13pure/03sequences/17arthimetic/principles.shtml [13] University of California, Riverside: http://math.ucr.edu/home/baez/physics/Quantum/casimir.html [14] Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/whdwar.html [15] Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/redgia.html [16] Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/pulsar.html APPEDIX A 4

ω Ω ( r, ∆ r, M ) ZPF

∆ω δr n PV, r , ∆ r , M

3

2.c . ∆ U PV( r , ∆ r , M ) h

ω PV n PV, r

∆ r, M

∆ω δr( 1 , r , ∆ r , M )

4

(A1) ω PV n PV, r , M

(A2)

G .M

ω PV n PV, r , M

n Ω ( r, ∆ r, M )

n PV 3 2 . c . G. M . .e r π .r

2 r .c

(A3)

ω Ω ( r, ∆ r, M ) ZPF ZPF

Where, “G” denotes the Universal Gravitation Constant.

6

ω PV( 1 , r , M )

(A4)