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

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

Abstract A number of tools to facilitate the experimental design process are presented. These include the development of a design matrix based on the unit amplitude spectrum, the derivation of Harmonic and Spectral Similarity Equations, Critical Phase Variance, Critical Field Strengths and Critical Frequency.

1

[email protected], [email protected].

1

1

ITRODUCTIO

1.1

GENERAL

In [1-10], a number of practical engineering tools for application to the Polarizable Vacuum (PV) model of gravity was derived by application of Buckingham Π Theory (BPT) and dimensional analysis techniques. BPT is a well-tested and experimentally verified method that relates a mathematical model to an Experimental Prototype2 (EP). The power of BPT to facilitate and articulate the derivation of the mathematical constructs defined in [6-10] has advanced theoretical boundaries in particle physics to a higher level. The tools derived have volunteered precise calculations leading to the determination of the mass-energy and radii of Leptons, Quarks and Bosons. Most notably, the radius prediction for the Proton has been experimentally verified by the SELEX Collaboration to astounding precision. [7] Two important aspects of the works derived are Critical Ratio “KR” and General Similarity Equations (GSE). “KR” is defined as the ratio of the sum of the magnitudes of the applied electromagnetic (EM) fields, to the magnitude of the background gravitational field. The General Modelling Equations (GME) exploit this definition to produce the “GSE” derived in [5]. 1.2

PRACTICAL METHODS

Practical engineering of the hypothesis to be tested as stated in [2] may be realised by application of the Equivalence Principle with respect to “KR”. Complete similarity occurs when |KR| = 1 and proportional similarity at |KR| ≠ 1, therefore it follows that “KR” may be used to represent relationships in terms of potential, acceleration and energy densities or any suitable measure in harmonic form. It shall be demonstrated that harmonic representation of “KR” in the Fourier domain, at complete dynamic, kinematic and geometric similarity (when |KR| = 1), leads to a useful engineering tool facilitating the experimental design process. Considering that the mathematical representations in [1-4] lead to the determination of physically verified fundamental sub-atomic characteristics as stated in [7], it follows that inertial mass might also be represented harmonically and potentially modified. 1.3

OBJECTVES

The formulations derived herein assist in the qualitative and quantitative experimental design process as follows, i. Harmonic representation of |KR| = 1 in the Fourier domain over an elemental displacement “∆r”, termed the Critical Harmonic Operator “KR H”. ii. Utilisation of “KR H” to formulate harmonic representations of various other physical variables for consideration in the experimental design process. iii. Utilisation of “KR H” to simplify “GSEx”, on a modal basis, to Harmonic Similarity Equations (HSEx). iv. Graphical visualisation of “HSEx” based on Complex Phasor Forms of the magnitude of the applied Electric Field Intensity “EA” and Magnetic Flux Density “BA”. v. Reduction of “HSEx” into simplified electromagnetic (EM) design consideration forms “HSEx R”. vi. Spectral Similarity Equations (SSE): these qualify and quantify the similarity of a singularly applied experimental EM source to the frequencies that inhabit the ambient Electro-GraviMagnetic (EGM) spectrum. [4] vii. Determination of the applied EM phase requirements with respect to the background gravitational field utilising “SSEx”. viii. Assess suitability of Maxwells Equations (ME) to experimental investigations utilising “SSEx”. 1.4

i. ii.

RESULTS The results obtained may be articulated as follows, Development of a design matrix based on “KR H”. Derivation of “HSEx R”, “SSEx”, Critical Phase Variance “φC”, Critical Field Strengths “EC” and “BC” (electric and magnetic field strengths respectively) and Critical Frequency “ωC”.

2

The experimental prototype represents the PV at the surface of the Earth by which all similarity conditions are referenced.

2

2

THEORETICAL MODELLIG3

Assuming complete dynamic, kinematic and geometric similarity between the EP and the mathematical model (|KR| =1) where the harmonic mode of the PV “nPV” approaches the harmonic cutoff mode “nΩ” [|nPV| → nΩ and |nΩ| < ∞], “KR” has many representations. One such representation incorporating the change in harmonic frequency modes across “∆r” shall be derived. The spectral characteristics of the EP may be articulated at the surface of the Earth assuming4, i. The Zero-Point-Field (ZPF) physically exists as a spectrum of frequencies and wave vectors. ii. The sum of all ZPF wave vectors at the surface of the Earth is coplanar with the gravitational acceleration vector. This represents the only vector of practical experimental consequence. iii. A modified Complex Fourier Series representation of “g” is physically real and is representative of the magnitude of the resultant ZPF wave vector. iv. A physical relationship exists between gravity, electricity and magnetism such that the physical interaction of applied EM fields with the PV, in accordance with the hypotheses to be tested [2] may be investigated and potentially modified. It was illustrated in [3] that, for an engineered change in “g” by application of BPT and the Equivalence Principle, a change in the PV may be described [as |nPV| → nΩ] by,

∆g ≡ ∆a PV

∆K 0( ω , X )

E PV k PV, n PV, t

2

B PV k PV, n PV, t

2

n PV, k PV

.

r n PV, k PV

i

(1)

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

Description Change of gravitational acceleration vector Change in PV acceleration vector Magnitude of PV electric field vector Magnitude of PV magnetic field vector Field frequency Harmonic wave vector of PV Denotes initial conditions of PV Engineered relationship function Magnitude of position vector from centre of mass

r

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

Table 1, Subsequently, considering only the resultant ZPF wave vector relating to “g” in a practical laboratory experiment, equation (1) may be usefully simplified by removing “kPV” notation and relating it to a generalised Fourier representation5 of constant “g” over “∆r” as |nPV| → nΩ by, E PV n PV, t ∆ K 0( ω , X ) n PV . r B n PV

2

G. M . PV n PV, t

2

2

i .

2

r

n PV

π . n PV

.e

i

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

(2)

Where, “∆ωδr” denotes the beat frequency across “∆r” as defined in [5].

3

All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 4 A spherical object with uniform mass distribution. 5 The Right Hand Side (RHS) of equation (2) is analogous to the form utilised in [4].

3

E PV n PV, t c

2

n PV

2

B PV n PV, t

2

n PV

(3)

G.M . ∆K 0( ω , X ) KR 2 r .c

(4)6

Substituting equation (3) and (4) into (2) yields the PV EM harmonic representation of the ideal value of the magnitude of “KR” for the complete reduction of “g” over “∆r” in a laboratory at the surface of the Earth7 as |nPV| → nΩ, 2. 1 . π .n PV .∆ω δr( 1 , r , ∆r , M ) .t .i K R( r , ∆ r , M ) i . e n PV π n PV (5) Where the maximum amplitude occurs at time index, t n PV, r , ∆r , M

1 2 . n PV .∆ω δr( 1 , r , ∆r , M )

(6)

Yields the unit amplitude spectrum analogous to the result previously found [4] as |nPV| → nΩ(r+∆r), K R n PV

3

MATHEMATICAL MODELLIG

3.1

DESIGN MATRIX

H

2 . π n PV

(7)

Utilising equation (7) a table of expressions for the magnitude of the amplitude spectrum of various experimental design considerate relationships may be formulated for complete dynamic, kinematic and geometric similarity between the EP and the mathematical model (|KR| =1) as |nPV| → nΩ , Description Engineered Relationship Function Refractive Index

Primitive Form

∆K 0 n PV, r , M , X

G .M

2.

K PV n PV, r , M

2 r .c

K PV e

Engineered Refractive Index Gravitational Potential Energy (per unit mass) Critical Factor8

Harmonic Form

G.M . ∆K 0( ω , X ) KR 2 r .c

K EGM K PV. e

2 . ∆K 0( ω , X )

G. M . m Ug

KR

r

G. M

m

r

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

K EGM n PV, r , M , K R

U g n PV, r , M

.

ε0

H

µ0

H

H

4

ΣKPV H →KPV

H

H

H

∆U PV( r , ∆r , M ) .K R n PV .

“∆K0(ω,X)” is defined as the engineered relationship function. [3] Where, “i” on the RHS of equation (2) represents complex number notation. 8 Representing the applied experimental field. [2,3] 7

H

K EGM r , M , K R .K R n PV

Table 1,

6

Result Σ∆K0 H →∆K0

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

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

H

∆K C n PV, r , ∆r , M

H

G.M . K R n PV 2 r .c

H

µ0 ε0

ΣKEGM H →KEGM ΣUg H →Ug Σ∆KC H →∆KC

Where, the permittivity and permeability of free space (“ε0” and “µ0” respectively) act as the impedance function9 such that, Variable ∆UPV(r,∆r,M) ∆KC(r,∆r,M) G

3.2

Description Change in energy density of PV Change in critical factor Gravitational constant Table 2,

Units Pa PaΩ m3kg-1s-2

ENGINEERING CONSIDERATIONS

Factors to be considered in experimental design configurations when applying equations defined in table (1) are as follows: i. The actual EM modes over “∆r” are subject to normal physical influences. The fundamental frequency mode will not exist within a Casimir experiment; hence the equivalent gravitational acceleration harmonic cannot exist. The relative contribution of the fundamental frequency mode to the gravitational acceleration vector “g” is trivial. ii. Optimal experimental conditions occur when the ratio of the applied Poynting Vector to the Impedance Function approaches unity. [3] 4

PHYSICAL MODELLIG

4.1

HARMONIC SIMILARITY EQUATIONS

A family of “HSEx” may be defined by relating the EP to the mathematical model on a modal basis, termed discrete similarity for |∆r| << ∞. Utilising “GSE1,2” derived in [5] yields “HSE1,2” as follows, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

i . EA k A, n A, t

2

2 ± c .B A k A , n A , t

2

.e

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

2 π . n PV. c . B A k A , n A , t

1, 2

(8)

2

Similarly, “HSE3” may be formed as follows, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

2 .1i.K PV( r , M ) .St α ( r , ∆r , M ) .e

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

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

3

(9)

Hence, “HSE4,5” becomes, HSE E A , B A , k A , n A , n PV, r , ∆r , M , t

2 4 .1i.K PV( r , M ) .St α ( r , ∆r , M ) .c .B A k A , n A , t .e 4, 5

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

2

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

2 ± c .B A k A , n A , t

2

(10)

Recognising that, i .e

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

1

(11)

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

1. 1, 2

E A k A, n A, t

2

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

2

2 ± c .B A k A , n A , t

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

3

2

.K

2

R n PV

H

(12)

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

.K

R n PV

H

(13)

2.

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

9

2. 4, 5

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

2

2 ± c .B A k A , n A , t

.K 2

The impedance function “ Z = µ 0 /ε 0 ” is independent of KPV in the PV representation.

5

R n PV

H

(14)

Where, [5] St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) . 2

∆ U PV( r , ∆ r , M )

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

4.2

3.M .c . 4.π

µ0 ε0

1 (r

(15) 1

∆ r)

3

3

r

(16)

Description Magnitude of applied electric field vector in complex form Magnitude of applied magnetic field vector in complex form Velocity of light in a vacuum Mass Table 3,

Units V/m T m/s kg

VISUALISATION OF HSEx OPERANDS

Visualisation of “HSE” operands10 provides valuable information regarding the differences between forms. For example, it shall be demonstrated that “HSE4,5” suggest constructive and destructive EM interference considerations. To achieve this, we shall utilise the following definitions for the applied EM fields in Complex Phasor Form, E A E 0 , n E, r , ∆ r , M , t

B A B 0 , n B, φ , r , ∆ r , M , t

1 . E 0 .e 2

2 .π .ω E n E , r , ∆r , M .t

1 . B 0 .e 2

π . i 2

(17) 2 .π .ω B n B , r , ∆r , M .t

π 2

φ .i

(18)

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

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

10

E rms

E A E 0 , n E, r , ∆ r , M , t

B rms

B A B 0 , n B, φ , r , ∆ r , M , t

2. 2

E0 2. 2

(19) B0

Description Applied electric field vector Applied magnetic field vector Amplitude of electric field vector Amplitude of magnetic field vector Harmonic mode number of the ZPF with respect to “EA” Harmonic mode number of the ZPF with respect to “BA” Relative phase variance between “EA” and “BA” Harmonic frequency of the ZPF with respect to “EA” Harmonic frequency of the ZPF with respect to “BA” Root-mean-square of “EA” Root-mean-square of “BA” Table 4,

The expression inside the magnitude notation on the RHS of equation (8-10).

6

(20) Units V/m T V/m T None None None Hz Hz V/m T

Equations (17,18) are functions in Complex Form and contain Real and Imaginary components. For visualisation purposes, only the Real component is required. Figure (1) includes a graphical representation of “EA” and “BA” for arbitrary illustrational values. The representations for “Re(HSE1,2)” have been accentuated for illustrational purposes by a large value of “φ” (180°). Typically, values of “0°” would be expected in accordance with classical EM propagation, or “90°” in accordance with Maxwell’s Equations11.

Re E A 1 .

V m

, 1 , R E , ∆r , M E , t

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

V m

, 1 .( T ) , 1 , 1 , 180 .( deg ) , 3 , R E , ∆r , M E , t

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

t

Electric Forcing Function Magnetic Forcing Function HSE 1 HSE 2

Figure 1, Figure (2) includes arbitrary illustrational values but also contains important information regarding “φ”. Exploratory graphical analysis demonstrates that “Re(HSE3)” is in-phase with “Re(HSE4)” and out-of-phase with “Re(HSE5)” for key values (0° and 90°) of “φ”. The significance of this being that “Re(HSE3)” is analogous to the Poynting Vector and implies that “Re(HSE4)” is representative of constructive EGM interference and “Re(HSE5)” is representative of destructive EGM interference. “HSE4,5” were formed from General Modelling Equation “1” and “2” (GME1,2) as described in [5]. “GME1” is proportional to a solution of the Poisson equation applied to Newtonian gravity, where the resulting acceleration is a function of the geometry of the energy densities. “GME2” is proportional to a solution of the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities of the EM field harmonics in a vacuum. Therefore, experimental investigations with the objective of reducing the local gravitational acceleration on a test bench, by means of EGM interference, should bias engineering designs governed by “HSE5”. However, designs favouring “HSE4” should not be completely discounted and should form part of any complete design process.

Re HSE 3 1 .

V m

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

V Re HSE 4 1 . , 1 .( T ) , 1 , 1 , 0 .( deg ) , 3 , R E , ∆r , M E , t m Re HSE 5 1 .

V m

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

HSE 3 HSE 4 HSE 5

Figure 2,

11

This shall be discussed in greater detail in proceeding sections.

7

t

4.3

REDUCTION OF HSEx

“HSEx” may be simplified by performing the appropriate substitution12 of equation (17-20). The simplified equations carry the subscript13 “R” and facilitate the investigation of the influence of “φ” on a modal basis. This becomes important in a practical sense because experimental investigations will involve “1” (or very few) applied forcing function frequencies. The reproduction of the entire background EGM spectrum would be technically difficult to achieve. Subsequently, experimental configurations will need to consider “φ” influences very carefully. Assuming the forcing function frequency for “EA” is equal to that of “BA” yields “HSEx R” as follows, HSE 1 φ , n PV

2 . ( cos ( 2 . φ ) π . n PV

R

HSE 2 φ , n PV

2 . ( cos ( 2 . φ ) π . n PV

R

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

1)

(21) 1)

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

π . n PV. E rms . B rms

2 . HSE 3 E rms , B rms , n PV, r , ∆ r , M HSE 4 E rms , B rms , n EM , φ , n PV, r , ∆ r , M , t

R

R

(23) R

cos 4 . π . n EM . t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t . sin 4 . π . n EM . t 2 . φ ... +1 2 . HSE 3 E rms , B rms , n PV, r , ∆ r , M

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

(22)

(24)

R

cos 4 . π . n EM . t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t . sin 4 . π . n EM . t 2 . φ ... + 1

(25)

Where, “nEM” denotes a common EM harmonic mode for “EA” and “EB”. 4.4

VISUALISATION14 OF HSEx R

Significant design information leading to complete dynamic, kinematic and geometric similarity with the background field (|KR| =1), may be obtained by visualisation of “HSEx R”. Assigning arbitrary values where required to analyse modelling behaviour facilitating the experimental design process yields, Figure (3) analysis: i. Similarity is maximised at “φ = 0°” and “φ = 90°”: Intuitively, this appears to agree with expectations: these phase angles are observed in classical vacuum EM wave propagation and Maxwell's Equations respectively. ii. Similarity is maximised as |nPV| → 1: This implies that a low frequency carrier wave encasing a high frequency Poynting Vector maximises similarity of the applied field with the background gravitational field. Intuitively, this appears to agree with expectation as the population of Photons in the ZPF is maximised at the fundamental harmonic as described in [6]. iii. HSE1 = HSE2 at “φ = 45°” and “φ = 135°”.

12

See appendix (A). Of the form “HSEx R”. 14 Alteration in notation is required for graphing purposes. It is a limitation of “MathCad 8 Professional” that axial arguments may not be written precisely in the form “HSE1 R(φ,1)” etc. 13

8

π

π

Harmonic Similarity

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

φ Phase Variance

Figure 3, Figure (4) analysis [Y-Axis is logarithmic]: iv. |HSE3 R| → 1 as |nPV| → nΩ ZPF: This is consistent with Poynting Vector characteristics described in [2,3]. Where,

Harmonic Similarity

Variable nΩ(r,∆r,M)ZPF RE ME

Description ZPF beat cut-off mode across “∆r” at “r” [5] Radius of the Earth Mass of the Earth Table 5,

Units None m kg

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

n PV Harmonic Mode

Figure 4, 4.5

2nd REDUCTION OF HSE4,5

Reducing the number of independent variables where possible can make obvious gains. To achieve this, we shall express “HSE4,5 R” as time averaged functions “HSE4A,5A R” explicitly in terms of “φ” and “HSE3 R” as follows15, HSE 4A E rms , B rms , φ , n PV, r , ∆ r , M HSE 5A E rms , B rms , φ , n PV, r , ∆ r , M

1 cos ( φ )

R

sin ( φ )

1

Hence, v. |HSE4,5 R| → 1 at “φ = 0°” and “φ = 90°” respectively. vi. |HSE4| = |HSE5| at “φ = 45°” and “φ = 135°”.

15

. HSE

R

Refer to appendix (A) for derivation.

9

. HSE

3 E rms , B rms , n PV, r , ∆ r , M

3 E rms , B rms , n PV, r , ∆ r , M

R

(26)

R

(27)

4.6

SPECTRAL SIMILARITY EQUATIONS

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

SSE

.

n Ω ( r, ∆ r, M )

1 ZPF

HSE n PV

(28)

Where, “nPV” has the odd harmonic distribution: -nΩ ZPF, 2 - nΩ ZPF …. nΩ ZPF Recognising that16, 1 n Ω ( r, ∆ r, M )

1 ZPF

ln 2 . n Ω ( r , ∆ r , M )

1

. n PV

n PV

n Ω ( r, ∆ r, M )

γ ZPF

1 ZPF

(29)

As nPV → nΩ ZPF and nΩ ZPF >>1: Substituting equation (29) into the “HSE” yields, SSE 1( φ , r , ∆ r , M )

2 . ( cos ( 2 . φ )

n Ω ( r, ∆ r, M )

2 . ( cos ( 2 . φ )

1) .

π

γ 1

ZPF

ln 2 . n Ω ( r , ∆ r , M ) n Ω ( r, ∆ r, M )

(30) γ

ZPF

1 ZPF

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

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

4.7

ln 2 . n Ω ( r , ∆ r , M ) ZPF

π

SSE 2( φ , r , ∆ r , M )

SSE 3 E rms , B rms , r , ∆ r , M

1) .

1

. SSE

cos ( φ ) 1 sin ( φ )

. SSE

3 E rms , B rms , r , ∆ r , M

3 E rms , B rms , r , ∆ r , M

(31)

(32)

(33)

(34)

CRITICAL PHASE VARIANCE

“φC” is defined as the phase difference between “EA” and “BA” for complete dynamic, kinematic and geometric similarity with the background EGM field “|SSEx| = 1”. Therefore, by analyses of the preceding figures or the appropriate transformation of equation (31-34), “φC” may be easily determined. For proportional solutions to the Poisson equation applied to Newtonian gravity where the resulting acceleration is a function of the geometry of the energy densities, “φC = 0°”. For proportional solutions to the Lagrange equation where the resulting acceleration is a function of the Lagrangian densities, “φC = 90°”.

16

Error “< 6.7x10-6(%)” at “nΩ ZPF > 106”.

10

4.8

CRITICAL FIELD STRENGTH

“EC” and “BC” are derived utilising the reciprocal harmonic distribution describing the EGM amplitude spectrum. Solutions to “|SSE4,5| = 1” represent conditions of complete dynamic, kinematic and geometric similarity with the amplitude of the background EGM spectrum. “EC” and “BC” denote RMS values satisfying the proceeding equation, SSE 4 E rms , B rms , 0 , r , ∆ r , M

SSE 5 E rms , B rms ,

π

, r, ∆ r, M

1

2

(35)

Where, E rms E C( r , ∆ r , M )

B rms

4.9

(36)

E C( r , ∆ r , M ) c

(37)

DC-OFFESTS

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

SSE 4 ( 1

DC) . E rms, B rms, 0 , r , ∆ r , M

DC) . E rms, ( 1

DC) . B rms, 0 , r , ∆ r , M

SSE 5 E rms, ( 1

SSE 5 ( 1

DC) . B rms,

DC) . E rms, ( 1

π

, r, ∆ r, M

2

DC) . B rms,

1 2

π

, r, ∆ r, M

2

(38) 1 4 (39)

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

MAXWELLS EQUATIOS

5.1

GENERAL

By considering “ME” in relation to the applied EM fields and the requirements of similarity, it is possible to deduce important design characteristics for further consideration. “ME” (in MKS units) for time-varying fields are as follows (“ρ” is the charge density and “J” is the vector current density), [11] ρ ∂B . ∂E ∇ .E ,∇ E , ∇ B 0, ∇ B µ 0.J ε 0.µ 0. ε 0 ∂t ∂t (40) Consequently as |SSE5| → 1, optimal similarity occurs when: 1. The divergence of “EA” is maximised. 2. The curl of “BA” is maximised. 5.2

CRITICAL FREQUENCY

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

11

c 2.∆ r

(41)

6

COCLUSIOS

A number of tools that facilitate the experimental design process are presented. These include the development of a design matrix based on the unit amplitude spectrum, the derivation of Harmonic and Spectral Similarity Equations, Critical Phase Variance, Critical Field Strengths and Critical Frequency. 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, “Derivation of the Photon mass-energy threshold” http://www.deltagroupengineering.com/Docs/Photon_Mass-Energy_Threshold.pdf [7] R. C. Storti, T. J. Desiato, “Derivation of fundamental particle radii (Electron, Proton & Neutron)” http://www.deltagroupengineering.com/Docs/Fundamental_Particle_Radii.pdf [8] R. C. Storti, T. J. Desiato, “Derivation of the Photon & Graviton mass-energies & radii”, http://www.deltagroupengineering.com/Docs/Photon-Graviton_Mass-Energy.pdf [9] R. C. Storti, T. J. Desiato, “Derivation of Lepton radii”, http://www.deltagroupengineering.com/Docs/Lepton_Radii.pdf [10] R. C. Storti, T. J. Desiato, “Derivation of Quark & Boson mass-energies & radii”, http://www.deltagroupengineering.com/Docs/Quark_&_Boson_Mass-Energies_&_Radii.pdf [11] Wolfram Research: http://scienceworld.wolfram.com/physics/MaxwellEquations.html

12

APPEDIX A

i . EA

2

2.

c BA

2

substitute , E A

E 0 .e

substitute , B A

B 0 .e

π . i 2

2 .π .n E .t

2 .π .n B .t

π

φ .i

2

i . ( exp( 2 . i .φ ) π . n PV

substitute , n E n B

2 2 π . n PV. c . B A

substitute , B 0

2 . B rms

substitute , E 0

2 . E rms

1)

substitute , E rms c . B rms

(A1)

simplify 1

i . ( exp( 2 . i . φ ) π . n PV

HSE 1 φ , n PV

2 i . EA

R

2.

( cos ( 2 . φ )

1)

2

π

2 . ( cos ( 2 . φ ) π . n PV

2 2 c .B A

1.

1 ) simplify

n PV

(A2)

1)

(A3)

substitute , E A

E 0 .e

substitute , B A

B 0 .e

π . i 2

2 .π .n E .t

π

2 .π .n B .t

φ .i

2

i . ( exp( 2 . i .φ ) π . n PV

substitute , n E n B

2.

2 π . n PV. c B A

2

substitute , B 0

2 . B rms

substitute , E 0

2 . E rms

1)

substitute , E rms c . B rms

(A4)

simplify 1

( exp( 2 .i . φ ) i . π . n PV

HSE 2 φ , n PV

2 . i . K PV( r , M ) . St α ( r , ∆ r , M )

substitute , E A

E 0 .e

substitute , B A

B 0 .e

2 .π .n E .t

2 .π .n B .t

R

1)

2 . ( cos ( 2. φ ) π . n PV

substitute , E 0

2 . E rms

2

(A5) (A6)

π . i 2 π 2

φ .i

substitute , n B n EM 2 . B rms

2)

2

n PV

1)

i . K PV( r , M ) .

substitute , B 0

( 2 . cos ( 2 . φ )

π

substitute , n E n B

π . n PV. E A . B A

1.

simplify

St α ( r , ∆ r , M ) . exp i . 4 . π . n . EM t π . n PV. E rms . B rms

π

φ

(A7)

simplify

i . K PV( r , M ) .

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

. exp i . 4. π . n . EM t

π

φ

simplify

π

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

2 4 .i . St α ( r , ∆ r , M ) . K PV( r , M ) . c . B A 2 π . n PV. E A . E A

substitute , E A

E 0 .e

substitute , B A

B 0 .e

2 .π .n B .t

π 2

2 . B rms

substitute , E 0

2 . E rms

substitute , c

n PV. E rms. B rms

(A8)

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

(A9)

φ .i

substitute , n E n B

substitute , B 0

R

K PV( r , M ) .

St α ( r , ∆ r , M )

π . i 2

2 .π .n E .t

2 . i . St α ( r , ∆ r , M ) .

substitute , n B n EM

2 2 c .B A

1.

K PV( r , M ) E rms . B rms .π . n PV. exp i . π . 4 .n EM . t 1

exp i . 4 . π . n EM . t

π

2.φ

. exp( i . φ )

E rms B rms

(A10)

simplify, factor 1

2.i .e

i .φ .

E rms . B rms . π . n PV. e

K PV( r , M ) . St α ( r , ∆ r , M )

i .π . 4 .n EM .t

1

e

i . 4 .π .n EM .t

π

2 .φ

simplify

1. π

2 2 . K PV( r , M ) .

St α ( r , ∆ r , M ) 2 2 2 B rms .E rms . n PV . cos 4 . π . n EM . t . cos 4 . π . n EM .t

13

2.φ

2

2

1

sin 4 . π . n EM . t . sin 4 . π . n EM . t

2.φ

(A11)

2 .HSE 3 E rms , B rms , n PV, r , ∆ r , M HSE 4 E rms , B rms , n EM , φ , n PV, r , ∆ r , M , t

substitute , E A

E0

substitute , B A

B 0 .e

π

2 .π .n B .t

2

2 .i . St α ( r , ∆ r , M ) . K PV( r , M ) .

substitute , B 0

2 . B rms

substitute , E 0

2 . E rms

substitute , c

(A12)

φ .i

substitute , n B n EM

2 2 c .B A

R

cos 4 . π . n EM .t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t .sin 4 .π . n EM . t 2 . φ ... +1

π . i 2

2 .π .n E .t

substitute , n E n B

2 4 .i . St α ( r , ∆ r , M ) . K PV( r , M ) . c . B A 2 π . n PV.E A . E A

.e

R

exp( i . φ ) E rms . B rms . π . n PV. exp i .π . 4 . n EM . t 1

exp i . 4 .π . n EM . t

π

2.φ

E rms B rms

(A13)

simplify, factor 1 i 2.i . e

.φ .

E rms . B rms . π . n PV. e

K PV( r , M ) . St α ( r , ∆ r , M )

i .π . 4 .n EM .t

1

e

i . 4 .π .n EM .t

π

simplify

2 .φ

1. π

2 2 . St α ( r , ∆ r , M ) .

K PV( r , M ) 2 2 2 B rms . E rms . n PV . cos 4 . π . n EM . t . cos 4 . π . n EM . t

2

2

2. φ

1

sin 4 . π . n EM . t . sin 4 . π . n EM . t

2 . HSE 3 E rms , B rms , n PV, r , ∆ r , M HSE 5 E rms , B rms , n EM , φ , n PV, r , ∆ r , M , t

R

2.φ

(A14)

R

cos 4 . π . n EM .t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t .sin 4 .π . n EM . t 2 . φ ... + 1

(A15)

T δr n PV, r , ∆ r , M HSE 4A E rms, B rms, φ , n PV, r , ∆ r , M T δr n PV, r , ∆ r , M ∆ω δr n PV, r , ∆ r , M

.

R

∆ω δr n PV, r , ∆ r , M

.

HSE 4 E rms, B rms, n EM , φ , n PV, r , ∆ r , M , t 0.( s )

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

R

dt R

(A16) ∆ω δr n PV, r , ∆ r , M

dt

T n , r, ∆ r, M . δr PV . HSE 1

cos 4 . π . n EM . t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t . sin 4 . π . n EM . t 2 . φ ... +1

cos ( φ )

2

3 E rms , B rms , n PV, r , ∆ r , M

R

2

(A17)

0.( s )

Recognising, .T

∆ω δr n PV, r , ∆ r , M

δr n PV, r , ∆ r , M

1

(A18)

Simplifying the integral yields the Time Averaged HSE4 at each frequency mode “nPV” as follows: HSE 4A E rms, B rms, φ , n PV, r , ∆ r , M

1 R

.HSE

cos ( φ )

3 E rms, B rms, n PV, r , ∆ r , M

(A19)

R

Hence, it follows that: T δr n PV, r , ∆ r , M HSE 5A E rms , B rms , φ , n PV, r , ∆ r , M

R

T δr n PV, r , ∆ r , M

∆ω δr n PV, r , ∆ r , M

.

∆ω δr n PV, r , ∆ r , M

.

HSE 5 E rms , B rms , n EM , φ , n PV, r , ∆ r , M , t 0.( s )

dt R

(A20)

K PV( r , M ) . St α ( r , ∆ r , M ) 2. π . n PV. E rms . B rms

i

dt

cos 4 . π . n EM . t . cos 4 . π . n EM . t 2 . φ ... + sin 4 . π . n EM . t . sin 4 . π . n EM . t 2 . φ ... + 1

. HSE

sin ( φ )

3 E rms , B rms , n PV, r , ∆ r , M

2

0.( s )

R

(A21)

Recognising that for “0 < φ < π”: i sin ( φ )

1 2

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

sin ( φ )

1 R

sin ( φ )

(A22) . HSE E 3 rms, B rms, n PV, r , ∆ r , M

(A23)

R

Subsequently, the solution representing HSE4,5A R equality may be stated as follows: HSE6A E rms, B rms, φ , n PV, r , ∆ r , M

R

HSE4A E rms, B rms, φ , n PV, r , ∆ r , M

14

R

HSE5A E rms, B rms, φ , n PV, r, ∆ r, M

R

(A24)