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

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

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

1

[email protected], [email protected].

1

1

ITRODUCTIO

1.1

GENERAL

The Polarizable Vacuum (PV) model provides a theoretical description of space-time that may be derived from the superposition of electromagnetic (EM) fields. It is conjectured that the space-time metric might be engineered utilising Electro-Gravi-Magnetics2 (EGM), where EM fields may be applied to affect the state of the PV and thereby facilitate interactions with the local gravitational environment. [1-4] This paper continues previous work leading to practical modelling methods of the PV based on the assumption that dimensional similarity exists between the space-time geometric manifold and applied EM fields. In accordance with Buckingham's Π Theory (BPT), experiments must be designed that test the hypothesis stated in [2]. 1.2

HARMONICS

The formulation herein advances the works [1-4] and facilitates the following additions to the global EGM construct, i. Derivation of the fundamental harmonic beat frequency across an elemental displacement “∆r”, IFF ∆r << r. This is evaluated to illustrate that the contribution of low frequency harmonics is trivial relative to high frequency harmonics when considering “g”. ii. Group velocities across “∆r”. iii. Formulation and development of the Critical Boundary “ωβ” leading to the proposition that the dominant bandwidth arising from the formation of beat spectrums is several orders of magnitude above the Tera-Hertz (THz) range, terminating at the ZPF beat cut-off frequency “ωΩ ZPF”. iv. The development of General Similarity Equations (GMEx) applicable to experimental investigations. v. The proposition that the modification of “g” is dominated by the magnitude of the applied magnetic field vector “BA”. vi. The proposition that the EGM Spectrum is an extension of the classical Electro-Magnetic Spectrum. 2

THEORETICAL MODELLIG

Fourier series may be applied to represent a constant function over an arbitrary period by the infinite summation of sinusoids. Since the PV model of gravitation is an isomorphic approximation of General Relativity (GR) in the weak field, it follows that Fourier series may present a useful tool by which to describe gravity as the number of harmonic frequency modes tends to infinity3. [2-4] 3

MATHEMATICAL MODELLIG4,5

3.1

INTRODUCTION

The spectral composition of the PV / Zero-Point Field (ZPF) is an important design consideration for experimental investigations. It was illustrated that the harmonic cut-off mode “nΩ” may be quantified by a system of equations. [4] Taking limits of “nΩ” as described in [4] yields the free space6 harmonic cut-off mode as follows, lim n ( r, M ) - Ω r ∞ →∞ (1)

2

Electro-Gravi-Magnetics (EGM) is based on Buckingham’s Π Theory as defined in [1]. The frequency spectrum of the PV may be postulated to range from -∞ < ωPV < ∞. [1-4] 4 All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 5 See appendix A for graphical representations of the modelling criteria covered in this section. 6 Free space refers to a flat space-time manifold where the magnitude of the acceleration vector is 0(m/s2). 3

2

Therefore, the spectral modelling characteristics of the PV / ZPF may be articulated as follows, i. The free space harmonic mode bandwidth is -∞ < nPV < +∞. ii. The magnitude of the free space harmonic cut-off mode tends to infinity (nΩ → ∞). iii. The fundamental harmonic frequency of free space tends to zero [nPV = 1, ωPV(nPV,r,M) → 0Hz]. iv. The presence of a planetary mass superimposed on the PV / ZPF alters the free space harmonic mode spectrum, described by equation (1), to -nΩ(r,M) ≤ nPV ≤ +nΩ(r,M). v. The fundamental and cut-off harmonic frequencies of the PV / ZPF for a planetary mass increases as “r” decreases according7 to: ωPV(1,r-∆r,M) > ω PV(1,r,M), ωΩ(r-∆r,M) > ωΩ(r,M) and nΩ(r-∆r,M) < nΩ(r,M) ωΛ(1,r,M) Hz ωPV(1,RE,M0) → 0 ωPV(1,RE,MM) ≈ 0.008 ωPV(1,RE,ME) ≈ 0.0358 ωPV(1,RE,MJ) ≈ 0.2445 ωPV(1,RE,MS) ≈ 2.4841

ωΩ(r,M) YHz ωΩ(RE,M0) < ∞ ωΩ(RE,MM) ≈ 196 ωΩ(RE,ME) ≈ 520 ωΩ(RE,MJ) ≈ 2x103 ωΩ(RE,MS) ≈ 9x103 Table 1,

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

Where, Variable nΩ(r,M) ω PV(nPV,r,M) ωΩ(r,M) nPV r ∆r M RE M0 MM ME MJ MS

Description Harmonic cut-off mode of PV Harmonic field frequency of PV Harmonic cut-off frequency of PV Harmonic frequency modes of PV Magnitude of position vector relative to the centre of mass of a planetary body Magnitude of change of position vector Mass of the planetary body Radius of the Earth Zero mass condition of free space Mass of the Moon Mass of the Earth Mass of Jupiter Mass of the Sun Table 2,

3.2

PHENOMENA OF BEATS [5]

3.2.1

FREQUENCY

Units None Hz Hz None m m kg m kg kg kg kg kg

It was illustrated in [4] that it is convenient to model a gravitational field at a mathematical point utilising Complex Fourier Series obeying an odd number harmonic distribution. Subsequently, it follows that a beat frequency “∆ωδr” spectrum forms across “∆r” since nΩ(r,M) ≠ nΩ(r ± ∆r,M). Hence, it is postulated that the change8 in frequency across “∆r” may be usefully approximated as follows9, ∆ω δr n PV, r , ∆ r , M

ω PV n PV, r

∆ r, M

ω PV n PV, r , M

(2)

10

The change in harmonic cut-off frequency “∆ωΩ” becomes, ∆ω Ω ( r , ∆r , M )

ω Ω( r

∆r , M )

ω Ω ( r, M )

Where, “ωΩ” represents the harmonic cut-off frequency of the PV. [4]

7

YHz = 1024(Hz). Also termed a beat. 9 The fundamental beat frequency occurs when nPV = 1 and may be expressed as ∆ωδr(1,r,∆r,M). 10 Also been termed the beat bandwidth of the PV across “∆r”. 8

3

(3)

3.2.2

WAVELENGTH

The change in harmonic wavelength “∆λδr” across “∆r” may be determined in a similar manner as follows, ∆λ δr n PV, r , ∆r , M

λ PV n PV, r

∆r , M

λ PV n PV, r , M

(4)

Where, λ PV n PV, r , M

c ω PV n PV, r , M

(5)

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

3.2.3

1

1 ω Ω( r

∆r , M )

ω Ω ( r, M )

(6)

GROUP

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

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

(7)

The terminating group velocity “∆vΩ” is the group velocity induced by the change in frequency at the highest harmonic mode “nΩ”. Since the number of modes varies significantly with “r”, the group velocity terminates with respect to the induced beat across “∆r” at the highest common11 mode number “nΩ(r,M)”. Subsequently, “∆vΩ” occurs at the lower harmonic cut-off mode and may be defined as follows, ∆v Ω ( r , ∆r , M ) ∆v δr n Ω ( r , M ) , r , ∆r , M (8) 3.2.3.2 ERROR Evaluating equation (7,8) reveals incrementally non-zero magnitudes at low harmonics tending to zero ([∆vδr],[∆vΩ]) → 0(m/s) as |nPV| → nΩ. However, the expected result is that the group velocity is exactly zero at all modes ([∆vδr],[∆vΩ]) = 0(m/s). However, if ∆r → ∞, then “∆vδr” is non-trivial and a mathematical statement has been made predicting the radiation of gravitational waves from the centre of mass of a planetary body. Therefore, we may consider the calculation of “∆vδr” and “∆vΩ” as being proportional measures of the mathematical representation error “RError” across “∆r”. It should be noted that the error revealed by equation (7,8) is introduced by the simplification that the magnitude of the amplitude of “nPV” is constant across “∆r”. Typically, for practical values of “∆r” at the surface of the Earth12, (RError ≈ ∆vδr ≈ ∆vΩ) << 1%.

11 12

Recalling that “nΩ” increases with “r”. Refer to table (4).

4

3.2.4

BEAT BANDWIDTH CHARACTERISTICS

3.2.4.1 FREQUENCY Thus far, it has been illustrated in [4] that an amplitude and frequency spectrum exists at each mathematical point over the domain 0 < |r| < ∞. The preceding body of work has defined certain characteristics, including change over the domain “∆r”. However, the variation in spectral bandwidth from “r” to “r+∆r” requires further consideration. Assuming the ZPF energy across “∆r” is equal to the change in the magnitude of the rest mass energy density influence “|∆UPV(r,∆r,M)|” yields, h . ω Ω ( r, ∆ r, M ) 3 ZPF 2.c

∆ U PV( r , ∆ r , M )

4

∆ω δr( 1 , r , ∆ r , M )

4

(9)

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

ω Ω ( r, ∆ r, M ) ZPF

3

2.c . ∆ U PV( r , ∆ r , M ) h

∆ω δr( 1 , r , ∆ r , M )

4

(10)

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

∆ω δr( 1 , r , ∆r , M )

(11)

3.2.4.2 MODES The ZPF beat cut-off mode “nΩ ZPF” corresponding to “ωΩ ZPF” may be determined utilising equation (12) developed previously in [4] as follows, G .M

ω PV n PV, r , M

n PV 3 2 . c . G. M . .e r π .r

2 r .c

(12)

Where, ωPV(nΩ ZPF,r,M) = ωΩ(r,∆r,M)ZPF and |nPV| = nΩ(r,∆r,M)ZPF n Ω ( r, ∆ r, M )

ω Ω ( r, ∆ r, M ) ZPF ZPF

ω PV( 1 , r , M )

(13)

3.2.4.3 CRITICAL RATIO “KR” is defined as the ratio of the applied fields to the background13 field by any suitable measure. [3] Consequently, “KR” in terms of the ratio of energy densities may be defined as the following simplification, KR

∆U ZPF U ZPF

,K R

3.3

CRITICAL BOUNDARY

3.3.1

FREQUENCY

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

4

4

ωβ

4

∆ω δr( 1 , r , ∆r , M )

4

(14)

The Critical Boundary “ωβ” represents the lower boundary of the ZPF spectrum yielding a specific proportional similarity value as follows, 13

Refers to the conditions of the space-time manifold at the surface of the Earth, prior to successful experimentation.

5

4

ω β r , ∆r , M , K R

ω Ω ( r , ∆r , M ) ZPF

4

K R . ω Ω ( r , ∆r , M ) ZPF

4

∆ω δr( 1 , r , ∆r , M )

4

(15)

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

3.3.2

ω Ω ( r , ∆r , M ) ZPF

ω β r , ∆r , M , K R

(16)

MODE The mode number of “ωβ” may be calculated by re-use of equation (12) as follows, n β r, ∆ r, M , K R

ω β r, ∆ r, M , K R ω PV( 1 , r , M )

(17)

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

3.4

n Ω ( r , ∆r , M )

ZPF

n β r , ∆r , M , K R

(18)

BANDWIDTH RATIO

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

Bandwidth Ratio

∆ω R( r , ∆r , M )

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

(19)

∆ω R R E , ∆r , M E

∆r Change in Radial Displacement

Figure 1, 4

PHYSICAL MODELLIG

4.1

GENERAL SIMILARITY EQUATIONS

4.1.1

OVERVIEW

It was illustrated in [2] that acceleration may be represented by the superposition of wavefunctions. The Primary Precipitant was decomposed to form General Modelling Equations “GMEx”. Therefore, for applied14 experimental fields, the change in “GMEx” is equal to the required change of the magnitude of the gravitational acceleration vector “g”. Storti et. al. stated in [2] that ∆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 14

Artificial fields commencing from zero strength.

6

acceleration is a function of the Lagrangian densities of the EM field harmonics in a vacuum. Assuming proportional15 similarity (|KR| ≠ 1) between the Experimental Prototype16 (EP) and the mathematical model, a family of General Similarity Equations “GSEx” may be defined where ∆GME1 ≠ ∆GME2 for all “∆r” as |nPV| → nΩ ZPF and +nΩ ZPF < +∞. 4.1.2

GSEx “GSE1,2” may be formed utilising the energy balancing equations as follows: ∆GME x

g 0

∆GME x

g 2 .g

(20) (21)

Where, N EA k A,n A,t ∆GME x

∆K 0( ω , X )

.

2

nA= N

2 .r

±c

2

N B A k A,n A,t

2

nA= N

(22)

G.M . ∆K 0( ω , X ) KR 2 r .c

(23)

17

Where, “∆K0(ω,X)” is the engineered relationship function as derived in [3], “kA” denotes the applied wave vector and the permittivity and permeability of free space, “ε0” and “µ0” respectively, act as the impedance function18. Substituting equations (22,23) into (20,21) and solving for “KR” yields the Critical Ratio explicitly in terms of applied fields as |nA| → nΩ ZPF such that |KR| → 1 as follows, N 2 2 .c .

B A k A,n A,t

2

nA= N

KR

N

N E A k A,n A,t

nA= N

2

±c

2.

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

2

(24)

19

Subsequently, proportional representations of similarity over the domain 1<|nA|
15

Relative to the initial state of the PV. The Experimental Prototype is the ambient gravitational field across “∆r”. 17 “X” denotes all variables, within the experimental environment, that influences experimental results and behaviour. This also includes all parameters that might otherwise be neglected, due to practical calculation limitations, in theoretical analysis. 18 The impedance function “ Z = µ 0 /ε 0 ” is independent of KPV in the PV representation. 16

19

Numerical investigations of the summing function described herein, performed using “MathCad 8 Professional”, has illustrated that acceptably constant functional behaviour is observed for harmonic modes greater than or equal to 501. ∴|KR| → 1 for [|nPV|,|nA|] ≥ 501.

7

N 2 2 .c .

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

B A k A,n A,t

2

nA= N N

1, 2

N E A k A,n A,t

2

±c

2.

nA= N

B A k A,n A,t

2

nA= N

(25)

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

ε0 ∆K C ∆K 1 , ∆K 2 . ∆U PV( r , ∆r , M ) µ0

(26)

Where, ∆K C ∆K 1 , ∆K 2

2

N

∆K 1( ω , r , E, D , X )

1

N

.

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

E A k A,n A,t

2

.

2

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

(27)

Substituting equations (27) into (26) when |KR| = 1 yields GSE3 as follows, GSE E A , B A , k A , n A , t , r , ∆r , M

1 3

.

K PV( r , M ) .∆U PV( r , ∆r , M )

ε0 µ0

N

N

.

EA k A,n A,t nA= N

2

.

B A k A,n A,t

2

nA= N

(28)

Such that, 2.

K PV( r , M ) e

G .M 2 r .c

(29)

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

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

4, 5

3

1, 2

(30)

Where, Variable ∆GMEx g EA(kA,nA,t) BA(kA,nA,t) ∆KC(∆K1,∆K2) c G

4.2

Description Change in applied acceleration vector Magnitude of gravitational acceleration vector Magnitude of applied electric field vector in complex form Magnitude of applied magnetic field vector in complex form Critical Factor Velocity of light in a vacuum Gravitational constant Table 3,

Units m/s2 m/s2 V/m T PaΩ m/s m3kg-1s-2

QUALITATIVE LIMITS

Theoretical qualitative behaviour may be obtained for “GSE1,2” by taking the limits of the RHS20 of equation (25) with respect to applied EM fields. By performing the appropriate substitutions21 the following results were obtained utilising the limit function within the “MathCad 8 Professional” environment. lim lim GSE E A , B A , k A , n A , t - B + 1, 2 0 EA ∞ A →0 (31) 20 21

Right Hand Side. Where |KR| → 1 as [|nPV|,|nA|] → nΩ ZPF.

8

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

1, 2

→2

(32)

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

2

2.

n PV, k PV

2

EA k A,n A,t nA= N

(33)

N B f k PV, n PV, t

2

2.

B A k A,n A,t

2

nA= N

n PV, k PV

(34)

As |nA| → nΩ ZPF, the superposition of applied wavefunctions describes the magnitudes of the electric and magnetic field vectors as constant (steady state) functions. Therefore, Maxwells Equations (in MKS units) may define the system characteristics as follows (“ρ” is the charge density and “J” is the vector current density), [6] ρ ∇ .E A , ∇ E A 0 , ∇ .B A 0 , ∇ B A µ 0 .J ε0 (35) Consequently as |nA| → nΩ ZPF, optimal similarity occurs when: 1. The divergence of “EA” is maximised. 2. The magnitude and curl of “EA” is minimised. 3. The magnitude and curl of “BA” is maximised. As the square root of the ratio of the sum of the applied field’s approach “c”, GSE1 approaches unity as follows, lim

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

2

B A k A,n A,t

2

1

nA= N

c

N nA= N

→1

(36)

Similarly, the square root of the ratio of the sum of the applied field’s influence on GSE2 may be expressed as follows, lim

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

c

E A k A,n A,t

2

B A k A,n A,t

2

2

nA= N N nA= N

→ |Undefined|

9

(37)

Consequently, characteristics of equations (31-37) are such that: 4. GSE1(EA,BA,kA,nA,t) qualitatively implies |KR| = 1 when |ΣEA2/ΣBA2| → c as |nA| → nΩ ZPF 5. GSE1(EA,BA,kA,nA,t) qualitatively implies use over the range 0 ≤ |GSE1(EA,BA,kA,nA,t)| < 2 6. GSE2(EA,BA,kA,nA,t) qualitatively implies use over the range: 0 ≤ |GSE2(EA,BA,kA,nA,t)| < 1 ∪ 1 < |GSE2(EA,BA,kA,nA,t)| < 2 The results presented above should not be taken as definitive mathematical solutions or experimental predictions. However, deeper consideration may suggest that GSE1 represents an expression biasing constructive EGM interference, whilst GSE2 biases destructive EGM interference with “g”. The “undefined” result indicated by equation (37) may suggest that the local space-time manifold cannot be totally flattened in the presence of applied EM fields. The applied fields represent energy contributions that inherently modify the geometry of the local space-time manifold. 5

METRIC EGIEERIG

5.1

POLARIZABLE VACUUM

Utilising GSE3, we may write22 the exponential metric tensor line element for the PV model representation of GR in the weak field limit analogous to the form specified in [3] as follows, ds

2

2 2 c .dt

µ υ g µυ.dx .dx

K EGM g 00

2 2 r .dθ

2 K EGM. dr

2 2 2 r .sin ( θ ) .dψ

(38)

1 K EGM

(39)

g 11 g 22 g 33 K EGM

(40)

Where, 2.

K EGM e

G .M . 1 2 r .c

1. 2

GSE 3 3 K PV . e

∆K 0( ω , X )

(41)

Note: i. KEGM is a function of the applied fields and constituent characteristics (EA,BA,kA,nA,t). ii. |nA| >> 1 5.2

DESIGN CONSIDERATIONS

5.2.1

RANGE FACTOR

The range factor “Stα(r,∆r,M)” is the product of ∆UPV(r,∆r,M) and the impedance function “Z”. It is a useful “at-a-glance” design tool that indicates the boundaries23 of the applied energy requirements for experiments and may be represented as follows, St α ( r , ∆r , M ) ∆U PV( r , ∆r , M ) .

µ0 ε0

(42)

We may determine specific limiting characteristics of the range factor for an ideal experimental solution, where the upper limiting value is defined by, 2

lim St ( r , 0 , M ) + α ∆r 0

3 .M .c . 4 .π

1 (r

∆r )

3

1 . µ0 3 ε0 r

St α ( r , 0 , M ) 0

(43)

The lower limiting value is defined by,

22

In terms of the applied Poynting Vector. The greater the magnitude of the range factor, the greater the magnitude of applied energy required for complete dynamic, kinematic and geometric similarity with the EP. 23

10

1

µ0 2

lim St α ( r , ∞ , M ) ∆r ∞

3 .M .c . 4 .π

1 . µ0 3 ε0 r

1 (r

∆r )

3

St α ( r , ∞ , M )

3 . . 2. ε 0 Mc 3 4 π .r

2

(44)

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

5.2.2

St α ( r , ∆r , M ) <

2 3 .M .c . µ 0 3 ε0 4 .π .r

(45)

SENSE CHECKS & RULES OF THUMB

For non-experimentally validated engineering undertakings, it is common practice to sense check predicted behaviour before proceeding. We may develop simple sense checks and rules of thumb by further considering the predicted mathematical results herein, in relation to other physical phenomena. For example, it is widely believed by proponents of the PV and ZPF models of gravity that the Compton Frequency of an Electron “ωCe” represents some sort of boundary condition. Subsequently, we may define the ratio of “∆ωZPF” to “ωCe” as the 1st Sense Check “Stβ” as defined by equation (46). This acts as an indicator regarding order-of-magnitude relationships and results. The Electron represents a fundamental particle in nature and it would seem inappropriate that Stβ >> 1 (∆ωZPF >> ωCe) as it would imply that the beat bandwidth of ZPF frequencies, over practical benchtop values of “∆r”, is much larger that the Compton frequency of an Electron, contradicting contemporary belief. Similarly, if Stβ → 0, then ωCe >> ∆ωZPF and would seem to imply that, assuming “ωCe” is representative of a natural gravitational boundary condition, proportional similarity (|KR| ≈ 1) by artificial means is not experimentally practical and the mathematical model derived to achieve similarity is inappropriate. Therefore, it follows that we might expect that 0 << Stβ < 1. Hence, St β ( r , ∆r , M )

∆ω ZPF( r , ∆r , M ) ω Ce

(46)

nd

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

∆ω Ω ( r , ∆r , M ) ω Ce

(47)

rd

The 3 Sense Check “Stδ” may be defined as the ratio of the harmonic cut-off modes across “∆r” (expected to be: “≈1”). n Ω ( r ∆r , M ) St δ ( r , ∆r , M ) n Ω ( r, M ) (48) Therefore it follows that, St β ( r , ∆r , M ) ∆ω R( r , ∆r , M ) St γ ( r , ∆r , M ) (49) The 4th Sense Check “Stε” may be defined in terms of “RError” across “∆r” as follows (expected to be: “≈1”), ∆v δr n PV, r , ∆r , M St ε n PV, r , ∆r , M ∆v Ω ( r , ∆r , M ) (50)

11

Hence,

Sense Check

RE

St β R E , ∆r , M E

St β r , ∆r , M E St γ r , ∆r , M E

St γ R E , ∆r , M E

r Radial Distance

Figure24 2,

Sense Check

N

N

St ε n PV , R E , ∆r , M E

n PV Harmonic

Figure 3, 6

EGIEERIG CHARACTERISTICS

6.1

BEAT SPECTRUM

Characteristics of the beat PV / ZPF spectrum, over ∆r = 1(mm), at the surface of the Earth may be approximated according25 to the following table, Characteristic Wavelength Change in Wavelength Change in Cut-Off Wavelength Group Velocity Terminating Group Velocity Representation Error Fundamental Beat Frequency Change in Cut-Off Frequency Beat Cut-Off Frequency Beat Cut-Off Mode Beat Bandwidth 24 25

Evaluated Approximation λPV(1,RE,ME) ≈ 8.4x106 (km) ∆λδr(1,RE,∆r,ME) ≈ 1.8 (m) ∆λΩ(RE,∆r,ME) ≈ 0 (m) ∆vδr(1,RE,∆r,ME) ≈ 1.3x10-11 (m/s) ∆vΩ(RE,∆r,ME) ≈ 1.3x10-11 (m/s) RError ≈ 1.3x10-9 (%) ∆ωδr(1,RE,∆r,ME) ≈ 7.5x10-12 (Hz) ∆ωΩ(RE,∆r,ME) ≈ 45 (PHz) ωΩ(RE,∆r,ME)ZPF ≈ 371 (PHz) nΩ(RE,∆r,ME)ZPF ≈ 1x1019 ∆ωZPF(RE,∆r,ME) ≈ 371 (PHz)

Y-Axis is logarithmic scale. PHz = 1015(Hz).

12

Critical Boundary Frequency Critical Boundary Mode Similarity Bandwidth Similarity Modes Bandwidth Ratio Bandwidth Ratio (∆r = 17mm) Range Factor

ωβ(RE,∆r,ME,50%) ≈ 312 (PHz) nβ(RE,∆r,ME,50%) ≈ 8.7x1018 ∆ωS(RE,∆r,ME,50%) ≈ 59 (PHz) ∆nS(RE,∆r,ME,50%) ≈ 1.7x1018 ∆ωR(RE,∆r,ME) ≈ 8.2 ∆ωR(RE,∆r,ME) ≈ 1 |Stα(RE,∆r,ME)| ≈ 88 (MPa MΩ)

Range Factor Upper Limit

|Stα(RE,∞,ME)| ≈ 2x105 (GPa GΩ)

st

1 Sense Check 2nd Sense Check 3rd Sense Check 4th Sense Check

6.2

Stβ(RE,∆r,ME) ≈ 4.8x10-4 Stγ(RE,∆r,ME) ≈ 5.8x10-5 Stδ(RE,∆r,ME) ≈ 1 Stε(nPV,RE,∆r,ME) ≈ 1 Table 4,

CONSIDERATIONS

Some of the factors to be considered in experimental design configurations may be articulated as follows: 1. The experimental design should attempt to maximise the applied energy density with the highest frequency conditions possible. 2. Optimal conditions occur approaching the ZPF beat cut-off mode “nΩ ZPF”. 3. EM modes within an experimental volume are subject to normal physical influences. The fundamental frequency mode will not exist within a Casimir experiment, as the pseudowavelength is too large. Hence, the equivalent gravitational acceleration harmonic cannot exist. 4. Optimal experimental conditions occur when the ratio of the applied Poynting Vector to the Impedance Function approaches unity. [2] 5. Numerical solutions26 to equation (15) indicate that greater than 99.99 (%) of the EGM beat spectrum occurs in the PHz range27. 6.3

EGM WAVE PROPAGATION

The gravitational effect generated by a specifically applied EM field harmonic may be conceptualised as a modified EM wave. Figure (5) and (6) depict the manner in which pseudo-wave propagation occurs. This has been termed EGM Wave Propagation and has 5 components as follows, i. The Electric Field Wave (magenta). ii. The Magnetic Field Wave (blue). iii. The Electro-Gravitic Coupling Wave (green). iv. The Magneto-Gravitic Coupling Wave (Red). v. The Poynting Vector indicated in Figure (5) and (6) as the wave propagation arrow.

Figure28 5,

26

Numerical approximations were performed using “MathCad 8 Professional” at default convergence, constraint and precision tolerance settings, to a display accuracy of 15 decimal places. 27 ∴ KR ≈ 1 when ωβ(RE,∆r,ME,99.99999999999999%) ≈ 312 (PHz) and ∆ωZPF(RE,∆r,ME) ≈ 371 (PHz). 28 Figure not to scale.

13

Figure28 6, 6.4

DOMINANT & SUBORDINATE BANDWIDTHS

The EGM spectrum is fictitious and is derived from the concept of similarity. However, practical benefits to facilitate understanding of the concepts presented herein may be realised by the articulation, in terms of applied experimental fields of the conventional representation of the EM spectrum. [8,9] The EGM spectrum represents all frequencies within the EM spectrum but may be simplified into two regimes. These have been termed the dominant and subordinate gravitational bandwidths (“∆ωEGM δ” and “∆ωEGM σ” respectively) as indicated in Figure (7).

Figure31 7, At the surface of the Earth, over practical benchtop values of “∆r”, “∆ωEGM δ” is responsible for significantly more than 99.99(%) of the spectral composition of “g”. Therefore, utilising table (1) we may speculatively re-define29 the classical EM spectral representation for frequencies of Gamma Rays “ωγ” at a mathematical point with displacement “r” as follows30, i. 105(PHz) > ωγ > 1(YHz) ii. ωg > 1(YHz) Where “ωg” represents the gravitational frequency of the applied experimental fields for complete dynamic, kinematic and geometric similarity with the background gravitational field at the surface of the Earth. 6.5

KINETIC & POTENTIAL

The EGM Spectrum may be considered a hybrid function of an amplitude and frequency distribution. The harmonic behaviour across an element “∆r” has been described in terms of: i. The Fourier Spectrum – termed the Potential Spectrum and is non-physical. ii. The ZPF Spectrum – termed the Kinetic Spectrum and is physical. Properties of the Fourier spectrum are such that wavefunction amplitude decreases as frequency increases, whereas properties of the ZPF spectrum dictate constant amplitude with increasing frequency. Consequently, merging the two distributions, as defined by equation (14), produces engineering properties and boundaries seemingly consistent with common-sense expectations. The Potential Spectrum has the advantage of being able to fictitiously represent ZPF behaviour at a mathematical point in addition to “∆r”. This is otherwise not possible due to the ZPF 29 30

By approximation for illustration purposes at the surface of the Earth. YHz = 1024(Hz)

14

being a physical manifestation of “g” and the constituent wavefunctions possess finite wavelengths. Appendix (A) contains visualisation of physical modelling characteristics of the ZPF Spectrum. 7

COCLUSIOS

The construct herein suggests that the delivery of EM radiation to a solid spherical test object may be used to modify its weight. Specifically, at high energy density and frequency, the gravitational spectral signature of the test object may undergo constructive or destructive interference. However, the frequency dependent conditions for gravitational similarity at the surface of the Earth are enormous: [ωβ ≈ 312(PHz) and30 ∆ωZPF ≈ 371(PHz)]. Summarising yields: i. The ZPF spectrum of free space is composed of an infinite number of modes “nPV”, with frequencies tending to 0(Hz), as illustrated in table (1). ii. The group velocity produced by the PV at a mathematical point and across practical values of “∆r” at the surface of the Earth is 0(m/s). Consequently, gravitational wavefunctions are not observed to propagate from the centre of a planetary body. iii. |∆UPV(r,∆r,M)| is proportional to ∆ωZPF(r,∆r,M). iv. “g” exists (at practical bench-top experimental conditions / dimensions) as a relatively narrow band of beat frequencies in the PHz Range. Spectral frequency compositions below this range31 are negligible [similarity ≈ 0(%)]. v. General Similarity Equations (GSEx) facilitate the construction of computational models to assist in designing optimal experiments. Moreover, they can readily be coded into “off-the-shelf” 3D-EM simulation tools to facilitate the experimental investigation process. vi. A solution for optimal experimental similarity utilising EM configurations exists when Maxwells Equations at steady state conditions are observed such that: 1. The divergence of “EA” is maximised. 2. The magnitude and curl of “EA” is minimised. 3. The magnitude and curl of “BA” is maximised. 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] Wolfram Research: http://scienceworld.wolfram.com/physics/BeatFrequency.html [6] Wolfram Research: http://scienceworld.wolfram.com/physics/MaxwellEquationsSteadyState.html [7] “MathCad 8 Professional” Reference Tables by MathSoft: http://www.mathcad.com/ [8] Wolfram Research: http://scienceworld.wolfram.com/physics/ElectromagneticRadiation.html [9] Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html - c1

31

Approximately less than 42(THz).

15

APPEDIX A PHYSICAL MODELLING CHARACTERISTICS For ∆r << r yields: 50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

Re ω β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure A1,

50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

Im ω β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure A2,

50 .%

ω β R E , ∆r , M E , 50 .% 100 .%

ω β R E , ∆r , M E , K R

0

0.5

1 KR Critical Ratio

Figure A3,

16

1.5

2

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

0

0.5

100 .% ∆ω S R E , ∆r , M E , 50 .% 1

1.5

2

KR Critical Ratio

Figure A4,

50 .%

100 .%

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

0

0.5

1

1.5

2

KR Critical Ratio

Figure A5, 50 .%

n β R E , ∆r , M E , 50 .% 100 .%

Re n β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure A6, 50 .%

n β R E , ∆r , M E , 50 .% 100 .%

Im n β R E , ∆r , M E , K R

0

0.5

1 KR Critical Ratio

Figure A7,

17

1.5

2

n β R E , ∆r , M E , 50 .% 100 .%

50 .%

n β R E , ∆r , M E , K R

0

0.5

1

1.5

2

KR Critical Ratio

Figure A8,

50 .%

100 .% ∆n S R E , ∆r , M E , 50 .%

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

0

0.5

1

1.5

2

KR Critical Ratio

Figure A9, 50 .%

100 .%

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

0

0.5

1 KR Critical Ratio

Figure A10,

18

1.5

2