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

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

Abstract An experimental prediction is developed considering gravitational acceleration “g” as a purely mathematical function. This expands potential experimental avenues in relation to the hypothesis to be tested presented in “Practical modelling methods of the polarizable vacuum I-III”. Subsequently, the construct herein presents the following: i. A pseudo-electromagnetic, pseudo-propagating transverse plane wave harmonic representation of gravitational fields at a mathematical point, arising from geometrically spherical objects, using modified Complex Fourier Series. ii. Characteristics of the Amplitude Spectrum based on (i). iii. Derivation of the fundamental harmonic frequency based on (i). iv. Characteristics of the frequency spectrum of an implied Zero-Point-Field (ZPF) based on (i) and the assumption that an electromagnetic (EM) relationship exists over a change in displacement across a practical benchtop test volume.

1

[email protected], [email protected].

1

1

ITRODUCTIO

1.1

GENERAL [1-3]

A metric engineering description was presented in [3] based on the principles of similarity and an engineering parameter, termed the critical ratio “KR”, has been formulated to indicate proportional experimental conditions. “KR” may be stated as the ratio of the applied electromagnetic (EM) fields to the induced change of gravitational field, in terms of energy densities. 1.2

HARMONICS

To further articulate the applicability of “KR”, a purely harmonic non-EM description of gravitational fields is developed. This acts to expand potential experimental avenues in relation to the hypothesis to be tested as stated in [2]. Subsequently, the construct herein presents the following, i. A pseudo-EM, pseudo-propagating transverse plane wave harmonic representation of gravitational fields at a mathematical point, arising from geometrically spherical objects of uniform mass distribution, using modified Complex Fourier Series. ii. Characteristics of the Amplitude Spectrum based on (i). iii. Derivation of the fundamental harmonic frequency based on (i). iv. Characteristics of the frequency spectrum of an implied Zero-Point-Field (ZPF) based on (i) and the assumption that an EM relationship exists over a change in displacement across a practical benchtop test volume. The proceeding construct obeys the following hierarchy, v. A pseudo-EM, pseudo-propagating transverse plane wave harmonic representation of the magnitude of gravitational acceleration “g” is developed in section 3.1. vi. The frequency spectrum of (v) is derived in section 3.2 by application of Buckingham Π Theory (BPT) and dimensional similarity developed in [1]. vii. The ZPF energy density is related to (vi) in section 3.3 based on 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 Polarizable Vacuum (PV) are derived in section 3.4 based on (vii). ix. A description of physical modelling criteria is presented in section 4. x. A set of sample calculations and illustrational plots are presented in section 5. 1.3

EXPERIMENTATION

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

THEORETICAL MODELLIG

2.1

TIME DOMAIN

Fourier series may be applied to represent a periodic function as an infinite trigonometric series in sine and cosine terms. It may also be applied to represent a constant function over an arbitrary period by the infinite summation of sinusoids. Since the PV model is a weak field isomorphic approximation of General Relativity (GR) and the frequency spectrum is postulated to range from -∞ < ω < ∞, it follows that Fourier Series may present a useful tool by which to describe gravity as the number of harmonic frequency modes tends to infinity. 2.2

DISPLACMENT DOMAIN

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

2

3

MATHEMATICAL MODELLIG2

3.1

CONSTANT ACCELERATION

Constant functions may be expressed as a summation of trigonometric terms. Subsequently, it is convenient to model a gravitational field utilizing modified Complex Fourier Series, 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” may be usefully represented by equation (1) as3 |nPV| → ∞, G. M . 2 . i . π .n PV .ω PV( 1 , r , M ) .t .i g( r , M ) e 2 π . n PV r n PV (1) Where, the wavefunction amplitude spectrum “CPV” is calculated to be, C PV n PV, r , M

G. M . 2

r

2 π . n PV

(2)

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

3.2

Description Field harmonic of PV Fundamental field harmonic of PV Harmonic frequency modes of PV Magnitude of position vector relative to the centre of mass Mass Gravitational constant Table 1,

Units Hz Hz None m kg m3kg-1s-2

FREQUENCY SPECTRUM

It was illustrated in [1] that dimensional similarity and the equivalence principle could be applied to represent the magnitude of an acceleration vector “aPV” as follows, a PV K 0 ω PV, r , E, B, X .

3 2 ω PV . r

c

(3)

Where, Variable K0(ωPV,r,E,B,X) ωPV E B c

Description Experimental relationship function4 Harmonic frequency modes of PV Magnitude of PV electric field vector Magnitude of PV magnetic field vector Velocity of light in a vacuum Table 2,

2

Units None Hz V/m T m/s

All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 3 Equation (1) represents the magnitude of a periodic square wave solution with constant amplitude. 4 “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.

3

In accordance with the harmonic representation of “g” illustrated by equation (1), “K0(ωPV,r,E,B,X)” is a frequency dependent experimental function. It was illustrated in [1] that K0(ωPV,r,E,B,X) = K0(X) = KPV-3/2. Hence, an expression for the frequency spectrum may be derived in terms of harmonic mode. This may be achieved by assuming the acceleration described by equation (3) is dynamically, kinematically and geometrically similar to the amplitude of the 1st harmonic (|nPV| = 1) as described by equation (2) as follows, aPV ≡ CPV(1,r,M)

(4)

Therefore, utilising equation (3) and (4), it follows that all frequency modes may be represented by, G .M

ω PV n PV, r , M

. . . . 2 c G M .e r π .r

n PV

3

2 r .c

(5)

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

Fundamental Frequency

RE

ω PV 1 , r , M E ω PV 1 , R E , M E

r Radial Distance

Figure 1, Where, Variable RE ME

3.3

Description Radius of the Earth Mass of the Earth Table 3,

Units m kg

ENERGY DENSITY

The gravitational field surrounding a homogeneous solid spherical mass may be characterised by its energy density. Assuming that the magnitude of this field is directly proportional to the massenergy density of the object, then the energy density “Uω” may be evaluated over the difference between successive odd frequency modes as follows, U ω n PV, r , M

U ω ( r, M ) .

n PV

2

4

4

n PV

(6)

Where, U ω ( r, M )

Variable Uω(nPV,r,M) h

h . 4 ω PV( 1 , r , M ) 3 2.c

Description Energy density per change in odd harmonic mode Planck’s Constant Table 4,

4

(7) Units Pa Js

3.4

SPECTRAL CHARACTERISTICS

3.4.1

CUT-OFF MODE AND FREQUENCY

Utilizing the approximate rest mass-energy density “Um” of a solid spherical object, as described by equation (8), an expression relating the terminating harmonic frequency mode to “r” and “M” may be derived as follows, 2 3.M .c U m( r , M ) 3 4.π .r (8) Assuming that |Um(r,M)| = |Uω(nPV,r,M)|, equation (8) may be related to equation (6) and solved for “|nPV|”. This is termed the harmonic cut-off mode “nΩ” as follows, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(9)

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

(10)

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

(11)

The derivation of equations (9-11) is based on the compression of the energy density to one change in odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordance with BPT. The subsequent application of these results to equation (1) acts to decompress the energy density over the Fourier domain yielding a highly precise reciprocal harmonic representation of “g”. Hence, “nΩ” and “ωΩ” may be graphically represented for the Earth as follows, RE n Ω R E, M E n Ω r, M E ω Ω r, M E ω Ω R E, M E

r Radial Distance

Cutoff Mode Cutoff Frequency

Figure 2, 3.4.2

ZERO-POINT-FIELD

The cross-fertilisation of the amplitude and frequency characteristics of the Fourier spectrum with the ZPF spectral energy density distribution is a useful tool by which to analyse expected Experimental Prototype (EP) characteristics. This may be achieved by graphing the ZPF Poynting Vector “Sω” as follows, S ω n PV, r , M c .U ω n PV, r , M (12)

5

ZPF Poynting Vector

S ω n PV , R E , M E

n PV Harmonic

Figure 3, 4

PHYSICAL MODELLIG

4.1

POLARIZABLE VACUUM

The spectral characteristics of an EP may be articulated for a spherical test object with uniform mass distribution at the surface of the Earth assuming, i. The 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 electricity, magnetism and gravity such that the physical interaction of the test object with the PV may be investigated and potentially modified utilizing the equations defined in the preceding section. 4.2

TEST VOLUMES

The application of modified Fourier Series to define the modes of oscillation of physical systems has been experimentally verified since its development by Joseph Fourier (1768-1830). [4] The representation developed in the preceding section is defined in the time domain but may also be applied over an arbitrary displacement domain “∆r” as appears in standard engineering texts for beams, membranes, strings, control systems and wave equations. [5,6] If we consider a small5 cubic test volume of length “∆r” to be filled with a large number of incremental displacement elements, frequency characteristics of the test volume may be hypothesised. Assuming each element within the test volume may be described by sinusoids of appropriate amplitude and frequency, it may be conjectured that the system interaction of the elements produces an amplitude and frequency spectrum consistent with a modified Fourier Series representation of “g” over “∆r”. The resultant wave vector at each frequency mode of the test volume is required to be coplanar with the gravitational acceleration vector for it to be representative of physical reality. Hence, only a line of action vertically downward through the cubic element is required for experimental consideration. Moreover, the mathematical representation of forces acting through the test volume is further simplified by approximating “g” as constant over the vertical dimension of the test volume. 4.3

TEST OBJECT

In accordance with PV / ZPF theory, test objects are assumed to produce a gravitational spectral signature6 in the same manner as the signature produced by the planetary masses. It is also assumed that the gravitational forces experienced by objects arise from its spectrum coupling to the background field. 5

Refers to an experimentally practical benchtop volume. Gravitational spectral signature is defined as the spectrum of amplitudes and frequencies unique to “r” and “M” by the application of modified Fourier series. 6

6

5

SAMPLE CALCULATIOS

5.1

BACKGROUND GRAVITATIONAL FIELD

5.1.1

FUNDAMENTAL FREQUENCY

The fundamental frequency mode of the PV at the surface of the Earth may be usefully approximated as follows, ωPV(1,RE,ME) ≈ 0.04 (Hz) (13) 5.1.2

MODE BANDWIDTH

An expression may be defined representing the mode bandwidth of the local gravitational field as follows, ∆ω PV( r , M ) ω Ω ( r , M ) ω PV( 1 , r , M ) (14) Assuming an ideal relationship between the mathematical model and the background gravitational field yields7, ∆ωPV(RE,ME) ≈ 520 (YHz) (15) 5.2

APPLIED EXPERIMENTAL FIELDS

5.2.1

MODE BANDWIDTH

Assuming a cubic element of length “∆r” possesses spectral attributes over the displacement domain, the permissible modes “Ν∆r” starting from “ωΩ” at “r” over “∆r” as |nPV| → nΩ may be approximated by, ∆r N ∆r( r , M ) ω Ω ( r , M ) . c (16) In figure (4), 1. The arrows pointing downwards represent a uniform gravitational field. 2. The arrows pointing upwards represent a uniformly applied system field. 3. The cube represents the experimental test volume of length “∆r”, with base residing at co-ordinates (0,0,r). 4. The square section represents an EM flux area. 5. “h” represents the vertical displacement above the EM flux area.

∆r

h -X

Y

X

-Y

Figure 4,

7

YHz = 1024 (Hz).

7

5.2.2

1. 2. 3. 4.

6

ENGINEERING CONSIDERATIONS The factors to be considered in experimental design configurations are as follows: Where possible, the experimental should attempt to maximise the applied energy density with preference towards the highest frequency bombardment possible. Optimal energy delivery conditions occur at the highest achievable frequencies tending towards the harmonic cut-off mode “nΩ”. Optimal experimental conditions occur when the ratio of the applied Poynting Vector to the Impedance Function approaches unity. [3] EM modes within the test volume are subject to normal physical influences. For example, the fundamental frequency mode cannot exist within a typical Casimir experiment; hence the equivalent gravitational acceleration harmonic cannot exist. The relative contribution of low harmonic mode numbers to “g” is trivial. COCLUSIOS

The construct herein suggests that the delivery of EM radiation to a solid spherical test object may be used to alter the weight of the object. If the test object is bombarded by EM radiation, at high energy density and frequency, the gravitational spectral signature of the test object may undergo constructive or destructive interference. 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] Lennart Rade, Bertil Westergren, “Beta Mathematics Handbook Second Edition”, Chartwell-Bratt Ltd, 1990, Page 470. [5] K.A. Stroud, “Further Engineering Mathematics”, MacMillan Education LTD, Camelot Press LTD, 1986, Programme 17. [6] Erwin Kreyszig, “Advanced Engineering Mathematics Seventh Edition”, John Wiley & Sons Inc., 1993, Ch 10 and 11.

8

APPEDIX A FORMULATIOS, DERIVATIOS, CHARACTERISTICS AD PROOFS Integrating the Spectral Energy Density equation stated in [2], ρ 0( ω )

2 .h .ω c

3

3

(A.1)

Yields, 2 .h . c

3

ω dω

3

1. h . 4 ω 2 c3

(A.2)

where, “ω ≡ ωPV”: utilising equation (5), n PV 3 2 .c .G.M . . K PV r π .r

ω PV

(5)

Yields a generalised frequency change representation according to, Uω

h . ωPV 3 2 .c

4 2

ωPV

4

(A.3)

Substituting equation (5) into (A.2) yields the generalised change in odd mode representation according to, U ω n PV, r , M

U ω( r , M ) .

n PV

2

4

4

n PV

(6)

where, h .G.M . 2 .c .G.M . 2 K PV 2 5 . πr π .c .r 3

U ω( r , M )

(A.4)

#ote: Equation (A.4) is a modified representation of equation (7). Subsequently, if:

9

U m( r , M )

3 .M .c

2

3 4 .π .r

(8)

And assuming: U m( r , M ) U m( r , M )

U ω( r , M ) .

U ω n PV, r , M

n PV

2

4

then, 4

n PV

(A.5)

Next, let: D

U m( r , M ) U ω( r , M )

(A.6)

Hence, D

n PV

2

4

4

n PV

(A.7)

Solving for “nPV” yields,

10

2 1

1 . 108.D 12

2

2 12. 768 81.D

1

3

1

48 12. 108.D

1 1

108.D 1

D

n PV 2

4

4

108.D

1. 24

solve , n PV, factor

n PV

2 12. 768 81.D

2 12. 768 81.D

1

3

3

1

48 24. 108.D

2 12. 768 81.D

3

2

2

2

2 12. 768 81.D

3

2

2

2 1

i . 3 . 108.D

2 12. 768 81.D

3

2

48.i . 3

1 1

108.D

2 12. 768 81.D

2 1

1 . 108.D

2 12. 768 81.D

2

3

2 1

3

1

48 24. 108.D

2 12. 768 81.D

2

3

2

1

i . 3 . 108.D

24

12. 768 81.D

2

2

3

48.i . 3

1 1

108.D

2 12. 768 81.D

3

2

(Eq. A.8) Analysing the structure of the preceding equation leads to simplification by assigning temporary definitions of “F” and “L” for use with equation (A.8). This approach is required to fully exploit the “MathCad 8 Professional” symbolic calculation environment and may be articulated as follows, Let: “F = 108D+12√(768+81D2)” and “F = L3”. Hence, an expression for “nPV” as a function of “L” may be defined by, 1. L 12

solve , n PV, factor D

n PV 2

4

4

n PV

substitute , 108.D

2 12. 768 81.D F 2

3

substitute , F L,

1 3

1

F L

,F

3

2

L

1

4 L

1. . i 3 24

1 . L 24

1

1. . i 3 24

1 . L 24

1

2 .i . 3

2

L

collect , L

11

2 .i . 3 L

2

(A.9)

Equation (A.9) is a simplifying intermediary step leading to the harmonic cut-off function “Ω(r,Μ)” subject to the redefinition of “L” as follows, Let: “L = Ω(r,M)” and “nΩ(r,M) = nPV + 2” hence, n Ω ( r , M ) n PV 2

Ω ( r, M )

4

12

Ω ( r, M )

1

2

(A.10)

Therefore, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(9)

Performing the appropriate substitutions of “D” into “L3 = 108D+12√(768+81D2)” for application to equation (9) yields, 3

Ω ( r, M )

108.

U m( r , M ) U ω( r , M )

12. 768 81.

U m( r , M )

2

U ω( r , M )

(10)

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

(11)

12