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

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

Abstract In Practical modelling methods of the polarizable vacuum - I, by application of Buckingham’s Π Theory, it was demonstrated how constant acceleration may be derived from a superposition of electromagnetic (EM) fields. An experimentally determined relationship function “K0(ω,X)” was predicted which couples gravitational acceleration to the intensity of an applied EM field, in agreement with The Equivalence Principle and the Polarizable Vacuum (PV) Model. The EM field was then decomposed into its constituent frequency modes and their respective intensities to show that their summation results in constant acceleration as the number of harmonic frequencies in the field tends to infinity. This paper is an extension of previous work, intended to present an hypothesis to be tested and to demonstrate how “K0(ω,X)” may be expressed, by decomposition, as two relationship functions that may be directly measured by experimentation. This results in two representations that are proportional to solutions of the Poisson and Lagrange equations. It is demonstrated that the ratio of the resulting relationship functions is proportional to the square of the magnitude of the resultant Poynting vector. This property, in conjunction with the orientation of the resultant Poynting vector, may be utilized as a practical design tool for engineering the PV by the application of “off-the-shelf” EM modelling software.

1

[email protected], [email protected].

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1

ITRODUCTIO

1.1

HYPOTHESIS TO BE TESTED

Previously, [1] it was illustrated that an experimentally determined relationship function “K0(ω,X)”, may be used to characterize the relationship between the magnitude of the acceleration vector “a”, and the energy densities of discrete frequency modes “N”, in an applied electromagnetic (EM) field. This relationship was shown to be equivalent to the Polarizable Vacuum (PV) Model approach to General Relativity (GR). [2-7] The Newtonian gravitational potential is described by an approximation to an exponential function [7], where it was shown that a weak gravitational field may be represented by a superposition of EM fields. [1] Historically, variations in the energy density are known to result in gravitation from the solutions of Poisson’s equation in Newtonian gravity. [8] An equivalent2 result is presented here in terms of the energy densities at the discrete frequency modes of the applied EM field. The relationship function “K0(ω,X)” may also be derived from the results of experiments that determine other relationship functions. It is demonstrated how these experimentally determined relationship functions may be found and used to address experimental design issues. To achieve this, experiments must be designed that test the following hypothesis, There are three key factors in achieving a local modification of the magnitude of the acceleration vector “a” in a gravitational field. These are; (i) An increase in the energy density of the EM field at specific frequency modes, (ii) The superposition of time varying EM fields at specific frequency modes and (iii), The Equivalence Principle, which indicates that an accelerated reference frame is equivalent to a uniform gravitational field. It has been speculated that if the frequency and phase of all modes in the PV where known, then by application of the appropriate EM fields, the interaction at those modes may facilitate destructive interference resulting in a complete cancellation of the local value of gravitational acceleration “g”. However, in the hypothesis to be tested, it is assumed that it is impossible to decrease the energy density of a gravitational field by applying an EM field to a region of space-time. Therefore, the hypothesis requires that in any practical experiment, it is only possible to increase the energy density at specific frequency modes. 1.2

WHAT IS DERIVED?

This paper derives three key design considerations. These are, I. The “α” forms, which are an inversely proportional description of how the energy density may result in an acceleration “ax(t)”. II. The “β” forms, which are a proportional description of how the energy density may result in an acceleration “ax(t)”. III. The critical factor “KC”: which is the ratio of the experimentally determined relationship functions “K1” and “K2”, to be presented in section 3? The key design considerations are derived from the hypothesis to be tested, which seeks to couple gravitational acceleration to electromagnetism, i.e. “g”, “E” and “B”. Dimensional analysis was utilized in [1], to demonstrate that coupling may exist between electromagnetism and gravity, by the application of Buckingham’s Π Theory. Analytical results herein, suggest that the square of the magnitude of the resultant Poynting Vector may be a useful design tool. The orientation and intensity of the Poynting Vector is commonplace functionality in many “off-the-shelf” EM simulation products. This provides a convenient platform from which to design practical laboratory benchtop experiments. The key design considerations are derived by identifying possible interpretations of equation (1) that impact the hypothesis to be tested, as illustrated in section 2. Equation (1) is then separated into subordinate elements based on these interpretations, as illustrated in section 3. The subordinate elements are then used to determine “KC” by solving for the ratio of the experimental relationship functions, “K1” and “K2”, defined in table (3) and (4).

2

Proportional to the ratio “ε0/µ0”.

2

2

THEORETICAL MODELLIG

2.1

PRIMARY PRECIPITANT The frequency domain precipitation3 [1] is as follows, N E( k , n , t ) a( t )

K 0( ω , X ) r

2

. n= N N B( k , n , t )

2

n= N

(1)

Where, Variable a(t) E(k,n,t) B(k,n,t) r

Units m/s2 V/m T m Hz

k

Description Magnitude of acceleration vector Magnitude of electric field vector in complex form Magnitude of magnetic field vector in complex form Magnitude of position vector 1. Field frequency 2. Independent scalar variable Harmonic frequency modes Magnitude of the harmonic wave vectors

K0(ω,X)

Experimental relationship4 function5

None

ω n, N

None m-1

Table 1, Equation (1) is termed the primary precipitant. It may be manipulated to alternate forms by incorporation6 and is biased to expression in terms of the energy density. It is a function representing constant acceleration by the superposition of EM waves. This remains true of all subsequent forms, as illustrated in appendix (B). In the PV representation, the value of the EM field at infinity replaces the fields in equation (1) as follows, E(k,n,t) → E0(k,n,t), B(k,n,t) → KPVΒ0(k,n,t), r → r0/√KPV and ω → ω0/√KPV, hence, N E 0( k , n , t )

a r0

2

K 0 ω 0, X . n= N N 3 r 0 . K PV 2 B 0( k , n , t ) n= N

3

(2)

Precipitation is defined as a result derived by the application of limits. [1] Under ideal conditions of complete dynamic, kinematic and geometric similarity between the experimental configuration and the resulting acceleration, the experimental relationship function has a value of unity. [1] 5 “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. 6 Incorporation is the redefinition of an as yet undetermined relationship function to include a variable contained within the equation under consideration {eg. equation (1) may be written as a=K0(r,ω,X)(E/B)2 }. 4

3

2.2

INTERPRETATIONS OF THE PRIMARY PRECIPITANT

The primary precipitant is subject to two interpretations. (i) It may be increased or decreased as a function of the magnitude of the electric field vector “E0(k,n,t)”, or (ii), it may be increased or decreased as a function of the magnitude of the magnetic field vector “B0(k,n,t)”. Subsequently, if the magnitude of the electric field vector “E0(k,n,t)” is constant in an experiment, then we may incorporate it into the experimental relationship function, as follows, α1

K 1 ω 0, r 0, E 0, D , X N 3 r 0 . K PV .

B 0( k , n , t )

2

n= N

(3)

Incorporating “B0(k,n,t)” represents the second interpretation of the primary precipitant. β1

K 2 ω 0, r 0, B 0, D , X . r 0 . K PV

3

N E 0( k , n , t )

2

n= N

(4)

Where, Variable

K1(ω0,r0,E0,D,X)

Description The subset formed, as N → ∞, by the method of incorporation applied to equation (2). The subset formed, as N → ∞, by the method of incorporation applied to equation (2). New experimental relationship function

K2(ω0,r0,B0,D,X)

New experimental relationship function

α1 β1

D

Experimental configuration factor7

c = c0 / KPV

Velocity of light in the PV

Units m/s2 m/s2 V2/m2 T-2 None m/s

Table 2, 3

MATHEMATICAL MODELLIG8

3.1

SEPARATION OF PRIMARY FORMS

The “α” and “β” forms, equation (3) and (4) respectively, may be used to generate subset expressions with respect to the hypothesis to be tested. The subsets have been termed the first and second alpha subsets (α1, α2), and the first and second beta subsets (β1, β2), to better characterize anticipated results. The “αx” and “βx” subsets, and experimental relationship functions “K1(ω0,r0,E0,D,X)” and “K2(ω0,r0,B0,D,X)”, may be formulated as follows,

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The experimental configuration factor is a specific value relating all design criteria. This includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy. 8 All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation.

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Var. / Eq. αx Eq.(3)

Description α1

K 1 ω 0, r 0, E 0, D , X N 3 r 0 . K PV .

B 0( k , n , t )

2

n= N

Eq.(5)

2

α2

K ω , r , E , D, X . 1 0 0 0

c0

N

r 0 . K PV

3

E 0( k , n , t )

2

n= N

Eq. (6)

N K 1 ω 0, r 0, E 0, D , X

K 0 ω 0, X .

E 0( k , n , t )

2

n= N

βx

Eq. (4) β1

K 2 ω 0, r 0, B 0, D , X . r 0 . K PV

3

Εq. (7)

N E 0( k , n , t ) n= N N

2

β 2 K 2 ω 0, r 0, B 0, D, X .

c0

.

r 0 . K PV

3

Eq. (8)

K 2 ω 0, r 0, B 0, D, X

2

B 0( k , n , t )

2

n= N

K 0 ω 0, X N B 0( k , n , t )

2

n= N

Formulation The “α1” subset may be formed, as N → ∞, by the method of incorporation applied to equation (2).

Units m/s2

The “α2” subset may be formed by direct substitution assuming a transverse EM wave relationship, c0=E0/B0, into “α1”. The experimental relationship function may be expressed, in terms of the magnitude of the electric field vector “E0(k,n,t)”, by relating “α1” to equation (2) and solving. The “β1” subset may be formed, as N → ∞, by the method of incorporation applied to equation (2). The “β2” subset may be formed may be formed by direct substitution assuming a transverse EM wave relationship, c0=E0/B0, into “β1”. The experimental relationship function may be expressed, in terms of the magnitude of the magnetic field vector “B0(k,n,t)”, by relating “β1” to equation (2) and solving.

m/s2

V2/m2

m/s2

m/s2

T-2

Table 3: Subsets, 3.2

GENERAL MODELLING EQUATIONS

The subsets, table (3), suggest two experimental avenues with respect to the hypothesis to be tested. These have been termed General Modelling Equation1 “GME1” and GME2, as follows, Description Form Equation GME1 (9) a1(r0) = ±½(β1 + β2) GME2 (10) a2(r0) = ±½(β1 - β2) Table 4: General Modelling Equations, 3.3

CRITICAL FACTOR

The resulting relationship functions may be characterized by the critical factor “KC”. It takes the form of a squared9 term and is a measure of the applied EM field intensity within the experimental test volume. “KC” may be derived from the ratio of the experimentally determined relationship functions “K1(ω0,r0,E0,D,X)” and “K2(ω0,r0,B0,D,X)”, by taking the limit as the number of harmonic frequency modes tends to infinity (N → ∞), N N K 1 ω 0, r 0, E 0, D , X 2 2 . 2 K C K 1, K 2 E 0( k , n , t ) B 0( k , n , t ) K 2 ω 0, r 0, B 0, D, X n= N n= N (11) 9

The left hand side of equation (11) “KC2”, is an arbitrary definition as a consequence of its units of measure, (PaΩ)2.

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It is also possible to formulate “KC” by utilising “a1(r0)” and “a2(r0)” in table (4) and solving for the ratio of experimental relationship functions. The permittivity and permeability of free space, “ε0” and “µ0” respectively, may be included to express equation (11) in units of energy density squared. 4

PHYSICAL MODELLIG

4.1

POYNTING VECTOR

Equation (11) illustrates that “KC” may be usefully approximated, by proportionality, to the magnitude of the resultant Poynting vector. Assuming a transverse EM wave relationship, the magnitude of the Poynting vector for EM waves at a particular frequency mode is, S = E H , where µ0H = B. [9] The intensity of each field mode is the power per unit area and the intensities of each harmonic frequency mode are summed to yield the total intensity. This indicates the intensity of the field passing through a surface of an experimental test volume. The resultant Poynting vector from a benchtop Experimental Prototype (EP) device may be oriented parallel to the position vector “r”, such that the acceleration acts to increase or decrease the local value of “g”. 4.2

POISSON & LAGRANGE

Table (4) defines two expressions that may be applied to experimental investigations and may illustrate modelling significance, as follows, N 2

a r0

±

β1

β2 2

K 2 ω 0, r 0, B 0, D, X . ± 3 2 .r 0 . K PV

N

N E 0( k , n , t )

2

2 c0 .

n= N

B 0( k , n , t )

2

n= N

E 0( k , n , t ) K 0 ω 0, X n = N . ± N 3 2 .r 0 . K PV 2 B 0( k , n , t )

2

c0

(12)

n= N

N 2

a r0

±

β1

β2 2

K 2 ω 0, r 0, B 0, D , X . ±

N

2 .r 0 . K PV

n= N

3

N E 0( k , n , t )

2

2.

B 0( k , n , t )

c0

n= N

2

E 0( k , n , t ) K 0 ω 0, X . n= N ± N 3 2 .r 0 . K PV 2 B 0( k , n , t ) n= N

2

c0

(13)

Equation (12) is proportional to a solution of the Poisson equation [8] applied to Newtonian gravity, where the resulting acceleration is a function of the geometry of the energy densities. Equation (13) 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. [9] This demonstrates that the relationship function “K2(ω0,r0,B0,D,X)” is the same in both instances. In forthcoming publications, these equations will be used to make predictions of the expected modification to the local value of gravitational acceleration “g”, in various experimental design configurations. 5

COCLUSIOS

In an experiment, the value of “KC” may be determined by direct measurement of the intensities at each harmonic frequency mode. In some hypothetical (EP) design configurations, the value of the experimental relationship functions “K1(ω0,r0,E0,D,X)” and “K2(ω0,r0,B0,D,X)” may also be determined by direct measurement of the EM field strength, at each harmonic frequency mode. If a resulting acceleration vector can also be measured, it may then be used to support or falsify the hypothesis to be tested, as presented in section 1.

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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] H.A. Wilson, “An electromagnetic theory of gravitation”, Phys. Rev. 17, 54-59 (1921). [3] R.H. Dicke, “Gravitation without a principle of equivalence”. Rev. Mod. Phys. 29, 363-376, 1957. See also R.H. Dicke, “Mach's principle and equivalence”, in Proc. of the Intern'l School of Physics “Enrico Fermi” Course XX, Evidence for Gravitational Theories, ed. C. Møller, Academic Press, New York, 1961, pp. 1-49. [4]. A.M. Volkov, A.A. Izmest'ev, and G.V. Skrotskii, “The propagation of electromagnetic waves in a Riemannian space”, Sov. Phys. JETP 32, 686-689 1971. [5] H. E. Puthoff, “Polarizable-Vacuum (PV) representation of general relativity”, gr-qc/9909037 v2, Sept,1999. [6] H. E. Puthoff, “Polarizable-vacuum (PV) approach to general relativity”, Found. Phys. 32, 927-943 (2002). [7] H. E. Puthoff, et. al., “Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight”, JBIS, Vol. 55, pp.137-144, astro-ph/0107316 v1, Jul. 2001. [8] G. Arfken, “Mathematical Methods For Phyisicists – Third Edition”, Academic Press, Inc. 1985 ISBN 0-12-059820-5. Ch. 1, pp. 77. [9] J.D. Jackson, Classical Electrodynamics, Third Edition, 1998, ISBN 0-471-30932-x, Ch. 6, Secs. 6.7-6.9, Ch. 12, Sec. 12.7.

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APPEDIX A Fourier series, representing the summation of trigonometric functions, may be applied to define a constant vector field “a” over the period 0 ≤ t ≤ 1/fΛ. A constant function is termed even due to symmetry about the “Y-Axis”, subsequently, the Fourier representation contains only Cosine terms and may be expressed in complex form. Therefore, let an arbitrary10 transverse EM plane wave be defined by, π .f Λ .n .t .i

Λ 0( k ) .e Where “k” and “n” denote wave vector and field harmonic respectively, such that, Λ ( k, n , t )

B( k , n , t )

Re( Λ ( k , n , t ) ) .( T )

(A1) (A2)

Where “T” denotes Tesla and “V/m” denotes volts per meter, E( k , n , t )

Im( Λ ( k , n , t ) ) .

V

EM Function

m

(A3)

1

2

fΛ

fΛ

Re( Λ ( k , 1 , t ) ) Re( Λ ( k , 2 , t ) ) Re( Λ ( k , 3 , t ) )

t Time

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

EM Function

Figure A1, 1

2

fΛ

fΛ

Im( Λ ( k , 1 , t ) ) Im( Λ ( k , 2 , t ) ) Im( Λ ( k , 3 , t ) )

t Time

1st Harmonic (Fundamental) 2nd Harmonic 3rd Harmonic

Figure A2,

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Phase (θ) has been excluded for simplicity. It has been numerically simulated that phase contributions may be usefully approximated to zero, when applied to equation (A1), for field harmonic values N ≥ 20 (approx.). Subsequently, as N → ∞ for a(t) to be constant, θ → 0.

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APPEDIX B All equation forms containing the squared sum of the magnitudes of the electric and / or magnetic field vectors result in constant values, by the superposition principle, as the system harmonic N → ∞. N ∑E( t )

E( k , n , t )

2

n= N

(B1)

N ∑B( t )

B( k , n , t )

2

EM Wave Function Superposition

n= N

(B2) 1

1

2 .f Λ

fΛ

∑E( t ) ∑B ( t )

t Time

Electric Field Magnitude Magnetic Field Magnitude

Figure B1 The EM Zero-point Field (ZPF), of the quantum vacuum is described in terms of the energy density per frequency mode, or “Spectral Energy Density”. ρ 0( ω )

2 .h .ω c

3

3

-34

(B3)

Where, “h” denotes Planck’s Constant [6.6260693x10 (Js)] and “ω” is in (Hz). The functions “E(k,n,t)2” and “B(k,n,t)2” used in equation (1) and throughout this paper, are proportional to the applied energy density at each frequency mode, with respect to experimental configuration.

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