PV), One Is Isomorphic to General Relativity (GR

PACS numbers: 03.03.+p, 03.50.De, 04.20.Cv, 04.25.Dm, 04.40.Nr, 04.50.+h

THE ELECTROMAGNETIC ZERO-POINT FIELD AND THE FLAT POLARIZABLE VACUUM REPRESENTATION

Todd J. Desiato1, Riccardo C. Storti2

June 18, 2003 v2

Abstract

There are several interpretations of the Polarizable Vacuum (PV). One is the variable speed of light (VSL) approach, that has been shown to be isomorphic to General Relativity (GR) within experimental limits. However, another interpretation is representative of flat geometry, in which intervals of time and distance are measured in local inertial reference frames where the speed of light remains constant. The Flat PV approach leads to variable impedance transformations, governed by the spectral energy content of the Quantum Vacuum’s Electromagnetic (EM) Zero-Point Field (ZPF). The EM ZPF consists of photons. An unlimited number of photons may occupy the same quantum state at an arbitrary set of coordinates. Therefore, the spectral energy of the ZPF may be varied smoothly, represented by a superposition of EM waves with a large number of photons per cubic wavelength. Utilizing the Flat PV representation, a family of frequency dependent solutions of Poisson’s equation are derived, that may be applied as tools for engineering the PV space-time metric, within the GR representation.

EXT-2004-112 18/06/2003

v1. Released v2. Revised Section 5, added tables 2 and 3.

1 [email protected] Delta Group Research, LLC. San Diego, CA. USA, an affiliate of Delta Group Engineering, P/L, Melbourne, AU. 2 [email protected] Delta Group Engineering, P/L, Melbourne, AU, an affiliate of Delta Group Research, LLC. San Diego, CA. USA,

1. INTRODUCTION

The first interpretation to be outlined in section 2 is the Polarizable Vacuum (PV) representation of General Relativity (GR).[1-7] This is a geometric representation of the space-time manifold, where variations in the speed of light are expressed by the refractive index “ K ”, which is also used to represent changes in the gravitational potential. It is illustrated that the Electric field vector, “ E ” and the Magnetic intensity vector “ H ”, are unaffected by gravitational fields, therefore, independent of “ K ”. This suggests it may be possible to modify the space-time metric to affect the motion of matter by immersion in a superposition of EM fields.[8-10] The second interpretation to be outlined in section 3 is the primary objective to be presented. It has been termed the Flat PV representation because the space-time manifold may be usefully approximated as “flat” in local inertial reference frames. Hence, the speed of light remains constant, thus preserving Lorentz invariance.3 In this representation, the spectral energy density of the EM ZPF is not uniform. Variations are described by relatively large numbers of photons per cubic wavelength as a function of the coordinates. It is illustrated that this leads to Impedance Transformations that are comparable to those typically found on transmission lines or wave guides, that permit propagation of EM waves. This approach is used in section 4 to illustrate that Planck’s constant “ = ”, the electrical 4 = ε impedance “ Zcoo1/ ” and the Gravitational impedance “ Gc/ ” vary proportionally with the impedance transformations, dependent upon the variable spectral energy density of the EM ZPF. These values are illustrated as integrals representing the total probability for finding photons within 1 cubic wavelength. It is demonstrated in section 5 that by normalizing this value, the physical constants are retained and the geometric PV representation of GR is restored. Section 6 demonstrates a sample calculation of a frequency dependent acceleration, derived from a spherically symmetric EM ZPF. This calculation is then used in section 7 to define a set of engineering modeling equations that may be used to engineer the PV. This demonstration results in four modeling equations, that permit the calculation of, i. The number of photons per cubic wavelength within a particular frequency range, as a function of the radial coordinate. ii. The upper limit cut-off frequency, given the number of photons per cubic wavelength and the lower limit cut-off frequency. iii. The frequency range given the upper and lower limit cut-off frequencies. This may be used for example, to represent Heisenberg uncertainty in the energy of a particle. iv. A family of frequency dependent, Newtonian gravitational potentials.

In the proceeding construct, the quantum of electric charge “ e ” is assumed to be constant and invariant to changes in the gravitational field. The units of “Coulombs” are represented by “C” and the SI unit system is used throughout, except in equation (24). The classical macroscopic representations of an EM field in a vacuum are as follows, [13]

= ε DEo (1) = µ BHo (2)

Where,

3 It has been illustrated in [11,12] that Lorentz invariance may be broken when a particle’s energy approaches the Planck Energy, “EP”. Invariance is restored at low energy where E/EP Æ0. 4 The symbol “Zo” is used to represent the impedance of free space, as is typically used in electrical engineering to express the impedance at the terminals of a two terminal transmission line or circuit. It does not refer to the Atomic Number, that also typically uses this symbolic character in particle Physics.

2 Value Description Units 2 D Charge Displacement C/m E Electric Field V/m 2 B Magnetic Flux Density T = Vs/m H Magnetic Intensity A/m ε o Permittivity F/m µ o Permeability H/m Table 1,

A “ ∆ ” denotes an incremental change in value.

2. 1ST INTERPRETATION – THE PV REPRESENTATION OF GR

In the PV representation of GR, [4,5] the speed of light is a variable that depends on the gravitational potential. Intervals of space and time are affected by gravity and it is assumed that K →1 where space-time is asymptotically flat at infinity. Equations (1) and (2) are expressed in terms of “ K ”, as follows,

= ε DEK o (3) = µ BHK o (4) Where,

2GM 2 2GM K =≈++e1rc ... (5) rc 2

It is implied that “ K ” varies as a function of the coordinates. However, the coordinate variables are typically not shown explicitly.[4-7] In this representation, Voltage “ ∆φ ” and Length “ ∆x ” are transformed proportionally by “1/ K ”. ∆φ ∆=φ o (6) K ∆x ∆=x o (7) K

∆φ ∆ = Where, “ o ” and “ xo ” are the values where “ K 1”, at infinity. Therefore, “ E ” is invariant with respect to a change in the gravitational field. It is independent of “ K ”, according to,

∆φ ∆φ =→o E ∆∆ (8) xxo

In this representation, intervals of length vary according to equation (7) and intervals of time vary according to, ∆=∆ ttKo (9)

This suggests that the PV representation of GR may be considered a geometric representation, isomorphic to GR in a weak gravitational field.[4-7] Since the total charge “ ∆Q ” is constant and electrical current is measured in Amperes, “A = C/s”, “ H ” is also invariant with respect to a change in the gravitational field.

3 ∆∆QQ =→H ∆∆ ∆ ∆ (10) xxttoo

∆∆=∆ ∆ As is the product xxttoo, which leads to the following interpretations,

i. Equation (8), suggests that intervals of length “ ∆x ” vary proportionally with changes in the Voltage potential “ ∆φ ” or Energy “ ∆E ”. For instance, this may be associated with the potentials that bond electrons to atoms. ii. Equation (9), suggests that “ ∆x ” varies inversely proportional to intervals of time, “ ∆t ". Subsequently, rulers appear to contract and clocks appear to run slow in a gravitational field, as in GR. iii. Equation (10), suggests that “ ∆x ” varies proportionally with changes in the electrical current “ ∆∆Qt/ ”. For instance, this may be associated with a change in the kinetic energy of electrons balancing the change in potential energy, analogous to the Bohr model of the atom. iv. Equation (10), also suggests that “ ∆t ” varies proportionally with changes in the charge per unit length “ ∆∆Qx/ ”. v. Considering equations (8), (9) and (10), “ ∆t ” varies inversely proportional to “ ∆φ ”. vi. By (i) and (v) the Energy-Time product, " ∆∆Et" is independent of “ K ”. Therefore, Planck’s constant, “ = ” remains invariant with respect to gravity. vii. The speed of light varies inversely proportional to “ K ”,

1 ∆x c ==o (11) 2 µε Kt∆ K K oo o

viii. The impedance is independent of “ K ”,

Kµ ZZ==o ε o (12) K o

ix. Energy " E " varies inversely proportional to “ K ”,

∆E ∆=E o (13) K

2 x. Energy Density varies proportionally with “ K ”. Therefore, in-falling matter is compressed according to the PV Metric, [4-7] and the Schwarzschild metric of GR where there are Tidal forces at work.5 [14]

∆ ∆∆EEK3/2 2 EK==oo ()333()() (14) ∆x K ∆x oo∆x

This interpretation suggests that in the PV representation of GR, “ ∆x ” and “ ∆t ” are measured by variable rulers and clocks. Where changes in voltage potential and current may result in changes to the

5 Free falling matter in a spherically symmetric gravitational potential is not in inertial motion. True inertial motion is only possible if the gravitational field is uniform and matter is not experiencing tidal forces. Tidal forces do work to compress the material as it falls in a spherically symmetric field. This is not true of inertial motion.[14]

4 scale of atomic bonds, because they depend on “ K ”. Thereby affecting the length of a ruler or the transition energy between different states of a Cesium clock. They are observable effects, [4-7] as they are in GR when comparing these intervals at different gravitational potentials. Thus, experimental observations result in the geometric interpretation and representations.[14,15] Equations (8), (9) and (10) indicate that it may be possible to modify the space-time metric and regulate its affects on matter, by use of superimposed EM fields that control the induced change in voltage, current and charge densities.[6,7,16] If it is possible to alter space-time and matter in this manner, then the effects of Special Relativity, specifically time dilation and length contraction may be controlled. Potentially making it possible to achieve superluminal velocities.[8-10,16,17] Evidence of this effect may be found in Quantum Electrodynamics (QED), where atoms in a strong electric field undergo a “Stark” frequency shift, [18] that increases the transition frequency and energy, as expected, based on the above results.

3. 2nd INTERPRETATION – THE FLAT PV REPRESENTATION

The Flat PV representation is intended as a heuristic tool with which to better understand the PV and GR, in terms of the EM ZPF. In this representation, the intervals “ ∆x ” and “ ∆t ”, do not change when measured in local inertial reference frames. However, they do change when comparing local measurements to those of events at a different gravitational potential. This type of measurement is said to be non- local.[14,15] The transformation from the Flat PV representation to the PV representation of GR will be illustrated in sections 4 and 5. In this representation, referring to Table 1, it is postulated that “ D ” and “ H ” remain invariant with respect to a change in the gravitational potential, whilst permitting “ E ” and “ B ” to vary. Therefore, equations (1) and (2) become, κ ED= ε (15) o = κµ BHo (16)

Where “κ ” represents a variable Impedance Transformation that takes the place of the variable refractive index. It is implied that “κ ” varies as a function of the coordinates in a gravitational field. This leads to the following interpretation,

i. Since “ ∆Q ” is constant and “ D ” is postulated to remain invariant with respect to gravity, expressed by “κ ”, it follows that Area is independent of “κ ”. κ ∆=∆ ii. By (i), it follows that Length is independent of “ ”, xxo . iii. Since “ H ” is postulated to be invariant with respect to gravity, the Length-Time product is κ ∆∆=∆ ∆ independent of “ ”, xt xoo t, consistent with section 2. κ ∆=∆ iv. Therefore, by (ii) and (iii) it follows that Time is independent of “ ”, tto . Hence, the geometry may be usefully approximated as flat. v. If “ ∆x ” and “ ∆t ” are unaffected, their ratio is constant in all inertial reference frames.

1 ∆x ==c µκ⋅ ε κ ∆ (17) oo/ t

vi. The impedance of the ZPF varies proportionally with “κ ”. Therefore, Impedance Transformations are required when traveling through space-time, as they are when modeling transmission lines or wave guides permitting the propagation of EM waves through varying polarizable mediums.[13,19]

κµ2 ZZ==o κ ε o (18) o

5 Therefore, Z κ = (19) Zo

vii. The Energy and Energy Density of in-falling matter varies proportionally with “κ ”, gaining ∆→∆κ ∆→∆κ mass and energy according to, mmo and EEo , as evidenced by the blue-shift of in-falling photons. viii. By (iv) and (vii), " ∆∆Et" is variable. This suggests that Planck’s constant may be varied proportionally with “κ ” and their product may be used to represent the variable energy per frequency mode, as a function of the coordinates. This shall be illustrated in section 4. ix. Since impedance varies proportionally with “κ ” and Volume remains unaffected, “κ ” may be usefully approximated by the number of photons contained in a cubic wavelength. This shall also be illustrated in section 4.

4. FREQUENCY DEPENDENCE

ρ ()ω The spectral energy density of the EM ZPF “ o ”, as derived in QED, [18] may be expressed λπω= using the dispersion relationship for EM waves, “ ω 2/c ”, as follows,

ωπ3 ρω()====4 o 23 3 (20) 2πλc ω

Utilizing this representation, Planck’s constant “ = ” may be expressed in integral form, as an integral over the Volume of 1 cubic wavelength.

λ3 1 ω ρω()dV =×⋅= ~ 1.05 10-34 J sec π ∫ o (21) 4 0

απ= 2 Substituting the Fine Structure constant “ eZo /4 = ”, the impedance may also be expressed in integral form, λ3 α ω ρω() ==⋅2 2 ∫ oodV Z ~ 377 Ohms J sec/C (22) e 0

2 = Substituting the square of the Planck Mass “ mcGp = / ”, the Gravitational impedance, [20] may also be expressed in integral form,

λ3 1 ω G ρω()dV =×⋅ ~ 2.23 10-19 J sec /kg 2 π 2 ∫ o (23) 4 mp 0 c

The approximate values shown are exemplary of the order of magnitude in each form. However, in Natural units where cG==== 1, equations (21), (22) and (23) have identical integral representations. The integral is interpreted as the Normalized Total Probability of finding a photon within 1 cubic wavelength. Consistent with “box normalization” when using periodic boundary conditions, [12,18] according to, λ3 1 ω ρω()dV = 1 π ∫ o (24) 4 0

6 Equation (24) suggests that the spectral energy density is a function of probability density and that any EM spectral energy density may be used to represent a continuum of values. Subsequently, a variable spectral energy density, “ ρ ()ω ” may be used as an engineering tool, to improve our understanding of gravitational fields. The spectral energy content is governed by the number of photons per cubic wavelength in each frequency mode, according to,

4πκω= () ρω()=↔= ρ κ 3 , o 1 (25) λω

Where the number of photons, “κω()” may be large, such that small changes have a negligible effect on the physical constants. The derivatives of “κω()” with respect to the coordinates determines the strength of the gravitational field. When its value is normalized to unity, Planck’s constant and the other physical constants retain their value and the GR representation of section 2 is restored. Therefore, the leading coefficients in equations (21), (22) and (23) are normalization factors for the total probability in units of energy, charge and mass, respectively.6 The frequency dependent version of equations (21), (22) and (23) may be expressed,

λ3 1 ω ρω()dV =⋅ κω ()= π ∫ (26) 4 0 λ3 α ω ρω()=⋅ κω () 2 ∫ dV Zo (27) e 0 λ3 1 ω G ρω()dV =⋅ κω () π 2 ∫ (28) 4 mp 0 c

Where “κω()” is a frequency dependent dimensionless variable. It represents the impedance transformations of the PV as a function of the coordinates and the number of photons per cubic wavelength.

Therefore, a gravitational field described by “κω()” may be quantized by the number of photons per cubic wavelength at each frequency mode.7

These equations may be used to express a fundamental description of the PV, in terms of a coherent superposition of EM waves and a large number of photons per cubic wavelength. Subsequently, the spectral energy density of the ZPF may be scaled proportionally to the mass-energy density generating the gravitational field.8 Solutions of Poisson’s equation may then be found that are typical of scalar Newtonian gravity, [24] and shall be illustrated in section 6.

5. PLANCK NORMALIZATION

It may be illustrated how the Fine Structure constant remains invariant in both representations. In α the PV representation of GR, “ Zo ” and “ = ” are unaffected by gravity, hence, “ ” is unaffected. Using the

6 Historically, energy, charge and mass were independently defined units of measure. 7 The may have relevance to Quantum Gravity Theories and should correspond at low energy, where E/EP Æ0. 8 This may be illustrated using Buckingham’s Π Theory.[8-10, 21-23]

7 κω() Flat PV representation, equations (26) and (27) illustrate that “ Zo ” varies proportionally with “κω()= ”. Therefore, “α ” remains unaffected. 2 Similarly, the square of the Planck Mass “ mp ” is the ratio of equations (26) and (28). Therefore it κω() = 2 is independent of “ ”, as is the Planck Energy, Emcpp. The invariance of “ Ep ” is consistent with [11,12]. However, the Planck Length and the Planck Time may be represented as variables in terms of “κω()”, using equations (26) and (28) as follows,

λ3 1 ω =G LdV→=⋅ρω() κω () p π ∫ 3 (29) 4 mcp 0 c λ3 1 ω =G tdV→=⋅ρω() κω () p π ∫ 5 (30) 4 Ecp 0

κω() Therefore, length and time at the Planck scale, “ Lp ” and “ t p ”, are not invariant to changes in “ ”. Unlike “ ∆x ” and “ ∆t ”, which are invariant in the local inertial reference frames of the Flat PV representation. It may be illustrated that when the normalized value of the integrals are used, as in equation (24), the Flat PV representation may be transformed to the PV representation of GR. The normalized representation, PV representation of GR and the Flat PV representation of the Planck scale are compared in the following tables,

Representation Normalized PV Representation Flat PV Values of GR Representation m κ = =c ===c p ===c Planck Mass mp mpK mmppκ G GK K κG

=G 3 κ 2 = ===GK 3/2 ===G κ Planck Length Lp LLKpK p LLppκ c3 c3 c3

=G 5 κ 2 = ===GK 5/2 ===G κ Planck Time t p ttKpK p ttppκ c5 c5 c5

5 5 E κ 5 = =c ===c p ===c Planck Energy E p E pK EEppκ G GK55/2 K κG Table 2,

PV Representation Flat PV Representation Normalized Values of GR Representation

m m m 2 mκ m Relative Mass o K = o K = o κ mp mmpK p mmppκ L L L L L o K = o κ = o Relative Length 2 κ Lp LLKpK p LLppκ t t t t t o K = o κ = o Relative Time 2 κ t p ttKpK p ttppκ

E E E 2 Eκ E Relative Energy o K = o K = o κ Ep EEpK p EEppκ Table 3,

8 2 Table 3 indicates that when comparing relative values, κ = K . Therefore, the refractive index may be determined directly from the number of photons per cubic wavelength, when normalizing to the Planck scale.

6. A NEWTONIAN CALCULATION OF “g”

Using a Newtonian approximation, frequency dependent solutions of Poisson’s equation may be found. At the low energy limit, the acceleration “ g ” at the surface of the Earth, may be represented by,

∞ 2GM 2 gdr=−∫ Earth =−9.830 m/s (31) r3 REarth

Where “ r ” is the radial distance measured from the center of mass of the Earth, “ REarth ” is the average radius of the Earth and “ M Earth ” is its rest mass. In terms of the approximate rest mass-energy density of the Earth,

Mc 2 Ur() = Earth Earth 4 (32) π R3 3 Earth equation (31) may be expressed by,

∞∞ 2GM 8πG ∫∫Earth dr→=− U() r dr g (33) rc323 Earth RREarth Earth

Assuming that the number of photons “κω()r, ” in each frequency mode as a function of “ r ”, may be expressed as the product of two functions, as follows,

κω()rXrW, = ()() ω (34) then “W ()ω ” may be usefully approximated as unity, uniformly distributed throughout the range of frequencies specified at each set of coordinates along “ r ”. Therefore, only the radial component, “ Xr()” need be considered. Other distribution functions may be substituted for “W ()ω ” and the integrated solutions illustrated below will differ accordingly. ρ ()ω However, Lorentz invariance of “ o ” is well known in QED. It is derived from its 3 proportionality to “ω ” [18] when “κ →1 ” at all frequency modes and coordinates in flat space-time. Therefore, where “κω()r, ” differs from unity and variations in the distribution functions of equation (34) lead to observable intensity differences between inertial reference frames, Lorentz invariance is broken. In the Newtonian approximation, the spectral energy density is dependent on the radial coordinate,

ρ ()ω → ()ρ ()ω Xr o (35) and on the frequency range defined by an initial and final frequency, applied as the limits of integration.

This yields the energy density “U zpf ” of the ZPF within those bounds, as follows,

ω f Ur(),,ωω = Xr()ρ (ωω ) d ifzpf ∫ o (36) ω i

9 The acceleration “ g ” is calculated by substituting equation (36) into equation (33),

∞∞ 88ππGG= () =−ωω44() ∫∫U r dr()fi X r dr (37) 338ccc2223Earth π RREarth Earth

∆=ωω − ω to yield the gravitational acceleration within the frequency range, fi,

L2 ∞ =−p ωω44 −() =− 2 gXrdr()fi∫ 9.830 m/s (38) 3π c2 REarth

23= Where LGcp = / is the normalized Planck Area. Equation (38) represents a family of solutions that may be derived for each band of frequency modes. The narrower the bandwidth, the greater the number of photons “ Xr()”, required to result in the value “ g ”. This suggests that “ Xr()>> 1 ” such that it may vary smoothly over the change in coordinates.

7. BASIC MODELING EQUATIONS To solve for the number of photons as a function of “ r ”, within a particular frequency range, take the derivative with respect to “ r ” on both sides of equation (38). This results in the first modeling equation of the ZPF and permits “κ ” to be calculated as a family of frequency dependent variables, as follows,

6π c5 M κω()rrR,∆=Earth , for > ωω44− 3 Earth (39) =()fir

The second modeling equation of the ZPF permits the final frequency cut-off to be calculated, κ ω given “ ” as a function of “ r ” and the initial frequency mode “ i ”, according to,

6π cM5 ωωκ()=+>4 Earth ω4 firrR, , i , for Earth (40) =κ r3

The third modeling equation of the ZPF is the bandwidth. It is defined as the difference between the initial and final frequency limits, according to,

∆=ωωκω()() ωκω −> rr,,ifii ,, , for rR Earth (41)

It is speculated that the bandwidth may be proportional to the Heisenberg uncertainty in the energy of subatomic particles. These particles may then be coupled to photons in the ZPF within their respective bandwidths. A general modeling equation may be derived that permits practical modeling of any value of acceleration “ a,()r ∆ω ”, by the application of superimposed EM fields9. Where “ Nr(),ω ” is an arbitrary function of the independent variables, representing the number of photons resulting from the applied EM fields. This number may be readily calculated from the Poynting vectors and intensity of the applied field sources, as was shown in [9,10].

9 A virtually identical result was also derived by Delta Group Engineering, P/L, Melbourne, AU, utilizing Buckingham’s Π Theory, based on dimensional analysis techniques.[8-10].

10 ω 4L2 r2 f a,()rNrddr∆=−ωωωωp () , 3 2 ∫∫ (42) 3πc ω r1 i

Dimensional similarity and Einstein’s Equivalence Principle assures that this acceleration may be used to express an effective, frequency dependent, Newtonian gravitational potential “ϕ ()r, ∆ω ”,

r2 ϕω()()rrdr,a,∆=∫ ∆ ω (43)

This represents a family of frequency dependent (or energy dependent) gravitational potentials, that are solutions to Poisson’s equation. It is the fourth modeling equation of the ZPF.10

8. CONCLUSIONS

In the PV representation of GR, [1-7] the speed of light varies inversely proportional to the refractive index “ K ”, whilst the impedance of the PV is independent of “ K ”, as shown in equations (11) and (12). This was demonstrated to be a geometric representation of space-time due to variations of the intervals “ ∆x ” and “ ∆t ” as a function of “ K ” and the gravitational potential.. It was demonstrated that in this representation, the Electric field “ E ” and the Magnetic intensity “ H ” are independent of “ K ”. This suggests that voltage and current vary proportionally with the interval “ ∆x ”, in a gravitational field. These variations may affect the scale of atoms. Therefore, it may be possible to control the space-time metric and regulate its affects on matter, by immersion in a superposition of EM fields.[8-10] Potentially, these fields may be used to regulate voltage, current and charge densities within matter. Thereby, controlling the effects of Special Relativity and making it possible to travel at superluminal velocities. A locally Flat representation of the PV was then presented, in which the speed of light remains constant whilst the impedance of the PV is transformed by a dimensionless variable “κ ”, representing the gravitational field. This suggests that density variations in the EM ZPF may be used to describe gravitational fields by the number of photons per cubic wavelength. This number is derived from the spectral energy density, integrated over the volume of a cubic wavelength, leading to a family of frequency dependent impedance transformations, “κω()r, ∆ ”. The integral representation of “κ ” has three forms that may be used to represent the variable κω() κω() electrical impedance “ Zo ”, gravitational impedance “ Gc/ ” and energy per frequency mode “κω()= ”. It followed that normalization of the total probability of finding a photon within 1 cubic wavelength, renormalizes “κ →1 ”, restoring the physical constants and the geometrical interpretation of space-time. The distribution function “κω()r, ∆ ”, as derived in (39), is frequency dependent. Since gravity appears to affect all observable matter and energy equally, it permits the assertion that the number of photons per cubic wavelength is large and uniformly distributed within the limits of observation at low energy conditions. This suggests that atomic and subatomic particles couple to the EM ZPF within a unique frequency range equivalent to the Heisenberg uncertainty in its local energy value. This may be associated

10 An analogous derivation was recently presented in [12], titled “Gravity’s Rainbow”. The authors effectively showed that space-time appears differently to particles of different energy that occupy different frequency bands. They call this the “Rainbow Metric”. Depending on the regime of interest, this new approach has been termed doubly Special Relativity or doubly General Relativity. It may be considered an alternative representation of what has been presented here.

11 with the fluctuation-dissipation relationship between particles and the Quantum Vacuum, as discussed extensively in QED.[18] Research currently being conducted suggests that this relationship may also lead to the origin of inertia, [25,26] and may be consistent with the Flat PV approach presented herein. A sample calculation deriving the terrestrial acceleration “ g ” from the ZPF was presented. This lead to general modeling equations that permit calculations of energy density, acceleration, photon density, frequency limits and gravitational potentials that represent a family of frequency dependent solutions of Poisson’s equation. These results may permit engineering of electromechanical devices, purposely designed to control the space-time metric by application of superimposed EM fields.[10,16,17]

Acknowledgment

This research was made possible by the collaborative efforts of Delta Group Engineering, P/L, Melbourne, AU, and Delta Group Research, LLC. San Diego, CA. USA.

12 References

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