_{1}

^{*}

Einstein theorized that a mass travels towards another mass, not because it is attracted by a force acting across a distance, but because it travels through space and time that is warped by masses and energy. Einstein postulated that this space-time fabric can have wave-like modes which have been measured by the LIGO experiment. A consistent model of the generation of space-time-fabric-modes by a light Photon is derived for slight space-time deformations. Each Photon generates a shower of very small amplitude space-time fabric modes. Each mode can have a number of energy quanta. The probability of a Photon generating a shower of space-time modes is much larger than the probability of all the space-time modes collecting and generating a Photon. Therefore, this process has a unique Arrow of Time. Similar to the energy quanta of displacement modes in an elastic medium which is called Phonons, the energy quanta of the space-time fabric modes are called gravity Phonons. Both are tensor waves. Gravity Phonons have spin angular momentum of 2 and propagate with the speed of light. At every step of these calculations, equations derived from the General Relativity Theory by scientists and verified by Astronomical observations or experiments are employed.

Einstein [

A light Photon will not interact to first order with flat space-time, the ordinary four-dimensional Minkowski space. The Earth-like mass is used to remove this symmetry and facilitate the interaction of the Photon with warped space-time.

Einstein also postulated the existence of wave modes in the space-time fabric. This has been verified by the Laser Interferometer Gravity-Wave Observatory LIGO experiment [

In this paper, equations derived from the General Relativity Theory (GRT) and verified by astronomical observations or experiments are employed at every step of the calculations.

Professor Paul Sutter of Ohio State University and Chief Scientist of the COSI Science Center [

This discussion involves space-time which permeates all space at all times, a light electromagnetic wave, and a large non-rotating mass. The mass used has the same value as the mass of the Earth. The derivation is in four steps:

FIRST, to model the affect of the slightly warped space-time on an electromagnetic wave, Einstein’s description [

SECOND, the affect of the energy of the light wave on the curvature of space-time is described by Einstein’s equation [

THIRD, a Taylor series expansion to second order of the time dilation equation is used for the derivation of the frequency ω of the light wave in the presence of gravity as a function of the frequency ω E of the light wave in the absence of gravity and the space-time mode frequency ω G . The gravitational blue shift of the frequency, in this case, of gamma rays emitted by iron F 57 e samples mounted a vertical elevation of 22.5 m apart has been measured by R. V. Pond and J. L. Sinder [

In the FOURTH step, Quantum Electrodynamics [

A momentum vector diagram illustrating the affect of the curvature of space-time on a light wave, and the affect of the energy of the light wave on the curvature of the space-time fabric to within a constant for slightly curved space-time fabric only, as shown in

This derivation is not valid in the vicinity of large masses such as Black Holes and Neutron Stars. It is also not applicable to the Early Universe, where the mass density was much larger than it is now, and the affect of gravity was larger.

The space-time fabric, described by the second rank metric tensor has a similarity to an elastic fabric, described by the fourth rank elastic constant tensor. The metric tensor that describes the curvature of the space-time fabric is itself a function of the coordinate components of space-time. The elastic constant tensor is constant. The elastic fabric mode energy quanta are called Phonons. The electromagnetic field mode and space-time mode interactions are similar to the quantized electromagnetic field and acoustic mode interaction [

The gravity Phonons are massless Bosons, that propagate with the speed of light. Light Photons are derived from a vector model, the Maxwell Electromagnetic Theory, and have a spin angular momentum of 1. The gravity Phonons are derived from a tensor model, the GRT, and have a spin angular momentum of 2. The space-time deformation caused by the mass acts as a cavity to determine the frequency of the space-time fabric modes.

Before Ludwig Boltzmann [

Therefore, the gravity Phonons derived here can be used for the calculations of statistical Thermodynamic properties of the space-time fabric such as its average energy, entropy, pressure, and temperature (2.72503˚K). The thermal energy of 0.2340247 meV of the Cosmic Background Radiation is much larger than the energy of a gravity Phonon of 0.59893 atto eV calculated here. The calculations of the Statistical Thermodynamic models are deferred to future publications.

A. D. Sakharov [

Each light Photon generates a shower of identical space-time modes, see

This Arrow of time is consistent with Arthur Stanley Eddington’s [

The reciprocal relationship between electromagnetic wave modes and gravity is similar to the reciprocal interaction between sound waves and gravity discussed by Nicolis and Penco [

An interesting approach to the interaction of Electromagnetic fields and deformations of the space-time fabric is discussed by Gerald E. Marsh [

introduction of a local inertial frame. In this paper, we also use a second-order expansion approach in our derivations.

Bryce C. DeWitt attempts to derive a Quantum theory of gravity from the GRT starting with a General Relativistic Lagrangian. His derivation is intended to be applicable for all gravity values. He wrote 3 papers [

The model derived in my paper corresponds approximately to DeWitt’s limit Hamiltonian H ∞ appearing in equation {4.9} on page 1119 of Quantum Theory of Gravity I. DeWitt is using the full tensor formulation of the GRT for his derivation. I use contractions with time-like unit vectors of the tensors in the GRT which results in a scalar formulation of the GRT. The contraction leaves only functions that were verified by astronomical observations or experiments because most of the tests that validated the GRT were based on calculations that reduced parameters of the GRT to scalars. This results in a description that is similar to an Empirical description of physical observations. Since my derivation is from equations validated by real physical observations and experiments it supersedes any difficulties that occur in the full quantization of the GRT.

Carlo Rovelli [

The FIRST attempt was to formulate a quantum field theory of the fluctuations of the metric on a classical flat Minkowski space. Even though Van Nieuwenhuizen, and others, found firm evidence of a non-renormalizability model, a search for an extension of the GRT using a renormalizable perturbation expansion was started. A high order derivative theory and supergravity theory converged successfully to string theory.

The SECOND was an attempt to construct a quantum theory in which the Hilbert space carries a representation of the operators corresponding to the full metric, or some functions of the metric, without a background metric to be fixed. The formal equations of the quantum theory were successfully formulated with loop quantum gravity.

The THIRD was an attempt to use some version of Feynman’s functional integral quantization to define the theory. This is somewhat similar to the approach used in this paper.

The paper “Potential Origin of a Quantitive Equivalence between Gravity and Light” by Michael A. Persinger [

The research described above and in references [

NOTE: This derivation is only valid for slightly curved space-time fabric.

The effect of the curvature of the space-time fabric on a light wave is to deflect it as Einstein [

The deflection of a light wave by the curvature of space-time due to the Solar mass is calculated in Einstein’s Chapter “On the Influence of Gravitation on the Propagation of Light” in section §4 “Bending of Light-Rays in the Gravitational Field” on page 108 and page 168, equation {74}. The deflection angle [

a ) θ ⊙ = 2 M ⊙ G c 2 R ⊙ b ) θ ⊙ = r ⊙ s s R ⊙ c ) r ⊙ s s R ⊙ ≡ S c h w a r z s c h i l d r a t i o (1)

Here M ⊙ is the solar mass, R ⊙ = 6.957 × 10 8 m is the radius of the Sun and r ⊙ s s = 2954.030552 m is its Schwarzschild [

r ⊙ s s = 2 M ⊙ G c 2 (2)

For the Sun the Schwarzschild ratio r ⊙ s s R ⊙ = 4.246127 × 10 − 6 and the deflection angle θ ⊙ = 0.8758266 arc seconds. It was first measured by F. W. Dyson, A. S. Eddington, and C. Davidson [^{6} m is the radius. For the Earth, the Schwarzschild ratio r s s R = 1.390642 × 10 − 9 . The deflection angle of a light beam θ = 0.286841 marc seconds which corresponds to slightly curved space-time.

The wave vector ω E c of the light wave in the absence of gravity, the wave vector ω c of the light wave subject to gravity, and change in the wave vector ∆k of the light wave due to the deflection by gravity form the large triangle in

a ) sin θ = c Δ k ω E b ) θ ≈ c Δ k ω E c ) r s s R ≈ c Δ k ω E d ) Δ k ≈ r s s ω E c R (3)

Dadhigh Naresh [

a ) c 4 8 π G E μ ν = T μ ν where b ) E μ τ = R μ τ − 1 2 g μ τ R (4)

E μ ν is the Einstein tensor derived from the Ricci tensor R μ ν . T μ τ is the electromagnetic part of the stress-energy tensor. Here g μ τ is the metric tensor. The Ricci tensor is derived by a contraction over the indices ρ and σ of the Riemann-Christoffel tensor B μ σ τ ρ . The curvature of space-time is described by the Riemann-Christoffel tensor. Equation (4) is a tensor equation. It can be reduced to a scalar equation for slightly curved space-time fabric only by contracting Equation (4) with a time like unit vector with components u μ .

a ) c 4 8 π G [ R μ τ u μ u τ − 1 2 g μ τ u μ u τ R ] = T μ τ u μ u τ b ) − c 4 8 π G R t t ≈ T t t c ) c 4 8 π G Curvature = Electromagneticenergydensityfunction (5)

Summation over repeated Greek indices is implied. No summation over the Latin indices t is implied. E tt and T tt are the tt components of the Einstein tensor and Stress Energy tensor. Here the T tt component of the Stress Energy tensor is equal to the electromagnetic energy density function. The timelike unit vector u μ is:

u μ = [ − 1 0 0 0 ] (6)

The development of Einstein’s equation resulting in Equation (5b) is described in detail by Lee Loveridge in “Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor and Scalar Curvature”, Reference 5. Loveridge gives an excellent description of the derivation of the Einstein equation from the concept of curvature of space-time, the Riemann-Christoffel Tensor, and the Ricci Tensor. Loveridge uses the sum over the coordinates of a large number of point particles in the vicinity of the original point particle under consideration. Summing over a large number of particles is equivalent to the contraction of the Riemann-Christoffel tensor to form the Ricci tensor. Instead of describing the motion of a single point particle one has now to describe the motion of a volume δ V of all the point particles in the vicinity of the original point particle. From Loveridge’s [

− c 4 4 π G [ 1 V 0 D 2 δ V c 2 d t 2 − 1 V 0 D 2 δ V Massonly c 2 d t 2 ] = Q ( t ) (7)

where Q ( t ) is the electromagnetic energy density function. Loveridge also derives Newton’s laws from the Einstein equation, Equation (4). The electromagnetic energy density function Q ( t ) in the real world is a function of space and time. A constant and uniform electromagnetic energy density would result in a constant, uniform and therefore, isotropic space-time fabric deformation. But the electromagnetic field does not interact with an isotropic space-time fabric. Q ( t ) can be expressed as a Fourier Integral.

− c 4 4 π G [ 1 V o D 2 δ V c 2 d t 2 − 1 V 0 D 2 δ V Massonly c 2 d t 2 ] = ∫ − ∞ ∞ d ψ ℏ Δ k ( ψ ) c V o q ( ψ ) exp ( j ψ t ) (8)

where j = − 1 . Here q ( ψ ) is a dimensionless Fourier Integral of the electromagnetic energy density function and ψ is a frequency. A detailed derivation of the electromagnetic stress tensor and its contraction is derived in Appendix A. Equation (A4b) is used in the argument of the Fourier integral on the right sides of Equations (8). Expressing the incremental volume δ V also as Fourier integrals.

a ) Forcurvedspace-time δ V = ∫ − ∞ ∞ d ψ V ( t ) exp ( j ψ t ) b ) Forcurvedspace-timeduetomassonly δ V Massonly = ∫ − ∞ ∞ d ψ V o exp ( j ψ t ) (9)

where V ( t ) is a very slow varying function of time. By substituting Equations (9) into Equation (8).

− c 2 4 π G [ D 2 d t 2 ∫ − ∞ ∞ V ( t ) V o exp ( j ψ t ) d ψ − D 2 d t 2 ∫ − ∞ ∞ exp ( j ψ t ) d ψ ] = ∫ − ∞ ∞ d ψ ℏ Δ k ( ψ ) c V o q ( ψ ) exp ( j ψ t ) (10)

Performing first, the differentiation of the terms in the square bracket of Equation (10). The volume amplitude V ( t ) in the first Fourier Integral in the square bracket of Equation (10) is a very slow varying function of time compared to the oscillation. Therefore, it is treated as a constant in the Fourier Integral. Taking the inverse Fourier Integral:

c 2 4 π G [ − V ¨ V o − 2 j ω G V ˙ V o + ω G 2 V V o − ω G 2 ] = ℏ Δ k ( ω G ) c V o q ( ω G ) (11)

where ω G is an oscillating frequency of a Fourier component of the incremental volume δ V . Therefore, ω G is also an oscillating frequency of the space-time fabric modes. Since the volume V ( t ) is a very slow varying function of time, the first term in the square bracket of Equation (11) has been neglected. The Volume V in the third term in the square bracket of Equation (11) was approximated by V o . Collecting the remaining terms of Equation (11).

− j c 2 ω G 2 π G V ˙ V o = ℏ Δ k ( ω G ) c V o q ( ω G ) (12)

Assuming the volume V has also a harmonic time dependence. Therefore, one can approximate the first time derivative V ˙ of V in Equation (12) by:

V ˙ ≡ j 2 π G ℏ q c 2 w R r s s (13)

where 2 π G ℏ c R r s s = 1.060763 × 10 − 43 m 3 / second and where the dimensionless constant w will have to be evaluated from observations. Substituting Equation (13) into Equation (12).

Δ k = w R ω G r s s c (14)

The disturbance of the space-time fabric with wave vector ω G c propagates in the opposed direction to the deflected wave, see

a ) ω G c w R r s s ≈ r s s ω E c R b ) ω G ω E ≈ r s s 2 w R 2 (15)

For the Earth r s s 2 R 2 = 1.938138404 × 10 − 18 is as expected, a very small number. Equations (3) and (14) describe the reciprocal relationship of the light wave and the space-time fabric. Equation (15b) describes the combined affect of the space-time curvature acting on the light wave and the energy of the light wave acting on the space-time curvature.

The electromagnetic energy density is employed for deriving a Hamiltonian of the electromagnetic mode and space-time fabric modes in second quantized form. The electromagnetic energy density has a quadratic form in the components F μ ν of the electromagnetic field tensor. Therefore, in order to derive the Hamiltonian, a quadratic form of a function of the small Schwarzschild ratio r s s R is obtained from a Taylor series expansion of the gravitational time dilation [

The other General Relativistic affect used, is that clocks run slower when closer to the center of gravity of a mass than at larger distances from the mass [

This change of frequency can be calculated from the time dilation. The time dilation was calculated by Einstein [

a)

Equation (16c) can also be calculated from the dilation of the oscillating period of a harmonic oscillator derived in “Relativistic Harmonic Oscillator” by Kirk T. McDonald [

This is the quadratic form of the function of the small Schwarzschild ratio

The smaller the distance R, the closer the light wave is to the center of gravity of a mass, and the stronger is the affect of gravity acting on the light wave. The smaller the distance R, the lower the oscillating frequency

Here

Substituting Equation (15b) into Equation (17) one obtains for the frequency ω of the light wave near the surface of an Earth like mass:

Putting the last term

The constant w in Equations (18) and (19) will have to be evaluated from observations.

Next, a second quantized Hamiltonian is derived from the square root of the energy

the square root of the energy

the energy

and the energy term

The Quantum Electrodynamics models described in “Quantum Electronics” by Amnon Yariv [

where

where the form factors

The form factors

Photon mode operators

The state raising and state lowering operators, which operate on the light Photon and gravity.

Phonon number state wave functions

The light Photon operators

The electromagnetic energy density u is described by John David Jackson [

where MKS units were used. The electromagnetic field tensor with components

Substituting Equations (24) into Equation (B3) of Appendix B for

Equation (26) represents the three terms of Equations (17) and (18a). Equation (26) contains three sub Hamiltonian operators, an Electromagnetic wave Hamiltonian operator

Equation (27b) implies that each Photon interacts with

For an Earth-like mass ^{8}. The quantity

The Interaction of the electromagnetic wave and gravity wave are restricted to the volume where both are present. This is the Interaction volume

Quantum mechanics is a probabilistic model of Nature. The probability that a light Photon is annihilated and

The total energy and momentum before and after the interaction must also be conserved.

where

Equation (31) is in agreement with Equation (17).

Fermi’s Golden Rule is employed to calculate the average rate

Here

The Bose-Einstein distribution function [

The following data is used:

^{15} radians per second, the frequency at the peak of the black body radiation curve for a solar temperature

^{11} m or 1.733526 A.U.

^{8} calculated from Equations (28c).

The top of the Earth atmosphere receives a Solar radiation energy density of 1361 W/m^{2}. This results in a Photon density ^{13} Photons/m^{3}.

Setting the value of the constant w equal to 1. By substituting the numerical values into equation 34 one obtains:

Einstein [

Einstein also postulated the existence of wave modes in the space-time fabric. This has been verified by the LIGO experiment [

Slightly curved space-time fabric implies that the escape velocity of a test mass from the surface of a mass under consideration is much less than the velocity of light. It also implies that the space-time fabric mode caused by the mass can expand with the velocity of light to a size that is much larger than the mass.

The curvature of space-time deflects an electromagnetic wave. The energy of the electromagnetic wave also contributes to the curvature of space-time. Thus, there is a reciprocal relationship between an electromagnetic wave mode and space-time fabric modes. Second quantization is employed to formulate the space-time fabric energy quanta. Each mode can have any number of energy quanta called gravity Phonons. The Phonon number state wave functions form a Hilbert space. The gravity Phonons propagate with the speed of light and are massless Bosons. Since the gravity Phonons are tensor waves, they have spin angular momentum 2. The shape of the deformation of space-time and the Photon frequency

As of this date, 29 October 2019, there now exists quantized models for all four forces, a Gravity Force for slightly deformed space-time fabric, the Electromagnetic Force, the Weak Nuclear Force and the Strong Nuclear Force of the standard model of Physics. The gravity Phonons can, among other applications, be used to calculate the Statistical Thermodynamic quantities of the space-time fabric such as its average energy, entropy, pressure, and temperature (2.72503˚K), etc. The space-time mode quanta have only an energy of 0.59893 atto eV and the thermal energy of the Cosmic Background Radiation (CBR) in the Universe at a temperature of 2.72503˚K has a thermal energy of 0.23482478 meV. Therefore, the ratio of the space-time mode quanta energy of equation 29 to the thermal energy of the CBR is only 2.55054 × 10^{−15}. Thus, the Ideal Gas Law should hold for the Universe modeled as a gas, with the galaxies as molecules.

It should be possible to verify this model by observations since its derivation is based on astronomical observations and experiments.

Each gravity Phonon has an energy of only 0.6 atto eV. The annihilation of a light Photon generates a shower of 36,000,000 identical space-time modes. Each space-time mode can have any number of energy quanta gravity Phonons. This interaction has two forms. The Photon can generate 36,000,000 space-time modes, or the 36,000,000 space-time modes can assemble and create a Photon. Quantum Mechanics is a probabilistic model of Nature. The probability of a Photon generating a large number of gravity Phonons is much larger than the probability of a large number of gravity space-time modes assembling and form a Photon. Therefore, this process exhibits a Unique Arrow of Time. That is, the Unique Arrow of Time is a consequence of a large number of gravity Phonons necessary to create a light Photon because gravity is such a weak force.

A quantized model of space-time modes and their properties for slightly curved space-time fabric only has been derived in this paper.

I thank my wife Marlene Danzig Kornreich for her suggestions to the text, and for making the text more understandable to a reader. I also thank her for editing and working together on this manuscript.

The author declares no conflicts of interest regarding the publication of this paper.

Kornreich, P. (2019) Light Induced Gravity Phonons. Journal of Modern Physics, 10, 1674-1695. https://doi.org/10.4236/jmp.2019.1014110

The electromagnetic field tensor described here is only valid for gravity-free space. However, since small terms describing slightly curved space-time fabric are employed with this field tensor, and any slightly curved space-time fabric affects included in the field tensor would result in a second order affect, which is neglected. The contravariant Electromagnetic field tensor with components

The electromagnetic stress tensor with components

Substituting Equations (A1) into Equation (A2)

Contracting the electromagnetic stress tensor

where the time-like unit vector

The electromagnetic stress tensor takes the form:

Substituting Equations (21) into Equation (25b) to obtain the total Hamiltonian H operator.

Multiplying out Equation (B1)

Collecting terms.