_{1}

^{*}

Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a model of space-time curvature modes and associated curvature quanta in slightly warped space-time generated by a light Photon is derived. Both a Schr ödinger and a Second Quantized representation of the space-time curvature mode quanta are calculated and are fourth rank tensors. The eigenvalues of these equations are radii of curvature, not energy. The Eigenfunctions are linear functions of the components of the tensor that describes the curvature of space-time.

Consistent with Einstein’s [

The space-time curvature modes called gravity Phonons [

The reason we don’t feel or hear the space-time curvature modes is that the curvature is very slight. Even the Sun with its huge mass produces only a slight curvature in space-time. If a sphere is fitted to the space-time curvature produced by the Sun, it would need a radius equal to the distance light travels in 5 days and 3.5 hours. The radius of this sphere is the radius of curvature that characterizes the modes.

Einstein [

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.

Because this space-time curvature mode model is only valid for slightly warped space-time, it is only scale-invariant within a limited range of space-time curvature amplitudes.

The space-time curvature mode quanta are fourth rank tensors, and massless Bosons, that propagate with the speed of light. Light Photons are vector components derived from the Maxwell Electromagnetic Theory and have a spin angular momentum of 1. The space-time curvature modes are fourth rank tensors derived from the GRT and have spin angular wave vector of 4. The example of gravity Phonons calculated here has a very large radius of curvature of

Note, that this is only a mathematical model that is valid for a limited range of Nature, and in this region of Nature it is only an approximation.

The gravity Phonons, like other wave quanta, can form entangled states [

The eccentric motion of moons and planets also produces space-time wave modes. For example, the space-time wave mode generated by the Moon’s eccentric orbit has a period of 27 days 7 hours, 12 minutes, and a wavelength of 7.071264665 × 10^{14} m or 27.3 Light Days. Because the wavelength is much larger than the 2.667 light seconds size of the system, the motion of the Lunar system can be modeled by Newtonian Classical Mechanics.

In the late part of the 19’th century it was thought that like sound, which is a mode of the atoms in a material, electromagnetism is a mode of a substance that was named Luminiferous Aether. The Michelson-Morley experiment [

The electromagnetic wave discussed in this paper is affected by the curvature of space-time, and the electromagnetic wave affects the curvature of space-time. If the Michelson-Morley experiment could have been performed at a higher accuracy, the interaction of the light and space-time would have been discovered. Light propagates differently in different directions due to the Earth not being an ideal sphere, and also due to the gravity effects of the Moon, the Sun, Jupiter, and the effect of other solar system objects. These effects are described by the GRT.

Sakharov [

Bryce C. DeWitt [

There are a large number of publications describing unsuccessful attempts to develop models of Quantized Gravity called Gravitons [

Professor Wytler C. dos Santos [

Professor Rovelli [

The Quantum Mechanical model of Gravity derived in this paper is based on the GRT which has been verified by observations and experiments rather than use postulated quantities.

Note: Numbers such as {8} in curly brackets denote equation numbers in the literature.

To describe the effect of the energy of the light wave on the curvature of space-time, Einstein’s Field Equation is used. In Einstein’s Field Equation, the Einstein tensor describes the curvature of space-time, and the Stress tensor describes the effect of the light wave energy density. The derivation is for slightly warped space-time only. The lightwave does not interact with flat isotropic Minkowski space to first order in the coordinate components.

The Metric tensor in curved space-time describes the transformation from one coordinate system to another coordinate system that is located at an infinitesimal distance away. The two coordinate systems are tilted with respect to each other. Thus, the Metric tensor describes the local tilt of the two coordinate systems. The components of the Metric tensor describe the effect of mass and energy. This is the curvature of space-time. Einstein called the Metric Tensor the Fundamental Tensor.

The components of the Metric tensor for isotropic Minkowski space are only constant for rectangular coordinates. For curvilinear coordinates, for the same isotropic space-time, the Metric tensor has components that are functions of the coordinates. Therefore, this derivation is restricted to rectangular coordinates. After the calculation, the result can be transformed into any arbitrary coordinates. The metric tensor has the following properties:

The Metric tensor components

The tensor components

A four-dimensional Riemann space with a signature _{4} and area S_{3} of a four-dimensional sphere is

where _{3} in four-

dimensions has dimensions of volume in three-dimensional space. Here R_{4} is the radius of the four-dimensional sphere which is not the same as the radius R of

the three-dimensional sphere. The Schwarzschild ratio

mass is 1.390705726 × 10^{−9}. The ratio of the radius R_{4} of a four-dimensional sphere to the radius R of the corresponding three-dimensional sphere is constant for all sizes of the sphere. Therefore, the Schwarzschild ratio is the same for the three or four-dimensional sphere.

For a mass M, modeled as a point mass, the radius r_{ss} is the distance from the center of the mass M to the point at which a test mass m has to have an escape velocity equal to the speed of light c. This distance is the Schwarzschild radius r_{ss}.

where

The g_{00} component of the Metric tensor can be approximated by the effect of the time dilation calculated by Einstein [_{00} from equation {70}. This equation for {g_{00}} is also the Schwarzschild metric [

where

mass. The distance vector with components

Einstein’s Field equation [

where

dimensions of curvature

The GRT has always given results in agreement with observations and experiments. Therefore the effect of the energy of the light wave on the curvature of space-time, predicted by the GRT must also be correct.

The Stress tensor has two parts. A part due to the energy density u_{E} of the light beam and a part due to the energy density u_{EM} of the interaction of the light beam and the mass.

The part of the Stress tensor that depends only on the energy density of the light beam is the Electromagnetic Stress tensor calculated in Appendix A. Without losing generality one can assume that the light electric field is aligned along the x_{1} axis and it can be assumed to be a plane wave. As is shown in Appendix A, Equation (A6), the Electromagnetic Stress Tensor [_{00}(Light), T_{30}(Light), T_{03}(Light), and T_{33}(Light) all equal, to within a plus or minus sign, to the Electromagnetic Energy density_{E} of the light beam is:

on top of the atmosphere, and _{00} = 1, e_{30} = −1, e_{03} = −1, e_{33} = 1, and all other components of this tensor are equal to zero. This term does not change with the direction of the light beam, it is symmetric, and depends on even powers of the coordinate components only. The effect of the Earth-like mass

Light electromagnetic wave with a Photon density n and wave vector

deflect it. The relativistic energy density

S_{3} of a four-dimensional sphere. Using Equation (4) for

By using Equation (8b) and the second term on the right of Equation (9c) to form the interaction energy density u_{EM} of the light beam and the Earth-like mass:

The energy density u_{EM} of the interaction of the light beam and the space-time curvature modes are not symmetric and can be a linear function of the coordinates. The much smaller energy density u_{E} of the light beam does not depend on the direction of the light beam. It is symmetric and depends to second-order on the coordinates. It has been neglected.

The radius R_{4} of the four-dimensional sphere is calculated from the radius _{3} equal to the three-dimensional volume of the sphere. Substituting Equation (10b) into Einstein’s Field Equation, Equation (7).

The Einstein Tensor with components

The Ricci tensor with components

The first Riemann-Christoffel tensor contravariant index describes the index of the components ρ of the vector V. The second Riemann-Christoffel tensor covariant index describes the turning point number μ. The row and column numbers ν and τ of the rotation matrices at the turning points provide another two covariant Riemann-Christoffel tensor indices. Thus, the Riemann-Christoffel tensor

The Riemann-Christoffel tensor is also equal to the commutation relation of two covariant derivatives [

This description has too much unnecessary data to just describe the curvature of the space-time. Certainly, the initial direction of the vector V with component indices ρ is irrelevant. It is, also, sufficient to describe the new direction acquired by the vector V at each turning point by a vector with index ν or τ, rather than describing the turning process by a matrix with indices ν and τ. This leaves the tensor describing the curvature of space-time with just two indices, μ and ν. Thus, the Riemann-Christoffel tensor describing the curvature of space-time can be reduced to a second-rank tensor, the Ricci tensor with indices μ and ν. The conversion of the fourth-rank Riemann-Christoffel tensor to the second-rank Ricci tensor, with components

The Riemann-Christoffel tensor was derived in Reference [

The Christoffel symbol

ratio

consisting of products of Christoffel symbols are of the order of 1.934 × 10^{−18}, and thus can be neglected which simplifies the calculation. The Christoffel symbols

where

The trace of the Ricci tensor can be calculated from Equation (16)

Substituting Equation (16) and Equation (17) into Equation (12).

Equation (18) for Einstein’s tensor can be written in terms of a fourth-rank tensor with components

Since the gradient vector with components _{33} component of the Einstein tensor derived from Equation (18) is:

After multiplying out Equation (20) all derivatives have to be moved to the left like in Equation (19). The differentiation operators

Equation (21) has dimensions of energy density. Equation (21b) is in the form of a Hamiltonian. The canonical variables are the momentum components

The terms

Equation (21b), derived from the GRT, bears a resemblance to a tensor form of Schrödinger’s equation. Here

curvature in German. By multiplying Equation (21b) by

tensor Schrödinger like equation for curvature instead of energy. The equation instead of having an energy Eigenvalue, as the conventional Schrödinger

equation, this equation has a curvature Eigenvalue

No summation over subscripts in brackets is not implied. This derivation of a Quantum Mechanical formulation is only valid for slightly curved space-time. Since this is a straightforward derivation from Einstein’s Field Equation no Gauge requirements are necessary. From Equation (21) the square of the reciprocal of the radius of curvature R_{EM} is:

where _{EM} = 5769.812126 Light Years.

Dividing Equation (21b) by

The

Similar to the derivation of the quantization of the Electromagnetic Field by G. M. Wysin [

The state raising and state lowering operators are dimensionless curvature operators. Here

Substituting Equation (27) into Equation (25) and dividing the result equation by

where

The gravity Phonon ground state from Equation (31) is:

The tensor wave functions that describe the components of the Metric tensor can be calculated from the Curvature Hamiltonians of Equation (22) and Equation (31). The Metric tensor components describe the curvature of space-time due to the Earth-like mass and the energy of the light beam.

Both a Schrödinger and second quantized formulations of space-time curvature quanta are derived. The tensor wave functions that describe the components of the Metric tensor can be calculated from the Curvature Hamiltonians. The Metric tensor components describe the effect of the Earth-like mass and the energy of the light beam.

Einstein [

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

A model of space-time curvature modes in slightly warped space-time generated by a light Photon is derived. Since an electromagnetic wave does not interact with isotropic space-time, to first order, an Earth-like mass is used to warp space-time to facilitate the Photon space-time curvature mode interaction. Each space-time curvature mode is a fourth-rank tensor and can have any number of curvature quanta, called gravity Phonons. Gravity Phonons have very large radii of curvature in agreement with gravity being a weak force. The gravity Phonons are fourth-rank tensors, massless Bosons, have a spin angular wave vector of 4, and propagate with the speed of light. Energy and momentum are conserved because curved space-time causes a change in energy and momentum of the electromagnetic wave, and the energy of the electromagnetic wave causes a change of the curvature of space-time. Because this model is only valid for slightly warped space-time, it is only scale invariant within a limited range of space-time curvature amplitudes.

A large number of publications describing unsuccessful attempts to develop models of Quantized Gravity called Gravitons have been published. A number of these models use Post Newtonian Maxwell’s Equations modified by gravitational potentials. These run into difficulty with Gauge Normalization resulting in terms that go to infinity. The other approach uses “Loop Quantum Gravity”. It uses a lattice of triangles with Plank distance and Plank time sides to calculate the gravity energy quanta. This results in huge energy quanta amplitudes, not consistent with gravity being a very weak force. Therefore, in order to derive a Quantum Mechanical model of Gravity, one has to calculate space-time curvature mode quanta rather than use Gravitational potentials or Plank space and time to formulate gravity energy quanta.

Space-time is only slightly warped throughout most of the Universe except in the vicinity of Black Holes and Neutron Stars. For example, space-time at the surface of the sun has a radius of curvature of approximately 5.165 Light Days. Thus, gravity Phonons are a suitable Quantum Mechanical description of most of space-time in the Universe.

The space-time curvature quanta, like electromagnetic wave quanta, can form entangled states [

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. (2020) Space-Time Curvature Mode Quanta. Journal of Modern Physics, 11, 1977-1992. https://doi.org/10.4236/jmp.2020.1112125

The electromagnetic field tensor described here is only valid for gravity-free space. Any additions to the electromagnetic field tenser due to the effect of the Earth-like mass are small parameters. When these small parts of the electromagnetic Tensor are multiplied by the small terms describing the space-time curvature one obtains very small second-order terms that can be neglected. The

The contravariant Electromagnetic field tensor with components

and where

Substituting Equation (A2) and Equation (A3) into Equation (A1a)

The Electric field and the Magnetic Flux density pseudo vectors of a Plane-wave propagating in the 3 direction are:

where the Electric field vector is in the 1 direction. Substituting Equation (A5) into Equation (A4)

where ε_{0} is the dielectric constant of free isotropic space.