# Vector theory of gravity: solution of dark energy problem

Anatoly A. Svidzinsky  Department of Physics & Astronomy, Texas A&M
University, College Station TX 77843-4242

###### Abstract

We propose an alternative theory of gravity which assumes that background
geometry of the Universe is fixed four dimensional Euclidean space and gravity
is a vector field $A_{k}$ in this space which breaks the Euclidean symmetry.
Direction of $A_{k}$ gives the time coordinate, while perpendicular directions
are spatial coordinates. Vector gravitational field is coupled to matter
universally and minimally through the equivalent metric $f_{ik}$ which is a
functional of $A_{k}$. We show that such assumptions yield a unique theory of
gravity, it is free of black holes and to the best of our knowledge it passes
all available tests. For cosmology our theory predicts the same evolution of
the Universe as general relativity with cosmological constant and zero spatial
curvature. However, the present theory provides explanation of the dark energy
as energy of longitudinal gravitational field induced by the Universe
expansion and yields, with no free parameters, the value of
$\Omega_{\Lambda}=2/3\approx 0.67$ which agrees with the recent Planck result
$\Omega_{\Lambda}=0.686\pm 0.02$. Such striking agreement with cosmological
data indicates that gravity has a vector, rather than tensor, origin. We
demonstrate that gravitational wave signals measured by LIGO are compatible
with vector gravity. They are produced by orbital inspiral of massive neutron
stars which can exist in present theory. We also quantize gravitational field
and show that quantum vector gravity is equivalent to QED and, thus, it is a
renormalizable theory. Vector gravity can be tested by making more accurate
measurement of time delay of radar signal traveling near the Sun, by improving
accuracy of the light deflection experiments or measuring polarization of
gravitational waves. Resolving supermassive object at the center of our Galaxy
with VLBA could provide another test of gravity and also shed light on the
nature of dark matter.

###### pacs:

04.20.-q, 04.50.Kd, 95.30.Sf

###### Contents

  1. I Introduction
  2. II Postulates of the vector theory of gravity
  3. III Equivalent metric
  4. IV Action for gravitational field
  5. V Equations for classical gravitational field
  6. VI Static gravitational field
  7. VII Post-Newtonian limit
  8. VIII Weak field limit
    1. VIII.1 Linearized gravitational field equations
    2. VIII.2 Energy density and energy flux in the classical limit
    3. VIII.3 Gravitational waves
  9. IX Cosmology
  10. X Contents of the Universe
  11. XI Quantization of Gravitational Field
    1. XI.1 Classical electrodynamics
    2. XI.2 Quantization of electromagnetic field in elementary photon theory
    3. XI.3 Photon as composite particle
      1. XI.3.1 Field quantization in the composite photon theory
    4. XI.4 Quantization of gravitational field
  12. XII Radiation of gravitational waves by system of masses
  13. XIII Neutron star mass limit in vector gravity
  14. XIV Tests of the theory of gravity
  15. XV Galactic centers and dark matter problem
  16. XVI Summary
  17. A Derivation of the equivalent metric
  18. B Derivation of gravitational field action
    1. B.1 Weak field limit
    2. B.2 Arbitrary gravitational field
  19. C Equations for gravitational field
  20. D Motion of particles in gravitational field
  21. E Motion of particles in static gravitational field
  22. F Equations for metric in post-Newtonian limit
  23. G Analogy of weak gravity in the classical limit with electromagnetism in medium with negative refractive index
  24. H Energy density and energy flux for weak classical gravitational field
  25. I Cosmological suppression of preferred frame and preferred location effects

##  I Introduction

A century ago, Albert Einstein completed the general theory of relativity
Eins15 . Einstein’s theory then became an accepted theory of gravity. In
general relativity the space-time geometry $g_{ik}$ (metric tensor) is the
gravitational field described by the action

$S_{\text{GR}}=-\frac{c^{3}}{16\pi G}\int
d^{4}x\sqrt{-g}g^{ik}R_{ik}-\int\rho\sqrt{g_{ik}\frac{dx^{i}}{dt}\frac{dx^{k}}{dt}}d^{4}x,$
(1)

where $G$ is the gravitational constant, $c$ is the speed of light and
$R_{ik}$ is the Ricci tensor. The second term in Eq. (1) describes interaction
between gravitational field and matter with the rest mass density
$\rho(t,\mathbf{r})$. Variation of the action (1) with respect to $g_{ik}$
yields Einstein equations

$R_{ik}=\frac{8\pi G}{c^{4}}\left(T_{ik}-\frac{1}{2}g_{ik}T\right),$ (2)

where $T_{ik}$ is the energy-momentum tensor of matter. Einstein equations (2)
are a consequence of the postulate that space-time geometry $g_{ik}$ is
gravitational field.

One should mention that so far general relativity was accurately tested only
at weak gravitational field Will93 ; Will06  and thus it is not a theory fully
confirmed experimentally. Observations of binary pulsars yet have not provided
an accurate test of general relativity at strong gravity. Even though neutron
stars in the binary systems are relativistic objects, they are sufficiently
well-separated and all aspects of their orbital behavior and gravitational
wave generation in general relativity are characterized only by their net
masses and angular momentum. As a result, observations of binary pulsars
tested Einstein equations for the weak time-dependent field and also the
equivalence principle which guarantees effacement of the bodies (relativistic)
internal structure. One should note, however, that strong internal
gravitational fields of neutron stars can affect orbital dynamics and
gravitational wave generation in alternative theories of gravity that violate
the strong equivalence principle Will93 ; Will06 ; Will14 . Only in this sense
observations of binary pulsars is a test of strong gravity.

Recent direct detection of gravitational waves from a binary “black hole”
merger by the LIGO team Abbo16 ; Abbo16a  is also not an accurate test of
strong gravity. Such detection was unable to constrain higher-order post-
Newtonian parameters with a reasonable accuracy Abbo16b ; Abbo16c . Obtained
bounds on relative deviations in the post-Newtonian parameters are of the
order of $O(1)$. We argue in Sec. XIV of this paper that the LIGO signal can
be interpreted in the framework of vector gravity as being produced by a
merger of massive neutron stars, rather than black holes, which yields
radiation waveform compatible with LIGO data. We also demonstrate that stable
neutron stars with a simple linear “causal” equation of state can have masses
upto about $35$M⊙ in vector gravity (see Sec. XIII). Neutron star mass can be
much larger if “causality” constraint on the equation of state is not imposed.
Moreover, as we show in Sec. XV, in vector gravity, compact objects composed
of dark matter can have masses exceeding billions solar masses.

In 1998, published observations of Type Ia supernovae by the High-Z Supernova
Search Team Ries98  followed in 1999 by the Supernova Cosmology Project Perl99
suggested that the expansion of the universe is accelerating. Since then,
these observations have been corroborated by several independent sources.
Measurements of the cosmic microwave background, gravitational lensing, and
the large-scale structure of the cosmos as well as improved measurements of
supernovae have been consistent with the Lambda-CDM model, which is the
current standard model of cosmology. According to this model, a mysterious
dark energy gives the main contribution to the Universe composition. Work done
in 2013 based on the Planck spacecraft observations of the cosmic microwave
background gave the most accurate estimate of about 68% of dark energy in the
Universe Planck14 .

There are several unwanted issues about general relativity. E.g., it is not
compatible with quantum theory; it can not explain why Universe is spatially
flat; it does not provide a mechanism of matter generation at the Big Bang;
and it can not explain the value of the cosmological term (mysterious dark
energy). It also predicts existence of singularities such as black holes when
a massive star collapses into a point with zero volume and infinite matter
density. One can argue that general relativity becomes invalid in the vicinity
of singularities and a quantum theory of gravity will remove them. In
contrast, the present theory is free of such singularities at the classical
level. Namely, the end point of a gravitational collapse is not a point
singularity but rather a stable star with a reduced mass. One should mention
that black holes have never been observed directly and “evidences” of their
existence are based on the presumption that general relativity describes
gravity for strong field. Until signatures of the event horizons are found the
existence of black holes will not be proven.

Here we propose an alternative theory of gravity which is a Lagrangian-based
vector field theory in fixed four dimensional Euclidean space. The present
vector theory is a metric theory of gravity Will93  which means that space-
time is endowed with a symmetric equivalent metric $f_{ik\text{ }}$ formed out
of the vector field and Euclidean metric. Matter and nongravitational fields
respond only to the space-time metric $f_{ik\text{ }}$. The world lines of
test bodies are geodesics of that metric and in local freely falling frames
the nongravitational laws of physics are those of special relativity. Our
theory is prior-geometric for it contains the fixed background Euclidean
geometry (Euclidean metric) and gravity is a dynamical four-vector field in
this geometry which generates the space-time curvature (equivalent metric
$f_{ik\text{ }}$).

We show that vector theory of gravity passes all available tests. At strong
field our theory substantially deviates from general relativity and yields no
black holes. For cosmology the present theory gives the same evolution of the
Universe as general relativity with cosmological constant and zero spatial
curvature. However, zero spatial curvature of the Universe is a solution of
our equations, while in general relativity the spatial curvature is a free
parameter. Moreover, the vector theory of gravity yields, with no free
parameters, the value of the cosmological constant
$\Omega_{\Lambda}=2/3\approx 0.67$ which agrees with the recent Planck result
$\Omega_{\Lambda}=0.686\pm 0.02$ Planck14 . Such striking agreement is a
strong argument in favor of the vector nature of gravity.

Physical explanation of the dark energy (cosmological term) in our theory is
the following. Expansion of the Universe yields change of spatial scale with
time which can be viewed as an increase of the distance between masses. This
generates matter current directed away from an observer. Such current induces
longitudinal vector gravitational field in a similar way as electric current
creates vector potential in classical electrodynamics. Average energy of the
longitudinal gravitational field induced by the Universe expansion is the
mysterious dark energy. Contrary to matter, it has negative energy density and
accelerates expansion of the Universe.

Vector gravity also suggests a mechanism of matter generation at the Big Bang
without involving an additional hypothetical field (inflaton). Namely, matter
was created at the expense of production of negative energy gravitons and the
gravitational field itself caused the stage of inflation and heated up the
Universe. According to the vector gravity, the total energy of the Universe
(which includes the energy of matter and gravitational field) is equal to
zero.

As we show in Sec. XI, quantization of vector gravity yields quantum theory
which is equivalent to QED. Therefore, quantum vector gravity is a
renormalizable theory and, thus, it can make meaningful physical predictions
at the quantum level. Compatibility of vector gravity with quantum mechanics
is another strong argument in its favor.

A remarkable feature of our theory is that equations for gravitational field
can be solved analytically for arbitrary static mass distribution (see Sec.
VI). If point masses are located at $\mathbf{r}_{1}$, $\mathbf{r}_{2}$, …
$\mathbf{r}_{N}$ then exact solution for the equivalent metric is

$f_{ik}=\left(\begin{array}[]{cccc}e^{2\phi}&0&0&0\\\ 0&-e^{-2\phi}&0&0\\\
0&0&-e^{-2\phi}&0\\\ 0&0&0&-e^{-2\phi}\end{array}\right),$ (3)

where

$\phi(\mathbf{r})=-\frac{m_{1}}{|\mathbf{r}-\mathbf{r}_{1}|}-\ldots-\frac{m_{N}}{|\mathbf{r}-\mathbf{r}_{N}|}$
(4)

and $m_{k}$ ($k=1,\ldots,N$) are constants determined by the value of masses.
Solution (3) is free of black holes: photons with a radial velocity component
can always escape from gravitationally compact objects. In recent years, the
evidence for the existence of ultra-compact supermassive objects at centers of
galaxies has become very strong. It is important to note that present solution
(3) not only argues that such objects are not black holes, but also can
explain quantitatively their observed properties and give us a hint about
composition of dark matter (see Sec. XV and Ref. Svid07 ).

Before we proceed with building the vector theory of gravity we discuss an
algorithm that we use to construct the theory. Classical electrodynamics is an
example of a successful field theory which is very well tested. It postulates
that electromagnetic field is a four dimensional vector $A_{k}$ in Minkowski
space-time. The conserved $4-$current density $j^{k}$ is the source of the
electromagnetic field which is coupled to $A_{k}$ through the Lorentz
invariant term in the action

$S_{\text{coupl}}=-\frac{1}{c^{2}}\int d^{4}xA_{k}j^{k}.$ (5)

Such postulate allows us to construct classical electrodynamics in a unique
way using symmetries of Eq. (5). Namely, conservation of current yields that
$S_{\text{coupl}}$ is invariant under the gauge transformation
$A_{k}\rightarrow A_{k}+\partial\psi/\partial x^{k}$. Action for the
electromagnetic field $S_{\text{field}}$ must possess the same symmetries as
the coupling term $S_{\text{coupl}}$, namely, it must be Lorentz and gauge
invariant. Such a requirement, together with the condition that
$S_{\text{field}}$ must be quadratic in field derivatives yields a unique
answer for $S_{\text{field}}$

$S_{\text{field}}=-\frac{1}{16\pi c}\int d^{4}x\left(\frac{\partial
A_{k}}{\partial x^{i}}-\frac{\partial A_{i}}{\partial
x^{k}}\right)\left(\frac{\partial A^{k}}{\partial x_{i}}-\frac{\partial
A^{i}}{\partial x_{k}}\right).$ (6)

Variation of the total action of the system
$S=S_{\text{field}}+S_{\text{coupl}}+S_{\text{matter}}$ with respect to
$A_{k}$ yields Maxwell equations and variation with respect to the particle
trajectories gives the Lorentz force. Thus, the whole classical
electrodynamics is uniquely assembled from the structure of the coupling term
(5).

In the next sections we will use the same algorithm to construct the classical
vector theory of gravity. Namely, first we postulate how the gravitational
field is coupled to matter. Then, using symmetries of the coupling term, we
uniquely assemble the total action and obtain the classical field equations
from the variational principle. One should note that symmetries of the
coupling term in vector gravity are very different from the symmetries of Eq.
(5). As a consequence, the vector theory of gravity substantially differs from
classical electrodynamics.

When we make a transition from the classical to quantum electrodynamics we
must make an additional assumption, namely, that the quantum of the vector
filed $A_{k}$, the photon, is an elementary particle. This postulate yields
that photon is a boson because fermion field does not transform as a vector.
However, a product of two fermion fields can transform as a vector. E.g., a
four-vector field $A_{k}$ can be created from a fermion-antifermion pair as
Bjor65

$A_{k}=\Psi^{+}\gamma_{0}\gamma_{k}\Psi,$

where $\gamma_{k}$ are $4\times 4$ gamma matrices and $\Psi$ is the fermion
field. In this regard there is an interesting proposal that the photon is not
an elementary particle but rather a composite particle formed of fermion-
antifermion pairs (the so called composite theory of photon). The idea that
the photon is a composite particle dates back to 1932, when Louis de Broglie
DB32  suggested that the photon is composed of neutrino-antineutrino pairs.
Recall that many composite bosons, such as Cooper pairs, atoms with total
integer spin, deuterons, pions, and kaons, are not perfect bosons because of
their internal fermion structure, however in the asymptotic limit they are
essentially bosons Perk02 . This suggests that photon could be a particle
composed of spin$-1/2$ fermions as well. Work on the composite theory of
photon continues to be of some interest Dvoe01 ; Perk02 ; Sen07 ; Perk13 ;
Perk14 .

The present paper deals with the vector gravity, rather than photons. However,
as we show, the measured value of the energy loss by binary stars due to
emission of gravitational waves can be explained in the vector gravity only if
the quantum of the vector gravitational field (the graviton) is a composite
particle formed of fermion-antifermion pairs. This fact makes a link between
vector gravity and the composite theory of photon. We quantize gravitational
field in Sec. XI assuming that graviton is a composite particle and obtain a
remarkable result that quantum vector gravity is equivalent to QED.

One of the motivations for the composite photon theory is lack of a well
defined wave function for a single photon. In particular, the authors of the
classic book on Quantum Optics Scul97  say: “There is, strictly speaking, no
such a thing as a photon wave function”. If this is the case, in order for QED
to be compatible with first quantization the photon can not be an elementary
particle, but rather a particle composed of fermions. Recall that for
fermions, described by the Dirac equation, the wave function is well defined.
Hence, the assumption that photon is composed of fermions seems natural. For
QED such assumption yields essentially no detectable experimental consequences
(see Sec. XI and Ref. Perk02 ). However, as we show in Sec. XI, for quantum
vector gravity the consequences are dramatic.

##  II Postulates of the vector theory of gravity

The present vector theory of gravity is based on four postulates:

1. Background geometry of the Universe is a fixed four dimensional Euclidean space with metric $\delta_{ik}=$diag$(1,1,1,1)$. Such space is completely isotropic and has no preferred directions.

2. In the four dimensional Euclidean space there is a dynamical $4-$vector field $A_{k}$ (the gravitational field) which breaks the symmetry. Namely, direction of $A_{k}$ is now preferred and this direction becomes the time coordinate. Directions perpendicular to $A_{k}$ are three spatial coordinates.

3. Vector gravitational field is coupled to matter and all nongravitational fields through the equivalent metric $f_{ik}$ which is an algebraic function of $A_{k}$ and the background Euclidean metric $\delta_{ik}$. Gravitational field couples universally and minimally to all the fields of the Standard Model by replacing everywhere the Minkowski metric $\eta_{ik}$ with the equivalent metric $f_{ik\text{ }}$ and replacing partial derivatives with covariant derivatives formed from $f_{ik}$. In particular, the trajectories of freely falling bodies are geodesics of the equivalent metric $f_{ik\text{ }}$. Action for a point particle with mass $m$ moving in the gravitational field reads

$S_{\text{matter}}=-mc\int\sqrt{f_{ik}dx^{i}dx^{k}},$ (7)

where $c$ is the speed of light. Action (7) has the same form as in general
relativity, however, the tensor gravitational field $g_{ik}$ of general
relativity is now replaced with the equivalent metric $f_{ik}$. One should
note that the Einstein equivalence principle is a consequence of the action
(7).

4\. The quantum of the vector field $A_{k}$ (the graviton) is not an
elementary particle, but rather it is a composite particle formed of massless
fermion-antifermion pairs. Emission and absorption of a graviton corresponds
to creation and annihilation of particle-antiparticle pairs.

For construction of the classical equations for the vector field $A_{k}$ it
does not matter whether the graviton is an elementary particle or a composite
particle. However, processes involving gravitons, e.g. emission of
gravitational waves, might depend on the graviton composition. The present
vector theory of gravity can quantitatively explain the energy loss by binary
stars orbiting each other provided that graviton is a composite particle.

The postulates 1-3 outlined above allow us to construct the classical vector
theory of gravity in a unique way using symmetries of the action
$S_{\text{matter}}$ (7). Such symmetries uniquely specify the structure of the
total action. We proceed with assembling the classical theory in the next
sections.

##  III Equivalent metric

In Appendix A we show that Einstein equivalence principle yields the following
unique expression for the equivalent metric $f_{ik}$ in terms of the vector
field $A_{k}$ and the background Euclidean metric $\delta_{ik}$

$f_{ik}=-\frac{\delta_{ik}}{A}+\left(A+\frac{1}{A}\right)\frac{A_{i}A_{k}}{A^{2}},$
(8)

where

$A=\sqrt{A_{i}A_{k}\delta^{ik}}.$

Throughout the paper we use the usual conventions. Namely, unless otherwise
noted, there is summation over repeated indices. Lower case Latin indices
($i$, $k$, $l$, …) label four dimensional coordinates (range $0$, $1$, $2$,
$3$), while lower case Greek letters $\alpha$ and $\beta$ denote spatial
coordinates (range $1$, $2$, $3$).

In Cartesian coordinate system if we chose $x^{0}-$axis along the direction of
$A_{k}$ the equivalent metric is diagonal and reads

$f_{ik}=\text{diag}\left(A,-\frac{1}{A},-\frac{1}{A},-\frac{1}{A}\right).$

Since $A_{k}$ is a dynamical variable one can make a transformation
$A_{k}\rightarrow F(A)A_{k}$, where $F$ is an arbitrary function of $A$. Such
a transformation changes the norm of $A_{k}$. It also modifies the expression
for the metric (8) and field equations. However, the physical answer, e.g.,
motion of particles in gravitational field is independent of how we normalize
the vector field. Thus, we can choose the field normalization in any suitable
way.

Instead of $A_{k}$, it is convenient to introduce new independent functions, a
scalar $\phi$ and a unit vector $u_{k}$, according to the relations

$A=e^{2\phi},\quad u_{k}=\frac{A_{k}}{A}.$

The vector $u_{k}$ has the unit norm

$u_{i}u_{k}\delta^{ik}=1.$

In terms of the unit vector $u_{k}$ and the scalar $\phi$ the equivalent
metric (8) reads

$f_{ik}=-e^{-2\phi}\delta_{ik}+2\cosh(2\phi)u_{i}u_{k},$ (9)

while metric $\tilde{f}^{ik}$ inverse to $f_{ik}$, defined as
$\tilde{f}^{ik}f_{im}=\delta_{m}^{k}$, is

$\tilde{f}^{ik}=-e^{2\phi}\delta^{ik}+2\cosh(2\phi)u^{i}u^{k},$ (10)

where

$u^{i}=\delta^{ik}u_{k}.$

In the present paper raising and lowering of the indices is performed using
Euclidean metric $\delta_{ik}$, unless otherwise stated. We denote determinant
of $f_{ik}$ as $f$. Equation (9) yields

$\sqrt{-f}=e^{-2\phi}.$

One should note that in the literature there were attempts to construct a
vector theory of gravity in background Minkowski (rather than Euclidean)
metric $\eta_{ik}=$diag$(1,-1,-1,-1)$ by Rastall Rast75 ; Rast77  and by the
author Svid09 . The equivalent metric obtained in such theories has a form
similar to our Eq. (9) in which Euclidean metric $\delta_{ik}$ is replaced
with $-\eta_{ik}$ and $\cosh(2\phi)$ is replaced with $\sinh(2\phi)$. However,
the main achievement of the present paper compared to the previous work on
vector gravity is the discovery of the correct way of building the action for
the gravitational field $S_{\text{gravity}}$ based on symmetries of the
coupling term $S_{\text{matter}}$. In the previous attempts to construct the
theory $S_{\text{gravity}}$ has not possessed symmetries of
$S_{\text{matter}}$ which yielded no success.

##  IV Action for gravitational field

Next we construct action of the system in terms of $\phi$ and $u_{k}$. The
total action for the gravitational field and matter is given by

$S=S_{\text{gravity}}+S_{\text{matter}},$ (11)

where $S_{\text{matter}}$ is the action of matter written in curvilinear
coordinates with the metric $f_{ik}$. We obtain the action for the
gravitational field $S_{\text{gravity}}$ using the requirement that symmetries
of $S_{\text{matter}}$ and $S_{\text{gravity}}$ must be the same.
$S_{\text{matter}}$ possesses the following symmetry: it is invariant under
coordinate transformations that leave background Euclidean metric
$\delta_{ik}$ intact. This symmetry is exact. In addition there are
approximate symmetries. For small deviations of $\phi$ from a constant value
$\phi_{0}$ and small deviations of $u_{k}$ from $(1,0,0,0)$ in the rescaled
coordinates

$x^{0}\rightarrow e^{-\phi_{0}}x^{0},\qquad x^{\alpha}\rightarrow
e^{\phi_{0}}x^{\alpha}$ (12)

the equivalent metric is given by

$f_{ik}=\eta_{ik}+\left(\begin{array}[]{cccc}h_{00}&h_{01}&h_{02}&h_{03}\\\
h_{01}&h_{00}&0&0\\\ h_{02}&0&h_{00}&0\\\
h_{03}&0&0&h_{00}\end{array}\right),$ (13)

where

$h_{00}=2(\phi-\phi_{0}),\qquad h_{0\alpha}=2\cosh(2\phi_{0})u_{\alpha},$ (14)

$\alpha=1$, $2$, $3$ and $\eta_{ik}=$diag$(1,-1,-1,-1)$ is the Minkowski
metric. For small deviations of $f_{ik}$ from the Minkowski metric we obtain
Land95

$\delta S_{\text{matter}}=-\frac{1}{2c}\int d^{4}x\sqrt{-f}T^{ik}\delta
f_{ik}\approx$ $-\frac{1}{2c}\int
d^{4}x\left(T^{00}h_{00}+2T^{0\alpha}h_{0\alpha}+T^{\alpha\alpha}h_{00}\right),$
(15)

where $T^{ik}$ is the energy-momentum tensor of matter.

One can see that in the rescaled coordinates the action $S_{\text{matter}}$ is
independent of the background cosmological field $\phi_{0}$. This is one of
the symmetries of $S_{\text{matter}}$. Another symmetry can be found by taking
into account Eq. (15) and approximate energy conservation in the Minkowski
metric

$\frac{\partial T^{00}}{\partial x^{0}}+\frac{\partial T^{0\alpha}}{\partial
x^{\alpha}}=0$

which yields that action $S_{\text{matter}}$ is approximately invariant under
the gauge transformation

$h_{00}\rightarrow h_{00}+2\frac{\partial\psi}{\partial x^{0}},\quad
h_{0\alpha}\rightarrow h_{0\alpha}+\frac{\partial\psi}{\partial x^{\alpha}}$
(16)

upto the terms quadratic in the mass velocity $V$. Here $\psi$ is an arbitrary
scalar function.

In addition, there is approximate Lorentz invariance. Namely, the line element
that enters the matter action (7) can be approximately written in the metric
(13) as

$ds^{2}=f_{ik}dx^{i}dx^{k}\approx$
$\eta_{ik}dx^{i}dx^{k}+h_{00}(dx^{0})^{2}+2h_{0\alpha}dx^{0}dx^{\alpha}+h_{00}d\mathbf{r}^{2}$
(17)

which is invariant (upto the terms of the order of $V^{3}/c^{3}$) under
transformation

$x^{0}\rightarrow\left(1+\frac{V^{2}}{2c^{2}}\right)x^{0}+\frac{1}{c}\mathbf{V\cdot
r},\quad\mathbf{r}\rightarrow\mathbf{r}+\frac{\mathbf{V}}{c}x^{0},$ (18)
$h_{00}\rightarrow
h_{00}\left(1+\frac{2V^{2}}{c^{2}}\right)-2\frac{V^{\alpha}}{c}h_{0\alpha},\quad
h_{0\alpha}\rightarrow h_{0\alpha}-2\frac{V^{\alpha}}{c}h_{00},$ (19)

where $\mathbf{V}$ is a constant (velocity) vector and $\alpha=1$, $2$, $3$.
In Eq. (17) $h_{0\alpha}$ and $d\mathbf{r}/dx^{0}$ are of the order of $V/c$.

Requirement that $S_{\text{gravity}}$ must also possess these symmetries,
namely, $S_{\text{gravity}}$ should be invariant under Euclidean
transformations; for small deviations from the background field be independent
of $\phi_{0}$ after scaling transformation (12); and approximately invariant
under the gauge (16) and low-velocity Lorentz (18), (19) transformations,
allows us to find $S_{\text{gravity}}$ uniquely111As a matter of fact, to fix
$S_{\text{gravity}}$ uniquely one should also use condition of gauge symmetry
of the action in a higher post-Newtonian order. We discuss this in Appendix
B.. In Appendix B we obtain that the gravitational field action in the
background four dimensional Euclidean space is

$S_{\text{gravity}}=\frac{c^{3}}{8\pi G}\int
d^{4}x\left[\frac{\partial\phi}{\partial x^{i}}\frac{\partial\phi}{\partial
x^{k}}\left(-\delta^{ik}+\left(1-3e^{-4\phi}\right)u^{i}u^{k}\right)\right.$
$+\cosh^{2}(2\phi)\frac{\partial u_{i}}{\partial x^{k}}\frac{\partial
u_{m}}{\partial x^{l}}\Big{(}\delta^{im}\delta^{kl}-\delta^{il}\delta^{km}-$
$\left.\left(1+e^{-4\phi}\right)\delta^{im}u^{k}u^{l}\Big{)}+2\left(1+e^{-4\phi}\right)\frac{\partial\phi}{\partial
x^{i}}\frac{\partial u_{m}}{\partial x^{k}}\delta^{im}u^{k}\right].$ (20)

where $G$ is the gravitational constant.

Action (20) is written in Euclidean metric, it has no free parameters and
serves as a foundation of the present theory of gravity. Our derivation of the
action (20) is unique and, hence, the classical vector theory of gravity is
also a unique consequence of the postulates 1-3.

##  V Equations for classical gravitational field

Taking variation of the total action (11) with respect to $\phi$ and unit
vector $u_{k}$ yields four equations for the classical gravitational field in
the background Euclidean space (see Appendix C for derivation details)

$\left[\delta^{mk}u^{i}-2\delta^{im}u^{k}+\left(1+3e^{-4\phi}\right)u^{m}u^{k}u^{i}\right]\frac{\partial^{2}\phi}{\partial
x^{m}\partial x^{k}}$
$+2\left[\delta^{im}+\left(3e^{-4\phi}-1\right)u^{m}u^{i}\right]\frac{\partial\phi}{\partial
x^{m}}\frac{\partial\phi}{\partial x^{k}}u^{k}$
$+2\left[e^{4\phi}\left(\delta_{l}^{k}\delta^{im}-\delta_{l}^{i}\delta^{mk}\right)+\delta_{l}^{i}\delta^{mk}-\delta_{l}^{m}\delta^{ik}\right]\frac{\partial\phi}{\partial
x^{k}}\frac{\partial u^{l}}{\partial x^{m}}$
$+\left[2\left(e^{4\phi}-2e^{-4\phi}-1\right)\delta_{l}^{i}u^{m}u^{k}-\left(1+3e^{-4\phi}\right)\delta_{l}^{m}u^{i}u^{k}\right.$
$\left.-\left(2e^{4\phi}+3e^{-4\phi}+1\right)\delta_{l}^{k}u^{m}u^{i}\right]\frac{\partial\phi}{\partial
x^{k}}\frac{\partial u^{l}}{\partial x^{m}}$

$+\cosh(2\phi)\left[e^{2\phi}\frac{\partial}{\partial
x^{k}}\left(\frac{\partial u^{k}}{\partial x_{i}}-\frac{\partial
u^{i}}{\partial
x_{k}}\right)+e^{-2\phi}u_{m}u^{i}\frac{\partial^{2}u^{m}}{\partial
x_{k}\partial x^{k}}\right.$
$+\left.2\cosh(2\phi)u^{k}u^{l}\frac{\partial^{2}u^{i}}{\partial x^{l}\partial
x^{k}}-\left(e^{2\phi}+2e^{-2\phi}\right)u^{m}u^{i}\frac{\partial^{2}u^{k}}{\partial
x^{k}\partial x^{m}}\right]$ $+2\cosh^{2}(2\phi)\left[\frac{\partial
u^{i}}{\partial x^{k}}\frac{\partial}{\partial
x^{m}}\left(u^{k}u^{m}\right)-\frac{\partial u_{k}}{\partial
x_{i}}\frac{\partial u^{k}}{\partial x^{l}}u^{l}\right.$
$\left.-\frac{\partial u^{k}}{\partial x^{m}}\frac{\partial u^{m}}{\partial
x^{k}}u^{i}+\left(1+2e^{-4\phi}\right)\frac{\partial u^{k}}{\partial
x^{m}}\frac{\partial u_{k}}{\partial x^{l}}u^{m}u^{l}u^{i}\right]$
$=\frac{8\pi G}{c^{4}}\left(T^{ik}-\frac{T}{2}\tilde{f}^{ik}\right)u_{k},$
(21)

where $T^{ik}$ is the energy-momentum tensor of matter, $T=T^{mk}f_{mk}$ is
the trace of the energy-momentum tensor and $\tilde{f}^{ik}$ is the metric
inverse to $f_{ik}$ given by Eq. (10). The right hand side of Eqs. (21) is the
source of the gravitational field.

Equations (21) for the scalar $\phi$ and the unit vector $u_{k}$ are the main
equations of the vector theory of gravity. They are written in Euclidean
metric which means that raising and lowering of indexes is carried out using
metric $\delta_{ik}=$diag$(1,1,1,1)$. Equations (21) play the same role in
vector gravity as Einstein equations in general relativity.

In our theory the motion of particles in gravitational field is described by
the same equation as in general relativity

$\frac{d^{2}x^{b}}{ds^{2}}=\frac{1}{2}\tilde{f}^{bl}\left[\frac{\partial
f_{ik}}{\partial x^{l}}-\frac{\partial f_{lk}}{\partial x^{i}}-\frac{\partial
f_{il}}{\partial x^{k}}\right]\frac{dx^{i}}{ds}\frac{dx^{k}}{ds},$ (22)

where $ds=\sqrt{f_{ik}dx^{i}dx^{k}}$. In Eq. (22) the metric $g_{ik}$ of
general relativity is replaced with the equivalent metric $f_{ik}$. We obtain
equation of particle motion in Appendix D.

Next we explore solutions of the classical gravitational field equations (21)
for various cases.

##  VI Static gravitational field

In this section we consider gravitational field produced by rest matter
distributed with density $\rho(\mathbf{r})$. For static field
$u_{k}=(1,0,0,0)$, scalar $\phi$ depends only on spatial coordinates
$\mathbf{r}$ and the equivalent metric (9) reads

$f_{ik}=\left(\begin{array}[]{cccc}e^{2\phi(\mathbf{r})}&0&0&0\\\
0&-e^{-2\phi(\mathbf{r})}&0&0\\\ 0&0&-e^{-2\phi(\mathbf{r})}&0\\\
0&0&0&-e^{-2\phi(\mathbf{r})}\end{array}\right),$ (23)

$\sqrt{-f}=e^{-2\phi}$, while the inverse metric is

$\tilde{f}^{ik}=\left(\begin{array}[]{cccc}e^{-2\phi}&0&0&0\\\
0&-e^{2\phi}&0&0\\\ 0&0&-e^{2\phi}&0\\\ 0&0&0&-e^{2\phi}\end{array}\right).$
(24)

For static field the energy-momentum tensor of matter has only one nonzero
component $T^{00}$ which depends on spatial coordinates. This component can be
found from the conservation equation

$T_{;k}^{ik}=0,$ (25)

where $";"$ stands for the covariant derivative with metric $f_{ik}$. Equation
(25) yields $T^{00}=\rho c^{2}e^{\phi}$ and $T=\rho c^{2}e^{3\phi}$, where
$\rho$ is the mass density which is independent of $\phi$.

For static gravitational field, Eqs. (21) reduce to a single equation for
$\phi(\mathbf{r})$

$\Delta\phi=\frac{4\pi G}{c^{2}}\rho e^{\phi}.$ (26)

In the Newtonian limit Eq. (26) yields $\Delta\phi=4\pi G\rho
e^{\phi_{0}}/c^{2}$ and, thus, $c^{2}\phi(\mathbf{r})$ has a meaning of
gravitational potential.

Exponential metric solution (23) is free of black holes for any mass
distribution and field strength. For a point mass $M$ located at $r=0$ Eq.
(26) leads to $c^{2}\Delta\phi=4\pi GM\delta(\mathbf{r})$ and has a solution
$\phi=-GM/c^{2}r$. For $N$ point masses located at $\mathbf{r}_{1},$ …,
$\mathbf{r}_{N}$ Eq. (26) yields

$\Delta\phi=4\pi\left[m_{1}\delta(\mathbf{r}_{1})+\ldots+m_{N}\delta(\mathbf{r}_{N})\right],$
(27)

where $m_{1}$, …, $m_{N}$ are positive constants. Solution of Eq. (27) is

$\phi(\mathbf{r})=-\frac{m_{1}}{|\mathbf{r}-\mathbf{r}_{1}|}-\ldots-\frac{m_{N}}{|\mathbf{r}-\mathbf{r}_{N}|}.$
(28)

We discuss motion of particles in static gravitational field in Appendix E.
For a star of mass $M$ and radius $R$ Eq. (28) reduces to $\phi(r)=-GM/c^{2}r$
($r\geq R$) and using Eq. (220) for energy conservation we obtain that escape
velocity for a particle from the stellar surface is

$v=c_{s}\sqrt{1-e^{2\phi(R)}},$ (29)

where $c_{s}=ce^{2\phi(R)}$ is the speed of light at the stellar surface (see
Eq. (228)). Equation (29) shows that escape velocity is always smaller then
$c_{s}$ ($c_{s}\leq c$).

One should note that exponential metric (23) was also obtained for static
field in some alternative theories of gravity Yilm58 ; Yilm71 ; Rose71 ;
Chan80 ; Chan80b ; Rast77 ; Svid09  and based on simple physical arguments in
Rast75 ; Lind81 ; Mart10 . Stability of compact stars in the exponential
metric has been investigated in the literature. It has been shown that
solution (23) predicts that stars do not collapse into a point singularity but
rather form stable compact objects with no event horizon and finite
gravitational redshift Robe99 .

##  VII Post-Newtonian limit

Post-Newtonian limit applies when the gravitational field is weak, and the
motion of the matter is slow. It is sufficiently accurate to encompass all
solar-system tests of gravity performed so far. In the post-Newtonian
formalism the metric is expanded in a small parameter $\epsilon$. The “order
of smallness” is determined according to the rules that matter velocity is of
order $V\sim\epsilon^{1/2}$ and gravitational constant $G\sim\epsilon$. A
consistent post-Newtonian limit requires determination of $g_{00}$ correction
through $O(\epsilon^{2})$, $g_{0\alpha}$ through $O(\epsilon^{3/2})$, and
$g_{\alpha\beta}$ through $O(\epsilon)$ Will93 .

We compare the vector theory of gravity with general relativity in the post-
Newtonian limit in the cosmological reference frame (the mean rest frame of
the Universe in which the Universe appears isotropic) for which background
equivalent metric $f_{ik}$ is diagonal and after rescaling coordinates reduces
to Minkowski metric. Let us consider small deviations $h_{ik}$ of the tensor
gravitational field $g_{ik}$ from the Minkowski metric $\eta_{ik}$

$g_{ik}=\eta_{ik}+h_{ik}.$

In the post-Newtonian limit of general relativity Land95

$h_{\beta}^{\alpha}=-h_{00}\delta_{\beta}^{\alpha}$ (30)

and the tensor gravitational field $g_{ik}$ is described by four independent
functions $h_{0k}$, so that metric is given by

$g_{ik}=\eta_{ik}+\left(\begin{array}[]{cccc}h_{00}&h_{01}&h_{02}&h_{03}\\\
h_{01}&h_{00}&0&0\\\ h_{02}&0&h_{00}&0\\\
h_{03}&0&0&h_{00}\end{array}\right).$ (31)

In the present vector theory of gravity in the post-Newtonian limit after
rescaling of coordinates in the cosmological reference frame the equivalent
metric $f_{ik}$ also has the form of Eq. (31) (see Eq. (13)). Therefore, in
both theories in the post-Newtonian limit the gravitational fields are
described by an equal number of independent functions which are coupled with
matter in the same way. Since, by construction of both theories, such coupling
uniquely specifies the total action and, hence, the field equations, the
general relativity and the vector theory of gravity are identical in the post-
Newtonian limit.

To convince the reader that this is indeed the case, in Appendix F we show
directly that in the post-Newtonian limit the general relativity and the
vector theory of gravity give the same equations for $h_{0k}$ in the
cosmological reference frame

$\frac{1}{2}\Delta h_{00}+\frac{3}{2}\frac{\partial^{2}h_{00}}{\partial
x^{0}\partial x^{0}}-\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}+\frac{1}{2}h_{00}\Delta h_{00}-\frac{1}{2}\left(\nabla
h_{00}\right)^{2}$ $=\frac{8\pi
G}{c^{4}}\left(T_{00}-\frac{1}{2}g_{00}T\right),$ $\frac{1}{2}\Delta
h_{0\alpha}-\frac{1}{2}\frac{\partial^{2}h_{0\beta}}{\partial
x^{\alpha}\partial x^{\beta}}+\frac{\partial^{2}h_{00}}{\partial x^{0}\partial
x^{\alpha}}=\frac{8\pi G}{c^{4}}T_{0\alpha}.$

Boundary conditions for $h_{0k}$ are also the same and, therefore, both
theories are equivalent in the post-Newtonian limit.

We made comparison of the theories in the cosmological reference frame in
which equivalent metric of the vector gravity after rescaling coordinates has
the form of Eq. (31). Since both theories are equivalent in such a frame in
the post-Newtonian limit they are also equivalent in a frame moving with non
relativistic velocity relative to the mean rest frame of the Universe. This is
the case because in vector gravity the equivalent metric $f_{ik}$, by its
definition, is also a tensor under general coordinate transformations. If
equations of vector gravity and general relativity give the same metric in one
reference frame then in any other frame the two metrics will coincide because
both of them transform in the same way under coordinate transformation from
one frame to another.

As a consequence, the vector theory of gravity, similarly to general
relativity, yields no post-Newtonian preferred frame and preferred location
effects. For vector gravity the ten post-Newtonian parameters introduced to
compare metric theories of gravity with each other Will93 ; Will06  have the
same values as in general relativity.

##  VIII Weak field limit

###  VIII.1 Linearized gravitational field equations

In this section we linearize equations for classical gravitational field in
the cosmological reference frame assuming that unit vector $u_{k}$ slightly
deviates from $(1,0,0,0)$. For small deviations of $\phi$ from a constant
value $\phi_{0}$ and $|u_{\alpha}|\ll 1$, keeping linear terms, Eqs. (21)
yield

$\Delta\phi+3e^{-4\phi_{0}}\frac{\partial^{2}\phi}{\partial x^{0}\partial
x^{0}}-2e^{-2\phi_{0}}\cosh(2\phi_{0})\frac{\partial^{2}u^{\beta}}{\partial
x^{0}\partial x^{\beta}}$ $=\frac{8\pi
G}{c^{4}}\left(T^{00}-\frac{T}{2}\tilde{f}^{00}\right),$

$\cosh(2\phi_{0})\left(e^{2\phi_{0}}\frac{\partial^{2}u^{\beta}}{\partial
x_{\alpha}\partial x^{\beta}}-e^{2\phi_{0}}\Delta
u^{\alpha}+e^{-2\phi_{0}}\frac{\partial^{2}u^{\alpha}}{\partial x^{0}\partial
x^{0}}\right)$ $-2\frac{\partial^{2}\phi}{\partial x^{\alpha}\partial
x^{0}}=\frac{8\pi G}{c^{4}}T^{\alpha 0}.$

In the rescaled coordinates

$x^{0}\rightarrow e^{-\phi_{0}}x^{0},\qquad x^{\alpha}\rightarrow
e^{\phi_{0}}x^{\alpha}$

the equivalent metric is given by

$f_{ik}=\eta_{ik}+\left(\begin{array}[]{cccc}h_{00}&h_{01}&h_{02}&h_{03}\\\
h_{01}&h_{00}&0&0\\\ h_{02}&0&h_{00}&0\\\
h_{03}&0&0&h_{00}\end{array}\right),$

where

$h_{00}=2(\phi-\phi_{0}),\quad h_{0\alpha}=2\cosh(2\phi_{0})u_{\alpha}$

and $\alpha=1$, $2$, $3$. In terms of $h_{0k}$ equations for the gravitational
field in the weak field limit read

$\Delta h_{00}+3\frac{\partial^{2}h_{00}}{\partial x^{0}\partial
x^{0}}-2\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}=\frac{16\pi G}{c^{4}}\left(T^{00}-\frac{T}{2}\right),$ (32)
$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)h_{0\alpha}+\frac{\partial^{2}h_{0\beta}}{\partial
x^{\alpha}\partial x^{\beta}}-2\frac{\partial^{2}h_{00}}{\partial
x^{\alpha}\partial x^{0}}=\frac{16\pi G}{c^{4}}T^{\alpha 0}.$ (33)

In these equations the energy-momentum tensor of matter $T^{ik}$ is written in
Minkowski metric. Equations (32) and (33) are invariant upto the terms of the
order of $V^{2}/c^{2}$ under Lorentz transformation

$\frac{\partial}{\partial x^{0}}\rightarrow\frac{\partial}{\partial
x^{0}}-\frac{\mathbf{V}}{c}\nabla,\quad\frac{\partial}{\partial\mathbf{r}}\rightarrow\frac{\partial}{\partial\mathbf{r}}-\frac{\mathbf{V}}{c}\frac{\partial}{\partial
x^{0}}+\frac{\mathbf{V}}{2c^{2}}\left(\mathbf{V}\frac{\partial}{\partial\mathbf{r}}\right),$
$h_{00}\rightarrow
h_{00}\left(1+\frac{2V^{2}}{c^{2}}\right)-2\frac{V^{\alpha}}{c}h_{0\alpha},\quad
h_{0\alpha}\rightarrow h_{0\alpha}-2\frac{V^{\alpha}}{c}h_{00},$
$T^{00}\rightarrow
T^{00}+T^{00}\frac{V^{2}}{c^{2}}+2\frac{V^{\alpha}}{c}T^{0\alpha},\quad
T^{\alpha 0}\rightarrow T^{\alpha 0}+\frac{V^{\alpha}}{c}T^{00}$

for which $h_{0k}$ and $T^{0k}$ transform as symmetric tensors with
$h_{\alpha\alpha}=h_{00}$, and $\mathbf{V}$ is a constant (velocity) vector.

Equation (22) yields that non relativistic motion of particles in weak
gravitational field is described by the following equation

$\frac{1}{c^{2}}\frac{dV^{\alpha}}{dt}=\frac{\partial h_{0\alpha}}{\partial
x^{0}}-\frac{1}{2}\frac{\partial h_{00}}{\partial
x^{\alpha}}-\left(\frac{\partial h_{0\beta}}{\partial
x^{\alpha}}-\frac{\partial h_{0\alpha}}{\partial
x^{\beta}}\right)\frac{V^{\beta}}{c}+\frac{\partial h_{00}}{\partial
x^{0}}\frac{V^{\alpha}}{c},$ (34)

where $V^{\alpha}=dx^{\alpha}/dt$ is the particle velocity.

In Appendix G we explore an analogy between weak gravity and electrodynamics
and show that equations for weak classical gravitational field are analogous
to Maxwell’s equations in a medium with negative refractive index.

###  VIII.2 Energy density and energy flux in the classical limit

In Appendix H we derive expression for the energy density and energy density
flux (Poynting vector) for the weak classical gravitational field interacting
with matter that moves with nonrelativistic velocity. We find the following
expression for the energy density

$w=-\frac{c^{4}}{32\pi G}\left[3\left(\frac{\partial h_{00}}{\partial
x^{0}}\right)^{2}-\left(\nabla
h_{00}\right)^{2}+\left(\frac{\partial\mathbf{h}}{\partial
x^{0}}\right)^{2}+\text{curl}^{2}\mathbf{h}\right]$ $+\rho
c^{2}+\frac{1}{2}\rho c^{2}h_{00}+\frac{1}{2}\rho V^{2},$ (35)

where $\rho$ is the mass density, $\mathbf{V}$ is the matter velocity and
three dimensional vector

$\mathbf{h}=h^{0\alpha}.$

The energy density flux is given by

$\mathbf{S=}\frac{c^{5}}{16\pi
G}\left[-\left(2\frac{\partial\mathbf{h}}{\partial x^{0}}+\nabla
h_{00}\right)\frac{\partial h_{00}}{\partial
x^{0}}+\frac{\partial\mathbf{h}}{\partial x^{0}}\times\text{curl
}\mathbf{h}\right]+\rho c\mathbf{V}.$ (36)

Equation of the energy conservation reads

$\frac{\partial w}{\partial x^{0}}+\text{div}\mathbf{S}=0.$

###  VIII.3 Gravitational waves

The weak field limit homogeneous equations for the classical gravitational
field

$\Delta h_{00}+3\frac{\partial^{2}h_{00}}{\partial x^{0}\partial
x^{0}}-2\frac{\partial^{2}h_{0\beta}}{\partial x^{\beta}\partial x^{0}}=0,$
(37) $\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)h_{0\alpha}+\frac{\partial^{2}h_{0\beta}}{\partial
x^{\alpha}\partial x^{\beta}}-2\frac{\partial^{2}h_{00}}{\partial
x^{\alpha}\partial x^{0}}=0$ (38)

have solutions describing waves propagating with the speed of light $c$.
Taking $\partial/\partial x^{0}$ from Eq. (37) and $(1/2)\partial/\partial
x^{\alpha}$ from Eq. (38) and adding them together we obtain

$\frac{\partial^{2}}{\partial x^{0}\partial x^{0}}\left(\frac{\partial
h_{00}}{\partial x^{0}}-\frac{1}{2}\frac{\partial h_{0\alpha}}{\partial
x^{\alpha}}\right)=0.$

Thus, for the time-dependent solutions describing gravitational waves

$\frac{\partial h_{00}}{\partial x^{0}}-\frac{1}{2}\frac{\partial
h_{0\alpha}}{\partial x^{\alpha}}=0$

and Eqs. (37), (38) reduce to separate wave equations for $h_{00}$ and
$h_{0\alpha}$

$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)h_{00}=0,$ $\left(\frac{\partial^{2}}{\partial
x^{0}\partial x^{0}}-\Delta\right)h_{0\alpha}=0.$

Field equations have two classes of solutions corresponding to transverse and
longitudinal waves. For transverse waves

$\frac{\partial h_{0\alpha}}{\partial x^{\alpha}}=0,\qquad h_{00}=0$

and, e.g., for a transverse wave propagating along the $x-$axis the equivalent
metric reads

$f_{ik}^{\text{tr}}=\eta_{ik}+\left(\begin{array}[]{cccc}0&0&h_{02}(t,x)&h_{03}(t,x)\\\
0&0&0&0\\\ h_{02}(t,x)&0&0&0\\\ h_{03}(t,x)&0&0&0\end{array}\right).$ (39)

According to Eqs. (35) and (36), the energy density and the energy density
flux of the transverse gravitational wave is

$w_{\text{tr}}=-\frac{c^{4}}{32\pi
G}\left[\left(\frac{\partial\mathbf{h}}{\partial
x^{0}}\right)^{2}+\text{curl}^{2}\mathbf{h}\right],$ (40)
$\mathbf{S}_{\text{tr}}\mathbf{=}\frac{c^{5}}{16\pi
G}\frac{\partial\mathbf{h}}{\partial x^{0}}\times\text{curl}\mathbf{(h}).$
(41)

Thus, graviton has negative energy in the classical description of the
gravitational field. This result has important implication for cosmology. If
graviton energy is negative and the matter energy is positive this suggests
that at the Big Bang matter was created at the expense of generation of the
negative energy gravitons. We address this issue in Sec. X. In Sec. XI we show
that in the quantum limit (present epoch) the graviton energy is positive. In
this limit the energy density and the energy density flux are given by the
same Eqs. (40) and (41) but with the opposite sign.

For a plane transverse wave

$\mathbf{h=h}_{0}\cos\left(\omega t-\mathbf{kr}\right)$

Eq. (41) yields

$\mathbf{S}_{\text{tr}}\mathbf{=-}\frac{c^{4}h_{0}^{2}\omega}{16\pi
G}\mathbf{k}\sin^{2}\left(\omega t-\mathbf{kr}\right),$

that is Poynting vector which gives the direction of the energy flow is
opposite to the wave vector $\mathbf{k}$. This is analogous to propagation of
electromagnetic waves in a medium with negative refractive index Vese . Thus,
vacuum for the classical vector gravitational field is left-handed.

For a longitudinal wave $h_{00}\neq 0$ and $\partial h_{0\alpha}/\partial
x^{\alpha}=2\partial h_{00}/\partial x^{0}$. For such wave propagating along
the $x-$axis the metric oscillates as

$f_{ik}^{\text{long}}=\eta_{ik}+\left(\begin{array}[]{cccc}h_{00}(t,x)&h_{01}(t,x)&0&0\\\
h_{01}(t,x)&h_{00}(t,x)&0&0\\\ 0&0&h_{00}(t,x)&0\\\
0&0&0&h_{00}(t,x)\end{array}\right).$ (42)

As we show in Sec. XII, binary stars orbiting each other do not emit
longitudinal gravitational waves. Quantum mechanical analysis of Sec. XI.4
yields the same answer.

One should mention that gravitational waves in the vector gravity
substantially differ from those in general relativity. In general relativity
the metric for weak plane gravitational waves propagating along the $x-$axis
in a properly chosen coordinate system reads Land95

$g_{ik}=\eta_{ik}+\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&0&h_{22}(t,x)&h_{23}(t,x)\\\
0&0&h_{23}(t,x)&-h_{22}(t,x)\end{array}\right),$ (43)

where $h_{23}$ and $h_{22}$ are small perturbations obeying the wave equation

$\square h_{22}=0,\qquad\square h_{23},$

where $\square$ is the d’Alembertian operator. By making proper change of
coordinates one can transform Eqs. (39) and (42) for plane waves in vector
gravity to

$f_{ik}^{\text{tr}}=\eta_{ik}+\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&h_{12}(t,x)&h_{13}(t,x)\\\ 0&h_{12}(t,x)&0&0\\\
0&h_{13}(t,x)&0&0\end{array}\right),$ (44)
$f_{ik}^{\text{long}}=\eta_{ik}+\left(\begin{array}[]{cccc}0&0&0&0\\\
0&-2h(t,x)&0&0\\\ 0&0&h(t,x)&0\\\ 0&0&0&h(t,x)\end{array}\right),$ (45)

which have different structure than general relativistic Eq. (43). Thus,
measuring polarization of gravitational waves with gravitational wave
detectors could provide a test of the vector gravity.

One should note, however, that both in general relativity and vector gravity
the polarization of gravitational waves emitted by orbiting binary stars is
transverse, that is wave produces motion of test particles in the plane
perpendicular to the direction of wave propagation. The reader might get an
impression that Eq. (44) does not describe a transverse wave. This illusion
appears because metric (44) is written in non orthogonal coordinates. The
original metric (39) is written in Cartesian orthogonal coordinate system and
clearly shows that the wave is transverse.

According to Eq. (34), a nonrelativistic particle (or mirrors of an
interferometer) will move under the influence of the gravitational wave (39)
with time-dependent velocity $V^{\alpha}=h_{0\alpha}c$. This yields that
signal of the LIGO-like interferometer with perpendicular arms in the $y$ and
$z$ directions will be proportional to $h_{02}-h_{03}$. The answer can be
easily obtained taking into account that photon hits the interferometer
mirrors at different times and the mirror velocity periodically changes with
time under the influence of the gravitational wave. On the other hand, for
gravitational wave in general relativity (43) the interferometer signal is
proportional to $h_{22}$.

##  IX Cosmology

In this section we apply our theory to evolution of the Universe. Cosmological
model of the Universe is an effective model which replaces a complicated
infinitely large spatially nonuniform system of moving matter with those that
has uniform density distribution and zero velocity. In such a model the unit
gravitational field vector is $u_{k}=(1,0,0,0)$ and the scalar $\phi$ depends
only on time. However, correct description of the Universe evolution requires
solution of the full system of spatially nonuniform equations for the
gravitational field and then averaging of the result over the large scales.
Symmetry arguments yield that such an averaging can give an effective
gravitational field action with an additional cosmological term of the form

$S_{\text{cosm}}=-c\Lambda\int d^{4}x\sqrt{-f},$ (46)

where $\Lambda$ is a constant independent of the gravitational field. The
structure of $S_{\text{cosm}}$ follows from the requirement that after
rescaling of coordinates (12) the action should be independent of the
background cosmological field. This is one of the symmetries of the full
action and the effective action must possess such a symmetry. The cosmological
term appears due to replacement of the exact equations with the equations for
the averaged metric which is spatially uniform and isotropic. Since the term
(46) is an effective, $\Lambda$ depends on the choice of coordinate system,
namely on the reference frame in which we perform the spatial averaging.

For cosmology, instead of $\phi$, it is convenient to use a scale (expansion)
factor

$a=e^{-\phi}$

as a variable. In terms of $a$ the equivalent metric (9) reads

$f_{ik}=\left(\begin{array}[]{cccc}\frac{1}{a^{2}}&0&0&0\\\ 0&-a^{2}&0&0\\\
0&0&-a^{2}&0\\\ 0&0&0&-a^{2}\end{array}\right),$ (47)

and $\sqrt{-f}=a^{2}$. Components of the energy-momentum tensor of matter can
be obtained from the conservation equation (25) and are given by $T^{00}=\rho
c^{2}/a$ and $T=\rho c^{2}/a^{3}$, where $\rho$ is a constant that has a
meaning of the matter density for $a=1$.

Spatial part of nonlinear gravitational field equations (21) ($i=\alpha=1,$
$2,$ $3$) gives a simple linear equation for $a(t,\mathbf{r})$

$\frac{\partial^{2}a}{\partial x^{\alpha}\partial t}=0.$ (48)

Solution of this equation with the initial condition
$a(t=0,\mathbf{r})=b(\mathbf{r})$ is

$a(t,\mathbf{r})=a(t)+b(\mathbf{r}),$ (49)

where $a(t)$ is an arbitrary function of time such that $a(0)=0$. If at the
Big Bang ($t=0$) the Universe was inhomogeneous then subsequent expansion
makes the spatially uniform term $a(t)$ in Eq. (49) much larger than
$b(\mathbf{r})$ which is time independent. Therefore, we can omit
$b(\mathbf{r})$ and treat metric as spatially uniform.

Thus, present theory predicts that shortly after Big Bang the Universe becomes
spatially flat and homogeneous on the large scales regardless of the initial
condition. This is not the case for general relativity and known as the
problems of large-scale homogeneity and flatness of the Universe. To resolve
these problems, cosmological models based on general relativity require stage
of inflation and introduction of additional hypothetical field, the inflaton,
responsible for inflation. In contrast, vector gravity does not need
additional fields.

Temporal part of Eqs. (21) ($i=0$) with the cosmological term yield the
following equation for $a(t)$

$-\frac{d}{dt}(a\dot{a})=\frac{8\pi
G}{3}\left(\frac{\rho}{2a^{3}}-\Lambda\right)$ (50)

which shows that matter decelerates expansion of the Universe, while the
$\Lambda-$term accelerates it (provided $\Lambda>0$). In Eq. (50) a dot
denotes derivative with respect to time $t$. Integration of Eq. (50) gives

$\dot{a}^{2}=\frac{8\pi
G}{3}\left(\frac{\rho}{a^{3}}+\Lambda\right)+\frac{C}{a^{2}},$ (51)

where $C$ is an integration constant which is proportional to the total energy
density of the Universe. Indeed, in the metric (47) the action (11) with the
additional cosmological term (46) reads

$S=-\int d^{4}x\left[\frac{3c}{8\pi G}a^{2}\dot{a}^{2}+c\Lambda
a^{2}+\frac{\rho c}{a}\right].$

Hence, the Lagrangian density is

$L=-\frac{3c^{2}}{8\pi G}a^{2}\dot{a}^{2}-c^{2}\Lambda a^{2}-\frac{\rho
c^{2}}{a}$ (52)

which yields the following conserved Hamiltonian density (the total energy
density) $w$

$w=\dot{a}\frac{\partial L}{\partial\dot{a}}-L=-\frac{3c^{2}}{8\pi
G}a^{2}\dot{a}^{2}+c^{2}\Lambda a^{2}+\frac{\rho c^{2}}{a}.$ (53)

Therefore, integration constant $C$ in Eq. (51) is

$C=-\frac{8\pi G}{3c^{2}}w.$ (54)

Observations indicate that $C=0$, or, at least, that in the present epoch the
term $C/a^{2}$ in Eq. (51) is small compared to the other terms. Since the
term $C/a^{2}$ evolves as $1/a^{2}$, while the matter term is proportional to
$1/a^{3}$ the total energy density in the early Universe must be equal to zero
with very high precision. Namely, for small $a$ the term $C/a^{2}$ becomes
very small as compared to the matter contribution. Thus, positive energy of
matter in the Universe is balanced by the negative energy of the gravitational
field giving zero net energy. This result can be considered as an
observational evidence that matter in the Universe has been produced at the
expense of generation of the gravitational field with negative energy.

On the other hand, in the metric (47), Einstein equations with the
cosmological constant term give our Eq. (51) with $C=0$. Indeed, Einstein
equations (2) with the extra cosmological term $\frac{8\pi G}{c^{2}}\Lambda
g_{ik}$ in the left hand side yield for $ik=\alpha\alpha$

$\frac{d}{dt}\left(a^{3}\dot{a}\right)=8\pi G\left(\frac{\rho}{2a}+\Lambda
a^{2}\right).$ (55)

Multiplying both sides of Eq. (55) by $a^{3}\dot{a}$ and integrating over time
we obtain

$\dot{a}^{2}=\frac{8\pi
G}{3}\left(\frac{\rho}{a^{3}}+\Lambda\right)+\frac{C_{1}}{a^{6}},$ (56)

where $C_{1}$ is an integration constant which, according to observations, is
equal to zero. For $C_{1}=C=0$ Eqs. (51) and (56) are identical. Thus, general
relativity and vector gravity predict the same evolution of the Universe which
agrees with available cosmological data for a certain value of $\Lambda$.
General relativity, however, does not predict the value of $\Lambda$.

To find the value of the cosmological constant $\Lambda$ in vector gravity we
must start from the full system of equations (21) for nonuniform matter
distribution without the cosmological term and average them over the large
scales. We perform this procedure for the linearized equations for which the
answer can be found exactly. Then we will match the nonlinear equation (50)
with the exact linearized equations (which do not have the $\Lambda-$term) and
obtain the value of $\Lambda$ in the effective cosmological model.

We calculate $\Lambda$ in our coordinate system, that is at the present time.
To do so we linearize Eq. (50) near $a=a_{\text{now}}$ and rescale time and
coordinates as $t\rightarrow a_{\text{now}}t$ and $x_{\alpha}\rightarrow
x_{\alpha}/a_{\text{now}}$. In the vicinity of the present time the equivalent
metric is

$f_{ik}=\left(\begin{array}[]{cccc}1+h_{00}&0&0&0\\\ 0&-1+h_{00}&0&0\\\
0&0&-1+h_{00}&0\\\ 0&0&0&-1+h_{00}\end{array}\right),$

where $h_{00}=-2(a-a_{\text{now}})/a_{\text{now}}$ and linearization of Eq.
(50) yields the following equation for $h_{00}$

$3\ddot{h}_{00}+16\pi G\Lambda=8\pi G\frac{\rho}{a_{\text{now}}^{3}}.$ (57)

On the other hand, the full system of linearized nonuniform equations without
cosmological term is (see Eqs. (32) and (33))

$\Delta h_{00}+3\frac{\partial^{2}h_{00}}{\partial x^{0}\partial
x^{0}}-2\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}=\frac{8\pi G}{c^{4}}T_{\text{now}}^{00},$ (58)
$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)h_{0\alpha}+\frac{\partial^{2}h_{0\beta}}{\partial
x^{\alpha}\partial x^{\beta}}-2\frac{\partial^{2}h_{00}}{\partial
x^{\alpha}\partial x^{0}}=\frac{16\pi G}{c^{4}}T_{\text{now}}^{0\alpha},$ (59)

where we took into account that for nonrelativistic matter
$\tilde{f}^{00}T=T^{00}$. Taking $2\partial/\partial x^{0}$ from both sides of
Eq. (58) and divergence from both sides of Eq. (59), adding them together and
using the continuity equation

$\frac{\partial T_{\text{now}}^{00}}{\partial x^{0}}+\frac{\partial
T_{\text{now}}^{0\alpha}}{\partial x^{\alpha}}=0$

we obtain

$\frac{\partial^{2}}{\partial x^{0}\partial x^{0}}\left(2\frac{\partial
h_{00}}{\partial x^{0}}-\frac{\partial h_{0\beta}}{\partial
x^{\beta}}\right)=0.$ (60)

Integration yields

$\frac{\partial}{\partial x^{0}}\left(2\frac{\partial h_{00}}{\partial
x^{0}}-\frac{\partial h_{0\beta}}{\partial x^{\beta}}\right)=F(\mathbf{r}),$
(61)

where $F(\mathbf{r})$ is a function of spatial coordinates.

Equation (60) has the following physical meaning. Change in time of the
spatial scale (given by $h_{00}$) can be viewed as motion of masses relative
to each other. This matter current produces longitudinal vector field
$h_{0\alpha}$ which, according to Eq. (60), has nonzero divergence. Relation
between $\partial h_{00}/\partial x^{0}$ and $\partial h_{0\beta}/\partial
x^{\beta}$ should be independent of what causes $h_{00}$ to change (expansion
of the Universe or motion of a nearby star). If time dependence of $h_{00}$ is
produced by a moving star then solution is bound and average of the full
derivative in the left hand side of Eq. (61) over time vanishes. Since
$F(\mathbf{r})$ is independent of time we obtain $F(\mathbf{r})=\left\langle
F(\mathbf{r})\right\rangle_{t}=0$ and, therefore, Eq. (61) reduces to

$\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}=2\frac{\partial^{2}h_{00}}{\partial x^{0}\partial x^{0}}.$ (62)

Equation (62) must be also valid for the evolution of the Universe.
Physically, expansion of the Universe changes spatial scale which can be
treated as an effective matter current directed away from a local observer.
Such current produces longitudinal vector field $h_{0\alpha}$ (according to
Eq. (62) change of $h_{00}$ with time induces $h_{0\alpha}$). This vector
field makes the third term in (58) nonzero which acts as the cosmological term
in the evolution equation. Since, according to Eq. (35), the energy density of
the longitudinal vector field is negative the cosmological term accelerates
expansion of the Universe.

Substituting Eq. (62) into Eq. (58) we find

$\frac{\partial^{2}h_{00}}{\partial x^{0}\partial x^{0}}=\Delta
h_{00}-\frac{8\pi G}{c^{4}}T_{\text{now}}^{00}.$ (63)

Next we average Eq. (63) over the large scales so that $T_{\text{now}}^{00}$
becomes spatially uniform. In the effective cosmological model the averaged
metric (47) is diagonal and depends only on time (see discussion after Eq.
(49)). Therefore, spatial averaging of $\Delta h_{00}$ must be equal to zero
and Eq. (63) yields

$\frac{\partial^{2}}{\partial x^{0}\partial x^{0}}\left\langle
h_{00}\right\rangle=-\frac{8\pi G}{c^{4}}\left\langle
T_{\text{now}}^{00}\right\rangle.$ (64)

On the other hand, spatial averaging of Eq. (62) gives that

$\left\langle\frac{\partial h_{0\beta}}{\partial
x^{\beta}}\right\rangle=2\frac{\partial}{\partial x^{0}}\left\langle
h_{00}\right\rangle$ (65)

is not equal to zero. This is consistent with our assumptions. Indeed, Eq.
(65) can be satisfied by choosing $h_{0\beta}$ as an odd function of
coordinates, $h_{0\beta}\propto x_{\beta}$. Thus, spatial averaging of
$h_{0\beta}$ in the local region yields zero and, hence, the averaged metric
$\left\langle h_{ik}\right\rangle$ is diagonal and spatially homogeneous.

Comparing Eq. (57) with Eq. (64) and taking into account that

$\left\langle T_{\text{now}}^{00}\right\rangle=\frac{\rho
c^{2}}{a_{\text{now}}^{3}}$

we find the following value for the cosmological constant

$\Lambda=\frac{2\rho}{a_{\text{now}}^{3}}.$

Thus, at the present time the cosmological term contribution (dark energy) is
twice as much as the energy of matter. Therefore, the ratio between the energy
density due to the cosmological constant and the critical density of the
Universe $\Omega_{\text{critical}}=3\dot{a}_{\text{now}}^{2}/8\pi
G=\rho_{\text{matter}}+\Lambda$ is

$\frac{\Lambda}{\Omega_{\text{critical}}}=\frac{2}{3}\approx 0.67.$ (66)

This is also the case in any other reference frame. That is an observer who
lives billion years before or after would find the same answer for
$\Lambda/\Omega_{\text{critical}}$ in his reference frame. Our prediction has
no free parameters and agrees with the recent Planck result which measured
$\Lambda/\Omega_{\text{critical}}$ and obtained $0.686\pm 0.02$ Planck14 .

In the present theory the $\Lambda-$term appears as a solution of equations.
It comes from the gravitational part (left hand side) of Eqs. (21) as a result
of averaging of the term $\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}$ over the large scales. Such average does not vanish in Eq. (58) if
we make the transition to the infinitely large size of the Universe properly.

The physical meaning of the dark energy becomes clear if we compare the exact
expression for the weak field limit Lagrangian density with those obtained
from the effective cosmological action containing the $\Lambda-$term. Keeping
only the relevant contributions the weak field limit Lagrangian density in the
rescaled coordinates reads (see Eq. (183))

$L=-\frac{3c^{2}}{32\pi G}\left(\dot{h}_{00}\right)^{2}-\frac{c^{2}}{32\pi
G}\mathbf{\dot{h}}^{2}-\rho c^{2},$

where $\mathbf{h}=h^{0\alpha}$ depends on time and spatial coordinates. On the
other hand, linearization of the effective cosmological Lagrangian (52) yields

$L=-\frac{3c^{2}}{32\pi G}\left(\dot{h}_{00}\right)^{2}-c^{2}\Lambda-\rho
c^{2}.$

Thus,

$\Lambda=\frac{1}{32\pi G}\left\langle\mathbf{\dot{h}}^{2}\right\rangle$

and dark energy is the average energy of the longitudinal part of the
gravitational field. From the perspective of a local observer the change in
the spatial scale caused by the Universe expansion is equivalent to a matter
current directed away from the observer. Such current generates the time-
dependent longitudinal field $\mathbf{h}$ which is analogous to generation of
the vector potential $\mathbf{A}$ by a current in classical electrodynamics.
The value of the current depends on the matter density and on the expansion
rate of the Universe which, in turn, is a function of the matter density.
Thus, the value of the cosmological constant $\Lambda$ is determined by the
averaged matter density in the reference frame of the observer (matter density
at the moment the observer measures $\Lambda$).

One should mention that quantization of the gravitational field involves only
radiative part of the field. Since dark energy originates from the non
radiative part it is a pure classical effect which can be described by the
classical field equations.

##  X Contents of the Universe

Vector theory of gravity explains the nature of dark energy as the energy of
longitudinal gravitational field induced by the Universe expansion. Such
energy is negative and accelerates expansion of the Universe. According to the
present theory, the Universe is made of matter (dark matter and ordinary
matter) and gravitational field (see Fig. 1). Observations indicate that the
total energy of the Universe is equal to zero (see discussion after Eq. (54)).

Figure 1: Contents of the Universe according to vector gravity. Universe is
made of matter ($85\%$ dark matter and $15\%$ ordinary matter) which has
positive energy and gravitational field which has negative energy. The total
energy of the Universe is equal to zero.

As we show in Sec. VIII.3, classical field equations yield that the graviton
has negative energy. This suggests that at the Big Bang matter was created at
the expense of generation of the negative energy gravitons. Since the pressure
is determined by the wave momentum, rather than energy, the gas of gravitons
has positive pressure $P$. For such a gas the equation of state reads

$P=-\frac{w_{g}}{3},$ (67)

where $w_{g}$ is the graviton energy density. Since the energy momentum tensor
of an isotropic gas is

$T^{ik}=\left(\begin{array}[]{cccc}w_{g}&0&0&0\\\ 0&P&0&0\\\ 0&0&P&0\\\
0&0&0&P\end{array}\right)$

we obtain that for the gas of gravitons

$T^{00}-\frac{1}{2}\tilde{f}^{00}T=\frac{1}{2}\left(w_{g}+3P\right)=0.$ (68)

According to Eqs. (21), the combination $T^{00}-\frac{1}{2}\tilde{f}^{00}T$ is
the source of gravitational field. This combination is also the source of
gravitational field in general relativity. Namely, it determines gravitational
mass of an object (Komar mass) in terms of the energy-momentum tensor. Since
for the primordial gravitational waves $T^{00}-\frac{1}{2}\tilde{f}^{00}T=0$
they do not produce gravitational field and, hence, they do not contribute to
the gravitational acceleration measured by an accelerometer. Equation (68)
also yields that primordial gravitons do not change the cosmological Eq. (50)
and, thus, they have no effect on expansion rate of the Universe.

According to the postulate $4$ of our theory the graviton is a composite
particle, namely it is composed of fermion-antifermion pairs. Since no more
than one fermion can occupy the same quantum state the matter generation at
the Big Bang has continued until fermion states were filled. The following
Universe expansion practically did not change the fermion occupation number
and states remain filled. These filled states form a vacuum in the present
epoch. As we show in the next section, for the filled vacuum the graviton
energy is positive and, thus, vacuum is stable. For the filled vacuum creation
of a graviton corresponds to creation of fermion-antifermion hole pairs out of
the filled fermion states.

Analogy with the composite photon theory Perk14  suggests that the fermion-
antifermion pairs that compose the graviton are coupled to matter with the
gravitational constant $G$, while a single fermion interacts with a much
weaker (perhaps zero) coupling constant. As a consequence, the energy scales
associated with the graviton and its constituent fermion are very different.
The Planck energy is the characteristic quantum energy scale for the graviton
and the corresponding Planck frequency is

$\omega_{\text{Pl}}=\sqrt{\frac{c^{5}}{\hbar G}}=1.8\times 10^{43}\text{
s}^{-1}\text{.}$

Perhaps at the Big Bang the gravitational waves have been generated upto the
Planck frequency. However, states of the constituent fermions are filled up to
a much higher frequency which is determined by their coupling constant. Such
frequency can not be predicted in the framework of the present theory.

Constituent fermions have both positive and negative energy states. The number
of negative energy states which are filled at the Big Bang is greater than
those with positive energy. So that the net energy of the graviton gas is
negative. Probably shortly after the Big Bang the negative energy of the
graviton gas has been transferred to the kinetic energy of the Universe
expansion (the first term in the right hand side of Eq. (53)) and the number
of negative and positive energy states that are filled became equal. This
subject, however, requires detail analysis of cosmological models and is
beyond the scope of our paper.

##  XI Quantization of Gravitational Field

Here we quantize gravitational field assuming that graviton is composed of
fermion-antifermion pairs. Since quantization procedure is similar to
quantization of electromagnetic field we start from a brief review of the
field quantization in electrodynamics.

###  XI.1 Classical electrodynamics

Classical electrodynamics is a vector field theory in four dimensional
Minkowski space-time. Electromagnetic field is a $4-$vector
$A^{k}=(A_{0},\mathbf{A})$ in this space-time, while electric current density
is a $4-$vector $j^{k}=(c\rho,\mathbf{j})$, where $\rho$ and $\mathbf{j}$ are
the electric charge and spatial current densities. The conserved $4-$current
density $j^{k}$ is coupled to $A_{k}$ through the Lorentz and gauge invariant
term in the action

$S_{\text{coupl}}=-\frac{1}{c^{2}}\int d^{4}xA_{k}j^{k}.$

The total action of the system is
$S=S_{\text{field}}+S_{\text{coupl}}+S_{\text{matter}}$, where action for free
electromagnetic field is

$S_{\text{field}}=-\frac{1}{16\pi c}\int d^{4}x\left(\frac{\partial
A_{k}}{\partial x^{i}}-\frac{\partial A_{i}}{\partial
x^{k}}\right)\left(\frac{\partial A^{k}}{\partial x_{i}}-\frac{\partial
A^{i}}{\partial x_{k}}\right)$

and for nonrelativistic motion of particles

$S_{\text{matter}}=\int dt\sum_{a}\frac{m_{a}\dot{r}_{a}^{2}}{2},$

where the sum is over all particles $a$ having positions $\mathbf{r}_{a}$,
masses $m_{a}$ and electric charges $q_{a}$. Electric current density is given
by

$\mathbf{j}=\sum_{a}q_{a}\mathbf{\dot{r}}_{a}\delta\left(\mathbf{r}-\mathbf{r}_{a}(t)\right).$

Particle momentum conjugate to $\mathbf{r}_{a}$ is

$\mathbf{p}_{a}=\frac{\partial
L}{\partial\mathbf{\dot{r}}_{a}}=m_{a}\mathbf{\dot{r}}_{a}+\frac{q_{a}}{c}\mathbf{A(r}_{a}),$

while momentum $\pi^{k}$ conjugate to the field $A_{k}$ reads

$\pi^{k}=\frac{\partial L}{\partial\dot{A}_{k}}=-\frac{1}{4\pi
c}\left\\{\begin{array}[]{c}0,\quad k=0\\\ E^{k},\quad
k=1,2,3\end{array},\right.$

where $L$ is the Lagrangian density and $\mathbf{E}=-\nabla
A^{0}-\partial\mathbf{A}/\partial x^{0}$ is the electric field.

Electromagnetic field $A_{k}$ has four real components. However, since
momentum $\pi^{0}$ conjugate to $A_{0}$ vanishes the time component $A_{0}$ is
not a dynamical field. This means that $A_{0}$ is not an independent degree of
freedom but rather it is a functional of $\mathbf{A}$ and electric charge
density $\rho$ Cohe87 . In addition, gauge symmetry implies that there are
only two independent physical degrees of freedom because one of the degrees of
freedom can be eliminated by gauge fixing. These two independent degrees of
freedom of the electromagnetic field corresponding to radiation are quantized,
their quantum is a photon.

Classical Hamiltonian of the system is a functional of $\mathbf{r}_{a}$,
$\mathbf{p}_{a}$, $A_{k}$ and $\pi_{k}$

$\mathcal{H}=\sum_{a}\mathbf{p}_{a}\cdot\mathbf{\dot{r}}_{a}+\int
d^{3}x\left(\pi^{k}\dot{A}_{k}-L\right)$ $=\int
d^{3}x\left[\frac{1}{8\pi}\left(E^{2}+B^{2}\right)-\frac{c}{4\pi}A_{0}\text{div}\mathbf{E}\right]$
$+\sum_{a}\left[\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-\frac{q_{a}}{c}\mathbf{A}\right)^{2}+q_{a}A_{0}\right],$

where $\mathbf{B}=$curl$\mathbf{A}$.

One can decompose $\mathbf{E}$ and $\mathbf{B}$ into longitudinal and
transverse parts

$\mathbf{E}=\mathbf{E}_{\text{lon}}+\mathbf{E}_{\text{tr}},\quad\mathbf{B}=\mathbf{B}_{\text{tr}}$

such that div$\mathbf{E}_{\text{tr}}=0$ and div$\mathbf{B}_{\text{tr}}=0$.
This decomposition can be done in a straightforward way if we write
$\mathbf{E}$ as a Fourier series

$\mathbf{E(r)}=\mathbf{\sum_{p}E(p)}e^{i\mathbf{p}\cdot\mathbf{r}}.$

Then

$\mathbf{E}_{\text{lon}}\mathbf{(r)}=\mathbf{\sum_{p}}\hat{p}(\hat{p}\cdot\mathbf{E(p))}e^{i\mathbf{p}\cdot\mathbf{r}},$

where $\hat{p}$ is a unit vector in the direction of $\mathbf{p}$, and

$\mathbf{E}_{\text{tr}}=\mathbf{E}-\mathbf{E}_{\text{lon}}.$

Having in mind that photon originates from the transverse electromagnetic
field we decompose 4-vectors $A_{k}$ and $j_{k}$ into transverse spatial part
$\mathbf{A}_{\text{tr}}$, $\mathbf{j}_{\text{tr}}$ and the remaining piece
which contains longitudinal and time-like components, e.g.

$A^{k}=A_{\text{tr}}^{k}+A_{{}_{\text{{l}-t}}}^{k}.$ (69)

Terms in the right hand side of Eq. (69) do not transform as 4-vectors, only
their sum does. In particular, for the spatial part of $A^{k}$ we have

$\mathbf{A}=\mathbf{A}_{\text{tr}}+\mathbf{A}_{{}_{\text{lon}}}.$

Decomposition of the field into the longitudinal and transverse components
decouples Hamiltonian into two independent pieces
$\mathcal{H}=\mathcal{H}_{\text{tr}}+\mathcal{H}_{\text{{l}-t}}$, where
Hamiltonian of the transverse field is

$\mathcal{H}_{\text{tr}}=\frac{1}{8\pi}\int
d^{3}x\left(E_{\text{tr}}^{2}+B^{2}\right)+\sum_{a}\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-\frac{q_{a}}{c}\mathbf{A}_{\text{tr}}\right)^{2}$
(70)

and

$\mathbf{E}_{\text{tr}}=-\frac{\partial\mathbf{A}_{\text{tr}}}{\partial
x^{0}}.$

For the longitudinal and transverse fields Maxwell equations also decouple. In
particular, for transverse field Maxwell equations read

$\text{curl}\mathbf{B}=\frac{4\pi}{c}\mathbf{j}_{\text{tr}}+\frac{1}{c}\frac{\partial\mathbf{E}_{\text{tr}}}{\partial
t},\quad\text{curl}\mathbf{E}_{\text{tr}}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial
t}.$ (71)

###  XI.2 Quantization of electromagnetic field in elementary photon theory

In conventional quantization of the electromagnetic field the
$\mathcal{H}_{{}_{\text{{l}-t}}}$ part of the Hamiltonian remains classical
Ginz87 . Part of the transverse field that corresponds to radiation (field
without sources) is quantized and is described by the following Hamiltonian
operator Cohe87

$\mathcal{\hat{H}}_{\text{tr}}=\sum_{\mathbf{p,}\mu=1,2}\hbar
cp\hat{A}_{\mathbf{p,}\mu}^{+}\hat{A}_{\mathbf{p,}\mu}+\sum_{a}\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-\frac{q_{a}}{c}\mathbf{\hat{A}}_{\text{tr}}(\mathbf{r}_{a})\right)^{2},$

where

$\mathbf{\hat{A}}_{\text{tr}}(\mathbf{r})=\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{2\pi\hbar
c}{pV}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}+\text{H.c.}\right),$

$\mathbf{\epsilon}_{\mathbf{p,}\mu}$ ($\mu=1,2$) are unit three-dimensional
polarization vectors perpendicular to the photon wave vector $\mathbf{p}$, $V$
is the photon volume and operators $\hat{A}_{\mathbf{p,}\mu}$ obey Bose-
Einstein commutation relations

$[\hat{A}_{\mathbf{p},\mu},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2$

and all other commutators are equal to zero. Operators
$\hat{A}_{\mathbf{p},\mu}^{+}$ and $\hat{A}_{\mathbf{p},\mu}$ describe
creation and annihilation of a spin $1$ photon with wave vector $\mathbf{p}$
and polarization $\mu$. Electric

$\mathbf{\hat{E}}_{\text{tr}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{2\pi\hbar
cp}{V}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right),$

and magnetic

$\mathbf{\hat{B}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{2\pi\hbar
c}{pV}}\left(\mathbf{p}\times\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right)$

field operators obey commutation relations

$[\hat{A}_{\text{tr}}^{\beta}(\mathbf{r}^{\prime}),\hat{E}_{\text{tr}}^{\alpha}(\mathbf{r})]\mathbf{=-}4\pi
i\hbar
c\delta^{\beta\alpha}\delta(\mathbf{r-r}^{\prime}),\quad\alpha,\beta=1,2,3,$
(72)
$[\hat{A}_{\text{tr}}^{\beta}(\mathbf{r}^{\prime}),\hat{B}_{\alpha}(\mathbf{r})]\mathbf{=}0,$
(73)
$[\mathcal{\hat{H}}_{0},\mathbf{\hat{E}}_{\text{tr}}]=-ic\hbar\text{\thinspace
curl}\mathbf{\hat{B},}$ (74)
$[\mathcal{\hat{H}}_{0},\mathbf{\hat{B}}]=ic\hbar\text{\thinspace
curl}\mathbf{\hat{E}}_{\text{tr}},$ (75)

where

$\mathcal{\hat{H}}_{0}=\sum_{\mathbf{p,}\mu=1,2}\hbar
cp\hat{A}_{\mathbf{p,}\mu}^{+}\hat{A}_{\mathbf{p,}\mu}.$ (76)

In the Heisenberg picture the Heisenberg equations of motion

$\frac{d\mathbf{\hat{E}}_{\text{tr}}}{dt}=\frac{i}{\hbar}[\mathcal{\hat{H}}_{\text{tr}},\mathbf{\hat{E}}_{\text{tr}}],\quad\frac{d\mathbf{\hat{B}}}{dt}=\frac{i}{\hbar}[\mathcal{\hat{H}}_{\text{tr}},\mathbf{\hat{B}}]$

yield Maxwell’s equations for the transverse field (71) in the operator form

$\text{curl}\mathbf{\hat{B}}=\frac{4\pi}{c}\mathbf{j}_{\text{tr}}+\frac{1}{c}\frac{\partial\mathbf{\hat{E}}_{\text{tr}}}{\partial
t},\quad\text{curl}\mathbf{\hat{E}}_{\text{tr}}=-\frac{1}{c}\frac{\partial\mathbf{\hat{B}}}{\partial
t}.$ (77)

One should emphasize that only a part of the transverse electromagnetic field
is quantized, namely the part that corresponds to radiation. For example,
magnetic field produced by stationary currents is transverse but is not
quantized and remains classical.

###  XI.3 Photon as composite particle

In the composite photon theory the elementary particle is a massless spin 1/2
fermion and photon is composed of the fermion-antifermion pairs. In free space
the Dirac equation for massless spin 1/2 fermion, described by a four-
component spinor $\Psi=\left(\begin{array}[]{c}\psi_{R}\\\
\psi_{L}\end{array}\right)$, reads

$\gamma^{\mu}\partial_{\mu}\Psi=0,$ (78)

where $\gamma^{\mu}$ are gamma matrices which can be written in terms of
2$\times$2 sub-matrices taken from the Pauli matrices and the 2$\times$2
identity matrix $I$. In the Weyl (chiral) basis the gamma matrices have the
form

$\gamma^{0}=\left(\begin{array}[]{cc}0&-I\\\
-I&0\end{array}\right),\quad\gamma^{1}=\left(\begin{array}[]{cc}0&\sigma_{x}\\\
-\sigma_{x}&0\end{array}\right),$
$\gamma^{2}=\left(\begin{array}[]{cc}0&\sigma_{y}\\\
-\sigma_{y}&0\end{array}\right),\quad\gamma^{3}=\left(\begin{array}[]{cc}0&\sigma_{z}\\\
-\sigma_{z}&0\end{array}\right)$

and Pauli matrices are

$\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

Recall that $\gamma^{\mu}$ are fixed under Lorentz transformations in the
forms given above. Lorentz invariance of the Dirac equation is achieved by
proper transformation of spinor $\Psi$ that counterbalances the transformation
of $\partial_{\mu}$. In the Weyl (or chiral) representation of the Dirac
matrices the Weyl spinors $\psi_{R}$ and $\psi_{L}$ do not mix under Lorentz
transformations.

Solutions of the Dirac equation are arbitrary superposition of four plane wave
spinors

$\Psi_{a}=u_{a}(\mathbf{p})e^{-ipct+i\mathbf{p}\cdot\mathbf{r}},\quad\Psi_{b}=u_{b}(\mathbf{p})e^{ipct+i\mathbf{p}\cdot\mathbf{r}},$
$\Psi_{c}=u_{c}(\mathbf{p})e^{ipct+i\mathbf{p}\cdot\mathbf{r}},\quad\Psi_{d}=u_{d}(\mathbf{p})e^{-ipct+i\mathbf{p}\cdot\mathbf{r}},$

where $\mathbf{p}$ is the wave vector, $p=|\mathbf{p}|$ and for $\mathbf{p}$
oriented along the positive direction of the $z-$axis

$u_{a}(\mathbf{p})=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right),\quad u_{b}(\mathbf{p})=\left(\begin{array}[]{c}0\\\ 1\\\
0\\\ 0\end{array}\right),$ $u_{c}(\mathbf{p})=\left(\begin{array}[]{c}0\\\
0\\\ 1\\\ 0\end{array}\right),\quad
u_{d}(\mathbf{p})=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ 1\end{array}\right).$

Energy of particles described by spinors $\Psi_{a}$ and $\Psi_{d}$ is positive
$\varepsilon=\hbar cp$, while fermions corresponding to spinors $\Psi_{b}$ and
$\Psi_{c}$ have negative energies $\varepsilon=-\hbar cp$. The helicity of a
particle is right-handed if the direction of its spin is the same as the
direction of its motion. It is left-handed if the directions of spin and
motion are opposite. $u_{a}$ and $u_{c}$ are right-handed spinors, while
$u_{b}$ and $u_{d}$ are left-handed spinors. Helicity of a massless particle
is Lorentz invariant.

General solution of the free Dirac equation can be written as

$\Psi=\sum_{\mathbf{p}}\left[a_{\mathbf{p}}u_{a}(\mathbf{p})e^{-icpt}+b_{\mathbf{p}}u_{b}(\mathbf{p})e^{icpt}\right.$
$+\left.c_{\mathbf{p}}u_{c}(\mathbf{p})e^{icpt}+d_{\mathbf{p}}u_{d}(\mathbf{p})e^{-icpt}\right]e^{i\mathbf{p}\cdot\mathbf{r}},$

where $a_{\mathbf{p}}$, $b_{\mathbf{p}}$, $c_{\mathbf{p}}$ and
$d_{\mathbf{p}}$ are arbitrary constants which transform as scalars under
Lorentz transformation.

Out of $u_{a}(\mathbf{p})$, $u_{b}(\mathbf{p})$, $u_{c}(\mathbf{p})$ and
$u_{d}(\mathbf{p})$ one can construct four linearly independent $4-$vectors in
$4-$dimensional space-time:

$u_{a}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{b}(\mathbf{p})=(0,1,-i,0),$
(79)
$u_{d}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{c}(\mathbf{p})=(0,-1,-i,0),$
(80)
$u_{b}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{b}(\mathbf{p})=u_{c}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{c}(\mathbf{p})=(1,0,0,-1),$
(81)
$u_{a}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{a}(\mathbf{p})=u_{d}^{+}(\mathbf{p})\gamma_{0}\gamma_{\mu}u_{d}(\mathbf{p})=(1,0,0,1).$
(82)

Other combinations give zero vectors. Vectors (79) and (80) are transverse,
while (81) and (82) are combinations of the longitudinal and time-like
vectors. Recall that wave vector $\mathbf{p}$ is chosen to be oriented along
the positive direction of the $z-$axis.

We map the fermion field into the real transverse field
$\mathbf{A}_{\text{tr}}$

$\mathbf{A}_{\text{tr}}(t,\mathbf{r})=\sum_{\mathbf{p}}\left[\mathbf{A}_{\mathbf{p,}\text{tr}}(t)e^{i\mathbf{p}\cdot\mathbf{r}}+\mathbf{A}_{\mathbf{p,}\text{tr}}^{\ast}(t)e^{-i\mathbf{p}\cdot\mathbf{r}}\right]$
(83)

such that $\mathbf{A}_{\text{tr}}(t,\mathbf{r})$ obeys Maxwell equations for
free transverse field. In terms of the Fourier components
$\mathbf{A}_{\mathbf{p,}\text{tr}}(t)$ this can be done in the following way

$\mathbf{A}_{\mathbf{p,}\text{tr}}(t)=A_{\mathbf{p},1}e^{-icpt}\mathbf{\epsilon}_{\mathbf{p,}1}+A_{\mathbf{p},2}e^{-icpt}\mathbf{\epsilon}_{\mathbf{p,}2},$
(84)

where $\mathbf{\epsilon}_{\mathbf{p,}1}$ and
$\mathbf{\epsilon}_{\mathbf{p,}2}$ are the spatial unit polarization vectors
of the left and right circularly polarized photons respectively. Equations
(79) and (80) indicate that $A_{\mathbf{p},1}$ and $A_{\mathbf{p},2}$ should
be chosen as

$A_{\mathbf{p},1}=\sum_{\mathbf{k\parallel
p}}F_{1}(\mathbf{p},\mathbf{k})a_{\mathbf{p}+\mathbf{k}}b_{-\mathbf{k}},$
$A_{\mathbf{p},2}=\sum_{\mathbf{k\parallel
p}}F_{2}(\mathbf{p},\mathbf{k})d_{\mathbf{p}+\mathbf{k}}c_{-\mathbf{k}},$

where summation is over all fermion states with wave vectors $\mathbf{k}$
parallel to $\mathbf{p}$.

In order for $\mathbf{A}_{\mathbf{p}\text{,tr}}$ to transform as a transverse
field under Lorentz transformations the spectral functions
$F_{1}(\mathbf{p},\mathbf{k})$ and $F_{2}(\mathbf{p},\mathbf{k})$ must be
scalars. Therefore they can depend only on the absolute values of the
$4-$vectors and their dot products. $F_{1}(\mathbf{p},\mathbf{k})$ is a factor
in front of $a_{\mathbf{p}+\mathbf{k}}b_{-\mathbf{k}}$ which is a product of
two fermion states with $4-$wave vectors $(p+k,\mathbf{p}+\mathbf{k})$ and
$(-k,-\mathbf{k})$. Absolute values of these $4-$vectors and their dot product
(with Minkowski metric) are equal to zero. Therefore, spectral function
$F_{1}(\mathbf{p},\mathbf{k})$ is independent of $\mathbf{p}$ and
$\mathbf{k}$. Similar arguments yield that $F_{2}(\mathbf{p},\mathbf{k})$ is
also a constant which can be chosen arbitrary. We choose
$F_{1}=F_{2}=1/\sqrt{N_{\parallel}}$, where $N_{\parallel}$ is the number of
fermion states with wave vectors parallel to $\mathbf{p}$, and assume that
$N_{\parallel}$ is independent of $\mathbf{p}$.

####  XI.3.1 Field quantization in the composite photon theory

Next we quantize the fermion field by replacing $a_{\mathbf{p}}$,
$b_{\mathbf{p}}$, $c_{\mathbf{p}}$ and $d_{\mathbf{p}}$ with operators that
obey canonical anticommutation relations

$\hat{a}_{\mathbf{p}}\hat{a}_{\mathbf{p}^{\prime}}^{+}+\hat{a}_{\mathbf{p}^{\prime}}^{+}\hat{a}_{\mathbf{p}}=\delta_{\mathbf{p},\mathbf{p}^{\prime}},\quad\hat{b}_{\mathbf{p}}\hat{b}_{\mathbf{p}^{\prime}}^{+}+\hat{b}_{\mathbf{p}^{\prime}}^{+}\hat{b}_{\mathbf{p}}=\delta_{\mathbf{p},\mathbf{p}^{\prime}},$
(85)
$\hat{c}_{\mathbf{p}}\hat{c}_{\mathbf{p}^{\prime}}^{+}+\hat{c}_{\mathbf{p}^{\prime}}^{+}\hat{c}_{\mathbf{p}}=\delta_{\mathbf{p},\mathbf{p}^{\prime}},\quad\hat{d}_{\mathbf{p}}\hat{d}_{\mathbf{p}^{\prime}}^{+}+\hat{d}_{\mathbf{p}^{\prime}}^{+}\hat{d}_{\mathbf{p}}=\delta_{\mathbf{p},\mathbf{p}^{\prime}}.$
(86)

All other anticommutators are equal to zero.

The second quantized Hamiltonian for the free fermion field is

$\mathcal{\hat{H}}_{0}=\sum_{\mathbf{p}}\left[\varepsilon_{a}(p)\hat{a}_{\mathbf{p}}^{+}\hat{a}_{\mathbf{p}}+\varepsilon_{b}(p)\hat{b}_{\mathbf{p}}^{+}\hat{b}_{\mathbf{p}}+\varepsilon_{c}(p)\hat{c}_{\mathbf{p}}^{+}\hat{c}_{\mathbf{p}}+\varepsilon_{d}(p)\hat{d}_{\mathbf{p}}^{+}\hat{d}_{\mathbf{p}}\right],$
(87)

where $\varepsilon_{a}(p)=\varepsilon_{d}(p)=\hbar cp$ and
$\varepsilon_{b}(p)=\varepsilon_{c}(p)=-\hbar cp$.

The free transverse field (83) (field without sources) now becomes an operator
which in the Schrödinger picture reads

$\mathbf{\hat{A}}_{\text{tr}}(\mathbf{r})=\sum_{\mathbf{p}}\left(\mathbf{\hat{A}}_{\mathbf{p,}\text{tr}}e^{i\mathbf{p}\cdot\mathbf{r}}+\mathbf{\hat{A}}_{\mathbf{p}\text{,tr}}^{+}e^{-i\mathbf{p}\cdot\mathbf{r}}\right),$
(88)

where

$\mathbf{\hat{A}}_{\mathbf{p,}\text{tr}}=\hat{A}_{\mathbf{p},1}\mathbf{\epsilon}_{\mathbf{p,}1}+\hat{A}_{\mathbf{p},2}\mathbf{\epsilon}_{\mathbf{p,}2}$

and

$\hat{A}_{\mathbf{p},1}=\frac{1}{\sqrt{N_{\parallel}}}\sum_{\mathbf{k\parallel
p}}\hat{a}_{\mathbf{p}+\mathbf{k}}\hat{b}_{-\mathbf{k}},$ (89)
$\hat{A}_{\mathbf{p},2}=\frac{1}{\sqrt{N_{\parallel}}}\sum_{\mathbf{k\parallel
p}}\hat{d}_{\mathbf{p}+\mathbf{k}}\hat{c}_{-\mathbf{k}}.$ (90)

Since fermion $a$ is right-handed the operator
$\hat{a}_{\mathbf{p}+\mathbf{k}}^{+}$ creates a spin $1/2$ fermion with spin
parallel to $\mathbf{p}+\mathbf{k}$ and energy $\varepsilon_{a}=\hbar c(p+k)$.
On the other hand, $\hat{b}_{-\mathbf{k}}^{+}$ creates a spin $1/2$
antifermion with spin antiparallel to $\mathbf{-k}$ (that is parallel to
$\mathbf{k}$) and negative energy $\varepsilon_{b}=-\hbar ck$. Thus, the
combination $\hat{b}_{-\mathbf{k}}^{+}\hat{a}_{\mathbf{p}+\mathbf{k}}^{+}$
with $\mathbf{k\parallel p}$ creates a fermion-antifermion pair with the total
energy $\varepsilon=\hbar cp$ and spin $1$ parallel to $\mathbf{p}$. Recall
that for left (right) circularly polarized photon the photon spin is parallel
(antiparallel) to the wave vector $\mathbf{p}$. Therefore, operator
$\hat{A}_{\mathbf{p},1}^{+}$ ($\hat{A}_{\mathbf{p},2}^{+}$) creates a left
(right) circularly polarized photon with spin $1$. According to Eqs. (89) and
(90), emission of a single photon corresponds to creation of $N_{\parallel}$
fermion-antifermion pairs with equal probability $1/N_{\parallel}$.

Equations (85) and (86) yield the following commutation relations for
operators $\hat{A}_{\mathbf{p},\mu}$ and $\hat{A}_{\mathbf{p},\mu}^{+}$

$[\hat{A}_{\mathbf{p},1},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}1}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$-\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{a}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}\hat{a}_{\mathbf{p}+\mathbf{k}}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{b}_{-\mathbf{k}^{\prime}}^{+}\hat{b}_{-\mathbf{k}}\right),$
(91)
$[\hat{A}_{\mathbf{p,}2},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}2}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$-\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{d}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}\hat{d}_{\mathbf{p}+\mathbf{k}}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{c}_{-\mathbf{k}^{\prime}}^{+}\hat{c}_{-\mathbf{k}}\right).$
(92)

Terms under the sum are written in the normal order, that is annihilation
operators are placed to the right of the creation operators. If the total
number of fermion states is very large compared to the number of occupied
states the commutation relations for the vector operator become exactly the
same as in the conventional quantum electrodynamics, namely, in the limit
$N_{\parallel}\rightarrow\infty$ we obtain

$[\hat{A}_{\mathbf{p},\mu},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2$
(93)

and all other commutators are equal to zero. Roughly, the correction term to
the Bose–Einstein commutation relations is of the order of the ratio of the
number of fermions in the system to the total number of fermion states in the
Universe. Such correction is negligible.

One should mention that the number of fermion states can be much larger than
the number of photon states. For example, for electromagnetic field in a
cavity only certain photon states satisfy boundary conditions. However, there
is no such constraint on the fermion states because fermions do not interact
directly with the cavity walls. The boundary condition constrains the total
sum in Eqs. (89) and (90), that is values of the photon wave vector
$\mathbf{p}$. However, there is no constraint on the values of the summation
index $\mathbf{k}$.

In the composite theory of the photon the transverse part of the classical
electromagnetic Hamiltonian (70) describing radiation field interacting with
electric charges is replaced by the operator

$\mathcal{\hat{H}}_{\text{tr}}=\mathcal{\hat{H}}_{0}+\sum_{a}\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-\frac{q_{a}}{c}\mathbf{\hat{A}}_{\text{tr}}(\mathbf{r}_{a})\right)^{2},$

where $\mathcal{\hat{H}}_{0}$ is the Hamiltonian for the free fermion field
(87) and $\mathbf{\hat{A}}_{\text{tr}}(\mathbf{r})$ is given by Eqs.
(88)-(90).

In the composite photon theory the electric and magnetic field operators

$\mathbf{\hat{E}}_{\text{tr}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{2\pi\hbar
cp}{V}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right),$
(94)
$\mathbf{\hat{B}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{2\pi\hbar
c}{pV}}\left(\mathbf{p}\times\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right)$
(95)

obey the same commutation relationships (72)-(75) as in the case of the
elementary photon theory. Namely, Eqs. (72) and (73) are a direct consequence
of Eq. (93), while relations (74) and (75) are satisfied because for the free
fermion field Hamiltonian (87), as in the case of the free photon Hamiltonian
(76), we obtain

$[\mathcal{\hat{H}}_{0},\hat{A}_{\mathbf{p},\mu}]=-\hbar
cp\hat{A}_{\mathbf{p},\mu},\quad\mu=1,2.$

Therefore, in the composite photon theory the Heisenberg equation of motion
also yields Maxwell’s equations for the transverse field (77). Thus, in the
limit $N_{\parallel}\rightarrow\infty$ the composite photon theory yields the
same Quantum Electrodynamics as the elementary photon theory and, in
particular, photons obey Bose-Einstein statistics. Despite the fermionic
nature of light the manner in which fermions interact with charged particles
leads to the simplified description of light in terms of composite photons.

###  XI.4 Quantization of gravitational field

In the vector theory of gravity for weak gravitational field and
nonrelativistic motion of masses the Lagrangian of the field interacting with
matter reads

$\mathcal{L}=\frac{c^{4}}{32\pi G}\int d^{3}x\left(-3\frac{\partial
h_{00}}{\partial x^{0}}\frac{\partial h_{00}}{\partial x^{0}}-\frac{\partial
h_{00}}{\partial x^{\alpha}}\frac{\partial h_{00}}{\partial
x^{\alpha}}-\frac{\partial h_{0\alpha}}{\partial x^{0}}\frac{\partial
h_{0\alpha}}{\partial x^{0}}\right.$ $+\left.\frac{\partial
h_{0\alpha}}{\partial x^{\beta}}\frac{\partial h_{0\alpha}}{\partial
x^{\beta}}-\frac{\partial h_{0\alpha}}{\partial x^{\beta}}\frac{\partial
h_{0\beta}}{\partial x^{\alpha}}+2\left[\frac{\partial h_{0\alpha}}{\partial
x^{0}}\frac{\partial h_{00}}{\partial x^{\alpha}}+\frac{\partial
h_{0\alpha}}{\partial x^{\alpha}}\frac{\partial h_{00}}{\partial
x^{0}}\right]\right)$
$+\sum_{a}\left(\frac{m_{a}\dot{r}_{a}^{2}}{2}-\frac{m_{a}}{2}c^{2}h_{00}+m_{a}c\mathbf{\dot{r}}_{a}\cdot\mathbf{h}\right),$
(96)

where $h_{0k}$ are components of the equivalent metric,
$\mathbf{h}=h^{0\alpha}$ and the sum is over all masses $m_{a}$ having
positions $\mathbf{r}_{a}$ and velocities $\mathbf{\dot{r}}_{a}$. Particle
momentum conjugate to $\mathbf{r}_{a}$ is

$\mathbf{p}_{a}=\frac{\partial\mathcal{L}}{\partial\mathbf{\dot{r}}_{a}}=m_{a}\mathbf{\dot{r}}_{a}+m_{a}c\mathbf{h(r}_{a}),$

while momentum $\Pi^{k}$ conjugate to the field $h_{0k}$ reads

$\Pi^{k}=\frac{\partial L}{\partial\dot{h}_{0k}}=-\frac{c^{3}}{16\pi
G}\left\\{\begin{array}[]{c}\pi^{0},\quad k=0\\\ E^{k},\quad
k=1,2,3\end{array},\right.$

where $L$ is the Lagrangian density,

$\pi^{0}=\text{div}\mathbf{h}+3\frac{\partial h_{00}}{\partial x^{0}},$

and

$\mathbf{E}=-\nabla h_{00}-\frac{\partial\mathbf{h}}{\partial x^{0}}.$

Classical Hamiltonian of the system is a functional of $\mathbf{r}_{a}$,
$\mathbf{p}_{a}$, $h_{0k}$ and $\Pi_{k}$

$\mathcal{H}=\sum_{a}\mathbf{p}_{a}\cdot\mathbf{\dot{r}}_{a}+\int
d^{3}x\Pi^{k}\dot{h}_{0k}-\mathcal{L}$ $=-\frac{c^{4}}{32\pi G}\int
d^{3}x\left(\mathbf{E}^{2}+\mathbf{B}^{2}\mathbf{+}\frac{1}{3}\left(\pi^{0}-\text{div}\mathbf{h}\right)^{2}-2h_{00}\text{div}\mathbf{E}\right)$
$+\sum_{a}\left(\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-m_{a}c\mathbf{h}\right)^{2}+\frac{m_{a}}{2}c^{2}h_{00}\right),$
(97)

where $\mathbf{B}=$curl$\mathbf{h}$. Transverse gravitational field
$\mathbf{h}_{\text{tr}}$ interacting with matter is described by the
Hamiltonian

$\mathcal{H}_{\text{tr}}=-\frac{c^{4}}{32\pi G}\int
d^{3}x\left(\mathbf{E}_{\text{tr}}^{2}+\mathbf{B}^{2}\right)+\sum_{a}\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-m_{a}c\mathbf{h}_{\text{tr}}\right)^{2},$
(98)

where

$\mathbf{E}_{\text{tr}}=-\frac{\partial\mathbf{h}_{\text{tr}}}{\partial
x^{0}}.$

Please note that the energy of the free classical gravitational field is
negative.

As in the case of electrodynamics, the field quantization procedure replaces
transverse gravitational field corresponding to radiation with the operator

$\mathbf{\hat{h}}_{\text{tr}}(\mathbf{r})=\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{8\pi
G\hbar}{pVc^{3}}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{h}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}+\text{H.c.}\right),$
(99)

where $\mathbf{\epsilon}_{\mathbf{p,}1}$ and
$\mathbf{\epsilon}_{\mathbf{p,}2}$ are spatial unit polarization vectors of
the right and left circularly polarized gravitons respectively. The non
radiative part of gravitational field is not quantized.

By analogy with the composite theory of the photon, we assume that graviton is
not an elementary particle but rather it is composed of fermion-antifermion
pairs. In the composite graviton model the part of the classical Hamiltonian
(98) describing radiation field interacting with matter is replaced with the
operator

$\mathcal{\hat{H}}_{\text{tr}}=\mathcal{\hat{H}}_{0}+\sum_{a}\frac{1}{2m_{a}}\left(\mathbf{p}_{a}-m_{a}c\mathbf{\hat{h}}_{\text{tr}}\right)^{2},$

where $\mathcal{\hat{H}}_{0}$ is the Hamiltonian of the free fermion field
(87) and $\mathbf{\hat{h}}_{\text{tr}}(\mathbf{r})$ is given by Eq. (99) in
which

$\hat{h}_{\mathbf{p},1}=\frac{1}{\sqrt{N_{\parallel}}}\sum_{\mathbf{k\parallel
p}}\hat{a}_{-\mathbf{k}}\hat{b}_{\mathbf{p}+\mathbf{k}},$ (100)
$\hat{h}_{\mathbf{p},2}=\frac{1}{\sqrt{N_{\parallel}}}\sum_{\mathbf{k\parallel
p}}\hat{d}_{-\mathbf{k}}\hat{c}_{\mathbf{p}+\mathbf{k}},$ (101)

$N_{\parallel}$ is the number of fermion states with wave vectors $\mathbf{k}$
parallel to $\mathbf{p}$ and summation is taken over all such states.

Since fermion $a$ is right-handed the operator $\hat{a}_{-\mathbf{k}}^{+}$
creates a spin $1/2$ fermion with spin parallel to $\mathbf{-k}$ and energy
$\varepsilon_{a}=\hbar ck$. On the other hand,
$\hat{b}_{\mathbf{p}+\mathbf{k}}^{+}$ creates a spin $1/2$ antifermion with
spin antiparallel to $\mathbf{p}+\mathbf{k}$ and negative energy
$\varepsilon_{b}=-\hbar c(p+k)$. Thus, the combination
$\hat{b}_{\mathbf{p}+\mathbf{k}}^{+}\hat{a}_{-\mathbf{k}}^{+}$ with
$\mathbf{k\parallel p}$ creates a fermion-antifermion pair with the total
negative energy $\varepsilon=-\hbar cp$ and spin $1$ antiparallel to
$\mathbf{p}$. The combination
$\hat{c}_{\mathbf{p}+\mathbf{k}}^{+}\hat{d}_{-\mathbf{k}}^{+}$ with
$\mathbf{k\parallel p}$ creates a fermion-antifermion pair with the same
energy $\varepsilon=-\hbar cp$ but with spin parallel to $\mathbf{p}$. If we
adopt the convention that for the left (right) circularly polarized graviton
the graviton spin is parallel (antiparallel) to the wave vector $\mathbf{p}$
then operator $\hat{h}_{\mathbf{p},1}^{+}$ ($\hat{h}_{\mathbf{p},2}^{+}$)
creates a right (left) circularly polarized graviton with spin $1$. Emission
of a single graviton corresponds to creation of $N_{\parallel}$ fermion-
antifermion pairs.

Operators $\hat{h}_{\mathbf{p},1}$ and $\hat{h}_{\mathbf{p},2}$ obey the
following commutation relations

$[\hat{h}_{\mathbf{p},1},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}1}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$-\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{b}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}\hat{b}_{\mathbf{p}+\mathbf{k}}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{a}_{-\mathbf{k}^{\prime}}^{+}\hat{a}_{-\mathbf{k}}\right),$
(102)
$[\hat{h}_{\mathbf{p,}2},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}2}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$-\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{c}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}\hat{c}_{\mathbf{p}+\mathbf{k}}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{d}_{-\mathbf{k}^{\prime}}^{+}\hat{d}_{-\mathbf{k}}\right),$
(103)

which also can be written in the form

$[\hat{h}_{\mathbf{p},1},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}1}^{+}]=-\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$+\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{b}_{\mathbf{p}+\mathbf{k}}\hat{b}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{a}_{-\mathbf{k}}\hat{a}_{-\mathbf{k}^{\prime}}^{+}\right),$
(104)
$[\hat{h}_{\mathbf{p,}2},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}2}^{+}]=-\delta_{\mathbf{p},\mathbf{p}^{\prime}}$
$+\frac{1}{N_{\parallel}}\sum_{\mathbf{k\parallel
p}}\sum_{\mathbf{k}^{\prime}\mathbf{\parallel
p}^{\prime}}\left(\delta_{\mathbf{k},\mathbf{k}^{\prime}}\hat{c}_{\mathbf{p}+\mathbf{k}}\hat{c}_{\mathbf{p}^{\prime}+\mathbf{k}}^{+}+\delta_{\mathbf{p}+\mathbf{k},\mathbf{p}^{\prime}+\mathbf{k}^{\prime}}\hat{d}_{-\mathbf{k}}\hat{d}_{-\mathbf{k}^{\prime}}^{+}\right).$
(105)

In Eqs. (102) and (103) the terms under the sum are written in the normal
order, while in Eqs. (104) and (105) the order is the opposite. At the moment
of Big Bang the fermion states are mostly empty, that is the total number of
fermion states is very large compared to the number of occupied states. In
this case in the limit $N_{\parallel}\rightarrow\infty$ Eqs. (102) and (103)
yield Bose–Einstein commutation relations for operators
$\hat{h}_{\mathbf{p},1}$ and $\hat{h}_{\mathbf{p},2}$

$[\hat{h}_{\mathbf{p},\mu},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2.$
(106)

All other commutators are equal to zero. At this stage of the Universe
evolution the graviton has negative energy $\varepsilon(\mathbf{p})=-\hbar cp$
which causes cosmological inflation.

Shortly after the Big Bang the fermion states become filled and remain filled
in the present epoch. Such filled states form a new vacuum. If the total
number of fermion states is very large compared to the number of empty states
we must use Eqs. (104) and (105) that in the limit
$N_{\parallel}\rightarrow\infty$ give

$[\hat{h}_{\mathbf{p},\mu},\hat{h}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=-\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2$
(107)

which differs from the Bose–Einstein commutation relations by the minus sign
in the right hand side. However, for operators

$\hat{A}_{\mathbf{p,}\mu}^{+}=\hat{h}_{-\mathbf{p},\mu},\quad\hat{A}_{\mathbf{p,}\mu}=\hat{h}_{-\mathbf{p},\mu}^{+}$

Eq. (107) yields Bose–Einstein commutation relations

$[\hat{A}_{\mathbf{p},\mu},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2,$
(108)

while Eq. (106) gives relations with the minus sign

$[\hat{A}_{\mathbf{p},\mu},\hat{A}_{\mathbf{p}^{\prime}\mathbf{,}\mu^{\prime}}^{+}]=-\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{\mu,\mu^{\prime}},\quad\mu,\mu^{\prime}=1,2.$
(109)

Operator $\hat{A}_{\mathbf{p},1}^{+}$ ($\hat{A}_{\mathbf{p},2}^{+}$) creates a
graviton with positive energy $\varepsilon(\mathbf{p})=\hbar cp$ and spin $1$
parallel (antiparallel) to $\mathbf{p}$. Thus, in the present epoch the
graviton energy is positive. Creation of such graviton corresponds to
annihilation of fermion-antifermion pairs out of the filled vacuum states
(creation of holes). In terms of $\hat{A}_{\mathbf{p},\mu}$ and
$\hat{A}_{\mathbf{p,}\mu}^{+}$ the field operator (99) reads

$\mathbf{\hat{h}}_{\text{tr}}(\mathbf{r})=\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{8\pi
G\hbar}{pVc^{3}}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}+\text{H.c.}\right),$

where now $\mathbf{\epsilon}_{\mathbf{p,}1}$ and
$\mathbf{\epsilon}_{\mathbf{p,}2}$ are spatial unit polarization vectors of
the left and right circularly polarized gravitons respectively.

Commutators of $\hat{A}_{\mathbf{p},\mu}$, gravitoelectric

$\mathbf{\hat{E}}_{\text{tr}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{8\pi
G\hbar
p}{Vc^{3}}}\left(\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right),$

and gravitomagnetic

$\mathbf{\hat{B}}(\mathbf{r})=i\sum_{\mathbf{p,}\mu=1,2}\sqrt{\frac{8\pi
G\hbar}{pVc^{3}}}\left(\mathbf{p}\times\mathbf{\epsilon}_{\mathbf{p,}\mu}\hat{A}_{\mathbf{p,}\mu}e^{i\mathbf{p}\cdot\mathbf{r}}-\text{H.c.}\right)$

field operators with the free field Hamiltonian (87) are the same as in
electrodynamics, namely

$[\mathcal{\hat{H}}_{0},\hat{A}_{\mathbf{p},\mu}]=-\hbar
cp\hat{A}_{\mathbf{p},\mu},\quad\mu=1,2,$
$[\mathcal{\hat{H}}_{0},\mathbf{\hat{E}}_{\text{tr}}]=-ic\hbar\,\text{curl}\mathbf{\hat{B},}$
$[\mathcal{\hat{H}}_{0},\mathbf{\hat{B}}]=ic\hbar\,\text{curl}\mathbf{\hat{E}}_{\text{tr}}.$

This result is independent of the commutation relation between
$\hat{A}_{\mathbf{p},\mu}$ and $\hat{A}_{\mathbf{p,}\mu}^{+}$. In addition,
the commutator

$[\hat{h}_{\text{tr}}^{\beta}(\mathbf{r}^{\prime}),\hat{B}_{\alpha}(\mathbf{r})]\mathbf{=}0,\quad\alpha,\beta=1,2,3$

remains the same. However, sign of the commutation relation between
$\mathbf{\hat{h}}_{\text{tr}}(\mathbf{r})$ and
$\mathbf{\hat{E}}_{\text{tr}}(\mathbf{r})$ depends on the vacuum state, namely

$[\hat{h}_{\text{tr}}^{\beta}(\mathbf{r}^{\prime}),\hat{E}_{\text{tr}}^{\alpha}(\mathbf{r})]\mathbf{=\mp}i\frac{16\pi
G\hbar}{c^{3}}\delta^{\beta\alpha}\delta(\mathbf{r-r}^{\prime}).$ (110)

The lower sign in Eq. (110) corresponds to Eq. (109), that is to the vacuum
with empty fermion states. The upper sign follows from Eq. (108) obtained for
the filled vacuum and is the same as in quantum electrodynamics.

In the Heisenberg picture the Heisenberg equations of motion yield Maxwell-
like equations for the transverse gravitational field in the operator form

$\text{curl}\mathbf{\hat{B}}=\pm\frac{16\pi
G}{c^{3}}\mathbf{j}_{\text{tr}}+\frac{1}{c}\frac{\partial\mathbf{\hat{E}}_{\text{tr}}}{\partial
t},\quad\text{curl}\mathbf{\hat{E}}_{\text{tr}}=-\frac{1}{c}\frac{\partial\mathbf{\hat{B}}}{\partial
t},$ (111)

where $\mathbf{j}_{\text{tr}}$ is the transverse part of the mass current
density

$\mathbf{j}=\sum_{a}m_{a}\mathbf{\dot{r}}_{a}\delta\left(\mathbf{r}-\mathbf{r}_{a}(t)\right).$

Hamiltonian (98) gives the following equation of motion of mass $m$ in
transverse gravitational field

$m\mathbf{\ddot{r}}=c^{2}\left[m\mathbf{E}_{\text{tr}}+\frac{m}{c}\left(\mathbf{\dot{r}}\times\mathbf{B}\right)\right].$
(112)

Comparison of Eqs. (111) and (112) with those of quantum electrodynamics
yields that quantum vector gravity, upto irrelevant numerical factor, is
equivalent to QED for the upper sign in Eq. (111), that is for the filled
vacuum (present epoch). Because of this equivalence, quantum vector gravity is
also a renormalizable theory.

The lower sign in Eq. (111) corresponds to the vacuum with empty fermion
states. This is the classical limit of the quantum vector gravity which
reproduces the classical weak field equations for the transverse field. In the
present epoch the fermion states are filled and we must take the upper sign in
Eq. (111). Thus, quantum mechanical analysis yields that evolution equations
describing gravitational radiation in the present epoch are different from
those that follow from the classical Lagrangian (96). Namely, in classical
equations describing radiation, $\mathbf{j}_{\text{tr}}$ must be taken with
the opposite sign.

One should mention that the difference in equations appears only when we are
dealing with the radiation part of the transverse field which is quantized.
Transverse gravitational field produced by stationary mass currents is not
quantized and is described by the same classical equations. As a consequence,
the Post-Newtonian limit of vector gravity is entirely classical and Post-
Newtonian equations are not modified by the quantum mechanical analysis. This
is also the case for Universe evolution after the end of the inflation stage.
In particular, dark energy comes from the classical longitudinal part of the
gravitational field.

Averaging the operator equations (111) over the state vector yields Maxwell-
like equations for the average fields $\mathbf{E}_{\text{tr}}$ and
$\mathbf{B}$. The averaged equations lead to the following expression for the
energy of the radiation field interacting with matter

$W_{\text{tr}}=\pm\frac{c^{4}}{32\pi G}\int
d^{3}x\left(\mathbf{E}_{\text{tr}}^{2}+\mathbf{B}^{2}\right)+\sum_{a}\frac{m_{a}\mathbf{\dot{r}}_{a}^{2}}{2}$
(113)

and the energy flux density (Poynting vector) of the radiation gravitational
field

$\mathbf{S}=\pm\frac{c^{5}}{16\pi G}\mathbf{E}_{\text{tr}}\times\mathbf{B}.$
(114)

Energy of the graviton in the classical limit (moment of the Big Bang) is
negative (lower sign in Eqs. (113) and (114)). In the present epoch the energy
is positive (upper sign). In this case the analogy of Eqs. (113) and (114)
with the corresponding expressions in electrodynamics

$W_{\text{tr}}=\frac{1}{8\pi}\int
d^{3}x\left(\mathbf{E}_{\text{tr}}^{2}+\mathbf{B}^{2}\right)+\sum_{a}\frac{m_{a}\mathbf{\dot{r}}_{a}^{2}}{2},\quad\mathbf{S}=\frac{c}{4\pi}\mathbf{E}_{\text{tr}}\times\mathbf{B}$
(115)

is obvious.

Finally we discuss quantization of the longitudinal gravitational waves (42)
which are described by equations

$\frac{\partial h_{00}}{\partial
x^{0}}+\frac{1}{2}\text{div}\mathbf{h}_{l}\mathbf{=}0,$ (116)
$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)\mathbf{h}_{l}=0.$ (117)

Classical analysis yields that such waves are not emitted or absorbed by
matter (see next Section). Quantum consideration gives the same answer.
Indeed, Eqs. (81), (82), (97), (116) and (117) suggest that longitudinal
gravitational waves must be quantized by replacing $\mathbf{h}_{l}$ and
$h_{00}$ with operators

$\mathbf{\hat{h}}_{l}(\mathbf{r})=\sum_{\mathbf{p}}\sqrt{\frac{32\pi
G\hbar}{3pVc^{3}}}\left(\hat{p}\hat{h}_{\mathbf{p}}e^{i\mathbf{p}\cdot\mathbf{r}}+\text{H.c.}\right),$
(118) $\hat{h}_{00}(\mathbf{r})=-\frac{1}{2}\sum_{\mathbf{p}}\sqrt{\frac{32\pi
G\hbar}{3pVc^{3}}}\left(\hat{h}_{\mathbf{p}}e^{i\mathbf{p}\cdot\mathbf{r}}+\text{H.c.}\right),$
(119)

where $\hat{p}$ is a unit vector in the direction of $\mathbf{p}$ and operator

$\hat{h}_{\mathbf{p}}=\frac{1}{\sqrt{N_{\parallel}}}\sum_{\mathbf{k\parallel
p}}\left(\hat{a}_{\mathbf{k}}\hat{a}_{\mathbf{p}+\mathbf{k}}^{+}+\hat{b}_{\mathbf{p}+\mathbf{k}}\hat{b}_{\mathbf{k}}^{+}+\hat{c}_{\mathbf{p}+\mathbf{k}}\hat{c}_{\mathbf{k}}^{+}+\hat{d}_{\mathbf{k}}\hat{d}_{\mathbf{p}+\mathbf{k}}^{+}\right)$
(120)

describes a composite particle with negative energy $\varepsilon(p)=-\hbar
cp$. Commutator of $\hat{h}_{\mathbf{p}}$ with the free field Hamiltonian (87)
is

$[\mathcal{\hat{H}}_{0},\hat{h}_{\mathbf{p}}]=\hbar cp\hat{h}_{\mathbf{p}}.$

It is easy to check that in the Heisenberg picture the Heisenberg equations of
motion involving the free field Hamiltonian (87) yield the free field Eqs.
(116) and (117) in the operator form

$\frac{\partial\hat{h}_{00}}{\partial
x^{0}}+\frac{1}{2}\text{div}\mathbf{\hat{h}}_{l}\mathbf{=}0,$ (121)
$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)\mathbf{\hat{h}}_{l}=0.$ (122)

In the limit $N_{\parallel}\rightarrow\infty$ we obtain commutation relations

$[\hat{h}_{\mathbf{p}},\hat{h}_{\mathbf{p}^{\prime}}^{+}]=[\hat{h}_{\mathbf{p}},\hat{h}_{\mathbf{p}^{\prime}}]=0$
(123)

for both empty and filled vacuum. $\hat{h}_{\mathbf{p}}$ and
$\hat{h}_{\mathbf{p}}^{+}$ also commute with the transverse field (graviton)
operators. Thus, if longitudinal waves are coupled with matter through the
operators $\hat{h}_{\mathbf{p}}$ and $\hat{h}_{\mathbf{p}}^{+}$ the commutator
of $\hat{h}_{\mathbf{p}}$ with the interaction Hamiltonian will be equal to
zero. As a result, the Heisenberg equations of motion involving the full
Hamiltonian will also give the free field equations (121) and (122) without
sources. This means that evolution of the longitudinal waves is not affected
by matter. Therefore, longitudinal waves are not produced by matter, at least
in the classical (empty vacuum) and quantum (filled vacuum) limits.
Commutation relations (123) indicate that longitudinal and time-like
components of the gravitational field behave as classical quantities.

##  XII Radiation of gravitational waves by system of masses

In this section we consider radiation of gravitational waves by a system of
stars moving with nonrelativistic velocities. Our analysis is also valid for
neutron stars which produce strong gravitational field in their vicinity. We
assume that stars have masses $M_{i}$ ($i=1,2,\ldots$) and move with
velocities $\mathbf{V}_{i}(t)\ll c$. Spacing between starts is large compared
to their dimension, however, the total size of the system is much smaller that
the radiation wavelength.

Strong gravitational field of neutron stars demands to keep nonlinear terms in
the equations for the gravitational field. Such terms are not confined to a
compact region, but extend over all space. However, nonlinear terms decay as
$1/r^{2}$ away from the star and can be large only in the vicinity of a
neutron star. We average the gravitational field equations (21) over the
volume large compared to the stellar size but much smaller than spacing
between stars. After such averaging the source of the gravitational field
becomes a sum of $\delta-$functions localized at the star positions. Nonlinear
terms in the stellar vicinity are subsumed into the $\delta-$function sources.
Nonlinear terms far from the star could give a small correction to the
solution in the wave zone of the order of $G^{2}$ which we neglect.

As a result, after averaging we obtain the following linear equations for the
gravitational field in the cosmological reference frame (cf. Eqs. (32) and
(33))

$\Delta h_{00}+3\frac{\partial^{2}h_{00}}{\partial x^{0}\partial
x^{0}}-2\frac{\partial^{2}h_{0\beta}}{\partial x^{0}\partial
x^{\beta}}=\frac{8\pi
G}{c^{2}}\sum_{i}M_{i}\delta\left(\mathbf{r}-\mathbf{r}_{i}(t)\right),$ (124)
$\left(\frac{\partial^{2}}{\partial x^{0}\partial
x^{0}}-\Delta\right)h_{0\alpha}+\frac{\partial^{2}h_{0\beta}}{\partial
x^{\alpha}\partial x^{\beta}}-2\frac{\partial^{2}h_{00}}{\partial
x^{\alpha}\partial x^{0}}$ $=\frac{16\pi
G}{c^{3}}\sum_{i}M_{i}^{\ast}V_{i}^{\alpha}\delta(\mathbf{r}-\mathbf{r}_{i}(t)),$
(125)

where $\mathbf{r}_{i}(t)$ are the radii vectors of the stars. In Eq. (124)
$M_{i}$ are the stellar masses measured by a distant observer. However, for
neutron stars, $M_{i}^{\ast}$ in Eq. (125) could differ from $M_{i}$ due to
strong-field effects and depend on the stellar equation of state. Next we show
that $M_{i}^{\ast}=M_{i}$ at least upto the second order in the stellar
velocity.

Taking $\partial/\partial x^{0}$ from Eq. (124) and $(1/2)\partial/\partial
x^{\alpha}$ from Eq. (125) and adding these equations together we obtain

$3\frac{\partial^{2}}{\partial x^{0}\partial x^{0}}\left(\frac{\partial
h_{00}}{\partial x^{0}}-\frac{1}{2}\frac{\partial h_{0\beta}}{\partial
x^{\beta}}\right)=$ $\frac{8\pi
G}{c^{2}}\sum_{i}\left(M_{i}\frac{\partial}{\partial
x^{0}}\delta(\mathbf{r}-\mathbf{r}_{i}(t))+\frac{1}{c}\frac{\partial}{\partial
x^{\alpha}}M_{i}^{\ast}V_{i}^{\alpha}\delta(\mathbf{r}-\mathbf{r}_{i}(t))\right).$
(126)

Using

$\frac{\partial}{\partial
x^{0}}\delta(\mathbf{r}-\mathbf{r}(t))=-\frac{1}{c}\frac{\partial}{\partial
x^{\alpha}}\left[V^{\alpha}\delta(\mathbf{r}-\mathbf{r}(t))\right]$

one can write Eq. (126) as

$3\frac{\partial^{2}}{\partial x^{0}\partial x^{0}}\left(\frac{\partial
h_{00}}{\partial x^{0}}-\frac{1}{2}\frac{\partial h_{0\beta}}{\partial
x^{\beta}}\right)=$ $\frac{8\pi
G}{c^{2}}\sum_{i}\left(M_{i}-M_{i}^{\ast}\right)\frac{\partial}{\partial
x^{0}}\delta(\mathbf{r}-\mathbf{r}_{i}(t)).$ (127)

The left hand side of Eq. (127) is of the order of $(V/c)^{3}$. This yields
that $M_{i}-M_{i}^{\ast}$ in the right hand side are of the order of
$(V/c)^{2}$.

Thus, with the required accuracy one can take $M_{i}^{\ast}=M_{i}$ in Eqs.
(124) and (125). Solution of Eqs. (124) and (125) satisfying the proper
boundary condition is given by the retarded potentials

$h_{00}=-\frac{2G}{c^{2}}\sum_{i}\frac{M_{i}}{\left|\mathbf{r}-\mathbf{r}_{i}(t_{i})\right|},$
(128)
$h_{0\alpha}=\frac{4G}{c^{3}}\sum_{i}\frac{M_{i}V_{i}^{\alpha}(t_{i})}{\left|\mathbf{r}-\mathbf{r}_{i}(t_{i})\right|},$
(129)

where $t_{i}$ is solution of the equation
$t=t_{i}+|\mathbf{r}-\mathbf{r}_{i}(t_{i})|/c$. Retarded potentials describe
outgoing waves with phase velocity directed away from the source. For solution
(128)-(129)

$\frac{\partial h_{00}}{\partial x^{0}}-\frac{1}{2}\frac{\partial
h_{0\beta}}{\partial x^{\beta}}=0.$

In Appendix I we provide an alternative derivation of the gravitational field
produced by an orbiting neutron star which is valid in the $V^{3}/c^{3}$ order
in the stellar velocity and arbitrary strength of the stellar gravitational
field $\phi$. The answer for the equivalent metric is given by Eqs.
(272)-(274). The analytical result includes both near and far field regions in
a single equation which is valid for arbitrary $\phi$ but omits retardation
effects. In the far field the answer reduces to Eqs. (275) and (276) of
Appendix I which match the retarded potentials (128) and (129) obtained here
using linearized equations. Such a match justifies omission of the nonlinear
terms in the present Eqs. (124) and (125).

Since $\sum_{i}M_{i}$ is constant, the time dependent part of $h_{00}$ in Eq.
(128) vanishes at large $r$ as $1/r^{2}$. Thus, $h_{00}$ does not contribute
to the emission of gravitational waves, as in the case of electromagnetic
radiation in classical electrodynamics.

In the present theory, a graviton is not an elementary particle, but rather it
is composed of fermion-antifermion pairs. Solution (129) is obtained in the
classical limit when vacuum corresponds to empty fermion states. However,
fermion states are filled in the present epoch and, as we showed in the
previous section, for filled vacuum the equations describing radiation of
gravitational waves must be modified by changing the sign of the mass current
to the opposite. As a consequence, solution (129) must be replaced with

$\mathbf{h}=\frac{4G}{c^{3}}\sum_{i}\frac{M_{i}\mathbf{V}_{i}(t_{i})}{\left|\mathbf{r}-\mathbf{r}_{i}(t_{i})\right|},$
(130)

where $\mathbf{h}=h^{0\alpha}=-h_{0\alpha}$. For filled vacuum emission of a
graviton corresponds to absorption of fermion-antifermion pairs out of the
filled vacuum states or creation of fermion-antifermion holes which propagate
away from the source. According to Eq. (113), for filled vacuum graviton has
positive energy and binary stars orbiting each other are loosing their energy
by emitting gravitational waves. Equation (130) has a very similar form to the
vector potential produced by moving charges $e_{i}$

$\mathbf{A}=\frac{1}{c}\sum_{i}\frac{e_{i}\mathbf{V}_{i}(t_{i})}{|\mathbf{r}-\mathbf{r}_{i}(t_{i})|},$
(131)

and, according to Eqs. (114) and (115), expressions for the Poynting vectors
of the radiation field in vector gravity and electrodynamics are also similar.
As a consequence, to obtain the answer for the power $P$ of emission of
gravitational waves by the system of masses we can apply the formula of
classical electrodynamics for the radiation of electromagnetic waves by a
system of charges

$P=\frac{2}{3c^{3}}\mathbf{\ddot{d}}^{2}+\frac{1}{180c^{5}}\dddot{D}_{\alpha\beta}^{2}+\frac{2}{3c^{3}}\mathbf{\ddot{m}}^{2},$

where

$\mathbf{d}=\sum e\mathbf{r}$

is the electric dipole moment of the system,

$D_{\alpha\beta}=\sum
e\left(3x_{\alpha}x_{\beta}-r^{2}\delta_{\alpha\beta}\right)$

is the quadrupole moment,

$\mathbf{m}=\frac{1}{2c}\sum e\left(\mathbf{r}\times\mathbf{V}\right)$

is the magnetic moment and the sum is over all charges in the system.

Comparison of Eqs. (130) and (131) yields that in order to obtain power loss
due to emission of gravitons one should in the electrodynamics equations
replace the electric charges with $e_{i}\rightarrow 4GM_{i}/c^{2}$. In
addition, Eq. (114) gives that gravitational energy density flux for the
radiation field is

$\mathbf{S}=-\frac{c^{5}}{16\pi G}\frac{\partial\mathbf{h}}{\partial
x^{0}}\times\text{curl}\mathbf{(h}),$ (132)

which is in a factor $c^{4}/4G$ greater than the corresponding expression for
the electromagnetic waves

$\mathbf{S}=\frac{c}{4\pi}\mathbf{E}_{\text{tr}}\times\mathbf{H}=-\frac{c}{4\pi}\frac{\partial\mathbf{A}}{\partial
x^{0}}\times\text{curl}\mathbf{(A}).$ (133)

Thus, $P$ for gravity must be also multiplied by $c^{4}/4G$.

Combining all factors together we finally obtain the following expression for
the power loss by the system of masses due to emission of gravitational waves

$P=\frac{8G}{3c^{3}}\mathbf{\ddot{d}}^{2}+\frac{G}{45c^{5}}\dddot{D}_{\alpha\beta}^{2}+\frac{2G}{3c^{5}}\mathbf{\ddot{L}}^{2},$
(134)

where we introduced the dipole moment of the system

$\mathbf{d}=\sum M\mathbf{r,}$

the quadrupole moment of masses

$D_{\alpha\beta}=\sum
M\left(3x_{\alpha}x_{\beta}-r^{2}\delta_{\alpha\beta}\right)$

and the net angular momentum

$\mathbf{L}=\sum M\left(\mathbf{r}\times\mathbf{V}\right).$

In these equations the summation is over all masses.

With the required accuracy, the stellar trajectories can be calculated in the
Newtonian limit. In such limit

$\mathbf{\dot{d}}=\sum_{i}M_{i}\mathbf{V}_{i}$

is the total linear momentum of the isolated system which is a conserved
quantity. Therefore, $\mathbf{\ddot{d}}$ vanishes and, hence, there is no
dipole radiation. For an isolated system the total angular momentum
$\mathbf{L}$ is also conserved and, thus, the last term in Eq. (134) also
vanishes. The fact that $M_{i}^{\ast}=M_{i}$ in Eqs. (124) and (125)
guarantees that inertial mass that determines the dipole moment is the same as
mass that generates gravitational waves. One should mention that possible
small deviation of $M_{i}^{\ast}$ from $M_{i}$ in our Eq. (125) due to strong-
field effects (which is of the order of $(V/c)^{2}$) might yield contribution
to the dipole radiation of the order of
$\mathbf{\ddot{d}}^{2}\propto(V/c)^{8}$, while
$\mathbf{\ddot{L}}^{2}\propto(V/c)^{10}$. These contributions are much smaller
than the quadrupole emission which is proportional to $(V/c)^{6}$.

As a result, the quadrupole radiation gives the dominant contribution to the
energy loss. The quadrupole term in Eq. (134) coincides with the Einstein’s
formula obtained in general relativity. The rate of loss of angular momentum
from a system of bodies emitting gravitational waves is also given by the same
equation as in general relativity

$\frac{dL_{\alpha}}{dt}=-\frac{2G}{45c^{5}}e_{\alpha\beta\gamma}\dddot{D}_{\beta\delta}\ddot{D}_{\delta\gamma},$
(135)

where $e_{\alpha\beta\gamma}$ is the Levi-Civita symbol. Equation (135) is
obtained in Land95  directly from the rate of energy loss by the system which
is given by the same quadrupole formula in both theories. As a consequence,
the present theory yields the same orbital decay of binary stars from
gravitational radiation as general relativity.

The energy flux at infinity carried out by gravitational radiation is balanced
by an equal loss of mechanical or orbital energy by the system $W$. This loss
of energy results in a decrease in the orbital period $T$ given by Kepler’s
third law Will93

$\frac{\dot{T}}{T}=-\frac{3}{2}\frac{\dot{W}}{W}.$

Such decrease in the orbital period has been measured for several binary
systems and agreed with the predictions of general relativity. Thus, it also
agrees with the vector theory of gravity.

In other alternative theories of gravity, while the inertial dipole moment may
remain uniform, the “gravity wave” dipole moment need not, because the mass
that generates gravitational waves depends differently on the internal
gravitational binding energy of each body than does the inertial mass Will93 ;
Will06 . In such theories, the additional form of gravitational radiation
damping (dipole radiation) could be significantly stronger for neutron stars
than the usual quadrupole damping. Our vector theory of gravity predicts no
dipole gravitational radiation because it satisfies the strong equivalence
principle at least to the post-Newtonian order.

##  XIII Neutron star mass limit in vector gravity

According to general relativity an object of nuclear density and more than
about $3M_{\odot}$ would be a black hole Frie87 ; Kalo96 . Here we examine the
neutron star upper mass limit in the vector theory of gravity. To calculate
the maximum mass of a neutron star, one must have an equation of state for
matter at high density which is very uncertain at present.

Taking $T_{k}^{i}$ as the energy momentum tensor of a perfect fluid, namely

$T_{0}^{0}=\varepsilon,\quad T_{\alpha}^{\alpha}=-P\text{,}$

all others are $0$, with energy density $\varepsilon=\varepsilon(r)$ and
pressure $P=P(r)$ the field equations for static gravitational field described
by the equivalent metric

$f_{ik}=\left(\begin{array}[]{cccc}e^{2\phi}&0&0&0\\\ 0&-e^{-2\phi}&0&0\\\
0&0&-e^{-2\phi}&0\\\ 0&0&0&-e^{-2\phi}\end{array}\right),$

become

$\Delta\phi=\frac{4\pi G}{c^{4}}\left(\varepsilon+3P\right)e^{-2\phi}.$

The energy-momentum relation

$T_{i;k}^{k}=0,$ (136)

where covariant derivative is taken using equivalent metric $f_{ik}$

$T_{i;k}^{k}=\frac{1}{\sqrt{-f}}\frac{\partial}{\partial
x^{k}}\left(\sqrt{-f}T_{i}^{k}\right)-\frac{1}{2}\frac{\partial
f_{kl}}{\partial x^{i}}T^{kl},$

yields equation of hydrostatic equilibrium

$\frac{\partial P}{\partial r}=-(P+\varepsilon)\frac{\partial\phi}{\partial
r},$

where $P$ is related to $\varepsilon$ by an equation of state

$P=P(\varepsilon).$

The boundary conditions at the stellar center $r=0$ and the surface $r=R$ read

$\phi^{\prime}(0)=0,\quad P(R)=0.$

Taking into account that outside the star $\phi(r)=-GM/rc^{2}$, continuity of
$\phi$ and $\phi^{\prime}$ yields additional boundary condition at the star
surface

$R\phi^{\prime}(R)=-\phi(R).$

Matching solution inside and outside the star yields expression for the
stellar mass $M$ in terms of $\phi(R)$

$M=-\frac{Rc^{2}}{G}\phi(R).$

It is now necessary to choose an equation of state. We take it in the form
which have been studied previously in general relativity Brec76

$P=b\left[\varepsilon-\varepsilon_{0}(\phi)\right],$ (137)

where $b$ is a constant and $\varepsilon_{0}(\phi)$ is the energy density
above which the equation of state becomes “stiff.”

Very often in literature the so called “causality” condition is imposed to the
equation of state: $dP/d\varepsilon\leq 1$, that is $b\leq 1$ in Eq. (137).
The condition requires that the speed of sound in the stellar matter $c_{s}$
can not exceed the speed of light $c$. However, though the last statement is
true it does not mean that we must impose the restriction $dP/d\varepsilon\leq
1$ to the possible equation of state. The point is that equations of
hydrodynamics, which result in $c_{s}=c\sqrt{dP/d\varepsilon}$, are derived
under the assumption of instantaneous interactions between particles (local
approximation). However, real interactions propagate with the speed of light.
As a result, if $dP/d\varepsilon>1$ the equations become substantially
nonlocal and static compressibility $dP/d\varepsilon$ no longer describes the
speed of sound. In this regime the speed of interaction propagation imposes
the restriction on the speed of compression waves in matter. As a consequence,
the speed of sound never exceeds the speed of light no matter what is the
value of the static compressibility $dP/d\varepsilon$.

One should mention that some authors also express caution about the
“causality” condition constraint on the equation of state. E.g., Kalogera and
Baym say Kalo96 : “The connection between the zero frequency sound velocity