Covariant theory
of gravitation
(CTG) is a theory of gravitation published by Sergey Fedosin in 2009. It
includes extended special theory of
relativity, Lorentzinvariant theory
of gravitation, metric theory of
relativity and Newtonian law of gravitation, and describes gravitation as a
physical force acting on the particles of matter. The matter, the gravitational
field, as well as other fields change such properties of wave quanta as their
propagation velocity and frequency of oscillations. Since the spacetime
measurements are carried out by means of waves, it follows that the observed
geometrical properties of spacetime depend on the sources of energymomentum in
the form of matter and fields which are present in the reference frame. This dependence is determined by the
field equations for the metric, forming a system of
partial differential equations. In CTG
gravitational field is a component of the general
field.
Just as the [[general relativity general theory of
relativity] (GTR) and some other alternative theories of gravitation, CTG
predicts change in the rate of time, the observed geometry of space, the
trajectories of falling bodies, propagation of light. However, there is a
difference between the predictions of GTR and CTG in the description of such
effects as gravitational time dilation, gravitational redshift of the
wavelength, signal delay in the gravitational field. This difference conforms
to the correction containing the fourth degree of the speed of light, within
the limits of which all the tests of GTR with respect to wave signals give the
same results as CTG. If gravitation in GTR is the consequence of the curvature
of spacetime by the sources of energymomentum, in CTG gravitation appears as
the result of the influence of gravitons on the matter within the framework of
the modernized Le Sage's theory of gravitation. The fluxes of gravitons also
affect the propagation of waves and hence the effective spacetime metric near
the sources of energymomentum, so in CTG geometry is secondary relative to the
physics of phenomena. In weak fields and at low velocities CTG turns into LITG. Since the equations of LITG are
similar to the equations of Maxwell's electrodynamics (see Maxwelllike gravitational equations) which
are successfully quantized, it allows you to quantize the equations of the gravitational
field of LITG in the framework of quantum gravitation.
Among the astrophysical applications, CTG as well as GTR,
based on the effect of light deflection in the gravitational field, predicts
the phenomenon of gravitational lensing, when there are multiple images of the
same remote astronomical object. CTG assumes gravitational emission from
particular accelerated massive bodies, and it can have dipole character
(whereas in GTR only quadrupole and multipole emissions are always considered).
Contents

The first step in development of CTG was presenting by
Fedosin the complete Lorentzinvariant
theory of gravitation (LITG) in the book in 1999. ^{[1]}
LITG is valid for inertial reference frames and describes all the gravitational
effects associated with delay of gravitation propagation and with the gravitational torsion field.
In 2002 the second book by Fedosin appeared, which was
devoted to the development of the theory of relativity. ^{[2]}
In this book the axioms of extended special
theory of relativity (ESTR) were formulated. In ESTR it is proved that the
constancy of the speed of light in all inertial reference frames, assumed by
the special theory of relativity (STR), is the result of the procedure of
spacetime measurements, in which twoway propagation of light (electromagnetic
wave) is always used. This leads to the averaging of the wave velocity in all
directions, regardless of the true speed of light and the velocity of the
reference frame, making the effective speed of light constant for each
observer. According to ESTR we can consider such an isotropic reference frame
in which the fluxes of gravitons have the same intensity from all sides. This
reference frame can be considered fixed relative to the ether as the medium consisting of the
fluxes of gravitons. In ESTR it is also shown that the theory of relativity as
the theory, which allows us to recalculate the results of measurements of the
coordinates, time and physical quantities from one frame to another, depends on
the wave representation, that is, on the type and the properties of the wave
used for spacetime measurements. This dependence is expressed in particular
through the Lorentz factor of the form containing the effective wave
velocity which depends on the properties of the medium
(e.g., the refractive index of the medium), and the velocity of
the reference frame as its average velocity in the period of the wave.
In his article in 2007 ^{[3]}
Fedosin draws a deep analogy between the electromagnetic and gravitational
fields, considering the similarity of their equations and the contribution that
the field as the sources of energymomentum must make in the result of
determining the spacetime metric through the HilbertEinstein equations. Another
article in 2008 ^{[4]} considers the violation of the
equivalence principle, which is the methodological basis of GTR, as applied to
the massenergy of the gravitational field. The following article deals with
the phenomenon of gravitation in the concept of gravitons (the Le Sage's theory
of gravitation) in the framework of the Theory of Infinite Hierarchical Nesting of Matter. ^{[5]}
Based on the concepts of gravitation as the force
interaction which arises from the action of gravitons and conforms to the
condition of Lorentz invariance in the inertial reference frames; the
conventionality of the constancy of the speed of light, following from the
measurement procedure; the dependence of the results of spacetime measurements
on the type and the properties of the used wave; the assumption about the same
propagation speed of gravitational and electromagnetic waves, based on the
model of the common ether and the assumed structure of the corresponding
photons, in his book in 2009 Fedosin develops CTG with the help of postulated
by him axioms of the metric theory of
relativity (MTR) and the covariant force equations of motion suitable for
all possible reference frames. ^{[6]} The structure
of CTG also includes the gravitational field equations of LITG, which are
generalized to any reference frames by means of replacing the metric tensor of
the flat Minkowski spacetime by the metric tensor of the curved spacetime, and
which use the operation of covariant differentiation.
Just as in GTR, in CTG the most difficult task is finding
exact solutions of the HilbertEinstein equations to determine the components
of the metric tensor. The solution of these equations in CTG is much more
complicated than in GTR, since in CTG, in contrast to GTR, the proper
gravitational field of the body is taken into account, which changes the metric
both inside and outside of the body. One of the exact solutions, determining
the metric tensor outside a single spherical body, was found by Fedosin. ^{[6]} Using this solution he described in the
framework of CTG the anomalous precession of the perihelion of planets,
including Mercury; the deflection of relativistic particles, radio signals and
the light of stars, passing close to the Sun's surface; the anomalous
acceleration of "Pioneers"; the gravitational redshift; the
gravitational time dilation; the effects associated with the spin, generating
gravitational torsion field.
The covariant theory of gravitation must be considered in
several aspects. On the one hand, CTG is the theory of the gravitational field.
On the other hand, CTG describes the interaction of the gravitational field
with the matter, and the gravitational force according to Newton second law
leads to acceleration of bodies. Besides, the gravitational field and other
sources of energymomentum influence the propagation of wave quanta, change
their velocity, energy and frequency. This leads to the effective curvature of
spacetime and to deviation of the form of the metric tensor from its value in
the flat Minkowski spacetime. In turn, the metric tensor and its derivatives
with respect to coordinates are involved in determining the gravitational force
and the quantities, characterizing the gravitational field and the motion of
bodies, from the perspective of the coordinate observer, in whose reference
frame the metric is calculated.
According to classical mechanics the motion of physical
bodies is described as the combination of free motion by inertia and some
deflection from it. The reasons for the deflection are various forces acting on
the body. According to Newton second law the force’s value is determined by the
product of the body mass and its acceleration. According to the Newtonian law
of gravitation between any two bodies the force of gravitational attraction
appears which is proportional to the masses of bodies and inversely
proportional to the square of the distance between them. Therefore, the
trajectory of the test particle near the massive body is deflected from a
straight line and the velocity changes due to the gravitational acceleration.
Due to the proportionality of the gravitational force to the gravitational mass
of test particles, the latter will move with the same acceleration near the
massive body. This means that the freefall trajectories of the particles
depend on the initial position and the initial velocity, but not on the mass of
the particles or their physical or chemical composition. This property of
gravitation is defined as the principle of universality of free fall (in GTR it
is called the weak equivalence principle) and is associated with the principle
of equivalence of the passive gravitational and inertial masses.
The last principle can be explained as follows. Due to
the equality of the gravitational acceleration, the masses of the test
particles on the surface of the massive body (for example, in the laboratory on
the Earth) can be determined simply by weighing (the more is the gravitation,
the greater is the mass, in this case, the passive gravitational mass). Then
any nongravitational forces, which give the test particles acceleration, equal
to the free fall acceleration on the Earth surface, can be equated by their
effect to the corresponding gravitational forces. Because in the Newton second
law for nongravitational forces there is the socalled inertial mass, then
from the equality of gravitational and nongravitational forces and
accelerations the equality follows of gravitational and inertial masses. In
other words, with the appropriate calibration of forces and accelerations both
masses can be equated to each other.
The Newtonian spacetime is characterized by Euclidean
geometry and the independence of the rate of time on the spatial coordinates
and the velocities of the bodies. Space and time depend neither on each other,
nor on the material bodies, nor on the motion of these bodies. Transformations
of the time and the coordinates from one frame to another are carried out by
means of the Galilean transformations, in which the measurements of time and
coordinates are carried out mechanically and not by means of electromagnetic
waves. To synchronize the clocks in each reference frame, they are transferred
from the origin of the reference frame to other points at infinitesimal
velocity. It is assumed that the speed of interaction transfer by means of the
force field is infinitely large.
The emergence in the early 20th century of the special
theory of relativity (STR) changed significantly the classical mechanics and
the theory of gravitation as the part of it, giving them relativistic form. The
Galilean transformations of classical mechanics were replaced by the Lorentz
transformations. Previously independent of each other, the space and the time
were combined into a single continuum, called Minkowski space. Mathematically
this was expressed in the transformation of the time coordinate (multiplied by
the speed of light to save the dimension) and the threedimensional position
vector of a point into the fourvector describing an event in the
fourdimensional Minkowski space. Scalar and 3vector physical quantities were
combined in the 4vectors and tensors, to obtain the value of which in
different inertial reference frames the Lorentz transformations should be used
(in general, in the presence of shifts and turns of reference frames Poincare
transformations should be used). Due to these innovations the mechanics became
consistent with electrodynamics and the concept of carrying out any spacetime
measurements by means of electromagnetic waves with finite velocity of
propagation. This allowed describing accurately the motion of particles even at
speeds, close to the speed of light, and the phenomena with the release of
energy, comparable to the rest energy.
In the theory of gravitation STR has led to the creation
of the Lorentzinvariant theory of gravitation
(LITG). In the inertial reference frames the force of gravitation must be
transformed in a Lorentzcovariant way as any other force. If there is a
physical system with a massive body and test particles, the observer can make
his inertial reference frame with the help of additional forces, which balance
the force of gravitation. For such observer the moving body creates not only
the gravitational field strength, but also the gravitational torsion field, influencing
additionally the moving test particles. The equations of LITG have relativistic
form and are similar to the Maxwell equations in electrodynamics. ^{[1]} The Newtonian law of gravitation is a special
case of LITG.
Clarification of SRT is done in the extended special theory of relativity, as
it was described above in the historic section.
Main source: Metric theory of relativity
The transition from the inertial to arbitrary reference
frames means transition from special (partial) relativity of inertial reference
frames to general relativity of accelerated reference frames. In case if a
force is acting on the reference frame reference, the frame begins to
accelerate and can no longer be considered inertial. In the presence of
acceleration the relation between the physical quantities in different
reference frames through the Lorentz transformation is inaccurate and requires
correction. The situation is more complicated when not only on the reference
frame but also to on all the matter in it the omnipresent force of gravitation
is acting. The example is the isolated massive body, near which the
trajectories of motion of test particles are deflected from the straight lines,
having a special name – the geodesic lines.
General relativity is contained in GTR in which
gravitational force and the difference of the geodesic lines from straight
lines are the consequence of the curvature of spacetime near the bodies.
Accordingly, the flat Minkowski space in the presence of gravitation looks like
pseudoRiemannian curved spacetime. In the free fall of particles near a
massive body the velocity of the particles and the gravitational acceleration
acting on them increase. Despite this, in GTR it is assumed that in the freely
falling reference frame the same laws hold as in the inertial Lorentz reference
frame. In this particular case general relativity differs not much from special
relativity, as indicated by some experiments with the propagation of light,
such as gravitational redshift. The assumption that freefalling frames are Minkowskian in GTR is called Einstein equivalence
principle. It means that the falling observer with the help of internal
experiments may not know whether he is falling in the uniform gravitational
field, or is moving by inertia without such a field. Obviously, this assumption
is only an idealization and in reality can be not satisfied. For example, in
the free fall of a charged test body by changing the acceleration of the fall
by the law of inverse square of the distance between the attracting center and
the body, there is electromagnetic emission, proportional to the charge of the
body, which is absent in the Lorentz inertial frames.
The common property of the gravitational field can be
considered the slowdown of the rate of electromagnetic clock in comparison with
the same clock outside the gravitational field. This follows from the reduction
of the speed of light as it approaches the massive bodies. The visible sizes of
bodies are also reduced in the direction of the gradient of the gravitational
field. In contrast to STR, in the presence of gravitation the position vector
is not a real 4vector, and the main role is played by 4vectors of
displacement (position shift). This means the impossibility to use the integral
Lorentz transformations for physical quantities. In particular, not the time
and the coordinates of events are subject to transformations from one frame to
another, but also the differentials of time and coordinates near these events.
In the Newtonian theory is the principle of equivalence
of accelerations: if all bodies in the reference frame are given the same
acceleration, then it is mechanically equivalent to the action of some uniform
gravitational field, creating at all points in space the same gravitational
acceleration. Einstein extended this principle to nonmechanical phenomena.
Based on this principle Fedosin determined the metric inside a uniformly
accelerated reference frame. ^{[6]} It allowed
finding the relation between the coordinates and time in the accelerated and
stationary reference frames. In particular it turned out that in the
accelerated reference frame the transverse dimensions visually decrease, in the
direction of the body’s acceleration they get longer, and at different points
of the accelerated frame the time flows differently with respect to the origin
of coordinates.
The main characteristic that specifies the geometry of curved
spacetime is the metric tensor. It can be used to calculate the Riemann
curvature tensor, as well as the connection coefficients, which determine the
parallel translation of the 4vector in curved spacetime. Since the components
of the metric tensor specify the angles between the unit vectors of the
coordinate axes of the reference frame, changing due to the curvature of
spacetime, in general case the metric tensor is the function of time and
coordinates. Due to its properties the metric tensor is included in the
equations of motion of test particles and wave quanta, is taken into account in
calculations of spacetime parameters and in measurements in the gravitational
field, as well as in recalculation of physical quantities from one reference
frame to another.
The generalization of general relativity is the metric theory of relativity (MTR), the
purpose of which is the expression of general relativity of phenomena in
different reference frames with the help of the metric. In MTR it is emphasized
that the geometry of spacetime is not absolute, it depends on the properties of
test particles and wave quanta used for the spacetime measurements and fixing
the metric. The metric can depend on the velocity of test particles, and can be
different for particles and wave quanta. By definition in MTR, the square of
the interval is equal to zero if it is connected with two close events on the
world line of test particles (wave quanta), used for spacetime measurements. In
this case, the speed of test particles (wave quanta) is included in the
expression for the 4vectors and tensors, so that physical quantities are
determined in the corresponding wave representation.
Theoretically, the propagation speed of light and
gravitational perturbations can be different, which can give two different
representations – for electromagnetic and gravitational waves, respectively. Just as in GTR, the metric in MTR is
found using the corresponding equations for the metric and depends on all the sources
of energymomentum available in the reference frame. An important difference of MTR from GTR is that
the gravitational field of the massive body, like any other field, becomes the
source of energymomentum and is involved in the determination of the metric.
If in GTR the dependence of the metric on time and coordinates as if gives rise
to gravitation, then in MTR this means that gravitation is not the consequence
of the curvature of spacetime, on the contrary, gravitation itself leads to
this dependence of the metric. In this case, the metric shows how under the
influence of energymomentum sources the difference of the phenomena occurs
from their form in the Minkowski space, in particular due to changing of the
velocity and frequency of electromagnetic waves and their deflection near
massive bodies, changing of the rate of time, etc. STR, ESTR and general
relativity in GTR are particular cases of MTR.
CTG includes three components:
The covariant gravitational field equations have the
form:
where is
the gravitational tensor; is the mass 4current (mass current
density), which generates the gravitational field; is the 4velocity of the matter unit in the curved
spacetime; is the 4vector of displacement; is
the differential of proper time; is the mass density in the frame at rest relative to the matter; is the gravitational
constant; is
the speed of gravitation, which is
assumed to be equal to the speed of light.
In contrast to Newtonian gravitational
theory, where the source of the gravitational force is assumed to be the mass
of bodies, in relativistic mechanics the mass density is part of the
stressenergy tensor, taking into account the energies of motion and pressure
in the matter units. This tensor is used in GTR as the source of
energymomentum to determine the metric inside the matter. In CTG as the
additional source the gravitational stressenergy tensor is used, which is not equal to zero even outside
the matter. As a result, the equation for the metric with condition can be written as follows: ^{[7]}
where is the Ricci tensor, which is the trace of the Riemann curvature tensor, is the scalar curvature, is the metric tensor, is the coefficient subject to be determined, is the acceleration stressenergy tensor, is the pressure stressenergy tensor, is the gravitational stressenergy tensor, is the stressenergy tensor of electromagnetic field.
Outside the
matter, in accordance with the procedure of energy and metric gauging, both the
cosmological constant and the scalar curvature, as well as tensors and become zero. ^{[8]} As a
result, the equation for the metric is simplified:
The equation of motion for the particles
has the following form:^{ [9]}
taking into account the expression for the
4vector of force density (see the 4force) through
the covariant derivative of the stressenergy tensor of the acceleration field
and through the operator of propertimederivative of the 4potential of the acceleration field in the
Riemannian space
where is the acceleration tensor, is the proper dynamic time of the
particle in its rest frame, is the tensor of electromagnetic field strengths, is the pressure field tensor, is the electromagnetic 4current
, is the density of the electric
charge of the matter unit in its rest frame.
In CTG it is considered that the ordinary
gravitational and electromagnetic forces are acting on the wave quanta in a
special way, the fields change their velocity and frequency more. This is due
to its proximity to zero of the rest mass and charge of the quanta, which leads
to nulling of the densities , , and correspondingly of the
4vectors and for quanta, and to reduction of the
action of forces on the quanta from the strengths of external fields.
Therefore, the covariant derivatives of
stressenergy tensors of the gravitational and electromagnetic fields which
specify the corresponding forces for quanta will be small. On the other hand,
for the electromagnetic waves the interval is set to zero: , which reflects the fact that these waves
are used for spacetime measurements (see also the third axiom of the metric
theory of relativity).
Since for the square of the interval the
relation holds: ,
then for the waves the differential of the proper time is also zero. If in the equation of
motion (1) we assume to be exactly equal to zero, then
in the equation uncertainty arises. It is to avoid this uncertainty by
multiplying equation (1) by the squared differential ,
and then dividing by the squared differential , where is the time parameter which marks the
position of the wave quantum in its trajectory. The right side of equation (1)
vanishes because of the presence of zero multiplier in the form of the
differential of the proper time , and for the electromagnetic waves the
equation of motion takes the following form:
The obtained equation of motion will have
the same form as in GTR for the waves on the zero geodesic line. In deriving
(2), it was taken into account that for solidstate point particles and wave
quanta, the 4potential of the acceleration field and the 4velocity are equal
to each other, . In addition, since , the following equality holds:
Just as GTR, CTG is the metric theory of gravitation. In
contrast to LITG, satisfying only the Lorentz transformations, the equations of
the gravitational field of CTG are written in covariant form and satisfy any
transformations possible for the reference frames. The equations of motion for
the particles and wave quanta are also covariant (covariance here means that
the equations are written in the tensor form suitable for any reference frames). Before finding the physical
quantities characterizing the gravitational field or the motion of test
particles, it is necessary to determine the metric tensor corresponding to the
distribution of the sources of energymomentum in this reference frame. For this purpose, the appropriate equations for the metric are used.
According to the axioms of CTG the source of the
gravitational field is the mass 4current , and the field itself is characterized by
the gravitational fourpotential ,
where is
the scalar potential and is the vector potential. Through the 4vector the antisymmetric gravitational tensor is determined in a
covariant way:
.
In turn, the tensor allows us to determine the gravitational stressenergy tensor:
.
In CTG gravitation is a real physical force which can be
explained in the framework of the Le Sage's theory of gravitation as the result
of the action of the fluxes of gravitons on the matter. Under the action of the
fluxes of gravitons near massive bodies the medium, in which the wave quanta
propagate, changes its properties so that the propagation velocity and the
frequency of the quanta become dependent on the gravitational potential.
The gravitational field equations of CTG are written in
the language of 4vectors and tensors of second rank. Due to the correspondence
principle in the weak field, these equations turn into the equations of LITG,
which are valid in the special theory of relativity (STR). In turn, for the fixed
bodies and with the zero vector gravitational potential, the equations of LITG
can be represented as one equation for the scalar gravitational potential,
which turns into the Poisson equation for the gravitational potential of
classical physics.
After we could write in the explicit form the axioms of
the general theory of relativity (GTR), it became possible to compare them with
the systems of axioms of the metric theory
of relativity (MTR) and CTG. ^{[}^{10}^{]} ^{[}^{11}^{]} It turns out that the equations of motion
of GTR are a special case of the equations of motion of CTG.
Comparison of the equations of motion for
particles in CTG and GTR shows their essential difference. The equation of motion in GTR for
particles in view of the electromagnetic field and its energymomentum tensor has
the form:
This equation of motion is not suitable
for the description of reaction propulsion. Meanwhile, if in the equation of
motion of the CTG (1) the mass density bring in the derivative, then for the condition for motion of solidstate particle in the equation will be the rate of
change of momentum density of matter due to changes in mass density.
If this density changes with time, its
derivative specifies the term for the reactive force, analogous to the
wellknown Meshcherskiy formula in classical mechanics for bodies
of variable mass. ^{[1}^{2}^{]} The
equation of motion of CTG has the meaning of the law of conservation of
energymomentum of matter which is under the action of forces in the
electromagnetic, gravitational and other fields. In contrast to this, the
equation of motion of GTR reflects only equivalence principle (the acceleration
of the fall is equal to the acceleration originating from the curvature of
spacetime and from the
nongravitational forces), and is
not connected with the law of conservation of energymomentum. Therefore, in
GTR there is no definite limiting transition to STR, i.e. to the case of weak
fields, which would be based on the principle of conformity and the laws of
conservation of physical quantities such as energy, momentum and angular
momentum. ^{[1}^{3}^{]}
In the presence of the moving matter and
propagating waves the solution of the equations of CTG becomes significantly
more complicated. Due to the motion of the sources of energymomentum and their
interaction with each other, the metric tensor in the considered reference
frame becomes dependent on time. This leads to the change of the motion of the
matter and waves, and the change of the field strengths, including due to the
contribution of the changing metric tensor. As a result, the gravitational
field equations, the equations for the metric and the equations of motion
become coupled and must be solved simultaneously. Since these equations contain
partial derivatives up to the second order, the exact solution is possible only
in some special cases.
For example, the metric near the massive
body, taking into account the energymomentum of its gravitational and
electromagnetic field, has been calculated by Fedosin. ^{[6]} ^{[1}^{4}^{]} The
contribution to the metric and to the total gravitational field from the test
particles is considered negligible, so that the motion of the particles is
regulated only by the gravitation of the massive body. The standard expression
for the square of the interval between two close points in all metric theories
is the following:
Substituting in this expression of the
metric tensor components, found for the space outside the isolated massive
body, in the fourdimensional spherical coordinates gives:
where is the time component of the metric
tensor.
The constants and can not be determined by solving
the equation for the metric, but their values can be found from the equations
of motion of particles and waves in any given form of the metric.
In the general case, for the solutions of
equations it is necessary, as in GTR, to use the numerical methods, the method
of small perturbations, the parameterized postNewtonian formalism (PPN
formalism) and other approximations. As a rule, the basic term of the given
approximations is determined by Newtonian gravitation, and the additions arise
from the general relativity of CTG (i.e., from the dependence of the results of
spacetime wave measurements on any sources of energymomentum). The feature of
PPN formalism is that it allows us to compare the various alternative theories
of gravitation.
The equations of motion of the matter, the
equations for determining the metric, the equations for the acceleration field and pressure field, gravitational and
electromagnetic fields can be derived from the principle of least
action.
In the case of the matter, continuously
distributed throughout the space volume, the action function for the matter in
the gravitational and electromagnetic fields in the covariant theory of
gravitation is given by: ^{[1}^{5}^{]} ^{[7]}
where is the Lagrange function or Lagrangian, is the time differential of the
used reference frame, is some coefficient, is the scalar curvature, is the cosmological constant, which
characterizes the energy density of the considered system as a whole, and
therefore is the function of the system, is the speed of light as the
measure of the propagation speed of the electromagnetic and gravitational
interactions,
is the electromagnetic 4potential,
where is the scalar potential and is the vector potential, – electric fourcurrent, is the electric
constant, – electromagnetic tensor, and are the
constants of acceleration field and pressure field, respectively, – 4potential of pressure
field,
is the invariant 4volume,
expressed through the differential of the time coordinate ,
through the product of the differentials of the spatial
coordinates, and through the square root of the determinant of the metric tensor, taken with the
negative sign.
Variations of the action function by the
metric tensor, by the coordinates, by the 4potentials of the field give the
EulerLagrange equations as the equations of motion of the metric, matter and fields.
Gravitational
AharonovBohm effect
Основная статья: Gravitational phase shift
The analysis
of the action function shows that it has the physical meaning of the function
describing the change of such intrinsic properties of bodies and reference
frames, as the rate of the proper time and the rate of increase of the phase
angle of periodic processes. For the gravitational and electromagnetic fields
difference of the clock in the weak field approximation is described by the
formulas: ^{[16]}
The clock 2,
which measures the time , is check one and the clock 1 measures the time and is under the influence of
additional 4field potentials or . Time points 1 and 2 within the
integrals indicate the beginning and the end of the field action.
The phase
shift for similar processes in the field and outside it, or occurring in
different states of motion is equal to:
The phase
shift, obtained due to the electromagnetic 4potential , is proved by the AharonovBohm effect. The
phase shift in the gravitational 4potential is also confirmed in the papers. ^{[17]}
^{[18]} ^{[}^{19]}
From the
above formulas for fixed clocks, located in the field close to each other at
points 1 and 3, the next equations are following:
This shows
that the rates of the clocks at the points with different potentials of the
field do not match. In the case of the gravitational field it gives gravitational time dilation.
With the help of the Legendre
transformation we can proceed from the known Lagrangian to the Hamiltonian in
the fourdimensional form. In the covariant theory of gravitation the
Hamiltonian is determined through the 4velocity, the scalar potentials and the
strengths of acceleration and pressure fields, of gravitational and
electromagnetic fields, taking into account the metric, and for the
continuously distributed matter it has the following form: ^{[7]}
where and are the scalar potentials of
acceleration and pressure fields, respectively.
If we introduce the 4vector of
generalized velocity with the covariant index:
then for the equation for the metric to
hold and to perform the calibration of the Hamiltonian, the following relation
is necessary:
Since the Hamiltonian specifies the
relativistic energy, it is included into the time component of the 4vector of
the Hamiltonian.
This 4vector can be written in the
contravariant form: where is the velocity of center of mass
of the system.
The fourmomentum of the system
is:
These effects include gravitational time dilation, gravitational
redshift of the wavelength, the signal delay in the gravitational field, the
deflection of light beams in the gravitational field of the Sun, and others.
Since the equation of motion of CTG for the wave quanta (2) almost coincides
with the corresponding equation of GTR, then the found metric coincides almost
exactly. The additional difference occurs due to the contribution to the metric
from the gravitational field, which is equal to and is included in . With the same degree of accuracy all the
effects of CTG, associated with the propagation of waves, give the same result
as GTR. For the waves from the equation of motion, the effect of the beam
deflection in the gravitational field and the gravitational time dilation it
follows that . Determination of the coefficient is
possible with the help of experiments on measuring the rate of time in the
gravitational field.
If we consider the electromagnetic field from the moving
charged particles, then it is characterized by the dipole, quadrupole, and
multipole emission. As a rule, the intensity of quadrupole emission and of the
subsequent multipoles is much less than the intensity of dipole emission. The
similar situation for gravitational emission takes place in CTG, as the
consequence of similarity of the equations of the electromagnetic and
gravitational fields and the vector character of the field sources. Meanwhile,
in GTR the dipole emission as such is absent, and the quadrupole and multipole
gravitational emissions are associated with the tensor sources of the field and
the metric oscillations propagating at the speed of light.
From the observations of the parameters of the orbits of
binary neutron stars and the speeds of their approaching the change of the total
energy of stars’ interaction due to the emission of gravitational waves is
estimated. ^{[}^{20}^{]} In such frames the emission can be only quadrupole, as the consequence of
stars’ rotation relative to their common center of mass. This conclusion
satisfies both CTG and GTR. Although the dipole and multipole gravitational
emissions can be calculated separately for each body, but in a closed frame the
total dipole emission of all the bodies of the frame tends to zero.
In some cases, in CTG the contributions from the
effective curvature of spacetime and from the forces arising from the ordinary
gravitational field and gravitational torsion field are combined. This leads to
different effects in the motion of test particles around massive bodies. Among
them, the precession of the perihelions of the orbits, the spin and orbital
LenseThirring effects, the geodetic precession, the effect of
"Pioneers", the approaching of the orbits of bodies due to the
emission of gravitational waves by them, etc.
The calculation of the finite motion of the test particle
around the massive body in the Kepler problem in CTG using the metric in the
square of the interval (3), given above, allows us to determine the constants and
comparing the results with the shift of the
perihelion of Mercury and other planets: where is
the quantity approximately equal to the velocity of the test particles in the
orbit. These values differ from the results of GTR, where for particles and
waves and there is no term in
the metric. The difference between CTG and GTR is due to the different
equations of motion for the particles (test bodies) and different metric.
During the rotation of the body there is the gravitational torsion field near it, the
main term of which is the dipole component of the torsion field proportional to
the spin (the proper angular momentum) of the body. In the formula for the
torsion field strength there is an inverse proportional dependence on the cube
of distance from the rotating body to the observation point, and on the square
of the velocity of the gravitation propagation. The latter indicates that the
torsion field is a relativistic effect and the consequence of the delay of the
change of the gravitational field during the motion of the bodies. Since these
effects are fully taken into account in STR, then to describe the interaction
of two fixed rotating bodies through the torsion field, in the first
approximation the formulas of LITG are sufficient, into which the formulas of
CTG turn in the weak field. ^{[1]} ^{[6]}
In particular, checking of the effect was carried out on
the Gravity Probe B satellite in 20042005 by
measuring the angular velocity of the precession of the gyroscope in the
torsion field of the Earth . If the gyroscope would always be located
only over the North Pole of the Earth, where the spin of the Earth and the radius vector of the distance from the center of the Earth to the satellite are parallel, the angular
velocity of the precession of the gyroscope would be equal to the maximum
value:
Under the condition of equality of the gravitation speed
and the speed of light, for the Gravity Probe B the value should be approximately equal to 0,0409 arc
seconds per year, or 6.28•10^{–15} rad/s. The same formula for the
effect is obtained in GTR, but later is was averaged over the entire orbit. ^{[}^{21}^{]} In GTR the effect of spinspin
interaction is called the LenseThirring spin effect or the Schiff effect, and
is assumed to be the consequence of dragging of the spin inertial reference
frames (framedragging). To describe the torsion field in GTR the socalled
gravitomagnetic field is often involved, see gravitoelectromagnetism.
If we give a test particle of some velocity of
the motion in its orbit around the rotating massive body with the spin ,
under the action of the torsion field from this spin the moment of force arises, changing the orbital angular
momentum of the particle:
where the force is equal: , denotes the mass of the particle, is the gravitational field strength
(gravitational acceleration) from the massive body, the radius vector of the distance is measured from the center of the rotating body to the test particle, and
the orbital angular momentum of the particle equals
If we express the torsion field through the spin of the body , then for the case of circular motion we
obtain the formula for the angular velocity of precession of the orbital plane
of the particle relative to the direction of the body’s spin:
This result, as it follows from GTR and the experiments,
should be doubled, because it does not take into account the spacetime metric
in the reference frame of the body. In this reference frame according to CTG
the gravitational field is rotating with the body and in the space there is the
torsion field, which makes its contribution to the metric as the source of energymomentum.
As a result, the metric near the rotating massive body differs from the metric
of a stationary body and by its form must resemble the Newman metric found in
GTR for the rotating and charged body. ^{[}^{22}^{]} In the Newman metric in comparison to the Reissner metric for a stationary
charged body, ^{[2}^{3}^{]} there is an additional quantity associated with the rotation of the body.
Therefore, in the metric of CTG near rotating massive body, taking into account
the energymomentum of the gravitational field of the body and the
energymomentum from rotation of the body, instead of the square of the
interval (3) and the coefficients of the type and in it there is a new form of the square of the interval and other
coefficients in the metric. By choosing their values in accordance with the
equation of motion of the test particle and the experiments on measuring the
orbital precession, it is possible to specify the form of the metric around a
rotating massive body from the perspective of CTG.
This type of precession is sometimes called the de Sitter
effect or the Fokker precession. This precession occurs during the orbital
motion of the test particle with the spin around the body, which may or may not
rotate (the presence of rotation of the body is shown as the absolute effect
and is expressed in the emergence of centripetal acceleration). The spin of the
particle tends to maintain its direction in space in any motion of the
particle. The parallel transfer of the spin of the particle in the orbit in the
curved spacetime around the massive body leads to the fact that the spin is
affected by the effective moment of force which changes its direction in space
and leads to the precession with some angular velocity.
Geodetic precession also occurs in LITG, where the
curvature of spacetime is not taken into account. From the perspective of the
observer in the reference frame of the rotating particle, the body moves around
the particle in some orbit, creating the torsion field. This torsion field acts
on the spin of the particle, creating the moment of force and the corresponding
spin precession of the particle. Both effects, from the spacetime curvature and
from the spinorbit interaction of torsion fields, depend on the same variables
and can be added. According to GTR and to the results of experiments, ^{[2}^{4}^{]} the contribution from the curvature of
spacetime is two times greater than the contribution from the interaction of
torsion fields. This gives the formula for the angular velocity of the
precession of gyroscope, which equivalent to the rotating test particle:
where is
the velocity of the motion of the gyroscope in the orbit, is the gravitational acceleration acting on the
gyroscope from the massive body, is the speed of gravitation propagation.
For accurate calculation of geodetic precession in CTG we
should use the form of the metric near the rotating massive body and with the
help of it calculate the orbital motion of the rotating test particle. As in
the case of the orbital LenseThirring effect, the indefinite coefficients in
the metric are subject to redefining in comparison with the experimental
results.
The difference of the methods of including the
gravitational field into equations for the metric and the discrepancy between
the equations of motion in CTG and GTR lead to the fact that in CTG it becomes
possible to explain the Pioneer anomaly. ^{[6]}
^{[2}^{5}^{]} This effect consists in the fact that in
measuring the frequency of wave signals from the spacecrafts on the Earth there
is difference from the predictions of GTR. According to CTG it is the consequence
of the inaccuracy of the equations of GTR. CTG predicts the difference in the
velocities of the spacecrafts, moving with the engines turned off and
decelerated by the attraction of the Sun, is of the order of several cm/s as
compared with the results of GTR in the Solar system limits. This difference in
the velocities, probably also manifested as a flyby
anomaly, ^{[2}^{6}^{]} apparently creates the Pioneer anomaly.
The equations of motion of the particles (1) and of the
wave quanta (2) in CTG different from each other by their form so that the wave
quanta seem not to be influenced by ordinary forces. This leads to the concept
of the dynamic proper time of the moving bodies, not coinciding with the time
determined by the wave (electromagnetic or gravitational) clock. The dynamic
proper time in the reference frame, which is at rest relative to the particle,
differs from the coordinate time of the reference frame, in which the motion of
the particle is considered, due to two effects. The first is associated with
the initial velocity of the particle and by its way of description is similar
to the Lorentz factor in STR. The second effect results from the action of the
gravitational field changing the initial velocity, and the total effect is
corrected by means of the metric. In polar coordinates, the proper time of the
particle is expressed through the metric coefficient and the radial and tangential velocities:
As it follows from the calculation of the motion of the
relativistic particle near the massive body with the mass , in the square of the interval (3) the
coefficient where is the velocity of the particle at infinity. This gives for the total
deflection angle of the relativistic particle from the rectilinear motion the
value ,
where is the impact parameter at infinity.
For nonrelativistic particles in their orbital motion as it was
described in the section on the perihelion of the planets. Given these
circumstances, with the typical orbital velocities of particles in the Solar
system, the contribution from the metric into the dynamic time of the particle
is small, and this time is almost entirely determined by the velocity of the
particle. The proper dynamic time of particles in CTG does not have much
importance, because in fact the time is always measured by electromagnetic
clock. For the clock, using the waves as the working matter, the use of the equation
of motion for the waves (2) gives the results similar to the results of GTR
(see above the effects associated with the propagation of waves). The wave
clock, except the gravitational potential, is also affected by the motion of
the clock, through the values of velocity and acceleration of its motion. As it
is shown in CTG, if we would use for measuring the time the wave clock, the
metric of which coincides with the metric of the test body, carrying this clock
(this happens in GTR due to the equivalence principle), then the time of this
clock would be the proper time of the test body only in the case, when the
direction of the waves in the clock and the direction of the velocity of the
test body lie in the same line.
One of the main consequences of CTG in respect of the
spacetime is the fact that in every system of bodies and test particles there
is its proper spacetime. If in GTR in the static case for one massive body and
one test particle the metric, which characterizes the spacetime, at each point
depends only on body mass, then in CTG the situation is different. In CTG the
metric depends on what is moving near the massive body, the metric is different
for the wave and particle, and depends on the properties of the test particle,
in particular, on its motion velocity. The dependence of the metric is realized
through the coefficients and ,
the values of which are determined by the properties of the studied
bodies, particles and waves. Thus, in CTG the concept of unified spacetime for
particles and waves is destroyed, which is typical of GTR. It also means the
inapplicability in CTG of the equivalence principle of GTR to describe the
motion of particles and waves. These consequences follow from the fact that in
CTG gravitation is a real physical force and not the result of curvature of the
unified spacetime as in GTR.
Changing of the concept of spacetime in CTG conforms to
the idea of the scale dimension which
determines the location of cosmic objects on the scale axis, and to the Theory
of Infinite Hierarchical Nesting of Matter.
At each basic matter level we can consider its proper gravitation (the examples
are the strong gravitation at the level
of atoms and the ordinary gravitation at the stars), and its proper spacetime,
and the rate of time at the lowest levels of matter increases. Due to SPФ symmetry, the equations of physics
remain the same if in the transition from one matter level to another, we shall
make in them the corresponding transformations of such physical quantities, as
mass, size and velocity. From this the relativity of spacetime follows, not
only from the point of view of the method of determining its properties by
definite measuring procedures, but also as the consequence of the location of
the reference frame on the scale ladder of matter.
Due to GTR in astrophysics such notions have become
familiar as gravitational lensing and microlensing, detectors of gravitational
waves, black holes, cosmological theories of the Universe. The phenomena
associated with waves in CTG almost exactly coincide with their description in
GTR. This concerns gravitational lensing, as the consequence of the deflection
of light beams from the distant source by some intermediate massive object
located in the way of the beams. However, the interpretation of black holes and
cosmological theories in CTG is different from the standard approach.
Main source: Black
hole
The most complete picture of black holes has been
developed in GTR. In this theory the proper gravitational field of the body
usually is not involved in determining the metric, the metric as the object of
geometric form defines gravitation and the gravitational force is the result of
the spacetime geometry. However, the question – why and how does the massive
body change the spacetime away from it – can not be answered by GTR. It is
common for GTR and the Newtonian gravitational model – in the latter the cause
of gravitation is also unknown, although there is its description as the
formula for the force. As a result, in GTR neither the maximum possible degree
of spacetime curvature actually realized in the nature is known, nor, correspondingly,
the maximum gravitational force.
If we assume that the speed of light is the limiting
speed of propagation of interactions, then this speed corresponds to the rest
energy of each body, which is also proportional to body mass (see the mass–energy
equivalence). Theoretically, in the formation of a black hole the massenergy
of its constituent matter should substantially decrease due to the contribution
of the negative massenergy of the gravitational field of the black hole. GTR
predicts for black holes the socalled event horizon and the singularity of
spacetime. It is assumed that the matter or emission, which are under the
horizon of the black hole, can not get outside of the horizon and must somehow
move inside the hole at relativistic velocities. From outside the black hole
must look as an allabsorbing dark object with the strong gravitational field.
In
CTG the description of objects with the gravitational field is made based on
the modernized Le Sage's theory of gravitation, in
which gravitation is generated by the fluxes of gravitons, penetrating all
bodies. The calculations allow us to deduce Newton's gravitational formula and
to estimate the spatial energy density of the fluxes of gravitons and their
penetrating ability in the matter. ^{[1]} ^{[5]} It becomes possible to understand
the origin of mass and inertia of a body, since the mass can be expressed
through the power of the energy flux of gravitons, interacting with the matter
of the body. ^{[2}^{7}^{]}
As gravitons the relativistic particles, photons and
neutrinos are assumed, generated by the matter at the lower scale levels of
matter. This is consistent with the essence of the Theory of Infinite Hierarchical Nesting of Matter,
according to which the objects similar to stars, white dwarfs and neutron
stars, create various relativistic particles and emissions at the same level of
matter, and these particles and emissions with sufficiently high density of
their energy lead to clustering and compression of the matter at a higher scale
level. Thus gravitons as the field quanta from the scattered matter generate
new compact objects, which in turn become the sources of new, more powerful
field quanta. In this process the density of gravitational energy, achieved in
material objects, decreases during the transition to more massive objects. In
the described picture there is no place for black holes in the traditional
sense. Here are some arguments against black holes:
In the present time the main cosmological models are the
models arising from the equations for the metric in GTR. As a rule it is
considered that the Universe is expanding after the Big Bang which happened in
the past, and the galaxies scatter from each other. Cosmological theories are
meant to describe the known experimental facts, such as the redshift of the
spectra of distant galaxies, homogeneity, isotropy and almost exact Euclidean
character of cosmic space and of the distribution of matter on large scales,
the distribution of concentrations of chemical elements in cosmos and the
nucleosynthesis, the structure and the forms of large galactic systems, the
isotropic cosmic microwave background radiation, the existence of dark matter,
etc.
In CTG the cosmological theories on the basis of GTR are
treated critically. In view of the Theory of Infinite Hierarchical Nesting of
Matter and the Le Sage's theory of gravitation, the Universe can be assumed
consisting of hierarchically related scale levels of matter. The observed part
of the Universe, called the Metagalaxy, according to the similarity of matter levels and the SPФ symmetry is similar in size and mass
to one of the objects of the metagalactic level of matter. At different scale
levels of matter the gravitational quanta and gravitational fields can act
which are different in energy density, penetrability and range of action. In
this case, applying the conclusions of GTR with respect to ordinary gravitation
to the supposedly homogeneous and infinite Universe is wrongful. Outside the
Metagalaxy we can expect voids in the distribution of matter, stretching up to
other similar objects. As for the experimental observations such as the
redshift of the spectra of distant galaxies or the background microwave
radiation, for all of them there are other explanations. For example, the
redshift and the Hubble constant can be connected with the absorption of the
energy of photons during their propagation in cosmological space. ^{[6]} ^{[}^{31}^{]} It is known that in the Big Bang theory the initial state of the Universe
is assumed the singularity of spacetime. Then GTR must explain not only the
emergence of the hypothetical state of singularity, but the reason for its
explosive instability. For the complete and selfconsistent explanation of this
problem in GTR quantum gravitation is considered necessary, which has not yet
been properly developed. In contrast to this in CTG neither black holes nor
singularities are required, which removes a number of problems in cosmology.
In the derivation of the equations of CTG from the principle
of least action we managed to show that the cosmological constant up to a
constant factor determines the massenergy density of the matter in the
Universe, without taking into account the contribution of massenergy of the
macroscopic gravitational and electromagnetic fields. This means that the
cosmological constant depends only on the fundamental microscopic fields acting
at the level of elementary particles. ^{[1}^{5}^{]}
According to the principle of equivalence of mass and energy, the inertial mass of an isolated object at rest can be found
through the energies of the matter and field associated with this object. To do
this in GTR it is necessary to sum all kinds of energy, including the rest
energy of matter, its internal energy and the energy of fields both inside and
outside the object. The sum of all the energies gives the relativistic energy,
which must be equal to the product of the object inertial mass and the squared speed of light. As
the mass density of the object increases due to reduction of the volume under
the influence of gravitation, the gravitational energy becomes more negative,
which according to GTR reduces both the relativistic energy and the mass of the
object, while the gravitational mass is
equal to intertial mass. Thus, the star must be less massive than the sum of the masses of all
the particles of which it consists.
In contrast to GTR, in CTG the other conclusions are made. This follows from the fact that the
gravitational energy must be part of the relativistic energy with the negative
sign due to the energy of matter in the potentials of field, and with the
positive sign in relation to the energy of the field associated with field
strengths. Are taken into account even the energy of matter in the acceleration field, in the pressure field and in the electromagnetic
field and the energies of these fields. ^{[}^{32}^{]}
As a result there is a difference between the gravitational mass and the
inertial mass of the system, so that the gravitational mass exceeds the
inertial mass. Considering the relativistic energy for the case of
spherically symmetrical collapse leads to four types of mass.
Gravitational mass is obtained equal to the mass ,
calculated as the integral of the density by volume. Inertial mass of the
system taking into account the particles and fields is , and the fourth mass is found from the condition of the
absence in the matter of the energy of macroscopic fields, and is obtained, for
example, after sputtering of matter, and removing it to infinity. With this
mass ratio is:
In GTR, the mass of the system on the
principle of equivalence is equal to the gravitational mass, and for the mass
ratio another expression obtained:
The article ^{[33]}
shows that the relativistic uniform system with continuous matter distribution
is characterized by five types of mass: the gauge mass is related to the cosmological
constant and represents the massenergy of the matter’s particles in the
fourpotentials of the system’s fields; the inertial mass ; the auxiliary mass is equal to the product of the
particles’ mass density by the volume of the system; the mass is the sum of the invariant
masses (rest masses) of the system’s particles, which is equal in value to the
gravitational mass . The relation for these masses
is as follows:
Relativistic uniform
system
Main source: Relativistic uniform system
In the gravitationallybound uniform
system, in the framework of CTG, we managed to calculate precisely the kinetic energy
of particles and to find the difference from the classical virial theorem,
taking into account the vector pressure field, acceleration
field and electromagnetic field. ^{[3}^{4}^{]} In particular, the ratio of the
kinetic energy to the energy of the forces, acting on the particles, turned out
to be equal to 0.6 instead of 0.5 in the classical case. Moreover, it was
proved that in the equation of motion the material derivative should be used,
because the velocity turned out to be the function of spatial coordinates.
An analysis of the integral theorem of generalized virial allows us to find
formulas for the radial component of the velocity of typical particles of the
system and for their rootmeansquare speed, without using the notion of
temperature. ^{[3}^{5}^{]} The relation between the theorem and the cosmological constant,
characterizing the physical system under consideration, is shown. The
difference is explained between the kinetic energy and the energy of motion,
the value of which is equal to half the sum of the Lagrangian and the
Hamiltonian.
The model allows us to estimate the particles’
velocity at the center of the sphere, the
corresponding Lorentz factor , the scalar potential of the pressure field; to find the
relationship between the field coefficients; to express the dependences of the
scalar curvature and the cosmological constant in the matter as functions of
the parameters of typical particles and field potentials. ^{[8]} Besides, comparison of the cosmological
constants inside a proton, a neutron star and in the observable Universe allows
us to explain the problem of the cosmological constant arising in the
LambdaCDM model.
In article, ^{[3}^{6}^{]}covariant formulas are derived for such additive integrals of system motion
as momentum, energy, fourmomentum, angular momentum, pseudotensor of angular
momentum, and also for the radiusvector of the center of the system momentum.
In a closed system, the integrals of motion are preserved, and the center of
the momentum moves at a constant speed. The difference between the
fourmomentum and the integral vector, obtained by integrating the equation of
motion through the energymomentum tensors of the fields, is shown. This
difference is associated with the difference of particles and fields as such.
The proton, neutron star and observable Universe are very close in their
properties to the relativistic uniform system. At the same time, they are
extremal objects in the sense that their gravitational field significantly
deviates from the form prescribed by the classical uniform system. ^{[3}^{7}^{]} For a neutron star, this
allows us to find the Lorentz factor for the motion of matter in the center of
the star, equal to 1.04. Similarly, for the proton, the Lorentz factor in the
center is 1.9. Analysis of the formula for the gravitational field allows us to
explain the weakening of the field at the boundaries of the Metagalaxy, which manifests
itself in the largescale cellular structure of the Universe.
Model of gravitational
equilibrium
This model is used to describe the
internal parameters of such cosmic objects as planets and stars. In contrast to
the polytropic model, relating the pressure and density by means of a certain
assumed power law, the model of gravitational equilibrium is the consequence of
the equations of the gravitational field, pressure field, acceleration field,
electromagnetic field and other fields, acting in the matter. The approach
under consideration allows us to find the distribution of the internal
pressure, temperature, and other parameters. ^{[3}^{8}^{]} The acceleration field coefficient η and the pressure field coefficient σ are the functions of the state of matter,
and their sum is close in magnitude to the gravitational constant G. For macroscopic objects the
gravitational field is the main component of the general field.
^{[3}^{9}^{]} ^{[40]}
NavierStokes equation
The phenomenological NavierStokes
equation describes the motion of the viscous fluid with regard to the dissipation
field. The gravitational and electromagnetic fields are
included in this equation, providing the socalled mass terms in the
expressions for the acting forces. It is possible to derive the NavierStokes
equation in a covariant way, taking into account the acceleration field
and the equation of the matter’s motion in the CTG. ^{[41]}
The viscosity effect is described in
this approach by the 4potential of the field of energy dissipation, dissipation
field tensor and dissipation stressenergy
tensor. A complete set of equations is presented, which
suffices to solve the problem of motion of viscous compressible and charged
fluid in the gravitational and electromagnetic fields.
References
Theories of
gravitation 

