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Lorentzinvariant theory of gravitation (LITG) is one of the alternatives to
general relativity in weak field approximation. The reason for its appearance
was at first the absence of Lorentz covariance in Newton's law of universal
gravitation. Subsequent development of LITG was stimulated by the presence of
problems existing in general relativity (GR). Although general relativity is
considered the most developed theory of gravitation, it has difficulty
explaining the fundamental nature of the fact of noninvariance
of gravitational field energy. In classical general relativity there are
problems describing the spinorbit interaction, the uniqueness of some results
and their consistency,^{[1]} impossibility of
constructing a quantum field model in a canonical way. LITG has the same
theoretical level as the electromagnetic theory of Maxwell. This follows from
the similarity of the basic equations of these theories, descriptions of field
with the two potentials and two strengths, the same degree of covariance under
coordinate transformations between two frames of reference (see also Maxwelllike gravitational equations). LITG
is the limit of covariant theory of
gravitation, when it is possible neglect the influence of gravitational
field on propagation of wave quanta and results of spacetime measurements. Gravitational field is considered at the same time as
one of the components of general field.
Contents

In one of his fundamental works ^{[2]}
Maxwell in 1865 suggested that gravitation could
be described by equations similar to equations of electromagnetism. However,
Maxwell used gravitational equations on the basis of mechanical analogies and
he could not understand the reason for the negativity of static gravitational
field energy and flux of gravitational energy, and therefore did not pursue
further the theory in this direction. Just as Weber modified Coulomb's law for electric
charges, so in 1870 Holzmüller ^{[3]}
and then Tisserand ^{[4]}
changed Newton's law, introducing the expression for the gravitational force
term depending on the relative velocity of two attracting particles. A
discussion of these innovations in the expression for the force can be found in
some works. ^{[5]} ^{[6]}
Apparently, one of the first scientists who described
mathematically perfect analogy between the electromagnetic and gravitational
theories, was Oliver Heaviside. Taking into account the coefficients
used in accordance with their system of physical units in his writings in 1893 ^{[7]} ^{[8]} Heaviside gave
correct expression for the curl of gravitational quantity, similar in the sense
to the magnetic field in electrodynamics. This quantity now determines the gravitational
torsion field and is
often referred to simply torsion, and in gravitoelectromagnetism,
if we consider it a part of general relativity in the limit of small field is
gravitomagnetic field strength.
Heaviside also introduced the vector of energy flux
density of gravitational field and defined the two components that make up the
total energy density of gravitational field, and then comes to the expression
for curl of gravitational field strength,
connecting it with the speed of change of torsion field. In second part of his
work ^{[9]} Heaviside applies his results to estimate
the total force between the Earth and the Sun, which includes the component of
force arising from the action of the Sun orbital torsion field on moving in its
orbit the Earth (if we consider only two gravitationally bound bodies, each of
them revolves around a common center of mass of the system on their own
orbits.) On the basis of possible disturbances in the Earth moving by the force
of Sun torsion, it concludes that the speed of gravity must be large, about the
speed of light. The fact that Heaviside came to LITG equations no surprise,
since he has given a modern form to Maxwell's equations in fourvector
differential equations (previously there were used 20 equations with 12 unknown
quantities).
In 1905, Poincaré in his article "On the dynamics of
electron» ^{[10]} asserts the need for Lorentz
covariance of gravitational force as a consequence of expansion of principle of
relativity not only for electromagnetic but also gravitational effects directly
in the inertial reference systems. This approach corresponds to the essence of
relativity principle of special relativity. Then Poincare considers parallel
motion of two bodies, fixed relative to each other in a frame of reference.
Based on the transformation of Lorentz group Poincaré
describes a number of invariants preserved under transformations, and discusses
their possible significance. In the Lorentz transformation the speed of light is presented as a
result of the procedure of spacetime measurements using electromagnetic waves.
This fact could induce Poincare to admit in the article that the speed of
gravity will equal the speed of light. Perhaps this would not have happened if the Poincare considered the
theory of relativity is not based on electromagnetic but gravitational waves
with their corresponding speed.
In papers of Poincare and Heaviside turns out that the
total force of gravitation has two components, one of which is proportional to
the vector distance to the attracting body, and the second component is
associated with the components of velocity vector of the body, taken at the
time when gravitational wave leaves the body. The second component of the
force, as noted by Poincare, behaves like a magnetic force in electrodynamics.
A detailed calculation of similar situation for two bodies was made in the book
^{[11]} as an illustration of LITG to describe
the motion of bodies. Derivation of Poincare about the origin of second
component of force fully confirmed, because without it are broken Lorentz
covariance and wellknown result of special theory of relativity (STR) about
time dilation in moving bodies (the change of force is accompanied by a change
in the length of unit time, given the fact that the force between objects in
each frame is defined as a change in momentum of the body per unit time). The
importance of Poincare contribution to the theory of gravitation is underlined
in the article. ^{[12]} Richard Gans
in his work ^{[13]} also comes to equations of
gravitation, like Maxwell's equations.
In 19081909 Minkowski published two papers on the
Lorentzinvariant theory of gravitation. ^{[14]} ^{[15]} For speed of gravity Minkowski takes a value equal
to the speed of light, and uses the same transformation of force as for Lorentz
force in electrodynamics.
The article of Sommerfeld ^{[16]} has
clarified several issues in LITG. Sommerfeld, in particular, transcribed
results of Poincare and Minkowski in 4vector formalism, and show their
similarities and differences. In 1910, there was also the paper of Lorenz.
^{[17]} The purpose of all of the above works was
primarily a representation of a modified Newton's law in its Lorentz invariant
form.^{ [18]}
In 1922 Felix Kottler ^{[19]}
displays a number of relations LITG in terms of vector and tensor algebra,
gives full expression for gravitational force and gravitational 4potential.
Unfortunately, these works have not been considered
sufficiently important, because it was thought that the results of LITG can be
deduced from general relativity in the weak field limit. In this case, it seems
LITG is some intermediate stage in the development of theory of gravitation.
Besides, LITG could not independently explain the shift of perihelion of
Mercury and other consequences of general relativity without using the idea of
space curvature and dependence of metric tensor on coordinates and time. ^{[20]} A typical example is a paper of J. J. Rawal and J.
V. Narlikar, ^{[21]} in
which considerations of Lorentz covariance gives the wave equation of LITG for
gravitational potential and the result is applied to the analysis of planetary
motion and gravitational redshift.
The idea that GR is probably not complete and not
sufficient to explain the full range of gravitational phenomena, was possibly
absent at that time (see below unclear questions in general relativity).
Among the works devoted to the development of LITG and retardation of gravitation in Newton's law, we can mention the article Whitrow and Morduch in 1960, ^{[22]} article of J. North, ^{[23]} articles of Kustaanheimo P., Nuotio V. S., ^{[24]} ^{[25]}, and the article Coster, H. G. L. and J. R. Shepanski. ^{[26] }
J. Carstoiu
introduced gravitational equations as Maxwelllike equations. ^{[27]}
In paper ^{[28]} are discussed
empirically equivalent theory of gravitation – standard general relativity,
Lorentzinvariant theory of gravitation, gravitational gauge theories such as
Lorentz's theory. Elements of theory LITG and some consequences described in
the book, ^{[29]} as well as in the articles. ^{[30]} ^{[31]} ^{[32]} ^{[33]} ^{[34]} ^{[35]} ^{[36]} The theory
of informatons by Antoine Acke
results in gravitational equations of LITG. ^{[37]}
The LITG equations are also presented
in the articles.
^{[38]} ^{[39]} ^{[40]}
In his studies professor at the University of West
Virginia Oleg D. Jefimenko, as
well as Heaviside and Poincare, considers a generalization of Newton's law of
universal gravitation, by introducing in the theory of second component of
gravitational field. This allows LITG satisfy the principle of causality and
makes it possible to describe the timedependent gravitational interactions. ^{[41]} ^{[42]} ^{[43]}
Full version of LITG was published also by Sergey G.
Fedosin, physicist and philosopher of Perm [1],
in 1999. LITG was built anew and independently from their predecessors, whose
works are seldom cited, and were therefore out of sight.
In analyzing the fundamentals and results of general
relativity, which is considered the modern theory of gravitation, may be found
the following points that require explanation or serious scientific
substantiation:
The above features of general relativity show that most
of problems of theory of gravitation may be removed by use of LITG with the
idea of using a metric similar to metric of general relativity, as a first
approximation to a more accurate theory of gravitational field. In this case,
general relativity becomes an extension of special relativity and has its
function in the case when the results of spacetime measurements are dependent
on existing in a system of electromagnetic and gravitational fields produced by
sources of charge and mass. If there were not of influence of gravitation on
propagation of light, similar to effects of deflection of electromagnetic waves
from the initial direction, changing the wavelength and speed of its
propagation, instead of general relativity would continue to operate special
relativity and would be valid LITG. As well as special relativity is not a
substitute of electrodynamics then general relativity can
not be instead of LITG or electrodynamics, which have arisen and exist
independently of general relativity. From the point of view of LITG,
EinsteinHilbert equations for metric are needed to determine the metric tensor
that defines effective properties of spacetime for a given energymomentum
distribution, and changes metric tensor of flat Minkowski space. After finding
the metric tensor from the equations for the metric, electrodynamics and LITG
are not just Lorentz covariant (it is a special case of covariance that take
place only in special relativity), but covariant for all the possible systems
of reference in which the metric can be found. It follows from the possibility
of writing the equations of these theories in the vector and tensor form. Then
LITG becomes the covariant theory of
gravitation (CTG).
A feature of LITG is that in it the force of gravitation,
in contrast to most other theories, including general relativity, is not a
consequence of curvature of spacetime, but a real physical force, determined by
covariant way in all reference frames. Taking into account the limited in value
speed of gravity and using the method calculating the field Liénard–Wiechert potential (Alfred Liénard,
1898, Emil Wiechert, 1900) inevitably leads to the
Lorentz covariance of gravitational field in the weak field limit and to the
need for torsion of gravitational field. LITG structure resembles the structure
of electromagnetic field theory, but a synthesis of GR and LITG ideas unlike
electromagnetism significantly alters the meaning and interpretation of the
theory of gravitation itself, resulting in a covariant theory of gravitation
(CTG). According to its position LITG is between the static Newton's
gravitational theory, not yet included the speed of propagation of gravity and
not calculated the force of gravitation in inertial reference frames, and
general theory of relativity, which considers phenomenon already in
noninertial reference systems through nonEuclidean geometry. LITG uses a
generalization of extended special theory
of relativity for gravitational phenomena.
Gravitational field equations in Euclidean space consist
of four vector differential equations for two strengths of the gravitational
field and can be regarded as Maxwelllike
gravitational equations. In International System of Units, these equations [2] are as follows:^{[7]} ^{[10]}
where:
These equations coincide in form with the equations
arising from general relativity in the weak field limit (see gravitoelectromagnetism). The torsion field lines are always closed, as in magnetic field, whereas the
gravitational field strength lines can escape
to infinity. As follows from the equations, the torsion is produced by the
motion of matter and change in time of gravitational field strength. When a
body rotates with constant angular speed torsion field
around the body has stationary character. When torsion field is changed over time, a vortical field of acceleration is
generated in space around the body. The total gravitational force acting on the
body has two components. One of them is normal force of Newton, which depends
on gravitational field strength and mass, while the other depends on the vector
product of velocity of the body and torsion field, which is in space at the
location of the body. Therefore, each body acts on other bodies, not only
through the gravitational field strength, but also through torsion created by
body rotation.
The expression for the gravitational force is as follows:
,
where:
This formula coincides with expression for the force of
general relativity in the weak field limit, ^{[67]} ^{[68]} although in some publications of general relativity
in the formula for force to the speed is factor of 2 due to the proposed doubling of the
mass for the field .
For the energy density, the vector of energy flux density (Heaviside vector), and the vector of momentum density of
gravitational field in LITG are obtained:
Main source: Gravitational
torsion field
For torsion outside of a rotating body from the field
equations by integrating over all points of the body can be derived the
formula:
,
where is angular momentum of rotation or spin of the body.
The torsion of gravitational field of a body at
rectilinear motion is:
where – speed of movement of the body, – field strength of the gravitational field of the body in the
point where the torsion is determined, and field
strength is taken in view of the delay distribution of the gravitational
perturbation.
In general, the torsion of an arbitrary point of moving
mass can be expressed in terms of gravitational field strength , which is produced by the point:
where is the unit vector directed from the point mass to the point where torsion
is determined, taken at an early time, dependent on the delay.
The formula for the torque acting on a rotating particle
with spin in torsion field ,
is written as follows:
Rotating particle can be considered like a top with spin . Under the action of the torque of torsion field particle will precess along the
field direction . This follows from the equations of
rotational motion:
.
Since the torque is perpendicular to the spin and torsion , then the same is true for the increment
of the spin for the time . Perpendicularity of and leads to the spin precession of a particle with angular velocity around direction of .
The last equality follows from the fact that , and ,
where is the angle between and , the angle
is measured from the projection of vector on the plane perpendicular to the vector , to the projection of vector on this plane.
In the presence of an inhomogeneous torsion field a
particle with spin will attract in the region of stronger field. From equations of LITG follow
the expression for such force:
The mechanical energy of a particle with spin in torsion
field will be:
The presence of torsion field in gravitational phenomena
leads to the effect of gravitational
induction.
If we introduce the notion scalar and vector potentials of gravitational field, then it is possible to express field
strength and torsion field :
As for the field strengths so for potentials themselves
in LITG true wave equations, depending on the mass density and mass current.
These wave equations are directly derived from the basic equations of the field
and have the form:
For the potentials is used gauge condition, which reduces
the degree of uncertainty:
The presence of wave equations for strengths and
potentials suggests that gravitational field propagates in the form of waves.
The speed of propagation of gravitational waves is assumed to be close to the
speed of light.
Scalar and vector potentials of gravitational field together form the gravitational fourpotential:
The wave equations for the potentials of gravitational
field can be expressed by one equation through D'Alembert operator , acting on fourpotential, and with a
constant factor equal to the fourmomentum density: ^{[69]}
where is a 4momentum density (current density of mass), which generates a
gravitational field, is fourvelocity, – mass density of matter in its rest
system.
When 4divergence of 4vector and are equal to zero, it is possible to set
gauge condition for potentials and continuity equation, respectively:
With the help of 4vector can be determined the antisymmetric gravitational
tensor:
In Minkowski space the components of this tensor are:
With the help of the tensor four vector equations of gravitational field are transformed into two
tensor equations:
,
.
The density of gravitational force is given by the
corresponding 4vector:
The tensor of gravitational field strengths allows
building of gravitational stressenergy
tensor:
where:
is the metric tensor in Minkowski spacetime.
Timelike components of the tensor are the energy density of gravitational field and the Heaviside vector, divided by speed of
gravity . The spacelike components form a threedimensional tensor of
gravitational stresses (of gravitational pressure), taken with the opposite
sign. Tensor is built from invariants of the tensor such that from it also possible to find the 4vector density of
gravitational force:
Thus, the theory of gravitational field can research
phenomena up to relativistic velocities of bodies.
For a single particle in a gravitational field Lagrangian
has the form:^{[11] }
,
where – invariant interval, – 4vector of particle displacement, – element of 3volume.
Time integral of the Lagrangian is a function of action,
by varying of which there are the Lagrange equations, which give equations of
motion of particles in gravitational field and equation for the field itself.
In particular, for a single particle is derived Newton's second law in
relativistic form, on which the rate of change of momentum of a particle with
time is equal to gravitational force.
From various experiments on the propagation of light near
massive bodies (see tests of general relativity) follow that gravitational
field of the bodies bends light rays, changing speed and frequency of
electromagnetic waves. This means that measured dimensions of the bodies and
time are dependent on their location in gravitational field, in particular from
the field potentials. Thus, there is a dependence of properties of spacetime of
used reference systems in gravitational field. Gravitation effectively distorts
the flat fourdimensional Minkowski world. To take this into consideration,
instead of the metric tensor in general is used the metric tensor .
The tensor of gravitational field strengths is still
determined by the 4potential:
The field equations in an arbitrary frame of reference
through the covariant derivatives of tensor gravitational field can be
expressed by two tensor equations:
,
Gauge condition for potentials and continuity equation
for the mass 4current are written respectively as follows:
Stressenergy tensor of gravitational field takes the
following form:
In general relativity is possible to calculate the metric
in the limit of weak fields inside a homogeneous sphere without internal
pressure, and without energymomentum of fields, when , where – a small supplement. For time components
of metric tensor follow ^{[11]}:
in static case, and
in dynamic case, where
These equations in terms of LITG mean that the additive
components to the metric tensor as it were twice as much dependent on the vector potential , than the components depend on the scalar gravitational potential . In general relativity, which are based
on the components of the metric tensor, or discover that being found from the
equations the vector potential in double "weaker" than the scalar
potential (this leads to LITG), or define a new vector potential in the form of . Hence, in some papers on gravitoelectromagnetism gravitational vector
potential, and after him, the vector of gravitomagnetic field twice differ,
respectively, on the magnitude of vector and the vector of torsion field , which are used in LITG. At the same time
it leads to a difference in two times in formula for the component of force
which is associated with gravitomagnetic field. There is a statement that the
gravitational mass for gravitomagnetic field two times higher than for
gravitoelectric field, as a consequence of tensor nature of the metric field of
general relativity. ^{[70]}
At the same time, in case of classical definition of
gravitational vector potential equations of general relativity in the weak
field limit coincide with the equations of LITG, ^{[71]}
^{[72]} ^{[73]} see also gravitoelectromagnetism. Indeed, in a weak
field LITG and GR operate effectively in Minkowski space and must be Lorentz
covariant. As a result, in publications of general relativity can be found at
least five different versions of formulas for the weak gravitational field and
the total force, ^{[74]} ^{[70]}
^{[75]} ^{[76]} so that
conclusions of general relativity for a weak field up to now is impossible to
recognize generally accepted.
As in general relativity, in formulas of LITG to account
for strongfield instead of the metric tensor is used metric tensor . As a result LITG transforms in covariant theory of gravitation (CTG).
However, the approach of LITG and CTG in relation of essence of gravitational
field is opposite to general relativity – if the geometry in general relativity
as it gives rise to gravitation, in LITG gravitational properties of bodies and
their surroundings alter the geometry of the world, which is observed and
measured by means of electromagnetic waves. The cause of gravitation in Le
Sage's theory of gravitation is in action of gravitons flows, ^{[77]}^{ }^{[78]}^{ }which is consistent with LITG, but contrary to a sense of general
relativity.
Due to the tensor in LITG and CTG automatically solve the
current problem of lack in general relativity of stressenergy tensor of gravitational
field. Tensor is involved in solving all the problems in finding the metric. Together
with the boundary conditions (for example, on the surface of massive bodies and
at infinity) it sets the conditions necessary for the correct identification of
reference systems, allowing to avoid the corresponding problem of general
relativity. For example, calculations made with respect to the contribution of
gravitational field energy in the metric, have shown that the additive is of
second order to the square of the speed of light and contains terms with fourth
power of the speed of light. ^{[11] [62]}
The general theory of relativity goes a step further with
respect to the theories of electromagnetism and gravitation field (LITG) – it
takes into account the fact that massenergy of fields affect
on the passage of time and values of the measured lengths by changing the speed
of electromagnetic waves (light), one way or otherwise used in measuring
instruments. At the same time, the electromagnetic wave is not the only one
that can be used for spacetime measurements. With the same success in terms of
theory it is possible to use gravitational waves. If their speed is not equal
to the speed of light, the content of the theory of relativity formulas is
different, since they include the speed of gravitational waves. It has been
shown in paper. ^{[79]}
According to LITG, gravitational field is a separate
physical field. A metric field consisting of the components of the metric
tensor and depends on time and coordinates of the point where it is defined, is
derived and is the total effect on the presentation of the mass density, the
pressure in it, the state of motion of the matter (speed, acceleration) as well
as available gravitational and electromagnetic fields and other possible values
of energymomentum. In contrast to general relativity, in LITG metric field is
not identified with the gravitational field, the metric field simply considered
necessary for a correct description of phenomena.
Since gravitational field is a vectorial field in LITG
(not a tensor, as in general relativity), then in LITG is allowed dipole
gravitational radiation. The power of this radiation for the case of a periodic
rotation of a body mass around the center of attraction is equal to:^{ [80]}
where is angular velocity of rotation, – radius of rotation.
From this formula it follows that a gravitational closed
system of a two bodies can emit only the quadrupole radiation (for each body,
the dipole radiation has the same power but is directed opposite to the other).
Indeed, the terms and in the formula for the emission of both bodies are equal to each other, and
the angular velocity can be considered the same. All of this correlates with
the observed absence of dipole gravitational radiation from close binary
neutron stars. It can be noted also that the above formula for the dipole
gravitational radiation corresponds to the formula for the dipole
electromagnetic radiation of rotating charge. It is known that if a system of
particles has the same charge to mass ratio for all particles then dipole
electromagnetic radiation is absent. ^{[81]} If in a system of two bodies on the stationary orbits one body is
charged, in addition to the gravitational electromagnetic radiation occurs. In
this case, we should expect not only quadrupole, but dipole gravitational
radiation from the system, since at their bodies there is a mismatch dipole
gravitational radiations.
According to LITG this phenomenon arises even in flat
Minkowski space between any two rotating objects, with its own angular momentum
or spin. Interpretation of the effect in LITG is so that rotating bodies create
around them torsion fields, which interact with each other in the same way as two
magnetic dipole. A similar interaction of the spins in general relativity is
called spinspin precession or spin Lense–Thirring precession or PughSchiff precession. This effect
relies a consequence of gravitomagnetic framedragging, that is dragging of freely
falling bodies near a massive rotating object. In general relativity,
gravitation is replaced by the curvature of space, so that the deviation of a
test particle from its normal geodesic line is due to the rotation of a massive
body and a corresponding change in the metric.
Due to the weakness of the effect it is desirable to have
at least one rotating body had great spin and therefore a large torsion field.
As such a body is convenient to take the Earth, and a second body – rapidly
spinning gyroscope in orbit around the Earth. Measuring the effect was carried
by satellite Gravity Probe B in 20042005. The formula for angular velocity of
precession of interaction of the spins in LITG is as follows:
and spin of the gyroscope precesses
about the direction of torsion field , which is created by the spin of the
Earth. The torsion field of the Earth as dipole field is given by:
where – spin (angular momentum) of the Earth, – distance from the center of Earth to the satellite, defined by the radius
of the Earth and height of the satellite (for Gravity Probe B the height was of 640 km.)
Torsion field in the motion of the satellite in orbit is
constantly changing, so for assessments of the precession is more convenient to
use the formula for the value of the effect in a constant field. Assume that
the gyroscope is kept just above the north pole of the Earth, where and are parallel, and the field has maximum. In this case, the formula for the
torsion field of the Earth is simplified, and angular velocity of precession is
equal to:
Under the condition of equality of the speed of gravity
and the speed of light, for Gravity Probe B value should be approximately equal to 0.0409 arcsecond per year or 6.28•10^{–15}
rad/sec. The same formula for effect, but after averaging along all the orbit,
is obtained in general relativity. ^{[45]}
In the motion of a test particle on a closed path around
a massive body with a spin, there is an effect of torsion field of the body
spin on the path of the particle. On the particle acts Lorentz force of
gravitation, creating a moment of force and causing a change in the direction
of the orbital angular momentum of the particle, i.e. orbital precession. The
equation of rotational motion of a particle is:
where force is equal to: , the quantities and denote the mass and velocity of the particle, and the orbital angular
momentum of the particle is
In the reference frame associated with the center of the
Earth, the vectors and are parallel to each other, and their cross product is zero. To calculate
the torsion field of the Earth, use the formula (1). For simplicity, assume
that the orbit of a particle purely circular, so that the radius vector of the
particle perpendicular to its velocity and . This gives:
It follows that the angular velocity of precession of
orbital angular momentum is equal to:
Accounting for the effect of gravitational field and the
Earth's rotation on spacetime metric in general relativity give the result,
that the angular velocity of precession of the orbital angular momentum becomes
more and approximately equal . ^{[82]} In
addition, the precession is not only for the orbital angular momentum of a test
particle, but also for the perihelion of its orbit. For satellites LAGEOS and
LAGEOS II angular velocity of precession of the nodes of orbit is obtained
about 0.031 arcsecond per year, with the distance from the satellite to the
Earth's surface of about 6000 km.
In geodetic effect, also called de Sitter effect are contributed two different phenomena. The first of
these may be called the spinorbit interaction. In the case of a gyroscope in
orbit around the Earth, this interaction can be understood as the effect of
torsion field from the orbital rotation of the Earth (relative to the reference
system rigidly connected to the center of mass of the gyroscope), on the spin
of the gyroscope. Earth rotates relative to the gyroscope with velocity , opposite to the direction of the
velocity of
the gyroscope relative to the Earth. The orbital torsion which is produced by
the Earth can be estimated by the formula:
where – field strength of Earth's gravitational field near the
gyroscope.
The angular velocity of the spinorbit precession will
be:
here was took into account that
The second term, making contributions to de Sitter effect
is related to the influence of gravitational field on the metric around the
Earth. The presence of the field leads to an effective curvature of spacetime,
which is expressed in an appropriate amendment to the metric tensor of flat
Minkowski space. ^{[44]} The magnitude of the
second term is two times more than As a result, the angular velocity
of de Sitter precession is:
Substituting the field strength of the Earth's gravitational field near
gyroscope where – the mass of the Earth, with condition for the gyroscope on the satellite Gravity Probe B
the angular velocity of precession is of the order of 6.6 arcsecond per year.
Refinement of LITG results and comparison with the
results of gravitational experiments carried out in covariant theory of gravitation. ^{[83]}
As it was shown in one paper, ^{[84]}
equivalence principle of general relativity does not hold with respect to the
massenergy of gravitational field itself. In particular, in the weak field
limit the gravitational massenergy of gravitational field of a stationary
body, and the inertial massenergy of field of moving with constant velocity
the same body does not coincide with each other. ^{[85]} A similar situation is known for
electromagnetic field and is called 4/3 problem. One possible explanation for
this is as follows. Most theories of gravitation, including LITG and general
relativity, only by mathematical language (with the help of symbols), or
geometrically, by means of spatial representations describe the phenomenon of
gravitation, without delving into its essence and not offering a specific
physical mechanism of interaction of gravitons with matter. So, general
relativity predicts black holes, based on the alleged large gravitational
force, able to effectively deal with the nuclear forces of repulsion of
nucleons in superdense matter of neutron stars and a more massive objects, and
compressing matter up to the state of gravitational singularity. These
assumptions lead to contradictions, like the fundamental unobservability of the
inner structure of black holes. In such cases, when the research reaches the
field carriers themselves and their interaction with matter, in the absence of
reliable data on the properties and energy density of field quantum conclusions
of the theories become inaccurate. The solution of such problems are expected
with the transition to quantum field theory level, which is difficult for
general relativity (see quantum gravity), but easier for LITG by the structure
of its equations, which coincides with the structure of the equations of
successfully quantized electrodynamics.
On the other hand, if to use Le Sage's theory of
gravitation as a model of gravitation, the difference in massenergy of the gravitational field of a stationary and
moving bodies could show the difference of relative rest and motion – in motion
the massenergy of the field increases 4/3 times by adding to the field energy
of work against the flow of gravitons, necessary to transfer the body from one
state of motion to another. The 4/3 problem can also
be solved for the whole system, if we introduce the two vector fields – acceleration field and pressure field. Both these fields, together
with the gravitational and electromagnetic field are combined in such a way
that the total massenergy of fields in the system vanishes. ^{[86]}
The articles ^{[61]} ^{[87]} within LITG equations (as well as the
equations of gravitomagnetism as
approximations of general relativity) clarify the relativistic expressions for
the energy and momentum of gravitational field inside and outside a homogeneous
sphere. The conclusion is that inequality of field massenergy, found from
gravitational energy and momentum of the field is an intrinsic property of the
field, contrary to the principle of equivalence between the gravitational and
inertial masses in general relativity. Analysis of the 4/3 problem, and ways to
include the mass of the field in common body mass leads to the following
expression: , where is
the rest energy, is
the negative energy of gravitational binding.
Meanwhile, in GR another expression is used: $~m=(E+E_{binding})/c^{2}$