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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]}
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. ^{[37]} ^{[38]} ^{[39]}
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 shows 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, ^{[63]} ^{[64]} 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: ^{[65]}
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. ^{[66]}
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, ^{[67]} ^{[68]} ^{[69]} 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, ^{[70]} ^{[66]} ^{[71]} ^{[72]} 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, ^{[73]}^{ }^{[74]}^{ }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] [58]}
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. ^{[75]}
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:^{ [76]}
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. ^{[77]} 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. ^{[41]}
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 . ^{[78]} 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. ^{[40]} 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.
As it was shown in one paper, ^{[79]} 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. ^{[80]} 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. ^{[81]}
The articles ^{[57]} ^{[82]} 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: $~m=(E+E_{binding})/c^{2}$