# Physics/Essays/Fedosin/Lorentz-invariant theory of gravitation

Lorentz-invariant theory of gravitation (LITG) is one of 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 of fundamental nature in explaining the fact of noninvariance of gravitational field energy. In classical general relativity there are problems describing spin-orbital interaction, uniqueness of some results and their consistency, impossibility of constructing a quantum field model in a canonical way.
LITG has the same theoretical level as electromagnetic theory of Maxwell. This follows from the similarity of 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 reference frames (see also Maxwell-like 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.

## Historical background

In one of his fundamental works Maxwell in 1865 used Newton's law to compare energy of gravitational interaction between two bodies and energy of interaction between two magnets. However, Maxwell, based on the mechanical analogies he used, could not understand the reason for negativity of both the energy of static gravitational field and flux of gravitational energy. This could be the reason why he did not describe gravitation with equations similar to the equations of electromagnetism.
Just as Weber modified Coulomb's law for electric charges, so in 1870 Holzmüller and then Tisserand changed Newton's law, introducing an expression for gravitational force term depending on relative velocity of two attracting particles. A discussion of these innovations in expression for the force can be found in some works.
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 Heaviside gave correct expression for curl of a gravitational quantity, similar in the sense to the magnetic field in electrodynamics. This quantity now determines the gravitational torsion field, 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 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 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 momentum of the system on their own orbits.) On the basis of possible disturbances in the Earth motion by the force of Sun torsion field, it concludes that the speed of gravity must be large, about the speed of light. The fact that Heaviside came to LITG equations is no surprise, since he has given a modern form to Maxwell's equations in four-vector differential equations (previously there were used 20 equations with 12 unknown quantities).
In 1905, Poincaré in his article "On the dynamics of electron» 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 procedure of space-time 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 Poincare had considered that the theory of relativity is not based on electromagnetic waves but gravitational waves with their corresponding speed.
In the papers of Poincare and Heaviside, it turns out that the total force of gravitation between two bodies has two components, one of which is proportional to vector distance to attracting body. The second component is associated with 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 a similar situation for two bodies was made in the book as an illustration of LITG to describe the motion of bodies. Derivation by Poincare about the origin of second component of the force is fully confirmed, because without it the Lorentz covariance is broken as well as the well-known 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's contribution to the theory of gravitation is underlined in the article.
Richard Gans in his work also comes to equations of gravitation, like Maxwell's equations.
In 1908-1909 Minkowski published two papers on the Lorentz-invariant theory of gravitation. 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 has clarified several issues in LITG. Sommerfeld, in particular, transcribed results of Poincare and Minkowski in 4-vector formalism, and showed their similarities and differences. In 1910, there was also the paper of Lorenz. The purpose of all of the above works was primarily a representation of a modified Newton's law in its Lorentz invariant form.
In 1922 Felix Kottler displays a number of relations LITG in terms of vector and tensor algebra, gives full expression for gravitational force and gravitational four-potential.
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 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.
A typical example is a paper of J. J. Rawal and J. V. Narlikar, in which considerations of Lorentz covariance gives the wave equation of LITG for gravitational potential and the result is applied to 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).

## Modern period

Among the works devoted to development of LITG and retardation of gravitation in Newton's law, we can mention the article Whitrow and Morduch in 1960, article of J. North, articles of Kustaanheimo P., Nuotio V. S.,, and the article Coster, H. G. L. and J. R. Shepanski. J. Carstoiu introduced gravitational equations as Maxwell-like equations.
In paper are discussed empirically equivalent theory of gravitation – standard general relativity, Lorentz-invariant theory of gravitation, gravitational gauge theories such as Lorentz's theory. Elements of theory LITG and some consequences described in the book, as well as in the articles.
The Theory of Informatons by Antoine Acke results in gravitational equations of LITG. The LITG equations are also presented in the articles.
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 time-dependent gravitational interactions.
Full version of LITG was published also by Sergey G. Fedosin, physicist and philosopher of Perm , in 1999. LITG was built anew and independently from their predecessors, whose works are seldom cited, and were therefore out of sight.

## Unclear questions in general relativity

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:
* GR is not the theory for using geometry as a subordinate science to explain the physical properties of interacting bodies but the theory in which spacetime geometry is at the main place, replacing the action of observed gravitational forces by curvature of spacetime.
* Motion along geodesics in general relativity is not the same for all possible test particles: for neutral particles near a massive charged and magnetized rotating body, and for charged particles trajectories do not coincide, despite the same spacetime metric. Thus called into question the universality of principle of geometrization of physical force and ability to define a single metric by means of test particles.
* There is a contrast for motion of two test particles in gravitational field, if one of them has not its own rotation, and the other has a spin. Description of test particle spin in the latter case is not unique in general relativity, as there are various distinct approaches for it. The difference between motions of test particles shall be greater, if they fall in gravitational field of a rotating massive body. In this case, the test particle with spin is subjected to gravitomagnetic field, leading to the spin precession of test particle (Pugh-Schiff precession). Such precession can not be determined experimentally in a test particle without spin, because of the lack of spin.
* As the basis of general relativity are used: a) the equivalence principle or the principle of universality of free fall (free-falling bodies at the same initial conditions have the same acceleration, independent of the mass and composition of matter of these bodies); b) the principle of local Lorentz invariance (time is not dependent on speed of clock); c) the principle of local invariance of location (time does not depend on when and where in space to make measurements). These principles are idealized and are suitable only for small-size particles that can not significantly change the gravitational field in which they fall. However, if the test bodies are massive enough, they change the metric of spacetime around itself, so that the geodesic lines of different bodies are no longer the same. In massive test bodies may appear substantial internal gravitational stresses and pressure of matter, so that time at different points of a test body will depend on location of clocks, and on speed of center of momentum of the body. Test bodies can be active, they can interact with each other, emit particles, etc. All this leads to a violation of the above principles and inability to use them as axioms of general relativity. If a test body is comparable to influence of the main body in a gravitational field, in which it is located, GR can not build an analytical solution for the metric of two bodies system, even for the case when the bodies are represented by points. In this case, it becomes difficult to deduce the conditions under which principle of equivalence can be considered as an effective one, taking into account gravitational binding energy and other properties of the test body.
* Even if we ignore the intrinsic properties of test particles and consider them as ideal, the principle of equivalence is not already running. The fact is that when particles move near massive bodies the particles have real dynamic gravitational perturbations, which depend on mass of test particles and material properties of massive bodies. An example is the lunar tides, so Earth's rotation is slowing down and lead to removal of the Moon from the Earth at a rate of about 3.82 cm per year.
* In general relativity, a long time had not been built the original system of axioms, which prevented delineate the boundaries of the theory and verify its authenticity in the assumptions. Axiomatization of general relativity was implemented on the basis of the axioms of metric theory of relativity (MTR) and covariant theory of gravitation (CTG). The results showed that general relativity is a special case of MTR, and equation of motion of general relativity is a special case of equation of motion of CTG.
* The gravitational field in GR is not defined as physical, but as a geometric object, since the corresponding stress–energy tensor of the field, taken as a covariant form in any frame of reference is missing. Instead is used pseudotensor, but even his form clearly is not defined – there are various options, such as by Einstein, by Landau–Lifshitz, and by Møller-Mickiewicz.
* In general relativity, is not known where and how much gravitational energy is concentrated. In this context general relativity can not be considered as an independent theory of gravitational field, with no idea of the most important characteristic of the field as the object of its study. In addition, in GR energy is determined locally, but there is difficulty in finding the total energy is taken over all space. The conservation laws in general relativity established only for two extreme idealized situations – for island systems, surrounded by empty flat Minkowski space, and for the universe as a whole. The absence of a general expression for the energy of gravitational field leads to the fact that contribution of the energy in the body mass is not accurate. Moreover, adopted in GR definition of body mass in a weak field limit (see Mass in general relativity#The Newtonian limit for nearly flat space-times), according to which gravitational mass decreases due to negative mass-energy of gravitational field does not coincide with definition of mass in LITG where gravitational mass of a body increases in its gravitational field.
* GR does not have a definite limit passage in special relativity, that is, in the case of weak fields, based on the correspondence principle and conservation laws of physical quantities such as energy, momentum and angular momentum.
* In general relativity, there is a problem with analysis of emergence and propagation of gravitational waves. Due to the lack of analytical solutions of general relativity in two-body problem, it becomes difficult to formulate the initial conditions for generation of gravitational waves by any two interacting sources.
* Gravitational field, considered as a metric field, has an exclusive status – all sources of energy-momentum make contributions to the field that follows from the extended to all phenomena of principle of equivalence of mass and energy in special relativity. However, in general relativity gravitational field itself, not necessarily directly affects other fields, such as electromagnetic field. This shows that gravitational field is determined by operational, conventional manner, appearing as a result of a geometric object.
* Gravitational field of a body itself is not considered when finding the metric field around the body, because calculation of metric inside a body in general relativity uses only stress-energy tensor of matter, pressure, and electromagnetic field of the body, which is zero outside the body (except for contribution from stress-energy tensor of electromagnetic field). But this approach is at odds with the fact that if a body located in gravitational fields from other sources, then the fields will inevitably change metric near the body, demonstrating dependence of metric on gravitational field itself, including own gravitational field of the body. Attempts have been made to take into account in GR self-action of metric field of a moving body on the metric around this body with the help of perturbation theory that can not be considered unique and satisfactory general description of the self-action.
* GR can not get only from its principles metric tensor around of an isolated solid body, because it does not have enough equations to calculate the metrics compared to the number of unknown quantities. For the result is used an approximation of a weak field to be able to compare equation of motion from GR with equation of motion and gravitational force of Newton's law, when all possible higher order corrections in metric are automatically reset to zero (Newton's law is only the first order approximation for gravitation). In this case, it turns out that the accuracy of metric tensor in GR is not higher than precision of classical Newton's law. The subsequent application of metrics obtained in this way makes it really only the first order correction to relativistic phenomena near massive bodies, making it difficult to compare results of general relativity with results of alternative theories of gravitation. This is particularly true for theories that differ from GR only in second order accuracy with respect to the square of the speed of light in metric.
* In general relativity, there is no generalization of well-known superposition principle for classical Newtonian potential and strength of gravitational field, since the metric of two bodies system as a personification of gravitational potential of general relativity is implicit function on metrics of both bodies, taken by themselves. Thus the N-body problem is complicated, and metric of solitary massive body has no direct connection with metrics of individual parts of the body.
* GR differs from other physical theories also in the sense that there arise significant problems when trying to find a quantum field theory expansion. To determine the nature of gravitation on microlevel is developed quantum gravity. Additional difficulty arises from the quantum uncertainty principle – if position and velocity of a particle is not determined exactly how to find gravitational field of a single particle and particle composition of the set?
* Limitation and failure of philosophical basis of general relativity appear when anyone attempt to view the internal structure of spacetime of predicted by general relativity black holes and singularities of space-time with unlimited energy density. The internal properties of these objects are obtained in principle unknowable and not verified by an external observer, since no information can go beyond their Schwarzschild radius. The very appearance of such objects is a consequence of limiting geometrization of gravitational field physics – in general relativity assumes that density of gravitational force and power of field energy are characterized only by a curvature of spacetime and therefore may be very large. But in GR there is no proof of existence of such a large force of gravitation, which would have been able to turn any matter into black hole mass, breaking nuclear forces of nucleons repulsion at the stage of formation of a neutron star. From a philosophical point of view, a theory can not be considered complete if it allowed inaccessible to knowledge objects or structures.
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, equations for metric are needed to determine the metric tensor that defines effective properties of spacetime for a given energy-momentum distribution, and changes metric tensor of flat Minkowski space. After finding the metric tensor from 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 equations of these theories in vector and tensor form. Then LITG becomes the covariant theory of gravitation (CTG).

## Description of theory

A feature of LITG is that in it the force of gravitation, in contrast to some 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 Lorentz covariance of gravitational field in a weak field limit and to the need for torsion of gravitational field. LITG structure resembles the structure of electromagnetic field theory, but synthesis of GR and LITG ideas unlike electromagnetism significantly alters the meaning and interpretation of 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 non-inertial reference systems through non-Euclidean geometry. LITG uses a generalization of extended special theory of relativity for gravitational phenomena.
According to metric theory of relativity, description of physical phenomena requires a preliminary choice of method of recording space-time relationships between physical quantities. At present, the optimal choice is electromagnetic-wave representation, when the means of determining distances and time intervals are electromagnetic devices. After the choice, all physical quantities are defined in such a way that they correspond to the speed of light as the speed of electromagnetic wave used for measurements. The LITG relationships presented below contain such gravitational physical quantities that are defined in the electromagnetic-wave representation.

## LITG equations

Gravitational field equations in Euclidean space consist of four vector differential equations and can be regarded as Maxwell-like gravitational equations. In International System of Units, these equations are as follows:
    $~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho ,$
    $~\nabla \cdot \mathbf {\Omega } =0,$
    $~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},$
    $~\nabla \times \mathbf {\Omega } ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right)={\frac {1}{c^{2}}}\left(-4\pi G\rho \mathbf {v} _{\rho }+{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),$
where:
* $~\mathbf {\Gamma }$ is the gravitational field strength,
* $~G$ – gravitational constant,
* $~\mathbf {\Omega }$ is the gravitational torsion field, of dimension as in the frequency,
* $~\mathbf {J}$ – mass current density, which creates the gravitational field and torsion field,
* $~\rho$ – mass density of moving matter,
* $~\mathbf {v} _{\rho }$ – mass flow velocity,
* $~c$ – speed of light.
These equations in their form almost exactly coincide with the equations following 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 field is produced by 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 changes over time, the strength of vortex gravitational field appears in space around the body. The total gravitational force acting on a body has two components. One of them is Newton force, which depends on gravitational field strength and mass, while the other depends on 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 gravitational field strength, but also through torsion field created by body motion or rotation.
The expression for the gravitational force is as follows:
    $~\mathbf {F} _{m}=m\left(\mathbf {\Gamma } +\mathbf {v} _{m}\times \mathbf {\Omega } \right)$,
where:
* $~m$ – mass of a particle, which experiences a force,
* $~\mathbf {v} _{m}$ – speed of the particle.
This formula coincides with expression for the force of general relativity in the weak field limit, although in some publications on general relativity the coefficient 2 is placed before the velocity $~\mathbf {v} _{m}$ in the formula for force due to a different way of determining the field $~\mathbf {\Omega }$.
For energy density and vector of energy flux density ( Heaviside vector) of gravitational field in LITG are obtained:
    $~u=-{\frac {1}{8\pi G}}\left(\Gamma ^{2}+c_{g}^{2}\Omega ^{2}\right),$
    $~\mathbf {H} =-{\frac {c^{2}}{4\pi G}}\mathbf {\Gamma } \times \mathbf {\Omega } .$

## Formulas for torsion fields

For torsion field outside of a rotating body from the field equations by integrating over all points of the body can be derived the formula:
    $~\mathbf {\Omega } ={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {r} /r)\mathbf {r} /r}{r^{3}}}$,
where $~\mathbf {L}$ is angular momentum of rotation or spin of the body.
The torsion field of a body at rectilinear motion is:
    $~\mathbf {\Omega } ={\frac {\mathbf {V} \times \mathbf {\Gamma } }{c^{2}}},$
where $~\mathbf {V}$ – velocity of body motion, $~\mathbf {\Gamma }$ – gravitational field strength of the body in point where torsion field $~\mathbf {\Omega }$ is determined, and field strength $~\mathbf {\Gamma }$ is taken in view of delay distribution of gravitational perturbation.
In general, the torsion field of an arbitrary point moving mass can be expressed in terms of gravitational field strength $~\mathbf {\Gamma }$, which is produced by the point:
    $~\mathbf {\Omega } ={\frac {1}{c}}\mathbf {e_{r}} \times \mathbf {\Gamma } ,$
where $~\mathbf {e_{r}}$ is unit vector directed from the point mass to the point where torsion field is determined, taken at an early time, dependent on the delay.
The formula for the torque acting on a rotating particle with spin $~\mathbf {L}$ in torsion field $~\mathbf {\Omega }$, is written as follows:
    $~\mathbf {K} ={\frac {1}{2}}\mathbf {L} \times \mathbf {\Omega } .$
Rotating particle can be considered like a top with spin $~\mathbf {L}$. Under the action of the torque $~\mathbf {K}$ of torsion field particle will precess along the field direction $~\mathbf {\Omega }$. This follows from the equations of rotational motion:
    $~\mathbf {K} ={\frac {d\mathbf {L} }{dt}}$.
Since the torque $~\mathbf {K}$ is perpendicular to the spin $~\mathbf {L}$ and torsion field $~\mathbf {\Omega }$, then the same is true for the increment of the spin $~d\mathbf {L}$ for the time $~dt$. Perpendicularity of $~\mathbf {L}$ and $~d\mathbf {L}$ leads to the spin precession of a particle with angular velocity $~\mathbf {w} =-{\frac {\mathbf {\Omega } }{2}}$ around direction of $~\mathbf {\Omega }$.
The last equality follows from the fact that $~\mathbf {K} ={\frac {d\mathbf {L} }{dt}}={\frac {1}{2}}\mathbf {L} \times \mathbf {\Omega }$, and $~w={\frac {dL}{L\sin Qdt}}={\frac {d\varphi }{dt}}$, where $~Q$ is the angle between $~\mathbf {\Omega }$ and $~\mathbf {L}$, the angle $~\varphi$ is measured from the projection of vector $~\mathbf {L}$ on the plane perpendicular to the vector $~\mathbf {\Omega }$, to the projection of vector $~\mathbf {L} +d\mathbf {L}$ on this plane.
In presence of an inhomogeneous torsion field a particle with spin $~\mathbf {L}$ will attract in the region of stronger field. From equations of LITG follow the expression for such force:
    $~\mathbf {F} ={\frac {1}{2}}\nabla \left(\mathbf {L} \cdot \mathbf {\Omega } \right).$
The mechanical energy of a particle with spin in torsion field will be:
    $~U=-{\frac {1}{2}}\mathbf {L} \cdot \mathbf {\Omega } .$
The presence of torsion field in gravitational phenomena leads to the effect of gravitational induction.

## Field potentials

If we introduce the notion scalar $~\psi$ and vector $~\mathbf {D}$ potentials of gravitational field, then it is possible to express field strength $~\mathbf {\Gamma }$ and torsion field $~\mathbf {\Omega }$:
    $~\mathbf {\Gamma } =-\nabla \psi -{\frac {\partial \mathbf {D} }{\partial t}},$
    $~\mathbf {\Omega } =\nabla \times \mathbf {D} .$
As for the field strengths so for potentials themselves in LITG true wave equations, depending on mass density and mass current. These wave equations are directly derived from the basic equations of gravitational field and have the form:
    $~\nabla ^{2}\mathbf {\Gamma } -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {\Gamma } }{\partial t^{2}}}=-4\pi G\nabla \rho -{\frac {4\pi G}{c^{2}}}{\frac {\partial \mathbf {J} }{\partial t}},$
    $~\nabla ^{2}\mathbf {\Omega } -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {\Omega } }{\partial t^{2}}}={\frac {4\pi G}{c^{2}}}\nabla \times \mathbf {J} ,$
    $~\nabla ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}=4\pi G\rho ,$
    $~\nabla ^{2}\mathbf {D} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {D} }{\partial t^{2}}}={\frac {4\pi G}{c^{2}}}\mathbf {J} .$
For the potentials is used gauge condition, which reduces degree of uncertainty:
    $~\nabla \cdot \mathbf {D} +{\frac {1}{c^{2}}}{\frac {\partial \psi }{\partial t}}=0.$
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.

## Equations in Minkowski four-dimensional world

Scalar $~\psi$ and vector $~\mathbf {D}$ potentials of gravitational field together form the gravitational four-potential:
    $~D_{i}=\left({\frac {\psi }{c}},-\mathbf {D} \right).$
The wave equations for potentials of gravitational field can be expressed by one equation through D'Alembert operator $~\Box$, acting on four-potential, and with a constant factor equal to mass four-current $~J^{i}$:
    $~\Box D^{i}=-{\frac {4\pi G}{c^{2}}}J^{i},$
where $~J^{i}=\rho _{0}u^{i}=\left({\frac {c\rho _{0}}{\sqrt {1-V^{2}/c^{2}}}},{\frac {\mathbf {V} \rho _{0}}{\sqrt {1-V^{2}/c^{2}}}}\right)=(c\rho ,\mathbf {J} )$ is mass four-current; $~\mathbf {J}$ is mass current density, which generates a gravitational field, $~u^{i}$ is four-velocity, $~\rho _{0}$ – mass density of matter in its rest system.
4-divergences of 4-vectors $~D^{i}$ and $~J^{i}$ are equated to zero and thus set the gauge condition for potentials and continuity equation, respectively:
    $~\partial _{i}D^{i}={\frac {1}{c^{2}}}{\frac {\partial \psi }{\partial t}}+\nabla \cdot \mathbf {D} =0.$
    $~\partial _{i}J^{i}={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.$
With the help of 4-vector $~D_{i}$ can be determined antisymmetric gravitational tensor:
    $~\Phi _{ik}=\partial _{i}D_{k}-\partial _{k}D_{i}.$
In Minkowski space the components of this tensor are:
    $~\Phi _{ik}={\begin{vmatrix}0&{\frac {\Gamma _{x}}{c}}&{\frac {\Gamma _{y}}{c}}&{\frac {\Gamma _{z}}{c}}\\-{\frac {\Gamma _{x}}{c}}&0&-\Omega _{z}&\Omega _{y}\\-{\frac {\Gamma _{y}}{c}}&\Omega _{z}&0&-\Omega _{x}\\-{\frac {\Gamma _{z}}{c}}&-\Omega _{y}&\Omega _{x}&0\end{vmatrix}}.$
With the help of tensor $~\Phi _{ik}$ four vector equations of gravitational field are transformed into two tensor equations:
    $~\partial _{n}\Phi _{ik}+\partial _{i}\Phi _{kn}+\partial _{k}\Phi _{ni}=0$,
    $~\partial _{k}\Phi ^{ik}={\frac {4\pi G}{c^{2}}}J^{i}$.
The density of gravitational force is given by the corresponding 4-vector:
    $~f_{i}=\Phi _{ik}J^{k}.$
The tensor $~\Phi _{ik}$ allows building of gravitational stress-energy tensor:
    $~U^{ik}={\frac {c^{2}}{4\pi G}}\left(-\eta ^{im}\Phi _{mr}\Phi ^{rk}+{\frac {1}{4}}\eta ^{ik}\Phi _{rm}\Phi ^{mr}\right),$
where:
    $~\eta ^{ik}={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}$
is the metric tensor in Minkowski spacetime.
Time-like components of tensor $~U^{ik}$ are the energy density of gravitational field and the Heaviside vector, divided by speed of light $~c$. The space-like components form a three-dimensional tensor of gravitational stresses (of gravitational pressure), taken with the opposite sign. Tensor $~U^{ik}$ is built from the tensor $~\Phi _{ik}$ in such way that from it also possible to find the 4-vector density of gravitational force:
    $~f^{i}=-\partial _{k}U^{ik}.$
Thus, the theory of gravitational field can research phenomena up to relativistic velocities of bodies.

## Lagrangian

For a single particle in a gravitational field Lagrangian has the form:
    $~L=-mc{\frac {ds}{dt}}-m{\frac {D_{k}dx^{k}}{dt}}+{\frac {c}{16\pi G}}\int {\Phi _{ik}\Phi ^{ik}{\frac {dx^{4}}{dt}}}=$
$=-mc^{2}{\sqrt {1-V^{2}/c^{2}}}-m(\psi -\mathbf {D\cdot V} )-{\frac {1}{8\pi G}}\int {(\Gamma ^{2}-c^{2}\Omega ^{2})}dx^{3}$,
where $~ds$ – invariant interval, $~dx^{k}$ – 4-vector of particle displacement, $~dx^{3}=dxdydz$ – element of 3-volume.
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.

## LITG equations in arbitrary reference frame

From various experiments on propagation of light near massive bodies (see tests of general relativity) follow that gravitational field of bodies bends light rays, changing speed and frequency of electromagnetic waves. This means that measured dimensions of 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 four-dimensional Minkowski world. To take this into consideration, instead of the metric tensor $~\eta ^{ik}$ in general is used the metric tensor $~g^{ik}$.
The tensor of gravitational field is still determined by the 4-potential:
    $~\Phi _{ik}=\partial _{i}D_{k}-\partial _{k}D_{i}.$
The field equations in an arbitrary reference frame through covariant derivatives of gravitational field tensor can be expressed by two tensor equations:
    $~\nabla _{n}\Phi _{ik}+\nabla _{i}\Phi _{kn}+\nabla _{k}\Phi _{ni}=0$,
    $~\nabla _{k}\Phi ^{ik}={\frac {4\pi G}{c^{2}}}J^{i}.$
Gauge condition for potentials and continuity equation for the mass 4-current are written respectively as follows:
    $~\nabla _{k}D^{k}=0.$
    $~\nabla _{k}J^{k}=0.$
Stress-energy tensor of gravitational field takes the following form:
    $~U^{ik}={\frac {c^{2}}{4\pi G}}\left(-g^{im}\Phi _{mr}\Phi ^{rk}+{\frac {1}{4}}g^{ik}\Phi _{rm}\Phi ^{mr}\right).$

## LITG and GR

In general relativity, one can calculate the metric in the limit of a weak field inside a homogeneous sphere without taking into account the internal pressure, energies and momenta of the fields, when $~g_{ik}=\eta _{ik}+h_{ik}$, where $~h_{ik}$ is a small addition. For time components of metric tensor follow: $~h_{00}={\frac {2\psi }{c^{2}}}$ in static case, and $~h_{0p}=-{\frac {4D_{p}}{c}}$ in dynamic case, where $~p=1{,}2{,}3.$ These equations in terms of LITG mean that the additive components to the metric tensor $~h_{0p}$ as it were twice as much dependent on the vector potential $~\mathbf {D_{p}}$, than the components $~h_{00}$ depend on the scalar gravitational potential $~\psi$. In general relativity, which starts from the components of the metric tensor, they either find that the vector potential found from the equations is twice as “weak” as the scalar potential (this leads to LITG), or they define a new vector potential in the form $~\mathbf {D} _{OTO}=2\mathbf {D}$.
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 $~\mathbf {D}$ and the vector of torsion field $~\mathbf {\Omega }$, 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. It is argued that such differences arise due to the tensor nature of the metric field of general relativity.
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, 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, 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 strong-field instead of the metric tensor $~\eta ^{ik}$ is used metric tensor $~g^{ik}$. 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, which is consistent with LITG, but contrary to a sense of general relativity.
Due to the tensor $~U^{ik}$ in LITG and CTG automatically solve the current problem of lack in general relativity of stress-energy tensor of gravitational field. Tensor $~U^{ik}$ 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 correct identification of reference systems, allowing to avoid corresponding problem of general relativity. For example, calculations made with respect to 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.
According to LITG, gravitational field is a separate physical field. A metric field consisting of the components of metric tensor and depends on time and coordinates of the point where it is defined, is derived and is the total effect on presentation of mass density, 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 energy and energy flux. 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 field, 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 $~m$ around center of attraction is equal to:
    $~W=-{\frac {2Gm^{2}w^{4}R^{2}}{3c^{3}}},$
where $~w$ is angular velocity of rotation, $~R$ – radius of rotation.
From this formula it follows that a gravitational closed system of 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 $~m_{1}R_{1}$ and $~m_{2}R_{2}$ in formula for 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 dipole gravitational radiation corresponds to the formula for 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.

## Gravitational experiments with torsion field
### Interaction of spins

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 spins in general relativity is called spin-spin precession or spin Lense–Thirring precession or Pugh-Schiff precession. This effect relies a consequence of gravitomagnetic frame-dragging, that is dragging of freely falling bodies near a massive rotating object. In general relativity, gravitation is replaced by curvature of space, so that deviation of a test particle from its normal geodesic line is due to rotation of a massive body and a corresponding change in the metric.
Due to 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 2004-2005. The formula for angular velocity of precession of interaction of the spins in LITG is as follows:
    $~\mathbf {w_{ss}} =-{\frac {\mathbf {\Omega } }{2}},$
and spin of the gyroscope precesses about the direction of torsion field $~\mathbf {\Omega }$, which is created by the spin of the Earth. The torsion field of the Earth as dipole field is given by:
    $~\mathbf {\Omega } ={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {r} /r)\mathbf {r} /r}{r^{3}}},\qquad \qquad (1)$
where $~\mathbf {L}$ – spin (angular momentum) of the Earth, $~r=R_{e}+h$ – distance from the center of Earth to the satellite, defined by the radius of the Earth $~R_{e}$ and height of the satellite $~h$ (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 value of the effect in a constant field. Assume that the gyroscope is kept just above the north pole of the Earth, where $~\mathbf {L}$ and $~\mathbf {r}$ are parallel, and the field has maximum. In this case, the formula for torsion field of the Earth is simplified, and angular velocity of precession is equal to:
    $~w_{ss}={\frac {GL}{2c^{2}r^{3}}}.$
For Gravity Probe B value $~w_{ss}$ 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.

### Orbital precession

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 orbital angular momentum of the particle, i.e. orbital precession. The equation of rotational motion of a particle is:
    $~{\frac {d\mathbf {L_{o}} }{dt}}=\mathbf {r} \times \mathbf {F} ,$
where force is equal to: $~\mathbf {F} =m\left(\mathbf {\Gamma } +\mathbf {V} \times \mathbf {\Omega } \right)$, the quantities $~m$ and $~\mathbf {V}$ denote the mass and velocity of the particle, and the orbital angular momentum of the particle is $~\mathbf {L_{o}} =m\mathbf {r} \times \mathbf {V} .$
In the reference frame associated with the center of the Earth, the vectors $~\mathbf {r}$ and $~\mathbf {\Gamma }$ 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 $~\mathbf {r} \cdot \mathbf {V} =0$. This gives:
    $~{\frac {d\mathbf {L_{o}} }{dt}}=m\mathbf {r} \times [\mathbf {V} \times \mathbf {\Omega } ]\approx m\mathbf {V} (\mathbf {r} \cdot \mathbf {\Omega } )=-{\frac {Gm}{c^{2}r^{3}}}\mathbf {V} (\mathbf {r} \cdot \mathbf {L} ).$
It follows that angular velocity of precession of orbital angular momentum is equal to:
    $~w_{o}={\frac {GL}{c^{2}r^{3}}}.$
Accounting for effect of gravitational field and the Earth's rotation on space-time metric in general relativity give the result, that the angular velocity of precession of orbital angular momentum becomes more and approximately equal $~2w_{o}$. 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.

### Geodetic effect

In geodetic effect, also called de Sitter effect are contributed two different phenomena. The first of these may be called the spin-orbital interaction. In the case of a gyroscope in orbit around the Earth, this interaction can be understood as 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 $~\mathbf {V_{E}}$, opposite to the direction of the velocity $~\mathbf {V_{g}}$ of the gyroscope relative to the Earth. The orbital torsion field which is produced by the Earth can be estimated by the formula:
    $~\mathbf {\Omega } ={\frac {\mathbf {V_{E}} \times \mathbf {\Gamma } }{c^{2}}},$
where $~\mathbf {\Gamma }$ – field strength of Earth's gravitational field near the gyroscope.
The angular velocity of the spin-orbital precession will be:
    $~\mathbf {w_{so}} =-{\frac {\mathbf {\Omega } }{2}}={\frac {\mathbf {V_{g}} \times \mathbf {\Gamma } }{2c^{2}}},$
here was took into account that $~\mathbf {V_{E}} =-\mathbf {V_{g}} .$
The second term, making contributions to de Sitter effect is related to influence of gravitational field on the metric around the Earth. The presence of the field leads to an effective curvature of space-time, which is expressed in an appropriate amendment to the metric tensor of flat Minkowski space. The magnitude of the second term is two times more than $~\mathbf {w_{so}} .$ As a result, the angular velocity of de Sitter precession is:
    $~\mathbf {w} ={\frac {3\mathbf {V_{g}} \times \mathbf {\Gamma } }{2c^{2}}}.$
Taking into account the field strength of the Earth's gravitational field near the gyroscope $~\Gamma =-{\frac {GM}{r^{2}}},$ where $~M$ is the mass of the Earth, for 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.

## Masses and energies

As it was shown in one paper, equivalence principle of general relativity does not hold with respect to the mass-energy of gravitational field itself. In particular, in the weak field limit mass-energy of gravitational field of a stationary body, and mass-energy of field of moving with constant velocity the same body does not coincide with each other. 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 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 fundamental unobservability of inner structure of black holes. In such cases, when the research reaches the field carriers themselves and their interaction with matter, in absence of reliable data on the properties and energy density of field quantum, conclusions of theories become inaccurate. The solution of such problems are expected with 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 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 mass-energy of gravitational field of a stationary and moving bodies could show the difference of relative rest and motion – in motion the mass-energy of the field increases 4/3 times by adding to the field energy of work against the flux 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 two vector fields – acceleration field and pressure field. Both these fields, together with gravitational and electromagnetic field are combined in such a way that the total mass-energy of fields in the system vanishes.
The articles within LITG equations (as well as equations of gravitomagnetism as approximations of general relativity) clarify relativistic expressions for energy and energy flux of gravitational field inside and outside a homogeneous sphere. The conclusion is that inequality of field mass-energy, found from gravitational energy and energy flux is an intrinsic property of the field, contrary to the principle of equivalence between gravitational and inertial masses in general relativity.
Analysis of the 4/3 problem, and ways to include the mass of field in body mass leads to the following expression: $~m=(E-E_{binding})/c^{2}$, where $~E$ is the rest energy, $~E_{binding}$ is the negative energy of gravitational binding. Meanwhile, in GR another expression is used: $~m=(E+E_{binding})/c^{2}$, see Mass in general relativity#The Newtonian limit for nearly flat space-times.
If to consider the virial theorem, from LITG it follows the formula for mass of a body at rest through energies of fundamental fields:
    $~m=-(U_{0g}+W_{0g}+U_{0}+W_{0})/2c^{2}$,
where $~U_{0g}$ – energy of strong gravitation, proposed as the basis of strong interaction in matter of nucleons and atoms, $~W_{0g}$ – energy of electromagnetic field in matter of nucleons and at the atomic level, $~U_{0}$ – energy of cosmological scale gravitation, $~W_{0}$ – energy of electromagnetic field in body matter and beyond.
In static case $~U_{0}<0$, $~W_{0}>0$, and body mass increases due to gravitational energy and decreases due to electrical energy. This conclusion is subject to correction, because it was found difference between gravitational mass and mass of a system consisting of particles and fields. Thus it turns out that the mass of the system can be constant, but the gravitational mass increases with decreasing size of the system due to contribution of gravitational energy. In contrast to general relativity, the gravitational mass of a body should be reduced by electric mass-energy if electric charge is transferred to the body.
A more accurate analysis of mass and energy was carried out in articles, where five types of masses of relativistic uniform system are defined:
    $~m'<M<m<m_{b}=m_{g}.$
Here, the gauge mass $~m'$ is related to cosmological constant and represents mass-energy of matter’s particles in four-potentials of the system’s fields; the inertial mass $~M$; the auxiliary mass $~m$ is equal to product of particles’ mass density by volume of the system; the mass $~m_{b}$ is the sum of mass-energies of moving particles of system, which is equal in value to gravitational mass $~m_{g}$.
The conclusion that as electric charge increases the system’s mass may decrease remains valid, however this applies not to gravitational mass $~m_{g}$, but to inertial mass $~M$ of the system.