Essay written for the Gravity Research
Foundation 2013 Awards for Essays on Gravitation. http://dx.doi.org/10.13140/RG.2.2.14384.97280

**Covariant theory of gravitation**

**Sergey G. Fedosin**

Sviazeva Str. 22-79, Perm, 614088,
Perm region, Russian Federation

e-mail intelli@list.ru

23 January 2013

The history of appearance and
characteristics of covariant theory of gravitation (CTG) and its difference
from the general theory of relativity (GTR) are described. CTG is developed as
an axiomatic theory based on five axioms using the language of 4-vectors and
tensors, and is derived from the principle of least action and through Hamilton
equations. In CTG gravitational radiation of any body is based on equation of
its motion and has a dipole component. In GTR we proceed from the tensor nature
of metric field and take into account all the bodies of considered material
system at once, and therefore the basis of gravitational radiation is only the
quadrupole component. The CTG Hamiltonian is obtained in two ways – either
through 4-velocity or through generalized 4-momentum.

Essay written for the Gravity
Research Foundation 2013 Awards for Essays on Gravitation.

**Introduction**

Covariant theory of gravitation (CTG) was introduced in 2009 [1] as an alternative to the general
theory of relativity (GTR). CTG generalizes equations of Lorentz-invariant theory of gravitation
(LITG) to non-inertial reference frames. LITG is valid only in case of inertial
frames of reference [2], [3], and is based on extended special theory of relativity
[4] and Le Sage’s theory of gravitation [5]. For transition from LITG to CTG it
is necessary not only to write equations of motion for field and substance in a
covariant form, but also to use metric
theory of relativity (MTR) [1].
In CTG gravitation is a real physical force, and gravitational field has its
own stress-energy tensor. In contrast, in GTR gravitation is reduced to
geometric curvature of spacetime and to metric field, the energy of which can
not be determined unambiguously.

**LITG**

According to axiomatics of
LITG [1], each moving mass creates around itself not only the gravitational field strength , but also gravitational
torsion field . The gravitational field equations
in LITG are similar by form to Maxwell equations for electromagnetic field (see
Maxwell-like gravitational equations
and selfconsistent gravitational constants),
and in this case corresponds to electric field strength and is similar to
magnetic field induction . With the help of and energy
density of gravitational field,
vector of field
energy flux (Heaviside
vector), gravitational
tensor (field strengths tensor) and gravitational stress-energy tensor
are determined. The connection between the two components and of gravitational field leads to effect of gravitational
induction.

In LITG not only strengths but
also potentials of gravitational field – scalar potential and vector potential are used. The scalar and vector potentials
form gravitational four-potential.
Each of the variables ,
,
and can be determined from corresponding wave
equation, the solution of which is integral over the spatial volume, taking
into account the delay time of gravitational effect propagation from the moving
mass to the point of observation.

Within LITG is put forth [6]
and the problem 4/3 is solved [7], which is associated with mass-energy of
gravitational field.

**CTG features**

The equations of CTG differ
from the equations of LITG since they are valid not only for Minkowski space,
but also for Riemannian space. In CTG the same idea as in GTR and other metric
theories is used: it is assumed that in any accelerated and therefore
non-inertial reference frame the spacetime is curved. The degree of spacetime
curvature at each point in space is characterized by curvature tensor which is
a function of metric tensor and depends on the stress-energy tensors of
substance and field at this point. Mathematically this is expressed in
Hilbert-Einstein equations for the metric. In contrast to GTR, in CTG in the
right side of this equation the stress-energy tensor of gravitational field is
included. Thus the gravitational field becomes an independent source of
spacetime curvature, as the substance and electromagnetic field. CTG can be
developed on the axiomatic basis [1], [8].

The structure of CTG includes
three types of equations – to determine metric, to calculate strengths and
potentials of gravitational and electromagnetic fields, and to find motion of
substance under action of fields and mechanical stresses. In the curved
spacetime the components of metric tensor become functions of spatial
coordinates and time, which allows us to describe additional changes of
dimensions and time intervals that occur in accelerated reference frames in comparison
with inertial frames. For writing the equations of CTG definitions of four-force, of operator of proper-time-derivative, of invariant
energy and principle of energies summation
are used.

All equations of CTG can be
derived from the principle of least action [9]. In this case the meaning of
cosmological constant is found – it is proportional to mass of substance of the considered system, taken
without including mass-energy of gravitational and electromagnetic fields. In
contrast to GTR, Hamiltonian in CTG contains terms related to energy of
gravitational field [10].

By introducing into theory of
generalized 4-velocity we can achieve more simple writing of
Lagrangian and Hamiltonian, besides the product sets the generalized 4-momentum. Another newly
introduced thing is the 4-vector , in which the Hamiltonian is
time-like component. Being written with the covariant index, the 4-vector is 4-vector of energy-momentum, setting
relativistic energy and momentum of substance taking into account contributions
from existing gravitational and electromagnetic fields.

**The consequences of CTG**

CTG
predicts the existence of three types of body mass: mass is the sum of masses of the body’s parts,
scattered at infinity, which allows us to exclude the contribution of
mass-energy of macroscopic gravitational
and electromagnetic fields from the body’s substance into the mass; mass determines the inertial and
gravitational properties of the body and is the relativistic mass; body’s mass and the substance density corresponding to
this mass are included in Lagrangian and Hamiltonian. In GTR two masses are
usually used, and , and mass is treated as the rest mass of the substance
without taking into account the energy of fields.

According
to CTG, in the absence of electromagnetic field the ratio between the masses is
as follows: , . This allows us to understand
the controversy arising in interpretation of the mass. In particular, in GTR we
proceed from the mass , adding to it the mass-energy
of gravitational field we obtain the mass . Since the mass-energy of the
gravitational field is negative, the ratio for the masses is: . In CTG we proceed from the
mass , from which we should
subtract the mass-energy of gravitational field in order to obtain the mass . This leads to inequality . The need to subtract the
field energies in CTG is proved in [7], [9], [11] for different situations.
Interestingly, in CTG the masses and can be measured experimentally, while the mass
must be calculated.

According to [1], for the case
of an isolated body at rest in the components of metric tensor there is an
additional term, which is inversely proportional to the fourth power of
gravitation propagation speed. The characteristic feature of solution for
static case is dependence of metric tensor on two constants that can be found
from comparison with experiment. Rotation of a body leads to emerging in the
metric tensor of an additional constant which depends on the angular momentum
of the body (a similar effect in GTR is seen in Kerr metric). Due to the
presence of indefinite constant coefficients in the solution for metric, CTG
can describe any gravitational phenomena and interactions of bodies, including
shift of perihelion of planets, gravitational redshift, time dilation in
gravitational field, deviation of motion of test particles and field quanta
near massive bodies, precession of orbits and gyroscopes etc.

Another consequence of the
theory is associated with the difference of equations of motion in GTR and CTG:
in GTR free motion of a test body in gravitational field is described as
4-acceleration of the body equal to zero. In CTG equation of motion has the
classical form: in the left side of the equation there is the total derivative
of momentum with respect to proper time, and in the right side there is the sum
of gravitational and electromagnetic forces. In [1] attention is drawn to the
fact that difference of equation of motion in CTG from equation in GTR is
sufficient to explain the "Pioneer" effect.

**Metric
theory of relativity**

According to [1] and [8], MTR
is based on five axioms and is considered as a theory, which includes the
special theory of relativity and the part of GTR which refers to the methods of
conversion of physical quantities from one reference frame to another. In MTR
it is assumed that spacetime measurements can be carried out not only by
electromagnetic waves but also by other waves, as well as by test particles.
These waves and particles can have different speeds of propagation and motion,
which affects the results of space and time measurements of the same events in
different reference frames. An example of this is Lorentz transformations,
which depend on the speed of electromagnetic waves (speed of light) as a
parameter. In GTR the metric tensor components are also dependent on the speed
of light. In MTR the generalization of principle of equivalence is used as the
principle of local equivalence of energy-momentum. All this means that the
spacetime metric in each frame depends entirely on the method of measurement,
including the speed of signal propagation and other properties of the signal
carrier. For example, measurement of metric around a massive body by spinless
test particles and particles with spin will give different results.

**Strong
gravitation**

According to infinite hierarchical nesting of matter
and SPФ symmetry,
at each basic level of matter its own gravitation is acting. It is assumed that
at the atomic level strong
gravitation is acting, the strong gravitational constant differs from the ordinary gravitational constant and equations of
strong gravitation are equations of CTG. Strong gravitation explains the cause
of rest energy, the structure and integrity of elementary particles, and strong
interaction between the particles [1], [12].

**References**

1. Fedosin S.G. Fizicheskie
teorii i beskonechnaia vlozhennost’ materii. (Physical
Theories and Infinite Hierarchical Nesting of Matter). – Perm’ : S.G. Fedosin,
2009-2012, 858 p. ISBN 978-5-9901951-1-0.

2. Fedosin S.G. Fizika i filosofiia podobiia ot preonov do
metagalaktik. (Physics and Phylosophy of Similarity From Preons
to Metagalaxies). – Perm’ : Style-MG, 1999, 544 p. ISBN 5-8131-0012-1.

3. Fedosin S.G. Electromagnetic and Gravitational Pictures of the World. Apeiron, 2007, Vol. 14,
No. 4, P. 385–413.

4. Fedosin S.G. Sovremennye problemy fiziki: v poiskakh novykh printsipov. Moskva: Editorial URSS, 2002, 192 pages. ISBN 5-8360-0435-8.

5.
Fedosin S.G. Model of Gravitational
Interaction in the Concept of Gravitons.
Journal of Vectorial Relativity, 2009, Vol. 4, No. 1, P.1–24.

6. Fedosin S.G. Mass, Momentum and Energy of Gravitational Field.
Journal of Vectorial Relativity, 2008, Vol. 3, No. 3, P.30–35.

7. Fedosin
S.G. Energy, Momentum, Mass and Velocity of a Moving Body.
vixra.org, 13 Jun 2011.

8. Fedosin
S.G. The General Theory of Relativity, Metric Theory of Relativity and
Covariant Theory of Gravitation: Axiomatization and Critical Analysis. vixra.org,
26 Mar 2011.

9. Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, 2012, Vol. 35, No. 1, P. 35–70.

10. Fedosin
S.G. The Hamiltonian in Covariant Theory of Gravitation. Advances
in Natural Science, 2012, Vol. 5, No. 4, P. 55–75.

11. Fedosin
S.G. The Principle
of Proportionality of Mass and Energy: New Version. Caspian
Journal of Applied Sciences Research, 2012, Vol. 1, No 13, P. 1–15.

12. Fedosin S.G. The Radius of the Proton in the Self-consistent Model. Hadronic Journal, 2012, Vol. 35, No. 4, P. 349–363.

Source: http://sergf.ru/esen.htm