International
Journal of Theoretical and Applied Physics (IJTAP), ISSN: 2250-0634, Vol. 4, No. I (2014), pp. 9 – 26.
The General Theory of Relativity,
Metric Theory of Relativity and Covariant Theory of Gravitation. Axiomatization
and Critical Analysis
Sergey G. Fedosin
Perm, Perm Region, Russia
e-mail intelli@list.ru
The axiomatization of general theory of relativity (GR) is done. The
axioms of GR are compared with the axioms of the metric theory of relativity
and the covariant theory of gravitation. The need to use the covariant form of
the total derivative with respect to the proper time of the invariant
quantities, the 4-vectors and tensors is indicated. The definition of the
4-vector of force density in Riemannian spacetime is deduced.
Keywords: general relativity; metric theory of relativity; covariant theory of
gravitation; axiomatization.
PACS: 04.20.Cv, 04.50.Kd, 04.90.+e,
11.10.Cd
Establishing of axiomatic foundations is considered as
an important stage in the development of any modern physical theory. This is
due to the fact that from a given complete set of mutually independent axioms
it is possible to deduce uniquely and unambiguously the whole theory. In
addition, on the basis of the axioms it is easy to define the scope of
applicability of the theory and its difference from alternative approaches.
Presented in 1915 by Albert Einstein [1] and David
Hilbert [2] the equations of general theory of relativity (GR) were based on
several principles and heuristic analogies, but were not axiomatized.
Mathematical apparatus available in GR made it possible to solve various
problems, which allowed the theory to become generally accepted model of
gravitation. The problem with the axiomatization of GR became acute in
mid-twentieth century, when it became clear that GR can not be quantized in the
same way, as electromagnetic theory.
In GR the tensor of gravitational field is also not
defined, which prevents from recognizing GR as a complete theory of
gravitational field. The equations of GR predict singularities with infinite
energy density, and black holes with such a magnitude of gravitation, that it
must hold within itself not only the matter, but even light quanta. However, in
the framework of GR it is apparently impossible to give the answer about the
real existence of such exotic objects.
For theoretical foundation of GR the following
principles are usually applied:
1) The principle of equivalence in different forms,
including:
1.1) The
equality of inertial and gravitational masses.
1.2) The
equivalence of inertial and gravitational accelerations in description of
phenomena in the infinitely small reference frame of the test particle.
1.3) The
equivalence of the state of free falling in any gravitational field and
inertial motion in the absence of a gravitational field, on condition that the
instantaneous velocity of falling is equal to velocity of inertial motion.
1.4) The
equivalence of the form of motion with the same initial conditions for any
uncharged and non-rotating test particles in a gravitational field regardless
of the structure and composition of their substance.
1.5) The equivalence
of physical phenomena for the free falling in the gravitational field of the
observer in his reference frame, understood as independence of the form of
phenomena on the fall velocity and the location in the gravitational field.
1.6) The equivalence
of effects of gravitation and deformation of spacetime; description of
gravitation through the metric tensor and its derivatives with respect to
coordinates and time.
2) The principle of motion along geodesics arising
from 1.1), 1.3) and 1.4).
3) The principle of distortion of spacetime by matter,
electromagnetic field and other non-gravitational fields.
4) The principle of linear relationship between the
curvature of spacetime and the energy-momentum of matter and nongravitational
fields (Hilbert-Einstein tensor equation for metric).
5) The principle of determining of force and of
equations of motion through the covariant derivative of stress–energy tensor.
6) Correspondence principle: in the weak field
the equations of GR turn into the classical Newton's equation of gravitation
and the metric of spacetime becomes the metric of Minkowski flat spacetime.
7) The principle of covariance: physical quantities
and equations of GR must be written in covariant form, independent on the
choice of the reference frame.
8) The principle of least action: the equations of GR
can be deduced from the variation of the four-volume integral of the Lagrangian
density.
It is most convenient to measure the metric in GR by
means of electromagnetic waves by determining the deflection of light rays and
the effect of time dilation of time of electromagnetic clock, depending on the
coordinates and time. From here we find the metric tensor that determines the
gravitational field. Therefore, in GR it is assumed that the rate of change and
propagation of gravitation equals the speed of light, at which the
electromagnetic wave is propagating at a given point of spacetime. The speed of
electromagnetic wave in the gravitational field depends on the coordinates and
time and is considered as the maximum speed of transfer of interactions. Metric
tensor in GR represents the gravitational field so that the covariance of the
metric tensor relative to the transformations of any reference frames
determines the covariance of the gravitational field.
After appearance of the metric theory of relativity
(MTR) and the covariant theory of gravitation (CTG) in 2009, which were
originally axiomatized [3], the need to conduct an axiomatization of GR
appeared in order to compare the physical bases of these theories with a
unified point of view. Axiomatization of GR can be also useful for comparison
with other alternative theories of gravitation.
Analysis of GR shows that it contains two closely
related components. The first of these is the general relativity of phenomena
in different reference frames. This part of the theory allows to link the
results of spacetime measurements of different observers and to recalculate the
physical quantities of one reference frame for another. The second part of GR
is the theory of gravitational field and its interaction with the matter. Both
parts of GR could be completely derived from the respective systems of axioms
[4]. Due of the merging of general relativity and the theory of gravitation in
GR in these systems of axioms, there is one common axiom that describes the
connection of the metric and the matter in the equation for calculating the
metric.
Convenience of comparing GR and CTG is due to the fact
that both these theories are metric tensor theories of gravitation. In contrast
to GR, in CTG the Lagrangian contains additional terms, which specify directly
the gravitational field [5]. The form of these terms is similar to the terms
determining the electromagnetic field. At the same time the strength tensor and
the stress-energy tensor of the gravitational field are introduced in
consideration. In CTG the transition from the Lagrangian to the Hamiltonian is
possible [6].
Axioms of general relativity in GR
1.
The properties of spacetime are defined by uncharged and noninteracting
test particles and waves and do not depend on the type of particles and waves.
2.
The characteristic of the spacetime is the symmetric metric tensor 
, which depends in general on the coordinates and the time.
With the help of the tensor various invariants
associated with 4-vectors and tensors are calculated.
3.
The square of the interval 
gives the square of
the length of the 4-vector differential of coordinates, which does not depend
on the choice of reference frame:
,
where the symbol 
denotes the total differential in curved spacetime.
Spatio-temporal measurement and fixing of the metric properties are carried out usually
by means of electromagnetic waves, the speed of which is considered equal to
the speed of light for local observers and does not depend on the speed of the radiating bodies. However, in the global coordinate
reference frame in the general case the speed of electromagnetic waves depends
on the coordinates and time, and decreases in the strong field. At the same time for the electromagnetic waves the
interval is always zero: 
.
4.
Physical properties of substance and fields, except the gravitational
field, are specified by the corresponding stress–energy tensors.
There is a mathematical function of the metric tensor (e.g. the
Hilbert-Einstein tensor on the left side of the equation for the metric) which is
proportional to the total stress–energy tensor
of matter and fields on the right side:
,
(1)
where 
− Ricci tensor, 
− scalar curvature, 
− gravitational
constant, 
− the speed of light, 
− stress–energy tensor of matter, 
− stress–energy
tensor of electromagnetic field and other
nongravitational fields. Using this equation, the connection is realized
between the geometric properties of spacetime, on the one hand, and the
physical properties of existing matter and non-gravitational field, on the other
hand.
5.
There are used additional conditions which determine the necessary for
the calculation ratios for the shifts and turns of the compared reference
frames, the velocities of their motion relative to each other, and taking into
account the symmetry properties of reference frames.
To derive the transformations linking the
differentials of the coordinates and time of any two frames of reference, we
use the condition of equality of intervals 
in axiom 3. The
interval is invariant for the calculation of which in each reference frame the
knowledge of the metric tensor, specified in axiom 2, is required. In addition,
according to axiom 5 there should be additional relations and connections
between these frames of reference. For example, the Lorentz transformations for
two inertial reference frames take into account: the location and relative
orientation of the reference frames; and their velocities relative to each
other; the symmetry of the transformations for the axes perpendicular to the
velocity of motion including the equal value of the speed of light.
The principle of equivalence can be considered as the consequence
of the independence of metric on the type and properties of test particles and
waves, as it is assumed in axiom 1. From axiom 4 it follows that the transition
from general relativity in GR to particular relativity in special theory of
relativity must be accompanied by tending to zero of the mass density and the
velocity of test particles, as well as the strengths of non-gravitational
fields acting on the particles. Taking in account axiom 5 it is enough to
obtain all the relations of special relativity.
Above we introduced the system of axioms of general relativity in GR
from our point of view. There are other attempts to justify the relativity in
GR. In particular, the article [7] is devoted to the problem of
axiomatic deriving of Lorentzian geometry of free fall and light propagation
that underlines the space-time of general relativity from compatible conformal
and projective structures on a four dimensional manifold.
Axioms of gravitational field in GR
1.
The properties of the gravitational field are set by the velocity of
propagation of gravitational interaction, equal to the speed of light and
depending in general on coordinates and time, as well as by non-degenerate
metric tensor of the second rank .
2.
The gravitational field is reduced to the geometric curvature
(deformation) of spacetime caused by the sources of matter and any
nongravitational field. The degree of curvature of spacetime is fixed by the
curvature tensor of the Riemann-Christoffel 
which is the function
of and its derivatives of
first and second order with respect to coordinates and time. With the help of
metric contraction, from the tensor 
the Ricci tensor and then scalar
curvature can be obtained.
3.
Gravitational acceleration is reduced to the gradients of the metric
tensor , i.e. to the rate of change of the components of the metric
tensor in space and time.
4.
The properties of matter and non-gravitational fields are determined by
the stress–energy
tensor 
.
5.
Relationship between the gravitational (metric) field, determined by the
metric tensor through the curvature
of spacetime, and the matter and field is defined by the Hilbert-Einstein
equations for the metric:
.
From Axiom 3 here the principle of equivalence can be deduced. The
covariant derivative acting on both sides of the equation for the metric in
axiom 5, converts them to zero. It defines the properties of the
Hilbert-Einstein tensor, and simultaneously sets the equation of motion of
matter.
Comparison of the theories of relativity
Axioms of the metric
theory of relativity (MTR) are [3]:
1.
The
properties of the spacetime manifold in a given frame depend on the properties
of test bodies and the waves, through which the spacetime measurements are
fulfilled in the frame of reference. The most important property of test bodies
and the waves is the speed 
of their
propagation, as it appears in formulas as a measure of the velocities of other
bodies and the delay of information in distance measurements.
2.
Geometric
properties of spacetime are fixed by a relevant mathematical object, which is a
function of spacetime coordinates of the reference frame. For a large class of
reference frames a suitable mathematical object is the non-degenerate
four-dimensional symmetric metric tensor of second rank , whose components are scalar products of unit vectors
of axes in the chosen reference frame. Tensor allows finding
any invariants that are associated with 4-vectors and tensors.
3.
The
square of the interval 
between two close
events, understood as the tensor contraction of the metric tensor with the product
of differentials of the coordinates 
, is an invariant, the measure of proper time 
of the moving
particle, and does not depend on the choice of reference frame:
.
The interval 
for two close events
is zero, if these events are related to the propagation of test bodies and the
waves, through which the spacetime measurements and fixing of metrics are
fulfilled.
4.
The
physical properties of any fields including the gravitational field in some
frame of reference are set by the corresponding stress–energy
tensors. There is a mathematical function of the metric tensor , found by certain rules and proportional to the total
stress–energy tensor of fields, acting in this
frame of reference. In the simplest case, such function is the tensor, in the
left part of the equation for metric [8]:
, (2)
where 
− the constant coefficient
depending on the type of test particles or waves, which is determined by
comparison with experiment or with the formulas of classical physics in the
weak-field or low-velocity approximation, and we believe that the speed of gravitation propagation
is equal to the speed of light,
− the stress–energy
tensor of acceleration field, which depends on the acceleration tensor 
:
.
− the stress–energy tensor of gravitational field. This
tensor is expressed through the tensor of gravitational field 
:
.
− the stress–energy tensor of pressure field, being found
through the pressure tensor 
:
.
Here 
and 
are some constants.
Equation (2) provides
the link between the geometric properties of the used spacetime manifold, on
the one hand, and the physical properties of the fields, on the other side. The
covariant derivative acting on both sides of the equation for the metric (2),
converts them to zero. It fixes the properties of the tensor in the left side of (2), and simultaneously sets the equation of
matter motion under the influence of fields.
5. There are additional
conditions which determine the ratios, necessary for the calculation, for the
shifts and turns of the compared reference frames, the
velocities of their motion relative to each other, and taking into account the
symmetry properties of reference frames.
From the comparison of the axioms of general relativity in GR with the
axioms of the metric theory of relativity the features of these theories follow
which are listed in Table 1.
Table 1
|
Features of theories |
General relativity in
GR |
Metric theory of
relativity |
|
Metric properties of
spacetime: |
Do not depend on the
type of test particles and waves |
Depend on the type of
test particles and waves |
|
Interval is equal to
zero: |
Only for electromagnetic waves |
For all test particles and waves, which are used for the spacetime
measurements and fixing of metric |
|
Sources of energy and momentum that define metric: |
Matter and any non-gravitational fields |
Any field including the gravitational field |
|
The principle of equivalence is understood as: |
Equivalence of phenomena in two reference frames of small size, one of
which is accelerated by the gravitational force, while the other receives the
same acceleration under the action of uniformly distributed non-gravitational
force of the same magnitude |
Equivalence of energy- momentum: “In accelerated reference frame the
metric does not locally depend on the type of the acting force causing this
acceleration, but depends on the configuration of this force in spacetime of
the reference frame defined by the stress-energy tensor” |
Equivalence of the acceleration, due to gravitation and inertial acceleration
under the action of uniformly distributed over the volume of the test body
non-gravitational force of the same value, leads to the equality of
gravitational and inertial masses. The homogeneity of the applied force means
that in the system of small size all parts of the system are accelerated
equally and the relative internal acceleration is absent. In this case, the
separate elements of the test body do not put pressure on each other and behave
as the test body moving by inertia in the absence of forces. The masses of
bodies can be found by weighing in relation to the standard mass in a
gravitational field, and the masses are proportional to the gravitational
forces. This implies the independence of the forms of motion of the falling
bodies on the mass and the composition of these bodies. Since at any point in
the gravitational field a falling body behaves in the same way as moving by
inertia (but with a change in velocity), it is assumed that in the falling body
Lorentz covariance takes place. Then the Lorentz covariance should be at any
point of the trajectory of the falling body and it does not depend on the
velocity, and the falling observer should not reveal by inner experiments the
acceleration of the movement. As a result, the equivalence principle leads to
the identification of the effect of the gravitational field of a massive body
with the effect of deformation of spacetime around the massive body. These are
the consequences of the equivalence principle in general relativity.
In the metric theory of relativity (MTR), instead of the principle of
equivalence of forces the principle of equivalence of energy-momentum is
considered. Indeed, from equation (2) for the metric in the MTR we can see that
the metric is completely determined by the sources in the form of stress–energy tensors of fields including the
gravitational field itself [3]. Only the energy-momentum of the system is
necessary to determine the metric and the equations of motion of a test body. If
two different interactions have the same dependence of the energy-momentum,
then the metric and the law of motion in both cases coincide. The equation of
general relativity for the metric (1) differs from equation (2) for the metric
in MTR by the fact that the right-hand side of (2) contains the stress–energy tensor of gravitational field 
. The contribution of this tensor in weak fields is small,
and the MTR metric is slightly different from the metric of general relativity.
However, in strong gravitational fields the tensor can not be ignored, since
there is significant self action of field on the source of field. Another
difference is that in (2) there is no stress–energy
tensor of matter and in the
MTO metric depends only on stress–energy tensors of fields. In contrast,
in GR, according to (1) we need stress–energy
tensor of matter 
to
determine the metric.
Besides, the scalar curvature 
is in (1) with a multiplier 
and in MTR in relation (2) the scalar curvature has a multiplier 
. This is
due to the calibration of the cosmological constant and relativistic energy in
the covariant theory of gravitation [8].
Comparison of the theories of gravitational field
Axioms of the covariant theory
of gravitation (CTG) in 4-dimensional vector-tensor formalism have the form
[3]:
1) The properties of gravitational
field are set by the velocity of propagation of gravitational influence 
, as
well as the scalar potential 
and the vector
potential 
.
2) The potentials of the gravitational
field can be combined into 4-vector of gravitational potential with lower
covariant index:
.
The speed of change of
the potentials in spacetime of chosen reference frame is determined by the
tensor of gravitational field, which is 4-rotor of 
:
,
where 
denotes the covariant
derivative , 
– the usual 4-indices,
so that in the case of Cartesian coordinates
, 
, 
, 
.
With an appropriate choice of field
potentials, the symmetry relation of potentials holds:
.
(3)
3) The properties of substance are set
by density 
in the comoving frame
of reference and by velocity 
.
4) In special relativity the quantities
and are combined into
4-vector of density of mass current or of momentum density:
,
where 
– 4-velocity of matter unit,
– the density of moving matter,
– 3-vector of mass
current density.
In Riemannian space 4-velocity is
given by: 
, and 
.
5) The relation between the
gravitational field and the matter can be expressed through the relationship of
4-vector of gravitational potential 
and mass 4-current 
, or
through the connection between the tensor 
and :
(4)
Features of gravitational field in GR and in the covariant theory of gravitation arising from their axioms are given in Table
2.
Table 2
|
Features of theories |
The theory of
gravitational field in GR |
Covariant theory of
gravitation |
|
Gravitational field
is: |
Metric tensor field,
which is characterized by the tensor |
Physical vector field,
which is characterized by the
4-potential |
|
Gauge: |
Contraction of the
metric tensor in the form |
The field 4-potentials are calibrated so that their divergence vanish |
|
The speed of the
gravitational field equals: |
The speed of light |
The speed of
propagation of gravitation (about the speed of light) [9] |
|
The connection between the
gravitational field and the matter in the absence of other fields: |
Through the Hilbert-Einstein tensor equations for the metric (1),
linking the function of the metric tensor and the stress–energy tensor of matter |
Through equation (4) for the potentials or strengths of the
gravitational field, and for mass 4-current |
|
Sources of energy and momentum that define metric: |
Matter and any non-gravitational fields |
Any fields including the gravitational field |
Despite the difference in systems of axioms of gravitational field in GR
and in CTG, we can show that the equation of motion of general relativity is a
special case of equation of motion of the CTG. As it was found in [3], the material derivative with respect to proper time in
general case can be written in the form of an operator using 4-velocity 
of the matter unit:
, (5)
where the symbol 
denotes the total
differential in curved spacetime, and 
is the covariant
derivative.
Operator (5) shall be applied only to 4-objects in spacetime, which are
scalars, 4-vectors and 4-tensors. 4-velocity 
is determined by the 4-vector 
and the invariant of proper time
. If,
however, we try to find 4-velocity through the coordinate value 
using (5) in the form 
, then
there is a discrepancy, because the value in Riemannian space is not a
4-vector.
By definition in CTG, the force density is the total rate of change of
the 4-vector of density of mass 4-current 
by the proper time in Riemannian
spacetime:
, (6)
where 
– Christoffel symbol.
On the other hand, the
expression for the force density acting on the matter unit from gravitational
and electromagnetic fields and pressure field is obtained by taking the covariant derivative in equation (2). Then the
left side of the equation for the metric (2) gives zero, and from the right
side of this equation it follows:
. (7)
where 
– tensor of electromagnetic field strengths,
– electromagnetic
4-current,
– electric charge
density of the matter unit in its rest referent frame.
Comparing (6) and (7) gives the
equation of motion of the matter unit in the
CTG under the influence of pressure and gravitational and electromagnetic
forces:
. (8)
Equation (8) allows taking fully into account the reactive force of Meshcherskiy [10], which appears due to changes of the density
of the matter unit. The mass density is part of the mass 4-current 
, from which in (8) the derivative with
respect to proper time is taken, which characterizes the reaction force in the
mechanics of bodies with variable mass.
To move to the formula
for the force in general relativity we should make the following
simplifications in (8): assume 
is equal to zero (in GR the gravitational field is the metric field
which does not have the property of self action, and therefore the
gravitational field in the right-hand side of equation (1) as the source of
curvature of spacetime is absent), and assume the mass density 
is constant with
respect to time and volume of test particle. In GR, is also not used pressure
tensor in the form, as we have defined it. Then the quantity in the left side of
(8) can be canceled, and from the mass 4-current it is possible to pass to the 4-velocity 
:
. (9)
In the simplest case the
motion of matter in the absence of electromagnetic fields is considered: 
, or in the absence of charges of matter particles: 
. Then the right side of the equation of motion (9) will be
zero and there is equation 
, i.
e. the 4-acceleration of a freely falling body in the gravitational field is
absent. Taking into account the relations for the 4-velocity 
and for the interval 
or 
, we obtain the standard equation of motion of GR for the
matter in the gravitational field in the form:
. (10)
For the propagation of
light it must be: 
. Consequently, in (9) the differential 
must tend to zero.
Further after multiplication in (9) by 
we have:
. (11)
For the first term in the left side
of (11) we can write down:
.
Setting now in (11) 
, we obtain zero in right side and arrive at the following:
.
We shall choose as the
proper time for the light quantum the parameter of time 
along the trajectory,
marking the location of the quantum in space, and divide the above equation by
the square of the differential 
:
. (12)
Equation (12) represents
the standard equation of motion for the light quantum in GR. As it was seen in
the derivation of (10) and (12) from equation (8), the equations of motion of
GR for particles and light are the consequence of the equation of motion of
CTG. In this regard, again the question arises, why in the solar system are
such phenomena discovered unexplained by GR, as the Pioneer anomaly [11] and
flyby anomaly [12]? One of the possible explanations is given in [3], which
emphasizes the difference between the equation of motion (10) of GR and the
equation of motion (8) of CTG.
Thus, from the system of axioms for general relativity in GR, and the
system of axioms for gravitational field in GR all the basic features of
general theory of relativity can be deduced. The axioms of GR are given in the form
that allows comparing them with the axioms of covariant theory of gravitation
(CTG) and metric theory of relativity (MTR). As a consequence, it turns out in
[4] that general relativity in GR is a special case of the MTR. As for the
axioms of gravitational field, in GR the principle of geometrization of
gravitation and the equivalence principle lead to the concept of metric tensor
field as the field of gravitation. In CTG the gravitational field is
characterized by the vector field of 4-potential and by built with its help
antisymmetric tensor field of strengths of gravitational field, which consists
of two components – the gravitational field strength and the torsion field. The principle of definition of the gravitational
field in CTG is similar to the definition of electromagnetic field, so that the
gravitational field of CTG is not less real than the electromagnetic field,
with which it refers to the fundamental fields. The latter means that the
electromagnetic and gravitational fields exist not only in researches available
in modern science, but according to the theory of infinite nesting of matter
they act at different levels of matter. In this case, the gravitational field
at the level of elementary particles leads to strong gravitation, and at the macro
level – to the normal gravitation [13].
The analysis of the equivalence principle in general relativity shows
that it is valid only in the infinitely small regions, in which approximation
of Lorentz covariance is possible. However, this approximation becomes
inaccurate in large enough areas where we can not neglect the curvature of
spacetime. For example, if the test particle is massive, its proper
gravitational field should be considered in the equation of motion of the
particle in an external gravitational field. This is because the metric of two
interacting bodies in a nonlinear manner depends on the values of the metric of
these bodies, taken separately from each other. Therefore general relativity,
which uses in calculation the principle of equivalence and the principle of
geometrization of gravitational field is only an intermediate theory on the way
of building more complete theory of relativity and deeper theory of
gravitational field, fully taking into account the interaction of gravitational
field with matter and other fields.
References
1.
Einstein A. – Berl. Ber., 1915, S. 778, 779,
844.
2.
Hilbert D. Grundlagen der Physik, 1 Mitt;
Gott, Nachr., 1915, math.-nat. Kl., S. 395.
3.
Fedosin S.G. Fizicheskie teorii i
beskonechnaia vlozhennost’ materii. – Perm, 2009. – 844 p. ISBN
978-5-9901951-1-0.
4.
Comments to the book: Fedosin S.G.
Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844
pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0 (in Russian).
5. Fedosin S.G. The Principle of Least Action in
Covariant Theory of Gravitation. Hadronic
Journal, 2012, Vol. 35, No. 1, P. 35 – 70.
6.
Fedosin S.G. The Hamiltonian in
covariant theory of gravitation. Advances in Natural Science, 2012, Vol. 5,
No. 4, P. 55 – 75.
7.
Ehlers
J., Pirani F. A. and Schild A. The geometry of free fall and light propagation.
in: General Relativity. Papers honor of J. L. Synge, editor: L. O'
Raifertaigh (Oxford: Clarendon Press, 1972), pp. 63-82.
8.
Fedosin S.G. About the cosmological constant, acceleration
field, pressure field and energy. vixra.org, 5 Mar
2014
9.
Kopeikin S., Efroimsky M. and Kaplan G. Relativistic
Celestial Mechanics in the Solar System, Wiley-VCH, 2011. XXXII, 860 Pages,
65 Fig., 6 Tab.
10.
Мещерский И.В. Соч.:
Работы по механике переменной массы, 2 изд., М., 1952.
11.
Nieto, Michael Martin (2006), The quest to
understand the Pioneer anomaly , EurophysicsNews, Vol. 37 (6), P. 30–34.
12. Anderson J.D., et. al. (7 March 2008), Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth, Phys. Rev. Lett., Vol. 100 (9), 091102(1-4).
13.
Fedosin S.G. Model
of Gravitational Interaction in the Concept of Gravitons. Journal of Vectorial Relativity, Vol. 4, No.
1, March 2009, P.1–24.
Source: http://sergf.ru/axen.htm