Jordan Journal of Physics. Vol. 9 (No. 1), pp. 1-30, (2016). http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html

**About the cosmological constant,
acceleration field, pressure field and energy**

**Sergey G.
Fedosin**

PO box 614088, Sviazeva str. 22-79,
Perm, Russia

E-mail: intelli@list.ru

Based on the condition
of relativistic energy uniqueness the calibration of the cosmological constant
was performed. This allowed us to obtain the corresponding equation for the
metric, to determine the generalized momentum,
the relativistic energy, momentum and the mass of the system, as well as
the expressions for the kinetic and potential energies. The scalar curvature at
an arbitrary point of the system equaled zero, if the substance is absent at
this point; the presence of a gravitational or electromagnetic field is enough
for the space-time curvature. Four-potentials of the acceleration field and
pressure field, as well as tensor invariants determining the energy density of
these fields, were introduced into the Lagrangian in order to describe the
system’s motion more precisely. The structure of the Lagrangian used is
completely symmetrical in form with respect to the 4-potentials of
gravitational and electromagnetic fields and acceleration and pressure fields.
The stress-energy tensors of the gravitational, acceleration and pressure
fields are obtained in explicit form, each of them can be expressed through the
corresponding field vector and additional solenoidal vector. A description of
the equations of acceleration and pressure fields is provided.

*Keywords:** cosmological constant;
4-momentum; acceleration field; pressure field; covariant theory of gravitation**.*

**PACS****:** 04.20.Fy, 04.40.-b, 11.10.Ef

**1. ****Introduction**

The most popular
application of the cosmological constant in the general theory of relativity (GTR) is
that this quantity represents the manifestation of the vacuum energy [1-2].
There is another approach to the cosmological constant interpretation,
according to which this quantity represents the energy possessed by any
solitary particle in the absence of external fields. In this case, including into the Lagrangian seems quite appropriate
since the Lagrangian contains such energy components, which should fully
describe the properties of any system consisting of particles and fields.

Earlier in [3-4] we
used such calibration of the cosmological constant, which allowed us to
maximally simplify the equation for the metric. The disadvantage of this
approach was that the relativistic energy of the system could not be determined
uniquely, since the expression for the energy included the scalar curvature. In
this paper we use another universal calibration of the cosmological constant,
which is suitable for any particle and system of particles and fields. As a
result, the energy is independent of both the scalar curvature and the cosmological
constant.

In GTR the
gravitational field as a separate object is not included in the Lagrangian, and
the role of a field is played by the metric itself. A known problem arising
from such an approach is that in GTR there is no stress-energy tensor of the
gravitational field.

In contrast, in the
covariant theory of gravitation the Lagrangian is used containing the term with
the energy of the particles in the gravitational field and the term with the
energy of the gravitational field as such. Thus, the gravitational field is
included in the Lagrangian in the same way as the electromagnetic field. In
this case, the metric of the curved spacetime is used to specify the equations
of motion as compared to the case of such a weak field, the limit of which is
the special theory of relativity. In the weak field limit a simplified metric
is used, which almost does not depend on the coordinates and time. This is
enough in many cases, for example, in case of describing the motion of planets.
However, generally, in case of strong fields and for studying the subtle
effects the use of metric becomes necessary.

We will note that the
term with the particle energy in the Lagrangian can be written in different
ways. In [5-6] this term contains the invariant , where is the mass density in the co-moving reference frame, is 4-velocity. The corresponding quantity in
[7] has the form . In [3] and [8]
instead of it the product is used, where is mass 4-current. In
this paper we have chosen another form of the mentioned invariant – in the form . The reason for this
choice is the fact that we consider the mass 4-current to be the fullest representative of the properties of substance
particles containing both the mass density and the 4-velocity. The mass
4-current can be considered as the 4-potential of the matter field. All the
other 4-vectors in the Lagrangian are 4-potentials of the respective fields and
are written with covariant indices. With the help of these 4-potentials tensor
invariants are calculated which characterize the energy of the respective field
in the Lagrangian.

**2. ****Action and its
variations in the principle of least action**

**2.1. ****The action function**

We use the following
expression as the action function for continuously distributed matter in the
gravitational and electromagnetic fields in an arbitrary frame of reference:

(1)

where – the Lagrange function or Lagrangian,

– the differential of the
coordinate time of the used reference
frame,

– a coefficient to be
determined,

– the scalar curvature,

– the cosmological constant,

– the 4-vector of
gravitational (mass) current,

– the mass density in the
reference frame associated with the particle,

– the 4-velocity of a point
particle, – 4-displacement, – interval,

– the speed of light as a
measure of the propagation velocity of electromagnetic and gravitational
interactions,

– the 4-potential of
the gravitational field, described by the scalar potential and the vector potential of this field,

– the gravitational constant,

– the gravitational tensor
(the tensor of gravitational field strengths),

– definition of the gravitational tensor with contravariant
indices by means of the metric tensor ,

– the 4-potential of
the electromagnetic field, which is set by the scalar potential and the vector potential of this field,

– the 4-vector of the
electromagnetic (charge) current,

– the charge density in the
reference frame associated with the particle,

– the vacuum
permittivity,

– the electromagnetic tensor
(the tensor of electromagnetic field strengths),

– the 4-velocity with a
covariant index, expressed through the metric tensor and the 4-velocity with a
contravariant index; it is convenient to consider the covariant 4-velocity locally averaged over the particle
system as the 4-potential of
the acceleration field , where and denote the scalar and vector potentials,
respectively,

– the acceleration tensor
calculated through the derivatives of the four-potential of the acceleration field,

– a function of
coordinates and time,

– the 4-potential of the
pressure field, consisting of the scalar potential and the vector potential , is the pressure in the reference frame
associated with the particle, the relation specifies the equation of the substance state,

– the tensor of the pressure
field,

– a function of
coordinates and time,

– the invariant 4-volume,
expressed through the differential of the time coordinate , through the
product of differentials of the
spatial coordinates and through the square root of the determinant of the metric tensor,
taken with a negative sign.

Action function (1)
consists of almost the same terms as those which were considered in [3]. The
difference is that now we replace the term with the energy density of particles
with four terms located at the end of (1). It is natural to assume that each
term is included in (1) relatively independently of the other terms, describing
the state of the system in one way or another. The value of the 4-potential of the set of matter units or point particles of the
system defines the 4-field of the system’s velocities, and the product in (1) can be regarded as the energy of
interaction of the mass current with the field of velocities.

Similarly, is the 4-potential of the gravitational field, and the
product defines the energy of interaction of the mass
current with the gravitational field. The electromagnetic field is specified by
the 4-potential , the source of the field is the electromagnetic
current , and the product of
these quantities is the density of the energy of interaction of
a moving charged substance unit with the electromagnetic field. The invariant
of the gravitational field in the form of the tensor product is associated with the gravitational
field energy and cannot be equal to zero even outside bodies. The same holds
for the electromagnetic field invariant . This follows from the
properties of long-range action of the specified fields. As for the field of velocities , the field should be
used to describe the motion of the substance particles. Accordingly, the field
of accelerations in the form of the tensor and the energy of this field associated with
the invariant refer to the accelerated motion of particles
and are calculated for those spatial points within the system’s volume where
the substance is located.

The last two terms in
(1) are associated with the pressure in the substance, and the product characterizes the
interaction of the pressure field with the mass 4-current, and the invariant is part of the stress-energy tensor of the
pressure field.

We will also note the
difference of 4-currents and – all
particles of the system make contribution to the mass current , and only charged
particles make contribution to the
electromagnetic current . This
results in difference of the fields’ influence –
the gravitational field influences any particles and the electromagnetic field
influences only the charged particles or the substance, in which by the field
can sufficiently divide with its influence the charges of opposite signs from
each other. The field of velocities , as well as the mass current , are
associated with all the particles of
the system. Therefore, the product describes that part of the particles’ energy,
which stays if we somehow "turn off " in the
system under consideration all the macroscopic gravitational and
electromagnetic fields and remove the pressure, without changing the field of
velocities or the mass current .

**2. 2. ****Variations
of the action function**

We will vary the action function in (1) term by term, then the total variation will be
the sum of variations of individual terms. In total there are 9 terms inside
the integral in (1). If we consider the quantity a constant
(a cosmological constant), then according to [7-9] the variation of the first
term in the action function (1) is equal to:

, (2)

where is the
Ricci tensor,

is the variation of the metric tensor.

According to [3] the variations of terms 2 and 3 in the action function
are as follows:

, (3)

, (4)

where is the variation of coordinates, which results in the variation of the
mass 4-current and in the variation of the electromagnetic 4-current
,

is the variation of the 4-potential of the
gravitational field,

and denotes the stress-energy tensor of the gravitational field:

. (5)

Variations of terms 4 and 5 in the action function according to [6-7],
[10] are as follows:

, (6)

, (7)

where is the variation of the 4-potential of the electromagnetic field,

and denotes the stress-energy tensor of the electromagnetic field:

. (8)

Variations of the other terms in the action function (1) are defined in
Appendices A-D and have the following form:

, (9)

, (10)

where is the variation of the 4-potential of the acceleration
field,

and denotes the stress-energy tensor of the field of accelerations:

. (11)

(12)

, (13)

where the
stress-energy tensor of the pressure field:

. (14)

In variation (10) in order to simplify the special case is considered,
when is a
constant, which does not vary by definition. According to
its meaning depends on the parameters of the system under consideration, and
therefore can have different values. The same should be said about .

**3. ****The**** motion equations of the field, particles
and metric**

According to the principle of least action, we should sum up all the
variations of the individual terms of the action function and equate the result
to zero. The sum of variations (2), (3), (4), (6), (7), (9), (10), (12) and
(13) gives the total variation of the action function:

. (15)

**3.1. ****The field equations**

When the system moves in spacetime, the variations , , , , and do not vanish, since it is supposed that it can occur only at the
beginning and the end of the process, when the conditions of motion are
precisely fixed. Consequently, the sum of the terms, which is located before
these variations, should vanish. For example, the variation occurs only in according to (6
) and in from
(7), then from (15) it follows:

.

From this we obtain the equation of the electromagnetic field with the
field sources:

or , (16)

where is the vacuum permeability.

The second equation of the electromagnetic field follows from the
definition of the electromagnetic tensor in terms of the electromagnetic
4-potential and from the antisymmetry properties of this tensor:

or , (17)

where is a Levi-Civita symbol or a completely antisymmetric
unit tensor.

The variation is present only in (3) and (4), so
that according to (15) we should obtain:

.

The equation of gravitational field with the field sources follows from
this:

or . (18)

If we take into account the definition of the gravitational tensor: , and take the covariant derivative of this
tensor with subsequent cyclic interchange of the indices, the following
equations are solved identically:

or . (19)

Equation (19) without the sources and equation (18) with the sources
define a complete set of gravitational field equations in the covariant theory
of gravitation.

Consider
now the rule for the difference of the second covariant derivatives with
respect to the covariant derivative of the electromagnetic 4-potential
:

With the
rule in mind, the applying of the covariant derivative to (16) and (18) gives the following:

.

.

This shows
that field tensors and lead to the divergence of the corresponding
4-currents in a curved space-time. Mixed curvature tensor and Ricci tensor vanish only
in Minkowski space. In this case, the covariant derivatives become the partial derivatives
and the continuity equation for the gravitational and electromagnetic
4-currents in the special theory of relativity are obtained:

, . (20)

We will note that in order to simplify the equations for the 4-potential of fields we
can use expressions which are called gauge conditions:

, . (21)

**3.2. ****The acceleration field
equations**

The variation of 4-potential
is included in (9) and (10), therefore according to (15) we should
obtain:

.

, or
. (22)

If we compare (18) and
(22), it turns out that the presence of the 4-vector of mass current not only leads to occurrence
of space-time gradient of the gravitational field in the system under
consideration, but is generally accompanied by changes in time or by 4-velocity
gradients of the particles that constitute this system. Besides the covariant
4-velocities of the whole set of particles forms the velocity field , the derivatives of which define the acceleration field
and are described by the tensor . As an example of a
system, where it can be clearly observed, we can take a rotating
partially-charged collapsing gas-dust cloud, held by gravity. An ordered acceleration
field occurs in the cloud due to the rotational acceleration and contains the
centripetal and tangential acceleration.

Due to its definition
in the form of a 4-rotor of , the following
relations hold for acceleration tensor :

or .
(23)

As we can see, the structure of equations (22) and
(23) for the acceleration field is similar to the structure of equations for
the strengths of gravitational and electromagnetic fields.

In the local geodetic reference frame the derivatives
of the metric tensor and the curvature tensor become equal to zero, the
covariant derivative becomes a partial derivative and the equations take the
simplest form. We will go over to this reference frame and apply the derivative
to (23) and make substitution
for the first and third terms using (22):

.

If we apply definition to 4-d’Alembertian , where is the d'Alembert operator, it will give:

.

Comparing with the
previous expression, we find the wave equation for the 4-potential :

.

On the other hand, after
lowering of the acceleration tensor indices we have from (22):

.

Comparing this equation
with equation for leads to the
expression:

,

where is some constant.

In an arbitrary reference frame we should specify the
obtained expressions, since in contrast to permutations of partial derivatives,
in case of permutation of the covariant derivatives from the sequence to the sequence some additional terms appear. In particular, if we use the relation:

,
(24)

then after substituting the expression in (22), the wave equation can be
written as follows:

. (25)

In the curved space operator acts differently on scalars, 4-vectors and 4-tensors, and
usually it contains the Ricci tensor. Due to condition (24), the Ricci tensor
is absent in (25), but the terms with the Christoffel symbols remain.

Equation (24) is a
gauge condition for the 4-potential , which is similar by
its meaning to gauge conditions (21) for the electromagnetic and gravitational
4-potentials. Both (24) and (25) will hold on condition that .

In Appendix E it will
be shown that the acceleration tensor includes the vector components and , based on which,
according to (E6), we can build a 4-vector of particles’ acceleration.

**3.3. ****The pressure field
equations**

To obtain the pressure field
equations we need to choose in (15) those terms which contain the variation . This variation is
present in (12) and (13), which gives the following:

.

or .
(26)

It follows from (26)
that the mass 4-current generates the pressure field in bodies, which can be
described by the pressure tensor . The same relations
hold for this tensor as for the tensors of other fields:

or . (27)

The wave equation for the
4-potential of the pressure field follows from (26) and (27):

.
(28)

Equations (26) and (28)
will be consistent at the same time if there is gauge condition of the pressure
4-potential:

,
(29)

where is some constant. As a
rule, these constants are chosen to be equal to zero.

The properties of the
pressure field are described in Appendix F, where it is shown that the pressure
tensor contains two vector components and , which determine the energy
and the pressure force, as well as the pressure energy flux.

**3.4. ****The equations of motion
of particles**

The variation that leads to the equations of motion of the
particles is present in (3), (6), (9) and (12). For this variation it follows from (15):

,

.

The left side of the
equation can be transformed, considering the expression for the 4-vector of mass current density and
the definition of the acceleration tensor :

. (30)

We used the relation , which follows from the equation , and the operator of proper-time-derivative as
operator of the derivative with respect to the proper time , where is a symbol of 4-differential in curved
spacetime, is the proper time [11]. Taking into account
(30) the equation of motion takes the form:

. (31)

We will note that the equations
of field motion (16) – (19), of the
acceleration field (22) and (23), of the pressure field (26) and (27) and the
equation of the particles’ motion (31) are differential equations, which are
valid at any point volume of spacetime in the system under consideration. In
particular, if the mass density in some point volume is zero, then all the
terms in (31) will be zero.

The quantity in the left side of (31) is the 4-acceleration
of a point particle, while the proper time differential is associated with the interval by relation: and the relation holds: . The first two terms
in the right side of (31) are the densities of the gravitational and electromagnetic
4-forces, respectively. It can be shown (see for example [3], [12]) that for
4-forces, exerted by the field on the particle, there are alternative
expressions in terms of the stress-energy tensors (5) and (8):

, . (32)

Similarly, the left side
of (31) with regard to (30) is expressed in terms of stress-energy tensor of
the acceleration field (11):

. (33)

To prove (33) we should
expand the tensor with the help of definition (11), apply the covariant derivative to the tensor products
and then use equations (22) and (23). Equation (33) shows that the
4-acceleration of the particle can be described by either the acceleration
tensor or the tensor .

For the pressure field we
can write the same as for other fields:

.
(34)

In (34) the pressure
4-force is associated with the covariant derivative of the stress-energy tensor
of the pressure field.

From (31) – (34) it follows:

or . (35)

In Minkowski space , where is present, 4-differetials become ordinary
differentials , ,

and the motion equation
(31) falls into the scalar and vector equations, while the vector equation
contains the total gravitational force with regard to the torsion field, the
electromagnetic Lorentz force and the pressure force:

, (36)

, (37)

where is the velocity of a point particle, is the gravitational field strength, is the charge density, is the electric field strength, is the pressure field strength, is the torsion field vector, is the magnetic field induction, is the solenoidal vector of the pressure
field.

If during the time the density does not change, it can
be put under the derivative’s sign. Then in the left side of (36) the quantity appears, where is the relativistic energy density. Similarly, in the left side of (37) the quantity appears, where is the mass 3-current density.

**3.5. ****The equations for the metric**

Let us consider action
variations (2), (3), (4), (6), (7), (9), (10), (12) and (13), which contain the
variation . The sum of all the
terms in (15) with the variation must be zero:

.

(38)

The equation for the
metric (38) allows us to determine the metric tensor by the known quantities characterizing the
substance and field. If we take the covariant derivative in this equation, the left side of the
equation vanishes on condition , and taking into
account (35) we obtain the following:

,

, (39)

where is a function of time and coordinates and the
scalar invariant with respect to coordinate transformations.

If we expand the scalar
products of vectors using the expressions:

, , (40)

.

then (39) can be written
as:

. (41)

If the system’s
substance and charges are divided to small pieces and scattered to infinity,
then there the external field potentials become equal to zero, since
interparticle interaction tends to zero, and at we obtain the following:

. (42)

Consequently, is associated with the particle’s proper scalar potentials and , the mass density and the pressure in the particle located
at infinity. Expression (41) can be considered as the differential law of
conservation of mass-energy: the greater the velocity of a point particle is, and the greater the
gravitational field potentials and , the electromagnetic
field potentials and , the pressure field
potentials and are, the more the mass
density differs from its value at infinity. For
example, if a point particle falls into the gravitational field with the potential , then the change in
the particle’s energy is described by the term . According to (41),
such energy change can be compensated by the change in the rest energy of the
particle due to the change . Since the
gravitational field potential is always negative, then the mass density and the pressure inside the point particle
should increase due to the field potential.

This is possible, if we
remember that the whole procedure of deriving the motion equations of
particles, field and metric from the principle of least action is based on the
fact that the mass and charge of the substance unit at varying of the
coordinates remain constant, despite of the change in the charge density,
substance density and its volume [7]. If the mass of a simple system in the
form of a point particle and the fields associated with it is proportional to , then according to
(41) the mass of such a system remains unchanged, despite of the change in the
fields, mass density and pressure . Conservation of the
mass-energy of each particle with regard to the mass-energy of the fields leads
to conservation of the mass-energy of an arbitrary system including a multitude
of particles and the fields surrounding them. We will remind that this article
refers to the continuously distributed matter, so that each point particle or a
unit of this matter may have its own mass density and its value .

We will now return to (38)
and take the contraction of tensors by means of multiplying the equation by , taking into account the relation , and then dividing all
by 2:

. (43)

where is the scalar curvature,
and it was taken into account that the contractions of tensors , , and are equal to zero.

In case if the
cosmological constant were known, based on (43)
we could find the scalar curvature .

In order to simplify
the equation (38) in [3] and [4] we introduced the gauge for , at which the following equation would hold, if we additionally
take into account the term with the pressure :

. (44)

In the gauge (44) the
equation for the metric (38) takes the following form, provided that , where is a constant of order of unity:

. (45)

We will note that if
from the right side of (45) we exclude the stress-energy tensor of the
gravitational field , replace the tensor with the stress-energy tensor
of the substance in the form , and also neglect the
tensor , then at we will obtain a typical equation for the metric used in the general
theory of relativity:

. (46)

The equation for the
metric (38) and the expression (39) must hold in the covariant theory of
gravitation, provided that . If in (39) we remove
the term , then we will obtain
an expression suitable for use in the general theory of relativity. In this
case, given that , instead of (39) we
obtain the following:

.
(47)

If in (47) we equate
the term with the energy of particles in the electromagnetic field (in the case
when the field is zero) to zero, then the sum of the rest energy density and
the pressure energy of each uncharged point particle must be unchanged. It
follows that the pressure change must be accompanied by a change in the mass
density. If the system contains the electromagnetic field with the 4-potential acting on the 4-currents generating them, then in the general case there must be inverse correlation
of the rest energy, pressure energy and the energy of charges in the
electromagnetic field.

Indeed, in the general
theory of relativity the mass density determines the rest energy density and
spacetime metric, which represents the gravitational field. In (47) the energy
of charges in the electromagnetic field is specified by the term
, and the mass density
and hence the metric are associated with this energy at a constant . On the other hand,
the metric is obtained from (46). Therefore, the occurrence of the
electromagnetic field influences the metric in two relations — in (47) the mass
density and the corresponding metric change, as well as in the equation for the
metric (46) the stress-energy tensor of the substance changes, while the stress-energy tensor of the
electromagnetic field also makes contribution to
the metric.

A well-known paradox of
general theory of relativity is associated with all of this — the
electromagnetic field influences the density, the mass of the bodies as the
source of gravitation, and the metric, while the gravitational field itself
(i.e. metric) does not influence the electrical charges of the bodies, which
are the sources of the electromagnetic field. Thus the gravitational and
electromagnetic fields are unequal relative to each other, despite the
similarity of field equations and the same character of long-range action.
Above we pointed out at the fact that the mass 4-current leads to the
gravitational field gradients, and the addition of the charge to this mass
current generates additional electromagnetic (charge) 4-current and the
corresponding electromagnetic field gradients, depending on the sign of the
charge. From this we can see that the gravitational field looks like a
fundamental, basic and indestructible field and the electromagnetic field
manifests as some superstructure and the result of the charge separation in the
initially neutral substance.

If we consider (44) to
be valid, then from comparison with (39) we see that the equation must be satisfied. Thus, when is considered as a
cosmological constant, we can use it to achieve simplification of the equation
for the metric (38) and bring it to the form of (45). At the same time the
relation (39) is symmetrical with respect to the contribution of the
gravitational and electromagnetic fields to the density, in spite of the
difference in fields. We will remind that in the equation of motion (31) both
fields also make symmetrical contributions to the 4-acceleration of a point
charge.

Although the gauge for in the form of (44) seems
the simplest and simplifies some of the equations, in Section 7 the necessity
and convenience of another gauge will be shown.

**4. ****Hamiltonian**

In this and the next
sections we rely on the standard approach of analytical mechanics. As the
coordinates it is convenient to choose a set of Cartesian coordinates: , , , .

Let us consider action
(1) and express the Lagrangian from it:

(48)

The integration in (48)
is carried out over the infinite three-dimensional volume of space and over all
the material particles of the system. We assume that the scalar curvature depends on the metric tensor, and the metric
tensor , the field tensors , , , , the density , the charge density and the pressure are functions of the coordinates and do not depend on the particle velocities.
Then the Lagrangian in its general form (48) depends on the coordinates, as
well as on the 4-potential of pressure and 4-potentials of the
gravitational and electromagnetic fields and .

We will divide
the first integral in the Lagrangian (48) to the sum of particular integrals,
each of which describes the state of one of the set of the system’s particles. We will take into
account also that the Lagrangian depends on the three-dimensional velocities of
the particles , where specifies the particle’s number, while the
velocity of any particle is part of only one corresponding particular integral.
If we denote by the second integral in (48), which is
associated with the energies of fields inside and outside the fixed physical
system and is independent of the particles’ velocities, then we can write for
the Lagrangian:

,

where is a particular Lagrangian of an arbitrary
particle.

We will
introduce now the Hamiltonian of the system as a function of generalized
three-dimensional momenta of the particles: . Under the
system’s generalized momentum we mean the sum of the generalized momenta of the
whole set of particles:

.

To find the Hamiltonian
we will apply the Legendre transformations to the system of particles:

, (49)

provided that

.
(50)

The
equality in (50) gives the definition of the generalized momentum
, and we can
see that the generalized momentum of an arbitrary particle equals . On the
other hand, the equations allow us to express the velocity of an arbitrary particle through its
generalized momentum . Then we
can substitute these velocities in (49) and determine only through .

In order to find in (50), in each
particular Lagrangian we should express and in terms of the
velocity and interval :

, , (51)

while and we introduce the
notation , where the
four-dimensional quantity is not a real 4-vector. With regard to the
definition of the 4-potential of the acceleration field , for each
particle we obtain:

.
(52)

In (48) the unit of
volume of the system in any particular integral can
be expressed in terms of the unit of volume in the reference frame associated with the particle in the following way:

. (53)

From this
formula in the weak-field limit in Minkowski space, when , it
follows that the volume of a moving particle is decreased in comparison with
the volume of a particle at rest. Given that , where is the proper time in the reference frame of the particle, the equality of 4-volumes in
different reference frames follows from (53):

.

This equation reflects
the fact that the 4-volume is a 4-invariant.

Under the above
conditions (40), (51), (52) and (53) can be written for the Lagrangian (48) as
follows:

(54)

as well as after partial
volume integration:

(55)

where is the mass of an arbitrary particle, is the particle’s charge. In (55) the scalar and vector
field potentials are averaged over the particle’s volume,
that means they are the effective potentials at the location of the
particle.

In operations with
3-vectors it is convenient to write vectors in the form of components or
projections on the spatial axes of the coordinate system using, for example,
instead of the velocity the quantity , where . Then
, , and the velocity derivative can be represented
as: . For the gravitational
vector potential in particular we obtain: .

With this in mind, from
(55) and (50) we find:

, . (56)

Based on this, we find
for the sums of the scalar products of 3-vectors by summing over the index :

. (57)

From (49) taking into
account (55) and (57) we have:

(58)

In (58) the Hamiltonian
contains the scalar curvature and the cosmological constant
. As it will be shown in
Section 6 about the energy, this Hamiltonian represents the relativistic energy
of the system. To make the picture complete we could also express the quantity in (58) through the generalized momentum . We have described
this procedure in [4].

For
continuously distributed substance the masses and charges of the particles in
(58) can be expressed through the corresponding integrals: , . Also
taking into account (53), in which we can substitute the expression
, where denotes the time component of
the 4-velocity of an arbitrary particle, from (58) we find:

(59)

If in (58)
we express the masses and charges of the particles through the mass density and the charge density from the standpoint of the arbitrary reference frame , then (58)
can be represented as follows:

(60)

We obtained
the Hamiltonian in an arbitrary reference frame , while in
(59) the densities in reference frames associated with the particles are used
and in (60) such particle densities are used, as they seem to be in .

**5. ****Hamilton’s equations**

Assuming
that the Hamiltonian depends on the generalized 3-momenta of particles : and the Lagrangian depends on 3-velocity of
particles : , where is a three-dimensional radius-vector of the
particle with the number , we will
take differentials of and , as well
as the differentials of both sides of equation (49):

(61)

(62)

. (63)

Substituting (61) and
(62) into (63), we find:

, ,
, ,

, , . (64)

The last equation in
(64) leads to (50) and gives the expression (56) for the generalized momentum of an arbitrary particle of the system in an
explicit form.

We will now apply the
principle of least action to the Lagrangian in the form , equating the action
variation to zero, when the particle moves from the time point to the time point .

(65)

In (65) it was assumed
that the time variation is equal to zero: . Partial derivatives
with variations , and lead to field equations
(16), (18) and (26). If we take into account the definition of velocity in the
second term in the integral (65): , then the integral for
this term is taken by parts. Then for the first and second terms in the
integral (65) we have the following:

. (66)

When varying the action, the variations are equal to zero only at the beginning and at
the end of the motion, that is when and . Therefore, for
vanishing of the variation it is necessary that the
quantity in brackets inside the integral (66) would be equal to zero. This
leads to the well-known Lagrange equations of motion:

. (67)

According to (64) , as well as . Let us substitute
this in (67):

.
(68)

Equation (68) together
with equation from (64)

represent the standard Hamiltonian equations describing the
motion of an arbitrary particle of the system in the gravitational and
electromagnetic fields and in the pressure field. According to (68), the rate
of change of the generalized momentum of the particle by the coordinate time is
equal to the generalized force, which is found as the gradient with respect to
the particle’s coordinates of the relativistic energy
of the system taken with the opposite sign. These equations are widely used not
only in the general theory of relativity, but also in other areas of theoretical
physics. We have checked these equations in [4] in the framework of the
covariant theory of gravitation by direct substitution of the Hamiltonian.

**6. ****The system’s energy**

We will consider a
closed system which is in the state of some stationary motion. An example would
be a charged ball rotating around its center of mass, which forms the system
under consideration together with its gravitational and electromagnetic fields
and the internal pressure. In such a system the energy should be conserved as a
consequence of lack of energy losses to the environment and taking into account
the homogeneity of time, i.e. the equivalence of the time points for the
system’s state.

The system’s
Lagrangian, taking into account the fields’ energy, has the form of (55). Due
to the stationary motion we can assume that within the system’s volume the
metric tensor , the scalar curvature , the 4-potentials of
the field , and of the pressure do not depend on time. But since any point
particle moves with the ball, then its location and velocity are changed, being
defined by the radius vector and velocity
, respectively. We may
assume that the Lagrangian of the system does not depend explicitly on time and
is a function of the form: . Now we will take the
time derivative of the Lagrangian, as it is done for example in [13], only not
for one but for a set of particles, and will apply (67):

.

The quantity in the
brackets is not time-dependent and is constant. This gives the definition of relativistic
energy as a conserved quantity for a closed system at stationary motion:

.
(69)

With regard to (64) and
(49), we find the following:

.
(70)

It turns out that the relativistic
energy can be expressed in a covariant form, since according to (70) the
formula for the energy coincides with the formula for the Hamiltonian in (49).

To calculate the
relativistic energy of the system with the substance, which is continuously
distributed over the volume, it is convenient to pass from the mass and charge
of the particle to the corresponding densities inside the particle. According
to (59) we obtain:

(71)

Using expression (71)
we can find the invariant energy of the system, for which we should use the
frame of reference of the center of mass and calculate the integral. In
addition, at a known velocity of the center of mass of the system in an
arbitrary reference frame we can calculate the
momentum of the system in . This can be clarified
as follows. We will define the invariant mass of the system taking into account
the mass-energy of the fields using the relation: , where is the speed of light as
a measure of the velocity of propagation of electromagnetic and gravitational
interactions. If the 4-displacement in has the form: , then for the
4-velocity of the system in we can write: . The 4-vector defines the 4-momentum, which contains the
relativistic energy and relativistic momentum . This gives the
formula for determining the momentum through the energy: , and, correspondingly, for the 4-momentum: .

In the reference frame , in which the system is at rest , , , and then , and also , that is in the
4-momentum in the reference frame only the time component is nonzero.

If we
multiply the 4-momentum by the speed of light, we will obtain the 4-vector of
the form , the time
component of which is the relativistic energy, equal in value to the
Hamiltonian. Thus we find the 4-vector, which in [4] was called the Hamiltonian
4-vector.

**7. ****The cosmological
constant gauge and the resulting consequences**

We will make
transformations and substitute (43) and (39) in (71):

(72)

If we choose the
condition for the cosmological constant in the form:

,
(73)

then the relativistic
energy (72) is uniquely defined, since the dependence on the constants and disappears:

(74)

We will remind that the
quantities and can have their own values for each particle of
matter. But on condition of (73) the expression for the relativistic energy
(74) becomes universal for any particle in an arbitrary system of particles and
their fields.

From (73) and (39) the
equation follows:

. (75)

In order to estimate
the value of the cosmological constant , it is convenient to
divide all of the system’s substance into small pieces, scatter them apart to
infinity and leave there motionless. Then the vector potentials of the fields
and pressure become equal to zero and the relation remains: . It follows that , just like in ( 42), is associated with the rest energy,
with the pressure energy and with the proper energy of the fields of the system
under consideration.

If in some volume there
are no particles and the mass density and the charge density are zero, then in this volume there must
remain the relativistic energy of the external fields:

. (76)

Based on (74) we can
express the energy of a small body at rest. For simplicity we will assume that
the body does not rotate as a whole and there is no motion of the substance
and charges inside of it (an ideal solid body without the intrinsic magnetic
field and the torsion field). Under such conditions the coordinate time of the
system becomes approximately equal to the proper time of the body: . Since the interval , then we obtain: . Since there are no spatial motion in any part of the body, we can write:

, .

With this in mind we
obtain from (74):

(77)

In the weak field limit
in (77) we can use , . The tensor product in the absence of
substance motion inside the ideal solid body vanishes. Using (F5) and (F6) we can write:

, , .

Besides in [4] it was
found that in the weak field for a motionless body in the form of a ball with
uniform density of mass and charge the following relations hold for the body’s
proper fields:

, ,

,

. (78)

According to (78) the potential
energy of the ball’s substance in the proper gravitational field which is
associated with the scalar potential is twice greater than the
potential energy associated with the field strength . The same is true for
the electromagnetic field with the potential and the strength both in the case of uniform
arrangement of charges in the ball’s volume and in case of their location on
the surface only. Substituting (78) into (77) in the framework of the special
theory of relativity gives the invariant energy of the system in the form of a
fixed solid spherical body with uniform density of mass and charge, taking into
account the energy of their proper potential fields:

(79)

This calculation is apparently not complete since in reality inside any
body there are particles, which cannot be as motionless as the body itself is.
Therefore in (79), in addition to the pressure and its gradient within the body
it is necessary to add the kinetic energy of motion of all the particles which
constitute the body.

**7.1. ****The metric**

Substituting (75) and
(39) into (43), we find the expression for the scalar curvature
:

, (80)

while , where is a constant of the
order of unity.

As it can be seen, the
scalar curvature is zero in the whole space outside the body. The equation does not mean however, that the spacetime is
flat as in the special theory of relativity, since the curvature of spacetime
is determined by the components of the Riemann curvature tensor.

We will now substitute
(75) into the equation for the metric (38):

(81)

that also can be written
using (39) as follows:

. (82)

If we take the covariant
derivative of (82), the left side of the equation vanishes due to the property
of the Einstein tensor located there. The right side, with regard to the
equation of motion (35) and provided that the metric tensor in covariant differentiation behaves as a
constant and is a constant, vanishes too.

In (81) we can use (80)
to replace the scalar curvature:

(83)

If we sum up (81) and
(83) and divide the result by 2 , we will obtain the
following equation for the metric:

, (84)

while with regard to (80) , according
to (35) , and as the property of the Einstein tensor.

In empty space
according to (84) the curvature tensor depends only on the
stress-energy tensors of the gravitational and electromagnetic fields and , so these fields
change the curvature of spacetime outside the bodies. We will note that in
equation (84) the cosmological constant and the tensor product of the type are missing. This fact makes determination of the metric tensor
components much easier.

If we compare (84) with
the Einstein equation (46), then two major differences will be found out — in
the right side of (84 ) stress-energy tensors , and are present, and in addition the coefficient
in front of the scalar curvature is two times less than in
(46).

**8. ****Компоненты**** ****энергии**

In Newtonian mechanics the relations for the Lagrangian and the total energy are known: