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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: