Solution of the Einstein equation and metric of

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Schwarzchild solution is considered as an approximation of this solution. Keywords: Einstein ..... of relativity http://arxiv.org/pdf/1404.3083v1. 11. A.Einstein.

Solution of the Einstein equation and metric of accelerated system V.B.Morozov

Metric of a uniformly accelerated system and the equivalent metric of a uniform stationary gravitational field were obtained. Exact solution of the Einstein gravitational equation for a localized mass as a Cauchy problem with asymptotically uniform boundary conditions was obtained. The energy momentum tensor of this metric is nonzero and is similar to the energy momentum tensor of a uniform electric field, taken with the opposite sign. Schwarzchild solution is considered as an approximation of this solution. Keywords: Einstein equation, energy momentum tensor of a field, uniform noninertial reference frame, rigorous solutions of the Einstein equation.

1.

Introduction

Einstein gravitational equation describes known empirical data correctly. However, difficulties with introduction of the energy momentum tensor to the theory appeared at the very beginning. Searching for a suitable solution of this problem was not simple. A compromise solution was finally found. It is important that the energy of the gravitational field interacting with the system of the matter's masses and fields is introduced for a case when the space-time is =(

asymptotically plain [1, 2]. If interspace is set

) +(

) +(

) , the metric

coincides asymptotically with the Minkowski space metric η at spatial infinity =η +

1

,

=

1

,



=

1

→ ∞,, that is:

.

With such approach, density of the gravitational field energy is not localizable. In this case, a plane metric describes a zero gravitational field and it is necessary to use additional assumptions, to introduce "newtonian tail" for obtaining a physically significant result - that is boundary conditions are beyond the geometrical theory. These difficulties are of fundamental nature. Is the known approach only possible? If there is other approach, will it lead to other solutions? It is usually supposed that limiting (initial or boundary) conditions do not affect results of solving specific problems considerably, including gravitational problems. However, that is quite untrue. Small changes in limiting (initial or boundary) conditions can lead to qualitative changes in solutions of partial differential equations (Cauchy problem). The well-known Hadamard's example [3] is a vivid confirmation of this. The classical close-range interaction theory assumes that there are stresses in the medium and/or in the space. A.Sommerfeld [4] expressed this idea quite definitely: "With respect to the statics of the electron we have familiarized ourselves till now only with the Lorentz force, acting

at the locus of the electron. A field concept cannot be satisfied herewith, however, but must follow up the transfer of force actions in vacuum, where there are no charges. This was Faraday's intimation when he spoke of lines of force as of elastic bands which transmit tension and compression. Maxwell was also here able to place Faraday's notions into clear mathematical focus. This was the origin of Maxwell's stress tensor, which may be expanded relativistically into a stress-energy tensor.” I know no reasons why the same cannot be done in the geometrical gravitation theory. In the Einstein's gravitation theory, motion of bodies is determined by the space-time geometry, namely by the metric tensor

. It can be seen that in case of purely geometrical

approach to the equations of motion, there is no need to introduce stresses in the theory, similar to stresses in the electromagnetic theory or in the mechanics of continua. This feature of such approach to the general theory of relativity has been noted in the paper [5]: "Gravitational fields can be set without introducing stresses and energy density" (it is highlighted by me, V.M.). On the other hand, in the paper [5], A.Einstein gave a simple example where tensions in vacuum not only can, but must be nonzero. In this example, he considered two motionless gravitating bodies connected with a rigid rod that prevented them from approaching. Einstein demonstrated that the rod tensions must be counterbalanced by the vacuum tensions of the gravitational field. "It should be also noted that the stresses

of those gravitational fields through which interaction of

many bodies occur cannot be zero in any way". However, in the final version of the gravitation theory, the most elementary of possible options was used. In fact, in the general case, energymomentum tensor of a field must be nonzero, that is at least the stress tensor, which is a component of the energy-momentum tensor, is nonzero. In [6], A. Einstein advanced the equivalence principle as a method of researching the relativistic gravitation theory. The first step in this direction was researching uniformly accelerated relativistic reference frame1 that, according to the equivalence principle, must be equivalent to the stationary uniform gravitational field. Metrics of uniform reference frames must be form-invariant in relation to the shift. Field uniformity at infinity as a boundary condition looks attractively. Besides, it is necessary to take into account the isotropy of space. However, to realize these principles, it is necessary to know the exact value of the metric describing the uniform field.

1

Uniformly accelerated reference frame is a frame all of whose points have identical and constant own

acceleration.

2.

Metric of noninertial relativistic reference frames

The careful and most complete research of noninertial reference frames was carried out by C.Møller [7]. Meanwhile, it is quite unnecessary to search for noninertial system as a coordinate function. It is possible to find the metric directly. Let us consider a one-dimensional motion on the Minkowsky plane ′. The metric of such plane: ′ =

′ −

′ .

Let us confine ourselves to "correct" systems , namely such systems in which sufficiently small rectangular area with the sides

′ and

′ can be considered to be uniform

pseudo-Euclidean (locally inertial systems). Then such neighborhood can be described by the distorted metric: =

.

(1)

is a scale factor, a velocity function of the systems ′ and

The parameter the initial system. If



in relation to

is a squared Lorentz factor, the metric (1) describes distortion of an

element of the Minkowsky plane of one reference frame in relation to another one (Lorentzian length contraction and Lorentzian time dilation). Let us proceed to noninertial reference frames. Such systems, with arbitrary values = ( , ), have the metric in the form (1). In particular, a stationary reference frame has the following metric: = ( )

− ( )

.

Let us consider a small area of three-dimensional space to be isotropic. Then this result can be applied to four dimensions2: = ( )

− ( ) (

+

+

In the most elementary case, ( ) is a linear function = 1− Besides,

= −

2

− 1−

2

(

+

). ( ). Then +

is the acceleration field potential where

is locally exact (within the limit

(2)

),

(3)

is acceleration. This metric

→ 0). Within the same limit, it coincides with the first

approximation of the solution of the Einstein gravitational equation [8]. Thus, within the general relativity theory, local equivalence of a uniform gravitational field and a uniformly accelerated system is proved. In the papers [9, 10], a similar result has been obtained in another way.

2

Such static metricss were considered by V.A.Fock [8].

3.

Time in uniformly accelerated systems and uniform gravitational

fields Let us choose two points

and

with coordinates

= 0 and

= > 0 in the

accelerated reference frame . Let ′ is an inertial reference frame accompanying the point Comparison of frequencies of "pendulum" of standard clock placed in the point same result point

.

will give the

in both reference frames. On the other hand, according to [6], observer in the

will see that the frequency (1 +

=

of identical clock in the point ⁄ )=

is different:

(1 − ⁄ ).

(4)

Einstein [11] gave an extremely simple proof of this relation that is of a universal character since its proofing requires nothing except for the relativity principle and the Doppler effect. Therefore, in the general relativity theory, this result remained unchanged [12] and found confirmation in the gravitational field experiments [13]. However, the relation (4) is exact neither for inertia fields nor for a gravitational field. ⁄ . However, the exact expression can be

The relation (4) is exact only within small

obtained also for finite interval. Let us break up the interval into

segments. Then, after

consecutive transformations (4), we obtain in the limit the exact expression: =

lim 1 +

(5)

=



Therefore, the exact value3 for the proper time intervals: =



(6)

.

In 1907, Einstein [6] emphasizing an approximate character of the expression (5) suggested such expression for finite

on the ground of space uniformity: "the choice of origin of

coordinates should not influence this relation". Surprisingly, this result was not even mentioned for a long time.

4.

Asymptotically non-Euclidean metrics.

The value of the scale factor (6) in the metric (2) allows to give an exact expression for interval of uniformly accelerated reference frame:

= 3

space result.



.

(7)

The same result can be obtained from covariant equation of motion movement in pseudo-Riemannian −

= 0 (§87 [12]) provided that

=

which once more emphasizes a generality of the

The system is uniform (form-invariant). Spatial shift of the origin of coordinates by some fixed value is equivalent to displacement of the point of contact with the pseudo-Euclidean space =

whose metric is



. In other words, we have simply displaced the observer to

another point. The same approach that has led us to the passage to the limit (5) can be applied to the locally exact solution of Einstein equation for uniform field [8]: = 1−

2

− 1+

2

(

+

+

).

The passage to the limit = lim 1 −

(

− lim 1 +





+

+

),

(8)

leads ultimately to the same metric (7). Therefore, the equivalence principle for uniform gravitational and inertial fields remains true. The metric (7) describes the pseudo-Riemannian space. Einstein tensor of this metric is nonzero and is of diagonal form: /





0 − 0 0

0 0 0 where

=

0 0

0 0 , 0

0

. Einstein gravitational equation for uniform field of accelerated system in vacuum

takes the form:

= where

8

,

(9)

is gravitation constant. It is precisely the equation form that corresponds to the general

relativity theory principles. According to these principles, the field (distortion of the Minkowski space) is caused not only by massive bodies, but also by fields including gravitational field. The equation for vacuum

= 0 has been suggested as an approximation of the Einstein equation

and should be considered as such in what follows. Now the energy-momentum tensor of a uniform acceleration field or gravitational field: /

− =

where

=



0 0 0

. The value and the sign of

gravitational field with intensity



0 − 0 0

0 0

0 0 , 0

(10)

0

coincides with the value of tension of the

obtained in the problem about attraction of two infinite

gravitating plates within the Newtonian limit. Within the same limit, density of gravitational field energy is −



(see Problem 1, Paragraph 106 [12]).

The tensor

in the origin of coordinates is similar to the energy-momentum tensor of a

uniform electric field. The sign of tensions that are the components of the tensor

is opposite

to the sign of tensions of the energy-momentum tensor of electromagnetic field. It becomes clear after comparison of the forces acting on a charged spherical shell with the forces acting on the same heavy shell. In both cases, the field is outside the shell, but in the first case the field stretches it, in the second case the field compresses it.

5.

Metric of localized mass field

In the general case, the Einstein equation sould be written in the following form: = where

8

(

+

),

(11)

is energy-momentum tensor of gravitating masses. In the standard Schwarzchild solution for the stationary problem about gravitational field =1+2 /

of localized mass (§100 [12]), the asymptotic value . Substitution of this value with the exact value = here , , ,



(sin

= exp 2 /

- spherical coordinates, the parameter

equation for vacuous space

=

=

/

results in the metric

)−



is used, where

,

(12)

is Schwarzschild radius. The

= 0 is untrue now. However, obtaining the metric (12) requires

no equation now, and we can find the energy-momentum tensor of gravitational field from the equation (11). Locally isotropic solution of the Einstein equation for localized mass at infinity should have the Newtonian potential

=

. Then, from the metric (7), the metric of spherically

symmetric problem follows: =



(13)

.

This metric satisfies the Einstein equation (11). It should be noted that the exact metrics obtained have a locally uniform acceleration or gravitational field. Therefore, it would be reasonable to supplement the principle of local uniformity of a usual space with a principle of local uniformity of a field. The Schwarzschild metric in isotropic coordinates [12] =

1−4 1+4

− 1+

4



should be considered as one of possible approximations of the metric (13). Inversely, from this metric form, the metric (13) can be obtained in another way by using the passage to the limit similar to the passage (8). In the space region occupied with a substance, the gravitational field energy can be neglected. The obtained solution of the usual Einstein equation is joined with the metric (12). For the metric (13), from the equation of motion



= 0, gravitational

acceleration in the stationary spherically symmetrical case can be found: = . For comparison, the intensity obtained from the Schwarzchild solution: =

1 1−2

/

(14)

.

The Figure allows to compare the curves of these fields with the Newton's law of gravitation. Besides, relativistic effects available for observation do not practically differ from the effects in Schwarzchild approximation.

6.

Conclusion

The obtained metrics includng the elementary metric (3) belong to pseudo-Riemannian spaces. Thus, non-Euclideanism of space shows itself even when there is no gravitation in noninertial systems. Judging by my own experience, I know that it is extremely difficult to accept it. Actually, it is not something new. For example, Einstein gave an example of such noninertial system - rotating disk [13] - more than once. Møller [7] ascertained that this noninertial reference frame was a space of negative curvature. It is important, that energy-momentum tensor of gravitational field with understandable and physically sensible terms appeared in solutions of the Einstein equation. One should not think that during a hundred years there were no serious attempts, except for Møller's papers, to comprehend noninertial reference frames. I would like to make special mention of the papers by V.I.Rodichev and his pupil S.A.Podocionov. Rodichev [14, 15] distinguished between transformation of coordinates and transformation of noninertial reference frames and obtained the metric of nonzero scalar curvature. Podocionov [16] also obtained the metric of nonzero curvature and suggested, as a solution of spherically symmetrical problem, the metric

=



(sin



)−

, but it leads to the results that differ

considerably from predictions of the general relativity theory. V.A. Fock's point of view deserves mentioning too. He researched the gravitational field metric similar to the metric (7) and obtained interesting results (§56 [8]).

My heated discussions with S.A.Podocionov and J.Foukzon initiated my interest to the problem of uniformly accelerated system and ultimately led to the above-stated results. I am also grateful to M.B.Belonenko, U.N.Eroshenko, D.B.Zotiev, M.G.Ivanov and V.A.Rubakov for their remarks and advices that affected my paper.

References 1. S.P.Novikov, I.A.Taimanov. Modern geometrical structures and fields, MTNMO, (2005). 2. L.D.Faddeev. The energy problem in Einstein’s theory of gravitation (Dedicated to the memory of V.A. Fock) Sov. Phys. Usp. 25 130-142 (1982). 3. S.L.Sobolev. Equations of mathematical physics. М.: Nauka, (1966) §2. 4. A.Sommerfeld. Electrodynamics. Academic Press, (1952) §31. 5. A.Einstein. Notiz zu Schrцdingers Arbeit "Die Gravitationsfeldes". Phys. Z., 19, 115-116, (1918).

Energiekomponenten

des

6. A.Einstein. Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jahrb. d. Radioaktivitat u. Elektronik, 4, 411-462 (1907). 7. C.Møller. Theory of Relativity Oxford University Press; 2nd edition, (1972) 8. V. A.Fock. "The Theory of Space, Time and Gravitation". Macmillan. (1964). 9. V.B.Morozov. Whether or not a Body Form Depends on Acceleration? http://arxiv.org/pdf/1305.5412v2. 10. V.B.Morozov. A note on the equivalence principle applicability to the general theory of relativity http://arxiv.org/pdf/1404.3083v1. 11. A.Einstein. Uber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes. Ann. Phys., 35, 898-908. (1911). 12. L.D.Landau, E.M.Lifshitz. The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann (1975). 13. A.Einstein. Über die spezielle und die allgemeine Relativitätstheorie (Gemeinverständlich). Druck und Verlag von Friedr. Vieweg & Sohn, Braunschweig, 1920. 14. V.I.Rodichev. Geometrical properties of reference frames. (in the book "Einsteinian collection", 1971). 15. V.I.Rodichev. Evolution of the reference frame concept and the Einstein's program. (in the book "Einsteinian collection", 1974). 16. S.A.Podocionov. Geometrical properties of noninertial reference frames in the relativistic mechanics. (in the book "Debatable questions of the theory of relativity and gravitation". М.: Nauka, 1982, PP. 95 - 103).

Figure. Dependence of the gravity force on the distance to the point mass: Newton's law dashed line; Schwarzchild solutions - dashed-dot line; exact solution - solid line. The rectangular shows a 50x scaled-up curve section.