The Modification of Einstein`S Gravitational Field Equation Following from the Energy Conservation

The modification of Einstein`s gravit ational field equation following from the energy conservation law

Roald Sosnovskiy Technical University, 194021, St. Petersburg, Russia E-mail:[email protected]

Abstract. The cause of an infringement in GR of a gravitational field energy conservation law is investigated . The equation of a gravitational field not contradicting to the energy conservation law is suggested. This equation satisfy to the Einstein ,s requirement of equivalence of all energy kinds as sources of a gravitational field. This equation is solved in paper for cosmic objects. It is showed, that results for some objects - for black holes and gravitating strings -essentialy differ from such for Ei n- stein,s equation, have the symple meaning and do not contradictions.

k Introduction. The paper is devoted to the problem of ti gravitational field energy. This problem takes special place in the relativity theory as on its i mportance as k on difficulty. Einstein's “energy - moment pseudotensor “ ti cannot be interpreted as the quantity adequately describing the energy - moment of a gravitational field. So, 0 for the elementary case, cylindrically symmetric st atic field, t0  0 , that is erroneous. In this case a pseudotensor Landau-Lifshits has also incorrect value [2], [5]. Many authors offered different variants pseudotensor [7], [9],[11], however all of them a p- peared unsatisfactory. In the review devoted to a problem of energy of a gravitational field [8], are in details considered and systematized attempts to find a satisfactory variant pseudoten- k sor ti . It was made the conclusion that it is impossible to find such pseudotensor describing density of the energy - moment. There is a generally accepted opinion, that the energy - moment of a gravitational field should be local, and that the local law of energy-moment conservation must be fulfilled [8]. In paper [8] are considered other approaches to the decision of a problem of the gravitational field energy - moment also. Attempts to refuse of pseudotensors use were made by introduction of additional connections, use of higher derivatives or a 4 - formalism, development of alternative theories of gravitation [8], [10], [12]. Ho w- ever, these attempts were unsuccessful. There is the opinion [3] that to descript an energy quantity it is difficult b e- cause, for it cannot to be measured. It is possible to agree with this opi nion, since these make difficulties in searching of alternative approaches. However, in [5] there was suggested the mental experiment, allowing find the gravitational field energy density in little volume. Used calculations based on the energy conservation law. This results used in presented paper in order to solve the gravitational field energy conse r- vations problem for static systems. Such problem could be solved in principle for sy s- tems having Killing`s vectors [4] including static systems. In the presented paper made attempt to use these results in order to solve a problem of conservation of e n- ergy – moment, at least for static systems.

1 In a case of the relative movement of material bodies, in all space there can be streams of energy. Localization of energy and t he description of fields in such systems demand using of more complicated approaches and in the present paper is not considered. The consideration is limited to static systems. In this paper are studied problems: -the analysis of the cause of a gravitational field energy conservation law i n- fringement -the modification of Einstein`s gravitational field equation -the solution of the modified equation for spherically and cylindrically sy m- metrical field -the solution of the system of a gravitatio nal field equation and a field masse density equation for cosmical objects. 2. The cause of the infringement of a gravitational field energy conserv a- tion law. The Einstein´s initial assumption in the time of an elaboration of the grav i- tational field equation was that Ricci tensor is equal 0 for free from material bodies part of space

R  0 (1) This equation is obvious for the total empty sp ace. However, it is not obvious for the free from matter part of a space, containing the matter in others parts. Then this assumption is equivalent to the assumption that gravitational field energy not generate gravitational field. However, such assumption contradict to requirement, that “the gravitational field energy must be equivalent in full to all others energy forms” [1]. From (1) follow Einstein`s equation [2]

k 1 8  G k R i  R  T i (2) 2 c 4 i Here Tk is an energy-moment density tensor of matter. The equation (2) is k asymmetrical with regard to matter and field energy. However, the pseudotensor ti it follows from so equation. Thus, just the as sumption about the gravitational field e n- ergy not generate such field is the cause of a breach of a gravitational field energy conservation law. 3. Modification of gravitation field equation. In order to remove the cause of a breach a gravitational field energy conservation law it is suggested attach (2) to form

k k 1 8  G k k G  R  R  (T  tˆ ) (3) i i 2 c 4 i i k Here tˆi is a tensor (not a pseudotensor) of the gravitational field energy- 0 k moment density and tˆ0 is a gravitational field energy density from [5]. The tensor tˆi 0 it is not possible to found from (3). The gravitational field energy density tˆ0 it is possible calculate from equation [5] 1   M f x , M 0  1    ln g   0 (4) 1  M 00  x  0 2  Here x1is the coordinate coinciding with trajectory of test particles falling with infinitesimal velocity (that is by test particles motion energy dispersio n); 1 dMf (x ,M0) is the gravitational field masse inside tube of trajectories in a layer dx1; dM0 is the increase of matter masse on the body surface inside a this tube; gik – is the metric tensor. It is obvious that 2 0 2 tˆ0  c  f x (5)

Here f x is the field mass density. Must be also fulfilled conservation laws

k 1  k 1 rs tî k    gtî  g rs itˆ  0 (6) ;  g x k 2 , and   g T k  tˆ k  i i  0 (7) x k k For each point of a field it is necessary to count g ik , M f , tensor tî . As will k be shouwn further, the equations (3)-(7) completely definie tî for static systems. k Tensor tî for moved systems is not calculated in this paper.. But, it is necessery to note, that its behaviour in such systems is normally. For observer,s coordinate sys-

M f tem moving relatively of field source with constant speed v : M fv  and v2 1 c2 0 0 tˆ0 , consequently tˆ0v  . For free falling observer s coordinate system, the gravi- v 2 1  c 2 0 tational field is absent and M fa  0,tˆ0a  0 . For material bodies system with a variable configuration cannot use directly of the equations (3), (4). Such systems not consider therefore. 4. The gravitational field energy -moment tensor for the spherically sym - metrical systems. For spherical symmetry the interval ds2 hat a form 2 i 2 ds  g ii dx  (8) where

dx0 = dt, dx1 = dr, dx2 =dθ, dx3 = dφ (9) 2 2 2 g22= r , g33 = - Sin θ · r (10) Equation (3) for vacuum with a gravitational field hat then the form

i i 1 8  G i G  R  R  tˆ (11) i i 2 c 4 i The solutions of (11) is [13] 1  8G r  g  1 r 2tˆ0 dr  (12) 11  c 4 r  0   0    1 8G r  g   exp rg tˆ1  tˆ0 dr (13) 00 g 4  11 1 0 11  c 0    The field mass density equal

M f r   r   ,1 (14) 4 r 2

where M f r is the field mass of a spherical area with radius r. If r 0 is the ra-

2GM 0 dius of a material body and M 0 is his mass, r  , then 0 c 2

3 r c 2 r 2tˆ0 dr  M  M r  0  0 f   (15) 0 4 and 1  2GMr g   1 , M(r) = M0 + Mf(r) (16) 11  c2r  From the conservation requirements (6), (7) follow 1 tˆ1  0 (17) and from (13)   r  2GM (r)  2G M f ,1dr  g  1 exp (18) 00 2  2    c r  c r  2GM r   0 r1    c 2 r  Then 1 g 00,1 2GM r  2GM r   2 2 1 2  (19) g 00 c r  c r  From (6) one can to obtain

2 3 rg 00,1 0 tˆ2  tˆ3   tˆ0 (20) 4g 00 5. The gravitational field energy -moment tensor for the cylindrically sym - metrical systems. For ds2 in (8) where dx0 = dt, dx1 =dr, dx2 = dφ, dx3 = dz (21) and by the assumption a g a  r  i g   g  r 2 ii,1  i g    11 1, , , ii   (22) g ii r  r0  one can to obtain

a1  0, ai  2, ai,1  0 (23) i i The Riccy covariant tensor [2] l l l m m l Rii  Г ii,l  Г il,i  Г ii Г lm  Г il Г im (24) From (22) and (24) the mixed Riccy tensor is equal 1  1  1  R i  a  i  a 2   i i.1 2  i 1 (25) 2r r  4 i  Then 1  1  R  R i   a 2    R1  i 2  i 1 1 (26) i r  4 i  1 For i = 1 the tensor component tˆ1 in consequence of conservation law (6) is equal 1 tˆ1  0 (27) and then from (3) 1 G1  R1  R  0 (28) 1 1 2 In consequence of (25), (26), (27) 2  ai  4 (29) i

4 If a gravitational field mass density μ(r) is known then is known a tensor co m- 0 ponent tˆ0 and from (11), (25), (26), (29) 1 8G G 0  a  μ(r) (30) 0 2r 0,1 c 2 2 M zf r c tˆ0  , (31) 0 2r If the matter cylinder radius is r 0 and his linear mass density is M z0 then an equation (30) give 8G a r  a r  M r M  (3 2) 0 0 0 c 2 z z0

M z r  M zf r M z0 (3 3)

Here M zf r  is the field mass of a cylindrical area with radius r. From [5] on the material cylinder surface a r  g r  4GM 0 0  00,1 0  z0 2 (34) r0 g 00 r0  c r0 Then (31) give

g 00,1 a r 4G  0  M  M 2  z0 2 zf  (35) g 00 r c r i 2 3 The tensor’s tˆi components tˆ2 ,tˆ3 one can to obtain from the conservation law (6) c 4 a c 4 a tˆ 2  2,1 ;tˆ3  3,1 (36) 2 16Gr 3 16Gr Here a2 , a3 are equal from (23),(29) to   2  a0 4a0 a2  1 1  (37) 2  2  a0    2  a0 4a0 a3  1 1  (38) 2  2  a0 

6.The solution of equations (3), (4) system for spherically symmetrical obects. Gravitatin mass of a field inside of the sphere equal: M (r) = M0 + Mf (r, M0). This mass creates a field outside of sphere. It is possible to write down the equation (4) in the form of

M 0 dM 1 g 00,r ~  dM   0 0 (39) dr 2 0 g 00 ~ ~ Here M 0 = 0÷M0 and g 00  g 00 r, M 0 . From (39) follows

 2G M f  M 0      dM 2 1 2 f M 0 c c r   ln  (40) dr 2r 2G  2G M r   1   c 2 r  For objects with spherical symmetry as argument x and functions depending on it quantity are used r 2GM M 1 dM x  y  u  pu  ln( ) ; 2 ; ; (41) r0 c r M 0 M 0 dx

5 For quantities in (41) this equation hat a form  1 1  y e  x  dy   y  y e x    0  dx  0 ln1  (42)  2 2  1 y  Boundary condition x=0, y=y 0. Relative value of mass of system on d istance r from the center of system equal M ry u   (43) M 0 r0 y0 If y0 « 1 that, representing a logarithm in (42) in the polynomial form, it is possible to lead this equation to a form

dM f 2G 2G 2  M M f  M (44) d 1 c2 0 4c2 0  r  The solution of this equation M 1    r 0  u   3  expy 1  (45) M 0  r  0 4     33 10 -6 For Sun M0 = 1,99·10 g, r0 = 6,96·10 см and y0 = 4,24·10 « 1. From the equation (45) follows, that at greater distances r » r0 (order of space horizon) M  0,99999 M 0 . 33 8 For the white dwarf Sirius B M 0 = 1,97·10 g, r0 = 5,42·10 см and y0 = 5,45·10-6 « 1. From (45) follows, that at such distances M  0,999986 M 0 . Thus, for these stars influence of field mass gravitation is extremely insignif i- cant. For black holes y0 =1and the formula (42) is used. Computer calculation is made with step dy = 0,001. As in the formula ( 42) relative quantities x, y, y0 contain only, results of this calculation are suitable for black holes of any sizes. On the 2 Fig.1.a dependence y =2GM/c r and u = M/M0 from x = ln (r/r0) is presented. One can see, at great values x M/M0 = 0,71, i.e. influence of gravitation of a field on full mass of system is essential.

1,1 M/Mo 1

0,9

0,8

0,7

0,6 0 0,4 0,8 1,2 ln(r/r0)

Fig.1. Relative mass u = M/M 0 as a function of ln (r/r0) for black holes

6 The density of a field energy from (14) can be presented in the form of tˆ0 1 du 0   x  0 exp( 3 ) (46) T0 3 dx 3M c 2 T 0  0 Here 0 3 - average density of energy in a spherical source of a field. 4r0 tˆ0 t  0 On Fig.2 dependence pu=du/dx and 0 from x is presented T0

-3 t,pu -2,5

-2

-1,5 pu -1

-0,5 t 0 0 0,4 0,8 1,2 1,6 ln(r/ro) tˆ0 t  0 pu  dM Fig.2.Relativ mass density 0 and M dx as a function of T0 0 x=ln(r/ro) for black holes. 0 From the diagram Fig.2 in a range ln (r/r0) = 0 ÷ 0,2 quantities tˆ0 and du/dx = dM/M0dx change approximately in 2 times. It essentially changes the metrics of space near to a black hole. The received results, apparently, are suitable for massive black object in the center of a galaxy which is considered in [16], [ 17] as a black hole.

Components g 00 , g 11 of metrical tensor can be calculated fro m (16), (17) by

M=M0·u. 7. The solution of equations (3), (4) system for cylindrically symmetrical obects. For cylindrically symmetric systems and at boundary conditions

dM zf r  r0 ,  0 this equation have a form dM z0

dM zf 1 1  ln g 00 r, M zf , M z0   g 00 r0 , M z0  (47) dM z0 2 2 The solution of this equation 2   1 c  GM z0 r  M zf   M z  1 exp 4 ln  (48) 2 0 r  c 2 r  8G ln   0  r0 For objects with cylindrical symmetry quantities are used r 4GM M x  b  b  z0 u  z c ln( ) ; 2 ; c (49) r0 c M z0 From the equation (48) using (49) it is possible to receive   e  xc  1  1  uc  1  (50) 2  xc 

7 du 1 1 x e  xc c  c 2 (51) dxc 2xc Relative mass uc = Mz/Mz0 as a function of xc = ln (r/r0) is presented on Fig.3. On can see, that a influence of gravitation field mass on a full mass is esse n- tially.

1 M/Mo 0,9

0,8

0,7

0,6

0,5 0 1 2 3 4 bln(r/ro)

Рис.3. Relative mass Mz/Mzo as a function of xc = b·ln (r/r0) for cylindrically symmetrical system. The relation of a field energy density to a energy density of the body is equal from (31) and (51): tˆ0 1 2x du 0   c 0 exp( ) (52) Tc0 2 b dxc 3M c 2 T 0  z0 Here co 3 is an average density of energy in a gravitating 4r0 0 tˆc0 r body. On the diagram Fig.4 dependence of quantity 0 from for value Mz0=1022 T0 r0 g/cm [18] i.e. b =3∙10-6 is resulted.

-0,25 t/T -0,2

-0,15

-0,1

-0,05

0 1 1.4 1,6 2 r/ro

0 tˆc0 r Fig.4. Relative energy 0 as a function of for cylindrically symmetric T0 r0 system.

Components g ii of metrical tensor can be calculated from (22), (23), (32),

(37), (38) by Mz=Mz0·uc.

8 8.The conclusion. Thus, the suggested decision of a problem of gravitational field energy conservation give correct results for important special cases of symme t- ric static systems. These decisions have simple sense and does not contain contr adic- tions

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