Energy-momentum pseudotensors of the gravitational field in the general theory of relativity

(2.44)

G In a region of center to the source liptically polarized source of the metric components we obtain

space with linear dimensions considerably less than the distance from its of the metric waves, the spherical wave (2.43) may be considered an elwave. Assuming, to be specific, that the region in question and the waves are located on the z axis (n 3 = l, nl = n 2 = 0), for the nonzero

(DII ~ _ _

q)22

__ 1__

=

[]')11

__/~522]

3ro L--

qo~2 -

~'

(2.45)

2 Oi2, 3ro

-

where r0 is the distance from the source to the center of the region in question. In the case of radiation of monochromatic waves, the second derivatives with respect to time of the components of the quadrupole moment can be written in the form

D n = d '~ exp [i ( k z - - Or)], b 22 =~ d 22 exp [i ( k z

-

o3t)],

-

~()12 = idI2 exp [i ( k z - - o3t)]. Therefore,

from expression

(2.45) we have

{pn ~_____ q022~___/'/0COS 2 4 exp [i ( k z - - mr)], (9~2~- iho sin 2ap exp [i ( k z - - cot)], w h e r e ho = 8 ~ ] / 4 (dr2) 2 @ (d 11-- d22)2',

2 d t~ tg 24---- a,,_~2,

(2.46

9

The degree of ellipticity of the polarization of the gravitational wave (2.46) is measured by the quantity tan2@. If tan 2 @ = 0 or co, then the wave is linearly polarized; if [tan2}[ = I, then the wave is circularly polarized. For other values of tan 2@ the wave is elliptically polarized. If t a n 2 ~ > 0, then the wave has right-hand polarization, while if t a n 2 @ < 0 it has left-hand polarization. For the nonzero

components

f~OlOl =

I~0113 :/~13~3

of the curvature =

wave

--- ]~2323 -----/~0223 =

in the case we consider

Rolo2=R132a=Ro213--

2c,

It should be noted that the magnitude of the components not depend on the dimensions of the region considered. 3.

Energy--Momentum

the General

Theory

Pseudotensors

Q -- 27

-- ~0202 =

of the Gravitational

we have

"'H ~ '

(2.47

)

qoi2. of the curvature

Field

tensor

(2.47)

does

in

of Relativity

Einstein considered that in the general theory of relativity the gravitational field with matter should possess some conservation law [32, p. 299]: "... by all means it should be required that matter and the gravitational field together satisfy laws of conservation of energy--momentum." In the opinion of Einstein this problem was completely resolved on the basis of "conservation laws" using the energy-momentum pseudotensor as the energy--momentum characteristic of the gravitational field. To obtain such "conservation laws" one usually bert--Einstein equations are written in the form 8nO g

then the left

side can be identically

t~k-

gik R

represented

c" [R i k - - +

8~0

gikR

[I I] proceeds

as follows.

If the Hil-

= __ gT~k

as the sum of two noncovariant

] g = ~ - xOl hikl+ gTik,

(3 ,, 1

quantities (3.2)

1715

where zik = Tki is the energy--momentum pseudotensor of the gravitational h ki~ = --hilk is the spin pseudotensor. Using identity equivalent form

(3.2), the Hilbert--Einstein equations

field and h ikl =

(3.1) can be written in another

- - g [T ik -t- ~ik]_= 0 hikz"

Ox---t

(3.3)

Because of the obvious equality

~ -d~~ hikt__O

(3.4)

-

the Hilbert--Einstein equations

(3.3) imply the differential conservation law

(3.5) Ox~ which from a formal point of view is analogous to the conservation law of energy-momentum in electrodynamics. In correspondence With this analogy the "energy flux" of gravitational radiation through dSe in the general theory of relativity is defined by the expression

d I -- c (-- g) ~ d S ~ . C h o o s i n g as a s u r f a c e o f i n t e g r a t i o n a s p h e r e o f r a d i u s r (dSa = - - r a d O n s ) , the "intensity of gravitational radiation" in an element of solid angle d~:

we o b t a i n

OI__I_= __ cr 2 ( _ g) x0=n~.

(3.6)

0R Relation (3.5) in the general theory of relativity is used also to obtain integral "conservation laws of energy--momentum" of a system consisting of matter and a gravitational field. For t ~ e x p r e s s i o n (3.5) is usually integrated [32] over some volume and it is assumed that ther~s no flux of matter across the surface bounding the volume of integration: _d

(3.7)

at I ( -- g) [To/@ x0q d V = -- ~ ( - - g) x~'dS=.

E i n s t e i n [ 3 2 , p. 645] assumed t h a t t h e r i g h t s i d e o f t h i s represents the energy loss of the material system".

relation

dE = ~ ( _ g) To=dS=. dt

for

i = 0 "surely

(3.8)

In the absence of "energy-momentum fluxes" of gravitational field across the surface bounding the volume of integration from expression (3.7), we obtain the law of conservation of "energy--momentum" of the system

P i = - 7-

( - - g)[To* + x o q d V = c o n s t .

With the help of the Einstein--Hilbert equations in the form

(3.3) relation

(3.9) (3.9) can be rewritten

/

1

pi=_ c ~ h~

= const.

(3.10)

In the opinion of Einstein [32, p. 652] the four quantities pi represent the energy (i = 0) and momentum (i = I, 2, 3) of the physical system. Here it is usually asserted [11]: "The quantities pl _ the four-momentum of the field and matter --have a completely definite meaning and are independent of the choice of reference system just to the extent required on the basis of physical considerations." However, as we shall show below, this assertion is incorrect. On the basis of the definition of energy--momentum of a system consisting of matter and gravitational field, in the general theory of relativity the inertial mass m of the system is introduced: 1716

1 ~ lz~176 m = g1 po~._~ - g1 S ( - - g) l T~176 + *~176dV --- ~r

Expressions analogous to relations Einstein equations in mixed components:

(3.5)-(3.11)

can also

(3.11)

be o b t a i n e d

by w r i t i n g

Hilbert--

V - g iT,. + ~,~1 = o , . ~ ~. The choice of the energy--momentum pseudotensor of the gravitational field depends to a large extent on the inclinations of authors and, as a rule, is realized on the basis of secondary properties. Thus, for example, choosing h ikl in the form

h~kl = ~ c~

~o

[_g

(g~kg,,,l

._gUgmk)],

(3.12)

we obtain the symmetric pseudotensor of Landau--Lifshits which contains only first derivatives of the metric tensor:

~,~=?~_~.{(2r%,rL_r,~r~.__r~,r%~)(g,,g~k_g,kg~q+g,,g"a~%trot" ls

t s + g~tg.~ (F.,P.~. + r .l. r , ,s

Choosing

- r m~n r l s - - r nes r m~l - - r m~l r"n s ]

ran-T-

i s t k - r . t, r ms, - r~,r.,) + g m.,g.. ( r .t , r .k. - r~,r~.)}.

+

(3.13)

[17] (3.14)

we arrive at Einstein's pseudotensor

c" C-----'7 {_zv,,,,rk~g

"~k'=

~-r~ll'k~,g + v t ~ r ~ g

16n O

+ (3.15)

_~I

~i

"as

9

r

ms~l

~n

~s

~ra

w h i c h c o i n c i d e d w i t h t h e c a n o n i c a l energy--momentum ( p s e u d o ) t e n s o r variant Lagrangian density of the gravitational field Lg-----~/--~--g

nl]l

obtained

from the nonco-

r~n ~s ~n lpl ] gs~9 I_J. a , ~ n z - - - t l, usJ

For

o k~ ,-1---gTd--o - ~ ' V - ~ - ~-.l 5 lO,gk~-- Omg~,l

(3.16)

we have the Lorentz pseudotensor

lj=

16~Q t ~

st~

which coincides with the canonical energy--momentum (pseudo)tensor obtained on the basis of the noncovariant method of infinitely small displacements from the covariant Lagrangian density of the gravitational field Lg = /----gR. We shall study the properties of the "energy--momentum" quantities introduced in Einstein's theory for the examples of the determination of the "inertial mass" of a spherically syrmnetric source and the computation of the radiation "intensity" of small perturbations of the metric. To be specific we shall carry out all computations on the basis of the symmetric pseudotensor of Landau--Lifshitz (3.13). 4.

Absurdity of the Definition of Inertial Mass in the General

Theory of Relativity The equality of inertial and gravitational mass of the mass body Einstein considered to be an exact law of nature which must be reflected in his theory. At present it is commonly assumed to be proved that in the general theory of relativity the gravitational mass (or, as it is sometimes called, the heavy mass) of a system consisting of matter and the gravitational field is equal to its inertial mass. Such an assertion is contained in the

1717