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THE INERTIAL MASS DEFINED IN THE GENERAL THEORY OF RELATIVITY HAS NO PHYSICAL MEANING

V.I. Denisov and A.A. Logunov

It is shown that the inertial mass introduced in the general theory of relativity depends on the choice of the three-dimensional coordinate system, so that it can take arbitrary values. This means that the inertial mass in Einstein's theory is devoid of any physical meaning. In addition, the expression for the inertial mass in Einstein's theory in the general case of an arbitrary three-dimensional coordinate system does not have a classical Newtonian limit, so that the general theory of relativity does not satisfy the principle of correspondence with Newton's theory.

Introduction
It is currently assumed that in the general theory of relativity the gravitational mass of a system is equal to its inertial m a s s . This a s s e r t i o n goes back to the studies of Einstein [1], Eddington [2], Tolman [3], and Weyl [4]. Subsequently, this theorem was "proved" with various modifications by a number of other authors [5-7]. Nevertheless, we feel it is necessary to return to this apparently resolved question and make a more detailed investigation.

1. The Gravitational

Mass in the

General Theory of Relativity

The gravitational mass M of an arbitrary physical system in rest as a whole relative to a Schwarzschild coordinate s y s t e m Galilean at infinity was defined by Einstein ([1], p. 660) as the quantity which multiplies the t e r m - 2 G / c 2 r in the asymptotic e x p r e s s i o n (r -~ oo) for the component g00 of the m e t r i c tensor of Riemannian s p a c e - t i m e : g00 = 1 - (2G/c2r)M.

A somewhat different definition of the gravitational mass was given by Tolman [3]:

r

M= ~ R,~

(1)

It follows directly from these definitions that the gravitational mass does not change under transformations of the three-dimensional coordinates, since both the component R] of the Ricci tensor and the component g00 of the m e t r i c t e n s o r t r a n s f o r m in this case as s c a l a r s .
In the case of static systems, the definitions of the gravitational mass given by Einstein and Tolman are equivalent. To see this, we write the component 1~ in the form
R0~ tr~r0x, Fo,, - -o-0~r o . , ,+1--0, ro.T~,. , - . ~r - r ~ -'j].

After identical transformations, we obtain from this

Ro~ ~__-_g...Oz~[~-egO.ro.ol-goa,-z~~

2

oz'

0, ~ [ ~ - g g o - r 0 . ~

(2)

Since the last three terms can be ignored for static systems, it follows from the expression fl) that

e j'eso~-':7r176

(3)

M = 4riG

Since the metric sufficiently far from a static system must be described with given accuracy by the Schwarzschild m e t r i c , the expression (3) b e c o m e s

Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 51, No. 2, pp. 163-170, May, 1982. Original article submitted May 29, 1981.

0040-5779/82/5102-0421.r

9 1982 Plenum Publishing Corporation

421

C 2

(4)

8~G ~

]/-g ~,x- J ' "

Since the integrand in (1) is a s c a l a r under all t r a n s f o r m a t i o n s of the t h r e e - d i m e n g i o n a I coordinate s y s t e m , the gravitational mass M will also be independent of the choice of the coordinates. In Schwarzschild c o o r d i n a t e s , we obtain f r o m the e x p r e s s i o n (4)

C 2
2G , . ~ t

C ~
2G ,~=t

Thus, in accordance with Tolman's definition, the gravitational mass of a static system is the factor multiplying the t e r m - 2 G / c 2 r in the a s y m p t o t i c e x p r e s s i o n for the component g00 of the m e t r i c t e n s o r of the Riemannian space-time. Therefore, the definitions of the gravitational mass given by Einstein and Tolman are equal for static systems.

2. Inertial Mass in the General Relativity
The concept of the inertial mass of a physical system in the general theory of relativity was intimately related by Einstein to the concept of energy of the s y s t e m ([1], p. 660): " . . . t h e quantity that we have interpreted as the energy also plays the role of i nert ial mass in accordance with the special theory of relativity." However, in the general theory of relativity it is not possible to introduce the concept of the energy of a system consisting of matter and the gravitational field, since in Einstein's theory matter and the gravitational field are characterized by quantities of different dimension: the physical characteristic of the gravitational field is the curvature tensor, i.e., a tensor of fourth rank, while the matter is characterized by the:energymomentum tensor, i.e., a tensor of second rank. Because of the difference between the ra,aks, general relativity does not in principle contain any conservation laws (apart from the Einstein equations themselves) linking the matter and the gravitational field [8]. Thus, the general theory of relativity was Constructed at the price of giving up the energy-momentum conservation laws for the matter and the gravitational field taken together.
Following the book of Landau and Lifshitz [5], who use more modern notation, let us consider the manner in which the concept of the energy of a system was introduced in the general theory of relativity by Einstein ([1], p. 528) and other authors [2-7, 9-11].
If Einstein's equations ([5], .~96) are written in the form
Ca =[.~.h_Te=]

then the left-hand side can be split in a noncovariant manner into two terms:

~' g[n,~ t g'~Rl =-s h'='+g~'~,

(6)

8-~t -T J 0='

w h e r e g~=-d~ is the e n e r g y - m o m e n t u m p s e u d o t e n s o r of the g r a v i t a t i o n a l field, and h ~ : h ~ = - h ~'~ is the spin pseudotensor:

h'~' = t6~d"o=" [-g(g'"g"'-g"r 1"

(7)

Substituting (6) in (5), we obtain

- g [r,~+r = ~ h~,.

(8)

By v i r t u e of the identity O2h~/Ox~Ox~:O , the Einstein equations (8) yield the differentia1 c o n s e r v a t i o n law

a~ [- g (r'~+~~) ]: 0 .

(9)

Integrating this r e l a t i o n o v e r a sufficiently l a r g e volume and a s s u m i n g t h e r e a r e no "energy vTfluxes through the surface bounding the volume of integration, an integral "energy-momentum conservation law for the s y s t e m " is usually obtained f r o m the e x p r e s s i o n (9):

d d--t-

(- g) [r'~176

(10)

422

From this there follow four quantities that do not depend on the time:

=

(_g)

(11)

C

By means of the Einstein equations (8), Eq. (11) can be rewritten in the f o r m

l~=-~-'~h'~

(12)

In Einstein's opinion ([1], p. 652), the four quantities pi r e p r e s e n t the e n e r g y (i = 0) and momentum (i = 1, 2, 3) of the physical s y s t e m . It is usually a s s e r t e d (see [5], p.283): "The quantities t)i (the 4momentum of field plus matter) have a completely definite meaning and are independent of the choice of the reference system to just the extent that is necessary on the basis of physical considerations."

On the basis of such a definition of the "energy and momentum" of the system consisting of matter and

the gravitational field, the concept of the inertial mass m of the system is introduced in the general theory of relativity:

_ ip0 t

m - c = 7 [o (-g)[T~176176

(13)

To calculate the inertial mass of the system, the Schwarzschild solution is generally used.
[n isotropic Cartesian coordinates, the m e t r i c of the Biemannian s p a c e - t i m e can be written in this ease in the form

+

..... 4rJ ' g~176

1[ I ~rJ'

(14)

where r=?x2+g2+z~, r~=2GM/c z. These coordinates a r e asymptotically Galilean, since in the limit r ~

go~

; g:,=-5:,[ l+O(rl---)] .

(15)

Using the eovariant components (14) of the m e t r i c , we obtain f r o m the expression (7)

h~176 16riG Ox~ [g,,g22go3g=~]. Substituting this e x p r e s s i o n in (12), noting that

d S ~ = . x~ ~ sin OdOd%

(1 6)

T

and integrating over an infinitely distant surface, we obtain

P~ =

t

c 3 6~G

l~im r~fJ

x~ r

Oa_xa~_.[--g.g~2g~g=~]siOndOd%

(17)

Thus, the component p0 does not depend on the c o m p o n e n t g00 of the metric t e n s o r of the Riemannian s p a c e time. Substituting the expressions (14) in Eq. (17) and noting that

a ](r)=---ax-~-lO(r)'rr x=xa:-r',

(18)

we obtain

P~

(19)

it was this equality of the "inertial mass" to the gravitational mass which led to the assertion that they are equal in the general theory of r e l a t i v i t y ([5], p. 334): ,,... p a = 0, t)o = Mc, a r e s u l t which was naturally to be expected. It is an expression of the equality of "gravitational" and "inertial" mass ("gravitational" mass i s the mass that determines the gravitational field produced by the body, the same mass that appears in the metric tensor in a gravitational field, or, in particular, in Newton's law; "inertial" mass is the mass that determines the ratio of energy2 and momentum of the body; in particular, the r e s t energy of the body is equal to this mass multiplied by c ). "

H o w e v e r , t h i s a s s e r t i o n of E i n s t e i n ([1], p.660) a n d o t h e r a u t h o r s [2-7, 9-11] i s i n c o r r e c t . As w i l l be s h o w n b e l o w , t h e " e n e r g y " (11) of t h e s y s t e m a n d , t h e r e f o r e , i t s " i n e r t i a l m a s s " h a v e no p h y s i c a l m e a n i n g , s i n c e t h e i r v a l u e

423

depends even on the choice of the three-dimensional

coordinate system.

The general theory of relativity does not in principle admit the introduction of a concept of inertial mass, since in Einstein's theory there are no integrals of the motion linking the matter and the gravitational field ( c h a r a c t e r i z e d by the c u r v a t u r e t e n s o r ) . The only f o r m a l c o n s e r v a t i o n law [8] in the g e n e r a l t h e o r y of relativity is provided by the Einstein equations themselves, which lead to integrals of the motion identically equal to zero, which precludes the introduction of an inertial mass.

3. The Concept of Inertial Mass is

Meaningless in the General Theory of Belativity

An e l e m e n t a r y r e q u i r e m e n t which a definition of inertial m a s s m u s t s a t i s f y is the condition that i~s value should be independent of the choice of the three-dimensional coordinate system, which is the case in any physical theory. However, in the general theory of relativity the definition (13) of the inertial mass does not satisfy this requirement.

We show, for example, that in the case of the Schwarzschild solution the inertial mass (13) may take all values depending on the choice of the system of spatial coordinates. For this, we go over from the three, dimensional Cartesian coordinates Xc to other coordinates x~, which are related to the old coordinates by

z2=x~'[t+!(r~) ],

(20)

where r~=]/x.2+y.2+z.*, ](r.) is an a r b i t r a r y nonsingular function satisfying the conditions

l(r~)>~O, liml(r~)=0, limr.~r.al(r~)-----O.

(21)

It is r e a d i l y seen that the t r a n s f o r m a t i o n (20) c o r r e s p o n d s to a change in the a r i t h m e t i z a t i o n of the points of ~

rr t h r e e - d i m e n s i o n a l s p a c e along the r a d i u s :

If the t r a n s f o r m a t i o n (20) is to have an i n v e r s e

arr and be a o n e - t o - o n e t r a n s f o r m a t i o n , it is n e c e s s a r y and sufficient that the condition

where ]'=al(r.)/Sr., hold. Then the Jacobian of the transformation is nonvanishing:

II {~Xa I I

(TT:~

In particular, all the imposed requirements are satisfied by the function

(22)

where and s are arbitrary nonvanishing numbers. Since in the given case

8rr _i.t_c~2-1/SGM[__~_i_(e2r ___~) exp(_s2r.) ]

'~rH'--

f c2ru

](r.)

is a monotonic function of r s.

](r~) It is readily seen that

is a non-negative nonsingular function in the

whole of space. The Jacobian of the t r a n s f o r m a t i o n in this c a s e is strictly g r e a t e r than unity: ]=(i+])~aro[

Or.>t. T h e r e f o r e , the t r a n s f o r m a t i o n (20) with the function ](r,) defined by the e x p r e s s i o n (22) has an

inverse and is one-to-one.

It is obvious that under the t r a n s f o r m a t i o n (20) the value of the gravitational m a s s (1) does not change.

We now calculate the value of the "inertial m a s s " (13) in the new c o o r d i n a t e s x~. formation law of the metric tensor,
g , ~ = = O- -x.c:~~O_xJ~~g,..~(xo ( x . ) ) , aXR ~ ~X~

Using the trans(23)

we find the components of the Schwarzschild metric in the new coordinates. As a result, we obtain

g~176t" 5r.(tq-,)'] /[_ t-F4r.(~_l_]------~]; g ~ = [ i - [ 4r.(t+,)] {--6~(i+])~--x='x~,[.(,). +~-~-, ( t + f ) ] } . (24)

The d e t e r m i n a n t of the m e t r i c t e n s o r (24) has the f o r m

424

g=-eoo r|[ i +,4r.(irqg-]) ~IJ ':(i+]) ~[ (i+])~+"r.2(f)Z+2r.f (i+]) ].

(25)

It should be noted e s p e c i a l l y that the m e t r i c (24) is a s y m p t o t i c a l l y Galilean:

lira g00=i; lira g~=-6:~.

In the s p e c i a l c a s e when the function [(r.) is defined by (22) and rH --) oo, the. m e t r i c of the Biemannian
space-time will have the asymptotic behavior

For the contravariant components of the metric (24), we have

g~176 g~a=-5~A+x.~x.~B,

(27)

where we have introduced the notation

rg ] - ~ A=(lq-/) -~ l-~ 4 r . ( i + / )

B=[r.(f')2+2]" (t+[)]{r. [t + r, ]" (1§
4r~ (1+[) J Substituting the e x p r e s s i o n s (27) and (25) in (12), we obtain

[ (l+])2+rZ(l,)~+2r.] , (t+1)] }j-'.

p,= C3 limrz~X:~" O ~__5~(1_4_])2[iq- rg 1 8 [(l+/)2+r.Z(/')2+2r.]'(l+]) ]+

x~x'----~[ l+/] 2[ l q- r,

r~,~ "

4r.(l+/)

] ' [r~(f)z-t:2r~]'(t+])]}"

By virtue of the relations (I8),

C ~
P~

{r.Z(]')~(l+])z [t +

rg

]

a

q-rg(l-~-/)z(lq--/--~r~ff)[

t-~

4

r

.

re
(l+

/

)

]

}

.

(28)

2G ,-,,.

4r~ (t-~])

Using the a s y m p t o t i c e x p r e s s i o n (21) for f, we finally obtain

P' = -C-a-- lira {r,+r: (1,)2}.

(29)

2G ~|

Thus, the "inertial m a s s " depends on the rate at which f ' tends to zero as rrl ~ ~. In particular, choosing
the function f(r.) in the f o r m (22), we obtain for the "inertial m a s s " f r o m the e x p r e s s i o n (29)

m=M(t+~').

(30)

I t f o l l o w s t h a t f o r t h e " i n e r t i a l m a s s " (13) o f t h e s y s t e m c o n s i s t i n g o f

matter and the gravitational

field in the general theory of relativity we can,

because the value of ~ is arbitrary,

obtain any preassigned number m >-M

d e p e n d i n g o n t h e c h o i c e o f t h e s p a t i a l c o o r d i n a t e s , although the gravitational m a s s M (1)

of this system and, therefore, all three effects in general relativity remain unchanged. We note also that

under more general transformations of the spatial coordinates that leave the metric asymptotically Galilean

the "inertial m a s s " (13) of the s y s t e m m a y take all p r e a s s i g n e d values, both positive and negative.

Thus, we see that in the general theory of relativity the "inertial mass,"

which was first introduced by Einstein and subsequently taken over by many

authors [2-7, 9-11], depends on the choice of the three-dimensional

coordinate

system, and it therefore has no physical meaning. Therefore, the assertion that the

inertial and gravitational masses are equal in Einstein's theory is also devoid of physical meaning. Such

equality holds only in a s m a l l c l a s s of t h r e e - d i m e n s i o n a l coordinate s y s t e m s , and since the "inertial" (13)

and gravitational (1) m a s s e s have different t r a n s f o r m a t i o n laws, they a r e no longer equal a f t e r transition to

different three-dimensionalcoordinate systems. In addition, the definition (13) of the inertial

mass in the general theory of relativity does not satisfy the principle of corre-

spondence with Newton's theory. Indeed, since the inertial mass m in Einstein's

theory depends on the choice of the three-dimensional

coordinate system, its

425

expression in the general case of an arbitrary three-dimensional

coordinate

system does not go over into the corresponding

expression of Newton's theory~

in which the inertial mass does not depend on the choice of the spatial coordi-

nates. Thus, in the general theory of relativity there is no classical Newtonian

limit and, therefore,

it does not satisfy the correspondence

principle.

This leads us to a s k why the m e a n i n g l e s s n e s s of the definition (11) of the "energy and m o m e n t u m " of a system and its "inertial" mass in the general theory of relativity has remained obscured until now~

This can only be explained by the fact that usually all calculations of the "energy, momentum, and inertial m a s s " have been made in a small class of three-dimensional coordinate systems in which the "inertial" and the gravitational mass are equal. *

In the s a m e c l a s s of c o o r d i n a t e s y s t e m s , the e x p r e s s i o n (13) for the inertial m a s s in the Newtonian approximation is equal to the corresponding expression in Newton's theory, which created the illusion that the general theory of relativity has a classical Newtonian limit. It was then apparently regarded as superfluous to consider the physical meaning of the inertial mass (13) introduced in the general theory of relativity.

We are very grateful to A. A. Vlasov, S. S. Gershtein, and A. N. Tavkhelidze for discussing the work.

LITERATURE CITED

1. A. Einstein, Collection of Scientific Works, Vol.1 [Russian translation}, Nauka, Moscow (1965). (The

original p a p e r s a r e as follows: Sitzungsber. I>reuss. Akad. W i s s . , 2, 1111 (1916) (p~ 528 ir~ the

Russian translation) and Sitzungsber. P r e u s s . Akad. W i s s . , 1, 448 (1918) (p. 652 and p. 660 ia the

Russian translation). 2. A. S. Eddington, The Mathematical Theory of Relativity, C.U.P., Cambridge (1923).

3. R. C. Tolman, Phys. Re,., 35, 875 (1930).

4. H. Weyl, Raum. Z. Materie, Springer-Verlag, Berlin (1923), ~37.

5. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed., Pergamon P r e s s , Oxford

(1975). 6. C. M~ller, The Theory of Relativity, Second Edition, Oxford University Press, London (1972).

7. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, San Francisco (1973).

8. V. I. Denisov and A. A. Logunov, Teor. Mat. Fiz., 45, 291 (1980).

9. R. C. Tolman, Relativity, Thermodynamics and Cosmology, Clarendon P r e s s , Oxford (1934).

10. W. Pauli, Theory of Relativity, Pergamon Press, Oxford (1958).

11. S. Weinberg, Gravitation and Cosmology, Wiley, New York (1972).

12. C. Mr

in: Gravitation and Topology [Russian translations], Mir, Moscow (1966), pp.60-61.

* Some aspects of this were made by Mr

[12], but he did not succeed in understanding ~he essence of the

matter and therefore failed to draw the appropriate conclusions, namely, that the inertial mass has no

physical meaning in the general theory of relativity and, therefore, that theory has no Newtoaian limit,

426