REVIEWS OF MODERN PH YSICS VOLUME 29, NUMBER 3 JULY, 1957

Gravitationa. . .&ie. .c oI': an Axia. .. .y SymriietricSystem in .first Approximation

NATHAN ROSEN AND HADASSAH SHAMIR

Israe/ IT4stit24te of Tech24ology, Haifa, Israel

HE question as to whether a material system canradiate gravitational waves, on the basis of the

6eld equations of the general theory of relativity, doesnot appear to have been answered conclusively. Thepresent work is intended as a 6rst step in looking forthe answer.

To investigate the possibility of a physical systemradiating gravitational waves it is desirable to choose asimple system, one with axial symmetry. If the fieldof such a system is described by means of a sphericalpolar coordinate system (x',x',x',x4) =—(Tp9, )p,t),thenby a suitable choice of coordinates one can satisfytwo conditions: (a) the metric tensor g„„is independentof the angle 22, (b) it is diagonal.

The 6eld equations in the empty space surroundingthe system, '

()Gtt v= ~tt, v 2 g&&~

ds'= —(1+p) dr2 T2(1+a)dl'—t2.—T' sin'8(1+ r)dy'+ (1+I4)dt', (2)

give a set of 7 equations, since G„„vanishes identicallyif (tl, v) = (1,3), (2,3), (3,4). These equations serve todetermine the 4 diagonal components of g„„.Among theequations there exist 3 identities, the contractedBianchi identities, except for the one with index 3,which is trivial.

The 6eld equations are nonlinear and dif6cult tosolve exactly. It is proposed to solve them by themethod of successive approximations. As a beginning,the first approximation is obtained here. I,et us writethe line element in the form

oil t411+p44+)144 (p22+t422)r2

1+-(pl —2o.l —t41) =0, (3c)

ctg8oil+ Tll+ (p22+ T22)+ . (p2 o2+2T2)

r2 r2

1 2(2—pl— 2o 1—3rl—) (—p —o) =—0) (3d)

r r2

1T12+t412——(P2+t42) —ctg8(ol —Tl) =0, (3e)

r

1o 14+ r 14 (2p4 o4——T4) = 0—, —(3f)

r

o 1+r 1——(2p —o —r) =0,r

(4a)

p2+ T2 ctg8(0' T) =—0. (4b)

Differentiating (4a), one obtains an equation which canbe put into the form

p24+ T24 C tg8 (IT4 —T4) =0. (3g)

Integrating (3f) and (3g) with respect to t and takingthe functions of integration to vanish, we get

where p, o, ~, and p are functions of r, 8, and t andare considered (along with their derivatives) to besmall of the first order. The linear approximation ofthe field equations has the following form (indexesdenoting partial differentiation):

2oil+ rll (pl 0 1 rl) =0,

r

while differentiating (4b) gives

(5a)

p12+ r, 2—ctgd(o, —rl) =0. (5b)1 ctg8

o44+ T44 (T22+t422) + (o2 2T2 t42)r2 r2

Subtracting (5b) from (3e) and integrating withrespect to 8 (with the function of integration taken tovanish), one gets

2(o 1+rl+2t41)+ ——(p o) =0, ——

r r2 1441 P 1 (P+14)=0— —

r(4c)

ctg8Tl1 t411+P44+ T44 (P2+t42)

r2

1+—(pl —2r 1—t41) =0, (3b) 2

y» —p» ——pi= o.r

(5c)' H. Dingle, Proc. Natl. Acad. Sci. U. S. 19, 559 (1933).

and from this by di6'erentiation one gets a relationwhich can be written

N. ROSEN AN D H. S HA M I R

Finally, by combining (3a), (3b), (3c), (3d), (5a), One obtainsand (5c), one gets T'„=—-', pS(r), (14a)

2 1 ctg8Pll+ Pl+ P22+ P2 P44

r r2 r2(6) i B(r)

T 34= GDP

Bs(14b)

that is, a wave equation for p.The general solution of (6) is, of course, well known.

It can be expressed, for example, as an expansion inspherical (zonal) harmonics. We limit ourselves hereto the term of lowest order that can be expected to givea nontrivial result, that corresponding to an axialquadrupole source at the origin. The method can beextended to multipoles of higher order. We also take thesolution to be sinusoidal in time and to represent anoutgoing wave. Using the complex form (in the endthe real part will be taken), the solution can be written

cos28Tll ————oPp= h (r),

4m. r2(15a)

cosP singTl2 ———cu2P —5 (r), (15b)

Transforming the components to the polar coordinatesystem and also expressing the three-dimensionaldelta function 8(r) in terms of the one-dimensionaldelta function 5(r), one gets the following nonvanishingcomponents:

P=Ch(2) j'2(cos8)e ' ' (7)

where the frequency or and C are constants, x=~r, and

(8)

3 cosVTl4 i(oP——— b(—r),

r'(15c)

with J (2:) denoting the Bessel function of order 22.

Once p has been found, the remaining unknowns canbe determined by means of the field equations.

The constant C is to be determined in terms of thestrength of the source. For this purpose we take a verysimple model of the source, in the form of an inhnites-imal system consisting of two massive particles situatedon the Z axis and connected by a nearly mass-lessspring. We assume that the energy-momentum-stresstensor has for its nonvanishing components in theCartesian coordinate system T'», T'34, and T'44, withx3=s. Representing the mass density of a particleby a Dirac delta function and neglecting the staticpart of the tensor, we take

826(r)T44 2p

Bs

T22 ———(u2P sin24M (r),4m

3 sin8 cos6T24 i(oP ——8 (—r),

4m- r'(15e)

3 5 cos20 —1T44 p 8(——r)—— (15f)

To take account of the source, (1) should be replacedby

(16)

in units such that 82rk/c2=1. If one follows throughthe steps previously used in obtaining the waveequation for p, (6), one finds that now the correspondingequation has the form

whereQu 5

0 (10) (17)3 ~5 cos8—4

~'p+~'p= p —+f(r) ~(r), —2m r'

and g(r) is the three-dimensional Dirac delta function.From (9) it follows that where f(r) is an arbitrary function. On the other hand,

the solution for p previously obtained, as given in~(l7),for small values of r can be written in the form

Tl'p v 02 V

which in 6rst approximation have the form

(12)

The remaining components of the tensor are determinedby integrating the equations of motion

a2(-I3iC & rJ

p~ std (

24) Bs

from which one gets

(18)

T333 T344 0)

T 43, 3—T 44, 4=0.

(13a)

(13b)

9iC 5 cos28 —j.+2p~ g(r)e—4+i

AX I ALLY SYM METR I C GRA VI TATI ONAL F I EL D 43i

Comparing (17) with (19) and using the expressionfor p given by (10), one finds that

'EM

C= — po.6~

(2o)

The solution for all four unknowns, expressed interms of real functions, is found to be given by

t = po—[j—2(x) sincot+ j—~(x) cosset&P2(cos~),6~

(21a)

o.= — po{[(3k~(x)+j2(x)) sincot+ (3k ~(x)127r

+j 2(x)) coscot]P&(cos6) —(k2(x)+ j2(x)) sincot

—(k ~(x)+j 2(x)) coscot), (21b)

GO

po{[(kp(x) —jp(x)) sincot+ (k p(x)12'

—j 2(x)) coscot]P2(cos8)+ (k2(x)+ j2(x)) sincot,

+(k, (x)+j,(x)) coscot), (21c)

u= ——po[(2g2(x)+ j,(x)) sin&et+(2q 2(x)6~

+j 2(x)) coscot]P&(cos8). (21d)

p, tc, and a+r are of a form corresponding to a quadru-pole wave, i.e., they depend on the angle throughP2(cos8). On the other hand, the dependence of o ro—nthe angle is given by sin'0.

We see then that, in the linear approximation, thefield equations admit a solution describing gravitationalwaves emitted by an oscillating physical system. Acalculation sho~s that these waves are real, that is,the Riemann-ChristoGel tensor is different from zero,so that they cannot be removed by a coordinatetransformation.

To calculate the rate of emission of energy by thematerial system, one can make use of the gravitationalenergy-momentum-stress pseudotensor tp. For this itis desirable to go over to a Cartesian coordinate system,since in that case the first-order terms in the metrictensor will give second-order terms in t„~, which will bethe terms of lowest order, and these will not depend onterms of higher order in the metric tensor. The calcula-tion is somewhat tedious, and the final result for therate of energy emission is

1'V= co po,

120m

in the present units, or

k&V =—-co'pp',

15c'

Here

(mP *

E2x)

1k„(x)=— j„(u)ctu,

x~„

(22a)

(22b)

in the usual units. This result agrees with that obtainedby others. '

It is planned to use the above solution as the startingpoint for a more accurate calculation. The interestingquestion is whether the exact equations have a solutiongoing over into the above for sufficiently weak fields.

ACKNOW LEDGMENTS

c7„(x)=x u 'j.(u)du. (22c)

In the limit of a static system (co=0) one finds that tc

is proportional to the Newtonian potential, as is tobe expected. It will be noted that in the above solution

In conclusion, we should like to express our indebted-ness to Professor G. Racah for helpful advice anddiscussion.

2 L. Landau and E. Lifshitz, Clcssica/ Theory of Fields (Addison-Wesley Press, Cambridge, 1951),p. 331.