SLOWLY ROTATING CONFIGURATIONS OF AN INCOMPRESSIBLE

FLUID IN THE BIMETRIC THEORY OF GRAVITATION

Ao Vo Sarkisyan, E~ Vo Chubaryan, and A~ S. Gevorkyan

In the framework of Rosen's bimetric theory of gravitation and in the first

approximation in the angular velocity, Rosen's equations are solved inside

and outside mass distributions. The most important integrated character-

istics of rotating configurations of incompressible fluid are calculated

for different values of the parameter 0 c = --in (I + Pc/PO ).

I. Rotating stellar configurations have been studied on a number of occasions in

the framework of Einstein's theory of gravitation.

In [I, 2], the most general solution to Einstein's equations was studied in the

case of small angular velocities (the ~ approximation), which is equivalent to the grav-

tational energy of the star being many times greater than the rotational energy: B =

~2/8~GPc << i (Pc is the density at the center of the configuration). In this approx-

imation, the diagonal components of the metric tensor outside the mass distribution keep

their Schwarzschild form, and there is also a nondiagonal component g03 = (iJ/r) sin 2 8,

where J is the total angular momentum. This approximation corresponds to considering

rotation of a sphere with allowance for Coriolis forces without change in its shape.

The solution depends on the two mass and angular momentum parameters and can obviously

be obtained from the Kerr solution [3] by expanding the latter in a series in the angular

momentum and retaining the terms linear in J.

In the present paper, we consider the rotation of stellar configurations in the

framework of Rosen's bimetric theory of gravitation [4]. The field equations inside and

outside an axisymmetric mass distribution are obtained in the ~ approximation. In this

approximation, we find analytic solutions to the equations outside the mass distribution.

The internal problem is solved numerically for stellar configurations of an incompressible

fluid~

2. For stationary rotation of gravitating gases and liquids, the coordinates of the

metric tensor in a spherical coordinate system depend on r, e, and the parameter ~ =

d~/dt, i~ gik = gik (r, e~ ~)~ Below, we use the notation

x ~ = t , x ~ = : r , x ~ = ~ , x 3 = ? . ( G = c = 1 ) .

It follows from the invariance of the interval

ds ~ =: gi;dx i dx k

under the coordinate transformations x 0, = --x 0, x 3' = --x 3 that

( 2 . 1 )

gox = go~- = g13 : g~3 = O.

Except for g o B ' all the nonvanishing components of the metric tensor are even func-

tions of ~~ With regard to g03 , it depends only on odd powers of the angular velocity.

This dependence on ~ follows from the invariance of the interval ds 2 under the transforma- tions x 0~ = --x 0' or x 3' = --x 3.

In the first approximation in the angular velocity, the diagonal components of the

metric tensor remain the same as in the spherically symmetric case:

m~d

2,I~ 2 r 2!x 2!~ . 2 g ~ o = e , g ~ = - - e , g ~ = - e , g 3 ~ = - - e sm o,

2T~ g o a . = - - ~ ~ sin cO, g n = O

State University~ Erevan. Translated from Astrofizika, Vol. 15, No. 4, pp. 657-669,

Oetober-December~ !979o Original article submitted July 13~ 1978.

434 0571-~I.2/~9,1504-0434507 ~ n T / , . 5 0 �9 1980 Pienu~ Publishing Corporation

- - �9 '~ 2 2,~ �9 2 ds ~ -~ e ~ d t 2 .... e ~ d r ~ e ~!~ (d~ ~ 4- s m ~ 0 d~ ) - - 20~e s m Od?dt .

T h e f l a t - s p a c e i n t e r v a l i s o b v i o u s l y

( 2 . 2 )

= . o 2 dS+ d t ~ - - d P - - r e (dO ~ +. sin+ @d? ).

T o i t t h e r e c o r r e s p o n d t h e C h r i s t o f f e l s y m b o l s

C~2 - - I "~,:~ = 1 , = ~, r h - - 0 , = - - s in 0 c o s O, I ' ~ = etgO. r

( 2 . 3 )

h a v e t h e f o r m T h e n o n v a n i s h i n g c o m p o n e n t s of t h e t e n s o r Np

"+: (, +) ++"'++'I - - + h . t - i . . . . . ' ? + - - - - r + e + ( + - I ` + - e + r r" p.2

"N~"+- I ~ : - - (I'1+I|o .-P- ~-r ltl'l - - 71~ ) - - I ")

I.

In ~ N <+- ' '+" 1 '+ v, ~"l +~o = - - r - - - - q't, + - - e : ,~+-0 sin" G - L _ _ r ~ 2 r +. r

] e2(It- ~)] r" I

+r r ~ e2(~_,l,)sin2 ~/[ t%~ 23 ~ ' J ' c tg~ ] ~ ~~-'' e2(--m't Sin2 0 ' 2 "

-=- ' , u - - - - - z 2 lht-- lh--- r r

H e r e we h a v e u s e d t h e n o t a t i o n O f / b x h ' = - f ~ .

B e a r i n g i n m i n d t h a t

~ = - - e sin'-' 6+ (,o 2] , u ~ : +.:u" u [e 2* 2" __ ~ ) - i '2

f o r t h e c o m p o n e n t s o f t h e e n e r g y - - m o m e n t u m t e n s o r we h a v e

+

(2 .4)

(2 .5)

T~=~, T/:-- T2+=T 3 o 3 - - P , T:~ = - - ( P -:- f,) (u~ 2 e 2~ (o~ =- ~ ) sin "~ 0. ( 2 . 6 )

3 . T o f i n d t h e u n k n o w n f u n c t i o n s o u t s i d e a n d i n s i d e t h e m a s s d i s t r i b u t i o n , we u s e t h e e q u a t i o n s

G1 a - - G ~ = - - 8r .k C, - e ) , Go ~ 3 G ~ = - - 8r, k ( 3 P + F,),

+

P z + ( P - ~ ? ) ~ O , N ~ 1 7 6 . . . . 7 N ~ , - - . ,

w h i c h i n t h e f2 a p p r o x i m a t i o n h a v e t h e f o r m

( 3 . 1 )

q~lz -L 2 , - - <t~ 1 = 4=k ( 3 P - - ~,),

r

2 ',~u -" - - % = - - 4 ~ k (f~ - - P ) ,

r

(3 .2)

( 3 . 3 )

PI 4- (P 4-, p) d~z = O, ( 3 . 4 )

[ 4 ~-~!('~,,--(Pl) lql-j-[ 2 (',b 1 ~l])--16=k(?+ P)]ql- q l l i - - - 7 - - -

r r

1 [qe2 + 3q+ c t g O] =: + 16 ~.k (? -+- P ) [F8V, F~. r -

( 3 . 5 )

H e r e we h a v e u s e d t h e u n c h a n g e d f o r m o f t h e d i a g o n a l c o m p o n e n t s o f t h e m e t r i c i n

o u r a p p r o x i m a t i o n a n d c h o s e n t h e f u n c t i o n ~ i n t h e f o r m

4 3 5

. . . . I'~3 q. ( 3 . 6 )

Outside the mass distribution, the first equations are identical to the equations

whose solutions are given in [4] and have the form

where

M M' r r

R R

e = r e , M ' = 4= k ( p - - P) r :dr , M = 4~ k ( ? ~ 3 P ) / d r , k :~ e

0 0

(3.7)

Equation (3,5) in this approximation takes the form

A r2qn @ (4r @ A ) q~ ~ - - q ~ q2~ @ 3q~ c t g 0 = 0, r

where

A = - -2(M~- M').

We seek the solution of Eq. ( 3 . 8 ) in the form

(3.8)

q--~ ~ Ql (r) p~l)(cos 0). t - -0

H e r e

p ( 1 ) , dP; ~cos 8) := d~

Bearing in mind that p~I)(7)

= cos e, Pl(7) is a Legendre polynomial.

is a solution of the equation

d~Y + 3 e r g o @ + ( l - 1 ) ( ~ , 2 ) g = o,

we obtain for Ql(r)

r~d'Q, +(4r4_A) + - - ( l - - 1 ) ( l + 2 ) Q,=0 . dr ~ " --~r

The substitution Ql = r -(I+2) u~ reduces this equation to the equation for the confluent

hypergeometric function~

Thus

/=I \ r /

We s h a l l see below t h a t i n the ~ approx imat ion on l y the f i r s t term o f the se r i es i s non- vanishing~ and t h e r e f o r e

(3.9)

~'~ = ~'~F F 2~ 4, -~- - 2 A ~ [e* 1) + + 1) l = ] / ( x ) , (3.10)

where x = A/2r.

4. To find the four unknown functions r ~, ~, and p inside the mass distribution;

we seek q~ as in the case of the external problem, in the form

CQ

q = ~ Q,(r) p~l)(cos 0), t==O

(4.1)

436

dr-----T+ --~-.~ 2(~,--O,) dQt ~ ("~.-O,)--16";k(f,+ P) Q~= �9 dr r 2

The last equation is homogeneous and its solution can be represented in the form Q1 = blql- The values of the constants b I and C 1 can be determined from the condition of continuity

of ~(r, 0) and its first derivative with respect to r on the boundary of the configurations.

These conditions give

b t q ~ ( R ) - k C t R - ' - 2 F ( l + l , 2(i + l ) , A ) - - - O , bldql I _ F c I { F ( I + I , 2 ( I + I ) ' A / R ) } " R:,2 = O. dr ~=R

The determinant of this system is nonzero and, therefore, the unique solution of the

system is b I = C 1 = 0. For 1 = I, the system does not have a trivial solution and b 1

and C 1 are nonzero, i.e., q E Ql(r): Thus, in the first approximation in the angular

velocity the metric component g03' like all the others, depends only on r. We represent

the solution to Eq. (4.2) in the form

q(r) : Bj(r) + Q(r), ( 4 . 4 )

where j ( r ) i s a s o l u t i o n o f t h e homoge ne ous e q u a t i o n and Q ( r ) a p a r t i c u l a r s o l u t i o n t o t h e i n h o m o g e n e o u s e q u a t i o n .

F o r c o n s t a n t B and C1, we o b t a i n

Ct = [ j ( R ) (dQ(R)/dr) - Q(R) (dj(R)/dr)] A 3 ' ] (R) ( #f (%)/d, ) - - / ( % ) (d] (R)/dr)

B= /(~") (dQ(R)/dr) - Q(R)(d/(zo)/dr), j (R) (d/(zo)/dr) - - / ( z o) ( d / ( R ) / d r )

where % ~ A/2R. The i n t e r n a l p r o b l e m was s o l v e d f o r t h e m o d e l o f an i n c o m p r e s s i b l e f l u i d :

I n E i n s t e i n ' s t h e o r y , t h i s p r o b l e m has an a n a l y t i c s o l u t i o n [ 5 - 6 ] . e q u a t i o n o f h y d r o d y n a m i c e q u i l i b r i u m f o r t h i s mode l g i v e s

( 4 . 5 )

( 4 . 6 )

o ( r ) = PO" I n t e g r a t i o n o f t h e

w h e r e

To f a c i l i t a t e t h e i n t e g r a t i o n ,

P = ?o(e- O__ 1),

0 (r) = �9 (r) - - q) (R).

we introduce the new functions

( 4 . 7 )

,tO d O v = 3" - el), -; : 3'? + 0, u = r . . . . . /2 ,

d,- dr

r 2 dv r 2 d~ 2 dr 2 dr

(4.8)

and go o v e r t o t h e d i m e n s i o n l e s s q u a n t i t i e s (R) �9 (RJ 't,l' (R-/)

x = r (4~[,0) '" e , y =: u (4~fq,) 1:2 e , z = c,, ( j ' = j (4~<,o) e '~CR~, Q' - Q{4r.[,t,) -I;2. : 4 . 9 )

Then the solutions of the equations must satisfy the initial and boundary conditions

v ( o ) - - z ~o) = o, ~ (o) = o~, x (o) = -~,

~ ( o ) = 2 x , O ( x o ) = o , P ( ~ ) = o ,

j ' (o) = 1, ~ (o) = o, ( / ( o ) -- xL

x.x (~o) ~ 2~ (xo) - y (xD = o. ( 4 . 1 0 )

5. The s y s t e m o f e q u a t i o n s ( 3 . 1 ) - ( 3 . 5 ) w i t h i n i t i a l a n d b o u n d a r y c o n d i t i o n s ( 4 . 1 0 ) was i n t e g r a t e d f o r a number o f v a l u e s o f t h e p a r a m e t e r s 0 c and Yc" The r e s u l t s o f t h e c a l c u l a t i o n s a r e p r e s e n t e d i n T a b l e s 1 and 4 and i n F i g s . 1 - 3 . The v a l u e s o f t h e

437

TABLE 1 . Values of the Functions ~(0)

and ~(0) at the Centers of the Configura-

tions and the Constants of Integration

Clx and B x for Different Values of the Parameter 8 c

- o - - �9 (o) ,? (o) c ~ , - - B x

0.01

0.02

O. 0253

0.05

0.1

O. 1625

0.2876

0.4

O. 693

1.0

1.5

2.0

3.0

4 . 0

0.0296

0.0598

0.0723

0.1434

0.2765

0.4365

0.7485

1.0124

1.7481

2.6100

4.3080

6.4410

12.81

15.01

0.0293

0.0517

0.0684

0.1300

0.2279

0.3141

0.4259

0.4912

0.5102

0.4408

0.1960

--0.1610

--1 .2400

--2.9900

0.007

0.032,

0.054

0.245

0.856

1.830

4.051

6.603

19.48

59.49

473.8

5.4-103

2.9.106

2.76.10 lo

0.0542

0.I080

0.1241

0,2309

0.3880

0.5276

0.6949

0.7927

0.9283

1.0004

1.0590

1.0864

1.1083

1.115

functions ~(0) and ~(0) and the constants Clx and B x as determined by the relations

3 --~-'~'( xo~

C: = - - Cr,, ( 4 7 9 o ) - I e , /3 = Bx (4.=~o)a2e | ( 5 . 1 )

are given in Table i.

At large distances, i.e., in the limit r + ~ , expanding g03 in a series in the small

parameter ~ =A/2r mud comparing the result with Papapetrou's solution [7] g03 = (2JG/r) sin 2 O, we obtain

and within the configurations

C, (5.2) J t o t = (4r '9o)- ' /2 48 I f 2 - '

TABLE 2 . M a i n I n t e g r a t e d C h a r a c t e r i s t i c s o f C o n f i g u r a t i o n s o n B a s i s o f t h e B i m e t r i c ( R ) a n d Einstein (E) Theories of Gravitation

6c R e o o r ( " . % ) IRtr (4=?0) ' MR(4r~?o )12 (4=Po) 1'2 coor t ",'0~ n 4- 1~2 R 1'2 M' R E r4_r. ~1/2 RE r (4npo)l/2 ME(4zpo)ll2

O. 242 4.65-10-3

O. 345 1.28.10 - 2

0,518 4.4 -10 -2

0.710 O. I04

1.364 O. 538

2.036 I . 320

.3.IOI 3.170

7.134 15.22

21.33 97.71

6.06- 10 ~ 1.62.104

! .80.105 4.98.107

jn(47.po) 3,-~

4.58.10 -3 1 .03. i0 -`4

1.23.10 -2 4.60.10 -4

4.20.10 -2 4 .14 ,10:3

8.87.10 - 2 1.64.10 - 2

o. 304 o. 244

o. 503 1.397

0.818 9.806

1.870 4.72- 10 ~

3.830 6.62.104

8.52.10 ~ 1.14-10 u

-4.87.106 [9.85.10 ",0

0.241

0. 335

0. 505

0.664

0.913

0. 977

1.100

1. 127

1.140

1.150

1.153

0. 242

0. 339

0.510

0. 702

1.030

1.132

1,364

1.377

1,465

1.492

1. 502

4.65,10 -3

1.25.10 -2

4.29.10 -2

9.77.10 -2

0.3112

O. 3 o 8

O. 442

0.477

0.493

O. 507

0.511

0 .0 t 0.236

0.02 0.318

- -0 .05 0.465

0 . ! 0.583

- 0 . 4 0.879

--O,69 t .251

�9 - - 1 ~0 ~ .970

- -1 .5 5.430

2.0 21.80

3.0 1.64.10 a

--4 0 1 2.58. i0 ~ l

]E(4~o)312 M R / M E

1.02-10 -4 1.0C

6.21.10 -4 1.021

2.72.10 - 3 1.026

1,21.10 -2 1.106

6.27.10 - 2 .1.729

0.159 3.317

0.172 7.178

O, 198 31. 908

0.219 198.2

0.228 3.19-104

0.231 9.75.10 ~

4 3 8

3 / f ~' \

/ x 0.3 i i \ \

/ /

t ' 2 /

/ l / ~\

I:::c~' 0.2 ll/~\\i .,.~

0.1

I t S / , ~ , , / ~ ~'~ I / 1 1 1 " i.f1"" t

0.2 0.6 1.0

r /R

F i g . 1 . D e p e n d e n c e o f goae*(R)(4r.po) 2 sin I 0

o n r / R f o r 0 c = - - 0 . 0 5 ( c u r v e 1 ) , 0 c = - - 0 . 1 6 5 ( c u r v e 2 ) , a n d 0 c = - - 0 . 2 8 7 ( c u r v e 3 ) . T h e c o n t i n u o u s c u r v e s c o r r e s p o n d t o t h e b i m e t r i c t h e o r y , a n d t h e b r o k e n c u r v e s t o E i n s t e i n ' s t h e o r y . A t s m a l l v a l u e s o f t h e p a r a m e t e r 0 c , t h e c o r r e s p o n d i n g c u r v e s i n t h e b i - m e t r i c t h e o r y a r e l o w e r t h a n t h e E i n s t e i n c u r v e s .

, 6 \\

R

'il -3 -2 -1

ec

Fig. 2 . D e p e n d e n c e Of i n M R a n d i n M E o n e c .

T h e c o n t i n u o u s a n d b r o - k e n c u r v e s c o r r e s p o n d t o t h e b i m e t r i c a n d E i n s t e i n c o n f i g u r a t i o n s , r e s p e c t i v e l y .

TABLE 3 . I n t e g r a t e d C h a r a c t e r i s t i c s

o f C o n f i g u r a t i o n s w i t h P0 = 2 . 8 4 - 1014 g / c m 3 a n d D i f f e r e n t V a l u e s o f

Pc

10"-32Pc R (km) (erg.cm -3)

0.0066

0.0128

0.0256

0.1278

0.2556

1.278

2.566

12.81

25.69

51.60

77.80

104.3

0.0764

0.1053

0.1488 0.3323

0.4703

1.053

1.488

3.265

4.703

6.510

7.967

9.191

M/M| l (kin 2 -M|

2.67.10 -7 6.24-10 -1~

7.20.10 -7 3.19.10 -9

2104.10 -6 1.80-I0 -~

2.22.10 -s 9.81.10 -7

6.01.10 - s 5.31.10 -6

6.90.10 .4 3.06.10 -4

2.39.10 -3 2.12.10 -a

2.13.10 -~-[ 0.091

6.11-11) -~1 0.54

0.1681 ] 2.849

0.2622 / 6.657

0.3706 [ 12.52

4 3 9

TABLE 4 . I n t e g r a t e d C h a r a c t e r i s t i c s o f C o n f i g u r a t i o n s w i t h P0 = 3 " 1 0 1 6 g / c m 3 a n d D i f f e r e n t V a l u e s o f Pc i n t h e B i m e t r i c (R) a n d E i n s t e i n ( E ) T h e o r i e s o f G r a v i t a t i o n

10.:37p~ R R jR [ jE (erg. cm -3) (kin) MR/]~-~ (km/.MC.O) R E (kin) MeM~ (kin2. M~) )

0; 552

01598

0;767

0.945

1.131

1. 328

1.534

1.752

1 ~980

2.220

2.700

i,802

1.855

2.072

2,254

2.416

2�9

2,753

2.957

3.174

3.420

3.849

0.274

0.298

0.384

0.472

0.584

0.688

0.811

0.932

1,059

1.303

1.688

0.094

0.!02

0.168

0.263

0.417

0,590

0.832

1.175

1.679

2.188

3.377

1.540

1. 572

1. 668

1.742

1.800

1.847

! .886

1.918

1.946

1. 969

2.005

0.231

0.245

0.293

0. 334

0.368 0. 398

0.423

0.446

0.465

0.485 O. 509

0 �9

0.062

0.084

0.105

0.125

0A52

0.185

0.215

0.253

0. 340

0.384

1.5

1.0

0.~

!

/

1.31 1.33 1.35

F i g . 3. Dependence o f M and J

R/2 on t h e p a r a m e t e r ~ = tan -I log Pc/1033 in the case P0 = 3"1016 g/cm3" The con- tinuous curves correspond to the bimetric theory.

440

X 3 = - - 3 : (Bj § ( s . 3 )

2 l In T a b l e 2 , we h a v e p l o t t e d t h e i n t e g r a t e d c h a r a c t e r i s t i c s o f v a r i o u s c o n f i g u r a -

t i o n s c a l c u l a t e d i n t h e f r a m e w o r k o f t h e b i m e t r i c t h e o r y a n d i n E i n s t e i n ' s . The m a s s e s and r a d i i i n E i n s t e i n ' s t h e o r y a r e g i v e n by

1 331~ [ 1 I RE ~1,2 ' 1 ]1,2 3 , , ,-1/21 1 �9 ( 5 . 4 ) = ~'~, = ~ -~ (~Po) [ - - (3 - - 2e ~ ) ;

M E 6 ] / 2 (4~P~ 1 (3 - - 2e~ z ,

I n t h e l a s t c o l u m n o f T a b l e 2 we h a v e g i v e n t h e v a l u e s o f t h e r a t i o s MR/M E . H e r e , b e - g i n n i n g w i t h e c = - - 0 . 6 9 3 , MR/M E i n c r e a s e s r a p i d l y a nd r e a c h e s MR/M E = 9 . 7 5 " 1 0 7 f o r e c = - -4 ( i t s h o u l d be n o t e d t h a t c o n f i g u r a t i o n s w i t h t e c l > 0 . 6 9 3 c a n no l o n g e r s e r v e a s m o d e l s f o r r e a l s y s t e m s , s i n c e h e r e P > p , i . e . , t h e c a u s a l i t y p r i n c i p l e i s v i o l a t e d ) .

The same r e s u l t s f o r s p h e r i c a l l y s y m m e t r i c c o n f i g u r a t i o n s were o b t a i n e d by R o s e n i n [ 8 ] .

I n [ 9 ] , a new e q u a t i o n o f s t a t e was r e c e n t l y o b t a i n e d f o r a r e a l b a r y o n g a s w i t h a l l o w a n c e f o r t h e p r e s e n c e o f a ~ - c o n d e n s a t e i n t h e n u c l e a r p l a s m a .

I t was f o u n d t h a t i n a d e f i n i t e r a n g e o f p r e s s u r e s t h e m a t t e r c a n be a s s u m e d t o b e - h a v e a s an i n c o m p r e s s i b l e f l u i d . T h u s , i n t h e r e g i o n P < 1034 e r g / c m 3 we a r e d e a l i n g w i t h i n c o m p r e s s i b l e n u c l e a r m a t t e r , and i n t h e i n t e r v a l 5"1036 e r g / c m 3 ~ P ~ 3 " 1 0 3 ? e r g / c m 3 w i t h an i n c o m p r e s s i b l e q u a r k f l u i d . W i t h a v i e w t o q u a l i t a t i v e s t u d y o f t h e p r o p e r t i e s o f s t e l l a r s y s t e m s i n t h e s e r e g i o n s , we h a v e g i v e n i n T a b l e s 3 and 4 t h e i n t e g r a t e d c h a r - a c t e r i s t i c s (M, R. J ) f o r R o s e n and E i n s t e i n c o n f i g u r a t i o n s i n t h e c a s e P0 = 2 " 8 4 " 1 0 1 4 g / c m 3 ' P0 = 3"101~ g / c m 3 ' and f o r d i f f e r e n t v a l u e s o f P.

The v a l u e s o b t a i n e d f o r t h e m a s s , r a d i u s , and moment o f i n e r t i a a nd t h e i r b e h a v i o r f o r P0 = 2"84"101~ g/cm3 a r e v i r t u a l l y i d e n t i c a l b e c a u s e a t s m a l l P (P << p) t h e two t h e o r i e s d i f f e r o n l y i n t h e s e c o n d t e r m s o f t h e p o s t - N e w t o n i a n a p p r o x i m a t i o n . But f o r P0 = 3"1016 g/cm3 (P j p ) ' t h e b i m e t r i c t h e o r y g i v e s , as one w o u l d e x p e c t , a mass o f t h e c o n f i g u r a t i o n t h a t i s l a r g e r t h a n t h e v a l u e o b t a i n e d i n g e n e r a l r e l a t i v i t y , and t h i s d i f - f e r e n c e i n c r e a s e s w i t h i n c r e a s i n g Pc - - ( ~ ) "

We t h a n k P r o f e s s o r G. S. S a a k y a n f o r h e l p f u l d i s c u s s i o n s .

LITERATURE CITED

1. A. G. Doroshkevieh, Ya. B. Zel'dov~ch, and I. D. Novikov, Zh. Eksp. Teor. Fiz., 4_99, 170

( 1 9 6 5 ) . 2 . V. T s . G u r o v i c h , A s t r o n . Z h . , 42 , 974 ( 1 9 6 5 ) . 3. R. P. K e r r , P h y s . Rev . L e t t . , 11 , 522 ( 1 9 6 3 ) . 4 . N. R o s e n , Ann. P h y s . ( N . Y . ) , 8_44, 455 ( 1 9 7 4 ) . 5. G. ft. A r u t y u n y a n and D. M. S e d r a k y a n , Uch. Zap . E r e v a n . Gos . U n i v . , 3 , 21 ( 1 9 6 9 ) . 6 . G. G. A r u t y u n y a n and D. M. S e d r a k y a n , A s t r o f i z i k a , 7 , 259 ( 1 9 7 1 ) . 7 . A. P a p a p e t r o u , P r o c . R. I r . A c a d . , 5_22, 11 ( 1 9 4 8 ) . 8. J . Rosen and N: R o s e n , A s t r o p h y s . J . , 212 , 605 ( 1 9 7 7 ) . 9 . G. S. S a a k y a n and L. Sh. G r i g o r y a n , A s t r o f i z i k a , 13, 669 ( 1 9 7 7 ) .

441