~'amitta, Vol. $, No, 6, 1977, pp. 512-517. © Printed in India.
The Kerr metric in cosmological background
P C VAIDYA
Department of Mathematics, Gujarat University, Ahmedabad 380009
(Presently Member, UPSC, Dholpur House, Shahjahan Road, lqew Delhi 110011) MS received 1 February 1977
Abstract. A metric satisfying Einstein's equations is given which in the vicinity of the source reduces to the well-known Kerr metric and which at large distances reduces to the Robertson-Walker metric of a homogeneous cosmological rr.odel. The radius of the event horizon of the Kerr black hole in the cosmological background is found out.
Keywords. Kerr metric; Kerr black holes; general relativity; expanding universe.
1. Introduction
The metric given by Kerr (1963) is recognised as the metric describing space-time exterior to a finite rotating body. Many attempts have been made to understand the source of this metric but the investigations are still inconclusive. On the other hand, the exterior metric itself has been very widely used in astrophysics in connection with the blacldlole theories.
The standard Kerr metric is described in flat background. Because of the poten- tial use of Kerr black holes in the story of the cosmological evolution of the universe, it would be interesting to find out what the Kerr metric looks like in the background of a homogeneous model of the u n i v e r s e rather than in the standard Minkowskian background. In the following, a metric is presented which in the vicinity of the source reduces to Kerr metric and which in the absence of the source leduces to Robertson-Walker metric.
In a series of papers (Vaidya 1972, 1973, 1974), we have developed a specific coordin3te system to describe the general Kerr-Schild (1965) metric and in parti- cular the Kerr metlic. This coordinate system is quite useful in the plesent work because it uses the Minkowskian time t as the time-like coordinate for the Kelr metric.
Take the Minkowskian background metric in the form
d s 2 = d t 9 - - d x 2 - d y 2 - - d z L (1)
In the physical 3-space given by t --- const., one can use rotating elIipsoidal polar coordinates (r, u, /~) in place of (x, y, z) given by
x = p cos/~ -- w sin/~, y = p sin p + w cos/~, (2)
p = r sin ct, z = r c o s ~ , w = a sin u, 512
T h e K e r r m e t r i c in c o s m o l o g i c a l b a c k g r o u n d 513~
so t h a t ( x 2 + y2)/(r~ + a ~) + zS/r 2 = 1,
i.e., surfaces r = c o n s t a n t are ellipsoids o f revolution. T h e n (1) will t r a n s f o r m to
d s 2 = dt ~ - - ( d r - - a sin s otdfl) s - - (r 2 + a s cos 2 ~t) (d0t 2 + sinS0t dfl2). ( 3 ) Using the r e t a r d e d null c o o r d i n a t e u = t - - r in place of" the space-like c o o r d i n a t e r, we write (3) in t h e f o r m
d s 2 = 2 ( d u + a s i n S ~ d / ~ ) d t - (r s + a2cos s ~) (d~# + sin s ~ dfl s)
- - ( d u + a sin ~ ~dfl) 2 (4),
r = t - - u .
T h e M i n k o w s k i a n metric (4) f o r m s the b a c k g r o u n d o f the K e r r metric w r i t t e n in the f o r m ( V a i d y a 1974)
ds s = 2 ( d u + a sin ~ ~d[3) dt - - (r 2 + a s cosSc0 (dc~ 2 + sinS~ dff 2)
- - [1 + 2 m r / ( r 2 + a z cos s ~)] (du + a sin2~tdfl) s. (5)
T h u s the t i m e c o o r d i n a t e t o f the K e r r metric (5) is t h e G a l i l i a n t i m e t o f (1).
I n the next s e c t i o n we begin with the static metric o f Einstein's universe in place o f (1), leave t u n a l t e r e d , b u t t r a n s f o r m (x, y, z) to r o t a t i n g ellipsoidal p o l a r coordinates a n d thus get the b a c k g r o u n d c o s m o l o g i c a l metric. I n the f o l l o w - ing section a K e r r - l i k e s o l u t i o n o f the field e q u a t i o n s o f Einstein will be given in this c o s m o l o g i c a l b a c k g r o u n d a n d in the last section the ellipsoid o f r e v o l u t i o n which describes t h e e v e n t h o r i z o n in the c o s m o l o g i c a l b a c k g r o u n d is presented.
2. T h e R o b e r t s o n - W a l k e r m e t r i c in r o t a t i n g c o o r d i n a t e s
Consider t h e metric o f t h e Einstein universe in t h e f o r m ds s = dt2 dxS d y s _ dz2 ( x d x + y d y + z d z ) 2
R ' - - (x 2 + y2 + z2) • (6) H e r e a g a i n we shall leave the cosmic time t u n a l t e r e d a n d c a r r y o u t t h e f o l l o w - ing t r a n s f o r m a t i o n f r o m (x, y, z ) t o spheroidal p o l a r coordinates (r, ,t, fl):
x = p c o s f l - - w s i n f l , y - ~ p s i n f l + w c o s f l (7) p = R sin ( r / R ) sin ~t, z ---- R sin ( r / R ) cos ~, w ---- a c o s ( r / R ) sin ~.
I t will be seen t h a t t r a n s f o r m a t i o n s (7) reduce to t r a n s f o r m a t i o n s (2) w h e n R ~ c~.
U n d e r (7), t h e Einstein's metric (6) t r a n s f o l m s t o
ds 2 = dt 2 - - dr 2 + 2 a sin s otd f l d r - - M 2 [1 - - ( a 2 / R s) sin 2 ~]-1 d~tS
- - [ M s + a ~ sin ~ ct] sin~tdfl "- (8)
M s = ( R 2 - - a 2) sin s ( r / R ) + a ~ cos s ~t. (9) A g a i n u s i n g t h e null c o o r d i n a t e u = t - r in p l a c e o f r, we express (8) in t h e f o r m
p--3
514 P C Vaidya
ds ~ = 2 (du + a sin ~ ~dfl) dt - - M 2 [(1 - - a 2 sin ~ ~ / Rz) -1 d~ ~ + sin ~ ~dfl ~]
(lO)
-- ( d u + a sin ~ ~dfl) z,
= d s 2 (say).
(10) is t h e metric o f Einstein's universe.
T u r n i n g to R o b e r t s o n - W a l k e r m e m c , one writes it in the f o r m
[ ( x d x +_ ydy_ + zdz) 2 ] (11)
ds2 : dz~ - - e"~') dxZ + dY 2 + dz~ + R 2 _ ( x ~ + y2 + z2) j One c a n replace the cosmic time z by t such t h a t
.[ e - t g~') dz---t a n d define the function F ( t ) b y g ( z ) = 2 F ( t ) . T h e n the R o b e r t s o n - W a l k e r metric becomes
( x d x + y d y + zdz) ~ ] ds2 = e ~ " ~ d r 2 - - d x 2 - - dY ~ - d z 2 - - R 2 ( x ~ + y2 + z~).] •
But we have already t r a n s f o r m e d the quadratic f o r m within t h e brackets o n the light h a n d side b y changing only the coordinates x, y, z in the physical space, leaving the time t unaltered; so t h a t when the same t r a n s f o r m a t i o n is now carried out in (11), t h e c o n f o r m a l factor e 2v") will r e m a i n unaffected and the quadratic f o r m within the brackets will transform to dsE ~ o f (10). H e n c e one can immedi- ately write d o w n t h e Robertson-Walker metric f o r e x p a n d i n g m o d e l o f the universe in the new coordinates in the f o r m
ds ~ = e ~v~t) ds~ 2 (12)
where dsz ~ stands f o r the metric (10) and F ( t ) is a n u n d e t e l m i n e d function o f t.
In the next section we present Kerr-like solutions o f Einstein's field equations in the h o m o g e n e o u s b a c k g r o u n d given by (10) a n d (12).
3 . K e r r - l i k e s o l u t i o n s
We first take the Einstein's universe given by (10) as t h e b a c k g r o u n d universe. T h e Kerr-metric in the cosmological b a c k g r o u n d o f Einstein's univelse is given by
ds ~ = 2 (du + a sin z *td[3) d t - (1 + 2 r a p ) ( d u + a s i n ~ d p ) 2
- - M2[(1 - - a ~ sin 2 ~/R~) -x d~ ~ + sin 2 ~ dfl ~] (13)
with M given b y (9) r - ~ t - - u r e = c o n s t a n t a n d
= R sin ( r / R ) cos tr]R)/M~. (14)
I f m is p u t equal to zero, the metric (13) reduces to the metric of the Einstein's universe (10). W h e n R ~ c~, it becomes the K e r r metric (5). The material con- tent o f the field described by this metric c a n be f o u n d f r o m Einstein's equations
R~ - - ½g~ R "-- - - 8 n [(p + p) v, v~ - - p g ~ ] ~ Ag~k.
Using the m e t h o d o f tetrads a n d the exterior calculus, one c a n see that the material c o n t e n t o f the metric (13) can be described as a perfect fluid with the following pressure and density
T h e K e r r m e t r i c in c o s m o l o g i c a l b a c k g r o t m d 515 87zp = - - 1 ( l _ 2 m p ) + A
R s
8 rtp = ~ (1 - - 2m/t) - - A. 3
T h e detailed calculations a r e lengthy b u t s t r a i g h t f o r w a r d a n d so a r e n o t given here.
O n e c a n n o w get t h e K e r r metric in the b a c k g r o u n d o f t h e e x p a n d i n g universe given by (12). T h e metric will be c o n f o r m a l to the metric (13), w i t h a little c h a n g e in the m u l t i p l y i n g f a c t o r o f m. This metric turns o u t to be
ds ~ -~ e 2F(=} [ 2 ( d u + a s i n 2 o t d f l ) d r - - ( 1 + 2 m p e - s v ) ( d u + asinSctdfl) s - - M 2 [(1 - - a s sin 2 a/RS) - I d a s + sin s ~dflS]] (15) p being a g a i n defined b y (14).
T h e metric (15) bears the s a m e relation to the metric (12) o f t h e e x p a n d i n g uni- verse as t h e metric (13) bears to the metric ( 1 0 ) o f Einstein's static universe.
H o w e v e r , t h e r e is a qualitative difference. The resultant effect o f the i s o t r o p i c e x p a n s i o n o f the c o s m i c fluid a n d t h e presence of the r o t a t i n g s o u r c e is t h a t the cosmic fluid in the vicinity o f the source exhibits a n i s o t r o p y in the pressure which diminishes as we go a w a y f r o m the source.
A t a p o i n t in t h e 3-space t = c o n s t a n t , if we c h o o s e 3 m u t u a l l y o r t h o g o n a I infinitesimal vectors 01, 0 s, 0 z, defined by
01 : e ~ ( d u + a sin s adfl)
0= = e v M ( 1 - - a s sin s :~/Rs) - t da 0 a = e v M s i n edfl
t h e n the pressures in the direction 0 s a n d 0 3 are equal (say, p) while the pressure in the direction 01 is q ~ p . T h e m a t h e m a t i c a l expressions f o r t h e pressures p a n d q a n d the density p o f the cosmic fluid at a point, as disturbed b y the pre- sence o f t h e K e r r s o u r c e a r e given in the appendix.
4. The event hmizon
T h e c o o r d i n a t e s y s t e m which we h a v e been using, t h o u g h very convenient f o r expressing the K e r r metric irt the b a c k g r o u n d of a h o m o g e n e o u s universe is n o t c o n v e n i e n t to locate the event h o r i z o n . This is because in o u r s y s t e m t h e null c o o r d i n a t e u (which is essentially a retarded time) replaces the space-like radial c o o r d i n a t e r. I n order, therefore, to discuss the event h o r i z o n we shall h a v e to replace u in the K e r r metric (5), in the Einstein-Kerr metric (13) a n d in the R o b e r t s o n - W a l k e r - K e r r metric (15) b y r t h r o u g h u = t - - r.
Let us t a k e the K e r r metric (5). W e first replace t h e c o o r d i n a t e u b y r (i.e., p u t du = d t - dr). T h e n since the event h o r i z o n is spheroid given b y a c o n s t a n t value o f r, in the t r a n s f o r m e d metric we put dr = 0 a n d thus get t h e metric o n the spheroid r = c o n s t a n t in t h e f o r m
516 P C V a i d y a
where
B . , 2 ~ m r d t ) * A M * ~
daZ = - - M Z d a 2 - - - ~ , sin • ( d r + --if- d t
(16)
M s = r ~ + a 2 c o s i ~ ,
A = r z - 2 m r + a 2 B : ( r ~ + a Z ) s - - A a s s i n s ~ .
It will be seen t h a t (16) agrees with the expression f o r d a ~ o b t a i n e d f r o m the f o r m o f K e r r metric given by Boyer a n d Lindquist (1967), when we put r = c o n s t a n t in it a n d so we get the event h o r i z o n o f the K e r r metric as the spheroid with r p a r a m e t e r given by
A = 0, i.e., r s - - 2 m r + a s ---- 0, i.e., "r = m + ( m s - - a2) t. (17) W e use t h e same procedure to find the spheroid representing the event h o r i z o n for t h e K e r r black hole immersed in the Einstein universe, We shall find t h a t it is given by r ~ - c o n s t a n t satisfying the e q u a t i o n
R 2 tan s ( r / R ) - - 2 m R t a n ( r / R ) + a s : 0
i.e., R t a n ( r ] R ) = m + (m s - aS) t. (18)
Since R is large compared to r, a c o m p a r i s o n o f (17) a n d (18) shows t h a t t h e effect o f the c u r v a t u r e o f the universe o n the radius o f the event h o r i z o n (i.e., o n the r-parameter o f the spheroid) is to reduce this radius by a fraction ~r (rS/Rs).
Again taking a K e r r black hole immersed in the expanding universe (15), we shall find t h a t the radius o f the event h o r i z o n is g i v e n by eq. (18) with m replaced by m e -zF, i.e., by
R 2 t a n s ( r / R ) - - 2 m e - S F R t a n ( r / R ) + a s = O. (19) This event h o r i z o n exists if a2/m 2 < e - w . I f me is the gravitational mass a n d J is the t o t a l angular m o m e n t u m o f the black hole t h e n o u r parameters a a n d m are given by m : Gmo/C 2, a ~ J / c m o a n d therefore the c o n d i t i o n o f existence o f a n event h o r i z o n becomes
C J / G m o 2 < e -~F"~ = e-a~*~.
(20)
N o w a t the present epoch e gtr~ is a l m o s t unity b u t f o r a n expanding universe model, in t h e early stages o f evolution e gt~ , ~ z "~ when z is small. T h u s e ut~ is m u c h larger t h a n unity at the earlier stages a n d so c o n d i t i o n s at the early stages o f evolution are m u c h m o r e favourable for the e inequality (20) t o be satisfied. Again for large values o f z, i.e., in far future, e otT~ will be m u c h less t h a n 1 a n d so conditions in t h e later stages o f evolution are m u c h less f a v o u r a b l e for the restriction (20) t o be satisfied. We m a y therefore draw a general conclusion t h a t during the earlier stages o f the evolution o f a n expanding universe the c o n d i t i o n s were far more f a v o u r a b l e for the existence o f K e r r black holes t h a n are at the present epoch a n d they will be m u c h less favourable at a distant future epoch.
The Kerr metric in cosmological background Appendix
W e use t h e f o l l o w i n g n o t a t i o n s e - p ~---T.
X ~ mR
sin(r/R)
c o s(r/R)
dT d X
X , = U r r . Define
T h e n
R(11) , Rc22) , R(14) , R(44) b y
R(ll) = - - ~- [(1 - - T 2
2T2mp)*(1/R ~) +
(1 +2T2mp)2(T./T)
- - 2T*m l: (TJT) (X, /X)]
.R(2~) = T2[(1 - -
2T2ml z) (--
2 / R 2) + (1 +2T2mp)(T./T)
- - (3 + 2 T Z m p )
( T,/T) 2 -- T2ml: (Tt/T) ( X,/ X) ]
R(14, = T2[(1 - - 2T2m/z) (1/R 2) - - 2 (1 +
T2mp) (TtdT) + 3 (Tt/T) 2]
.R(,4, = - - 2 T 2 [(1/-R2) +
(rt,/T)].
- - 1
8 ~ p : ~ R ( U ) - - .R(22) - - A x
8 7rp = - - R(14) + A,
8 ~ r q : - - x 1 R (xl) + R ( m ) + A ,
where x is t h e p o s i t i v e r o o t o f R t 4 ~ ) x 2 - .R(xl) ~ O.
517
References
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K e r r R P a n d Schild A 1965 AItL Conv. Relativita Generale. cd. G B a r b e r a . p. 222 Vaidya P C 1972 Tensor 24 315
Vaidya P C 1973 Tensor 27 276
Vaidya P C 1974 Proc. Camb. Phil. Soc. 75 383
