Kerr metric in the deSitter background

Kerr metric in the deSitter background

P C VAIDYA

Department of Mathematics, Gujarat University, Ahmedabad 380009, India

Abstract. In addition to the Kerr metric with cosmological constant A several other metrics are presented giving a Kerr-like solution of Einstein's equations in the background of deSitter universe. A new metric of what may be termed as rotating deSitter space-time---a space-time devoid of matter but containing null fluid with twisting null rays, has been presented. This metric reduces to the standard deSiRer metric when the twist in the rays vanishes. Kerr metric in this background is the immediate generalization of Schwarz,~hild's exterior metric with cosmological constant.

Keywords. Kerr metric; deSitter universe.

1. Introduction

In an earlier paper (Vaidya 1977, referred to hereafter as I) we have considered Kerr metric in cosmological background, the background metric being the Robertson- Walker metric. In the present paper we single out the background universe as empty deSitter space-time for several reasons. One reason is that though deSitter metric can be expressed as a particular case of the general Robertson-Walker metric, the deSitter space-time has features which are geometrically distinct from Robertson-Walker models. Again the simple deSitter space-time representing an expanding, curved and yet empty open universe is the immediate generalization of Minkowski flat space-time and has very similar properties as a background for physical phenomena.

Schwarzschild's exterior metric in deSitter background is the well-known Schwarzschild's solution with cosmological constant A

re) (

r ~ dt 2 - 1 -r R2,] dr2 - r2 d~')2 (1) d D 2 = d 0 2 + s i n 2 0 dip 2 and A = R-- ~. 3

On parallel lines we may expect that Kerr metric in deSitter background will be the Kerr solution with cosmological constant which has been derived by several authors earlier (Carter 1968; Demianski 1973; Frolov 1974). However we shall see that one can have other (non-equivalent) forms for Kerr metric in deSitter background.

In the next section we shall derive the known form of Kerr-metric with cosmological constant. In the third section we shall write down a simple form of metric for anti deSitter space-time (A negative) and get a Kerr-like solution in this background.

In the last section we first derive the metric for what can be termed as rotating deSitter space-time--a space-time devoid of matter but containing null fluid with 151

152

P C Vaidya

twisting null rays. This metric reduces to the s t a n d a r d deSitter metric when the twist in the rays vanishes. T h e K e r r metric in this b a c k g r o u n d is the immediate generalization o f Schwarschild's exterior metric (1) with cosmological constant.

2. Kerr metric with A

We begin with the standard deSitter metric

R 2 ,] dP2 - ~6 2 d[~ 2

A = 3/R 2,

d f l 2 = d 0 2 -t" sin e 0 dip 2,

and carry out the t r a n s f o r m a t i o n p2 = r , 2 / [ l +

(r,2/R2)].

It then transforms to

( ( ] ⁽²⁾

(2) shows that the deSitter metric can be put in a f o r m e o n f o r m a l to Einstein- U n i v e r s e - - m e t r i c with negative curvature ( -

1/RZ).

We rewrite (2) in terms o f the usual Cartesian coordinates (t, x, y, z) as

ds2 = R2 + (x 2 + y2 + zZ) dr2 - dx2 - dY 2 - dz2 "~

(xdx

+ ~ l y + zdz) 2 R 2 + (x 2 + y~ + z 2)

(3)

and then follow the steps initiated in ! to carry out the t r a n s f o r m a t i o n s to spheroidal polar coordinates

(t, r, ~, [3)

by the substitutions t = t

r r

x = R sinh ~ sin e cos [1 - a cosh ~ sin e sin [1,

r r

y = R sinh ~ sin ct sin [1 + a cosh ~ sin 0t cos [1, z = R sinh ~ cos ~. r

T h e deSitter metric then takes the f o r m

ds~ = 2 (dr - dr + a sin 2 e d[1) dt - (dr - dr + a sin2e d[1) 2 - M 2 [ ( I + ~-~a2 sin 2 0 t ) - 1 dat: + sin 2 ~,dfl2]

M 2 = ( R 2 -I- a 2) sinh 2 r + a2 COS2 at.

/ (

(4)

(4a)

(5)

Following the scheme o f I, one can write d o w n the Kerr-metric in the deSitter background as

ds~ = ds~ - 2m~ (dt - dr + a sin 2 e dfl) 2, (6a) r 3 r

~ = R sinh ~ cosh ~-, m = constant.

If we transform our spheroidal polar coordinates to the conventional Boyer-Lindquist type coordinates, (6) is transformed to Kerr metric with cosmological constant as given by Demianski (1973). The explicit transformation equations are given in appendix (A).

Alternatively using a result recently obtained by Taub (1981) and the form o f Kerr metric in the background o f Einstein's universe as given in I one can verify that (6) is K e r r metric in the background ofdeSitter universe. The verification is given in appendix (B).

3. Anti-deSitter background

When the cosmological constant A is negative so that one can write A = -

3/R 2

the space-time represented by deSitter metric is known in literature as anti-deSitter space- time. A surprising result is that the following axially symmetric metric, conformal to the usual deSitter metric, represents anti-deSitter space-time

d r 2 _ r 2 dot 2 d s 2 --" r 2 c o s " 0t L ~ ~ - b2 J

- r 2 sin 2 0tdfl2]. (7)

It satisfies Rik = Agik with A = -

3/R 2,

b 2 being an undetermined constant. Now it is known (Hawking and Ellis 1973) that in a certain definite space-time region anti- deSitter space-time metric is conformal to Einstein universe metric. Thus one can transform (7) to the form

R2 F dx 2 _ dy 2 _ dz 2

(xdx + ydy +

zdz) 21

d s 2 =

~- L dr2

^- b 2 _ (x2 + y2 + z2) .j"

is)

One can now use the method o f transforming to spheroidal polar coordinates initiated in I and used in § 2 above to get the Kerr metric in the background (7). However we shall not write down the resulting metric here. Instead, we note a simple particular case o f (7) or (8) obtained by choosing the undetermined constant b -* oo. The background anti-deSitter metric (7) or (8) then simplifies to the following plane symmetric metric

R 2

ds 2 = ~-y [dt 2 - dx 2 - dy 2 - dz2]. (9)

The Kerr metric in this background is given by

ds 2 = ~ 2 ( d t - d r + a s i n 2 e d f l ) d t - l + r 2 + a 2 c o s 2 e

x (dr - dr + a sin 2 e dp) 2 - (r 2 + a 2 cos 2 e) (de 2 + sin 2 e dfl2)]

(10)

3

154

P C Vaidya

It can again be verified that for the metric (10)

Rik = Agik,

A = - 3/R 2, m = c o n s t a n t (10) gives a very simple metric for the Kerr-like gravitational field satisfying Rik = Agik with A a n o n - z e r o negative constant. T h a t A has to be n o n - z e r o is a c o n d i t i o n f o r the existence o f the b a c k g r o u n d metric o f anti-deSitter space-time given by (7) o r (9).

O n e can show that (10) satisfies Rik = Agik by the m e t h o d given in appendix (B).

4. deSitter space-time with twisting null rays In I it was shown that the M i n k o w s k i metric

ds 2 = dt 2 _ dx 2 _ d y 2 - dz 2

can be transformed in ellipsoidal c o o r d i n a t e s to the f o r m

ds 2 = 2 (du + a sin 2 ctdfl)dt - (du + a sin 2 ot dfl) 2 - (r 2 + a 2 cos 2 0 t ) d ~ 2 (1 l ) d ~ 2 = d~t 2 + sin 2 ~t dfl 2, r = t - u,

a n d that (11) f o r m s the b a c k g r o u n d metric for the K e r r field. O u r aim is to get an immediate generalization o f (11) which will take us f r o m the M i n k o w s k i b a c k g r o u n d to deSitter b a c k g r o u n d . Let us therefore begin with the Schwarzschild's exterior metric in deSitter b a c k g r o u n d

i.e.

the metric ( 1 ) a b o v e a n d i n t r o d u c e the retarded time u in place o f the c o o r d i n a t e t. T h e e q u a t i o n defining u is

( 1 2 m r 2 ) t3u t~u

r R 2 - & r + ~ - = 0 ' a solution o f which is

I( "Y'

u = t - 1 dr.

r

Using u as a time co-ordinate in place o f t one t r a n s f o r m s (1) to

ds2 = 2dudr + ( l 2m

r R-2 du2 - r 2 di)2" r 2 ) (12) But in o u r scheme as used in § 2 o n w a r d s we d o n o t use u in place o f t , but we use u in place o f r. A n d one can d o this because u is a null coordinate. We return to a time-like c o o r d i n a t e T by the substitution u = T - r and use (T, u, ~t, fl) as c o o r d i n a t e s

i.e.

replace r by T - u . Metric (12) then takes the f o r m

ds 2 = 2du T - 1 + - - r + ~ - du2 - r2 dL'12" (13) This is the Schwarzschild's exterior metric in deSitter b a c k g r o u n d .

T h e f o r m (11) o f the M i n k o w s k i metric (which is the b a c k g r o u n d metric o f K e r r solution) suggests an immediate generalization o f (13) to the following

2mr r 2 + y2 ,~

ds 2 = 2 (du + 0 sin e dfl) dt - 1 + ~ + - - ~ 5 - - ) (du + g sin e dfl) 2

r

-- H (r 2 +

y2)

dl-.~2, (14)

# = g(a), y = y(a), H = H (~),A = ~ 3 a n d d D 2 = da 2 + sin 2 ~, dfl 2.

It m a y be noted that if A = 0, with g = a sin ~,, y = - a cos a a n d H = 1 (14) gives us the K e r r metric. T o determine these functions when A ~: 0 we further note that the metric (14) is in the f o r m which we have termed a K e r r - N U T metric ( V a i d y a et al

1976) viz

ds 2 = 2 (du + g sin a dfl) d x - 2 L ( d u + a sin ~, dfl) 2 - M 2 df~ 2 (15) a n d so we can use the tetrad formalism developed in that p a p e r to w o r k o u t the physics o f metric (14). Using the tetrad

01 = du + # sin a dfl, 02 = M d a , 03 = M sin ~, dfl, 04 = d x - LO 1

(so that (15) takes the f o r m ds 2 = 20104 - (02) 2 - (03) 2) the tetrad c o m p o n e n t s R(ab) o f the Ricci tensor are also recorded there.

It can be verified that if one chooses

g d ~ = dy, H = f / ( - y ) , with 2 f = ( t ~ g / ~ ) + g c o t o ~ , (16) one finds that for the metric (14)

R(12) = R(13) = R ( 2 3 ) = R(24) = R ( 3 4 ) = R(44) = 0,

R~14) = A, R~22) = - A + [ 2 + ~ A y2 _ yG ] (r 2 + y 2 ) - 1 = R 33,

2y 3

g ~ , ) - 3 A [ ( a 2 / f ) + 2 Y ] (r2 + Y 2 ) - I ' A = ~ - "

In the a b o v e G is defined by

l 0 [ l O f \ - ] . Of 2 f G = o 2 ~-~ + ~ y y ~ y y ) J + z ~Ty - 2"

It is clear that if we c h o o s e

8~a = ~ A [ ( 0 2 / f ) + 2y] (r 2 + y 2 ) - , and

G y - 2 - 4 ~ A y2 = 0, (17)

we shall have R~k = AOlk - - 8 r C ~ i ~k, where ~ is a null vector defined by

~i d xl = du + 0 sin ~tdfl.

E q u a t i o n s (16) a n d (17) are the three e q u a t i o n s which determine the three u n k n o w n functions o f ~t in the metric (14).

I f A = 3 / R 2 = 0, (16) a n d (17) are satisfied by f = - y = + a cos 0t, O = a sin a, a being constant. T h e metric (14) then b e c o m e s the K e r r metric.

T h e other simple case is A 4: 0, m = 0. T h e n . r 2 q - y 2 " ~

ds 2 = 2(du + g sin a dfl) dt - l . ~ y - j ( d u + o s i n o c d f l ) 2

_ ( f / _ y) (r 2 + y2)dl)2 (18)

156

P C Vaidya

d a

with dy = gd~, 2 f = ~ + g cot ~t and f =

f(y)

satisfying

Gy - 2 - (4y2/R t)

= 0. (19)

One can interpret this simple solution as a rotating deSitter space-time because (i) it represents a universe devoid o f matter (ii) its metric (18) reduces to the usual deSitter metric when the rotation parameter g = 0. It has an additional feature that it is pervaded by a unidirectional flow o f null radiation which arises solely due to rotation.

As a matter o f fact if one solves (19) correctly upto the first power o f 1/R t

(i.e.

upto first power o f A), one can show that

2at(3 cos 20t - 1) 87ta =

Rt(r t + a t

cos t ~)'

s o t h a t

/2 M 2 a s i n 0 t d ~ t d f l = 0, M 2 =

( f / - y ) ( r 2 + y 2)

i.e.

the net outflow o f null radiation across the 2-space with metric M2(d0t 2 + sin e ~t dfl e) is zero. Thus there is no net loss o f energy due to this flowing radiation. The expanding nature o f 3-space and the rotation introduced in it, together, so to say, lead to a churning o f gravitational energy in deSitter space-time which flows out from a cone o f semiangle cos-1 (1/x/3) and with the axis coinciding with the axis o f rotation and returns through the rest o f the surface.

With this interpretation (18) becomes the metric o f deSitter space-time with rotating null rays and metric (14)

2m r r 2 -I- y2 '~

ds 2 = 2 (du + 0 sin ~t dfl) dt - 1 + r2-T-~y 2 + W ) (du + g sin ,y dfl) 2

r

_ ( f / _ y) (r t + y2) (dot2 + sin 2 r, dfl 2)

with d y = 0d~t a n d f =

f(y)

given by (19) becomes Kerr metric in the background o f this rotating deSitter universe. And this last metric is easily seen to be a simple generalization o f Schwarzschild's exterior metric in deSitter background.

A p p e n d i x A

Let (L O, ~p, -t) be the Boyer-Lindquist type coordinates, then the required transform- ation equations are

~- -- R tanh ~ , T = t + dr ( a 2 s i n 2 0 t ) -1/2 cos 0 = cos ~t 1 -~ ~-T

f R + 2mlz

(a2 + R2)tP =

R 2 f l - a t - a A dr,

where 2m/~ is defined by (6a) and

r r 3 r

A = (R 2 + a 2) sinh 2 ~ + a 2 - 2mR sin ~ cosh ~ -

Appendix B

Taub (1981) has shown that if a space-time 1~ has metric given by

g ~ = g ~ + 2H i. l~, (B.1)

where g~, is the metric for an arbitrary space V, H is a scalar field over Vand l~ is a null, geodetic and shear-free vector field in V, then l~ is also null geodetic and shear-free with respect to g~,. He has further shown that if the space-time Vwith the metric tensor g~v is conformal to space-time ~ with metric tensor ~ that is, if

g~, = exp (2~)g%, (s.2)

and if lu is null, geodetic and shear-free in V, it is also one in V.

Using (B.I) and (B.2) one can write

~ . , = exp (2a) (g~, +

2Hlfl,)

= exp (2a)O~, + 2Hol~l, (B.3)

with Ho = H exp (2a). Taub has also worked out expressions for the Ricci tensor R~ for V in terms of R~, of V and the scalars H, a and their derivatives.

One can see that our metric (6) has the metric tensor of the form given by (B.3) with exp (2~r) = cosh 2 1 + R-T sin2 ~ ,

g~, is the metric tensor obtained from

dso 2 = g . , d x " d x ~ of (4a) x I = r , x 2 = u , x 3 = f l , x 4 = t , r 3 r

Ho = - r a p = - m R sinh ~ c o s h ~ and

(a2)

i~ = 1 + ~-~ sin2~, ~., ~. dx" = d t - dr + a sin 2 ot dfl.

It may be noted that ~ is a geodetic, shear-free null congruence and since ot~ ~ = 0, l~

is also a similar congruence in V.

With these substitutions and following Taub's calculations one can verify that

References

-Ga~er B 1968 Phys. Leu. A26 399 Demianski M 1973 Acta Astron. 23 221 Frolov V P 1974 Teor. Mat. Fiz. 21 213

158 P C Vaidya

Hawking S W and Ellis G F R 1973 Laroe scale structure of space time (Cambridge: University press) p. 131 Taub A H 1981 Ann. Phys. 134 326

Vaidya P C, Patel L K and Bhatt P V 1976 Gen. Rel. Gray. 7 701 Vaidya P C 1977 Pramana g 512