A N I S O T R O P I C S P H E R E S IN G E N E R A L R E L A T I V I T Y

(Letter to the Editor)

HECTOR RAGO

Departamento de Fisica, Faeultad de Ciencias, Universidad de Los Andes, Venezuela

and

International Center for Theoretical Physics, Trieste, Italy

(Received 21 March, 1991)

Abstract. A prescription originally conceived for perfect fluids is extended to the case of anisotropic pressures. The method is used to obtain exact analytical solutions of the Einstein equations for spherically- symmetric self-gravitating distribution of anisotropic matter. The solutions are matched to the Schwarzschild exterior metric.

1. Introduction

Recently there has been increasing interest in the interior solutions of Einstein field equations corresponding to fluid distributions with anisotropic pressures (Bowers and Liang, 1974; Letelier, 1980; Bayin, 1982; Cosenza et al., 1981; Herrera and Ponce de Leon, 1985; Ponce de Leon, 1987; Maharaj and Maartens, 1989). These solutions are of interest in relativistic astrophysics, as models of massive compact objects. Indeed, theoretical progress in the physics of ultradense matter suggests that various physical phenomena may induce anisotropy, at least in certain density range (Ruderman, 1972; Canuto, 1973).

In this paper we present a procedure to obtain such solutions. The method is a generalization of one originally developed to find the general static solution for isotropic fluid spheres (cf. Berger et aI., 19878), and later applied to the charged perfect fluid case (cf. Patifio and Rago, 1989) to spheres with slow differential rotation (Diaz and Pullin, t988), and to certain types of time-dependent metrics (cf. Gaete and Hojman, 1990).

The approach in this paper is based on the introduction of two arbitrary functions: the anisotropic function which measures the degree of anisotropy and a generating function. These functions determine the relevant physical variables as well as the metrical coefficients.

This paper is organized as follows. In Section 2 we present the field equations and the precriptions used to obtain the solutions. In Section 3 the junction conditions and some examples illustrating the method are given.

2. The Generating Function

We consider a static equilibrium distribution of anisotropic matter which is spherically- symmetric, so the energy-momentum tensor is T~ = diag(p, - p , - p j_, - p • Here p

Astrophysics and Space Science 183: 333-338, 1991. @ 1991 Kluwer Academic Publishers. Printed in Belgium.

334 H. RAGO

is the energy density, p the radial pressure, and p• the tangential pressure. We adopt standard Schwarzschild coordinates (t, r, 0, q~) with respect to which the line element assumes the form

ds z = eV dt 2 - e ~ dr ~ - r 2 (dO s + sin20d~p2), (1)

where v and 2 are unknown functions of the radial coordinate r. The resulting Einstein equations are

8~p e - Z ( ~ 2 5 ) 1 = - - + rS , (2)

8~p = e + r 2 , (3)

e - z ( v '2 v ' - 2 ' v'2"~

8 p• +--2 + r 2 ) (4)

where a prime denotes differentiation with respect to r. If we use Equations (3) and (4), or equivalently the conservation equation T;~# = 0, we obtain the equation of hydrostatic equilibrium

_ v' 2 dp (p +p) + - (p• - p ) . (5) dr 2- r

Equation (2) can be formally integrated to give

e - a = 1 - 2m(r)/r , (6)

where the mass function re(r) defined as

re(r) = f 4~pr 2 dr (7) . I

0

is the mass inside a sphere of radius r as seen by a distant observer. Substitution of Equations (5) and (6) into Equation (3) gives

i l , , l -4(p~ up__) 2p' + (8) 8zcp + 1/r 2 = (1 - zm/r ) Lr2(P § r(p + p ) ~ "

Let us consider now a dimensionless function g(r) defined by

1 - 2m(r) /r g(r) = (9)

8zcpr 2 + 1

In terms of g, Equation (8) can be rewritten as

- - - ( 1 - g + c o ) ( 1 - g - r g ' ) ( 1 0 ) 8~p' +(1 3g rg ' ) (1 g + o ) ) 8~p+ rg(1 + g - co) r3g(1 + g - co)

ANISOTROPIC SPHERES IN GENERAL RELATIVITY 335

where we introduced the anisotropic function

4 (p - p • co(r) - g ( r ) . (11)

(p + p)

I f co = 0, the isotropic case is recovered. It should be stressed that for given g(r) and og(r), the physical variables p, p, and p • as well as the metric coefficients e ~ and e v can

be obtained as known functions of r. Indeed, Equation (10) can be integrated to give

8 ~ p ( r ) = e x p [ - f B d r ] [ p o + f C e x p ( f B d r ) d r ] , (12)

where Po is an integration constant and B(r) and C(r) are

8 ( 0 = (1 - 3 g - rg ' ) (1 - g + co)

rg(1 + g - co)

C(r) = - (1 - g + c o ) ( 1 - g - r g ' )

r3g(1 + g - co)

The energy density is obtained from Equation (7) and the definition of g. A short calculation gives

1 ( , ) 8 7 r p = ( 1 - g ) ~ - 8 r t ( 3 p + p ' r ) g - 8np + ~ rg . (13)

Once p(r) and p are obtained, the tangential pressure can be calculated by means of Equation (11) to yield

CO P• = - - - ( p + P ) + p . (14)

4g

Finally the metric coefficients can be obtained from Equations (5), (6), and (9). The final expressions are

e -x = g(8rcpr 2 + 1) (15)

and

e V = - - exp - - dr , (16) r rg

3. T h e S o l u t i o n s

The function g must satisfy some general requirements if the associated solution is to be physically meaningful. I f we assume a non-divergent pressure at the origin, the existence of a local Minkowskian space e ~ ~ 1, via Equation (15), that g(r) ~ 1 as r --, 0.

336 H. RAGO

In fact, the simplest choice g = 1 gives,us Minkowski flat space-time. One can verify

that g s ( r ) = 1 - 2 M / r with Po = 0 in Equation (12) corresponds to Schwarzschild

vacuum solution, irrespectively of the choice of the anisotropic function. If the interior

solution is to be matched smoothly onto the Schwarzschild metric at the surface r = r o we must demand continuity of the generating function at r o

2M g(r) lr=o = gs ( ro ) = 1 - - - (17)

r0

We observe that there is no junction condition imposed to the anisotropic function co.

Let us consider a less trivial case defining

g(r) = 1 + ar e (18)

and

co(r) = - ar 2 , (19)

where a is a constant. This choice should led to a physically reasonable model, since

function g(r) ,-~ 1 as r ,-~ 0. The value of the constant a is fixed by the junction con-

dition (17), giving

2 M a - (20)

r o

Equation (12) gives a vanishing radial pressure

p(r ) = 0 ; (21)

while the energy density is found to be

M D - 4 3 ' (22)

5z~r6

where we used Equations (13) and (19). The tangential pressures which holds this

homogeneous sphere is calculated from Equation (14) to be

p ( ar2 ~

P J- = 4 \ l - ~ a r 2 , / " (23)

Finally, the metric coefficients are

e - ~ = 1 - 837rr2p, (24)

A 2 e v = . ( 2 5 )

1 2M \2/2 r2)

ANISOTROPIC SPHERES IN GENERAL RELATIVITY 337

Configurations supported by tangential pressures were considered in another context by Lemaitre (t933). We note that Equation (23) implies that r o > 2M, so that the boundary is outside the gravitational radius of the distribution.

As a second example consider the choice

g(r) = b , (26)

co(r) = constant. (27)

If we feed back these values into Equations (12)-(14) we get for the physical variables, the following expressions,

V 8rcp = po r - D + - - , (28)

t.2

1 8 ~ p = ( 1 - b - b V ) = + p o b ( D - 3)r - D ,

r ~ (29)

8rip+ : I V - c o (1+ V ) ( 1 - b ) ] ~ + I 1 - co ] P ; 4b ~ (1 + bD - 3b) ; (30)

and the metrical coefficients are

e v = A2r(1 - b)/b, (31)

e - ~ P o b r - • + b V = -- + b; (32) r 2

where

D = (1 - 3 b ) ( 1 - b + co)

b(1 + b - co)

V = ( b - 1 ) ( 1 - b + c o )

b2 - 6b - bco + co + 1

This solution is an anisotropic generalization of Tolman V solution (Tolman, 1939) to which it reduce if co = 0. Note a slight change in notation, his n corresponding to (1 - b)/2b.

A c k n o w l e d g e m e n t s

The author is indebted to Professor Abdus Salam for hospitality at the Inter- national Centre for Theoretical Physics, Trieste, Italy. Also the author wishes to thank Marinela Silva B. for stimulating discussions. This work was partially supported by Consejo de Desarrollo Cientifico y Humanistico de la Universidad de los Andes (C.D.C.H.T.U.L.A.), Venezuela, under project C-379-89-05.

338 H. RAGO

References

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