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Higher-dimensional relativistic-fluid spheres

L. K. PATEL(1), (2), N. P. MEHTA(2) and S. D. MAHARAJ(1)

(1) Department of Mathematics and Applied Mathematics, University of Natal

Private Bag X10, Dalbridge 4014, South Africa

(2) Department of Mathematics, Gujarat University - Ahmedabad 380009, India (ricevuto il 22 Ottobre 1996; approvato il 26 Novembre 1996)

Summary. — We consider the hydrostatic equilibrium of relativistic-fluid spheres for a D-dimensional space-time. Three physically viable interior solutions of the Einstein field equations corresponding to perfect-fluid spheres in a D-dimensional space-time are obtained. When D 44, they reduce, respectively, to the Tolman IV solution, the Mehra solution and the Finch-Skea solution. The solutions are smoothly matched with the D-dimensional Schwarzschild exterior solution at the boundary

r 4 a of the fluid sphere. Some physical features and other related details of the

solutions are briefly discussed. A brief description of two other new solutions for higher-dimensional perfect-fluid spheres is also given.

PACS 04.20.Jb – Exact solutions.

1. – Introduction

Multi-dimensional space-time is now an active field of research in its attempt to unify gravitation with other forces of nature (Sherk and Schwarz [1]). In view of recent developments in superstring theory (Schwarz [2], Weinberg [3]), and ten-dimensional Yang-Mills supergravity in its field theory limit, it becomes much more important to study the theories in space-time of more than 4 dimensions.

The implications of theories in which the dimension of the space-time is greater than 4 have been discussed by several investigators. What is required from the viewpoint of physics is direct observational evidence supporting theories of higher dimensions. There are many authors who have studied the exact solutions of Einstein field equations in 4-dimensional space-time over the past eighty years (Kramer et

al. [4]). But the number of articles dealing with the higher-dimensional exact solutions

of Einstein field equations is relatively small.

Myers and Perry [5] have obtained the higher-dimensional generalizations of the Schwarzschild, Reissner-Nordström and Kerr metrics. Shen and Tan [6, 7] have presented Wyman solutions and Gonzales-Dias [8] solution in higher dimensions. Dianyan [9] has discussed Schwarzschild-de Sitter, Reissner-Nordström-de Sitter and Kerr-de Sitter metrics in higher dimensions. Krori et al. [10] have presented a

(2)

higher-dimensional generalization of the well-known Schwarzschild interior solution. Tikekar [11] has obtained D-dimensional generalization of a class of metrics for superdense stars derived by Vaidya and Tikekar [12]. Such solutions are believed to be of physical relevance possibly at the extremely early times before the universe underwent the compactification transitions. Cosmological models in higher dimensions prescribing material content in the form of a perfect fluid have been discussed by Chatterjee et al. [13, 14].

The main purpose of the present investigation is to derive three new physically significant exact higher-dimensional perfect-fluid solutions of Einstein field equations which can be smoothly matched with the higher-dimensional Schwarzschild exterior solution across the boundary of the fluid sphere. Two newer higher-dimensional sol-utions describing the interior fields of perfect-fluid spheres will also be briefly mentioned. We take the dimension of the space-time under consideration to be D 4n13. Therefore, n 4 1 gives the corresponding 4-dimensional solution.

2. – The metric and the field equations

The line element for a static spherically symmetric (n 1 3)-dimensional space-time can be expressed in the form

ds2

4 egdt2

2 eldr2

2 r2[ du2

11 sin2u1du221 sin2u1sin2u2du231 R 1

(1)

1sin2u1sin2u2Rsin2undu2n 11] ,

where l and g are functions of the radial coordinate r. The coordinates are named as

x0

4 t , x14 r , x24 u1, xj4 uj 21 ( j 43, 4, R, n12). The surviving components of

the Ricci tensor for the metric (1) can be obtained by a routine calculation. These are given by

.

`

`

/

`

`

´

R004 eg 2l

y

2 1 2

g

g 91 1 2g 8 2

h

1 1 4l 8 g82 (n 11) g8 2 r

z

, R114 1 2

g

g 91 1 2g 8 2

h

2 1 4 l 8 g82 (n 11) l8 2 r , R224 2 1 2re 2l(l 82g8)2ne2l(el 2 1 ) ,

Ri 12, i124 Ri 11, i11sin2

ui, i 41, 2, R, n .

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Here and in what follows a prime indicates differentiation with respect to r.

We assume that the matter content of the space-time is a perfect fluid whose energy-momentum tensor is given by

Tik4 (p 1 r) vivk2 pgik, vivi4 1 ,

(3)

where p, r and vi, respectively, denote the fluid pressure, the material density and the

(3)

The Einstein field equations for non-empty space-time are

Rik2

1

2Rgik4 28 pTik, (4)

where Tik is given by (3). We use comoving coordinates and, consequently, we have

avi

4 (e2gO2, 0 , 0 , R , 0 ) , vi4 (egO2, 0 , R , 0 ) .

(5)

The field equations (4) can be expressed in the form

Rik4 28 p

y

(p 1r) vivk2

1

(n 11)(r 2p) gik

z

. (6)

In view of the results (2) and (5), the field equations (6) reduce to the following system of three equations: g 91 1 2g 8 2 2 l 8 g8 2 2 g 8 r 2 nl 8 4 1 2 n r2 (e l 2 1 ) 4 0 , (7) 8 pr 4 (n 11) e 2ll 8 2 r 1 n(n 11) 2 r2 ( 1 2e 2l) , (8) 8 pp 4 (n 11) e 2lg 8 2 r 1 n(n 11) 2 r2 (e 2l 2 1 ) . (9)

When n 4 1, the space-time becomes 4-dimensional and the above equations (7)-(9) reduce to the usual governing equations for the hydrostatic equilibrium of a static spherically symmetric relativistic star in 4 dimensions.

Thus we have a system of three equations (7), (8) and (9) for the four unknown functions l, g, p and r. In order to obtain an explicit solution of the above system we have to place one additional restriction on the behaviour of these four functions.

The most natural way would be to choose an equation of state (i.e. a relation between p and r). This ensures that the resulting solution would be of physical interest. But such a procedure is almost impossible to carry through because it involves complicated non-linear expressions. For this reason, the usual procedure is to choose the auxiliary relation in such a way that the differential equation (7) becomes solvable. This procedure is not very satisfactory because all the solutions obtained in this way will not be physically significant.

In the next three sections, we shall introduce this additional assumption in three different ways and shall obtain exact solutions of the above system of eqs. (7)-(9). We shall also examine briefly the physical plausibility of these solutions.

Any physically acceptable interior solution must satisfy the following boundary conditions: i) At the surface of the sphere it should match with the (n 1 3)-dimensional

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Schwarzschild exterior solution given by the line element ds2 4

g

1 2 2 m rn

h

dt 2 2

g

1 2 2 m rn

h

21 dr2 2 2r2[ du2

11 sin2u1du221 R 1 R sin2u1sin2u2R 1 sin2undu2n 11] ,

where m is a constant and M 4 nm represents the total mass of the sphere.

ii) The pressure p must be finite at the centre r 4 0 and it must vanish at the surface of the sphere.

3. – Higher-dimensional Tolman IV solution

A number of new static solutions of Einstein field equations for perfect-fluid spheres in 4 dimensions had been obtained by Tolman [15]. These solutions are helpful in the study of stellar structure. He obtained the fourth solution by introducing the mathematical assumption

eg

g 8 O2r4constant .

(11)

As we are interested in deriving a higher-dimensional version of Tolman IV solution, we shall also continue to work with the assumption (11). Equation (11) can be easily integrated. The solution can be expressed in the form

eg

4 B2

g

1 1 r

2

A2

h

,

(12)

where A and B are arbitrary constants of integration. Substitution of erfrom (12) in (7) yields the differential equation

Z 82 2]n12n(r 2 OA2) 1 (n11)(r4OA4)( Z r ( 1 1r2 OA2)]n1 (n11)(r2 OA2)( 4 2 2 n( 1 1r2OA2) r]n1 (n11)(r2 OA2)( , (13)

where Z 4e2l. The general solution of the above linear differential equation is

e2l 4 (n 2r 2 OR2)( 1 1r2 OA2) n 1 (n11)(r2 OA2) , (14)

where R is a constant of integration. To ensure the regularity of the metric we assume that r

2

R2E n .

The physical parameters r and p for the above solution can be obtained from (8) and (9). They are given by

16 pr (n 11) 4 2(n 2r2 OR2) A2 [n 1 (n11)(r2 OA2) ]2 1 n 1 (n12)(A2OR2) 1 (n12)(r2OR2) A2 [n 1 (n11)(r2 OA2) ] (15)

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and 16 pp (n 11) 4 n 2 (n12)(r2 OR2) 2n(A2 OR2) A2[n 1 (n11)(r2OA2) ] . (16)

From the result (15) it is evident that r is always positive. The equation of state of the fluid can be obtained from (15) and (16) by eliminating r2. But the result is not very enlightening and we shall not give it here.

From the results (15) and (16) one can easily obtain 16 pA2 (n 11) dp dx 4 2 n[ 1 OR21 (n 1 1 ) OA2] [n 1 (n11)(xOA2) ]2 , (17) 16 pA2 (n 11) dr dx 4 (18) 4 2(nOR 2 )](n14)1

(

(n 11) x

)

OA2 ( 1

(

n(n 11)

)

OA2

)

](n 1 4 ) 1

(

(n 11) x

)

OA2 ( [n 1 (n11)(xOA2) ]3 and dp dr 4 n 1 (n11)(xOA2) n 141 (n11)(xOA2) , (19)

where x 4r2. From the result (19) it is clear that dp

dr is always less than 1 for any x. The results (17) and (18) indicate that dp

dx and dr

dx are negative. Consequently, p and

r are monotonically decreasing functions of r. They attain their maximum values at the

centre r 4 0.

Applying the boundary conditions i) and ii) of sect. 2 to the above solution, we find the constants A, B and m in terms of the boundary radius a. They are given by

.

`

/

`

´

B2 4 n 2a 2 OR2 n 1 (n11)(a2OA2) , A2 R2 4 1 2 (n 12) n a2 R2 , 2 m an 4 (a2 OA2)( 1 1a2 OR2) 1a2 OR2 n 1 (n11)(a2OA2) . (20)

Clearly, m is positive. It is clear from result (20) that, for a physically meaningful solution, we must have

0 E a 2 R2 G n n 12 . (21)

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The central values p0and r0of p and r are given by 8 pp04

g

n 11 2

h

g

1 A2 2 1 R2

h

, 8 pr04 (n 11)(n12) 2 n

g

1 A2 1 1 R2

h

. (22) For R2

D A2, r0 and p0 are positive. It is easy to check that r0F p0. The physical

requirement r02 3 p0F 0 gives 0 E a 2 R2 G n 1 12n . (23)

If the inequality (23) is satisfied, then it is easy to see that the inequality (21) is also satisfied. The inequality (23) gives a restriction on the possible size of the fluid sphere. The geometry of our higher-dimensional Tolman IV solution is described by the line element ds24 B2

g

1 1 r 2 A2

h

dt 2 2 [n 1 (n11)(r 2 OA2) ] (n 2r2OR2)( 1 1r2OA2)dr 2 2 (24) 2r2[ du2

11 sin2u1du221 R 1 sin2u1sin2u2Rsin2undu2n 11] ,

where the constants A and B are given by (20).

Two limiting cases of the above solution are worth mentioning:

Case i): Let R tend to infinity. In this case we have

.

`

/

`

´

eg 4 B2

g

1 1 R 2 A2

h

, e 2l 4 n( 1 1r 2 OA2) n 1 (n11)(r2 OA2) , 16 pr (n 11) 4 n(n 12)1n(n11)(r2 OA2) A2 [n 1 (n11)(r2 OA2) ]2 , 16 pp (n 11) 4 n A2 [n 1 (n11)(r2 OA2) ] . (25)

From the above result it is clear that p cannot be made zero at the boundary. Consequently, we cannot match the solution with the higher-dimensional Schwarzschild exterior solution at the boundary. But pressure and density are always positive and tend to zero when r tends to infinity. In this case we get a higher-dimensional generalization of the cosmological solution discussed by Thomas [16]: when n 4 1, the results (25) agree with those of Thomas. It is interesting to note that the density and pressure are always positive: r2pF0 and r is a decreasing function of r.

Case ii): Let us consider the case in which A tends to infinity. In this case we obtain e2l 4 1 2 r 2 nR2 , e g 4 1 , 16 pp 4 2 (n 11) R2 , 16 pr 4 (n 11)(n12) nR2 . (26)

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Here we have r 1

g

1 1 2

n

h

p 40. To get positive pressure, we have to introduce the

cosmological constant in the field equations. This case gives a higher-dimensional version of Einstein’s static universe. When n 4 1, we obtain the usual Einstein universe. Here also the solution is of cosmological nature and it cannot be matched with the higher-dimensional Schwarzschild exterior solution.

4. – Higher-dimensional analogue of Mehra solution

Mehra [17] has obtained an exact solution of Einstein’s field equations which describes the interior field of a gaseous-fluid sphere in 4 dimensions. He assumed that the density distribution ia given by

8 pr 48pr0

g

1 2

r2 a2

h

,

(27)

where r0 is the central constant density and a is the boundary radius of the sphere.

Obviously, the density at the boundary of the sphere is zero.

We take the density distribution r in a slightly more general form than (27). We assume that

8 pr 48pr0

g

1 2K

r2

a2

h

, K G1 ,

(28)

where K is a constant. When K 4 1, we get the Mehra result.

Substitution for r from (28) in the result (8) leads to a differential equation for the metric potential el

. On using x 4r

2

a2 as the independent variable, this differential

equation gives on integration

e2l 4 1 2 16 pr0a 2 (n 11)

k

x n 12 2 Kx2 n 14

l

. (29)

Substituting e2lfrom (29) into (7) we obtain the differential equation

y

1 2 16 pr0a 2 (n 11)

g

x (n 12) 2 Kx2 (n 14)

h

z

( 2 g 91g8 2 ) 2 (30) 22 g 8 r

y

1 2 16 pr0Ka2x2 (n 11)(n14)

z

1 64 pr0nKa2x (n 11)(n14) 4 0 . Note that in this section x stands for r

2

a2 . The substitution

egO24 y

(31)

reduces eq. (30) to the form 2 d 2 y dx2

y

1 2 16 pr0a2 (n 11)

g

x n 12 2 Kx2 n 14

h

z

1 (32) 1dy dx

y

16 pr0a2 (n 11)

g

2 xK n 14 2 1 n 12

h

z

1 8 pr0nKa2 (n 11)(n14)y 40 .

(8)

Again by substituting ez4 x 2 (n 14) 2 K(n 12) 1

y

x 2 2 (n 14) x K(n 12) 1 (n 11)(n14) 16 pr0a2K

z

1 O2 (33)

in eq. (32) we find that eq. (32) reduces to the simple form d2y

dz2 1

n

4 y 40 . (34)

The solution of eq. (34) is

egO24 y 4 c1cos kn z 2 1 c2sin kn z 2 , (35)

where c1and c2are constants of integration.

The pressure p for the above solution can be determined from (9). It is given by 16 pp n 11 4 2 16 pr0 (n 11)

g

1 n 12 2 Kx n 14

h

1 (36) 14 kn a

y

4 pr0K (n11)(n14)

z

1 O2

k

1216 pr0a 2 n11

g

x n12 2 Kx2 n14

hl

1 O2

u

c22c1tan

(

knO2

)

z c11c2tan

(

knO2

)

z

v

. The boundary conditions i) and ii) of sect. 2, when applied to the above solution, lead to m 4 8 pr0a n 12 (n 11)

k

1 n 12 2 K n 14

l

, (37) c24

y

1 2 16 pr0a2]n 1 4 2 K(n 1 2 )( (n 11)(n12)(n14)

z

1 O2 sin kn 2 z11 (38) 1 a n 12

y

4 rp0n K(n 11)(n14)

z

1 O2 ]n 1 4 2 K(n 1 2 )( cos kn 2 z1 and c14

y

1 2 16 pr0a2]n 1 4 2 K(n 1 2 )( (n 11)(n12)(n14)

z

1 O2 cos kn 2 z12 (39) 2 a n 12

y

4 pr0n K(n 11)(n14)

z

1 O2 ]n 1 4 2 K(n 1 2 )( sin kn 2 z1, where z14 log

y

2 K(n 12)2 (n14) 2 K(n 12) 1

{

1 2 (n 14) K(n 12) 1 (n 11)(N14) 16 pr0a2K

}

1 O2

z

. (40)

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Therefore, the metric potentials eg and el for the above interior solution are given by egO24

y

1 2 16 pr0a 2 ]n 1 4 2 K(n 1 2 )( (n 11)(n12)(n14)

z

1 O2 cos kn 2 (z12 z) 2 (41) 2 a (n 12)

y

4 pr0n K(n 11)(n14)

z

1 O2 ](n 1 4 ) 2 K(n 1 2 )( sin kn 2 (z12 z) and (29), respectively, where z and z1are given by (33) and (40), respectively.

From eqs. (36) and (40) it is clear that the interior solution is real only if

a2G K(n 11)(n12) 2 4 pr0(n 14) , m G 4 K(n 12) a n ]n 1 4 2 K(n 1 2 )( 2(n 14)2 . (42)

The relations (42) give an upper limit of the possible size for given density and of the mass for given radius.

When n 4 1, we get the 4-dimensional fluid sphere solution which is a generalization of the Mehra solution. Further, if K 4 1, we recover Mehra solution. For the sake of brevity the other details of the above solution are omitted.

5. – Higher-dimensional Finch-Skea solution

Finch and Skea [18] have obtained an exact realistic interior solution for a perfect-fluid sphere in 4 dimensions which can be used as an exact relativistic model of a superdense star. The aim of this section is to give a D-dimensional generalization of this solution.

For this section we shall use the notations

Cr2

4 x , e2l

4 Z , egO24 y ,

(43)

where C is an arbitrary constant. In these notations the system of equations (7)-(9) reduces to 4 x2Zd 2 y dx2 1 2 x 2 dy dx dZ dx 1 n

k

1 2Z1x dZ dx

l

y 40 , (44) 8 pr C(n 11) 4 n( 1 2Z) 2 x 2 dZ dx , (45) and 8 pp C(n 11) 4 2 Z y dy dx 1 n(Z 21) 2 x . (46)

Finch and Skea [18] have introduced the assumption

Z 4 (11x)21

(47)

(10)

their solution we shall also continue to work with the above assumption for the function

Z. Using Z given by (47), the differential equation (44) can be integrated in terms of

elementary functions. Its solution can be expressed in the form

y 4egO24

(

c 22 c1v kn) cos (kn v) 1 (c11 c2v kn) sin (kn v) , (48) where v2 4 1 1 x (49)

and c1and c2are arbitrary constants of integration.

Using the results (47) and (48) in eqs. (8) and (9) one can find the pressure p and the density r. They are given by

8 pr C(n 11) 4 2 1nv2 2 v4 (50) and 8 pp C(n 11) 4 n 2 v2

y

1 1bv kn 1

(

b 2v kn

)

tan

(

kn v

)

1 2bv kn 1

(

b 1v kn

)

tan

(

kn v

)

z

, (51) where b 4 c1 c2 . (52)

The physical requirement r D 0 demands that the constant C should be positive. The equation of state of the fluid may be obtained in the form p 4p(r) by substituting

v24 1

32 pr[Cn(n 11)1 ]C

2

n2(n 11)21 128 prC(n 1 1 )(1 O2]

into eq. (51). The equation of state has a complicated form but the above analysis shows that it can be expressed in terms of elementary functions.

Now applying the boundary conditions i) and ii) of sect. 2 to the above solution, we can determine the constants m, b and c2. They are given by

m 4C2nO2(v2

a2 1 )((n 12)O2)O2 va2, va24 1 1 Ca2,

(53)

b 4 [kn vatan (kn va) 21]O[vakn 1 tan (vakn) ]

(54) and

c24 6[va( 1 2bvakn) cos (kn va) 1va(b 1vakn) sin (kn va) ]21.

(11)

In the 4-dimensional case, i.e. (n 4 1), the above three relations reduce to

.

/

´

mkC 4 (va22 1 )3 O2O2 va2, b 4 [va tan va2 1 ] O[va1 tan va] ,

c24 6 1 2 v2 a (vacos va1 sin va) . (56)

The relations (56) agree with those obtained by Finch and Skea [18]. The final form of the metric of our interior solution is

ds24 c22[ ( 1 2bv kn) cos (kn v)1 (b1v kn) sin (kn v) ]2dt22

(57)

2( 1 1 Cr2) dr2

2 r2[ du2

11 sin2u1du221 R 1 sin2u1sin2u2Rsin2undu2n 11] ,

where v2

411Cr2 and the constants b and c2 are given by (54) and (55), respectively.

With the help of the results (50) and (51) one can easily obtain 16 p (n 11) dr dv 4 22 C (nv21 4 ) v5 , (58) 16 pv3 n(n 11) dp dv [ (b 1v kn) tan (kn v)2 (b kn v21) ] 2 4 (59)

4 22 C [ 11b tan (kn v) ] [ bnv2tan (kn v)1b tan (kn v) v kn tan (kn v)2kn bv111nv2]

and dp dr (nv2 1 4 ) nv2 [ (b 1v kn) tan (kn v)112bv kn] 2 4 (60) 4 ] 1 1 b tan (kn v)([ tan (kn v)( bnv21 b 1 v kn) 2 bv kn 1 nv21 1 ] . For a physically significant interior solution we must have dp

dr E 1 . From the result (58) it is clear that dr

dv E 0 and, consequently, the density r is a decreasing function of r. It attains its maximum value at the centre.

The central values r0 and p0 of the density and pressure are given by

16 pr0 (n 11) 4 C(n 1 2 ) (61) and 16 pp0 (n 11) 4 C [ b kn 111 (b2kn) tan kn] [ 1 2b kn 1 (b1kn) tan kn] . (62)

The physical requirement p0F 0 gives the inequality

kntan kn 21

kn 1tan kn E b . (63)

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If kn(n 11)2tankn F0, then p0G r0 requires

b E 1 1 (n11) kn tan kn

kn (n 11)2tan kn . (64)

If kn(n 11)2tankn E0, then p0G r0 requires

1 1 (n11) kn tan kn kn (n 11)2tan kn E b . (65)

Therefore, if kn(n 11)2tankn D0, then kntan kn 21

kn 1tan kn E b E

1 1 (n11) kn tan kn kn (n 11)2tan kn (66)

or, if kn(n 11)2tankn E0, then

b Dmax

{

kn tan kn 21

kn 1tan kn

, 1 1 (n11) kn tan kn kn (n 11)2tan kn

}

(67)

If ra denotes the value of r at the boundary r 4 a, we have

8 pra4 C (n 11)(nv2 a1 2 ) 2 va4 . (68)

We shall denote the ratio ra

r0

by m. One can easily check that m is less than 1. Also we have va24 n 1

k

n2 1 8 m(n 1 2 ) 2 m(n 12) . (69)

For a given n and m, one can obtain b,

g

dp dr

h

0

,

g

dp dr

h

a

and va2from (54), (60) with r 4 0,

(60) with r 4 a and (69), respectively. For given n, the limits for the constant b can be obtained from (66) or (67). For a 5-dimensional case (n 4 2), we have verified numerically that our solution remains physically valid in the range 0 .5 GmG0.9. The numerical estimates of various quantities are given in table I.

TABLEI. m v2 a va b ( dpOdr)0 ( dpOdr)a 0.5 0.6 0.7 0.8 0.9 1.618034 1.4201329 1.2022971 1.1625919 1.0732123 1.2720196 1.1916933 1.0964931 1.0782356 1.0359596 3.4875119 2.2332014 1.4842359 1.3820004 1.176082 0.7672567 0.5898206 0.4274915 0.4198910 0.3709081 0.7733601 0.5024520 0.4537703 0.5707571 0.5366061

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6. – Concluding remarks

In the previous three sections we have discussed three higher-dimensional generalizations of three well-known solutions describing interior fields of perfect-fluid spheres. Many other 4-dimensional fluid sphere solutions can also be generalized to higher dimensions. We briefy discuss two of them.

First, let us consider the higher-dimensional version of Tolman VI solution. Following Tolman [15] we assume that

e2l

4 const 4 ( 2 2 k2)21,

(70)

where k is a constant.

Using e2l given by (70) in eq. (7), we can perform the integration for the metric

function eg. The solution can be expressed in the form

egO2

4 ArM2 BrN, M 412

k

1 1n(k22 1 ) , N 411

k

1 1n(k22 1 ) , (71)

where A and B are constants of integration. The physical parameters r and p are given by

.

`

/

`

´

16 pr 4 n(n 11) r2

g

1 2k2 2 2k2

h

, 16 pp 4 (n 11) r2( 2 2k2)

y

A

]

1 2

k

1 1n(k2 2 1 )

(

22 B

]

1 1

k

1 1n(k2 2 1 )

(

2 A 2Br2k1 1n(k22 1 )

z

. (72)

For real square root we must have 1 2n1 E k2. The positivity of density requires

k2 E 1 . Thus we get 1 2 1 n E k 2 E 1 .

The boundary conditions i) and ii) of sect. 2 at the boundary r 4 a for the above solution determine the constants m, A and B as

.

`

/

`

´

m 4 a n 2

g

1 2k2 2 2k2

h

, B 4

]

1 2

k

1 1n(k2 2 1 )

(

2 4( 2 2k2)1 O2 ] 1 1 n(k22 1 )(1 O2a1 1k1 1n(k22 1 ) , A 4

]

1 1

k

1 1n(k 2 2 1 )

(

2 4( 2 2k2)1 O2] 1 1 n(k22 1 )(1 O2a1 2k1 1n(k22 1 ) . (73)

It is clear that the density and pressure are infinite at the centre. This is an undesirable feature of the solution. When n 4 1, we recover the usual Tolman VI solution.

Secondly, let us consider the higher-dimensional analogue of Adler [19, 20] solution. Adler’s solution is an important solution which can be used as an exact relativistic model of a neutron star. Following him, let us assume the form of the metric function eg

as

egO24 A 1 Br2, (74)

(14)

where A and B are arbitrary constants. Using egO2 given by (74) in eq. (7) we get a

first-order differential equation for e2l. Its solution is given by

e2l

4 1 1 Dr2]nA 1 (n 1 2 ) Br2(22 O(n 1 2 ), (75)

where D is a constant of integration. The physical parameters p and r for the above solution are given by

(76) 16 pp(A 1Br2

)]nA1 (n12) Br2

(2 O(n12)4

4 4 B(n 1 1 )]nA 1 (n 1 2 ) Br2(2 O(n12)1 D(n 1 1 )]nA 1 (n 1 4 ) Br2( , (77) 16 pr]nA1 (n12) Br2

)((n 14)O(n12)4 2n(n 1 1 ) D]A(n 1 2 ) 1 (n 1 4 ) Br2

( . The positivity of the central density r0implies that the constant D should be negative.

The boundary conditions i) and ii) of sect. 2 at r 4 a for the above solution determine the constants A, B and D in terms of mass parameter m. They are given by

.

`

/

`

´

2 e a2]nA 1 (n 1 2 ) Ba 2 (2 O(n12)4 2D , A 4 2 2 (n14) e 2k1 22e , B 4 ne 2 a2 k1 22e , e 4 m an . (78)

For the sake of brevity we omit the other details of the above solution.

There are few exact solutions to Einstein field equations for higher-dimensional fluid spheres. Here we have made an attempt to obtain more exact solutions for higher-dimensional fluid spheres so that our understanding of higher-dimensional physics may be improved.

* * *

LKP would like to thank the Department of Mathematics and Applied Mathematics, University of Natal (South Africa) for hospitality and Gujarat University (India) for granting leave of absence during the period of this work. NPM wishes to thank the Government of Gujarat for financial support.

R E F E R E N C E S

[1] SCHERK J. and SCHWARZJ. H., Phys. Lett. B, 82 (1979) 60. [2] SCHWARZ J. H., Superstrings (World Scientific, Singapore) 1985.

[3] WEINBERG S., Strings and Superstrings (World Scientific, Singapore) 1986.

[4] KRAMER D., STEPHANI H., HERLT E. and MACCALLUM M. A. H., Exact Solutions of

Einstein’s Field Equations (Cambridge University Press, Cambridge) 1980.

[5] MYERS R. C. and PERRY M. T., Ann. Phys. (N.Y.), 172 (1986) 304. [6] SHEN Y. G. and TAN Z. Q., Phys. Lett. A, 137 (1989) 96.

[7] SHEN Y. G. and TAN Z. Q., Phys. Lett. A, 142 (1989) 341. [8] GONZALES-DIAZ P. F., Lett. Nuovo Cimento, 32 (1981) 161. [9] DIANYAN X., Class. Quantum. Grav., 5 (1988) 871.

(15)

[11] TIKEKAR R., Indian Math. Soc., 61 (1995) 37.

[12] VAIDYA P. C. and TIKEKAR R., J. Astrophys. Astron., 3 (1982) 325. [13] CHATTERJEE S., BANERJEE A. and BHUI B., Phys. Lett. A, 149 (1990) 91. [14] CHATTERJEE S. and BHUI B., Mon. Not. R. Astron. Soc., 247 (1990) 57. [15] TOLMAN R. C., Phys. Rev., 55 (1939) 365.

[16] THOMAS V. O., Master of Philosophy Dissertation, Sardar Patel University, India (1992). [17] MEHRAA. L., J. Austr. Math. Soc., 6 (1966) 153.

[18] FINCH M. R. and SKEA J. E. F., Class. Quantum. Grav., 6 (1989) 467. [19] ADLER R., J. Math. Phys., 15 (1974) 727.

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