A general relativistic solution for the space time generated by a spherical shell with constant uniform density

Abstract

In this paper we present a general relativistic solution for the space time generated by a spherical shell of uniform density. The Einstein's field equations are solved for a distribution of matter in the form of a spherical shell with inner radius a and outer radius b and with uniform constant density p . We first consider the region which contains matter (a < r < b ). As the metric has to be spherically symmetric we take the metric in the form ds2 =ev c2dt2 -eA.dr2 -r2d0.2, where d0.2 = (dB2 +sin2B drjJ2 ), A and v are functions of r as in Adler, Bazin and Schiffer1 where the space time metric for a spherically symmetric distribution of matter in the form of sphere of uniform density has been worked out. Solving the field equations, we o btain eA. = 1 ( r2 1 EJ --+ R2 -r and 2 Here R 2 = �� , where c and K are the velocity of the light and the gravitational 87rKp constant respectively and A , B and E are constants to be determined. Let the metric for the matter free regions be ds2 =ev c2dt2 -eA.dr2 -r2d0.2, where as before from spherical symmetry A and v are functions of r . Solving the field equations, we o btain, e" and e'' in the form e' �� (I : 7) and e" �� n(l + ��), for the regions 0 < r <a, andr >b. where D and G are constants. For the region 0 < r <a, whenr = 0, the metric should be regular. So G = 0. Hence the metric for the region 0 < r <a is ds2 = Dc2dt2 - dr2 -r2d0.2• For the region r > b, the metric should be Lorentzian at in finity. So D = 1. Hence the metric for the exterior matter free region is ds' = ( 1 + ��}'dt'- (I +l��r 2 -r2dn2 • Then we can write the metric for the space-time as ds2 = D c2 dt2 - dr2 -r2dQ2 , whenO < r <a, 2 Applying the boundary conditions at r = a and r = b , we have when a �� r �� b , and whenr> b. - !! - - (b3 - a3) E- 2 G R - R 2 ' (i ) __ (ii) 157 where r 2(a3 - r3 + rR2 Y ( - 9a 6 J;- 3a3rYz R2 + 2rh R 4 ) f Yz dr = --------,-- Yz.,--------'------------'----- (1 - C + _a_3 -) 2 r% ( a3 - r3 + rR2 ) 2 (-27a9 + 27a6r3 - 2 7a6rR2 + 4a3 R6 - 4r3 R6 + 4rR6) R2 R2 r rR2 tP -J; a3 + r3 (-I + :: ) ] F(f/Jim)= fV - msin2 e) dB tP ( )Yz ff ff and E(f/J I m)= f 1 -m sin 2 e dB , -- < fjJ <- 0 2 2 0 are the Elliptic integrals of first kind and second kind respectively, where fjJ =Arcsin (- ;+r3) (r3 - r2) and Here r1 =The first root of ( - 1 + R2 r2 + a3r3 )r2 =The second root of ( - 1+R2 r2 +a3r3. ). r3 =The third root of ( - 1+R2 r2 +a3r3). Furthermore we know that the potential fjJ of a shell of inner radius a1 and outer radius b1 and constant uniform density in Newtonian gravitation is given by fjJ = 2ffKp(a12 -b12) ,;. 2ffKp 2 4ffKp 3 2 b 2 'f'=-3-r +�� a1 - ffKP 1 fjJ = _ 4ffKp (a13 - b13) Using the fact that g00 = ( 1 + ����). (for example in Adler, Bazin and Schiffer1 )we find that the constants a,, b1 in Newtonian gravitation and D can be written in the form __ (iii) __ (iv) D =(I+ 3(a�;,b/ l} Hence the final form of the metric is O<r<a 2 b <r. where A,B,a1 and b1 are given by the equations (i), (ii), (iii) and (iv) respectively.