A metric which represents a sphere of constant uniform density comprising electrically counterpoised dust

Abstract

Following the authors who have worked on this problem such Bonnor et.al 1•2 , Wickramasuriya3 and we write the metric which represents a sphere of constant density p = -1-, with suitable units, as ds2 = 47Z" (e(: ))2 c2 dt2 - ( e(r )Y ( dr2 + r2 dQ 2) ds2 = ( 1 B)' c'dT' - ( D + !)' (dR' + R2dQ') D+-R O��r��a A<R where dQ2 = (dB2 +sin2 BdrjJ2), B(r) is the Emden function satisfying the Emden equation4 with n = 3 . Since the metric has to be Lorentzian at infinity, we can take D = 1 . However, there is an important difference between the above authors and us as they had taken the same coordinate r in both regions, and as a result A = a. In general these coordinates do not need to be the same. In this particular case the coefficients of dQ2 are not of the same form in the above two metrics and that forces us to take two different coordinates rand R. In our approach r=a in the matter-filled region corresponds to R = A in the region without matter. Applying the boundary conditions at r = a or R = A , we have, B1(a )cdt = 1 (I +��) cdT => .!!! = e(a) dT (1+ ��) (i) -2 ( ) -2 ( B (e(a) )3 B' a cdt = ) ( B )3 -7 cdT 1+-A => _dt = _-_B--'(,e(-'-a��-- )Y---=----�(ii) dT A'B'(a{l + ��)' (1+ B => dr A ) -=-dR --B(a) _____ (iii) From (i) and (ii), we have e(a1 = - B (B(a )Y 3 (t+ A) A'll'(a{l+ ��) (1+ B ) From (iv), ( ) = !!_ (vi) Ba A __ (v) Using equation (vi) in equation (v), we have B �� -A'(:: } '(a)�� -a2ll'(a ) . Substituting the value of B in equation (iv), B(a )a = ( 1 + ��) A = A+ B =A- a2B'(a) =>A= aB(a)+ a2B'(a) . Then the metric becomes ds2 = 1 c2 dt2 - (e( )Y (dr2 + r2 dQ2) (e(r )Y r dsz = 1 cz dT z - (1- (a2B '(a))J 2 ( dRz + R z dQ z ) (t _(a'��(a))J' R where A=(ae(a )+ a2B'(a ))