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 ))
