ARS 2008
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Item On the Schwarzschild singularity(University of Kelaniya, 2008) Rajapaksha, R.L.R.A.S.; de Silva, N.The Schwarzschild ABSTRACT metric d<2�� (1- 2; }'dt' (I-�� t'-r' (d02+sin2 Od��' ) appears to behave badly near r = 2m, where gtt becomes zero, and grr tends to infinity1• There is a pathology in the line element that is due to a pathology in the space-time geometry itself. The womsome region of Schwarzschild metric, r = 2m , IS called the "event horizon". It is also called the "Schwarzschild singularity"1• There are many coordinate systems that have been found to overcome the Schwarzschild singularity1,2. By using the Schwarzschild metric in Schwarzschild coordinates, m Eddington-Filkelstein coordinates and m Kruslal-Szekeres coordinates, we have obtained some expressions for geodesics to check the behavior of a test particle at r = 2m , and in the two regions, the region outside r = 2m and the region inside r = 2m . W h h h · 11 h d . · · · ak dr e ave s own t at m. a t e coor mate systems It IS consistent to t e - < 0 ds when r > 2m and dr > 0 when r < 2m. The coefficient of dr2 becomes negative ds when r < 2m , making r a time like coordinate in that region. Thus r has to increase in this region. Further I: I becomes greater than c , the speed of light when r = 2mk , where k is a constant that depends on the initial condition, in the case of Schwarzschild coordinates and Eddington-Finkelstein coordinates, and there is jump at 2 . dr r = m, In - ds from -cl to cl , where l is a constant. These results suggest that once the particle crosses the event horizon at r = 2m it tends to remain there as dr > 0 , when r < 2m in all the three coordinate systems. ds A transformation of coordinate does not change this fact and we may suggest that the particle does not cross the event horizon, making it more than a mere coordinate singularity. The fact that I: I becomes greater than c m the neighborhood of r = 2m at least m two coordinate systems also suggest that the particle 1s changed physically around r = 2m . Hence we may say that the singularity at r = 2m is a physical singularity and not merely a coordinate singularity.Item A metric which represents a sphere of constant uniform density comprising electrically counterpoised dust(University of Kelaniya, 2008) Wimaladharma, N.A.S.N.; de Silva, N.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 .!!! = 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 ))Item A general relativistic solution for the space time generated by a spherical shell with constant uniform density(University of Kelaniya, 2008) Wimaladharma, N.A.S.N.; de Silva, N.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 b. where D and G are constants. For the region 0 < 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 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