Browsing by Author "Wimaladharma, N.A.S.N."

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    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
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    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 ))
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    A metric which represents a sphere of constant uniform density comprising electrically counterpoised dust,
    (Faculty of Graduate Studies, University of Kelaniya, 2008) Wimaladharma, N.A.S.N.; De Silva, Nalin.
    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 .!!! = 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) 154 Proceedi11gs of the A1111Ua/ Research Symposium 2008- Faculty of Graduate Studies U11iversity of Kela11iya 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 ))
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    The red shifts of pulses of light which are emitted at a point on the surface of a sphere and at a point inside of the sphere comprising electrically counterpoised dust with constant uniform density as observed by an observer in a large distance away in the exterior region
    (University of Kelaniya, 2013) Wimaladharma, N.A.S.N.; de Silva, N.; Hewageegana, P.S.
    A sphere, comprising a special kind of matter, with electrically counterpoised dust in which all the elastic forces have been cancelled out has been considered. A static spherically symmetric solution to Einstein’s field equations has been found using a new set of boundary conditions. In introducing these new boundary conditions, we assume that the radial coordinates in and out of the sphere need not be the same and we are guided by the notion of what may be called proper distances and proper times of two observers on either side of the sphere .In these new boundary conditions we replace ordinary partial derivatives by generalized partial derivatives in curvilinear coordinates. Then the solution takes the form  2 2 2  2 2 2 2 2 1                             dr r d l r c dt l r ds   0  r  a   2 2 2 2 2 2 2 2 2 2 1 1 1                   dR R d R A c dT R A ds R A  where          l a l a A  2 2 ,       l r  is the solution of the Lane-Emden equation y r lx dx dy x dx d x        , 1 2 3 2 , l is a constant of dimension length , a is the coordinate radius of the sphere. In our approach r  a in the matter-filled region corresponds to R  Ain the region without matter, outside the sphere. The red shift of a pulse of light emitted at a point on the surface of the sphere as observed by an observer who is at a large distance in the exterior region of the sphere is calculated. This valueequals to                                 l a l a l a l a    when the observer is at infinity. The comparison of this value with the value for the red shift obtained using the metric derived using the standard (Lichernowicz) boundary conditions which says that the metric coefficients and their partial derivatives are continuous across the boundary of the sphere when the observer is at infinity is also done. It is shown that the values obtained for the red shifts are the same irrespective of the boundary conditions used. The red shift of a pulse of light emitted at a point inside of the surface of the sphere as observed by an observer who is at a large distance in the exterior region of the sphere is also calculated and it is shown that the value obtained is different from the value obtained using the metric derived using standard (Lichernowicz) boundary conditions.
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    The velocity of a particle relative to an observer instantaneously at rest coinciding with the point through which the particle passes in a spherical distribution of matter comprising electrically counterpoised dust with constant uniform density
    (University of Kelaniya, 2013) Wimaladharma, N.A.S.N.; de Silva, N.; Hewageegana, P.S.
    A sphere comprising a special kind of matter, electrically counterpoised dust in which all the elastic forces have been cancelled out, has been considered. A static spherically symmetric solution to Einstein’s field equations has been found using a new set of boundary conditions. In introducing these new boundary conditions, we assume that the radial coordinates in and out of the sphere need not be the same and we are guided by the notion of what may be called proper distances and proper times of two observers on either side of the sphere. In these new boundary conditions we replace ordinary partial derivatives by generalized partial derivatives in curvilinear coordinates. Then the solution takes the form  2 2 2  2 2 2 2 2 1                             dr r d l r c dt l r ds   0  r  a   2 2 2 2 2 2 2 2 2 2 1 1 1                dR R d R A c dT R A ds R A  where          l a l a A  2 2 ,     l r  is the solution of the Lane-Emden equation y r lx dx dy x dx d x          , 1 2 3 2 , l is a constant of dimension length , a is the coordinate radius of the sphere . In our approach r  a in the matter-filled region corresponds to R  A in the region without matter, outside the sphere.The velocity of a particle relative to an observer instantaneously at rest coinciding with the point through which the particle passes has been calculated for this metric. Using these values, a minimum value for a measure of energy with which the particle has to be projected at the center of the sphere, to reach infinity has been calculated to be               l a l a l a c   where c is the velocity of the light. A minimum value for a measure of energy with which the particle has to be projected at the center of the sphere, to reach infinity has also been calculated for metric derived using standard (Lichernowicz) boundary conditions which says that the metric coefficients and their partial derivatives are continuous across the boundary of the sphere. It is shown that we have the same value irrespective of boundary conditions used. Also a minimum value for a measure of energy with which the particle has to be projected at the center of the sphere, to reach the exterior region of the sphere has been calculated to be       l a c  . The comparison of this value with the value obtained for the metric derived using standard (Lichernowicz) boundary conditions is also done and it is shown that these two values are the same irrespective of the boundary conditions used.