Graduate Studies
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Item Effects of the Cosmological Constant on Energy and Angular Momentum of a Particle Moving in a Circle with Respect to the Schwarzschild - de Sitter Metric in Comparison with the Schwarzschild Metric(Faculty of Graduate Studies, University of Kelaniya, 2015) Jayakody, J.A.N.K.; de Silva, L.N.K.; Hewageegana, P.S.Considering the Schwarzschild - de Sitter space-time, many authors have explored a range of cosmological events and effects. But, the effects of the cosmological constant () on energy and angular momentum in the Schwarzschild – de Sitter space-time are not studied in depth in comparison to the Schwarzschild space-time. In this study, we obtain the expressions for total energy per unit rest mass ( ) and for angular momentum per unit rest mass ( ) not only in the Schwarzschild - de Sitter space-time but also in the Schwarzschild space-time considering a particle moving in a circular path. Then, we discuss the conditions for the possibility of circular orbits. Finally, we plot the graphs for and for against the coordinate radius of the circle for different low and high values of the central mass ( ) for positive and negative cosmological constants for the Schwarzschild - de Sitter space-time in comparison with the Schwarzschild space-time. Also, we plot the graphs for when is negative. Considering the plotted graphs, we conclude that the effects introduced by the cosmological constant on and are negligible with the present value of the cosmological constant. But, for higher cosmological constant values, the effects on and are known to be significant. However, affects and indeed when a particle moves in a circle. According to this study, positive creates a repulsive field and when it is negative it creates an attractive field. Accordingly, in the nonappearance of a central mass there is no possibility of circular motion when is positive as a repulsive field would not give rise to circular motion. In the case of the Schwarzschild - de Sitter space-time and for a particle moving in a circle are less (greater) than that in the case of the Schwarzschild space-time when is positive (negative).Item Cosmological constant in gravitational lensing(University of Kelaniya, 2011) Jayakody, J.A.N.K.; de Silva, L.N.K.Consider the Schwarzschild de Sitter Metric, 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 ( sin ). 3 3 GM r GM r ds c dt dr r d d rc rc (1) The constant term 2 2GM c is recognized as the Schwarzschild radius ( s r ), and typically it is replaced by a constant term2m, where 2 1 2 s GM m r c and then the equation (1) can be written as follows. 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 ( sin ). 3 3 m r m r ds c dt dr r d d r r (2) is the cosmological constant. The null-geodesic equation in Schwarzschild-de Sitter metric can be written as, 2 2 2 2 2 2 2 3 2 2 0 3 E l l u l u ml u c , [1] (3) where E is the energy, l is the orbital angular momentum, is the cosmological constant, 1 u r and . du u d Differentiating (3) with respect to , 2 u(uu 3mu ) 0. (4) Neglecting the solution,u 0 which implies u = constant, the equation of a light ray trajectory can be written as, 2 uu 3mu . (5) The zeroth order solution and the first order solution of the equation (5) that represent the light ray trajectory are respectively given below. 0 0 1 u cos r [2], (6) 2 2 2 0 0 0 1 2 cos cos 3 3 u r r r [2], (7) where 3m. In general, in the literature, it is assumed that (7) is a solution of equation (3) without considering the limitations imposed. In this paper we discuss conditions under which (7) is a solution of equation (3). Now the orbital angular momentum, 0 l pr where p is the linear momentum. The linear momentum, E p c . Therefore, 0. E l r c (8) Substituting (7) and (8) in (3), we have, 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 3 2 2 2 2 2 0 0 0 1 2 1 2 sin sin cos cos cos 3 3 3 2 1 2 + cos cos 0. 3 3 3 3 E l l c r r r r r l l r r r (9) By simplifying the above equation and since l 0 we obtain the following equation, 3 3 3 3 2 2 2 2 4 6 5 3 6 6 6 6 5 5 5 0 0 0 0 0 0 0 2 4 4 4 4 0 0 0 8 4 2 4 4 cos cos cos cos cos cos 27 9 9 27 3 3 3 2 0 2 2 3 cos cos 3 2 r r r r r r r m r r r 2 2 2 2 2 4 6 5 6 6 6 6 5 2 0 0 0 0 0 3 2 4 5 5 4 4 4 0 0 0 0 0 8 4 2 cos cos cos cos 3 3 18 . 4 4 2 2 1 cos cos cos cos 3 2 m m m m m r r r r r m m m r r r r r (10) From (10) it is clear that the solution given by (7) of equation (3) is valid only if is a constant of order m2, and as we neglect terms of order 2 and above we are justified in assuming (7) as a solution of equation (3). However, it turns out that this particular solution is valid only if is a constant of order 2 or more in m. If is a non zero constant and of order one in m, the solution (7) is not valid and we have to seek other solutions.Item The Effects Introduced by the Gravitational Redshift into the Redshift-Apparent Magnitude Relationship in Cosmology(University of Kelaniya, 2007) Jayakody, J.A.N.K.; de Silva, L.N.K.The redshift-apparent magnitude relationship 111 for nearby objects is concerned with the cosmological redshift. In the derivations of this relationship the gravitational redshift is not considered yet in depth. But for objects which are having very strong gravitational fields, the gravitational redshift ought to be considered. Then, the redshift-apparent magnitude relationship could be affected due to the gravitational redshift. In this study, the redshift-apparent magnitude relationship is derived for combined cosmological and gravitational redshifts. The quasars have considerably large redshifts and they are very distant objects. However the logarithm of the cosmological redshift verses apparent magnitude curves do not fit with observations in the case of the quasars. Therefore, it is important to find a cosmological model which fits with the observed properties of quasars. We have attempted to find such cosmological model, assuming that the redshift of the source has a gravitational component as well. With this assumption, the logarithm value of the red shifts against the apparent magnitudes for different values of the gravitational redshift and for different values of the deceleration parameter have been plotted for different zero pressure cosmological models. According to the present study, the effect of gravitational redshift on the redshiftapparent magnitude relationship is very small. Within this limitation, the cosmological model with the parameters, q0' >+I, CJ'0 = 0, k = + 1, A > 0 and q0' = 75 fits best with the quasars having taken into consideration the acceleration of the Universe predicted by the supernovae observations 121· 131. Here q0. is the acceleration parameter, CJ'0 is the density parameter, k is the space curvature constant and A is the cosmological constant. Keywords: gravitational redshift, cosmological redshift, apparent magnitude, quasars, deceleration parameterItem Path of a light ray near a body with cosmological constant(Research Symposium 2009 - Faculty of Graduate Studies, University of Kelaniya, 2009) Jayakody, J.A.N.K.; de Silva, L.N.K.Emitted light rays from a very distant and bright source are deflected between the source and the observer when they pass near a massive body with an enormous gravity. As a result the massive body such as a cluster of galaxies have an ability to perform as a gravitational lens. In recent times, some authors [1] have found that the cosmological constant , affects the phenomenon of gravitational lensing. In this paper, we have corrected an expression for the total deflection angle which was published in 2008 of our first paper regarding this subject [2] . Considering the effect of the cosmological constant, we have also found two equations for the path of a light ray when it passes near a massive object with a very high gravitational influence.Item Null Geodesics in de-Sitter Universe(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Jayakody, J.A.N.K.; de Silva, L.N.K.