Annual Research Symposium (ARS)
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Item Concept of Mass in General Relativity(University of Kelaniya, 2012) Perera, K.L.M.M.; de Silva, L.N.K.The General Theory of Relativity formulated by Albert Einstein, is a widely accepted description of “gravitation” in modern physics. We discuss the concept of mass in General Relativity using the interior Schwarzschild solution. Here we explain that there occurs an error in the mass, given by the defect , as a result of using many radial markers in the interior Schwarzschild solution, as explained in text books. However, there is a weakness in the method as the ‘mass’ of the body is evaluated at two points namely at infinity and locally at the body. In this study, we introduce a new method of calculating mass defect at the same point at infinity, using the redshift as observed at infinity.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 On the velocity of waves in Quantum Mechanics(University of Kelaniya, 2011) de Silva, L.N.K.It is generally believed that in Classical Physics it is the group velocity of a wave that carries information from one point to another point in space. The group velocity under normal circumstances for classical waves turns out to be less than that of light and the phase velocity though could be greater than the velocity of light, is not believed to carry information. However, in the case of Quantum Mechanical de Broglie waves corresponding to particles we could obtain an expression for the phase velocity in terms of the momentum (hence the velocity of the particle or the group velocity of the de Broglie wave), which is of more significance as far as Quantum Mechanical particles (systems) are concerned. Consider a particle of mass m moving with velocity v in a frame of reference F, and suppose that it exhibits Quantum Mechanical properties. If E is the energy of the particle and p= mv is its momentum in F then the de Broglie wave length and the corresponding frequency are given by λ = h / p and ω = u / λ respectively where u is the phase velocity of the de Broglie wave. Since E = h ω we have E=up. Substituting these in the relativistic equation E2/c2 = p2 + m02c2 we have p2 (u2/c2 -1) = m02c2 and m02 v2(u2/c2 -1)/(1-v2/c2)= m02c2. These equations imply that u>c and uv=c. It can be seen that the group velocity turns out to be v, the velocity of the particle. Thus the phase velocity u of the de Broglie wave is c/v= mc/p, in terms of p the momentum of the particle. Now let the frame F be moving with velocity w in a frame of reference F1 in the same direction as that of v. Then the velocity v1 of the particle in F1 is given by the addition formula v1 = (v+w) / (1+vw/c2) . If u1 is the phase velocity of the de Broglie wave as observed in F1 then u1v1=c2. This gives u1= (u+w)/(1+uw/c2) for the phase velocity of the de Broglie wave in frame F1 agreeing with the usual special relativistic law of addition of velocities. For photons both the phase velocity and group velocity turn out to be c for any frequency, and for particles when vItem කාළාම සූත්රයට අනුව කාළාම සූත්රය යොදා ගැනීම(University of Kelaniya, 2011) de Silva, L.N.K.Item The Sehwarzschild Space-Time in the Background of the Flat Robertson-Walker Space-Time(University of Kelaniya, 2007) Senevirathne, K.W.P.B.; de Silva, L.N.K.The Schwarzschild space-time is well known in describing the gravitational field of an object in an otherwise empty universe. The Schwarzschild space-time was derived by Karl Schwarzschild ( 1916) considering the merger of the Schwarzschild space-time with the Lorentz metric as the boundary (!)_ However, the Loremtz metric cannot be used in investigations of non empty large scale space-times, the whole universe being one such case. Thus, the cosmologists use the Robertson-Walker space-times, in describing the universe (2. -'i. As a result it becomes necessary to investigate the gravitational field of an object in the background of the Robertson-Walker space-time, We have studied the merger of the isotropic Schwarzschild space-time with the flat Robertson-Walker space-time. In this scenario, the flat Robertson-Walker space-time was considered for simplicity. The expressions for the radial coordinates r11 and rJl at the merger of the flat Robertson-Walker space-time and the isotropic Schwarzschild space-time were derived in terms of the scale factor R(t) and a constant R* and found to be given by An analytic expression for the time coordinate ( t) of the Schwarzschild space-time was obtained in the case of the de-Sitter universe, l = 2T0 In[- 1 !R' _l where To is the reciprocal of the Hubble constant (2'. 2~ I( - Jf?(t) J Schwarzschild Flat Robertson-Walker space-time space-time Figure: The radial coordinates and the time coordinates of the Schwarzschild space-time and the t1at Robertson Walker space-time at the merger The derived expressions for the radial coordinates '~, and rJI imply that an object in the universe begins to communicate with the "outside world" after a particular time, before which r11 and rfl are negative. At this particular time, R(t) approaches the constant R* and r,, , rfl tend to infinity. It could be said that the object comes into existence as far as the rest of the universe is concerned at this particular instant. The values of r11 and rf.i decrease with increase of time. When the time coordinate of the Schwarzschild space-time tends to infinity, rfl achieves the value (;) , the value of the Schwarzschild radius in isotropic coordinates.Item The Resultant Red-Shift of a Source in the Case of the Merger of the Schwarzschild Space-Time with the Flat Robertson-Walker Space-Time(University of Kelaniya, 2007) Senevirathne, K.W.P.B.; de Silva, L.N.K.The red-shift of a source in the space can be described by considering the Schwarzschild space-time (I) (gravitational red-shift) or the Robertson-Walker space-time (2) (cosmological red-shift). When obtaining expressions for the red-shift, the path of light particles or photons plays an important role (3l. In the case of merger of the isotropic Schwarzschild space-time with the flat Robertson-Walker space-time, it is not meaningful to discuss the path of light particles or photons in the isotropic Schwarzschild space-time or the flat Robertson-Walker space-time separately. We have considered the red-shift of a source as observed by an observer on the other side of the merger. The expressions for the radial coordinates, derived by the authors (4l, at the merger of the isotropic Schwarzschild space-time and the flat Robertson-Walker spacetime were used. Schwarzschild space-time Path of the photon Robertson-Walker space-time Path of the photon Photon Figure: Radial motion of a photon with the source in the flat Robertson-Walker space-time When the source is located in the flat Robertson-Walker space-time, the observer is considered to be in the isotropic Schwarzschild space-time and vise versa. The expressions for the gravitational red-shift and the cosmological red-shift of the source were derived, and the resultant red-shift of the source was obtained from these expressions.Item Cosmological Models with Both Acceleration and Deceleration(University of Kelaniya, 2007) Katugampala, K.D.W.J.; de Silva, L.N.K.Since Perlmutter and others (1997) & ( 1998) 12 observed that the universe expand with an acceleration, many models involving dark energy have been proposed to explain this phenomenon. In this paper \ve present a family of cosmological models with both acceleration and deceleration . We write Einstein's Field Equations in general relativity in the form, The 1\ term introduced by Einstein himself gives rise to a field that repels particles and objects rather than to one that attracts them. Hemantha and de Silva (2003)&(2004) 3'4 modified the field equations so that what is conserved is not the energy momentum of matter and radiation but the energy momentum of matter and radiation and the energy of the 1\ field, which they considered as the "dark energy". They obtained the equations, .. 2 kc 2 R2 2R Kp=I\C +-+-+- R2 R2 R 3k 3R2 Kp = - 1\ - R 2 - R 2 c 2 ' where • denotes differentiation with respect to cosmic time t .The above equations lead to . . ( pJR . 1\ 3p+--+p+--=0 c 2 R K As the density p(t) has to be a positive quantity we can show that k = 1, is the only possible value of k that satisfies the above equations. We assume that a family of solutions of above equations for R, can be written in the form, R =a+ b1 coswt + b3 cos3wt Using the boundary conditions, we have * R = 0 at t = 0 . • •• 7( * and R = 0, R = 0, at t = -, (point of inflection) 2 R = -b3 (1 - cos3 OJt) . Recent observations 5 have led to the approximate value 2 for the ratio of dark energy 3 matter density ( p) [~ J , p {:.=/(! and to the value 1.6 for the redshift [ ;1 "'0 :,, J , at the onset of acceleration. Taking this redshift to be a constant I wl=- 2 a family of solutions can be found for different ratios of dark energy to matter. Similarly keeping the ratio of dark energy to matter as 2 we find that a family of solutions can be 3 obtained for different values for the above redshift. Though there is no solution when the redshift is 1.6, there is a solution when its value is 1.3, which is good enough considering the uncertainties associated with measurements. The age of the universe is estimated 6 to be 13.7 billion years. Then taking the present value of the cosmic time t as 13.7 billion years, we find b 3 = - 8. 3 3 x 1 026 em , OJ = 5.16 X 1 o-IS rad r 1 ' when the above redshift is 1.3. The graphs for these values are given below. It is seen that R(t) has both acceleration and deceleration. Radius of the universe x1o"' Density of the homogeneous universe ~ 10 2' Density , R<:t) 0 06 0 8 1 1 2 I 4 0 5 Cosmic timet 2 x10 ·s Cosmic timet • ,··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 A model to explain interference patterns using probability density distribution(Research Symposium 2009 - Faculty of Graduate Studies, University of Kelaniya, 2009) Harshani, P.G.T.; de Silva, L.N.K.Item Some cosmological models with inflation, acceleration and deceleration(Research Symposium 2009 - Faculty of Graduate Studies, University of Kelaniyar, 2009) Katugampala, K.D.W.J.; de Silva, L.N.K.