Annual Research Symposium (ARS)
Permanent URI for this communityhttp://repository.kln.ac.lk/handle/123456789/154
Browse
8 results
Search Results
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.Item 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 Matrix representation of mixed numbers and quaternions(Research Symposium 2009 - Faculty of Graduate Studies, University of Kelaniya, 2009) Hansameenu, W.P.T.; de Silva, T.P.; de Silva, L.N.K.