Browsing by Author "de Silva, L.N.K."

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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.
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(uu 3mu )  0. (4) Neglecting the solution,u  0 which implies u = constant, the equation of a light ray trajectory can be written as, 2 uu  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.
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 • ,··
Deflection of light with cosmological constant
(University of Kelaniya, 2008) Jayakody, J.A.N.K.; de Silva, L.N.K.
A beam of light is deflected when it passes near a massive object such as the Sun or a black hole, due to the gravitational influence of the massive body. Without the cosmological constant, the total deflection angle of light is 28 = 4m (11, where r0 is the closest approach of ro the light ray from the centre of the massive body and m = GAf . Here G is the gravitational c constant and M is the mass of the central object. Taking into account the Schwarzschild de Sitter geometry, some authors [21 have found recently, that the cosmological constant contributes to the deflection of light. In this paper, we study the effect of the cosmological constant on the deflection of light when it passes near a massive object. The null-geodesic equation in Schwarzschild-de-sitter metric where can be written as 2m Ar2 3 f(r)=1----[ -l r 3 ' where l and h are constants, which can be written in another form of u" + u = 3mu2 (11. The zeroth order solution and the first order solution of the light ray trajectory are respectively given below. 1 u0 = -cos [IJ ro 1 E 2 2E u=-cos"'---cos "'+--['1 where E =3m.Now, assume a second order solution for the light ray trajectory in the following form. 1 £ 2 2£ 2 u = -cos th ---cos th + --+ £ w 'I' 2 'I' 0 ' r0 3r0 3r0 􀁑 where w = w(r0,). We find that the solution up to the second order for the light ray trajectory can be written as, 1 E 2 2£ r0 sin [ 5£ 2 E 2 • ( 5£ 2 A J ] u = -cos --- 2 cos +-- 2 + --4 ---4 sm2 + --4 +- cot . r0 3r0 31() 2 6r0 12r0 9r0 3 n n Taking the limits as, r 􀀋 00' u 􀀋 O, <1> 􀀋 2 + 8 '<1> 􀀋 -2-8 we find the two asymptotes of the trajectory, which give the total deflection angle as, [􀀉 _ 8£ 2 + 2A102 ] 28 = 3ro 9ro 2 3 (1-􀁒+ Ar0 33r0 2£ J A is shown to be a function of E 2 , and substituting A= E 2 k in the above equation, and neglecting the terms ofe , of order higher than four , we find, 28 = 4£ - 3 2£ 3 - 16£ 4 k + 64£ 4 3r0 27r􀁓3 27 81r0 4 • Writing the above equation in terms of A, we obtain, Substituting E = 3m, 28 =4m _ 16m2 A_ 32m3 + 64m4 ro 3 ro 3 ro 4 , which is the angle of deflection in the presence of the cosmological constant, neglecting fifth and higher order terms of £ .
Derivation of Equations that satisfy Dirac’s Equation and invariance of transformations similar to Guage Transformations under mixed numbers
(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Hansameenu, W.P.T.; de Silva, L.N.K.; de Silva, T.P.
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 parameter
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).
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.
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.
A new Cosmological model that including inflation, deceleration, acceleration and deceleration again
(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Katugampola, K.D.W.J.; de Silva, L.N.K.
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.
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 v
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.
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.
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.
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.
කාළාම සූත්‍රයට අනුව කාළාම සූත්‍රය යොදා ගැනීම
(University of Kelaniya, 2011) de Silva, L.N.K.