Graduate Studies

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    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).
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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.
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    Electronic Energy States in Nano Particles
    (University of Kelaniya, 2012) Jayalalani, J.A.D.; Hewageegana, P.S.; Siripala, W.P.
    An electron in the conduction band of a particle is nearly free to move inside the particle and this situation can be pictured as an “Electron inside a finite depth potential well”. The energy equations for this system can be derived by applying the “Time Independent Schrödinger equation” and corresponding boundary conditions in terms of one and three dimensions. In this study we have employed computer software and numerical root finding methods to obtain the numerical values of the legitimate energy states as it is more reliable than the conventional graphical methods. According to these numerical solutions, we could demonstrate that the number of allowed energy states and the spacing between adjacent levels inside a nano particle depend on both particle size and the magnitude of the attractive potential. Further, “Quantum tunneling effect” is significant when the particle size is below 20 nm and lowering the magnitude of the attractive potential, would extend the wave function far beyond its boundary. The energy levels obtained by employing the computer software and numerical root finding methods to the energy equation were plotted and compared with reported experimental observations and they are in good agreement. The most interesting size dependent property related to the semiconducting nano particles is that, we can obtain every colour of the visible spectrum by changing the size within the nano range, while the composition is unchanged.
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    A study of temperature and salinity variation with depth in salt pans at Palaviya in the North-Western region of Sri Lanka
    (University of Kelaniya, 2000) Hewageegana, P.S.; Amarasekara, C.D.; Jayakodi, J.P.R.; Punyasena, M.A.