Studies of Spinning Particles in Curved Spacetimes

Abstract

We investigate the motion of spinning particles, such as Dirac fermions, in curved spacetimes by pseudo-classical mechanics models in which the spin degrees of freedom are characterized in terms of anti-commuting Grassmann variables. The work of this thesis has been organized in 7 chapters along with an introduction at the very outset and a discussion at the end of the thesis. In the Introduction we present a short account of the work of investigating pseudo-classical spinning point particles in the four-dimensional curved spacetimes of general relativity. We review the relevant formulations and the equations of motion for the theory in chapters 1 and 2. We generalize the Killing equations in the spinning space, i.e., the configuration space of the spinning particles, spanned by the usual position coordinates and spin variables. We describe the symmetries and the corresponding constants of motion in terms of the solutions of the generalized Killing equations. Spinning space can have fermionic symmetries along with the standard kind of symmetries. The standard or generic symmetries exist for any spacetime metric gµv (x) (chapter 1 ), while the new nongeneric symmetries depend on the specific form of spacetime and are generated by the square root of bosonic constants of motion other than the Hamiltonian (chapter 2). These formalisms have been exploited in chapters 4-7 to study the pseudo-classical spinning point particles in particular types of black hole spacetimes. In chapter 3 we investigate geodesic motion of pseudo-classical spin one half point particles in the purely de Sitter spacetime and asymptotically de Sitter Schwarzschild spacetime. We apply the formalism of chapter 1 to solve the equations of motion of the pseudo-classical spinning particles in a plane. In chapter 4 we investigate geodesic motions of pseudo-classical spinning point particles in the Euclidean Taub-NUT space. Applying the formalism of chapters 1 and 2, we describe the symmetries and derive the corresponding constants of motion. The spinning space is found to admit hidden supersymmetries, generated by the mysterious Killing-Yano tensors. We use these formalisms to analyze the motion of the pseudo-classical spinning particles on a cone and plane. In chapter 5 we analyze the geodesic motion of pseudo-classical spinning particles in the NUT-Taub spacetime. We find that spinning spacetime admits new fermionic supersymmetries along with generic symmetries. The corresponding constants of motion have been obtained and the motion has been described on a cone and on a plane. In chapter 6 we study geodesic motion of pseudo-classical spinning particles in the Taub-NUT-de Sitter spacetime, which is the Taub-NUT spacetime generalized with cosmological constant. We obtain the conserved quantities for spinning space and describe the motion of the pseudo-classical Dirac fermion on a cone and plane. Geodesic motions of pseudo-classical spinning point particles are analyzed for the generalized NUT spacetime in chapter 7. Both types of symmetries, generic and nongeneric, are found to exist in the spinning space. We obtain the corresponding conserved quantities and investigate the motion of the pseudo-classical spinning particles on a cone and plane. Finally, we present the concluding remarks of this work in the discussion, at the end of the thesis.