Aspects of General Relativity: Pseudo-Finsler Extensions, Quasi-Normal Frequencies and Multiplication of Tensorial Distributions

Abstract

This thesis is based on three different projects, all of them are directly linked to the classical general theory of relativity, but they might have consequences for quantum gravity as well. The first chapter deals with pseudo-Finsler geometric extensions of the classical theory, these being ways of naturally representing high-energy Lorentz symmetry violations. In this chapter we prove a certain type of “no-go” result for significant number of theories. This seems to have important consequences for the question of whether some weaker formulation of Einstein’s equivalence principle is sustainable, if (at least) certain types of Lorentz violations occur. The second chapter deals with the problem of highly damped quasi-normal modes related to different types of black hole spacetimes. First, we apply to this problem the technique of approximation by analytically solvable potentials. We use the Schwarzschild black hole as a consistency check for our method and derive many new and interesting results for the Schwarzschild-de Sitter (SdS) black hole. One of the most important results is the equivalence between having a rational ratio of horizon surface gravities and periodicity of quasinormal modes. By analysing the complementary set of analytic results derived by the use of monodromy techniques we prove that all our theorems almost completely generalize to all the known analytic results. This relates to all the types of black holes for which quasi-normal mode results are currently known. The third chapter is related to the topic of multiplication of tensorial distributions. We focus on an alternative approach to the ones presently known. The new approach is fully based on the Colombeau equivalence relation, but technically avoids the Colombeau algebra construction. The advantage of this approach is that it naturally generalizes the covariant derivative operator into the generalized tensor algebra. It also operates with much more general concept of piecewise smooth manifold, which is in our opinion natural to the language of distributions.