A geometric approach to the analysis of black holes and spacetime invariants

Abstract

The application of geometric techniques for the problem of characterizing spacetimes is explored for a number of topics. We examine the properties of black holes through a geometric definition based on curvature invariants, and explore the nature of such special objects. Through applications of the Cartan-Karlhede algorithm for determining the local equivalence of spacetimes, we prescribe and explore methods of discussing surfaces such as black hole horizons and photon surfaces in Einstein's theory of general relativity, black holes in alternative theories of gravity such as teleparallel and Brans-Dicke theory, and the occurrence of naked singularities in such theories. In addition to the analytical exploration of these topics, a numerical application is given and discussed, displaying the usefulness of the invariant characterization and Cartan scalars in numerical problems. The emphasis of this thesis is on the application of frame methods of analyzing spacetimes in theories of gravity, and through them developing invariant definitions of special surfaces in these spacetimes.