Equivalência conforme de espaços-tempos

Abstract

In this thesis we study the conformal equivalence problem applied to space-times. Initially we present a review of Topology, Differential Geometry, Riemannian Geometry and a brief résumé of Lie Groups and the theory of differentiable fibre bundles, using a coordinate-free approach. To understand the Equivalence Problem and its solution it is necessary to study a wide range of areas of mathematics: conformal transformations between metrics and their respective curvatures; Weyl Geometry and the relation between its curvature tensors and the Riemannian analogues; the Equivalence Problem for coframes; the G-equivalence problem for coframes; and the Equivalence Problem for Riemannian geometries. All these subjects are treated in the body of the thesis. After this preparation, the Conformal Equivalence Problem between metrics is defined and solved for arbitrary signature and dimension. We specialise this solution to Lorentzian geometries using spinor formalism which is then used to study space-times with a non-trivial isotropy group. At the end of the thesis, we use the method to determine under what conditions a Petrov type III space-time has maximal conformal symmetry, and obtain some explicit solutions and metrics with these characteristics.