Classification of Weyl and Ricci Tensors

Abstract

The theory of General Relativity was formulated by Albert Einstein and introduced a set of equations: Einstein field equations. Since then, many exact solutions of these equations have been found. An important result in the study of exact solutions to Einstein equations is the classification of space-times according to the Weyl tensor (Petrov classification) and the Ricci tensor. Such classifications help to group the set of solutions that share some geometric properties. In this thesis, we review all the basic concepts in differential geometry and tensor calculus, from the definition of smooth manifold to the Riemann curvature tensor. Then we introduce the concept of Lorentz spaces and the tetrad formalism, which is very useful in the field of exact solutions to Einstein equations. After that, the concept of bivector is introduced and some of their main properties are analyzed. Using bivector formalism, we set the algebraic problem that derives in the Petrov classification. Moreover, we study principal null directions, that allow to make an analogous classification. Finally, we classify second order symmetric tensors and, from this, we can classify space-times according to their energy tensor.