Lovelock Theories as extensions to General Relativity

Abstract

In this thesis we will study Lovelock Theories, that is, some extensions to General Relativity with particularly good properties, for example, giving second-order differential equations and having Levi-Civita connection as a solution of firstorder formalism. Despite their advantages, these theories had never been studied so deeply and in this thesis we will present several new results. First of all, we explain basic concepts and set the mathematical base. In second chapter, we study the Einstein-Hilbert action. We will see that the solution to the metric-affine formalism is not only the Levi-Civita connection, but a set of connection that we will call Palatini connections. In third chapter, we talk about general properties of every Lovelock Theory, especially about projective invariance, which explains why Palatini connections are solutions of these theories. Finally, we study the Gauss-Bonnet action and we give a non-trivial solution of metric-affine formalism that is physically distinguishable of Levi-Civita, hence demonstrating the non-equivalence between metric and metric-affine formalisms.