On symmetries of exact solutions of Einstein field equations

Abstract

In this dissertation, symmetries are studied in the context of general relativity. Initially, we address the problem of separability of the Hamilton-Jacobi equation for the geodesic Hamiltonian, a particular Hamiltonian function constructed from the metric of the spacetime that gives rise to the geodesic equation. In this scenario, the existence of classes of coordinate systems that separate the Hamilton-Jacobi equation, the so-called separability structures, turns out to be intimately connected to the existence of symmetries. In fact, this study leads to the most general form taken by the metric tensor in n dimensions containing m ≤ n rank-2 Killing tensors in involution with each other and r = n − m commuting Killing vector fields. The close relationship between the notion of separability structures and the existence of symmetries is manifest in this framework. In particular, we show that the existence of a separability structure enables the complete integrability of the geodesic motion. Then, a study on symmetries from the point of view of the action of continuous group on differential manifolds is conducted. A review on groups, Lie groups and Lie algebras is provided, and a study on spaces admitting the particular case of a separability structure with m = 2 is done under the light of these tools. Finally, equipped with all this knowledge, starting with the most general fourdimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. We assume that the gauge eld inherits the symmetries of the background and is aligned with the principal null directions of the spacetime. In particular, generalizations of the Kerr-NUT-(A)dS spacetime containing nonabelian gauge elds as source of matter are obtained.