Renormalizaci\'on conforme de teor\'ias escalar-tensor de la gravedad

Abstract

In this thesis, we investigate the method of conformal renormalization applied to theories with degrees of freedom beyond the metric ones. Specifically, we examine this method in the presence of a scalar field. To do this, as part of a review, we revisit the action principle of General Relativity and Einstein's equations, in addition to re-examining the conditions for this theory to have a well-defined variational principle when Dirichlet boundary conditions are imposed. We then explore various methods for calculating conserved charges in asymptotically flat spaces. In asymptotically anti-de Sitter spaces, we study two renormalization schemes that are relevant to this work. To motivate the method used here, we observe that Conformal Gravity is finite for spaces that are asymptotically anti-de Sitter, as demonstrated in [Grumiller, 2014]. This guiding principle gives us a clue as to how conformal symmetry may be related to renormalization in spacetimes with such asymptotics. The basis of this construction is the extension of a covariant tensor under Weyl rescalings composed of the metric and the scalar field as proposed in [Oliva and Ray, 2011]. This extension ensures that the conformal weight of this tensor is equal to that of the Weyl tensor. We extend this realization by considering tensor-scalar theories with conformal symmetry, coupled with the Einstein-AdS action written in the MacDowell-Mansouri form. Despite the fact that the Einstein-AdS sector breaks conformal symmetry, we show that the entire theory can still be renormalized if the scalar field has an appropriate decay when considering asymptotically anti-de Sitter solutions. Finally, we study black hole-type solutions, calculating their Hawking temperature and the Euclidean on-shell action, explicitly demonstrating that the latter is finite for asymptotically anti-de Sitter spaces.