On the characterization of static spacetimes with positive cosmological constant

In this thesis we study static vacuum spacetimes. These are very special solutions of the Einstein Field Equations in General Relativity, where the Lorentzian structure disappears and we are left with the study of a system of PDEs on a Riemannian manifold. Although they represent the simplest examples of spacetimes, their study is by no means trivial. Our main focus will be on spacetimes with positive cosmological constant, even though we will provide a general overview of the other cases as well. Our main contribution is the introduction of a new notion of mass (which will be called virtual mass) on vacuum static spacetimes with positive cosmological constant. We will show the plausibility of our definition, by proving that the virtual mass satisfies properties analogous to the well known Positive Mass Theorem and Riemannian Penrose Inequality for Riemannian manifolds with nonnegative scalar curvature. As a consequence, we will prove a uniqueness theorem for the Schwarzschild– de Sitter spacetime. As we will discuss, this result shares some similarities with the well known Black Hole Uniqueness Theorem for the Schwarzschild spacetime.