Some global causal properties in certain classes of spacetimes

Abstract

Causality is a specific tool of Lorentzian Geometry, with a clear physical motivation, which has played a central role in proving important theorems about the global structure of spacetimes. Causality conditions are classified in terms of the so called causal ladder, whose steps determine how these conditions are logically related. Each of these levels presents some specific properties, standing out at the top one, which is occupied by the condition of global hyperbolicity. In fact, it is believed that any physical spacetime must be globally hyperbolic (roughly, this is the content of the strong cosmic censorship hypothesis), and then, will admit a global splitting in terms of a Cauchy surface, on which the Einstein equations can be posed as an initial value problem.

Causality theory also provides a boundary construction for the very general class of strongly causal spacetimes, namely, the so-called causal boundary or just c-boundary. This boundary is less commonly used in General Relativity than the conformal one, because some classical spacetimes present a simple conformal boundary with quite a few of interesting properties. However, beyond such examples, there is no a general way to ensure that the conformal boundary exists. In contraposition, the c-boundary is not only conformally invariant but also intrinsic and it can be computed systematically; such properties make it more suitable in general situations. This includes the holographic principle, which original started at a restricted situation concerning the conformal boundary of Anti-de Sitter spacetime for the AdS/CFT correspondence. The main purpose of this memory is to improve our knowledge about the c-boundary and the causal ladder of some important classes of spacetimes.