Maximal extension of conformally flat globally hyperbolic space-times

Abstract

The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal extension. Since the causal structure of a space-time is invariant under conformal changes of metrics we may generalize this notion of maximality to the conformal setting. In this article we focus on conformally flat space-times of dimension greater or equal than 3. In this case, by a Lorentzian version of Liouville theorem, these space-times are locally modeled on the Einstein space-time. In the first part we use this fact to prove the existence and uniqueness of the maximum extension for globally hyperbolic conformally flat space-times. In the second part we find a causal characterization of globally hyperbolic conformally flat maximal space-times whose developing map is a global diffeomorphism.