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. Choquet-Bruhat and Geroch proved (Commun Math Phys 14:329–335, 1969) 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’s theorem, these space–times are locally modeled on the Einstein space–time. In the first part of the article 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 give a causal characterization of globally hyperbolic conformally flat maximal space–times whose developing map is a global diffeomorphism.