A note on non-compact Cauchy surfaces

Abstract

It is shown that if a spacetime has a non-compact Cauchy surface Σ, then its causal structure is completely determined by the class of compact subsets of Σ of the forms J−(p)∩Σ and J+(p)∩Σ. Since the causal structure determines its metric structure up to a conformal factor, this implies that the sets J−(p)∩Σ and J+(p)∩Σ determine the conformal structure of a globally hyperbolic spacetime. In this way, we can encode the conformal structure of the spacetime into its Cauchy surface and we get another method for reconstructing spacetime.