Lorentzian Cheeger-Gromov convergence and temporal functions

Abstract

Uniqueness (up to isometries) and existence of limits are studied in the context of Cheeger-Gromov convergence of spacetimes. To address the non-compactness of the vector isometry group in the semi-Riemannian setting, standard pointed convergence is strengthened to anchored convergence, which in the Lorentzian case requires the convergence of a timelike direction. This allows one to construct a local isometry between neighborhoods of the basepoints, which can be extended globally under geodesic completeness or just inextensibility. In spacetimes, by using Cauchy temporal functions as both strengthening of anchors and tools to ``Wick rotate'' metrics, a special notion of convergence for globally hyperbolic spacetimes (including those with timelike boundaries) is introduced. After revisiting the tools related to time functions and studying their connections with Sormani-Vega null distance, the machinery of Riemannian Cheeger-Gromov theory becomes applicable. In particular, several results of independent interest are obtained, including local regularity of time functions up to rescaling, global and local characterizations of $h$-steep functions, independence of steepness and $h$-steepness for temporal functions, compatibility of both conditions for Cauchy temporal functions, and stability of the latter.