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 linear 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 the neighborhoods of the basepoints, which can be extended globally when one of the limits is geodesically complete for a single causal type of geodesics and the other one is inextensible; a Misner-type example shows that inextensibility of both limits is not enough. In spacetimes, by using Cauchy temporal functions both as a strengthening of the anchors and as a tool 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 the Sormani-Vega null distance, the machinery of Riemannian Cheeger-Gromov theory becomes applicable. In particular, several results of independent interest are obtained, including 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 the stability of the latter.