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.

arXiv commentThe part of inextensibility in Theorem 1.1 is ammended and enlarged with Example 5.29, Appendix 1 substituted by Remark 3.5, and minor modifications