Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Burgos, Saúl, Flores, José L., Sánchez, Miguel
Institutions: Universidad de Granada, Universidad de Málaga
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G121809054, G1219052735, G1320060331, G161246385, G1799245145, G1826204064, G1855990336, G21826877, G2193086574, G2262748287, G2309174360, G2660406951, G3250484376, G3480869486, G425668148, G4495373907, G4644199935, G4648751995, G5080475149, G5421974903, G567365627, G5805502524, G6779418066, G7084143925, G8638682022, G897484643
OpenAlex ID: W4417080236
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