Communications in Analysis and Geometry

Volume 31 (2023)

Number 6

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

Pages: 1469 – 1528

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n6.a5

Authors

Stephanie B. Alexander (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il.,U.S.A.)

Melanie Graf (Department of Mathematics, University of Washington, Seattle, Wa., U.S.A.; and Department of Mathematics, University of Hamburg, Germany)

Michael Kunzinger (Faculty of Mathematics, University of Vienna, Austria)

Clemens Sämann (Department of Mathematics, University of Toronto, Ontario, Canada; and Faculty of Mathematics, University of Vienna, Austria)

Abstract

We study generalizations of Lorentzian warped products with onedimensional base of the form $I \times {}_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of $\href{https://doi.org/10.1007/s10455-018-9633-1}{[39]}$, displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

Received 27 January 2020

Accepted 15 June 2021

Published 9 August 2024