Lorentzian length spaces

TL;DR

This paper develops a Lorentzian analogue of length spaces using the time separation function, establishing synthetic curvature bounds and broadening the scope of Lorentzian geometry to include low-regularity metrics and quantum gravity models.

Contribution

It introduces a new framework for Lorentzian length spaces, generalizing classical results and incorporating synthetic curvature bounds based on triangle comparison.

Findings

Established fundamental properties of Lorentzian length spaces.

Defined synthetic curvature bounds analogous to Alexandrov and CAT(k) spaces.

Applied theory to low-regularity Lorentzian metrics and quantum gravity models.

Abstract

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The r\^ole of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT$(k)$-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.