Abstract
We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle 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.
Highlights
Metric geometry, and in particular the theory of length spaces, is a vast and very active field of research that has found applications in diverse mathematical disciplines, such as differential geometry, group theory, dynamical systems and partial differential equations
The purpose of this work is to lay the foundations for a synthetic approach to Lorentzian geometry that is similar in spirit to the theory of length spaces and that, in particular, allows one to introduce curvature bounds in this general setting
One may extend the validity of these results to their natural maximal range, in particular to settings where the Lorentzian metric is no longer differentiable, or even to situations where there is only a causal structure not necessarily induced by a metric
Summary
In particular the theory of length spaces, is a vast and very active field of research that has found applications in diverse mathematical disciplines, such as differential geometry, group theory, dynamical systems and partial differential equations. Busemann—a pioneer of length spaces— introduced so-called timelike spaces in [11], but his approach was too restrictive to even capture all smooth (globally hyperbolic) Lorentzian manifolds Another closely related work is due to Kronheimer and Penrose [27], who studied the properties and interdependences of the causal relations in complete generality. We consider closed cone structures following the recent fundamental work of Minguzzi [37] and give a brief outlook on potential applications in certain approaches to quantum gravity, namely causal Fermion systems and the theory of causal sets To conclude this introduction, we introduce some basic notions and fix notations.
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