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Global hyperbolicity and manifold topology from the Lorentzian distance

A. Bykov, E. Minguzzi

TL;DR

The paper develops manifold-topology-independent characterizations of global hyperbolicity by translating causality and compactness into Lorentzian distance data. It introduces $d$-distinction and $d$-reflectivity, showing their equivalence to standard causality notions in smooth settings and extending the framework to Lorentz-Finsler spaces and low-regularity contexts. The central results show that a Lorentzian distance $d$ encodes enough structure to recover the manifold topology (the Alexandrov topology) and to identify Lorentzian isometries via distance-preserving maps; in particular, globally hyperbolic spacetimes correspond to Lorentzian metric spaces with finite, continuous $d$. These developments unify smooth, Finslerian, and abstract causal structures under a distance-based viewpoint, with potential implications for quantum spacetime models where the manifold is not presupposed.

Abstract

In this work, we seek characterizations of global hyperbolicity in smooth Lorentzian manifolds that do not rely on the manifold topology and that are inspired by metric geometry. In particular, strong causality is not assumed, so part of the problem is precisely that of recovering the manifold topology so as to make sense of it also in rough frameworks. After verifying that known standard characterizations do not meet this requirement, we propose two possible formulations. The first is based solely on chronological diamonds and is interesting due to its analogies with the Hopf-Rinow theorem. The second uses only properties of the Lorentzian distance function and it is suitable for extension to abstract `Lorentzian metric' frameworks. It turns out to be equivalent to the definition of `Lorentzian metric space' proposed in our previous joint work with S. Suhr, up to slightly strengthening weak $d$-distinction to `future or past $d$-distinction'. The role of a new property which we term `$d$-reflectivity' is also discussed. We then investigate continuity properties of the Lorentzian distance and the property of $d$-reflectivity in non-smooth frameworks. Finally, we establish a result of broader interest: the exponential map of a smooth spray is $C^{1,1}$ (smooth outside the zero section). Additionally, we derive a Lorentz-Finsler version of the Busemann-Mayer formula and demonstrate that, in strongly causal smooth Finsler spacetimes, the Finsler fundamental function can be reconstructed from the distance. As a consequence, distance-preserving bijections are shown to be Lorentz-Finsler isometries in the conventional smooth sense.

Global hyperbolicity and manifold topology from the Lorentzian distance

TL;DR

The paper develops manifold-topology-independent characterizations of global hyperbolicity by translating causality and compactness into Lorentzian distance data. It introduces -distinction and -reflectivity, showing their equivalence to standard causality notions in smooth settings and extending the framework to Lorentz-Finsler spaces and low-regularity contexts. The central results show that a Lorentzian distance encodes enough structure to recover the manifold topology (the Alexandrov topology) and to identify Lorentzian isometries via distance-preserving maps; in particular, globally hyperbolic spacetimes correspond to Lorentzian metric spaces with finite, continuous . These developments unify smooth, Finslerian, and abstract causal structures under a distance-based viewpoint, with potential implications for quantum spacetime models where the manifold is not presupposed.

Abstract

In this work, we seek characterizations of global hyperbolicity in smooth Lorentzian manifolds that do not rely on the manifold topology and that are inspired by metric geometry. In particular, strong causality is not assumed, so part of the problem is precisely that of recovering the manifold topology so as to make sense of it also in rough frameworks. After verifying that known standard characterizations do not meet this requirement, we propose two possible formulations. The first is based solely on chronological diamonds and is interesting due to its analogies with the Hopf-Rinow theorem. The second uses only properties of the Lorentzian distance function and it is suitable for extension to abstract `Lorentzian metric' frameworks. It turns out to be equivalent to the definition of `Lorentzian metric space' proposed in our previous joint work with S. Suhr, up to slightly strengthening weak -distinction to `future or past -distinction'. The role of a new property which we term `-reflectivity' is also discussed. We then investigate continuity properties of the Lorentzian distance and the property of -reflectivity in non-smooth frameworks. Finally, we establish a result of broader interest: the exponential map of a smooth spray is (smooth outside the zero section). Additionally, we derive a Lorentz-Finsler version of the Busemann-Mayer formula and demonstrate that, in strongly causal smooth Finsler spacetimes, the Finsler fundamental function can be reconstructed from the distance. As a consequence, distance-preserving bijections are shown to be Lorentz-Finsler isometries in the conventional smooth sense.

Paper Structure

This paper contains 14 sections, 41 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Available characterizations
  3. A manifold-topology-independent characterization
  4. Lorentzian distance function
  5. Globally hyperbolic Lorentz-Finsler spaces
  6. Distance preservation implies Lorentzian isometry
  7. The Lorentz-Finsler case
  8. The positive signature Finsler case
  9. Low regularity
  10. Equivalence of distinction/reflectivity concepts with their $d$-version
  11. Proper Lorentz-Finsler spaces
  12. Lorentzian length spaces
  13. Lorentzian distance in abstract setting
  14. Conclusions

Key Result

Theorem 1.1

A smooth spacetime $(M,g)$ is globally hyperbolic if and only if the Lorentzian distance has the following properties In this case $T$ is the manifold topology and the causal relation is given by

Figures (1)

  • Figure 1: A figure illustrating the proof of Thm. \ref{['coox']} under the assumption $q\ne x$. We have $(q,x)\in D$, $I^+(q)=I^+(x)$, $I^-(x)=I^-(r)$, so past and future distinction are violated. The figure suggests $I^+(r)\cap \sigma_k\ne \emptyset$ for any sufficiently large $k$, but this is just due to its bidimensionality.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Corollary 2.4
  • Definition 2.5
  • ...and 65 more