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Time functions on Lorentzian length spaces

Annegret Burtscher, Leonardo García-Heveling

TL;DR

This work extends core time-function results from smooth spacetimes to the broad framework of Lorentzian (pre-)length spaces, establishing sharp equivalences with the causal ladder: $K$-causality characterizes the mere existence of time functions, causal continuity aligns with averaged-volume time functions, and global hyperbolicity is captured by the existence of Cauchy time functions and Cauchy sets. The authors develop and combine limit-curve theorems, approximating structures, and measure-based volume constructions to operate without a manifold setting. Their approach unifies and generalizes classical results (Geroch, Hawking–Sachs, Dieckmann) to non-smooth, non-manifold spacetimes, enabling robust definitions of time functions in singular or discrete causal spaces. The results have potential implications for convergence theories and quantum gravity frameworks where spacetime may lack smooth manifold structure. Overall, the paper provides a comprehensive causal-theoretic blueprint for time-function existence and global hyperbolicity in a broad, abstract spacetime setting.

Abstract

In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by $K$-causality, show that a modified notion of Geroch's volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

Time functions on Lorentzian length spaces

TL;DR

This work extends core time-function results from smooth spacetimes to the broad framework of Lorentzian (pre-)length spaces, establishing sharp equivalences with the causal ladder: -causality characterizes the mere existence of time functions, causal continuity aligns with averaged-volume time functions, and global hyperbolicity is captured by the existence of Cauchy time functions and Cauchy sets. The authors develop and combine limit-curve theorems, approximating structures, and measure-based volume constructions to operate without a manifold setting. Their approach unifies and generalizes classical results (Geroch, Hawking–Sachs, Dieckmann) to non-smooth, non-manifold spacetimes, enabling robust definitions of time functions in singular or discrete causal spaces. The results have potential implications for convergence theories and quantum gravity frameworks where spacetime may lack smooth manifold structure. Overall, the paper provides a comprehensive causal-theoretic blueprint for time-function existence and global hyperbolicity in a broad, abstract spacetime setting.

Abstract

In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by -causality, show that a modified notion of Geroch's volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

Paper Structure

This paper contains 16 sections, 35 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lorentzian pre-length spaces
  3. Basic definitions and properties
  4. Limit curve theorems
  5. Approximating Lorentzian pre-length spaces
  6. Causality conditions and time functions
  7. Time functions and $K$-causality
  8. Proof of Theorem \ref{['Kthm']}
  9. Volume time functions
  10. Averaged volume functions
  11. Averaged volume functions as generalized time functions
  12. Continuity of averaged volume functions
  13. Global hyperbolicity and Cauchy time functions
  14. Definitions and main result
  15. Properties of Cauchy sets
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose $X$ is a second countable, locally compact Lorentzian length space. Then $X$ is $K$-causal if and only if $X$ admits a time function.

Figures (2)

  • Figure 1: The sets $I^+(p)$ (blue, without boundary) and $J^+(p) = I^+(p) \cup \{p\}$ for a point $p$ in Example \ref{['example1']}.
  • Figure 2: The sets $I^-(p)$ (dark blue) and $I^-_r(p)$ (light and dark blue) for some point $p$ in Minkowski spacetime, with $d$ the Euclidean distance.

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: KuSa
  • Definition 2.2: KuSa
  • Lemma 2.3: Push-up KuSa
  • Definition 2.4: KuSa
  • Remark 2.5
  • Definition 2.6: KuSa
  • Definition 2.7: ACS
  • ...and 86 more