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Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds

Adam Rennie, Ben Whale

TL;DR

The paper addresses the problem of controlling Lorentzian lengths along limit curves when the underlying distance is incomplete. It develops a generalized limit-curve theorem for incomplete metric spaces and then leverages null distances from suitable time functions to translate compact-region length control into global Lorentzian length bounds. Key contributions include a local-to-global framework for limit curves in incomplete spaces, a length-control mechanism via null distances, and verification that null distances from regular cosmological time and surface functions on globally hyperbolic manifolds satisfy the required regularity conditions. The findings extend classical results to incomplete settings and provide practical tools for analyzing Lorentzian geometry with incomplete metrics, with direct applications to Sormani–Vega’s null distance and related time-function constructions.

Abstract

We prove a limit curve theorem for incomplete metric spaces. Our main application is to Sormani and Vegas' null distance, where our results give strong control on the Lorentzian lengths of limit curves. We also show that regular cosmological time functions and the surface function of a Cauchy surface in a globally hyperbolic manifold define such a null distance.

Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds

TL;DR

The paper addresses the problem of controlling Lorentzian lengths along limit curves when the underlying distance is incomplete. It develops a generalized limit-curve theorem for incomplete metric spaces and then leverages null distances from suitable time functions to translate compact-region length control into global Lorentzian length bounds. Key contributions include a local-to-global framework for limit curves in incomplete spaces, a length-control mechanism via null distances, and verification that null distances from regular cosmological time and surface functions on globally hyperbolic manifolds satisfy the required regularity conditions. The findings extend classical results to incomplete settings and provide practical tools for analyzing Lorentzian geometry with incomplete metrics, with direct applications to Sormani–Vega’s null distance and related time-function constructions.

Abstract

We prove a limit curve theorem for incomplete metric spaces. Our main application is to Sormani and Vegas' null distance, where our results give strong control on the Lorentzian lengths of limit curves. We also show that regular cosmological time functions and the surface function of a Cauchy surface in a globally hyperbolic manifold define such a null distance.

Paper Structure

This paper contains 11 sections, 28 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Some Lorentzian geometry
  4. Cosmological time, surface functions and the null distance
  5. Smooth approximation of Lipschitz functions
  6. Limit curve theorems for complete spaces
  7. Lorentzian length control
  8. What goes wrong in incomplete spaces
  9. The limit curve theorem for incomplete metric spaces
  10. Length control over limit curves via null distances
  11. Null distances of surface functions

Key Result

proposition 2.6

sormani2016null Let $f:M\to\mathbb{R}$ be a generalised time function. If $f$ is locally anti-Lipschitz then $d(\cdot,\cdot; f)$ is a distance. If, in addition, $f$ is continuous then $d(\cdot,\cdot; f)$ induces the manifold topology.

Figures (1)

  • Figure 1: An illustration of Example \ref{['ex_convergence_in_incomplete']}. The circle represents the removed origin in $\mathbb{R}^2$. The dashed lines represent the $x$ and $y$ axis. The remaining curves present the images of three of the $\gamma_i$'s. The intersection of these curves with the $x$ axis tends to the removed origin as $i\to\infty$.

Theorems & Definitions (60)

  • definition 2.1: Causal, timelike and null curves
  • definition 2.2: Cosmological time, andersson1998cosmological
  • definition 2.3: Surface function
  • definition 2.4: sormani2016null
  • definition 2.5
  • proposition 2.6
  • definition 2.7: Clarke's generalised directional derivative, clarke1990optimization
  • definition 2.8: Clarke's generalised gradient, clarke1990optimization
  • Theorem 2.9: clarke1990optimization
  • Theorem 2.10: czarnecki2006approximation
  • ...and 50 more