Null Distance and Temporal Functions

TL;DR

This paper explores the properties of null distance functions in Lorentzian geometry, establishing bounds on null distances in terms of Riemannian distances on spacelike hypersurfaces and confirming a conjecture relating level set diameters to Big Bang singularities.

Contribution

It proves that null distances are bounded by Riemannian distances on level sets of temporal functions and confirms a conjecture linking shrinking level set diameters to spacetime singularities.

Findings

Null distance is bounded by Riemannian distance on level sets.

Confirmed a conjecture relating shrinking level set diameters to Big Bang singularities.

Established properties of null distances for smooth temporal functions.

Abstract

The notion of null distance was introduced by Sormani and Vega as part of a broader program to develop a theory of metric convergence adapted to Lorentzian geometry. Given a time function $\tau$ on a spacetime $(M,g)$, the associated null distance $\hat{d}_\tau$ is constructed from and closely related to the causal structure of $M$. While generally only a semi-metric, $\hat{d}_\tau$ becomes a metric when $\tau$ satisfies the local anti-Lipschitz condition. In this work, we focus on temporal functions, that is, differentiable functions whose gradient is everywhere past-directed timelike. Sormani and Vega showed that the class of $C^1$ temporal functions coincides with that of $C^1$ locally anti-Lipschitz time functions. When a temporal function $f$ is smooth, its level sets $M_t = f^{-1}(t)$ are spacelike hypersurfaces and thus Riemannian manifolds endowed with the induced metric $h_t$. Our main result establishes that, on any level set $M_t$ where the gradient $\nabla f$ has constant norm, the null distance $\hat{d}_f$ is bounded above by a constant multiple of the Riemannian distance $d_{h_t}$. Applying this result to a smooth regular cosmological time function $\tau_g$ -- as introduced by Andersson, Galloway, and Howard -- we prove a theorem confirming a conjecture of Sakovich and Sormani (arXiv:2410.16800, 2025): if the diameters of the level sets $M_t = \tau_g^{-1}(t)$ shrink to zero as $t \to 0$, then the spacetime exhibits a Big Bang singularity, as defined in their work.