Null Distance and 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.