Null Distance and Temporal Functions

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 $τ$ on a spacetime $(M,g)$, the associated null distance d̂_τ is constructed from and closely related to the causal structure of M. While generally only a semi-metric, d̂_τ becomes a metric when $τ$ 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 C¹ temporal functions coincides with that of C¹ locally anti-Lipschitz time functions. When a temporal function f is smooth, its level sets Mₜ = f⁻¹(t) are spacelike hypersurfaces and thus Riemannian manifolds endowed with the induced metric hₜ. Our main result establishes that, on any level set Mₜ where the gradient ∇ f has constant norm, the null distance d̂f is bounded above by a constant multiple of the Riemannian distance dhₜ. Applying this result to a smooth regular cosmological time function τg -- 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 Mₜ = τg⁻¹(t) shrink to zero as t → 0, then the spacetime exhibits a Big Bang singularity, as defined in their work.