Existence of uniform Temple charts and applications to null distance

In this paper, we prove that Temple’s cylindrical future null coordinate charts can be constructed uniformly and we estimate the gradients of their optical functions. We then apply these charts to study a spacetime $(N,g)$ that has been converted into a definite metric space $(N,{\widehat{d}}_{\tau })$, where ${\widehat{d}}_{\tau }$ is the null distance of Sormani and Vega defined using a weak temporal function $\tau$. In particular, we prove that $(N,{\widehat{d}}_{\tau })$ is a rectifiable metric space, where the causal structure is locally encoded by $\tau$ and ${\widehat{d}}_{\tau }$. As a consequence, applying a classical theorem of Hawking and following a technique developed by Sakovich and Sormani, we can prove a Lorentzian isometry theorem, generalizing our earlier result.