The null distance encodes causality

A Lorentzian manifold endowed with a time function, $\tau$, can be converted into metric space using the null distance, $\hat{d}_\tau$, defined by Sormani and Vega. We show that if function is proper regular cosmological as studied Andersson, Galloway Howard, also Wald Yip, or if, more generally, it satisfies anti-Lipschitz condition of Chru\'sciel, Grant Minguzzi, then causal structure encoded distance in following sense: $$ \hat{d}_\tau(p,q)=\tau(q)-\tau(p) \iff q \textrm{ lies future } p. As consequence, dimension $n+1$, $n\ge 2$, we prove there bijective map between two such spacetimes, $F: M_1\to M_2$, which preserves $ \tau_2(F(p))= \tau_1(p)$ for any p \in M_1$, $\hat{d}_{\tau_2}(F(p),F(q))=\hat{d}_{\tau_1}(p,q)$ $p,q\in isometry them, $F_*g_1=g_2$. This yields canonical procedure allowing us to convert spacetimes unique spaces structures functions. will applied our upcoming work define Spacetime Intrinsic Flat Convergence.