The null distance encodes causality

A Lorentzian manifold, N, endowed with a time function, τ, can be converted into a metric space using the null distance, d̂τ, defined by Sormani and Vega [Classical Quant. Grav. 33(8), 085001 (2016)]. We show that if the time function is a regular cosmological time function as studied by Andersson, Galloway, and Howard [Classical Quant. Grav. 15(2), 309–322 (1998)], and also by Wald and Yip [J. Math. Phys. 22, 2659–2665 (1981)], or if, more generally, it satisfies the anti-Lipschitz condition of Chruściel, Grant, and Minguzzi [Ann. Henri Poincare 17(10), 2801–2824 (2016)], then the causal structure is encoded by the null distance in the following sense: for any p ∈ N, there is an open neighborhood Up such that for any q ∈ Up, we have d̂τ(p,q)=τ(q)−τ(p) if and only if q lies in the causal future of p. The local encoding of causality can be applied to prove the global encoding of causality in a variety of settings, including spacetimes N where τ is a proper function. As a consequence, in dimension n + 1, n ≥ 2, we prove that if there is a bijective map between two such spacetimes, F : M1 → M2, which preserves the cosmological time function, τ2(F(p)) = τ1(p) for any p ∈ M1, and preserves the null distance, d̂τ2(F(p),F(q))=d̂τ1(p,q) for any p, q ∈ M1, then there is a Lorentzian isometry between them, F∗g1 = g2. This yields a canonical procedure allowing us to convert large classes of spacetimes into unique metric spaces with causal structures and time functions. This will be applied in our upcoming work to define spacetime intrinsic flat convergence.