The null distance encodes causality

A Lorentzian manifold, N, endowed with a time function, tau, can be converted into a metric space using the null distance, d circumflex expressionccent (tau), 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 Chrusciel, 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 is an element of N, there is an open neighborhood U-p such that for any q is an element of U-p, we have d circumflex expressionccent (tau)(p, q) = tau(q) - tau(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 T 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 : M-1 -> M-2, which preserves the cosmological time function, tau(2)(F(p)) = tau(1)(p) for any p is an element of M-1, and preserves the null distance, d circumflex expressionccent tau(2) (F(p), F(q)) = d circumflex expressionccent tau(1) (p, q) for any p, q is an element of M-1, then there is a Lorentzian isometry between them, F*g(1) = g(2). 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.