Summary: A Lorentzian manifold, \(N\), endowed with a time function, \(\tau\), can be converted into a metric space using the null distance, \(\hat{d}_\tau\), defined by C. Sormani and C. Vega [Classical Quantum Gravity 33, No. 8, Article ID 085001, 29 p. (2016; Zbl 1337.83015)]. We show that if the time function is a regular cosmological time function as studied by L. Andersson et al. [Classical Quantum Gravity 15, No. 2, 309–322 (1998; Zbl 0911.53039)], and also by R. M. Wald and P. Yip [“On the existence of simultaneous synchronous coordinates in spacetimes with spacelike singularities”, J. Math. Phys. 22, No. 11, 2659–2665 (1981; doi:10.1063/1.524844)], or if, more generally, it satisfies the anti-Lipschitz condition of P. T. Chruściel et al. [Ann. Henri Poincaré 17, No. 10, 2801–2824 (2016; Zbl 1351.83006)], then the causal structure is encoded by the null distance in the following sense: for any \(p\in N\), there is an open neighborhood \(U_p\) such that for any \(q\in U_p\), we have \(\hat{d}_\tau(p, q) = \tau(q) - \tau(p)\) 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 \(\tau\) is a proper function. As a consequence, in dimension \(n + 1\), \(n \geq 2\), we prove that if there is a bijective map between two such spacetimes, \(F: M_1\rightarrow M_2\), which preserves the cosmological time function, \(\tau_2(F(p)) = \tau_1(p)\) for any \(p \in M_1\), and preserves the null distance, \(\hat{d}_{\tau_2}(F(p), F(q)) = \hat{d}_{\tau_1}(p, q)\) for any \(p, q\in M_1\), then there is a Lorentzian isometry between them, \(F_\ast g_1 = g_2\). 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.
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