The Null distance encodes causality

Abstract

A Lorentzian manifold endowed with a time function, $\tau$, can be converted into a metric space using the null distance, ${\hat{d}}_{\tau }$, defined by Sormani and Vega. We show that if the time function is a proper regular cosmological time function as studied by Andersson, Galloway and Howard, and also by Wald and Yip, or if, more generally, it satisfies the anti-Lipschitz condition of Chruściel, Grant and Minguzzi, then the causal structure is encoded by the null distance in the following sense: $${\hat{d}}_{\tau }(p,q)=\tau (q)-\tau (p)⟺q\text{ lies in the causal future of }p.$$ As a consequence, in dimension $n+1$, $n\ge 2$, we prove that if there is a bijective map between two such spacetimes, $F:M_1\to M_2$, 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 such 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.

arXiv comment24 pages, 4 figures

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• Global Hyperbolicity through the Eyes of the Null Distance
• On the asymptotic assumptions for Milne-like spacetimes
• Volume Singularities in General Relativity
• Differentiability and Other Properties of the Cosmological Volume Function
• A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes