The Null distance encodes causality

TL;DR

This paper demonstrates that the null distance on certain Lorentzian manifolds encodes their causal structure, enabling a canonical conversion of spacetimes into metric spaces with preserved causality and time functions.

Contribution

It establishes conditions under which the null distance encodes causality and proves that maps preserving this distance and the cosmological time induce Lorentzian isometries.

Findings

Null distance encodes causal structure under specific conditions.

Bijective maps preserving null distance and cosmological time are Lorentzian isometries.

Framework for converting spacetimes into metric spaces with causal structure.

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 Chru\'sciel, Grant and Minguzzi, then the causal structure is encoded by the null distance in the following sense: $$ \hat{d}_\tau(p,q)=\tau(q)-\tau(p) \iff q \textrm{ lies in the causal future of } p. $$ 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, $F_*g_1=g_2$. 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.