Properties of the Null Distance and Spacetime Convergence

Abstract

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

57 pages, 9 figures, comments welcome. v2: removed parts of sections 3.3 and 3.4 but the main results and other sections are not affected, minor changes throughout. v3: Rewrote and expanded background section, rewrote Theorem 1.3, referee comments addressed. To appear in IMRN

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