Introducing Various Notions of Distances between Space-Times

Abstract

We introduce the notion of causally-null-compactifiable space-times which can be canonically converted into a compact timed-metric-spaces using the cosmological time of Andersson-Howard-Galloway and the null distance of Sormani-Vega. We produce a large class of such space-times including future developments of compact initial data sets and regions which exhaust asymptotically flat space-times. We then present various notions of intrinsic distances between these space-times (introducing the timed-Hausdorff distance) and prove some of these notions of distance are definite in the sense that they equal zero iff there is a time-oriented Lorentzian isometry between the space-times. These definite distances enable us to define various notions of convergence of space-times to limit space-times which are not necessarily smooth. Many open questions and conjectures are included throughout.

v1: 135 pages with wide margins and line spacing, color images, (presented and shared in a reading seminar), v2: ready to submit to a journal for evaluation after adding remarks and citations, fixing typos and figures, (following the feedback by the team reading the paper), v3: added refs, Kuratowski maps corrected to Frechet maps,