Diamond-Shaped Regions as Microcosmoi

TL;DR

This paper introduces a geometric method to construct a global time function for diamond-shaped regions in any spacetime, linking it to conformal symmetries in Minkowski space and Lorentzian distances generally.

Contribution

It provides an intrinsic geometric construction of a global time function applicable to arbitrary spacetimes, extending known Minkowski results.

Findings

Time function expressed via Lorentzian distance

Flow of associated diffeomorphisms matches conformal subgroup in Minkowski

Asymptotic behavior similar to Minkowski spacetime

Abstract

We give a geometrically intrinsic construction of a global time function for relatively compact diamond-shaped regions in arbitrary spacetimes. In the case of Minkowski spacetime, the flow of diffeomorphisms associated to a suitably normalized gradient of this time function becomes the conformal isotropy subgroup of the diamond. In full generality, this time function is elegantly expressed in terms of the Lorentzian distance function, and it has an asymptotic behavior at large absolute times similar to the one in Minkowski spacetime.