Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes

TL;DR

This paper proves that in globally hyperbolic maximal conformally flat spacetimes, the indecomposable past and future sets are domains of injectivity of the developing map, and maximal spacetimes have IPs/IFs conformally equivalent to Minkowski domains.

Contribution

It provides a new proof of completeness for these spacetimes and characterizes maximal IPs/IFs as conformally equivalent to Minkowski domains.

Findings

IPs/IFs are domains of injectivity of the developing map.

In maximal spacetimes, IPs/IFs are conformally equivalent to Minkowski domains.

The proof simplifies previous arguments on completeness.

Abstract

Let M be a globally hyperbolic conformally spacetime. We prove that the indecomposable past/future sets (abbrev. IPs/IFs) -in the sense of Penrose, Kronheimer and Geroch -of the universal cover of M are domains of injectivity of the developing map. This relies on the central observation that diamonds are domains of injectivity of the developing map. Using this, we provide a new proof of a result of completeness by C. Rossi, which notably simplifies the original arguments. Furthermore, we establish that if, in addition, M is maximal, the IPs/IFs are maximal as globally hyperbolic conformally flat spacetimes. More precisely, we show that they are conformally equivalent to regular domains of Minkowski spacetime as defined by F. Bonsante.