Static- and Stationary-complete Spacetimes: Algebraic and Causal Structures

TL;DR

This paper analyzes the global algebraic and causal structures of static and stationary spacetimes, introducing new tools like a causality detection measure and classifying spacetimes based on their fundamental cocycle behavior.

Contribution

It introduces a new causality detection measure and provides a complete classification of stationary spacetimes based on their fundamental cocycle, advancing understanding of their global properties.

Findings

A new causality detection measure for stationary spacetimes.

Complete classification of stationary spacetimes via the fundamental cocycle.

Analysis of how quotient operations affect global hyperbolicity.

Abstract

This is intended as an analysis of the global properties of static and stationary spacetimes with complete (timelike) Killing field, with particular attention to quotients by group actions. This is presented in terms of algebraic structures which are fairly simple for the static case and more involved for the stationary case; the most important tool, the fundamental cocycle, is a cohomological class for static spacetimes but of somewhat looser structure in the stationary case. In particular: (1) A new measurement, similar to the spacetime interval in Minkowski space, is devised for detecting whether two points are causally related in a stationary spacetime; this proves very useful for analysis. (2) All stationary spacetimes are categorized by how they behave with respect to the fundamental cocycle; this enables a complete characterization of global causality properties. (3) It is shown how these tools determine whether global hyperbolicity of a stationary spacetime is inherited by its quotients. (4) Examples are examined in detail, a large range including both ones of mathematical curiosity and ones of physical interest, such as cosmic strings in flat, accelerated, Schwarzschild, Kerr, and other backgrounds.