K-causality and degenerate spacetimes

TL;DR

This paper investigates the properties of the causal relation $K^+$ in Lorentzian spacetimes, especially those with topology changes and degeneracies, revealing its robustness and connections to causal continuity.

Contribution

It provides a detailed analysis of $K^+$ in topology-changing Morse spacetimes, extending previous results to cases with degeneracies and linking it to causal continuity.

Findings

$K^+$ coincides with the Seifert relation in stably causal spacetimes

$K^+$ is robust under degeneracies and topology changes

Causal continuity can be characterized using $K^+$ in general spacetimes

Abstract

The causal relation $K^+$ was introduced by Sorkin and Woolgar to extend the standard causal analysis of $C^2$ spacetimes to those that are only $C^0$. Most of their results also hold true in the case of spacetimes with degeneracies. In this paper we seek to examine $K^+$ explicitly in the case of Lorentzian topology changing Morse spacetimes containing isolated degeneracies. We first demonstrate some interesting features of this relation in globally Lorentzian spacetimes. In particular, we show that $K^+$ is robust and that it coincides with the Seifert relation when the spacetime is stably causal. Moreover, the Hawking and Sachs characterisation of causal continuity translates into a natural expression in terms of $K^+$ for general spacetimes. We then examine $K^+$ in topology changing Morse spacetimes both with and without the degeneracies and find further characterisations of causal continuity.