Causal continuity in degenerate spacetimes

TL;DR

This paper explores the causal structure of degenerate spacetimes constructed from Morse functions, confirming a conjecture that causal discontinuity relates to Morse index in these geometries.

Contribution

It introduces a method to analyze causal structures in degenerate spacetimes and verifies a conjecture linking causal discontinuity to Morse index.

Findings

Causal discontinuity occurs if and only if the Morse index is 1 or n-1.

Verification of Borde and Sorkin's conjecture in specific geometries.

Analysis of causal structure near degeneracies in Morse-function-based spacetimes.

Abstract

A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkin's conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1.