Topology change with Morse functions: progress on the Borde-Sorkin conjecture

TL;DR

This paper investigates the causal properties of topology-changing spacetimes constructed via Morse functions, proving a special case of a conjecture and proposing a refined version based on heuristic arguments.

Contribution

It proves a specific case of the Borde-Sorkin conjecture on Morse spacetimes and suggests a refined conjecture based on heuristic reasoning.

Findings

Proved a special case of the Borde-Sorkin conjecture.

Argued heuristically that the original conjecture may be false.

Formulated a refined version of the conjecture.

Abstract

Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.