Morse index and causal continuity. A criterion for topology change in quantum gravity

TL;DR

This paper investigates the relationship between Morse index critical points and causal continuity in topology-changing spacetimes within quantum gravity, establishing a criterion linking Morse indices to causal discontinuities.

Contribution

It proves a conjecture that Morse functions with index 1 or n-1 critical points lead to causally discontinuous geometries, while others can be causally continuous.

Findings

Critical points of index 1 or n-1 imply causal discontinuity.

Absence of such critical points allows for causally continuous Morse geometries.

Supports the conjecture relating Morse index and causal structure in quantum gravity.

Abstract

Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse points in topology changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n-1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n-1, then there exist associated Morse geometries which are causally continuous.