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

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.