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

arXiv commentSmall changes in v3: typos fixed, added reference to Yodzis in introduction. To appear in Adv. Theor. Math. Phys. 23 pages, 8 figures

References in corpus (9)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Forks in the Road, on the Way to Quantum Gravity
• Feynman Propagator for a Free Scalar Field on a Causal Set
• Limit curve theorems in Lorentzian geometry
• Scalar Field Theory on a Causal Set in Histories Form
• Topology Change and Causal Continuity
• Topology change in quantum gravity
• K-causality and degenerate spacetimes
• Quantum fields on semi-globally hyperbolic space-times

Cited by in corpus (2)

• Morse-Bott inequalities, Topology Change and Cobordisms to Nothing
• A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes