Topology Change in Classical General Relativity

This paper clarifies some aspects of Lorentzian topology change, and it extends to a wider class of spacetimes previous results of Geroch and Tipler that show that topology change is only to be had at a price. The scenarios studied here are ones in which an initial spacelike surface is joined by a connected “interpolating spacetime” to a final spacelike surface, possibly of different topology. The interpolating spacetime is required to obey a condition called causal compactness, a condition satisfied in a very wide range of situations. No assumption is made about the dimension of spacetime. First, it is stressed that topology change is kinematically possible; i.e., if a field equation is not imposed, it is possible to construct topology-changing spacetimes with non-singular Lorentz metrics. Simple 2-dimensional examples of this are shown. Next, it is shown that there are problems in such spacetimes: Geroch’s closed-universe argument is applied to causally compact spacetimes to show that even in this wider class of spacetimes there are causality violations associated with topology change. It follows from this result that there will be causality violations if the initial (or the final) surface is not connected, even when there is no topology change. Further, it is shown that in dimensions ≥ 3 causally compact topology-changing spacetimes cannot satisfy Einstein’s equation (with a reasonable source); i.e., there are severe dynamical obstructions to topology change. This result extends a previous one due to Tipler. Like Tipler’s result, it makes no assumptions about geodesic completeness; i.e., it does not † Permanent address: Long Island University, Southampton, NY 11968, and High Energy Theory Group, Brookhaven National Laboratory, Upton, NY 11973. ∗ Electronic mail: [email protected] permit topology change even at the price of singularities (of the standard incomplete-geodesic variety). Brief discussions are also given of the restrictions that are placed on the source in this result, of the possibility of creating exotic differentiable manifolds, and of ways in which the results of this paper might be circumvented. PACS numbers: 02.40.-k, 04.20.Gz, 04.20.-q