You Can't Get There from Here: Constraints on Topology Change

Summary

Abstract

Recent studies of topology change and other topological effects have been typically initiated by considering semiclassical amplitudes for the transition of interest. Such amplitudes are constructed from riemannian or possibly complex solutions of the Einstein equations. This simple fact limits the possible transitions for a variety of possible matter sources. The case of riemannian solutions with strongly positive stress-energy is the most restrictive: no possible solution exists that mediates topology change between two or more boundary manifolds. Restrictions also exist for riemannian solutions with negative or indefinite stress-energy sources: all boundary manifolds must admit a metric with nonnegative curvature. This condition strongly restricts the possible topologies of the boundary manifolds given that most manifolds only admit metrics with negative curvature. Finally, the ability to construct explicit examples of topology changing instantons relies on the existence of a symmetry or symmetries that simplify the relevant equations. It follows that initial data with symmetry cannot give rise to a nonsymmetric solution of the Einstein equations. Moreover, analyticity properties of the Einstein equations strongly suggest that in general, complex solutions encounter the same topological restrictions. Thus the possibilities for topology change in the semiclassical limit are highly limited, indicating that detailed investigations of such effects should be carried out in terms of a more general construction of quantum amplitudes.