Four-dimensional quantum topology changes of space-times

Topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant are investigated. As Riemannian manifolds, which describe quantum tunnelings of space–time, constant negative curvature solutions of the Einstein equation, i.e., hyperbolic geometries are considered. Using four dimensional polytopes, one can explicitly construct hyperbolic manifolds with topologically nontrivial boundaries which describe topology changes. These instantonlike solutions are constructed out of 8‐cells, 16‐cells, or 24‐cells and have several points at infinity called cusps. The hyperbolic manifolds are noncompact because of the cusps but have finite volumes. Topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds are evaluated and it was found that the more complicated the topology changes, the more likely are suppressed.