Four Dimensional Quantum Topology Changes of Spacetimes

TL;DR

This paper explores quantum topology changes in four-dimensional spacetimes using hyperbolic geometries and polytopes, analyzing their amplitudes and suppression factors in quantum cosmology with a negative cosmological constant.

Contribution

It introduces explicit constructions of hyperbolic manifolds with non-trivial boundaries to model topology change in quantum cosmology, using polytopes like 8-cell, 16-cell, and 24-cell.

Findings

Topology change amplitudes depend on the volume of hyperbolic manifolds.

More complex topology changes are more suppressed.

Finite volume hyperbolic manifolds with cusps are used to model quantum tunneling.

Abstract

We investigate topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant. As Riemannian manifolds which describe quantum tunnelings of spacetime we consider constant negative curvature solutions of the Einstein equation i.e. hyperbolic geometries. Using four dimensional polytopes, we can explicitly construct hyperbolic manifolds with topologically non-trivial boundaries which describe topology changes. These instanton-like solutions are constructed out of 8-cell's, 16-cell's or 24-cell's and have several points at infinity called cusps. The hyperbolic manifolds are non-compact because of the cusps but have finite volumes. Then we evaluate topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds. We find that the more complicated are the topology changes, the more likely are suppressed.