Tunelling with a Negative Cosmological Constant

TL;DR

This paper explores how hyperbolic geometry and gravitational instantons relate to quantum cosmology, proposing new instantons, analyzing their entropies, and discussing implications for universe creation and perpetual motion scenarios.

Contribution

It introduces new gravitational instantons in hyperbolic geometry and examines their role in universe creation, entropy, and topological complexity.

Findings

Gravitational instantons relate to hyperbolic topological entropy.

Universes with negative cosmological constant cannot be created from nothing with supergravity.

Wormholes may serve as perpetual motion machines.

Abstract

The point of this paper is see what light new results in hyperbolic geometry may throw on gravitational entropy and whether gravitational entropy is relevant for the quantum origin of the univeres. We introduce some new gravitational instantons which mediate the birth from nothing of closed universes containing wormholes and suggest that they may contribute to the density matrix of the universe. We also discuss the connection between their gravitational action and the topological and volumetric entropies introduced in hyperbolic geometry. These coincide for hyperbolic 4-manifolds, and increase with increasing topological complexity of the four manifold. We raise the questions of whether the action also increases with the topological complexity of the initial 3-geometry, measured either by its three volume or its Matveev complexity. We point out, in distinction to the non-supergravity case, that universes with domains of negative cosmological constant separated by supergravity domain walls cannot be born from nothing. Finally we point out that our wormholes provide examples of the type of Perpetual Motion machines envisaged by Frolov and Novikov.