There is no $R^3 X S^1$ vacuum gravitational instanton

TL;DR

This paper proves that there are no nontrivial vacuum gravitational instantons with topology R^3 x S^1, establishing that such solutions must be flat and thus constraining models in quantum and classical gravity.

Contribution

It demonstrates that all vacuum gravitational instantons with topology R^3 x S^1 are necessarily flat, ruling out nontrivial solutions and impacting theories in quantum and classical gravity.

Findings

All such instantons have the same asymptotic structure as the Schwarzschild instanton.

If the Ricci tensor is non-negative, the manifold must be flat.

No nontrivial vacuum gravitational instanton exists with this topology.

Abstract

Gravitational instantons, solutions to the euclidean Einstein equations, with topology $R^3 XS^1$ arise naturally in any discussion of finite temperature quantum gravity. This Letter shows that all such instantons (irrespective of their interior behaviour) must have the same asymptotic structure as the Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor of the manifold is non-negative it must be flat. One special case is when the Ricci tensor vanishes; hence one can conclude that there is no nontrivial vacuum gravitational instanton. This result has uses both in quantum and classical gravity. It places a significant restriction on the instabilities of hot flat space. It also can be used to show that any static vacuum Lorentzian Kaluza-Klein solution is flat.