Self-dual Gravitational Instantons in Conformal Gravity: Conserved Charges and Thermodynamics

TL;DR

This paper explores non-Einstein gravitational instantons in conformal gravity, analyzing their topological and thermodynamic properties, and identifying boundary conditions that recover Einstein solutions within this broader class.

Contribution

It introduces new non-Einstein instanton solutions in conformal gravity, computes their topological and Noether charges, and studies their thermodynamics and boundary conditions.

Findings

Finite topological and Noether charges for instantons.

Existence of geometries with weakened AlAdS asymptotics.

Neumann boundary condition selects Einstein solutions.

Abstract

We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary condition that, as it happens in the asymptotically AdS sector, selects the Einstein solution out of the solutions of conformal gravity. All the geometries present non-vanishing Hirzebruch signature and Euler characteristic, being single-centered instantons. We compute the topological charges as well as the Noether charges of the Taub-NUT/Bolt and Eguchi-Hanson spacetimes, which happen to be finite. This enables us to study the thermodynamic properties of these geometries.