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.
Highlights
Conformal gravity is a very interesting theory which has been considered in many different contexts
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
Among all the contexts in which four-dimensional conformal gravity has been considered, we find that it has been studied as an ultraviolet completion of Einstein gravity [2], the theory is anomalous as its gauge symmetry gets broken by loop corrections
Summary
Conformal gravity is a very interesting theory which has been considered in many different contexts. [27], Maldacena has observed a remarkable connection between classical fourdimensional conformal gravity and Einstein gravity and used it to show that the former theory, when supplemented with an appropriate boundary condition, led to obtain the semiclassical wave function of the Universe of asymptotically de Sitter spacetime This is achieved by removing the ghost mode that conformal gravity exhibits by imposing the mentioned boundary condition. We will discuss a generalization of the Euclidean Taub-NUT [57,58] and Taub-bolt [59] solutions to the case of conformal gravity.2 This family of solutions presents self-dual geometries and includes the most general static, spherically symmetric black hole solutions as particular cases.
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