Taub-NUT Black Holes in Third order Lovelock Gravity

TL;DR

This paper explores Taub-NUT solutions in third order Lovelock gravity, revealing their dependence on base space geometry and identifying specific conditions for extremal and non-extremal solutions in eight dimensions.

Contribution

It provides the first detailed analysis of Taub-NUT solutions in third order Lovelock gravity, highlighting the dependence on base space geometry unlike in Einstein gravity.

Findings

Non-extremal NUT solutions exist only for specific base spaces in 8D.

Lovelock gravity admits extremal NUT solutions with particular base spaces.

The dependence of NUT solutions on base space geometry differs from Einstein gravity.

Abstract

We consider the existence of Taub-NUT solutions in third order Lovelock gravity with cosmological constant, and obtain the general form of these solutions in eight dimensions. We find that, as in the case of Gauss-Bonnet gravity and in contrast with the Taub-NUT solutions of Einstein gravity, the metric function depends on the specific form of the base factors on which one constructs the circle fibration. Thus, one may say that the independence of the NUT solutions on the geometry of the base space is not a robust feature of all generally covariant theories of gravity and is peculiar to Einstein gravity. We find that when Einstein gravity admits non-extremal NUT solutions with no curvature singularity at $r=N$, then there exists a non-extremal NUT solution in third order Lovelock gravity. In 8-dimensional spacetime, this happens when the metric of the base space is chosen to be $\Bbb{CP}^{3}$. Indeed, third order Lovelock gravity does not admit non-extreme NUT solutions with any other base space. This is another property which is peculiar to Einstein gravity. We also find that the third order Lovelock gravity admits extremal NUT solution when the base space is $T^{2}\times T^{2}\times T^{2}$ or $S^{2}\times T^{2}\times T^{2}$. We have extended these observations to two conjectures about the existence of NUT solutions in Lovelock gravity in any even-dimensional spacetime.