Taub-NUT/bolt black holes in Gauss-Bonnet-Maxwell gravity

We present a class of higher-dimensional solutions to Gauss-Bonnet-Maxwell equations in $2k+2$ dimensions with a $U(1)$ fibration over a $2k$-dimensional base space $\mathcal{B}$. These solutions depend on two extra parameters, other than the mass and the Newman-Unti-Tamburino charge, which are the electric charge $q$ and the electric potential at infinity $V$. We find that the form of metric is sensitive to geometry of the base space, while the form of electromagnetic field is independent of $\mathcal{B}$. We investigate the existence of Taub-Newman-Unti-Tamburino/bolt solutions and find that in addition to the two conditions of uncharged Newman-Unti-Tamburino solutions, there exist two other conditions. These two extra conditions come from the regularity of vector potential at $r=N$ and the fact that the horizon at $r=N$ should be the outer horizon of the black hole. We find that for all nonextremal Newman-Unti-Tamburino solutions of Einstein gravity having no curvature singularity at $r=N$, there exist Newman-Unti-Tamburino solutions in Gauss-Bonnet-Maxwell gravity. Indeed, we have nonextreme Newman-Unti-Tamburino solutions in $2+2k$ dimensions only when the $2k$-dimensional base space is chosen to be ${\mathbb{C}\mathbb{P}}^{2k}$. We also find that the Gauss-Bonnet-Maxwell gravity has extremal Newman-Unti-Tamburino solutions whenever the base space is a product of 2-torii with at most a 2-dimensional factor space of positive curvature, even though there a curvature singularity exists at $r=N$. We also find that one can have bolt solutions in Gauss-Bonnet-Maxwell gravity with any base space. The only case for which one does not have black hole solutions is in the absence of a cosmological term with zero curvature base space.

Figure 1

$F_{\mathrm{nut}}(r)$ versus $r$ with $\mathcal{B}={\mathbb{C}\mathbb{P}}^2$ for $N=1$, $\alpha =0.1$, $\Lambda =-2$, and $q=q_{\mathrm{crit}}=24.815$ (continuous line), $q<q_{\mathrm{crit}}$ (dotted line) and $q>q_{\mathrm{crit}}$ (bold line).

Figure 2

$F_{\mathrm{nut}}(r)$ versus $r$ with ${\mathcal{B}}_B=T^2\times T^2$ for $N=1$, $\alpha =0.1$, $\Lambda =-2$, and $q_{\mathrm{crit}}=11.322$ (continuous line), $q<q_{\mathrm{crit}}$ (dotted line) and $q>q_{\mathrm{crit}}$ (bold line).

Figure 3

$F_{\mathrm{nut}}(r)$ versus $r$ with ${\mathcal{B}}_B=T^2\times S^2$ for $N=1$, $\alpha =0.1$, $\Lambda =-2$, and $q_{\mathrm{crit}}=19.973$ (continuous line), $q<q_{\mathrm{crit}}$ (dotted line) and $q>q_{\mathrm{crit}}$ (bold line).