Extremal black holes inD=4Gauss-Bonnet gravity

Abstract

We show that four-dimensional Einstein-Maxwell-dilaton-Gauss-Bonnet gravity admits asymptotically flat black hole solutions with a degenerate event horizon of the Reissner-Nordstr\om type ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$. Such black holes exist for the dilaton coupling constant within the interval $0\ensuremath{\le}{a}^{2}l{a}_{\mathrm{cr}}^{2}$. Black holes must be endowed with an electric charge and (possibly) with magnetic charge (dyons) but they cannot be purely magnetic. Purely electric solutions are constructed numerically and the critical dilaton coupling is determined ${a}_{\mathrm{cr}}\ensuremath{\simeq}0.488\text{ }219\text{ }703$. For each value of the dilaton coupling $a$ within this interval and for a fixed value of the Gauss-Bonnet coupling $\ensuremath{\alpha}$ we have a family of black holes parametrized by their electric charge. The relation between the mass, the electric charge, and the dilaton charge at both ends of the allowed interval of $a$ is reminiscent of the Bogomol'nyi-Prasad-Sommerfield condition for dilaton black holes in the Einstein-Maxwell-dilaton theory. The entropy of the dilaton-Gauss-Bonnet extremal black holes is twice the Bekenstein-Hawking entropy.