Charged spherically symmetric black holes in scalar-tensor Gauss–Bonnet gravity

Abstract

We derive a novel class of four-dimensional black hole (BH) solutions in Gauss–Bonnet (GB) gravity coupled with a scalar field in presence of Maxwell electrodynamics. In order to derive such solutions, we assume the ansatz for metric potentials. Due to the choice of the ansatz of the metric, the Reissner Nordström gauge potential cannot be recovered because of the presence of higher-order terms which are not allowed to be vanishing. Moreover, the scalar field is not allowed to vanish. If it vanishes, a function of the solution results undefined. Furthermore, it is possible to show that the electric field is of higher-order in the monopole expansion: this fact explicitly comes from the contribution of the scalar field. Therefore, we can conclude that the GB scalar field acts as non-linear electrodynamics creating monopoles, quadrupoles, etc in the metric potentials. We compute the invariants associated with the BHs and show that, when compared to Schwarzschild or Reissner–Nordström space-times, they have a soft singularity. Also, it is possible to demonstrate that these BHs give rise to three horizons in AdS space-time and two horizons in dS space-time. Finally, thermodynamic quantities can be derived and we show that the solution can be stable or unstable depending on a critical value of the temperature.