Charged black holes in a generalized scalar-tensor gravity model

We study 4-dimensional charged and static black holes in a generalized scalar-tensor gravity model, in which a shift symmetry for the scalar field exists. For vanishing scalar field the solution corresponds to the Reissner-Nordström (RN) solution, while solutions of the full scalar-gravity model have to be constructed numerically. We demonstrate that these black holes support Galilean scalar hair up to a maximal value of the scalar-tensor coupling that depends on the value of the charge and can be up to roughly twice as large as that for uncharged solutions. The Hawking temperature TH of the hairy black holes at maximal scalar-tensor coupling decreases continuously with the increase of the charge and reaches TH=0 for the highest possible charge that these solutions can carry. However, in this limit, the scalar-tensor coupling needs to vanish. The limiting solution hence corresponds to the extremal RN solution, which does not support regular Galilean scalar hair due to its AdS2×S2 near-horizon geometry.