Spontaneous scalarization of charged black holes at the approach to extremality

Abstract

We study static, spherically symmetric and electrically charged black hole solutions in a quadratic Einstein-scalar-Gauss-Bonnet gravity model. Very similar to the uncharged case, black holes undergo spontaneous scalarization for sufficiently large scalar-tensor coupling γ – a phenomenon attributed to a tachyonic instability of the scalar field system. While in the uncharged case, this effect is only possible for positive values of γ, we show that for sufficiently large values of the electric charge Q two independent domains of existence in the γ-Q-plane appear: one for positive γ and one for negative γ. We demonstrate that this new domain for negative γ exists because of the fact that the near-horizon geometry of a nearly extremally charged black hole is AdS2×S2. This new domain appears for electric charges larger than approximately 74% of the extremal charge. For positive γ we observe that a singularity with diverging curvature invariants forms outside the horizon when approaching extremality.