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Spontaneous scalarization of charged black holes at the approach to extremality. (English) Zbl 1416.83082
Summary: 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 $$\gamma$$ – 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 $$\gamma$$, we show that for sufficiently large values of the electric charge $$Q$$ two independent domains of existence in the $$\gamma$$-$$Q$$-plane appear: one for positive $$\gamma$$ and one for negative $$\gamma$$. We demonstrate that this new domain for negative $$\gamma$$ exists because of the fact that the near-horizon geometry of a nearly extremally charged black hole is $$\mathrm{AdS}_2 \times S^2$$. This new domain appears for electric charges larger than approximately 74% of the extremal charge. For positive $$\gamma$$ we observe that a singularity with diverging curvature invariants forms outside the horizon when approaching extremality.

##### MSC:
 83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories 83C57 Black holes
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