Stability of scalarized charged black holes in the Einstein–Maxwell–Scalar theory

We analyze the stability of scalarized charged black holes in the Einstein–Maxwell–Scalar (EMS) theory with quadratic coupling. These black holes are labelled by the number of $$n=0,1,2,\ldots $$ , where $$n=0$$ is called the fundamental black hole and $$n=1,2,\ldots $$ denote the n-excited black holes. We show that the $$n=0$$ black hole is stable against full perturbations, whereas the $$n=1,2$$ excited black holes are unstable against the $$s(l=0)$$ -mode scalar perturbation. This is consistent with the EMS theory with exponential coupling, but it contrasts to the $$n=0$$ scalarized black hole in the Einstein–Gauss–Bonnet–Scalar theory with quadratic coupling. This implies that the endpoint of unstable Reissner-Nordström black holes with $$\alpha >8.019$$ is the $$n=0$$ black hole with the same q. Furthermore, we study the scalarized charged black holes in the EMS theory with scalar mass $$m^2_\phi =\alpha /\beta $$ .

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