Angular and radial stabilities of spontaneously scalarized black holes in the presence of scalar-Gauss-Bonnet couplings

Abstract

We study the linear stability of spontaneously scalarized black holes (BHs) induced by a scalar field $\varphi$ coupled to a Gauss-Bonnet (GB) invariant $R_{\mathrm{GB}}^2$. For the scalar-GB coupling $\xi (\varphi )=(\eta /8)({\varphi }^2+\alpha {\varphi }^4)$, where $\eta$ and $\alpha$ are constants, we first show that there are no angular Laplacian instabilities of even-parity perturbations far away from the horizon for large multipoles $l\gg 1$. The deviation of angular propagation speeds from the speed of light is largest on the horizon, whose property can be used to put constraints on the model parameters. For $\alpha \gtrsim -1$, the region in which the scalarized BH is subject to angular Laplacian instabilities can emerge. Provided that $\alpha \lesssim -1$ and $-1/2<\alpha {\varphi }_0^2<-0.1155$, where ${\varphi }_0$ is the field value on the horizon with a unit of the reduced Planck mass $M_{\mathrm{Pl}}=1$, there are scalarized BH solutions satisfying all the linear stability conditions throughout the horizon exterior. We also study the stability of spontaneously scalarized BHs in scalar-GB theories with a nonminimal coupling $-\beta {\varphi }^{2}R/16$, where $\beta$ is a positive constant and $R$ is a Ricci scalar. As the amplitude of the field on the horizon approaches an upper limit $|{\varphi }_0|=4/\sqrt{\beta }$, one of the squared angular propagation speeds $c_{\Omega -}^2$ enters the instability region $c_{\Omega -}^2<0$. So long as $|{\varphi }_0|$ is smaller than a maximum value determined for each $\beta$ in the range $\beta >5$, however, the scalarized BHs are linearly stable in both angular and radial directions.

21 pages, 5 figures

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