Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity

Scalar-tensor theories of gravity where a new scalar degree of freedom couples to the Gauss-Bonnet invariant can exhibit the phenomenon of spontaneous black hole scalarization. These theories admit both the classic black hole solutions predicted by general relativity as well as novel hairy black hole solutions. The stability of hairy black holes is strongly dependent on the precise form of the scalar-gravity coupling. A radial stability investigation revealed that all scalarized black hole solutions are unstable when the coupling between the scalar field and the Gauss-Bonnet invariant is quadratic in the scalar, whereas stable solutions exist for exponential couplings. Here, we elucidate this behavior. We demonstrate that, while the quadratic term controls the onset of the tachyonic instability that gives rise to the black hole hair, the higher-order coupling terms control the nonlinearities that quench that instability and, hence, also control the stability of the hairy black hole solutions.

Figure 1

Bound scalar field solutions (with no nodes) in the quartic theory with $\zeta =-(3/2)\eta$. Solutions for which the effective potential $V_{\mathrm{eff}}$ defined in Eq. (15) is (is not) positive definite correspond to the solid (dashed) line. The horizontal line (marked by a circle) corresponds to the constant ${\phi }_{+}=\sqrt{3/2}$ solution, whereas the vertical line corresponds to the solutions of the quadratic sGB. For $\eta =0$, the theory reduces to that of scalar field minimally coupled to gravity. No-hair theorems force $\phi$ to have (any) constant value. See text for a detailed description.

Figure 2

$Q–M$ diagram for scalarized solutions in quartic sGB gravity with $n=0$, considering different values for $\zeta /\eta$. For comparison, we also show the solutions for exponential sGB gravity [cf. Eq. (5)]: for small $Q/{\overline{\eta }}^{1/2}$ (i.e., small scalar field amplitudes), the curve overlaps with the case $|\zeta /\eta |=1.5$, as it should. The vertical line represents the scalarization threshold $\eta ={\eta }_{\mathrm{thr}}$. Solutions to the left (right) of the vertical dotted line are stable (unstable). The inset shows additional illustrative curves showing the behavior near the scalarization threshold ${\eta }_{\mathrm{thr}}$. Curves with $\zeta /\eta \lesssim -0.8$ are always to the left of the scalarization threshold.

Figure 3

Eigenfrequencies of the unstable modes ${\omega }_I$ as function of the coupling $\eta$ for different $\zeta /\eta$ and for the Schwarzschild BH in sGB gravity. The Schwarzschild BH is unstable for $\eta \gtrsim 0.726$, as indicated by the linear analysis outlined in Sec. 3. For $\eta \lesssim 0.726$ we can see the instability time scale for the nodeless solutions for the cases $\zeta /\eta >-0.5$. The inset zooms-in into these frequencies.