Scalar Hairy Black Holes in Four Dimensions are Unstable

We present a numerical analysis of the stability properties of the black holes with scalar hair constructed by Herdeiro and Radu. We prove the existence of a novel gauge where the scalar field perturbations decouple from the metric perturbations, and analyze the resulting quasinormal mode spectrum. We find unstable modes with characteristic growth rates which for uniformly small hair are almost identical to those of a massive scalar field on a fixed Kerr background.

Figure 1

The imaginary part of $\varpi$ around HBHs, computed with $m=2$, as a function of $\mu M$—each curve contains a family of HBHs with a fixed value of ${\mu }_H/T_H$.

Figure 2

The ratio ${\varpi }_H/{\varpi }_K$, as a function of $\mu M$—each curve contains a family of HBHs with a fixed value of ${\mu }_H/T_H$ and Kerr BHs with the same $J{\mu }^2$ and $M\mu$ as the HBHs.