Stability and quasinormal modes of the massive scalar field around Kerr black holes

TL;DR

The paper analyzes the stability and quasinormal modes of a massive scalar field around Kerr black holes. It solves the radial equation using the Frobenius/Leaver continued-fraction method, with λ(ω) computed numerically, to obtain QNMs across a range of μM values. The results show all detected modes are damped for μM not large, indicating stability under non-superradiant boundary conditions, while the high-damping spectrum aligns with the massless case and quasi-resonances emerge at larger μ. These findings extend stability insights beyond Beyer’s inequality, and reveal that Kerr black holes exhibit quasi-resonances similar to Schwarzschild in the massive-scalar perturbation regime examined. Overall, the work confirms Kerr stability against small-to-moderate mass massive-scalar perturbations and clarifies the high-damping behavior of the spectrum.

Abstract

We find quasinormal spectrum of the massive scalar field in the background of the Kerr black holes. We show that all found modes are damped under the quasinormal modes boundary conditions when $μM$ is not large, thereby implying stability of the massive scalar field. This complements the region of stability determined by the Beyer inequality for large masses of the field. We show that, similar to the case of a non-rotating black holes, the massive term of the scalar field does not contribute in the regime of high damping. Thereby, the high damping asymptotic should be the same as for the massless scalar field.

Figures (3)

• Figure 1: $Im \omega$ as a function of azimutal number $m$ for $\ell = 6$, $a=0.3$, $n=0$, $\mu =0.1$.
• Figure 2: $Im \omega$ as a function of azimutal number $m$ for $\ell = 6$, $a=0.3$, $n=0$, $\mu =0.1$.
• Figure 3: $Re \omega$ vs $Im \omega$ for $n=1$, $a=0.5$, $\ell = m = 0$ for different values of $\mu$