Radial Oscillations of Scalar Hair in Black Hole Bombs

TL;DR

The paper addresses a nonlinear mechanism by which a stable Reissner–Nordström black hole evolves into a hairy state and can trigger a black hole bomb without artificial confinement. It develops an Einstein–Maxwell–scalar model with a self-interacting massive complex scalar, and analyzes the coupled dynamics in spherical Painlevé–Gullstrand coordinates. A key finding is that scalar-hair quantities oscillate with a common complex frequency $\omega = \omega_R + i\omega_I$, and both $\omega_R$ and $\omega_I$ scale linearly with the gauge coupling $q$; the horizon radius and hair mass also increase with $q$, while a critical initial charge $Q_c$ delineates bomb-triggering regimes. This work demonstrates a purely nonlinear route to black hole hair and bombs without mirrors or AdS, with implications for generalizations to rotating black holes and other hair types, and suggests perturbative approaches for deeper insight.

Abstract

Recent research has revealed a novel nonlinear mechanism, distinct from the linear superradiant instability, which triggers the black hole bomb phenomenon. Introducing a massive complex scalar field with nonlinear self-interactions drives the Reissner-Nordström black hole to shed substantial energy, thereby triggering a black hole bomb. Radial oscillations in the scalar hair profile are observed during this process. In this paper, we further reveal that physical quantities associated with scalar hair exhibit identical oscillation patterns during the evolution of the black hole-scalar field system. Moreover, the oscillation frequency exhibits a linear dependence on the gauge coupling constant of the scalar field with other parameters fixed. Meanwhile, the horizon radius of hairy black holes and the mass within the horizon increase monotonically with the gauge coupling constant. We have also identified a critical initial charge value that distinguishes hairy solutions that trigger black hole bombs from those that do not.

Figures (6)

• Figure 1: The evolution of the scalar field energy $E_{\psi},$ black hole mass $M_{B},$ scalar field charge $Q_{\psi},$ black hole charge $Q_{h}$ and total charge $Q$ starting from an RN black hole with $M_{0}=1,Q=0.9M_{0}$ under the perturbation (\ref{['eq:IScalar']} ) with $p=0.26$ when $qM_{0}=4$.
• Figure 2: The ultimate stable distributions of the scalar field value $|\psi|$ and of the energy density $\rho_{\psi}.$ Initial parameters: $M=M_{0},Q=0.9M_{0},p=0.26,qM_{0}=4$.
• Figure 3: The snapshots of the energy density $\rho_{\psi}$ of the scalar field during the evolution. Showing one and a half oscillation period s, Panels 1 and 3 correspond to maxima, while Panels 2 and 4 correspond to a minima. The black regions in the lower left corner represent the region inside the apparent horizon at the corresponding times. Initial parameters: $M=M_{0},Q=0.9M_{0},p=0.26,qM_{0}=4$.
• Figure 4: Time evolution of the scalar field value $|\psi|$ and energy density $\rho_{\psi}$ at radial positions $r=3.5r_{h0},4r_{h0},4.5r_{h0},5r_{h0},5.5r_{h0},6r_{h0},$ where $r_{h0}=1.43589$ is the horizon radius. Initial parameters: $M=M_{0},Q=0.9M_{0},p=0.26,qM_{0}=4.$
• Figure 5: Variation of the scalar field oscillation frequency $\omega_{R}$(real part) and decay coefficient $\omega_{I}$(imaginary part) versus both initial charge $Q$ and gauge coupling constant $q.$ Other initial parameters: $M=M_{0},p=0.26$ (fixed).