Scalar perturbation around a rotating Kalb-Ramond BTZ black hole

TL;DR

The paper analyzes scalar perturbations of a rotating BTZ black hole in a Kalb–Ramond background, showing that the radial Klein–Gordon equation reduces to a general Heun equation. Quasinormal modes are computed under generic Robin boundary conditions via a Wronskian matching of Heun solutions, with the nonzero KR parameter $\ell$ substantially shifting the spectrum and breaking left–right symmetry as angular momentum grows. A flux-based analysis in ingoing EF coordinates demonstrates that superradiant instability occurs for the fundamental left-branch mode when the Robin parameter crosses a threshold $\xi_c$, with this threshold and the instability window being strongly modulated by $\ell$. The work highlights the KR field as a key factor in black hole perturbations, providing a framework to distinguish genuine superradiance from bulk AdS instabilities and guiding future studies of higher-spin perturbations and scalar clouds.

Abstract

We investigate the scalar perturbation of a newly proposed Kalb-Ramond (KR) BTZ-like black hole. After the separation of variables for the Klein-Gordon equation, we find that the radial part reduces to the general Heun equation. Using the Heun function, we compute quasinormal modes (QNMs) subject to generic Robin boundary conditions, which shows that the KR parameter substantially modifies the QNM spectrum and only the fundamental mode on the left branch has an instability. To ascertain whether the instability is superradiant, we further analyze how the KR field changes the energy and angular momentum fluxes. Our results show that the KR parameter shifts the threshold and the range of the Robin coupling parameter where the superradiance occurs, underscoring the importance of the KR field in modeling black hole perturbations.

Figures (4)

• Figure 1: Trajectories in the real part $\omega_R$ and imaginary parts $\omega_I$ of selected QNM frequencies with $n=0$ as the coupling $\xi$ varies for a KR BTZ black hole with different KR parameters $\ell=-0.03$, $0$, $0.03$ with $\mu^2=-0.65$, $m=1$, $\lambda=1$ and $M=34$. Panels from left to right correspond to angular momenta $j=0, 10$, and $30$. Filled circles and triangles denote solutions satisfying Dirichlet and Neumann boundary conditions, respectively.
• Figure 2: Real and imaginary parts of some QNM frequencies as a function of KR parameter $\ell$ with $\mu^2=-0.65$, $m=1$, $\lambda=1$, $M=34$ and $j=30$. The left two panels display the real and imaginary parts of the QNMs for Dirichlet boundary conditions ($\xi=0$), the middle panels for Robin boundary conditions ($\xi=\pi/4$), and the right panels for Neumann boundary conditions ($\xi=\pi/2$).
• Figure 3: Real and imaginary parts of some QNM frequencies as a function of $\xi/\pi$ for a KR BTZ black hole with different KR parameters $\ell=-0.03$, $0$, $0.03$ with $\mu^2=-0.65$, $m=1$, $\lambda=1$, $M=34$ and $j=30$. The filled circles and triangles denote QNMs under Dirichlet boundary conditions and Neumann boundary conditions, respectively. The squares represent QNMs with the zero imaginary part, and from left to right the corresponding thresholds are $\xi_c/\pi=0.561393$, $0.563514$ and $0.566042$.
• Figure 4: Energy and angular momentum fluxes as functions of $\xi/\pi$ for a KR BTZ black hole with different KR parameters $\ell=-0.03$, $0$, $0.03$ with $n=0$, $\mu^2=-0.65$, $m=1$, $\lambda=1$ and $M=34$. The left panels are $j=30$ and right panels are $j=10$. Here, the thresholds of $\ell=0$ for $j=30$ are $\xi_{c1}=0.563514$ and $\xi_{c2}=0.575424$, and the right ones of $\ell=0$ for $j=10$ are $\xi_{c1}=0.560693$ and $\xi_{c2}=0.561089$.