Charged Dirac perturbations on Reissner-Nordström black holes in a cavity: quasinormal modes with Robin boundary conditions

TL;DR

We address the problem of charged Dirac perturbations around Reissner-Nordström black holes inside a mirror-like cavity. The authors derive the charged Dirac equations and two Robin-type boundary conditions from the vanishing energy flux principle, and compute the quasinormal spectra analytically in limiting regimes and numerically across the full parameter space using matrix, pseudospectral, and direct integration methods. They uncover a symmetry between the two boundary-condition branches, show that the near-horizon and charge-coupled asymptotics fix the spectra in characteristic ways, and report an anomalous decay where excited modes can outlive the fundamental mode for large field charge $qQ$. The results reinforce the robustness of the vanishing energy flux boundary conditions for black holes in cavities and illuminate nuanced differences from AdS or asymptotically flat setups, with potential extensions to rotating black holes.

Abstract

We investigate charged Dirac quasinormal spectra on Reissner-Nordström black holes in a mirror-like cavity. For this purpose, we first derive charged Dirac equations, and \textit{two} sets of Robin boundary conditions following the vanishing energy flux principle. The Dirac spectra are then computed both analytically and numerically. Our results reveal a symmetry hidden in the Dirac spectra between two boundary conditions. Moreover, when the cavity is placed close to the event horizon $r_+$, we identify that, in the neutral background the Dirac spectra asymptote to $-(3/8+N/2)i$ [$-(1/8+N/2)i$] for the first [second] boundary condition; while in the charged background the real part of charged Dirac spectra asymptote to $qQ/r_+$ for both boundary conditions; where $N$ is the overtone number, $q$ and $Q$ are charges for the field and for the background. In particular, we uncover a striking anomalous decay pattern, $i.e.$ the excited modes decay \textit{slower} than the fundamental mode, when the charge coupling $qQ$ is large. Our results further illustrate the robustness of vanishing energy flux principle, which are applicable not only to anti-de Sitter black holes but also to black holes in a cavity.

Figures (8)

• Figure 1: (color online) A comparison for the imaginary part of Dirac QNMs between analytic (dashed lines) and numeric (solid lines) results, with the first (red) and second (blue) boundary conditions, in terms of the location of the cavity $r_m$. Notice that this figure is made with semilogarithmic coordinates.
• Figure 2: (color online) The neutral Dirac quasinormal spectra of the first (solid) and second (dashed) boundary conditions, with fixed $\lambda=1$ and for $N=0$ (red), $N=1$ (green) and $N=2$ (blue), in terms of mirror radius $r_m$.
• Figure 3: (color online) The neutral Dirac quasinormal spectra of the first (solid lines with circle dots) and second (solid lines with square dots) boundary conditions, with fixed $r_m=10$ (left), $r_m=2.5$ (right) and for $\lambda=1$ (red), $\lambda=2$ (green) and $\lambda=3$ (blue), in terms of the overtone number $N$.
• Figure 4: (color online) The real (left) and imaginary (right) parts of neutral Dirac quasinormal spectra of the first (solid lines with square dots) and second (dashed lines with circle dots) boundary conditions, with fixed $r_m=10$ and for $N=0$ (red), $N=1$ (green) and $N=2$ (blue), with respect to the separation constant $\lambda$.
• Figure 5: (color online) QNMs of neutral Dirac fields in terms of the background charge $Q$, for the cases of $r_m=10$ (left) and $r_m=2.5$ (right), with the first (solid) and second (dashed) boundary conditions. Note that the real and imaginary parts of Dirac quasinormal spectra are shown in the first and second rows and, in the third row we display the imaginary part of QNMs with respect to the real part, by varying $Q$ from $0$ (pink dot) to $0.995$ (pink square). Here we consider the first three modes, $i.e.$$N=0$ (red), $N=1$ (green), $N=2$ (blue), with fixed $\lambda=1$.