First-Order Axial Perturbation of the Reissner-Nordström Metric In a Possible Chern-Simons Gravity Background

TL;DR

This work analyzes axial perturbations of Reissner-Nordström black holes within Chern-Simons modified gravity, deriving a Schrödinger-like radial equation for $R(r)$ and exploring its behavior across $r\to\infty$, $r\to 0$, and $r_-<r<r_+$. Through analytical limiting analyses and numerical solutions, the study finds that the electromagnetic field suppresses perturbations at higher $Q/M$, while higher multipoles $l$ exhibit resonance-like radial modes and a striking symmetry in the extremal limit tied to the AdS$_2\times S^2$ near-horizon geometry; a WKB quantization condition also emerges for high-$l$ modes. The analysis shows that the CS scalar $\Theta$ must be constant at leading order, effectively nullifying observable CS effects in this perturbative setup unless symmetry-breaking backgrounds are present. These results have implications for the stability and dynamical properties of charged black holes in CS gravity and point to potential observational signatures in gravitational-wave contexts, while setting constraints on CS effects in this regime.

Abstract

We axial perturbations of Reissner-Nordström black holes within the framework of Chern-Simons modified gravity, a theory with includes parity violation. We derive the governing equations for the perturbations, focusing on the radial function $R(r)$ and its behavior across distinct regions: near the singularity ($r \rightarrow 0$), between the inner and outer Reissner-Nordström horizons ($r_-< r< r_+$), and in the asymptotic regime ($r \rightarrow \infty$). Using a combination of analytical and numerical methods, we analyze the solutions for varying black hole charge-to-mass ratios ($Q/M$) and angular momentum parameters ($l$). Key findings include the suppression of perturbations by the electromagnetic field for higher $Q/M$; the emergence of radial resonance-like behavior for specific $l$ values; and a high degree of symmetry for solutions in the extremal limit ($Q/M \sim 1$), attributed to the AdS$_2 \times S^2$ near-horizon geometry. The WKB approximation is employed to study high-$l$ regimes, revealing quantized radial modes and singular behavior in the extremal limit. Additionally, we explore the role of boundary conditions and the Chern-Simons scalar field $Θ$, showing that consistency demands a constant field (and thus no actually observable Chern-Simons effects) in this perturbative framework. These results provide insights into the stability and dynamical properties of charged black holes under axial perturbations.

Figures (16)

• Figure 1: This plot shows the solutions for three different $Q/M$ ratios when $l = 1$. On the left is the case for a Schwarzschild BH, where $Q \sim 0$; in the center is the case of a typical Reissner-Nordstrom BH, for which $Q / M = 0.5$; and on the right is the near-extremal Reissner-Nordstrom case, where $Q / M \sim 1$. Note that at in the true extremal limit, $Q/M=1$, there is no region between the horizons, because they degenerate to $r_{+}=r_{-}$.
• Figure 2: This plot shows the solution for the three different $Q/M$ ratios when $l = 2$.
• Figure 3: This plot shows the solution for the three different $Q/M$ ratios when $l = 4$.
• Figure 4: This plot shows the solution for the three different $Q/M$ ratios when $l = 8$.
• Figure 5: This plot shows the solution for the three different $Q/M$ ratios when $l = 16$.