First-Order Axial Perturbation of the Reissner–Nordström Metric Against a Possible Parity-Violating Gravity Background

We study axial perturbations of Reissner–Nordström black holes within the general framework of parity-violating modified gravity theories. We derive the governing equations for a class of frame-dragging perturbations, focusing on the symmetry structure and radial dependence of the perturbed metric component, describing its behavior across three distinct regions: near the singularity (r→0), between the inner and outer Reissner–Nordström horizons (r−<r<r+), and in the asymptotic exterior regime (r→∞). 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∼1), attributed to the AdS2× S2 near-horizon geometry. The WKB approximation is employed to study the high-l regime, revealing quantized radial resonance modes and singular behavior in the extremal limit. Additionally, we explore the role of boundary conditions and the possibility of a Chern–Simons field Θ as the source of the parity violation, showing that consistency and the behavior of the perturbations under time reversal demand a constant field (and thus no actually observable Chern–Simons effects) at leading order. These results provide a basis for further analysis of the stability and dynamical properties of charged black holes in parity-violating theories, with potential experimental signatures in gravitational wave observations.