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

Abstract

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 ($r\to 0$), between the inner and outer Reissner-Nordström horizons ($r_{-}<r<r_{+}$), and in the asymptotic exterior regime ($r\to \infty$). 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 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 $\Theta$ 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.

52 pages

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