Exceptional lines of Reissner-Nordström-de Sitter black hole surrounded by a thin shell of matter

Abstract

We study the exceptional line (EL) in the quasinormal modes (QNMs) of Reissner-Nordström-de Sitter black hole surrounded by a static thin shell of matter. For a conformally scalar perturbation, we derive the QNM condition by matching the interior and exterior solutions across the shell and show that higher overtones are particularly sensitive to variations of the shell and background parameters. A mode permutation between two QNMs reveals an exceptional point (EP). After extending the parameter space, this degeneracy forms a continuous EL. We show that the spectral response near the line is intrinsically directional. For a perturbation in parameter space $ε\widehat{\mathbf u}$, the QNM splitting is $|ω_+-ω_-| =C_{\widehat{\mathbf u}}\sqrtε+o(\sqrtε)$, with $C_{\widehat{\mathbf u}}=(\widehat{\mathbf u}^{T}\mathbf{K}\widehat{\mathbf u})^{1/4}$ and $\mathbf{K}$ is so-called spectral sensitivity anisotropy matrix. The tangent direction of EL is a null direction of $\mathbf{K}$, so that the leading square-root splitting vanishes along the exceptional line, whereas the two principal directions in the normal plane exhibit different sensitivities. Furthermore, the square-root branch structure makes conventional linear QNM parametrizations singular near an EL. We therefore construct an EL adapted parametrization which incorporates both the local geometry of the line and the nonanalytic QNM splitting.