Scalar QNM spectra of Kerr and Reissner-Nordstr\"om revealed by eigenvalue repulsions in Kerr-Newman

Recent studies of the gravito-electromagnetic frequency spectra of Kerr-Newman (KN) black holes have revealed two families of quasinormal modes (QNMs), namely photon sphere modes and near-horizon modes. However, they can only be unambiguously distinguished in the Reissner-Nordstr\"om (RN) limit, due to a phenomenon called eigenvalue repulsion (also known as level repulsion, avoided crossing or the Wigner-Teller effect), whereby the two families can interact strongly near extremality. We find that these features are also present in the QNM spectra of a scalar field in KN, where the perturbation modes are described by ODEs and thus easier to explore. Starting from the RN limit, we study how the scalar QNM spectra of KN dramatically changes as we vary the ratio of charge to angular momentum, all the way until the Kerr limit, while staying at a fixed distance from extremality. This scalar field case clarifies the (so far puzzling) relationship between the QNM spectra of RN and Kerr black holes and the nature of the eigenvalue repulsions in KN, that ultimately settle the fate of the QNM spectra in Kerr. We study not just the slowest-decaying QNMs (both for $\ell=m=0$ and $\ell=m=2$), but several sub-dominant overtones as well, as these turn out to play a crucial role understanding the KN QNM spectra. We also give a new high-order WKB expansion of KN QNMs that typically describes the photon sphere modes beyond the eikonal limit, and use a matched asymptotic expansion to get a very good approximation of the near-horizon modes near extremality.