More nonlinearities. I. Electromagnetic and gravitational mode mixing in neutron star-black hole mergers

We investigate the possibility of electromagnetic fields leaving imprints on gravitational wave signals from neutron star-black hole mergers, specifically in the context of extreme mass ratio inspirals. Using black hole perturbation theory in the context of a minimally coupled Einstein-Maxwell system on a Schwarzschild background, we demonstrate that electromagnetic quasinormal modes (QNMs) can excite gravitational QNMs with frequencies that are linear or quadratic in the electromagnetic QNMs, at the first level of mixing. Moreover, we then study the electromagnetism-gravity coupling by approximating the Regge-Wheeler and Zerilli potentials with Dirac delta functions. In this example, we examine gravitational perturbations induced by the electromagnetic field of an ideal dipole radially free falling toward the black hole, building on calculations from a companion paper [F. Aly and D. Stojkovic, Phys. Rev. D 111, 104083 (2025)]. Our results show that both linear and quadratic electromagnetic QNMs appear in gravitational perturbations. In addition, linear gravitational QNMs are also excited due to the electromagnetic source, with their amplitudes depending on the details of the electromagnetic and gravitational potentials, analogous to gravitational mode mixing analysis. Furthermore, at late stages, gravitational perturbations might exhibit polynomial tails induced by electromagnetic perturbations. This article sets the stage for future numerical investigations aimed at identifying such modes in various scenarios.