Coupling between gravitational and electromagnetic perturbations on Kerr Spacetime

We extend our previous Schwarzschild metric-based studies of gravitational--electromagnetic (GEM) coupling to rotating black holes by working directly in a curvature-based Newman--Penrose/Teukolsky framework on Kerr spacetime. Within a minimally coupled Einstein--Maxwell system, we derive explicit quadratic electromagnetic source terms for the spin-$-2$ Teukolsky equation, providing a foundation for future numerical studies of GEM interactions in the framework of black-hole spectroscopy. Moreover, we give order-of-magnitude arguments showing that GEM quadratic quasinormal modes (QQNMs) can become relevant in a range of charged and magnetized astrophysical scenarios. Finally, we show through a brief dilaton-theory example that the GEM QQNM spectrum is sensitive to how gravity couples to electromagnetism, thereby providing a model-based way to test minimal coupling and to constrain hidden $U(1)$ sectors with gravitational-wave observations.