More nonlinearities. II. A short guide to first- and second-order electromagnetic perturbations in the Schwarzschild background

We study second-order electromagnetic perturbations in the Schwarzschild background and derive the effective source terms for the Regge-Wheeler equation, which are quadratic in first-order gravitational and electromagnetic perturbations. In addition to the induced mixed quadratic modes, we find that linear gravitational modes are also excited, with amplitudes dependent on the electromagnetic potential. A toy model involving a Dirac delta function potential demonstrates mixing of linear gravitational and electromagnetic perturbations with frequencies ${\ensuremath{\omega}}^{(1)}$ and ${\mathrm{\ensuremath{\Omega}}}^{(1)}$, resulting in the second-order quasinormal mode (QNM) mixing in the electromagnetic field at ${\mathrm{\ensuremath{\Omega}}}^{(2)}={\mathrm{\ensuremath{\Omega}}}^{(1)}+{\ensuremath{\omega}}^{(1)}$. This complements prior work in Aly et al. [Companion paper, Phys. Rev. D 111, 104082 (2025).] on the second-order gravitational perturbation mixing and highlights potential applications in multimessenger astrophysics for systems observed by LIGO-Virgo-KAGRA and upcoming LISA. We also study first-order perturbations due to a point charge and show it could be reduced to a one-dimensional path integral. Within the toy model, we investigate the first-order electromagnetic perturbation due to a radially free-falling single charge $q$ and radial dipole moment $p=q\ensuremath{\eta}$, employing semianalytical and numerical methods. For the dipole case, we show that the QNM perturbation is excited with a nearly constant amplitude. Future work will focus on incorporating mixing in more realistic potentials and exploring a numerical approach in the context of rotating spacetimes.