More Nonlinearities? II. A Short Guide of 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 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 ${\omega }^{(1)}$ and ${\mathrm{\Omega }}^{(1)}$, resulting in the second-order QNM mixing in the electromagnetic field at ${\mathrm{\Omega }}^{(2)}={\mathrm{\Omega }}^{(1)}+{\omega }^{(1)}$. This complements prior work in \cite{aly2024nonlinearities} on the second-order gravitational perturbation mixing and highlights potential applications in multi-messenger astrophysics for systems observed by LIGO-Virgo-KAGRA (LVK) 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 $p=q\eta$, employing semi-analytical 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 numerical approach in the context of rotating spacetimes.