More Nonlinearities? II. A Short Guide of First- and Second-Order Electromagnetic Perturbations in the Schwarzschild Background

TL;DR

This work develops a framework for second-order electromagnetic perturbations in the Schwarzschild background within the Regge-Wheeler formalism, deriving effective sources that are quadratic in first-order EM and gravitational perturbations and showing how EM QNMs can be excited through mixing with gravitational modes. A Dirac delta potential toy model is used to illustrate the mixing mechanism and to demonstrate that second-order EM QNMs follow frequencies ${\Omega^{(2)}} = {\Omega^{(1)}} \pm {\omega^{(1)}}$, with amplitudes modulated by the gravitational and EM potentials. The authors also derive first-order EM perturbations from point charges, reducing the time-domain solution to a one-dimensional path integral and analyzing ideal dipole configurations through semi-analytical and numerical methods. Numerical results for a radially falling dipole show a nearly constant QNM amplitude and a nondecaying flat component, highlighting the limitations of the delta-approximation and motivating future work with more realistic potentials and rotating spacetimes. The findings have potential implications for multi-messenger astrophysics, providing a pathway to detect electromagnetic imprints of gravitational dynamics in LVK-era and LISA-era observations and offering tests of minimal coupling in GR or its extensions.

Abstract

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 \( ω^{(1)} \) and \( Ω^{(1)} \), resulting in the second-order QNM mixing in the electromagnetic field at \( Ω^{(2)} =Ω^{(1)} + ω^{(1)} \). 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 \( p = q η\), 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.

Figures (21)

• Figure 1: Comparison of gravitational and electromagnetic QNMs in Schwarzschild spacetime. Gravitational frequencies $\omega^{(1)}_{ln}$ for $l = 2, 3$ and $n = 0, 1$ are compared with electromagnetic frequencies $\Omega^{(1)}_{ln}$ for $l = 1, 2$ and $n = 0, 1$. For the quadratic QNMs, the real part of the frequencies is given by the sum $\Omega^{(2)}_{(i \times j)R} = \Omega^{(1)}_{(i)R} \pm \omega^{(1)}_{(j)R}$ of the linear modes, while the imaginary part, representing the reciprocal of the decay time, is given by $\Omega^{(2)}_{(i \times j)I} = \Omega^{(1)}_{(i)I} + \omega^{(1)}_{(j)I}$ as a sum of the corresponding linear terms. The blue squares and red dots represent the electromagnetic and gravitational QNMs, respectively, while cyan rhombuses indicate the quadratic QNMs.
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5: