Coupling between gravitational and electromagnetic perturbations on Kerr Spacetime

Abstract

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.

Figures (2)

• Figure 1: Frequencies of linear and quadratic QNMs for various spin values of the final black hole, $a$, ranging from $0$ to $0.99$ (with 10 evenly spaced values). The modes are categorized into gravitational, GG, and GEM modes: $G (2,2,0)$, $G (2,2,1)$, $GG (1,1,0) \times (1,1,0)$, $GG (1,1,0) \times (1,1,1)$, $GEM (1,1,0) \times (1,1,0)$, and $GEM (1,1,0) \times (1,1,1)$.
• Figure 2: Frequencies of the polar gravitational QNM $(2,0)$ and the polar GEM QQNM $(1,0)\times(1,0)$ on a static charged background with charge-to-mass ratio $v = Q/M = 0.6$ in Einstein--Maxwell--dilaton theory, shown as functions of the dilaton coupling $\eta \in [0,1.52]$. The stars denote the corresponding QNMs predicted by minimally coupled Einstein--Maxwell theory at $\eta = 0$ for both modes. The electromagnetic sector exhibits a much stronger $\eta$-dependence than the gravitational one, so that the GEM quadratic mode inherits an enhanced sensitivity to the dilaton coupling at the level of the spectrum. This figure is intended as a proof of concept for the coupling dependence of GEM QQNMs; whether such differences are observationally accessible depends on the achievable signal-to-noise ratio, detector sensitivity, and systematic uncertainties, and we do not attempt a detailed forecast here. The data used in this figure are taken from Pacilio_2018QNM_EMD_notebook.