More Nonlinearities? Electromagnetic and Gravitational Mode Mixing in NSBH Mergers

Abstract

We investigate the possibility of electromagnetic fields leaving imprints on gravitational wave (GW) signals from Neutron Star-Black hole (NSBH) mergers, specifically in the context of extreme mass ratio inspirals (EMRIs). Using black hole perturbation theory (BHPT) in the context of a minimally coupled Einstein-Maxwell system, we demonstrate that electromagnetic quasi normal modes(QNMs) can excite gravitational QNMs with frequencies that are linear or quadratic in the electromagnetic QNMs, at first level of mixing. Moreover, We then study the electromagnetism-gravity coupling by approximating the Regge-Wheeler and Zerilli potentials with Dirac delta functions. In this example, we examine gravitational perturbations induced by the electromagnetic field of an ideal dipole radially free fall towards the blackhole, building on calculations from a companion paper [1]. Our results show that both linear and quadratic electromagnetic QNMs appear in gravitational perturbations. In addition, linear gravitational QNMs are also excited due to the electromagnetic source, with their amplitudes depending on the details of the electromagnetic and gravitational potentials, analogous to gravitational mode mixing analysis. Furthermore, at late stages, gravitational perturbations might exhibit polynomial tails induced by electromagnetic perturbations. This article sets the stage for future numerical investigations aimed at identifying such modes in various scenarios.

Figures (3)

• Figure 1: This diagram illustrates the iterative solution of the perturbed Einstein-Maxwell equations, assuming subdominant EM effects. Starting with the EM source $J^{\mu}$, we solve for the EM perturbation $\Phi$ and compute the EM stress-energy tensor $T^{\mu\nu}_{EM}$. This acts as a source for gravitational perturbations, producing GEM modes ${{}_{EM}\psi_{G}}$. If $T^{\mu\nu}_{EM}$ and the gravitational stress-energy tensor $T^{\mu\nu}_{GG}$ are comparable at some order $k$, the total gravitational perturbation $\psi$ will be a combination of GEM modes ${{}_{G}\psi_{EM}}$ and GG modes ${{}_{G}\psi_{GG}}$.
• Figure 2: Comparison of G and EM QNMs in Schwarzschild spacetime. G frequencies $\omega_{ln}$ for $l = 2, 3$ and $n = 0, 1$ are compared with EM frequencies $\Omega_{ln}$ for $l = 1, 2$ and $n = 0, 1$. For instance, the GG, quadratic QNMs, the real part of the frequencies is given by the sum $\omega_{(i\times j)R}=\omega_{(i)R} \pm \omega_{(j)R}$ of the linear modes, while the imaginary part, representing the reciprocal of the decay time, is the sum $\omega_{(i\times j)I}=\omega_{(i)I} + \omega_{(i)I}$ of the corresponding linear terms. The blue squares and red dot refer to the electromagnetic and gravitational QNMs while green triangles and rhombuses refer to the quadratic electromagnetic and gravitational QNMs respectively.
• Figure 3: This figure illustrates the QNM region $B$ in gradient Red and the flat region $B-A$ in light Blue, where the source consists of an ideal dipole with charges $q$ and $-q$ represented by the green solid and dotted curves, respectively. These curves asymptotically approach an ingoing null geodesic with $v = 23.6 M$, shown as the blue dashed line with a slope of -45 degrees. The support region is chosen for a event $(t,x)$ with $t = 100 M$--for late-time scenario-- while $x$ lies in the asymptotically flat region at $x = 40 M$. The dipole's worldlines follow a characteristic behavior as they near the potential peak, transitioning toward null geodesics for late times. The geometry and dynamics depicted here are essential for understanding the electromagnetic QNMs generated by the dipole.