Ringdown signatures in the Ernst-Wild geometry: modeling Kerr black holes immersed in a magnetic field

TL;DR

The paper probes how an external, weak magnetic field alters Kerr black hole ringdown by studying scalar quasinormal modes in the Ernst-Wild geometry through a perturbative expansion in the spin $\tilde{a}$ and field $\tilde{B}$. Using Leaver's continued fraction method, it computes the EW QNM spectrum and develops an interpolation (EW1) to form a ringdown template, subsequently testing it against LVK data with pyRing. The analysis finds no evidence for magnetized remnants within current sensitivity, yielding indicative bounds on $\tilde{B}_f$ and demonstrating that environmental effects can be treated as nuisance parameters in ringdown analyses. The work lays a framework for incorporating magnetospheric environments into gravitational-wave ringdown modeling and points to future improvements via higher-order spin, tensor perturbations, and nonlinear dynamics.

Abstract

We analyze the quasinormal mode spectrum for Kerr black holes surrounded by an asymptotically uniform magnetic field, modeled with the Ernst-Wild geometry. A perturbative expansion in both the rotation parameter $a$ and the magnetic field $B$ allows the analysis of perturbations with Kerr-like asymptotics well inside the Melvin radius, and we obtain the spectrum for a variety of scalar quasinormal modes over a range of parameters using the continued fraction method. We then interpolate the low-lying mode spectrum to construct an Ernst-Wild template for the ringdown, and use the LIGO-Virgo-KAGRA analysis tool pyRing to assess the impact of the magnetosphere on the extraction of ringdown signatures from several observed binary black hole mergers.

Figures (8)

• Figure 1: The real (left) and imaginary (right) parts of the leading scalar $s=0$ and tensor $s=-2$ Kerr quasinormal mode frequencies $\omega_{s {\ell m n}}$ are shown as a function of the rotation parameter $\tilde{a}$ computed via the continued fraction technique.
• Figure 2: The real (left) and imaginary (right) parts of the $(\ell, m, n)=\{(2,2,0), (2,2,1)\}$ scalar quasinormal mode frequencies for the Kerr solution as a function of $\tilde{a}$, compared to the corresponding modes of the linearized Ernst-Wild geometry (for $B \rightarrow 0$). The differences for the $\{(3,2,0),(4,2,0)\}$ modes are very similar.
• Figure 3: Quasinormal mode frequencies of the linearized Ernst-Wild geometry are presented for $0 \leq \tilde{a} , \tilde{B} \leq 0.3$ in increments of 0.01, exhibiting the $\omega_{\ell -m n} = - \omega^*_{\ell m n}$ pairing for $\ell, m = \pm 1, \pm 2$ that results from axial symmetry. Starting from the bottom of the plots, as $\tilde{a}$ increases the lines move in opposite directions horizontally while as $\tilde{B}$ increases the values become less damped and move vertically upwards. These modes, including both positive and negative branches, were computed with the continued fraction methods described in Section \ref{['sec:QNM_computation']}.
• Figure 4: The real (left) and imaginary (right) parts of the $(\ell, m, n)=\{(2,2,0), (2,2,1)\}$ scalar quasinormal mode frequencies for the Ernst solution as a function of $\tilde{B}$, compared to the corresponding modes within the linearized fit to Ernst-Wild spectrum in (\ref{['eq:fit']}) (for $\tilde{a} \rightarrow 0$). The differences for the $\{(3,2,0),(4,2,0)\}$ modes are very similar.
• Figure 5: Posterior distributions for the remnant mass $M_f$, spin $\tilde{a}_f$, and magnetic field $\tilde{B}_f$ resulting from the analysis of four gravitational wave signals of binary mergers from the LVK catalogue: GW150914, GW190521_074359, GW190727_060333 and GW200224_222234. The upper and lower limits on the estimated parameters bound the $90\%$ credible interval and are shown with the dashed purple lines.