Numerical computation of electromagnetically sourced nonlinear tails

TL;DR

This paper addresses nonlinear tails in black-hole ringdown driven by electromagnetic perturbations, focusing on second-order gravitational perturbations in Schwarzschild spacetime. It solves the inhomogeneous Bardeen-Press-Teukolsky equation using horizon-penetrating hyperboloidal coordinates within the Newman-Penrose formalism and reconstructs Maxwell scalars to form a quadratic source. The authors find that for multipoles with $l\ge 4$, the nonlinear tails decay as $t^{-2l-2}$ at fixed radius and as $u^{-l-3}$ at null infinity, largely independent of initial data or mode coupling, indicating a potentially observable nonlinear contribution to ringdown. For $l=2,3$, the tails depend on the initial electromagnetic data, highlighting a nuanced distinction from higher multipoles. The results have implications for multi-messenger observations and motivate extending the analysis to Kerr spacetime, with advanced numerical techniques like AnMR and time-symmetric integration enhancing long-time stability and accuracy.

Abstract

Amazingly, recent studies indicate that nonlinear effects are of great significance for modelling black hole ringdown. Transient electromagnetic events in the astrophysical environment are typically high energetic, potentially responsible for some nonlinearities in ringdown. Motivated by the desire to understand these nonlinearities, we solve the inhomogeneous Bardeen-Press-Teukolsky equation numerically, and find second-order gravitational tails induced by an electromagnetic source. Our results suggest that the second-order tails of curvature perturbations with multipole numbers $l\geq4$ decay as $t^{-2l-2}$ at fixed spatial position and $u^{-l-3}$ in retarded-time $u$ at null infinity, slower than their linear counterparts, which can play a role in multi-messenger observations.

Figures (10)

• Figure 1: Comparison between the collocation points $\{R_i^{\text{Cheb}}\}$ and $\{R_i^{\text{AnMR}}\}$. Here, we set $\kappa=6$ and $N=N^\prime=32$.
• Figure 2: Spectral coefficients $c_n$ of $\{{}_{-1}\hat{\psi}^{[33]}(T,R_i)\}$ and $\{{}_{-2}\hat{\psi}^{[55]}(T,R_i)\}$ for both the grids $\{R_i^{\text{Cheb}}\}$ and $\{R_i^{\text{AnMR}}\}$ when a steep gradient of ${}_{-1}\hat{\psi}^{[33]}$ occurs at about $T=260M$. Here, we take $N=N^\prime=256$.
• Figure 3: LPIs of second-order perturbation ${}_{-2}\hat{\psi}^{[l]}$ for $l=4,5,6$ suggest a power law of the form $T^{-2l-2}$ at $\mathcal{H}^+$ and finite radii, and $T^{-l-3}$ at $\mathcal{I}^+$.
• Figure 4: Waveform of gravitational perturbation ${}_{-2}\hat{\psi}$ and its parent electromagnetic perturbation ${}_{-1}\hat{\psi}$, as well as a gravitational perturbation ${}_{-2}\hat{\psi}$ solved by the source-free BPT equation. Here, the same ingoing initial data are set for all the perturbative variables. The source-driven tail dominates over the source-free tail, supporting the breakdown of linear perturbation theory at late times.
• Figure 5: LPIs of second-order perturbation ${}_{-2}\hat{\psi}^{[l]}$ when $l=2$ (upper panels) and $l=3$ (lower panels) for ${}_{-1}\hat{\psi}$ with the compact (left panels) and non-compact (right panels) support initial data, extracted at future null infinity $\mathcal{I}^+$, future event horizon $\mathcal{H}^+$ and finite radii, respectively.