Source-Driven Tails in Kerr Spacetime: Nonlinear effects in Late-Time Behavior

TL;DR

This paper studies late-time tails of Kerr spacetime perturbations driven by nonlinearity-inspired outgoing sources, using long-duration, GPU-accelerated time-domain simulations of a scalar field and examining both sub-extremal and extremal spins. The main finding is that the sourced tail exponents satisfy $p_{\\ell\\ell'}^{\\text{sourced}} = p_{\\ell\\ell'}^{\\text{Price}} + 1$ across observed multipoles, indicating a universal slowdown due to nonlinear effects; a gravitational-wave case with $(\\ell,\\;m)=(4,4)$ driven by a $(2,2)^2$ source confirms related behavior. The study employs the Teukolsky framework in Kerr, augments it with a nonlinearity-inspired $1/r^{2}$ source, and extracts power-law indices from late-time data at finite $r$ and at null infinity $\\mathscr{I}^+$ for $\\ell' \\le 4$. For extremal Kerr, horizon tails align with Price-like behavior driven by initial data, suggesting a breakdown of the conformal symmetry in the presence of an outgoing source. The results have implications for gravitational-wave template accuracy and motivate further nonlinear analyses toward full nonlinear gravitational predictions in black-hole ringdown.

Abstract

We present the long-duration time-domain simulations of scalar-field tails in Kerr spacetimes driven by \emph{outgoing} multipolar sources. Extending the recent work in the literature from Schwarzschild to rotating black holes, we evolve sources with $\ell'=\{0,1,2,3,4\}$ on backgrounds with dimensionless spin $a/M=\{0.0, 0.8, 1.0\}$ and extract the late-time decay rates of measured modes $\ell\le4$ for a nonlinearity-inspired outgoing source with a $1/r^2$ fall-off. In all cases we find the inverse power-law index $p_{\ell\ell'}$ to be larger than the source-free Price law values by one unit, i.e. $p^{\text{sourced}}_{\ell\ell'} = p^{\text{Price}}_{\ell\ell'} + 1$. We also include a power-law index value computation for a similar source-driven gravitational wave case $(\ell,m)=(4,4)$ and confirm closely related results in the recent literature.

Figures (2)

• Figure 1: Log-log plot of different angular projections $\ell=0,2,4$ of the scalar field $\Psi^{(2)}$ vs. $u$ at $\mathscr{I}^+$ for $\ell'=4$ source for a Kerr black hole with $a/M=0.8$.
• Figure 2: Log–log plot of the $(\ell,m)=(4,4)$ angular projection of $|\Psi^{(2)}|$ vs. $t, u$ with $(\ell',m')=(2,2)$ squared source and $a/M=0.8$.