Nonlinear tails in the Kerr black hole ringdown

Abstract

Power law tails induced by nonlinearities of General Relativity (``sourced'' or ``nonlinear'' tails) were recently shown to dominate the late time waveform of Schwarzschild black hole ringdowns. We extend the analytical results regarding such nonlinear tails from Schwarzschild to Kerr black holes by studying the Teukolsky equation. Using a far field approximation to the radial Green's function, we analytically derived the tail power law to be $t^{-\ell-β-s}$ for spin-weight $s \neq 0$, harmonic mode $(\ell m)$ and source decay $r^{-β}$. We numerically confirmed these results for $β= 0, 1$. We also demonstrate the dynamical formation of such nonlinear tails for a massless scalar by numerically solving the Teukolsky equation. In all numerical results, Kerr black hole nonlinear tails have the same power laws as that for Schwarzschild black holes, as expected from the Minkowski nature of the spacetime in the far field region.

Figures (4)

• Figure 1: Integration region for the response integral \ref{['eq:response_integral']}. The Green's function is nonzero in both green and blue region, but the far field approximation is only valid in the blue region. Rectangular regions $R_1$ and $R_2$ have nonzero contributions to the integral, and analytical evaluation is possible only in $R_2$. For $s>0$, contribution from $R_2$ dominates the full integral.
• Figure 2: Evolution of $|\tilde{\psi}_{\ell m}(t,r_*)|$ for $(\ell m)=(22), (32)$ and $a/M = 0.1, 0.2, 0.4, 0.8$. The tail index for $(\ell m) = (22)$ curves (left panel) is $-1$, and that for $(\ell m) = (32)$ curves are $-2$. One can see that changing the rotation parameter $a$ does not affect the tail power law index, but does lead to different early oscillation patterns (quasinormal modes).
• Figure 3: Evolution of $|\tilde{\psi}_{1 1}(t,r_*)|$ for $\beta = 0, 1, 2$ and $a/M = 0, 0.4$. The tail power laws are $t^0$, $t^{-1}$ and $t^{-4}$ for $\beta = 0, 1, 2$, respectively. It is clear that rotation do not affect the tail power law indices or amplitudes.
• Figure 4: Comparison of tails dynamically formed due to a cubic coupling. The power laws for the curves in the left (right) panel is $t^{-4}$ and $t^{-5}$, consistent with the $t^{-\ell-1}$ analytical prediction. On the left panel, the blue and yellow curves overlap, and the amplitudes scale like $(a/M)^2$. On the right panel, the amplitudes scale like $\lambda^1 (a/M)^2$.