High-order gravitational late-time tails in Kerr spacetime

Abstract

We calculate high-order late-time tails of the retarded Green function of the Teukolsky equation for linear field perturbations of (subextremal) Kerr spacetime. We calculate these tails at a fixed spheroidal harmonic $\ell$ and azimuthal number $m$ up to the first three orders for the field point: at finite radius (away from the event horizon) for large Boyer-Lindquist time $t$; along the future event horizon $\mathscr{H}^+$ for large ingoing Eddington-Finkelstein coordinate $v$; and along future null infinity $\mathscr{I}^+$ for large outgoing Eddington-Finkelstein coordinate $u$. We obtain the tail powers for generic integer field spin $s$ and the tail coefficients specifically for gravitational ($s=-2$) perturbations. Our asymptotics include the known leading power-law (generic) tails, respectively,$t^{-2\ell-3}$, $e^{im\Omega_H v}v^{-2\ell-3-b}$ (where $b=1$ for $s>0, m=0$ and $b=0$ otherwise, and where $\Omega_H$ is the angular velocity of the event horizon) and $u^{-\ell+s-2}$, as well as their higher-order logarithmic corrections: $t^{-2\ell-5}\ln t$, $e^{im\Omega_H v}v^{-2\ell-5-b}\ln v$ and $u^{-\ell+s-3}\ln u$ (as well as $u^{-\ell+s-4}\ln^2 u$). Since we obtain the high-order expansions for modes for generic $\ell$ and $m$, we can readily infer the explicit expansions of the {\it full} retarded Green function for $s=-2$ (and its decay powers for generic integer $s$). We obtain the late-time asymptotics from small-frequency expansions of the Fourier modes of the retarded Green function in the frequency domain. Accordingly, we also provide small-frequency expansions of various quantities of interest in the scattering theory. We also attach two notebooks which provide expansions for specific values of $s$: one notebook provides them to the first three leading orders for generic $\ell$ and the other one to arbitrary order for specific values of $\ell$.