Late-time Kerr tails revisited

Abstract

The decay rate of late time tails in the Kerr spacetime have been the cause of numerous conflicting results, both analytical and numerical. In particular, there is much disagreement on whether the decay rate of an initially pure multipole moment $\ell$ is according to $t^{-(2\bar{\ell }+3)}$, where $\bar{\ell }$ is the least multipole moment whose excitation is not disallowed, or whether the decay rate is according to $t^{-n}$, where $n=n(\ell )$. We do careful 2+1D numerical simulations, and explain the various results. In particular, we show that pure multipole outgoing initial data in either Boyer--Lindquist on ingoing Kerr coordinates on the corresponding slices lead to the same late time tail behavior. We also show that similar initial data specified in terms of the Poisson spherical coordinates lead to the simpler $t^{-(2\bar{\ell }+3)}$ late time tail. We generalize the rule $n=n(\ell )$ to subdominant modes, and also study the behavior of non--axisymmetric initial data. We discuss some of the causes for possible errors in 2+1D simulations, demonstrate that our simulations are free of those errors, and argue that some conflicting past results may be attributed to them.

16 pages, 26 figures, 2 tables; Many changes compared with previous version

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