Intermediate behavior of Kerr tails
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The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rates of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed ‘splitting.’ We discuss far-field ‘splitting’ in the full field and near-horizon ‘splitting’ in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the ‘splitting’ behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field ‘splitting’ is explained by competition between projected modes. The near-horizon ‘splitting’ is due to excitation of lower multipole modes that back excite the multipole mode for which ‘splitting’ is observed. In both cases ‘splitting’ is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter-mode excitation effect that we study.
KeywordsBlack hole perturbation theory Wave equations Tail decay Hyperboloidal compactification
We thank Gábor Zs Tóth and István Rácz for discussions. AZ is supported by the NSF Grant PHY-1068881, and by a Sherman Fairchild Foundation grant to Caltech. GK acknowledges research support from NSF Grant Nos. PHY-1016906, PHY-113566 and PHY-1303724. LMB is supported by a NASA EPSCoR RID grant and by NSF Grants PHY-0757344, PHY-1249302, DUE-0941327 and DUE-1300717. Initial work on this project was done while LMB was at the University of Alabama in Huntsville. Most of the data presented in this work were generated on the Air Force Research Laboratory CONDOR supercomputer. GK also acknowledges support from AFRL under CRADA No. 10-RI-CRADA-09.
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