Late–time Kerr tails: generic and non–generic initial data sets, “up” modes, and superposition Lior M. Burko 1 , 2 and Gaurav Khanna 3 $^{1}$Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA $^{2}$Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA $^{3}$Physics Department, University of Massachusetts at Dartmouth, N. Dartmouth, Massachusetts 02747, USA (Dated: October 18, 2010) Three interrelated questions concerning Kerr spacetime late–time scalar–field tails are considered numerically, specifically the evolutions of generic and non–generic initial data sets, the excitation of “up” modes, and the resolution of an apparent paradox related to the superposition principle. We propose to generalize the Barack–Ori formula for the decay rate of any tail multipole given a generic initial data set, to the contribution of any initial multipole mode. Our proposal leads to a much simpler expression for the late–time power law index. Specifically, we propose that the late–time decay rate of the Y$_{ℓm}$ spherical harmonic multipole moment because of an initial Y$_{ℓ}$ $^{′}$m multipole is independent of the azimuthal number m , and is given by t $^{$_{−}$n}$, where n = ℓ ′ + ℓ + 1 for ℓ < ℓ ′ and n = ℓ ′ + ℓ + 3 for ℓ ≥ ℓ $^{′}$. We also show explicitly that the angular symmetry group of a multipole does not determine its late–time decay rate. PACS numbers: 04.70.Bw, 04.25.Nx, 04.30.Nk I. INTRODUCTION AND SUMMARY Perturbations of black holes decay first with complex frequencies, known as the quasinormal modes of of the black hole, taken over at late times by the tails of the perturbation field— decaying as power–laws of time. The decay rate of the late–time tails in the spacetime of spinning black holes has been the subject of much debate and some confusion. Much of the debate in the literature was focused on the late–time decay rate of an initially pure azimuthal hexadecapole ( ℓ ′ = 4, m = 0) scalar field perturbation. All authors agree that at late times the field is dominated by its monopole moment, and that the decay rate would be according to an inverse power of time, but different claims were made as to the value of the power–law exponent. Specifically, there were claims for decay rate along fixed Boyer–Lindquist r according to t $_{−}$3 [1], t $_{−}$5 [2, 3], and even t $_{−}$5 . 5 [4]. In the last couple of years the behavior of the power–law tails has been clarified in a number of papers [5–7] (and the reader is referred to the detail therein), yet there are some remaining interesting detail waiting to be unveiled. In particular, it was shown in [7] that the disagreement between the t $_{−}$3 and t $_{−}$5 behaviors can be attributed to the use of different initial data sets (see also [6]), but not to the use of different slicing conditions as was suggested in [5]. Most importantly, Boyer–Lindquist slicing is not unique in terms of the resulting tails as had been argued, but rather belongs to an equivalency class of slicing conditions (including also ingoing Kerr slicing), all having similar tail structure properties. For the evolution of scalar fields on a fixed background of a Kerr black hole, the value of the power-law index, namely the exponent n in the late-time decay rate t $^{$_{−}$n}$, was given by Hod [2] as n = ℓ ′ + m + 1 for even ℓ ′ − m > 2, n= ℓ ′ + m + 2 for odd ℓ ′ − m > 2, and n = 2 ℓ ′ + 3 for ℓ ′ − m < 2, where ℓ ′ is the multipole moment of the initial perturbation and m is its azimuthal number. This expression for n is rather complicated and notably depends on the azimuthal number m . We argue that the added tacit requirement that it is the decay rate of the slowest decaying mode that is of usual interest masks the inherent simplicity of n in the Hod formula. Specifically, we propose that the decay rate of the ℓ mode because of an initial ℓ ′ mode is independent of m , and is given by n = ℓ ′ + ℓ + 1 for ℓ < ℓ ′ and n = ℓ ′ + ℓ + 3 for ℓ ≥ ℓ $^{′}$. Notice that this proposal is consistent with Hod’s formula and with [7] when they are applicable, and generalizes them also to all other cases. It is therefore a proposal for the general formula for the decay rate of any multipole moment of the tail because of any multipole moment of the initial data. It is striking that this more general formula is simpler than the formula for the decay rate of the least damped mode, that was the question that [2] was addressing. We believe that this simple, general formula will provide insight into a more complete understanding of the tail phenomenon in spinning black hole spacetimes. Our general formula allows up to address some questions left open also in the analysis of Barack and Ori [8]. Barack and Ori studied analytically the late–time tails for generic families of initial data (only assuming compactly supported outgoing initial pulses). Instead of carrying the analysis in the frequency domain —which turns out to be complicated on a Kerr background because the frequency dependence of the spheroidal harmonics implies that separation of the two angular variables depends on the frequency— Barack and Ori analyzed the evolution of perturbations in the time domain. Barack and Ori found the decay rate of the ℓ multipole given generic initial data, which they considered to be arXiv:1001.0541v2 [gr-qc] 25 Oct 2010