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data in which all multipoles are present. In fact, because the smallest multipole contribution to the initial data turns out as we show below to dominate the decay rate of the ℓ multipole at late times, Barack and Ori found the decay rate of an even (odd) mode ℓ excites by the scalar field monopole (dipole) mode of the initial data. The Barack–Ori formula for the decay rate of the ℓ multipole has not been confirmed by numerical simulations, which we do below for the first time. Our proposed formula also generalizes it to any initial multipole, not just the monopole (or dipole) as in [8]. 

Consider next the question of the decay rate of a certain multipole moment ℓ . In the Schwarzschild spacetime the decay rate depends only the the value of ℓ , specifically it is t $_{−}$n where n = 2 ℓ + 3. Most notably, the decay rate is independent of the history of the mode. This is no longer the case in the Kerr spacetime: the decay rate of a multipole ℓ does depend on how that mode came into existence, specifically, it depends on both ℓ, ℓ $^{′}$. In the Schwarzschild case the dependence is only on ℓ because of the degeneracy ℓ ′ = ℓ and the absence of mode couplings (“spherical harmonics are eigenfunctions of the wave operator”) . Therefore, in the Schwarzschild case one may say that “a multipole is a multipole,” and consequently its decay rate is intimately linked to its angular distribution and the associated symmetry group. Specifically, in Schwarzschild there is a one–to–one relation of the symmetry group of the mode in question and the mode’s late–time decay rate, such that the symmetry determines the time evolution of the mode. Previous studies of tail behavior in Kerr have suggested that it is not correct to say in Kerr that “a multipole is a multipole,” although the authors are not aware of explicit statements in this regard. We bring here new evidence that that is indeed the case, and the tail decay rate has nothing to do with the symmetry group of the mode in question. This insight may be helpful in the understanding of cases where more than one channel contributes to a mode of a certain ℓ value, although each channel contributes so that the decay rate is different, in what might be construed as an apparent violation of the superposition principle. The realization that in Kerr one may no longer correctly argue that “a multipole is a multipole” completely resolves this apparent violation. 

The organization of this Paper is as follows: In Section II we describe the numerical code and the type of initial data sets that we use. Specifically, we describe the code development that was necessary in order to support our proposal for the tail decay-rate formula. In this Paper we discuss three related questions: First, in Section III we address the difference in the time evolution of late–time tails between generic and non–generic initial data sets. Generic initial data sets were first considered by Barack and Ori [8], but their results have not been confronted with numerical simulations. In addition to verifying the results of [8], we also discuss the meaning of genericity of initial data sets. As we demonstrate, not any mixture of all multipole modes is in fact generic, and one needs to exclude mixtures that are the outcome of the evolution of non-generic initial data, although such mixtures exhibit similar properties to mixtures that represent bona fide generic data sets. Then, in Section IV, we consider an aspect of late–time tails that has not been studied before, specifically the decay rate of excited higher multipole modes, which we call “up”–excited modes, namely the case ℓ > ℓ $^{′}$. The decay rate of “up” excited modes generalized the Barack–Ori result in [8] to some interesting cases: the case of non–generic initial data in which (only) the lowest dynamically allowed mode is not present, and the case in which only one multipole mode is present in the initial data. 

## II. NUMERICAL CODE 

## A. Initial data sets 

Linearized perturbations of Kerr black holes are described by Teukolsky’s master equation, given in Boyer–Lindquist coordinates for a scalar field ( s = 0) ψ , known as the Teukolsky function, by 

where M, a are the black hole’s mass and spin angular momentum per unit mass, respectively, and the horizon function Δ = r 2 − 2 M r + a $^{2}$. To solve Eq. (1) numerically, we define, following [9], b ( r, θ ) := ( r 2 + a $^{2}$) / Σ where Σ 2 = ( r 2 + a $^{2}$)2 a $^{2}$Δ sin 2 θ . 

− We next define the ‘momentum’ Π of the field φ defined by ψ = e imϕ φ , according to 

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