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the late - time tails of black holes have been studied in much detail since price s seminal work @xcite . the formulation of the problem is a straightforward one : place an observer in a circular orbit around a black hole , and have her measure at late times a generic perturbation field , that had compact support at some initial time . it is generally accepted that the observer measures the late - time perturbation field to drop off as an inverse power law of time , specifically as @xmath3 . it is the value of @xmath5 that has been controversial in the literature , with some conflicting results reported .    in the case of a schwarzschild black hole , @xmath6 , where @xmath7 is the multipole moment of the initial perturbation field . namely , if the initial ( compactly supported ) perturbation field has the angular dependence of @xmath8 , the angular dependence remains unchanged (  spherical harmonics are eigenvectors of the laplacian operator " ) , and the decay rate of the field is governed by the @xmath7 value of the initial perturbation . these results remain unchanged also for a reissner  nordstrm black hole @xcite ( including the extremal case @xcite ) , that like the schwarzschild black hole is spherically symmetric . notably , a generic perturbation is a linear superposition of @xmath7-modes , so that the tail is dominated by the slowest damped mode , or by the lowest value of @xmath7 .    for rotating black holes conflicting values of @xmath5 have been reported . obviously , the observer measures a well defined decay rate . of the various conflicting values in the literature some are wrong ( i.e. , they report a value inconsistent with the outcome of the initial value problem ) , while other results are correct ( in the same sense , of reporting the right value for the initial value problem ) . as it turns out , different initial value formulations may lead to different decay rate for the tails . while none of these is  more correct " _ sensu stricto _ than others , it is important to understand the relationship of different initial value sets , so that the understanding may lead to some insights . one of the correct answers to the question of the value of @xmath5 was described , e.g. , in @xcite : the late - time field decays according to the same rule as in the spherically - symmetric case , namely according to @xmath1 , where @xmath9 if @xmath10 is even , and @xmath11 if @xmath10 is odd , independently of the initial value of @xmath7 . below , we dub this tail as tail a. " the reason for this decay rate is that spherical harmonic modes do not evolve independently on the background of a rotating black hole (  spherical harmonics are _ not _ eigenvectors of the laplacian operator " ) . starting with an initial value for @xmath7 , infinitely many other @xmath7 modes are excited , respecting only the dynamical constraints ( @xmath12 , where @xmath13 is the spin of the field , and @xmath14 ) and the equatorial symmetry of the initial data ( such that even and odd modes do not mix ) . another correct answer to the question was discussed recently in @xcite and in @xcite . according to this result , @xmath15 if @xmath16 , @xmath17 if @xmath18 is even , and @xmath19 if @xmath18 is odd . this result is dubbed below as  tail b. " evidently , this result is inconsistent with that of tail a. the tail b behavior was first predicted in @xcite . while either the tail a or the tail b behavior is a solution to _ some _ initial value problem , these problems are different enough to lead to markedly different late  time evolutions . in this paper we present some explanation for these different results , that resolves the apparent discrepancy at least for some cases . this paper s motivation is to explain some of the historical discrepancies , in what we believe is an illuminating manner . specifically , finding the reasons for the controversy in the literature teaches us important lessons . in particular , we consider two results , conflicting both with each other and with the results of this paper : the numerical results of @xcite , and the analytical model of @xcite . more precisely , we consider the simplest case for which there is a disagreement in the literature , specifically the case of a massless scalar field ( @xmath20 ) perturbation of a kerr black hole , with initial @xmath21 and @xmath22 . for this case , the late time tail drops as @xmath23 according to tail a , because the monopole @xmath24 ) mode of the scalar field can and is excited , and this mode s drop off rate obeys the @xmath25 rule . according to tail b , the late  time decay rate is @xmath26 . the ( numerical ) results of @xcite suggest a drop off rate of @xmath27 that does not appear to converge to the decay rate of tail b. the poisson analytical model ( of a globally weakly curved spacetime ) predicts a drop off rate of @xmath26 if carried over to the kerr case @xcite , which may suggest a tail b behavior . however , as we show below , the correct behavior of such tails should be that of tail a.    we discuss in detail the tails developing from initial data specified on ingoing kerr silces of @xcite . it was suggested in @xcite that the tail a results of @xcite are due to the slicing condition . we show that this is not the case . in fact , one can find initial condition sets with the same slicing condition as in @xcite ( slicing corresponding to the ingoing kerr coordinates ) that result in either tail a or tail b decay rates , in just the same way as with boyer lindquist slicing . in fact , although the boyer  lindquist and ingoing kerr time slices do not coincide , we find that similar initial data lead to the same tail behavior in either . we therefore suggest that different slicing conditions may fall into equivalency classes , and boyer  lindquist and ingoing kerr slices belong to the same class . we propose that this is the case because of the properties of the transformation from boyer  lindquist and ingoing kerr coordinates .    in the case of @xcite , we attribute the ( numerically stable ! ) result of a drop off rate of @xmath27 to an intermediate tail being misidentified as an asymptotic tail . as we show , the traditional representation of the tail in a log  log plot of the field as a function of time is susceptible to lead to an  optical illusion , " that may support this misidentification . specifically , taking initial conditions that lead to a tail a decay rate , we show what may be interpreted as a tail falling off as @xmath27 , similar to what is found in @xcite , except that if the integration is carried on further in time the intermediate behavior is taken over by the true asymptotic behavior , or a tail falling off as @xmath23 . a similar effect can also be found for the tail b case . the reason @xcite mistook an intermediate tail for an asymptotic one is that @xcite chose parameters that make it harder to distinguish the two using the crude tools used in @xcite . specifically , we show that the farther out the initial pulse , the longer the intermediate tail . reference @xcite chose the initial pulse very far out , which resulted in a protracted intermediate tail . while this choice made the problem severe enough , it was even further exacerbated by making the initial pulse outgoing and by taking the kerr parameter @xmath28 . either choice makes the beginning of the tail regime postponed . as the integration time in @xcite is rather short , we conclude that the tail analyzed in @xcite is an intermediate one , and that if the integration time were longer , @xcite too would have found a @xmath23 tail ( for tail a initial data ) or a @xmath26 tail ( for tail b initial data ) . ( in view of our argument below , we comment that the code of @xcite was fourth order angularly , which we find to be satisfactory for the relatively short runs done by @xcite . ) how can one then avoid misinterpreting intermediate tails as asymptotic ones ? we provide a criterion , that we believe can be used for the identification of the asymptotic tail . by our criterion , a tail is asymptotic if the local power index @xcite is linear in @xmath29 both locally and globally , in a sense that we make precise .    in the case of @xcite , we explicitly follow the proposal of poisson , and evolve initial data that are a pure spherical harmonic mode with respect to  spherical " coordinates , not the boyer  lindquist coordinates , that are  spheroidal " in the language of @xcite . according to @xcite , such initial data should lead to a tail b behavior when applied to a kerr background . we show that this is not the case , and tail a behavior is found . specifically , starting with @xmath21 and @xmath22 , poisson predicts the asymptotic tail to drop off as @xmath26 . in @xcite , such time evolution is demonstrated in a globally weak  curvature spacetime ( that has no symmetries ) , and it is conjectured that such behavior is carried over to the case of a kerr black hole . we show that this conjecture is not realized : instead , the evolution is according to tail a , and the drop off rate is according to @xmath23 . very recently , gleiser , price , and pullin ( henceforth gpp ) have proposed a theoretical model that expands the wave equation in ( even ) powers of the black hole s spin rate @xmath30 @xcite . the advantage of the gpp approach is that the zeroth order wave operator is identical to the spherically  symmetric operator , so that it can be solved numerically using a simple 1 + 1d code . each iteration in the gpp approach gives the field to the next order in @xmath31 , using the field of the previous iteration as the source term of a spherically symmetric wave operator . gpp argue that their approach is superior to the 2 + 1d approach , because it frees the numerics from the subtleties of the 2 + 1d simulations , and because the 1 + 1d numerics are inherently more reliable . we agree with these claims . the gpp argument implies that delicate cancellations exactly cancel not only the leading tail term ( in an expansion in @xmath29 ) , but also the leading subdominant mode @xcite ( for an initial @xmath21 ; for other initial choices more subdominant modes may need to be cancelled ) . gpp further argue that turning off the coupling term at very late times destroys the exact cancellation , so that the tail returns to the  naive " decay rate . it is quite remarkable that even if this is done at extremely late times , so that the hexadecapole field is much smaller than the monopole field , it still does not lose its ability to exactly cancel out its two leading terms ( in @xmath29 ) . notably , gpp imply that different slicing conditions may be responsible for the apparently conflicting results of numerical simulations . indeed , in @xcite the slicing condition was that associated with ingoing kerr coordinates , in @xcite the slicing was that of the kerr schild coordinates , and in @xcite boyer linquist slicing was used . gpp argue that the translation of an initially pure @xmath21 mode , say , in some other slicing condition ( e.g. , ingoing kerr or kerr  schild ) into boyer lindquist slicing results in a mixed state of modes for the boyer lindquist slicing , and the existing lower @xmath7 ( specifically , the quadrupole and monopole ) modes on the initial time slice are the ones responsible for the tail structure . according to gpp , among all slicing conditions the boyer lindquist one is special , as only in this slicing an initial pure @xmath21 mode leads to a tail of @xmath26 , and in all other slicing conditions an initially pure @xmath21 mode leads to a @xmath23 tail . we disagree with this viewpoint , as we find that also for ingoing kerr slicing the tail decays like @xmath26 . more recently , tiglio , kidder , and teukolsky ( tkt ) @xcite performed careful 2 + 1d simulations of the axisymmetric kerr tail problem using a pseudospectral code . the pseudospectral approach has an advantage over the finite differencing approach , in that the convergence is exponential in the former , and only a power law in the latter . consequently , in the pseudospectral approach the noise evolution problem that permeates finite differencing computations is nearly nonexistent . tkt studied two slicing cases : kerr  schild slicing and boyer lindquist slicing . tkk found the tail a behavior in the former ( consistently with the conclusions of @xcite that were obtained in 3 + 1d ) and the tail b behavior in the latter ( consistently with gpp ) . tkt correctly point out the main reason for the difference between tail a and tail b behaviors that are found in boyer lindquist initial data and on boyer lindquist slicing , specifically that the zero  field momentum " of the former is in fact a mixed state of pure multipole modes . we consider this question in great detail , and show that the momentum condition is a very special combination of states , as described below . we discuss in detail the noise evolution problem that may cause a tail b masquerade as a tail a. that is , numerical noise evolves on the grid the same way that the physical signal does , so that the tail a behavior of the noise evolution overwhelms the tail b behavior of the physical field . we then show that with care this phenomenon can be easily identified . we next discuss how initial data with the same slicing condition could lead to either tail a or tail b behavior . as shown first by gpp and then by tkt , initial data that are a pure multipole of the angular boyer lindquist coordinates in boyer lindquist slicing lead to tail b behavior . when one sets the initial conditions by requiring that the field s  momentum " @xmath32 ( see below for definitions ) vanishes , the tail s behavior is that of tail a. indeed , the condition on the momentum implies that the field s multipole structure is changed , as the field s velocity now is proportional to the field multiplied by a function of the angular coordinate @xmath33 . restricting attention to low black hole spin values ( @xmath34 ) , this function is proportional to @xmath35 . therefore , one would expect that in addition to an initial hexadecupole at @xmath36 one would have now also all other ( even ) multipoles at @xmath37 . an existing monopole on the initial slice at @xmath37 would then lead to a tail that has the same decay rate as a pure multipole that behaves like tail a , which is at @xmath37 . however , we show that the tail is at @xmath38 , not @xmath37 . the explanation is that the leading contribution ( in @xmath31 ) does not contribute to the monopole , so that on the initial slice the monopole is at @xmath38 , not at @xmath37 . evolution of this mode then can dominate at late times , showing a tail that is both at @xmath38 and has the tail a behavior . nevertheless , this is not the only contribution to this tail . the initial data contains also a quadrupole at @xmath37 , which also evolves a @xmath38 monopole because of the mode excitation effect . however , as in this case the quadrupole has an @xmath7 value that differs from the monopole s by only @xmath39 , the behavior is again consistent with that of tail a. therefore , the tail a behavior has three sources : a pre - existing monopole at @xmath38 , a pre - existing quadrupole at @xmath37 , and a pre - existing hexadecapole at @xmath36 , all three behaving consistently with tail a.    we further extend the numerical study to non - axisymmetric modes , and find that the tail a and tail b behaviors occur for such modes consistently with their behavior in the axisymmetric case , i.e. , they obey the aforementioned decay rate rules . we then extend the argument for tail b behavior to subdominant modes . specifically , we argue that starting with an initial mode @xmath7 , the decay rate of an ( allowed ) excited mode @xmath40 is given by @xmath41 if @xmath42 , and @xmath43 otherwise . this rule includes that of @xcite as a particular case , when @xmath44 . in particular , we consider the axisymmetric case of @xmath45 and @xmath46 ( which is a subdominant mode , as also the monopole is excited ) . the quadrupole mode @xmath46 then indeed decays with @xmath47 as predicted by our proposal . we are hopeful that this paper may put some order in a confused state of affairs , in which the literature includes side by side correct results and incorrect ones , without a proper explanation of the incorrect ones , and also apparently conflicting correct results without a proper explanation of the sources for the apparent conflict . the organization of this paper is as follows . in section [ reliability ] we discuss the reliability of the 2 + 1d approach in calculating tails , and how carelessness can result in an incorrect value for the field . in section [ a2 ] we discuss the decomposition of the monopole projection of the tail in powers of @xmath31 , and in section [ optical_illusion ] we discuss one possible pitfall that can cause a correct field be misinterpreted . in section [ s_ingoing_kerr ] we study tails in ingoing kerr coordinates , and in section [ poisson_data ] we analyze a pure mode in  spherical " coordinates @xcite , and show that it leads to the same tail decay rate as in  spheroidal " coordinates . the solution of the teukolsky equation in 2 + 1d is delicate and sensitive , and can lead to qualitatively wrong results if proper care is not taken . in section [ optical_illusion ] below we show that even when the numerically computed field is the correct one , careless interpretations may lead to incorrect conclusions . in this section we first demonstrate how carelessness may lead to qualitatively wrong behavior of the solution , and then we demonstrate the reliability of the approach when proper care is taken .      the standard of scientific computation in the finite difference approach has been second order algorithms for many years . indeed , second order convergence typically implies the solution is both consistent and stable to a practical level , so that the solution is accurate enough without having to reduce the typical grid spacing to extremely low levels that would make the code too costly computationally . normal application of the dahlquist theory ( extended to pdes ) then means that if the discrete formula approximating the differential equation has locally a positive order of accuracy ( convergence ) , and that if numerical errors do not grow unboundedly ( stability ) , then the solution is convergent , in the sense that as the grid spacing goes to zero , the solution would approach the correct one . as we show below , the tail problem involves the dahlquist theory with a twist . early codes to calculate the late  time kerr tail were therefore designed to be second  order in all coordinates , i.e. , temporally , radially , and with respect to the angular coordinate @xmath33 @xcite . ( in the 2 + 1d approach the field is decomposed into @xmath48 modes , so that one does not have numerical derivatives with respect to @xmath48 . ) while this approach seems very natural , it can mislead into qualitatively wrong solutions . consider first a schwarzschild black hole , excited by an azimuthally symmetric ( @xmath22 ) field with @xmath21 . the decay rate of the price tail is incontrovertible in this case : as the background is spherically symmetric , spherical harmonic modes evolve independently , and the decay rate is therefore @xmath49 . indeed , 1 + 1d codes readily verify this result .    ) for three different angular resolutions ( @xmath50 ( solid ) , @xmath51 ( dashed ) , and @xmath52 ( dotted ) ) as a function of time . the lower panel on the right is the convergence order as a function of the time at @xmath53 . the initial data are for a schwarzschild black hole ( @xmath54 ) , and are a gaussian centered at @xmath55 with width @xmath56 . the radial resolution is @xmath57 , and the temporal resolution is chosen to be @xmath58 , which satisfies the courant condition . the initial data are a pure multipole with @xmath21.,width=326 ]    the numerical solution with a second - order 2 + 1d numerical code may lead however to the erroneous conclusion that the decay rate is @xmath23 : in fig . [ 2nd_1 ] we show the late  time field as a function of time for three choices of the angular grid spacing , and for fixed radial and temporal grid spacings . as can be readily inferred from the upper panel in fig . [ 2nd_1 ] , the decay rate is @xmath23 , which is definitely wrong . the origin of this wrong result is indicated in the same figure : the field values themselves reduce considerably in amplitude under refinement of the grid ( almost as the second power of the ratio of the grid parameters ) . the signal seen is therefore not the physical field , but an evolved numerical noise (  leakage of multipoles " ) . our interpretation is that while the physical signal indeed drops like @xmath49 , it is swamped by the evolved numerical noise , which has a monopole component . the latter drops like @xmath23 , so that any very long time evolution must be dominated by numerical noise . refinement of the grid parameter can postpone the numerical - noise dominated regime , but can not utterly eliminate it : refinement of the grid parameter reduces the level of the numerical noise , so that if the grid is fine enough , the noise level at a particular value of the time reduces to below the level of the physical field . however , as the former drops more slowly than the latter , at very late times the field is dominated by numerical noise ( see fig . [ 6th_1 ] below ) .    here enters the twist on the dahlquist theory : the numerical noise does not grow unboundedly . in fact , it even drops quadratically with the grid spacing , and drops with time . it does so , notwithstanding , slower than the physical signal , so that the signal may be dominated by numerical noise even though the regular dahlquist criteria are apparently satisfied .      while a second order code may in principle be sufficient to see physical tails ( for very fine grids ) , an alternative approach is to increase the order of the angular differentiation . we first show how the same initial data and grid resolutions lead to qualitatively different results in schwarzschild , and then we discuss kerr . figure [ 6th_1 ] shows the sixth order convergence of the field , for the same parameters that were shown in fig . [ 2nd_1 ] with the second order code . the field is not made of evolved noise (  leakage of multipoles " ) , as it clearly converges to a non - zero value under grid refinement . notably , the field s amplitude even _ increases _ slightly under grid refinement . the hallmark of noise signals is that their amplitude _ decreases _ under grid refinement , as is indeed evident in fig . [ 2nd_1 ] . most importantly , the price tail drops as @xmath49 , as it should . we emphasize that these conclusions are true up to some maximal time value . for longer times the residual monopole noise , that drops slower than the physical field , will inevitably dominate . simple extrapolations reveal that the signal noise will dominate over the physical signal for @xmath59 for our choice of angular resolution . longer evolutions than that will be dominated by noise evolution . as noted above , it is not difficult to separate the physical signal regime from the noise signal regime by doing global convergence tests . it is therefore crucial _ for any tail simulation _ that convergence tests are done throughout the entire computational domain , and not limited just to short integration times ( despite the temptation to save computer time  e.g. , see @xcite ) .     for the sixth order code . the upper panel shows the @xmath60 norm for the full field ( integrated over a sphere at @xmath55 ) in three resolutions . the three curves are nearly indistinguishable on this plot s scale . the upper plot also shows the monopole projection of the field , which is pure evolved numerical noise for resolution of @xmath61 . this figure also shows the tails local power index as a function of time ( lower left ) and the convergence order of the three resolution shown on the upper panel ( lower right).,width=326 ]    in fig . [ 6th_2 ] we demonstrate the sixth order angular convergence of kerr evolutions . this figure summarizes the main parts for our claim that our tail results are reliable : the code is convergent to a non - zero value , and even increases in magnitude with grid refinement ( so that the signal seen is not an evolved noise signal ) , and the convergence order is global , throughout the computational domain . notice that in fig . [ 6th_2 ] we study the convergence with the monopole projection of the total field . this projection is more sensitive to the created and evolved monopole numerical noise than the full field , say , as we can test its convergence also at times earlier than the instant the monopole dominates the total field for the first time . figure [ 6th_3 ] depicts the ( @xmath60 norm of the ) full field and its monopole projection . the parameters are chosen so that intermediate tails are minimized , so that computation time is used efficiently . the monopole projection amplitude is many order of magnitudes higher than typical  multipole leakage " noise levels ( cf .  with fig . [ 6th_1 ] ) . indeed , it is the result of physical multipole excitations , not noise generation . below , we also bring evidence that the observed tail is indeed asymptotic , so that we are not misled by an intermediate tail masquerading as an asymptotic one .     for the monopole projection of the full field using the sixth order code for kerr . the angular resolutions are @xmath62 ( solid ) , @xmath63 ( dashed ) , and @xmath64 ( dotted ) . the initial data are an outgoing gaussian pulse located at @xmath55 with width of @xmath56 with compact support on the boyer  lindquist @xmath65 hypersurface of a kerr black hole with @xmath66 . the initial multipole is a pure boyer lindquist @xmath21 . the radial resolution is @xmath67 , and the temporal resolution is chosen so that @xmath58.,width=326 ]    . shown are the @xmath60 norm of the full field ( solid curve ) and the field s monopole projection ( dashed curve ) . for both , the angular resolution is @xmath50 and the radial resolution is @xmath57 . the temporal resolution is chosen so that @xmath58 . , width=326 ] gpp argue that when expanded in powers of @xmath31 , the tail would be at @xmath38 . when a mode @xmath68 exists at any iteration order ( in @xmath31 ) , at the next iteration three modes will result : @xmath69 ( if @xmath70 ) , @xmath68 , and @xmath71 . the mode excitation scheme is shown in fig . [ 6th_2_1 ] . in the gpp approach , this behavior comes about because of the term @xmath72 appearing in the source term ( see @xcite for detail ) . this way , starting with a pure @xmath21 mode , there is just one way to create a monopole field at @xmath38 , but three ways to create a monopole field at @xmath73 . we study two classes of initial data sets : in class a the field @xmath74 is a pure multipole ( in practice , we focus attention here mostly on the azimuthal @xmath21 mode ) and the field s  momentum " @xmath75 , where @xmath76 . here , @xmath77 , @xmath78 , and @xmath79 ( for details see , e.g. , @xcite ) . the @xmath80 coordinates are the boyer lindquist ones . class a initial data lead to tail a behavior ( i.e. , to @xmath81 ) , with the tail at @xmath38 . in class b initial data we choose the field @xmath74 in the same way as in class a , but replace the condition @xmath75 with @xmath82 or @xmath83 or a linear combination thereof ( with coefficients that are independent of @xmath84 ) . class b initial data lead to tail b behavior ( i.e. , to @xmath85 ) , with the tail at @xmath38 . ( the momentary stationary data case @xmath83 is unique , and leads to @xmath86 in a similar way to the schwarzschild case @xcite . ) the different sets of initial data are summarized in table [ table2 ] . .the initial data classes used in the text . unaccented coordinates mean any of the coordinate systems described in table [ table1 ] . see the text for more detail . [ cols="^,^,^",options="header " , ]     [ table1 ]    a spherical harmonic mode in boyer lindquist coordinates with a specific multipole @xmath7 and a specific azimuthal mode @xmath87 is therefore also a spherical harmonic mode in ingoing kerr coordinates with the same values for @xmath88 , and the difference is only in a different function of the radial coordinate . one may therefore expect that initial data specified in ingoing kerr coordinates would lead to the same tail behavior as in boyer  lindquist coordinates . one may object to this expectation by stating that the constant @xmath89 slices in the boyer lindquist case are not the same as the constant @xmath90 slices in the ingoing kerr case . while this difference is certainly existing and the two hypersurfaces are not the same , we find that it does not change the tail behavior .    ) mode in ingoing kerr coordinates on a @xmath91 slice . here , @xmath92 . the initial data radial profile is a gaussian with @xmath93 and @xmath94 . the grid resolution is @xmath95 and @xmath96 . the dashed curve is the carve @xmath97 which is added for reference . the local power index is calculated for three radial resolutions : @xmath98 ( dotted ) , @xmath99 ( dashed ) , and @xmath96 ( solid).,width=326 ]    class a initial data were studied in detail in @xcite . we therefore present here the results for class b initial data . we find that starting with a pure @xmath21 azimuthal mode ( in ingoing kerr coordinates on a @xmath91 slice ) the tail has tail b behavior , similar to that of boyer  lindquist coordinates and boyer lindquist slicing . figure [ ingoing_kerr ] shows the field and the local power index for these data . we infer from the simulation data that the tail index is @xmath100 , a deviation of @xmath101 from the tail b prediction of 5 . the ingoing kerr results are obtained using a code based on the penetrating teukolsky code ( ptc ) @xcite . the ptc is written on a fixed ingoing kerr coordinate grid , which means that grid resolution becomes increasingly poor approaching the event horizon . while simple fixed mesh refinement methods can resolve this difficulty , we were able to obtain the tail index using the following method : we first find the local power index for a number of time values for different choices of the radial ( and temporal ) resolution . we then extrapolate the local power indices to infinite radial resolution , and then extrapolate the sequence of local power indices at different values of the time ( and with infinite resolutions ) to infinitely late times . the result is rather close to the expected value of 5 , but suffers from somewhat higher numerical error because of the two extrapolation cycles . as noted above , the deviation of the tail index from the tail b value is at the order of @xmath102 . we therefore conclude that class b initial data in the ingoing kerr case lead to tail b behavior . we therefore contend that initial time slices , even when are distinct , fall into equivalency classes in terms of the type of tail behavior that emerges ( recall that on a given slice multipoles can also mix because of coordinate transformations ) . the boyer lindquist and ingoing kerr slices belong to the same equivalency class , in the sense that one finds the same tail b behavior for class b initial data . schild slices belong to a distinct equivalency class , and class b initial data lead to tail a behavior . we conjecture that the boyer  lindquist and ingoing kerr slices belong to the same equivalency class because the time slices in either are related by a transformation that does not involve the polar angular coordinate and that depends only on the temporal and azimuthal angular coordinate . one may therefore conjecture that other slicing conditions that differ from the boyer lindquist slicing condition by similar transformations will behave accordingly . poisson has made the argument that if class b initial data are specified in so - called  spherical coordinates , " rather than in boyer lindquist coordinates , which poisson calls  spheroidal , " mode evolution is changed @xcite . in particular , poisson proposes that in kerr spacetime such initial data lead to tail b behavior . in a spacetime with global weak curvature ( no black hole , but no assumptions on symmetry ) , poisson data indeed lead to such late  time behavior this may be surprising , as a pure mode of poisson coordinates may be written as a mixed state of boyer  lindquist modes . this mixed state of modes effectively behaves as class a initial data , so that the intuitively expected behavior is that of tail a.    clearly , the assumption of globally weak curvature affects the tails , and shifts them from their tail a values to those reported by poisson . this could be the case if the field reflects off the regular center in the poisson spacetime , so that the leading order part of the field ( in @xmath29 ) is cancelled out and one is left with the next order in @xmath29 , or if the monopole part of the field is canceled after the reflection , leaving only the dipole and higher moments present @xcite . the following question arises : what are the tail indices for  pure poisson " class b data in a kerr spacetime ? specifically , if one sets at @xmath65 a pure class b hexadecapole mode with respect to the poisson spherical " coordinates , does the late  time tail decay as @xmath103 , as appropriate for tail a , or as @xmath104 as appropriate for tail b ( and as predicted by @xcite ) ? as we are showing here , the answer is clearly @xmath103 . mode , @xmath105 ( solid curve ) , and its projections on boyer lindquist @xmath7 modes @xmath106 : dashed ( @xmath107 ) , dash - dotted ( @xmath46 ) , and dotted ( @xmath108 ) . here , @xmath66 . the initial data radial profile is a gaussian with @xmath109 and @xmath110 . the abscissa is @xmath111 or @xmath112 as is appropriate.,width=326 ]    initial data that describe a pure poisson " mode are a linear combination of boyer  lindquist modes , as one may expand such a mode in the function space of boyer  lindquist spherical harmonics . importantly , as the transformation from boyer lindquist to poisson coordinates does not involve the time , boyer lindquist slices are identical to poisson slices . therefore , one may use a code in which the wave equation is written in boyer lindquist coordinates to simulate the evolution of a pure poisson " mode , if one writes the initial data as the appropriate linear combination of boyer lindquist modes . this may be done as follows : denote the boyer lindquist coordinates as @xmath80 and the poisson spherical " coordinates as @xmath113 . as shown in @xcite , the coordinate transformation is @xmath114 where @xmath115 is a measure of the  ellipticity " of the boyer lindquist coordinates , which in practice equals the kerr black hole s spin @xmath30 . to specify a  pure poisson " mode , we take the field to be @xmath116 after choosing the radial function to be a gaussian of width @xmath117 centered on @xmath118 . the initial pure poisson " field @xmath119 can be expanded in the function space of boyer lindquist legendre polynomials as @xmath120 as shown in @xcite , the functions @xmath121 are complicated . ( in fact , only the coefficients for the _ inverse _ transformation are found in @xcite ( for @xmath21 and @xmath108 ) as a power series in the  ellipticity " @xmath115 . ) in fig . [ pd1 ] we show the functions @xmath105 , and @xmath106 for @xmath122 ( in general , these functions are non - vanishing for all values of @xmath40 , respecting the parity symmetry ) , for a high value for the  ellipticity , " specifically for @xmath123 . to complete the initial data problem , we also need to specify at @xmath65 the time derivative of the field . for class b initial data we choose @xmath124 , and for class a initial data we choose @xmath75 . figure [ poisson_classb ] depicts the full field and its multipole projections and the local power indices for the full field and its monopole projection as functions of time for class b initial data . the late time behavior is clearly that of tail a , with the monopole s decay rate at infinite time extrapolated to @xmath125 , within @xmath126 from the predicted value of @xmath127 .     mode , and its boyer lindquist @xmath122 modes , and the local power indices for the full field and for the boyer lindquist monopole projection . here , @xmath66 . the ( class b ) initial data are those of fig . [ pd1].,width=326 ]    in fig . [ pd2 ] we consider class a initial data . again , it suggests that the tail field drops of as @xmath103 for a scalar field with initial data that are a  pure poisson " hexadecapole mode . specifically , we show this to be the case more precisely , using the criterion based on simultaneous local and global behavior of the local power index . comparison of the evolution of the evolution of the  pure poisson " @xmath21 mode with the evolution of its boyer lindquist @xmath107 projection suggests that the monopole projection @xmath108 at the initial time is important at late times . the @xmath108 mode is being excited during the evolution also when it is not present at the initial time , and eventually it dominates the field at late times . however , the generated @xmath108 mode starts dominating the total field only at a later time , as its amplitude is lower than that of the monopole part of the full @xmath21 mode . the latter mode includes a monopole component already at the initial time , and this part of the field starts dominating the total field at an earlier time than the monopole component that is excited . for this reason the asymptotic @xmath21 mode is much stronger than the asymptotic tail of initial data that include only the @xmath107 projection of the @xmath21 mode , even though at the initial time they are almost identical , as can be seen in fig . [ pd1 ] . mode ( panel a ) , and its boyer lindquist @xmath122 modes ( panels b , c , and d , respectively ) ( solid curves ) , and the evolution of the boyer lindquist @xmath107 projection of the  pure poisson " mode and its boyer lindquist projections ( dashed curves ) . here , @xmath66 . the initial data are those of fig . [ pd1].,width=326 ]     projection of the class a  pure poisson " @xmath21 mode . the initial data are those of fig . [ pd1].,width=326 ]    to show that indeed the late  time tail is dominated by the evolution of the @xmath108 projection of the original  pure poisson " @xmath21 mode at @xmath65 , we present in fig . [ pd3 ] the evolution of this mode . comparison with fig . [ pd2 ] indeed shows that the late time field of the full @xmath21 mode is dominated by the evolution of its monopole projection , not the monopole mode that is excited during the evolution because of mode coupling .     for the class a  pure poisson " mode @xmath21 ( solid curve ) and for its @xmath108 boyer lindquist projection ( dashed curve ) . the initial data are those of fig . [ pd1 ] , and the field evolution is presented in fig . [ pd2].,width=326 ]    in fig . [ pd4 ] we show the behavior of the local power indices for the full field of the class a  pure poisson " @xmath21 mode , and for its monopole @xmath108 projection . figure [ pd4 ] strongly suggests that the full field , and also the monopole projection , drop off at late times as @xmath103 . indeed , the tail seen in figs . [ pd3 ] and [ pd4 ] is asymptotic , and the local power index satisfies the ansatz ( [ ansatz ] ) both locally and globally simultaneously . as noted in @xcite , it is not surprising that starting with the initial data of a @xmath128 mode , the @xmath129 mode is excited . figure [ pd2 ] ( in conjunction with the conclusion that the tail is asymptotic ) shows that when starting with a  pure poisson " mode , the excited monopole mode is insignificant for the asymptotic tail problem , as it is swamped by the evolution of the monopole projection of the initial data . above , we showed that starting with class b initial data of a pure boyer lindquist @xmath130 , the late time tail drops off according to @xmath26 , in accordance with tail b behavior , consistently with the results of @xcite . in that case , there is no monopole field present at @xmath65 , and all the monopole field present at late times is due to the excitation of the @xmath131 mode due to mode coupling . the excited mode that has tail b behavior is overwhelmed by the evolution of the monopole mode existing at the initial time that has tail a behavior . most importantly , we choose the initial data to be a pure spherical harmonic mode in the so - called  spherical " coordinates . the late time tail still decays according to tail a behavior , in contrast with the tail b expectations based on @xcite . in the case presented , as the initial mode is azimuthal , the asymptotic tail drops off as @xmath23 . the authors wish to thank richard price and jorge pullin for invaluable discussions . early work on this problem was done with elspeth allen at the university of utah . lmb is supported in part by nasa / gsfc grant no . ncc580 and by nasa / ssc grant no . gk is grateful for research support from the umd college of engineering rsi fund and scea . many of the simulations were performed on nsf s teragrid infrastructure ( under grant number tg - phy060047 t ) and the hpc consortium s facilities .
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 @xmath0 is according to @xmath1 , where @xmath2 is the least multipole moment whose excitation is not disallowed , or whether the decay rate is according to @xmath3 , where @xmath4 . 
 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 @xmath1 late time tail . 
 we generalize the rule @xmath4 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 .