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the general solution for an asymptotically flat black hole in the einstein - maxwell theory is given by the kerr - newman solution . it describes an electrically charged rotating black hole with its three parameters : mass , charge and angular momentum . a systematic study of the fields dynamic in the vicinity of a black hole is essential for understanding black - hole evaporation , quasinormal modes and stability . the linear dynamics of a charged massive scalar field in the background of a charged black hole is characterized by the two dimensionless parameters @xmath0 and @xmath1 . a black hole is not believed to be formed possessing considerable electric charge , and , once it is formed , it undergoes a rather quick discharging @xcite . yet , even if a black hole has a very small charge of order @xmath7 , the parameter @xmath1 need not be small . in addition , a charge induced by an external magnetic field , may be formed at the surface of an initially neutral , but rotating black hole @xcite . thus , the complete analysis of a massive charged scalar field dynamics should include consideration of the whole range of values @xmath8 and @xmath0 . in this work we shall study the stability and evolution of perturbation of a massive charged scalar field in the kerr - newman background in terms of its _ quasinormal modes _ and _ asymptotic tails _ at late times . it is believed that if the quasinormal modes are damped , the system under consideration is stable , though a rigorous mathematical proof of stability is usually complicated and sometimes includes a nontrivial analysis of the initial value problem . by now , quite a few papers have been devoted to scalar field perturbations in the black - hole background , yet , while the behavior of the massless neutral scalar field is indeed very well studied , the quasinormal modes of charged fields was studied only in the regime @xmath9 , @xmath10 @xcite , except for @xcite , where the wkb estimation for quasinormal modes ( qnms ) of a massless charged scalar field around the reissner - nordstrm black hole was given in the regime @xmath11 , where @xmath12 is the multipole number . the complete analysis of quasinormal modes ( allowing also to judge about stability ) for a massive charged scalar field _ for arbitrary values _ @xmath8 and @xmath0 has been lacking so far not only for the kerr - newman , but even for reissner - nordstrm solutions . here we shall solve this problem by adopting the two numerical methods of computation ( the frobenius method and the time - domain integrations ) based on convergent procedures , which allow us to find quasinormal modes accurately and with no restriction on the parameters of the system . perturbation of a charged massive field in the background of a rotating charged black hole has rich physics , because there are a number of phenomena which must be taken into consideration :    * _ superradiance _ , that is the amplification of waves with particular frequencies reflected by a black hole , if it is rotating @xcite or electrically charged @xcite . thus , there will be the two regimes of superradiance for kerr - newman black holes , owing to charge and rotation @xcite . * _ superradiant instability _ of bound states around black holes owing to the massive term , which creates a local minimum far from the black hole , so that the wave will be reflected repeatedly and can grow . it is essential that this instability occurs under the bound states boundary condition , which differ from the quasinormal modes ones @xcite . * _ quasiresonances . _ when a field is massive , quasinormal modes with arbitrarily long lifetimes , called quasiresonances , appear , once some critical value of mass of the field is achieved @xcite . when the damping rate goes to zero , the quasinormal asymptotically approach the bound state , but still remain quasinormal modes for whatever small but nonzero damping . @xcite . * _ instability of the extremal black holes _ that apparently occurs for fields of any spin and both for extremal reissner - nordstrm and kerr black holes @xcite , and therefore , must be expected for the extremal kerr - newman solution as well . however , in the linear approximation this instability develops only on the event horizon and can not be seen by an external observer . * _ mode branching_. quasinormal modes of kerr black holes were believed to be completely studied until a few months ago when an interesting observation has been made @xcite . it was shown that , for the near - extremal rotation there are two distinct sets of damped quasinormal modes , which merge to a single set in the exactly extremal state @xcite .    here , through the numerical analysis of quasinormal modes and asymptotic tails we have shown that a massive charged scalar field is stable in the vicinity of the kerr - newman black hole , in spite of the instability of the corresponding bound states . we found that at some values of the field s charge @xmath3 quasinormal modes may behave qualitatively differently from those of the neutral field : the fundamental mode ( dominating at late times ) may have arbitrarily small real part ( real oscillation frequency ) which appears in the time domain as a very short period of quasinormal ringing consisting of damped oscillations and the quick onset of asymptotic power - law tails . in addition , we generalized earlier results on mode branching of massless fields around nearly extremal kerr black holes to the massive fields and kerr - newman solutions . an analytic formula has been obtained for large @xmath1 . the paper is organized as follows . in sec ii the basic formulas for a charged massive scalar field in the kerr - newman background is given . the wave equation is reduced to the schrdinger - like form with an effective potential . sec iii describes the numerical methods which we used : the frobenius methods , two schemes of time - domain integration ( for neutral and charged fields ) together with the method for extraction of frequencies from the time - domain profiles , called the prony method , and the wkb approach . we have related separately perturbations of nonextremal black holes ( sec iv ) and nearly and exactly extremal ones ( sec . v ) , as near extremal black holes shows new phenomena , such as mode branching . in sec . vi we discuss some technical difficulties which appear when one considers higher overtones of a charged scalar field or approach closely to the extremal state , keeping @xmath3 non zero . in sec vii we summarize the results obtained . in the boyer - lindquist coordinates the kerr - newman metric has the form @xmath13 ^ 2,\end{aligned}\ ] ] where @xmath14 @xmath15 and @xmath5 is the black - hole charge , @xmath4 is its mass . the electromagnetic background of the black hole is given by the four - vector potential , @xmath16 we shall parameterize the metric by the following three parameters : the event horizon @xmath17 , the inner horizon @xmath18 , and the rotation parameter @xmath19 , @xmath20 the black hole s mass and charge are then @xmath21    a massive charged scalar field satisfies the klein - gordon equation , @xmath22 where @xmath3 and @xmath2 are the field s charge and mass respectively . one separates variables by the following ansatz @xmath23 where @xmath24 obeys the following equation @xmath25 and @xmath26 is the separation constant . this equation can be solved numerically for any value of @xmath27 in the same way as the equation for a massive scalar field in the kerr black - hole background @xcite . let us note that , when @xmath28 , eq . ( [ angularpart ] ) can be reduced to the well - known equation for the spheroidal functions . in this case , for any fixed value of @xmath27 the separation constant @xmath26 can be found numerically using the continued fraction method @xcite . when the effective mass is not zero , the separation constant @xmath29 can be expressed , in terms of the eigenvalue for spheroidal functions @xmath30 @xcite , as @xmath31 when @xmath32 , one has @xmath33 . for nonzero values of @xmath19 , the separation constant can be enumerated by the integer multipole number @xmath34 . the radial function satisfies a schrdinger equation , @xmath35 where @xmath36 is the tortoise coordinate , @xmath37 and the effective potential is @xmath38    the asymptotics of the effective potential near the event horizon and at spatial infinity are @xmath39 where we fix the sign of @xmath40 such that @xmath41 is of the same sign as @xmath42 . note , that @xmath43 and @xmath42 can have different signs . this corresponds to the superradiant regime in which one has @xmath44    by definition , quasinormal modes ( qnms ) are proper oscillation frequencies which correspond to purely incoming wave at the event horizon and purely outgoing wave at infinity , so that no incoming waves from either of the `` infinities '' are allowed . thus , the boundary conditions for the qnms can be written as follows @xmath45 here , we shall briefly relate the three numerical methods used for finding quasinormal frequencies :    * leaver method , which is based on a convergent procedure and , thereby , allowing one to find qn modes accurately , * wkb method ( accurate in the regime of high multipole numbers ) , * time - domain integration which includes contribution of all modes , and , together with the prony method , usually allows extracting a few lower dominant frequencies from a time - domain profile . we shall see that in some ranges of parameters , when one method becomes slowly convergent or inapplicable , the other can be used , so that the use of a few alternative methods is necessary here not only for an additional checking , but also for getting the complete picture of quasinormal modes and stability in the full range of parameters . equation ( [ radialpart ] ) has an irregular singularity at spatial infinity and four regular singularities at @xmath46 , @xmath47 and @xmath48 . the appropriate frobenius series is determined as @xmath49 where @xmath50 and @xmath51 is the hawking temperature @xmath52 the function @xmath53 must be regular at the horizon and spatial infinity , so that the series in the vicinity of the event horizon @xmath54 satisfies both these requirements . this series converges everywhere outside the event horizon ( @xmath55 ) . when boundary conditions ( [ boundary - conditions ] ) are satisfied , that is when @xmath27 is a quasinormal ( qn ) frequency , the series convergence also at the spatial infinity @xcite . the coefficients @xmath56 satisfy the three - term recurrence relation . @xmath57 where @xmath58 , @xmath59 , @xmath60 can be found in an analytic form .    by comparing the ratio of the series coefficients @xmath61we obtain an equation with a convergent _ infinite continued fraction _ on its right side : @xmath62which can be solved numerically by minimizing the absolute value of the difference between its left- and right - hand sides . equation ( [ continued_fraction ] ) has an infinite number of roots , but the most stable root depends on @xmath63 . the larger number @xmath63 corresponds to the larger imaginary part of the root @xmath27 @xcite . as we study qnms with slower decay rate , we usually choose @xmath64 . in order to improve convergence of the infinite continued fraction for nonzero mass of a scalar field , we use the nollert procedure @xcite .      as for the study of mode branching we are also interested in modes with high multipole numbers , the wkb approach is useful here , which is accurate in the eikonal regime @xmath65 and usually provides very good accuracy at moderate @xmath66 . the wkb formula for calculation of qnms has the following form : @xmath67 where @xmath68 and @xmath69 are the values of the effective potential ( [ effective - potential ] ) and its second derivative with respect to the tortoise coordinate @xmath36 at the potential s peak . the terms @xmath70 depend on higher derivatives of @xmath71 at its maximum , and @xmath63 labels the overtones . the wkb approach was developed by schutz and will @xcite and later extended to higher orders @xcite . since there is implicit dependence on @xmath27 ( either through the separation constant @xmath26 for nonvanishing rotation or due to the @xmath1 coupling in the effective potential ( [ effective - potential ] ) ) , one has to search for the roots of equation ( [ wkbformula ] ) by minimizing the absolute value of the difference between its left and right sides .      for nonrotating black holes we are able to construct a time - dependent profile of the wave function at a fixed @xmath36 . the recently found instability of the exactly extremal reissner - nordstrm black hole @xcite makes it important to check stability of the nearly and exactly extremal black holes . even when the frobenius method finds no growing quasinormal modes , one could think that the growing modes were simply missed in the frequency domain at the stage of search for the roots of the equation with continued fractions eq . ( [ continued_fraction ] ) , or , that a different boundary condition should be imposed at the event horizon in the extremal case . in order to eliminate both these suspicions , we will use the numerical characteristic integration method in time domain . here , two different schemes of integration were used for _ neutral _ and _ charged _ fields .    * a scheme for a neutral field . * the wave equation can be written in time - dependent form as follows : @xmath72 the technique of integration of the above wave equation in the time domain was developed in @xcite . the method uses the light - cone variables @xmath73 , @xmath74 , so that the wave equation reads @xmath75 the initial data are specified on the two null surfaces @xmath76 and @xmath77 . acting by the time evolution operator @xmath78 on @xmath79 and taking account of ( [ light - cone ] ) , one finds @xmath80 where one introduced letters to mark the points as follows : @xmath81 , @xmath82 , @xmath83 , and @xmath84 . equation ( [ integration - scheme ] ) allows us to calculate the values of @xmath79 inside the rhombus , which is built on the two null - surfaces @xmath85 and @xmath86 , starting from the initial data specified on them . as a result we can find the time profile data @xmath87 in each point of the rhombus . the time - domain integration includes a contribution from all overtones , and , thus , missing some mode is excluded . this method is based on the scattering of the gaussian wave on the potential barrier and therefore does not specify the boundary condition on the event horizon , so that the potentially missed instability due to possibly different boundary conditions must be also discarded .    * a scheme for a charged field . * when the scalar field is charged , an extra term containing the first derivative in time appears . therefore , a different integration scheme is required . we shall use the finite difference scheme proposed in @xcite . first we rewrite the wave - like equation ( [ radialpart ] ) for the reissner - nordstrm black hole ( @xmath32 ) in the time - dependent form @xmath88 where @xmath89 following @xcite , one can derive the evolution of @xmath79 in an isosceles triangle with the base on the axis @xmath36 , where initial conditions are imposed . then , one has @xmath90 indexes @xmath91 and @xmath92 enumerate , respectively , the coordinates @xmath93 and @xmath36 of the grid : @xmath94 we choose the initial conditions again as a gaussian distribution whose maximum is near the maximum of the effective potential . since von neumann stability conditions require @xmath95 , in this scheme we chose @xmath96 . in order to achieve convergence we decrease @xmath97 . note , that as @xmath1 grows the convergence becomes seemingly slower , requiring smaller @xmath97 , what increases the computation time . when @xmath98 this scheme is reduced to the one above for the neutral field , yet , the codes for both schemes differ , so that letting @xmath98 in the mathematica code for the second scheme leads to a much longer computing than the first scheme . * prony method for mode extraction . * once a time domain profile is found , one can extract dominant frequencies from it with the help of the prony method . we fit the profile data by superposition of damped exponents @xmath99 and look for the convergence of the obtained frequencies at the increasing @xmath100 . we shall show that although the fit works well for the neutral scalar field , it can not be effectively used once modes with very small @xmath42 dominate in the spectrum at late times . in the regime of large @xmath1 and nonextremal @xmath5 , the frobenius method allows us to find an approximate analytic expression for the quasinormal frequencies . when @xmath101 one can observe that @xmath102 , @xmath103 , and @xmath104 . then , we can rewrite eq . ( [ continued_fraction ] ) as @xmath105 considering @xmath106 , from ( [ continued_fraction_qq ] ) we find that @xmath107 . we observe that @xmath108 when @xmath109 . then , we can write down @xmath26 as @xmath110 and find the asymptotic formula for the qnms @xmath111 here @xmath112 is the overtone number . from the data given in fig . [ figasymptot ] one can find that @xmath113 for @xmath114 . in order to find @xmath115 as a function of @xmath12 and @xmath116 we have analyzed numerically the asymptotic behavior of the eigenvalue of ( [ angularpart ] ) and found that @xmath117 + 2,\ ] ] where the brackets denote the integer part .    when @xmath5 is not very close to its extremal value , the above analytical formula ( in the limit of vanishing rotation ) can be verified by the time - domain integration for a few lower modes , as it is shown in fig . [ figworotation ] . there one can see that the analytic formula ( [ asymptotical_frequency ] ) works very well already at moderate values of @xmath1 . nevertheless , even though we are able to obtain a stable time - domain profile for any values of @xmath3 and @xmath5 , the prony method does not converge for high overtones as well as near extremal black holes . we shall discuss this in detail in the last section . for @xmath32 ( [ asymptotical_frequency ] ) coincides with the asymptotic formula found in @xcite in the regime @xmath118 with the help of the wkb approximation . unlike @xcite in our calculations one does not need to be limited by the regime @xmath119 . it is well known that due to a symmetry of the kerr - newman metric ( @xmath120 , @xmath121 ) quasinormal modes of a neutral scalar field come in within a degenerate pair @xmath122 when the scalar field is charged , in addition to ( [ degeneration ] ) one needs to take @xmath123 for the symmetry to be restored . that is why we study only qnms with positive real parts . the degeneration ( [ degeneration ] ) does not exist as soon as the coupling @xmath1 is not zero ( figs .  [ figworotation ] and [ figrotation ] ) . then , at negative values of @xmath1 , the mode with a positive real part has slower decay rate than the one with the negative one ( while for positive @xmath1 the situation is opposite ) . moreover , when the absolute value of @xmath1 increases ( figs . [ figworotation ] and [ figrotation ] ) , the real oscillation frequency ( given by @xmath42 ) approaches zero and then the mode `` disappears '' from the spectrum at @xmath1 larger than some critical value . such disappearing of a mode in some range of parameters is not unusual and happens , for example , for arbitrarily long living modes ( quasiresonances ) of a massive neutral scalar field @xcite . we observe that at asymptotically late times , a neutral massless scalar field decays as @xmath124 in concordance with @xcite , while for the charged massless field , the dominant asymptotical tail is @xmath125 as it was first found in @xcite . " note , that in the regime of large @xmath8 the correction formula to ( [ chargedtail ] ) was reported in @xcite , which might be correct , and , then , should appear at larger values of @xmath8 than those we considered in the numerical time - domain evolution here . the late - time tails in the background of the extremally charged black hole obey the same law ( see fig . [ figtailsextr ] ) . at intermediately late times , _ a massive charged _ scalar field decay as ( see fig .  [ figasympttailsmassive ] ) @xmath126 while at asymptotically late times , the decay law is @xmath127    thus , the power laws of a massive charged scalar field s decay at intermediate and asymptotically late times do not depend on multipole @xmath12 or charges @xmath3 and @xmath5 . the same law was obtained in @xcite for the massive dirac field . we suppose thereby that the asymptotic behavior of the charged massive fields does not depend on spin . the decaying time - domain profiles for various values of the parameters @xmath3 , @xmath5 and @xmath2 show that no instability exist for nonextremal reissner - nordstrm black holes . the complementary frequency domain data gives no indication of growing modes when the rotation is not vanishing . therefore , we conclude that there are no signs of instability for a massive charged scalar field in the nonextremal kerr - newman background under the quasinormal modes boundary conditions . we shall measure all quantities in units of @xmath128 , that is we take @xmath129 . in order to keep staying near the extremal state , it is sufficient to keep @xmath130 close to @xmath128 . here we shall start from the generalization of level plots of @xcite , which show the branching of modes . in @xcite it was found that , in the near - extremal kerr limit , zero - damped mode ( zdm ) satisfies @xmath131 where @xmath132 , @xmath133 .    therefore , in fig . [ fig1 ] the same multipole @xmath12 and azimuthal @xmath116 numbers were chosen as in @xcite . in fig . [ fig1 ] it is shown the logarithm of the absolute value of difference between the left and right hand sides of the continued fraction equation ( [ continued_fraction ] ) as a function of real and imaginary parts of @xmath27 . in fig .  [ fig1 ] we can see that at some near - extremal @xmath130 , and @xmath19 , which starts from @xmath134 until its extremal value , the new branch of modes appears : the two branches of modes , which take place for high rotation ( @xmath135 ) , merge into a single one at slower rotation ( @xmath136 ) . as it is expected , for small values of black hole s charge @xmath5 , the obtained in fig . [ fig1 ] zero - damped mode is close to the one described by eq . ( [ eqhod2 ] ) . the mode branching is owing to extremal rotation and not owing to extremal charge . indeed , a neutral scalar field around the near - extremal reissner - nordstrm black hole shows no mode branching . we have checked that the time - domain profile of scalar and gravitational perturbations of the exactly extremal reissner - nordstrm black hole consists of damped quasinormal oscillations , what proves stability of the extremal reissner - nordstrm solution from the point of view of an external observer , because 1 ) the profile includes contribution from all the modes , so that none can be missed , 2 ) the method of calculations does not imply any specific boundary conditions at the event horizon . this does not contradict a special instability of the extremal reissner - nordstrm black holes @xcite as this instability develops only on the event horizon and can not be observed by an external observer @xcite . .fundamental quasinormal modes ( @xmath114 ) of the scalar field for the extremal kerr - newman black hole ( @xmath137 ) . [ cols="^,^",options="header " , ]     we have also showed that the decay of a scalar field ( be it charged or neutral , massive or massless ) is dominated by the power - law tails ( see table ( [ tail - summary ] ) ) which are the same for the nonextremal black hole as for the extremal one . here we have considered an already formed black hole and the perturbation propagating outside of it , while the modeling of collapse of charged matter leads to different result for the late - time decay of the extremal reissner - nordstrm black hole @xcite . thus we complement earlier results on late - time tails of neutral massive @xcite and massless fields @xcite in the schwarzschild and reissner - nordstrm backgrounds . in addition , when @xmath138 the continued fraction equation ( [ continued_fraction ] ) indicates presence of some new roots in the frequency domain , which do not exist for the neutral field , and whose real parts are proportional to @xmath139 . these roots exist for any value of @xmath5 and @xmath19 ( and nonzero @xmath3 ) , but correspond to higher `` overtones '' for weakly charged black hole ( fig . 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so far analysis of the quasinormal spectrum of a massive charged scalar field in the black hole background has been limited by the regime of small @xmath0 and @xmath1 , where @xmath2 , @xmath3 ( @xmath4 , @xmath5 ) are mass and charge of the field ( black hole ) . here 
 we shall present a comprehensive picture of quasinormal modes , late - time tails and stability of a massive charged scalar field around kerr - newman black holes for any physically meaningful values of the parameters . 
 we shall show that despite presence of the two mechanisms of superradiance ( owing to black hole s rotation and charge ) and the massive term creating growing bound states , there is no indication of instability under quasinormal modes boundary conditions . 
 we have shown that for some moderate values of @xmath1 dominant quasinormal modes may have _ arbitrarily small real oscillation frequencies _ 
 @xmath6 . 
 an analytic formula for the quasinormal modes has been derived in the regime of large @xmath1 . the larger the field s charge , the sooner asymptotic tails dominate in a signal , making it difficult to extract quasinormal frequencies from a time - domain profile . 
 analytic expressions for intermediate and asymptotically late - time tails have been found for the reissner - nordstrm black hole . for the near extremal kerr - newman black holes we have obtained a more general picture of the _ mode branching _ found recently for massless fields ( arxiv:1212.3271 ) in the kerr background .