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motivated by the beliefs that one can directly identify a black hole existence by comparing qnms with the gravitational waves observed in the universe , and the study of qnms may lead to a deeper understanding of the thermodynamic properties of black holes in loop quantum gravity @xcite @xcite , as well as the qnms of anti - de sitter black holes have a direction interpretation in terms of the dual conformal field theory @xcite @xcite @xcite , the investigation of qnms in black hole spacetimes becomes appealing recent years . since regge and wheeler @xcite first presented the idea of the qnms and chandrasekhar and detweller @xcite worked out the numerically calculation of the qnms for the schwarzschild black hole , a great deal of effort has been contributed to compute the qnms of the black holes @xcite-@xcite for fields of integer spin , such as the scalar , electromagnetic and gravitational perturbations .    on the contrary , the study of the qnms of the dirac field is very limited @xcite@xcite@xcite . cho @xcite studied the qnms of the dirac field of the schwarzschild black hole with the help of the third - order wkb method , and zhidenko @xcite extended the study to the case of the schwarzschild - de sitter black hole by using the sixth - order wkb method . we @xcite investigated the dirac qnms of the reissner - nordstrm de sitter black hole using the pshl - teller potential approximation . nevertheless , in these papers the study of the dirac qnms were limited by using wkb or pshl - teller potential approximative methods . the reason to obstruct study of the dirac qnms is that for static black holes the standard wave - equations @xmath2 possesses the special potentials @xmath3 which are the functions of @xmath4 @xcite@xcite@xcite , where @xmath5 is a function related to metric , say @xmath6 for the schwarzschild black hole . so , it is very hard to calculate dirac qnms by using the known numerical methods ( except wkb or pshl - teller approximative methods ) since we have to expand the potentials as a series at boundary of the field , such as at the event horizon . however , castello - branco _ et al _ @xcite evaluated the dirac qnms of the schwarzschild black hole by using the continued fraction approach in which they expanded the wave function as a series of @xmath7 , and they found that the spacing for imaginary part is not @xmath8 , as it takes place for scalar , electromagnetic , and gravitational perturbations , but @xmath9 . recently , we @xcite@xcite found that the wave functions and potentials of the dirac field in the static spacetimes can be expressed as new forms , and the new wave functions are related to the standard wave functions @xmath10 in eq . ( [ sta ] ) in a simple way @xcite . starting from the new wave functions and potentials , we @xcite@xcite showed that the dirac qnms of the schwarzschild anti - de sitter and reissner - norstrm anti - de sitter black holes can easily be worked out by using horowitz - hubeny approach @xcite . we have checked that the new potentials also present correct quasinormal frequencies for the scalar and gravitational perturbations in the schwarzschild black hole . the main purpose of this paper is to study dirac qnms of the schwarzschild black hole by using the new kinds of wave functions and potentials with the help of the continued fraction and hill - determinant approaches . the results obtained here are different from castello - branco s one @xcite ( if taking @xmath11 , we can only obtain results @xmath12 in ref . @xcite . that is to say we do not find  specific modes " . ) . we are sure that we do not miss any quasinormal frequency because the hill - determinant approach gives all roots for @xmath13 . the organization of this paper is as follows . in sec . 2 the decoupled dirac equations and corresponding wave equations in the schwarzschild spacetime are obtained by using newman - penrose formalism . in sec . 3 the numerical approach to computing the dirac qnms is introduced . in sec . 4 the numerical results for the dirac qnms in the schwarzschild black hole are presented . the last section is devoted to a summary . the dirac equations @xcite are @xmath14 where @xmath15 is covariant differentiation , @xmath16 and @xmath17 are the two - component spinors representing the wave function , @xmath18 is the complex conjugate of @xmath19 , and @xmath20 is the particle mass . in the newman - penrose formalism @xcite the equations become @xmath21 for the schwarzschild spacetime @xmath22 where m represents the black hole mass , the null tetrad can be taken as @xmath23 with @xmath24 then , if we set @xmath25 where @xmath26 and @xmath27 are the energy and angular momentum of the dirac particles , after tedious calculation eq . ( [ np ] ) can be simplified as @xmath28 with @xmath29    we can eliminate @xmath30 ( or @xmath31 ) from eqs . ( [ aa1 ] ) and ( [ aa2 ] ) and obtain @xmath32s_s=0,\end{aligned}\ ] ] here and hereafter we take @xmath33 for the case @xmath30 ( @xmath34 ) and @xmath35 for @xmath31 ( @xmath36 ) , and @xmath37 and @xmath38 . the angular equation ( [ ang ] ) can be solved exactly and @xmath39 , where @xmath40 is the quantum number characterizing the angular distribution . so , for both cases @xmath33 and @xmath35 we have @xmath41    we will focus our attention on the massless case in this paper . therefore , we can eliminate @xmath42 ( or @xmath43 ) from eqs . ( [ dd2 ] ) and ( [ dd3 ] ) to obtain a radial decoupled dirac equation for @xmath44 ( or @xmath42 ) . then , introducing an usual tortoise coordinate @xmath45 and resolving the equation in the form @xmath46 we obtain the wave equation @xmath47 where @xmath48+\frac{s^2 r^4}{4}\left[\frac{d}{d r}\left(\frac{\delta}{r^4}\right)\right]^2+is \omega r^2\frac{d}{d r}\left(\frac{\delta}{r^4}\right)+\frac{\lambda^2 \delta}{r^4}.\end{aligned}\ ] ] although the form of the potential is more complicated than standard wave - equations @xmath49 with the potentials @xmath50 @xcite@xcite@xcite , we will see that we can easily work out the dirac quasinormal frequencies of the schwarzschild black hole from eqs . ( [ wave ] ) and ( [ poten ] ) by using continued fraction and hill - determinant approaches . for the schwarzschild black hole , the qnms are defined to be modes with purely ingoing waves at the event horizon and purely outgoing waves at infinity @xcite . then , the boundary conditions on wave function @xmath51 at the horizon @xmath52 and infinity @xmath53 can be expressed as @xmath54 where @xmath55 is the surface gravity of the black hole .    a solution to eq . ( [ wave ] ) that has the desired behaviour at the boundary can be written in the form @xmath56 if we take @xmath57 , the sequence of the expansion coefficient @xmath58 is determined by a three - term recurrence relation staring with @xmath59 : @xmath60 the recurrence coefficient @xmath61 , @xmath62 , and @xmath63 are given in terms of @xmath64 and the black hole parameters by @xmath65 with @xmath66    the three terms recurrence relation ( [ rec ] ) can be used to find quasinormal frequencies by using both the continued fraction and hill - determinant methods which are introduced in the following .    * [ the continued fraction approach ] : * the ratio of the series coefficients @xmath67 is finite and can be determined in two ways : @xmath68 using eqs . ( [ rec ] ) and ( [ an ] ) we can obtain an equality between two continued fractions , one of infinite length , and the other finite : @xmath69=   \left[\frac{\alpha_m\gamma_{m+1}}{\beta_{m+1}- }   \frac{\alpha_{m+1}\gamma_{m+2}}{\beta_{m+2}- }   \frac{\alpha_{m+2}\gamma_{m+3}}{\beta_{m+3}-} ... \right ] ,   ~~(m=1,2 ... ) . \end{aligned}\ ] ] this leads to a simple method to find quasinormal frequencies  define a function which returns the value of the continued fraction for an initial guess at the frequency . then , use a root finding routine to find the zeros of this function in the complex plane . the frequency for which this happens is a quasinormal frequency . the @xmath70th quasinormal frequency is usually found to be numerically the most stable root of the @xmath70th inversion @xcite . the success of the root - finding technique depends greatly on the initial guess given for the root . however , which initial approximation we shall take is a difficult work , especially at the beginning . fortunately , we can easily find the low overtone quasinormal frequencies by means of the hill - determinant method . * [ the hill - determinant method ] : * the condition that nontrivial solutions of eq . ( [ rec ] ) exist is given by the requirement that the hill - determinant vanish @xcite@xcite . the determinant can be easily seen in this case to be of the form @xmath71 if @xmath72 stand for the @xmath73 determinant of @xmath74 , then the @xmath72 can be concisely expressed in terms of a difference relation @xcite @xmath75 these equations are polynomials in @xmath26 and can be solved at each order . however , the further requirement that the waves be purely outgoing at infinity and obey the stability conditions restrict the frequencies to complex plane with negative imaginary parts which are just the quasinormal frequencies we look for . the method is conceptually simpler than the continued fraction method of leaver . however , the method looses accuracy for the higher overtones since one must evaluate the determinant at values of order @xmath76 ( for giving @xmath76 , one require solving polynomials of the order of @xmath77 in @xmath26 ) very much higher than the overtone number @xmath70 of the quasinormal modes . also , the method does not have any control over the stability of the root search algorithm . we have checked that both the methods present the same results for @xmath78 to @xmath79 and @xmath80 to @xmath81 . in this section we represent the numerical results of the dirac quasinormal frequencies of the schwarzschild black hole obtained by using the numerical approaches just outlined in the previous section . the results will be organized into two subsections : the fundamental quasinormal modes and high overtones . the fundamental quasinormal frequencies ( n=0 ) for @xmath82 to @xmath83 are listed in the table ( [ table1 ] ) . ( [ fig1 ] ) shows that @xmath84 as a function of @xmath85 . from the table and figure we know that the fundamental quasinormal frequencies become evenly spaced for large @xmath85 and the spacing is given by @xmath86    .[table1 ] the fundamental dirac quasinormal frequencies of the schwarzschild black hole for @xmath82 to @xmath83 . [ cols="^,^,^,^,^,^,^,^",options="header " , ]      first 21 dirac quasi - normal frequencies of the schwarzschild black hole for @xmath82 to @xmath79 . ] dependence of the real parts of the quasinormal frequencies on @xmath85 for the schwarzschild black hole , which shows that @xmath85 has the surprising effect of increasing @xmath87 . ] dependence of the imaginary parts of the quasinormal frequencies on @xmath85 for the schwarzschild black hole , which shows that @xmath85 almost does not affect imaginary part , @xmath88 , especially for the lower overtones . ]     graphs of @xmath89 versus overtone number @xmath70 for @xmath90 . the top figure is drawn for @xmath91 which shows that @xmath92 tends to zero for large @xmath70 . the bottom one is for @xmath93 which shows that @xmath94 for large @xmath70.,title="fig : " ] +   graphs of @xmath89 versus overtone number @xmath70 for @xmath90 . the top figure is drawn for @xmath91 which shows that @xmath92 tends to zero for large @xmath70 . the bottom one is for @xmath93 which shows that @xmath94 for large @xmath70.,title="fig : " ] the wave equations for the dirac fields in the schwarzschild black hole spacetime are obtained by means of the newman - penrose formulism . then , the quasinormal frequencies corresponding to the dirac field perturbation in the black hole spacetime are evaluated by using continued fraction and hill - determinant approaches . three interesting results obtained in this paper are listed in the following : ( i ) the fundamental quasinormal frequencies become evenly spaced for large @xmath85 and the spacing is given by @xmath95 . ( ii ) @xmath85 has the surprising effect of increasing real part of the quasinormal frequencies , but it almost does not affect imaginary part , especially for the low overtones . ( iii ) the quasinormal frequencies also become evenly spaced for large @xmath70 and the spacing for imaginary part is @xmath8 , which is same as that of the scalar , electromagnetic , and gravitational perturbations .    in future papers in this series the new wave functions and potentials will be applied to study of dirac qnms in other black hole spacetimes , such as the charged reissner - norstrm black hole and the rotating kerr black hole . this work was supported by the national natural science foundation of china under grant no . 10275024 and under grant no . 10473004 ; the fanedd under grant no . 200317 ; and the srfdp under grant no .
the quasinormal modes ( qnms ) associated with the decay of dirac field perturbation around a schwarzschild black hole is investigated by using continued fraction and hill - determinant approaches . 
 it is shown that the fundamental quasinormal frequencies become evenly spaced for large angular quantum number and the spacing is given by @xmath0 . 
 the angular quantum number has the surprising effect of increasing real part of the quasinormal frequencies , but it almost does not affect imaginary part , especially for low overtones . 
 in addition , the quasinormal frequencies also become evenly spaced for large overtone number and the spacing for imaginary part is @xmath1 which is same as that of the scalar , electromagnetic , and gravitational perturbations . 
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