Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges
Owen Pavel Fernández Piedra, Fidel Sosa, José L. Bernal-Castillo, Yulier Jimenez
aa r X i v : . [ h e p - t h ] J un November 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2
International Journal of Modern Physics Dc (cid:13)
World Scientific Publishing Company
OBJECT PICTURE, QUASINORMAL MODES AND LATE TIMETAILS OF FERMION PERTURBATIONS IN STRINGY BLACKHOLES WITH U(1) CHARGES
OWEN PAVEL FERN ´ANDEZ PIEDRA
Grupo de Estudios Avanzados, Departamento de F´ısica, Facultad de Ingenier´ıa, Universidad deCienfuegos, Carretera a Rodas,km 4, Cuatro Caminos, Cienfuegos, [email protected]
FIDEL SOSA NU ˜NEZ, JOSE BERNAL CASTILLO, YULIER JIMENEZ SANTANA
Grupo de Estudios Avanzados, Departamento de F´ısica, Facultad de Ingenier´ıa, Universidad deCienfuegos, Carretera a Rodas,km 4, Cuatro Caminos, Cienfuegos, Cuba
The aim of the present report is the study of massless fermion perturbations outside five-dimensional stringy black holes with U(1) charges. The Dirac equation was numericallysolved to obtain the time profiles for the evolving fermion fields, and the quasinormalfrequencies at intermediate times are computed by numerical Prony fitting and semi-analytical WKB expansion at sixth order. We also computed numerically the late-timepower law decay factors, showing that there are in correspondence with previously re-ported results for the case of boson fields in higher dimensional odd space-times. Thedependence of quasinormal frequencies with U(1) compactification charges are studied.
Keywords : black holes; quasinormal modes
1. Introduction
The study of test field perturbations in a black hole background has been an interest-ing subject of research since the pioneer work of Regge and Wheeler 1 , , , , , , , , ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al theories. In this sense, a carefull analysis of the quasinormal spectrum of thesesolutions provides a way to fix the parameters of the black hole and consequentlyof string/M theories 17 , , ,
20. Also fermion perturbations of the gravitationalbulk field equations for asymptotically AdS string/M theory black hole solutionsturn out to be fundamental matter fields in the boundary conformal field theory,providing an important class of objects relevant for condensed matter physics in aframe of the AdS/CFT correspondence 22 , , / , ,
19, and in this sense it describes a family of compact objects comingfrom string/M theory.The structure of the paper is the following. After the introduction, in Section 2we presents the general line element considered in the paper and write the funda-mental equations to use for a study of Dirac perturbations in five dimensional space-times, obtaining a general expression for the effective potential that describes thetest field propagation. Section 3 is devoted to the numerical study of the evolutionof the considered perturbation, as well as to present numerical and semi-analyticalsixt order WKB results obtained for the quasinormal frequencies and its dependenceon the charges of the black hole background, including the limit of large angularmultipole numbers. In section 4 we present the numerical results concerning therelaxation of the perturbation at very late times, and propose an analytical formfor the decay factors. Section 5 is devoted to the conclusions.
2. Fundamental equations
In five dimensions, there exist a family of black hole solutions parametrized by threeindependent charges, that can be obtained by the intersection of three 2-branes ata point, or from the intersection of a 2-brane and a 5-brane with a boost, i.e, witha momentum along the common string 19 ,
20. Upon toroidal compactification, themetric reads in both cases as: ds = − A ( r ) dt + B ( r ) dr + C ( r ) d Ω (1)where d Ω denotes the line element of the round 3-sphere and A ( r ) = f ( r ) − / (cid:18) − r H r (cid:19) B ( r ) = f ( r ) / (cid:18) − r H r (cid:19) − (2) C ( r ) = r f ( r ) / where the function f ( r ) is defined as f ( r ) = (1 + r H Q r )(1 + r H Q r )(1 + r H Q r ) . (3)ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges In the above solution if at least one of the charges Q i is zero, the hipersurface r = 0 is space-time singularity covered by the event horizon of the five dimensionalblack hole located at r H . Moreover the case in which all the charges are non-zerocorresponds to a regular black hole with an event horizon at r = r H and an innerhorizon at r = 0.The evolution of a masless spin- fermion field in a five dimensional curvedbackground is described by the Dirac equation: / ∇ Ψ = 0 (4)where / ∇ = Γ µ ∇ µ is the Dirac operator that acts on the five-spinor Ψ, Γ µ are thecurved space Gamma matrices, and the covariant derivative is defined as ∇ µ = ∂ µ − ω abµ γ a γ b , with µ and a being tangent and space-time indices respectively,related by the basis of orthonormal one forms ~e a ≡ e aµ . The associated conectionone-forms ω abµ ≡ ω ab obey d~e a + ω ab ∧ ~e b = 0, and γ a are flat space-time gammamatrices related with curved-space ones by Γ µ = e µa γ a . They form a Clifford algebrain five dimensions, i.e, they satisfy the anti-conmutation relations γ a , γ b = − η ab ,with η = − g µν = Ω ˜ g µν , (5)the five-spinor ψ and the Dirac operator transforms as 25 , ψ = Ω − ˜ ψ, (6)and / ∇ ψ = Ω − ˜ / ∇ ˜ ψ, (7)For a line element in the form ds = ds + ds , where ds = g ab ( x ) dx a dx b and ds = g mn ( y ) dy m dy n , the Dirac operator / ∇ satisfies the direct sum decomposition / ∇ = / ∇ x + / ∇ y . (8)Thus, for two conformally related metrics, the validity of massless Dirac equation inone implies the validity of the same equation in the other. We use this fact to solvethe Dirac equation in the curved space with the line element (1) by performing suc-cesive conformal transformations that isolate the metric components that depend ofthe angular variables, and applied each time the direct sum decomposition of Diracoperator, until to obtain an equivalent problem in a spacetime of the form M × P ,where M is two-dimensional Minkowsky spacetime in ( t, r ∗ ) coordinates (where r ∗ is the tortoise coordinate defined by dr ∗ = q BA dr ) and P is the metric describ-ing the 3-sphere, in which the spectrum of massless Dirac operator is known. Thisprocedure is general and has been applied previously to four dimensional stringlyblack hole by one of the authors 21.ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al
The above method allow us to obtain, for each component of a Dirac spinor inthe manifold M defined as: ˜ ϕ (+) ℓ = (cid:18) iζ ℓ ( t, r ) χ ℓ ( t, r ) (cid:19) , (9)the following equations: i ∂ζ ℓ ∂t + ∂χ ℓ ∂r ∗ + Λ ℓ χ ℓ = 0 (10)and i ∂χ ℓ ∂t − ∂ζ ℓ ∂r ∗ + Λ ℓ ζ ℓ = 0 (11)where Λ ℓ ( r ) = r AC (cid:18) ℓ + 32 (cid:19) (12)This equations can be separated to obtain: ∂ ζ ℓ ∂t − ∂ ζ ℓ ∂r ∗ + V + ( r ) ζ ℓ = 0 (13)and ∂ χ ℓ ∂t − ∂ χ ℓ ∂r ∗ + V − ( r ) χ ℓ = 0 (14)where: V ± = ± d Λ ℓ dr ∗ + Λ ℓ . (15)The above equations gives the temporal evolution of Dirac perturbations outside theblack hole spacetime 21 ,
16. As the potentials V + and V − are supersymmetric to eachother in the sense considered by Chandrasekhar in ? , then ζ ℓ ( t, r ) and χ ℓ ( t, r ) willdevelop similar time evolutions and will have the same spectra, both for scatteringand quasi-normal. At this point it should be stressed that for the spinor ˜ ϕ ( − ) ℓ , wehave these two potentials again. In the following we will work with equation (13) andwe eliminate the subscript (+) for the effective potential, defining V ( r ) ≡ V + ( r ).For the stringy black hole solution considered in this report we have for theeffective potencial V ( r ) the following expression: V ( r ) = λ ℓ f − r (cid:18) − r H r (cid:19) " − (cid:18) − r H r (cid:19) " − rf ′ f (cid:18) r H r (cid:19) − (16)where λ ℓ = ℓ + .Figure (1) shows the effective potential for different multipole numbers ℓ in thecase of stringy black holes with Q = Q = Q = 1. The form of the effectivepotential is similar for other values of compactification charges, and as we can see,this assures the stability of the solution under fermion perturbations, due to itsdefinite positive character.ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges (cid:144) r H V H r L Fig. 1.
Effective potencial for a (4+1)-stringy black hole with ℓ from 0(bottom) to 5(top) and Q = Q = Q = 1 .
3. Time evolution of Dirac perturbations and quasinormal modes
To integrate the equation (13) numerically we use the technique developed by Gund-lach, Price and Pullin 2.The obtained results in the case of massless Dirac fields in (4 + 1)-dimensionalstringy black hole background can be observed as the time-domain profiles showedin Figures (2) to (5). In such profiles r = 3 r H and the time is measured in units ofblack hole event horizon.
10 20 50 100 20010 - - (cid:144) r H Ψ ¤ Fig. 2.
Logaritmic plots of the time-domain evolution of ℓ = 0 and Q = Q = Q = 1 masslessDirac perturbations. As is easily seen, the time evolution of Dirac perturbations outside five dimen-sional stringy black holes follows the usual dynamics for fields in other black holeovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al
10 20 50 100 20010 - - - - (cid:144) r H Ψ ¤ Fig. 3.
Logaritmic plots of the time-domain evolution of ℓ = 1 and Q = Q = Q = 1 masslessDirac perturbations.
10 20 50 100 20010 - - - - - (cid:144) r H Ψ ¤ Fig. 4.
Logaritmic plots of the time-domain evolution of ℓ = 2 and Q = 0 , Q = Q = 1 massless Dirac perturbations. backgrounds. After a first transient stage strongly dependent on the initial condi-tions and the point where the wave profile is computed, we observe the characteristicexponential damping of the perturbations associate with the quasinormal ringing,followed by a so-called power law tails at asymptotically late times.To compute the quasinormal frequencies that dominated at intermediate times,we assume for the function ζ ℓ ( t, r ) in equation (13) the time dependence: ζ ℓ ( t, r ) = Z ℓ ( r ) exp( − iω ℓ t ) (17)ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges
10 20 50 100 20010 - - - - - (cid:144) r H Ψ ¤ Fig. 5.
Logaritmic plots of the time-domain evolution of ℓ = 3 and Q = 0 , Q = Q = 1 massless Dirac perturbations. Then, the function Z ℓ ( r ) satisfy the Schrodinger-type equation: d Z ℓ dr ∗ + (cid:2) ω − V ( r ) (cid:3) Z ℓ ( r ) = 0 (18)The quasinormal modes are solutions of the wave equation (13) with the specificboundary conditions requiring pure out-going waves at spatial infinity and pure in-coming waves on the event horizon. Thus no waves come from infinity or the eventhorizon.The quasinormal frequencies were computed using two different methods. Thefirst method uses directly the numerical data obtained previously, and fit this databy superposition of damping exponents. This numerical fitting scheme, known asProny method, allow us to obtain very accurate results for the fundamental andfirst overtones 3 ,
4. For higher overtones is very difficult to be implemented, becausewe need to do a fitting with a great number of exponentials, and also we need toknow very well the particular time in which quasinormal ringing begins, a difficultpoint to be solved in general. For this reason in the following we only presents thequasinormal frequencies, using this method, for the first two overtone number.The second method that we employed is a semi-analytical approach to solveequation (18) with the required boundary conditions, based in a WKB-type ap-proximation, that can give accurate values of the lowest ( that is longer lived )quasinormal frequencies, and was used in several papers for the determination ofquasinormal frequencies in a variety of systems 5 , , , , , , , , , , , ℓ and different set of charges. We also present the resultfor a particular stringy black holes with Q = Q = Q = 1 in Figure (6). As itis observed, the sixth order WKB approach gives results in agreement with thoseobtained by fitting the numerical integration data using Prony technique. As it isovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al
Table 1.
Dirac quasinormal frequencies ωr H in(4+1)-stringy black hole with Q1=0.5, Q2=1,Q3=1.8 for ℓ = 0 to ℓ = 4 . The frequencies aremeasured in units of the black hole horizon ra-dious r H . ℓ n Sixth order WKB Prony0 0 0 . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . . − . i . − . i . − . i -4 0 1 . − . i . − . i . − . i . − . i . − . i -4 3 1 . − . i -Table 2. Dirac quasinormal frequencies ωr H in (4+1)-stringy black hole with Q1=0, Q2=1,Q3=1 for ℓ = 0 to ℓ = 4 . The frequencies aremeasured in units of the black hole horizon ra-dious r H . ℓ n Sixth order WKB Prony0 0 0 . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i -4 0 1 . − . i . − . i . − . i . − . i . − . i − . − . i − expected the oscillation frequency increases for higher multipole and fixed overtonenumbers. Increasing ℓ the magnitude of the negative imaginary part of the funda-mental overtone ( n = 0) increases while for higher overtones the opposite situationarises. For a fixed angular number ℓ , the real part of the oscillation frequencies de-creases as the overtone number increases, and the magnitude of the imaginary partincreases. Then, modes with higher overtone numbers decay faster than low-lyingones. As we expected for stability, all quasinormal frequencies calculated in thiswork have a well defined negative imaginary part.Figures (7) and (8) show the dependence of the quasinormal modes with com-pactification charges. As one of the charges increases, leaving fixed the other two,the real part and the absolute value of the imaginary part of the quasinormal fre-quencies decreases, but the rates of decreasing are different. Then, as the chargesovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges ææ àà ìì òòò ôôôôççááíí óóóõõõ õ { =
34 2 1 0 3 4210 n = - n = - n = - n = - n = n = n = n = - - - - - Ω I m Ω Fig. 6.
Massles Dirac quasinormal frequencies of (4+1)-stringy black holes with Q = Q = Q = 1 . æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò òô ô ô ô ô ô ô ô ô ô ô { = { = { = { = { = n = R e Ω Fig. 7. Dependence upon charge Q i , i = 1 , , Q j = 1, j = i . The resultscorrespond to the first overtone for multipolar number from ℓ = 0 to ℓ = 4. increases, the modes labeled by the same angular multipole numbers quickly be-comes less damped and are long lived.An interesting case arises for large angular multipole numbers because in thislimit the first order WKB approximation becomes exact and we can obtain ananalytical result. In this limit the effective potential (16) can be written as U ( r ) = ℓ ∆( r ) (19)where ∆( r ) = Ξ( r ) r f ( r ) and Ξ( r ) = 1 − r H r . Then the first order WKB approximationgives for the quasinormal frequencies in this limit the result: ω = ℓ ∆( r m ) − iℓ ( n + 12 ) s − d ∆( r ) dr ∗ | r = r m , (20)ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò òô ô ô ô ô ô ô ô ô ô ô { = { = { =
2, 3, 4 - - - - - I m Ω Fig. 8. Dependence upon charge Q i , i = 1 , , Q j = 1, j = i . Theresults correspond to the first overtone for multipolar number from ℓ = 0 to ℓ = 4. being r m the point in which the asymptotic effective potential (19) reach its peak.This value can be determined as the maximum root of the equation − Ξ( r ) (cid:20) f ( r ) + r df ( r ) dr (cid:21) + rf ( r ) d Ξ( r ) dr = 0 . (21)In the particular case of equal charges Q = Q = Q = Q we obtain the result: r m = r H q Q + p Q + Q (22)
4. Late Time Tails
Another important point to study is the relaxation of the perturbing fermion fieldoutside the black hole at late times 27 ,
28. It is a known result that in higher di-mensional Schwarzschild black hole neutral massless boson fields had a late-timebehavior dominated ( for a fixed r and each multipole moment ℓ ) by a factor t − (2 ℓ + D − for odd D -dimensions and t − (2 ℓ +3 D − for even D -dimensions 29.To study the late-time behavior, we numerically fit the profile data obtained inthe appropriate region of the time domain, to extract the power law exponents thatdescribe the relaxation. As a test of our numerical fitting scheme, we obtained thepower law exponents for the massless Dirac field considered in this paper in thespace-time corresponding to higher dimensional Schwarzschild black hole. As weexpected, the results obtained are consistent with the power law falloff mentionedin the previous paragraph.Figures (9) and (10) show two representative examples of the results obtained,for a five dimensional stringy black hole with one charge zero and the other two Q = Q = 1. In general, all our numerical results indicate that, in five dimensions,the decay in the Dirac case is governed by factors of the form t − (2 ℓ +3) . This resultis in complete correspondence with the general result obtained for bosons fields inhigher dimensional space times: if D is the dimension of the space time then theovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges d = { = ~ t - ~ t -
100 200 3001501 ´ - ´ - ´ - ´ - ´ - ´ - t (cid:144) r H Ψ ¤ Fig. 9.
Tail for ℓ = 0 and Q = 0 , Q = Q = 1 . The power-law coefficients were estimated fromnumerical data represented in the dotted line. The full red line is the possible analytical result. d = { = ~ t - ~ t - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - t (cid:144) r H Ψ ¤ Fig. 10.
Tail for ℓ = 4 and Q = 0 , Q = Q = 1 . The power-law coefficients were estimatedfrom numerical data represented in the dotted line. The full red line is the possible analyticalresult. late time tail is described by the power-low falloff Ψ ∼ t − (2 ℓ + D − for odd D
29. Itseems that the fermion or boson character of the test field have no influence in thelate falloff of the perturbations also in higher dimensions. Then, we can concludethat, outside five dimensional stringy black holes, as well as Schwarzschild blackhole, the massless Dirac field shows identical decay at late times.However, we remark at this point that this dependance is only a result consistentwith our numerical data. A simple analytical argument to support this late timebehaviour do not exist, in contrast to the case for boson fields, in which the generalform of the effective potential is suitable to expand for large values of the tortoisecoordinate 29 and then extract the above power law behavior directly from thisasymptotic expansion. The problem related with the analytical determination ofthe decay factors for fermion perturbations in higher dimensional stringy blackovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2 Owen Pavel Fern´andez Piedra et. al holes remains open.
5. Concluding remarks
In this report we considered the evolution of massless Dirac test field in the spacetime corresponding to five dimensional black hole solutions coming from intersectingbranes in string theories.After the initial transient epoch, the evolutions is dominated by quasinormalmodes, and at late times by a power-low falloff. We computed the quasinormalfrequencies for different values of compactification charges using two different ap-proaches, 6th-order WKB formula and time domain integration with Prony fittingof the numerical data, obtaining by both methods very close numerical results. Theresults for the dependence of the quasinormal frequencies with charges appears tobe universal for all values of this parameter.We also computed the decay factors for the late time relaxation of fermionperturbations in five dimensional stringy black holes and show that the result issimilar for those obtained for boson field in higher odd dimensional space-times.It should be interesting to study the case of more higher dimensions, whenremains open if the fermion decay picture at late times obey the same power lawbehaviour of boson fields. Another important question is the analytical study of thislate time decay factors, taking into account that for potentials typical of fermionfields do not exist a simple analytical argument to approach this problem, as in thecase of boson fields.Stringy black holes obtained by intersection of branes are known for dimensionsup to D=9, and then it would be interesting to study the evolution of test fields inthis higher dimensional backgrounds. In future reports we will complete this studiesto gain a more complete knowledge about the evolution of fermion as well as bosonperturbations in this interesting physical systems.
Acknowledgments
We are grateful to Professor Elcio Abdalla and Dr. Jeferson de Oliveira at IFUSP,Brazil, for useful discussions about quasinormal modes and Universidad de Cien-fuegos for technical support.
References
1. T. Regge and J. A. Wheeler, Phys. Rev. , 1063 (1957).2. C. Gundlach, R. H. Price, J. Pullin,
Phys. Rev. , 883 (1994);3. E. Berti, V. Cardoso, J. A. Gonzales and U. Sperhake , Phys. Rev.
D75 , 124017(2007).4. A. Zhidenko,
PhD Thesis , arXiv:0903.3555.5. B. Shutz and C. M. Will,
Astrophys. J. Lett.
L33 , 291 (1985).6. S. Iyer and C. M. Will,
Phys. Rev.
D35 , 3621 (1987).7. R. A. Konoplya,
Phys. Rev.
D68 , 024018 (2003). ovember 16, 2018 22:31 WSPC/INSTRUCTION FILE stringy5dijmpdv2
Object picture, quasinormal modes and late time tails of fermion perturbations in stringy black hole with U(1) charges
8. R. A. Konoplya,
J. Phys. Stud. , 93 (2004).9. R. A. Konoplya and E. Abdalla, Phys. Rev.
D71 , 084015 (2005).10. M.I. Liu, H. I. Liu and Y. X. Gui,
Class. Quantum Grav. , 105001 (2008).11. J. F. Chang, J. Huang and Y. G. Shen Int. J. Theor. Phys. , 2617 (2007).12. R. A. Konoplya Phys. Lett.
B550 , 117 (2002).13. H. Kodama, R. A. Konoplya and A. Zhidenko, arXiv:0904.2154.14. R. A. Konoplya and A. Zhidenko,
Phys. Lett.
B644 , 186 (2007).15. E. Abdalla, O. P. F. Piedra and J. de Oliveira, C. Molina Phys. Rev. D , 064001(2010)16. H. T. Cho, A. S. Cornell, J. Doukas and W. Naylor Phys. Rev. D , 584 (1996)18. M. Cvetic and D. Youm, Nucl. Phys. B , 249 (1996)19. M. Cvetic and A. A. Tseytlin, Nucl. Phys. B , 181 (1996)20. G. T. Horowitz, J. M. Maldacena and A. Strominger, Phys. Lett. B , 151 (1996)21. Owen P.F Piedra and J. de Oliveira, Class. Quantum Grav.
L28 , 085023 (2011)22. D. T. Son and A. O. Starinets, Ann. Rev. Nucl. Part. Sci. , 95 (2007)23. S. S. Gubser and A. Karch, Ann. Rev. Nucl. Part. Sci. , 145 (2009)24. S. A. Hartnoll, Class. Quant. Grav. , 224002 (2009)25. G. W. Gibbons and A. R. Steif, Phys. Lett. B
13 (1993)26. G. Gibbons and M. Rogatko, Phys. Rev. D , 044034 (2008). bibitemchandra S.Chandrasekar, The Mathematical theory of Black Holes , ( Clarendon, Oxford, 1983).27. R. H. Price,
Phys. Rev. D5 , 2419 (1972).28. R. H. Price and L. M. Burko, Phys. Rev.
D70 , 084039 (2004).29. V. Cardoso, O.J.C. Dias and J. P. S. Lemus,
Phys. Rev.