Stability Analysis and Area Spectrum of 3-Dimensional Lifshitz Black Holes
SStability Analysis and Area Spectrum of 3-Dimensional Lifshitz Black Holes
Bertha Cuadros-Melgar
Department of Physics, National Technical University of Athens,Zografou Campus GR 157 73, Athens, Greece andPhysics Department, University of Buenos Aires FCEN-UBA and IFIBA-CONICET,Ciudad Universitaria, Pabell´on 1, 1428, Buenos Aires, Argentina
Jeferson de Oliveira and C. E. Pellicer
Instituto de F´ısica, Universidade de S˜ao Paulo, C.P.66318, CEP 05315-970, S˜ao Paulo, Brazil
In this work, we probe the stability of a z = 3 three-dimensional Lifshitz black hole by using scalarand spinorial perturbations. We found an analytical expression for the quasinormal frequencies ofthe scalar probe field, which perfectly agree with the behavior of the quasinormal modes obtainednumerically. The results for the numerical analysis of the spinorial perturbations reinforce theconclusion of the scalar analysis, i.e. , the model is stable under scalar and spinor perturbations. Asan application we found the area spectrum of the Lifshitz black hole, which turns out to be equallyspaced. PACS numbers: 04.50.Kd, 04.70.-s, 04.25.Nx
I. INTRODUCTION
Some decades ago Regge and Wheeler began a pio-neering study of a small perturbation in the backgroundof a black hole in order to get information of the stabil-ity of this object [1], a problem that was continued byZerilli [2]. The oscillations found in these studies are notnormal modes due to the emission of gravitational waves;thus, their frequencies are complex and, as a result, theoscillations are damped.The terminology quasinormal mode (QNM) and quasi-normal frequency (QNF), aiming to name these newmodes and their frequencies, was first pointed out byVishveshwara [3] and Press [4]. Although initially stud-ied in black hole backgrounds, the concept of QNM ap-plies to a much broader class of systems. The QNMsof black holes were first numerically calculated by Chan-drasekhar and Detweiler [5] showing that the amplitudeis dominated by a ringing characteristic signal at interme-diate times. The QNMs are independent of the particularinitial perturbation that excited them. The frequenciesand damping times of the oscillations depend only on theparameters of the black hole and are, therefore, the “foot-prints” of this structure. Soon, the connection of QNMsto astrophysics was established by noting that their ex-istence can lead to the detection of black holes throughthe observation of the gravitational wave spectrum. Theinterest in QNMs has motivated the development of nu-merical and analytical techniques for their computation(see [6–8] for a review). Also, the study of the quasinor-mal spectrum gives information about the stability as-pects of black hole solutions using probe classical matterfields (scalar, electromagnetic, spinorial) evolving in thegeometry without backreacting on the spacetime back-ground. Much has been done in that direction, not onlyin four dimensions [9][10], but also in two [11], and inmore than four [12].Aside from the study of the stability of the solutions, the QNFs are important in the context of the gauge-gravity correspondence, whose most celebrated exampleis the duality between the type IIA-B string theory in
AdS × S spacetime and the four-dimensional supersym-metric Yang-Mills theory [13]. Such a correspondence canbe generalized for those cases in which there is an eventhorizon in the gravity side. In this case the Hawking tem-perature of the black hole is related to the temperature ofa thermal field theory defined at the anti-de Sitter (AdS)boundary. Also, as a consequence of the correspondence,the quasinormal spectrum corresponds to the poles ofthermal Green functions [14], more precisely, the inverseof the imaginary part of the fundamental quasinormalfrequency can be interpreted as the dual field theory re-laxation time [15].Another interesting application of QNMs appears inthe context of black hole thermodynamics. Some decadesago Bekenstein [16] suggested that the horizon area of ablack hole must be quantized, so that the area spectrumhas the form A n = γn ¯ h , with γ a dimensionless con-stant to be determined. The first proposal to calculatethis constant through QNMs was made by Hod [17]. Ac-cordingly, the real part of the asymptotic quasinormalmode can be seen as a transition frequency in the semi-classical limit, and its quantum emission causes a changein the mass of the black hole, which is related to thearea. In this way, the constant γ for a Schwarzschildblack hole was determined as γ = 4 ln 3. Later, Kun-statter [18] repeated the calculation quantizing the adia-batic invariant I = (cid:82) dE/ω ( E ) via the Bohr-Sommerfeldquantization and using the real part of the QNF as thevibrational frequency. The result was exactly the same asHod’s. However, recently Maggiore [19] pointed out thatQNMs should be described as damped harmonic oscilla-tors, thus, the imaginary part of the QNF should not beneglected, and the proper physical frequency is the mod-ule of the entire QNF. Moreover, when considering thequantization of the adiabatic invariant, the frequency to a r X i v : . [ h e p - t h ] J a n be considered is that corresponding to a transition be-tween two neighboring quantum levels. With this iden-tification, the constant γ for a Schwarzschild black holebecomes γ = 8 π , a result that coincides with the valuecalculated by other methods [20]. The consequences ofHod’s and Maggiore’s proposals were promptly studiedin several spacetimes [21, 22].In this paper, we are interested in the study of thestability of the z = 3, three-dimensional Lifshitz blackhole found in the context of the so-called new massivegravity (NMG) [23]. Moreover, as an application of ourQNM results we aim to calculate the area spectrum ofthis black hole.NMG is a novel parity-preserving, unitary [24], power-counting super-renormalizable [25], three-dimensionalmodel describing the propagation of a massive positive-energy graviton with two polarization states of helicity ± i.e. , they exhibit the anisotropicscale invariance, t → λ z t , (cid:126)x → λ(cid:126)x , where z is the dy-namical critical exponent. Specifically, we deal with thesolutions found for the particular case of z = 3 and a pre-cise value of the mass parameter [23]. The general classof these solutions are important in the context of gauge-gravity duality [30, 31] and were also investigated in otherbackground theories [32–35]. No stability study of blackholes with Lifshitz asymptotics in three dimensions in thescenario of NMG has been performed yet. We aim to givesome contribution to this issue by considering the QNFof scalar and spinorial matter fields in the probe limit, i.e , there are no backreaction effects upon the asymp-totically Lifshitz black hole metric. Spinor fields havebeen extensively studied in general relativity [36][37], andtheir quasinormal frequencies have also been considered[39][38][40].The paper is organized as follows. In section 2 we in-troduce the Lifshitz black holes and discuss their causalstructure. Sections 3 and 4 are dedicated to the studyof stability under scalar and spinorial perturbations withspecial emphasis on the massless spinor for the latter. Insection 5 we present the numerical analysis for both kindsof perturbations showing the QNMs and the correspond- ing QNFs computed in each case. Section 6 is devoted tothe calculation of the area spectrum of these black holesas an application of our quasinormal spectrum. Finally,we discuss our results and conclude in section 7. II. LIFSHITZ BLACK HOLES IN THREEDIMENSIONS
In this section we review the black hole solutions wewill consider within this paper, and we comment some oftheir features.The NMG theory [26] is defined by the (2 + 1)-dimensional action, S = 116 πG (cid:90) d x √− g (cid:20) R − λ − m (cid:18) R µν R µν − R (cid:19)(cid:21) , (1)where m is the so-called “relative” mass parameter, and λ is the three-dimensional cosmological constant. Definingthe dimensionless parameters, y = m l and w = λl , itis found that the theory exhibits special properties at thepoints y = ± /
2. When looking for black hole solutionswith Lifshitz asymptotics, it is precisely at the point y = − / w = − /
2, with Lifshitz scaling z = 3, where thefield equations turn out to be solved by [23] ds = − a ( r ) ∆ r dt + r ∆ dr + r dφ , (2)where a ( r ) = r l , (3)and ∆ = − M r + r l , (4)with M an integration constant and l = − λ . Also,the NMG admits as a solution, the well-known Ba˜nados-Teitelboim-Zanelli (BTZ) black hole with the dynamicalcritical exponent z = 1. As we shall see below in more de-tail, this metric (2) exhibits a regular single event horizonlocated at r = r + = l √ M and a spacetime singularity at r = 0. Besides, the surface r = r + acts as a one-waymembrane for physical objects as we can see from thenorm of a vector χ normal to a given surface s . Since s has to be null in order to be a one-way membrane, thenorm of χ must be null as well, i.e. , g rr = 0, which occursat r = r + .From the behavior of the Kretschmann invariant forthe metric (2), R µνλσ R µνλσ = − l r (cid:2) r − r r + 91 r (cid:3) , (5)we see that, for r → r + R µνλσ R µνλσ → − l , (6)and for r → R µνλσ R µνλσ → ∞ . (7)Thus, the black hole solution has a genuine spacetimesingularity at the origin r = 0 and an event horizon at r = r + . Nevertheless, to see if the singularity is timelike,spacelike, or null we have to construct the Penrose-Carterdiagram. First of all, we must remove the coordinatesingularity at r = r + . Rewriting the metric in terms ofnull coordinates ( U, V ) U = e r ( t + r ∗ ) , V = − e − r ( t − r ∗ ) . (8)where r ∗ is the tortoise coordinate shown in the nextsection, we get ds = − (cid:18) rr + (cid:19) (cid:16) r + r (cid:17) e − r + r dU dV , (9)which is manifestly regular at r = r + .Finally, to construct the Penrose-Carter diagram(Fig.1) we use the following set of null coordinates T = ˜ U + ˜ V , X = ˜ U − ˜ V , (10)with ˜ U = arctan( U ) and ˜ V = arctan( V ). FIG. 1. Penrose-Carter diagram for the Lifshitz black hole.The singularity at r = 0 is light-like and covered by a regularevent horizon r + . From this diagram we see that the spacetime singu-larity is located at r = 0, as previously observed from the behavior of the Kretschmann invariant. Moreover,it is light-like and covered by a regular event horizon at r = r + . III. SCALAR PERTURBATION
In this section, we analyze the behavior of a scalar fieldperturbation in the background of a three-dimensionalLifshitz black hole.The scalar field obeys the Klein-Gordon equation, (cid:50)
Φ = 1 √− g ∂ M (cid:0) √− gg MN ∂ N (cid:1) Φ = m Φ , (11)where m is the mass of the field Φ. Performing the de-composition Φ( t, r, φ ) = Ψ( t, r ) e iκφ , (12)The Klein-Gordon equation takes the form, − ∂ t Ψ + r l (cid:18) − M l r (cid:19) (cid:18) r l − M r (cid:19) ∂ r Ψ++ r l (cid:18) − M l r (cid:19) ∂ r Ψ −− r l (cid:0) m r + κ (cid:1) (cid:18) − M l r (cid:19) Ψ = 0 . (13)Even though this equation has an analytical solution,as we will see in what follows, it is also useful to check thenumerical results. With this goal we further decomposeΨ = X ( t, r ∗ ) / √ r , where the tortoise coordinate r ∗ isgiven by r ∗ = l (cid:20) − M / l arccoth (cid:18) rl √ M (cid:19) + 1 M l r (cid:21) . (14)In this way the Klein-Gordon equation reduces to − ∂ t X + ∂ r ∗ X = V ( r ) X , (15)where V ( r ) is the scalar effective potential given by V ( r ) = (cid:18) l + m l (cid:19) r − (cid:18) M l + M m l − κ l (cid:19) r ++ (cid:18) M l − M κ l (cid:19) r . (16)Now let us come back to the issue of finding an exactsolution for Eq.(13). We set the time dependence of thefield Ψ( t, r ∗ ) as R ( r ) e − iωt and redefine the radial coordi-nate as r = l √ M /y . Thus, Eq.(13) turns out to be ∂ y R + y − y (1 − y ) ∂ y R − − l (1 − y ) (cid:20) − ω y M (1 − y ) + m y + κ M l (cid:21) R = 0 , (17)whose solution is given in terms of Heun confluent functions, R ( y ) = C y α (1 − y ) β/ HeunC (cid:18) , α, β, − β , α κ M , y (cid:19) ++ C y − α (1 − y ) β/ HeunC (cid:18) , − α, β, − β , α κ M , y (cid:19) , (18)where C and C are integration constants, while α = √ m l , and β = − i lω/M / .Imposing the Dirichlet condition at infinity we set C = 0. In order to apply the boundary condition ofin-going waves at the horizon we use the following con-nection formula [41],HeunC(0 , b, c, d, e, z ) = c Γ(1 − b )Γ( c )Γ(1 + c + k )Γ( − b − k ) HeunC(0 , c, b, − d, e + d, − z ) ++ c Γ(1 − b )Γ( − c )Γ(1 − c + s )Γ( − b − s ) (1 − z ) − c HeunC(0 , − c, b, − d, e + d, − z ) . (19)This formula connects a solution around the singular reg-ular point z = 0 to the corresponding solution about thesingular regular point z = 1 of the confluent Heun equa-tion given by z ( z − H (cid:48)(cid:48) + [( b + 1)( z −
1) + ( c + 1) z ] H (cid:48) + ( dz − (cid:15) ) H = 0 . (20)The parameters k and s are obtained from k + ( b + c + 1) k − (cid:15) + d/ , (21) s + ( b − c + 1) s − (cid:15) + d/ , (22) and (cid:15) is related to e as (cid:15) = − bc − c − b − e . (23)Thus, near y = 1 Eq.(18) can be written as R ( y → ≈ ξ (1 − y ) β/ Γ(1 + α )Γ( β )Γ( α − k )Γ(1 + β + k ) ++ ξ (1 − y ) − β/ Γ(1 + α )Γ( − β )Γ( α − s )Γ(1 − β + s ) , (24)with ξ i as constants. As we are looking for quasinor-mal frequencies with negative imaginary parts, whichgive stable solutions, we find that for β < β + k ) → ∞ . Thus, the quasinormal frequenciesare ω = 2 i M / l (cid:34) N + (cid:112) m l − (cid:114) m l + κ M + (3 + 6 N ) (cid:112) m l + 6 N ( N + 1) (cid:35) , (25)where N is a positive integer. The imaginary part of thefundamental frequency ( N = 0) is negative provided that (cid:114) m l + κ M + 3 (cid:112) m l > (cid:112) m l . (26) While the asymptotic frequency ( N → ∞ ) is given by ω ∞ = − √ − i M / l N < . (27)Thus, since the imaginary part of the quasinormal fre-quencies is negative provided that the parameters respectthe relation (26), we can conclude that the model is sta-ble under scalar perturbations. IV. SPINORIAL PERTURBATION
In this section, we are going to consider a spinorial fieldas a perturbation in the spacetime given by the three-dimensional Lifshitz black hole. We analyze the covariantDirac equation for a two component spinor field Ψ withmass µ s . This equation is given by iγ ( a ) e µ ( a ) ∇ µ Ψ − µ s Ψ = 0 , (28)where Greek indices refer to spacetime coordinates( t, r, φ ), and the Latin indices enclosed in parenthesesdescribe the flat tangent space in which the triad basis e µ ( a ) is defined. The spinor covariant derivative ∇ µ isgiven by ∇ µ = ∂ µ + 18 ω ( a )( b ) µ (cid:2) γ ( a ) , γ ( b ) (cid:3) , (29) where ω ( a )( b ) µ is the spin connection, which can be writ-ten in terms of the triad e µ ( a ) as ω ( a )( b ) µ = e ( a ) ν ∂ µ e ( b ) ν + e ( a ) ν Γ νµσ e σ ( b ) , (30)where Γ νµσ are the metric connections. The γ ( a ) denotesthe usual flat gamma matrices, which can be taken interms of the Pauli ones. In this work we take γ (0) = iσ , γ (1) = σ , and γ (2) = σ .We can write the triad basis e µ ( a ) for the metric (2) asfollows: e ( a )0 = √ a ( r )∆ r δ ( a )0 , e ( a )1 = r √ ∆ δ ( a )1 ,e ( a )2 = rδ ( a )2 , (31)and the metric connections,Γ = ddr (cid:34) ln (cid:18) a ( r )∆ r (cid:19) / (cid:35) , Γ = ddr (cid:34) ln (cid:18) r ∆ (cid:19) / (cid:35) , Γ = ∆2 r ddr (cid:20) a ( r )∆ r (cid:21) , Γ = − ∆ r , Γ = 1 r . (32)Using these quantities it is straightforward to writedown the expressions for spin connection components.In the present case, we have only two nonvanishing com-ponents, ω (0)(1)0 = 12 (cid:112) a ( r ) ddr (cid:18) a ( r )∆ r (cid:19) , ω (1)(2)2 = − √ ∆ r . (33) At this point we are able to write the Dirac equationfor the two component spinorΨ = (cid:18) Ψ ( t, r, φ )Ψ ( t, r, φ ) (cid:19) , (34)which turns to be the set of coupled differential equations ir (cid:112) a ( r )∆ ∂ t Ψ + i √ ∆ r ∂ r Ψ + ir ∂ φ Ψ + i (cid:20) a ( r ) (cid:48) ∆ a ( r ) r + ∆ (cid:48) r √ ∆ (cid:21) Ψ − µ s Ψ = 0 , (35) − ir (cid:112) a ( r )∆ ∂ t Ψ + i √ ∆ r ∂ r Ψ − ir ∂ φ Ψ + i (cid:20) a ( r ) (cid:48) ∆ a ( r ) r + ∆ (cid:48) r √ ∆ (cid:21) Ψ − µ s Ψ = 0 . (36)In order to simplify our problem, we redefine Ψ andΨ as Ψ = i [ a ( r )∆] / e − iωt + imφ Φ + ( r ) , Ψ = [ a ( r )∆] / e − iωt + imφ Φ − ( r ) , (37)and the tortoise coordinate as in the scalar case (14), ddr ∗ = ∆ (cid:112) a ( r ) r ddr . (38) Thus, we can rewrite Eqs.(35) and (36) as ∂ r ∗ Φ − − iω Φ − = i (cid:112) a ( r )∆ r ( ˆ m − iµ s r ) Φ + , (39) ∂ r ∗ Φ + + iω Φ + = i (cid:112) a ( r )∆ r ( ˆ m + iµ s r ) Φ − , (40)where m = i ˆ m .Furthermore, we define a new function θ , a new rescal-ing for the spinorial components R ± , and a new tortoise coordinate ˆ r ∗ through the expressions, θ = arctan( µ s r ˆ m ) , -11 -10 -9 -8 -7 -6 -5 -4 -3
0 5 10 15 20 25 ψ ( t ) t M=1.0M=1.2M=1.4M=1.6M=1.8M=2.0 FIG. 2. Decay of scalar field with mass m = 1 and l = 1 fordifferent values of black hole mass M . Φ ± = e ± iθ/ R ± ( r ) , ˆ r ∗ = r ∗ + 12 ω arctan( µ s r ˆ m ) . In this way Eqs.(39) and (40) become( ∂ ˆ r ∗ ± iω ) R ± = W R ∓ , (41)where W is the so-called superpotential, W = i (cid:112) a ( r )∆ ( ˆ m + µ s r ) / r ( ˆ m + µ s r ) + µ s ˆ m √ a ( r )∆2 ω . (42)Notice that when a ( r ) = 1, Eq.(42) reduces to the BTZsuperpotential [42].Finally, letting X ± = R + ± R − we have (cid:0) ∂ r ∗ + ω (cid:1) X ± = V ± X ± , (43)where V ± are the superpartner potentials, V ± = W ± dWd ˆ r ∗ , (44)which in the case of a massless spinor ( µ s = 0) reducesto V ± = (cid:18) − m Ml ∓ mMl (cid:112) r − M l (cid:19) r + (cid:18) m l ± ml (cid:112) r − M l (cid:19) r . (45) V. NUMERICAL ANALYSIS
In this section, we numerically solve Eqs.(15) and (43),which correspond to the scalar and massless spinorialperturbations, respectively. Although in the scalar casewe found an analytical solution and the corresponding QNF, our motivation to perform the numerical analysisis to verify the applicability of certain numerical meth-ods in asymptotically Lifshitz spacetimes. In particular,it would be interesting to check if the Horowitz-Hubenymethod [15] works well when finding the QNF.Using the finite difference method, we define ψ ( r ∗ , t ) = ψ ( − j ∆ r ∗ , l ∆ t ) = ψ j,l , V ( r ( r ∗ )) = V ( − j ∆ r ∗ ) = V j , andrewrite Eqs.(15) and (43) as − ψ j,l +1 − ψ j,l + ψ j,l − ∆ t + ψ j +1 ,l − ψ j,l + ψ j − ,l ∆ r ∗ − V j ψ j,l + O (∆ t ) + O (∆ r ∗ ) = 0 , (46)which can be rearranged as ψ j,l +1 = − ψ j,l − + ∆ t ∆ r ∗ ( ψ j +1 ,l + ψ j − ,l ) + (cid:18) − t ∆ r ∗ − ∆ t V j (cid:19) ψ j,l . (47)The initial conditions ψ ( r ∗ ,
0) = f ( r ∗ ) and ˙ ψ ( r ∗ ,
0) = f ( r ∗ ) define the values of ψ j,l for l = 0 and l = 1 andwe use Eq. (47) to obtain the values of ψ j,l for l > j = 0 we impose Dirichlet boundary conditions since V ( r ∗ ) tends to infinity as r ∗ tends to zero. The numerical solution is stable if∆ t ∆ r ∗ + ∆ t V MAX < , (48)where V MAX = V is the largest value of V j in our domain.This condition is verified in all cases studied here.Now we are going to analyze the potential for the scalarcase. By rewriting Eq.(16) in terms of a new variable z = r , we obtain V ( r ) = zl (cid:20)(cid:18)
74 + m l (cid:19) z − (cid:18)
52 + m l − κ l z h (cid:19) z h z + (cid:18) − κ l z h (cid:19) z h (cid:21) , (49)where z h = r h . The parable in brackets tends to infinityas long as (cid:0) + m l (cid:1) >
0, which is consistent with theBreitenlohner-Freedman-type bound for the present case.The roots of this polynomial potential are given by z = 0 ,z + = z h , (50) z − = z h (cid:34) − κ l z h + m l (cid:35) . (51) -10 -9 -8 -7 -6 -5 -4 -3
0 5 10 15 20 25 ψ ( t ) t K=0K=2K=4K=6K=8K=10 FIG. 3. Decay of scalar field with mass m = 1 and l = 1varying the azimuthal parameter κ . If m l > −
1, we see that z − < z + . Thus, going backto the original variable r , the roots of the potential are r = 0 with double multiplicity, r = √ z − and r = r h ( r = −√ z − and r = − r h are excluded as r > r h is the biggest root and V ( r ) tends to ∞ when r tends to ∞ , the potential is positive-definite in theregion ( r h , ∞ ). Therefore, the quasinormal modes for m l > − z = 3 Lifshitz black hole is stable under -3-2.5-2-1.5-1-0.5 0 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ω I M AnalyticNumericHorowitz-Hubeny
FIG. 4. Imaginary part of scalar quasinormal frequencies. Wedisplay the results obtained using different approaches. scalar perturbations. Moreover, according to Fig.4, thenumerical results have a very good agreement with theanalytical calculation.Figure 4 shows that the Horowitz-Hubeny methodgives unreliable results. In [48], it is argued that thefrequencies do not converge as required by the method,and that may be explained by ill-conditioned polynomi-als. However, in this work, the frequencies converge, butthey do not agree with the analytic expression and withthe results from finite difference method. In [49], theauthors find cases where this method does not work ei-ther, and they do so by comparing the results with othermethods. For instance, they find that the method is un-reliable for dimensions bigger than 6. Even in the originalwork [15] the method is unreliable for small black holes,and there is no clear explanation for this limitation. Inour case the asymptotic behavior of the black hole understudy might play an important role in the convergenceof the method. Nevertheless, a general criteria for theconvergence of the Horowitz-Hubeny method remains anopen question.In the case of the massless spinorial perturbation thesuperpartner potentials (45) can be written as V ± = 1 l (cid:20) ( ml ) r ( r − r ) ± ( ml ) r (2 r − r ) (cid:113) r − r (cid:21) , (52) and their derivative turns to be V (cid:48)± = 1 l ( ml ) r (2 r − r ) ± ( ml ) r ( r − r ) (cid:113) r − r + r r − r (cid:113) r − r . (53)We can see that outside the event horizon V + ispositive-definite if ml >
0, and lim r →∞ V + ( r ) = −∞ if ml <
0. Whereas V − is positive-definite if ml <
0, andlim r →∞ V − ( r ) = −∞ if ml >
0. Moreover, we noticethat if ml = 0, we have a free-particle case. The decay-ing behavior of the massless spinor is given in Figs.5-6.Thus, we conclude that the z = 3 Lifshitz black hole isstable under massless spinorial perturbations. -11 -10 -9 -8 -7 -6 -5 -4 -3
0 5 10 15 20 25 30 35 40 ψ ( t ) t m=1m=3m=7m=10 FIG. 5. Decay of massless spinor with l = 1 and black holemass M = 1 . m . VI. AREA SPECTRUM
One of the applications of our results for the quasinor-mal frequencies is the relation they have with the areaspectrum of a black hole. According to Maggiore [19],the proper physical frequency of the damped harmonicoscillator equivalent to the black hole quasinormal mode -11 -10 -9 -8 -7 -6 -5 -4 -3
0 5 10 15 20 25 ψ ( t ) t m=1m=3m=7m=10 FIG. 6. Decay of massless spinor with l = 1 and black holemass M = 1 . m . is given by ω p = (cid:113) ω R + ω I , (54)where ω R and ω I stand for the real and imaginary partof the asymptotic QNF, respectively. Thus, using (27)we have ω p = 2( √ − M / l N . (55)According to Myung et al. [43], the Arnowitt-Deser-Misner (ADM) mass of the Lifshitz black hole we arestudying is given by M = M . (56)Applying Maggiore’s method, we calculate the adiabaticinvariant I as I = (cid:90) d M ∆ ω = (cid:90) M ∆ ω dM , (57)where ∆ ω is the change of proper frequency between twoneighboring modes, i.e. ,∆ ω = 2( √ − M / l . (58)Thus, I = lM / ( √ − , (59)which is quantized via Bohr-Sommerfeld quantization inthe semiclassical limit. Recalling that the horizon areaof the black hole is given by A = 2 πr + , with r + = l √ M ,and using (59), we arrive at the result, A = 2 π ( √ − n ¯ h , (60)with n an integer number. Therefore, we see that thehorizon area for the z = 3 Lifshitz black hole is quantizedand equally spaced.This result would not be expected for a theory contain-ing higher order curvature corrections since, in general,black hole solutions in such theories do not have a propor-tional relation between their entropy and area, and con-sequently, both of them (if any) should not be quantizedwith an equally spaced spectrum for large quantum num-bers [21, 44, 45]. However, it was already demonstratedthat the z = 3, three-dimensional Lifshitz black hole hasan entropy proportional to its horizon area [43, 46]. Thus,our result (60) also states that the entropy should bequantized with a spectrum evenly spaced. Nevertheless,we should stress that a generalization of this result forLifshitz black holes should wait for the calculation of thearea spectra of other black holes of such a type. Solelythese studies can give a definite answer on this subject. VII. CONCLUDING REMARKS
We have studied the stability of the three-dimensionalLifshitz black hole under scalar and spinorial pertur-bations in the probe limit through the computation ofquasinormal modes. In addition, we have found the eventhorizon area quantization as an application of the resultsfor quasinormal modes using Maggiore’s prescription.Regarding the stability, we have not found unstablequasinormal modes in the range of parameters that wehave considered; all the frequencies have a negative imag-inary part indicating that the modes are damped andthus, the perturbations decay, leaving the system stableagainst this particular sort of probe fields .In the case of a scalar probe field, such results totallyagree with the analytical expressions for the quasinormalfrequencies; they show a very large imaginary part and a very small real part. These modes are almost purelyimaginary. We have implemented two different numeri-cal methods in order to obtain the quasinormal frequen-cies and modes: the finite difference and the Horowitz-Hubeny methods. The former allows us to obtain thetemporal behavior of the fields showing all the stages ofthe decay, while the latter gives only the frequencies val-ues. As explained in section V, the Horowitz-Hubenymethod failed in the calculation of the scalar frequenciesas it can be observed in Fig.4. On the contrary, the fi-nite difference method has a very good agreement withthe analytical expression (25). Apart from the numeri-cal factor, the asymptotic scalar frequency found in thepresent work is the same as the one calculated in thehydrodynamic limit of the scalar perturbations in thecontext of gauge-gravity duality[47].Regarding the spinorial perturbation, our numericalresults show that the probe massless spinor decays and,thus, the z = 3 Lifshitz black hole is stable also underspinorial perturbations.As a by-product we also obtained the area spectrumof this black hole by means of the application of Mag-giore’s method using our results for the scalar asymptoticquasinormal frequencies. Equation (60) shows that thehorizon area is quantized and equally spaced. Further-more, in light of the conclusions shown in [43, 46], thecorresponding entropy should also have an evenly spacedspectrum.Finally, although we have demonstrated the stabilityof the z = 3, three-dimensional Lifshitz black hole underscalar and spinor perturbations, we should stress that thedefinite answer on stability should come from the gravita-tional perturbations, in particular, from the tensor partof the metric perturbation. It is well known that Einsteingravity in three dimensions has no propagating degreesof freedom, however, the massive versions of the theory, e.g. , NMG, allow the propagation of gravitational waves.Albeit this subject deserves further study, the calculationof metric perturbations is a formidable task that is outof the scope of the present paper. The analysis is notdead easy because the perturbation equation is a fourthorder differential equation. Thus, some other techniquesneed to be used together with the traditional QNM anal-ysis [50]. This study will be discussed elsewhere. ACKNOWLEDGMENTS
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