Stability Analysis and Area Spectrum of 3-Dimensional Lifshitz Black Holes

0.57 MB·English

by  Bertha Cuadros-Melgar

Stability 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 and
Physics 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 Sa˜o Paulo, C.P.66318, CEP 05315-970, S˜ao Paulo, Brazil
Inthiswork,weprobethestabilityofaz=3three-dimensionalLifshitzblackholebyusingscalar
and spinorial perturbations. We found an analytical expression for the quasinormal frequencies of
the scalar probe field, which perfectly agree with the behavior of the quasinormal modes obtained
numerically. The results for the numerical analysis of the spinorial perturbations reinforce the
conclusionofthescalaranalysis,i.e.,themodelisstableunderscalarandspinorperturbations. As
2 anapplicationwefoundtheareaspectrumoftheLifshitzblackhole,whichturnsouttobeequally
1 spaced.
0
2 PACSnumbers: 04.50.Kd,04.70.-s,04.25.Nx

n
a
J I. INTRODUCTION the QNFs are important in the context of the gauge-
gravity correspondence, whose most celebrated example
3
Some decades ago Regge and Wheeler began a pio- is the duality between the type IIA-B string theory in
] neering study of a small perturbation in the background AdS5×S5spacetimeandthefour-dimensionalsupersym-
h of a black hole in order to get information of the stabil- metricYang-Millstheory[13]. Suchacorrespondencecan
t
- ity of this object [1], a problem that was continued by be generalized for those cases in which there is an event
p
Zerilli [2]. The oscillations found in these studies are not horizoninthegravityside. InthiscasetheHawkingtem-
e
h normalmodesduetotheemissionofgravitationalwaves; peratureoftheblackholeisrelatedtothetemperatureof
[ thus, their frequencies are complex and, as a result, the athermalfieldtheorydefinedattheanti-deSitter(AdS)
oscillations are damped. boundary. Also, asaconsequenceofthecorrespondence,
2
Theterminologyquasinormalmode(QNM)andquasi- the quasinormal spectrum corresponds to the poles of
v
6 normal frequency (QNF), aiming to name these new thermal Green functions [14], more precisely, the inverse
5 modes and their frequencies, was first pointed out by of the imaginary part of the fundamental quasinormal
8 Vishveshwara [3] and Press [4]. Although initially stud- frequency can be interpreted as the dual field theory re-
4 ied in black hole backgrounds, the concept of QNM ap- laxation time [15].
.
0 plies to a much broader class of systems. The QNMs Another interesting application of QNMs appears in
1 of black holes were first numerically calculated by Chan- thecontextofblackholethermodynamics. Somedecades
1 drasekhar and Detweiler [5] showing that the amplitude ago Bekenstein [16] suggested that the horizon area of a
1
isdominatedbyaringingcharacteristicsignalatinterme- black hole must be quantized, so that the area spectrum
:
v diatetimes. TheQNMsareindependentoftheparticular has the form A = γn¯h, with γ a dimensionless con-
n
i initial perturbation that excited them. The frequencies stant to be determined. The first proposal to calculate
X
anddampingtimesoftheoscillationsdependonlyonthe thisconstantthroughQNMswasmadebyHod[17]. Ac-
r
a parametersoftheblackholeandare,therefore,the“foot- cordingly, the real part of the asymptotic quasinormal
prints” of this structure. Soon, the connection of QNMs mode can be seen as a transition frequency in the semi-
to astrophysics was established by noting that their ex- classicallimit,anditsquantumemissioncausesachange
istence can lead to the detection of black holes through in the mass of the black hole, which is related to the
the observation of the gravitational wave spectrum. The area. In this way, the constant γ for a Schwarzschild
interest in QNMs has motivated the development of nu- black hole was determined as γ = 4ln3. Later, Kun-
merical and analytical techniques for their computation statter [18] repeated the calculation quantizing the adia-
(cid:82)
(see [6–8] for a review). Also, the study of the quasinor- batic invariant I = dE/ω(E) via the Bohr-Sommerfeld
mal spectrum gives information about the stability as- quantization and using the real part of the QNF as the
pects of black hole solutions using probe classical matter vibrationalfrequency. Theresultwasexactlythesameas
fields (scalar, electromagnetic, spinorial) evolving in the Hod’s. However, recently Maggiore [19] pointed out that
geometry without backreacting on the spacetime back- QNMs should be described as damped harmonic oscilla-
ground. Much has been done in that direction, not only tors, thus, the imaginary part of the QNF should not be
in four dimensions [9][10], but also in two [11], and in neglected, andtheproperphysicalfrequencyisthemod-
more than four [12]. ule of the entire QNF. Moreover, when considering the
Aside from the study of the stability of the solutions, quantization of the adiabatic invariant, the frequency to
2
be considered is that corresponding to a transition be- ingQNFscomputedineachcase. Section6isdevotedto
tween two neighboring quantum levels. With this iden- the calculation of the area spectrum of these black holes
tification, the constant γ for a Schwarzschild black hole as an application of our quasinormal spectrum. Finally,
becomes γ = 8π, a result that coincides with the value we discuss our results and conclude in section 7.
calculated by other methods [20]. The consequences of
Hod’s and Maggiore’s proposals were promptly studied
in several spacetimes [21, 22]. II. LIFSHITZ BLACK HOLES IN THREE
DIMENSIONS
In this paper, we are interested in the study of the
stability of the z = 3, three-dimensional Lifshitz black
hole found in the context of the so-called new massive In this section we review the black hole solutions we
gravity (NMG) [23]. Moreover, as an application of our willconsiderwithinthispaper,andwecommentsomeof
QNM results we aim to calculate the area spectrum of their features.
this black hole. The NMG theory [26] is defined by the (2 + 1)-
dimensional action,
NMGisanovelparity-preserving,unitary[24],power-
counting super-renormalizable [25], three-dimensional 1 (cid:90) √ (cid:20) 1 (cid:18) 3 (cid:19)(cid:21)
model describing the propagation of a massive positive- S = d3x −g R−2λ− R Rµν − R2 ,
16πG m2 µν 8
energy graviton with two polarization states of helicity
(1)
±2 in a Minkowski vacuum, whose linearized version is
wheremistheso-called“relative”massparameter,andλ
equivalent to the Pauli-Fierz theory for a massive spin-2
is the three-dimensional cosmological constant. Defining
fieldinthreedimensions. TheactionofNMGconsistsof
the dimensionless parameters, y = m2l2 and w = λl2, it
a “wrong sign” Einstein-Hilbert term plus a quadratic
isfoundthatthetheoryexhibitsspecialpropertiesatthe
curvature term given by a precise combination of the
points y = ±1/2. When looking for black hole solutions
Riccitensorandthecurvaturescalar,whichintroducesa
withLifshitzasymptotics, itispreciselyatthepointy =
mass parameter [26]. As with other massive gravity the-
−1/2, w =−13/2, with Lifshitz scaling z =3, where the
ories, NMG also admits black hole-type solutions with
field equations turn out to be solved by [23]
several asymptotics and additional parameters [27, 28].
Even though this last feature could challenge the usual ∆ r2
Einstein-Hilbert gravity, it is seen that the definition of ds2 =−a(r) dt2+ dr2+r2dφ2, (2)
r2 ∆
massinthisnewtypeofblackholesisaconservedcharge
where
computed from a combination of the black hole parame-
ters, which satisfies the first law of thermodynamics. A r4
study of QNMs in these static new type of black holes a(r)= , (3)
l4
has been performed in [29].
and
Theblackholeswetakeintoaccountforourstudyare
asymptotically Lifshitz, i.e., they exhibit the anisotropic r4
scale invariance, t → λzt, (cid:126)x → λ(cid:126)x, where z is the dy- ∆=−Mr2+ l2 , (4)
namical critical exponent. Specifically, we deal with the
solutionsfoundfortheparticularcaseofz =3andapre- with M an integration constant and l2 = −13. Also,
2λ
cise value of the mass parameter [23]. The general class theNMGadmitsasasolution, thewell-knownBan˜ados-
of these solutions are important in the context of gauge- Teitelboim-Zanelli (BTZ) black hole with the dynamical
gravityduality[30,31]andwerealsoinvestigatedinother criticalexponentz =1. Asweshallseebelowinmorede-
background theories [32–35]. No stability study of black tail,thismetric(2)ex√hibitsaregularsingleeventhorizon
holeswithLifshitzasymptoticsinthreedimensionsinthe located at r =r+ =l M and a spacetime singularity at
scenarioofNMGhasbeenperformedyet. Weaimtogive r = 0. Besides, the surface r = r+ acts as a one-way
some contribution to this issue by considering the QNF membrane for physical objects as we can see from the
of scalar and spinorial matter fields in the probe limit, norm of a vector χ normal to a given surface s. Since s
i.e, there are no backreaction effects upon the asymp- has to be null in order to be a one-way membrane, the
totically Lifshitz black hole metric. Spinor fields have normofχmustbenullaswell,i.e.,grr =0,whichoccurs
beenextensivelystudiedingeneralrelativity[36][37],and at r =r+.
their quasinormal frequencies have also been considered From the behavior of the Kretschmann invariant for
[39][38][40]. the metric (2),
The paper is organized as follows. In section 2 we in-
troduce the Lifshitz black holes and discuss their causal R Rµνλσ =− 4 (cid:2)8r4 −48r2r2+91r4(cid:3) , (5)
structure. Sections 3 and 4 are dedicated to the study µνλσ l4r4 + +
ofstabilityunderscalarandspinorialperturbationswith
we see that, for r →r
specialemphasisonthemasslessspinorforthelatter. In +
section5wepresentthenumericalanalysisforbothkinds 204
ofperturbationsshowingtheQNMsandthecorrespond- RµνλσRµνλσ →− l4 , (6)
3
and for r →0 the behavior of the Kretschmann invariant. Moreover,
it is light-like and covered by a regular event horizon at
RµνλσRµνλσ →∞. (7) r =r+.
Thus, the black hole solution has a genuine spacetime
III. SCALAR PERTURBATION
singularity at the origin r = 0 and an event horizon at
r =r . Nevertheless, toseeifthesingularityistimelike,
+
spacelike,ornullwehavetoconstructthePenrose-Carter Inthissection,weanalyzethebehaviorofascalarfield
diagram. First of all, we must remove the coordinate perturbation in the background of a three-dimensional
singularity at r = r . Rewriting the metric in terms of Lifshitz black hole.
+
null coordinates (U,V) The scalar field obeys the Klein-Gordon equation,
U =er+3(t+r∗), V =−e−r+3(t−r∗). (8) (cid:50)Φ= √1−g∂M(cid:0)√−ggMN∂N(cid:1)Φ=m2Φ, (11)
where r is the tortoise coordinate shown in the next
∗ where m is the mass of the field Φ. Performing the de-
section, we get
composition
ds2 =−1(cid:18) r (cid:19)6(cid:16)1+ r+(cid:17)2e−2rr+dUdV , (9) Φ(t,r,φ)=Ψ(t,r)eiκφ, (12)
4 r r
+
The Klein-Gordon equation takes the form,
which is manifestly regular at r =r .
+
Finally, to construct the Penrose-Carter diagram r4 (cid:18) Ml2(cid:19)(cid:18)5r3 (cid:19)
−∂2Ψ+ 1− −3Mr ∂ Ψ+
(Fig.1) we use the following set of null coordinates t l6 r2 l2 r
T =U˜ +V˜, X =U˜ −V˜ , (10) +r8 (cid:18)1− Ml2(cid:19)2∂2Ψ−
l8 r2 r
with U˜ =arctan(U) and V˜ =arctan(V). −r4 (cid:0)m2r2+κ2(cid:1)(cid:18)1− Ml2(cid:19)Ψ=0.(13)
l6 r2
Even though this equation has an analytical solution,
aswewillseeinwhatfollows,itisalsousefultocheckthe
numerical results. With this goal we further decompose
√
Ψ = X(t,r )/ r, where the tortoise coordinate r is
∗ ∗
given by
(cid:20) (cid:18) (cid:19) (cid:21)
1 r 1
r =l4 − arccoth √ + . (14)
∗ M3/2l3 l M Ml2r
In this way the Klein-Gordon equation reduces to
−∂2X+∂2 X =V(r)X, (15)
t r∗
where V(r) is the scalar effective potential given by
(cid:18) 7 m2(cid:19) (cid:18)5M Mm2 κ2(cid:19)
V(r)= + r6− + − r4+
4l8 l6 2l6 l4 l6
(cid:18)3M2 Mκ2(cid:19)
+ − r2. (16)
FIG. 1. Penrose-Carter diagram for the Lifshitz black hole. 4l4 l4
Thesingularityatr=0islight-likeandcoveredbyaregular
event horizon r . Now let us come back to the issue of finding an exact
+
solution for Eq.(13). We set the time dependence of the
From this diagram we see that the spacetime singu- fieldΨ(t,r∗)√asR(r)e−iωt andredefinetheradialcoordi-
larity is located at r = 0, as previously observed from nate as r =l M/y. Thus, Eq.(13) turns out to be
4
y2−3 l2 (cid:20) ω2y4 m2 κ2 (cid:21)
∂2R+ ∂ R−− − + + R=0, (17)
y y(1−y2) y (1−y2) M3(1−y2) y2 Ml2
whose solution is given in terms of Heun confluent functions,
(cid:18) β2 α2 κ2 (cid:19)
R(y)=C y2+α(1−y2)β/2HeunC 0,α,β,− , + ,y2 +
1 4 4 4M
(cid:18) β2 α2 κ2 (cid:19)
+C y2−α(1−y2)β/2HeunC 0,−α,β,− , + ,y2 , (18)
2 4 4 4M
where C and C are integration constants, while α = C = 0. In order to apply the boundary condition of
√ 1 2 1
4+m2l2, and β =−ilω/M3/2. in-going waves at the horizon we use the following con-
Imposing the Dirichlet condition at infinity we set nection formula [41],
c Γ(1−b)Γ(c)
HeunC(0,b,c,d,e,z)= 1 HeunC(0,c,b,−d,e+d,1−z)+
Γ(1+c+k)Γ(−b−k)
c Γ(1−b)Γ(−c)
+ 2 (1−z)−cHeunC(0,−c,b,−d,e+d,1−z). (19)
Γ(1−c+s)Γ(−b−s)
Thisformulaconnectsasolutionaroundthesingularreg- and (cid:15) is related to e as
ular point z =0 to the corresponding solution about the
bc c b
singular regular point z =1 of the confluent Heun equa- (cid:15)=− − − −e. (23)
tion given by 2 2 2
Thus, near y =1 Eq.(18) can be written as
Γ(1+α)Γ(β)
z(z−1)H(cid:48)(cid:48)+[(b+1)(z−1)+(c+1)z]H(cid:48)+(dz−(cid:15))H =0. R(y →1)≈ξ (1−y2)β/2 +
1 Γ(α−k)Γ(1+β+k)
(20)
The parameters k and s are obtained from Γ(1+α)Γ(−β)
+ξ (1−y2)−β/2 , (24)
2 Γ(α−s)Γ(1−β+s)
with ξ as constants. As we are looking for quasinor-
i
mal frequencies with negative imaginary parts, which
give stable solutions, we find that for β < 0 we need
k2+(b+c+1)k−(cid:15)+d/2=0, (21) Γ(1+β +k) → ∞. Thus, the quasinormal frequencies
s2+(b−c+1)s−(cid:15)+d/2=0, (22) are
(cid:34) (cid:114) (cid:35)
M3/2 (cid:112) 3 κ2 (cid:112)
ω =2i 1+2N + 4+m2l2− 7+ m2l2+ +(3+6N) 4+m2l2+6N(N +1) , (25)
l 2 2M
where N is a positive integer. The imaginary part of the While the asymptotic frequency (N →∞) is given by
fundamentalfrequency(N =0)isnegativeprovidedthat
√ M3/2
ω =−2( 6−2)i N <0. (27)
∞ l
Thus, since the imaginary part of the quasinormal fre-
(cid:114)
3 κ2 (cid:112) (cid:112) quenciesisnegativeprovidedthattheparametersrespect
7+ m2l2+ +3 4+m2l2 >1+ 4+m2l2.
2 2M the relation (26), we can conclude that the model is sta-
(26) ble under scalar perturbations.
5
IV. SPINORIAL PERTURBATION where ω (a)(b) is the spin connection, which can be writ-
µ
ten in terms of the triad e µ as
(a)
Inthissection,wearegoingtoconsideraspinorialfield
as a perturbation in the spacetime given by the three-
dimensionalLifshitzblackhole. Weanalyzethecovariant ωµ(a)(b) =eν(a)∂µe(b)ν +eν(a)Γνµσeσ(b), (30)
Dirac equation for a two component spinor field Ψ with
mass µ . This equation is given by whereΓν arethemetricconnections. Theγ(a) denotes
s µσ
the usual flat gamma matrices, which can be taken in
terms of the Pauli ones. In this work we take γ(0) =iσ ,
2
iγ(a)e(a)µ∇µΨ−µsΨ=0, (28) γ(1) =σ1, and γ(2) =σ3.
Wecanwritethetriadbasise µ forthemetric(2)as
(a)
where Greek indices refer to spacetime coordinates follows:
(t,r,φ), and the Latin indices enclosed in parentheses
describe the flat tangent space in which the triad basis √
e(a)µ is defined. The spinor covariant derivative ∇µ is e0(a) = a(rr)∆δ0(a), e1(a) = √r∆δ1(a),
given by
e (a) =rδ (a), (31)
2 2
∇ =∂ + 1ω (a)(b)(cid:2)γ ,γ (cid:3) , (29) and the metric connections,
µ µ 8 µ (a) (b)
d (cid:34) (cid:18)a(r)∆(cid:19)1/2(cid:35) d (cid:34) (cid:18)r2(cid:19)1/2(cid:35) ∆ d (cid:20)a(r)∆(cid:21) ∆ 1
Γ0 = ln , Γ1 = ln , Γ1 = , Γ1 =− , Γ2 = .
01 dr r2 11 dr ∆ 00 2r2dr r2 22 r 12 r
(32)
Using these quantities it is straightforward to write At this point we are able to write the Dirac equation
down the expressions for spin connection components. for the two component spinor
In the present case, we have only two nonvanishing com-
ponents, (cid:18)Ψ (t,r,φ)(cid:19)
Ψ= 1 , (34)
Ψ (t,r,φ)
2
√
(cid:18) (cid:19)
1 d a(r)∆ ∆
ω (0)(1) = , ω (1)(2) =− (3.3) whichturnstobethesetofcoupleddifferentialequations
0 2(cid:112)a(r)dr r2 2 r
√
ir i ∆ i i (cid:20)a(r)(cid:48)∆ ∆(cid:48) (cid:21)
∂ Ψ + ∂ Ψ + ∂ Ψ + + √ Ψ −µ Ψ =0, (35)
(cid:112)a(r)∆ t 2 r r 2 r φ 1 4 a(r)r r ∆ 2 s 1
√
ir i ∆ i i (cid:20)a(r)(cid:48)∆ ∆(cid:48) (cid:21)
− ∂ Ψ + ∂ Ψ − ∂ Ψ + + √ Ψ −µ Ψ =0. (36)
(cid:112)a(r)∆ t 1 r r 1 r φ 2 4 a(r)r r ∆ 1 s 2
In order to simplify our problem, we redefine Ψ and Thus, we can rewrite Eqs.(35) and (36) as
1
Ψ2 as (cid:112)
a(r)∆
∂ Φ −iωΦ =i (mˆ −iµ r)Φ , (39)
r∗ − − r2 s +
Ψ =i[a(r)∆]1/4e−iωt+imφΦ (r), (cid:112)a(r)∆
1 + ∂ Φ +iωΦ =i (mˆ +iµ r)Φ , (40)
Ψ =[a(r)∆]1/4e−iωt+imφΦ (r), (37) r∗ + + r2 s −
2 −
where m=imˆ.
and the tortoise coordinate as in the scalar case (14), Furthermore,wedefineanewfunctionθ,anewrescal-
ing for the spinorial components R , and a new tortoise
±
coordinate rˆ through the expressions,
∗
(cid:112)
d ∆ a(r) d µ r
= . (38) θ =arctan( s ),
dr∗ r2 dr mˆ
6
1 µ r
rˆ =r + arctan( s ).
10-3 ∗ ∗ 2ω mˆ
M=1.0
M=1.2
M=1.4 In this way Eqs.(39) and (40) become
10-4 M=1.6
M=1.8
M=2.0 (∂ ±iω)R =W R , (41)
10-5 rˆ∗ ± ∓
where W is the so-called superpotential,
10-6
(cid:112)
i a(r)∆(mˆ2+µ 2r2)3/2
ψ(t) 10-7 W = s √ . (42)
r2(mˆ2+µ2r2)+ µsmˆ a(r)∆
s 2ω
10-8
Notice that when a(r) = 1, Eq.(42) reduces to the BTZ
10-9 superpotential [42].
10-10 Finally, letting X =R ±R we have
± + −
10-11 (cid:0)∂2 +ω2(cid:1)X =V X , (43)
0 5 10 15 20 25 rˆ∗ ± ± ±
t
where V are the superpartner potentials,
FIG. 2. Decay of scalar field with mass m=1 and l =1 for ±
different values of black hole mass M.
dW
V =W2± , (44)
± drˆ
∗
which in the case of a massless spinor (µ = 0) reduces
s
Φ =e±iθ/2R (r), to
± ±
(cid:18) m2M mM(cid:112) (cid:19) (cid:18)m2 2m(cid:112) (cid:19)
V = − ∓ r2−Ml2 r2+ ± r2−Ml2 r4. (45)
± l4 l5 l6 l7
V. NUMERICAL ANALYSIS QNF, our motivation to perform the numerical analysis
is to verify the applicability of certain numerical meth-
Inthissection,wenumericallysolveEqs.(15)and(43), ods in asymptotically Lifshitz spacetimes. In particular,
which correspond to the scalar and massless spinorial it would be interesting to check if the Horowitz-Hubeny
perturbations, respectively. Although in the scalar case method [15] works well when finding the QNF.
we found an analytical solution and the corresponding
Using the finite difference method, we define ψ(r ,t) = ψ(−j∆r ,l∆t) = ψ , V(r(r )) = V(−j∆r ) = V , and
∗ ∗ j,l ∗ ∗ j
rewrite Eqs.(15) and (43) as
ψ −2ψ +ψ ψ −2ψ +ψ
− j,l+1 j,l j,l−1 + j+1,l j,l j−1,l −V ψ +O(∆t2)+O(∆r2)=0, (46)
∆t2 ∆r2 j j,l ∗
∗
which can be rearranged as
∆t2 (cid:18) ∆t2 (cid:19)
ψ =−ψ + (ψ +ψ )+ 2−2 −∆t2V ψ . (47)
j,l+1 j,l−1 ∆r2 j+1,l j−1,l ∆r2 j j,l
∗ ∗
The initial conditions ψ(r ,0)=f (r ) and ψ˙(r ,0)= solution is stable if
∗ 0 ∗ ∗
f (r ) define the values of ψ for l = 0 and l = 1 and
1 ∗ j,l ∆t2 ∆t2
we use Eq. (47) to obtain the values of ψj,l for l > 1. ∆r2 + 4 VMAX <1, (48)
At j = 0 we impose Dirichlet boundary conditions since ∗
V(r )tendstoinfinityasr tendstozero. Thenumerical
∗ ∗ whereV =V isthelargestvalueofV inourdomain.
MAX 1 j
7
This condition is verified in all cases studied here. z =r2, we obtain
Nowwearegoingtoanalyzethepotentialforthescalar
case. By rewriting Eq.(16) in terms of a new variable
z (cid:20)(cid:18)7 (cid:19) (cid:18)5 κ2l2(cid:19) (cid:18)3 κ2l2(cid:19) (cid:21)
V(r)= +m2l2 z2− +m2l2− z z+ − z2 , (49)
l8 4 2 z h 4 z h
h h
where z =r2. The parable in brackets tends to infinity
as longhas (cid:0)7h+m2l2(cid:1) > 0, which is consistent with the 0
4 Analytic
Breitenlohner-Freedman-typeboundforthepresentcase. Numeric
Horowitz-Hubeny
The roots of this polynomial potential are given by
-0.5
z =0,
0
z =z , (50)
+ h -1
(cid:34) 3 − κ2l2 (cid:35)
z =z 4 zh . (51)
− h 74 +m2l2 ωI -1.5
-2
10-3
K=0
K=2
K=4 -2.5
10-4 KK==68
K=10
-3
10-5 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
M
10-6 FIG.4. Imaginarypartofscalarquasinormalfrequencies. We
ψ(t) display the results obtained using different approaches.
10-7
10-8
10-9 scalar perturbations. Moreover, according to Fig.4, the
numerical results have a very good agreement with the
analytical calculation.
10-10
0 5 10 15 20 25
t Figure 4 shows that the Horowitz-Hubeny method
gives unreliable results. In [48], it is argued that the
FIG. 3. Decay of scalar field with mass m = 1 and l = 1
varying the azimuthal parameter κ . frequencies 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, but
If m2l2 >−1, we see that z <z . Thus, going back they do not agree with the analytic expression and with
− +
to the original variable r, the roots of the potential are the results from finite difference method. In [49], the
√
r = 0 with double multiplicity, r = z and r = r authors find cases where this method does not work ei-
√ − h
(r = − z and r = −r are excluded as r > 0). Then, ther, andtheydosobycomparingtheresultswithother
− h
since r is the biggest root and V(r) tends to ∞ when methods. For instance, they find that the method is un-
h
r tends to ∞, the potential is positive-definite in the reliablefordimensionsbiggerthan6. Evenintheoriginal
region (r ,∞). Therefore, the quasinormal modes for work [15] the method is unreliable for small black holes,
h
m2l2 >−1 are necessarily stable [15]. and there is no clear explanation for this limitation. In
Thenumericalresultsregardingthedecayofthescalar ourcasetheasymptoticbehavioroftheblackholeunder
field are shown in Figs.2-3, and the comparison between study might play an important role in the convergence
thenumericalandanalyticalresultsisdisplayedinFig.4. of the method. Nevertheless, a general criteria for the
Our results reinforce the conclusion already found ana- convergenceoftheHorowitz-Hubenymethodremainsan
lytically; the z = 3 Lifshitz black hole is stable under open question.
8
In the case of the massless spinorial perturbation the and their derivative turns to be
superpartner potentials (45) can be written as
1 (cid:20) (cid:113) (cid:21)
V = (ml)2r2(r2−r2)±(ml)r2(2r2−r2) r2−r2 ,
± l8 + + +
(52)
  
1  (cid:113) 2r2−r2 
V±(cid:48) = l8 (ml)2r(2r2−r+2)±(ml)2r(r2−r+2) r2−r+2 +r3(cid:113) +  . (53)
 r2−r2 
+
We can see that outside the event horizon V is
+
positive-definite if ml > 0, and limr→∞V+(r) = −∞ 10-3 m=1
if ml<0. Whereas V is positive-definite if ml<0, and m=3
− m=7
lim V (r) = −∞ if ml > 0. Moreover, we notice 10-4 m=10
r→∞ −
that if ml = 0, we have a free-particle case. The decay-
ing behavior of the massless spinor is given in Figs.5-6. 10-5
Thus, we conclude that the z = 3 Lifshitz black hole is
stable under massless spinorial perturbations. 10-6
ψ(t) 10-7
10-3 m=1 10-8
m=3
m=7
10-4 m=10 10-9
10-5 10-10
10-6 10-11
0 5 10 15 20 25
t
ψ(t) 10-7
FIG. 6. Decay of massless spinor with l = 1 and black hole
10-8 massM =1.5fordifferentvaluesoftheazimuthalparameter
m.
10-9
10-10
is given by
10-11
0 5 10 15 20 25 30 35 40 (cid:113)
t ωp = ωR2 +ωI2, (54)
FIG. 5. Decay of massless spinor with l = 1 and black hole
where ω and ω stand for the real and imaginary part
massM =1.0fordifferentvaluesoftheazimuthalparameter R I
of the asymptotic QNF, respectively. Thus, using (27)
m.
we have
√ M3/2
ω =2( 6−2) N. (55)
p l
According to Myung et al. [43], the Arnowitt-Deser-
VI. AREA SPECTRUM
Misner (ADM) mass of the Lifshitz black hole we are
studying is given by
Oneoftheapplicationsofourresultsforthequasinor-
mal frequencies is the relation they have with the area M2
spectrum of a black hole. According to Maggiore [19], M= . (56)
2
the proper physical frequency of the damped harmonic
oscillator equivalent to the black hole quasinormal mode Applying Maggiore’s method, we calculate the adiabatic
9
invariant I as a very small real part. These modes are almost purely
(cid:90) dM (cid:90) M imaginary. We have implemented two different numeri-
I = = dM, (57) cal methods in order to obtain the quasinormal frequen-
∆ω ∆ω
cies and modes: the finite difference and the Horowitz-
Hubeny methods. The former allows us to obtain the
where∆ω isthechangeofproperfrequencybetweentwo
temporal behavior of the fields showing all the stages of
neighboring modes, i.e.,
the decay, while the latter gives only the frequencies val-
√ M3/2 ues. As explained in section V, the Horowitz-Hubeny
∆ω =2( 6−2) . (58)
method failed in the calculation of the scalar frequencies
l
as it can be observed in Fig.4. On the contrary, the fi-
Thus, nite difference method has a very good agreement with
the analytical expression (25). Apart from the numeri-
lM1/2
I = √ , (59) cal factor, the asymptotic scalar frequency found in the
( 6−2) present work is the same as the one calculated in the
hydrodynamic limit of the scalar perturbations in the
which is quantized via Bohr-Sommerfeld quantization in
context of gauge-gravity duality[47].
the semiclassical limit. Recalling that the horizon area
√
Regarding the spinorial perturbation, our numerical
of the black hole is given by A=2πr , with r =l M,
+ + results show that the probe massless spinor decays and,
and using (59), we arrive at the result,
thus, the z = 3 Lifshitz black hole is stable also under
√
spinorial perturbations.
A=2π( 6−2)n¯h, (60)
As a by-product we also obtained the area spectrum
with n an integer number. Therefore, we see that the of this black hole by means of the application of Mag-
horizonareaforthez =3Lifshitzblackholeisquantized giore’smethodusingourresultsforthescalarasymptotic
and equally spaced. quasinormal frequencies. Equation (60) shows that the
Thisresultwouldnotbeexpectedforatheorycontain- horizon area is quantized and equally spaced. Further-
ing higher order curvature corrections since, in general, more, in light of the conclusions shown in [43, 46], the
blackholesolutionsinsuchtheoriesdonothaveapropor- correspondingentropyshouldalsohaveanevenlyspaced
tional relation between their entropy and area, and con- spectrum.
sequently, both of them (if any) should not be quantized Finally, although we have demonstrated the stability
withanequallyspacedspectrumforlargequantumnum- of the z =3, three-dimensional Lifshitz black hole under
bers [21, 44, 45]. However, it was already demonstrated scalarandspinorperturbations,weshouldstressthatthe
that the z =3, three-dimensional Lifshitz black hole has definiteansweronstabilityshouldcomefromthegravita-
anentropyproportionaltoitshorizonarea[43,46]. Thus, tional perturbations, in particular, from the tensor part
our result (60) also states that the entropy should be ofthemetricperturbation. ItiswellknownthatEinstein
quantized with a spectrum evenly spaced. Nevertheless, gravity in three dimensions has no propagating degrees
we should stress that a generalization of this result for of freedom, however, the massive versions of the theory,
Lifshitzblackholesshouldwaitforthecalculationofthe e.g.,NMG,allowthepropagationofgravitationalwaves.
area spectra of other black holes of such a type. Solely Albeitthissubjectdeservesfurtherstudy,thecalculation
these studies can give a definite answer on this subject. of metric perturbations is a formidable task that is out
of the scope of the present paper. The analysis is not
dead easy because the perturbation equation is a fourth
VII. CONCLUDING REMARKS order differential equation. Thus, some other techniques
needtobeusedtogetherwiththetraditionalQNManal-
We have studied the stability of the three-dimensional ysis [50]. This study will be discussed elsewhere.
Lifshitz black hole under scalar and spinorial pertur-
bations in the probe limit through the computation of
quasinormalmodes. Inaddition,wehavefoundtheevent
horizonareaquantizationasanapplicationoftheresults ACKNOWLEDGMENTS
for quasinormal modes using Maggiore’s prescription.
Regarding the stability, we have not found unstable We thank E. Abdalla and A. M. da Silva for enlight-
quasinormal modes in the range of parameters that we ening discussions and remarks. We also thank E. Pa-
haveconsidered;allthefrequencieshaveanegativeimag- pantonopoulosandG.Giribetforreadingthemanuscript
inary part indicating that the modes are damped and and pointing out useful suggestions, and J. Oliva for
thus, the perturbations decay, leaving the system stable interesting comments. This work was supported by
against this particular sort of probe fields . Fundac¸˜ao de Amparo `a Pesquisa do Estado de S˜ao
In the case of a scalar probe field, such results totally Paulo (FAPESP-Brazil), State Scholarships Founda-
agreewiththeanalyticalexpressionsforthequasinormal tion(IKY-Greece),andConsejoNacionaldeInvestiga-
frequencies; they show a very large imaginary part and cionesCient´ıficasyT´ecnicas(CONICET-Argentina).
10
[1] T. Regge and J. A.Wheeler, Phys. Rev. 108, 1063-1069 [24] M. Nakasone and I. Oda, Prog. Theor. Phys. 121, 1389
(1957). (2009). S. Deser, Phys. Rev. Lett. 103, 101302 (2009).
[2] F. J. Zerilli, Phys. Rev. D 2, 2141-2160 (1970). [25] I. Oda, JHEP 0905, 064 (2009).
[3] C. V. Vishveshwara, Nature 227, 936-938 (1970). [26] E. Bergshoeff, O. Hohm, and P. Townsend, Phys. Rev.
[4] W. H. Press, Astrophys. J. 170, L105-L108 (1971). Lett. 102, 201301 (2009).
[5] S.ChandrasekharandS.Detweiler,Proc.R.Soc.London [27] E.Bergshoeff,O.Hohm,andP.Townsend,Phys.Rev.D
A 344, 441 (1975). 79, 124042 (2009).
[6] K. D. Kokkotas, B. G. Schmidt, Liv. Rev. Rel. 2, 2 [28] G. Clement, Class. Quant. Grav. 26, 105015 (2009);
(1999). J. Oliva, D. Tempo, and R. Troncoso, JHEP 0907, 011
[7] H.P.Nollert,Class.Quant.Grav.16,R159-R216(1999). (2009); G. Giribet, J. Oliva, D. Tempo, and R. Tron-
[8] E.Berti,V.Cardoso,andA.O.Starinets,Class.Quant. coso, Phys. Rev. D 80, 124046 (2009); A. Ghodsi,
Grav. 26, 163001 (2009). M. Moghadassi, Phys. Lett. B 695, 359 (2011).
[9] B. Wang, C. -Y. Lin, E. Abdalla, Phys. Lett. B481, 79- [29] Y. Kwon, S. Nam, J. D. Park, S. H. Yi, Class. Quant.
88 (2000). Grav. 28, 145006 (2011).
[10] B. Wang, E. Abdalla, R. B. Mann, Phys. Rev. D65, [30] S. Kachru, X. Liu, and M. Mulligan, Phys. Rev. D 78,
084006 (2002). 106005 (2008).
[11] V. Cardoso, J. P. S. Lemos, Phys. Rev. D63, 124015 [31] T. Azeyanagi, W. Li, and T. Takayanagi, JHEP 0906,
(2001). 084 (2009).
[12] E. Abdalla, O. P. F. Piedra, J. de Oliveira, Phys. Rev. [32] R. Mann, JHEP 06, 075 (2009).
D81, 064001 (2010). [33] G.Bertoldi,B.Burrington,andA.Peet,Phys.Rev.D80,
[13] J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998); 126003 (2009).
Int. J. Theor. Phys. 38, 1113 (1999). [34] U. Danielsson and L. Thorlacius, JHEP 0903, 070
[14] D. Birmingham, I. Sachs, S. N. Solodukhin, Phys. Rev. (2009).
Lett. 88, 151301 (2002). [35] K.BalasubramanianandJ.McGreevy,Phys.Rev.D80,
[15] G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62, 104039 (2009).
024027 (2000). [36] F., Finster, J. Smoller, S.-T. Yau, Phys. Rev. D 59,
[16] J. D. Bekenstein, Lett. Nuovo Cim. 11, 467 (1974), 104020, (1999).
[arXiv:gr-qc/9710076]. [37] F., Finster, N. Kamaran, J. Smoller, S.-T. Yau, Adv.
[17] S. Hod, Phys. Rev. Lett. 81, 4293 (1998). Math. Phys. 7, 1, (2003).
[18] G. Kunstatter, Phys. Rev. Lett. 90, 161301 (2003). [38] H. T. Cho, A. S. Cornell, J. Doukas, W. Naylor, Phys.
[19] M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008). Rev. D75, 104005 (2007).
[20] A. J. M. Medved, Mod. Phys. Lett. A 24, 2601 (2009); [39] H. T. Cho, Phys. Rev. D68, 024003 (2003); J. Jing,
K.Ropotenko,Phys.Rev.D80,044022(2009);T.Pad- Phys. Rev. D71, 124006 (2005).
manabhan and A. Patel, [arXiv:gr-qc/0309053]. [40] O. P. F. Piedra, J. de Oliveira, Class. Quant. Grav. 28,
[21] P. Gonzalez, E. Papantonopoulos, and J. Saavedra, 085023 (2011).
JHEP 1008, 050 (2010). [41] A.Ya.Kazakov,J.Phys.A:Math.Gen.39,2339(2006).
[22] S. W. Wei, R. Li, Y. X. Liu, and J. R. Ren, JHEP [42] V.CardosoandJ.P.S.Lemos,Phys.Rev.D63,124015
0903, 076 (2009); E. Vagenas, JHEP 0811, 073 (2008); (2001).
A.J.M.Medved,Class.Quant.Grav.25,205014(2008); [43] Y. S. Myung, Y. W. Kim, and Y. J. Park, Eur. Phys. J.
W. Li, L. Xu, and J. Lu, Phys. Lett. B 676, 177 (2009); C70, 335-340 (2010).
S.Fernando,Phys.Rev.D79,124026(2009);S.W.Wei [44] D. Kothawala, T. Padmanabhan, S. Sarkar, Phys. Rev.
andY.X.Liu,[arXiv:0906.0908];D.Kothawala,T.Pad- D 78, 104018 (2008).
manabhan, and S. Sarkar, Phys. Rev. D 78, 104018 [45] E. Abdalla, L. A. Correa-Borbonet, Phys. Rev. D 65,
(2008); R. Banerjee, B. Majhi, and E. Vagenas, Phys. 124011(2002);E.Abdalla,L.A.Correa-Borbonet,Mod.
Lett. B 686, 279-282 (2010); A. Lopez-Ortega, Phys. Phys. Lett. A 16, 2495-2504 (2001).
Lett.B682,85(2009);M.SetareandD.Momeni,Mod. [46] R.-G. Cai, Y. Liu, Y.-W. Sun, JHEP 0910, 080 (2009);
Phys. Lett. A 26, 151-159 (2011); B. Majhi, Phys. Lett. O. Hohm, E. Tonni, JHEP 1004, 093 (2010); H. Gonza-
B 686, 49 (2010); Y. Kwon and S. Nam, Class. Quant. lez, D. Tempo, R. Troncoso, JHEP 11, 066 (2011).
Grav. bf 27, 125007 (2010); Y. S. Myung, Phys. Lett. [47] E.Abdalla,J.deOliveira,A.Lima-SantosandA.B.Pa-
B 689,42 (2010); S. W. Wei, Y. X. Liu, K. Yang, and van, [arXiv:1108.6283].
Y.Zhong,Phys.Rev.D81,104042(2010);M.R.Setare, [48] E. Abdalla, C. E. Pellicer, J. de Oliveira, A. B. Pavan,
Class.Quant.Grav.21,1453-1458(2004);M.R.Setare, Phys. Rev. D82, 124033 (2010).
Phys. Rev. D 69, 044016 (2004). [49] J. Morgan, V. Cardoso, A. S. Miranda, C. Molina,
[23] E.Ayo´n-Beato,A.Garbarz,G.Giribet,andM.Hassa¨ıne, V. T. Zanchin, JHEP 0909, 117 (2009).
Phys. Rev. D 80, 104029 (2009). [50] Y. S. Myung, Y.-W. Kim, T. Moon, Y.-J. Park, Phys.
Rev. D 84, 024044 (2011).