High frequency quasi-normal modes for black-holes with generic singularities

0.19 MB·English

by  Saurya Das

hep-th/0410209
High frequency quasi-normal modes for black-holes with generic singularities
Saurya Das∗
Department of Physics, University of Lethbridge,
4401 University Drive, Lethbridge, Alberta T1K 3M4, CANADA
S. Shankaranarayanan†
HEP Group, The Abdus Salam International Centre for Theoretical Physics,
Strada costiera 11, 34100 Trieste, Italy.
(Dated: February 1, 2008)
Wecomputethehighfrequencyquasi-normalmodes(QNM)forscalarperturbationsofspherically
symmetricsinglehorizonblack-holesin(D+2)-space-timedimensionswithgenericcurvaturesingu-
laritiesandhavingmetricsoftheformds2 =ηxp(dy2 dx2)+xqdΩ2 nearthesingularityx=0. The
− D
real part of the QN frequencies is shown to be proportional to log[1+2cos(π[qD 2]/2)] where
−
5 theconstantofproportionality isequaltotheHawkingtemperaturefornon-degenerateblack-holes
0 and inverse of horizon radius for degenerate black-holes. Apart from agreeing with the QN fre-
0 quenciesthat havebeencomputedearlier, ourresultsimplythat thehorizon area spectrumfor the
2 generalsphericallysymmetricblack-holesisequispaced. Applyingourresults,wealsofindtheQNM
n frequencies for extremal Reissner-Nordstr¨om and various stringy black-holes.
a
J PACSnumbers: 04.30.-w,04.60.-m,04.70.-s,04.70.Dy

4
2
I. INTRODUCTION tum of the area is

3
(D 1)lnk
v
Quasi-normal modes (QNM) are classical perturba- ω = − =(lnk)T , (2)
H
9 4πr
tionswithnon-vanishingdampingpropagatinginagiven h
0
gravitational background subject to specific boundary
2 where r is the horizon and T is the Hawking temper-
0 conditions. The frequency and damping of these oscilla- ature. h H
1 tions depend only on the parameters characterizing the
Hod[3]noticed,basedonthenumericalresult[7],that
4 black hole and are completely independent of the par-
the QNM spectrum for the Schwarzschildblack-hole has
0 ticular initial configurationthat caused the excitation of
/ a frequency whose real part numerically approached Eq.
h such vibrations. Over the last three decades, QNM have
(2) with k = 3 in the limit of infinite imaginary fre-
t been ofinterestdue to their observationalsignificancein
- quency. This was later confirmed analytically by Motl
p the detection of gravitational waves. (For a review, see
[8]for4-dimensionalSchwarzschild,whoshowedthatthe
e Ref. [1])
h QN frequencies have the following structure:
During the last few years there has been renewed in-
:
v terest in QNM for the following two reasons. First, in
1
i estimating the thermalization time-scales in connection ω =2πiT n+ +T ln(3)+ (n−1/2). (3)
X QNM H 2 H O
with the AdS/CFT conjecture [2]. Secondly, and more (cid:18) (cid:19)
r
a importantly, it has been conjectured recently that the
Eqs. (1) and (3) were used by Dreyer [4] recently to
real part of the QN frequencies for black-hole perturba-
argue that the Barbero-Immirzi parameter appearing in
tions(ω )correspondtotheminimumenergychange
QNM loop quantum gravity is ln(3)/(2√2π) and that the spin
of a black-hole undergoing quantum transitions [3, 4].
network states in the theory are dominated by j = 1
Earlier, Bekenstein [5, 6] had conjectured using semi- m
representations of the SU(2) group1. Together, this also
classical arguments that the black-hole area spectrum is
ledtothemicroscopicentropyoftheblack-holetobeone
equispaced and is of the form
quarter its horizon area (in Planck units). Turning the
argument around, (1) and (3) can also be used to vin-
A =4 log(k)ℓ2 s s=1,2, , (1)
s Pl ··· dicate the point of view that horizon area is equispaced
[6, 13].
where k is an integer to be determined and ℓ is
Pl Subsequently,MotlandNeitzke[14](seealsoRef.[15])
the Planck length. This implies that for a (D +
used a more flexible and powerful approach, called the
2) dimensional Schwarzschild black-hole, the energy
−
ω = ∆M emitted when the black-hole looses one quan-
1 Note,however,thattheabovevalueoftheBarbero-Immirizipa-
rameterdoesnotagreewiththatpredictedbytherecentanalyses
∗E-mail:[email protected] by Domagala and Lewandowski [9], and Meissner [10]. See also
†E-mail:[email protected] Refs. [11,12]
2
monodromy method, and were able to compute high- the metric on unit SD and
frequencyQNMandtheasymptoticvalueoftherealpart dr
oftheQNfrequenciesforaD-dimensionalSchwarzschild. x= , (6)
f(r)g(r)
Naturally, attempts were made to compute the QN fre- Z
quenciesforahostofotherblack-holes. Therehavebeen denotes the tortoise coordpinate. As it can be seen, the
quite a bit of effort to understand the physics underly- line-element(5)factorizesintothe productoftwospaces
ing the real (see, for instance, Refs. [16–21]) and the 2 SD, where 2 is the 2-dimensional space-time
M × M
imaginary part of the high-frequency QNM (see, for in- with Minkowskian topology.
stance, Refs. [22–36]), and there appears to be pieces of In order for the line-element (4) to describe a static
evidence in favour of as well as against the predictions black-hole,thespace-timemustcontainasingularity(say
referred to above. Thus, it is important to explore the atr =0)andhavehorizons. Inthiswork,wewillassume
QN frequencies for other black-holes and to probe their thatthe space-timecontainsoneevent-horizonatr and
h
full implications. thatitisasymptoticallyflat. However,wedonotassume
Even though the monodromy approach does not re- anyspecificformoff(r),g(r)andρ(r). Intherestofthis
quire the full knowledge of the space-time – except at section, we discuss generic properties of the space-time
the singularity, horizons and spatial asymptotic infinity near the horizon (r =r ) and the singularity (r =0).
h
– allthe previousanalyseshavebeenrestrictedtospace-
times whose line-element is known for the whole of the
manifold including the recent work by Tamaki and No- A. Horizon structure
mura [37]. In this work, we compute the QN frequen-
ciesfor(D+2) dimensionalsphericallysymmetricsingle Fortheline-element(4),thereexistsatime-likeKilling
−
horizon black-holes with generic singularities and near- vector field ξµ(xν) which can be expressed as ξµ(xν) =
horizonproperties(whichincludetheonesthathavebeen (1,0, ,0)[41]. Substituting Iµ =ξµ into the definition
···
already explored). Near the horizon, we assume that of the surface gravity, i. e.,
the spherically symmetric metric takes the form of the
1
Rindler while close to the singularity we use the form of κ2 = Iµ,νI (7)
µ,ν
−2
Szekeres-Iyer metric [38–40]. Using the Monodromy ap-
proach, we show that (i) the imaginary part of the high we obtain [41]
frequency QNM are discrete and uniformly spaced and
(ii) the real part depends on the horizon radius, space- 1 g(r)df(r)
κ= . (8)
time dimension and power-lawindex of SD near the sin- 2 sf(r) dr !
gularity. We also show that the real part of the high r=rh
frequency QNM has a logarithmic dependence whose ar- Using the property that the event horizon is a null hy-
gument need not necessarily be an integer. In order to persurface, the location of the horizon is determined by
illustrate this fact, we consider specific black-holes and the conditiongµν∂µN∂νN =0. Forthe line-element(4)
obtain their QN frequencies. N is a function of r characterizing the null hypersurface
The rest of the paper is organized as follows. In the which gives g(rh) = 0. Thus, the location of the event
next section, we discuss generic properties of the space- horizon is given by the roots of the above equation.
time near the horizon and the singularity. In Sec. (III), Since κ = constant and g(r) = 0 at the event hori-
we obtain the QN frequencies for a general static spher- zon, using the definition of surface gravity from Eq.
ically symmetric black-holes. In Sec. (IV), we apply our (8), we have the condition f(rh)/g(rh) = H(rh) where
general results to specific black-holes, reproducing ear- H(rh) = 0. Using the property that f(r) and g(r) are
6
lier resultsandobtaining new ones. Finally, we conclude smoothfunctions, wehavethe followingrelationforgen-
in Sec. (V) summarising our results and speculating on eral black-holes:
future directions.
f(r)/g(r)=H(r) (9)
whereH(r) is a smoothfunction andisnon-vanishingat
II. SPHERICALLY SYMMETRIC BLACK-HOLE the event-horizon.
In order to obtain the line-element near the horizon,
we make the coordinate transformation (t,r) (t,γ),
We startwith the general(D+2) dimensionalspher- →
− which is defined by
ically symmetric line element:
1 1 d f
r
dr2 γ f, dγ = dr, (10)
ds2 = f(r)dt2+ +ρ2(r)dΩ2 , (4) ≡ κ 2κ √f
− g(r) D p
where κ is given by (8). Note that the horizon (r ) is at
= f(r) dt2+dx2 +ρ2(r)dΩ2 , (5) h
− D γ =0. The line-element (4) becomes
where f(r), g(r) and(cid:2)ρ(r) are arb(cid:3)itrary (continuous, dif- f κ2
ds2 = κ2γ2dt2+4 dγ2+ρ2(r)dΩ2 , (11)
ferentiable) functions of the radial coordinate r, dΩ2 is − g (d f)2 D
D r
3
and hence, near the horizon, we have [40])investigatedalargeclassoffour-dimensionalspher-
ically symmetric space-times with power-law singulari-
ds2 →−κ2γ2dt2+dγ2+ρ2(r0)dΩ2D. (12) ties. These space-times practically encompass all known
sphericallysymmetricsolutionsoftheEinsteinequations
For space-times with single non-degenerate horizon (like suchasSchwarzschild(-deSitter),Reissner-Norstro¨mand
Schwarzschild for which κ = 0), f(r),g(r) can be ex- other type of metrics with null singularities. In Sec.
6
panded around rh as (IV), we will show that certain stringy black-holes and
higherdimensionalGibbons-Maedatypeblack-holes[42–
f(r)=f′(r ) (r r ); g(r)=g′(r ) (r r ), (13)
h h h h 44] also fall into this class.
− −
Szekeres-Iyer had shown that near the singularity,
using (8), we have
the spherically symmetric metric (4) takes the following
form(s):
1
κ= g′(r )f′(r ), (14)
h h
2
p ds2 r→0 ηr2p/qdy2 4ηr2(p−q+2)/qdr2+r2dΩ2 ,(21)
and using the relation (6), we have ≃ − q2 D
= ηxp(dy2 dx2)+xqdΩ2 , (22)
x=c ln(r r ), (15) − D
0 h
−
where y = βt, β > 0, η = 1,0, 1 correspond to space-
where c0 is a constant and is given in Table (I). like, null and time-like singulari−ties respectively and p,q
In the case of (D+2) dimensional Schwarzschild, we are constants and capture the dominant behavior near
−
have the singularity. Note that the notation of t and r is
adapted to the case of η = 1 where the singularity is
r D−1
h −
f(r)=g(r)=1 , (16) time-like and t is time. However,we will continue to use
− r
this notation even for space-like singularities where t is
(cid:16) (cid:17)
where r is related to the black-hole mass (M) by the actually space-like. The line-element (22) clearly shows
h
relation M = (DΩ rD−1)/(16πG ), G being that near the singularity the product spaces have dif-
D h D+2 D+2
the (D+2) dimensional Newton’s constant and Ω = ferent singularity structure. The curvature invariants –
D
(2π(D+1)/2/Γ−[(D+1)/2]. Using (8), we have RicciandKretschmannscalars–fortheline-element(21)
go as
(D 1)
κ= − . (17) ax−p+bx2−q
2rh R = , (23)
x2
For space-times with degenerate horizon,such as the ex- cx−2p+dx−2q+4+ex−p−q+2
R Rabcd = , (24)
tremal Reissner-Nordstro¨m(RN), we have abcd x4
(rD−1 rD−1)2 where a(p,q),...,e(p,q) = (1). The form of the in-
f(r)=g(r)= − h (18) O
r2(D−1) variants show that the Szekers-Iyer line-element indeed
describe the spherically symmetric space-time near the
where singularity.
Comparing Eqs. (4, 21), we have
r =(8πG M)/(D Ω ) (19)
h D+2 D
1 4η
For these space-times using the relation (6), we have f(r)= ηβ2r2p/q; = r2(p−q+2)/q; ρ(r)=r .
− g(r) −q2
(near the horizon)
(25)
Substituting the above expressions in Eq. (6), we have
x c log(r r )+ (rn) (20)
0 h
≃ − O
r2/q
where c0 is a constant and is given in Table (I). See x= . (26)
β
Table (I) and also Sec. (IV) for properties of the stringy
black-holes.
Note that nearthe curvaturesingularity,the tortoise co-
ordinate depends on q but not on p. This will be crucial
in obtaining the real part of the high frequency QNM.
B. Generic power-law singularities
In Table (I), we have given the values of p,q for various
black-holes.
The analysisofMotl-Neitzke [14] depends crucially on
For example, in the case of (D + 2)-dimensional
the behavior of the metric near the singularity. Thus,
Schwarzschild:
in order to assess the generality of the result, one needs
to understand the generality of the space-time singulari- 1 D 2 rD
p= − ; q = ; x= , (27)
ties. A decade ago, Szekeres-Iyer [38, 39] (see also, Ref. D D β
4
whereasfor(D+2)-dimensional(non-)extremalReissner- at r = 0,r , and . Assuming that the space-time is
h
∞
Nordstro¨m: asymptotically flat at radial infinity and using the rela-
tion(13)nearthehorizon,itiseasytoshownthatr
D 1 2 r2D−1 →∞
and r = r are irregular and regular singular points of
p= 2 − ; q = ; x= . h
− 2D 1 2D 1 β the differential equation (29) respectively. In the case of
(cid:18) − (cid:19) −
(28) power-law singularities as discussed in the previous sec-
tion, in order for r = 0 to be a regular singular point of
thedifferentialequation(29),itcanbeeasilyshownthat
III. QUASI-NORMAL MODES FOR STATIC p,q must satisfy the following conditions:
BLACK-HOLES
q >0 and p q+2>0 (31)
In this section, we obtain the high (imaginary) fre- Theaboveconditionswillbeu−sefulinreducingthegener-
quency QNM, corresponding to the scalar gravitational alized Regge-Wheeler potential (30) near the singularity
perturbations,forthegeneralsphericallysymmetric(D+ similar to the near-origin form of the potential derived
2) dimensionalblack-holesdiscussedintheprevioussec- by Motl and Neitzke [14]. Using the above conditions
−
tion. andx continuedto the whole complex plane (sayz), Eq.
(29)is anordinarydifferentialequationwithregularsin-
gular points at r =0, r and an irregular singular point
h
A. Scalar Perturbations
at r = . Thus, by the general theory of differential
∞
equations[46],anysolutionof(29)inthephysicalregion
The perturbations of a (D + 2)-dimensional static extends to a solution on the r plane. However, this so-
−
black-holes (4) can result in three kinds – scalar, vec- lutionmaybemulti-valuedaroundthesingularpointsat
tor and tensor – of gravitational perturbations (see for r =0, r [14].
h
example,Ref. [45]). The higherdimensionalscalargrav-
itational effective potential, which is of our interest in
thiswork,correspondtothewell-knownfour-dimensional
B. Computation of quasi-normal modes
Regge-Wheelerpotential. Theevolutionequationforthe
scalar gravitational perturbations follows directly from
the massless, minimally coupled scalar field propagating QNM are solutions to the differential equation (29)
in the line element (4), i. e., whose frequency is allowedto be complex. In the caseof
asymptotically flat space-times, which is of our interest,
2 Φ 1 ∂ √ ggµν∂ Φ =0, themodesarerequiredtohavepurelyoutgoingboundary
µ ν
≡ √ g − conditions both at the horizon and in the asymptotic
−
d2R(r) (cid:0) (cid:1) region, i. e.,
+ ω2 V(r) R(r)=0, (29)
dx2 −
R(x) e±iωx as x . (32)
(cid:2) (cid:3)
where Φ(xµ) = ρ(r)−D/2R(r) exp(iωt)Y and ∼ →∓∞
lm1...mD−1
V(r) is the higher dimensional analog of the Regge- Inorderfortheblack-holetobestable,themodesshould
Wheeler potential and is given by decay in time, hence (ω) > 0. In the monodromy ap-
ℑ
proach [14], unlike the earlier approaches, the authors
l(l+D 1) D
V(r) = − f(r)+ ρ(r)−D2 f(r)g(r) analytically continued x (in the complex plane z), in-
r2 2 stead of ω, and introduced the boundary conditions as
(cid:18) (cid:19)
d ρ(r)D−22 dρ(r) f(r)g(r) . p (30) tahpeprporaocdhu,cwteωnzeed→to±k∞no,wintshteeasdoluoftioxn→of E±q∞. .(29In)ntehaisr
× dr dr
(cid:18) (cid:19) r =0, r and compare their monodromies.
p h
The analysis of Motl and Neitzke requires the extension For the general spherically symmetric space-time –
of Eq. (29) beyond the physical region r < r < . In with the power-law singularity at the origin and generic
h
∞
order to perform the analysis we need to know the na- horizon structure – the generalized Regge-Wheeler po-
ture of the singularity of the differential equation (29) tential (30) near r =0 and r =r is
h
V(r) r→rh l(l+D−1)f′(r )+ D f′(r )g′(r )ρ′(rh) (r r )+ [(r r )2], (33)
≃ ρ2(r ) h 2 h h ρ(r ) − h O − h
(cid:20) h h (cid:21)
r→0 qβ 2 D D 2 r−4/q ηβ2l(l+D 1)r2(p−q)/q. (34)
≃ 2 2 2 − q − −
(cid:18) (cid:19) (cid:18) (cid:19)
5
Following points are worth noting regarding the above ( (ωz) ). Now from Eq.(26) and the relation
ℑ → −∞
result: z = 0, it follows that the the rotation of the contour on
(i) Using the conditions (31) in the second equation the r plane is by an angle 3πq/2, i.e. r rexp(i3πq/2).
→
above, it is clear that near the origin the first term in This translates to a rotation of 3πq/2 2/q = 3π in
×
the RHS dominates the second term. Hence, it would the z plane. That is: z exp(3iπ)z. Further, using
→
suffice to consider first term in the rest of the analysis. J (zeimπ)=eimνπJ (z), we have (as ωz )
ν ν
→−∞
Rewriting the potential near the origin in-terms of z, we
R(z) A e5iα+ +A e5iα− e−iωz (41)
have ∼ + −
Dq qD 1 + (cid:0)A+e7iα+ +A−e7iα−(cid:1)e+iωz .
r→0
V[r[z]] = 2 . (35)
8 2 − z2 Then, from Eqs. (cid:0)(40,41), following [14(cid:1)], we obtain the
(cid:18) (cid:19)
monodromy about the specified contour around the sin-
(ii) It is also interesting to note that the generalized gularity to be
Regge-Wheelerpotentialneartheorigindependsonlyon
q, D. From the line-element (22), this implies that the A+e5iα+ +A−e5iα− . (42)
potential depends on the singularity structure of the SD A+eiα+ +A−eiα−
space and not on 2. Eliminating the constants A using the constraint (39),
M ±
(iii) Remarkably, the generalized Regge-Wheeler poten- we get
tial near the origin is similar to the form of the near-
origin potential derived by Motl and Neitzke [14] except e6iα+ e6iα− sin(3πν) π
− = =1+2cos [Dq 2] .
for the coefficients2. In view of these observations, the − e2iα+ e2iα− sin(πν) 2 −
monodromy calculation [14] should carry through rela- − (cid:16) ((cid:17)43)
tively unchanged for the general spherically symmetric We can evaluate the monodromy of R(z) by choosing a
metrics with power-law singularities. In the rest of the contour which passes through the horizon. The mode
section,wedescribethiscalculationforthegeneralspher- function near the horizon (as z ) is given by
→−∞
icallysymmetricmetricsandobtainthehigh(imaginary)
R(z) e+iωz exp[ic ωln(x x )] . (44)
frequency QNM. [We follow the notation of Motl and ∼ ∼ 0 − h
Neitzke closely to provide easy comparison.] Thus, the monodromy by choosing the contour near the
Substituting the generalized Regge-Wheeler potential horizon is exp(4πωc ). Equating the two monodromies,
0
(35) in the differential equation (29), we get we obtain
i 1 1 π
R(z)=A+c+√ωzJ+ν(ωz)+A−c−√ωzJ−ν(ωz), (36) ω = n+ log 1+2cos [Dq 2] ,
QNM 2c 2 ±4πc 2 −
0 (cid:20) (cid:21) 0
where ν = (Dq 2)/4, while the products c A and h (cid:16) ((cid:17)4i5)
+ +
−
c A represent constant coefficients. Following [14], we where n is an integer. This is the main result of our
− −
will choose the “normalization factors” (denoted by c ) paper, regarding which we would like to stress a few
±
so that points. First,theaboveresultisvalidforageneralspher-
icallysymmetricspace-timeswhichisasymptoticallyflat
c±√ωzJ±ν(ωz)∼2cos(ωz−α±) as ωz →∞, and has a single horizon at r = rh. Second, it is clear,
(37) fromtheaboveexpression,thattheimaginarypartofthe
with highfrequencyQNMarediscreteandareequallyspaced.
π Third,therealpartofthehighfrequencyQNMhasalog-
α [1 2ν] . (38)
± ≡ 4 ± arithmic dependence and has a prefactor which depends
on r (See Table I). Lastly and more importantly, it is
h
From Eqs. (36) and (37), as well as the boundary condi-
clearfromtheaboveexpressionthatthe argumentofthe
tion (32) in-terms of ωz we get the following constraint
logarithm is not always an integer. It is a non-negative
integer only if
A e−iα+ +A e−iα− =0 (39)
+ −
Dq 2
and obtain the asymptotic form for R(z) as a) − π =2jπ where j (46)
2 ∈Z
(cid:18) (cid:19)
R(z) A e+iα+ +A e+iα− e−iωz as ωz . Dq 2 mπ
∼ + − →∞ b) − π = where m . (47)
(40) 2 3 ∈Z
(cid:2) (cid:3) (cid:18) (cid:19)
We follow the contour from the the negative imaginary
For the case (a), the real part of high frequency QNM
axis of z ( (ωz) ) to the positive imaginary axis
ℑ → ∞ is proportional to log(3) for all j. For the case (b), the
real part of high frequency QNM is (i) proportional to
log(2)form= 1, 5, 7, ,and(ii)blowsupform=
± ± ± ···
2, 4, 8, . Itisalsointerestingtonotethatthereal
2 The above behaviour of the Regge-Wheeler potential was also ± ± ± ···
partoftheQNfrequenciesvanishforallhalfoddintegers
observed in Ref. [47]. Other consequences of this universal be-
haviourwillbeexploredinRef. [48]. in Eq. (46).
6
IV. APPLICATION TO SPECIFIC found in Refs. [51–53], with or without matter fields 3 :
BLACK-HOLES
−1
M M
ds2 = 1 e2λr dt2+ 1 e2λr dr2 ,
In the previous section, we have obtained the high − − λ − λ
(cid:18) (cid:19) (cid:18) (cid:19)
frequency QNM for a spherically symmetric black hole (53)
with a generic singularity. As we have shown, the real where λ is a constant and M can be interpreted as the
part of the high frequency QNM is not necessarily pro- mass of the black-hole. It has a horizon at:
portional to ln(3) as in the case of (D+2)-dimensional
1 λ
Schwarzschild. In order to illustrate this fact, we take r = ln , (54)
specific examples and obtain their QNM. 2λ M
(cid:18) (cid:19)
whose Hawking temperature is
A. (D+2) dimensional Schwarzschild
λ
−
T = . (55)
H
2π
As noted earlier, for these black-holes
(Note that it is independent of M.) D = 0 for these
1 2
black-holesrendersqirrelevantforQNM.QNfrequencies
4πc = and q = . (48)
0
TH D can thus be read-off from (45):
Thus from (45),
ω =2πiT (n+m+1) or 2πiT (n m), m .
QNM H H − ∈Z
(56)
1
ω =2πiT n+ T log3 , (49) It is clear from the above expression that the real part
QNM H 2 ± H
(cid:20) (cid:21) of the high QN frequencies for generic 2-dimensional
(stringy) black-holes vanish. In Ref. [21], the au-
as found by previous authors [14].
thors have obtained high-frequency QNM for a generic
2-dimensionaldilatongravity. Ourresultisinagreement
withtheresultstheircaseforh(φ)=1whichcorresponds
B. (D+2) dimensional extremal
Reis−sner-Nordstr¨om to pure 2-dimensional gravity.
From Table I, we see:
D. 4-dimensional Stringy Black-Holes
Dr 2
h
c = and q = . (50)
0 (D 1)2 2D 1 The line-element of the 4-dimensional generalization
− −
of a charged black-hole solution [42, 43] is given by (4)
Thus where
i(D 1)2 1 r r (1−α2)/(1+α2)
ω = − n+ h 0
QNM 2Dr 2 f(r) = g(r)= 1 1 ;
h (cid:20) (cid:21) − r − r
(D 1)2 π(D 1) r(cid:16) α2/(1+(cid:17)α(cid:16)2) (cid:17)
− log 1+2cos − .(51) ρ(r) = r 1 0 . (57)
± 4πDrh (cid:20) (cid:18) 2D−1 (cid:19)(cid:21) − r
(cid:16) (cid:17)
Thus, for example in four dimensions (D =2), we get: rh,r0 are relatedto the physical mass and charge by the
relation
i 1 1
ωQNM = 4rh (cid:20)n+ 2(cid:21)± 8πrh log2 . (52) M = rh + 1−α2 r0; Q= rhr0 1/2 . (58)
2 1+α2 2 1+α2
Note that the logarithmic nature of the real part of the (cid:20) (cid:21) (cid:20) (cid:21)
QNfrequencypersistsalthoughitscoefficientisnolonger Note that α=0 reduces it to the familiar 4-dimensional
the Hawking temperature. The realpartis in agreement Reissner-Nordstro¨msolution,whereasforα=1the line-
with the argument of Motl and Neitzke [14], although element takes the form of the charged stringy black-hole
their analysis is for non-extremal Reissner-N¨ordstrom. solutionof[43]. Theabovesolutionhasaregularhorizon
Note however, that the above result appears to disagree atr . Foranynon-zerovalueofα,theinner-horizon(r )
h 0
with that stated in Ref. [49] and [50].
C. 2-dimensional stringy black-holes −1
3 The transformations η = −tanh−1 1− Me2λr /λ and
λ
x± =exp(−λ(r′±t))(where exp(2λr′)h=(−λ2expi(2λr))/(1−
Let us consider the generic 2-dimensional black-hole
µexp(2λr)/λ)convertthemetric(53)intotheformsassumedin
solutioninstringtheory,whichencompassesthesolutions [51]and[53]respectively.
7
is a space-like singularity. Thus, we will focus on the where (2π)4V and 2πR are the volume andradius of the
situation where α=0 which corresponds to a black-hole T4 andS1 respectivelyand g is the stringcoupling. The
6
with a singular horizon. above solution has a regular event horizon at r . When
h
It can be easily shown that the above black-holehas a all the three charges are nonzero, the surface r = 0 is a
non-zero Hawking temperature given by: smoothinnerhorizon. Whenatleastoneofthechargesis
zero(sayσ =0), the eventhorizonremains,howeverthe
1 r0 (1−α2)/(1+α2) surface r = 0 becomes singular. Thus, we will focus on
T = 1 . (59)
H
4πr − r the situation where σ =0 which corresponds to a black-
h (cid:18) h(cid:19)
hole with a singularity at r =0. The surface gravity for
Near the singularity (r r ), we have [ǫ r r ]:
→ 0 ≡ − 0 this resultant black-hole is
(1−α2)/(1+α2)
rh ǫ κ=(r coshα coshγ)−1 . (66)
f(r) = 1 , h
− r r
(cid:18) 0(cid:19) (cid:18) 0(cid:19)
Near the singularity, we have
1/(1+α2)
ρ(r) = r ǫα2 ,
0 f(r)= k r2/3; g(r)= l r2/3;
x = (cid:16)(r0 ǫ)2α(cid:17)2/(1+α2) . (60) ρ(r)=−m r1/3; x= m−√kl r. (67)
Thus, the 4-dimensional stringy black-hole line-element (cid:16) (cid:17)
near the singularity becomes Thus, the 5-dimensional stringy black-hole line-element
near the singularity becomes
ds2 r→r0 hx(1−α2)/(2α2) dt2 dx2 + xdΩ2, (61)
≃ − ds2 r→0 sx2/3 dt2 dx2 +x2/3dΩ2 (68)
where h = h(r0,rh,α) = (cid:2)(1). Comp(cid:3)aring the above ≃ −
O
line-element with (22), it follows that: where k,l,m,s=k,l,m(cid:2),s (r ,α,γ(cid:3))= (1). Comparing
h
O
1 α2 with (22), it follows that:
p= − ; q =1. (62)
2α2 2 2
p= ; q = . (69)
Thus, from (45), we get: 3 3
1 Thus, from (45), we get:
ω =2πiT n+ T log(3) . (63)
QNM H 2 ± H
(cid:20) (cid:21) 1
ω =2πiT n+ T log(3) . (70)
The above result is in agreement with Ref. [37]. QNM H 2 ± H
(cid:20) (cid:21)
We would like to clarify the following point regarding
E. 5-dimensional Stringy black-holes our result: As we had mentioned earlier, the analy-
sis is strictly valid for single-horizon black-hole space-
times. In obtaining the high-frequency QNMs for the
Theline-elementforRRcharged5-dimensionalstringy
5-dimensionalstringy black-holes,we have assumedthat
black-hole[44]formedbywrappingD1andD5braneson
T4 S1 is given by (4) where oneofthechargestothezero(σ =0). Note,however,for
× these stringy black-holes with a regular inner horizon, it
f(r)=F−2/3 1 rh2 ; ρ(r)=F1/6r; has been shown in Ref. [54] that the frequencies deter-
− r2 minedbythemonodromies(ofthetwohorizons)coincide
(cid:18) (cid:19)
r2 with the QNM of the near horizon BTZ metric and that
g(r)=F−1/3 1 h , (64) these are the ones that are relevant for quantum gravity
− r2
(cid:18) (cid:19) (the so called ‘non-quasinormal modes’). It will be in-
and terestingtocloselyexaminetherelationbetweenthetwo
r2sinh2α r2sinh2γ r2sinh2σ results.
F = 1+ h 1+ h 1+ h .
r2 r2 r2
(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21)
V. DISCUSSION
Theblack-holecarriesthreeU(1)charges,whicharepro-
portional to the number of D1 branes, D5 branes and
− −
open string momentum along the compact dimension In this paper, we have computed the high frequency
common between these branes, are related to the black- QNM for scalar perturbations of spherically symmetric
hole parameters as: singlehorizonandasymptoticallyflatblack-holesin(D+
2) dimensional space-times. We have computed these
Vr2 r2 −
Q = h sinh2α ; Q = h sinh2γ; modes using the monodromy approach [14]. We have
1 5
2g 2g shown that the real part of these modes depends on the
R2Vr2 horizon radius (r ), dimension (D) and the power-law
n = h sinh2σ , (65) h
2g2 index (q) of SD near the singularity.
8
TABLE I: The table gives the list of physical quantities for different black-holes. k is the surface gravity of the black-hole; c
0
is the constant which appears in the near-horizon relation between x and r; p and q are the power-law index for 2 and SD
M
respectively,and(Dq 2)/2isthequantitywhichdeterminestherealpartofQNfrequency( [ω ]). Thepropertiesofthe
− ℜ QNM
various black-holes (BH) are discussed in Secs. (II,IV).
Space-Time Near horizon properties Near singularity properties [ω ]
ℜ QNM
κ c p q (Dq 2)/2
0
−
Non-degenerate √f′(rh)g′(rh) 1 = 1 p>q 2 q>0 Any real value 1 log 1+2cos π(Dq−2)
2 2κ 4πTH − 4πc0 2
Horizons h (cid:16) (cid:17)i
(D + 2)-dimens. (D2r−h1) Dr−h1 1−DD D2 0 THlog3
Schwarzchild
1−α2
4-dimensional 2r1h 1− rrh0 1+α2 21κ 12−αα22 1 0 THlog3
stringy BH h i
5-dimensional coshrαh−c1oshγ 21κ −23 23 0 THlog3
stringy BH
2-dimensional 1 1 1 0 1 0
2 −
stringy BH
(D + 2)-dimens. 0 D r 2 D−1 2 1−D 1 log2
(D−1)2 h − 2D−1 2D−1 2D−1 8πrh
Degenerate RN (cid:16) (cid:17) (cid:16) (cid:17)
We have also shown that the real part of the high fre- andℓ is the (D+2) dimensionalPlancklengthin this
Pl −
quency modes has a logarithmic dependence, although case. Equating with the minimum area quantum in loop
the argument of the logarithm is not necessarily an in- quantumgravity,namely∆A=8πℓ2 γ j (j +1)and
m m
Pl
teger. However, when we applied our result to spe- subsequently setting D =2, we get the following predic-
p
cific examples and obtained their QNM, the argument tion for the Immirzi parameter:
turned out to be an integer. In particular, for (D+2)-
dimensional Schwarzschild and 4,5-dimensional stringy log[1+2cos(π(q 1))]
γ = − , (73)
black-holes, we found that the real part of the QN 2π j (j +1)
m m
frequencies is proportional to ln(3). However for 4-
p
dimensional Reissner-Nordstro¨m it is proportional to where j denotes the representation of SU(2) for the
m
ln(2) and vanishes in the case of 2-dimensional stringy spin-network states in loop quantum gravity (for super-
black-holes. ItwouldbeinterestingtocomputetheQNM symmetric spin networksthe areaspectrum is derivedin
for black holes with multiple horizons as well as those Ref. [55]). Assuming that the number of points where
which are not spherically symmetric or asymptotically thespin-networkstatespuncturethe horizonisgivenby:
flat and also to compare our results with others that are
related to a generic class of black holes [21]. N =A/∆A (74)
In the light of the above results, let us re-examine the
argumentpresentedin[4]. Forblack-holeswithnon-zero one gets the microscopic entropy of the black-hole, as
Hawking temperature, it follows from (45) that the real the logarithm of the dimension of the boundary Hilbert
part of QN frequencies is of the form: space, to be
π(Dq 2) S = Nlog(1+2j )
(ω )=T log 1+2cos − . (71) m
QNM H
ℜ 2 A ln(1+2j )
(cid:20) (cid:18) (cid:19)(cid:21) = m (75)
Identifyingtheabovewiththeminimumchangeofblack- 4ℓ2 log[1+2cos(π(q 1))]
Pl −
hole mass (due to Hawking radiation or accretion), we
Thus,fortheaboverelationtoagreewiththeBekenstein-
get,usingthefirstlawofblack-holethermodynamicsand
Hawking entropy, one must have:
the area law for entropy:
π(Dq 2) j =cos(π(q 1)) . (76)
∆A=4ℓ2 log 1+2cos − , (72) m −
Pl 2
(cid:20) (cid:18) (cid:19)(cid:21) Even though, for q = 1, (76) agrees with the prediction
where A = Ω rD is the horizon area of the black-hole ofRef. [4],onewouldliketohaveabetterunderstanding
D h
9
of the above result. More importantly, the above result Suchequispacedspectrumhasbeenverifiedearlierinsev-
is valid only for non-extremal black-holes. Since κ = eralother approaches,albeitwith differentproportional-
0 for the extremal black-holes, the real part of the QN ity constants and often with a ‘zero-point’ or ‘ground
frequencies (71) cannot be related to the temperature state’ energy, interpreted as a Planck-sized remnant left
implying that the relation for the change in the horizon over when the black-hole evaporates (see for e.g. [56–
area(72)is no morevalid. We hope to addressthis issue 58] and references therein). It is also interesting to note
elsewhere. from (52) and the relation r M1/(D−1) that the re-
h
∼
Finally,following[13]wefindthattheadiabaticinvari- lation I A continues to hold for extremal Reissner-
∝
ant in our case is: No¨rdstrom.
dE We end with a couple of caveats: (i) The area spec-
I = truminLoopQuantumGravityisnotequispacedingen-
ω
Z QNM eral [59, 60]. However, it is equispaced in the large area
−1
π(Dq 2) dE limit,aswellasifoneusesadifferentrepresentation[61].
= log 1+2cos −
2 T (ii) Strictly speaking, the QNM are only associatedwith
(cid:18) (cid:20) (cid:18) (cid:19)(cid:21)(cid:19) Z H
π(Dq 2) −1 transitions of short durations. Thus, other transitions
= log 1+2cos − S , (77) can modify the equispacedarea spectrum obtained here.
2
(cid:18) (cid:20) (cid:18) (cid:19)(cid:21)(cid:19) Wehopetofurtherexaminetheseandrelatedissueselse-
where.
which again confirms Bekenstein’s conjecture that hori-
zon area (and hence black-hole entropy) is an adiabatic
invariant. The crucial ingredient in the above is:
(ω ) T . (78)
QNM H
ℜ ∝ Acknowledgments
This, along with the Bohr-Sommerfeld quantization rule
I =n implies the equispaced nature of the horizonspec-
This work was supported by the Natural Sciences and
trum:
EngineeringResearchCouncil of Canada. We would like
to thank G.Kunstatter, A.J.M. Medved,J.M.Natario
A n ℓD , (79)
∝ Pl and R. Schiappa for useful comments. SS would like to
thank the Department of Physics, University of Leth-
with the following proportionality constant:
bridge,Canadafor hospitality where this workwas com-
−1 pleted. SS would also like to thank Martin O’Loughlin
π(Dq 2)
4 log 1+2cos − (80) and F. Hussain for discussions.
× 2
(cid:18) (cid:20) (cid:18) (cid:19)(cid:21)(cid:19)
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