Overspinning BTZ black holes with test particles and fields

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by  Koray Düztaş

Overspinning BTZ black holes with test particles and fields
Koray Du¨zta¸s∗
Physics Department, Eastern Mediterranean University,
Famagusta, Northern Cyprus, Mersin 10, Turkey
It has been claimed that in a test of an asymptotically AdS version of weak cosmic censorship
conjecture by attempting to overspin a BTZ black hole with test particles, one finds that it is not
possible to spin-up the black hole past its extremal limit. The result of this analysis is restricted
to the case where the initial black hole is extremal. We extend this analysis to find that massive
test particles can overspin the black hole, if we start with a nearly extremal black hole, instead.
Wealso considertheinteraction of theBTZ black hole with test fields. Weshow that overspinning
of nearly extremal black holes is possible whether or not there is superradiance for the field. If
there is superradiance, overspinning occurs in a narrow range of frequencies bounded below by the
7 super-radiantlimit. However,ifthereisnosuperradianceforthefield,overspinningbecomesgeneric
1 and also applies to extremal black holes. This is in analogy with theKerr case.
0
2
PACSnumbers: 04.70.Bw,04.20.Dw

n
a
J I. INTRODUCTION high angular momentum can also be absorbed. Backre-
5 action effects become negligible as frequency is lowered
2 The weak form of the cosmic censorship conjec- below the superradiant limit. For that reason violation
ture(wCCC) states that the gravitational collapse of a of wCCC by neutrino fields is generic. (see [19] )
]
c body alwaysends up ina black hole ratherthana naked The three dimensional black hole solution in anti-de
q singularity. The conjecture was proposed by Penrose in Sitter backgrounds which was derived by Ban˜ados,Teit-
- 1969 [1] in order to ensure that, the singularities which elboim, and Zanelli (BTZ) [20], has many similarities
r
g inevitably form as a result of gravitational collapse are with the Kerr metric. No-hair theorem holds, i.e. a
[ hidden behind the event horizons without any access to BTZ black hole is fully characterised by its ADM-mass,

distant observers. This way, the deterministic nature of and angular momentum. Moreover, a rotating BTZ
1
v generalrelativitycanbepreservedsincedistantobservers black hole has an inner and outer event horizon like a
1 donotencounteranyeffectspropagatingoutofsingular- Kerr black hole. These properties enable similar tests of
4 ities. wCCC to be employed on charged or neutral BTZ black
2 Over many decades, it has not been possible to estab- holes[21–24]. Thoughtexperimentsinvolvingasymptoti-
7 lish a concrete proof of wCCC. An alternative thought callyAdSblackholesinfourandhigherdimensionswere
0
procedure was developed by Wald [2] to test the sta- also constructed [25–27], though the notion of distant
.
1 bility of event horizons of black holes interacting with observers in AdS space-times may not be well defined.
0 test particles or fields. In the first of these thought ex- In this work we first apply a test of wCCC in the case
7
periments Wald started with an extremal Kerr-Newman of BTZ black holes interacting with massive test parti-
1
blackhole,andshowedthatparticleswithenoughcharge cles. ThispartismotivatedbytherecentworkbyRocha
:
v and angular momentum to exceed extremality are not andCardoso[23],wherethey conclude thataBTZblack
Xi captured by the black hole. Though, some of the simi- holecannotbespun-uppastitsextremallimit. However,
lar attempts involving test particles have been success- their analysis is restricted to the case where the initial
r
a ful to produce naked singularities [3–7], later backreac- blackholeisextremal. We showthatthe samelineofin-
tioneffectswereemployedtocompensatefortheseviola- vestigationleadstooverspinningifwestartwithanearly
tions [8–11]. Thought experiments involving scalar and extremal black hole, instead of an extremal one.
electromagnetic fields proved that extremal Kerr black We also challenge wCCC by sending in test fields to
holes cannot be overspun [12–14], however it is possi- a BTZ black hole. We follow a similar method to our
ble to overspin nearly extremal Kerr black holes by us- recent work on overspinning Kerr black holes [15]. Su-
ing integer spin test fields in a narrow frequency range perradiance plays a key role, and the results turn out to
bounded below by the superradiant limit [15]. In princi- be analogous to the Kerr case.
ple, this overspinning can also be invalidated by backre- Theoutline ofthe paperis asfollows: Inthe introduc-
actioneffectssincetheresultsareanalogoustotheparti- tion we review the general properties of BTZ spacetime
clecase[4]. TheonlygenericviolationinKerrcaseisdue andthe previous workof Rocha andCardoso. In section
to neutrino fields [16–18], andit also applies to extremal (II) we apply a thought experiment which extends the
blackholes. Since neutrino fields do notexhibit superra- work of Rocha and Cardoso to include nearly extremal
diantbehaviour,lowfrequencymodescarryingrelatively blackholes. Insection(III)wechallengewCCCbysend-
ing in test fields to both nearly extremal and extremal
black holes. Finally we comment on the results and the
∗ [email protected] possibilitytofix theviolationsofwCCCbybackreaction
2
effects. where (T(τ),R(τ),Φ(τ)) are the coordinates of the par-
ticle, and the dot indicates derivation with respect to
proper time τ. (α = 1,0 for timelike and null geodesics
A. The BTZ solution respectively.) Imposing the geodesic to be future di-
rected, T˙ ≥ 0, leads to ER2−(1/2)JL ≥ 0 outside the
The BTZ black hole is a solution of vacuum Einstein event horizon, via (6). At the horizon, this condition
equations in (2+1) dimensions, with a negative cosmo- implies that
logical constant Λ = −1/ℓ2. The line element is given
by L 2r+2
≤ (8)
ds2 =−N2dt2+N−2dr2+r2 Nφdt+dφ 2 (1) E J
AtthefinalstatethedimensionlessspinoftheBTZblack
where N = N(r) and Nφ = Nφ(r)(cid:0)are the usu(cid:1)al lapse hole becomes
and shift functions defined by
Jfin J0+m0L
r2 J2 J j ≡ = (9)
N2 =−M + + , Nφ =− (2) Mfinℓ ℓ(M0+m0E)
ℓ2 4r2 2r2
Mass and angular momentum parameters of the BTZ In the limit m0 → 0, the backreaction on the geometry
metric are denoted by M and J as usual. Like the Kerr tends to zero, the system reduces to a black hole and a
case, the metric has two singularities at point test particle. For that reason, RC Taylor expand
(9) around j0.
2r±2 J2
=M 1± 1− (3)
ℓ2 r M2ℓ2! j =j0− m0E j0− L (10)
M0 Eℓ
(cid:18) (cid:19)
(3) implies that the metric (1) defines a black hole sur-
roundedbyaneventhorizonprovidedthatthe massand For an extremal black hole, j0 =1 by definition, and (8)
the angular momentum parameters satisfy impliesandL/E ≤ℓ. Substitutingthesein(10)onefinds
that j ≤ 1. Therefore RC concluded that an extremal
|J|≤Mℓ (4)
BTZ black hole cannot be spun-up past extremality.
B. Extremal BTZ black holes and test particles
II. NEARLY EXTREMAL BTZ BLACK HOLES
AND TEST PARTICLES
One can test the validity of wCCC in the interaction
of a BTZ black hole with test particles or fields via (4).
An alternative approach to test the validity of wCCC
Atestparticleorafieldhasnegligibleeffectontheback-
is to start with a nearly extremal black hole instead of
ground geometry, it is only expected to slightly perturb
an extremal one. This procedure was first employed by
the mass and angular momentum parameters. Initially
Hubeny [3] who showed that it is possible to overcharge
we start with a black hole satisfying (4), after the in-
a nearly extremal Reissner-Nordstro¨m black hole as one
teraction with test particles or fields there exists a new
neglects backreactioneffects. Asimilar analysiswas car-
configuration with perturbed parameters J and M. If
riedoutbyJacobsonandSotiriou [4]tooverspinanearly
the new configuration also satisfies (4), wCCC remains
extremal Kerr black hole by throwing in test particles.
valid. Ifthefinalstateviolates(4)wCCCisalsoviolated.
Later backreaction effects were considered for this case
Theblackholeisoverspunintoanakedsingularity,since
tocompensatethe overspinningofthe blackhole[9]. We
an event horizon cannot exist if (4) is not satisfied.
also considered integer spin test fields interacting with
In that respect, a thought experiment to test the va-
Kerr black holes [15], and showed that, though an ex-
lidity of wCCC was recently constructed by Rocha and
tremal black hole cannot, a nearly extremal black hole
Cardoso (RC) [23]. In this section we summarise their
can actually be overspun by test fields with a frequency
results. RC considera black hole with initialparameters
slightly above the superradiant limit.
M0 and J0. Initially the black hole satisfies (4), so that In this section we carryouta similaranalysis to check
j0 ≡ J0/(M0ℓ) ≤ 1. They envisage a particle incident the validity of wCCC in the interaction of a nearly ex-
on the black hole from infinity with mass m0, angular tremal BTZ black hole with a test particle. We adopt
momentum δJ = m0L, and energy δM = m0E. The thenotationofRC.Letusparametriseanearlyextremal
geodesic equations lead to
BTZ black hole as
2ER2−JL
T˙ = (5) J
2R2N2 j0 = =1−2ǫ2 (11)
R˙2 =−αN2+E2−L2/ℓ2+R−2(L2M −JEL) (6) Mℓ
2EJℓ2−4ℓ2LM +4LR2 where ǫ ≪ 1. (This parametrization was also used
Φ˙ = 4R2ℓ2N2 (7) in [4] and [15].) We work up to second order in ǫ so
3
J2/(M2ℓ2)=1−4ǫ2. This leads to phenomenon canbe roughlydefined as the amplification
of a wave as it scatters off the black hole. For a BTZ
2r+2 black hole the angularvelocityof the horizonis givenby
=M(1+2ǫ) (12)
ℓ2 ΩH = J/2r+2, and the limiting frequency for superradi-
ance is ωs = nΩH. If there is superradiance for a test
Thus,foranearlyextremalblackhole(8)takesthe form
field incident on the BTZ black hole from infinity, it will
L Mℓ2(1+2ǫ) occur when the frequency of the field satisfies ω <ωs.
≤ =ℓ 1+2ǫ+2ǫ2+O(ǫ3) (13) Whether or not superradiance occurs when test fields
E Mℓ(1−2ǫ2)
scatteroffBTZblackholes,isanopenproblem. Naively,
(cid:0) (cid:1) from Kerr analogy one would expect it to occur for
For ǫ = 0 (13) reduces to (8), and j0 = 1−2ǫ2 → 1. bosonic test fields, and to be absent for fermionic fields.
Thus, the results of RC for the extremal BTZ black hole
This expectation was justified by Carlip who argued
are recovered. For a nearly extremal black hole let us
that superradiance exists for scalar fields on BTZ back-
first choose L/(Eℓ) arbitrarily close to (1 + 2ǫ + 2ǫ2).
ground [28]. However, Ortiz recently argued that su-
Using (11), the expression (10) for the final value of the
perradiance does not exist for scalar fields on BTZ
dimensionless spin takes the form
background, for vanishing boundary conditions at infin-
ity [29].
j =1−2ǫ2+(2ǫ+4ǫ2)m0E (14) Letus now considera nearly extremalBTZ black hole
M0 withinitialparametersJ0/(M0ℓ)=1−2ǫ2,asin(11). A
Choosing δM = m0E > ǫM0 leads to j > 1 so that the test field is incident on the black hole from infinity. For
BTZ black hole is spun-up past extremality. (With that overspinning to occur we require Jfin >Mfinℓ; that is
choice the term 4ǫ2(m0E)/(M0) is neglected) Actually
we have to choose δM ∼ ǫM0 to justify the test parti- J0+δJ >(M0+δE)ℓ (17)
cle approximation. To summarise the BTZ black hole is
spun-uppastextremalitywithtwochoices: m0E >∼ǫM0, vLieotlautsincghtohoeseteδstEfi∼eldMap0ǫprfooxrimthaetiionnc.idUensitngfie(l1d6,)wwiethmoauyt
and L/(Eℓ)<∼1+2ǫ+2ǫ2. rewrite (17). (Note that M0ℓ−J0 =2ǫ2M0ℓ.)
n
M0ǫ>2ǫ2M0ℓ+ℓM0ǫ (18)
III. OVERSPINNING BTZ BLACK HOLES ω
WITH TEST FIELDS
Equation (18) implies that the BTZ black hole will be
overspun if the frequency of the incoming field is below
InarecentworkweevaluatedthevalidityofwCCCby
the critical value:
sending in a packet of waves with dominant frequency ω
n
anddominantazimuthalwavenumberm scatteringoffa ω < ≡ωc (19)
Kerrblackhole[15]. Inthissectionwecarryoutasimilar ℓ(1+2ǫ)
analysis for the BTZ black hole. We consider massless
The critical frequency ωc is slightly larger than the su-
fields incident on the black hole from infinity that may
perradiant limit ωs = nΩH for a nearly extremal black
be bosonic or fermionic. The BTZ metric admits two
hole. This will become manifest if we write ωs in the
Killing vectors ∂/∂t, and ∂/∂φ like the Kerr metric, so
form
an incident wave mode has the form
J n n
ϕ(r,t,φ)=e−iωteinφf(r) (15) ωs = < =ωc (20)
Mℓ ℓ(1+2ǫ) ℓ(1+2ǫ)
(cid:18) (cid:19)
(Note that, in BTZ spacetime ω still denotes frequency, where we have used (12). Since our initial black hole
but the azimuthal wave number is referred to with the satisfies J < Mℓ, ωs is smaller than ωc. Therefore if
letter n.) We consider a test field which has negligible the frequency of the incident field is in the range ωs <
impact on the background geometry. This field will lead ω < ωc and δE ∼ M0ǫ for the field, the sub-extremal
to perturbations δM and δJ in the mass and angular BTZ black hole will be overspun whether or not there is
momentum parameters as it scatters off the BTZ black superradiance for the field. If there is no superradiance
hole. Ourthoughtexperimentinvolvesamonochromatic for the incoming field the allowed range of frequencies
wave which represents many quanta of energy ¯hω and will be extended to 0<ω <ωc.
angular momentum ¯hn. The perturbations of mass and Let us also consider the case where our initial black
angular momentum are related by hole is extremal. By definition J0 =M0ℓ. The condition
for overspinning reduces to
δJ =(n/ω)δE (16)
δJ >ℓδE (21)
where δE =δM for the black hole.
′ ′
Inathoughtexperimentinvolvingtestfieldsoneofthe Let us choose a field with δE ∼ M0ǫ where ǫ ≪ 1.
most critical issues is superradiance. The superradiance Using (16) one finds that for overspinning to occur, the
4
frequency of the incoming field should satisfy coincides with the superradiant limit therefore, if there
is superradiancefor the fieldanextremalblackhole can-
n notbeoverspun. Ifthereisnosuperradianceforthefield,
ω < (22)
ℓ overspinningalsoappliestoextremalblackholes,andthe
allowed range of frequencies is extended to 0<ω <ωc.
This value coincides with the superradiant limit for an Backreactioneffectswereneglectedinthisanalysis. In
extremal BTZ black hole. Therefore if there is superra- principle, backreaction effects can be employed to com-
diance for the incoming field, the challenging modes will pensate for the overspinning of BTZ black holes by test
not be absorbed by the black hole so an extremal black particles, and test fields that exhibit superradiant scat-
hole cannotbe overspun. Howeverif there is no superra- tering. However,ifthereisnosuperradianceforthefield,
dianceforthefield,themodeswithω <(n/ℓ)willalsobe the violation of wCCC becomes generic and also applies
absorbed, then overspinning will also apply to extremal to extremal black holes. The reason for that is the fact
black holes. thatthe ratioδJ/δE will growwithout boundandback-
reactioneffectswillbecomenegligibleasthe frequencyω
is lowered below the superradiant limit. This is in anal-
ogy with the Kerr case, where the overspinning due to
IV. CONCLUSIONS
neutrino fields is generic.
Superradiance can turn into an instability with a mir-
In this work we tested the validity of wCCC in the in- ror mechanism that reflects the radiation backwards,
teractions of test particles and fields with a BTZ black causing it to be amplified for multiple times. However,
hole. For the particle case, Rocha and Cardoso had pre- as far as the scattering of test fields is concerned, su-
viously concluded that it was not possible to spin-up a perradiance acts as a cosmic censor as it prevents the
BTZblackholepastextremality. However,theiranalysis absorptionofdangerousmodeswithlowenergyandhigh
onlyappliestoextremalblackholes. Wehaveshownthat angularmomentum. Inotherwordsitworksinfavourof
the same line of investigationleads to overspinning if we the stability of the event horizon, therefore the stability
start with a nearly extremal black hole instead. For the of the black hole. In the absence of superradiance one
caseoffields,wefound thatanearlyextremalblackhole can always find a frequency sufficiently low, so that the
can be overspun whether or not there is superradiance ratio δJ/δE is sufficiently high to overspin an extremal
forthefield,providedthatthefrequencyoftheincoming ora nearlyextremalblackhole. Fromthat pointofview
field is in the range ωs < ω < ωc, and δE ∼ M0ǫ for superradiance is necessary for the stability of the black
the field. The critical frequency for extremal black holes hole.
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