Thermodynamics of extremal rotating thin shells in an extremal BTZ spacetime and the extremal black hole entropy

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by  José P. S. Lemos

Thermodynamics of extremal rotating thin shells in an extremal BTZ spacetime and
the extremal black hole entropy
Jos´e P. S. Lemos,1,∗ Masato Minamitsuji,1,† and Oleg B. Zaslavskii2,‡
1Centro Multidisciplinar de Astrof´ısica - CENTRA, Departamento de F´ısica,
Instituto Superior T´ecnico - IST, Universidade de Lisboa - UL,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal
2Department of Physics and Technology, Kharkov V. N. Karazin National University,
4 Svoboda Square, Kharkov 61022, Ukraine, and Institute of Mathematics and Mechanics,
Kazan Federal University, 18 Kremlyovskaya St., Kazan 420008, Russia
(Dated: February 23, 2017)
In a (2+1)-dimensional spacetime with a negative cosmological constant, the thermodynamics
7 and theentropy of an extremal rotating thin shell, i.e., an extremal rotating ring, are investigated.
1
TheouterandinnerregionswithrespecttotheshellaretakentobetheBan˜ados-Teitelbom-Zanelli
0
(BTZ) spacetime and the vacuum ground state anti-de Sitter (AdS) spacetime, respectively. By
2
applying the first law of thermodynamics to the extremal thin shell, one shows that the entropy

b of the shell is an arbitrary well-behaved function of the gravitational area A+ alone, S = S(A+).
e Whenthethinshellapproachesitsowngravitationalradiusr+ andturnsintoanextremalrotating
F BTZ black hole, it is found that the entropy of the spacetime remains such a function of A+,
both when the local temperature of the shell at the gravitational radius is zero and nonzero. It
1
is thus vindicated by this analysis, that extremal black holes, here extremal BTZ black holes,
2
have different properties from the corresponding nonextremal black holes, which have a definite

A
] entropy, the Bekenstein-Hawking entropy S(A+) = 4G+, where G is the gravitational constant. It
h is argued that for extremal black holes, in particular for extremal BTZ black holes, one should set
t A
- 0≤S(A+)≤ 4G+,i.e.,theextremalblackholeentropyhasvaluesinbetweenzeroandthemaximum
p A
Bekenstein-Hawking entropy +. Thus, rather than having just two entropies for extremal black
e 4G
A
h holes, as previous results have debated, namely, 0 and +, it is shown here that extremal black
4G
[ holes, in particular extremal BTZ black holes, may have a continuous range of entropies, limited

by precisely those two entropies. Surely, the entropy that a particular extremal black hole picks
2
must depend on past processes, notably on how it was formed. A remarkable relation between the
v
thirdlaw ofthermodynamicsandtheimpossibility for amassivebodytoreachthevelocity oflight
8
is also found. In addition, in the procedure, it becomes clear that there are two distinct angular
4
velocities for the shell, the mechanical and thermodynamic angular velocities. We comment on
3
the relationship between these two velocities. In passing, we clarify, for a static spacetime with a
2
thermalshell, themeaningoftheTolman temperatureformula at agenericradiusandat theshell.
0
.
1 PACSnumbers: 04.70.Dy,04.40.Nr.
0 Keywords: Quantumaspectsofblackholes,thermodynamics,three-dimensionalblackholes,spacetimeswith
7 fluids
1
:
v
I. INTRODUCTION dimensional spacetime.
i
X An important property of black holes is their entropy,
through it one can grasp the microscopic intrinsic ele-
r
a The Ban˜ados-Teitelboim-Zanelli (BTZ) spacetime [1] ments of a spacetime. For extremal BTZ black holes,
is the spacetime of a (2+1)-dimensional rotating black i.e.,blackholesforwhichthegravitationalradiusisequal
hole in a negative cosmological constant Λ background, to the Cauchy radius, r = r , or the angular momen-
+ −
being thus anasymptoticallyanti-de Sitter (AdS) space- tum is equal to the mass, =mℓ, it was found through
J
time with length scale ℓ = 1/√ Λ. It obeys a no hair topological arguments in the Euclidean sector that the
−
theorem [2], i.e., the black hole is characterized by its entropy S is [3]
gravitational radius r and its Cauchy radius r or,
+ − S =0. (1)
equivalently, by its mass m and its angular momentum
. It is thus a simple (2+1)-dimensionalspacetime, and Other studies in string theory [4] suggested that the
J
as such it provides a way to test and check many differ- extremal BTZ black hole entropy is the Bekenstein-
entpropertiesoftheKerrblackholesintheusual(3+1)- Hawking entropy, namely,
A
+
S = , (2)
4G
whereA =2πr istheeventhorizonarea,actuallyhere
∗Electronicaddress: [email protected] + +
†Electronicaddress: [email protected] aperimeter,r+ isthegravitationalradius,Gisthethree-
‡Electronicaddress: [email protected] dimensionalgravitationalconstant,andweuseunitssuch
2
that the velocity of the light, the Planck constant, and that can be either zero or the Bekenstein-Hawking en-
the Boltzmann constant are set to one. Thus, the black tropy, was indeed found first in (3+1)-dimensionalblack
holeentropyforextremalBTZblackholesisnotasettled holes. For extremal (3+1) black holes, it was found in
issue. On the other hand, for nonextremal BTZ black one approach based on the horizon topology [26] (see
holes,i.e.,blackholesforwhichtheangularmomentumis also [3]) that the entropy is zero S = 0, see Eq. (1).
lessthanthemass, <mℓ,orr >r ,theentropyS is The otherapproach,basedonstringtheorycalculations,
+ −
J
precisely and uniquely given by the Bekenstein-Hawking yields that the entropy of extremal black holes is given
entropy ofEq. (2). For further studies on the thermody- by the Bekenstein-Hawking entropy, Eq. (2) [27, 28], see
namics and entropy of BTZ black holes see [5–11]. also [29–38]. On the other hand, for (3+1)-dimensional
As the concept of entropy is originally based on quan- nonextremal black holes, the entropy is the original un-
tum properties of matter, it would be useful to study ambiguous Bekenstein-Hawking entropy, S = A+ of
4G
whether the black hole thermodynamics could emerge Eq. (2) [39–41].
from thermodynamics of collapsing matter, when we Here, we pursue further the problem of the entropy
compressmatter within its owngravitationalradius. So, of an extremal BTZ black hole by using a shell, an ex-
in orderto understandbetter the physicsat the horizon, tremal rotating thin shell. This is important in order
a promising setting is a thin shell, i.e., a thin ring, in to gain new insights into the entropy and other physical
a (2+1)-dimensional spacetime that is compressed qua- relevantquantitiesfromspacetimesthatpossessrotation
sistatically to its own gravitational radius. Outside the and angular momentum.
ring,thespacetimehastheBTZform,insideit,thespace The paper is organized as follows. In Sec. II, we dis-
timeisthegroundstateoftheBTZspacetime,i.e.,azero cuss the mechanics of an extremal rotating thin shell in
mass locally AdS spacetime. One can calculate the en- (2+1)dimensions with a cosmologicalconstant. The ex-
tropy of this matter ring system for any ring radius R, terior spacetime to the shell is the BTZ spacetime. In
in particular, when the ring is compressed to its gravi- Sec.III,westudythefirstlawofthermodynamicsapplied
tational radius r , R = r . This has been done in the for such a thin shell, derive the thermodynamic entropy
+ +
nonrotating BTZ case [12] and in the rotating nonex- ofthe thinshell, andshow thatthe entropyis a function
tremal BTZ case [13] where the entropic properties of of the gravitational area A only, S = S(A ). We also
+ +
theringatthegravitationalradiuswerededuced. Inpar- analyzetheequationofstateforthetemperatureandfor
ticular, it was found that the entropy of the ring is the the angular velocity of the shell. In Sec. IV, we consider
Bekenstein-Hawking entropy given in Eq. (2), provided the extremal shell with zero local temperature and take
that the shell’s temperature coincides with the Hawking the limit to its gravitational radius, obtaining thus the
temperature of the corresponding black hole. Still lack- properties of the corresponding extremal black hole. In
ing is the study of the thermodynamics and the entropy Sec.V,weconsidertheextremalshellwithsomenonzero
oftheextremalBTZringcase,thatmightemulatethedi- constant local temperature and take again the limit to
rectcalculationofthe entropyofanextremalBTZblack its gravitational radius, obtaining also the properties of
hole. For other studies related to the properties of mat- this extremalblack hole. In Sec. VI, we discuss the non-
ter systems, in particular, rotational properties in BTZ trivialrelationbetweenmechanicalandthermal,angular
backgrounds see [14–18]. and linear, velocities and compare the nonextremal and
The fact that shells reflect black hole properties was extremal cases. In Sec. VII, we give some concluding re-
found in (3+1) Reissner-Nordstro¨m asymptotically flat marks. IntheAppendixA,weclarifythemeaningofthe
spacetimeswithanelectricshellwheretheentropyprop- Tolman temperature formula at a generic radius and at
ertiesofzerocharge,i.e.,Schwarzschild[19],nonextremal the shell for a spacetime with a thermal shell.
[20], and extremal [21, 22], black holes were reproduced,
making these shells a very useful setting. Related to it
therewerethestudiesoftheentropyforquasiblackholes II. THIN SHELLS IN A (2+1)-DIMENSIONAL
[23, 24] and of quasistatic collapse of matter [25]. In the EXTREMAL BTZ SPACETIME
nonextremal case these studies found that at the gravi-
tational radius of the shell the spacetime, and thus the A. Outer and inner spacetimes
corresponding black hole, has the Bekenstein-Hawking
entropy of Eq. (2), where in the (3+1)-dimensional case Weconsidergeneralrelativitywithacosmologicalcon-
A+ = 4πr+2. On the other hand, for extremal shells the stant Λ in a (2+1)-dimensional spacetime. We also as-
entropy at the gravitationalradius and thus the entropy sumethatΛ<0,sothatthespacetimeisasymptotically
ofthecorrespondingextremalblackhole,canbeanywell- AdS, with curvature length scale ℓ=1/√ Λ. Through-
behaved function of the gravitationalradius r , or since −
+ out this paper, we work in units where the velocity of
A = 4πr2, the entropy can be any well-behaved func-
+ + light, the Planck constant, and the Boltzmann constant
tionofthe gravitationalareaA , S =S(A ). Soamong
+ + are set to unity. G denotes the gravitational constant in
many other values, it can be zero as in Eq. (1) or A+ as (2+1) dimensions.
4G
in Eq. (2). Weintroduceatimelikeshell,i.e.,aring,inthe(2+1)-
The ambiguity in the entropy of extremal black holes, dimensional spacetime, with radius R, which divides the
3
spacetimeintotheouterandinnerregionslabelledby(o) Therefore,theouterextremalregion(o)doesnotcontain
and (i), respectively [12, 13] (see also [18]). We assume aneventhorizonr =r ,exceptinthecaseR=r . One
+ +
that the spacetime is vacuum everywhere off the shell. can also define the area A, here a perimeter, of the shell
Outside the shell (r > R), the spacetime is described as
by the extremal rotating BTZ solution, while inside the
A=2πR, (10)
shell (r < R), the spacetime is the ground state of the
BTZ solution and is locally AdS. One can express the
so that Eq. (9) is written as
line element for the inner and outer regions by
A A . (11)
ds2(I) = −f(I)(r)dt2I +g(I)(r)dr2 ≥ +
+ r2 dφ ω (r)dt 2, (3) An important quantity is the redshift function k at
− (I) (I) some coordinate outer radius r, k f (r) which in
(o)
≡
where t is the time coo(cid:0)rdinate, (r,φ) are(cid:1)the radial and our case is
p
azimuthal coordinates, I = o/i refers to the outer or
r r2
inner regioninrelationto the shell, respectively,andthe k r ,r = 1 + . (12)
functions f , g and ω read + ℓ − r2
(I) (I) (I) (cid:16) (cid:17)
(cid:0) (cid:1)
This functiongets a value equalto one atthe coordinate
r 2 r2 2 1
f(o)(r)= ℓ 1− r+2 , g(o)(r)= f (r), r =r0 given by
(o)
(cid:16)ω(o(cid:17))(r(cid:16))= ℓrr+22,(cid:17) (4) r0 = 2ℓ +r 2ℓ 2+r+2, (13)
(cid:16) (cid:17)
r 2 1 where k(r ,r ) = 1. It is also of interest to dis-
f (r)= , g (r)= , + 0
(i) ℓ (i) f (r) play the redshift function at the position of the shell,
(i)
(cid:16) (cid:17)
ω(i)(r)=0. (5) k ≡ f(o)(R), which in our case is
The subscript (I) in the time coordinate t indicates p R r2
(I) k r ,R = 1 + . (14)
that in general the time coordinate of the outer region + ℓ − R2
t(o) differs from that of the inner region t(i). The radius This function ge(cid:0)ts a va(cid:1)lue eq(cid:16)ual to on(cid:17)e when the shell is
r isthegravitationalradiusofthespacetime. Intheex-
+
at the position
tremal case, the case we consider here, the gravitational
radius r is equal to the Cauchy radius r , r = r ,
+ − + − ℓ ℓ 2
and we stick to the usual notation r for such a radius. R = + +r2, (15)
+ 0 2 2 +
The gravitational radius r becomes the horizon radius r
+ (cid:16) (cid:17)
if the solution is an extremal black hole or an object on wherek(r ,R )=1,andR is alwaysgreaterthanboth
+ 0 0
the verge of becoming an extremal black hole. In the r and ℓ.
+
extremal case the radius r is given by
+
r2 =4Gℓ2m, (6)
+ B. Spacetime at the junction: Properties of the
where m is the asymptotic Arnowitt-Deser-Misner extremal shell
(ADM) mass. The radius r can also be written as
+
r2 = 4Gℓ upon using that the spacetime angular mo- 1. Metric and rotation of the extremal shell
+ J
mentum and m are related in the extremal case by
J
The shell dynamics and its matter content are deter-
=mℓ. (7)
J mined by the Israel junction conditions. The first junc-
We assume m>0. A gravitational or horizon radius r tion condition ensures the uniqueness of the induced ge-
+
corresponds to a gravitational or horizon area A+, here ometry on the shell, at R, hab = ha(ob) = h(aib), where
a perimeter, given by a,b = t,φ. The second junction condition determines
theenergy-momentumtensorofmatteronthethinshell,
A =2πr . (8)
+ + S ,thatcompensatesthejumpoftheextrinsiccurvature
ab
Theinnerregion(i)correspondstothegroundstatevac- tensor across the shell.
uum solution, i.e., m = 0 and = 0. In the junction As the outer spacetime is rotating while the inner
J
between the outer extremal BTZ spacetime and the in- spacetime is static, in order to match these two regions,
ner vacuum AdS spacetime, at the radius R, there is a the shell at r =R must corotate with the outer BTZ re-
stationary thin shell. We assume that the shell’s charac- gion. We introduce a coordinate system corotating with
ter is alwaystimelike and the shell is locatedoutside the the shell by adopting a new angular coordinate dψ such
event horizon, that
R r . (9) dψ =dφ ω (R)dt . (16)
+ (I) (I)
≥ −
4
The line element given in Eq. (3) is then written as 3. Relation between global extremal spacetime quantities
and local extremal shell quantities
ds2 = f (r)dt2 +g (r)dr2
(I) − (I) (I) (I)
+ r2 dψ o (r)dt 2, (17) The spacetime quantities m and are related to the
− (I) (I) J
shell quantities M and J. From Eqs. (6) and (26), one
where we have introdu(cid:0)ced (cid:1) finds thatthe localproper mass of the shell M is related
to the ADM mass m by
o (r)=ω (r) ω (R), (18)
(I) (I) (I)
−
so that at the position of the shell o (R) = 0 and the MR
(I) m= . (28)
line element is diagonal. Also, from Eqs. (4) and (5), ℓ
r2 From Eq. (27), one sees that the angular momentum of
ω (R)= + , (19)
(o) ℓR2 the shell J is independent of the position of the shell R,
and from Eqs. (6) and(7), we see that it is identical to
and
the angular momentum of the outer BTZ spacetime,
ω (R)=0, (20)
(i)
=J. (29)
respectively. The induced line element on the shell at J
R uniquely determined by the first junction condition is We would like to emphasize that in our case the inner
given by regionisa(2+1)-dimensionalspacetimelocallyAdSand
it has zero ADM mass and zero angular momentum. In
ds2 = dτ2+R2dψ2, R=R(τ). (21)
R − the more complex case that the region inside the shell
The proper time on the shell τ is defined by contains instead a BTZ black hole, then the total ADM
mass and angular momentum of the outer spacetime de-
dτ = f (R)dt2 g (R)dR2 finedatinfinitywouldincludeinadditiontheADMmass
(o) (o)− (o)
and angular momentum of the interior black hole.
q
= f (R)dt2 g (R)dR2. (22)
(i) (i)− (i)
q
Sinceweareinterestedinaquasistaticprocesswealways
III. FIRST LAW OF THERMODYNAMICS,
assume that dR =0 and d2R =0. ENTROPY OF AN EXTREMAL ROTATING
dτ dτ2
THIN SHELL IN A BTZ SPACETIME, AND THE
EQUATIONS OF STATE
2. Energy-momentum tensor of the extremal shell
A. First law of thermodynamics of an extremal
The rotating thin shell is supported by an imperfect rotating thin shell in a BTZ spacetime
fluid with an energy-momentum tensor Sa , such that
b
the nonzero components are Sττ = σ, Sψψ = p, and Now,weturntothe thermodynamicsofthethinshell.
Sτψ =j, where σ, p, andj, represent−the energy density Following [13], we assume that the rotating shell has
of the shell, the pressure in the shell, and the angular a thermodynamic angular velocity Ω and is in thermal
momentum flux density of the shell, respectively. The equilibrium, with local temperature T and entropy S.
second junction condition gives The entropy S of a system can be expressed as a func-
tion of the state independent variables. One can take as
r2
σ = + , (23) state independent thermodynamic variables for the thin
8πGℓR2
shellthepropermassM,theareaoftheshell A,andthe
r2
j = + , (24) angular momentum of the shell J. Thus, the entropy of
8πGℓR theshellisafunctionofthesequantitiesthroughthefirst
r2 law of thermodynamics which reads
p = + . (25)
8πGℓR2
TdS =dM +pdA ΩdJ. (30)
Thus, σ = p = j/R. The extremal rotating shell obeys −
both the weak and dominant energy conditions [18].
To obtain the entropy S, in general we have to specify
Definingthelocalpropermassandangularmomentum
the three equations of state T(M,A,J), p(M,A,J), and
of the shell by M = 2πRσ and J = 2πRj, respectively,
Ω(M,A,J), namely, the temperature, pressure, and an-
and using Eqs. (23) and (24), we obtain
gularvelocityequationsofstate,respectively. Toproceed
r2 in this direction, define the inverse local temperature of
M = 2πRσ = + , (26) the shell as
4GℓR
r2 1
J = 2πRj = + . (27) β = . (31)
4Gℓ T
5
Note further that in a (2+1)-dimensional spacetime the where
area A of the shell, actually a perimeter in common us-
r β
+
age, is given by Eq. (10) so that it is mathematically s(r+)= 1 V , (39)
2GℓR −
equivalent to the position R except for the trivial factor
(cid:0) (cid:1)
2π, i.e., we can make use of the variable R instead of A. is actually the integrability condition for Eq. (37) and
UsingEq.(31),thefirstlawofthermodynamics(30)now where β and V are arbitrary functions of (r+,R), but
reads in the (M,R,J) variables s(r+) is an arbitrary function of r+ alone. Thus, the
relation(38)indicatesthattheentropyS oftheshellisa
dS =βdM +2πβpdR βΩdJ. (32) function of the gravitational radius of the shell r only,
+
−
S =S(r ), R>r , (40)
We need to give the equations of state for β(M,R,J), + +
p(M,R,J), and Ω(M,R,J).
wherewehavesetaconstantofintegrationtozero. Since
one can trade r for A trivially through Eq. (8), we
+ +
write Eq. (40) in the more visual form
B. Entropy of an extremal rotating thin shell in a
BTZ spacetime S =S(A ), A>A . (41)
+ +
Dependingonthechoiceofs(r )inEq.(39),theentropy
We can make progress using first, for the time being, +
S(r ), orequivalentlyS(A ),oftherotatingshellinthe
the equation of state for the pressure p. Through the + +
BTZ spacetime can take a wide range of values.
junction condition, i.e., through the spacetime mechan-
Now, in the variables (r ,R) (or if one prefers,
ics, p is indeed fixed by Eq. (25). Changing to the vari- +
(A ,A)), which are the natural variables for calculation
ables M and R, and using Eqs. (25) and (26), valid for +
inthisproblem,onehastworemainingfunctionsofstate,
anextremalshell,onefindsthattheequationofstatefor
β(r ,R) and V(r ,R). These two functions of state,
the pressure p can be written as + +
β(r ,R)andV(r ,R),arearbitraryaslongastheyobey
+ +
the thermodynamicconstraint(39). The extremalrotat-
M
p(M,R)= . (33) ing shell is in this sense quite special.
2πR
Also, one can still take advantage of Eqs. (26) and (27).
Through these equations, we obtain C. Equations of state for the inverse temperature β
and for the rotational velocity V
J =MR. (34)
1. Inverse temperature β equation of state
Equation (34) gives that J, M, and R are not indepen-
dent, with dJ = RdM +MdR. Putting Eqs. (33) and Now,weprepareagivenextremalshellatagenericra-
(34) intothe firstlaw Eq.(32),we obtainthe differential dius R. The unique integrability condition (39) is quite
of entropy dS = β 1 ΩR d(MR). It is then useful to general and does not impose per se a restriction on the
R −
define the the thermodynamic rotational velocity of the temperature distribution. However,thermodynamic sys-
(cid:0) (cid:1)
shell V by tems have to obey the Tolman formula for the tempera-
ture or, equivalently, for the inverse local temperature.
V =ΩR, 0 V 1, (35) The Tolman temperature formula, by its very mean-
≤ ≤
ing, states that the coordinate dependence of the tem-
where the constraint V 1 ensures that the maximum
perature obeys some restriction. The Tolman temper-
≤
velocity is the velocity of light. Then, the entropy of the
ature formula for spherical systems of the type we are
extremal rotating shell obeys
using here is T(r) = T /k(r) where the dependence on
0
r, the local radial coordinate, exists only in the redshift
β
dS = 1 V d(MR). (36) factor k, see Appendix A for a thorough discussion on
R −
the Tolman formula for shells. The quantity T may de-
0
(cid:0) (cid:1)
r2 pend on the parameters of the system. In our case, the
Clearly, from Eq. (26), i.e., MR = + , it is natural
4Gℓ gravitationalradius r+ and the radius of the shell R are
to pass from the variables MR to the variable r . In
+ examples of such parameters, such that T = T (r ,R).
0 0 +
this variable r , the differential of entropy Eq. (36) can
+ For a given shell’s position, fixed R, we can consider the
further be reduced to
Tolmanformulainthewholeouterspace. Forsomeother
R, the configuration changes, and for that new configu-
r β
+
dS = 1 V dr+. (37) ration we can consider again the Tolman formula in the
2GℓR −
new whole outer space. More precisely, suppose that we
(cid:0) (cid:1)
It is manifest that Eq. (37) has to be written as havea shell atposition R with intrinsic gravitationalra-
dius r at some given temperature. The local tempera-
+
dS =s(r )dr , (38) ture of the spacetime at some specific radius r is T, say.
+ +
6
The Tolman formula relates the local temperature T to b = b(r ), and b is a function of r alone [13]. On the
+ +
a constant temperature parameter T . In the case under otherhand,whenonehasanextremalshellabinitio,one
0
discussion, T has the meaning of the local temperature finds b = b(r ,R), see Eq. (46), i.e., in this more com-
0 +
at r = r , where k(r ,r ) = 1 (see Eq. (13)). In brief, prehensivecase, bis afunctionnotonlyofr but alsoof
0 0 + +
theTolmanformulastatesthatT atr,T(r),isafunction R.
ofatemperatureT ,attheradiusr wherek(r ,r )=1. The difference comes of course from the different in-
0 0 0 +
T isitselfafunctionofthecharacteristicsofthesystem, tegrability conditions arising in the nonextremal and
0
r and R in our case, extremal cases. For an ab initio extremal shell, the
+
only integrability condition (39) is too general and gives
T0 =T0(r+,R). (42) b = b(r+,R) as in Eq. (46). For a nonextremal shell,
the threeintegrabiltyconditionsareveryrestrictive,and
Thus, dividing by the redshift function at r given in
when one takes the extremal limit, the memory of this
Eq. (12), the Tolman temperature formula in full is
restrictiveness remains, so b=b(r ).
+
T (r ,R)
0 +
T(r ,R,r)= . (43)
+
k(r ,r)
+ 2. Rotational velocity V equation of state
Inverting this equation, and using Eq. (31), yields the
required Tolman formula for the inverse temperature β, With the choice for the inverse temperature equation
i.e., of state (46), we find from Eq. (39) that the rotational
velocity of the shell has the functional form
β(r ,R,r)=b(r ,R)k(r ,r), (44)
+ + +
R r2
where V(r ,R)= g(r ,R) + , (47)
+ ℓk(r ,R) + − R2
+
1 (cid:16) (cid:17)
b(r ,R)= . (45)
+ T (r ,R) where we have defined
0 +
Inthisway,oneinterpretsb(r ,R)astheinversetemper- 2Gℓ2
+ g(r ,R)=1 s(r ). (48)
+ +
atureattheradiusr =r0 forwhichk =1. Conversely,β − r+b(r+,R)
is the inverse temperature at r, blueshifted or redshifted
with factor k from the inverse temperature at the posi- Foranabinitioextremalshell,g hasadependenceonr+
tion where k =1. as well as on R. For a nonextremal shell and taking the
Now,onthe shell, r =R,sothe Tolmanformulathere limit to the extremal shell [13] the corresponding func-
isβ(r+,R,r=R)=b(r+,R)k(r+,r =R),orsimplifying tion depends only on r+. As in the equation of state for
the notation, the inversetemperature, this comes fromthe verydiffer-
ent integrability conditions in each case. Note that g in
β(r ,R)=b(r ,R)k(r ,R). (46) Eq.(48)correspondstocinEq.(59)of[13]withg =cr2,
+ + + +
but now here g in general has the dependence on R as
If the shell happens to be at R = R0, where R0 = 2ℓ + wellas r+ due to the different integrability conditions as
2 discussed.
ℓ +r2, then k(r ,R ) = 1, see Eq. (15), and so
2 + + 0 With the definition of g in Eq. (48), we have from
r
the(cid:16)re(cid:17)β(r ,R )=b(r ,R ). Eq. (39) the useful formula
+ 0 + 0
It is important to note that the formula (46) for
r
ab initio extremal shells is more comprehensive, and s(r )= + b(r ,R) 1 g(r ,R) , (49)
so different, than the one found for extremal shells + 2Gℓ2 + − +
formed from taking the limit of nonextremal shells (cid:0) (cid:1)
which shows that, although b and g are functions of r
+
[13]. In [13] it was found that for nonextremal
andR,theircombinationisafunctionofr alone. Also,
+
shells, with radius R and gravitational and Cauchy
from the definition of g in Eq. (48), we have another
radii r and r , the following Tolman equation at the
+ − useful formula,
shell’s radius R, found from the integrability conditions,
holds, i.e., β(r+,r−,R) = b(r+,r−)k(r+,r−,R), where R
V r ,R =1 1 g(r ,R) . (50)
k r+,r−,R = Rℓ 1− Rr+22 1− Rr−22 is the redshift (cid:0) + (cid:1) − ℓk(r+,R)(cid:0) − + (cid:1)
r
fu(cid:0)nction in(cid:1)the none(cid:16)xtremal(cid:17)c(cid:16)ase. Ta(cid:17)king then, from We see that the velocity V 1 when g 1, i.e., when
tahnednnoonteixntgretmhaatl skherl+l,,trh−e,Rextr=emkalrs+he,lRl liminittrh+is→limr−it tbh→ere∞isaaccdoirrdecintgretmo Earqk.a(b4l8e)→i,notrerTe0st→ing0r.e→Ilnattiohnisbreestwpeecetn,
where k = k r ,R is given in Eq. (14), one finds the unattainability of the absolute zero of temperature
+ (cid:0) (cid:1) (cid:0) (cid:1)
β(r ,R) = b(r )k(r ,R) [13]. Note the difference: the and the impossibility for a material body to reach the
+ + +
(cid:0) (cid:1)
limit of a nonextremal shell to an extremal shell gives velocity of light.
7
D. Explicit computation of the entropy of the shell Let us then send the extremal shell to its own gravi-
tational radius R = r . In doing so we are taking the
+
extremal black hole limit. Since the entropy differential
For the explicit computation of the entropy S of the
for the shell depends only on r through the function
shell, see Eq. (40) (or Eq. (41)), we have to specify +
s(r ) that is arbitrary, see Eq. (39), we see that the en-
the equationsb(r ,R)andg(r ,R)whichdeterminethe +
+ +
tropy of the extremal shell in the extremal black hole
thermodynamic properties of the shell. In this paper,
limit is given by
wedonotproceedinthiswaybut,instead,focusonEqs.
(46)and(47)andstudytheparticularcasesforwhichwe
S =S(r ), R=r , (51)
can take the limit to the extremal black hole, R r . + +
+
→
or, in terms of the horizon area if one prefers,
IV. ENTROPY IN THE EXTREMAL BTZ S =S(A ), A=A . (52)
+ +
BLACK HOLE LIMIT: EXTREMAL THIN SHELL
WITH ZERO LOCAL TEMPERATURE T AT THE Thisistheextremalblackholelimitofanextremalshell.
GRAVITATIONAL RADIUS This type of configuration, a matter system at its own
gravitational radius, is called a quasiblack hole [23, 24].
We willnowstudy the extremalblackholelimitinthe Thus the entropy of the extremal black hole can be any
sense that we take quasistatically the extremal shell to well-behavedfunctionofr ,orA ,whichdependsonthe
+ +
its own gravitationalradius, R=r . constitutionofmatterthatcollapsestoformtheextremal
+
In this procedure of going quasistatically to the grav- BTZ black hole. Depending on the choice of s(r+), in
itational radius R = r+, we have to prepare in advance turn of β and V, we can obtain any function of r+, or
the shell. In first place, we put the shell at some radius A+, for the entropy S of the extremal black hole. This
R>r andinadditionchoosethe functionsβ andV,or result is quite different from the nonextremal case dis-
+
b and g, in an appropriate manner. After doing this, we cussed in [13], where the entropy of the shell for which
take the shell to R=r . In second place, we stipulate b the temperature coincides with the Hawking tempera-
+
andsoβ. WeknowthattheHawkingtemperatureT for ture can only take the form of the Bekenstein-Hawking
H
aBTZblackholeismeasuredatr0,i.e.,T0 =TH,seee.g., entropy S(A+)= A4G+.
[6]. For an extremal black hole, this temperature is zero Our preceding calculations and discussion were exact.
TH = 0. We assume that the equality T0 = TH is valid Now, we can speculate on ways to constrain the entropy
for our shell since otherwise the backreaction of quan- function S(A ) for the extremal black hole. For the
+
tum fields would destroy it when the shell approaches nonextremal black holes, the entropy is S(A ) = A+.
+ 4G
the horizon. Now, the temperature T0 is precisely re- This expression is found when one takes the shell to its
lated to our b, b(r+,R)=1/T0(r+,R), see Eq.(45). But owngravitationalradiusandassumesthattheshelltakes
sinceT0 =TH =0,wehave,forashellatradiusR,toset the Hawking temperature. In this case, the pressure at
b= . Thus,fromEq.(44),β(r+,R,r)= andinpar- the shell blows up, p [12, 13]. This blowing up of
ticul∞ar β(r+,R,r = R) β(r+,R) = . T∞he tempera- the pressure can be in→ter∞preted as the excitation of all
≡ ∞
tureattheshelliszero. Wecannowchangetheradiusof possible degrees of freedom and the corresponding black
the shell quasistatically, and the same rationale applies, hole takes the Bekenstein-Hawking entropy, the maxi-
since we always want T0 =TH =0, i.e., b= . In third mum possible entropy. Taking the extremal limit from a
∞
place,wefindgandV. FromEq.(49),wefindthatwhen nonextremal black hole, one finds that in this particular
b = then g = 1. Let us suppose that we start with limit the extremal black hole entropy is the Bekenstein-
∞
a configuration in which g is not equal to 1 exactly and Hawking entropy [12, 13] (see also [23, 24]). Thus, this
b is large but finite. Then, 1 g = s(2Gℓ2/r+)/b, for suggests that the maximum entropy that an extremal
−
some well-specified s. We are interested in the limit in black hole can take is the Bekenstein-Hawking entropy.
which b , g 1, V 1. To trace in more detail Therefore, in this sense, the range of values for the en-
→ ∞ → →
this limit, we can choose g as close to 1 as we want and tropy of an extremal black hole is
T as small as we like, i.e., b as large as we like. In the
0
πr
end, keeping the product fixed in Eq. (49) (for a given +
0 S(r ) , (53)
+
r ), we can take the limit of g to 1 and T to zero, i.e., ≤ ≤ 2G
+ 0
b to infinity. We see that the shell at R > r has been
+ or, in terms of A ,
prepared with T = 0, i.e., b = , and g = 1, such that +
0
β2Gr+=ℓ2b 1a−ndgV==s1a.nSdinscoe 2VrG+ℓ=Rβ∞11, t−heVshe=llsr,otwatitehs wthiuths 0 S(A+) A+ . (54)
∞(cid:0) (cid:1) (cid:0) (cid:1) ≤ ≤ 4G
the velocity of light precisely in this limit. The shell is
now correctly prepared. Having made the correctprepa- The result (51), or equivalently (52), is a quite sim-
rationsonthe shell,andaslongasR r andimposing ilar result to the case of the extremal charged shell in
+
≥
that Eq. (49) is always obeyed for some fixed s(r ), we a (3+1) Reissner-Nordstro¨m spacetime [21, 22]. As for
+
can send it to its gravitationalradius r . the extremal electrically charged shells [21, 22], we then
+
8
constrained the entropy function S(r ), or S(A ), for Hawking entropy S(r ) = A+ as the maximal entropy.
+ + + 4G
the extremal black hole. For the nonextremal Reissner- This suggests that the range of values for the entropy of
Nordstro¨m black holes, the entropy is given by the the extremalblack hole in the (3+1)dimensions is given
Bekenstein-Hawkingformula. Inthiscase,whentheshell by Eq.(54) [21], as in the case of the rotating black hole
is taken to its own gravitational radius, the pressure at in the (2+1) dimensions considered here.
the shell diverges, p as k−1 (see (54) of [13]), and
→ ∞
the spacetime is assumed to take the Hawking temper-
ature. Thus all possible degrees of freedom on the shell In Table I, we summarize the thermodynamic proper-
are excited and the black hole formed as the limit of ties of the extremal thin shell at its own gravitational
the shellto its gravitationalradiustakesthe Bekenstein- radius with zero local limiting temperature.
T0 b T β Backreaction V Entropy
A
0 ∞ 0 ∞ Finite 1 0≤S(A+)≤ 4G+
TABLE I: The extremal shell with zero local temperatureat its own gravitational radius.
So, we have found through a thin shell approach that where T¯ is independent of R. Then, from Eq. (43), the
0
the(2+1)-dimensionalextremalrotatingBTZblackhole temperature T is T(r ,R,r) = T¯ (r )k(r+,R). At the
has an entropy S = S(A ) as we had found for the + 0 + k(r+,r)
+ shell r =R, one gets
(3+1)-dimensional extremal Reissner-Nordstro¨m electri-
cally charge black hole [21, 22], and again suggested T(r ,R)=T¯ (r ). (56)
+ 0 +
0 S(A ) A+, see Eq. (54). The extremal black hole
en≤tropy +was≤di4sGcussed originally in a (3+1)-dimensional In terms of the inverse temperatures Eq. (55) translates
black hole context. It was found in [26] that the en- into
tropy of an extremal (3+1)-dimensional black hole is ¯b(r )
+
zero,S =0,Eq.(1). This proposalwassubstantiatedby b(r ,R)= , (57)
+
k(r ,R)
topologyarguments. The(2+1)-dimensionalmetricdoes +
not containone of the angle coordinateswhen compared
where
to the (3+1)-dimensional metric, but this is unessential
1
since the main arguments concern the topology of the ¯b(r ) , (58)
(τ,r) submanifold, where τ it is the Euclidean time, + ≡ T¯0(r+)
≡
see also [3] for (2+1)-dimensional BTZ black hole. This
is independent of R. Then, from Eq. (44), the local in-
reasoningis purely classical,andinclusionof the backre-
actionduetoquantumfieldscandestroythispicture. On verse temperature β at r is β(r+,R,r) = ¯b(r+)kk((rr++,,Rr)).
theotherhand,theproposalputforwardbystringtheory Likewise, at the shell, r=R, one gets
leads to S = A+, Eq. (2), i.e., to a Bekenstein-Hawking
entropy for ex4tGremal black hole [27, 28], see [2] for the β(r+,R)=¯b(r+) (59)
BTZ black hole. Our conclusion that 0 S(A ) A+,
≤ + ≤ 4G is constant and finite.
seeEq.(54),incorporatesboththeS =0andtheS = A+ Combining (48) with (57), we find that the rotational
4G
results. velocity equation of state can be written as
1 g(r ,R)=k(r ,R)(1 g¯(r )), (60)
+ + +
− −
V. ENTROPY IN THE EXTREMAL BTZ
BLACK HOLE LIMIT: EXTREMAL THIN SHELL where we have defined
WITH NONZERO LOCAL TEMPERATURE T AT
2Gℓ2
THE GRAVITATIONAL RADIUS
g¯(r ) 1 s(r ), (61)
+ ≡ − r ¯b(r ) +
+ +
Now, we take again the extremal black hole limit but
which is independent of R and is assumed to be in the
assume that the extremal shell has another equation of
range 0<g¯(r )<1.
state with a nonzero local temperature. Specifically, at +
We now take the limit to the gravitational radius of
anyr >r , we considerthe following temperatureequa-
+ the shell, R r . In this limit, k 0. So, suppos-
tion of state, ing T¯ finite,→whi+ch we do, we see fr→om Eq. (55) that
0
T (r ,R)=T¯ (r )k(r ,R), (55) T (r ,R) = 0. Since T¯ is finite, the local temperature
0 + 0 + + 0 + 0
9
T at the shell is nonzero but finite, since T = T¯ from the entropy of this extremal shell in the extremal BTZ
0
Eq. (56), and hence the quantum backreaction remains black hole limit also obeys
finite even when the shell is taken to its gravitational
radius R = r . Since T is finite β given in Eq. (59) is
+ A
+
alsofinite, andfromEq.(39), wefind thatthe shellwith 0 S(A+) . (62)
≤ ≤ 4G
nonzero local temperature rotates with thermal velocity
less than the velocity of light V < 1. Finally, the en-
tropyoftheshellobtainedbyintegratingEq.(38)iswell In Table II, we summarize the thermodynamic prop-
behaved and can take any function of r , or A . Thus, erties of the extremal thin shell at its own gravitational
+ +
using the same arguments as before, we can write that radius with a local temperature T =T¯ .
0
T0 b T β Backreaction V Entropy
A
0 ∞ Nonzero and Finite finite <1 0≤S(A+)≤ 4G+
TABLE II: The extremal shell with nonzero local temperature at its own gravitational radius.
VI. DISCUSSION ON THE ANGULAR, AND From Eqs.(65) and (66) one deduces that an observer
THE CORRESPONDING LINEAR, VELOCITIES comoving with the shell has ψ = constant. Another ob-
OF ROTATING THIN SHELLS serveron the shell moving with respect to this comoving
observerhasangularvelocityω¯ withrespecttotheproper
A. Mechanical angular velocities time on the shell τ given by
dψ
It is instructive to rewrite the formula for the line ele- ω¯ = . (67)
dτ
ment outside of the shell (see Eqs. (3) and (4)) as
Thissameobserverhasanangularvelocityωwithrespect
ds2(o) =−f(o)(r)dt2(o)+g(o)(r)dr2+r2 dφ−ω(o)(r)dt(o) 2. to t(o) given by ω = ddt(ψo) = ddt(τo)ddψτ = kω¯, where here
(63) k k(r ,R) to simplify the notation, i.e.,
(cid:0) (cid:1) +
Now, define ≡
dψ
ω = =kω¯. (68)
ωR ≡ω(o)(R). (64) dt(o)
Static observers sitting at infinity, r = , have an AdS
Now, from Eq. (63), the coordinate φ is the angular co-
∞
metric, since the BTZ metric turns into an asymptoti-
ordinate defined at infinity. Define then the angular ve-
cally AdS metric at infinity. These observers do not ro-
locity ω of an observer on the shell as seen by the co-
∞
tate relative to this AdS spacetime. Thus, observers sit-
ordinate φ and in terms of t as
(o)
ting at infinity see a rotation of the shell with ω . Here
R
we keep the discussion quite general, for the extremal dφ
r2 ω∞ = . (69)
BTZ case ω is given in Eq. (19), i.e., ω = + . dt
R R ℓR2 (o)
At the shell r =R, the metric (63) becomes
We can now give a relation between ω, ω , and ω ,
∞ R
ds2R =−dτ2+R2dψ2, R=R(τ), (65) or between ω¯, ω∞, ωR, and k. Clearly, from Eq. (66),
where τ is the proper time at the shell and in terms of dψ dφ
= ω , (70)
R
dt is dτ = f (R)dt = k(r ,R)dt (see also dt dt −
(o) (o) (o) + (o) (o) (o)
(14)), and we have chosen a new angular coordinate ψ,
p so that, using Eqs. (68) and (69) in (70), we find
ψ =φ ω t , (66)
R (o)
− ω =ω ω . (71)
∞ R
−
such that the metric is displayed as diagonal (see also
Eq. (16)). The angular velocity ω defined in Eq. (64), If we prefer to use ω¯, i.e., to use the proper time coor-
R
appears thus quite naturally in Eq. (66) and is one of dinate τ, then from the help of Eqs. (67) and (68) we
a number of interesting mechanical angular velocities in find
this problem. Letusdisplaytheothers,thatwename ω¯, ω ω
∞ R
ω, and ω . ω¯ = − . (72)
∞ k
10
Expression(72) is quite general. One canspecialize. For On the other hand, the quantity ω¯ represents the an-
instance, the special choice of ω¯ for which a shell’s ob- gular velocity of the effective perfect fluid that fills the
server detects no angular momentum flux density was shell, i.e., it is obtained from geometry and mechanics,
given by Eq. (39) of [18], i.e., ω¯ = r− R2−r+2 . Note namely, by gluing two metrics, the BTZ metric and the
r+R R2−r−2 zero mass BTZ metric on the different sides of the shell,
r
thatω is annotatedasΩ in[18], the angularvelocityof calculating the corresponding energy-momentum tensor
R
the shell with respect to infinity. We use here the nota- anddeterminingtheangularvelocityoftheeffectivefluid
tion Ω for a quite different quantity, the thermodynamic in terms of which the vanishing angular momentum flux
angular velocity on the shell, see Eq. (30). The linear is observed[18]. The samerationaleappliesofcoursefor
velocities corresponding to ω¯, ω, and ω are v¯ = ω¯R, thecorrespondinglinearvelocities,V =ΩRandv¯=ω¯R,
∞
v = ωR, and v = ω R. Angular and linear velocities i.e., they are conceptually different.
∞ ∞
share the same properties. So, a priori, it is not obvious at all, whether or not,
It is also instructive and important to point out the and under what conditions, these two velocities can be
analogywiththeblackholecase. Inthiscontext,Eq.(72) identified. It is thus instructive to compare this issue for
is similar to the expression for the angular velocity of extremal and nonextremal shells.
the heat bath surrounding a (2+1)-dimensional rotating
black hole,
2. Extremal shell
ω ω
bh zamo
ω = − , (73)
hb k For an extremal shell the situation is interesting and
different from what one would expect.
where ωhb is the heat bath angular velocity, ωbh is the Indeed, the two velocities Ω and ω¯, or V and v¯, do
black hole angular velocity, ωzamo is the angular veloc- not need to coincide at all, as the integrability condition
ity of a zero angular momentum observer (ZAMO), and (39) does not restrict the form of the function Ω in the
k in this formula is the redshift function at the ZAMO extremal case. We will explain this fact now.
radius, see Eq. (13) of [5]. So, at a first glance, one may The thermodynamic angular velocity Ω presents new
identify ω¯ ωhb, ω∞ ωbh, and ωR ωzamo. There features. As it is argued above in Sec. IV, when T = 0,
≡ ≡ ≡
is, however, an important difference between Eqs. (72) then we must select
and (73). For a black hole, the quantity ω enters both
hb 1 1
the mechanical and thermodynamic relations (see [5] for Ω= , T = =0, (74)
R β
details), so the mechanical and thermodynamic angular
velocities coincide. However, for a shell the relationship so that,
between both angular velocities is much more subtle as
we now discuss. V =1, (75)
in the extremal limit under discussion, to ensure the
finiteness of s(r ). This is the extremal black hole limit
+
B. Mechanical and thermodynamic angular and from an extremal thin shell with zero local temperature
linear velocities for a rotating shell: The
T, see also Table I. However, when T = 0, see Sec. V,
nonextremal and extremal cases 6
any thermodynamic angular velocity Ω obeying
1 1
1. The problem Ω< , T = =0, (76)
R β 6
Considerationofthermalshellsin(2+1)dimensionsin i.e.,
the present paper as well in previous ones [18] (see also
V <1, (77)
[13])revealedinterestingsubtleties otherwisehidden. At
first sight, the shell thermodynamic angular velocity Ω issuitableprovidedthatT =β−1 remainsnonzerointhe
thatappearsinthefirstlawofthermodynamics,Eq.(30),
extremal black hole limit R r , see Eq. (39). This is
+
should be immediately identified with the shell mechan- →
the extremalblackholelimit fromanextremalthinshell
ical angular velocity of a rotating shell fluid ω¯, Eq. (72),
with nonzero local temperature T, see also Table II.
inthesamewayasthequantityω isboththemechani-
hb Themechanicalangularvelocityω¯,onthe otherhand,
calandthermodynamicangularvelocityintheblackhole
is given by
case, as discussed above (see [5]).
However,the two angular velocities, namely, the ther- 1
ω¯ = , (78)
modyamic angular velocity Ω and the mechanical veloc- R
ity ω¯, are indeed conceptually different, they have dif-
as is seen from Eq. (82). As a result, the linear mechan-
ferent physical meanings. The quantity Ω is the quan-
ical velocity, v¯=ω¯R, is
tity ascribed to a thermodynamic system as a whole, it
is calculated from a pure thermodynamic approach [13]. v¯=1, (79)