Black Hole Thermodynamics and Hamilton-Jacobi Counterterm

0.19 MB·English

by  Luzi Bergamin

MIT–CTP–3887
MPP–2007–153
Black Hole Thermodynamics and Hamilton-Jacobi
Counterterm
8 Luzi Bergamin
0
0 ESAAdvancedConcepts Team,ESTEC,Keplerlaan1,2201AZNoordwijk,The
Netherlands
2
E-mail: [email protected]
n
a
J
Daniel Grumiller
4
Massachusetts Institute ofTechnology, 77Massachusetts Ave,Cambridge,MA

02139, USA
]
h
E-mail: [email protected]
t
-
p
e Robert McNees
h
PerimeterInstituteforTheoreticalPhysics,31CarolineStreetNorth,Waterloo,
[
OntarioN2L2Y5,Canada

2 E-mail: [email protected]
v
0
4 Ren´e Meyer
1
MaxPlanckInstitutfu¨rPhysik,Fo¨hringerRing6,80805Mu¨nchen,Germany
4
. E-mail: [email protected]
0
1
7 Abstract. We review the construction of the universal Hamilton-Jacobi
0 countertermfordilatongravityintwodimensions,derivethecorrespondingresult
: intheCartanformulationandelaboratefurtheruponblackholethermodynamics
v and semi-classical corrections. Applications include spherically symmetric black
i holesinarbitrarydimensionswithMinkowski-orAdS-asymptotics,theBTZblack
X
holeandblackholesintwo-dimensionalstringtheory.
r
a
1. Introduction
There are numerous applications in physics where an action
I [φ]= dnx (φ, φ) (1)
bulk bulk
L ∇
ZM
has to be supplemented by boundary terms
I [φ]=I [φ]+ dn−1x (φ, φ, φ). (2)
tot bulk boundary ⊥ k
L ∇ ∇
Z∂M
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 2
Here and denote the normal and parallel components of the derivative with
⊥ k
∇ ∇
respect to the boundary ∂ . The simplest example is quantum mechanics, where
M
t1
I [q,p]= dt [ qp˙ H(q,p)] (3)
bulk
− −
Zt0
has to be supplemented by a boundary term
I [q,p]=I [q,p]+qpt1 (4)
tot bulk |t0
ifDirichletboundaryconditionsareimposedonthe coordinate,δq =0. Inaddition
|ti
to this “Gibbons-Hawking-York”boundaryterm one canaddanother boundary term
Γ[q,p]=I [q,p] (t,q)t1 (5)
tot −F |t0
which depends only on quantities held fixed at the boundary. This seems to be
a superfluous addition, as it does neither change the equations of motion nor the
variational principle (as opposed to the “Gibbons-Hawking-York” boundary term),
but in some applications such a term is crucial and determined almost uniquely from
consistencyrequirements: symmetriesandaccessibilityoftheclassicalapproximation.
One such application is the Euclidean path integral for black holes (BHs), which
provides a convenient shortcut to BH thermodynamics.
2. Hamilton-Jacobi counterterm in two-dimensional gravity
Theessentialfeaturesanddifficultiesarisealreadyinlowdimensions. Fortransparency
we focus ontwo-dimensional(2D) models. The bulk actionfor 2D dilatongravity[1],
1
I [g,X]= d2x√g XR U(X) ( X)2 2V(X) (6)
bulk
−16πG − ∇ −
2 ZM h i
has to be supplemented by a Gibbons-Hawking-York boundary term
1
I [g,X]=I [g,X] dx√γXK , (7)
tot bulk
− 8πG
2 Z∂M
if Dirichlet boundary conditions are imposed on the dilaton field X and the induced
metric at the boundary γ. The meaning of all symbols is standard and our notation
is consistent with [2]. In addition one could add another boundary term
1
Γ[g,X]=I [g,X] dx√γ (X, X,γ, γ) . (8)
tot − 8πG F ∇k ∇k
2 Z∂M
ICT[γ,X]
| {z }
We demonstrate now why such a term is needed and show that it is determined
essentially uniquely from consistency requirements: symmetries and accessibility of
the classical approximation.
2.1. Symmetries
Diffeomorphism covariance along the boundary requires that in (8) transforms as
F
a scalar. Since there are no scalar invariants constructed from γ in one dimension,
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 3
can be reduced to = (X, X). Another simplification arises if we restrict
k
F F F ∇
ourselves to isosurfaces of the dilaton field, which is sufficient for our purposes. Then
X is constant along the boundary, X =0, so that we are left with a function
k
∇
= (X). (9)
F F
Symmetry requirements have reduced the dependence on four variables in (8) to a
dependence on only one variable, X.
2.2. Accessibility of the classical approximation
While symmetrieshelptoreducetheambiguitiesin theydonotexplainwhysucha
F
term is needed inthe firstplace. To this endwe consider the Euclideanpath integral,
1
= DgDX exp I[g,X] . (10)
Z − ~
Z (cid:16) (cid:17)
The path integral is evaluated by imposing boundary conditions on the fields and
then performing the weighted sum over all relevant space-times ( ,g) and dilaton
M
configurations X. In the classical limit it is dominated by contributions from
stationarypointsoftheaction. Thiscanbeverifiedbyexpandingitaroundaclassical
solution
1
I[g +δg,X +δX]=I[g ,X ]+δI[g ,X ;δg,δX]+ δ2I[g ,X ;δg,δX]+... (11)
cl cl cl cl cl cl cl cl
2
where δI and δ2I are the linear and quadratic terms in the Taylor expansion. The
saddle point approximation of the path integral
1 1
exp I[g ,X ] DδgDδX exp δ2I[g ,X ;δg,δX] (12)
Z ∼ − ~ cl cl − 2~ cl cl
(cid:16) (cid:17) Z (cid:16) (cid:17)
is defined if:
(i) The on-shell action is bounded from below, I[g ,X ]> .
cl cl
−∞
(ii) The first variation vanishes on-shell, δI[g ,X ;δg,δX] = 0 for all variations δg
cl cl
and δX preserving the boundary conditions.
(iii) ThesecondvariationhasthecorrectsignforconvergenceoftheGaussianin(12).
Thelastconditionactuallymeansconsistencyofthesemi-classicalapproximation,and
we shall not discuss it here. Instead, we focus on the first two conditions. Both are
violated for typical BH solutions of (7) if the boundary is located in the asymptotic
region X :
→∞
(i) The on-shellactionbehavesasI [g ,X ]=2M/T S w(X )/T,
tot cl cl cl
→∞ − − →∞
and lim w(X) for most models of interest.2
X→∞
→∞
(ii) The first variation of the action receives a boundary contribution
δI dx√γ πabδγ +π δX =0 (13)
tot ab X
(cid:12)on−shell ∼Z∂M 6
becausetheproduct(cid:12)(cid:12)πabδγabisnon-vanishi(cid:2)ng: thevariationo(cid:3)ftheinducedmetric
γ does not fall off sufficiently fast to compensate for the divergence of the
ab
momenta πab.
2 T istheHawkingtemperatureandS theBekenstein-Hawkingentropy. Botharedeterminedfrom
themassM andthefunctionsQ(X)andw(X)definedintheAppendix.
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 4
We emphasize that an ad-hoc subtraction I [g ,X ] := I [g ,X
ren cl cl tot cl cl
→ ∞ →
]+w(X )/T is inconsistent: while it leads to a finite on-shell action it does
cl
∞ → ∞
not address the second problem. Both can be solved by choosing (X) adequately.
F
Since the on-shell action solves the Hamilton-Jacobi equation one can expect
cancellationsifalso isasolutiontotheHamilton-Jacobiequation. Thus,ourguiding
F
principle is to demand that be a solution of the Hamilton-Jacobi equation (see the
F
nextSectionfordetails). ThismethodwasappliedfirsttotheWittenBHandtotype
0Astringtheory[3]andlatergeneralizedto generic2Ddilatongravity[2]. The result
is
(X)= (w(X)+c)e−Q(X) (14)
F −
q
where c is an integration constant. It can be absorbed into a redefinition of w
(cf. Appendix) and reflects the freedom to choose the ground state of the system.
Thus, without loss of generality we can set it to zero and finally obtain a consistent
action [2]
1
Γ[g,X]= d2x√g XR U(X) ( X)2 2V(X)
−16πG − ∇ −
2 ZM h i (15)
1 1
dx√γXK+ dx√γ w(X)e−Q(X).
− 8πG 8πG
2 Z∂M 2 Z∂M q
The classical approximation is now well-defined because
(i) Γ[g ,X ]=M/T S is finite.
cl cl
→∞ −
(ii) δΓ[g ,X ;δg,δX]=0 for all δg and δX preserving the boundary conditions.
cl cl
Moreover, as opposed to (7) the action (15) is consistent with the first law of
thermodynamics. Perhapsthemostremarkablepropertyofthe countertermin(15)is
its universality: while usually different subtraction methods are employed depending
on whether spacetime is asymptotically flat, AdS or neither of both, our result is
not sensitive to the asymptotics. This universality does not appear to exist in higher
dimensions or even in 2D if ‘standard’ subtraction methods are used [4].
3. Cartan formulation
In many applications a first order formulation in terms of Cartan variables is
advantageous [1]. Therefore we now derive the Hamilton-Jacobi counterterm in this
formulation. The corresponding action (we set 8πG =1)
2
1
IFO[X,Ya,e ,ω]= YaDe +Xdω+ǫ( U(X)YaY +V(X))
tot a − a 2 a
ZM(cid:20) (cid:21) (16)
i e
k
+ Xω + Xdln ,
k 2 e¯
Z∂M(cid:20) k(cid:21)
contains the Cartan variables ω and e , as well as the scalar fields X and Ya (we
a
use a complexified dyad, e¯ = e∗, cf. [5] for the details of our notation). As (16) is
classically equivalent to (7) (cf. the Appendix of Ref. [5]) it will suffer from the same
problems as described in Section 2.2. We follow the same strategy as in the second
order formulation [2,3] to find the corresponding Hamilton-Jacobi counterterm IFO,
CT
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 5
which, using the arguments in Section 2.1, can be reduced to
ΓFO[X,Ya,e ,ω]=IFO[X,Ya,e ,ω] dxk 2e e¯ (X) . (17)
a tot a − k kF
Z∂M
p
ICFTO[X,eke¯k]
Thevariationoftheactionproducestheequationso|fmotion{pzlusthebo}undaryterm3
dxk Yδe¯ +Y¯δe¯ δXω . (18)
k k k
−
Z∂M
(cid:2) (cid:3)
To cancelit we assignDirichlet boundary conditions to X, e and e¯ . As in [2,3]it is
k k
possible to write the momenta which are not fixed at the boundary,
δIFO δIFO δIFO
ω = tot , Y = tot , Y¯ = tot , (19)
k − δX δe¯ δe
(cid:12)on−shell k (cid:12)on−shell k (cid:12)on−shell
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
as variations of the(cid:12) on-shell action. The Ha(cid:12)milton constraint (cid:12)
Ye¯ +Y¯e
ω k k +U(X)YY¯ +V(X)=0, (20)
− k 2e e¯
k k
follows from a standard constraint analysis of the first order action (16). By
construction the Hamilton-Jacobi counterterm must be a solution of this constraint.
Replacing in (19) the on-shell action IFO by the counterterm IFO, plugging this
tot CT
into the Hamilton constraint (20) and exploiting that IFO depends solely on the
CT
combination e e¯ establishes
k k
δIFO δIFO δIFO 2
CT CT +U(X)e e¯ CT +V(X)=0. (21)
δX δ(e e¯ ) k k δ(e e¯ )
k k (cid:18) k k (cid:19)
This functional differential equation for the counterterm by virtue of the Ansatz (17)
simplifies to
d
2(X)+U(X) 2(X)+2V(X)=0. (22)
dXF F
The solution of this first order ordinary differential equation is given by4
(X)= (w(X)+c)e−Q(X). (23)
F −
q
Thiscoincideswith(14)andthusweconcludethattheHamilton-Jacobicounterterms
in second and first order formalisms are identical, as might have been anticipated on
general grounds. Setting again c=0, the consistent first order action is
1
ΓFO[X,Ya,e ,ω]= YaDe +Xdω+ǫ( U(X)YaY +V(X))
a a a
− 2
ZM(cid:20) (cid:21) (24)
i e
+ Xω + Xdln k + dxk 2e e¯ w(X)e−Q(X).
k 2 e¯ k k
Z∂M(cid:20) k(cid:21) Z∂M q
p
3 Contributions emergingfromthelogarithmintheboundaryterm aredroppedasweassumethat
the boundary is an isosurface of the dilaton and that there is no boundary of the boundary. The
coordinatealongtheboundary,xk,canbethoughtofasEuclideantimeτ.
4 There is a sign ambiguity since (22) yields only F2(X). The sign choice in (23) gives an action
withaconsistentclassicallimit.
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 6
4. Black hole thermodynamics and further applications
An immediate consequence of our result (15) is the Helmholtz free energy [2]
1 2M X
Fc(Tc,Xc)= w(Xc)e−Q(Xc) 1 1 h Tc, (25)
8πG2 −s − w(Xc)!− 4G2
q
which is related to the on-shell action in the usual way, F =T Γ . Here X denotes
c c c c
the value ofthe dilatonfield atthe locationof acavity wallin contactwith a thermal
reservoir, while X denotes the value of the dilaton at the BH horizon. The local
h
temperature T is related to the Hawking temperature T by the standard Tolman
c
factor,T =T/ ξ(X). AllotherquantitiesaredefinedintheAppendix. Theentropy,
c
p
∂F X A
c h h
S = = = , (26)
− ∂T 4G 4G
c(cid:12)Xc 2 eff
(cid:12)
(cid:12)
is in agreement with the Bekenstein-H(cid:12)awking result. Here Ah = 1 because we are in
2D, and G = G /X . For dimensionally reduced models (26) can be interpreted
eff 2 h
also from a higher-dimensional perspective: A X and G G , with the same
h h eff 2
∝ ∝
proportionality constants. The result (26) is well-knownand was obtained by various
methods [6]. However, the free energy (25) contains a lot of additional information
and allows a quasi-local treatment of BH thermodynamics (where applicable in
agreement with [7]), including stability considerations. For an extensive study of
thermodynamical properties and more Refs. we refer to [2].
The class of BHs described by the action (15) or (24) is surprisingly rich
(cf. e.g. table 1 in [8]), and includes spherically symmetric BHs (like Schwarzschild
or Schwarzschild-AdS) in any dimension, spinning BHs in three dimensions [9] and
stringBHsintwodimensions[10,11]. Asanexampleweconsidernowtheexactstring
BH[11],andreviewsomeofitsproperties. Itstarget-spaceaction[12]isgivenby(15)
with 8πG =1 and the potentials
2
ρ
U(X)= , V(X)= 2b2ρ. (27)
−ρ2+2(1+ 1+ρ2) −
Here the canonical dilaton X is relatepd to a new field ρ by
X =ρ+arcsinhρ, (28)
and the parameter b is related to the level k and α′ by α′b2 =1/(k 2). In order for
−
the background following from (15), (27) and (28) to be a solution of string theory
it must satisfy the condition D 26+6α′b2 = 0. Because the target space here is
−
two-dimensional, D = 2, requiring the correct central charge fixes the level at the
critical value k =9/4. Following [13], we vary k by allowing for additional matter
crit
fields that contribute to the total central charge, so that k [2, ) is possible. The
∈ ∞
Witten BH arises in the limit k . Since it is not possible to place an abrupt
→ ∞
cut-offonthe space-timefields instringtheorywehavetoconsiderthe limitX
c
→∞
in the Helmholtz free energy (25),
2
FESBH = b 1 arcsinh k(k 2) (29)
− − k −
r
p
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 7
and its thermodynamical descendants. It is straightforward to show that (29) leads
to a positive specific heat for any k (2, ), and that it vanishes in the limit k 2
∈ ∞ →
in accordance with the third law.
We comment now briefly on the inclusion of semi-classical corrections from
fluctuations ofmasslessmatter fields ona givenBHbackground. We thereforeaddto
the classical action (6) the Polyakovaction,
1
Isemi =I +c d2x√g ψR+ ( ψ)2 , (30)
bulk bulk 2 ∇
ZM (cid:20) (cid:21)
where we have introduced an auxiliary field ψ which fulfills the on-shell relation
(cid:3)ψ = R, and the constant c depends on the number and type of massless matter
fields. Obviously, the addition of (30) requires a reconsideration of boundary issues.
One possibility is to demand that ψ is a function of X [14]. Then the action (30)
reduces to a standard dilaton gravity action (we set 8πG =1)
2
1
Isemi = d2x√g XˆR (U(X)+c(ψ′(X))2)( X)2 2V(X) , (31)
bulk −2 − ∇ −
ZM h i
upon introducing a redefined dilaton Xˆ = X 2cψ(X). Therefore, as long as the
−
assumption ψ = ψ(X) is meaningful, the discussion of boundary terms in Section 2
is still valid. We note in this context that the large X expansion of the exact string
BH(27),(28)canbeinterpretedasasemi-classicalcorrectiontotheWittenBH,with
ρ playing the role of the unperturbed dilaton X and ln(2ρ)/(2c) playing the role
−
of the auxiliary field ψ. This is consistent with the fact that the conformal factor ψ
scaleslogarithmicallywiththedilatonandconcurswithsemi-classicalcorrections[15]
tothe specificheatofthe WittenBH,whichalsoshowqualitativeagreementwiththe
specific heat of the exact string BH. Another, more general, possibility is to treat ψ
as an independent field. In that case boundary issues have to be reconsidered. We
expect them to be relevant whenever the boundary term
dx√γψnµ∂µψ (32)
Z∂M
doesnotfalloffsufficientlyfastattheasymptoticboundary(nµistheoutwardpointing
unit normal). Since ψ typically scales logarithmically with X this happens for all
models where w(X) grows linearly or faster than X. Interestingly, the Witten BH
is precisely the limiting case where this issue is of relevance. More recently semi-
classical corrections were considered in the context of large AdS BHs [16]. There the
issue is complicated because the matter fields couple non-minimally to the dilaton.
Since the results of [16] agree with ours in the large X limit only,5 it would be
interesting to analyze the counterterms using the Hamilton-Jacobi method, possibly
by adapting the strategy described and applied in [17]. This would also allow to
reconsider the path integral quantization of 2D dilaton gravity with matter in the
presenceofboundaries[18]andtoclarifytheroleoftheHamilton-Jacobicounterterm
for observables beyond thermodynamical ones.
For further applications and an outlook to future research we refer to the
discussion in Section 7 of [2].
5 The counterterm, (4.5) in [16], for finite values of r differs from (14), which yields F(r2) ∼
rp1+r2/ℓ2=1/ℓ(r2+ℓ2/2+...).
Black Hole Thermodynamics and Hamilton-Jacobi Counterterm 8
Acknowledgments
The content of this proceedings contribution was presented by one of us at the
conference QFEXT07 in Leipzig, and DG would like to thank Michael Bordag as
well as Boris, Irina, Mischa and Sascha Dobruskin for the kind hospitality.
The work of DG is supported in part by funds provided by the U.S. Department
of Energy (DoE) under the cooperative research agreement DEFG02-05ER41360.
DG has been supported by the Marie Curie Fellowship MC-OIF 021421 of the
European Commission under the Sixth EU Framework Programme for Research
and Technological Development (FP6). The research of RM was supported by
DoE through grant DE-FG02-91ER40688Task A (Brown University) and Perimeter
InstituteforTheoreticalPhysics. ResearchatPerimeterInstituteissupportedthrough
the government of Canada through Industry Canada and by the province of Ontario
through the Ministry of Research and Innovation.
Appendix A. Definitions of w and Q
The classical solutions of the equations of motion
1
X =X(r), ds2 =ξ(r)dτ2+ dr2, (A.1)
ξ(r)
with
2M
∂ X =e−Q(X), ξ(X)=w(X)eQ(X) 1 , (A.2)
r
− w(X)
(cid:18) (cid:19)
are expressed in terms of two model-dependent functions,
X X
Q(X):=Q + dX˜U(X˜), w(X):=w 2 dX˜V(X˜)eQ(X˜). (A.3)
0 0
−
Z Z
Here Q and w are constants, and the integrals are evaluated at X. Notice that w
0 0 0
andtheintegrationconstantM contributetoξ(X)inthesamemanner. Together,they
represent a single parameter that has been partially incorporated into the definition
of w(X). By definition they transform as w e∆Q0w and M e∆Q0M under the
0 0
→ →
shiftQ Q +∆Q . Thisensuresthatthefunctions(A.2)transformhomogeneously,
0 0 0
→
allowingQ tobeabsorbedintoarescalingofthecoordinates. Therefore,thesolution
0
is parameterized by a single constant of integration. With an appropriate choice of
w we can restrict M to take values in the range M 0 for physical solutions. The
0
≥
functionwisinvariantunderdilatondependentWeylrescalingsofthemetric,whereas
Qtransformsinhomogeneously. Allclassicalsolutions(A.1)exhibitaKillingvector∂ .
τ
With Lorentzian signature each solution X of ξ(X) = 0 therefore leads to a Killing
h
horizon. The Hawking temperature is given by the inverse periodicity in Euclidean
time, T =w′(X )/(4π).
h
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