Entanglement Entropy of two-dimensional anti-de Sitter black holes
Mariano Cadoni∗
Dipartimento di Fisica, Università di Cagliari, and INFN sezione di Cagliari,
Cittadella Universitaria 09042 Monserrato, ITALY
Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the
anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole
mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas
the subleading term behaves as lnSBH . This subleading term has the universal form typical for
the entanglement entropy of physical systems described by effective conformal fields theories (e.g.
one-dimensional statistical models at the critical point). The well-known form of the entanglement
entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our
result and is related with the entanglement entropy of a black hole with negative mass.
Quantum entanglement is a fundamental feature of quantum systems. It is related to the existence of correlations
between parts of the system. The degree of entanglement of a quantum system is measured by the entanglement
entropy Sent. In quantum field theory (QFT), or more in general in many body systems, we can localize observable
and unobservable degrees of freedom in spatially separated regions Q and R. Sent is then defined as the von Neumann
entropy of the system when the degrees of freedom in the region R are traced over, Sent = −TrQρ̂Q ln ρ̂Q, where the
trace is taken over states in the observable region Q and the reduced density matrix ρ̂Q = TrRρ̂ is obtained by tracing
the density matrix ρ̂ over states in the region R.
Investigation of the entanglement entropy (EE) has become relevant in many research areas. Apart from quantum
information theory, the field that gave birth to the notion of entanglement entropy, it plays a crucial role in condensed
matter systems, where it helps to understand quantum phases of matter (e.g spin chains and quantum liquids)[1, 2,
3, 4, 5]. Entanglement (geometric) entropy is also an useful concept for investigating general features of QFT, in
particular two-dimensional conformal field theory (CFT) and the Anti-de Sitter/conformal field theory (AdS/CFT)
correspondence [6, 7, 8, 9, 10, 11, 12] . Last but not least entanglement may held the key for unraveling the mystery
of black hole entropy [13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
We will be mainly concerned with the entanglement entropy of two-dimensional (2D) CFT and its relationship with
the entropy of 2D black holes. It is an old idea that black hole entropy may be explained in terms of the EE of the
quantum state of matter fields in the black hole geometry [13]. The main support to this conjecture comes from the
fact that both the EE of matter fields and the Bekenstein-Hawking (BH) entropy depend on the area of the boundary
region. On the other hand any attempt to explain the BH entropy as originating from quantum entanglement has to
solve conceptual and technical difficulties.
The usual statistical paradigm explains the BH entropy in terms of a microstate gas. This is conceptually different
from the EE that measures the observer’s lack of information about the quantum state of the system in a inaccessible
region of spacetime. Moreover, the EE depends both on the number of species ns of the matter fields, whose
entanglement should reproduce the BH entropy, and on the value of the UV cutoff δ arising owing to the presence
of a sharp boundary between the accessible and inaccessible regions of the spacetime. Conversely, the BH entropy is
meant to be universal, hence independent from ns and δ. Some conceptual difficulties can be solved using Sakharov’s
induced gravity approach [23, 24, 25], but the problem of the dependence on ns and δ still remains unsolved.
In this letter we will show that in the case of two-dimensional AdS black hole these difficulties can be completely
solved. We will derive an expression for the black hole EE that in the large black hole mass limit reproduces exactly
the BH entropy. Moreover, we will show that the subleading term has the universal behavior typical for CFTs and in
particular for critical phenomena. The reason of this success is related to the peculiarities of 2D AdS gravity, namely
the existence of an AdS/CFT correspondence and the fact that 2D Newton constant can be considered as wholly
induced by quantum fluctuations of the dual CFT.
Most of the progress in understanding the EE in QFT has been achieved in the case of 2D CFT. Conformal
invariance in two space-time dimension is a powerful tool that allows us to compute the EE in closed form. The
entanglement entropy for the ground state of a 2D CFT originated from tracing over correlations between spacelike
separated points has been calculated by Holzhey, Larsen and Wilckzek [6]. Introducing an infrared cutoff Λ the
spacelike coordinate of our 2D universe will belong to C = [0,Λ[. The subsystem where measurements are performed
is Q = [0,Σ[, whereas the outside region where the degrees of freedom are traced over is R = [Σ,Λ[. Because of the
contribution of localized excitations arbitrarily near to the boundary the entanglement entropy diverges. Introducing
http://arxiv.org/abs/0704.0140v2
an ultraviolet cutoff δ, the regularized entanglement entropy turns out to be [6]
Sent =
c+ c̄
, (1)
where c and c̄ are the central charges of the 2D CFT. The expression (1) emphasizes the characterizing features of
the entanglement entropy, namely subadditivity and invariance under the transformation which exchanges the inside
and outside regions
Σ → Λ− Σ. (2)
Moreover, Sent is not a monotonic function of Σ, but increases and reaches its maximum for Σ = Λ/2 and then
decreases as Σ increases further. This behavior has an obvious explanation. When the subsystem begins to fill most
of the universe there is lesser information to be lost and the entanglement entropy decreases.
Let us now consider 2D AdS black holes. As classical solutions of a 2D gravity theory they are endowed with a
non-constant scalar field, the dilaton Φ. In the Schwarzschild gauge the 2D AdS black hole solutions are [26],
ds2 = −
dt2 +
dr2, Φ = Φ0
, (3)
where the length L is related to cosmological constant of the AdS spacetime (λ = 1/L2), Φ0 is the dimensionless 2D
inverse Newton constant and a is an integration constants related to the black hole mass M and horizon radius rh by
. (4)
The thermodynamical, Bekenstein-Hawking, entropy of the black hole is [26]
SBH = 2πΦ0a = 2π
2Φ0ML, (5)
whereas the black hole temperature is T = a/2πL. Setting a = 0 in Eq. (3) we have the AdS black hole ground state
( in the following called AdS0) with zero mass, temperature and entropy. The AdS black hole (3) can be considered
as the thermalization of the AdS0 solution at temperature a/2πL [26].
It has been shown that the 2D black hole has a dual description in terms of a CFT with central charge [27, 28, 29, 30]
c = 12Φ0. (6)
The dual CFT can have both the form of a 2D [29, 30] or a 1D [27, 28] conformal field theory. This AdS2/CFT2 ( or
AdS2/CFT1) correspondence has been used to give a microscopical meaning to the thermodynamical entropy of 2D
AdS black holes. Eq. (5) has been reproduced by counting states in the dual CFT.
In Ref. [16] (see also Refs. [24, 25, 31]) it was observed that in two dimensions black hole entropy can be ascribed
to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields. On
the other hand the AdS2/CFT2 correspondence, and in particular Eq. (6), tells us that the 2D Newton constant is
induced by quantum fluctuations of the dual CFT. It follows that the black hole entropy (5) should be explained as
the entanglement of the vacuum of the 2D CFT of central charge given by Eq. (6) in the gravitational black hole
background (3).
At first sight one is tempted to use Eq. (1) to calculate the entanglement entropy of the vacuum of the dual CFT.
The exterior region of the 2D black hole can be easily identified with the region Q, whereas the black hole interior
has to be identified with the R region where the degrees of freedom are traced over. There are two obstacles that
prevents direct application of Eq. (1). First, Eq. (1) holds for a 2D flat spacetime, whereas we are dealing with a
curved 2D background. Second, the calculations leading to Eq. (1) are performed for spacelike slice Q, whereas in
our case the coordinate singularities at r = rh (the horizon) and r = ∞ (the timelike asymptotic boundary of the
AdS spacetime) do not allow for a global notion of spacelike coordinate (a coordinate system covering the whole black
hole spacetime in which the metric is non-singular and static). Owing to these geometrical features, in the black hole
case we cannot give a direct meaning to both the measures Σ and (Λ−Σ) of the subsystems Q,R. As a consequence
invariance under the transformation (2) is meaningless in the black hole case.
The second difficulty can be circumvented using appropriate coordinate system and regularization procedure, the
first using instead of Eq. (1) the formula derived by Fiola et al. [16], which gives the EE of the vacuum of matter
fields in the case of a curved gravitational background.
σ= ∞σ=ε 
FIG. 1: Regularized euclidean instanton corresponding to the 2D AdS black hole in the coordinate system (t, σ) covering
only the black hole exterior. The euclidean time is periodic. The point σ = ∞ correspond to the black hole horizon. σ = 0
corresponds to the asymptotic timelike boundary of AdS2.
In the coordinate system used to define the vacuum of scalar fields in AdS2, the 2D black hole metric (3) is [26]
ds2 =
sinh2(aσ
−dt2 + dσ2
. (7)
The coordinate system (t, σ) covers only the black hole exterior. The black hole horizon corresponds to σ = ∞
where the conformal factor of the metric vanishes. The asymptotic r = ∞ timelike conformal boundary of the AdS2
spacetime is located at σ = 0, where the conformal factor diverges.
The entanglement entropy of the CFT vacuum in the curved background (7) can be calculated, using the formula
of Ref. [16] as the half line entanglement entropy seen by an observer in the 0 < σ < ∞ region. From the CFT
point of view the AdS black hole has to be considered as the AdS0 vacuum seen by the observer using the black hole
coordinates (7) [26]. Moreover, this observer sees the the AdS0 vacuum as filled with thermal radiation with negative
flux [26]. It follows that the black hole entanglement entropy is given by the formula of Ref. [16] with reversed sign,
ent = −
ρ(σ = 0)− ln
, (8)
where ρ defines the conformal factor of the metric in the conformal gauge (ds2 = exp(2ρ)(−dt2+dσ2)), c is the central
charge given by Eq. (6) and δ,Λ are respectively UV and IR cutoffs. Notice that in Eq. (8) we have only contributions
from only one sector (e.g. right movers) of the CFT. In Ref. [29, 30] it has been shown that the 2D AdS black hole
is dual to an open string with appropriate boundary conditions. These boundary conditions are such that only one
sector of the CFT2 is present. The same is obviously true for the AdS2/CFT1 realization of the correspondence
[27, 28].
The conformal factor of the metric (7), hence the entanglement entropy (8) blows up on the σ = 0 boundary of the
AdS spacetime. The simplest regularization procedure that solves this problem is to consider a regularized boundary
at σ = ǫ. Notice that ǫ plays the role of a UV cutoff for the coordinate σ, which is the natural spacelike coordinate
of the dual CFT. ǫ is an IR cutoff for the coordinate r, which is the natural spacelike coordinate for the AdS2 black
hole. The regularized euclidean instanton corresponding to the black hole (7) is shown in figure (1). The regularizing
parameter ǫ can be set equal to the UV cutoff, δ = ǫ. Moreover, the regularized boundary is at finite proper distance
from the horizon so that ǫ acts also as IR regulator, making the presence of the IR cutoff Λ in Eq. (8) redundant. It
follows that the regularized EE is given by S
ent = −
ρ(ǫ)− ln ǫ
, which using equations (7) and (4) becomes
ent =
. (9)
As a check of the validity of our formula we note that in the case of AdS0 (rh = 0) the entanglement entropy vanishes.
The AdS/CFT correspondence enable us to identify the cutoff ǫ as the UV cutoff of the CFT : ǫ ∝ L. The
proportionality factor can be determined by requiring that the analytical continuation of Eq. (9) is invariant under
the transformation (2) (see later). This requirement fixes ǫ = πL. With this position we get
ent =
. (10)
This formula is our main result, it gives the entanglement entropy of the 2D AdS black hole. This entanglement
entropy has the expected behavior as a function of the horizon radius rh or, equivalently, of the black hole mass M .
ent becomes zero in the AdS0 ground state, rh = 0 (M = 0), whereas it grows monotonically for rh > 0 (M > 0).
In order to compare the black hole EE (10) with the BH entropy (5) let us consider the limit of macroscopic black
holes, that is the limit a → ∞ or equivalently rh >> L or also M >> 1/L. Expanding Eq. (10) and using Eqs. (4)
and (6) we get
ent = 2π
2Φ0ML− Φ0 lnLM +O(1) = SBH − 2Φ0 lnSBH +O(1). (11)
We have obtained the remarkable result that the leading term in the large mass expansion of the black hole en-
tanglement entropy reproduces exactly the Bekenstein-Hawking entropy. Moreover, the subleading term behaves as
the logarithm of the BH entropy and describes quantum corrections to SBH . It is an universally accepted result
that the quantum corrections to the BH entropy behave as lnSBH [32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. However,
there is no general consensus about the value of the prefactor of this term. For the microcanonical ensemble this
term has to be negative, whereas there are positive contributions coming from thermal fluctuation. Equation (11)
fixes the prefactor of lnSBH in terms of the 2D Newton constant. This result contradicts some previous results
supporting a Φ0-independent value of the prefactor. Our result is consistent with the approach followed in this
paper, which considers 2D gravity as induced from the quantum fluctuations of a CFT with central charge 12Φ0.
The first (Bekenstein-Hawking) term in Eq. (11) is the induced entanglement entropy, whereas the second term,
−(c/6) ln(rh/L), is determined by the conformal symmetry. It gives the entanglement entropy (1) of a CFT in 2D flat
spacetime with central charge 12Φ0 and Σ = rh in the limit Σ << Λ [6]. The subleading term in Eq. (11) represents
therefore an universal behavior shared with other systems described by 2D QFTs, such as one-dimensional statistical
models near to the critical point (with the black hole radius rh corresponding to the correlation length) or free scalars
fields [7, 9].
Eq. (10) shows a close resemblance with the CFT entanglement entropy (1). Eqs. (10) and (1) differs in two
main points: the absence in the black hole case of something corresponding to the measure of the whole space (the
parameter Λ in Eq. (1)) and the appearance of hyperbolic instead of trigonometric functions. These are expected
features for the entanglement entropy of a black hole. They solve the problems concerning the application of formula
(1) to the black hole case. For a black hole one cannot define a measure of the whole space analogue to Λ. For
static solutions the coordinate system covers only the black hole exterior. The appearance of hyperbolic instead of
trigonometric functions allows for monotonic increasing of S
ent (rh), eliminating the unphysical decreasing behavior
of Sent(Σ) in the region Σ > Λ/2.
It is interesting to see how Eq. (1) can be obtained as the analytic continuation rh → irh of our formula (10),
i.e by considering an AdS black hole with negative mass. The analytically continued black hole solution is given
by Eq. (3) with a2 < 0. In the conformal gauge the solution reads now ds2 = [a2/ sin2(aσ/L)](−dt2 + dσ2). The
range of the spacelike coordinate, corresponding to 0 < r < ∞, is now 0 < σ < πL/2a. Regularizing the solution
at σ = 0 by introducing the cutoff ǫ we get the euclidean instanton shown in Fig. (2). In terms of the 2D CFT
we have to trace over the degrees of freedom outside the spacelike slice ǫ < σ < πL/2a. The related entanglement
entropy can be calculated using the formula of Ref. [16] in the case of a spacelike slice with two boundary points:
Sent = −c/6[ρ(ǫ)+ρ(πL/2a)− ln(δ/Λ)]. Applying this formula to the case of the black hole solution of negative mass,
identifying ǫ in terms of the IR cutoff Λ, ǫ = πL2/Λ, and redefining appropriately the UV cutoff δ, we get
Sent =
. (12)
σ= πL/2a    σ=ε 
FIG. 2: Regularized euclidean instanton corresponding to the 2D AdS black hole with negative mass. The euclidean time
is periodic. The point σ = πL/2a corresponds to the black hole singularity at r = 0. σ = 0 corresponds to the asymptotic
timelike boundary of AdS2.
Thus, the entanglement entropy of the 2D CFT in the curved background given by the AdS black hole of negative
mass has exactly the form given by Eq. (1) with the horizon radius rh playing the role of Σ. Notice that the presence
of the factor π in the argument of the sin-function is necessary if one wants invariance under the transformation
(2). The requirement that equation (12) is the analytic continuation of Eq. (10) fixes, as previously anticipated, the
proportionality factor between ǫ and L in the calculations leading to Eq. (10).
In this letter we have derived a formula for the entanglement entropy of 2D AdS black holes that has nice striking
features. The leading term in the large black hole mass expansion reproduces exactly the BH entropy. The subleading
term has the right lnSBH , behavior of the quantum corrections to the BH formula and represents an universal term
typical of CFTs. Analytic continuation to negative black hole masses give exactly the entanglement entropy of 2D
CFT with the black hole radius playing the role of the measure of the observable spacelike slice in the CFT. Our
results rely heavily on peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and
on the fact that 2D Newton constant arises from quantum fluctuation of the dual CFT. The generalization of our
approach to higher dimensional gravity theories is therefore far from being trivial. A related problem is the form of
the coefficient of the lnSBH term. In the 2D context our result, stating that this coefficient is given in terms of the
2D Newton constant (or equivalently the central charge of the dual CFT) is rather natural. For higher dimensional
gravity theories this is again a rather subtle point.
I thank G. D’Appollonio for discussions and valuable comments.
∗ Electronic address: mariano.cadoni@ca.infn.it
[1] G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett. 90 (2003) 227902 [arXiv:quant-ph/0211074].
[2] A. R. Its, B. Q. Jin, V. E. Korepin, J. Phys. A 38 (2005) 2975 [arXiv:quant-ph/0409027].
[3] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404 [arXiv:hep-th/0510092].
[4] J. I. Latorre, C. A. Lutken, E. Rico and G. Vidal, Phys. Rev. A 71 (2005) 034301 [arXiv:quant-ph/0404120].
[5] V. E. Korepin, Phys. Rev. Lett. 92 (2003) 964021.
[6] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424 (1994) 443 [arXiv:hep-th/9403108].
mailto:mariano.cadoni@ca.infn.it
http://arxiv.org/abs/quant-ph/0211074
http://arxiv.org/abs/quant-ph/0409027
http://arxiv.org/abs/hep-th/0510092
http://arxiv.org/abs/quant-ph/0404120
http://arxiv.org/abs/hep-th/9403108
[7] P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406 (2004) P002 [arXiv:hep-th/0405152].
[8] H. Casini and M. Huerta, Phys. Lett. B 600 (2004) 142 [arXiv:hep-th/0405111].
[9] D. V. Fursaev, Phys. Rev. D 73 (2006) 124025 [arXiv:hep-th/0602134].
[10] S. N. Solodukhin, Phys. Rev. Lett. 97 (2006) 201601 [arXiv:hep-th/0606205].
[11] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96 (2006) 181602 [arXiv:hep-th/0603001].
[12] S. Ryu and T. Takayanagi, JHEP 0608 (2006) 045 [arXiv:hep-th/0605073].
[13] G. ’t Hooft, Nucl. Phys. B 256 (1985) 727.
[14] L. Bombelli, R. K. Koul, J. H. Lee and R. D. Sorkin, Phys. Rev. D 34 (1986) 373.
[15] V. P. Frolov and I. Novikov, Phys. Rev. D 48 (1993) 4545 [arXiv:gr-qc/9309001].
[16] T. M. Fiola, J. Preskill, A. Strominger and S. P. Trivedi, Phys. Rev. D 50 (1994) 3987 [arXiv:hep-th/9403137].
[17] F. Belgiorno and S. Liberati, Phys. Rev. D 53 (1996) 3172 [arXiv:gr-qc/9503022].
[18] S. Hawking, J. M. Maldacena and A. Strominger, JHEP 0105 (2001) 001 [arXiv:hep-th/0002145].
[19] J. M. Maldacena, JHEP 0304 (2003) 021 [arXiv:hep-th/0106112].
[20] R. Brustein, M. B. Einhorn and A. Yarom, JHEP 0601 (2006) 098 [arXiv:hep-th/0508217].
[21] R. Emparan, JHEP 0606 (2006) 012 [arXiv:hep-th/0603081].
[22] P. Valtancoli, arXiv:hep-th/0612049.
[23] T. Jacobson, arXiv:gr-qc/9404039.
[24] V. P. Frolov, D. V. Fursaev and A. I. Zelnikov, Nucl. Phys. B 486 (1997) 339 [arXiv:hep-th/9607104].
[25] V. P. Frolov and D. V. Fursaev, Phys. Rev. D 56 (1997) 2212 [arXiv:hep-th/9703178].
[26] M. Cadoni and S. Mignemi, Phys. Rev. D 51 (1995) 4319 [arXiv:hep-th/9410041].
[27] M. Cadoni and S. Mignemi, Phys. Rev. D 59 (1999) 081501 [arXiv:hep-th/9810251].
[28] M. Cadoni and S. Mignemi, Nucl. Phys. B 557 (1999) 165 [arXiv:hep-th/9902040].
[29] M. Cadoni and M. Cavaglia, Phys. Lett. B 499 (2001) 315 [arXiv:hep-th/0005179].
[30] M. Cadoni and M. Cavaglia, Phys. Rev. D 63 (2001) 084024 [arXiv:hep-th/0008084].
[31] L. Susskind and J. Uglum, Phys. Rev. D 50 (1994) 2700 [arXiv:hep-th/9401070].
[32] D. V. Fursaev, Phys. Rev. D 51 (1995) 5352 [arXiv:hep-th/9412161].
[33] R. B. Mann and S. N. Solodukhin, Nucl. Phys. B 523 (1998) 293 [arXiv:hep-th/9709064].
[34] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255 [arXiv:gr-qc/0002040].
[35] S. Carlip, Class. Quant. Grav. 17 (2000) 4175 [arXiv:gr-qc/0005017].
[36] A. Ghosh and P. Mitra, Phys. Rev. Lett. 73 (1994) 2521 [arXiv:hep-th/9406210].
[37] S. Mukherji and S. S. Pal, JHEP 0205 (2002) 026 [arXiv:hep-th/0205164].
[38] M. R. Setare, Phys. Lett. B 573 (2003) 173 [arXiv:hep-th/0311106].
[39] M. Domagala and J. Lewandowski, Class. Quant. Grav. 21 (2004) 5233 [arXiv:gr-qc/0407051].
[40] A. J. M. Medved, Class. Quant. Grav. 22 (2005) 133 [arXiv:gr-qc/0406044].
[41] D. Grumiller, arXiv:hep-th/0506175.
http://arxiv.org/abs/hep-th/0405152
http://arxiv.org/abs/hep-th/0405111
http://arxiv.org/abs/hep-th/0602134
http://arxiv.org/abs/hep-th/0606205
http://arxiv.org/abs/hep-th/0603001
http://arxiv.org/abs/hep-th/0605073
http://arxiv.org/abs/gr-qc/9309001
http://arxiv.org/abs/hep-th/9403137
http://arxiv.org/abs/gr-qc/9503022
http://arxiv.org/abs/hep-th/0002145
http://arxiv.org/abs/hep-th/0106112
http://arxiv.org/abs/hep-th/0508217
http://arxiv.org/abs/hep-th/0603081
http://arxiv.org/abs/hep-th/0612049
http://arxiv.org/abs/gr-qc/9404039
http://arxiv.org/abs/hep-th/9607104
http://arxiv.org/abs/hep-th/9703178
http://arxiv.org/abs/hep-th/9410041
http://arxiv.org/abs/hep-th/9810251
http://arxiv.org/abs/hep-th/9902040
http://arxiv.org/abs/hep-th/0005179
http://arxiv.org/abs/hep-th/0008084
http://arxiv.org/abs/hep-th/9401070
http://arxiv.org/abs/hep-th/9412161
http://arxiv.org/abs/hep-th/9709064
http://arxiv.org/abs/gr-qc/0002040
http://arxiv.org/abs/gr-qc/0005017
http://arxiv.org/abs/hep-th/9406210
http://arxiv.org/abs/hep-th/0205164
http://arxiv.org/abs/hep-th/0311106
http://arxiv.org/abs/gr-qc/0407051
http://arxiv.org/abs/gr-qc/0406044
http://arxiv.org/abs/hep-th/0506175
	Acknowledgments
	References

  Using the AdS/CFT correspondence we derive a formula for the entanglement
entropy of the anti-de Sitter black hole in two spacetime dimensions. The
leading term in the large black hole mass expansion of our formula reproduces
exactly the Bekenstein-Hawking entropy S_{BH}, whereas the subleading term
behaves as ln S_{BH}. This subleading term has the universal form typical for
the entanglement entropy of physical systems described by effective conformal
fields theories (e.g. one-dimensional statistical models at the critical
point). The well-known form of the entanglement entropy for a two-dimensional
conformal field theory is obtained as analytic continuation of our result and
is related with the entanglement entropy of a black hole with negative mass.

Entanglement Entropy of two-dimensional anti-de Sitter black holes
Mariano Cadoni∗
Dipartimento di Fisica, Università di Cagliari, and INFN sezione di Cagliari,
Cittadella Universitaria 09042 Monserrato, ITALY
Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the
anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole
mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas
the subleading term behaves as lnSBH . This subleading term has the universal form typical for
the entanglement entropy of physical systems described by effective conformal fields theories (e.g.
one-dimensional statistical models at the critical point). The well-known form of the entanglement
entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our
result and is related with the entanglement entropy of a black hole with negative mass.
Quantum entanglement is a fundamental feature of quantum systems. It is related to the existence of correlations
between parts of the system. The degree of entanglement of a quantum system is measured by the entanglement
entropy Sent. In quantum field theory (QFT), or more in general in many body systems, we can localize observable
and unobservable degrees of freedom in spatially separated regions Q and R. Sent is then defined as the von Neumann
entropy of the system when the degrees of freedom in the region R are traced over, Sent = −TrQρ̂Q ln ρ̂Q, where the
trace is taken over states in the observable region Q and the reduced density matrix ρ̂Q = TrRρ̂ is obtained by tracing
the density matrix ρ̂ over states in the region R.
Investigation of the entanglement entropy (EE) has become relevant in many research areas. Apart from quantum
information theory, the field that gave birth to the notion of entanglement entropy, it plays a crucial role in condensed
matter systems, where it helps to understand quantum phases of matter (e.g spin chains and quantum liquids)[1, 2,
3, 4, 5]. Entanglement (geometric) entropy is also an useful concept for investigating general features of QFT, in
particular two-dimensional conformal field theory (CFT) and the Anti-de Sitter/conformal field theory (AdS/CFT)
correspondence [6, 7, 8, 9, 10, 11, 12] . Last but not least entanglement may held the key for unraveling the mystery
of black hole entropy [13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
We will be mainly concerned with the entanglement entropy of two-dimensional (2D) CFT and its relationship with
the entropy of 2D black holes. It is an old idea that black hole entropy may be explained in terms of the EE of the
quantum state of matter fields in the black hole geometry [13]. The main support to this conjecture comes from the
fact that both the EE of matter fields and the Bekenstein-Hawking (BH) entropy depend on the area of the boundary
region. On the other hand any attempt to explain the BH entropy as originating from quantum entanglement has to
solve conceptual and technical difficulties.
The usual statistical paradigm explains the BH entropy in terms of a microstate gas. This is conceptually different
from the EE that measures the observer’s lack of information about the quantum state of the system in a inaccessible
region of spacetime. Moreover, the EE depends both on the number of species ns of the matter fields, whose
entanglement should reproduce the BH entropy, and on the value of the UV cutoff δ arising owing to the presence
of a sharp boundary between the accessible and inaccessible regions of the spacetime. Conversely, the BH entropy is
meant to be universal, hence independent from ns and δ. Some conceptual difficulties can be solved using Sakharov’s
induced gravity approach [23, 24, 25], but the problem of the dependence on ns and δ still remains unsolved.
In this letter we will show that in the case of two-dimensional AdS black hole these difficulties can be completely
solved. We will derive an expression for the black hole EE that in the large black hole mass limit reproduces exactly
the BH entropy. Moreover, we will show that the subleading term has the universal behavior typical for CFTs and in
particular for critical phenomena. The reason of this success is related to the peculiarities of 2D AdS gravity, namely
the existence of an AdS/CFT correspondence and the fact that 2D Newton constant can be considered as wholly
induced by quantum fluctuations of the dual CFT.
Most of the progress in understanding the EE in QFT has been achieved in the case of 2D CFT. Conformal
invariance in two space-time dimension is a powerful tool that allows us to compute the EE in closed form. The
entanglement entropy for the ground state of a 2D CFT originated from tracing over correlations between spacelike
separated points has been calculated by Holzhey, Larsen and Wilckzek [6]. Introducing an infrared cutoff Λ the
spacelike coordinate of our 2D universe will belong to C = [0,Λ[. The subsystem where measurements are performed
is Q = [0,Σ[, whereas the outside region where the degrees of freedom are traced over is R = [Σ,Λ[. Because of the
contribution of localized excitations arbitrarily near to the boundary the entanglement entropy diverges. Introducing
http://arxiv.org/abs/0704.0140v2
an ultraviolet cutoff δ, the regularized entanglement entropy turns out to be [6]
Sent =
c+ c̄
, (1)
where c and c̄ are the central charges of the 2D CFT. The expression (1) emphasizes the characterizing features of
the entanglement entropy, namely subadditivity and invariance under the transformation which exchanges the inside
and outside regions
Σ → Λ− Σ. (2)
Moreover, Sent is not a monotonic function of Σ, but increases and reaches its maximum for Σ = Λ/2 and then
decreases as Σ increases further. This behavior has an obvious explanation. When the subsystem begins to fill most
of the universe there is lesser information to be lost and the entanglement entropy decreases.
Let us now consider 2D AdS black holes. As classical solutions of a 2D gravity theory they are endowed with a
non-constant scalar field, the dilaton Φ. In the Schwarzschild gauge the 2D AdS black hole solutions are [26],
ds2 = −
dt2 +
dr2, Φ = Φ0
, (3)
where the length L is related to cosmological constant of the AdS spacetime (λ = 1/L2), Φ0 is the dimensionless 2D
inverse Newton constant and a is an integration constants related to the black hole mass M and horizon radius rh by
. (4)
The thermodynamical, Bekenstein-Hawking, entropy of the black hole is [26]
SBH = 2πΦ0a = 2π
2Φ0ML, (5)
whereas the black hole temperature is T = a/2πL. Setting a = 0 in Eq. (3) we have the AdS black hole ground state
( in the following called AdS0) with zero mass, temperature and entropy. The AdS black hole (3) can be considered
as the thermalization of the AdS0 solution at temperature a/2πL [26].
It has been shown that the 2D black hole has a dual description in terms of a CFT with central charge [27, 28, 29, 30]
c = 12Φ0. (6)
The dual CFT can have both the form of a 2D [29, 30] or a 1D [27, 28] conformal field theory. This AdS2/CFT2 ( or
AdS2/CFT1) correspondence has been used to give a microscopical meaning to the thermodynamical entropy of 2D
AdS black holes. Eq. (5) has been reproduced by counting states in the dual CFT.
In Ref. [16] (see also Refs. [24, 25, 31]) it was observed that in two dimensions black hole entropy can be ascribed
to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields. On
the other hand the AdS2/CFT2 correspondence, and in particular Eq. (6), tells us that the 2D Newton constant is
induced by quantum fluctuations of the dual CFT. It follows that the black hole entropy (5) should be explained as
the entanglement of the vacuum of the 2D CFT of central charge given by Eq. (6) in the gravitational black hole
background (3).
At first sight one is tempted to use Eq. (1) to calculate the entanglement entropy of the vacuum of the dual CFT.
The exterior region of the 2D black hole can be easily identified with the region Q, whereas the black hole interior
has to be identified with the R region where the degrees of freedom are traced over. There are two obstacles that
prevents direct application of Eq. (1). First, Eq. (1) holds for a 2D flat spacetime, whereas we are dealing with a
curved 2D background. Second, the calculations leading to Eq. (1) are performed for spacelike slice Q, whereas in
our case the coordinate singularities at r = rh (the horizon) and r = ∞ (the timelike asymptotic boundary of the
AdS spacetime) do not allow for a global notion of spacelike coordinate (a coordinate system covering the whole black
hole spacetime in which the metric is non-singular and static). Owing to these geometrical features, in the black hole
case we cannot give a direct meaning to both the measures Σ and (Λ−Σ) of the subsystems Q,R. As a consequence
invariance under the transformation (2) is meaningless in the black hole case.
The second difficulty can be circumvented using appropriate coordinate system and regularization procedure, the
first using instead of Eq. (1) the formula derived by Fiola et al. [16], which gives the EE of the vacuum of matter
fields in the case of a curved gravitational background.
σ= ∞σ=ε 
FIG. 1: Regularized euclidean instanton corresponding to the 2D AdS black hole in the coordinate system (t, σ) covering
only the black hole exterior. The euclidean time is periodic. The point σ = ∞ correspond to the black hole horizon. σ = 0
corresponds to the asymptotic timelike boundary of AdS2.
In the coordinate system used to define the vacuum of scalar fields in AdS2, the 2D black hole metric (3) is [26]
ds2 =
sinh2(aσ
−dt2 + dσ2
. (7)
The coordinate system (t, σ) covers only the black hole exterior. The black hole horizon corresponds to σ = ∞
where the conformal factor of the metric vanishes. The asymptotic r = ∞ timelike conformal boundary of the AdS2
spacetime is located at σ = 0, where the conformal factor diverges.
The entanglement entropy of the CFT vacuum in the curved background (7) can be calculated, using the formula
of Ref. [16] as the half line entanglement entropy seen by an observer in the 0 < σ < ∞ region. From the CFT
point of view the AdS black hole has to be considered as the AdS0 vacuum seen by the observer using the black hole
coordinates (7) [26]. Moreover, this observer sees the the AdS0 vacuum as filled with thermal radiation with negative
flux [26]. It follows that the black hole entanglement entropy is given by the formula of Ref. [16] with reversed sign,
ent = −
ρ(σ = 0)− ln
, (8)
where ρ defines the conformal factor of the metric in the conformal gauge (ds2 = exp(2ρ)(−dt2+dσ2)), c is the central
charge given by Eq. (6) and δ,Λ are respectively UV and IR cutoffs. Notice that in Eq. (8) we have only contributions
from only one sector (e.g. right movers) of the CFT. In Ref. [29, 30] it has been shown that the 2D AdS black hole
is dual to an open string with appropriate boundary conditions. These boundary conditions are such that only one
sector of the CFT2 is present. The same is obviously true for the AdS2/CFT1 realization of the correspondence
[27, 28].
The conformal factor of the metric (7), hence the entanglement entropy (8) blows up on the σ = 0 boundary of the
AdS spacetime. The simplest regularization procedure that solves this problem is to consider a regularized boundary
at σ = ǫ. Notice that ǫ plays the role of a UV cutoff for the coordinate σ, which is the natural spacelike coordinate
of the dual CFT. ǫ is an IR cutoff for the coordinate r, which is the natural spacelike coordinate for the AdS2 black
hole. The regularized euclidean instanton corresponding to the black hole (7) is shown in figure (1). The regularizing
parameter ǫ can be set equal to the UV cutoff, δ = ǫ. Moreover, the regularized boundary is at finite proper distance
from the horizon so that ǫ acts also as IR regulator, making the presence of the IR cutoff Λ in Eq. (8) redundant. It
follows that the regularized EE is given by S
ent = −
ρ(ǫ)− ln ǫ
, which using equations (7) and (4) becomes
ent =
. (9)
As a check of the validity of our formula we note that in the case of AdS0 (rh = 0) the entanglement entropy vanishes.
The AdS/CFT correspondence enable us to identify the cutoff ǫ as the UV cutoff of the CFT : ǫ ∝ L. The
proportionality factor can be determined by requiring that the analytical continuation of Eq. (9) is invariant under
the transformation (2) (see later). This requirement fixes ǫ = πL. With this position we get
ent =
. (10)
This formula is our main result, it gives the entanglement entropy of the 2D AdS black hole. This entanglement
entropy has the expected behavior as a function of the horizon radius rh or, equivalently, of the black hole mass M .
ent becomes zero in the AdS0 ground state, rh = 0 (M = 0), whereas it grows monotonically for rh > 0 (M > 0).
In order to compare the black hole EE (10) with the BH entropy (5) let us consider the limit of macroscopic black
holes, that is the limit a → ∞ or equivalently rh >> L or also M >> 1/L. Expanding Eq. (10) and using Eqs. (4)
and (6) we get
ent = 2π
2Φ0ML− Φ0 lnLM +O(1) = SBH − 2Φ0 lnSBH +O(1). (11)
We have obtained the remarkable result that the leading term in the large mass expansion of the black hole en-
tanglement entropy reproduces exactly the Bekenstein-Hawking entropy. Moreover, the subleading term behaves as
the logarithm of the BH entropy and describes quantum corrections to SBH . It is an universally accepted result
that the quantum corrections to the BH entropy behave as lnSBH [32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. However,
there is no general consensus about the value of the prefactor of this term. For the microcanonical ensemble this
term has to be negative, whereas there are positive contributions coming from thermal fluctuation. Equation (11)
fixes the prefactor of lnSBH in terms of the 2D Newton constant. This result contradicts some previous results
supporting a Φ0-independent value of the prefactor. Our result is consistent with the approach followed in this
paper, which considers 2D gravity as induced from the quantum fluctuations of a CFT with central charge 12Φ0.
The first (Bekenstein-Hawking) term in Eq. (11) is the induced entanglement entropy, whereas the second term,
−(c/6) ln(rh/L), is determined by the conformal symmetry. It gives the entanglement entropy (1) of a CFT in 2D flat
spacetime with central charge 12Φ0 and Σ = rh in the limit Σ << Λ [6]. The subleading term in Eq. (11) represents
therefore an universal behavior shared with other systems described by 2D QFTs, such as one-dimensional statistical
models near to the critical point (with the black hole radius rh corresponding to the correlation length) or free scalars
fields [7, 9].
Eq. (10) shows a close resemblance with the CFT entanglement entropy (1). Eqs. (10) and (1) differs in two
main points: the absence in the black hole case of something corresponding to the measure of the whole space (the
parameter Λ in Eq. (1)) and the appearance of hyperbolic instead of trigonometric functions. These are expected
features for the entanglement entropy of a black hole. They solve the problems concerning the application of formula
(1) to the black hole case. For a black hole one cannot define a measure of the whole space analogue to Λ. For
static solutions the coordinate system covers only the black hole exterior. The appearance of hyperbolic instead of
trigonometric functions allows for monotonic increasing of S
ent (rh), eliminating the unphysical decreasing behavior
of Sent(Σ) in the region Σ > Λ/2.
It is interesting to see how Eq. (1) can be obtained as the analytic continuation rh → irh of our formula (10),
i.e by considering an AdS black hole with negative mass. The analytically continued black hole solution is given
by Eq. (3) with a2 < 0. In the conformal gauge the solution reads now ds2 = [a2/ sin2(aσ/L)](−dt2 + dσ2). The
range of the spacelike coordinate, corresponding to 0 < r < ∞, is now 0 < σ < πL/2a. Regularizing the solution
at σ = 0 by introducing the cutoff ǫ we get the euclidean instanton shown in Fig. (2). In terms of the 2D CFT
we have to trace over the degrees of freedom outside the spacelike slice ǫ < σ < πL/2a. The related entanglement
entropy can be calculated using the formula of Ref. [16] in the case of a spacelike slice with two boundary points:
Sent = −c/6[ρ(ǫ)+ρ(πL/2a)− ln(δ/Λ)]. Applying this formula to the case of the black hole solution of negative mass,
identifying ǫ in terms of the IR cutoff Λ, ǫ = πL2/Λ, and redefining appropriately the UV cutoff δ, we get
Sent =
. (12)
σ= πL/2a    σ=ε 
FIG. 2: Regularized euclidean instanton corresponding to the 2D AdS black hole with negative mass. The euclidean time
is periodic. The point σ = πL/2a corresponds to the black hole singularity at r = 0. σ = 0 corresponds to the asymptotic
timelike boundary of AdS2.
Thus, the entanglement entropy of the 2D CFT in the curved background given by the AdS black hole of negative
mass has exactly the form given by Eq. (1) with the horizon radius rh playing the role of Σ. Notice that the presence
of the factor π in the argument of the sin-function is necessary if one wants invariance under the transformation
(2). The requirement that equation (12) is the analytic continuation of Eq. (10) fixes, as previously anticipated, the
proportionality factor between ǫ and L in the calculations leading to Eq. (10).
In this letter we have derived a formula for the entanglement entropy of 2D AdS black holes that has nice striking
features. The leading term in the large black hole mass expansion reproduces exactly the BH entropy. The subleading
term has the right lnSBH , behavior of the quantum corrections to the BH formula and represents an universal term
typical of CFTs. Analytic continuation to negative black hole masses give exactly the entanglement entropy of 2D
CFT with the black hole radius playing the role of the measure of the observable spacelike slice in the CFT. Our
results rely heavily on peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and
on the fact that 2D Newton constant arises from quantum fluctuation of the dual CFT. The generalization of our
approach to higher dimensional gravity theories is therefore far from being trivial. A related problem is the form of
the coefficient of the lnSBH term. In the 2D context our result, stating that this coefficient is given in terms of the
2D Newton constant (or equivalently the central charge of the dual CFT) is rather natural. For higher dimensional
gravity theories this is again a rather subtle point.
I thank G. D’Appollonio for discussions and valuable comments.
∗ Electronic address: mariano.cadoni@ca.infn.it
[1] G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett. 90 (2003) 227902 [arXiv:quant-ph/0211074].
[2] A. R. Its, B. Q. Jin, V. E. Korepin, J. Phys. A 38 (2005) 2975 [arXiv:quant-ph/0409027].
[3] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404 [arXiv:hep-th/0510092].
[4] J. I. Latorre, C. A. Lutken, E. Rico and G. Vidal, Phys. Rev. A 71 (2005) 034301 [arXiv:quant-ph/0404120].
[5] V. E. Korepin, Phys. Rev. Lett. 92 (2003) 964021.
[6] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424 (1994) 443 [arXiv:hep-th/9403108].
mailto:mariano.cadoni@ca.infn.it
http://arxiv.org/abs/quant-ph/0211074
http://arxiv.org/abs/quant-ph/0409027
http://arxiv.org/abs/hep-th/0510092
http://arxiv.org/abs/quant-ph/0404120
http://arxiv.org/abs/hep-th/9403108
[7] P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406 (2004) P002 [arXiv:hep-th/0405152].
[8] H. Casini and M. Huerta, Phys. Lett. B 600 (2004) 142 [arXiv:hep-th/0405111].
[9] D. V. Fursaev, Phys. Rev. D 73 (2006) 124025 [arXiv:hep-th/0602134].
[10] S. N. Solodukhin, Phys. Rev. Lett. 97 (2006) 201601 [arXiv:hep-th/0606205].
[11] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96 (2006) 181602 [arXiv:hep-th/0603001].
[12] S. Ryu and T. Takayanagi, JHEP 0608 (2006) 045 [arXiv:hep-th/0605073].
[13] G. ’t Hooft, Nucl. Phys. B 256 (1985) 727.
[14] L. Bombelli, R. K. Koul, J. H. Lee and R. D. Sorkin, Phys. Rev. D 34 (1986) 373.
[15] V. P. Frolov and I. Novikov, Phys. Rev. D 48 (1993) 4545 [arXiv:gr-qc/9309001].
[16] T. M. Fiola, J. Preskill, A. Strominger and S. P. Trivedi, Phys. Rev. D 50 (1994) 3987 [arXiv:hep-th/9403137].
[17] F. Belgiorno and S. Liberati, Phys. Rev. D 53 (1996) 3172 [arXiv:gr-qc/9503022].
[18] S. Hawking, J. M. Maldacena and A. Strominger, JHEP 0105 (2001) 001 [arXiv:hep-th/0002145].
[19] J. M. Maldacena, JHEP 0304 (2003) 021 [arXiv:hep-th/0106112].
[20] R. Brustein, M. B. Einhorn and A. Yarom, JHEP 0601 (2006) 098 [arXiv:hep-th/0508217].
[21] R. Emparan, JHEP 0606 (2006) 012 [arXiv:hep-th/0603081].
[22] P. Valtancoli, arXiv:hep-th/0612049.
[23] T. Jacobson, arXiv:gr-qc/9404039.
[24] V. P. Frolov, D. V. Fursaev and A. I. Zelnikov, Nucl. Phys. B 486 (1997) 339 [arXiv:hep-th/9607104].
[25] V. P. Frolov and D. V. Fursaev, Phys. Rev. D 56 (1997) 2212 [arXiv:hep-th/9703178].
[26] M. Cadoni and S. Mignemi, Phys. Rev. D 51 (1995) 4319 [arXiv:hep-th/9410041].
[27] M. Cadoni and S. Mignemi, Phys. Rev. D 59 (1999) 081501 [arXiv:hep-th/9810251].
[28] M. Cadoni and S. Mignemi, Nucl. Phys. B 557 (1999) 165 [arXiv:hep-th/9902040].
[29] M. Cadoni and M. Cavaglia, Phys. Lett. B 499 (2001) 315 [arXiv:hep-th/0005179].
[30] M. Cadoni and M. Cavaglia, Phys. Rev. D 63 (2001) 084024 [arXiv:hep-th/0008084].
[31] L. Susskind and J. Uglum, Phys. Rev. D 50 (1994) 2700 [arXiv:hep-th/9401070].
[32] D. V. Fursaev, Phys. Rev. D 51 (1995) 5352 [arXiv:hep-th/9412161].
[33] R. B. Mann and S. N. Solodukhin, Nucl. Phys. B 523 (1998) 293 [arXiv:hep-th/9709064].
[34] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255 [arXiv:gr-qc/0002040].
[35] S. Carlip, Class. Quant. Grav. 17 (2000) 4175 [arXiv:gr-qc/0005017].
[36] A. Ghosh and P. Mitra, Phys. Rev. Lett. 73 (1994) 2521 [arXiv:hep-th/9406210].
[37] S. Mukherji and S. S. Pal, JHEP 0205 (2002) 026 [arXiv:hep-th/0205164].
[38] M. R. Setare, Phys. Lett. B 573 (2003) 173 [arXiv:hep-th/0311106].
[39] M. Domagala and J. Lewandowski, Class. Quant. Grav. 21 (2004) 5233 [arXiv:gr-qc/0407051].
[40] A. J. M. Medved, Class. Quant. Grav. 22 (2005) 133 [arXiv:gr-qc/0406044].
[41] D. Grumiller, arXiv:hep-th/0506175.
http://arxiv.org/abs/hep-th/0405152
http://arxiv.org/abs/hep-th/0405111
http://arxiv.org/abs/hep-th/0602134
http://arxiv.org/abs/hep-th/0606205
http://arxiv.org/abs/hep-th/0603001
http://arxiv.org/abs/hep-th/0605073
http://arxiv.org/abs/gr-qc/9309001
http://arxiv.org/abs/hep-th/9403137
http://arxiv.org/abs/gr-qc/9503022
http://arxiv.org/abs/hep-th/0002145
http://arxiv.org/abs/hep-th/0106112
http://arxiv.org/abs/hep-th/0508217
http://arxiv.org/abs/hep-th/0603081
http://arxiv.org/abs/hep-th/0612049
http://arxiv.org/abs/gr-qc/9404039
http://arxiv.org/abs/hep-th/9607104
http://arxiv.org/abs/hep-th/9703178
http://arxiv.org/abs/hep-th/9410041
http://arxiv.org/abs/hep-th/9810251
http://arxiv.org/abs/hep-th/9902040
http://arxiv.org/abs/hep-th/0005179
http://arxiv.org/abs/hep-th/0008084
http://arxiv.org/abs/hep-th/9401070
http://arxiv.org/abs/hep-th/9412161
http://arxiv.org/abs/hep-th/9709064
http://arxiv.org/abs/gr-qc/0002040
http://arxiv.org/abs/gr-qc/0005017
http://arxiv.org/abs/hep-th/9406210
http://arxiv.org/abs/hep-th/0205164
http://arxiv.org/abs/hep-th/0311106
http://arxiv.org/abs/gr-qc/0407051
http://arxiv.org/abs/gr-qc/0406044
http://arxiv.org/abs/hep-th/0506175
	Acknowledgments
	References
