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arXiv:gr-qc/0002040v3 14 Jun 2000
Logarithmic correction to the Bekenstein-Hawking entropy
Romesh K. Kaul and Parthasarathi Majumdar∗
The Institute of Mathematical Sciences, Chennai 600 113, India.
The exact formula derived by us earlier for the entropy of a four dimensional non-rotating black
hole within the quantum geometry formulation of the event horizon in terms of boundary states of
a three dimensional Chern-Simons theory, is reexamined for large horizon areas. In addition to the
semiclassical Bekenstein-Hawking contribution proportional to the area obtained earlier, we find a
contribution proportional to the logarithm of the area together with subleading corrections that
constitute a series in inverse powers of the area.
The derivation of the Bekenstein-Hawking (BH) area law for black hole entropy from the quantum geometry
approach [1] (and also earlier from string theory [2] for some special cases), has lead to a resurgence of interest in
the quantum aspects of black hole physics in recent times. However, the major activity has remained focussed on
confirming the area law for large black holes, which, as is well-known, was obtained originally on the basis of arguments
of a semiclassical nature. The question arises as to whether any essential feature of the bona fide quantum aspect
of gravity, beyond the domain of the semiclassical approximation, has been captured in these assays. Indeed, as has
been most eloquently demonstrated by Carlip [3], a derivation of the area law alone seems to be possible on the basis
of some symmetry principle of the (semi)classical theory itself without requiring a detailed knowledge of the actual
quantum states associated with a black hole. The result seems to hold for arbitrary number of spatial dimensions, so
long as a particular set of isometries of the metric is respected. That quantum gravity has a description in terms of
spin networks (or for that matter, in terms of string states in a fixed background) appears to be of little consequence in
obtaining the area law, although these proposed underlying structures also lead to the same behaviour via alternative
routes, in the semiclassical limit of arbitrarily large horizon area.
Although there is as yet no complete quantum theory of gravitation, one would in general expect key features
uncovered so far to lead to modifications of the area law which could not have been anticipated through semiclassical
reasoning. Thus, the question as to what is the dominant quantum correction due to these features of quantum gravity
becomes one of paramount importance. Already in the string theory literature [4] examples of leading corrections to
the area law, obtained by counting D-brane states describing special supersymmetric extremal black holes (interacting
with massless vector supermultiplets) have appeared. This has received strong support recently from semiclassical
calculations in N = 2 supergravity [5] supplemented by ostensible stringy higher derivative corrections which are
incorporated using Wald’s general formalism describing black hole entropy as Noether charge [6]. However, the
geometrical interpretation of these corrections remains unclear. Further, there are subtleties associated with direct
application of Wald’s formalism which assumes a non-degenerate bifurcate Killing horizon, to the case of extremal
black holes which have degenerate horizons. Moreover, the string results do not pertain to generic (i.e., non-extremal)
black holes of Einstein’s general relativity, and are constrained by the unphysical requirement of unbroken spacetime
supersymmetry.
In this paper, we consider the corrections to the semiclassical area law of generic four dimensional non-rotating
black holes, due to key aspects of non-perturbative quantum gravity (or quantum geometry) formulated by Ashtekar
and collaborators [7]. In [1], appropriate boundary conditions are imposed on dynamical variables at the event horizon
considered as an inner boundary. These boundary conditions require that the Einstein-Hilbert action be supplemented
by boundary terms describing a three dimensional SU(2) Chern-Simons theory living on a finite ‘patch’ of the horizon
with a spherical boundary, punctured by links of the spin network bulk states describing the quantum spacetime
geometry interpolating between asymptopia and the horizon. On this two dimensional boundary there exists an
SU(2) Wess Zumino model whose conformal blocks describe the Hilbert space of the Chern-Simons theory modelling
the horizon. An exact formula for the number of these conformal blocks has been obtained by us [8] two years ago,
for arbitrary level k and number of punctures p. It has been shown that in the limit of large horizon area given by
arbitrarily large k and p, the logarithm of this number duly yields the area law. Here we go one step further, and
calculate the dominant sub-leading contribution, as a function of the classical horizon area, or what is equivalent, as
a function of the BH entropy itself.
On purely dimensional grounds, one would expect the entropy to have an expansion, for large classical horizon area,
in inverse powers of area so that the BH term is the leading one,
∗email: kaul, partha@imsc.ernet.in
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Sbh
=SBH +
∞
?
n=0
CnA−n
H
(1)
where, AHis the classical horizon area and Cnare coefficients which are independent of the horizon area but dependent
on the Planck length (Newton constant). Here the Barbero-Immirzi parameter [9] has been ‘fitted’ to the value which
fixes the normalization of the BH term to the standard one. However, in principle, one could expect an additional
term proportional to ln AH as the leading quantum correction to the semiclassical SBH. Such a term is expected on
general grounds pertaining to breakdown of na¨ive dimensional analysis due to quantum fluctuations, as is common
in quantum field theories in flat spacetime and also in quantum theories of critical phenomena. We show, in what
follows, that such a logarithmic correction to the semiclassical area law does indeed arise from the formula derived
earlier [8] and derive its coefficient.
We first briefly recapitulate the derivation [8] of the general formula for the number of conformal blocks of the
SU(2)k Wess Zumino model on a punctured 2-sphere appropriate to the black hole situation. This number can be
computed in terms of the so-called fusion matrices N
ij
[10]
r
NP =
?
{ri}
N
r1
j1j2
N
r2
r1j3
N
r3
r2j4
...... N
jp
rp−2jp−1
(2)
Diagrammatically, this can be represented as shown in fig. 1 below.
jp
r2r1 rp-2
j2j3
jp-1
j1
FIG. 1. Diagrammatic representation of composition of spins ji for SU(2)k
Here, each matrix element N
conformal field theory fusion algebra for the primary fields [φi] and [φj]
r
ij
is 1 or 0, depending on whether the primary field [φr] is allowed or not in the
(i,j,r = 0,1/2,1,....k/2):
[φi] ⊗ [φj] =
?
r
N
r
ij[φr] . (3)
Eq. (2) gives the number of conformal blocks with spins j1,j2,...,jpon p external lines and spins r1,r2,...,rp−2on
the internal lines.
We then use the Verlinde formula [10] to obtain
N
r
ij
=
?
s
SisSjsS†r
S0s
s
,(4)
where, the unitary matrix Sij diagonalizes the fusion matrix. Upon using the unitarity of the S-matrix, the algebra
(2) reduces to
NP =
k/2
?
r=0
Sj1 rSj2 r...Sjp r
(S0r)p−2
. (5)
Now, the matrix elements of Sijare known for the case under consideration (SU(2)kWess-Zumino model); they are
given by
Sij =
?
2
k + 2sin
?(2i + 1)(2j + 1)π
k + 2
?
, (6)
where, i, j are the spin labels, i, j = 0,1/2,1,....k/2. Using this S-matrix, the number of conformal blocks for the
set of punctures P is given by
?p
?
NP =
2
k + 2
k/2
?
r=0
l=1sin
?(2jl+1)(2r+1)π
(2r+1)π
k+2
k+2
??p−2
?
sin
?
. (7)
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Eq. (7) thus gives the dimensionality of the SU(2) Chern-Simons states corresponding to a three-fold bounded by a
two-sphere punctured at p points. The black hole microstates are counted by summing NPover all sets of punctures
P, Nbh=?
We are however interested only in the leading correction to the semiclassical entropy which ensues in the limit of
arbitrarily large AH. To this end, recall that the eigenvalues of the area operator [7] are given by
{P}NP. Then, the entropy of the black hole is given by Sbh = logNbh.
AH = 8πβ l2
Pl
p
?
l=1
[jl(jl+ 1)]
1
2,(8)
where, lPl is the Planck length, jl is the spin on the lth puncture on the 2-sphere and β is the Barbero-Immirzi
parameter [9]. Clearly, the large area limit corresponds to the limits k → ∞ , p → ∞. Now, from eq. (8), it
follows that the number of punctures p is largest for a given AHprovided all spins jl =
horizon area, we obtain the largest number of punctures p0as
1
2. Thus, for a fixed classical
p0 =
AH
4l2
Pl
β0
β
, (9)
where, β0= 1/π√3. In this approximation, the set of punctures P0with all spins equal to one-half dominates over
all other sets, so that the black hole entropy is simply given by
Sbh
= ln NP0 ,(10)
with NP0being given by eq. (7) with jl= 1/2.
Observe that NP0can now be written as
NP0=
2p0+2
k + 2
[F(k,po) − F(k,p0+ 2)] (11)
where,
F(k,p) =
[1
2(k+1)]
?
ν=1
cosp
?
νπ
k + 2
?
. (12)
The sum over ν in eq. (12) can be approximated by an integral in the limit k → ∞ , p0 → ∞, with appropriate
care being taken to restrict the domain of integration; one obtains
F(k,p0) ≈
?k + 2
π
? ?π/2
0
dx cosp0x ,(13)
so that,
NP0≈
2p0+2
π(p0+ 2)B (p0+ 1
2
,
1
2) , (14)
where, B(x,y) is the standard B-function [11]. Using well-known properties of this function, it is straightforward to
show that
ln NP0= p0ln2 −
3
2ln p0 − ln (2π)
+ O(p−2
0) .
−
5
2p−1
0
(15)
Substituting for p0as a function of AH from eq. (9) and setting the Barbero-Immirzi parameter β to the ‘universal’
value β0ln2 [1], one obtains our main result
Sbh = SBH −
3
2ln
?SBH
ln2
?
+ const. + ··· ,(16)
where, SBH= AH/4l2
Pl, and the ellipses denote corrections in inverse powers of AH or SBH.
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Admittedly, the above calculation is restricted to the leading correction to the semiclassical approximation. It has
been done for a fixed large AHby taking the spins on all the punctures to be 1/2 so that we have the largest number
of punctures. But it is not difficult to argue that the coefficient of the lnAH term is robust in that inclusion of spin
values higher than 1/2 do not affect it, although the constant term and the coefficients of sub-leading corrections
with powers of O(A−1
H) might get affected. The same appears to be true for values of the level k away from the
asymptotic value which we have assigned it above: the coefficient of the lnAH is once again unaffected. Thus, the
leading logarithmic correction with coefficient -3/2 that we have discerned for the black hole entropy is in this sense
universal. Moreover, although we have set β = β0 ln2 in the above formulae, the coefficient of the lnAH term is
independent of β, a feature not shared by the semiclassical area law.
It is therefore clear that the leading correction (and maybe also the subleading ones) to the BH entropy is negative.
One way to understand this could be the information-theoretic approach of Bekenstein [12]: black hole entropy
represents lack of information about quantum states which arise in the various ways of gravitational collapse that
lead to formation of black holes with the same mass, charge and angular momentum. Thus, the BH entropy is the
‘maximal’ entropy that a black hole can have; incorporation of leading quantum effects reduces the entropy. The
logarithmic nature of the leading correction points to a possible existence of what might be called a ‘non-perturbative
fixed point’. That this happens in the physical world of four dimensions is perhaps not without interest.
Recently, the zeroth and first law of black hole mechanics have been derived for situations with radiation present
in the vicinity of the horizon, using the notion of the isolated horizon [13]. Our conclusions above for the case of
non-rotating black holes hold for such generalizations [15] as well. Note however that while, the foregoing analysis
involves SU(2)k Chern Simons theory, for large k this reduces to a specific U(1) theory presumably related to the
‘gauge fixed’ classical theory discussed in [13]. The charge spectrum of this U(1) theory is discrete and bounded from
above by k. The SU(2) origin of the theory thus provides a natural ‘regularization’ for calculation of the number of
conformal blocks.
Note Added: After the first version of this paper appeared in the Archives, it has been brought to our attention
that corrections to the area law in the form of logarithm of horizon area have been obtained earlier [14] for extremal
Reissner-Nordstrom and dilatonic black holes. These corrections are due to quantum scalar fields propagating in fixed
classical backgrounds appropriate to these black holes. The coefficient of the lnAH term that appears in ref. [14] is
different from ours. This is only expected, since in contrast to ref. [14], our corrections originate from non-perturbative
quantum fluctuations of spacetime geometry (for generic non-rotating black holes), in the absence of matter fields.
Thus, this correction is finite and independent of any arbitrary ‘renormalization scale’ associated with divergences
due to quantum matter fluctuations in a fixed classical background.
We thank Prof. A. Ashtekar for many illuminating discussions and Prof. R. Mann for useful correspondence.
[1] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998).
[2] A. Strominger and C. Vafa, Phys. Lett. B379, 99 (1996).
[3] S. Carlip, Class. Quant. Grav. 16, 3327 (1999).
[4] J. Maldacena, A. Strominger and E. Witten, Jour. High Energy Phys. 12, 2 (1997).
[5] B. de Wit, Modifications of the area law and N = 2 supersymmetric black holes, hep-th/9906095 and references therein.
[6] R. Wald, Phys. Rev. D48, 3427 (1993); T. Jacobson, G. Kang and R. Myers, Phys. Rev. D49, 6587 (1994); V. Iyer and
R. Wald, Phys. Rev. D50, 846 (1995)..
[7] A. Ashtekar, Lectures on Non-perturbative Canonical Gravity, World Scientific, 1991; A. Ashtekar and J. Lewandowski in
Knots and Quantum Gravity, ed. J. Baez, Oxford University Press, 1994; Class. Quant. Grav. 14, A55 (1997); J. Baez,
Lett. Math. Phys. 31, 213 (1994); C. Rovelli and L. Smolin, Nucl. Phys. B331, 80 (1990); Nucl. Phys. B442, 593 (1995).
See also references quoted in A. Ashtekar, Interface of General Relativity, Quantum Physics and Statistical Mechanics:
Some Recent Developments, gr-qc/9910101.
[8] R. Kaul and P. Majumdar, Phys. Lett. B439, 267 (1998).
[9] F. Barbero, Phys. Rev. D54, 1492 (1996); G. Immirzi, Nucl. Phys. Proc. Suppl. 57, 65 (1997).
[10] P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, 1997, p 375.
[11] E. Whittaker and G. Watson, Modern Analysis, Cambridge, 1962.
[12] J. Bekenstein, Phys. Rev. D7, 2333 (1973).
[13] A. Ashtekar, C. Beetle and S. Fairhurst, Mechanics of isolated horizons, gr-qc/9907068, and references therein.
[14] R. Mann and S. Solodukhin, Nucl. Phys. 523B, 293 (1998) and references therein.
[15] A. Ashtekar, private communication.
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