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General Logarithmic Corrections to Black Hole Entropy

Saurya DasPhysics Dept., The University of Winnipeg and

Winnipeg Institute for Theoretical Physics,

515 Portage Avenue, Winnipeg, Manitoba R3B 2E9, CANADA

EMail: saurya@theory.uwinnipeg.ca

Parthasarathi MajumdarThe Institute of Mathematical Sciences,

C.I.T. Campus, Chennai - 600 113, INDIA

EMail: partha@imsc.ernet.in

Rajat K. BhaduriDept. of Physics and Astronomy,

Mc Master University,

Hamilton, Ontario L8S 4M1, CANADA

EMail: bhaduri@mcmail.cis.mcmaster.ca

We compute leading order corrections to the entropy of any thermodynamic system due to smallstatistical fluctuations around equilibrium. When applied to black holes, these corrections are shownto be of the form k ln(Area). For BTZ black holes, k = 3/2, as found earlier. We extend the resultto anti-de Sitter Schwarzschild and Reissner-Nordstrom black holes in arbitrary dimensions. Finallywe examine the role of conformal field theory in black hole entropy and its corrections.

I. INTRODUCTION

Since it is more or less certain that black holes much larger than the Planck scale have entropy proportional to itshorizon area [15], it is important to investigate what the leading order corrections are, as one reduces the size of theblack hole. There have been several attempts in this direction. For example, in non-perturbative quantum generalrelativity (NPQGR) (aka Quantum Geometry) [6], the density of microstates has been computed for non-rotatingasymptotically flat (AF) 4d black holes [3,7]. In this approach, the black hole horizon is treated as a boundary ofspacetime which is quantized within the NPQGR formalism. The dimensionality of Hilbert space of this theory,which is computed by counting the conformal blocks of a well-defined 2d CFT, namely 2d SU(2)k Wess-Zuminomodel, gives the density of states. For large black holes, the logarithm of this density of states is seen to be preciselythe Bekenstein-Hawking entropy together with corrections : 3/2 ln(Area) [8,9]. Similar logarithmic corrections werefound for quantum Schwarzschild black holes in [10] and for BTZ, string theoretic and all other black holes whosemicroscopic degrees of freedom are described by an underlying conformal field theory in [11] using a corrected versionof the asymptotic Cardy formula. More recently, precisely such corrections have been obtained for three dimensionalBTZ black holes by computing the exact partition function [12]. Logarithmic corrections to black hole entropy werealso obtained in [13] by considering matter fields in black hole backgrounds, in [14] in the context of string-black holecorrespondence, in [15] for dilatonic black holes and in [16] using Rademacher expansion of the partition function.Thus, logarithmic corrections to Bekenstein-Hawking entropy appear to be a generic property of black holes. Thequestion is why are they always logarithmic.We try to address this question here. In this paper, we show that logarithmic corrections to thermodynamic entropy

arise in all thermodynamic systems when small stable fluctuations around equilibrium are taken into account. Thestability condition is equivalent to the specific heat being positive, so that the corresponding canonical ensemble isstable. These fluctuations determine the prefactor in the expression for the density of states of the system, whoselogarithm gives the corrected entropy of the system (similar to the entropy correction due to the exact prefactor inCardy formula, ref. [11]). As shown in section (II), these corrections are always logarithmic in nature, and whenapplied to black holes in section (III), they indeed predict the form k ln(Area). Thus logarithmic corrections toBekenstein-Hawking entropy can be interpreted as corrections due to small thermal fluctuations of the black holearound its equilibrium configuration. Note that this prescription applies to all black holes with positive specificheat. Also note that this analysis simply uses macroscopic black hole properties such as expressions of entropy andtemperature in terms of its mass and charge etc, but does not use properties of the underlying microscopic theoryof gravity (CFT, quantum geometry, string theory etc). In this sense, the corrections are indeed universal, and will

1

apply to almost all black hole spacetimes, irrespective of whether they arise in Einstein gravity or some other versionsof general relativity, whether they are supersymmetric or not.When applied to BTZ black holes, the prefactor turns out to be k = 3/2, as found in [11]. Corrections to entropy

of AF Schwarzschild black holes cannot be found in this way, since the specific heat of the latter is negative andits canonical ensemble is unstable [17,18]. However, the problem can be circumvented by considering sphericallysymmetric black holes which are aymptotically anti-de Sitter, and not AF. Then, once again, the leading orderentropy corrections are logarithmic in nature. Similarly, for Reissner-Nordstrom black holes in arbitrary spacetimedimensions, there exists a range of mass and electric charge, for which the specific heat is positive. In this case too,logarithmic corrections result.Finally, in section (IV) we examine thermodynamic entropy as an exact function of the inverse temperature

(which is not necessarily the equilibrium temperature), namely S = S(). We note that the function suggested byCFT [11], and suitable generalizations thereof, give rise to identical logarithmic corrections (including coefficients) asfound in the earlier section from pure thermodynamic considerations.It may be noted that in the analysis of this paper, black holes are still considered large (macroscopic) compared

to the Planck scale. This ensures that the logarithmic terms are indeed much smaller compared to the Bekenstein-Hawking term, and can thus be regarded as corrections.

II. CANONICAL AND MICROCANONICAL ENTROPY

We consider a canonical ensemble with partition function

Z() =

0

(E)eEdE , (1)

where T = 1/ is the temperature in units of the Boltzmann constant kB. The density of states can be obtained from(1) by doing an inverse Laplace transform (keeping E fixed) [19,20]:

(E) =1

2i

c+ici

Z()eEd =1

2i

c+ici

eS()d , (2)

where

S() = lnZ() + E (3)

is the exact entropy as a function of temperature, not just its value at equilibrium. It is formally defined as the sumof entropies of subsystems of the thermodynamical system, which are small enough to be themselves in equilibrium.The complex integral can be performed by the method of steepest descent around the saddle point 0(= 1/T0),such that S0 := (S()/)=0 = 0. T0 is the equilibrium temperature, such that the usual equilibrium relationE = ( lnZ()/)=0 is obeyed. Expanding S() about = 0, we get

S = S0 +1

2( 0)2S0 + , (4)

where S0 := S(0) and S

0 := (2S()/2)=0 . Substituting (4) in (2) :

(E) =eS0

2i

c+ici

e1/2(0)2S

0 d . (5)

By choosing c = 0, and putting ( 0) = ix, where x is a real variable, we obtain, for S0 > 0,

(E) =eS02S0

. (6)

Note that the density of states (E) and S0 have dimensions of inverse energy and energy squared respectively.Henceforth, we set the Boltzmann constant kB to unity. The logarithm of the density of states (E) is then themicrocanonical entropy

S := ln (E) = S0 12lnS0 + (higher order terms). (7)

2

Here S is the corrected microcanonical entropy at equilibrium, obtained by incorporating small fluctuations aroundthermal equilibrium. This is to be distinguished from the function S(), which is the entropy at any temperature (andnot just at equilibrium). The above formula is applicable to all thermodynamic systems considered as a canonicalensemble, including black holes. For the latter, S0 is the Bekenstein-Hawking area law. It is seen that the correctionsolely depends on the quantity S0 . Moreover, the second term in Eq.(7) appears with a negative sign, as conjecturedin general in [9] from the holographic point of view. Now, we will estimate S0 , without assuming any specific form ofS(). From Eq.(3), it follows that

S() =1

Z(2Z()

2) 1

Z2(Z

)2 . (8)

This means that S0 is nothing but the fluctuation squared of energy from the equilibrium, i.e.,

S0 =< E2 > < E >2 , (9)

where, by the definition of 0, E =< E >= ( lnZ/)=0 . It immediately follows thatS0 = T

2C (10)

where C (E/T )T0 is the dimensionless specific heat. The above result also follows from a detailed fluctuationanalysis of a stable thermodynamic system [21].Substituting for S0 from (10) in (7), we get:

S = ln = S0 12ln(C T 2

)+ (11)

This is our master formula, and for dimensional correctness, it is understood that the quantity within the logarithmis divided by k2B. As mentioned earlier, the above correction arises for any thermodynamic system satisfying the firstlaw. We would like to apply this to the particular case of black holes [22]. To do this, we would replace T TH , theHawking temperature. Then we calculate C for the specific black holes.Before that, let us determine the limit of validity of the formula (11). We assume the quantum fluctuations of the

thermodynamic quatities under consideration, e.g. S, , are small. Denote such a generic variable by y. Then theabove restriction translates to y y. Then from the uncertainty relation for quasi-classical systems [23]

Ey yand approximating y = y/ (where is relaxation time scale), it follows that:

E >> 1/ .

Then, using E = TS, we get:

S >>1

T.

Finally, assuming S 1, we get from above [21]:

T >>1

(12)

which sets the limit that (11) ceases to be valid for very low temperatures, lower than a characteristic value 1/ . Forblack holes, is presumably of order Planck length, the only natural length (or time) scale relevant to the system.Note that the above inequality shows that for black holes very close to extremality (T 0), the fluctuation analysisceases to be valid due to large quantum fluctuations. This is consistent with earlier results which showed that nearextremal black holes are highly quantum objects, irrespective of their size [24].

III. APPLICATION TO BLACK HOLES ENTROPY

A. BTZ Black Hole:

First let us consider a non-rotating BTZ black hole in 3-dimensions with metric [25]:

3

ds2 = (r2

2 8G3M

)dt2 +

(r2

2 8G3M

)1dr2 + r2d2 . (13)

Its Bekenstein-Hawking entropy and Hawking temperature are given by

S0 =2r+4G3

(14)

TH =r+22

=

[G322

]S0 (15)

where r+ =8G3M is horizon radius (G3 = 3-dimensional Newtons constant), M being mass of the black hole and

is related to the cosmological constant by: = 1/2. The first law in this case is:dM = THdS0 (16)

Since there are no work terms, the specific heat in this case is simply a total derivative :

C =dM

dTH= S0 =

[22

G3

]TH (17)

Plugging in (15) and (17) in (11), we get.

S = ln = S0 12ln(S0S

20

)+

= S0 32lnS0 + . (18)

The quantity in square brackets in (15) is a fixed constant independent of black hole parameters, and hence can beignored for S0 >> 1 (macroscopic black holes). First, note that the correction term is proportional to the logarithmof horizon area. Secondly, the proportionality constant 3/2 is exactly that found in [11]. As claimed earlier, thisshows that logarithmic corrections of this type can be interpreted as those due to thermal fluctuations of the blackhole around its equilibrium.Introduction of angular momentum is straightforward. It can be shown that for rotating BTZ black holes far away

from extremality, the formulae (15) - (17) are modified as :

TH =

[G322

]S0 +O

(J2

M3/2

)

C :=

(THM

)1

J

= S0 +O(

J2

M3/2

)

where J is the angular momentum, and the specific heat C is now calculated keeping J fixed. Thus the entropy is,

S = ln = S0 32lnS0 (J

2/M3/2)

2S0+ (19)

Thus to leading order the correction (18) is reproduced, and sub-leading angular momentum dependent terms con-tribute further polynomial corrections of order 1/S0, the inverse of Bekenstein-Hawking entropy. Such smaller poly-nomial corrections were also predicted in [8,9] in the context of quantum geometry. For near-extremal (very lowtemperature) situations, the corrections of above type are no longer valid for reasons stated at the end of the lastsection, as the necessary condition (12) is violated.

B. Anti-de Sitter Schwarzschild:

First let us consider a d-dimensional Schwarzschild black hole with metric:

ds2 = (1 16GdM

(d 2)d2rd3)dt2 +

(1 16GdM

(d 2)d2rd3)1

dr2 + r2d2d2 , (20)

4

(where Gd = d dimensional Newtons constant, d2d2 is the metric on unit Sd2 and d2 is the area of that unitsphere). Its entropy and temperature are :

S0 =d2 r

d2+

4Gd(21)

TH =(d 3)4r+

(22)

The mass is related to the horizon radius as :

M =(d 2)(d 3)d2

16Gdrd3+ (23)

with which one can write

TH =

(d 24

)(d2)/(d3)(d24GdM

)1/(d3). (24)

Again, the first law (16) is obeyed, and

C =dM

dTH= (d 2)S0 (25)

Since the specific heat is negative, one cannot directly apply the fluctuation correction formula (11) to theSchwarzschild black hole. One way to circumvent this problem is to consider anti-de Sitter Schwarzschild blackholes, whose asymptotic structure is that of anti-de Sitter space, instead of asymptotically flat Schwarzschild blackholes. We will see that for black holes with large masses, the presence of a small negative cosmological constant is sufficient to make the specific heat positive and render (11) applicable. The microcanonical ensemble for theSchwarzschild black hole is of course well defined, and we will return to this issue later.We parametrize the cosmological constant in d-dimensions in terms of the length scale as: = (d1)(d2)/22.

The corresponding Einsteins equations admit of black holes solutions which are asymptotically anti-de Sitter andwhose metric is given by:

ds2 = (1 16GdM

(d 2)d2rd3 +r2

2

)dt2 +

(1 16GdM

(d 2)d2rd3 +r2

2

)1dr2 + r2d2d2 . (26)

Here the parameter M is the conserved charge associated with the timelike Killing vector (mass) [26]. In terms ofthe horizon radius r+, the entropy formula is same as (21), whereas the Hawking temperature is :

TH =(d 1)r2+ + (d 3)2

42r+=

[d 34

(d24Gd

)1/(d2)] [1 +

(d 1)r2+(d 3)2

]S1/(d2)0 . (27)

Although the function r+ = r+(M) is not explicit, C can still be calculated using :

C =(d 2)d2rd3+

4Gd

[ lnTHr+

],

which yields [27] :

C =

[(d 2)(d 1)r

2+/

2 + (d 3)(d 1)r2+/2 (d 3)

]S0 . (28)

First, note that the limit reproduces the (negative) Schwarzschild value (25) of C. However, for

< r+

d 1d 3 , (29)

C changes signature and becomes positive. In the limit of large ( 0), C approaches a positive finite value:

5

C = (d 2)S0 . (30)

Consequently, the canonical ensemble becomes stable against small fluctuations around equilibrium [18]. In otherwords, introduction of a negative cosmological constant remedies the otherwise unstable Schwarzschild canonicalensemble.Let us calculate the leading order corrections in this limit, although one can do so for any satisfying (29). From

(11), (27) and (30), we get:

S = S0 d2(d 2) lnS0 + (31)

Thus, once again the leading corrections are logarithmic. It is clear that logarithmic corrections result for all valuesof lying in the range (29). Also note from (29) that for black hole masses much greater than the Planck mass, r+ ishuge and the range of || corresponding to C > 0 is given by the expression:

|| > (d 2)(d 3)2r2+

0 . (32)

This means that the introduction of a very small negative cosmological constant is sufficient to make the presentformalism applicable.

C. Reissner-Nordstrom :

For a d-dimensional Reissner-Nordstrom black hole with metric [28]

ds2 = (1 16M

(d 2)d2rd3 +16Q2

(d 2)(d 3)r2(d3))dt2

+

(1 16M

(d 2)d2rd3 +16Q2

(d 2)(d 3)r2(d3))1

dr2 + r2d2d2 , (33)

the entropy and temperature are given by:

S0 =d2 r

d2+

4Gd(34)

TH =(d 3)2 rd2+

(8GdM

(d 2)d2

)2 2GdQ

2

(d 2)(d 3) (35)

where Q is the electric charge. Using the formula for horizon radius:

rd3+ =8GdM

(d 2)d2 +(

8GdM

(d 2)d2

)2 2GdQ

2

(d 2)(d 3) , (36)

the specific heat is :

C =

(d 2d 3

)S0 F2(M,Q) (37)

and the temperature far away from extremality (r+ >> r):

TH =

[d 34

(d24Gd

)1/(d2)]S1/(d2)0 F1(M,Q) (38)

where

6

F1(M,Q) = 1 2Gd(d 2d 3

d2Q

16GdM

)2

F2(M,Q) =

(8piGdM

(d2)d2

)2 2GdQ2(d2)(d3)

8piGdM(d2)d2

(d2)(d3)(

8piGdM(d2)d2

)2 2GdQ2(d2)(d3)

.

When the electric charge is very small, constant F1 is of order unity, F2 is negative and C is close to the (negative)Schwarzschild result (25), i.e. C (d 2)S0. However, in the range:[

2(d 2)2d(8)2(d 3)Gd

]Q2 < M2 0 and the leading order entropy correction from (11) is:

S = S0 (d 4)2(d 2) lnS0 + ln

(F 21F2

)+ . (40)

It is interesting to note that the fluctuation corrections in this case vanish for d = 4.As for BTZ, close to extremality, TH 0, and the assumptions made in the thermodynamic analysis breaks down,

as can be seen from (12). Thus (11) ceases to be valid too.Since the string theoretic black holes in 5 and in 4 dimensions, whose entropy was reproduced by D-brane state

counting, fall into the class of Reissner-Nordstrom black holes, we do not consider them separately.

IV. CFT AND EXACT ENTROPY FUNCTION S() :

Thus far, we have seen that logarithmic leading order corrections to entropy are generic to all statistical mechanicalsystems near its equilibrium due to small thermal fluctuations. The quantity S0 was sufficient to calculate thesecorrections, and the former can be expressed in terms of known thermodynamic quantities C and T .In other words, an exact entropy function S(), although would be sufficient to get above corrections, was not at

all necessary. In this section, we will attempt to reproduce these corrections by postulating an exact entropy functionS(), which follows from CFT and may also perhaps follow from other quantum theories of gravity. Thus, on theone hand, it will provide a consistency check between the derivations of the previous sections and corrections derivedfrom certain microscopic quantum gravitational principles, and on the other hand, it will provide an indication as towhat kind of entropy functions can be expected from underlying theories of quantum gravity.Let us return to Eq. (7), which as mentioned before, predicts leading correction to the equilibrium entropy in terms

of S0 , the second derivative of the exact entropy function S(), evaluated at the equilibrium temperature 0. Asstated before, if we wish to specialize to black holes, we must first make the following identifications:

0 =1

TH, (41)

S0 = S(0) =AH4Gd

(42)

where the first relation follows from the assumption that the black hole is at equilibrium at Hawking temperature.It is clear that this set of information is inadequate to determine the functional form of S0 , which is independent ofboth S0 and 0. Geometrically, the quantities 0 and S0 represent the abscissa and ordinate of the extremum of theparabola which approximates the actual curve near extremum. The slope S0 on the other hand is still an independentquantity.To determine S0 , we may assume any specific form of the function S(), which can be well approximated by the

parabolic form (4). The only stipulation is that such a form must admit of an extremum at some value 0 of . If weassume an underlying CFT, then from modular invariance of the partition functions of (1+1)-dimensional conformalfield theories it follows that S() is of the following form (see argument of exponential function in Eq(2.7) in [11] ):

S() = a +b

(43)

7

where a, b are constants. However, to test whether this form is absolutely essential, and whether more generalfunctional forms affect results significantly, we will assume a more general form which also admits of a saddle point,and hence the thermodynamic limit. We choose such a function:

S() = am +b

n(44)

wth m,n, a/b > 0. The form dictated by CFT is a special case (m = 1 = n) of the above. Later we will show that(44) is also consistent with assumptions made in the context of Euclidean black hole thermodynamics [17]. A constantadditive term that could have been included may be absorbed in the definition of S0. The above has an extremum at

0 =

(nb

ma

)1/(m+n) 1

TH. (45)

Expanding around 0, by computing the second derivative, we get:

S() = (anbm)1/(m+n)

+1

2(an+2bm2

)1/(m+n)( 0)2 + , (46)

where

= (n/m)m/(m+n) + (m/n)n/(m+n)

= (m+ n)m(n+2)/(m+n)n(m2)/(m+n)

are constants. Comparing with (4), we find:

S0 = (anbm)

1/(m+n), (47)

S0 = (an+2bm2

)1/(m+n). (48)

Next, we invert these equations to get a, b in terms of S0 and S

0 :

a =(m2)/2

m/2(S0 )

m/2S(m2)/20 (49)

b =

[(n+2)/2

]n/2S(n+2)/20 (S

0 )n/2

. (50)

These values of a and b can now be substituted in (45) to obtain:

0 =1

TH=( nm

)1/(m+n)S0S0

. (51)

The last relation can be inverted to express S0 in terms of S0 and TH :

S0 =

[(

) ( nm

)2/(m+n)]S0T

2H . (52)

This is the relation we sought, since it completely specifies S0 (and hence entropy corrections) in terms of Bekenstein-Hawking entropy and Hawking temperature, and is a direct consequence of our assumed form of exact entropy function(44). Again, the factors in square brackets are independent of black hole parameters and can be dropped. Substitutingin (7), we get:

S = ln = S0 12ln(S0T

2H

)+ smaller terms (53)

This is the generic formula for leading order corrections to Bekenstein-Hawking formula. We note that the leadingorder correction is independent of m,n and the latter has an effect only in the sub-leading terms. We will return tothis issue later. Now let us apply (53) to specific black holes.

BTZ :

8

From (17), we see that S0 = C for BTZ. Thus, the correction (53) becomes identical to (11) ! Or, by explicitcalculation using the relation (15) between temperature and entropy of BTZ black hole, we can write (53) as:

S = ln = S0 12ln(S0S

20

)+

= S0 32lnS0 + (54)

and get back the logarithmic correction for BTZ black hole entropy found in [11] and by the general thermodynamicanalysis (18), including the right coefficient 3/2 in front. Thus, the general form of the function S() in (44) issufficient to arrive at this correction, and not just the specific m = 1 = n case as dictated from CFT. This makes thecorrection quite robust.As before, angular momentum can be introduced quite easily, and the results continue to be valid so long as the

black hole is away from extremality.

AdS-Schwarzschild :Again for Schwarzschild-anti-de Sitter black holes, in the positive specific heat domain (29), it follows from (28)

that C = (d 2)S0 to leading order. Then, from (53), we get:

S = ln = S0 12ln(CT 2H

)+ (55)

which agrees with the fluctuation correction formula (11). Hence, the logarithmic correction term will have identicalcoefficient as in (31).

Reissner-Nordstrom :The generalization to charged black holes is also straightforward. From (37), we see that C S0F2(M,Q). Thus,

from (53) we get:

S = ln = S0 12ln(CT 2H

)+ lnF2 + (56)

which is identical to (40) to leading order.

V. DISCUSSIONS

In this paper, we assumed that a black hole is a thermodynamic system in equilibrium at Hawking temperature,and systematically analyzed its small statistical fluctuations around equilibrium. These fluctuations give rise to anon-trivial multiplicative factor in front of the expression for the density of states, whose logarithm gives the correctedentropy formula. The latter consists of the zeroth order term which is the Bekenstein-Hawking entropy AH/4Gd andan additional correction term which is logarithmic for all black holes. This shows that logarithmic corrections toBekenstein-Hawking entropy arise due to statistical fluctuations around thermal equilibrium of the black hole. Thisalso explains as to why leading correction terms have always been found to be logarithmic. Next, we explicitlyevaluated this logarithmic correction term for various black holes.Finally, we showed that identical correction terms result if one assumes a certain form of the entropy function S()

which is valid away from equilibrium, and which is consistent with conformal field theory. This consistency on the onehand lends additional support to certain underlying theories of black hole dynamics that have been used previouslyand on the other hand gives an indication of the nature of entropy function expected from other microscopic theories.Now, it may be mentioned that to demonstrate the instability of Schwarzschild black holes, it was assumed in [17]

that the exact entropy function S() is obtained from the well known relation at equilibrium (i.e. the area law)S(0), by simply substituting 0 . While this is sufficient for the area law to be valid at Hawking temperature,it is certainly not necessary. However, here we will show that the above assumption is perfectly consistent with theentropy function (44) that we chose in the last section.Again consider BTZ black holes. Using (15) and 0 = 1/TH, we get at equilibrium :

S(0) =22

G3

1

0. (57)

9

Following [17] if we assume that the above continues to hold for any , then we get :

S() =22

G3

1

. (58)

Also from (15) and the expression r+ =8G3M, we can write:

0 =2G3M

. (59)

This implies that for BTZ black hole mass much greater than the Planck mass, 1/0 0. Also forM/MPl , letus assume that the above inequality holds for temperatures slightly away from equilibrium (since for truly macroscopicblack holes, fluctuations are expected to be small). Then, from Eq.(44) it follows that :

limM/MP l

S() =b

n. (60)

Comparing (58) and (60) we find:

n = 1

& b =()2

G3.

In other words, the assumed relation (60) is perfectly consistent with the our general chosen form (44) in the limitof macroscopic black holes. This can also be seen from the fact that if (60) holds with n = 1, then it impliesS0 3 S30 , which using (7) in turn implies the required corrected entropy

S = S0 32lnS0 +

Similar conclusions follow for AdS-Schwarzschild black hole in d-dimensions. In this case, for large black holes, itagain follows from (44) that S() b/n. On the other hand, using (27) and assuming as before that the formulacontinues to be valid away from equilibrium, it follows that:

S() =

(42

d 1)d2

d24Gd

(d2) . (61)

Comparing, we find:

n = d 2

& b =

(42

d 1)d2

d24Gd

(d2) .

This shows that the more general form (44) with m,n 6= 1 is relevant for black holes other than BTZ. The above alsoimplies S0 d0 Sd/(d2)0 and

S = S0 d2(d 2) lnS0 +

as found earlier in (31).The above examples lend support to the entropy function (44). However, since the exponents m,n have no effect on

leading order corrections (implying the CFT case m = n = 1 is sufficient), it is tempting to speculate that conformalfield theory underlies all black hole descriptions in nature as was proposed in [4].Finally, let us comment on the role of the current analysis to entropy of d = 4 Schwarzschild black holes. The log

corrections found in [8,9] clearly does not fall into this category, since as remarked earlier, the instability of canonicalensemble for Schwarzschild black holes implies that its corrections should be calculated in the microcanonical picture,whereby the energy (and hence area) of the black hole is held fixed. This is precisely what was done in [8,9], wherethe area was held fixed during counting of the dimension of Hilbert space of states of spin networks. Presumably,similar analysis should be done for all black holes which have negative specific heat. Thus, the computation done hereand that in [8,9] actually complement each other and show that logarithmic corrections are indeed universal. Since

10

stability of a Schwarzschild black hole can also be ensured by confining it in a box [29], it would be interesting to seewhat the entropy corrections are in that case. Also, one can try to apply our corrections after taking into accounthigh temperature degrees of freedom, in which case the specific has been claimed to be positive [30]. It may be notedthat the entropy correction found here has the interesting property

S(A1 +A2) > S(A1) + S(A2)which is valid only if the coefficient in front of the log correction is negative, as in our case. It is also interestingto note that under certain assumptions, when black holes are thought to be composed of mutually non-interactingbut distinguishable particles in the grand canonical ensemble with fluctuating particle number, the entropy of theresultant system turns out to be proportional to area, followed by log area term [31], although in the latter case thelog correction occurs with a positive sign. Also, it might be useful to understand the implications of these correctionsfor the holographic hypothesis. We hope to report on it in the future.

Acknowledgements

SD and PM thank G. Kunstatter for useful discussions and interesting comments. SD thanks S. Carlip and V.Husain for useful correspondence which included several interesting comments and suggestions, A. Dasgupta forbringing to his attention refs. [27] and [29] and J. Bland for comments. PM thanks T. R. Govindarajan, R. Kaul,R. B. Mann, M. V. N. Murthy and D. Page for helpful comments. He also thanks the Physics Departments at theUniversities at Winnipeg and Waterloo for hospitality. This work was supported in part by the Natural Sciences andEngineering Research Council of Canada.

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