Eur. Phys. J. C (2010) 67: 205–211DOI 10.1140/epjc/s10052-010-1279-5

Regular Article - Theoretical Physics

Quantum corrections to the entropyof charged rotating black holes

M. Akbar1,a, K. Saifullah2,b

1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Rawalpindi, Pakistan2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Received: 12 October 2009 / Revised: 11 January 2010 / Published online: 9 March 2010© Springer-Verlag / Società Italiana di Fisica 2010

Abstract Hawking radiation from a black hole can beviewed as quantum tunneling of particles through the eventhorizon. Using this approach we provide a general frame-work for studying corrections to the entropy of black holesbeyond semiclassical approximations. Applying the proper-ties of exact differentials for three variables to the first lawthermodynamics, we study charged rotating black holes andexplicitly work out the corrections to entropy and horizonarea for the Kerr–Newman and charged rotating BTZ blackholes. It is shown that the results for other geometries likethe Schwarzschild, Reissner-Nordström and anti-de Sitter–Schwarzschild spacetimes follow easily.

1 Introduction

Hawking’s ground-breaking work [1–3] on black hole evap-oration and information loss is based on the idea that a pairof particles is created just inside the event horizon and fromthis pair the positive energy particle tunnels out of the holeand appears as Hawking radiation. The negative energy par-ticle tunnels inwards and results in decrease of the massof the black hole. The energy of the particle changes signas it crosses the horizon. These particles follow trajectorieswhich cannot be explained classically. This process of par-ticle evaporation from black holes is thus a phenomenonof quantum tunneling of particles through the event hori-zon [4, 5]. This contributes to the change in mass, angularmomentum and charge of the black hole, which change itsthermodynamics as well. The particle travels in time so thatthe action becomes complex and the dynamics of the out-going particle is governed by the imaginary part of this ac-tion. This action has been calculated using the radial null

a e-mail: makbar@camp.nust.edu.pkb e-mail: saifullah@qau.edu.pk

geodesics [4, 5] or the so-called Hamilton–Jacobi method[6] for various spacetimes. Using the method of the radialnull geodesics Hawking radiation as a tunneling process hasbeen analyzed for the Kerr and Kerr–Newman black holes[7]. They have done the analysis using transformed forms ofthese metrics and they do not consider the question of quan-tum corrections at all. The Hamilton–Jacobi method hasbeen used to calculate quantum corrections to the Hawkingtemperature and the Bekenstein–Hawking area law for theSchwarzschild, anti-de Sitter–Schwarzschild and Kerr blackholes [6]. These corrections have been worked out for theuncharged BTZ black hole as well [8].

In this paper we extend this analysis to black holes thatare not spherically symmetric and are charged and rotating.For this purpose we set up a criterion for exactness of dif-ferential of black hole entropy, from the first law of ther-modynamics for three parameters (mass, angular momen-tum and charge), so that entropy can be written in a con-venient form. Using this we provide a sufficiently generalframework for calculating quantum corrections beyond thesemiclassical limit of entropy. As a result of quantum effectsthe famous Bekenstein–Hawking area law is also modified.We apply this to the Kerr–Newman and the charged rotat-ing BTZ black holes, in particular. We find that the leadingcorrection term is logarithmic, which is consistent with theresults found using quantum geometry techniques and fieldtheoretic methods. The other terms involve ascending pow-ers of inverse of the area. Further, we show that the earlierinvestigations for finding (quantum) corrections are recov-ered as special cases of the present study.

The paper is organized as follows. In the next section webriefly provide motivation for studying quantum correctionsof black hole entropy. In Sect. 3 we set up a criterion forexactness of differentials in three variables and apply thisto the first law of thermodynamics. In Sects. 4 and 5 weapply the above theory to the cases of the Kerr–Newman

206 Eur. Phys. J. C (2010) 67: 205–211

and charged rotating BTZ black holes, respectively. At theend we give conclusions.

2 Quantum corrections for entropy of black holes

In semiclassical treatment the radial trajectories of a mass-less particle in a spacetime described by the metric, gμν ,have the wave function

φ(r, t) = e− i�S(r,t), (1)

which satisfies the Klein–Gordon equation

− �2

√−g∂μ

[gμν√−g∂ν

]φ = 0. (2)

When a particle with positive energy crosses the horizonand tunnels out, it escapes to infinity and appear as Hawk-ing radiation. However, when a particle with negative energytunnels inwards it is absorbed by the black hole and as a re-sult the mass of the black hole decreases. The essence of thequantum tunneling argument for Hawking radiation is thecalculation of the imaginary part of the action, which for ablack hole of radius r and momentum pr is defined by [4, 5]

Im S = Im∫ rout

rin

pr dr

= Im∫ rout

rin

∫ pr

0dp′

r dr. (3)

If we use Hamilton’s equation

r = dH

dpr

∣∣∣∣r

, (4)

in the above action, we get

Im S = Im∫ rout

rin

∫ H

0

dH ′

rdr. (5)

For the Schwarzschild spacetime, for example, rin = 2M

and rout = 2(M − ω), where ω denotes the energy. We ex-pand the action S(r, t) in powers of �

S(r, t) = S0(r, t) + �S1(r, t) + �2 S2(r, t) + · · · , (6)

such that S0 gives the semiclassical value and higher or-der terms correspond to quantum corrections. An analysissimilar to the one done earlier [6] shows that the correctionterms Si are of the order of �

i and are proportional to thesemiclassical contribution S0. The Klein–Gordon equationand the dimensional analysis suggest that the constants ofproportionality for charged rotating black holes have the di-mensions of (r2+ + a2)−i , where r+ represents the horizon.

In order to make the corrections dimensionless, we write theseries expansion as

S = S0

[

1 +∑

i

αi�i

(r2+ + a2)i

]

. (7)

This expansion will be used later to work out quantum cor-rections in entropy and horizon area in different geometries.

3 Exact differentials in three variables and the first lawof thermodynamics

We will be using the first law of thermodynamics in three pa-rameters, for which we need the analysis of differentials inthree variables. The three-dimensional differential of a func-tion f ,

df (x, y, z) = A(x,y, z) dx + B(x, y, z) dy

+ C(x, y, z) dz (8)

is exact if the following conditions hold:

∂A

∂y= ∂B

∂x,

∂A

∂z= ∂C

∂x,

∂B

∂z= ∂C

∂y. (9)

Here we have

∂f

∂x= A,

∂f

∂y= B,

∂f

∂z= C. (10)

For df to be exact (8) will have the solution of the form

f (x, y, z) =∫

Adx +∫

B dy +∫

C dz

−∫ (

∂

∂y

(∫Adx

))dy

−∫ (

∂

∂z

(∫Adx

))dz

−∫ (

∂

∂z

(∫B dy

))dz

+∫

∂

∂z

(∫ (∂

∂y

(∫Adx

))dy

)dz. (11)

We apply this criterion to the first law of thermodynamicsfor charged rotating black holes, which for three parametersM,J,Q, the mass, angular momentum and charge of theblack hole, respectively, can be written in the form

dM = T dS + Ω dJ + Φ dQ, (12)

where, T is the temperature, S entropy, Ω angular velocityand Φ electrostatic potential of the black hole. Or we canrewrite it as

dS(M,J,Q) = 1

TdM − Ω

TdJ − Φ

TdQ. (13)

Eur. Phys. J. C (2010) 67: 205–211 207

Written in this way the A,B,C of conditions (9) will bereplaced by 1/T , −Ω/T , −Φ/T , in which case M,J,Q

will play the role of x, y, z, respectively. In other words, wenote that in order for dS to be an exact differential (13) thefollowing conditions must be satisfied:

∂

∂J

(1

T

)= ∂

∂M

(−Ω

T

), (14)

∂

∂Q

(1

T

)= ∂

∂M

(−Φ

T

), (15)

∂

∂Q

(−Ω

T

)= ∂

∂J

(−Φ

T

). (16)

Now we can use (11) to write the entropy for the blackhole in integral form as

S(M,J,Q) =∫

1

TdM −

∫Ω

TdJ −

∫Φ

TdQ

−∫ (

∂

∂J

(∫1

TdM

))dJ

−∫ (

∂

∂Q

(∫1

TdM

))dQ

+∫ (

∂

∂Q

(∫Ω

TdJ

))dQ

+∫

∂

∂Q

(∫ (∂

∂J

(∫1

TdM

))dJ

)dQ.

(17)

We will see in the next sections that this provides a conve-nient way to calculate quantum corrections for entropy ofcharged rotating black holes.

4 Entropy corrections for the Kerr–Newman black hole

We want to calculate quantum corrections to entropy beyondthe semiclassical limit for charged rotating black holes. Thecorresponding Bekenstein–Hawking area law will also un-dergo corrections and will be modified. For this purpose, wewill use the criterion described above for exactness of differ-ential of entropy. We first consider the Kerr–Newman blackhole, which in the Boyer–Lindquist coordinates (t, r, θ,φ)

is given by

ds2 = −2

ρ2

(dt − a sin2 θ dφ

)2

+ ρ2

2dr2 + ρ2 dθ2

+ sin2 θ

ρ2

(a dt − (

r2 + a2)dφ)2

,

where

(r)2 = (r2 + a2) − 2Mr + Q2,

ρ2(r, θ) = r2 + a2 cos2 θ,

a = J

M.

To obtain the horizons for this metric, we put grr = 0, andget

r± = M ±√

M2 − a2 − Q2. (18)

The Hawking temperature is defined as

T =(

�

4π

)(r+ − r−r2+ + a2

), (19)

which in our case takes the form

T =(

�

2π

) √M4 − J 2 − Q2M2

M(2M2 − Q2 + 2√

M4 − J 2 − Q2M2). (20)

The angular velocity, Ω = a/(r2+ +a2), for the above metrictakes the form

Ω = J

M(2M2 − Q2 + 2√

M4 − J 2 − Q2M2), (21)

and the electrostatic potential, Φ = r+Q/(r2+ + a2), be-comes

Φ = Q(M2 + √M4 − J 2 − Q2M2)

M(2M2 − Q2 + 2√

M4 − J 2 − Q2M2). (22)

It can easily be verified that the above quantities for theKerr–Newman black hole satisfy conditions (14)–(16). ThusdS is an exact differential and we can use the integral form(17) to work out the semiclassical entropy for this blackhole. In order to do this note that the expressions for T , Ω

and Φ as given above satisfy

∂

∂J

∫ (dM

T

)= −Ω

T, (23)

∂

∂Q

∫ (dM

T

)= −Φ

T, (24)

∂

∂Q

∫ (−ΩdJ

T

)= −Φ

T. (25)

If we use these in (17), it reduces to

S(M,J,Q) =∫

dM

T

= 2π

�

∫M(2M2 − Q2 + 2

√M4 − J 2 − Q2M2)

√M4 − J 2 − Q2M2

dM

= π

�

(2M2 − Q2 + 2

√M4 − J 2 − Q2M2

), (26)

208 Eur. Phys. J. C (2010) 67: 205–211

which is the standard black hole entropy given in the litera-ture [9, 10]. Using (18) this can also be written as

S = π

�

(r2+ + a2). (27)

Our aim now is to calculate quantum corrections to thisformula. In order to do this we need to use the correctedform of the Hawking temperature [6], which in our case isgiven by

Tc = T

(1 +

∑ αi�i

(r2+ + a2)i

)−1

, (28)

where αi correspond to higher order loop corrections to thesurface gravity of black holes K = 2πT , and the modifiedform of the surface gravity [11, 12] due to quantum effectsbecomes

K = K0

(1 +

∑

i

αi�i

(r2+ + a2)i

)−1

. (29)

Now in the first law of thermodynamics (13) temperature,T , will be replaced by the corrected temperature, Tc. If weinclude the correction terms the conditions (14), (15), and(16) take the following form:

∂

∂J

(1

T

)(1 +

∑ αi�i

(r2+ + a2)i

)

= ∂

∂M

(−Ω

T

)(1 +

∑ αi�i

(r2+ + a2)i

), (30)

∂

∂Q

(1

T

)(1 +

∑ αi�i

(r2+ + a2)i

)

= ∂

∂M

(−Φ

T

)(1 +

∑ αi�i

(r2+ + a2)i

), (31)

∂

∂Q

(−Ω

T

)(1 +

∑ αi�i

(r2+ + a2)i

)

= ∂

∂J

(−Φ

T

)(1 +

∑ αi�i

(r2+ + a2)i

). (32)

These expressions replace the earlier conditions for exact-ness of the entropy. It is not difficult to verify these rela-tions, for all orders, by using from (20), (21), and (22). Ingeneral, quantum corrections to the temperature are accom-panied by corrections to the angular velocity and the electro-static potential, in which case they always make dS exact. Itis interesting to note that in our case we do not require cor-rections for Ω and Φ . This is due to the particular functionalform of (28). Thus the corrected form of entropy also satis-fies the exactness criteria for differentials in three variables

and, therefore, can be written in the following integral form:

S(M,J,Q)

=∫

1

T

(1 +

∑ αi�i

(r2+ + a2)i

)dM

−∫

Ω

T

(1 +

∑ αi�i

(r2+ + a2)i

)dJ

−∫

Φ

T

(1 +

∑ αi�i

(r2+ + a2)i

)dQ

−∫ (

∂

∂J

(∫1

T

(1 +

∑ αi�i

(r2+ + a2)i

)dM

))dJ

−∫ (

∂

∂Q

(∫1

T

(1 +

∑ αi�i

(r2+ + a2)i

)dM

))dQ

+∫ (

∂

∂Q

(∫Ω

T

(1 +

∑ αi�i

(r2+ + a2)i

)dJ

))dQ

+∫

∂

∂Q

(∫ (∂

∂J

(∫1

T

×(

1 +∑ αi�

i

(r2+ + a2)i

)dM

))dJ

)dQ. (33)

This is the corrected and modified form of (17). These com-plicated integrals can be simplified by employing the exact-ness criterion described above. Using an argument similar tothe one adopted to write the integral in (26) above, we findthat (33) reduces to

S(M,J,Q) =∫

1

T

(1 +

∑ αi�i

(r2+ + a2)i

)dM, (34)

which can be written in the expanded form as

S(M,J,Q) =∫

1

TdM +

∫α1�

T (r2+ + a2)dM

+∫

α2�2

T (r2+ + a2)2dM

+∫

α3�3

T (r2+ + a2)3dM + · · ·

= I1 + I2 + I3 + I4 + · · · , (35)

where the first integral I1 has been evaluated in (26). Wework out the other integrals one by one after substitutingvalues from (18) and (20). Thus

I2 = 2πα1

∫M dM

√M4 − J 2 − Q2M2

. (36)

Eur. Phys. J. C (2010) 67: 205–211 209

This can be integrated by substituting M2 −Q2/2 = w, say,so that after some steps it yields

I2 = πα1 ln∣∣2M2 − Q2 + 2

√M4 − J 2 − Q2M2

∣∣, (37)

which is nothing but

I2 = πα1 ln(r2+ + a2). (38)

Let us evaluate the k-th integral Ik where k = 3,4, . . . .

Ik =∫

αk−1�k−1 dM

T (r2+ + a2)k−1= 2π

�

×∫

αk−1�k−1M dM

√M4 − J 2 − Q2M2[2M2 − Q2 + 2

√M4 − J 2 − Q2M2]k−2

= παk−1�k−2

2 − k

[2M2 − Q2 + 2

√M4 − J 2 − Q2M2

]2−k, k > 2,

(39)

or, in terms of r+, this can be written as

Ik = παk−1�k−2

(2 − k)(r2+ + a2)k−2, k > 2. (40)

Thus the entropy including the correction terms becomes

S = π

�

(r2+ + a2) + πα1 ln

(r2+ + a2)

+ · · · +∑

k>2

παk−1�k−2

(2 − k)(r2+ + a2)k−2+ · · · . (41)

Note that this is a rather general formula for entropy cor-rections. If we put the charge Q = 0, we recover the cor-rections for the case of the Kerr black hole [6]. Further,if the angular momentum is also put equal to zero we getresults for the Schwarzschild black hole (a = Q = 0). Inthis case the proportionality constants in the above series in-volve inverse powers of square of the mass, M . On the otherhand if only the angular momentum vanishes (i.e. a = 0),we get corresponding corrections for Reissner–Nordströmblack hole, in which case the power series involve the chargeQ also, in addition to M , which are the only two parametersfor this object.

If we use the Bekenstein–Hawking area law relating en-tropy and horizon area, A, by

S = A

4�, (42)

where the area in our case is

A = 4π(r2+ + a2), (43)

from (41) we obtain

S = A

4�+ πα1 lnA − 4π2α2�

A− 8π3α3�

2

A2− · · · , (44)

which gives quantum corrections for the area law. Note thatthe first term is the usual semiclassical area, the secondterm is the logarithmic correction found earlier [13, 14] byfield theoretic arguments. The issue of the value of αi ’s ishighly disputatious. There are different interpretations foundin the literature. The prefactor of the logarithmic term, α1,for example, is given as −3/2 in Ref. [15], and −1/2 inRefs. [16, 17]. Similarly, some authors [18] take it to be apositive integer, while others find it even to be zero [19].

5 Entropy corrections for the charged rotating BTZblack hole

Next we consider the (1 + 2) Bañados–Teitelboim–Zanelli(BTZ) black hole [20], when it is charged and rotating. Themetric can be written as [21]

ds2 = −(

−M + r2

l2+ J 2

4r2− π

2Q2 ln r

)dt2

+(

−M + r2

l2+ J 2

4r2− π

2Q2 ln r

)−1

dr2

+ r2(

dφ − J

2r2dt

)2

, (45)

where M is the mass, J the angular momentum, Q thecharge of the black hole, and 1/l2 = −Λ is the negative cos-mological constant.

If r+ and r− are the event horizon and the inner horizon,respectively, then they must satisfy

−M + r2

l2+ J 2

4r2− π

2Q2 ln r = 0. (46)

We will not find roots of this expression to obtain r+ as wasdone easily in the case of uncharged BTZ black hole [8],where the entropy depended only upon two parameters M

and J . Here we will circumvent the difficulty of finding theroots by using the results obtained above on exactness of theentropy function for three parameters to work out the inte-grals involved in the corrections. Secondly, instead of sub-stituting the value of r+ directly into different expressionswe will be using chain rules of differentiation. Let us write

f (r) = −M + r2

l2+ J 2

4r2− π

2Q2 ln r. (47)

The angular velocity of a rotating black hole is [22]

Ω =[

− gφt

gφφ

−√(

gφt

gφφ

)2

− gtt

gφφ

]

r=r+

. (48)

For the BTZ black hole this becomes

Ω = − gφt

gφφ

∣∣∣∣r=r+

= J

2r2

∣∣∣∣r=r+

. (49)

210 Eur. Phys. J. C (2010) 67: 205–211

The event horizon is related with temperature T by [9, 23]

T = �f ′(r)4π

∣∣∣∣r=r+

, (50)

where f ′(r) denotes the derivative of f with respect to r .The electric potential is given by [21]

Φ = −∂M

∂Q

∣∣∣∣r=r+

= −πQ ln r+. (51)

With these thermodynamic quantities the BTZ black holesatisfy the first law of thermodynamics of the form (12).Therefore, the semiclassical entropy becomes

S =∫

1

TdM = 8πl2

�

∫r3+ dM

4r4+ − J 2l2 − πQ2l2r2+. (52)

In order to evaluate this integral, we note that from (46) wecan write

dM =(

2r+l2

− J 2

2r3+− πQ2

2r+

)dr+. (53)

Using this in the integral in (52) we readily get

S = 4πr+�

, (54)

which is the standard formula for entropy in this geometry.Now we calculate quantum corrections to the entropy. We

first note that for the case of three-dimensional BTZ blackhole the correction terms are proportional to (r+)−i , so thatthe corrected action can be written as

S = S0

[1 +

∑ αi�i

(r+)i

]. (55)

We use the differential of the entropy dS from (13) and us-ing the exactness criteria obtain an integral expression of theform (33). This expression is simplified as before, so that weobtain the following form which includes quantum correc-tions to the entropy as well:

S(M,J,Q) =∫

1

T

(1 +

∑ αi�i

(r+)i

)dM

= I1 + I2 + I3 + I4 + · · · , (56)

where the uncorrected semiclassical term I1 is given by (54).For the correction terms we proceed as follows. The integralI2 is given by

I2 = α1�

∫1

r+TdM. (57)

Using (50) and (53) this simplifies to

I2 = 4πα1

∫1

r+dr+ = 4πα1 ln r+. (58)

Similarly,

I3 = α2�2∫

1

r2+TdM = −4α2�π

r+, (59)

and so on. Thus the series becomes

S = 4πr+�

+ 4πα1 ln r+ − 4α2�π

r+− · · · . (60)

Putting 8G3 = 1, where G3 is the three-dimensionalNewton gravitational constant, the area formula is

A = 2πr+. (61)

If we include G3 this becomes

A = 16πG3r+, (62)

and the above series takes the form

S = A

4�G3+ 4πα1 lnA − 64α2�π2G3

A− · · · . (63)

Note that if we put Q = 0, the results for the unchargedBTZ black holes [8] are recovered.

6 Conclusion

One way to describe Hawking radiation is by the processof quantum tunneling thereby particles cross event horizonsand traverse “forbidden” trajectories. The positive energyparticles tunnel out of the event horizon, whereas, the nega-tive energy particles tunnel in and result in black hole evap-oration. Thus a system which is stable classically becomesunstable quantum mechanically, and this changes the ther-modynamics of the system as well. We have used this analy-sis to study quantum corrections in entropy for charged androtating black holes. For this purpose we use the criterionfor exactness of differential of entropy. With the help ofthis procedure the formidable integrals involving quantumcorrections become manageable and we obtain power seriesfor entropy and Bekenstein–Hawking horizon area of blackholes for a quite general framework. We first applied thisto the Kerr–Newman black holes. The first term obtained inthe power series is the semiclassical value, while the lead-ing correction term is logarithmic as has been found usingother methods. The other terms involve ascending powers of(r2+ +a2)−1. This reduces to (r2+)−1 for non-rotating objectslike Reissner–Nordström black hole. If the charge and angu-lar momentum both become zero, we obtain results for theSchwarzschild black hole, in which case the power seriesinvolve mass only as it is the only macroscopic parameterfor this object. This is true for the entropy corrections of theanti-de Sitter–Schwarzschild spacetimes also. An important

Eur. Phys. J. C (2010) 67: 205–211 211

feature of our approach is that we have obtained the cor-rections in entropy without requiring the corrections in theangular momentum and the electrostatic potential.

Another significant application of the procedure is forthe three-dimensional BTZ black hole, when it is chargedand rotating. Here without having to find the horizon wehave been able to calculate quantum corrections beyond thesemiclassical approximation by using the exactness criterionfor the entropy differential. In this case also the modifiedBekenstein–Hawking area law has been derived. The lead-ing order correction term is again found to be logarithmic inthis case.

Acknowledgements K.S. acknowledges a research grant from theHigher Education Commission of Pakistan.

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