Cent. Eur. J. Phys. • 7(3) • 2009 • 521-526DOI: 10.2478/s11534-009-0011-2

Central European Journal of Physics

Rediscussion of charged dilaton-axion black holeentropy

Research Article

Chunyan Wang∗, Yuanxing Gui†

School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian, Liaoning, 116024, P. R. China

Received 15 August 2008; accepted 23 November 2008

Abstract: We rediscuss the entropy of a charged dilaton-axion black hole for both the asymptotically flat and non-flatcases by using the thin film brick-wall model. This improved method avoids some drawbacks in the originalbrick-wall method such as the small mass approximation, neglecting the logarithm term, and taking theterm L3 as the contribution of the vacuum surrounding the black hole. The entropy we obtain turns out tobe proportional to the horizon area of the black hole, conforming to the Bekenstein-Hawking area-entropyformula for black holes.

PACS (2008): 04.; 04.62.+v; 04.70.-s

Keywords: charged dilaton-axion black hole • thin film brick-wall model • entropy • cut-off© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

It has now been three decades since Bekenstein origi-nally suggested that black holes carry an entropy S thatis proportional to the surface area A of their event horizon[1–3]. Soon after Bekenstein’s suggestion, Hawking dis-covered that when one considers the evolution of quantumfields around black holes, the black holes indeed radiatethermally [4, 5]. Moreover, the temperature of the thermalradiation, referred to as the Hawking temperature, deter-mines the exact relation between the surface area of theblack holes’ horizon and their entropy, viz. thatS = A4 , (1)

∗E-mail: wangchunyan5800@yahoo.cn†E-mail: guiyx@dlut.edu.cn

with the area to be ’measured’ in units of the square of thePlanck length. This Bekenstein-Hawking’s area-entropylaw is expected to apply to all black hole solutions of theEinstein equations.Ever since Bekenstein’s suggestion and Hawking’s discov-ery, a variety of approaches have been proposed to under-stand the microscopic origin of black hole entropy [6–10].One of their approaches has been the semi-classical ap-proach originally due to ’t Hooft [10], often referred to asthe brick-wall model (BWM). In this approach, the blackhole geometry is assumed to be a fixed classical back-ground in which quantum fields propagate, and the blackhole entropy is identified with the statistical mechanicalentropy arising from the thermal bath of quantum fieldspropagating outside the event horizon, evaluated in theWenzel-Kramers-Brillouin (WKB) approximation. But dueto the infinite blue-shifting of the modes in the vicinity ofthe black hole horizon, the density of states of the matterfields diverges. Hence, for this model to be viable, it is521

Rediscussion of charged dilaton-axion black hole entropy

necessary to introduce by hand a cut-off which is the or-der of the Planck length above the horizon, and the scalarfield must vanish within some fixed distance outside thehorizon. This approach has successfully led to the deriva-tion of the proportionality of the black hole entropy tothe horizon area by identifying the black hole entropywith thermal entropy of ambient quantum fields raised tothe Hawking temperature [11–31]. It was recently clarifiedthat in this model, back reaction is small enough so that itis perfectly legitimate to neglect it, and that this model isactually self-consistent as a semi-classical theory [32, 33].Though popular, in order to make the final entropy resultconform to the Bekenstein-Hawking area-entropy formula,the BWM has some drawbacks such as the small mass ap-proximation (to integrate the free energy), neglecting thelogarithm term (the quantum correction term is neglectedwithout any expression), and taking the term (L3) as thecontribution of the vacuum surrounding the black hole.Since then, the BWM has been improved to the thin filmbrick-wall model [34–36]. The improved method retainsthe idea of the original BWM by introducing two cut-offparameters, and gives more thermal characteristics of theblack hole, especially the relation to the event horizon ofthe black hole. In the thin film BWM, the only contri-butions to the free energy and entropy of the black holeare by a thin film near the event horizon. Therefore, thedrawbacks of the original model can be avoided. As an

effective approach for calculating the entropy of a blackhole, the thin film BWM has been employed to study manytypes of spaces, for instance, the Vaidya black hole [34, 35],Schwarzschild-de Sitter black hole [36–38], Kerr-de Sitterblack hole [39] and others [40, 41].Recently, Sourav Sur, et al., have given new asymp-totically flat and non-flat black hole solutions [42] forEinstein-Maxwell-scalar field systems inspired by low en-ergy string theory. In [43], Tanwi Ghosh and SoumitraSenGupta have investigated the entropy of this black holeby using the original BWM; however, there are many cal-culations which look unnatural. In this paper, we will con-sider this charged dilaton-axion black hole for the asymp-totically flat and non-flat cases again, and apply the thinfilm BWM to calculate the entropy. The electromagneticfield and the dilaton-axion fields are not considered, andthese fields are treated as background classical fields insemi-classical approximation [44].2. Entropy of asymptotically flatdilaton-axion black hole

The line element of the metric for the asymptotically flatdilaton-axion black hole is given [42] by

ds2 = − (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n dt2 + (r − r0)2−2n(r + r0)2n(r − r+)(r − r−) dr2 + (r + r0)2n(r − r0)2n−2 dθ2 + (r + r0)2n(r − r0)2n−2 sin2 θdφ2, (2)

various parameters for the asymptotically flat case aregiven asr± = m0 ±

√m20 + r20 − 18

(K1n + K21− n

), (3)

r0 = 116m0(K1n −

K21− n), 0 < n < 1, (4)

m0 = M − (2n− 1)r0,M = 116r0

(K1n −

K21− n)+ (2n− 1)r0, (5)

K1 = 4n [4r20 + 2kr0(r+ + r−) + k2r+r−] ,K2 = 4(1− n)r+r−, (6)

where M is the mass of the black hole and k = 1 forthe asymptotically flat case. The r± are the constants of

integration, and in order that at least one horizon exists,one must have r+ > r0. It is obvious that there is a point(curvature) singularity at r = r0, and such a singularitymust obviously be naked unless there is a horizon at leastat r+. In this paper, we only consider the special casewhere there is a regular black hole horizon at rH = r+.Now we calculate the entropy of such a charged dilaton-axion black hole for the asymptotically flat case using thethin film BWM. In this setting, we consider a massive andminimally coupled quantum scalar field that is propagat-ing in the the line element Eq. (2). Such a field satisfiesthe Klein-Gordon equation1√−g

∂∂xµ

(√−ggµν ∂

∂xν

)Ψ = u2Ψ, (7)where Ψ is the scalar field.Following the thin film BWM, we need to introduce a thinlayer cut-off near the event horizon such that the boundary

522

Chunyan Wang, Yuanxing Gui

conditionΨ = 0 for r ≤ rH + ε (8)

is satisfied. Another cut-off is introduced at rH + ε + δ,where we haveΨ = 0 for r ≥ rH + ε + δ, (9)

where ε and δ are the positive infinitesimal cut-off factorand the infinitesimal thin layer, respectively.

In the spherically symmetric space, the scalar field can bedecomposed as Ψ = f(r)Ylm(θ, φ) exp(−iEt). Then sub-stituting Eq. (2) into Eq. (7) and separating angular andtime variables, we obtain the radial equation

[ (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n]−1

E2f(r)− [u2 + (r − r0)2n−2(r + r0)2n l(l+ 1)] f(r)+ [ (r + r0)2n(r − r0)2n−2

]−1 ∂∂r

[ (r + r0)2n(r − r0)2n−2 (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n ∂∂r f(r)

] = 0. (10)Letting f(r) = exp (iS(r)), where S(r) is a complex function, we can obtain the radial wave number kr(r, l, E) by the WKBapproximation

k2r = (∂S(r)

∂r

)2 = E2 − (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n(u2 + l(l+ 1) (r − r0)2n−2(r + r0)2n

)( (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n

)2 . (11)According to the canonical assembly theory, the free energy of the quantum scalar field at inverse temperature β iswritten as

F = 1β∑E

ln (1− e−βE) . (12)We take the semi-classical approach and assume that the energy state is continuous. The summation can be written asquadrature

F = 1β

∫ ∞0 dEg(E) ln (1− e−βE) = −∫ ∞0

Γ(E)dEeβE − 1 , (13)where g(E) = dΓ(E)

dE is the density of states, and Γ(E) is the microscopic state number of the systematic energy less thanor equal to E . Under the condition of the semi-classical quantum, Γ(E) becomesΓ(E) =∑

l,m

nr(E, l,m) = ∫l(2l+ 1)dl 1π

∫rkr(r, E, l)dr. (14)

Therefore, the free energy of the system can be rewritten asF = − 1

π

∫ ∞0

dEeβE − 1∫rdr∫l(2l+ 1)dlA(r) = − 23π

∫ ∞0

dEeβE − 1∫rdr

(r + r0)2n(r − r0)2n−2[ (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n]2B(r), (15)

whereA(r) =

[E2 − (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n

(u2 + l(l+ 1) (r − r0)2n−2(r + r0)2n

)] 12

(r − r+)(r − r−)(r − r0)2−2n(r + r0)2n, (16)

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Rediscussion of charged dilaton-axion black hole entropy

B(r) = [E2 − (r − r+)(r − r−)(r − r0)2−2n(r + r0)2n u2] 32, (17)

From the previous equation, we can see that it is very simple to calculate the integration with respect to l. The upperlimit of the integration is taken so that k2r is non-negative, and the lower limit is naturally zero. We take only the freeenergy of a thin layer near the horizon: rH + ε ≤ r ≤ rH + ε + δ (where the thickness of the thin layer δ is much

less than the cut-off factor ε or they are the same order of infinitesimal), thus the coefficient (r−r+)(r−r−)(r−r0)2−2n(r+r0)2n before u2in Eq. (17) naturally approaches zero without the small mass approximation artificially introduced in the original BWM.Then the integration of free energy near the horizon rH is given byF = 23π π415β4

[ (rH + r0)6n(rH − r0)6−6n(rH − r−)2( 1ε + δ −

1ε

)] = − 2π345β4[ (rH + r0)6n(rH − r0)6−6n(rH − r−)2 · δ

ε(ε + δ)]. (18)

Thus, the entropy of the system can be obtained from the following formula:S = β2 ∂F

∂β = 8π345β3[ (rH + r0)6n(rH − r0)6−6n(rH − r−)2 · δ

ε(ε + δ)], (19)

where β is the inverse Hawking temperature calculated from the metric Eq. (2) as follows:β = 1

T = 4π [ (r0 + rH )2n(rH − r0)2−2n(rH − r−)]. (20)

Substituting the expression of β into the entropy expression Eq. (19), we obtainS = π90β3 · 16π2 [(rH + r0)2n(rH − r0)2−2n]2(rH − r−)2 ·

[(rH + r0)2n(rH − r0)2−2n] · δε(ε + δ) = AH4 T90 · δ

ε(ε + δ) , (21)where AH denotes the area of the asymptotically flat dilaton-axion black hole horizon, and AH = 4π(rH+r0)2n(rH−r0)2−2n.From Eq. (21), it is easy to see that the entropy of theasymptotically flat dilaton-axion black hole is 1/4 timesthe area of its horizon when we choose the appropriaterelation between the thickness δ of the thin layer and thedistance ε of the cut-off as follows:

δε(ε + δ) = 90

T . (22)The resulting entropy is satisfactory and agrees with theBekenstein-Hawking area-entropy of Eq. (1).3. Entropy of asymptotically non-flat dilaton-axion black holeUsing the thin film BWM, we have successfully calculatedthe entropy of the asymptotically flat dilaton-axion blackhole in Section 2. In this section, we will extend ourcalculations to asymptotically non-flat space-time.The line element of the metric for the asymptotically non-flat dilaton-axion black hole can be written [42] as

ds2 = − (r − r+)(r − r−)r2 ( 2r0

r

)2n dt2 + r2 ( 2r0r

)2n(r − r+)(r − r−)dr2

+ r2(2r0r

)2ndθ2 + r2

(2r0r

)2n sin2 θdφ2, (23)where

r± = ( 11− n)[

M ±√M2 − (1− n)K24

], (24)

K1 = 4n [4r20 + 2kr0(r+ + r−)] ,K2 = 4(1− n)r+r−, (25)

and the other parameters are the same as in the asymp-totically flat case. k = 0 for the asymptotically non-flatcase.524

Chunyan Wang, Yuanxing Gui

The wave equation for asymptotically non-flat space-time is obtained as (r − r+)(r − r−)

r2 ( 2r0r

)2n−1E2f(r)−

l(l+ 1)r2 ( 2r0

r

)2n + u2 f(r) + 1

r2 ( 2r0r

)2n ∂∂r

r2 (2r0r

)2n (r − r+)(r − r−)r2 ( 2r0

r

)2n ∂∂r f(r)

= 0. (26)

The radial wave number kr is given by the WKB approxi-mation

k2r =

E2 − (r − r+)(r − r−)r2 ( 2r0

r

)2n l(l+ 1)r2 ( 2r0

r

)2n + u2

(r − r+)(r − r−)r2 ( 2r0

r

)2n

2 . (27)

The expressions of free energy for the asymptotically non-flat case can be calculated as follows:F = 1

β∑E

ln(1− e−βE)= − 1

π

∫ ∞0

dEeβE − 1∫rdr∫l(2l+ 1)dlA(r)

= − 23π∫ ∞

0dEeβE − 1

∫rdr

r2 ( 2r0r

)2n (r − r+)(r − r−)

r2 ( 2r0r

)2n

2B(r),

where

A(r) =E2 − (r − r+)(r − r−)

r2 ( 2r0r

)2n(l(l+ 1)r2( 2r0

r )2n + u2)12

(r − r+)(r − r−)r2 ( 2r0

r

)2n,

(28)B(r) =

E2 − (r − r+)(r − r−)r2 ( 2r0

r

)2n u2

32. (29)

Similarly as in the asymptotically flat case, we take onlythe free energy of a thin layer near the horizon: rH + ε ≤r ≤ rH + ε + δ. It is easy to integrate E and l, while theintegration over l must make Eq. (27) significant. After

integration, the free energy of the system becomesF = 23π π415β4

r6H( 2r0rH

)6n(rH − r−)2 ·

( 1ε + δ −

1ε

)= − 2π345β4

r6H( 2r0rH

)6n(rH − r−)2 · δ

ε (ε + δ) .

(30)

Thus, the entropy of the system can be obtained:S = 8π345β3

r6H( 2r0rH

)6n(rH − r−)2 · δ

ε(ε + δ) . (31)

The expression for β for the asymptotically non-flat metricEq. (23) isβ = 1

T = 4π r2H

( 2r0rH

)2n(rH − r−)

. (32)Substituting Eq. (33) into the expression for entropy inEq. (32), we have

S = π90β3 · 16π2[r2H( 2r0rH

)2n]2(rH − r−)2 ·

[r2H(2r0rH

)2n]· δε(ε + δ) = AH4 T90 δ

ε(ε + δ) , (33)where AH denotes the area of the asymptotically non-flatdilaton-axion black hole horizon, and AH = 4πr2H ( 2r0

rH

)2n.From Eq. (34), similarly, it is easy to see that the entropyof the asymptotically non-flat dilaton-axion black hole is1/4 times the area of its horizon when we choose the ap-propriate relation between the two parameters δ and ε asfollows:δ

ε(ε + δ) = 90T . (34)

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Rediscussion of charged dilaton-axion black hole entropy

4. Conclusion

In summary, we have successfully calculated the entropyfor dilaton-axion coupled black holes both for the asymp-totically flat and non-flat cases by using the thin filmBWM. It is obvious that the results are satisfactory andconform to the Bekenstein-Hawking area-entropy formulafor black holes, when we choose an appropriate relationbetween two infinitesimal parameters, the cut-off ε andthe thickness of the thin layer δ. At the same time, thecalculation is reasonable without certain approximationsand unnatural neglecting of terms necessary when usingthe conventional BWM.Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No. 10573004.References

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