Int J Theor Phys (2011) 50:2805–2810DOI 10.1007/s10773-011-0779-x

The Quantum Formulation of Black Hole Area Spectrumand Entropy Spectrum

Juan Yang

Received: 24 November 2010 / Accepted: 28 March 2011 / Published online: 12 April 2011© Springer Science+Business Media, LLC 2011

Abstract In this paper, the area spectrum of this static BTZ black hole in different cases(rotating, non-rotating, and extreme) is investigated. The final results show that the spec-tral formulation is 2πn� when this black hole is non-rotating. For the black hole in othertwo different cases, its spectrum is angular momentum-dependent. Unexpectedly, their areaspectra are both equally spaced. What is more, the entropy spectrum is also calculated viathe method put forward by Chen et al. However, it is demonstrated that the well known area-entropy law is greatly changed. Above all, the entropy spectrum of this non-rotating BTZblack hole is also equally spaced.

Keywords Static BTZ black hole · Area spectrum · Entropy spectrum · Area-entropy law

1 Introduction

Maggiore [1] has proposed that the black hole is always perturbed by the near field, sothe black hole can be expressed as a collection of damped harmonic oscillators. The quasi-normal modes are just resulting from the perturbations, which comply with the linear equa-tions of motion constrained by the condition that the solution of the linear equations nearthe event horizon is purely ingoing and the spatial infinite solution is purely outgoing. Sothe quasi-normal modes are special ones, which possess complex frequencies. Recently,the quasi-normal mode of a black hole has attracted many people’s attention. Because thequasi-normal modes are important physical quantities, which may essentially characterizethe space time structure and some properties (including mass, charge, and angular momen-tum) of a black hole. What is more, the quasi-normal mode has been used to investigate thelinear stability of a black hole [2–6]. Hod [7, 8] is the first to put forward that the real part ofthe derived quasi-normal mode frequency can be regarded as a transition frequency in semi-classical limit. The QNM (quasi-normal mode) frequency of a black hole in different gravity

J. Yang (�)Institute of Theoretical Physics, China West Normal University, Nanchong, Sichuan 637009, People’sRepublic of Chinae-mail: yangjuangu@163.com

2806 Int J Theor Phys (2011) 50:2805–2810

(including Schwarzschild black hole, Dilaton Black Hole, and others) is correctly produced[6, 9, 10]. It is showed that their frequencies are formed by the decreasing relaxation timeterm (the imaginary part) and the real part. In physics, it factually gives rise to the real partof the frequency corresponding to the oscillation rate and the imaginary part correspondingto the damping rate.

In Bekenstein’s opinion [11–13], the horizon area of a black hole should be quantized ina quantum gravitational theory. Fortunately, The QNM frequency has provided ones with adirect way for calculating the spectrum of a black hole. Later Kunstatter [14, 15] obtainedthe quantified horizon area by introducing an adiabatic invariant I , which perfectly connectsthe energy E and the vibration frequency ω(E) of a system. In the case of n � 1, ω(E)

only depends on the imaginary part of complex frequency of the quasi-normal mode. Notlong ago, some results magnified that the horizon area spectrum was equally spaced. Here aquestion arises that whether all kinds of black holes have equally spaced area spectrum, nomatter they are rotating, charged, or extreme ones. Consequently, the results in references[16–18] state that the area spectra of both Kerr and Schwarzschild black holes are not onlyequally spaced, but also have the same universal formulation when some approximation istaken into consideration. Nevertheless, the question about if the approximation is reasonablepartly remains unsolved.

The purpose of this paper is to verify that whether the area spectrum of the BTZ blackhole in any case (rotating, non-rotating, extreme) is the same without considering certain ap-proximation. In the end, their entropies are also discussed in order to uncover some specialfeatures of the BTZ black hole. Although these results are just extension of those by pub-lished in reference [18], some important information may be showed with these comparison.So the study on this aspect is meaningful. The nature unit c = G = 1 is chosen throughoutthis paper. The reminders of this paper are structured as follows. In section two, the areaspectrum of the rotating BTZ black hole will be studied via the method deviating from theusual approach. In the section three, the area spectra of the non-rotating BTZ black hole andthe extreme BTZ black hole will be obtained respectively. In section four, the entropy ofthe BTZ black hole in different cases will be investigated, which may cast a shadow to thewell-known Bekenstein area-entropy law. Then some conclusions will be concluded in thesection six.

2 The Area Spectrum of the Rotating BTZ Black Hole

Firstly, we will introduce the metric describing D-dimension space-time as [19]

ds2 = − 1

l2(r2 − r2

+)dt2 + l2dr2

r2 − r2++ r2d�2

d−2. (1)

Where l and d�2d−2 stand for the AdS radius and the metric of a smooth base manifold �d−2

respectively. With exact calculation, the value of l can be defined as 1. The event horizonof this line element is located at r = r+. On the other hand, this metric can get back to theusual black hole solution with the choice of different values of the dimension. For example,the metric corresponds to the static BTZ black hole for d = 3, while for d = 4, the metricis obtained through solving the field equations in conformal gravity. In one word, the metricis one solution of conformal gravity in even dimensions, while others are the ones of othergravities such as Lovelock gravity.

Int J Theor Phys (2011) 50:2805–2810 2807

The quasi-normal mode frequency of massive scalar perturbation of this black hole isgiven by [19]

ω = −√

Q −(

d − 3

2

)2r2+l2

− ir+l

(2n + 1 +

√(d − 1

2

)2

+ m2l2

). (2)

Where, Q is the Eigen-value of the Laplace operator on �d−2.In order to get the horizon area spectrum, we should use the defined adiabatic invariant

as

I =∫

dE

�ω. (3)

Then via Bohr-Sommerfeld quantization I = n�, the quantization of the event horizon areacan be easily got. Following Maggiore’s idea, the frequency of oscillation should behave

as ω =√

ω2R + ω2

I . However, in the highly-damped limit, the required frequency is onlycorrelated with the imaginary part of the frequency of oscillation.

�ω ∼ |�ωI | = 2r+/l. (4)

For the static BTZ black hole, the horizon is given by [19–22]

r± = l

[M

2(1 ±

√1 − (J/Ml)2)

]1/2

, (5)

and the Hawking temperature is given by

TH = �(r2+ − r2−)

2πl2r+. (6)

Where M is the mass of this static BTZ black hole, and J = Ma is the angular momentum.What is more, the area of this black hole is defined as A = 2πr+. In this section, dE shouldbe replaced with,

dE = dM − �dJ. (7)

Where � = J

2r2+is the angular velocity. Substituting (5) and (4) into (3), we can get

I =∫

dM

(2M + √4M2 − 4J 2/l2)

12

− 2JdJ

l2(2M + √4M2 − 4J 2/l2)

32

= √2(1 + √

1 − a2/l2 − a2/l2)

(1 + √1 − a2/l2)3/2

√M. (8)

When l = 1,

I = √2√

M(1 +√

1 − a2)1/2 − √2√

M(1 +√

1 − a2)−1/2

= A√

1 − a2

π(1 + √1 − a2)

. (9)

Strictly speaking, here the relationship A = 2nπ� is no longer reliable. Nevertheless, in areference [18], a correspondence between the classical property and the quantum property

2808 Int J Theor Phys (2011) 50:2805–2810

of the Hydrogen is suggested. Namely, transition frequencies at large quantum numbersshould be equal to classical oscillation frequencies. Then even for rotating BTZ black hole,this formulation of the adiabatic invariant (I = ∫

dM�ω

) is still remained. Later, utilizing theleft and right transition frequencies corresponding to each quasi-normal mode, the obtainedresult shows that the spacing level of the horizon area of this rotating BTZ black hole isindeed 2π�. Obviously, the result we get in this paper deviates from the conclusions inreferences [18, 19]. How can we give explanation for the inconsistency between the newcalculation and the old one?

Firstly, from (8), we can notice that the area spectrum of this rotating BTZ black hole isdependent on the angular momentum per mass a. When this black hole is slowly rotating,namely, a � 1, the spacing level of the horizon area is approximately 2π�.

On the other hand, expanding (8), we have

A = 1√1 − a2

πn� + πn� = 2πn� + const, (10)

if the little constant term on the right of (10) is neglected, there is

A ∼ 2πn�. (11)

As we all know, in certain cases, the result derived in this paper can exactly get back to theconclusions given in [18, 19].

3 Two Special Cases

In the above section, the general rotating BTZ black hole is discussed. Without the lossof generalization, two special cases also should be taken into consideration, namely, non-rotating BTZ black hole and extreme BTZ black hole. Similarly, the exact calculation l = 1is employed in this section.

3.1 Non-rotating BTZ Black Hole

When we choose the dimension d = 3, the horizon resulting from the static BTZ black holeis given by (5). When this BTZ black hole is non-rotating, the event horizon just can beexpressed as r+ = l

√M with dE = dM . We have

I =∫

dM

2√

M= √

M = n�. (12)

So the quantization of the event horizon of this non-rotating BTZ black hole can be derivedas

�A = 2π�, or A = 2π� • n. (13)

Remarkably, this spacing level isn’t dependent on the parameters of this black hole. Further-more, it is natural to find that (12) is consistent with (9) when a = 0.

Int J Theor Phys (2011) 50:2805–2810 2809

3.2 Extreme BTZ Black Hole [23]

In the extreme condition J = Ml, dE should be replaced with dM − �dJ . Then the ex-pected horizon area quantization is obtained as

I =∫

dM√2M

− dJ√2M

=∫

(1 − a)√2M

dM = √2√

M(1 − a). (14)

The area spectrum of this black hole in such case is derived as

A = n�π

1 − a. (15)

It is showed that this area spectrum is also dependent upon the angular momentum per mass.Fortunately, it is still equally spaced.

4 The Entropy Spectrum of the BTZ Black Hole

In this section, the entropy spectrum of this BTZ black hole will be investigated. By themethod of tunneling, it is well known that Bekenstein-Hawking area law S = A/4 is reliableand accurate for most black holes. Then this relationship between the area and the entropywill be further justified. Generally, the change of energy of this static BTZ black hole shouldbe dM − �dJ . Via the first law of thermodynamic, we can get as

T �S = dM − �dJ = �E = �ω�. (16)

Then the entropy change of this black hole is calculated as

�S = 2π(1 + √1 − (J/M)2)√

1 − (J/M)2. (17)

When this black hole is non-rotating in the case J → 0, the entropy spectrum takes the formas

S = 4π • n. (18)

This result is consistent with those resulting from other methods. While for the extreme caseJ = Ml, it leads to be the zero temperature limit, so this black hole just reaches the well-known hyper-mixed state in this condition. At the same time, comparing (13) with (18),it is demonstrated that the so-called Bekenstein-Hawking area law should contain certainmodifications. Previously, this method has successfully been applied to study the entropyspectrum of near extreme black holes [24].

5 Conclusion

In this paper, the area spectrum of this BTZ black hole is obtained in different cases. Ob-viously, the black hole in different cases (rotating, non-rotating, extreme) has different areaspectral formulation. Nevertheless, all of them are equally spaced. For rotating and extremeblack holes, their area spectra are angular momentum- dependent. Fortunately, the area spec-trum for rotating BTZ black hole can get back to the idea (all area spectra are the same

2810 Int J Theor Phys (2011) 50:2805–2810

as A ∼ 2πn� no matter the black hole is rotating, non-rotating or extreme) supported byYongjoon Kwon et al. [16, 17] in some extent or in the case of neglecting some little terms.In essence, the key idea behind this paper is to calculate the area spectrum of this rotatingBTZ black hole without adopting the left and right QNM frequencies. This method used inthis paper seems to be much more direct and accurate. Then some corrected terms of thearea spectrum are easily found.

In the end, their entropies are also calculated. The results demonstrate that the entropiesfor both non-rotating BTZ black hole and rotating BTZ black hole are equally spaced. Forthe extreme black hole, the investigation of its entropy becomes meaningless due to theexistence of the zero temperature. Furthermore, through comparing (10), (13) with (17),(18), it is not surprising to find that the entropy and area satisfy the new relationship S = 2A,which is different from the usual formula S = A/4. Previously, it has only been pointed outthat the former Bekenstein-Hawking area law is no longer reliable because of higher-orderquantum corrections for black hole in Einstein-Gauss-Bonnet gravity. How to explain thisunusual phenomenon mentioned above? It is in progress.

Acknowledgements This work is supported by the Natural Science Foundation of China with GrantNos. 10773008, and the College Student Technological Innovation Foundation of China West Normal Uni-versity Nos. 42710072.

References

1. Maggiore, M.: Phys. Rev. Lett. 100, 141301 (2008)2. Konoplya, R.A., Zhidenko, A.: Nucl. Phys. B 777, 182 (2007)3. Konoplya, R.A., Zhidenko, A.: Phys. Rev. Lett. 103, 161101 (2009)4. Konoplya, R.A., Zhidenko, A.: Phys. Rev. D 78, 104017 (2008)5. Becar, R., Lepe, S., Saavedra, J.: Phys. Rev. D 75, 084021 (2007)6. López-Ortega, A.: Int. J. Mod. Phys. D 18, 1441–1459 (2009)7. Hod, S.: Phys. Rev. Lett. 81, 4293 (1998)8. Hod, S.: Phys. Lett. B 666, 483–485 (2008)9. Chen, B., Xu, Z.b.: Phys. Lett. B 675, 246 (2009). arXiv:0901.3588v2

10. López-Ortega, A.: Rev. Mex. Fis. 56, 44–53 (2010)11. Bekenstein, J.D.: Lett. Nuovo Cimento 11, 467 (1974)12. Bekenstein, J.D., Mukhanov, V.F.: Phys. Lett. B 360, 7–12 (1995)13. Bekenstein, J.D.: In: Piran, T., Ruffini, R. (eds.) Proceedings of the Eight Marcel Grossmann Meeting,

pp. 92–111. World Scientific, Singapore (1999)14. Kunstatter, G.: Phys. Rev. Lett. 90, 161301 (2003)15. Daghigh, R.G., Kunstatter, G., Ostapchuk, D., Bagnulo, V.: Class. Quantum Gravity 23, 5101–5115

(2006)16. Kwon, Y., Nam, S.: Class. Quantum Gravity 27, 125007 (2010)17. Kwon, Y., Nam, S.: in press. arXiv:1006.4713v118. Myung, Y.S.: Phys. Lett. B 689, 42–44 (2010)19. Oliva, J., Troncoso, R.: in press. arXiv:1003.2256v120. Banados, M., Teitelboim, C., Zanelli, J.: Phys. Rev. Lett. 69, 1849 (1992)21. Birmingham, D., Sachs, I., Solodukhin, S.N.: Phys. Rev. Lett. 88, 151301 (2002)22. Cardoso, V., Lemos, J.P.S.: Phys. Rev. D 63, 124015 (2001)23. Crisostomo, J., Lepe, S., Saavedra, J.: Class. Quantum Gravity 21, 2801 (2004)24. Chen, D.Y., Yang, H.T., Zu, X.T.: Eur. Phys. J. C 69, 289–292 (2010)