7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

1/6

Int J Theor Phys (2012) 51:37213726

DOI 10.1007/s10773-012-1255-y

Corrected Entropy of BTZ Black Holes

Hoda Farahani Jafar Sadeghi Hassan Saadat

Received: 5 April 2012 / Accepted: 3 July 2012 / Published online: 25 July 2012

Springer Science+Business Media, LLC 2012

Abstract In this paper, corrected entropy of a class of BTZ black holes, include charge and

rotation, studied. We obtain corrected Bekenstein-Hawking entropy and find that effect of

charge viewed at order A2, and effect of rotation viewed at order A6, therefore Q and Jdont have contribution in corrected entropy lower than the second order. We also write the

first law of black hole thermodynamics for the case of charged rotating BTZ black hole.

Keywords Corrected entropy BTZ black hole Thermodynamics

1 Introduction

The concept of black hole entropy proposed by Bekenstein [1, 2]. It is found that the grav-

itational surface of a black hole is proportional to the black hole temperature, and the event

horizon area is proportional to the entropy. Then, Hawking radiation related the quantum

mechanics, thermodynamics and general relativity [3]. For the case of black hole with two

horizons, the outer horizon is related with the Hawking radiation while the inner horizon isrelated with the absorption process [47]. According to the Hawking formulation, which

is based on quantum field theory, the entropy is given by S= A/4, where Plank-units= c = G = 1 is used and A denotes the area of the event horizon. Also, the Hawking

H. Farahani () J. SadeghiSciences Faculty, Department of Physics, Mazandaran University, P.O. Box 47416-95447, Babolsar,

Iran

e-mail: h.farahani@umz.ac.ir

J. Sadeghi

e-mail: pouriya@ipm.ir

H. Saadat

Islamic Azad University, Marvdasht Branch, Department of Physics, P.O. Box 71555-477, Marvdasht,

Iran

e-mail: hsaadat2002@yahoo.com

mailto:h.farahani@umz.ac.irmailto:pouriya@ipm.irmailto:hsaadat2002@yahoo.commailto:hsaadat2002@yahoo.commailto:pouriya@ipm.irmailto:h.farahani@umz.ac.ir

7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

2/6

3722 Int J Theor Phys (2012) 51:37213726

radiation from a black hole may be interpreted as a quantum tunneling of a particle though

the event horizon [8]. The above expression may be modified by quantum correction,

S= A4+ ln A

4+O

1

A+ , (1)where the is a dimensionless parameter which is depend on background geometry. In

the Ref. [9] the corrected entropy of a general spherically symmetric black hole though

the generalized law of the thermodynamics discussed, and two kinds of explicit spherically

symmetric black holes (Schwarzschild black hole and Anti-de Sitter Schwarzschild black

hole) considered to obtain corrected entropy of a general spherically symmetric black hole.

Now, in this paper we would like to consider another example of BTZ black holes. BTZ

black hole discovered by Banados, Teitelboim and Zanelli (BTZ) [10, 11] and served as a

useful model for realistic black hole physics [12]. Significant picture of this kind of black

hole is about thermodynamics of higher-dimensional black holes. Also the charged BTZ

black hole [13] is a solution of (2 + 1) dimensional gravity with a negative cosmologicalconstant = 1/ l2. BTZ black holes are asymptotically anti-de Sitter [14, 15], thereforehave usage in AdS/CFT [16]. Furthermore, the rotating BTZ black hole studied in the Ref.

[17].

In this paper we will discuss the corrected entropy of BTZ, charged BTZ and rotating

BTZ black holes by using the generalized law of the thermodynamics. First of all we review

several kinds of BTZ black holes. Then, in Sect. 3 we write the first law of the black hole

thermodynamics for the general case of charged rotating BTZ black hole. The main section

of this paper is the Sect. 4 where the corrected entropy represented for three special cases of

BTZ, charged BTZ and rotating BTZ black holes. Finally in Sect. 5 we give conclusion and

discuss about possible future works.

2 BTZ Black Holes

As we know the black hole solution of BTZ in (2+1) space-time are given by the followingline element,

ds2 =dt2 + dr2

+ r2d2, (2)

where the function is depend on kind of BTZ black hole and represented explicitly in thenext subsections.

2.1 BTZ

In the simplest case, the function is given by the following expression,

=M+ r2

l2, (3)

where M stands for Arnowwitt-Deser-Misner (ADM) mass. =

0 gives the radius of black

hole horizon,

rh =

Ml, (4)

which yields to the following expression of ADM mass,

M= r2h

l2. (5)

7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

3/6

Int J Theor Phys (2012) 51:37213726 3723

The BekensteinHawking entropy associated with the black hole is given by,

S0 = 4rh =A

4. (6)

Combining (5) and (6) gives the following expression of ADM mass in terms of entropy,

M= S20

(4l)2. (7)

Finally the Hawking temperature obtained as the following,

T= rh2l2

=

M

2l. (8)

2.2 Charged BTZ

In the case of charged BTZ black hole the function is given by the following expression[18],

=M+ r2

l2 Q

2

2ln

r

l, (9)

where Q denotes black hole charge. In that case the ADM mass modified as the following,

M= r2h

l2 Q

2

2ln

rh

l, (10)

where rh is the largest root of = 0, which represents outer horizon. In this case the en-tropy obtained similar to the relation (6), so combining (6) and (10) yields to the following

expression,

M= S20

(4l)2 Q

2

2ln

S0

4l. (11)

Also the Hawking temperature obtained as the following,

T= 14

2rh

l2 Q

2

2rh

. (12)

It is possible to write Hawking temperature as T= /2 , where surface gravity is given

by,

= rhl2 Q

2

4rh. (13)

2.3 Rotating BTZ

Rotating BTZ black hole described by the line element (2) under replacement ofr2d2 r2(d J

2r2dt)2 with the following lapse function [19, 20],

=

M+

r2

l2 +J2

4r2. (14)

In the special case of M= 1 and J= 0 the BTZ metric may be recognized as the ordinaryAdS space. Radius of black hole horizon obtained as the following,

rh = l

M2

1+

1

J

Ml

2. (15)

7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

4/6

3724 Int J Theor Phys (2012) 51:37213726

In that case the ADM mass modified as the following,

M= r2h

l2+ J

2

4r2h. (16)

Finally the Hawking temperature obtained as the following,

T= 14

2rh

l2 J

2

2r3h

. (17)

3 First Law

It is found that black holes have non-zero entropy which emit a thermal radiation propor-

tional to surface gravity at the horizon. When the black hole has other properties as angularmomentum J and electric charge Q, these quantities are related with the ADM mass through

the following first law of black hole thermodynamics,

dM= T dS+dJ+ dQ, (18)

where,

T= 14

2rh

l2 Q

2

2rh J

2

2r3h

, (19)

is the Hawking temperature of a charged rotating BTZ black hole, and

= MJ

= J2r2h

, (20)

is the angular velocity and also,

= MQ

=Q ln rhl

, (21)

is the electric potential. Finally one can obtain,

M= S20(4l)2

+ 42J2S20

Q22

ln S0

4l. (22)

4 Corrected Entropy

In the Ref. [9] the corrected entropy extracted for the case of spherically symmetric black

holes. Now, in the similar way, we write expression of corrected entropy of BTZ black holes

as the following,

S= S0 + S1 + S2 +

=

1

T

1+

i

i

M

idM. (23)

Following we examine above formulation for three separate cases of BTZ black holes.

7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

5/6

Int J Theor Phys (2012) 51:37213726 3725

4.1 BTZ

By using the relations (7) and (8) in (23) we obtain,

S0 = dMT = 4rh = A4 , (24)S1 =

1

T

MdM= 41l ln

A

4, (25)

S2 =

22

T MdM= 42

2l2

rh. (26)

So, under assumption of 41l , (24), (25) and (26) satisfy the relation (1).

4.2 Charged BTZ

Similarly, by using the relations (10) and (12) in (23), also assumption of small charge, we

obtain the following expressions,

S0 = 4rh =A

4, (27)

S1 = lnA

4+O

1

A2, (28)so the Q-dependent term includes in O( 1

A2). and,

S2 =O

1

A

+O

1

A3

. (29)

So, the Q-dependent term includes in O( 1A3

). Therefore, we can see that the logarithmic

correction has not affected by the black hole charge, hence (1) satisfied.

4.3 Rotating BTZ

Finally, by using the relations (16) and (17) in (23), also assumption of small rotation, we

obtain,

S0 =A

4, (30)

S1 = lnA

4+O

1

A6

, (31)

so the J-dependent term includes in O(1

A6 ). and,

S2 =O

1

A

+O

1

A7

. (32)

So, the J dependent term includes inO( 1A7

). Therefore, we found that the rotation parameter

has no effect on the logarithmic terms of corrected entropy (1).

7/28/2019 art-3A10.1007-2Fs10773-012-1255-y.pdf

6/6

3726 Int J Theor Phys (2012) 51:37213726

5 Conclusion

By using method of the recent work [9] we extracted corrected entropy of a class of BTZ

black holes include charge and rotation. It is known that at the second order the correction

term is logarithmic. We concluded that the BTZ, charged BTZ and rotating BTZ black holessatisfy this logarithmic behavior in the second order. In order to study such behavior, first

of all we reviewed several kinds of BTZ black holes. Then we discussed the first law of

black hole thermodynamics. By extracting the corrected entropy we found that the effect of

black hole charge Q is of order O( 1A2

) and, so the black hole charge has no effect on the

logarithmic term. In the similar way the rotational parameter viewed in orderO( 1A6

). Hence,

the parameter J has no effect on the logarithmic term. For the future study one can choose

another kind of black holes such as Godel black holes [21, 22] to extract corrected entropy.

References

1. Bekenstein, J.D.: Lett. Nuovo Cimento 4, 737 (1972)

2. Bekenstein, J.D.: Phys. Rev. D, Part. Fields 7, 2333 (1973)

3. Hawking, S.W.: Commun. Math. Phys. 43, 199 (1975)

4. Zhao, Z., Zhu, J.Y., Liu, W.B.: Chin. Phys. Lett. 16, 698 (1999)

5. Zhao, Z.: Int. J. Theor. Phys. 38, 1539 (1999)

6. Wu, S.Q., Cai, X.: Nuovo Cimento B 115, 143 (2000)

7. Wu, S.Q., Cai, X.: Int. J. Theor. Phys. 39, 2215 (2000)

8. Parikh, M.K., Wilczek, F.: Phys. Rev. Lett. 85, 5042 (2000)

9. He, T.-m., Yang, J.-b., Wu, F.-j.: Corrected entropy of a general spherically symmetric black hole. Int. J.

Theor. Phys. 51(7), 22202226 (2012). doi:10.1007/s10773-012-1101-210. Banados, M., Teitelboim, C., Zanelli, J.: Phys. Rev. Lett. 69, 1849 (1992)

11. Banados, M., Henneaux, M., Teitelboim, C., Zanelli, J.: Phys. Rev. D 48, 1506 (1993)

12. Carlip, S.: Class. Quantum Gravity 12, 2853 (1995)

13. Martinez, C., Teitelboim, C., Zanelli, J.: Phys. Rev. D61, 104013 (2000)

14. Rahaman, F., Banerjee, A., Radinschi, I.: A new class of stable (2+ 1) dimensional thin shell wormhole.Int. J. Theor. Phys. 51(6), 16801691 (2012). doi:10.1007/s10773-011-1045-y

15. Ulu Dogru, M., et al.: Energy and momentum of higher dimensional black holes. Int. J. Theor. Phys.

51(5), 15451554 (2012). doi:10.1007/s10773-011-1032-3

16. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math.

Phys. 2, 231 (1998)

17. Sadeghi, J., Farahani, H., Pourhassan, B., Noorbakhsh, S.M.: Cosmic string in the BTZ black hole back-

ground with time-dependent tension. Phys. Lett. B 703, 1419 (2011)18. Alexis, E., Rubio, L.: Thermodynamics of charged BTZ black holes and effective string theory. Acta

Phys. Pol. B 39, 1349 (2008)

19. Sadeghi, J., Setare, M.R.: BTZ black hole in non-commutative spaces. Int. J. Theor. Phys. 46(4), 817

(2007)

20. Sadeghi, J., Farahani, H., Pourhassan, B., Noorbakhsh, S.M.: Cosmic string in the BTZ black hole back-

ground with time-dependent tension. Phys. Lett. B 703, 1419 (2011)

21. Gimon, E., Hashimoto, A.: Black holes in Godel universes and pp waves. Phys. Rev. Lett. 91(2), 021601

(2003)

22. Wu, S.-Q.: General nonextremal rotating charged Godel black holes in minimal five-dimensional gauged

supergravity. Phys. Rev. Lett. 100, 121301 (2008)

http://dx.doi.org/10.1007/s10773-012-1101-2http://dx.doi.org/10.1007/s10773-011-1045-yhttp://dx.doi.org/10.1007/s10773-011-1032-3http://dx.doi.org/10.1007/s10773-011-1032-3http://dx.doi.org/10.1007/s10773-011-1045-yhttp://dx.doi.org/10.1007/s10773-012-1101-2