Physics Letters B 656 (2007) 217–220

www.elsevier.com/locate/physletb

Charged rotating BTZ black hole and thermodynamic behaviorof field equations at its horizon

M. Akbar a, Azad A. Siddiqui b,∗

a Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of EME College, Peshawar road, Rawalpindi, Pakistanb Department of Basic Science and Humanities, EME College, National University of Sciences and Technology, Peshawar road, Rawalpindi, Pakistan

Received 17 August 2007; received in revised form 17 September 2007; accepted 22 September 2007

Available online 4 October 2007

Editor: T. Yanagida

Abstract

In this Letter, we study different cases of the charged rotating BTZ black hole with reference to their horizons. For the existence of these casesconditions on mass, charge and angular momentum of the black hole are obtained. It is also shown that the Einstein field equations for the chargedrotating BTZ black hole at the horizon can be expressed as first law of thermodynamics, dE = T dS + Ω dJ + Φ dq + Pr dA.© 2007 Elsevier B.V. All rights reserved.

PACS: 04.70.Dy; 97.60.Lf

1. Introduction

The relationship between the Einstein field equations and thefirst law of black hole thermodynamics was first presented byJacobson [1]. Padmanabhan [2] made the key development byestablishing a general formalism for understanding the thermo-dynamics of horizons in spherically symmetric spacetimes andshowed that it is possible to write the Einstein field equations inthe form of first law of thermodynamics arising from the virtualdisplacement of the horizon normal to itself as

(1)T dS = dE + P dV.

Paranjape, Sarkar and Padmanabhan [3] studied this approachfor a more general Lanczos–Lovelock theories of gravity andfound that it is possible to interpret the field equations near hori-zon for spherically symmetric spacetimes as a thermodynamicidentity (1). The Lanczos–Lovelock gravity contains higherderivative terms which can be thought as quantum correctionsto the Einstein gravity, therefore, it may be expected that the

* Corresponding author.E-mail addresses: ak64bar@yahoo.com (M. Akbar), azad@ceme.edu.pk

(A.A. Siddiqui).

0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.09.053

thermodynamic interpretation of the field equations holds evenif one includes possible quantum corrections to the Einstein–Hilbert action. Kothawala, Sarkar and Padmanabhan [4] furtherstudied this approach for stationary axis-symmetry horizonsand time dependent evolving horizons and found, in both cases,that the field equations near horizon can be expressed as a ther-mal identity. In Refs. [5,6] it has been shown that the Einsteinfield equations at apparent horizon of the Friedmann universecan also be expressed as a thermal identity.

These thermodynamic interpretations of gravitational dy-namics at horizons need further investigation for understandingit at a deeper level [2,7]. In recent years, (2 + 1)-dimensionalBTZ (Banados–Teitelboim–Zanelli) black holes have drawn alot of attention as simplified models for exploring conceptual is-sues relating to the black hole thermodynamics (see, e.g., [8,9]).

Recently, the thermodynamic interpretation of field equa-tions for static as well as non-static BTZ black hole near horizonis presented in [10]. In this Letter we have extended the thermo-dynamic study for charged rotating BTZ (CR-BTZ) black hole.After describing the CR-BTZ black hole in the following sec-tion, which has also been discussed in Ref. [11], we will discussthe thermodynamic interpretation of the Einstein field equationsin Section 3 and finally the conclusion is given in Section 4.

218 M. Akbar, A.A. Siddiqui / Physics Letters B 656 (2007) 217–220

2. Charged rotating BTZ black hole

Consider Einstein field equations for (2 + 1)-dimensionalspacetime with negative cosmological constant Λ = −l−2

(2)Gab + Λgab = πTab (a, b = 0,1,2),

where Gab and Tab are the Einstein and stress-energy tensorsrespectively. For the electro-vacuum stress-energy tensor, thestandard CR-BTZ solution of the above field equations (2) isgiven by the following metric [12,13]

(3)ds2 = −f (r) dt2 + dr2

f (r)+ r2

(dφ − J

2r2dt

)2

,

where the lapse function f (r), in the units such that the (2+1)-dimensional gravitational constant G3 = 1

8 , is given by

(4)f (r) = −M + r2

�2+ J 2

4r2− π

2Q2 ln r.

Here M and J are the integration constants which respectivelycorrespond to the mass and angular momentum, and Q is thecharge carried by the black hole.

Horizons of the CR-BTZ metric (3) are roots of the lapsefunction. Depending on these roots there are three cases of theCR-BTZ black hole, which we name (analogous to the ReissnerNordstrom or Kerr–Newman spacetimes) as:

(i) Usual CR-BTZ black hole when two distinct real rootsexist.

(ii) Extreme CR-BTZ black hole in case of two repeated realroots.

(iii) Naked CR-BTZ singularity when no real root exists.

It is hard to find exact expressions for the roots of the lapsefunction, however we analyze the function qualitatively in orderto obtain conditions on M , J and Q for the existence of abovementioned three cases as follows:

Extremal points of the function are determined using dfdr

= 0,

which gives r2 = �2(πQ2 ± √π2Q4 + 16J 2/�2 )/8. Exclud-

ing the negative sign as it gives r2 < 0, we have a unique

extremal point r = rmin = �

√(πQ2 + √

π2Q4 + 16J 2/�2 )/8(here again the negative root is excluded). This extremal point

is a minima as d2f

dr2 > 0 and the minimum value of the functionis

f (rmin) = −M +πQ2 +

√π2Q4 + 16J 2

�2

8

+ 2J 2

�2(πQ2 +√

π2Q4 + 16J 2

�2 )

(5)− π

4Q2 ln

[�2

8

(πQ2 +

√π2Q4 + 16J 2

�2

)].

Further, the value of f (r) → ∞ both as r → 0 or r → ∞. Nowconsider the cases

(a) f (rmin) < 0,

Fig. 1. Lapse function f (r) for Q = 3, M = 3 and J = 2 is shown which cutsthe r-axis at two distinct points r = r± . Here f (rmin) is negative. �2 = 1 in allfigures.

(b) f (rmin) = 0,(c) f (rmin) > 0.

Case a. In this case f (r) intersects the r-axis at two points andhence we have two distinct real roots at r = r± (see Fig. 1) andf (rmin) < 0 implies

M >πQ2 +

√π2Q4 + 16J 2

�2

8+ 2J 2

�2(πQ2 +√

π2Q4 + 16J 2

�2 )

(6)− π

4Q2 ln

[�2

8

(πQ2 +

√π2Q4 + 16J 2

�2

)].

Notice that the above inequality (6) is the general condition forthe existence of two horizons for the CR-BTZ black hole. Itsparticular cases (i) for Q = 0 and J �= 0 reduces to M > J/�

which is a known condition for the non-static BTZ black hole

[10] and (ii) for J = 0 and Q �= 0 it reduces to M >πQ2

4 [1 −ln(

π�2Q2

4 )] which is also known for the static charged BTZblack hole [11].

Case b. This corresponds to the extreme CR-BTZ black hole.In this case we have two real repeated roots at r = r− = r+(see Fig. 2) and M is equal to the right-hand side of the in-equality (6). The particular cases (i) Q = 0 and J �= 0 implies

M = J/� and (ii) J = 0 and Q �= 0 implies M = πQ2

4 [1 −ln(

π�2Q2

4 )].

Case c. This corresponds to the naked CR-BTZ singularity. Inthis case there are no real roots (see Fig. 3) and M is less thanthe right-hand side of the inequality (6).

M. Akbar, A.A. Siddiqui / Physics Letters B 656 (2007) 217–220 219

Fig. 2. Lapse function f (r) for Q = 1, M = 1 and J = 0.28175 is shown whichtouches the r-axis at one point r = r− = r+ = rmin. Here f (rmin) ≈ 0.

Fig. 3. Lapse function f (r) for Q = 1, M = 1 and J = 5 is shown which doesnot intersect the r-axis. Here f (rmin) is positive.

2.1. Heat capacity of the CR-BTZ black hole

The heat capacity at the horizon of the CR-BTZ black hole,using relation CJ,Q = (∂M/∂T )J,Q, is obtained as

(7)CJ,Q = 4πr+(

4r4+ − J 2�2 − πQ2�2r2+4r4+ + 3J 2�2 + πQ2�2r2+

).

The heat capacity is positive if the term in the numerator ofEq. (7) is positive, i.e.,

(8)4r4+ − J 2�2 − πQ2�2r2+ > 0.

Completing square on left-hand side of the inequality (8) andsimplifying we get

(9)r+ > rmin.

Therefore CJ,Q > 0 if r+ > rmin, which holds for the usualCR-BTZ black hole case showing its stability. For the extremeCR-BTZ black hole case r+ = rmin and CJ,Q = 0. Notice thatfor Q = 0 both, the expression for the heat capacity given byEq. (7) and condition (9) reduce to the expression and condi-tion obtained by Cai et al. [14] for the non-static BTZ blackhole.

3. Thermodynamic interpretation of the Einstein fieldequations

In this section we discuss thermodynamic properties of theCR-BTZ black hole at the event horizon located at r = r+. Thishorizon is associated with temperature T , entropy S, angularmomentum J and electric potential Φ as

T = 1

4π

df

dr

∣∣∣∣r=r+

, S = 4πr+,

J = 2r2+Ω = 2r2+∂M

∂J

∣∣∣∣r=r+

,

(10)Φ = ∂M

∂Q

∣∣∣∣r=r+

= −πQ ln r+,

where Ω is the angular velocity. The mass M at r = r+ is givenby

(11)M = r2+�2

+ Ω2r2+ + Φ

2Q.

Note that the thermodynamic quantities T , S, J , Q and M obeythe first law of thermodynamics dM = T dS + Ω dJ + Φ dQ.

Taking differential of Eq. (11) we have

(12)dM = 2r+�2

dr+ + 2Ω2r+ dr+ + Φ

2dQ.

The (1, 1)-component of the field equation (2), for the met-ric (3), for a general f (r) can be written in the form

(13)J 2 + 2r3f ′(r)

4r4− 1

�2= πT 1

1 ,

where T 11 is the (1, 1)-component of the stress-energy tensor

which corresponds to the radial pressure (Pr = −T 11 ) of the

source. Eq. (13) when evaluated at r = r+ can be written as

(14)J 2

4r3++ 1

2f ′(r+) − r+

�2= −Pr(πr+).

The above equation (14) describes the dynamics of the space-time near horizon. In order to determine a possible thermody-namic interpretation of this equation near horizon, consider avirtual displacement dr+ of the horizon and multiply it on bothsides of the equation. Then the resulting equation can be rewrit-ten as

220 M. Akbar, A.A. Siddiqui / Physics Letters B 656 (2007) 217–220

J 2

2r3+dr+ + f ′(r+)

4πd(4πr+) − 2r+dr+

�2

(15)= −Pr(2πr+ dr+).

Now using Eq. (12) and value of J = 2Ωr2+ (dJ = 4Ωr+ dr+)in the above equation we have

4Ω2r+ dr+ + f ′(r+)

4πd(4πr+) + Φ

2dQ − dM

(16)= −Pr d(πr2+

).

The first two terms on the left-hand side of Eq. (16) are trivialto recognize as Ω dJ and T dS, respectively. As discussed inRef. [11], if we similarly identify M + Φ

2 Q as the total energy E

and A as the area enclosed by the horizon, then Eq. (16) can bewritten as

(17)dE = T dS + Ω dJ + Φ dQ + Pr dA,

which is identical to the first law of thermodynamics. Hence thefield equations near horizon of CR-BTZ black hole behave likea thermal system satisfying the first law of thermodynamics.

4. Conclusion

In this Letter, we have discussed the CR-BTZ black holeparticularly in context with its horizons. The presence of loga-rithmic term in the lapse function makes it difficult to find exactexpressions for the horizons. However, by qualitative analysisof the lapse function, we are able to obtain conditions on M ,J , and Q for the existence of naked singularity, extreme andusual CR-BTZ black hole cases. It is also shown that the partic-ular cases, (i) J = 0 and (ii) Q = 0, of our results reduce to theearlier known results for charged and rotating BTZ black holesrespectively.

The expression for the heat capacity, CJ,Q, for the CR-BTZblack hole is obtained. It is shown that the heat capacity is pos-itive definite for the usual CR-BTZ black hole and zero for theextreme case.

As mentioned in the introduction, the field equations forspherically symmetric geometry, in case of Einstein and

Lanczos–Lovelock theories of gravity can be expressed asT dS = dE + P dV . For the Kerr–Newman geometry, whichis axially symmetric, the field equations were shown to holdthe identity T dS = dM −Ω dJ +Φ dQ. Here we have furtherfollowed this approach for the (2 + 1)-dimensional CR-BTZblack hole. It is shown that near horizon the Einstein fieldequations can be expressed as a first law of thermodynamicsdE = T dS + Ω dJ + Φ dQ + Pr dA, where E = M + Φ

2 Q isthe total energy of the black hole. These thermal identities indi-cate intrinsic thermodynamic properties of spacetime horizons.

Acknowledgements

M. Akbar would like to acknowledge the research facili-ties provided by NUST Centre for Advanced Mathematics andPhysics during this work. Authors are also grateful to the ref-eree for his/her comments which have significantly improvedquality of the paper.

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