Research Article On Thermodynamics of Charged and ...In this paper, we study thermodynamics of...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013, Article ID 371084, 7 pageshttp://dx.doi.org/10.1155/2013/371084

Research ArticleOn Thermodynamics of Charged and Rotating AsymptoticallyAdS Black Strings

Ren Zhao,1,2 Mengsen Ma,1,2 Huaifan Li,1,2 and Lichun Zhang1,2

1 Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China2Department of Physics, Shanxi Datong University, Datong 037009, China

Correspondence should be addressed to Ren Zhao; [email protected]

Received 14 March 2013; Revised 17 June 2013; Accepted 4 July 2013

Academic Editor: Tapobrata Sarkar

Copyright © 2013 Ren Zhao et al.This is an open access article distributed under theCreativeCommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we study thermodynamics of cylindrically symmetric black holes and calculate the equation of states and heatcapacity of charged and rotating black strings. In the process, we treat the cosmological constant as a thermodynamic pressureand its conjugate quantity as a thermodynamic volume. It is shown that, when taking the equivalence between the thermodynamicquantities of black strings and the ones of general thermodynamic system, the isothermal compressibility and heat capacity ofblack strings satisfy the stability conditions of thermodynamic equilibrium and no divergence points exist for heat capacity. Thus,we obtain the conclusion that the thermodynamic system relevant to black strings is stable and there is no second-order phasetransition for AdS black holes in the cylindrically symmetric spacetime.

1. Introduction

Black hole physics, especially black hole thermodynamics,refers to many fields such as theories of gravitation, statisticalphysics, particle physics, and field theory, which makes theprofound and fundamental connection between the theories,andmuch attention has been paid to the subject. It can be saidthat black hole physics has become the laboratory of manyrelevant theories. The pioneering works of Bekenstein andHawking have openedmany interesting aspects of unificationof quantum mechanics, gravity, and thermodynamics. Theseare known for the last forty years [1–5]. The black holethermodynamics has the similar forms to the general ther-modynamics, which attracted great attention. In particularthe case with negative cosmological constant (AdS case) hasconcernedmany physicists [6–22]. Asymptotically, AdS blackhole spacetimes admit a gauge duality description and aredescribed by dual conformal field theory. Correspondingly,one has a microscopic description of the underlying degreesof freedomat hand.This duality has been recently exploited tostudy the behavior of quark-gluon plasmas and for the qual-itative description of various condensed matter phenomena[12].

Recently, the studies on black hole thermodynamics inspherically symmetric spacetime by considering cosmolog-ical constant as the variable have got many attentions [12–16, 23–25]. In the previous works on the AdS black hole,cosmological constant corresponds to pressure in generalthermodynamic system, the relation is [12, 13, 15]

𝑃 = −1

8𝜋Λ =

3

8𝜋

1

𝑙2, (1)

and the corresponding thermodynamic volume is

𝑉 = (𝜕𝑀

𝜕𝑃)

𝑆,𝑄𝑖 ,𝐽𝑘

. (2)

In [16], the relation between cosmological constant andpressure is given in the higher dimensional AdS sphericallysymmetric spacetime, which supplies the basis for the studyon the black hole thermodynamics in AdS spherically sym-metric spacetime.

Theoretically, if we consider black holes in AdS spacetimeas a thermodynamic system, the critical behaviors and phasetransitions should also exist. Until now the statistical originof black hole thermodynamics is still unclear. Therefore,

2 Advances in High Energy Physics

the search for the connection between kinds of thermo-dynamic quantities in AdS spacetime is meaningful, whichmay help to understand the entropy, temperature, and heatcapacity of black holes and to build the consistent theory forblack hole thermodynamics.

In this paper, we generalize the works of [12–19] andresearch the charged and rotating cylindrically symmetricspacetime. According to (1), we analyzed the thermodynamicproperties of charged and rotating black string, calculated theheat capacity, and discussed the critical behaviors and phasetransition of black string.

2. Rotating Charged Black Strings

The asymptotically AdS solution of the Einstein-Maxwellequations with cylindrical symmetry can be written as [26–28]

𝑑𝑠2= −Ξ2(𝑓 (𝑟) −

𝑎2𝑟2

Ξ2𝑙4)𝑑𝑡2+

𝑑𝑟2

𝑓 (𝑟)

− 2𝑎Ξ𝑙

𝑟(𝑏 −

𝑙

𝑟𝜆2)𝑑𝑡 𝑑𝜙

+ [Ξ2𝑟2− 𝑎2𝑓 (𝑟)] 𝑑𝜙

2+

𝑟2

𝑙2𝑑𝑧2,

𝐴𝜇= −Ξ

𝑙𝜆

𝑟(𝛿0

𝜇−

𝑎

Ξ𝛿2

𝜇) ,

(3)

where

𝑓 (𝑟) =𝑟2

𝑙2−

𝑏𝑙

𝑟+

𝜆2𝑙2

𝑟2, Ξ

2= 1 +

𝑎2

𝑙2. (4)

𝑎, 𝑏, and 𝜆 are the constant parameters of the metric. Theentropy, mass, electric charge, and angular momentum perunit length of black string are

𝑆 =𝜋Ξ𝑟2

+

2𝑙, 𝑀 =

1

8(3Ξ2− 1) 𝑏,

𝑄 =Ξ𝜆

2, 𝐽 =

3

8Ξ𝑏𝑎,

(5)

where 𝑟+is the location of the event horizon of black hole,

which satisfies 𝑓(𝑟+) = 0. The Hawking temperature, angular

velocity, and electric potential of black string are

𝑇 =3𝑟4

+− 𝜆2𝑙4

4𝜋Ξ𝑙2𝑟3+

, Ω+=

𝑎

Ξ𝑙2, Φ =

𝜆𝑙

Ξ𝑟+

. (6)

Expressing the mass per unit length of black string as thefunction of entropy 𝑆, angular momentum 𝐽, electric charge

𝑄, and pressure𝑃 (cosmological constant 𝑙), from (4) and (5),we have

𝑄2=

𝑙2𝑆2

𝜋2𝑟4+

(𝑏𝑟+

𝑙−

𝑟4

+

𝑙4) ,

𝑄2𝜋2

𝑙𝑆2+

1

𝑙3=

𝑏

𝑟3+

,

𝑏 =

𝑟3

+(𝑆2+ 𝑄2𝜋2𝑙2)

𝑙3𝑆2,

4𝐽𝜋𝑟2

+

3=

(𝑄2𝜋2𝑙2+ 𝑆2)

𝑙2𝑆𝑟3

+𝑎,

(7)

𝑟2

+=

2𝑙𝑆

9𝜋(𝑆2 + 𝑄2𝜋2𝑙2)2

× [√16𝐽4𝑆2𝑙2𝜋6 + 81(𝑆2 + 𝑄2𝜋2𝑙2)4

− 4𝐽2𝑆𝑙𝜋3]

=2𝑙𝑆

9𝜋(𝑆2 + 𝑄2𝜋2𝑙2)2𝑌,

(8)

where 𝑌 = √16𝐽4𝑆2𝑙2𝜋6 + 81(𝑆2 + 𝑄2𝜋2𝑙2)4− 4𝐽2𝑆𝑙𝜋3.

From this we can get

𝑀 =1

8[12𝑙2𝑆2

𝜋2𝑟4+

− 1] 𝑏

=1

8[12𝑙2𝑆2− 𝜋2𝑟4

+

𝜋2𝑟4+

]

(𝑆2+ 𝑄2𝜋2𝑙2)

𝑙3𝑆2𝑟3

+

=1

8𝜋2𝑆2𝑙3(𝑆2+ 𝑄2𝜋2𝑙2) [

12𝑙2𝑆2− 𝜋2𝑟4

+

𝑟+

]

=3

√8𝜋3𝑙3𝑆

[

[

3 × 81(𝑆2+ 𝑄2𝜋2𝑙2)4

− 𝑌2

81(𝑆2 + 𝑄2𝜋2𝑙2)2√𝑌

]

]

=3

√8𝜋3𝑙3𝑆

[

[

2 × 81(𝑆2+ 𝑄2𝜋2𝑙2)4

+ 8𝐽2𝑆𝑙𝜋3𝑌

81(𝑆2 + 𝑄2𝜋2𝑙2)2√𝑌

]

]

,

(9)

where 𝑌2 = −8𝐽2𝑆𝑙𝜋3𝑌 + 81(𝑆

2+ 𝑄2𝜋2𝑙2)4. From (9), we can

find that the thermodynamic quantities of black string satisfythe first law of thermodynamics as

𝑑𝑀 = 𝑇𝑑𝑆 + Φ𝑑𝑄 + Ω𝑑𝐽 + 𝑉𝑑𝑃. (10)

From (10), one can deduce

𝑇 = (𝜕𝑀

𝜕𝑆)

𝐽,𝑄,𝑃

, Ω = (𝜕𝑀

𝜕𝐽)

𝑆,𝑄,𝑃

,

Φ = (𝜕𝑀

𝜕𝑄)

𝐽,𝑆,𝑃

, 𝑉 = (𝜕𝑀

𝜕𝑃)

𝐽,𝑄,𝑆

,

Advances in High Energy Physics 3

(𝜕𝑀

𝜕𝑙)

𝑆,𝑄,𝐽

= −9

2𝑙√8𝜋3𝑙3𝑆

× [

[

2 × 81(𝑆2+ 𝑄2𝜋2𝑙2)4

+8𝐽2𝑆𝑙𝜋3𝑌

81(𝑆2 + 𝑄2𝜋2𝑙2)2√𝑌

]

]

+

3 (𝑆2+ 𝑄2𝜋2𝑙2)

√𝑌√8𝜋3𝑙3𝑆

× [8𝑄2𝜋2𝑙 −

(𝑆2+ 𝑄2𝜋2𝑙2)

𝑌

𝜕𝑌

𝜕𝑙

+4𝐽2𝑆𝜋3

81(𝑆2 + 𝑄2𝜋2𝑙2)3

× (2𝑌 + 𝑙𝜕𝑌

𝜕𝑙) −

32𝐽2𝑆𝑙2𝜋5𝑄2𝑌

81(𝑆2 + 𝑄2𝜋2𝑙2)4] ,

(11)

where 𝜕𝑌/𝜕𝑙 = (81 × 4𝑄2𝜋2𝑙(𝑆2+ 𝑄2𝜋2𝑙2)3− 4𝐽2𝑆𝜋3𝑌)/√𝑌.

From (1), one can derive the corresponding “thermody-namic” volume of black string as

𝑉 = (𝜕𝑀

𝜕𝑃)

𝑆,𝑄,𝐽

=𝑏𝜋

2𝑙2Ξ2−

2𝜋𝜆2𝑙3

3𝑟+

. (12)

From (12), one can get

Ξ2=

𝑙𝑉

𝜋𝑟3+

(𝑉 −2𝜋𝑙3𝑄2

𝑟+

)

±𝑙

𝑟2+

√1

𝜋2𝑟2+


𝑟+

)

2

+16𝑙2𝑄2

3.

(13)

Because of 𝑄 = 0, only the plus sign is kept, namely,

Ξ2=

𝑙𝑉

𝜋𝑟3+


𝑟+

)

+𝑙

𝑟2+

√1

𝜋2𝑟2+


𝑟+

)

2

+16𝑙2𝑄2

3.

(14)

From Ξ2= 1 + 𝑎

2/𝑙2, we can derive

𝑎2=[[

[

𝑙𝑉

𝜋𝑟3+


𝑟+

)

+𝑙

𝑟2+

√1

𝜋2𝑟2+


𝑟+

)

2

+16𝑙2𝑄2

3− 1

]]

]

𝑙2.

(15)

From 𝐽 = (3/8)Ξ𝑏𝑎, we get

𝐽2=

9

64Ξ2(𝑟3

+

𝑙3+

4𝑄2𝑙

𝑟+Ξ2

)

2

𝑎2. (16)

3. Thermodynamics of Charged Black String

In this section, we discuss thermodynamics of static chargedblack string. When 𝑎 = 0, Ξ = 1. From (12), one can get

𝑉 =𝜋𝑟3

+

2𝑙−

2𝜋𝑄2𝑙3

3𝑟+

,

𝑇 =3𝑟+

4𝜋𝑙2−

𝑙2𝑄2

𝜋𝑟3+

.

(17)

From this, we obtain

(𝜕𝑃

𝜕𝑉)

𝑇,𝑄

= (3

4𝜋𝑙2+

3𝑙2𝑄2

𝜋𝑟4+

)

× ((𝜋𝑟3

+

2𝑙2+

2𝜋𝑄2𝑙2

𝑟+

) × (3

4𝜋𝑙2+

3𝑙2𝑄2

𝜋𝑟4+

)

− (3𝜋𝑟2

+

2𝑙+

2𝜋𝑄2𝑙3

3𝑟2+

) ×(3𝑟+

2𝜋𝑙3+

2𝑄2𝑙

𝑟3+

))

−1

×3

4𝜋𝑙3

=3/4𝜋𝑙2+ 3𝑙2𝑄2/𝜋𝑟4

+

14𝑄4𝑙4/3𝑟5+− (15𝑟3

+/8𝑙4 + 𝑄2/𝑟

+)

3

4𝜋𝑙3.

(18)

From (18), when 14𝑄4𝑙4/3𝑟5

+> 15𝑟

3

+/𝑙4+𝑄2/𝑟+, (𝜕𝑃/𝜕𝑉)

𝑇,𝑄>

0, the thermodynamic system is unstable. When

14𝑄4𝑙4

3𝑟5+

<15𝑟3

+

𝑙4+

𝑄2

𝑟+

, (19)

(𝜕𝑃/𝜕𝑉)𝑇,𝑄

< 0, the thermodynamic system is stable. When𝑄 → 0, (𝜕𝑃/𝜕𝑉)

𝑇,𝑄< 0, the thermodynamic system is

stable. Heat capacity at constant pressure is

𝐶𝑃,𝑄

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑃,𝑄

= 𝑇[𝜋𝑟+

3/4𝜋𝑙 + 3𝑙3𝑄2/𝜋𝑟4+

] . (20)

According to (20), if 𝑇 > 0, namely, 3𝑟+/4𝑙2> 𝑄2𝑙2/𝑟3

+, 𝐶𝑃,𝑄

will be greater than zero, which fulfills the stable condition


10 20 30 40 50

−0.5

0.5

1.0

1.5

r4+

CV/Q

l

Figure 1: Plots of the heat capacity at constant 𝑉 and 𝑄 versus the𝑟4

+. The horizontal axis started from 4/3, because of 𝑇 > 0. The

intersection point of 𝐶𝑉,𝑄

with the horizontal axis is 20/3, whichshows that when 𝑟

4

+> (20/3)𝑄

2𝑙4 holds up, 𝐶

𝑉,𝑄> 0.

of thermodynamic equilibrium.The heat capacity at constantvolume is

𝐶𝑉,𝑄

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑉,𝑄

= 𝑇((𝜋𝑟+

𝑙(3𝑟5

++ 12𝑄

2𝑙4𝑟+

9𝑙𝑟4++ 4𝑄2𝑙5

) −𝜋𝑟2

+

2𝑙2)

× ((3

4𝜋𝑙2+

3𝑄2𝑙2

𝜋𝑟4+

) × (3𝑟5

++ 12𝑄

2𝑙4𝑟+

9𝑙𝑟4++ 4𝑄2𝑙5

)

−(3𝑟+

2𝜋𝑙3+

2𝑄2𝑙

𝜋𝑟3+

))

−1

)

= 𝑇10𝜋𝑄

2𝑙3𝑟2

+− 3𝜋𝑟

6

+/2𝑙

28𝑄4𝑙6/𝜋𝑟3+− 45𝑟5+/4𝜋𝑙2 − 6𝑄2𝑟

+𝑙2/𝜋

= (3𝑟+

4𝑙2−

𝑄2𝑙2

𝑟3+

)10𝜋𝑄

2𝑙3𝑟2

+− 3𝜋𝑟

6

+/2𝑙

28𝑄4𝑙6/𝑟3+− 45𝑟5+/4𝑙2 − 6𝑄2𝑟

+𝑙2.

(21)

We can plot the curve of 𝐶𝑉,𝑄

, which shows that only whenthe condition 𝑟

4

+> (20/3)𝑄

2𝑙4 holds up, 𝐶

𝑉,𝑄> 0 will work.

See Figure 1.From (20) and 𝑇 > 0, when 𝑟

4

+> (4/3)𝑄

2𝑙4, 𝐶𝑃,𝑄

will be greater than zero, which fulfills the stable conditionof thermodynamic equilibrium. When 𝑟

4

+= (4/3)𝑄

2𝑙4,

the Hawking temperature of heat capacity is zero whichcorresponds to the extreme case. However, for the 𝐶

𝑉,𝑄only

the condition 𝑇 > 0 is not enough. One needs more strictcondition 𝑟

4

+> (20/3)𝑄

2𝑙4, under which the 𝐶

𝑉,𝑄is greater

than zero. This suggests that the thermodynamic system ofcharged black strings does not have the first-order phasetransition only when 𝑟

4

+> (20/3)𝑄

2𝑙4. On the other hand, the

second-order phase transition points of the thermodynamicsystem of charged black strings turn up when heat capacitiesdiverge. In this charged black string spacetime, under thegiven condition 𝑟

4

+> (20/3)𝑄

2𝑙4, 𝐶𝑉,𝑄

and 𝐶𝑃,𝑄

are always

greater than zero, which suggests that the second-order phasetransition of black string will not happen. Whether thephase transition exists when the condition breaks out will bediscussed later.

4. Thermodynamics of Rotating Black String

In this section, we discuss thermodynamics of stationaryrotating black string. When 𝑄 = 0, from (12) and (14), wehave

Ξ2=

2𝑙𝑉

𝜋𝑟3+

, 𝑉(2𝑙𝑉

𝜋𝑟3+

− 1) =32𝜋𝐽2𝑙3

9𝑟3+

,

𝑉 =𝜋𝑟3

+

4𝑙+

𝜋

2

√𝑟6

+

4𝑙2+

64𝐽2𝑙2

9,

(22)

𝑇 =3𝑟2

+√𝑟+

4𝑙2√2𝜋𝑙𝑉

. (23)

From (22) and (23), we deduce

(𝜕𝑙

𝜕𝑉)

𝑇,𝐽

=

3 (20𝑙𝑉 − 8𝜋𝑟3

+) 𝑙

5𝜋 (9𝑉𝑟3++ 32𝜋𝐽2𝑙3) − 30𝑙𝑉2

=5𝑙𝑉 − 2𝜋𝑟

3

+

5𝑉.

(24)

Thus,

(𝜕𝑃

𝜕𝑉)

𝑇,𝐽

=

3 (2𝜋𝑟3

+− 5𝑙𝑉)

20𝜋𝑙3𝑉. (25)

From (22), we have 2𝑙𝑉 > 𝜋𝑟3

+, so (𝜕𝑃/𝜕𝑉)

𝑇,𝐽< 0, which

satisfies the condition of thermodynamic equilibrium. Wecan derive the heat capacities of rotating string at constantpressure and constant volume as follows:

𝐶𝑉,𝐽

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑉,𝐽

= 𝑇((1

6𝑟2+𝑉(

𝜋𝑉

2𝑙𝑟+

)

1/2

(2𝑉2

𝜋−

32𝐽2𝑙2

3)

−1

2𝑙(𝜋𝑉𝑟+

2𝑙)

1/2

)

× (5

8𝑙2𝑉√2𝜋𝑙𝑉𝑟+

(2𝑉2

𝜋−

32𝐽2𝑙2

3)

−15𝑟5/2

+

8𝑙3√2𝜋𝑙𝑉

)

−1

)

= 𝑇2𝜋𝑙𝑉√2𝜋𝑙𝑉𝑟

+

15𝑟3+

=𝜋

10

𝑉

𝑙,


𝐶𝑃,𝐽

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑃,𝐽

= 𝑇((1

6𝑟2+𝑉(

𝜋𝑉

2𝑙𝑟+

)

1/2

(4𝑙𝑉

𝜋− 𝑟3

+) +

1

2(𝜋𝑟+

2𝑙𝑉)

1/2

)

×(5

8𝑙2𝑉√2𝜋𝑙𝑉𝑟+

(4𝑙𝑉

𝜋− 𝑟3

+)

−3𝑟5/2

+

8𝑙2𝑉√2𝜋𝑙𝑉

)

−1

)

=1

2(𝜋𝑟+𝑉

2𝑙)

1/2

(2𝑙𝑉 − 𝜋𝑟

3

+

5𝑙𝑉 − 2𝜋𝑟3+

) .

(26)

From (22), we have 𝑙𝑉 > 𝜋𝑟3

+/2, so 𝐶

𝑃,𝐽> 0, which satisfies

the condition of thermodynamic equilibrium. The second-order phase transition points of thermodynamic systems willappear when heat capacities diverge. According to (26), theheat capacities do not have divergent points; therefore, thesecond-order phase transition of rotating black string alsocannot happen.

5. Thermodynamics of Charged and RotatingBlack String

In this section, we discuss thermodynamics of static chargedand rotating black string. The location 𝑟

+of event horizon

satisfies

𝑟2

+

𝑙2−

𝑏𝑙

𝑟+

+𝜆2𝑙2

𝑟2+

= 0, (27)

where

𝑟+=

1

2{𝛾1/2

+ [−𝛾 + 2(𝛾2− 4𝜆2𝑙4)1/2

]

1/2

} ,

𝛾 =

{

{

{

𝑏2𝑙6

2+ [(

𝑏2𝑙6

2)

2

+ (4𝜆2𝑙4

3)

2

]

1/2

}

}

}

1/3

+

{

{

{

𝑏2𝑙8

2− [(

𝑏2𝑙8

2)

2

+ (4𝜆2𝑙4

3)

2

]

1/2

}

}

}

1/3

.

(28)

For discussion purpose andwithout loss of generality, we take𝑎(𝐽) and 𝑄 to be small quantities relative to 𝑀 or 𝑙 and 𝑟

+,

namely, 𝑎2/𝑙2 ≪ 1, 𝑟2+≫ 𝑎2. From (16), we have

𝐽 ≈3

8(𝑟3

+

𝑙3+

4𝑄2𝑙

𝑟+

)𝑎, (29)

and when 𝑄 is small

𝑎 ≈8𝑙3

3𝑟3+

𝐽. (30)

From (12), we can get the approximate value of volume

𝑉 ≈𝜋𝑟3

+

2𝑙−

2𝜋𝑄2𝑙3

3𝑟+

+32𝜋𝑙6

9𝑟6+

𝐽2. (31)

According to (6), we can obtain the approximate Hawkingtemperature

𝑇 ≈3𝑟+

4𝜋𝑙2(1 −

32𝑙4

𝑟6+

𝐽2) −

𝑄2𝑙2

𝜋𝑟3+

. (32)

From this, we can deduce

(𝜕𝑃

𝜕𝑉)

𝑇,𝐽,𝑄

≈3/4𝜋𝑙2+ 120𝑙

2𝐽2/𝜋𝑟6

++ 3𝑄2𝑙2/𝜋𝑟4

+

16𝑙3𝐽2/𝑟6+− (15𝑟3

+/8𝑙4 + 𝑄2/𝑟

++ 12𝐽2/𝑟3

+)

×3

4𝜋𝑙3.

(33)

From (32), when requiring 𝑇 > 0, the following equationshould be satisfied:

3𝑟+

4𝑙2>

24𝑙2𝐽2

𝑟5+

+𝑄2𝑙2

𝑟3+

. (34)

From (33), when

15𝑟3

+

8𝑙4>

16𝑙3𝐽2

𝑟6+

− (12𝐽2

𝑟3+

+𝑄2

𝑟+

) , (35)

we have (𝜕𝑃/𝜕𝑉)𝑇,𝐽,𝑄

< 0, which satisfies the condition ofthermodynamic equilibrium. Substituting (34) into (33), wecan get 15𝑟3

+/8𝑙4> 16𝑙3𝐽2/𝑟6

+− 3𝑟3

+/4𝑙4, or

21𝑟3

+

8𝑙4>

16𝑙3𝐽2

𝑟6+

≈9𝑎2

4𝑙3. (36)

From (28), one can deduce 𝑟+/𝑙 ∝ (4𝑀)

1/3> 𝑎, 𝑟2

+≫ 𝑎2;

thus, (36) is satisfied.In order to show the relation between 𝑃 and𝑉 clearly, we

plot the 𝑉-𝑃 curve. According to (1), (31), and (32), we candepict the 𝑉-𝑃 curve of charged and rotating black strings(Figure 2).

From this figure, we know that the𝑉-𝑃 curves of chargedand rotating black strings are smooth and continuous; there-fore, under the condition of isothermality the first-order andsecond-order phase transitions caused by the variation ofpressure or volume do not exist.

The approximate expression of entropy is

𝑆 =𝜋Ξ𝑟2

+

2𝑙≈

𝜋𝑟2

+

2𝑙(1 +

𝑎2

2𝑙2) ≈

𝜋𝑟2

+

2𝑙(1 +

32𝑙4

9𝑟6+

𝐽2) . (37)


1.2

1.0

0.8

0.6

0.4

0.2

0.00 50 100 150

V

P

Figure 2: Plots of the pressure 𝑃 versus the volume 𝑉. The curvescorrespond to the parameters 𝑇 = 1, 𝐽 = 0.1, and 𝑄 = 0.0, 0.5, 1.0.The three curves roughly coincide.

The heat capacity of charged and rotating black strings atconstant pressure and constant volume is

𝐶𝑉,𝐽,𝑄

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑉,𝐽,𝑄

≈ 𝑇((5𝜋2𝑄2𝑙

3−

𝜋2𝑟4

+

𝑙3−

22𝜋2𝑙4𝐽2

3𝑟5+

+40𝜋2𝑙𝐽2

9𝑟2+

)

×(−33𝑟3

+

8𝑙4−

𝑄2

𝑟+

+16𝑙3𝐽2

𝑟6+

−12𝐽2

𝑟3+

)

−1

) ,

𝐶𝑃,𝐽,𝑄

= 𝑇(𝜕𝑆

𝜕𝑇)

𝑃,𝐽,𝑄

= 𝑇

𝜋𝑟+/𝑙 − (64𝜋𝑙

3/9𝑟5

+) 𝐽2

3/4𝜋𝑙2 + 120𝑙2𝐽2/𝜋𝑟6++ 3𝑄2𝑙2/𝜋𝑟4

+

.

(38)

From (35), one can deduce𝜋2𝑟4+/𝑙3> 22𝑙4𝐽2/3𝑟5

+−(40𝑙𝐽

2/9𝑟2

++

5𝑄2𝑙/3), 33𝑟3

+/8𝑙4> +16𝑙

3𝐽2/𝑟6

+−(12𝐽

2/𝑟3

++𝑄2/𝑟+); therefore,

𝐶𝑉,𝐽,𝑄

> 0, 𝐶𝑃,𝐽,𝑄

> 0. (39)

Thuswe can consider the charged and rotating black strings asa thermodynamic system and the system can satisfy the stableconditions of equilibrium under the assumption of small 𝑎and𝑄, because the second order phase transition points of thethermodynamic system turn upwhen heat capacities diverge.According to (39), the heat capacities are always greater thanzero, which suggests that the second-order phase transitionof black string will not happen when 𝑎 and 𝑄 are smallquantities.

6. Conclusion

In this paper, we study the thermodynamic properties ofcharged and rotating black strings in cylindrically symmetricAdS spacetime. Like the spherically symmetric case for the

charged and rotating black strings we take the cosmologicalconstant to correspond to the pressure in general thermody-namic system. The relation is (1). We consider the identifica-tion, because when solving Einstein equations the cosmolog-ical constant 𝑙 is independent of the symmetry of spacetimeunder consideration and the pressure in thermodynamicsystem also has nothing to do with the surface morphology.Thus the relation (1) should also be appropriate to the chargedand rotating cylindrically symmetric spacetime.

On the basis of (1), we analyze the corresponding ther-modynamic quantities for charged and rotating black strings.We find that, under some conditions, the heat capacitiesare greater than zero and (𝜕𝑃/𝜕𝑉)

𝑇,𝐽,𝑄< 0, which satisfy

the stable condition of thermodynamic equilibrium. Thus,when the system is perturbed slightly and deviates fromequilibrium, some process will appear automatically andmakes the system restore equilibrium.

Compared with the works of [12–14], it is found that thethermodynamic properties of black holes in spherically sym-metric spacetime are different from the ones of black holesin cylindrically symmetric spacetime, specially that the heatcapacities of black holes in cylindrically symmetric spacetimedo not have divergent points; thus, no second-order phasetransition occurs and no critical phenomena similar to Vander Waals gas occur. At present, the problem cannot beexplained logically and it deserves further discussion.

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grant nos. 11075098 and11175109), the Young Scientists Fund of the National NaturalScience Foundation of China (Grant no. 11205097), theNatural Science Foundation for Young Scientists of ShanxiProvince, China (Grant no. 2012021003-4), and the ShanxiDatong University Doctoral Sustentation Fund (nos. 2008-B-06 and 2011-B-04), China.

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