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
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+
(𝑉 −2𝜋𝑙3𝑄2
𝑟+
)
2
+16𝑙2𝑄2
3.
(13)
Because of 𝑄 = 0, only the plus sign is kept, namely,
Ξ2=
𝑙𝑉
𝜋𝑟3+
(𝑉 −2𝜋𝑙3𝑄2
𝑟+
)
+𝑙
𝑟2+
√1
𝜋2𝑟2+
(𝑉 −2𝜋𝑙3𝑄2
𝑟+
)
2
+16𝑙2𝑄2
3.
(14)
From Ξ2= 1 + 𝑎
2/𝑙2, we can derive
𝑎2=[[
[
𝑙𝑉
𝜋𝑟3+
(𝑉 −2𝜋𝑙3𝑄2
𝑟+
)
+𝑙
𝑟2+
√1
𝜋2𝑟2+
(𝑉 −2𝜋𝑙3𝑄2
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
4 Advances in High Energy Physics
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,
Advances in High Energy Physics 5
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)
6 Advances in High Energy Physics
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.
References
[1] J. D. Bekenstein, “Black holes and the second law,” Lettere AlNuovo Cimento Series 2, vol. 4, no. 15, pp. 737–740, 1972.
[2] J. D. Bekenstein, “Generalized second law of thermodynamicsin black-hole physics,”Physical ReviewD, vol. 9, no. 12, pp. 3292–3300, 1974.
[3] J. M. Bardeen, B. Carter, and S. W. Hawking, “The four lawsof black hole mechanics,” Communications in MathematicalPhysics, vol. 31, no. 2, pp. 161–170, 1973.
[4] S. W. Hawking, “Black hole explosions?” Nature, vol. 248, no.5443, pp. 30–31, 1974.
[5] S. W. Hawking, “Particle creation by black holes,” Communica-tions in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975.
[6] R. Banerjee, S. K. Modak, and S. Samanta, “Second order phasetransition and thermodynamic geometry in Kerr-AdS blackholes,” Physical Review D, vol. 84, no. 6, Article ID 064024, 8pages, 2011.
[7] R. Banerjee and D. Roychowdhury, “Critical behavior of Born-Infeld AdS black holes in higher dimensions,” Physical ReviewD, vol. 85, no. 10, Article ID 104043, 14 pages, 2012.
Advances in High Energy Physics 7
[8] R. Banerjee and D. Roychowdhury, “Critical phenomena inBorn-Infeld AdS black holes,” Physical Review D, vol. 85, no. 4,Article ID 044040, 10 pages, 2012.
[9] R. Banerjee and D. Roychowdhury, “Thermodynamics of phasetransition in higher dimensional AdS black holes,” Journal ofHigh Energy Physics, vol. 2011, article 4, 2011.
[10] R. Banerjee, S. Ghosh, and D. Roychowdhury, “New type ofphase transition in Reissner Nordstrom-AdS black hole and itsthermodynamic geometry,” Physics Letters B, vol. 696, no. 1-2,pp. 156–162, 2011.
[11] R. Banerjee, S. K. Modak, and S. Samanta, “Glassy phase tran-sition and stability in black holes,” European Physical Journal C,vol. 70, no. 1, pp. 317–328, 2010.
[12] D. Kubiznak and R. B. Mann, “𝑃 − 𝑉 criticality of charged AdSblack holes,” Journal of High Energy Physics, vol. 2012, no. 7,article 33, 2012.
[13] B. P. Dolan, D. Kastor, D. Kubiznak, R. B.Mann, and J. Traschen,“Thermodynamic volumes and isoperimetric inequalities for desitter black holes,” Physical Review D, vol. 87, no. 10, Article ID104017, 14 pages, 2013.
[14] S. Gunasekaran, D. Kubiznak, and R. B.Mann, “Extended phasespace thermodynamics for charged and rotating black holesand Born-Infeld vacuum polarization,” Journal of High EnergyPhysics, vol. 2012, article 110, 2012.
[15] M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, “Blackhole enthalpy and an entropy inequality for the thermodynamicvolume,” Physical Review D, vol. 84, no. 2, Article ID 024037, 17pages, 2011.
[16] S. W. Wei and Y. X. Liu, “Critical phenomena and thermody-namic geometry of charged Gauss-Bonnet AdS black holes,”Physical Review D, vol. 87, no. 4, Article ID 044014, 14 pages,2013.
[17] A. Fatima and K. Saifullah, “Thermodynamics of charged androtating black strings,” Astrophysics and Space Science, vol. 341,no. 2, pp. 437–443, 2012.
[18] M. Akbar, H. Quevedo, K. Saifullah, A. Sanchez, and S. Taj,“Thermodynamic geometry of charged rotating BTZ blackholes,” Physical Review D, vol. 83, no. 8, Article ID 084031, 10pages, 2011.
[19] A. Belhaj, M. Chabab, H. El Moumni, and M. B. Sedra, “Onthermodynamics of AdS black holes in arbitrary dimensions,”Chinese Physics Letters, vol. 29, no. 10, Article ID 100401, 2012.
[20] M. H. Dehghani and S. Asnafi, “Thermodynamics of rotatingLovelock-Lifshitz black branes,” Physical Review D, vol. 84, no.6, Article ID 064038, 8 pages, 2011.
[21] S. Bellucci and B. N. Tiwari, “Thermodynamic geometry andtopological Einstein-Yang-Mills black holes,” Entropy, vol. 14,no. 6, pp. 1045–1078, 2012.
[22] J. Shen, R. Cai, B. Wang, and R. Su, “Thermodynamic geometryand critical behavior of black holes,” International Journal ofModern Physics A, vol. 22, no. 1, pp. 11–27, 2007.
[23] B. Dolan, “The cosmological constant and the black holeequation of state,” Classical and Quantum Gravity, vol. 28, no.12, Article ID 125020, 2011.
[24] B. P. Dolan, “Pressure and volume in the first law of black holethermodynamics,” Classical and Quantum Gravity, vol. 28, no.23, Article ID 235017, 2011.
[25] B. P. Dolan, “Compressibility of rotating black holes,” PhysicalReview D, vol. 84, no. 12, Article ID 127503, 3 pages, 2011.
[26] J. P. S. Lemos and V. T. Zanchin, “Rotating charged black stringsand three-dimensional black holes,” Physical Review D, vol. 54,no. 6, pp. 3840–3853, 1996.
[27] M. H. Dehghani, “Thermodynamics of rotating charged blackstrings and (A)dS/CFT correspondence,”Physical ReviewD, vol.66, no. 4, Article ID 044006, 6 pages, 2002.
[28] R. G. Cai and Y. Z. Zhang, “Black plane solutions in four-dimensional spacetimes,” Physical Review D, vol. 54, no. 8, pp.4891–4898, 1996.
