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Research Article On Thermodynamics of Charged and ...In this paper, we study thermodynamics of...
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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 Β© 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 [1β5]. 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 [6β22]. 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 [12β16, 23β25]. In the previous works on the AdS black hole,cosmological constant corresponds to pressure in generalthermodynamic system, the relation is [12, 13, 15]
π = β1
8πΞ =
3
8π
1
π2, (1)
and the corresponding thermodynamic volume is
π = (ππ
ππ)
π,ππ ,π½π
. (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,
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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 [12β19] 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 [26β28]
ππ 2= βΞ2(π (π) β
π2π2
Ξ2π4)ππ‘2+
ππ2
π (π)
β 2πΞπ
π(π β
π
ππ2)ππ‘ ππ
+ [Ξ2π2β π2π (π)] ππ
2+
π2
π2ππ§2,
π΄π= βΞ
ππ
π(πΏ0
πβ
π
ΞπΏ2
π) ,
(3)
where
π (π) =π2
π2β
ππ
π+
π2π2
π2, Ξ
2= 1 +
π2
π2. (4)
π, π, and π are the constant parameters of the metric. Theentropy, mass, electric charge, and angular momentum perunit length of black string are
π =πΞπ2
+
2π, π =
1
8(3Ξ2β 1) π,
π =Ξπ
2, π½ =
3
8Ξππ,
(5)
where π+is the location of the event horizon of black hole,
which satisfies π(π+) = 0. The Hawking temperature, angular
velocity, and electric potential of black string are
π =3π4
+β π2π4
4πΞπ2π3+
, Ξ©+=
π
Ξπ2, Ξ¦ =
ππ
Ξπ+
. (6)
Expressing the mass per unit length of black string as thefunction of entropy π, angular momentum π½, electric charge
π, and pressureπ (cosmological constant π), from (4) and (5),we have
π2=
π2π2
π2π4+
(ππ+
πβ
π4
+
π4) ,
π2π2
ππ2+
1
π3=
π
π3+
,
π =
π3
+(π2+ π2π2π2)
π3π2,
4π½ππ2
+
3=
(π2π2π2+ π2)
π2ππ3
+π,
(7)
π2
+=
2ππ
9π(π2 + π2π2π2)2
Γ [β16π½4π2π2π6 + 81(π2 + π2π2π2)4
β 4π½2πππ3]
=2ππ
9π(π2 + π2π2π2)2π,
(8)
where π = β16π½4π2π2π6 + 81(π2 + π2π2π2)4β 4π½2πππ3.
From this we can get
π =1
8[12π2π2
π2π4+
β 1] π
=1
8[12π2π2β π2π4
+
π2π4+
]
(π2+ π2π2π2)
π3π2π3
+
=1
8π2π2π3(π2+ π2π2π2) [
12π2π2β π2π4
+
π+
]
=3
β8π3π3π
[
[
3 Γ 81(π2+ π2π2π2)4
β π2
81(π2 + π2π2π2)2βπ
]
]
=3
β8π3π3π
[
[
2 Γ 81(π2+ π2π2π2)4
+ 8π½2πππ3π
81(π2 + π2π2π2)2βπ
]
]
,
(9)
where π2 = β8π½2πππ3π + 81(π
2+ π2π2π2)4. From (9), we can
find that the thermodynamic quantities of black string satisfythe first law of thermodynamics as
ππ = πππ + Ξ¦ππ + Ξ©ππ½ + πππ. (10)
From (10), one can deduce
π = (ππ
ππ)
π½,π,π
, Ξ© = (ππ
ππ½)
π,π,π
,
Ξ¦ = (ππ
ππ)
π½,π,π
, π = (ππ
ππ)
π½,π,π
,
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Advances in High Energy Physics 3
(ππ
ππ)
π,π,π½
= β9
2πβ8π3π3π
Γ [
[
2 Γ 81(π2+ π2π2π2)4
+8π½2πππ3π
81(π2 + π2π2π2)2βπ
]
]
+
3 (π2+ π2π2π2)
βπβ8π3π3π
Γ [8π2π2π β
(π2+ π2π2π2)
π
ππ
ππ
+4π½2ππ3
81(π2 + π2π2π2)3
Γ (2π + πππ
ππ) β
32π½2ππ2π5π2π
81(π2 + π2π2π2)4] ,
(11)
where ππ/ππ = (81 Γ 4π2π2π(π2+ π2π2π2)3β 4π½2ππ3π)/βπ.
From (1), one can derive the corresponding βthermody-namicβ volume of black string as
π = (ππ
ππ)
π,π,π½
=ππ
2π2Ξ2β
2ππ2π3
3π+
. (12)
From (12), one can get
Ξ2=
ππ
ππ3+
(π β2ππ3π2
π+
)
Β±π
π2+
β1
π2π2+
(π β2ππ3π2
π+
)
2
+16π2π2
3.
(13)
Because of π = 0, only the plus sign is kept, namely,
Ξ2=
ππ
ππ3+
(π β2ππ3π2
π+
)
+π
π2+
β1
π2π2+
(π β2ππ3π2
π+
)
2
+16π2π2
3.
(14)
From Ξ2= 1 + π
2/π2, we can derive
π2=[[
[
ππ
ππ3+
(π β2ππ3π2
π+
)
+π
π2+
β1
π2π2+
(π β2ππ3π2
π+
)
2
+16π2π2
3β 1
]]
]
π2.
(15)
From π½ = (3/8)Ξππ, we get
π½2=
9
64Ξ2(π3
+
π3+
4π2π
π+Ξ2
)
2
π2. (16)
3. Thermodynamics of Charged Black String
In this section, we discuss thermodynamics of static chargedblack string. When π = 0, Ξ = 1. From (12), one can get
π =ππ3
+
2πβ
2ππ2π3
3π+
,
π =3π+
4ππ2β
π2π2
ππ3+
.
(17)
From this, we obtain
(ππ
ππ)
π,π
= (3
4ππ2+
3π2π2
ππ4+
)
Γ ((ππ3
+
2π2+
2ππ2π2
π+
) Γ (3
4ππ2+
3π2π2
ππ4+
)
β (3ππ2
+
2π+
2ππ2π3
3π2+
) Γ(3π+
2ππ3+
2π2π
π3+
))
β1
Γ3
4ππ3
=3/4ππ2+ 3π2π2/ππ4
+
14π4π4/3π5+β (15π3
+/8π4 + π2/π
+)
3
4ππ3.
(18)
From (18), when 14π4π4/3π5
+> 15π
3
+/π4+π2/π+, (ππ/ππ)
π,π>
0, the thermodynamic system is unstable. When
14π4π4
3π5+
<15π3
+
π4+
π2
π+
, (19)
(ππ/ππ)π,π
< 0, the thermodynamic system is stable. Whenπ β 0, (ππ/ππ)
π,π< 0, the thermodynamic system is
stable. Heat capacity at constant pressure is
πΆπ,π
= π(ππ
ππ)
π,π
= π[ππ+
3/4ππ + 3π3π2/ππ4+
] . (20)
According to (20), if π > 0, namely, 3π+/4π2> π2π2/π3
+, πΆπ,π
will be greater than zero, which fulfills the stable condition
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4 Advances in High Energy Physics
10 20 30 40 50
β0.5
0.5
1.0
1.5
r4+
CV/Q
l
Figure 1: Plots of the heat capacity at constant π and π versus theπ4
+. The horizontal axis started from 4/3, because of π > 0. The
intersection point of πΆπ,π
with the horizontal axis is 20/3, whichshows that when π
4
+> (20/3)π
2π4 holds up, πΆ
π,π> 0.
of thermodynamic equilibrium.The heat capacity at constantvolume is
πΆπ,π
= π(ππ
ππ)
π,π
= π((ππ+
π(3π5
++ 12π
2π4π+
9ππ4++ 4π2π5
) βππ2
+
2π2)
Γ ((3
4ππ2+
3π2π2
ππ4+
) Γ (3π5
++ 12π
2π4π+
9ππ4++ 4π2π5
)
β(3π+
2ππ3+
2π2π
ππ3+
))
β1
)
= π10ππ
2π3π2
+β 3ππ
6
+/2π
28π4π6/ππ3+β 45π5+/4ππ2 β 6π2π
+π2/π
= (3π+
4π2β
π2π2
π3+
)10ππ
2π3π2
+β 3ππ
6
+/2π
28π4π6/π3+β 45π5+/4π2 β 6π2π
+π2.
(21)
We can plot the curve of πΆπ,π
, which shows that only whenthe condition π
4
+> (20/3)π
2π4 holds up, πΆ
π,π> 0 will work.
See Figure 1.From (20) and π > 0, when π
4
+> (4/3)π
2π4, πΆπ,π
will be greater than zero, which fulfills the stable conditionof thermodynamic equilibrium. When π
4
+= (4/3)π
2π4,
the Hawking temperature of heat capacity is zero whichcorresponds to the extreme case. However, for the πΆ
π,πonly
the condition π > 0 is not enough. One needs more strictcondition π
4
+> (20/3)π
2π4, under which the πΆ
π,πis greater
than zero. This suggests that the thermodynamic system ofcharged black strings does not have the first-order phasetransition only when π
4
+> (20/3)π
2π4. 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 π
4
+> (20/3)π
2π4, πΆπ,π
and πΆπ,π
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 π = 0, from (12) and (14), wehave
Ξ2=
2ππ
ππ3+
, π(2ππ
ππ3+
β 1) =32ππ½2π3
9π3+
,
π =ππ3
+
4π+
π
2
βπ6
+
4π2+
64π½2π2
9,
(22)
π =3π2
+βπ+
4π2β2πππ
. (23)
From (22) and (23), we deduce
(ππ
ππ)
π,π½
=
3 (20ππ β 8ππ3
+) π
5π (9ππ3++ 32ππ½2π3) β 30ππ2
=5ππ β 2ππ
3
+
5π.
(24)
Thus,
(ππ
ππ)
π,π½
=
3 (2ππ3
+β 5ππ)
20ππ3π. (25)
From (22), we have 2ππ > ππ3
+, so (ππ/ππ)
π,π½< 0, which
satisfies the condition of thermodynamic equilibrium. Wecan derive the heat capacities of rotating string at constantpressure and constant volume as follows:
πΆπ,π½
= π(ππ
ππ)
π,π½
= π((1
6π2+π(
ππ
2ππ+
)
1/2
(2π2
πβ
32π½2π2
3)
β1
2π(πππ+
2π)
1/2
)
Γ (5
8π2πβ2ππππ+
(2π2
πβ
32π½2π2
3)
β15π5/2
+
8π3β2πππ
)
β1
)
= π2πππβ2ππππ
+
15π3+
=π
10
π
π,
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Advances in High Energy Physics 5
πΆπ,π½
= π(ππ
ππ)
π,π½
= π((1
6π2+π(
ππ
2ππ+
)
1/2
(4ππ
πβ π3
+) +
1
2(ππ+
2ππ)
1/2
)
Γ(5
8π2πβ2ππππ+
(4ππ
πβ π3
+)
β3π5/2
+
8π2πβ2πππ
)
β1
)
=1
2(ππ+π
2π)
1/2
(2ππ β ππ
3
+
5ππ β 2ππ3+
) .
(26)
From (22), we have ππ > ππ3
+/2, so πΆ
π,π½> 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 π
+of event horizon
satisfies
π2
+
π2β
ππ
π+
+π2π2
π2+
= 0, (27)
where
π+=
1
2{πΎ1/2
+ [βπΎ + 2(πΎ2β 4π2π4)1/2
]
1/2
} ,
πΎ =
{
{
{
π2π6
2+ [(
π2π6
2)
2
+ (4π2π4
3)
2
]
1/2
}
}
}
1/3
+
{
{
{
π2π8
2β [(
π2π8
2)
2
+ (4π2π4
3)
2
]
1/2
}
}
}
1/3
.
(28)
For discussion purpose andwithout loss of generality, we takeπ(π½) and π to be small quantities relative to π or π and π
+,
namely, π2/π2 βͺ 1, π2+β« π2. From (16), we have
π½ β3
8(π3
+
π3+
4π2π
π+
)π, (29)
and when π is small
π β8π3
3π3+
π½. (30)
From (12), we can get the approximate value of volume
π βππ3
+
2πβ
2ππ2π3
3π+
+32ππ6
9π6+
π½2. (31)
According to (6), we can obtain the approximate Hawkingtemperature
π β3π+
4ππ2(1 β
32π4
π6+
π½2) β
π2π2
ππ3+
. (32)
From this, we can deduce
(ππ
ππ)
π,π½,π
β3/4ππ2+ 120π
2π½2/ππ6
++ 3π2π2/ππ4
+
16π3π½2/π6+β (15π3
+/8π4 + π2/π
++ 12π½2/π3
+)
Γ3
4ππ3.
(33)
From (32), when requiring π > 0, the following equationshould be satisfied:
3π+
4π2>
24π2π½2
π5+
+π2π2
π3+
. (34)
From (33), when
15π3
+
8π4>
16π3π½2
π6+
β (12π½2
π3+
+π2
π+
) , (35)
we have (ππ/ππ)π,π½,π
< 0, which satisfies the condition ofthermodynamic equilibrium. Substituting (34) into (33), wecan get 15π3
+/8π4> 16π3π½2/π6
+β 3π3
+/4π4, or
21π3
+
8π4>
16π3π½2
π6+
β9π2
4π3. (36)
From (28), one can deduce π+/π β (4π)
1/3> π, π2
+β« π2;
thus, (36) is satisfied.In order to show the relation between π andπ clearly, we
plot the π-π curve. According to (1), (31), and (32), we candepict the π-π curve of charged and rotating black strings(Figure 2).
From this figure, we know that theπ-π 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
π =πΞπ2
+
2πβ
ππ2
+
2π(1 +
π2
2π2) β
ππ2
+
2π(1 +
32π4
9π6+
π½2) . (37)
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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 π versus the volume π. The curvescorrespond to the parameters π = 1, π½ = 0.1, and π = 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
πΆπ,π½,π
= π(ππ
ππ)
π,π½,π
β π((5π2π2π
3β
π2π4
+
π3β
22π2π4π½2
3π5+
+40π2ππ½2
9π2+
)
Γ(β33π3
+
8π4β
π2
π+
+16π3π½2
π6+
β12π½2
π3+
)
β1
) ,
πΆπ,π½,π
= π(ππ
ππ)
π,π½,π
= π
ππ+/π β (64ππ
3/9π5
+) π½2
3/4ππ2 + 120π2π½2/ππ6++ 3π2π2/ππ4
+
.
(38)
From (35), one can deduceπ2π4+/π3> 22π4π½2/3π5
+β(40ππ½
2/9π2
++
5π2π/3), 33π3
+/8π4> +16π
3π½2/π6
+β(12π½
2/π3
++π2/π+); therefore,
πΆπ,π½,π
> 0, πΆπ,π½,π
> 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 πandπ, 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 π and π 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 π 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 (ππ/ππ)
π,π½,π< 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 [12β14], 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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