Physics Letters B 753 (2016) 470–475
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The extended thermodynamic properties of a topological
Taub–NUT/Bolt–AdS spaces
Chong Oh Lee
Department of Physics, Kunsan National University, Kunsan 573-701, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received
12 November 2015
Received
in revised form 13 December 2015
Accepted
16 December 2015
Available
online 18 December 2015
Editor:
M. Cveti
ˇ
c
We consider higher dimensional topological Taub–NUT/Bolt–AdS solutions where a cosmological constant
is treated as a pressure. The thermodynamic quantities of these solutions are explicitly calculated.
Furthermore, we find these thermodynamic quantities satisfying the Clapeyron equation. In particular,
a new thermodynamically stable region for the NUT solution is found by studying the Gibbs free energy.
Intriguingly, we also find that like the AdS black hole case, the G − T diagram of the Bolt solution has
two branches which are joined at a minimum temperature. The Bolt solution with the large radius, at
the lower branch, becomes stable beyond a certain temperature while the Bolt solution with the small
radius, at the upper branch, is always unstable.
© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
It has been found that the area of the event horizon of a black
hole is proportional to its physical entropy in search for similar-
ities
between black hole physics and thermodynamics [1]. It has
been successively suggested that by using the thermodynamic re-
lationship
between the thermal energy, temperature, and entropy,
the first law of black hole thermodynamics can be expressed in
the similar forms to the first law of standard thermodynamics
[2]. It has been also found that there is a phase transition in the
Schwarzschild–AdS black hole through investigations of complete
analogy between black hole system and standard thermodynamic
system [3]. Since then, it has been studied for the phase transitions
and critical phenomena in a variety of black hole solutions [4–7]
and
extensively investigated in various thermodynamic issues of
black hole in higher dimensional AdS space [8–18].
It
has been recently suggested that by considering (d + 1)-
dimensional
AdS black holes, the thermodynamic pressure p is
given by
p =−
1
8π
=
u(2u +1)
8πl
2
, (1.1)
in units where G = c =
¯
h = k
B
= 1, and u is of the form d + 1 =
2u +2with a positive integer u. Several series of relevant investi-
gations
have been performed [19–42].
E-mail address: cohlee@kunsan.ac.kr.
Recently, it has been shown that the thermodynamic volume in
the Taub–NUT–AdS case can be negative. This negative thermody-
namic
volume may be interpreted in that the environment (uni-
verse)
applies work to the system (Taub–NUT–AdS black hole) in
the process of the Taub–NUT–AdS black hole formation, while the
positive thermodynamic volume may be interpreted as applying
the work on the environment (universe) by the system (the whole
black hole) considering the process of forming the black hole [30].
They also have found that there is the first order phase transition
from Taub–NUT–AdS to Taub–Bolt–AdS with considering the phase
structure of these black holes [36]. This issue has been studied for
the Kerr–Bolt–AdS case [35] and investigated extensively in higher
dimensional NUT/Bolt case. In these higher dimensional cases, it
has been particularly found that the Taub–NUT–AdS solution has a
thermodynamically stable range as a function of the temperature
for any odd u and there is the transition from Taub–NUT–AdS to
Taub–Bolt–AdS for all odd u only [43].
Furthermore,
it has been shown that this higher dimensional
NUT/Bolt case with a discrete parameter k can be generalized
[44]. In the context of the extended thermodynamics, the ther-
modynamic
properties of the case k = 1 have been studied [43]
only.
Thus, it would be interesting to be a similar discussion of
the generalizations in the k = 0, −1 topological solutions. More
intriguingly, their thermodynamic phase structure would be inves-
tigated
through exploring the behavior of the Gibbs free energy
since understanding its behavior is essential for uncovering pos-
sible
thermodynamic phase transitions. In this paper, we address
these questions.
http://dx.doi.org/10.1016/j.physletb.2015.12.050
0370-2693/
© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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