Physics Letters B 738 (2014) 294–300
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The extended thermodynamic properties of 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
19 August 2014
Received
in revised form 18 September
2014
Accepted
22 September 2014
Available
online 2 October 2014
Editor:
M. Cveti
ˇ
c
We investigate the extended thermodynamic properties of higher-dimensional Taub–NUT/Bolt–AdS spaces
where a cosmological constant is treated as a pressure. We find a general form for thermodynamic
volumes of Taub–NUT/Bolt–AdS black holes for arbitrary dimensions. Interestingly, it is found that the
Taub–NUT–AdS metric has a thermodynamically stable range when the total number of dimensions is a
multiple of 4 (4, 8, 12, ...). We also explore their phase structure and find the first order phase transition
holds for higher-dimensional cases.
© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
1. Introduction
Thermodynamic properties of black holes have been studied for
a long time since it was first noticed that the area of the event
horizon of a black hole is proportional to its physical entropy [1].
It was found that their physical quantities are expressed in terms
of the thermal energy U , temperature T , and entropy S and the
first law of black hole thermodynamics has the similar forms to
the first law of standard thermodynamics [2].
It
has been also studied in complete analogy with standard
thermodynamic systems. For example, it was found that there is
a phase transition in the Schwarzschild–AdS black hole [3]. Since
then, the phase transitions and critical phenomena in a variety of
black hole solutions have been studied [4–7]. In particular, they
have been used for investigation of various thermodynamic issues
including higher dimensional black hole with negative cosmologi-
cal
constant Λ [8–18].
As
widely known, thermodynamic quantities in black hole
physics, the black hole mass M, surface gravity κ and the horizon
area A correspond to the thermodynamic quantities of a physical
system, thermal energy U , temperature T , and entropy S respec-
tively.
However, comparing the first law of standard thermodynam-
ics
with the pressure and its conjugate to the first law of black
hole thermodynamics in parallel, their counterparts in black hole
physics are not quite captured. Recently, there have been new and
interesting developments in the question, meaning that the cos-
mological
constant Λ can be treated as the thermodynamic pres-
sure p
E-mail address: cohlee@kunsan.ac.kr.
p =−
1
8π
Λ =
u(2u +1)
8πl
2
, (1.1)
in units where G = c =
¯
h = k
B
= 1, and the total number of di-
mensions
(d + 1) = 2u + 2is even for some integer u. In addition,
the black hole mass M is defined as the enthalpy rather than the
thermal energy U . This framework is quite natural since when
the cosmological constant is considered as pressure, its conjugate
quantity becomes a dimension of volume. A lot of investigations
have been performed in this direction [19–38].
Very
recently this issue was generalized to Taub–NUT/Bolt–AdS
spaces in [30,36] and to Kerr–Bolt–AdS spaces in [35]. Interestingly,
they found the thermodynamic volume in Taub–NUT–AdS metric
can be negative. In the context of enthalpy, the positive thermo-
dynamic
volume may be understood as applying the work on the
environment (universe) by the system (the whole black hole) con-
sidering
the process of forming the black hole. In addition, the
negative thermodynamic volume may be understood in that the
environment (universe) applies work to the system (Taub–NUT–
AdS
black hole) in the process of the Taub–NUT–AdS black hole
formation [30]. They also found that there is the first order phase
transition from Taub–NUT–AdS to Taub–Bolt–AdS through explor-
ing
the phase structure of a NUT solution and a Bolt solution [36].
However,
their works on the thermodynamic properties of black
hole of the Taub–NUT–AdS metric have only shown the progress
in the four-dimensional cases [30,35,36]. Furthermore it is well
known that their thermodynamic properties are depending on odd
u (the total number of dimensions: 4, 8, 12, ...) or even u (the to-
tal
number of dimensions: 6, 10, 14, ...). Here we will investigate
extensively thermodynamic properties in the higher-dimensional
Taub–NUT/Bolt–AdS spaces. It will particularly show that the Taub–
NUT–AdS
solution has a thermodynamically stable range as a func-
tion
of the temperature for any odd u. We also will demonstrate
http://dx.doi.org/10.1016/j.physletb.2014.09.046
0370-2693/
© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.