hep-th/yymmnnn

The Extended Thermodynamic Properties
of Taub-NUT/Bolt-AdS spaces
Chong Oh Lee

Department of Physics, Kunsan National University,
Kunsan 573-701, Korea
<EMAIL_ADDRESS>
###### Abstract

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.

##  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 to various
thermodynamic issues including higher dimensional black hole with negative
cosmological constant $\Lambda$ [8]-[18].

As widely known, thermodynamic quantities in black hole physics, the black
hole mass $M$, surface gravity $\kappa$ and the horizon area $A$ correspond to
the thermodynamic quantities of a physical system, thermal energy $U$,
temperature $T$, and entropy $S$ respectively. However, comparing the first
law of standard thermodynamics 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 cosmological
constant $\Lambda$ can be treated as the thermodynamic pressure $p$

$\displaystyle p=-\frac{1}{8\pi}\Lambda=\frac{u(2u+1)}{8\pi l^{2}},$ (1.1)

in units where $G=c=\hbar=k_{B}=1$, and the total number of dimensions
$(d+1)=2u+2$ is 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 dimensions 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 thermodynamic volume may be understood as applying the
work on the environment (universe) by the system (the whole black hole)
considering 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
exploring 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, $\ldots$)
or even $u$ (the total number of dimensions: 6, 10, 14, $\ldots$). 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 function of the
temperature for any odd $u$. We also will demonstrate that there is the
transition from Taub-NUT-AdS to Taub-Bolt-AdS for all odd $u$ only.

The paper is organized as follows: In the next section we investigate
thermodynamic properties in Taub-NUT/Bolt-AdS spaces for any $u$. We find that
thermodynamic quantities such as the entropy, the enthalpy, the specific heat,
the temperature, and the thermodynamic volume. We discuss their phase
structure and their instability. In the last section we give our conclusion.
In Appendix A it is shown to be explicit forms for their thermodynamic
quantities.

##  2 Taub-NUT/Bolt-AdS black hole

We start by considering the $(d+1)$-dimensional-Taub-NUT with negative
cosmological constant. The general solution in the Euclidean section, for a
$U(1)$ fibration over a series of the space ${\cal M}^{2}$ as the base space
$\bigotimes_{i=1}^{u}{\cal M}^{2}$, is given by [39]-[45] (for the generalized
versions of the issue, see e.g., [46, 47])

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
f(r)\left\\{dt_{E}+2N\sum_{i=1}^{u}\cos(\theta_{i})d\phi_{i}\right\\}^{2}$
(2.2) $\displaystyle+$
$\displaystyle\frac{dr^{2}}{f(r)}+(r^{2}-N^{2})\sum_{i=1}^{u}\Biggr{\\{}d\theta_{i}^{2}+\sin^{2}(\theta_{i})d\phi_{i}^{2}\Biggr{\\}},$

with the metric function $f(r)$

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{r}{(r^{2}-N^{2})^{u}}\int^{r}\left\\{\frac{(a^{2}-N^{2})^{u}}{a^{2}}\right.$
(2.3) $\displaystyle+$
$\displaystyle\left.\frac{(2u+1)(a^{2}-N^{2})^{u+1}}{l^{2}a^{2}}\right\\}da-\frac{2mr}{(r^{2}-N^{2})^{u}}.$

Here, $N$ represents a NUT charge for the Euclidean section, $l$ is a
cosmological parameter, $m$ is a geometric mass. The NUT solution occurs when
solving $f(r)|_{r=N}$=0. The inverse of the temperature $\beta$ is obtained by
requiring regularity in the Euclidean time $t_{E}$ and radial coordinate $r$
[39]-[45]

$\displaystyle\beta=\left.\frac{4\pi}{f^{\prime}(r)}\right|_{r=N}=4(u+1)\pi
N,$ (2.4)

where $\beta$ is the period of $t_{E}$.

Using counter term subtraction method, we get the regularized action [39]-[45]

$\displaystyle I_{\rm
NUT}=\frac{(4\pi)^{u}N^{2u-1}(2uN^{2}-l^{2})}{16\pi^{\frac{3}{2}}l^{2}}\hskip
8.53581pt\Gamma(\frac{1}{2}-u)\Gamma(u+1)\beta,$ (2.5)

where the gamma function $\Gamma(t)$ is defined as

$\displaystyle\Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}dx.$ (2.6)

Solving (2.4) for the mass at $r=N$, the NUT mass is given as

$\displaystyle m_{\rm
NUT}=\frac{N^{2u-1}\bigg{\\{}l^{2}-2(u+1)N^{2}\bigg{\\}}}{\sqrt{\pi}(2u-1)l^{2}}\Gamma(\frac{3}{2}-u)\Gamma\left(u+1\right),$
(2.7)

and the entropy is found to be [39]-[45]

$\displaystyle S_{\rm
NUT}=\frac{(4\pi)^{u}N^{2u-1}\biggr{\\{}2u(2u+1)N^{2}-(2u-1)l^{2}\biggr{\\}}}{16\pi^{\frac{3}{2}}l^{2}}\Gamma(\frac{1}{2}-u)\Gamma(u+1)\beta,$
(2.8)

by the Gibbs-Duhem relation $S=\beta M-I$ where $M$ is the conserved mass

$\displaystyle M=u(4\pi)^{u-1}m.$ (2.9)

As mentioned in the previous section, its conjugate variable has dimension of
volume when the negative cosmological constant $\Lambda$ is treated as
thermodynamic pressure $p$ (1.1). In the context of the thermodynamic process,
we should think of the energy required to create the system $pV$ as well as
the energy of the system $U$ i.e., whole forming the system, which is quite
captured by the enthalpy. Thus, the conserved mass $M$ (2.9) is identified
with enthalpy $H$ rather than internal energy $U$

$\displaystyle M\equiv H=U+pV,$ (2.10)

which leads to

$\displaystyle H_{\rm NUT}$ $\displaystyle=$ $\displaystyle
u(4\pi)^{u-1}\left\\{\frac{N^{2u-1}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{\sqrt{\pi}(2u-1)}\right.$
$\displaystyle\left.-\frac{16\sqrt{\pi}(u+1)N^{2u+1}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{u(2u-1)(2u+1)}p\right\\}.$

To check whether we perform the computation correctly, we can reproduce the
temperature through

$\displaystyle T_{\rm NUT}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial H}{\partial S}\right|_{p}$ (2.12)
$\displaystyle=$ $\displaystyle u(4\pi)^{u-1}\left.\frac{\partial N}{\partial
S}\right|_{p}\left\\{\frac{N^{2u-2}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{\sqrt{\pi}}\right.$
$\displaystyle\left.-\frac{16\sqrt{\pi}(u+1)N^{2u}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{u(2u-1)}p\right\\}$
$\displaystyle=$ $\displaystyle\frac{1}{4(u+1)\pi N},$ (2.14)

since, from Eq. (2.8):

$\displaystyle\left.\frac{\partial S}{\partial N}\right|_{p}$ $\displaystyle=$
$\displaystyle
u(u+1)(4\pi)^{u}N\left\\{\frac{N^{2u-2}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{\sqrt{\pi}}\right.$
$\displaystyle\left.-\frac{16\sqrt{\pi}(u+1)N^{2u}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{u(2u-1)}p\right\\}^{-1}.$

One the other hand, employing thermal relation $C=-\beta\partial_{\beta}S$,
the specific heat is given as

$\displaystyle C_{\rm
NUT}=\frac{u(4\pi)^{u-1}N^{2u-1}\bigg{\\{}2(u+1)(u+1)N^{2}-(2u-1)l^{2}\bigg{\\}}}{2\sqrt{\pi}l^{2}}\Gamma(\frac{1}{2}-u)\Gamma(u+1)\beta.$
(2.16)

The thermodynamic volume is also obtained as

$\displaystyle V_{\rm NUT}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial H}{\partial p}\right|_{S}$ $\displaystyle=$
$\displaystyle
2(-2\sqrt{\pi})^{2u-1}(u+1)N^{2u+1}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)$
$\displaystyle+\left.\frac{\partial N}{\partial
p}\right|_{S}\left\\{\frac{u(2u-1)(2u+1)(2\sqrt{\pi}N)^{2u-2}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)}{4\sqrt{\pi}}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left.\frac{\sqrt{\pi}u(2u-1)(2u+1)(2\sqrt{\pi}N)^{2u-2}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)}{4}p\right\\}$
$\displaystyle=$
$\displaystyle-\frac{u(4\pi)^{u}N^{2u+1}}{2\sqrt{\pi}}\Gamma(-\frac{1}{2}-u)\Gamma(u).$
(2.18)

From this result (2.18), we can read the thermodynamic volumes of the NUT
solution are negative for all odd $u$ since $\Gamma(-\frac{1}{2}-u)$ produces
a positive sign for all odd $u$ only, and $\Gamma(u)$ does a positive sign for
any $u$ due to the property of Gamma function (Table 1 in Appendix A shows
their explicit volume forms for four ($u=1$) to ten ($u=4$) dimensions).
Also, due to dimensional scaling arguments, the generalized Smarr formula is
given as

$\displaystyle\frac{1}{2}H-\frac{u}{2u-1}TS+\frac{1}{2u-1}pV=0,$ (2.19)

which is precisely matched with that of static $d$-dimensional black holes
with negative cosmological constant [10, 19, 20].
Finally, the internal energy of Taub-NUT is obtained as

$\displaystyle U_{\rm
NUT}=\frac{u(2u+1)}{8\pi}\left\\{(2\sqrt{\pi}N)^{2u-1}-\frac{(2u+1)(2\sqrt{\pi}N)^{2u-1}N^{2}}{l^{2}}\right\\}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)$
(2.20)

by an thermal relation $U=H-pV$.

Figure 1: Plot of the available bolt radii $r_{\rm B}$ as a function of $N$
for $u$ (from left to right) 1 (the dimension of spacetime $2u+2=4$) to 6 (the
dimension of spacetime $2u+2=14$) for cosmological parameter $l=1$.

Requiring $f(r)|_{r=r_{\rm B}>N}$ and $f^{\prime}(r)|_{r=r_{\rm
M}}=\frac{1}{N(u+1)}$, the Bolt solution occurs. In Taub-Bolt-AdS metric, the
inverse of the temperature, the action, and the mass are respectively

$\displaystyle\beta=\left.\frac{4\pi}{f^{\prime}(r)}\right|_{r=r_{\rm
B}}=\frac{4\pi l^{2}r_{\rm B}}{l^{2}+(2u+1)(r_{\rm B}-N^{2})},$ (2.21)
$\displaystyle I_{\rm Bolt}$ $\displaystyle=$
$\displaystyle\frac{(4\pi)^{u-1}}{4l^{2}}\left[\frac{(2u+1)(-1)^{u}N^{2u+2}}{r_{\rm
B}}\right.$ (2.25) $\displaystyle+\sum_{i=0}^{u}\left(\begin{array}[]{l}u\\\
i\end{array}\right)\left.(-1)^{i}N^{2i}r_{\rm
B}^{2u-2i}\left\\{\frac{l^{2}}{(2u-2i-1)r_{\rm B}}-\frac{(2u+1)(u-2i+1)r_{\rm
B}}{(2u-2i+1)(u-i+1)}\right\\}\right]\beta,$ $\displaystyle m_{\rm Bolt}$
$\displaystyle=$
$\displaystyle\frac{1}{2l^{2}}\left\\{\sum_{i=0}^{u}\left(\begin{array}[]{l}u\\\
i\end{array}\right)\frac{(-1)^{i}N^{2i}l^{2}r_{\rm
B}^{2u-2i-1}}{(2u-2i-1)}\right.$ (2.31)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left.(2u+1)\sum_{i=0}^{u+1}\left(\begin{array}[]{l}u+1\\\
~{}~{}~{}i\end{array}\right)\frac{(-1)^{i}N^{2i}r_{\rm
B}^{2u-2i+1}}{2u-2i+1}\right\\}.$

Here the bolt radii $r_{\rm B}$ is

$\displaystyle r_{\rm
B}=\frac{l^{2}\pm\sqrt{l^{4}+(2u+1)(2u+2)^{2}N^{2}[(2u+1)N^{2}-l^{2}]}}{(2u+1)(2u+2)N},$
(2.32)

and its reality requirement implies

$\displaystyle N\leq\frac{l}{\sqrt{2(u+1)(2u+1)[u+1+\sqrt{u(u+2)}]}}=N_{\rm
max},$ (2.33)

where $N_{\rm max}$ denotes the maximum magnitude of the NUT charge. As shown
in Fig.1, we plot the available bolt radius for the various dimension of
spacetime. It is shown that the $N_{\rm max}$ grows up as the dimension of
spacetime increases. Thus, the magnitude of radii of small bolts in $r_{\rm
B}$ (2.32) increases in the case of the higher dimension.

Using parallel way as in the case of the NUT solution, the enthalpy $H_{\rm
Bolt}$, the entropy $S_{\rm Bolt}$, and thermodynamics volume $V_{\rm Bolt}$
for the Bolt solution yields respectively

$\displaystyle H_{\rm Bolt}$ $\displaystyle=$
$\displaystyle\frac{u(4\pi)^{u-1}}{2}\left\\{\sum_{i=0}^{u}\left(\begin{array}[]{l}u\\\
i\end{array}\right)\frac{(-1)^{i}N^{2i}r_{\rm B}^{2u-2i-1}}{(2u-2i-1)}\right.$
(2.39)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{8\pi}{u}\sum_{i=0}^{u+1}\left(\begin{array}[]{l}u+1\\\
~{}~{}~{}i\end{array}\right)\frac{(-1)^{i}N^{2i}r_{\rm
B}^{2u-2i+1}}{(2u-2i+1)}p\right\\},$ $\displaystyle S_{\rm Bolt}$
$\displaystyle=$
$\displaystyle\frac{(4\pi)^{u-1}}{4}\left[\sum_{i=0}^{u}\left(\begin{array}[]{l}u\\\
i\end{array}\right)\frac{(2u-1)(-1)^{i}N^{2i}r_{\rm
B}^{2u-2i-1}}{2u-2i-1}\right.$ (2.45)
$\displaystyle+\left.\left\\{\sum_{i=0}^{u}\left(\begin{array}[]{l}u\\\
i\end{array}\right)\frac{8\pi(2u^{2}+3u-2i+1)(-1)^{i}N^{2i}r_{\rm
B}^{2u-2i+1}}{u(u-i+1)(2u-2i+1)}+\frac{8\pi(2u-1)N^{2u+2}}{ur_{\rm
B}}\right\\}p\right]\beta,$

$\displaystyle V_{\rm Bolt}$ $\displaystyle=$
$\displaystyle\frac{\pi^{u}(2r_{\rm B})^{u-1}}{2u+1}\Bigg{\\{}-2(N^{2}-r_{\rm
B}^{2})\left(1-\frac{N^{2}}{r_{\rm B}^{2}}\right)^{u}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r_{\rm
B}^{2}\left(\frac{N}{r_{\rm B}}\right)^{2u+1}{\rm B}\left(\frac{N^{2}}{r_{\rm
B}^{2}},\frac{1}{2}-u,u+1\right)\Bigg{\\}},$

where the incomplete beta function ${\rm B}\left(x,a,b\right)$ is defined as

$\displaystyle{\rm B}\left(x,a,b\right)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt.$
(2.47)

Employing the thermal relation $U=H-pV$, the internal energy of AdS-Taub-Bolt
$U$ is given as

$\displaystyle U_{\rm
Bolt}=\frac{(4\pi)^{u-1}}{4l^{2}}uN^{2u-1}\Bigg{\\{}(2u+1)N^{2}-l^{2}\Bigg{\\}}{\rm
B}\left(\frac{N^{2}}{r_{\rm B}^{2}},\frac{1}{2}-u,u+1\right).$ (2.48)

Let us start with the action difference of Taub-NUT-AdS and Taub-Bolt-AdS in
order to study the phase structure of the Taub-NUT/Bolt-AdS system. Their
action difference, ${\cal I}_{D}$ is defined as

$\displaystyle I_{\rm Bolt}-I_{\rm NUT}\equiv{\cal I}_{D}$ $\displaystyle=$
$\displaystyle\frac{(4\pi)^{u}}{8r_{\rm B}l^{2}}\left[2N(N^{2}-r_{\rm
B}^{2})(u+1)\left(1-\frac{N^{2}}{r_{\rm B}^{2}}\right)^{u}\right.$
$\displaystyle+(2uN^{2}-l^{2})\left\\{(u+1)N^{2u}{\rm
B}\left(\frac{N^{2}}{r_{\rm B}^{2}},\frac{1}{2}-u,u+1\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-\frac{2N^{2u}}{\sqrt{\pi}}\Gamma(\frac{1}{2}-u)\Gamma(2+u)\right\\}\right].$

(a) (b)

Figure 2: (a) Plot of the action difference as a function of $N$ for $u$ (from
left to right) 1 (the dimension of spacetime $2u+2=4$) to 6 (the dimension of
spacetime $2u+2=14$) for cosmological parameter $l=1$ and pressure $p=3$. (b)
Plot of the six-dimensional action difference ${\cal I}_{6}$ as a function of
$N$ for cosmological parameter $l$ (from left to right) 1 to 5 for pressure
$p=3$.

As shown in Fig. 2, taking the cosmological parameter $l$ and the pressure $p$
as fixed parameters, the action difference ${\cal I}_{D}$ becomes negative as
$N$ increases. This means that there is the first order phase transition from
Taub-NUT-AdS to Taub-Bolt-AdS at a critical NUT charge. Note that there is the
transition to Taub-Bolt-AdS with $r_{\rm B,+}$ only since the right vertical
solid curves come from the larger branch of the two branches of bolts where
$r_{\rm B,+}$ represents large bolt radii in (2.32). As shown in Fig 2. (a),
the curves of ${\cal I}_{D}$ move more to the left as the dimension of
spacetime increases. In Fig 2. (b), it is shown that the curves of ${\cal
I}_{6}$ move more to the right as the pressure decreases. The pressure $p$
becomes zero as the cosmological parameter $l$ goes to infinity since $p$ is
inversely proportional to $l$ i.e., $p={u(2u+1)}/{(8\pi l^{2})}$. In such
limit, curves of the action difference as a function of $N$ correspond to each
dashed line in Fig. 2.

Figure 3: For the NUT solution, plot of thermodynamically stable range of $p$
as a function of $T$ for given dimension of spacetime (two red dashed curves
for $u=1$, two brown dashed curves for $u=3$, and two blue dashed curve for
$u=5$, respectively) and for the Bolt solution, plot of $p$ as a function of
$T$ for given dimension of spacetime (red solid curve for $u=1$, brown solid
curve for $u=3$, and blue solid curve for $u=5$, respectively).

Requiring both the entropy and the specific heat are positive, we find the
following thermodynamically stable range of $p$ for the NUT solution

$\displaystyle\frac{\pi
u(2u-1)(u+1)}{T^{2}}<p<\frac{\pi(2u-1)(u+1)^{2}}{T^{2}}.$ (2.50)

Note that there is thermodynamically stable range of $p$ for all odd $u$ only,
since both the entropy and specific heat at a given temperature are not
positive for all even $u$ (see e.g., Fig 5. (b)). It is shown in Fig. 3 that
such stable range (2.50) is given as the shaded regions between the two curves
on the left.

After obtaining $N$ with a change of sign in the action difference ${\cal
I}_{D}$ (2) through solving ${\cal I}_{D}=0$ for $N$, and substituting into
the inverse of the temperature (2.21) we may deduce an expression for the
critical temperature $T_{c}$

$\displaystyle T_{c}=\frac{\sqrt{p}}{\sqrt{2\pi}\alpha(u+1)\sqrt{u(2u+1)}},$
(2.51)

by employing the pressure (1.1). Here at $T_{c}$ a phase transition from Taub-
NUT-AdS to Taub-Bolt-AdS occurs and $\alpha$ is constant. For example, for
$u=1$, the transition temperature $T_{c}$ reduces to the result in [36]

$\displaystyle\alpha=\frac{\sqrt{5}-\sqrt{2}}{6},$ (2.52)

and when $u>1$ (beyond four-dimensional spacetime), $\alpha$ is numerically
written in Table 3 in Appendix A since it is difficult to solve the equation
with the analytic method for a complicated high-order polynomial equation
${\cal I}_{D}=0$. It is shown in Fig. 3 that the rightmost solid curves (red
solid curve for $u=1$, brown solid curve for $u=3$, and blue solid curve for
$u=5$, respectively) denote the lines of transition between the Taub-NUT-AdS
and Taub-Bolt-AdS phases for a given dimension of spacetime and this
transition occurs at $T_{c}$ (2.51).

(a) (b)

Figure 4: (a) Plot of the $(2u+2)$-dimensional entropy $S_{u(odd)}$ as a
function of temperature $T$ for odd $u$ (red solid curve for $u=1$, brown
solid curve for $u=3$, and blue solid curve for $u=5$, respectively) for
pressure $p=3$. (b) Plot the eight-dimensional entropy $S_{8}$ (brown solid
curve) and specific heat $C_{8}$ (black solid curve) as a function of
temperature $T$ for pressure $p=3$.

(a) (b)

Figure 5: (a) Plot of the $(2u+2)$-dimensional entropy $S_{u(even)}$ as a
function of temperature $T$ for even $u$ (orange solid curve for $u=2$, green
solid curve for $u=4$, and purple solid curve for $u=6$, respectively) for
pressure $p=3$. (b) Plot of the six-dimensional entropy $S_{6}$ (orange solid
curve) and specific heat $C_{6}$ (black solid curve) as a function of
temperature $T$ for pressure $p=3$.

In Fig 4.and Fig 5., dashed lines represent solutions for higher actions. As
shown in Fig 4. (b) and Fig 5. (b), the entropy is zero at $T_{0}$

$\displaystyle T_{0}=\frac{\sqrt{p}}{4\sqrt{5\pi}}~{}~{}~{}(u=3)~{}{\rm
and}~{}T_{0}=\frac{\sqrt{p}}{3\sqrt{3\pi}}~{}~{}~{}(u=2),$ (2.53)

and the specific heat is zero at $T_{h}$

$\displaystyle T_{h}=\frac{\sqrt{p}}{2\sqrt{15\pi}}~{}~{}~{}(u=3)~{}{\rm
and}~{}T_{h}=\frac{\sqrt{p}}{3\sqrt{2\pi}}~{}~{}~{}(u=2),$ (2.54)

where at $T_{h}$ the entropy has maximum value (odd $u$) or minimum (even $u$)
(for the generalized form of $T_{0}$ and $T_{h}$, see Table 3 in Appendix A).
Any NUT solution in AdS space is thermodynamically stable for all odd $u$
since both the entropy $S_{u(odd)}$ and the specific heat $C_{p,u(odd)}$
become positive between $T_{0}$ and $T_{h}$ where $S_{u(odd)}$ and
$C_{p,u(odd)}$ denote the $(2u+2)$-dimensional entropy for odd $u$ and the
$(2u+2)$-dimensional specific heat for odd $u$ at constant pressure (see e.g.,
Fig. 4 (b) for $u=3$, and [36] for $u=1$). It is also found that the
thermodynamic nature of the Taub-NUT/Bolt-AdS system dramatically changes due
to odd $u$ or even $u$, i.e., there is stable Bolt solution only for all even
$u$. Thus, here focusing on all odd $u$, we discuss this phase transition to
Taub-Bolt-AdS in detail. In Fig 4. (b), the brown solid curve on the left is
the eight-dimensional entropy of the NUT solution, the black solid curve is
the eight-dimensional specific heat of the NUT solution, and the brown solid
curve on the right is the eight-dimensional entropy of the Bolt solution with
the large bolt radii $r_{\rm B,+}$. Thus possible profiles of the entropy are
classified by the value of the temperature $T$: (i) when $T<T_{0}$ and for the
NUT solution, $S_{8}$ is negative but $C_{p,8}$ is positive as shown in Fig 4.
(b). (ii) when $T_{0}<T<T_{h}$ and for the NUT solution, both $S_{8}$ and
$C_{p,8}$ are positive as shown in Fig 4. (b). (iii) when $T_{h}<T<T_{c}$ and
for the NUT solution, $S_{8}$ is positive but $C_{p,8}$ is negative as shown
in Fig 4. (b). (iv) when $T>T_{c}$ and for the Bolt solution, $S_{8}$ is
positive as shown in Fig 4. (b)., and $C_{p,8}$ is positive. Here we do not
draw the plot of the specific heat as the function of temperature $T$ since
the specific heat $C_{p,8}$ is always positive in the case of Bolt solution
with the $r_{\rm B,+}$. As pointed out in [30, 36], in the extended
thermodynamics the profiles above may be interpreted as follows: For (i), the
negative entropy comes from some net heat outflow during the creation of the
Taub-NUT black hole. For (ii), the stable Taub-NUT black hole occurs and, in
particular, since the thermodynamic volume
$\scriptsize{-\frac{1}{4\pi}\left(\frac{\sqrt{2u-1}}{2\sqrt{p}}\right)^{2u+1}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)}$
at $T_{0}$ and the thermodynamic volume
$\scriptsize{-\frac{1}{4\pi}\left(\frac{\sqrt{u(2u^{2}+u-1)}}{2(u+1)\sqrt{p}}\right)^{2u+1}\Gamma(-\frac{1}{2}-u)\Gamma(u+1)}$
at $T_{h}$, the thermal stable range of the thermodynamic volume is given as

$\displaystyle-\frac{1}{24\sqrt{\pi}p^{\frac{3}{2}}}<V_{\rm
NUT,4}<-\frac{1}{48\sqrt{2\pi}p^{\frac{3}{2}}}~{}~{}~{}({\rm for}~{}u=1),$
(2.55)

and

$\displaystyle-\frac{25\sqrt{5}}{112\sqrt{\pi}p^{\frac{7}{2}}}<V_{\rm
NUT,8}<-\frac{675\sqrt{15}}{14336\sqrt{\pi}p^{\frac{7}{2}}}~{}~{}~{}({\rm
for}~{}u=3),$ (2.56)

where it can be seen that the thermal stable range of the thermodynamic volume
still increases with increasing dimensionality. For (iii), the Taub-NUT black
hole develops an instability due to the negative specific heat. For (iv), the
stable Taub-Bolt black hole finally occurs.

##  3 Conclusion

In the context of the extended thermodynamics, we considered the higher
dimensional Tabu-NUT/Bolt-AdS solution and obtained their thermal quantities
such as the enthalpy, the entropy, the specific heat, and the thermodynamic
volume and so on.

We investigated their phase structure through their thermal quantities, and
found out that the thermodynamic nature of the Taub-NUT/Bolt-AdS system
dramatically changes due to odd or even $u$. In particular, it was found out
that there existed the first order phase transition from the Taub-NUT-AdS to
the Taub-Bolt-AdS for all odd $u$.

Finally, we also explored a proportional behavior of their thermodynamic
quantities with increasing dimensionality and found that the maximum magnitude
of the NUT charge, the magnitude of radii of small bolt, and the thermal
stable range of a thermodynamic volume grow up.

## Appendix

##  A Thermodynamic quantities for NUT/Bolt solution

$D$ | $H_{\rm NUT}$ | $V_{\rm NUT}$ | $U_{\rm NUT}$
---|---|---|---
4 | $N(1-\frac{32N^{2}\pi}{3}p)$ | $-\frac{8N^{3}\pi}{3}$ | $N(1-\frac{3N^{2}}{l^{2}})$
6 | $-\frac{32}{3}N^{3}\pi(1-\frac{24N^{2}\pi}{5}p)$ | $\frac{128N^{5}\pi^{2}}{15}$ | $-\frac{32}{3}N^{3}(1-\frac{5N^{2}}{l^{2}})\pi$
8 | $\frac{384}{5}N^{5}\pi^{2}(1-\frac{64N^{2}\pi}{21}p)$ | $-\frac{1024N^{7}\pi^{3}}{35}$ | $\frac{384}{5}N^{5}(1-\frac{7N^{2}}{l^{2}})\pi^{2}$
10 | $-\frac{16384}{35}N^{7}\pi^{3}(1-\frac{20N^{2}\pi}{9}p)$ | $\frac{32768N^{9}\pi^{4}}{315}$ | $-\frac{16384}{35}N^{7}(1-\frac{9N^{2}}{l^{2}})\pi^{3}$
n | $\begin{array}[]{c}u(4\pi)^{u-1}\bigg{\\{}\frac{N^{2u-1}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{\sqrt{\pi}(2u-1)}\\\ -\frac{16\sqrt{\pi}(u+1)N^{2u+1}\Gamma\left(\frac{3}{2}-u\right)\Gamma\left(u+1\right)}{u(2u-1)(2u+1)}p\bigg{\\}}\end{array}$ | $-\frac{u(4\pi)^{u}N^{2u+1}}{2\sqrt{\pi}}\Gamma(-\frac{1}{2}-u)\Gamma(u)$ | $\begin{array}[]{c}\frac{u(2u+1)}{8\pi}\bigg{\\{}(2\sqrt{\pi}N)^{2u-1}\\\ -\frac{(2u+1)(2\sqrt{\pi}N)^{2u-1}N^{2}}{l^{2}}\bigg{\\}}\\\ \times\Gamma(-\frac{1}{2}-u)\Gamma(u+1)\end{array}$

Table 1: Summary of thermodynamic quantities for Taub-NUT-AdS

$D$ | $H_{\rm Bolt}$ | $V_{\rm Bolt}$ | $U_{\rm Bolt}$
---|---|---|---
4 | $\frac{r_{\rm B}^{2}+N^{2}}{2r_{\rm B}}+\frac{4\pi(r_{\rm B}^{4}-6N^{2}r_{\rm B}^{2}-3N^{4})}{3r_{\rm B}}p$ | $\frac{4\pi r_{\rm B}}{3}(r_{\rm B}^{2}-3N^{2})$ | $\frac{r_{\rm B}^{2}+N^{2}}{2r_{\rm B}}(1-\frac{3N^{2}}{l^{2}})$
6 | $\begin{array}[]{c}\frac{4\pi(r_{\rm B}^{4}-6N^{2}r_{\rm B}^{2}-3N^{4})}{3r_{\rm B}}\\\ +\frac{16\pi^{2}(r_{\rm B}^{6}-5N^{2}r_{\rm B}^{4}+15N^{4}r_{\rm B}^{2}+5N^{6})}{5r_{\rm B}}p\end{array}$ | $\frac{16\pi^{2}r_{\rm B}}{15}(3r_{\rm B}^{4}-10N^{2}r_{\rm B}^{2}+15N^{4})$ | $\frac{4\pi(r_{\rm B}^{4}-6N^{2}r_{\rm B}^{2}-3N^{4})}{3r_{\rm B}}(1-\frac{5N^{2}}{l^{2}})$
8 | $\begin{array}[]{c}\frac{24\pi^{2}(r_{\rm B}^{6}-5N^{2}r_{\rm B}^{4}+15N^{4}r_{\rm B}^{2}+5N^{6})}{5r_{\rm B}}\\\ +\frac{64\pi^{3}}{35r_{\rm B}}(5r_{\rm B}^{8}-28N^{2}r_{\rm B}^{6}\\\ +70N^{4}r_{\rm B}^{4}-140N^{6}r_{\rm B}^{2}-35N^{8})p\end{array}$ | $\begin{array}[]{c}\frac{64\pi^{3}r_{\rm B}}{35}(5r_{\rm B}^{6}-21N^{2}r_{\rm B}^{4}\\\ +35N^{4}r_{\rm B}^{2}-35N^{6})\end{array}$ | $\begin{array}[]{c}\frac{24\pi^{2}}{5r_{\rm B}}(r_{\rm B}^{6}-5N^{2}r_{\rm B}^{4}\\\ +15N^{4}r_{\rm B}^{2}+5N^{6})\\\ \times(1-\frac{7N^{2}}{l^{2}})\end{array}$
10 | $\begin{array}[]{c}\frac{128\pi^{3}}{35r_{\rm B}}(5r_{\rm B}^{8}-28N^{2}r_{\rm B}^{6}\\\ +70N^{4}r_{\rm B}^{4}-140N^{6}r_{\rm B}^{2}-35N^{8})\\\ +\frac{256\pi^{4}}{63r_{\rm B}}(7r_{\rm B}^{10}-45N^{2}r_{\rm B}^{8}+126N^{4}r_{\rm B}^{6}\\\ -210N^{6}r_{\rm B}^{4}+315N^{8}r_{\rm B}^{2}+63N^{10})p\end{array}$ | $\begin{array}[]{c}\frac{256\pi^{4}r_{\rm B}}{315}(35r_{\rm B}^{8}-180N^{2}r_{\rm B}^{6}\\\ +378N^{4}r_{\rm B}^{4}-420N^{6}r_{\rm B}^{2}+315N^{8})\end{array}$ | $\begin{array}[]{c}\frac{128\pi^{3}}{35r_{\rm B}}(5r_{\rm B}^{8}-28N^{2}r_{\rm B}^{6}\\\ +70N^{4}r_{\rm B}^{4}+40N^{6}r_{\rm B}^{2}\\\ -35N^{8})(1-\frac{9N^{2}}{l^{2}})\end{array}$
n | $\scriptsize{\begin{array}[]{c}\frac{u(4\pi)^{u-1}}{2}\bigg{\\{}{\sum_{i=0}^{u}\left({u\atop i}\right)}\frac{(-1)^{i}N^{2i}r_{\rm B}^{2u-2i-1}}{(2u-2i-1)}\\\ +\frac{8\pi}{u}\sum_{i=0}^{u+1}\left({u+1\atop i}\right)\frac{(-1)^{i}N^{2i}r_{\rm B}^{2u-2i+1}}{(2u-2i+1)}p\bigg{\\}}\end{array}}$ | $\scriptsize{\begin{array}[]{c}\frac{\pi^{u}(2r_{\rm B})^{u-1}}{2u+1}\bigg{\\{}-2(N^{2}-r_{\rm B}^{2})(1-\frac{N^{2}}{r_{\rm B}^{2}})^{u}\\\ +r_{\rm B}^{2}(\frac{N}{r_{\rm B}})^{2u+1}{\rm B}(\frac{N^{2}}{r_{\rm B}^{2}},\frac{1}{2}-u,u+1)\bigg{\\}}\end{array}}$ | ${\scriptsize\begin{array}[]{c}\frac{(4\pi)^{u-1}}{4l^{2}}uN^{2u-1}\\\ \times\\{(2u+1)N^{2}-l^{2}\\}\\\ \times{{\rm B}(\frac{N^{2}}{r_{\rm B}^{2}},\frac{1}{2}-u,u+1)}\end{array}}$

Table 2: Summary of thermodynamic quantities for Taub-Bolt-AdS

$D$ | ${\cal I}_{D}$ | $T_{0}$ | $T_{h}$ | $T_{c}$
---|---|---|---|---
4 | $\begin{array}[]{c}-\frac{2N\pi(r_{\rm B}-N)^{2}}{r_{\rm B}l^{2}}(r_{\rm B}^{2}+2Nr_{\rm B}+3N^{2}-l^{2})\end{array}$ | $\frac{\sqrt{p}}{2\sqrt{\pi}}$ | $\frac{\sqrt{p}}{\sqrt{2\pi}}$ | $0.84077\sqrt{p}$
6 | $\begin{array}[]{c}\frac{4N\pi^{2}(r_{\rm B}-N)^{3}}{r_{\rm B}l^{2}}\bigg{\\{}3r_{\rm B}^{3}+9Nr_{\rm B}^{2}+13N^{2}r_{\rm B}+15N^{3}\\\ -(r_{\rm B}+3N)l^{2}\bigg{\\}}\end{array}$ | $\frac{\sqrt{p}}{3\sqrt{3\pi}}$ | $\frac{\sqrt{p}}{3\sqrt{2\pi}}$ | $0.57253\sqrt{p}$
8 | $\begin{array}[]{c}-\frac{64N\pi^{3}(r_{\rm B}-N)^{4}}{5r_{\rm B}l^{2}}\bigg{\\{}5r_{\rm B}^{4}+20Nr_{\rm B}^{3}+36N^{2}r_{\rm B}^{2}\\\ +44N^{3}r_{\rm B}+35N^{4}\\\ -(r_{\rm B}^{2}+4Nr_{\rm B}+5N^{2})l^{2}\bigg{\\}}\end{array}$ | $\frac{\sqrt{p}}{4\sqrt{5\pi}}$ | $\frac{\sqrt{p}}{2\sqrt{15\pi}}$ | $0.46324\sqrt{p}$
10 | $\begin{array}[]{c}\frac{64N\pi^{4}(r_{\rm B}-N)^{5}}{7r_{\rm B}l^{2}}\bigg{\\{}35r_{\rm B}^{5}+175Nr_{\rm B}^{4}+390N^{2}r_{\rm B}^{3}\\\ +550N^{3}r_{\rm B}^{2}+55N^{4}3r_{\rm B}+315N^{5}\\\ -(5r_{\rm B}^{3}+25Nr_{\rm B}^{2}+47N^{2}3r_{\rm B}+35N^{3})l^{2}\bigg{\\}}\end{array}$ | $\frac{\sqrt{p}}{5\sqrt{7\pi}}$ | $\frac{\sqrt{p}}{2\sqrt{35\pi}}$ | $0.40000\sqrt{p}$
n | $\scriptsize{\begin{array}[]{c}\frac{(4\pi)^{u}}{8r_{\rm B}l^{2}}\bigg{[}2N(N^{2}-r_{\rm B}^{2})(u+1)(1-\frac{N^{2}}{r_{\rm B}^{2}})^{u}\\\ +(2uN^{2}-l^{2})\bigg{\\{}\\\ (u+1)N^{2u}{\rm B}(\frac{N^{2}}{r_{\rm B}^{2}},\frac{1}{2}-u,u+1)\\\ -\frac{2N^{2u}}{\sqrt{\pi}}\Gamma(\frac{1}{2}-u)\Gamma(2+u)\bigg{\\}}\bigg{]}\end{array}}$ | $\frac{\sqrt{p}}{\sqrt{\pi}(u+1)\sqrt{2u-1}}$ | $\frac{\sqrt{p}}{\sqrt{\pi}\sqrt{u(2u^{2}+u-1)}}$ | $\frac{\sqrt{p}}{\sqrt{2\pi}\alpha(u+1)\sqrt{u(2u+1)}}$

Table 3: Summary of thermodynamic quantities for exploring the phase structure

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