# Thermodynamics of Kerr-AdS black holes
in the restricted phase space

Zeyuan Gao, Xiangqing Kong and Liu Zhao
School of Physics, Nankai University, Tianjin 300071, China
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
and<EMAIL_ADDRESS>Correspondence author.

###### Abstract

The thermodynamics for Kerr-AdS black hole in four dimensions is revisited
using the recently proposed restricted phase space formalism, which includes
the central charge $C$ of the dual CFT and the chemical potential $\mu$, but
excludes the pressure and the conjugate volume, as thermodynamic variables.
The Euler relation holds automatically, and the first order homogeneity of the
mass and the zeroth order homogeneity of the intensive variables are made
explicit. Thermodynamic processes involving each pair of conjugate variables
are studied in some detail, with emphasis on the scaling properties of the
equations of states. It turns out that the thermodynamic behavior of the Kerr-
AdS black hole is very similar to that of the RN-AdS black hole studied
earlier. In particular, it is found that, there is a first order supercritical
phase equilibrium in the $T-S$ processes at fixed nonvanishing angular
momentum, while at vanishing angular momentum or at fixed angular velocities,
there is always a non-equilibrium transition from a small unstable black hole
state to a large stable black hole state. Moreover, there is a Hawking-Page
phase transition in the $\mu-C$ processes. Due to the complicatedness of the
Kerr metric, the exact critical point and the Hawking-Page temperature are
worked out explicitly only in the slow rotating limit, however the
characteristic thermodynamic properties do not rely on the slow rotating
approximation.

##  1 Introduction

Black hole thermodynamics has been an important and active subject of study
ever since the pioneering works of Bekenstein [1, 2] and Bardeen, Carter,
Hawking [3, 4]. For both historical and technical reasons, the study of black
hole thermodynamics can be subdivided into two major stages or formalisms,
i.e. the traditional and the extended phase space (EPS) stages/formalisms. The
traditional formalism concerns mainly with the establishment of thermodynamic
relations and the evaluation of thermodynamic quantities in various black hole
solutions. Among other things, the Smarr relation [5] and the Wald method [6]
played some important roles. The Hawking-Page transition [7] has also been
found in the development of the traditional formalism. The extended phase
space formalism initiated in [8] opened a new era for the thermodynamics of
black hole in AdS spacetime by introducing and extra $(P,V)$ pair of state
variables, where $P$ is related to the cosmological constant via
$P=-\Lambda/8\pi G$. This triggered a great number of subsequent works, see
[9, 10, 11, 12, 13, 14, 15, 16, 17] for examples. These developments are
mainly concentrated in the thermodynamic behaviors, especially the $P-v$
criticalities. A most recent variant of the extended phase space formalism is
developed by Visser [18], with the aid of certain considerations about the
AdS/CFT correspondence [19]. The major innovation in Visser’s formalism is the
inclusion of the CFT central charge $C$ and the conjugate chemical potential
$\mu$ as novel thermodynamic parameters. Meanwhile, the volume and the
pressure are changed into that of the CFT, i.e. $\mathcal{V}\sim L^{d-2}$ with
$L$ denoting the AdS radius, and $\mathcal{P}$ determined via the CFT equation
of states (EOS) $E=(d-2)\mathcal{PV}$ where $d$ is the dimension of the bulk
spacetime. Please note that the idea of introducing the chemical potential and
central charge (or the square of the number of colors in the CFT) as new
thermodynamic variables has been explored earlier in [20, 21, 22, 23, 24],
however, Visser’s work surpasses the previous works in that the $(P,V)$
variables are changed into $(\mathcal{P,V})$, so that, for charged rotating
AdS black holes, the first law takes the form

$\mathrm{d}E=T\mathrm{d}S-\mathcal{P}\mathrm{d}\mathcal{V}+\tilde{\Phi}\mathrm{d}\tilde{Q}+\Omega\mathrm{d}J+\mu\mathrm{d}C,$

which is accompanied with an Euler-like relation

$\displaystyle E=TS+\tilde{\Phi}\tilde{Q}+\Omega J+\mu C,$ (1)

where $\tilde{\Phi}$ and $\tilde{Q}$ are the properly rescaled electric
potential and electric charge.

Visser’s formalism is a grand framework in the sense that it captures
thermodynamic properties of both the black hole in the bulk and of the CFT on
the boundary. By replacing $(P,V)$ with $(\mathcal{P,V})$,  the total mass of
the spacetime restores its original interpretation as internal energy, which
conforms to the traditional formalism of black hole thermodynamics. Meanwhile
there is no need to introduce the thermodynamic volume $V$111In contrast, $V$
is required in the EPS formalism in order that the first law holds. Moreover,
$V$ is also connected with the Killing potential [8] and is known to obey the
so-called reverse isoperimetric inequality [25, 26]. Even though, since $V$ is
not connected with the geometric volume, it is still puzzling why $V$ plays
some role in black hole thermodynamics.. By fixing the central charge $C$,
Visser’s formalism leads to a thermodynamic description for the CFT which is
holographically dual to the AdS black hole in the bulk [27, 28].

Eq.(1) implies that $E$ is a first order homogeneous function in
$(S,\tilde{Q},J,C)$ but not in $\mathcal{V}$. However, if black hole
thermodynamics is indeed thermodynamics in the standard sense as described in
any textbooks, e.g.[29], the internal energy needs to be a first order
homogeneous function in all extensive variables. Moreover, both the original
EPS formalism and Visser’s variant suffer from the “ensemble of theories”
issue in the sense that varying cosmological constant (or AdS radius) implies
changing the underlying gravity theory.

To avoid the above mentioned issues, an alternative restricted version of
Visser’s formalism called the restricted phase space (RPS) formalism for AdS
black holes is proposed by us in [30], which is studied in detail in the
example case of four dimensional RN-AdS black hole in Einstein-Maxwell theory,
and some interesting thermodynamic behaviors such as the first order phase
equilibrium at supercritical temperatures (i.e. supercritical phase
equilibrium) and the Hawking-Page transitions are revealed. The present work
is a continuation of [30] to the case of Kerr-AdS black hole in Einstein
gravity. It will be shown that, in spite of the different geometries, the
thermodynamic behaviors for the Kerr-AdS black hole is very similar to that of
the RN-AdS in the RPS formalism. This leads us to guess that there might be
some universality lying behind the RPS formalism. Moreover, the study of black
hole thermodynamics in the RPS formalism may also be helpful in further
understanding the AdS/CFT correspondence, as will be briefly discussed at the
end of this paper. Please note that the thermodynamics and phase structures
for Kerr-AdS black hole in the extended phase space formalism is studied in
[31]. A comparison between the results of [31] and ours may help to understand
the differences between the extended and the restricted phase space
formalisms.

Since the RPS formalism allows for a variable Newton constant, it seems
necessary to clarify the differences between the variable cosmological
constant and Newton constant. In essence, the cosmological constant is a part
of the Lagrangian density in the Einstein-Hilbert action, while the Newton
constant is simply an overall factor in the total action. This is true not
only for pure Einstein gravity, but also for Einstein-Maxwell theory, provided
the electric charge $Q$ is appropriately rescaled (as did in [18]), which is
allowed because $Q$ is an integration constant. Therefore, varying
cosmological constant implies changing the corresponding field equation, while
varying Newton constant does not have this effect. One can think of a variable
cosmological constant as being corresponding to different gravity theories,
while a variable Newton constant to the same underlying theory.

This paper is organized as follows. In Section 2 we briefly introduce the
Kerr-AdS spacetime with emphasis on the values of thermodynamic quantities and
the proof of validity for the first law and Euler relation. In Section 3, we
rewrite the black hole mass as a macro state function in extensive variables
and represent the intensive variables in terms of the equations of states. In
this process, the correct homogeneity behaviors become evident. Section 4 is
devoted to the study of three typical types of thermodynamic processes, which
constitute the major context of this work. In this section, the thermodynamic
behaviors very similar to the RN-AdS case are rediscovered. Finally, in
Section 5, we present a brief summary of the results and make some further
discussions.

##  2 Kerr-AdS black hole in 4-dimensions and the RPS formalism

The metric for Kerr-AdS black hole in 4-dimensions is given as follows,

$\displaystyle\mathrm{d}s^{2}=-\frac{\Delta_{r}}{\rho^{2}}\left(\mathrm{d}t-\frac{a\sin^{2}}{\Xi}\theta\mathrm{d}\phi\right)^{2}+\frac{\rho^{2}}{\Delta_{r}}\mathrm{d}r^{2}+\frac{\rho^{2}}{\Delta_{\theta}}\mathrm{d}\theta^{2}+\frac{\sin^{2}\theta\Delta_{\theta}}{\rho^{2}}\left(a\mathrm{d}t-\frac{r^{2}+a^{2}}{\Xi}\mathrm{d}\phi\right)^{2},$

where

$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,\qquad\Delta_{r}=\left(r^{2}+a^{2}\right)\left(1+\frac{r^{2}}{\ell^{2}}\right)-2Gmr,$
$\displaystyle\Xi=1-\frac{a^{2}}{\ell^{2}},\qquad\qquad\quad\Delta_{\theta}=1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta.$

The mass $M$ and angular momentum $J$ are related to the parameters $m,a$ via

$\displaystyle M=\frac{m}{\Xi^{2}},\quad J=\frac{am}{\Xi^{2}},$

where $m$ can be solved from the equation $\Delta_{r}(r_{+})=0$, with $r_{+}$
representing the radius of the event horizon. This enables us to write $M$ and
$J$ as a function of the parameters $r_{+},a,G$,

$\displaystyle
M=\frac{1}{2\Xi^{2}r_{+}G}\left(r_{+}^{2}+a^{2}+\frac{r_{+}^{4}}{\ell^{2}}+\frac{a^{2}r_{+}^{2}}{\ell^{2}}\right),\quad
J=\frac{a}{2\Xi^{2}r_{+}G}\left(r_{+}^{2}+a^{2}+\frac{r_{+}^{4}}{\ell^{2}}+\frac{a^{2}r_{+}^{2}}{\ell^{2}}\right).$
(2)

The other thermodynamic parameters are given as

$\displaystyle T$
$\displaystyle=\frac{r_{+}}{4\pi(r_{+}^{2}+a^{2})}\left(1+\frac{a^{2}}{\ell^{2}}+\frac{3r_{+}^{2}}{\ell^{2}}-\frac{a^{2}}{r_{+}^{2}}\right),$
$\displaystyle S$
$\displaystyle=\frac{\pi(r_{+}^{2}+a^{2})}{G\Xi},\quad\Omega=\frac{a\Xi}{r_{+}^{2}+a^{2}}+\frac{a}{\ell^{2}}.$

Moreover, the RPS formalism introduces an extra pair of thermodynamic
variables, i.e.

$\displaystyle C$ $\displaystyle=\frac{\ell^{2}}{G},\quad\mu=\frac{M-TS-\Omega
J}{C}.$ (3)

It can be seen that, except for $\mu$, all other thermodynamic quantities must
be non-negative. Let us mention that the chemical potential can also be
defined independent of the other thermodynamic quantities using the AdS/CFT
dictionary $Z_{\rm CFT}=Z_{\rm Gravity}$, because the Gibbs free energy $W=\mu
C=-T\log Z_{\rm CFT}=-T\log Z_{\rm Gravity}$, wherein $Z_{\rm
Gravity}=\exp(-\mathcal{A}_{E})$, $\mathcal{A}_{E}$ represents the Euclidean
action evaluated at the black hole configuration. For details, see [32, 33,
34].

The RPS formalism differs from Visser’s formalism in that the AdS radius,
$\ell$, is fixed as a constant. Using the above quantities (which are all
viewed as functions of $r_{+},a$ and $G$), it can be checked straightforwardly
that the first law of thermodynamics holds,

$\displaystyle\mathrm{d}M=T\mathrm{d}S+\Omega\mathrm{d}J+\mu\mathrm{d}C,$ (4)

and, from eq. (3), the Euler relation is also satisfied explicitly,

$\displaystyle M=TS+\Omega J+\mu C.$ (5)

These two equations are fundamental to the RPS formalism.

Please be reminded that, although the variables $\mu,C$ are borrowed from the
dual CFT, they can be actually understood as the chemical potential and the
effective number $N_{\rm bulk}$ of microscopic degrees of freedom of the black
hole in the bulk. In other words, one may introduce the new rules $\mu_{\rm
CFT}=\mu_{\rm bulk},\,C=N_{\rm bulk}$ into the holographic dictionary. For
simplicity of symbols we shall keep the original $(\mu,C)$ notations rather
than replacing them with $(\mu_{\rm bulk},\,N_{\rm bulk})$, but please always
bear in mind that what we are studying is the thermodynamics of the black
holes in the bulk, rather than that of the dual CFT. Let us also stress that,
the bulk variables $\mu,C$ are only determined up to an arbitrary but opposite
constant rescaling: the changes $\mu\to\lambda^{-1}\mu,\,C\to\lambda C$ with
constant $\lambda$ will not bring any harm to eqs. (4) and (5)222Further
studies indicate that our formalism also works for non-AdS black holes without
a holographic dual. In those cases, the conjugate variables $(\mu,\,N)$ are
defined in a way similar to eq.(3), where $N$ takes the place of $C$, with
$\ell$ replaced by an arbitrarily chosen constant length scale. For details,
see [35, 36]..

##  3 Equations of states and homogeneity

In order to analyze the thermodynamic properties, we need to rewrite the mass
as well as the variables $T,\Omega,\mu$ as functions of the extensive
variables $S,J,C$. This is accomplished in the following steps. First, let us
rewrite $a,G$ in terms of $(J,M,C)$,

$\displaystyle a$ $\displaystyle=\frac{J}{M},\quad G=\frac{\ell^{2}}{C}.$ (6)

Next, we can solve $r_{+}$ from the expression for $S$, yielding

$\displaystyle r_{+}^{2}=\frac{S\left(\ell^{2}M^{2}-J^{2}\right)}{\pi
CM^{2}}-\frac{J^{2}}{M^{2}}.$ (7)

Finally, inserting eqs. (6) and (7) into eq.(2), we get an algebraic equation
for $M$, whose solution reads

$\displaystyle M=\frac{\sqrt{\pi C+S}\sqrt{S^{3}+\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)}}{2\pi^{3/2}\ell\sqrt{C}\sqrt{S}}.$ (8)

Using eq. (8) and the first law, we can easily obtain the equations of states
(EOS)

$\displaystyle T=\left(\frac{\partial M}{\partial S}\right)_{J,C}=\frac{4\pi
CS^{3}+3S^{4}+\pi^{2}C^{2}\left(S^{2}-4\pi^{2}J^{2}\right)}{4\pi^{2}\ell
S^{3/2}\sqrt{C(\pi C+S)}\cdot\sqrt{S^{3}+\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)}},$ (9)
$\displaystyle\Omega=\left(\frac{\partial M}{\partial
J}\right)_{S,C}=\frac{2\pi^{3/2}J\sqrt{C(\pi
C+S)}}{\ell\sqrt{S}\cdot\sqrt{S^{3}+\pi C\left(4\pi^{2}J^{2}+S^{2}\right)}},$
(10) $\displaystyle\mu=\left(\frac{\partial M}{\partial
C}\right)_{S,J}=\frac{\pi^{2}C^{2}\left(4\pi^{2}J^{2}+S^{2}\right)-S^{4}}{4\pi^{3/2}\ell
C^{3/2}\sqrt{S(\pi C+S)}\cdot\sqrt{S^{3}+\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)}}.$ (11)

We can see that $M$ and all the intensive variables are now given as functions
of the extensive variables $S,J,C$. It is evident that $M$ scales as
$M\to\lambda M$, while $T,\Omega,\mu$ are not rescaled, if the independent
variables scale as $S\to\lambda S,J\to\lambda J,C\to\lambda C$. This proves
the first order homogeneity of $M$ and zeroth order homogeneity of
$T,\Omega,\mu$ in $S,J,C$. Moreover, in accordance with the first law (4) and
the Euler relation (5), we can write down the Gibbs-Duhem equation

$\mathrm{d}\mu=-\mathcal{S}\mathrm{d}T-\mathcal{J}\mathrm{d}\Omega,$

where $\mathcal{S}=S/C,\mathcal{J}=J/C$ are both zeroth order homogeneous
functions in $S,J,C$. The Euler and Gibbs-Duhem relations and the various
homogeneity properties for the internal energy and intensive variables are of
essential importance in standard thermodynamics [29]. However, these relations
and the corresponding homogeneity behaviors are absent in other formalisms for
black hole thermodynamics.

Eqs. (9)-(11) provide three algebraic relations for six state parameters.
Therefore, a macro state for the Kerr-AdS black hole can be determined by only
three of the six variables $(T,S),(\Omega,J),(\mu,C)$. Moreover, each of the
EOS involves only four of the state variables. A thermodynamic process is
characterized by the continuous change of any of the state variables appearing
in the EOS.

Before dwelling into the study of the thermodynamic processes, let us remind
that there is an upper bound for $J$,

$\displaystyle J\leq J_{\rm max}=\frac{S\sqrt{\pi^{2}C^{2}+4\pi
CS+3S^{2}}}{2\pi^{2}C},$

due to the requirement of non-negativity for $T$. This is the famous Kerr
bound rewritten in terms of the macro state variables.

##  4 Thermodynamic processes

In this section we study the thermodynamic processes for Kerr-AdS black holes
in the RPS formalism. Since the EOS (9)-(11) are quite complicated, a generic
thermodynamic process may correspond to an arbitrary curve on a three-
dimensional hypersurface in the space of macro states parametrized by the
extensive variables $(S,J,C)$. Therefore, it looks impossible to make a
complete analysis for all admissible processes.

In the following, we shall restrict ourselves to only three kinds of specific
processes, i.e. $T-S$, $\Omega-J$ and $\mu-C$ processes. These are the
simplest processes involving only one pair of conjugate intensive-extensive
variables. Even for these simple cases, the exact result is still very
involved and unillustrative. To proceed, we have two choices, i.e. 1) by
resorting to numeric approaches which is more precise but less illustrative,
and 2) by making some reasonable approximation, e.g. consider only the black
hole states in the slow rotating limit (i.e. $a$ very small) and present
analytic results in this approximation. We choose to proceed in the second
choice when necessary.

###  4.1 $T-S$ processes in the slow rotating limit

One of the characteristic properties for black hole thermodynamics is encoded
in the $T-S$ curves at fixed $J$, which contains a first order phase
transition which becomes second order at the critical point. This is true in
either the EPST or the RPS formalisms, the latter has only been tested in the
special example case of RN-AdS black hole in four spacetime dimensions. The
present work will contribute a second example case for the RPS formalism.

The critical point on the $T-S$ curve at fixed $J$ is characterized by the
inflection point which obeys the following equations,

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{J,C}=0,\quad\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{J,C}=0.$ (12)

In the full form of the EOS (9), these equations are so complicated that it
takes pages to write down the solution which is still in implicit form.
Therefore it is better to resort to the slow rotating approximation.

By expanding $M=M(S,J,C)$ and all the EOS into power series in $J$ and keep
terms up to order $J^{2}$, we get the following equations,

$\displaystyle M(S,J,C)$ $\displaystyle=\frac{C\left[\pi
C\left(2\pi^{2}J^{2}+S^{2}\right)+S^{3}\right]}{2\pi^{3/2}\ell(CS)^{3/2}},$
(13) $\displaystyle T(S,J,C)$ $\displaystyle=\frac{\pi
CS^{2}+3S^{3}-6\pi^{3}CJ^{2}}{4\pi^{3/2}\ell\sqrt{CS^{5}}},$ (14)
$\displaystyle\Omega(S,J,C)$
$\displaystyle=\frac{2\pi^{3/2}J}{\ell}\sqrt{\frac{C}{S^{3}}},$ (15)
$\displaystyle\mu(S,J,C)$ $\displaystyle=\frac{\pi
C\left(2\pi^{2}J^{2}+S^{2}\right)-S^{3}}{4\pi^{3/2}\ell(CS)^{3/2}}.$ (16)

It is remarkable that, at this order of approximation, the first law (4) still
holds exactly, and the homogeneity behaviors are still correct.

Using eq. (14), we can solve the approximate critical point equations (12)
analytically, yielding the critical parameters

$\displaystyle S_{c}=\frac{\pi C}{96}\left(\sqrt{2689}-31\right),\quad
J_{c}=\frac{C}{768}\sqrt{\frac{1}{30}\left(3683\sqrt{2689}-180701\right)}\,.$
(17)

Accordingly, the critical temperature reads

$\displaystyle
T_{c}=\frac{1}{\pi\ell}\sqrt{\frac{3}{2}}\left(\frac{32197\sqrt{2689}-1449979}{160\left(\sqrt{2689}-31\right)^{5/2}}\right).$
(18)

We can also introduce the Helmholtz free energy

$F(T,J,C)=M(S,J,C)-TS,$

which, in the slow rotating limit, takes the value

$\displaystyle F(T,J,C)=\frac{\pi
C^{3/2}S^{5/2}\left(10\pi^{2}J^{2}+S^{2}\right)-{C^{1/2}S^{11/2}}}{4\pi^{3/2}\ell
CS^{4}},$ (19)

where $S$ should be understood as being implicitly determined by eq. (14). At
the critical point, we have

$\displaystyle
F_{c}=\frac{5\left(71989-1291\sqrt{2689}\right)}{1024\sqrt{6}\left(\sqrt{2689}-31\right)^{3/2}}\frac{C}{\ell}.$
(20)

Eqs. (17), (18) and (20) contain complicated expressions involving big
integers which do not seem to make too much sense. It is better to rewrite
these relations in terms of the approximate float point numbers, giving

$\displaystyle S_{c}\approx 0.68250\,C,\quad\,\,\,\,J_{c}\approx 0.02411\,C,$
$\displaystyle T_{c}\approx 0.26939\,\ell^{-1},\quad F_{c}\approx
0.10556\,\ell^{-1}C.$

Now we can re-express the EOS (14) and the free energy in terms of the
relative parameters

$\displaystyle s=\frac{S}{S_{c}},\quad
j=\frac{J}{J_{c}},\quad\tau=\frac{T}{T_{c}},\quad f=\frac{F}{F_{c}}.$

Using approximate float point coefficients, the results read

$\displaystyle\tau(s,j)$
$\displaystyle=\left[(0.41305s+0.63377)s^{2}-0.04682j^{2}\right]{s^{-5/2}},$
(21) $\displaystyle f(t,j)$
$\displaystyle=\left[(1.10389-0.23982s)s^{2}+0.13593j^{2}\right]{s^{-3/2}},$
(22)

where $s$ in eq. (22) is determined implicitly via eq. (21). Let us emphasize
that, in terms of the relative parameters, the EOS and the expression for the
free energy is completely independent of the central charge $C$. This implies
that the same thermodynamic behavior is shared for black holes with any
central charge. The same phenomenon also appears in standard thermodynamics
for ordinary matter and is known as the law of corresponding states, in which
the role of central charge is replaced by the number of particles..

[width=.48]TS-kerr [width=.48]FT-kerr

Figure 1: $T-S$ and $F-T$ curves at fixed $J>0$

Figure 1 presents the plots of the $T-S$ and $F-T$ curves at fixed $J>0$. From
the swallow tail structure on the $F-T$ curves we can see that there is a
first order phase transition for $0<J<J_{c}$, $T>T_{c}$. The transition
temperature $T_{\rm trans}$ at which the first order phase transition occurs
corresponds to the root (crossing point) of the swallow tail. The $T-S$ curves
indicate that, when the first order phase transition occurs, there are three
black hole states with the same temperature, angular momentum and central
charge, but with different entropies. Among these, the states with the
smallest and the largest entropies/event horizon radii are both stable, while
the medium sized black hole state is unstable. Thus the phase equilibrium is
basically an equilibrium between the stable small and large black states. When
$T_{\rm trans}=T_{c}$, the swallow tail disappears and the phase transition
becomes second order. The same characteristic properties were also seen in the
case of RN-AdS black hole in the RPS formalism.

Notice that, in plotting the above curves, there is no need to worry about the
upper bound for $J$, because the figures only cover a small portion of the
positive $T$ values.

Notice also that, the above discussion applies only to the cases $J>0$. When
$J=0$, the black hole falls back to Schwarzchild-AdS, and the $T-S$ and $F-T$
curves becomes quite different. In essence, the stable small black hole states
disappear completely, and the $T-S$ curve contains a single minimum. Using
eq.(14), we can easily find that the minimum is located at

$S_{\rm min}=\frac{\pi C}{3},\quad T_{\rm min}=\frac{\sqrt{3}}{2\pi\ell}.$

Then the EOS (14) and the free energy equation (19) can be rescaled into

$\displaystyle\hat{\tau}=\frac{\hat{s}^{2}(\hat{s}+1)}{2\hat{s}^{5/2}},\quad\hat{f}=\frac{(3-s)s^{1/2}}{2},$
(23)

where $\hat{s}=S/S_{\rm min},\,\hat{\tau}=T/T_{\rm min}$ and $\hat{f}=F/F_{\rm
min}$ with $F_{\rm min}=C/(6\sqrt{3}\ell)$. The corresponding $T-S$ and $F-T$
curves are depicted in Figure 2. For any $T>T_{\rm min}$, there are two black
hole states with the same $T,C$ but different $S$, of which the small one is
unstable and the large one stable. The transitions from small to large black
holes should occur at any $T>T_{\rm min}$ without any equilibrium condition.
This is in sharp contrast to the first order equilibrium phase transition
which occur only at some specific temperature for each fixed nonvanishing $J$,
as described previously.

[width=.48]TSj0 [width=.48]FTj0

Figure 2: $T-S$ and $F-T$ curves at $J=0$

We can also consider the $T-S$ process at fixed $\Omega$. To do so, we need to
insert eq. (15) into (14), obtaining an expression $T=T(S,\Omega,C)$. It turns
out that there is a single extremum for $T$ (which is a minimum) in the
resulting equation located at

$\displaystyle S_{\rm ex}=\frac{2\pi C}{3\ell(2-\ell^{2}\Omega^{2})},\quad
T_{\rm
ex}=\frac{1}{2\pi\ell}\sqrt{\frac{3}{2}}\left(2-\ell^{2}\Omega^{2}\right)^{1/2}.$

Introducing the new relative variables $\tilde{s}=\frac{S}{S_{\rm
ex}},\,\tilde{\tau}=\frac{T}{T_{\rm ex}}$, the $T-S$ EOS can be rewritten in
the form

$\displaystyle\tilde{\tau}=\frac{\tilde{s}^{2}(\tilde{s}+1)}{2\tilde{s}^{5/2}},$
(24)

which is free of both $C$ and $\Omega$. It is interesting that the
$\tilde{\tau}-\tilde{s}$ relation given in (24) is identical to the
$\hat{\tau}-\hat{s}$ relation given in (23).

For better understanding of the $T-S$ processes at fixed $\Omega$, it is
necessary to look at the behavior of the $\mu-T$ relation $\mu=\mu(T,\Omega)$,
where $\mu$ is to be viewed as Gibbs free energy divided by the central
charge. To get the $\mu-T$ relation, we can insert eq. (15) into (16) to
replace $J$ with $\Omega$, but the dependent on $T$ is still implicit in the
resulting relation. However, it can be easily find that $\mu$ is peaked
precisely at $S=S_{\rm ex}$ with the peak value given by

$\mu_{\rm
ex}=\frac{1}{3\sqrt{6}\,\ell}\left(2-\ell^{2}\Omega^{2}\right)^{-1/2}.$

Introducing the new variable

$\tilde{m}=\frac{\mu}{\mu_{\rm ex}},$

the desired $\mu-T$ relation can be written in the form

$\displaystyle\tilde{m}=\frac{\tilde{s}(3-\tilde{s})}{2\tilde{s}^{1/2}},$ (25)

where $\tilde{s}$ is implicitly given by eq. (24) in terms of $\tilde{\tau}$.
It is remarkable that both eq. (24) and (25) are not explicitly dependent on
$\Omega$. This may be viewed as a law of corresponding states at an enhanced
level.

Using eqs. (24) and (25), we present the $T-S$ and $\mu-T$ curves in Figure 3.
Besides the non-equilibrium transitions from the unstable small black hole
state to the stable large black hole state at $T>T_{\rm ex}$ which is quite
similar to the $J=0$ case discussed above, there is also a Hawking-Page
transition occurring at some $T=T_{\rm HP}>T_{\rm ex}$ signified by
$\mu(T_{\rm HP},\Omega)=0$. The Hawking-Page temperature $T_{\rm HP}$
corresponds to $\tilde{s}=3$, which, by use of eq. (24), reads

$T_{\rm HP}=\frac{2\sqrt{3}}{3}T_{\rm ex}.$

[width=.48]TSomega [width=.48]muTomega

Figure 3: $T-S$ and $\mu-T$ curves at fixed $\Omega$

Before moving on to the next subsection, let us stress that the purpose for
using the slow rotating limit is solely due to the demand for explicit
analytical expressions for the critical point parameters and the Hawking-Page
temperature. The same thermodynamic behaviors persists without this
approximation, but the corresponding analysis can only be worked out using
numeric method. Since the numerics are not as illustrative as the analytic
expressions in the slow rotating limit, we omit the details completely.

###  4.2 $\Omega-J$ processes at fixed $S$

Next let us look at the $\Omega-J$ processes at fixed $S$. According to eq.
(15), $\Omega$ is simply proportional to $J$ at fixed $(S,C)$ in the slow
rotating limit, the corresponding curve is just a segment of a straight line
beginning from the origin of the $\Omega-J$ plane and ending at $J=J_{\rm
max}$. In order to see more features of the $\Omega-J$ processes, it is better
to go back to the full form of the EOS (10). Recall that, as extensive
variables, $S$ and $J$ must be both proportional to $C$, thus we can
substitute $S=\mathcal{S}C$ and $J=\mathcal{J}C$ into eq. (10), yielding

$\displaystyle\Omega=\frac{2\pi^{3/2}}{\ell}\left(\frac{\mathcal{S}+\pi}{\mathcal{S}}\right)^{1/2}\frac{\mathcal{J}}{\left[4\pi^{3}\mathcal{J}^{2}+\mathcal{S}^{2}(\mathcal{S}+\pi)\right]^{1/2}},$

where $\mathcal{S},\mathcal{J}$ are both intensive quantities which are
independent of $C$.

In terms of the variable $\mathcal{J}$, the upper bound for $J$ becomes

$\displaystyle\mathcal{J}_{\rm
max}=\frac{\mathcal{S}\left(3\mathcal{S}^{2}+4\pi\mathcal{S}+\pi^{2}\right)^{1/2}}{2\pi^{2}}.$

This equation determines the end point of each $\Omega-J$ curve.

[width=.48]OmegaJ

Figure 4: $\Omega-J$ curves at fixed $S$

Figure 4 presents the $\Omega-J$ curves at different values of $S$. Each curve
corresponds to a different value of $S$, and they all start from the origin
and end at $J=J_{\rm max}$. There is neither inflection point nor extremum on
the $\Omega-J$ curves, which indicate that there can be no phase transition on
the $\Omega-J$ plane in the space of macro states. Notice that for $J$ very
small, each $\Omega-J$ curve can be approximated by a segment of a straight
line starting from the origin. This is the main feature captured by the slow
rotating limit expression (15).

###  4.3 $\mu-C$ processes at fixed $(S,J)$

Unlike the $T-S$ and $\Omega-J$ processes studied in the previous two
subsections, the $\mu-C$ processes call for the variation of the absolute
value of $C$, therefore it seems that the laws of corresponding states will
not apply here. However, as will be seen below, the study of $\mu-C$ process
can still be carried out in a manner in which certain scaling behavior plays a
prominent role.

Although we may choose to proceed with the full EOS (11), the corresponding
analytical analysis is too cumbersome and unillustrative. Therefore, we resort
to the slow rotating limit again by studying only the approximate EOS (16). It
can be seen that there is a single extremum on each $\mu-C$ curve at fixed
$(S,J)$, which corresponds to a maximum of $\mu$. The extremal point is
located at

$\displaystyle C_{\rm
max}=\frac{3S^{3}}{\pi\left(2\pi^{2}J^{2}+S^{2}\right)},\quad\mu_{\rm
max}=\frac{(2\pi^{2}J^{2}+S^{2})^{3/2}}{6\sqrt{3}\ell S^{3}}.$

Notice that $\mu_{\rm max}$ is positive for any nonvanishing choice of $S,J$.
Introducing the dimensionless variables

$c=\frac{C}{C_{\rm max}},\quad m=\frac{\mu}{\mu_{\rm max}},$

the EOS (16) becomes

$\displaystyle m=\frac{3c-1}{2c^{3/2}}.$ (26)

Two important remarks are in due here:

1\. The single equation (26) describes the $\mu-C$ processes at any fixed
values of $S$ and $J$, which clearly indicates some scaling properties;

2\. The very same equation (26) has also appeared in the description of
$\mu-C$ processes for RN-AdS black holes in the RPS formalism [30]. The
appearance of the same reduced $\mu-C$ EOS in both RN-AdS and Kerr-AdS cases
may not be a coincidence. There seems to be some universality lying behind.

[width=.5]muC-kerr

Figure 5: $\mu-C$ curve at fixed $(S,J)$

For completeness, we reproduce the $\mu-C$ curve for the Kerr-AdS case as
presented in Figure 5. Let us remind that at $C=\frac{1}{3}C_{\rm max}$, $\mu$
becomes zero, which signifies a Hawking-Page transition. A better description
for the Hawking-Page transition has already been presented in Figure 3.

##  5 Concluding remarks

The study of Kerr-AdS black hole in the RPS formalism has revealed several
remarkable features which are common to the case of RN-AdS case studied in our
previous work. In particular, we would like to summarize the following
features:

1)

The thermodynamics in the RPS formalism conforms to the standard description
of traditional extensive thermodynamics. The mass of the black hole plays the
role of internal energy and is a homogeneous function of the first order in
all extensive variables. Consequently, the Euler relation and Gibbs-Duhem
equation hold perfectly. Notice that, the absence of pressure-volume variables
leaves no room for defining the enthalpy – the variation of which, by
definition, equals the isobaric exchange of heat [29] – for black holes in our
formalism, as opposed to the EPS formalism.

2)

Each $T-S$ process at fixed nonvanishing angular momentum contains a first
order equilibrium phase transition at certain supercritical temperature. At
the critical temperature the phase transition becomes second order. Further,
at subcritical temperatures, the phase transition disappears completely. On
the contrary, at vanishing angular momentum or at fixed angular velocities,
there is always a non-equilibrium transition from a small unstable black hole
state to a large stable black hole state. The same behaviors are also found in
the case of RN-AdS with the variables $(\Omega,J)$ replaced by
$(\hat{\Phi},\hat{Q})$ [30], in spite of the fact that the underlying
geometries are very different.

3)

The $\Omega-J$ processes in Kerr-AdS (just like the $\hat{\Phi}-\hat{Q}$
processes in RN-AdS) at fixed $S$ are trivial in the sense that there are no
phase transitions in such processes.

4)

The Hawking-Page transition always appear for AdS black holes in the RPS
formalism, and there seems to be some universality in the $\mu-C$ processes
which needs some further exploration.

5)

For those who felt uncomfortable with variable Newton constant, let us mention
that $C$ can be kept fixed, in which case the first law in our formalism falls
back to that of the traditional formalism for black hole thermodynamics, which
is analogous to the ordinary thermodynamics for closed systems. In such cases,
the thermodynamic processes described in subsections 4.1, 4.2 are not
affected, but the $\mu-C$ processes described in subsection 4.3 will be
forbidden. Let us stress that, even when $C$ is kept fixed, the chemical
potential $\mu$ is still meaningful, and the Euler relation (5) still holds.
Moreover, the Euler relation (5) and the Smarr relation

$M=2(TS+\Omega J)$

are two distinct mass formulae [36]. One can even find a third mass formula

$M=2\mu C$

in our formalism, and all these mass formulae can be easily generalized to
higher dimensional rotating black holes with arbitrarily many permissible
rotation parameters [36].

It should be emphasized that, although some of the features for Kerr-AdS are
described only in the slow rotating limit, the same features are actually
maintained in the full exact cases. The only complexity in the full cases is
in the exact solution of the critical point and/or characteristic parameters,
however the characteristic features of the thermodynamic processes have been
verified by use numerical techniques.

The close similarity between the thermodynamic behaviors of the RN-AdS and
Kerr-AdS black holes forces us to make the conjecture that there might be some
hidden universality properties lying behind the RPS formalism. Therefore,
making further explorations about the behaviors of other black holes or black
holes from other gravity models are of utmost importance.

Among the various thermodynamic behaviors discussed so far, particular
attentions need to be paid toward the existence of the phase transitions at
supercritical temperatures. In ordinary non-gravitational thermodynamic
systems, higher temperature means more intensive random motions of individual
particles which disfavors the formation of symmetry breaking order. Therefore
phase transitions appear mostly at subcritical temperatures. However, in
gravitational systems, higher temperature also means higher thermal energy
which in turn produces gravity. Therefore, in gravitational systems, higher
temperature also implies stronger gravitation which may favor the formation of
order. The supercritical phase transitions observed in the RPS formalism of
AdS black hole thermodynamics may simply be signifying that thermal
gravitation overcomes random motion.

The last point to be noticed is the correspondence to the CFT side. As we have
done in subsection 4.3, the thermodynamic process with varying $C$ is a normal
thermodynamic process on the black hole side, however, it implies changing the
dual CFT. This reveals a hidden property of the AdS/CFT correspondence. On the
macroscopic level, the AdS/CFT correspondence is a correspondence between a
given macro state of the AdS black hole and one of the dual CFT. If the macro
state of the AdS black hole changes, the corresponding macro state might need
to be found in a different dual CFT. The same is true if one considers
thermodynamic processes on the CFT side, as did in Ref.[28]. There, a
thermodynamic process with varying volume corresponds to black hole states
with variable cosmological constant, which implies an ensemble of
gravitational theories, of which each member has a different cosmological
constant. It is clear that such processes are meaningful on the CFT side but
not on the gravity side. We end the paper by stressing once again that the
AdS/CFT correspondence is only a correspondence between macro states but not
between macro processes if the theories on both sides are fixed.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under the grant No. 11575088.

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