# Quantum measure as a necessary ingredient in quantum gravity and modified
gravities

Vladimir Dzhunushaliev<EMAIL_ADDRESS>Department of Theoretical
and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Academician J. Jeenbaev Institute of Physics of the NAS of the
Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan  International
Laboratory for Theoretical Cosmology, Tomsk State University of Control
Systems and Radioelectronics (TUSUR), Tomsk 634050, Russia  Vladimir Folomeev
<EMAIL_ADDRESS>Academician J. Jeenbaev Institute of Physics of the NAS of
the Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan
International Laboratory for Theoretical Cosmology, Tomsk State University of
Control Systems and Radioelectronics (TUSUR), Tomsk 634050, Russia

###### Abstract

We suggest commutation relations for a quantum measure. In one version of
these relations, the right-hand side takes account of the presence of
curvature of space; in the simplest case, this yields the action of general
relativity. We consider the cases of the quantization of the measure on spaces
of constant curvature and show that in this case the commutation relations for
the quantum measure are analogues of commutation relations in loop quantum
gravity. It is assumed that, in contrast to loop quantum gravity, a
triangulation of space is a necessary trick for quantizing such a nonlocal
quantity like a measure; in doing so, the space remains a smooth manifold. We
consider the self-consistent problem of the interaction of the quantum measure
and classical gravitation. It is shown that this inevitably leads to the
appearance of modified gravities. Also, we consider the problem of defining
the Euler-Lagrange equations for a matter field in the background of a space
endowed with quantum measure.

measure, quantization, modified gravity, Euler-Lagrange equations

##  I Introduction

Modified gravities lead to various interesting consequences, among which are
the Starobinsky inflation [1], the explanation of the present accelerated
expansion of the Universe [2], etc. But the following fundamental question
arises as to how from all possible versions of modified gravities Nature chose
the one? The truth is that there is a continuum of such versions, i.e., it is
not even a countable set, and it is evident that if Nature chose any variant
of them, this means that it differs somehow from other ones. If one compares
this situation with general relativity, the answer is very simple: general
relativity is the simplest geometrical theory of gravitation with second-order
field equations and with the Lagrangian proportional to the scalar curvature.
The answer to the above question can be as follows: modified gravities are an
approximate description of quantum gravity based on general relativity, and
various versions of modified gravities are realized in different physical
situations. In this paper, following this assumption, we show that the
quantization of the measure leads to modified gravities whose form depends on
the particular physical situation.

There are not so many investigations devoted to a measure as a physical
quantity. The following studies in this direction may be noted. In Refs. [3,
4, 5], instead of a measure determined by a metric, one introduces the so-
called dynamical measure $d\phi^{1}\wedge d\phi^{2}\wedge d\phi^{3}\wedge
d\phi^{4}$ with the scalar fields $\phi^{i},i=1,2,3,4$, which are determined
using some dynamical equations. In Ref. [6], a comparison of modified measure
theories (an approach to modified theories of gravity) with the theory of
causal fermion systems (an approach to unify quantum theory with general
relativity) is carried out. In Ref. [7], the options for defining a volume
element $\sigma$, which can be used for physical theories and defined as a
“dilaton” field, are discussed.

Modified gravities are very fascinating for many reasons, and one of them is
that they may serve as alternative models of dark energy. In particular, in
Ref. [8], modified theory of gravity is suggested within which the terms with
positive powers of the curvature support the inflationary epoch, while the
terms with negative powers of the curvature serve as effective dark energy,
supporting the current cosmic acceleration. In Ref. [9], braneworld cosmology
for a domain wall embedded in the charged anti-de Sitter-Schwarzschild black
hole of the five-dimensional Einstein-Gauss-Bonnet-Maxwell theory is
considered. In Ref. [10], the bound on the compactness of a stellar object in
pure Lovelock theories of arbitrary order in arbitrary spacetime dimensions
involving an electromagnetic field is studied. General reviews on modified
gravities can be found in Refs. [2, 11, 12].

Let us also call attention to Refs. [13, 14] where modified gravities
containing a nonstandard interaction between matter and gravity are under
consideration. Depending on the choice of the parameters, such an interaction
can lead to a big bang, an expanding universe with no beginning, a
cosmological turnaround, an eternally contracting universe, a big crunch, a
Big Rip avoidance, and a cosmological bounce.

In the present paper we suggest commutation relations for a quantum measure.
In Sec. II, we consider the commutation relations whose right-hand side takes
account of the presence of curvature of the spacetime by introducing the
integration of some function (depending on a metric) over the region which is
the intersection of the regions appearing on the left-hand side of the
commutation relations. To write out these commutation relations, it is
necessary to triangulate the space, as it is suggested in loop quantum gravity
(LQG). But in contrast to LQG, we, however, assume here that the space remains
smooth, and the quantized measure, as well as LQG, leads to discrete lengths,
areas, and volumes. In Sec. III, we study the self-consistent problem of the
interaction of the quantized measure and classical gravitation. In Sec. IV, we
find out how the field equations change in the background of a space endowed
with quantum measure.

##  II Quantization of the measure

To begin with, recall some results obtained in Ref. [15] where the following
procedure of quantization of the measure is suggested:

  * •

The triangulation of spacetime is introduced, as in LQG: the spacetime is
divided into four-simplices. Each four-simplex contains tetrahedrons,
triangles, and segments.

  * •

The measure $\mu$ is introduced, which in general is not associated with a
metric.

  * •

The operator of this measure $\hat{\mu}$ is introduced,

$\widehat{\mu(\mathcal{A})}=\int\limits_{\mathcal{A}}\widehat{d\mu},$ (1)

for which the commutation relations are imposed and where the region
$\mathcal{A}$ can be tetrahedrons, triangles, and segments belonging to a
four-simplex.

Note that here, in contrast to LQG, the triangulation of spacetime is
introduced as an auxiliary procedure that enables one to quantize such a
nonlocal quantity like a measure. The spacetime by itself remains a smooth
manifold which, according to the above statement, is triangulated.

Here, we would like to consider generalized commutation relations for the
operator of the measure (1),

$\left[\widehat{\mu\left({\mathcal{A}_{1}}\right)},\widehat{\mu\left({\mathcal{A}_{2}}\right)}\right]=il_{\text{Pl}}^{\alpha}\widehat{\mu\left(\mathcal{A}_{1}\cap\mathcal{A}_{2}\right)}.$
(2)

Here the exponent $\alpha$ equalizes the dimensions of the left- and right-
hand sides of the commutation relation (2), and $l_{\text{Pl}}$ is the Planck
length. Such a definition enables one to take account of a possibility when
the regions $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ may have different
dimensions. This quantization suggests that the operators
$\widehat{\mu\left({\mathcal{A}_{1}}\right)}$ and
$\widehat{\mu\left({\mathcal{A}_{2}}\right)}$ commutate if the regions
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ corresponding to them do not coincide.
If the dimensions coincide, then, by choosing as regions the lengths, areas,
and volumes of the corresponding borders of simplexes on which the space is
triangulated, we get the results known in LQG: the quantization of lengths,
areas, and volumes (see, e.g., the textbooks [16, 17]).

The quantization (2) does not depend on a point of spacetime on which some
metric can be given; this will lead to the fact that the quantized measure (in
particular, like lengths, areas, and volumes in LQG) will not depend on the
fact where the region $\mathcal{A}$ is located. As a quite deduced
generalization of the commutation relation (2), one can consider an idea that
the commutation relations for the quantum measure must depend on the location
of the region $\mathcal{A}$ in space:

$\left[\widehat{\mu\left({\mathcal{A}_{1}}\right)},\widehat{\mu\left({\mathcal{A}_{2}}\right)}\right]=i\gamma_{0}l_{\text{Pl}}^{\alpha}\int\limits_{\mathcal{A}_{1}\cap\mathcal{A}_{2}}f_{g}\;\widehat{d\mu}.$
(3)

Here $f_{g}$ is a function depending on the metric $g$ and $\gamma_{0}$ is
some dimensional constant. The characteristic size of triangulation cells is
determined by the right-hand side of the commutation relations (3), and this
size should not be less than the corresponding Planck values. This implies
that
$\left\langle\gamma_{0}\int\limits_{\mathcal{A}_{1}\cap\mathcal{A}_{2}}f_{g}\;\widehat{d\mu}\right\rangle\gtrsim
l_{\text{Pl}}^{\beta},$ where $\beta$ equalizes the dimensions of the right-
and left-hand sides of this inequality. In order for quantum gravity to smooth
out of singularities, it would be also logical to require that
$f_{g}\rightarrow\infty$ as $R\rightarrow\infty$. Since the metric $g$ is a
tensor and the function $f_{g}$ is a scalar, this function will depend on
scalar invariants of the metric, such as, for example,
$R,R_{\mu\nu}^{2},R_{\mu\nu\rho\sigma}^{2},\cdots$, although more complicated
variants are also possible.

Such a quantization results in interesting consequences. It is logical to
choose the function $f_{g}$ in such a way that, when the scalar invariants
tend to infinity, the least possible lengths, areas, and volumes would
increase. In such case, when they tend to infinity, for instance when
$R\rightarrow\infty$, $l_{\text{min}},S_{\text{min}}$, and $V_{\text{min}}$
would also increase such that these singularities are smoothed out. This means
that our physical system becomes a self-consistent one in the sense that the
quantized measure and metric affect each other.

##  III Self-consistent problem for a quantum measure and classical gravity

Keeping in mind that the measure is quantized, the gravitational action has
the following form:

$S_{g}=-\frac{1}{16\pi G}\int R\left\langle\widehat{d\mu}\right\rangle,$ (4)

where $\left\langle\widehat{d\mu}\right\rangle$ is a quantum averaged measure,
on account of the commutation relations (2) or (3), and $R$ is the scalar
curvature for the metric $g$. According to the Radon-Nikodym theorem (A.1),
the measures $\left\langle\widehat{d\mu}\right\rangle$ and
$d\mu_{g}=\sqrt{-g}\,d^{4}x$ are related by

$\left\langle\widehat{d\mu}\right\rangle=\mathcal{F}(g)d\mu_{g}.$ (5)

The function $\mathcal{F}(g)$ is called the Radon-Nikodym derivative, and it
depends on the metric $g$. Taking this relation into account, the
gravitational action takes the form

$S_{g}=-\frac{1}{16\pi G}\int R\mathcal{F}(g)d\mu_{g}=-\frac{1}{16\pi G}\int
F(g)d\mu_{g}.$ (6)

Here the function $F(g)$ is a Lagrangian of modified gravity. Thus the self-
consistent problem for a quantum measure and general relativity leads to
modified gravity. Analogous to the arguments for the function $f_{g}$ from Eq.
(3), we can say that the function $F(g)$ will depend on scalar invariants of
the metric, such as, for example,
$R,R_{\mu\nu}^{2},R_{\mu\nu\rho\sigma}^{2},\cdots$, although more complicated
variants are possible as well. This means that
$F(g)=F\left(R,R_{\mu\nu}^{2},R_{\mu\nu\rho\sigma}^{2},\cdots\right)$. An
explicit form of this function and its dependence on the scalar invariants
will depend on the particular physical situation: this can be usual modified
gravity $F(R)$, or Gauss-Bonnet gravity, or Weyl gravity, etc. This important
observation leads to the conclusion that a fixed version of modified gravity
is not a fundamental theory of gravitation but it is some approximation for a
description of a situation in some particular physical contexts;  that is, for
some conditions, this, for example, can be $F(R)$ modified gravity, for other
conditions – Weyl gravity, etc.

The function $\mathcal{F}$ must be dimensionless, and this results in the fact
that the corresponding expression contains new dimensional constants. With the
idea in hand that modified gravities are some approximations to quantum
gravity, the appearance of the new dimensional constants in modified gravities
is very similar to the effect of dimensional transmutation in quantum field
theory [18]. The relationship between nonperturbative quantization and the
effect of dimensional transmutation was discussed in Ref. [19].

Consistent with physical arguments, we may suppose that close to a singularity
the function $\mathcal{F}\gg 1$. In turn, for pure general relativity, we must
have $\mathcal{F}=1$.

Thus, the self-consistent problem of the interaction of the quantum measure
and classical gravitation is the pair: the commutation relations (2) or (3)
and field equations coming from the action (6).

To conclude this section, we may mention the following possibility of a self-
consistent interaction of the quantum measure and gravitation. In the
commutation relations (3), one can choose the function $f_{g}$ to be dependent
on the Einstein action,

$\left[\widehat{\mu\left({\mathcal{A}_{1}}\right)},\widehat{\mu\left({\mathcal{A}_{2}}\right)}\right]=i\gamma_{0}l_{\text{Pl}}^{\alpha}l_{0}^{2}\int\limits_{\mathcal{A}_{1}\cap\mathcal{A}_{2}}R\widehat{d\mu},$
(7)

where $l_{0}$ is some dimensional constant possibly different from
$l_{\text{Pl}}$. For example, $l_{0}^{-2}$ can be the average value of the
curvature $\left\langle R\right\rangle$ in the region
$\mathcal{A}_{1}\cap\mathcal{A}_{2}$.

Then the self-consistent problem of the interaction of the quantum measure and
gravitation can be formulated as follows: determine the operator algebra,
given by the commutation relations (7), in such a way that the right-hand side
of these commutation relations would be extremal in the following sense:

$\delta
S=\delta\left\langle\int\limits_{\Omega}R\;\widehat{d\mu}\right\rangle=0,$ (8)

where $\left\langle\ldots\right\rangle$ denotes the averaging over a quantum
state describing some quantum measure and classical metric satisfying the
principle of least action (8).

In the presence of matter in (8), it is necessary to add the corresponding
Lagrangian $L_{m}$:

$\delta S=\delta\left\langle\int\limits_{\Omega}\left(-\frac{R}{16\pi
G}+L_{m}\right)\widehat{d\mu}\right\rangle=0.$

On account of Eqs. (5) and (6) and the arguments given after Eq. (6), the
principle of least action (8) can be rewritten in the form

$\delta
S=\delta\left\langle\int\limits_{\Omega}F(g)\;\widehat{d\mu}\right\rangle=\delta\int\limits_{\Omega}F(g)\left\langle\widehat{d\mu}\right\rangle=0.$

Let us now give the simplest solutions to this self-consistent problem with
the commutation relations (7) for spaces of constant curvature
$R=\text{const}$.

###  III.1 $R=0$.

In this case the commutation relations (7) have the form

$\left[\widehat{\mu\left({\mathcal{A}_{1}}\right)},\widehat{\mu\left({\mathcal{A}_{2}}\right)}\right]=0.$

This means that in flat space the measure is classical and the field equations
(10) (which are under consideration in Sec. IV) for matter in the background
of a quantized measure do not change.

###  III.2 $R=\text{const}\neq 0$.

In this case the commutation relations (7) take the form

$\left[\widehat{\mu\left({\mathcal{A}_{1}}\right)},\widehat{\mu\left({\mathcal{A}_{2}}\right)}\right]=i\gamma_{0}l_{\text{Pl}}^{\alpha}l_{0}^{2}R\,\widehat{\mu\left(\mathcal{A}_{1}\cap\mathcal{A}_{2}\right)},$

where $R=\text{const}$. These are the commutation relations known in LQG; this
means that on spaces of constant curvature all results of LQG are reproduced.

##  IV Matter in the background of a quantized measure

Consider now the question of how the field equations change in the background
of a space endowed with quantized measure. In analogy to (4) and (6), the
action for matter takes the form

$S_{m}=\int L_{m}\left\langle\widehat{d\mu}\right\rangle=\int
L_{m}\mathcal{F}(g)d\mu_{g}=\int
L_{m}\mathcal{F}(R,R_{\mu\nu}^{2},\cdots)d\mu_{g}=\int
L_{m,\text{mod}}d\mu_{g}.$ (9)

Variation of the action (9) with respect to $\phi^{A}$ leads to the following
modified Euler-Lagrange equations:

$\frac{\partial}{\partial x^{\mu}}\frac{\partial
L_{m,\text{mod}}}{\partial\phi^{A}_{,\mu}}-\frac{\partial
L_{m,\text{mod}}}{\partial\phi^{A}}=0,$ (10)

where $\phi^{A}$ are matter fields with a generalized index $A$. It is evident
that the presence of the quantized measure in the form of the factor
$\mathcal{F}$ modifies the Euler-Lagrange field equations. According to the
discussion given in Sec. III, in high curvature regions, the Euler-Lagrange
equations differ considerably from the similar equations in flat space.

Modified theories with the action (9) have been considered in Refs. [13, 14]
using the Lagrangian

$L=\frac{R}{8\pi}-F_{\mu\nu}\phi^{,\mu}\phi^{,\nu},$

where the function $F_{\mu\nu}$ describes a nonminimal derivative coupling
between the scalar field $\phi$ and gravitation. In those papers, two cases
are under consideration:
$F_{\mu\nu}=g_{\mu\nu}+\kappa_{1}g_{\mu\nu}R+\kappa_{2}R_{\mu\nu}$ and
$F_{\mu\nu}=\epsilon g_{\mu\nu}+\kappa G_{\mu\nu}$, where $G_{\mu\nu}$ is the
Einstein tensor. The results of calculations within these theories indicate
that such interactions enable one, for example, to avoid a cosmological
singularity and a Big Rip.

Note also that in the flat space limit $R\rightarrow 0$ the right-hand side of
the commutation relations (7) becomes equal to zero, and the measure $\mu$
becomes classical; this enables one to choose the function $\mathcal{F}(g)=1$
in Eq. (5), and then (9) becomes the action for usual unmodified matter.

##  V Conclusions

We have considered the quantization of the measure and suggested the
commutation relations for operators of the measure. From the physical point of
view, it is quite natural to assume that a measure and a metric must somehow
affect each other. To realize this idea, we have considered the self-
consistent problem of the influence of the quantum measure and classical
gravity on each other and shown in this context that such a self-consistent
problem leads to modified theories of gravitation. It is important that in the
regions where gravitational forces differ strongly the modified gravities will
be different.

Along the lines suggested here, we have also considered the influence of the
quantum measure on the Euler-Lagrange equations for matter fields. It became
clear that in the regions with strong gravitational fields these equations
differ considerably from similar equations in flat space.

Thus, in the present paper, we have

  * •

suggested the commutation relations for the operator algebra of measure;

  * •

considered the self-consistent problem of the interaction of the quantized
measure and classical gravitation;

  * •

considered one of versions of the commutation relations whose right-hand side
contains the action of general relativity;

  * •

solved the self-consistent problem of the interaction of the quantized measure
and classical gravitation in the simplest case of spaces of constant
curvature;

  * •

shown that the quantized measure leads to modified gravities;

  * •

considered the problem of field equations for an arbitrary matter field in the
background of a space endowed with quantum measure.

In the context of the present study, there appear numerous problems, requiring
further investigations: (a) Is it possible, using the technique developed in
LQG, to get spectra of lengths, areas, and volumes in a space endowed with a
metric and with the use of the commutation relations (3) which take account of
this metric? (b) Are the commutation relations (2) and (3) mathematically
consistent in the case when the regions $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ have different dimensions? (c) Do the commutation relations
(2) and (3) lead to smoothing out of singularities? (d) What is the
relationship between the functions $f_{g}$ and $\mathcal{F}$? etc.

Another interesting question arising in this context concerns a possible
relationship between Weyl gravity and an approximate description of a self-
consistent interaction of the quantum measure and classical gravitation near
singularities. The reason is that Weyl gravity is conformally invariant and it
may not feel the appearance of singularities, as shown in Ref. [20]; this can
be related to smoothing out of the singularities while quantizing a measure.

The next interesting question: are there maximum lengths, areas, and volumes
arising while quantizing a measure? In LQG, the answer to this question is
negative, but we know that in quantum mechanics situations with finite
spectrum of some physical quantity are possible; therefore, a priori, one
cannot rule out such a situation while quantizing a measure interacting with
gravitation (metric).

## Acknowledgments

The work was supported by the Science Committee of the Ministry of Science and
Higher Education of the Republic of Kazakhstan (Grant No. AP14869140, “The
study of QCD effects in non-QCD theories”).

##  Appendix A Radon-Nikodym theorem

Let us introduce the definition of a measure:

######  Definition A.1.

Let $X$ be a set and $\Sigma$ a $\sigma$-algebra over $X$. A function $\mu$
from $\Sigma$ to the extended real number line is called a measure if it
satisfies the following properties:

  * •

Non-negativity: For all $E$ in $\Sigma$, we have $\mu(E)\geq 0$.

  * •

Null empty set: $\mu(\varnothing)=0$.

  * •

Countable additivity (or $\sigma$-additivity): For all countable collections
$\left\\{E_{k}\right\\}_{k=1}^{\infty}$ of pairwise disjoint sets in $\Sigma$,

$\mu\left(\bigcup_{k=1}^{\infty}E_{k}\right)=\sum_{k=1}^{\infty}\mu(E_{k}).$

The pair $\left(X,\Sigma\right)$ is called a measurable space. A triple
$\left(X,\Sigma,\mu\right)$ is called a measure space. The Radon-Nikodym
theorem involves a measurable space $(X,\Sigma)$ on which two $\sigma$-finite
measures are defined, $\mu$ and $\nu$.

######  Theorem A.1 (the Radon-Nikodym theorem [21]).

If $\nu\ll\mu$ (that is, if $\nu$ is absolutely continuous with respect to
$\mu$), then there exists a $\Sigma$-measurable function $f:X\to[0,\infty)$,
such that for any measurable set $\mathcal{A}\subseteq X$,

$\nu(\mathcal{A})=\int\limits_{\mathcal{A}}f\,d\mu.$ (11)

The function $f$ appearing here is called the Radon-Nikodym derivative and is
designated as $\frac{d\nu}{d\mu}$. This theorem implies that any two measures
can be interrelated by some function.

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# Thermodynamics of 5D Charged Rotating Black Holes:
A Counterterms Treatment

Adel Awad${}^{\,a,b}$ and Hassan ElSayed${}^{\,c,d,}$111Corresponding author;
<EMAIL_ADDRESS>

a Department of Physics, Faculty of Science, Ain Shams University, Cairo
11566, Egypt
b Centre for Theoretical Physics, The British University in Egypt, P.O. Box
43,
El Sherouk City, Cairo 11837, Egypt
c Department of Physics, University Connecticut, Storrs, CT 06269-3046, USA
d Department of Physics, School of Sciences and Engineering, American
University in Cairo,
P.O. Box 74, AUC Avenue New Cairo, Cairo, Egypt

###### Abstract

We use the counterterms subtraction method to calculate various
thermodynamical quantities for charged rotating black holes in five-
dimensional minimal gauged supergravity [1]. Specifically, we analyze certain
issues related to the first law and Smarr’s relation in the presence of a
conformal anomaly. Among the bulk quantities calculated are the on-shell
action, total mass, and angular momenta of the solution. All these quantities
are consistent with previous calculations made using other methods. For the
boundary theory, we calculate the renormalized stress tensor, conformal
anomaly, and Casimir energy. Using the Papadimitriou-Skenderis analysis [2],
we show that the mass calculated via the counterterms method satisfies the
first law of black hole thermodynamics. To discuss extended thermodynamics, we
extend the definition of the thermodynamic volume to cases with conformal
anomalies using a procedure similar to that of Papadimitriou-Skenderis. We
show that this volume correctly accounts for extra terms due to boundary
metric variation. This shows that the mass and volume calculated using
counterterms satisfy Smarr’s relation as well as the first law.

Keywords: Black holes, thermodynamics, holography, Smarr’s relation,
thermodynamic volume.

###### Contents

  1. 1 Introduction
  2. 2 The 5D Charged Rotating AdS Solution
  3. 3 Thermodynamics: Background Method
  4. 4 Thermodynamics: Counterterms Method
    1. 4.1 Action Calculation
    2. 4.2 Mass and Angular Momenta Calculations
  5. 5 Holographic Stress Tensor and Anomaly
    1. 5.1 CFT Stress Tensor and Conformal Anomaly
    2. 5.2 Casimir Energy
  6. 6 The First Law and the Counterterms Method
    1. 6.1 The First Law in the Presence of a Conformal Anomaly
    2. 6.2 Extended Thermodynamics and Volume Calculation
    3. 6.3 The First Law in Extended Thermodynamics
  7. 7 Conclusion

##  1 Introduction

The discovery of the anti-de sitter/conformal field theory (AdS/CFT)
correspondence [3] invoked lots of interest in asymptotically AdS solutions in
general relativity. This duality reveals how various gravitational solutions
in the AdS side encode important information about a specific gauge field
theory on the boundary in their semi-classical gravitational action. A strong
interest in studying black hole solutions in AdS5 spacetimes then followed
after studying a specific example of this duality, namely, the equivalence
between string theory on AdS${}_{5}\times S^{5}$ and four-dimensional ${\cal
N}=4$, Super Yang-Mills (SYM) theory on the boundary. This duality and other
similar ones allow one to map certain five-dimensional AdS solutions to
certain states in the boundary field theory. Furthermore, it shows that bulk
boundary quantities work as sources to CFT quantities, e.g., the boundary
metric works as a source for the boundary energy-momentum tensor. In this
duality as well as in a Euclidean path-integral formalism we have certain
boundary conditions which fix bulk boundary quantities. For example, AdS black
hole thermodynamics with its possible phase transitions are investigated for a
fixed boundary metric as is discussed in [4, 5].

The AdS/CFT correspondence relates a strong-coupling regime in the boundary
with a weak-coupling regime in the bulk. The bulk partition function can be
approximated [6] using saddle points leading to

$\mathcal{Z}_{\text{grav}}\simeq\exp(-I),$ (1)

where $I$ is the on-shell gravitational action. However, in AdS spacetimes,
the action as a volume integral diverges as we take the radial coordinate $r$
to infinity. There are two main techniques often used to regulate this action:
the background subtraction method (for example, see[7] and references therein)
and the counterterms subtraction method [8]. Background subtraction works by
defining a reference background spacetime, then calculating the finite action
as the difference between the action of the spacetime and that of the
background metric. Since both metrics have the same diverging asymptotic
region, this subtraction method leaves a finite action. But a disadvantage of
this technique is that the choice of a proper background reference is not
always obvious or clear [9, 10, 8]. In addition, sometimes certain quantities
which are common between the spacetime and its background (such as the
conformal anomaly and vacuum energy) cancel out in this procedure [11].

The counterterms subtraction method [9, 10, 8] works by adding certain
covariant boundary counterterms to the action which cancel action divergences
exactly. These counterterms are thus inherent to the spacetime boundary and
there is no need for a background or a reference spacetime. Moreover, the
employment of the counterterms subtraction technique allows one to compare
some important quantities between the bulk and the boundary. For instance, the
trace of the Brown-York quasilocal stress tensor [12] resulting from the
action of the counterterms technique is related to the conformal anomaly on
the boundary. Another example is the total energy/mass found using the
counterterms technique, which includes a non-vanishing contribution as the
mass parameter $m$ is sent to zero. In the context of the AdS/CFT
correspondence, this contribution which is the spacetime background energy is
interpreted as the Casimir energy of the boundary CFT. These two phenomena
appear particularly in odd-dimensional bulk theories, or even-dimensional
boundary field theories. For example, see [10, 11] and references therein.

The counterterms method was criticized in [7] where the authors argued that
the first law of black hole thermodynamics,

$dM=TdS+\sum_{i}\Omega_{i}dJ_{i}+\Phi dQ,$ (2)

is not satisfied through this procedure. Several authors have presented
resolutions to this apparent violation [13, 6, 2]. Here we are particularly
interested in the resolution presented in [2], which we briefly explain below.

It is well known that quantum corrections can break classical conformal
symmetries in even-dimensional field theories leading to conformal anomalies.
The conformal or Weyl transformation on the boundary is a subset of the bulk
diffeomorphisms. This diffeomorphism is a Penrose-Brown-Hennaux (PBH)
transformation [14, 15]. In the case of a non-vanishing conformal anomaly, a
PBH transformation will not leave the gravitational on-shell action invariant.
The action has a relation to other quantities through the Gibbs-Duhem given by

$I=\beta{M}-TS-\sum_{i}\Omega_{i}J_{i}-\Phi Q.$ (3)

Now by varying this equation, one is expected to get variations of all terms,
but instead, we get

$\delta I=\beta\delta{M}.$ (4)

In other words, this variation affects only the mass, but not the other
quantities in the first law, since entropy, electric charge $Q$, and angular
momenta $J_{a}$ and $J_{b}$ can be expressed in terms of integrals over the
horizon [2]. In $\S$4.2 we show that the counterterms mass is the sum of two
terms, the first is localized in the black hole region (depends on the mass
parameter and electric charge) while the second term is the background energy
of the spacetime which is not restricted to the black hole region. The flux of
the latter term depends on the boundary metric. An important boundary
condition for the variation problem [2] is to keep the boundary metric fixed
by utilizing a PBH or boundary Weyl transformation. Therefore, one can use a
Weyl transformation to cancel the boundary metric variation of the mass term
in the first law. As a result, the first law is satisfied. To summarize, the
first law is not violated by the counterterms mass, but one should be careful
about how to compute variations of bulk quantities since those must be done at
a fixed boundary metric as the AdS/CFT requires.

In this work, we use the counterterms subtraction method to calculate the
renormalized on-shell action for the general rotating charged AdS solution
presented in [1] as well as its mass and angular momenta. We show that these
quantities satisfy the first law and the Gibbs-Duhem relation. Going to
extended thermodynamics [16, 17] where we allow the cosmological constant to
vary and act as a pressure, a naive calculation of the first law shows that it
is not satisfied. We show that a similar issue exists with the volume defined
in extended thermodynamics [16, 17]. Meaning that varying this quantity does
not leave the boundary metric fixed and one needs to use a compensating PBH
term to keep the volume fixed. This modification is important to satisfy the
first law as well as the generalized Smarr’s formula in counterterms context
with a conformal anomaly.

The rest of the paper is organized as follows: in $\S$2 we present the black
hole solution and its thermodynamic quantities as in [1]. In $\S$3 we show
that these quantities satisfy the standard thermodynamic relations: the first
law (2) and the Gibbs-Duhem relation. In $\S$4 we use the counterterms
subtraction method to calculate finite expressions for the action, mass, and
angular momenta of the solution. In $\S$5 we calculate the renormalized stress
tensor and conformal anomaly of the CFT from the dual gravitational theory and
compare those results to the field theory calculations on rotating Einstein
Universe. Also, we calculate the Casimir energy of the CFT and compare it to
the background energy of the bulk theory which was calculated using the
counterterms subtraction method. In $\S$6 we show that the quantities
calculated using counterterms subtraction satisfy the first law in regular and
extended thermodynamics. We study the effect of a Weyl transformation on the
boundary and use this to show that the mass calculated using the counterterms
method satisfies the first law of thermodynamics. Furthermore, we check that
our expressions for action and mass satisfy the Gibbs-Duhem relation. Finally,
we discuss the effect of boundary variations on Smarr’s formula and propose a
new modification to the thermodynamic volume in the presence of a conformal
anomaly. This new volume satisfies the first law in extended phase-space as
well as Smarr’s formula.

##  2 The 5D Charged Rotating AdS Solution

The Einstein-Hilbert-Chern-Simons Lagrangian in five-dimensions has the
following form [1]

$\mathcal{L}=(R^{2}+12g^{2})\star\mymathbb{1}-\frac{1}{2}\star F\wedge
F+\frac{1}{3\sqrt{3}}F\wedge F\wedge A.$ (5)

The first term on the right-hand side is the gravitational Einstein-Hilbert
Lagrangian in AdS5, the second is the Maxwell Lagrangian and the third is the
Chern-Simons Lagrangian in five dimensions. The latter is required in five-
dimensional gauged supergravity [18].

In this letter we are interested in studying the general non-extremal rotating
black holes in minimal five-dimensional gauged supergravity [1] which has the
following metric:

$\begin{split}ds^{2}=&-\frac{\Delta_{\theta}[(1+g^{2}r^{2})\rho^{2}\text{d}t+2q\nu]\text{d}t}{\Xi_{a}\Xi_{b}\rho^{2}}+\frac{2q\nu\omega}{\rho^{2}}+\frac{f}{\rho^{4}}\left(\frac{\Delta_{\theta}\text{d}t}{\Xi_{a}\Xi_{b}}-\omega\right)^{2}+\frac{\rho^{2}\text{d}r^{2}}{\Delta_{r}}+\frac{\rho^{2}\text{d}\theta^{2}}{\Delta_{\theta}}\\\
&+\frac{r^{2}+a^{2}}{\Xi_{a}}\sin^{2}\theta\,\text{d}\phi^{2}+\frac{r^{2}+b^{2}}{\Xi_{b}}\cos^{2}\theta\,\text{d}\psi^{2},\end{split}$
(6)

where

$\displaystyle\nu=b\sin^{2}\theta\,\text{d}\phi+a\cos^{2}\theta\,\text{d}\psi,\quad\omega=a\sin^{2}\theta\frac{\text{d}\phi}{\Xi_{a}}+b\cos^{2}\theta\frac{\text{d}\psi}{\Xi_{b}},$
(7)
$\displaystyle\Delta_{r}=\frac{(r^{2}+a^{2})(r^{2}+b^{2})(1+g^{2}r^{2})+q^{2}+2abq}{r^{2}}-2m,$
(8)
$\displaystyle\Delta_{\theta}=1-a^{2}g^{2}\cos^{2}\theta-b^{2}g^{2}\sin^{2}\theta,\quad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta,$
(9) $\displaystyle\Xi_{a}=1-a^{2}g^{2},\quad\Xi_{b}=1-b^{2}g^{2},$ (10)
$\displaystyle f=2m\rho^{2}-q^{2}+2abqg^{2}\rho^{2}.$ (11)

The constant $g=1/\ell$, where $\ell$ is the AdS radius, is not to be confused
with the determinant of the metric.

The coordinate system in (6) is $(t,r,\theta,\phi,\psi)$. The ranges of the
last three coordinates is such that they cover a three-sphere. The range for
$\theta$ is between $0$ and $\pi/2$, while that for $\phi$ and $\psi$ is
between $0$ and $2\pi$. The electromagnetic four-potential is given in [1] by

$A=\frac{\sqrt{3}q}{\rho^{2}}\left(\frac{\Delta_{\theta}\text{d}t}{\Xi_{a}\Xi_{b}}-\omega\right).$
(12)

And the electric charge is given by

$Q=\frac{\sqrt{3}\pi q}{4\Xi_{a}\Xi_{b}}.$ (13)

In four spatial dimensions, the black hole has two possible rotation axes. The
corresponding angular velocities are denoted $\Omega_{a}$ (in the
$\phi$-direction) and $\Omega_{b}$ (in the $\psi$-direction). They are given
in [1] by

$\displaystyle\Omega_{a}$
$\displaystyle=\frac{a\left(r_{+}^{2}+b^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+bq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq},\quad\Omega_{b}=\frac{b\left(r_{+}^{2}+a^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+aq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}.$
(14)

Evidently, the metric in (6) has axial symmetries in $\phi$ and $\psi$. The
Killing vectors associated with these symmetries are $\partial_{\phi}$ and
$\partial_{\psi}$. The angular momenta were subsequently calculated in [1]
using the Komar integral,

$J_{a}=\frac{1}{16\pi}\int_{S^{3}}\star\text{d}\left(\partial_{\phi}\right),\quad
J_{b}=\frac{1}{16\pi}\int_{S^{3}}\star\text{d}\left(\partial_{\psi}\right),$
(15)

yielding

$\displaystyle J_{a}$
$\displaystyle=\frac{\pi\left[2am+qb\left(1+a^{2}g^{2}\right)\right]}{4\Xi_{a}^{2}\Xi_{b}},\quad
J_{b}=\frac{\pi\left[2bm+qa\left(1+b^{2}g^{2}\right)\right]}{4\Xi_{b}^{2}\Xi_{a}}.$
(16)

Note that the integrals in (15) directly lead to finite results and do not
need regularization.

##  3 Thermodynamics: Background Method

In this section we review the thermodynamics of the spacetime under
consideration using background-subtraction calculations. For this solution,
the temperature and entropy are given by

$\displaystyle
T=\frac{r_{+}^{4}\left[g^{2}\left(a^{2}+b^{2}+2r_{+}^{2}\right)+1\right]-(ab+q)^{2}}{2\pi
r_{+}\left[\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq\right]},$
(17) $\displaystyle
S=\frac{\pi^{2}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right]}{2\Xi_{a}\Xi_{b}r_{+}}.$
(18)

The total energy/mass was calculated in [1] by integration of the first law
(2). This mass is given by222The mass was denoted in [1] by $M$ but we will
use here the notation $M_{0}$ to distinguish this mass from that which we will
derive using the counterterms method in $\S$4.2.

$M_{0}=\frac{\pi m\left(2\Xi_{a}+2\Xi_{b}-\Xi_{a}\Xi_{b}\right)+2\pi
abg^{2}q\left(\Xi_{a}+\Xi_{b}\right)}{4G\Xi_{a}^{2}\Xi_{b}^{2}}.$ (19)

It is worth noting that this expression for the mass matches the one that was
found using the ADM calculation in [19, 6].

Using the above mass, it is straightforward to check the validity of the first
law

$dM_{0}=TdS+\Omega_{a}dJ_{a}+\Omega_{b}dJ_{b}+\Phi dQ.$ (20)

The electric potential is found from

$\displaystyle\Phi$
$\displaystyle=\xi^{a}A_{a}|_{r\to\infty}-\xi^{a}A_{a}|_{r\to r_{+}}$
$\displaystyle=\frac{\sqrt{3}qr_{+}^{2}}{\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq},$
(21)

which is in agreement with [20].

To check the first law, one needs to verify the following equation

$\frac{\partial M_{0}}{\partial\alpha}d\alpha=T\frac{\partial
S}{\partial\alpha}d\alpha+\Phi\frac{\partial
Q}{\partial\alpha}d\alpha+\Omega_{a}\frac{\partial
J_{a}}{\partial\alpha}d\alpha+\Omega_{b}\frac{\partial
J_{b}}{\partial\alpha}d\alpha,$ (22)

for the parameters, $\alpha=r_{+},q,a,b$, since the mass is a function of
these variables. Direct evaluation of these expressions shows that the first
law is verified.

Gibbs Free Energy and Action Calculation

Since the integral of (5) is divergent as $r\to\infty$, the background
subtraction method instructs us to start by taking a cut-off at a large value
$R$. We then specify a background metric by taking $m\to 0,q\to 0$ in (6),
which gives the Kerr-AdS5 spacetime. We compute the integral of this action at
the cut-off value $R$, then subtract it from our previous calculation with the
original metric to cancel the divergences. The action using the background
subtraction method was given in [19] by

$I_{0}=\frac{\pi\beta}{4\Xi_{a}\Xi_{b}}\left[m-\left(g^{2}\left(a^{2}+r^{2}\right)\left(b^{2}+r^{2}\right)\right)-\frac{q^{2}r^{2}}{\left(a^{2}+r^{2}\right)\left(b^{2}+r^{2}\right)+abq}\right].$
(23)

It is straightforward to verify that

$I_{0}=\beta G_{0},$ (24)

where the Gibbs free energy is given by333We use the subscript “0” to refer to
the Gibbs free energy calculated from $M_{0}$ to distinguish it from that
calculated from the counterterms mass.

$G_{0}(T,\Omega_{a},\Omega_{b},\Phi)=M_{0}-TS-\Omega_{a}J_{a}-\Omega_{b}J_{b}-\Phi
Q.$ (25)

This is the Gibbs-Duhem relation, one of the few important thermodynamic
relations used to check the consistency of black hole thermodynamics. To check
that $G_{0}=G_{0}(T,\Omega_{a},\Omega_{b},\Phi)$, one might vary $G_{0}$ to
obtain

$dG_{0}=-SdT-J_{a}d\Omega_{a}-J_{b}d\Omega_{b}-Qd\Phi,$ (26)

where

$\displaystyle\left(\frac{\partial G}{\partial
T}\right)_{\Omega_{a},\Omega_{b},\Phi}=-S,\hskip 36.135pt\left(\frac{\partial
G}{\partial\Omega_{a}}\right)_{T,\Phi,\Omega_{b}}=-J_{a},$
$\displaystyle\left(\frac{\partial
G}{\partial\Omega_{b}}\right)_{T,\Phi,\Omega_{b}}=-J_{b},\hskip
36.135pt\left(\frac{\partial
G}{\partial\Phi}\right)_{T,\Omega_{a},\Omega_{b}}=-Q.$ (27)

Notice that the above quantum-statistical relation which we call the Gibbs-
Duhem relation is going to be the same for the background and counterterms
methods apart from a modification in mass and action calculations. As we have
mentioned in the Introduction, in the counterterms method, this relation plays
an important role to prove the validity of the first law.

A Generalized Smarr’s Formula

For asymptotically AdS spaces, in order to construct a generalized Smarr’s
formula, one must allow the cosmological constant to vary, leading to the
following form of Smarr’s formula [21]

$\displaystyle
M_{0}=\frac{3}{2}\left(TS+\Omega_{a}J_{a}+\Omega_{b}J_{b}\right)+\Phi
Q-PV_{0},$ (28)

where the cosmological constant plays the role of a pressure with some
thermodynamic volume as a conjugate variable [16, 17], or

$\displaystyle P=-\frac{\Lambda}{8\pi}=\frac{3g^{2}}{4\pi},$ (29)

And $V_{0}$ is the “thermodynamic volume” $V_{0}$ conjugate to $P$. Following
the prescription in [21], we find

$\displaystyle V_{0}$ $\displaystyle=\left(\frac{\partial M_{0}}{\partial
P}\right)_{S,Q,J_{a},J_{b}}$
$\displaystyle=\frac{\pi^{2}}{2\Xi_{a}\Xi_{b}}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+\frac{2}{3}abq\right]+\frac{2\pi}{3}\left(aJ_{a}+bJ_{b}\right),$
(30)

in accordance with [20]. With some algebraic manipulations, it is
straightforward to show that the above Smarr’s formula (28) is satisfied.

##  4 Thermodynamics: Counterterms Method

As mentioned in the introduction the need to use the counterterms method is to
obtain a finite on-shell gravitational action. This method gives a non-
ambiguous procedure to regularise the action in a manner independent of any
other spacetime (i.e. a background spacetime). This allows us to make some
important connections between the bulk gravity and the dual CFT in four
dimensions. As we will see in the coming sections, this method enables us to
calculate quantities such as the vacuum expectation value (vev) of the stress-
tensor, the conformal anomaly, as well as the Casimir energy of the boundary
field theory. Our aim is to calculate the gravitational finite action first,
then use it to calculate the vev of the field theory stress tensor through
calculating the quasilocal stress tensor, or the Brown-York stress-tensor [12]
(BY) (using eq. (58). For details see for example[10, 11]). The BY tensor is
also used to calculate conserved quantities such as the mass and angular
momenta.

###  4.1 Action Calculation

The action of the Lagrangian presented in eq.(5) is composed of the following
terms

$I_{\text{non-ren}}=I_{\text{EH}}+I_{\text{GH}}+I_{\text{EM}}.$ (31)

Calculating the electromagnetic part444See also, [22], one finds

$I_{\text{EM}}=\frac{-\pi\beta}{4\Xi_{a}\Xi_{b}}\left[\frac{q^{2}r_{+}^{2}}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}\right],$
(32)

While the Einstein-Hilbert action calculation leads to

$I_{\text{EH}}=\beta\left[\frac{\pi g^{2}r^{4}}{4\Xi_{a}\Xi_{b}}+\frac{\pi
g^{2}r^{2}\left(a^{2}+b^{2}\right)}{4\Xi_{a}\Xi_{b}}-\frac{\pi
g^{2}r_{+}^{2}\left(a^{2}+b^{2}+r_{+}^{2}\right)}{4\Xi_{a}\Xi_{b}}\right]\Bigg{|}_{r\to+\infty}.$
(33)

And the Gibbons-Hawking action calculation produces

$\displaystyle I_{\text{GH}}$ $\displaystyle=\beta\bigg{[}-\frac{\pi
g^{2}r^{4}}{\Xi_{a}\Xi_{b}}+\frac{15\pi
r^{2}\left(\Xi_{a}+\Xi_{b}+3/4\right)}{24\Xi_{a}\Xi_{b}}+\frac{\pi\left(a^{4}g^{2}-8a^{2}b^{2}g^{2}-9a^{2}+b^{4}g^{2}-9b^{2}+24m\right)}{24\Xi_{a}\Xi_{b}}\bigg{]}\bigg{|}_{r\to+\infty}.$
(34)

Summing the previous three expressions we get the following divergent action

$\displaystyle I_{\text{non-ren}}=$ $\displaystyle
I_{\text{EH}}+I_{\text{GH}}+I_{\text{EM}}$ $\displaystyle=$
$\displaystyle\left[-\frac{3\pi\beta g^{2}r^{4}}{4\Xi_{a}\Xi_{b}}+\beta
r^{2}\left(\frac{\pi
g^{2}\left(a^{2}+b^{2}\right)}{4\Xi_{a}\Xi_{b}}+\frac{\pi\left(-15a^{2}g^{2}-15b^{2}g^{2}-18\right)}{24\Xi_{a}\Xi_{b}}\right)\right]\Bigg{|}_{r\to+\infty}+\text{finite
terms}.$ (35)

Now let us calculate the counterterms and verify that they cancel these
divergences. The expression for these counterterms is given in [8] by

$\displaystyle
I_{\mathrm{ct}}=\frac{1}{8\pi}\int_{\partial\mathcal{M}}\text{d}^{n}x\sqrt{h}\left[\frac{n-1}{\ell}+\frac{\ell}{2(n-2)}\mathcal{R}+\frac{\ell^{3}}{2(n-4)(n-2)^{2}}\left(\mathcal{R}_{ab}\mathcal{R}^{ab}-\frac{n}{4(n-1)}\mathcal{R}^{2}\right)+\ldots\right],$
(36)

where $\mathcal{R}_{ab}$ and $\mathcal{R}$ are the Ricci tensor and Ricci
scalar of the boundary metric, respectively, and we recall that $\ell=1/g$.
These counterterms were calculated in $D=n+1$ dimensions [10, 8]. The extra
terms denoted by “$...$” are needed in higher dimensions. Evaluation of the
counterterms in (36) yields

$I_{\text{ct}}=\beta\left[\frac{3\pi g^{2}r^{4}}{4\Xi_{a}\Xi_{b}}+\frac{3\pi
r^{2}\left(a^{2}g^{2}+b^{2}g^{2}+2\right)}{8\Xi_{a}\Xi_{b}}\right]\Bigg{|}_{r\to+\infty}+\text{finite
terms}.$ (37)

The final expression for the Euclidean action from the counterterms
subtraction method is found by adding the term in (37) to the divergent action
in (4.1), which gives the regulated action

$\displaystyle\begin{split}I_{\text{ren}}=&\frac{\pi\beta}{96g^{2}\Xi_{a}\Xi_{b}}\big{[}\Xi_{a}^{2}+\Xi_{b}^{2}+7\Xi_{a}\Xi_{b}+24g^{2}\big{(}m-a^{2}b^{2}g^{2}-g^{2}r_{+}^{2}(r_{+}^{2}+a^{2}+b^{2})\big{)}\big{]}\\\
&-\frac{\pi\beta}{4\Xi_{a}\Xi_{b}}\left[\frac{q^{2}r_{+}^{2}}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}\right].\end{split}$
(38)

This action reduces to that of the Reissner-Nordström black hole in [18] in
the limit $a,b\to 0$. We can also re-express the gravitational part of the
action as

$\displaystyle I_{\text{ren}}^{\text{grav}}=\beta M_{\text{BG}}^{\text{Kerr-
AdS}_{5}}+\frac{\pi\beta\left[m-g^{2}(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})\right]}{4\Xi_{a}\Xi_{b}},$

where

$M_{\text{BG}}^{\text{Kerr-
AdS}_{5}}=\frac{\pi\left[9\Xi_{a}9\Xi_{b}+(\Xi_{a}-\Xi_{b})^{2}\right]}{96g^{2}\Xi_{a}\Xi_{b}}$
(39)

is the background energy of the Kerr-AdS5 black hole [2]. Our result takes the
exact same form as the one calculated by Papadimitriou and Skenderis [2] for a
non-charged rotating black hole.

The difference here of course is that the charge does make an appearance in
the $\beta$ term. Nevertheless, it is interesting that the gravitational
action maintains the same expression in terms of $\beta$. This also
automatically shows that our result reduces to that in [2] since, in the
absence of an electric charge, the temperature takes the same form as that in
[2], and the electromagnetic action vanishes.

###  4.2 Mass and Angular Momenta Calculations

The calculation of conserved charges is based on the Brown-York quasilocal
charge definition [12]. The Brown-York quasilocal stress tensor is given by

$\displaystyle T^{ab}$ $\displaystyle=\frac{-2}{\sqrt{|h|}}\frac{\delta
I_{\text{ren}}}{\delta h_{ab}}$ $\displaystyle=\frac{1}{8\pi
G}\left[K^{ab}-h^{ab}K+g(n-1)h^{ab}-\frac{\mathcal{G}^{ab}}{g(n-2)}\right],$
(40)

where $h_{ab}$, $K_{ab}$, and $\mathcal{G}_{ab}$ are the metric, extrinsic
curvature, and Einstein tensors on the induced boundary, respectively. Notice
that $D=n+1$ is the spacetime dimensions. A charge associated with a Killing
vector $K^{a}$ is defined by

$Q[K]=\int_{S^{3}}\text{d}^{3}x\sqrt{\sigma}u^{a}T_{ab}K^{b}.$ (41)

Here $u_{a}=-N\nabla_{a}t$, while $N$ is the lapse function and $\sigma$ is
the spacelike metric that appear in the ADM-like decomposition of the metric

$ds^{2}=-N\text{d}t^{2}+\sigma_{ab}(\text{d}x^{a}+N^{a}\text{d}t)(\text{d}x^{b}+N^{b}\text{d}t).$
(42)

In the above expression $N^{a}$ is the shift vector. The Killing vector
associated with the mass is the timelike Killing vector $\chi=\partial_{t}$.
Plugging this in (41) initially yields a slightly complicated expression for
the mass. To simplify the long calculation let us write the mass as the sum of
four terms:

$M=M_{\text{nr1}}+M_{\text{nr2}}+M_{\text{ct1}}+M_{\text{ct2}}.$ (43)

The first two terms are divergent; they result from the original non-
regularized components of the Brown-York quasilocal stress tensor. The last
two terms are those arising from the counterterms action. The first of those
four terms is given by

$\displaystyle M_{\text{nr1}}=$ $\displaystyle\frac{\pi
m\left[-3g^{2}\left(a^{2}+b^{2}\right)+a^{2}g^{2}\Xi_{b}+b^{2}g^{2}\Xi_{a}-2\Xi_{a}\Xi_{b}+6\right]}{8G\Xi_{a}^{2}\Xi_{b}^{2}}$
$\displaystyle+\frac{\pi
q\left[abg^{2}\left(-3g^{2}\left(a^{2}+b^{2}\right)+a^{2}g^{2}\Xi_{b}+6\right)+ab^{3}g^{4}\Xi_{a}+2abg^{2}\Xi_{a}\Xi_{b}\right]}{8G\Xi_{a}^{2}\Xi_{b}^{2}}$
$\displaystyle+\frac{\pi\left[2a^{2}b^{2}g^{2}\Xi_{a}\Xi_{b}+2g^{2}r^{2}(a^{2}+b^{2})\Xi_{a}\Xi_{b}+2g^{2}r^{4}\Xi_{a}\Xi_{b}\right]}{8G\Xi_{a}^{2}\Xi_{b}^{2}},$
(44)

where we have reinstated the gravitational constant $G$ (originally a factor
in the denominator of the Brown-York quasilocal stress tensor) for future
need. The second term is given by

$\displaystyle M_{\text{nr2}}=$
$\displaystyle\frac{-\pi}{24G\Xi_{a}\Xi_{b}}\Big{[}a^{4}-g^{2}+3r^{2}\big{(}5g^{2}\left(a^{2}+b^{2}\right)+6\big{)}+8a^{2}b^{2}g^{2}+9a^{2}-b^{4}g^{2}+9b^{2}$
$\displaystyle+24g^{2}r^{4}-24m\Big{]}.$ (45)

The first part of the counterterms contribution is

$\displaystyle M_{\text{ct1}}=$
$\displaystyle\frac{\pi}{32g^{2}G\Xi_{a}^{2}\Xi_{b}^{2}}\Big{[}12g^{2}\left(a^{2}g^{2}\Xi_{b}+a^{2}g^{2}+b^{2}g^{2}-2\right)\left(abg^{2}q+m\right)+\Xi_{a}\Big{(}\Xi_{b}\big{(}-a^{4}g^{4}+a^{2}g^{2}$
$\displaystyle\quad\left(11b^{2}g^{2}+18g^{2}r^{2}+9\right)+24abg^{4}q-b^{4}g^{4}+9b^{2}\left(2g^{4}r^{2}+g^{2}\right)+3\big{(}8g^{4}r^{4}+4g^{2}r^{2}$
$\displaystyle\quad-1\big{)}\big{)}+12b^{2}g^{4}\left(abg^{2}q+m\right)\Big{)}\Big{]}.$
(46)

And lastly, the second contribution from the counterterms gives

$M_{\text{ct2}}=-\frac{3\pi\left[a^{2}g^{4}\left(b^{2}+r^{2}\right)+g^{2}r^{2}\left(b^{2}g^{2}-2\right)-1\right]}{16g^{2}G\Xi_{a}\Xi_{b}}.$
(47)

The addition of all these terms gives the mass via the counterterms
subtraction method,

$\displaystyle M=\frac{\pi m\left(2\Xi_{a}+2\Xi_{b}-\Xi_{a}\Xi_{b}\right)+2\pi
abg^{2}q\left(\Xi_{a}+\Xi_{b}\right)}{4G\Xi_{a}^{2}\Xi_{b}^{2}}+\frac{\pi\left[9\Xi_{a}\Xi_{b}+(\Xi_{a}-\Xi_{b})^{2}\right]}{96Gg^{2}\Xi_{a}\Xi_{b}}.$
(48)

The first term on the RHS is just the black hole mass calculated in [1] by
integrating the first law. The second term is the background energy of the
spacetime: it is the value that the total energy reduces to in the absence of
the black hole (when $m=0$, $q=0$). We note that this background energy is the
same as that in the Kerr-AdS5 black hole solution [23]. In $\S$5.2 we will
show that this is exactly equal to the Casimir energy of the dual CFT on the
boundary. Furthermore, when the two rotation parameters are set to zero, this
vacuum energy in (51) reduces to the background energy of the pure non-
rotating AdS5 spacetime given in [10] by

$M_{\text{BG}}\Big{|}_{a=b=0}=\frac{3\pi}{32g^{2}G}.$ (49)

The full expression for the mass calculated via the counterterms method can
now be written as

$\displaystyle M=M_{0}+M_{\text{BG}},$ (50)

with

$M_{\text{BG}}=\frac{\pi\left[9\Xi_{a}\Xi_{b}+(\Xi_{a}-\Xi_{b})^{2}\right]}{96Gg^{2}\Xi_{a}\Xi_{b}}.$
(51)

The final quantities to calculate using the counterterms method are the
angular momenta. To get $J_{a}$ (respectively $J_{b}$) we plug in (41) the
Killing vector $\partial_{\phi}$ (respectively $\partial_{\psi}$). The angular
momentum in the $\phi$-direction is given by

$\displaystyle J_{a}$
$\displaystyle=\int_{S^{3}}\frac{\sin^{3}\theta\cos\theta\left[am+b\left(a^{2}g^{2}q+\frac{1}{2}q\Xi_{a}\right)\right]}{2\pi\Xi_{a}^{2}\Xi_{b}}\text{d}\theta\text{d}\phi\text{d}\psi$
$\displaystyle=\frac{\pi\left[2am+qb\left(1+a^{2}g^{2}\right)\right]}{4\Xi_{a}^{2}\Xi_{b}}.$
(52)

And we find a similar result for $J_{b}$,

$\displaystyle J_{b}$
$\displaystyle=\frac{\pi\left[2bm+qa\left(1+b^{2}g^{2}\right)\right]}{4\Xi_{a}\Xi_{b}^{2}}.$
(53)

These results are equal to the angular momentum calculated in [1] using the
Komar integral.

There are some intriguing features of this solution worth discussing, some of
which are indeed unique. Before we discuss these features let us take some
limits to check the above mass/total energy expression. By setting $q=0$ and
$a=b=0$, we reduce this solution to the Schwarzschild mass,

$M\big{|}_{q=0,a=b=0}=\frac{3\pi m}{4}.$

This means that the parameter $m$ has the usual interpretation of a “mass
parameter”. Yet there seems to be an unusual aspect of this solution: when
this parameter vanishes, the black hole’s mass does not vanish not only
because of the background energy but also due to contributions from the
electric charge, $q$.

$M\big{|}_{m=0}=\frac{\pi
abg^{2}q(\Xi_{a}+\Xi_{b})}{2\Xi_{a}^{2}\Xi_{b}^{2}}+M_{\text{BG}}.$ (54)

Indeed, the charge contributes to the total mass of the solution! We do not
know any other solution that shares this property with the solution presented
here. For instance, we find the mass of a static charged black hole in five-
dimensional anti-de Sitter spacetime and the neutral Kerr-AdS solutions have
no charge contributions in their mass expressions. Therefore, the physical
significance of the expression in (48) when $m$ vanishes seems a bit puzzling.
However, one can show that there are no horizons, or real values for $r_{+}$,
if $m=0$ while $q\neq 0$. In other words, the case $m=0,q\neq 0$ has a naked
singularity.

Another puzzling feature appears in the angular momenta expression in
equations (4.2), and (53). The angular momenta carry some dependence on the
electric charge. This feature has no analog in four dimensions. While the mass
is non-vanishing if $m$ goes to zero, there are also non-vanishing angular
momenta $J_{a}$ and $J_{b}$. Furthermore, if we keep $m\neq 0$ and set $a=0$
we still have a non-null value for $J_{a}$! The same thing happens if we
$m\neq 0$ and $b=0$: we still have a non-null value for $J_{b}$! Of course,
this does not happen in the Kerr-AdS5 solution where the angular momenta are
given [2] by

$\displaystyle J_{a}$ $\displaystyle=\frac{\pi am}{2\Xi_{a}^{2}\Xi_{b}},\quad
J_{b}=\frac{\pi bm}{2\Xi_{b}^{2}\Xi_{a}}.$ (55)

One is then left to wonder if these new features can produce some interesting
phenomena for these black holes.

##  5 Holographic Stress Tensor and Anomaly

One of the predictions of the AdS/CFT correspondence is that the conformal
anomaly calculated from the conformal field theory on the boundary should
exactly match the trace of the gravitational stress tensor (BY tensor) [10,
11], where the identification $G^{-1}\leftrightarrow 2N^{2}g^{3}/\pi$ is to be
made. Another prediction is the matching of the dual field theory Casimir
energy with the background energy obtained as we set $m=0$ and $q=0$ in the
mass expression. In this section we will aim to verify these predictions as
well as calculate the vev of the renormalized CFT stress tensor predicted by
the duality.

###  5.1 CFT Stress Tensor and Conformal Anomaly

It is important to notice here that the metric in eq.(6) is asymptotically AdS
with a conformal boundary (with $\mathbb{R}\times S^{3}$ topology). The
induced metric at the boundary is given by

$ds_{\text{Boundary}}^{2}=g^{2}r^{2}\left[-\text{d}t^{2}+\frac{2a\sin^{2}\theta}{\Xi_{a}}\text{d}t\text{d}\phi+\frac{2b\cos^{2}\theta}{\Xi_{b}}\text{d}t\text{d}\psi+\frac{\text{d}\theta^{2}}{g^{2}\Delta_{\theta}(\theta)}+\frac{\sin^{2}\theta}{g^{2}\Xi_{a}}\text{d}\phi^{2}+\frac{\cos^{2}\theta}{g^{2}\Xi_{b}}\text{d}\psi^{2}\right],$
(56)

which is nothing but the rotating Einstein Universe. The CFT or the boundary
metric is found by removing the divergent conformal factor $g^{2}r^{2}$ from
the above metric [11]

$ds_{\text{BG}}^{2}=-\text{d}t^{2}+\frac{2a\sin^{2}\theta}{\Xi_{a}}\text{d}t\text{d}\phi+\frac{2b\cos^{2}\theta}{\Xi_{b}}\text{d}t\text{d}\psi+\frac{\text{d}\theta^{2}}{g^{2}\Delta_{\theta}(\theta)}+\frac{\sin^{2}\theta}{g^{2}\Xi_{a}}\text{d}\phi^{2}+\frac{\cos^{2}\theta}{g^{2}\Xi_{b}}\text{d}\psi^{2}.$
(57)

The bulk stress tensor is related to the expectation value of the renormalized
CFT stress tensor $\langle\hat{T}_{ab}\rangle$ [11] by

$\sqrt{-\gamma}\,\gamma_{ab}\langle\hat{T}^{bc}\rangle=\lim_{r\rightarrow+\infty}\sqrt{-h}\,h_{ab}T^{bc}.$
(58)

Therefore, we expect the trace of the gravitational tensor to be related to
the CFT’s stress tensor through the following factor

$\lim_{r\to+\infty}\sqrt{h/\gamma}=g^{4}r^{4}.$ (59)

Evaluation of the trace of (4.2) gives

$T^{a}{}_{a}=-\frac{\left(a^{2}-b^{2}\right)\Big{[}3g^{2}\left(a^{2}-b^{2}\right)\cos^{4}\theta-2\cos^{2}\theta\left(a^{2}g^{2}-2b^{2}g^{2}+1\right)-b^{2}g^{2}+1\Big{]}}{8\pi
gGr^{4}}.$ (60)

Multiplying this by the conformal factor $g^{4}r^{4}$ yields

$\displaystyle
g^{4}r^{4}T^{a}{}_{a}=-\frac{g^{3}\left(a^{2}-b^{2}\right)\Big{[}3g^{2}\left(a^{2}-b^{2}\right)\cos^{4}\theta-2\cos^{2}\theta\left(a^{2}g^{2}-2b^{2}g^{2}+1\right)-b^{2}g^{2}+1\Big{]}}{8\pi
G}.$ (61)

This result matches that found in [23] for the Kerr-AdS5 solution. This is
expected since the difference between the Kerr-AdS5 metric and that in (6) is
a term proportional to the charge parameter $q$, which vanishes at the
boundary.

We now look at the expectation value of the stress tensor of the dual CFT. The
details that we followed in calculating and renormalizing this tensor are
outlined in Chapter 6 of ref. [24]. Following the notation in ref. [23], the
renormalized stress tensor is given by

$\langle\hat{T}_{ab}\rangle=-\frac{1}{16\pi^{2}}\sum_{s}\left(\frac{1}{9}\alpha^{s}H_{ab}^{(1)}+\beta^{s}H_{ab}^{(3)}\right),$
(62)

where the summation is over the possible fields of the theory, with
$s=0,\frac{1}{2},1$ standing for scalar, Weyl spinor, and U(1) gauge fields,
respectively. If we calculate the expectation value over a vacuum state then
$H_{ab}^{(4)}$ will vanish since it is the vacuum expectation value of the
stress tensor in flat spacetime.

Furthermore, one can choose a regularization scheme in which the $\alpha^{s}$
coefficients vanish. For more details on this, the reader can refer to ref.
[25]. The values of the remaining $\beta^{s}$ coefficients are given [25] by

$\displaystyle\beta^{0}=-\frac{1}{2880\pi^{2}}N^{0},\quad\beta^{\frac{1}{2}}=-\frac{1}{2880\pi^{2}}N^{\frac{1}{2}},\quad\beta^{1}=-\frac{1}{2880\pi^{2}}N^{1},$
(63)

where $N^{i}$ is the number of fields of spin $i$. These numbers are [11]
$N^{0}=6N^{2}$, $N^{\frac{1}{2}}=4N^{2}$ and $N^{1}=N^{2}$. The tensor
$H_{ab}^{(3)}$ is given in [24] by

$H_{ab}^{(3)}=\frac{1}{12}\mathcal{R}^{2}\gamma_{ab}-\mathcal{R}^{cd}\mathcal{R}_{cadb},$
(64)

where $\gamma_{ab}$ is the CFT metric tensor and $\mathcal{R}_{abcd}$,
$\mathcal{R}_{ab}$ and $\mathcal{R}$ are the Riemann tensor, Ricci tensor, and
Ricci scalar of the CFT, respectively.

We list below the non-vanishing components of the CFT stress tensor:

$\displaystyle\langle\hat{T}_{tt}\rangle=$
$\displaystyle-\frac{495N^{2}g^{4}}{1952\pi^{2}\Xi_{a}\Xi_{b}}\Big{[}\Big{(}g^{2}\left(a^{2}+b^{2}\right)+g^{2}(a^{2}-b^{2})\cos(2\theta)-2\Big{)}\Big{(}g^{2}(a^{2}-b^{2})\big{(}20\cos(2\theta)\left(g^{2}\left(a^{2}+b^{2}\right)-2\right)$
$\displaystyle+7g^{2}(a^{2}-b^{2})\cos(4\theta)\big{)}-24g^{2}\left(a^{2}+b^{2}\right)+g^{4}\left(5a^{4}+14a^{2}b^{2}+5b^{4}\right)+24\Big{)}\Big{]},$
$\displaystyle\langle\hat{T}_{\theta\theta}\rangle=$
$\displaystyle\frac{495N^{2}g^{2}}{976\pi^{2}\left(g^{2}\left(b^{2}-a^{2}\right)\cos^{2}\theta-b^{2}g^{2}+1\right)}\Big{[}g^{2}\left(a^{2}-b^{2}\right)\Big{(}3g^{2}\left(a^{2}-b^{2}\right)\cos(4\theta)+4\cos(2\theta)\big{(}g^{2}\left(a^{2}+b^{2}\right)$
$\displaystyle-2\big{)}\Big{)}+8g^{2}\left(a^{2}+b^{2}\right)+g^{4}\big{(}-7a^{4}+6a^{2}b^{2}-7b^{4}\big{)}-8\Big{]},$
$\displaystyle\langle\hat{T}_{\phi\phi}\rangle=$
$\displaystyle\frac{495N^{2}g^{2}\sin^{2}\theta}{976\pi^{2}\Xi_{a}}\Big{[}g^{2}(a^{2}-b^{2})\Big{(}7g^{2}(a^{2}-b^{2})\cos(4\theta)-4\cos(2\theta)\big{(}g^{2}\left(3a^{2}-5b^{2}\right)+2\big{)}\Big{)}+8g^{2}\big{(}a^{2}+b^{2}\big{)}$
$\displaystyle+g^{4}\big{(}5a^{4}-18a^{2}b^{2}+5b^{4}\big{)}-8\Big{]},$
$\displaystyle\langle\hat{T}_{\psi\psi}\rangle=$
$\displaystyle\frac{495N^{2}g^{2}\cos^{2}\theta}{976\pi^{2}\Xi_{b}}\Big{[}g^{2}\left(a^{2}-b^{2}\right)\Big{(}7g^{2}\left(a^{2}-b^{2}\right)\cos(4\theta)+4\cos(2\theta)\big{(}g^{2}\left(5a^{2}-3b^{2}\right)-2\big{)}\Big{)}+8g^{2}\big{(}a^{2}+b^{2}\big{)}$
$\displaystyle+g^{4}\big{(}5a^{4}-18a^{2}b^{2}+5b^{4}\big{)}-8\Big{]}.$ (65)

It is important to state here that the above CFT stress tensor matches the one
predicted by the duality, which is calculated using Brown-York tensor and the
above relation in eq. (58). Indeed, this is one of the important non-
perturbative silent checks of this duality. As a result, evaluating the trace
of this stress tensor yields the expected conformal anomaly

$\displaystyle\langle\hat{T}^{a}{}_{a}\rangle=-\frac{\left(a^{2}-b^{2}\right)N^{2}g^{6}\left[3g^{2}\left(a^{2}-b^{2}\right)\cos^{4}\theta-2\cos^{2}\theta\left(a^{2}g^{2}-2b^{2}g^{2}+1\right)-b^{2}g^{2}+1\right]}{4\pi^{2}}.$
(66)

This identification is exact upon remembering that, in the AdS/CFT
correspondence, $2N^{2}g^{3}/\pi$ is translated to $G^{-1}$ at the gravity
side and vice versa. We have therefore shown that the CFT stress tensor –
which is calculated from the counterterm-regulated action at the gravity side
– is exactly equal to the CFT boundary stress tensor calculated at the field
theory side. This also implies an equality between the conformal anomalies on
both sides.

###  5.2 Casimir Energy

Another consequence of the matching of the above two stress tensors – the one
predicted by gravity and that calculated on the field theory side – is the
matching between the vacuum energies, (51) on the gravity side and the Casimir
energy $E_{\text{Casimir}}$ on the field theory side. The Casimir energy is
found using the formula ([11])

$E_{\text{Casimir}}=\sum_{s=0,\frac{1}{2},1}N^{s}\int_{S^{3}}\mathrm{d}^{3}x\sqrt{\sigma}\,\chi^{a}\langle\hat{T}_{ab}^{s}\rangle
u^{b}.$ (67)

Here the summation is again over the possible fields of the theory. $\chi^{a}$
and $u^{a}$ are the timelike Killing vector and unit normal vector, and
$\sigma_{ab}$ is the foliation metric of the conformal boundary. The conformal
foliation metric is found using

$\sigma_{ab}=g^{2}r^{2}\left(g_{ab}+u_{a}u_{b}\right),$ (68)

whose determinant is given by

$\sigma=\frac{\left(a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta+r^{2}\right)\sin^{2}\theta\cos^{2}\theta}{g^{6}r^{2}\Delta_{\theta}\Xi_{a}\Xi_{b}}.$
(69)

Direct evaluation of the integral in (67) followed by some simplifications
gives

$E_{\text{Casimir}}=\frac{N^{2}g\left[9\Xi_{a}\Xi_{b}+(\Xi_{a}-\Xi_{b})^{2}\right]}{48\Xi_{a}\Xi_{b}}.$
(70)

Making the identification $\pi/(2Gg^{3})\leftrightarrow N^{2}$, it is easy to
see that the Casimir energy in (70) is identical to the background energy of
the bulk spacetime in (51),

$\displaystyle E_{\text{Casimir}}=M_{\text{BG}}.$ (71)

##  6 The First Law and the Counterterms Method

As was discussed in [2], introducing surface counterterms to regulate the on-
shell gravitational action in five dimensions induces an anomalous Weyl
transformation on the boundary.555Notice that this happens when the rotation
parameters are different. This is how the bulk encodes the boundary Weyl
anomaly. Let us denote the radial spacelike boundary at infinity by
$\partial\mathcal{M}$. It was found [2] that a Weyl factor $\delta\sigma$
changes the renormalized action by

$\delta_{\sigma}I_{\text{ren}}=\int_{\partial\mathcal{M}}\text{d}^{n}x\sqrt{-h}\mathcal{A}\delta\sigma,$
(72)

in the presence of a conformal anomaly $\mathcal{A}$.

Now, we have seen that in order to verify the first law, we need to vary
various quantities, such as the entropy, angular momenta, and charge with
respect to $r_{+}$, $q$, $a$ and $b$ (see for example eq. (22)). But since the
boundary metric is expressed in terms of $a$ and $b$, variation of these
parameters will change the boundary metric that should remain fixed. We will
see (eq. (82)) that variations of $a$ and $b$ have the effect of a Weyl
transformation on the boundary metric. The entropy, angular momenta, and
electric charge are expressed as surface integrals of fluxes whose sources are
localized in the black hole region. In that sense, they can be written as
integrals over the horizon [2]. Thus they will not be affected by variations
of the boundary metric. This is not the case for the mass. As we saw in eq.
(48), the total mass is the sum of a term that depends on the mass parameter
($M_{0}$) plus the background energy ($M_{\text{BG}}$). The latter is
certainly affected by conformal transformations of the boundary. Another
quantity that is affected by the boundary conformal transformation is the
renormalized action, which contains contributions from surface integrals on
the boundary.

The main idea is to subtract from the LHS of the first law the variation of
the mass that results from varying the boundary metric. If we denote this mass
variation by $\delta_{\sigma}M$, the correct form of the first law [2] is thus
given by

$\displaystyle dM$
$\displaystyle=\delta_{\sigma}M+TdS+\Omega_{a}dJ_{a}+\Omega_{b}dJ_{b}+\Phi
dQ.$ (73)

Note that the variations of the renormalized action $I_{\text{ren}}$ and mass
are not independent since (see eq. (76) below)

$I_{\text{ren}}=\beta(M-TS-\Omega_{a}J_{a}-\Omega_{b}J_{b}-\Phi Q),$ (74)

so

$\delta_{\sigma}I_{\text{ren}}=\beta\delta_{\sigma}M.$ (75)

We argue here that, by a similar analysis, the volume term (see eq. (30))
needs to be modified due to extra terms arising from boundary variation when
the AdS radius $1/g$ is varied as well.

In conclusion, we expect the action and mass calculated from the counterterms
method to verify the quantum statistical relation, the first law, and Smarr’s
formula. But in order to show the latter two, we must account for extra terms
which result from a Weyl transformation of the boundary when certain
parameters are varied

###  6.1 The First Law in the Presence of a Conformal Anomaly

We begin by noting that the counterterms action (38) and mass (48) satisfy the
so-called quantum statistical relation

$I_{\text{ren}}=\beta G(T,\Omega_{a},\Omega_{b},\Phi).$ (76)

We will adopt the same procedure that was done in [2]. The first step is to
transform the boundary metric into a more canonical asymptotic form. To do so
we use the coordinates $\bar{r}$ and $\bar{\theta}$ defined by

$\displaystyle r$
$\displaystyle=\bar{r}\left[1+\frac{\hat{\Delta}_{\bar{\theta}}}{4g^{2}\bar{r}^{2}}+\frac{\hat{\Delta}_{\bar{\theta}}}{16g\bar{r}}\left(1+\hat{\Xi}_{a}+\hat{\Xi}_{b}-2\hat{\Delta}_{\bar{\theta}}\right)+\mathcal{O}\left(\frac{1}{\bar{r}^{6}}\right)\right],$
(77) $\displaystyle\theta$
$\displaystyle=\bar{\theta}+\frac{1}{16g\bar{r}}\left(1-\hat{\Delta}_{\bar{\theta}}\right)\hat{\Delta}_{\bar{\theta}}^{\prime}-\frac{1}{32g^{6}\bar{r}^{6}}\left(1-\hat{\Delta}_{\bar{\theta}}\right)\hat{\Delta}_{\bar{\theta}}^{\prime}\left(1+\hat{\Xi}_{a}+\hat{\Xi}_{b}+3\hat{\Delta}_{\bar{\theta}}\right)+\mathcal{O}\left(\frac{1}{\bar{r}^{8}}\right).$
(78)

The functions $\hat{\Delta}_{\theta}$, $\hat{\Xi}_{a}$ and $\hat{\Xi}_{b}$ are
given by

$\displaystyle\hat{\Delta}_{\theta}$
$\displaystyle=1-\Delta_{\theta},\quad\hat{\Xi}_{a}=1-\Xi_{a},\quad\hat{\Xi}_{b}=1-\Xi_{b}.$
(79)

The conformal boundary metric in terms of the new coordinates is

$d\bar{s}^{2}_{\text{BG}}=-\text{d}t^{2}+\frac{2a\sin^{2}\bar{\theta}}{\Xi_{a}}\text{d}t\text{d}\phi+\frac{2b\cos^{2}\bar{\theta}}{\Xi_{b}}\text{d}t\text{d}\psi+\frac{1}{g^{2}\Delta_{\bar{\theta}}}\text{d}\bar{\theta}^{2}+\frac{\sin^{2}\bar{\theta}}{g^{2}\Xi_{a}}\text{d}\phi^{2}+\frac{\cos^{2}\bar{\theta}}{g^{2}\Xi_{b}}\text{d}\psi^{2}.$
(80)

We now consider the variation of this metric under infinitesimal variations of
$a$ and $b$. We switch to a new coordinate system
$(t,\bar{r},\bar{\theta}^{\prime},\phi^{\prime},\psi^{\prime})$ given by

$\tan^{2}\bar{\theta}=\left(1+\frac{\delta\Xi_{a}}{\Xi_{a}}-\frac{\delta\Xi_{b}}{\Xi_{b}}\right)\tan^{2}\bar{\theta}^{\prime},\quad\phi=\phi^{\prime}-g^{2}t\delta
a,\quad\psi=\psi^{\prime}-g^{2}t\delta b.$ (81)

In terms of these coordinates, the resulting variation of the metric in (80)
is given [2] by

$d\bar{s}_{\text{BG}}^{2}\rightarrow\left(1-\frac{\delta\Xi_{a}}{\Xi_{a}}\sin^{2}\bar{\theta}-\frac{\delta\Xi_{b}}{\Xi_{b}}\cos^{2}\bar{\theta}\right)d\bar{s}_{\text{BG}}^{2}.$
(82)

We use the value of the conformal anomaly that we calculated in (66) to find
the variation of the action resulting from the Weyl factor in (82):

$\displaystyle\delta_{\sigma}I_{\text{ren}}$
$\displaystyle=\int_{[0,\,\beta]\times
S^{3}}\text{d}^{4}x\sqrt{-h}\mathcal{A}\delta\sigma$ (83)
$\displaystyle=\frac{\pi\beta}{96g^{2}G}\delta\left(\frac{\Xi_{a}}{\Xi_{b}}+\frac{\Xi_{b}}{\Xi_{a}}\right).$
(84)

This form is in agreement with [2], again with the caveat that the electric
charge appears in $\beta$. We also explicitly verify that

$\displaystyle\delta_{\sigma}I_{\text{ren}}$
$\displaystyle=\beta\delta_{\sigma}M$ $\displaystyle=\beta\delta
M_{\text{Casimir}},$ (85)

which is expected, given the result in (76). The first law can then be written
as666Recall that $M=M_{0}+M_{\text{BG}}=M_{0}+M_{\text{Casimir}}$.

$dM=dM_{0}+dM_{\text{Casimir}}=\delta
M_{\text{Casimir}}+TdS+\Omega_{a}dJ_{a}+\Omega_{b}dJ_{b}+\Phi dQ.$ (86)

We have already seen that the first term on the left-hand side equals the sum
of all but the first term on the right-hand side. The Casimir energy is only a
function of $a$ and $b$ and does not depend on $r_{+}$ or $q$. The term
$dM_{\text{Casimir}}$ is hence just the variation of the Casimir energy with
respect to $a$ and $b$. Using (6.1) this term can be calculated from

$\delta M_{\text{Casimir}}=\beta^{-1}\int_{[0,\,\beta]\times
S^{3}}\text{d}^{4}x\sqrt{-h}\mathcal{A}\delta\sigma.$ (87)

###  6.2 Extended Thermodynamics and Volume Calculation

Allowing the cosmological constant to vary leads to another treatment called
extended thermodynamics [16, 17]. In this treatment we are adding another pair
of thermal quantities, namely the pressure, $P={3g^{2}/4\pi}$, and its
conjugate thermodynamic volume

$V=\left(\frac{\partial M}{\partial P}\right)_{S,Q,J_{a},J_{b}}.$ (88)

This pair is supposed to obey Smarr’s relation (28) as in [17], but it is
straightforward to check that the volume in (88) does not satisfy this
relation. In this section we analyze the reason behind this contradiction and
show how it can be fixed. We are unaware of any similar attempts to address
this issue for this or any other black hole solutions in the literature.

The relation in (88) can equivalently be written as

$V=\frac{\partial g}{\partial P}\left(\frac{\partial M}{\partial
g}\right)_{S,Q,J_{a},J_{b}}.$ (89)

It is easy to see that the last term on the RHS induces a variation in the
boundary metric analogous to variations with respect to $a$ and $b$ in
$\S$6.1. We thus need to add a compensating term to the thermodynamic volume
to account for this variation. Let us go back to the metric in (80). We now
switch to a new coordinate system
$\left(t,\bar{r},\bar{\theta}^{\prime\prime},\phi^{\prime\prime},\psi^{\prime\prime}\right)$,
where

$\tan^{2}\bar{\theta}=\left(1+\frac{\delta\Xi_{a}}{\Xi_{a}}-\frac{\delta\Xi_{b}}{\Xi_{b}}\right)\tan^{2}\bar{\theta}^{\prime\prime},\quad\phi=\phi^{\prime\prime}-a^{2}t\delta
g,\quad\psi=\psi^{\prime\prime}-b^{2}t\delta g.$ (90)

We emphasize that the variations of $\Xi_{a}$ and $\Xi_{b}$ are now with
respect to $g$, and the rotation parameters $a$ and $b$ are kept fixed
throughout this subsection. With this in mind, it is easy to see that the
variation of the metric follows the form

$d\bar{s}_{\text{BG}}^{2}\rightarrow\left(1-\frac{\delta\Xi_{a}}{\Xi_{a}}\sin^{2}\bar{\theta}-\frac{\delta\Xi_{b}}{\Xi_{b}}\cos^{2}\bar{\theta}\right)d\bar{s}_{\text{BG}}^{2}.$
(91)

Let us denote the Weyl factor in this section by $\delta\tilde{\sigma}$. We
define an effective thermodynamic volume $\bar{V}$ that takes care of the
extra terms in the mass that arise from the transformation (91) by

$\displaystyle\bar{V}$ $\displaystyle=\left(\frac{\partial M}{\partial
P}\right)_{S,Q,J_{a},J_{b}}+\frac{\partial(\delta_{\tilde{\sigma}}M)}{\partial
P}.$ (92)

This is a generalized definition of the volume that reduces to the usual
definition in (30) and (88) in the absence of a conformal anomaly and still
satisfies Smarr’s formula in the presence of the latter. Furthermore, this
definition satisfies Smarr’s formula even if we use the ADM [19] or the
Kounterterms method [26]. These two techniques give the mass expression in
(19), resulting in the last term on the RHS vanishing. Expression (92) will
hence give the same quantity in (30), which we have already verified that it
satisfies Smarr’s formula if the mass is given by (19).

In AdS5, it is straightforward to verify that transformation (91) leads to a
variation of the mass given by

$\displaystyle\delta_{\tilde{\sigma}}M$
$\displaystyle=\beta^{-1}\delta_{\tilde{\sigma}}I_{\text{ren}}$
$\displaystyle=\beta^{-1}\int_{[0,\,\beta]\times
S^{3}}\text{d}^{4}x\sqrt{-h}\mathcal{A}\delta{\tilde{\sigma}}$
$\displaystyle=\frac{\pi}{96g^{2}G}\left[\frac{\delta\Xi_{a}(\Xi_{a}-\Xi_{b})(\Xi_{a}+\Xi_{b})}{\Xi_{a}{}^{2}\Xi_{b}}-\frac{\delta\Xi_{b}(\Xi_{a}-\Xi_{b})(\Xi_{a}+\Xi_{b})}{\Xi_{a}\Xi_{b}{}^{2}}\right]$
$\displaystyle=\frac{\pi}{96g^{2}G}\delta\left(\frac{\Xi_{a}}{\Xi_{b}}+\frac{\Xi_{b}}{\Xi_{a}}\right).$
(93)

The first line is just eq. (83), and we used relation (6.1) to arrive at the
second line. We now calculate the volume in (92) for our solution:

$\displaystyle\bar{V}=V_{0}-\frac{\pi^{2}\left[-9g^{2}\left(a^{2}+b^{2}\right)+g\left(a+7a^{2}b^{2}+b\right)+9\right]}{72g\Xi_{a}\Xi_{b}}.$
(94)

It is straightforward to show that the expression in (94) satisfies Smarr’s
formula with the mass calculated via the counterterms method,

$\displaystyle
M=\frac{3}{2}\left(TS+\Omega_{a}J_{a}+\Omega_{b}J_{b}\right)+\Phi Q-P\bar{V}.$
(95)

Alternatively, one can define the thermodynamic volume by (88) and add a
compensating term to Smarr’s formula,

$\displaystyle
M=\frac{3}{2}\left(TS+\Omega_{a}J_{a}+\Omega_{b}J_{b}\right)+\Phi
Q-\left(PV+P\frac{\partial(\delta_{\tilde{\sigma}}M)}{\partial P}\right).$
(96)

###  6.3 The First Law in Extended Thermodynamics

We discuss here the first law in extended thermodynamics for the general black
hole in five dimensions. In extended thermodynamics, the mass/energy $M$ is
nothing but the enthalpy

$H=U+P\bar{V}.$ (97)

The internal energy is thus found from

$U=M-P\bar{V}.$ (98)

With this, the first law in terms of $U$ can be written as

$dU=TdS+\sum_{i}\Omega_{i}dJ_{i}+\Phi dQ-Pd\bar{V}.$ (99)

It is easy to show that

$\displaystyle\frac{\partial U}{\partial r_{+}}$
$\displaystyle=T\frac{\partial S}{\partial
r_{+}}+\sum_{i}\Omega_{i}\frac{\partial J_{i}}{\partial
r_{+}}+\Phi\frac{\partial Q}{\partial r_{+}}-P\frac{\partial\bar{V}}{\partial
r_{+}},$ $\displaystyle\frac{\partial U}{\partial q}$
$\displaystyle=T\frac{\partial S}{\partial q}+\sum_{i}\Omega_{i}\frac{\partial
J_{i}}{\partial q}+\Phi\frac{\partial Q}{\partial
q}-P\frac{\partial\bar{V}}{\partial q},$ $\displaystyle\frac{\partial
U}{\partial g}$ $\displaystyle=T\frac{\partial S}{\partial
g}+\sum_{i}\Omega_{i}\frac{\partial J_{i}}{\partial g}+\Phi\frac{\partial
Q}{\partial g}-P\frac{\partial\bar{V}}{\partial g}.$ (100)

When we vary with respect to $c=a,b$ we find

$\frac{\partial U}{\partial c}=T\frac{\partial S}{\partial
c}+\sum_{i}\Omega_{i}\frac{\partial J_{i}}{\partial c}+\Phi\frac{\partial
Q}{\partial c}-P\frac{\partial\bar{V}}{\partial c}+\frac{\partial
M_{\text{BG}}}{\partial c}.$ (101)

Eqs. (6.3) and (101) can be combined in the form

$dU=TdS+\sum_{i}\Omega_{i}dJ_{i}+\Phi dQ-
Pd\bar{V}+\delta_{\sigma}M_{\text{BG}},$ (102)

where the variation $\delta_{\sigma}$ is understood to be w.r.t. to $a$ and
$b$ but not $g$.

This completes our investigation of the extended thermodynamics of the general
charged rotating black hole in five dimensions where we saw that Smarr’s
relation as well as the first law are satisfied using the counterterms method.

##  7 Conclusion

We used the counterterms subtraction method to calculate various physical
quantities of the charged rotating black holes in AdS5 introduced in [1]. We
showed that the resulting quantities satisfy the known thermodynamic relations
in the cases of a varying and fixed cosmological constant, i.e., in extended
and regular thermodynamics. All these quantities satisfy the Gibbs-Duhem
relation, the first law, and Smarr’s relation.

Quantum corrections break conformal symmetries in four-dimensional field
theories, producing conformal anomalies. This classical symmetry on the
boundary is realized as a subset of the bulk diffeomorphisms which is called
the Penrose-Brown-Hennaux (PBH) transformation [14, 15]. For boundary field
theories with a non-vanishing conformal anomaly, a PBH transformation will not
leave the gravitational on-shell action invariant. As was generally argued in
[2], this variation affects only the mass, but not the other quantities in the
first law. This is because these other quantities can be written as integrals
over the horizon.

We showed that the mass calculated from the counterterms method can be written
as a sum of two terms, one containing the mass parameter and electric charge,
and another which is nothing but the background energy of the spacetime. The
background energy is not restricted to the black hole region and its flux
depends on the boundary. An important boundary condition for the AdS/CFT
correspondence (or for the variation problem in [2]) is to keep the boundary
metric fixed by utilizing a PBH or boundary Weyl transformation. As a result,
one can use a Weyl transformation to cancel the variation in the mass term
that results from varying the boundary metric. This leaves the first law
satisfied.

In this work we used the counterterms subtraction method to calculate the
renormalized on-shell action for the general rotating charged AdS solution
presented in [1], as well as its mass and angular momenta. We showed that
these quantities satisfy the first law and the Gibbs-Duhem relation. Going to
extended thermodynamics [16, 17] where we allow the cosmological constant to
vary and act as a pressure, a naive calculation of first law shows that it is
not satisfied. We showed that a similar issue exists with the volume defined
in the extended-thermodynamics treatment [16, 17], meaning varying this
quantity does not leave the boundary metric fixed and one needs to use a
compensating PBH term to keep the volume fixed. This modification is important
to satisfy the first law in this case as well as the generalized Smarr’s
formula when the counterterms subtraction method is used in the presence of a
conformal anomaly.

We calculated the renormalized stress tensor and conformal anomaly of the CFT
living on the boundary as predicted by the AdS/CFT duality, i.e., from the
gravity side. We showed that these quantities coincide with the quantities
calculated on the field theory side on a rotating Einstein Universe.
Furthermore, we calculated the Casimir energy in the CFT and verified that it
exactly matches the background energy of the bulk theory.

Finally, we showed that the calculation of the thermodynamic volume induces a
Weyl transformation on the boundary metric, which adds extra terms to the
calculated expression. We have generalized the definition of the thermodynamic
volume in a way that accounts for these extra terms in the presence of a
conformal anomaly. We have shown that our definition leads to a volume term
that satisfies Smarr’s relation as well as the first law in extended
thermodynamics when the counterterms method is used. It would be interesting
to use the quantities calculated here to study various phase transitions for
the five-dimensional charged rotating solution [1], which are expected to have
a rich structure.

## Acknowledgments

This work was partially supported by research support grant
RSG1-2018-FY18-FY19 from the American University in Cairo;
id:10.13039/501100009229.

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