# Rotating hairy black holes and thermodynamics from gravitational decoupling

Subhash Mahapatra, Indrani Banerjee

Department of Physics and Astronomy, National Institute of Technology
Rourkela, Rourkela - 769008, India
mahapatrasub<EMAIL_ADDRESS>
###### Abstract

We study the method of extended gravitational decoupling in obtaining static
black hole solutions satisfying Einstein’s equations with a tensor vacuum. The
source has quite generic characteristics and satisfies the strong energy
condition. The stationary, axisymmetric counterpart of the static metric is
obtained by applying the Newman-Janis and Azreg-Aïnou algorithms. The
thermodynamics of the rotating solution is studied and the expressions of
various thermodynamic quantities are derived. The dependence of the
temperature, free energy and specific heat on the horizon radius is studied
for various values of the hairy parameter and the black hole spin. Such a
study reveals that small hairy black holes are thermodynamically more stable
compared to large hairy black holes, and that the horizon radius and
temperature range for which the rotating hairy black holes can be in
thermodynamic equilibrium with the surroundings depends non-trivially on the
hairy parameters. We further discuss the first law of black hole
thermodynamics for the hairy case and discuss its implications.

##  1 Introduction

Black holes, one of the most intriguing predictions of general relativity,
have received ample observational confirmations over the past five decades.
This includes indirect observations in the electromagnetic domain from
satellites like RXTE, Chandra, NuStar, etc as well as direct detections like
gravitational waves from the LIGO/VIRGO collaboration [1, 2, 3] and the black
hole image by the Event Horizon Telescope (EHT) [4, 5, 6, 7, 8, 9]. This
proves that black holes are not only objects of theoretical interest but also
have observational relevance. The celebrated no-hair theorem [10, 11, 12, 13,
14, 15, 16, 17, 18] propounds that black holes should not carry any charge
other than their mass, angular momentum and electric charge. However, the no-
hair theorem has not been proved in the rigorous mathematical sense and there
exist several counterexamples where the no-hair theorem is violated, see [19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42] for a necessarily incomplete selection. In fact, black holes
with scalar hairs have been extensively explored as scalar fields play a
pivotal role in particle physics and early universe cosmology [43, 44] and are
often invoked to explain the dark sector [45, 46]. Further, black holes can
harbour charges associated with inner gauge symmetries and they are also known
to possess soft quantum hair [47]. With the recent release of black hole
images and the discovery of gravitational waves, the scope to test the no-hair
theorem has been further enhanced [41, 42].

Apart from introducing various fields, one can evade the no-hair theorem by
other means as well. This includes exploring the method of minimal [48, 49]
and extended gravitational decoupling [49, 50, 51, 52, 53]. In this procedure,
one considers a known solution of Einstein’s equations with a given source and
then considers including the additional source that in turn modifies the
metric. The Schwarzschild metric is generally considered to be the known
background. In the case of minimal geometric deformation, one modifies only
the $g_{rr}$ component of the metric [48, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76], whereas in the
extended gravitational decoupling both the $g_{rr}$ and the $g_{tt}$
components of the metric are modified from the Schwarzschild solution. The
modification is done in such a way that the additional source satisfies
Einstein’s equations and the conservation equation with the deformed metric.
In this method, Einstein’s equations with the additional source completely
decouple from Einstein’s equations in the vacuum which is solved to obtain the
deformed metric. In order to obtain the deformed metric, apart from the
Einstein’s equations the additional source needs to satisfy certain other
conditions: (i) the strong energy condition, (ii) should well behave
everywhere outside the horizon and not lead to any additional singularity, and
(iii) the corresponding hairy metric should possess a well-defined event
horizon.

With these conditions, novel static, spherically symmetric and asymptotically
flat metrics were derived by the method of extended gravitational decoupling
in [52]. In particular, the obtained hairy solutions were not only regular
with well-defined event horizons but also satisfied physically relevant and
important energy conditions everywhere outside the event horizon. These
solutions were constructed in terms of two new parameters $\\{\alpha,\ell\\}$
(associated with the hair), which could greatly modify the properties of the
black hole. For instance, the static hairy solution can become extremal (as
opposed to the Schwarzschild case), thereby eminently changing the thermal
properties of the black hole. Similarly, the combination $\ell_{0}=\alpha\ell$
quantifies the increase in the black hole entropy from its Schwarzschild
value. These observations indicate substantial differences in the
thermodynamic as well as structural properties of the hairy black holes
compared to the Schwarzschild case. The hairy black hole solutions thus
obtained can also have interesting theoretical implications and observational
consequences [77, 52, 53]. Let us also mention that, progress also has been
made in recent years to put constraints on the hairy parameters from
experimental observations. The EHT observations related to shadows have been
used to set the lower limits for $\ell$ for a given choice of $\alpha$. For
instance, for $\alpha=1.0$, the lower limits are $\ell>0.35M$ ($1\sigma$) and
$\ell>0.15M$ ($2\sigma$) [78, 79].

Since astrophysical black holes are rotating in nature it is more realistic
and relevant to consider the stationary, axisymmetric counterpart of the
aforesaid deformed metric. This is generally accomplished by applying the
Newman-Janis algorithm [80] to the static, spherically symmetric seed metric.
The Newman-Janis algorithm has some ambiguities associated with the procedure
of complex coordinate transformation [81] which is circumvented by the method
suggested by Azreg-Aïnou which demands that the final rotating metric must be
written in the Boyer-Lindquist form and satisfy the gravitational field
equations [82, 83]. The stationary, axisymmetric solution can also be obtained
by the method of gravitational decoupling which has been discussed in [53].
Here the axisymmetric metric assumes the Gurses-Gursey form with a general
mass function $\tilde{m}(r)$. Interestingly, the Einstein tensor computed with
this metric depends linearly on the derivatives of $\tilde{m}(r)$. This
enables one to split $\tilde{m}(r)$ as $\tilde{m}(r)=m(r)+\alpha m_{s}(r)$
where $m(r)$ is associated with the axisymmetric seed metric while $m_{s}(r)$
represents the deformation in the mass function due to the presence of the
additional energy momentum tensor.

In this work, we perform this analysis and obtain the stationary, axisymmetric
counterpart of the gravitational decoupling inspired hairy black holes by
applying the Azreg-Aïnou formalism and find analytic expressions of rotating
hairy black holes satisfying various energy conditions outside the event
horizon. We provide a detailed discussion on the rotating solution for the
case in which the hair satisfies the strong energy condition, whereas a brief
but analogous and straightforward discussion for the case in which the hair
satisfies the dominant energy condition is presented in the Appendix. The
obtained solution makes sure that the source satisfies the energy conditions
(strong or dominant) everywhere outside the horizon. This is important
considering that it is always physically desirable to have the obtained
gravity solution with additional matter fields respecting the relevant energy
conditions. Moreover, we also find the form of complexified null tetrads which
leads to the same rotating hairy solution, however now, from the standard
Newman-Janis procedure.

Once the stationary, axisymmetric counterpart of the seed metric is obtained,
we study the horizon structure and the extremality conditions associated with
the rotating black hole. We next explore the curvature invariants like the
Ricci scalar and the Kretschmann scalar which are finite everywhere except at
the origin at $r=0$. Finally, we investigate the thermodynamics of the
rotating black hole solution thus derived. We Euclideanize the metric and
present the expressions for entropy, temperature and angular velocity. We
further find the expression of quasilocal energy in the slow rotating
approximation and find corrections due to hair. At the horizon, the quasilocal
energy is found to be (twice) the irreducible mass, whereas asymptotically it
reduces to the Arnowitt-Deser-Misner (ADM) mass. The variation of the Hawking
temperature with the horizon radius is presented for different values of spin
$a$ and parameters $\\{\alpha,\ell\\}$. This enables us to comment on the
stability of the black holes which can be further established when we analyze
the free energy and specific heat of the system. Our analysis reveals that
small hairy black holes for which the temperature increases with the horizon
radius are more stable compared to large hairy black holes for which the
temperature decreases with the horizon radius. This result is further
confirmed by plotting the free energy and specific heat with the horizon
radius for various values of $\alpha$, $\ell$ and $J$. The specific heat is
positive for small black holes and negative for the larger ones thereby
establishing that small black holes are indeed thermodynamically more stable.
We further analyse the free energy in the fixed momentum canonical ensemble
and find that small black holes always have smaller free energy compared to
large black holes at a fixed temperature. Our study further suggests that the
horizon radius and temperature range for the stable rotating hairy black holes
depend non-trivially on the hairy parameters. Finally, we also present the
first law of black hole thermodynamics for the hairy case both for rotating
and non-rotating black holes.

The paper is organized in the following way: In Section 2 we present the
static, spherically symmetric hairy black hole solutions obtained by the
method of extended gravitational decoupling on the seed Schwarzschild metric.
The stationary, axisymmetric counterparts of the static black hole solutions
are derived using the Newman-Janis method/Azreg-Aïnou approach in Section 2.3.
Section 3 is dedicated to investigating the thermodynamics of the aforesaid
black hole solution which enables us to understand the thermodynamic stability
of these black holes. Finally, we conclude with a summary of our results with
some scope for future work in 4.

##  2 A brief review of the static hairy solutions

In this section, we review the hairy black hole solutions obtained in [52]
using the extended gravitational decoupling (EGD) technique. The essential
idea of the EGD technique is to consider the seed solutions of the Einstein
field equations and make a consistent transformation of the metric functions
such that the total energy-momentum tensor separates out into subparts that
are then computationally more tractable. To see how this works, let us start
with Einstein field equation with source $\tau_{\mu\nu}$,

$\displaystyle
G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa^{2}{\tau}_{\mu\nu}\,,$
(2.1)

where $\kappa^{2}=8\pi G_{N}$, and $G_{N}$ is the Newton’s constant. We
consider the situation where the static, spherically symmetric solution of
(2.1) has been worked out and is given by,

$\displaystyle
ds^{2}=e^{g_{0}(r)}dt^{2}-e^{f_{0}(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,,$ (2.2)

where the form of $g_{0}(r)$ and $f_{0}(r)$ are known and the above metric is
termed as the seed metric.

We next consider adding new fields or hair in the system such that the total
energy momentum tensor becomes,

$\displaystyle\tilde{\tau}_{\mu\nu}=\tau_{\mu\nu}+\chi_{\mu\nu}\,,$ (2.3)

When the source term $\chi_{\mu\nu}$ associated with the hair is added, the
new static and spherically symmetric solution of Einstein’s equations assumes
the form,

$\displaystyle
ds^{2}=e^{g(r)}dt^{2}-e^{f(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,.$ (2.4)

The above metric is obtained by extended geometric deformation of the seed
metric such that,

$\displaystyle g(r)=g_{0}(r)+\alpha\mathcal{G}(r)\,,$ $\displaystyle
e^{-f(r)}=e^{-f_{0}(r)}+\alpha\mathcal{F}(r)\,,$ (2.5)

where $\mathcal{G}(r)$ and $\mathcal{F}(r)$ are respectively associated with
the geometric deformation of the $g_{tt}$ and $g_{rr}$ components of the seed
metric while $\alpha$ is a parameter which keeps track of the deformation.
Now, for a general static and spherically symmetric metric of the form (2.4),
the Einstein equation reduces to

$\displaystyle\kappa^{2}\left(\tau_{0}^{\ 0}+\chi_{0}^{\
0}\right)=\frac{1}{r^{2}}-e^{-f(r)}\left(\frac{1}{r^{2}}-\frac{f^{\prime}(r)}{r}\right)\,,$
$\displaystyle\kappa^{2}\left(\tau_{1}^{\ 1}+\chi_{1}^{\
1}\right)=\frac{1}{r^{2}}-e^{-f(r)}\left(\frac{1}{r^{2}}+\frac{g^{\prime}(r)}{r}\right)\,,$
$\displaystyle\kappa^{2}\left(\tau_{2}^{\ 2}+\chi_{2}^{\
2}\right)=-\frac{e^{-f(r)}}{4}\left(2g^{\prime\prime}(r)+g^{\prime
2}(r)-f^{\prime}(r)g^{\prime}(r)+2\frac{g^{\prime}(r)-f^{\prime}(r)}{r}\right)\,,$
(2.6)

where the prime denotes the derivative with respect to $r$, and
$\tilde{\tau}_{3}^{\ 3}=\tilde{\tau}_{2}^{\ 2}$ due to the spherical symmetry.

The transformation in Eq. (2.5) ensures that the two sources $\tau_{\mu\nu}$
and $\chi_{\mu\nu}$ are decoupled completely such that the set of equations
containing the source $\chi_{\mu\nu}$ are given by,

$\displaystyle\kappa^{2}\chi_{0}^{\
0}=-\alpha\left(\frac{\mathcal{F}(r)}{r^{2}}+\frac{\mathcal{F}^{\prime}(r)}{r}\right)\,,$
$\displaystyle\kappa^{2}\chi_{1}^{\ 1}+\alpha
Z_{1}=-\alpha\mathcal{F}(r)\left(\frac{1}{r^{2}}+\frac{g^{\prime}(r)}{r^{2}}\right)\,,$
$\displaystyle\kappa^{2}\chi_{2}^{\ 2}+\alpha
Z_{2}=-\alpha\frac{\mathcal{F}(r)}{4}\left(2g^{\prime\prime}(r)+g^{\prime
2}(r)+\frac{2g^{\prime}(r)}{r}\right)-\alpha\frac{\mathcal{F}^{\prime}(r)}{4}\left(g^{\prime}(r)+\frac{2}{r}\right)\,,$
(2.7)

where

$\displaystyle Z_{1}=\frac{e^{-f_{0}(r)}\mathcal{G}^{\prime}(r)}{r}\,,$
$\displaystyle
Z_{2}=e^{-f_{0}(r)}\left(2\mathcal{G}^{\prime\prime}(r)+\mathcal{G}^{\prime
2}(r)+\frac{2\mathcal{G}^{\prime}(r)}{r}+2g_{0}^{\prime}(r)\mathcal{G}^{\prime}(r)-f_{0}^{\prime}(r)\mathcal{G}^{\prime}(r)\right)\,.$
(2.8)

Therefore, it might be possible to compute the deformed hairy metric solution
if the deformation functions $\mathcal{G}$ and $\mathcal{F}$ can be computed
from the tensor $\chi_{\mu\nu}$ using Eq. (2.7).

The conservation of the total energy momentum tensor can be written as,

$\displaystyle\tilde{\nabla}_{\sigma}\tilde{\tau}^{\sigma}_{\alpha}=0\,,$
$\displaystyle\tilde{\nabla}_{\sigma}(\tau^{\sigma}_{\alpha}+\chi^{\sigma}_{\alpha})=0\,,$
(2.9)

where the covariant derivative $\tilde{\nabla}$ is computed with the deformed
metric (2.4). Note that the energy momentum tensor $\tau^{\sigma\alpha}$ is
separately conserved

$\displaystyle\nabla_{\sigma}\tau^{\sigma}_{\alpha}=0\,,$ (2.10)

the covariant derivative being calculated with the seed metric (2.2).

In case of extended gravitational decoupling $\tau^{\sigma\alpha}$ is not
conserved with respect to the deformed metric and is given by,

$\displaystyle\tilde{\nabla}_{\sigma}\tau^{\sigma}_{\alpha}={\nabla}_{\sigma}\tau^{\sigma}_{\alpha}-\alpha\frac{\mathcal{G}^{\prime}(r)}{2}(\tau^{0}_{0}-\tau^{1}_{1})\delta^{1}_{\alpha}=\alpha\frac{\mathcal{G}^{\prime}(r)}{2}(\tau^{0}_{0}-\tau^{1}_{1})\delta^{1}_{\alpha}\,.$
(2.11)

It is important to note that for minimal gravitational decoupling
$\mathcal{G}(r)=0$ such that the energy momentum tensor $\tau^{\mu}_{\nu}$ is
also conserved with respect to the deformed metric as well (see (2.11)).

Applying the above procedure for the simplest case of interest, namely
$\tau_{\mu\nu}=0$ and $\chi_{\mu\nu}\neq 0$, gives us the Schwarzschild
solution as the seed metric having,

$\displaystyle e^{g_{0}(r)}=e^{-f_{0}(r)}=1-\frac{2M}{r}\,.$ (2.12)

The conditions $\tau_{\mu\nu}=0$ and $\chi_{\mu\nu}\neq 0$ correspond to the
physical situation where the Schwarzschild vacuum is filled with a tensor
vacuum $\chi_{\mu\nu}$. The corresponding hairy solution can be obtained by
solving Eq. (2.7) for the two deformation functions $\mathcal{G}$ and
$\mathcal{F}$ for the given tensor vacuum $\chi_{\mu\nu}$.

For simplicity and to proceed further, one assumes that the deformed metric
satisfies the condition:

$\displaystyle e^{g(r)}=e^{-f(r)}\,.$ (2.13)

The above condition when applied in Eq. (2.6) immediately gives rise to the
following equation of state:

$\displaystyle\tilde{p}_{r}=-\tilde{\rho}\,.$ (2.14)

Using Eqs. (2.13) and (2.14) one obtains the following relation between the
deformation functions,

$\displaystyle\alpha\mathcal{F}(r)=\left(1-\frac{2M}{r}\right)\left(e^{\alpha\mathcal{G}(r)}-1\right)\,,$
(2.15)

such that the line element of the deformed metric (2.4) becomes

$\displaystyle
ds^{2}=\left(1-\frac{2M}{r}\right)e^{\alpha\mathcal{G}(r)}dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}e^{-\alpha\mathcal{G}(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,.$ (2.16)

Therefore, one is required to find the deformation function $\mathcal{G}$ and
the three components of $\chi_{\mu\nu}$, which must satisfy the Eq. (2.7). To
proceed further, we need to impose physically motivated constraints on
$\mathcal{G}(r)$ and $\chi_{\mu\nu}$. As in [52], we now discuss several
solutions of (2.16) based on different imposed conditions.

###  2.1 Kiselev black hole

The gravitational decoupling procedure can also be used to generate the
Kiselev black hole solution. For instance, let us apply the following
condition on the perturbed energy momentum tensor:

$\displaystyle\chi^{0}_{0}=a\chi^{1}_{1}+b\chi^{2}_{2}\,,$ (2.17)

where $a$ and $b$ are constants. Using the above condition in (2.7) one
obtains the following form of the deformed metric given in (2.16),

$\displaystyle
ds^{2}=\bigg{(}1-\frac{2\tilde{M}}{r}+\frac{l^{n}}{r^{n}}\bigg{)}dt^{2}-\bigg{(}1-\frac{2\tilde{M}}{r}+\frac{l^{n}}{r^{n}}\bigg{)}^{-1}dr^{2}-r^{2}d\Omega^{2}\,,$
(2.18)

where $\tilde{M}=M+\frac{l_{0}}{2}$, $n=\frac{2}{b}(a-1)$, while $l_{0}$ and
$l$ are constants proportional to $\alpha$. Imposing the dominant energy
condition on the energy momentum tensor and invoking asymptotic flatness
demand that $1\leq n\leq 2$. When $n=1$, one retrieves the Schwarzschild
solution while $n=2$ yields a metric similar to the Reissner Nordström
solution with traceless $\chi_{\mu\nu}$. The horizon radius is obtained by
solving for $g^{rr}(r=r_{h})=0$, which in the present case requires one to
solve for $r_{h}$ in the equation,

$\displaystyle r_{h}^{n}-2\tilde{M}r_{h}^{n-1}+l^{n}=0\,.$ (2.19)

###  2.2 Hairy black hole with strong energy condition

Another route to obtain solution for (2.15) is given by imposing strong energy
condition on $\chi_{\mu\nu}$ in the region of spacetime accessible to an
observer outside the horizon,

$\displaystyle\tilde{\rho}+\tilde{p}_{r}+2\tilde{p}_{\theta}\geq
0,~{}~{}~{}\tilde{\rho}+\tilde{p}_{r}\geq
0,~{}~{}~{}\tilde{\rho}+\tilde{p}_{\theta}\geq 0\,.$ (2.20)

Then the conditions in (2.13) and (2.14) imply,

$\displaystyle\chi_{2}^{\ 2}\leq 0,~{}~{}~{}\chi_{0}^{\ 0}\geq\chi_{2}^{\
2}\,.$ (2.21)

Together with Eq. (2.7), the above two constraint conditions can be
reformulated as,

$\displaystyle S_{1}(r)\equiv(r-2M)h^{\prime\prime}(r)+2h^{\prime}(r)\geq
0\,,$ $\displaystyle S_{2}(r)\equiv
r(r-2M)h^{\prime\prime}(r)+4Mh^{\prime}(r)-2h(r)+2\geq 0\,,$ (2.22)

where $h(r)=e^{\alpha\mathcal{G}(r)}$ has been used. Demanding the black hole
solution with a proper horizon at $r\sim 2M$ and that the spacetime solution
approaches the standard Schwarzschild solution in the asymptotically flat
$r\gg 2M$ regime, the following form of the deformed metric is obtained,

$\displaystyle ds^{2}=\bigg{(}1-\frac{2\mathcal{M}}{r}+\alpha
e^{\frac{-r}{\mathcal{M}-\alpha\ell/2}}\bigg{)}dt^{2}-\bigg{(}1-\frac{2\mathcal{M}}{r}+\alpha
e^{\frac{-r}{\mathcal{M}-\alpha\ell/2}}\bigg{)}^{-1}dr^{2}-r^{2}d\Omega^{2}\,,$
(2.23)

where the shift mass term $\mathcal{M}=M+\alpha\ell/2$, due to black hole hair
is introduced. This is the mass that will be observed by an asymptotic
observer at infinity. The effective density and tangential pressure are
respectively given by,

$\displaystyle\tilde{\rho}=-\tilde{p}_{r}=\chi_{0}^{\ 0}=\frac{\alpha
e^{-r/M}}{\kappa^{2}Mr^{2}}(r-M)\,,$
$\displaystyle\tilde{p}_{\theta}=-\chi_{2}^{\ 2}=\frac{\alpha
e^{-r/M}}{2\kappa^{2}Mr^{2}}(r-2M)\,.$ (2.24)

The horizon radius can be obtained by solving for $g^{rr}=0$ which leads to,

$\displaystyle r_{h}-2\mathcal{M}+\alpha r_{h}e^{-r_{h}/M}=0\,.$ (2.25)

For the horizon radius $r_{h}\geq 2M$, the hairy solution satisfies the strong
energy condition. Moreover, consideration of the extremal case, i.e.,
$r_{h}=2M$ in (2.25) provides a lower bound on $\ell$ which is given by,

$\displaystyle\ell\geq\frac{2M}{e^{2}}\,.$ (2.26)

From the above discussion, it is therefore clear that the metric solution
obtained in Eq. (2.23) corresponds to a consistent hairy black hole solution
with parameter $\\{M,\alpha,\ell\\}$. The parameter $\ell$ denote the charge
that can be associated with the primary hair. With the hairy black holes in
hand, we can now try to find the axisymmetric counterpart of this solution. In
the process, we will also generate the axisymmetric counterpart of the Kiselev
black hole (2.18).

###  2.3 Generating rotating solutions from the static metric

In this section, we generate stationary, axisymmetric solutions of the
gravitational field equations from a given static, spherically symmetric
solution of the same. The Newman-Janis algorithm is generally used for this
purpose. However, there is an ambiguity in the algorithm regarding the
complexification procedure which has been discussed and addressed in [81]. In
this paper, we follow the method suggested by [82] to generate the rotating
metric. We consider the following static metrics discussed in the last section
as the seed metric on which we apply the complex coordinate transformation,

$\displaystyle
ds^{2}=\bigg{(}1-\frac{2\tilde{M}}{r}+\frac{l^{n}}{r^{n}}\bigg{)}dt^{2}-\frac{dr^{2}}{\bigg{(}1-\frac{2\tilde{M}}{r}+\frac{l^{n}}{r^{n}}\bigg{)}}-r^{2}d\Omega^{2}\,,$
(2.27) $\displaystyle ds^{2}=\bigg{(}1-\frac{2\mathcal{M}}{r}+\alpha
e^{-r/(\mathcal{M}-\alpha
l/2)}\bigg{)}dt^{2}-\frac{dr^{2}}{\bigg{(}1-\frac{2\mathcal{M}}{r}+\alpha
e^{-r/(\mathcal{M}-\alpha l/2)}\bigg{)}}-r^{2}d\Omega^{2}\,.$ (2.28)

To see how the algorithm works we consider a general static metric,

$\displaystyle
ds^{2}=e^{2\nu(r)}dt^{2}-e^{2\lambda(r)}dr^{2}-\gamma(r)(d\theta^{2}+sin^{2}\theta
d\phi^{2})\,.$ (2.29)

We next write the metric in advanced null coordinates,

$\displaystyle
ds^{2}=e^{2\nu(r)}du^{2}+2e^{\nu(r)+\lambda(r)}dudr-\gamma(r)(d\theta^{2}+sin^{2}\theta
d\phi^{2})\,.$ (2.30)

The contravariant components of the metric in (2.30) can be written in terms
of null tetrads,

$\displaystyle
g^{\mu\nu}=l^{\mu}n^{\nu}+l^{\nu}n^{\mu}-m^{\mu}\bar{m}^{\nu}-m^{\nu}\bar{m}^{\mu}\,,$
(2.31)

where,

$\displaystyle l^{\mu}$ $\displaystyle=\delta^{\mu}_{r}\,,$ (2.32)
$\displaystyle n^{\mu}$
$\displaystyle=e^{-\lambda-\nu}\delta^{\mu}_{u}-\frac{1}{2}e^{-2\lambda}\delta^{\mu}_{r}\,,$
(2.33) $\displaystyle m^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2\gamma(r)}}\bigg{(}\delta^{\mu}_{\theta}+\frac{i}{sin\theta}\delta^{\mu}_{\phi}\bigg{)}\,,$
(2.34)

such that,

$\displaystyle
l^{\mu}l_{\mu}=n^{\mu}n_{\mu}=m^{\mu}m_{\mu}=0,~{}~{}~{}~{}~{}l^{\mu}n_{\mu}=-m^{\mu}\bar{m}_{\mu}=1,~{}~{}~{}~{}l^{\mu}m_{\mu}=n^{\mu}m_{\mu}=0\,.$
(2.35)

Let us consider now a complex coordinate transformation,

$\displaystyle x^{\prime\mu}=x^{\mu}+iy^{\mu}(x^{\nu})\,.$ (2.36)

A special case of the above transformation is known as the Newman-Janis
transformation, which is given by,

$\displaystyle r^{\prime}=r+iacos\theta;~{}~{}~{}u^{\prime}=u-iacos\theta\,.$
(2.37)

Denoting the null tetrads by the shorthand,

$\displaystyle V^{\mu}_{a}\equiv(l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu})\,,$
(2.38)

we then transform the null tetrads according to Newman-Janis transformation,
such that,

$\displaystyle V^{\prime\mu}_{a}=\frac{\partial x^{\prime\mu}}{\partial
x^{\nu}}V^{\nu}_{a}\,.$ (2.39)

After the Newman-Janis transformation the null tetrads assume the form,

$\displaystyle l^{\prime\mu}$ $\displaystyle=\delta^{\mu}_{r}\,,$
$\displaystyle n^{\prime\mu}$
$\displaystyle=\sqrt{\frac{\Upsilon(a,r,\theta)}{\Gamma(a,r,\theta)}}\delta^{\mu}_{u}-\frac{\Upsilon(a,r,\theta)}{2}\delta^{\mu}_{r}\,,$
$\displaystyle m^{\prime\mu}$
$\displaystyle=\frac{1}{\sqrt{2\Phi(a,r,\theta)}}\bigg{[}\delta^{\mu}_{\theta}+ia\sin\theta(\delta^{\mu}_{u}-\delta^{\mu}_{r})+\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\bigg{]}\,,$
(2.40)

such that there are the following restrictions on $\Upsilon$, $\Gamma$ and
$\Phi$,

$\displaystyle\lim_{a\to
0}\Upsilon(a,r,\theta)=e^{-2\lambda(r)},~{}~{}~{}\lim_{a\to
0}\Gamma(a,r,\theta)=e^{2\nu(r)},~{}~{}~{}\lim_{a\to
0}\Phi(a,r,\theta)=\gamma(r)\,.$ (2.41)

The original Newman-Janis approach has a certain level of ambiguity associated
with the complexification of $r$, which in turn determines the forms of
$\Upsilon$, $\Gamma$ and $\Phi$. The complexification procedure is completely
arbitrary and requires clever guess work. Consequently, the resultant
axisymmetric solution thus obtained often cannot be cast to the Boyer-
Lindquist form or fail to satisfy the original field equations. This is
because $\Gamma(r,\theta,a),\Upsilon(r,\theta,a),\Phi(r,\theta,a)$ are fixed
by the complexification procedure and there remain no free parameters or
functions to act on to achieve the transformation to Boyer-Lindquist
coordinates.

Since it is difficult to guess the right complexification procedure before
hand, Azreg-Aïnou proposed to keep $\Upsilon$, $\Gamma$ and $\Phi$ as general
functions of $r$ and $\theta$. This should be contrasted with the original
Newman-Janis algorithm where $\Upsilon$, $\Gamma$ and $\Phi$ respectively are
derived from $e^{-2\lambda(r)}$, $e^{2\nu(r)}$ and $\gamma(r)$. Demanding that
the metric must be cast in the Boyer-Lindquist form which fixes $\Upsilon$ and
$\Gamma$ in terms of the components of the static seed metric. The form of
$\Phi$ is fixed by demanding that the axisymmetric metric thus obtained is a
solution of the original gravitational field equations. Below we explicitly
show how the algorithm proposed by Azreg-Aïnou works in generating stationary
axisymmetric solutions from a static seed metric.

Using (2.40) we can construct the stationary, axisymmetric form of the metric
in advanced null coordinates $(u,r,\theta,\phi)$,

$\displaystyle g^{uu}(r,\theta)$
$\displaystyle=-\frac{a^{2}\sin^{2}\theta}{\Phi}\,,~{}~{}~{}~{}~{}~{}g^{ur}(r,\theta)=\sqrt{\frac{\Upsilon}{\Gamma}}+\frac{a^{2}\sin\theta}{\Phi}\,,~{}~{}~{}~{}~{}~{}g^{u\phi}(r,\theta)=-\frac{a}{\Phi}\,,$
$\displaystyle g^{rr}(r,\theta)$
$\displaystyle=-\Upsilon-\frac{a^{2}\sin^{2}\theta}{\Phi}\,,~{}~{}~{}~{}~{}~{}g^{r\phi}(r,\theta)=\frac{a}{\Phi}\,,$
$\displaystyle g^{\theta\theta}(r,\theta)$
$\displaystyle=-\frac{1}{\Phi}\,,~{}~{}~{}~{}~{}~{}g^{\phi\phi}(r,\theta)=-\frac{1}{\Phi\sin^{2}\theta}\,.$
(2.42)

We invert (2.42) to obtain the covariant components of the metric in Eddington
Finkelstein coordinates,

$\displaystyle g_{uu}(r,\theta)$
$\displaystyle=\Gamma\,,~{}~{}~{}~{}~{}~{}g_{ur}(r,\theta)=\sqrt{\frac{\Gamma}{\Upsilon}}\,,~{}~{}~{}~{}~{}~{}g_{u\phi}(r,\theta)=a\sin^{2}\theta\bigg{(}\sqrt{\frac{\Gamma}{\Upsilon}}-\Gamma\bigg{)}\,,$
$\displaystyle g_{r\phi}(r,\theta)$
$\displaystyle=-a\sin^{2}\theta\sqrt{\frac{\Gamma}{\Upsilon}}\,,~{}~{}g_{\theta\theta}(r,\theta)=-\Phi\,,~{}~{}g_{\phi\phi}(r,\theta)=-\sin^{2}\theta\biggl{[}\Phi+a^{2}\sin^{2}\theta\bigg{(}2\sqrt{\frac{\Gamma}{\Upsilon}}-\Gamma\bigg{)}\biggr{]}\,.$
(2.43)

In order to write the rotating solution in the Boyer-Lindquist form, we
consider the coordinate transformation where,

$\displaystyle
du=dt-\frac{(K+a^{2})dr}{e^{-2\lambda}\gamma+a^{2}};~{}~{}~{}~{}~{}~{}d\phi=d\varphi-\frac{adr}{e^{-2\lambda}\gamma+a^{2}}\,,$
(2.44)

where,

$\displaystyle K(r)=\gamma(r)e^{-\nu(r)-\lambda(r)}\,.$ (2.45)

In the Boyer-Lindquist coordinates, the axisymmetric metric has only one off-
diagonal term i.e, $g_{t\phi}$. Therefore, after applying the coordinate
transformation (2.44), we put the other off-diagonal components to zero. From
this we obtain the expression for $\Upsilon(r,\theta)$ and $\Gamma(r,\theta)$
as,

$\displaystyle\Gamma(r,\theta)=\frac{(e^{-2\lambda(r)}\gamma(r)+a^{2}\cos^{2}\theta)\Phi(r,\theta)}{(K(r)+a^{2}cos^{2}\theta)^{2}}\,,$
$\displaystyle\Upsilon(r,\theta)=\frac{e^{-2\lambda(r)}\gamma(r)+a^{2}cos^{2}\theta}{\Phi(r,\theta)}\,.$
(2.46)

Using (2.44) and (2.46), we obtain the final form of the stationary,
axisymmetric metric in Boyer-Lindquist coordinates,

$\displaystyle
ds^{2}=\Gamma(r,\theta)dt^{2}-\frac{\Phi(r,\theta)}{e^{-2\lambda(r)}\gamma(r)+a^{2}}dr^{2}+2a\sin^{2}\theta\bigg{[}\frac{K(r)-e^{-2\lambda(r)}\gamma(r)}{\left(K(r)+a^{2}cos^{2}\theta\right)^{2}}\bigg{]}\Phi(r,\theta)dtd\varphi$
$\displaystyle-\Phi(r,\theta)d\theta^{2}-\Phi(r,\theta)\sin^{2}\theta\bigg{[}1+a^{2}\sin^{2}\theta\frac{2K(r)+a^{2}\cos^{2}\theta-e^{-2\lambda(r)}\gamma(r)}{\left(k(r)+a^{2}cos^{2}\theta\right)^{2}}\bigg{]}d\varphi^{2}\,.$
(2.47)

The only unknown quantity in (2.47) is the functional form of
$\Phi(r,\theta)$. This is determined from the Einstein field equations with
source $\tilde{\tau}_{\mu\nu}$. It is important to note that if we compute the
Einstein tensor from (2.47), it has the non-zero off-diagonal component
$G_{r\theta}$. Since $\tilde{\tau}_{r\theta}$ is zero, therefore $G_{r\theta}$
should vanish to retain consistency with the Einstein field equations. This
gives rise to the following differential equation for $\Phi(r,\theta)$,

$\displaystyle\frac{\Phi^{2}}{\Sigma^{2}}(-48a^{2}r\sin
2\theta\Phi^{2}-24\Phi_{,r}\Phi_{,\theta}\Sigma^{2}+4\Phi_{,r\theta}\Sigma^{2})=0\,,$
(2.48)

where $\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}\theta$. Solving this equation
gives,

$\displaystyle\Phi(r,\theta)=r^{2}+a^{2}cos^{2}\theta=\Sigma(r,\theta)\,.$
(2.49)

Having obtained the form of $\Phi(r,\theta)$, we now substitute the form of
the metric components (2.27) and (2.28) in (2.45) and (2.46) to obtain their
corresponding rotating counterparts.

First of all we note that for our main cases of interest (Kiselev and hairy
black holes), the $g_{tt}$ and $g_{rr}$ components of the static black holes
are inverse of each other, i.e. $e^{2\nu}=e^{-2\lambda}$. Thus,

$\displaystyle K(r)=\gamma(r)\,.$ (2.50)

Moreover, we also have $\gamma(r)=r^{2}$. We now have all the ingredients to
write down the axisymmetric counterpart of static black holes in (2.27) and
(2.28). A general axisymmetric metric has the form,

$\displaystyle
ds^{2}=g_{tt}(r,\theta)dt^{2}+2g_{t\varphi}(r,\theta)dtd\varphi+g_{\varphi\varphi}(r,\theta)d\varphi^{2}+g_{rr}(r,\theta)dr^{2}+g_{\theta\theta}(r,\theta)d\theta^{2}\,.$
(2.51)

For the Kiselev static black hole metric in (2.27), we have

$\displaystyle g_{tt}(r,\theta)$
$\displaystyle=\frac{r^{2}\big{(}1-2\tilde{M}/r+l^{n}/r^{n}\big{)}+a^{2}\cos^{2}\theta}{\Sigma}\,,$
$\displaystyle g_{t\varphi}(r,\theta)$
$\displaystyle=\frac{ar^{2}\sin^{2}\theta}{\Sigma}\bigg{(}\frac{2\tilde{M}}{r}-\frac{l^{n}}{r^{n}}\bigg{)}\,,$
$\displaystyle g_{\varphi\varphi}(r,\theta)$
$\displaystyle=-\Sigma\sin^{2}\theta\bigg{[}1+a^{2}\sin^{2}\theta\frac{r^{2}(1+2\tilde{M}/r-l^{n}/r^{n})+a^{2}\cos^{2}\theta}{\Sigma^{2}}\bigg{]}\,,$
$\displaystyle g_{rr}(r,\theta)$
$\displaystyle=-\frac{\Sigma}{a^{2}+r^{2}\big{(}1-2\tilde{M}/r+l^{n}/r^{n}\big{)}}\,,$
$\displaystyle g_{\theta\theta}(r,\theta)$ $\displaystyle=-\Sigma\,.$ (2.52)

Similarly, for the static hairy black hole metric in (2.28), we have

$\displaystyle g_{tt}(r,\theta)$
$\displaystyle=\frac{r^{2}\big{(}1-2\mathcal{M}/r+\alpha
e^{-r/(\mathcal{M}-\alpha l/2)}\big{)}+a^{2}\cos^{2}\theta}{\Sigma}\,,$
$\displaystyle g_{t\varphi}(r,\theta)$
$\displaystyle=\frac{ar^{2}\sin^{2}\theta}{\Sigma}\bigg{(}\frac{2\mathcal{M}}{r}-\alpha
e^{-r/(\mathcal{M}-\alpha l/2)}\bigg{)}\,,$ $\displaystyle
g_{\varphi\varphi}(r,\theta)$
$\displaystyle=-\Sigma\sin^{2}\theta\bigg{[}1+a^{2}\sin^{2}\theta\frac{r^{2}(1+2\mathcal{M}/r-\alpha
e^{-r/(\mathcal{M}-\alpha l/2)})+a^{2}\cos^{2}\theta}{\Sigma^{2}}\bigg{]}\,,$
$\displaystyle g_{rr}(r,\theta)$
$\displaystyle=-\frac{\Sigma}{a^{2}+r^{2}\big{(}1-2\mathcal{M}/r+\alpha
e^{-r/(\mathcal{M}-\alpha l/2)}\big{)}}\,,$ $\displaystyle
g_{\theta\theta}(r,\theta)$ $\displaystyle=-\Sigma\,.$ (2.53)

Notice that all the metric coefficients, except the $g_{\theta\theta}$ term,
receive corrections due to hair. As expected, this new rotating hairy solution
reduces to the Kerr geometry for $\alpha=0$, and approaches the Schwarzschild
geometry asymptotically with mass $\mathcal{M}$.

The above stationary, axisymmetric solution can also be obtained by the method
of gravitational decoupling as done in [53]. In this method the axisymmetric
solution assumes the Gurses-Gursey form with a general mass function
$\tilde{m}(r)$. Interestingly, the Einstein tensor evaluated with the
aforesaid metric turns out to possess only linear derivatives of
$\tilde{m}(r)$, although it depends non-linearly on the spin parameter
$\tilde{a}$. Thus, the mass function can be split in the form
$\tilde{m}(r)=m(r)+\alpha m_{s}(r)$, where $m(r)$ represents the mass function
of the axisymmetric seed metric while the deformed metric corresponding to the
additional energy momentum tensor is obtained by adding $m_{s}(r)$ to $m(r)$.
The rotating hairy metric so obtained takes the same form as in Eq. (2.53).
These results for the hairy rotating metric from two seemingly different
methods not only emphasize the correctness of the obtained rotating solution
but also suggest that these two methods complement each other well to generate
rotating solutions in the present context.

At this point, to avoid any confusion, we would like to emphasize the
difference between the various mass parameters $\\{M,\mathcal{M},\tilde{M}\\}$
appearing in Eqs. (2.52) and (2.53). The mass parameter $M$ corresponds to the
usual Schwarzschild mass, i.e. of the non-hairy black hole. Whereas the
parameter $\mathcal{M}=M+\alpha\ell/2$ corresponds to the ADM mass of the
hairy black hole, which reduces to the Schwarzschild mass $M$ when the hairy
parameter $\alpha$ goes to zero. On the other hand, the parameter $\tilde{M}$
arises in the discussion of the Kiselev black only. Accordingly, it does not
appear and play any role in the discussion of the hairy black holes below.

Figure 1:  The behaviour of $\Delta(r)$ as a function of $r$ for different
values of $\alpha$. Here $\mathcal{M}=1$, $\ell=0.4$, $a=0.1$ are used. The
black, red, green, blue, brown and cyan curves correspond to $\alpha=0$,
$0.2$, $0.4$, $0.6$, $0.8$ and $1.0$ respectively.

Figure 2: The behaviour of $\mathcal{M}$ vs $a$ for the extremal case. Here
$\ell=0.4$ is used. The black, red, green, blue, brown and cyan curves
correspond to $\alpha=0$, $0.2$, $0.4$, $0.6$, $0.8$ and $1.0$ respectively.

It is interesting to analyse the geometric properties of the rotating hairy
solution and contrast them with the Kerr case. The horizon radius can now be
obtained by solving for $g^{rr}=0$, which leads to

$\displaystyle a^{2}+r_{h}^{2}-2\mathcal{M}r_{h}+\alpha
r_{h}^{2}e^{-r_{h}/(\mathcal{M}-\alpha\ell/2)}=0\,.$ (2.54)

Unfortunately, Eq. (2.54) cannot be solved analytically, however, it is
straightforward to solve it numerically. We find that, like the Kerr geometry,
there are two horizon radii: outer and inner horizons. Both these horizon
radii depend nontrivially on $\alpha$ and $\ell$. In particular, the inner
horizon radii always increase with $\alpha$, whereas the outer horizon radii
decrease with it. This is shown in Figure 2, where the points where
$\Delta(r)$ changes sign indicate the horizon. It is important to point out
one subtlety that one can also consider some particular values of $\ell$ for
which the outer horizon radii also increase with $\alpha$. However, we found
that for all such values the strong energy condition gets violated outside the
outer horizon. For the Kerr geometry, we must have $M=\mathcal{M}\geq a$ to
avoid the naked singularity. The equality condition $\mathcal{M}=a$
corresponds to the extremal case where the outer and inner horizon radii
become equal. We find that the $\mathcal{M}$ vs $a$ relation for the extremal
hairy case depends nontrivially on $\alpha$. In particular, the slope of $a$
vs $\mathcal{M}$ curve is always greater than one. This is illustrated in
Figure 2. This indicates that the maximum angular velocity that the hairy
black hole can attain to avoid the naked singularity depends nontrivially on
$\alpha$, and in particular, one should always have $\mathcal{M}>a$ in order
to mask the singularity for the hairy case.

Figure 3:  The profile of Ricci scalar for different values of $\alpha$ in the
plane $\theta=\pi/2$. Here $\mathcal{M}=1$, $\ell=0.4$, and $a=0.1$ are used.
The black, red, green, blue, brown and cyan curves correspond to $\alpha=0$,
$0.2$, $0.4$, $0.6$, $0.8$, and $1.0$ respectively.

Figure 4: The profile of Kretschmann scalar for different values of $\alpha$
in the plane $\theta=\pi/2$. Here $\mathcal{M}=1$, $\ell=0.4$, and $a=0.1$ are
used. The black, red, green, blue, brown and cyan curves correspond to
$\alpha=0$, $0.2$, $0.4$, $0.6$, $0.8$, and $1.0$ and respectively.

We can further evaluate the scalar invariants. The Ricci scalar is
proportional to $\Sigma^{-1}$ and has a simple form

$\displaystyle R=\frac{\alpha\left(2M^{2}-4Mr+r^{2}\right)}{M^{2}\Sigma}\,,$
(2.55)

whereas the Kretschmann scalar $R_{\mu\nu\rho\lambda}R^{\mu\nu\rho\lambda}$ is
proportional to $\Sigma^{-6}$. The exact expression of the Kretschmann scalar
is too long to reproduce here and is not very illuminating, therefore we skip
reproducing it here. The rotating hairy solution exhibits a ring singularity
in the plane $\theta=\pi/2$ at $r=0$. Importantly, there are no additional
singularities in the hairy case than those already present in the Kerr case.
Near $r=0$, both invariants have positive curvatures. This is shown in Figures
4 and 4. Interestingly, there can be special points in the hairy spacetime
where the Ricci scalar vanishes. This happens at $r=2M\pm\sqrt{2}M$, when the
numerator of (2.55) goes to zero. These points are moreover $\alpha$
independent, as is shown in Figure 4.

From the Einstein tensor, we can further evaluate the density and pressure
components of the energy momentum tensor in the rotating case. These are given
by,

$\displaystyle\tilde{\rho}=\frac{r^{2}\left(1-e^{-2\lambda}-r(e^{-2\lambda})^{\prime}\right)}{\kappa^{2}\Sigma^{2}}=\frac{\alpha
r^{2}e^{-\frac{r}{M}}}{\kappa^{2}M\Sigma^{2}}(r-M)\,,$
$\displaystyle\tilde{p}_{r}=-\tilde{\rho}\,,$
$\displaystyle\tilde{p}_{\theta}=-\tilde{p}_{r}+\frac{1}{\kappa^{2}\Sigma}\left(\frac{r^{2}(e^{-2\lambda})^{\prime\prime}}{2}+2r(e^{-2\lambda})^{\prime}+(e^{-2\lambda})-1\right)$
$\displaystyle~{}~{}~{}=\frac{\alpha
r^{2}e^{-\frac{r}{M}}}{\kappa^{2}M\Sigma^{2}}(r-M)+\frac{\alpha
e^{-\frac{r}{M}}}{2\kappa^{2}M^{2}\Sigma}(2M^{2}-4Mr+r^{2})\,,$
$\displaystyle\tilde{p}_{\varphi}=\tilde{p}_{\theta}\,.$ (2.56)

We can immediately notice that these expressions reduce to the standard static
expressions (2.24) when $a\rightarrow 0$ or $\theta=\pi/2$. The physical ring
singularity nature of the spacetime at $r=0$ is further reflected in the
density and pressure components. Notice that these components, written in
terms of ADM mass $\mathcal{M}$, depend on the parameter $\ell$. This allows
us to choose suitable values of hairy parameters $\alpha$ and $\ell$ such that
the strong energy condition is satisfied everywhere outside the horizon in the
rotating case as well. For example, for
$\\{\mathcal{M}=1,\alpha=0.2,a=0.1\\}$, we must have $\ell\gtrsim 0.3$ to
satisfy the strong energy condition. This is shown in Figure 5, where we can
explicitly see that both $\tilde{p}_{\theta}$ and
$\tilde{\rho}+\tilde{p}_{\theta}$ are positive outside the horizon. In
general, compared to the static spacetime, a higher value of $\ell$ is needed
to satisfy the strong energy condition when the rotation parameter $a$
increases.

Figure 5: The profile of $\tilde{p}_{\theta}$ and
$\tilde{\rho}+\tilde{p}_{\theta}$. Here
$\\{\mathcal{M}=1,\ell=0.3,\alpha=0.2\\}$ are used. The vertical dashed line
corresponds to the horizon radii.

Here we like to end this section by mentioning that the axisymmetric solution
can also be obtained using the standard Newman-Janis algorithm. However, this
algorithm involves a certain level of arbitrariness in the complexification of
null tetrads. Indeed, most of the time, there is no unique way of
complexifying all the terms in the tetrad in the same way. We can still,
however, generate a consistent rotating hairy metric in our case using the
Newman-Janis algorithm. This required choosing a particular form of the
complexified null tetrads. In Appendix B, we will briefly mention these
complexified null tetrads for the hairy black hole.

##  3 Thermodynamics of the rotating hairy solution

Having obtained the rotating solution of the hairy black hole, we now move on
to discuss its thermodynamic properties. For this purpose, we first recast the
metric elements of (2.53) in the following form

$\displaystyle
g_{tt}=\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\,,~{}~{}~{}g_{rr}=-\frac{\Sigma}{\Delta}\,,~{}~{}~{}g_{\theta\theta}=-\Sigma\,,$
$\displaystyle
g_{t\varphi}=\frac{a\sin^{2}\theta(r^{2}+a^{2}-\Delta)}{\Sigma}\,,~{}~{}~{}g_{\varphi\varphi}=-\frac{\sin^{2}\theta\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)}{\Sigma}\,,$
$\displaystyle\Delta(r)=a^{2}+r^{2}-2\mathcal{M}r+\alpha
r^{2}e^{-r/(\mathcal{M}-\alpha\ell/2)}\,.$ (3.1)

The Bekenstein-Hawking entropy of the black hole can be computed from the area
of the horizon, which for the above metric is given by

$\displaystyle S_{BH}=\frac{1}{4G_{N}}\int d\theta
d\varphi\sqrt{g_{\theta\theta}g_{\varphi\varphi}}=\int d\theta
d\varphi\sin\theta\sqrt{\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)}\,,$ (3.2)

since at the horizon $\Delta(r_{h})=0$, we get

$\displaystyle S_{BH}=\frac{\pi\left(r_{h}^{2}+a^{2}\right)}{G_{N}}\,.$ (3.3)

Notice that in $S_{BH}$ the hairy parameter $\alpha$ does not appear
explicitly, implying that the hairy black hole entropy expression remains the
same as in the non-hairy Kerr case. However, $S_{BH}$ does depend implicitly
on $\alpha$ through $r_{h}$.

Similarly, the temperature of the rotating black hole can be obtained from the
standard Euclidean procedure [84]. For this purpose, we follow the procedure
outlined in [85] and note that the above spacetime enjoys a pair of orthogonal
vectors

$\displaystyle\xi=\partial_{t}+\frac{a}{a^{2}+r^{2}}\partial_{\varphi}\,,~{}~{}~{}\tilde{\xi}=a\sin^{2}\theta\partial_{t}+\partial_{\varphi}\,.$
(3.4)

The vector $\xi$ is null on the horizon $\xi^{2}(r_{h})=0$ and is time-like
everywhere outside the horizon $r\geq r_{h}$, whereas the vector $\tilde{\xi}$
is spacelike for $r\geq r_{h}$. The corresponding one-forms dual to $\xi$ and
$\tilde{\xi}$ are

$\displaystyle d\omega=\frac{a^{2}+r^{2}}{\Sigma}(dt-a\sin^{2}\theta
d\varphi),~{}~{}~{}d\tilde{\omega}=\frac{a^{2}+r^{2}}{\Sigma}(-\frac{a}{a^{2}+r^{2}}dt+d\varphi)\,.$
(3.5)

We now consider the Euclideanization of the rotating hairy metric. The
standard procedure [84] suggests the change of time variable $t=i\tau$, along
with the transformation of the rotating parameter $a=i\hat{a}$. With these
transformations, the Euclidean vectors $\xi$ and $\tilde{\xi}$ and their one
forms $d\omega$ and $d\tilde{\omega}$ reduce to

$\displaystyle\xi=\partial_{\tau}-\frac{\hat{a}}{r^{2}-\hat{a}^{2}}\partial_{\varphi}\,,~{}~{}~{}\tilde{\xi}=\hat{a}\sin^{2}\theta\partial_{\tau}+\partial_{\varphi}\,,$
$\displaystyle
d\omega=\frac{r^{2}-\hat{a}^{2}}{\hat{\Sigma}}(d\tau-\hat{a}^{2}\sin^{2}\theta
d\varphi),~{}~{}~{}d\tilde{\omega}=\frac{r^{2}-\hat{a}^{2}}{\hat{\Sigma}}\left(\frac{\hat{a}}{r^{2}-\hat{a}^{2}}d\tau+d\varphi\right)\,,$
(3.6)

where $\hat{\Sigma}=r^{2}-\hat{a}^{2}\cos^{2}\theta$. The Euclidean metric now
simply reduces to

$\displaystyle
ds_{E}^{2}=-\frac{\hat{\Delta}\hat{\Sigma}}{\left(r^{2}-\hat{a}^{2}\right)^{2}}d\omega^{2}-\frac{\hat{\Sigma}}{\hat{\Delta}}dr^{2}-\hat{\Sigma}\left(d\theta^{2}+\sin^{2}\theta
d\tilde{\omega}^{2}\right)\,,$ (3.7)

where $\hat{\Delta}=r^{2}-\hat{a}^{2}-2\mathcal{M}r+\alpha
r^{2}e^{-r/(\mathcal{M}-\alpha\ell/2)}$, which also give the radius of the
horizon $r=\hat{r}_{h}$. Introducing a new radial coordinate $x$ near the
horizon

$\displaystyle\hat{\Delta}=\eta(r-\hat{r}_{h})=\frac{\eta^{2}x^{2}}{4}\,,$
$\displaystyle\eta=\frac{\hat{r}_{h}^{2}+\hat{a}^{2}}{\hat{r}_{h}}+\alpha\hat{r}_{h}e^{-\frac{\hat{r}_{h}}{\mathcal{M}-\alpha\ell/2}}\left(1-\frac{\hat{r}_{h}}{\mathcal{M}-\alpha\ell/2}\right)\,,$
(3.8)

the Euclidean metric (3.7) near the horizon takes the form

$\displaystyle
ds_{E}^{2}=-\hat{\Sigma}_{h}\left(dx^{2}+\frac{\eta^{2}x^{2}}{4(\hat{r}_{h}^{2}-\hat{a}^{2})^{2}}d\omega^{2}\right)-\hat{\Sigma}_{h}\left(d\theta^{2}+\sin^{2}\theta
d\tilde{\omega}^{2}\right)\,,$ (3.9)

where $\hat{\Sigma}_{h}=\hat{r}_{h}^{2}-\hat{a}^{2}\cos^{2}\theta$. On the
horizon surface $\mathcal{H}$, one can further introduce a well defined
coordinate $\psi=\varphi+\frac{\hat{a}\tau}{\hat{r}_{h}^{2}-\hat{a}^{2}}$,
giving metric on the horizon

$\displaystyle
ds_{\mathcal{H}}^{2}=\hat{\Sigma}_{h}\left(d\theta^{2}+\sin^{2}\theta
d\tilde{\omega}^{2}\right)=\hat{\Sigma}_{h}\left(d\theta^{2}+\frac{\left(\hat{r}_{h}^{2}-\hat{a}^{2}\right)^{2}}{\hat{\Sigma}_{h}^{2}}\sin^{2}\theta
d\psi^{2}\right)\,.$ (3.10)

Notice that the points $\psi$ and $\psi+2\pi$ should be identified on the
horizon $\mathcal{H}$ for the regularity of the metric (3.10) at points
$\theta=0$ and $\theta=\pi$. The metric (3.9) can be rewritten as

$\displaystyle
ds_{E}^{2}=-\hat{\Sigma}_{h}ds_{C_{2}}^{2}-ds_{\mathcal{H}}^{2}\,,$ (3.11)

where $ds_{C_{2}}^{2}$ is the metric on two dimensional disk $C_{2}$

$\displaystyle
ds_{C_{2}}^{2}=dx^{2}+\frac{\eta^{2}x^{2}}{4(\hat{r}_{h}^{2}-\hat{a}^{2})^{2}}d\omega^{2}\,.$
(3.12)

Considering the above metric with fixed $(\theta,\psi)$ and introducing a new
angle coordinate $\chi=\tau-\hat{a}\sin^{2}\theta\varphi$, the metric on disk
takes the form

$\displaystyle
ds_{C_{2}}^{2}=dx^{2}+\frac{\eta^{2}x^{2}}{4\hat{\Sigma}_{h}^{2}}d\chi^{2}=dx^{2}+x^{2}d\left(\frac{\eta}{2\hat{\Sigma}_{h}}\chi\right)^{2}\,.$
(3.13)

This metric takes the same form as a plane in polar coordinate if
$\eta/(2\hat{\Sigma}_{h})$ has the period $2\pi$. Otherwise, there is a
conical singularity at $x=0$. One must therefore identify the points $\chi$
and $\chi+4\pi\hat{\Sigma}_{h}\eta^{-1}$ to avoid the conical singularity.
Since this must be true independently of the coordinate $\theta$ on the
horizon $\mathcal{H}$, one must also identify the points $(\tau,\varphi)$ with
$(\tau+\beta_{h},\varphi-\Omega_{h}\beta_{h})$, where
$\beta_{h}=4\pi\hat{\Sigma}_{h}/\eta$ is inverse Hawking temperature of the
black hole and $\Omega=\hat{a}/(\hat{r}_{h}^{2}-\hat{a}^{2})$ is the angular
velocity. Now, analytically continuing the results back to the real values
$a$, we have the temperature and angular velocity of the hairy rotating black
hole as

$\displaystyle T_{h}=\frac{1}{\beta_{h}}=\frac{r_{h}^{2}-a^{2}+\alpha
r_{h}^{2}e^{-\frac{r_{h}}{\mathcal{M}-\alpha\ell/2}}\left(1-\frac{r_{h}}{\mathcal{M}-\alpha\ell/2}\right)}{4\pi
r_{h}\left(r_{h}^{2}+a^{2}\right)}\,,$
$\displaystyle\Omega_{h}=\frac{a}{r_{h}^{2}+a^{2}}\,.$ (3.14)

Note that the Hawking temperature gets a correction due to hair, and it
reduces to the standard Kerr temperature when the hair parameter $\alpha$ goes
to zero. The expression of angular velocity, like the entropy, does not depend
explicitly on $\alpha$ and remains the same as in the Kerr case. Therefore,
the angular velocity of the horizon is again given by
$\Omega_{h}=-g_{t\varphi}/g_{\varphi\varphi}$, as is expected. The angular
momentum of the black hole is similarly given by

$\displaystyle J=a\mathcal{M}\,.$ (3.15)

We now discuss the notion of quasilocal energy in the hairy rotating black
hole and its relation with the mass parameter $\mathcal{M}$. For this purpose,
we use the Brown-York definition of quasilocal energy [86]. To obtain it, one
first considers foliations of four-dimensional manifold $\mathcal{N}$ by
spacelike constant time hypersurfaces $\Gamma$. The spacetime is spatially
bounded by the three-dimensional surface $B^{(3)}$ ($r=$ constant surface),
and the intersection of $\Gamma$ with the boundary $B^{(3)}$ is a two-
dimensional surface $B^{(2)}$ with induced metric $\sigma_{ab}$, i.e.
$B^{(2)}$ is a $r=$ constant hypersurface within $\Gamma$. The quasilocal
energy is then given as the proper surface integral

$\displaystyle
E=\varepsilon-\varepsilon_{0}=\frac{1}{\kappa^{2}}\int_{B^{(2)}}\sqrt{\sigma}(\mathcal{K}-\mathcal{K}^{0})\,,$
(3.16)

where $\mathcal{K}$ is the trace of the extrinsic curvature of $B^{(2)}$ as
embedded in $\Gamma$ and $\mathcal{K}^{0}$ represents the trace of extrinsic
curvature for some reference spacetime 111The $\mathcal{K}$ here should not be
confused with the $K$ appeared in the previous section.. To compute the
quasilocal energy for asymptotically flat solutions, the reference space is
generally taken to be a flat three-dimensional slice $E^{(3)}$ of flat
spacetime. For our hairy black hole, the trace of the extrinsic curvature
$\mathcal{K}$ of $r=$ constant surface within the $t=$ constant hypersurface
is given by

$\displaystyle\mathcal{K}=-\sqrt{\frac{\Delta}{\Sigma}}\frac{\partial_{r}\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}{2\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}\,,$
$\displaystyle=-\frac{\sqrt{r^{2}-2\mathcal{M}r+a^{2}+\alpha
r^{2}e^{-\frac{r}{M}}}}{2\sqrt{r^{2}+a^{2}\cos^{2}\theta}}\frac{\left[4r(r^{2}+a^{2})-a^{2}\sin^{2}\theta\Delta^{\prime}(r)\right]}{\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}\,.$ (3.17)

Using $\sqrt{\sigma}=\sin\theta\sqrt{(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}$ for the induced metric on the two-sphere $B^{(2)}$, the
unreferenced Brown-York energy for the hairy rotating black hole within a
sphere of radius $r$ can be evaluated as

$\displaystyle\varepsilon=\frac{1}{8\pi}\int\sqrt{\sigma}\mathcal{K}=\frac{1}{8\pi}\int
d\theta d\varphi\sin\theta\sqrt{(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}\mathcal{K}\,,$
$\displaystyle=-\frac{r}{8}\sqrt{1-\frac{2\mathcal{M}}{r}+\frac{a^{2}}{r^{2}}+\alpha
e^{-\frac{r}{M}}}\int_{0}^{\pi}d\theta\sin\theta\frac{\left[4r(r^{2}+a^{2})-a^{2}\sin^{2}\theta\Delta^{\prime}(r)\right]}{\sqrt{\left(r^{2}+a^{2}\cos^{2}\theta\right)\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}}\,.$ (3.18)

Unfortunately, the above equation in general can not be integrated in closed
form. Therefore, one has to rely on some approximation for which the above
integration can be performed. A physically motivated and particularly
interesting approximation is a slow rotation limit i.e. $a/r\ll 1$, in which
case a closed form expression can be obtained. In this approximation, the
unreferenced Brown-York energy reduces to

$\displaystyle\varepsilon=-\frac{r}{2}\sqrt{1-\frac{2\mathcal{M}}{r}+\frac{a^{2}}{r^{2}}+\alpha
e^{-\frac{r}{M}}}\int_{0}^{\pi}d\theta\sin\theta\left[1-\frac{a^{2}\cos^{2}\theta}{2r^{2}}+\frac{a^{2}\sin^{2}\theta}{2r^{2}}\left(-\frac{\mathcal{M}}{r}+\frac{\alpha
re^{-\frac{r}{M}}}{2M}\right)\right]\,,$
$\displaystyle=-r\sqrt{1-\frac{2\mathcal{M}}{r}+\frac{a^{2}}{r^{2}}+\alpha
e^{-\frac{r}{M}}}\left[1-\frac{a^{2}}{6r^{2}}\left(1+\frac{2\mathcal{M}}{r}-\frac{\alpha
re^{-\frac{r}{M}}}{M}\right)\right]\,.$ (3.19)

Notice that for large $r$, we have $\varepsilon\rightarrow\mathcal{M}-r$,
which is divergent when $r\rightarrow\infty$. This suggests the need for a
subtraction term, i.e the quasilocal energy of a reference spacetime, to
renormalize the unreferenced quasilocal energy. Obtaining the Brown-York
energy for reference spacetime is, however, more difficult [87, 88, 89]. For
this one first needs to find a two-dimensional surface isometric to $B^{(2)}$,
which is consistently embedded in a three-dimensional flat space $E^{(3)}$. If
this $E^{(3)}$ is described by the coordinates
$\\{\mathcal{R},\Theta,\Phi\\}$, then

$\displaystyle
ds^{2}=d\mathcal{R}^{2}+\mathcal{R}^{2}d\Theta^{2}+\mathcal{R}^{2}\sin^{2}\Theta
d\Phi^{2}\,.$ (3.20)

The desired two-dimensional surface isometric to $B^{(2)}$ can be defined by
$\mathcal{R}=f(\Theta)$, where $f$ is a function of the azimuthal angle
$\Theta$. Its intrinsic metric or curvature can be evaluated from (3.20). For
this purpose, we first assume that $\Theta=\Theta(\theta)$ and $\Phi=\varphi$,
which makes $\mathcal{R}$ is a function of $\theta$ on the two-surface. The
line element on it takes the form

$\displaystyle
ds^{2}=\left[\dot{\mathcal{R}}^{2}+\mathcal{R}^{2}\dot{\Theta}^{2}\right]+\mathcal{R}^{2}\sin^{2}\Theta
d\Phi^{2}\,,$ (3.21)

where an overdot denotes derivative with respect to $\theta$. Requiring the
above line element to be isometric to (3.1) gives us the following coupled
differential equations for $\mathcal{R}$ and $\Theta$,

$\displaystyle\dot{\mathcal{R}}^{2}+\mathcal{R}^{2}\dot{\Theta}^{2}=r^{2}+a^{2}\cos^{2}\theta\,,$
$\displaystyle\mathcal{R}^{2}\sin^{2}\Theta=\frac{\sin^{2}\theta\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)}{r^{2}+a^{2}\cos^{2}\theta}\,.$ (3.22)

These two coupled equations need to be solved to get $\mathcal{R}$ and
$\Theta$ in terms of $\theta$. Unfortunately, these equations are again
difficult to solve in a closed form for a general value of $a$. Therefore,
once again we resort to the slow rotating approximation to get the analytic
results. In this approximation, the above equations can be combined to obtain
the following first-order equation 222Here the $\dot{\mathcal{R}}^{2}\simeq
O(a^{4}/r^{4})$ term is neglected in the calculation.

$\displaystyle\frac{d\Theta}{\sin\theta}=\frac{d\theta}{\sin\theta}\left[1-\frac{a^{2}\sin^{2}\theta}{2r^{2}}\left(1+\frac{2\mathcal{M}}{r}-\alpha
e^{-\frac{r}{M}}\right)\right]\,,$ (3.23)

which can be solved to give

$\displaystyle\sin\Theta=\sin\theta\left[1+\frac{a^{2}\cos^{2}\theta}{2r^{2}}\left(1+\frac{2\mathcal{M}}{r}-\alpha
e^{-\frac{r}{M}}\right)\right]\,.$ (3.24)

Putting this back into (3.22) and simplifying, we can obtain the solution for
$\mathcal{R}$

$\displaystyle\mathcal{R}=r\left[1+\frac{a^{2}\sin^{2}\theta}{2r^{2}}-\frac{a^{2}\cos
2\theta}{2r^{2}}\left(\frac{2\mathcal{M}}{r}-\alpha
e^{-\frac{r}{M}}\right)\right]\,.$ (3.25)

From these equations, we can also find $\mathcal{R}$ in terms of $\Theta$

$\displaystyle\mathcal{R}=r\left[1+\frac{a^{2}\sin^{2}\Theta}{2r^{2}}-\frac{a^{2}\cos
2\Theta}{2r^{2}}\left(\frac{2\mathcal{M}}{r}-\alpha
e^{-\frac{r}{M}}\right)\right]\,,$ (3.26)

which would allow us to write the intrinsic metric on two-surface, as embedded
in flat space, as

$\displaystyle ds^{2}\approx
r^{2}\left[1+\frac{a^{2}\sin^{2}\Theta}{r^{2}}-\frac{a^{2}\cos
2\Theta}{r^{2}}\left(\frac{2\mathcal{M}}{r}-\alpha
e^{-\frac{r}{M}}\right)\right]\left(d\Theta^{2}+\sin^{2}\Theta
d\Phi^{2}\right)\,.$ (3.27)

This metric describes a surface in flat three-dimensional space $E^{(3)}$
whose intrinsic curvature equals the intrinsic curvature of a $r=$ constant
surface in the rotating hairy space. It can be used to compute the extrinsic
curvature of the reference flat spacetime, which in turn can be used to
compute the reference contribution to the quasilocal energy in Eq. (3.16). A
long but direct computation in the slow rotating approximation gives us 333The
general expressions to compute the trace of the extrinsic curvature
$\mathcal{K}^{0}$, corresponding to two-dimensional surfaces embedded in a
three-dimensional axisymmetric space, can be found in [87, 88].

$\displaystyle\varepsilon_{0}=\frac{1}{8\pi}\int d\Theta
d\Phi\sqrt{\sigma}\mathcal{K}^{0}=-r\left[1+\frac{a^{2}}{3r^{2}}\left(1+\frac{\mathcal{M}}{r}-\frac{\alpha
e^{-\frac{r}{M}}}{2}\right)\right]\,.$ (3.28)

The quasilocal energy obtained by subtraction (3.28) from (3.19). The result
is

$\displaystyle
E=r\left[1-\sqrt{1-\frac{2\mathcal{M}}{r}+\frac{a^{2}}{r^{2}}+\alpha
e^{-\frac{r}{M}}}\right]+\frac{a^{2}}{3r}\left(1+\frac{\mathcal{M}}{r}-\frac{\alpha
e^{-\frac{r}{M}}}{2}\right)$
$\displaystyle+\frac{a^{2}}{6r}\left(1+\frac{2\mathcal{M}}{r}-\frac{\alpha
re^{-\frac{r}{M}}}{M}\right)\sqrt{1-\frac{2\mathcal{M}}{r}+\frac{a^{2}}{r^{2}}+\alpha
e^{-\frac{r}{M}}}\,.$ (3.29)

This gives the quasilocal energy of a slowly rotating hairy spacetime within a
sphere of radius $r$. In the asymptotic limit $r\rightarrow\infty$, we have

$\displaystyle E\rightarrow\mathcal{M}+\mathcal{O}(a^{2}/r^{2},\alpha
re^{-r/M})\,,$ (3.30)

indicating that $\mathcal{M}$ is indeed the ADM energy of the hairy system.

We can further use (3.30) to compute the energy at the horizon (whenever $a\ll
r_{h}$ or $a\ll\mathcal{M}$). In the slow rotating approximation, at
$r=r_{h}$, we have

$\displaystyle
E(r=r_{h})=r_{h}\left[1+\frac{a^{2}}{3r_{h}^{2}}\left(1+\frac{\mathcal{M}}{r_{h}}-\frac{\alpha
e^{-\frac{r_{h}}{M}}}{2}\right)\right]\,,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=r_{h}\left[1+\frac{a^{2}}{2r_{h}^{2}}+\mathcal{O}(a^{4}/r_{h}^{4})\right]\,.$
(3.31)

Notice that the above expression of horizon energy is exactly similar to the
Kerr case [87]. Moreover, we have

$\displaystyle E(r=r_{h})\simeq\sqrt{\frac{\text{Horizon
Area}}{4\pi}}=2\mathcal{M}_{irr}\left[1+\mathcal{O}(a^{4}/r_{h}^{4})\right]\,,$
(3.32)

where $\mathcal{M}_{irr}$ is the irreducible mass of the hairy rotating black
hole and is proportional to the square root horizon area [90]. Therefore, in
the hairy rotating case as well, to leading order in the slow rotating
approximation, quasilocal energy at the black hole horizon is twice the
irreducible mass.

Figure 6:  The variation of Hawking temperature with respect to the horizon
radius for different values of $\alpha$. Here $J=0.3$ and $\ell=0.4$ are used.
The black, red, green, blue, brown, and cyan curves correspond to $\alpha=0$,
$0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ respectively.

Figure 7: The variation of Hawking temperature with respect to the horizon
radius for different values of $\ell$. Here $J=0.3$ and $\alpha=0.2$ are used.
The black, red, green, blue, brown, and cyan curves correspond to $\ell=0.3$,
$0.4$, $0.5$, $0.6$, $0.7$ and $0.8$ respectively.

We now turn our attention to analysing the thermodynamic properties and
stability of the rotating hairy black hole. Here we mainly concentrate on the
fixed angular momentum canonical ensemble. The variation of Hawking
temperature with respect to the horizon radius for different values of
$\alpha$ and $\ell$ is shown in Figures 7 and 7. Notice that for finite
$\alpha$ and $\ell$ there are two black hole branches at a fixed temperature.
The first branch for which the temperature increases with radius is stable and
appears for small $r_{h}$, whereas the second branch for which the temperature
decreases with radius is unstable and appears for large $r_{h}$. The
thermodynamic stable/unstable nature of the small/large black hole branches
will be further confirmed from the free energy and specific heat analysis
shortly.

The thermodynamic variation of the hairy rotating black hole remains
qualitatively similar to that of the Kerr case. In particular, with finite
$\alpha$, there are again two black hole branches, with the smaller branch
being the stable one. Also, there exists a maximum temperature above which the
black hole solution does not exist. This maximum temperature is moreover a
hair-dependent quantity. The complete dependence of this maximum temperature
on the hair parameters will be discussed shortly. Notice also that the $r_{h}$
vs $T_{h}$ profiles for different values of $\ell$ almost overlap with each
other, suggesting a limited thermodynamic dependence of hairy black holes on
$\ell$.

Figure 8:  The variation of free energy $\mathcal{G}$ with respect to the
Hawking temperature for different values of $\alpha$. Here $J=0.3$ and
$\ell=0.4$ are used. The black, red, green, blue, brown, and cyan curves
correspond to $\alpha=0$, $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ respectively.
In all these curves, the horizon radius $r_{h}$ increases from left to right
and then up.

Figure 9: The variation of free energy $\mathcal{G}$ with respect to the
Hawking temperature for different values of $\ell$. Here $J=0.3$ and
$\alpha=0.2$ are used. The black, red, green, blue, brown, and cyan curves
correspond to $\ell=0.3$, $0.4$, $0.5$, $0.6$, $0.7$, and $0.8$ respectively.
In all these curves, the horizon radius $r_{h}$ increases from left to right
and then up.

The free energy, again in the fixed angular momentum canonical ensemble, can
further disclose the equilibrium thermodynamic stability of the rotating hairy
black hole. The Gibbs free energy is given by

$\displaystyle\mathcal{G}=\mathcal{M}-T_{h}S_{BH}\,,$ (3.33)

and its global minimum yields the state of a system for a fixed $(T_{h},J)$.
In Figures 9 and 9, we have shown the behaviour of the Gibbs free energy for
different values of $\alpha$ and $\ell$. We see that the smaller black hole
branch has a lower free energy compared to the larger black hole branch at all
temperatures. This suggests that the smaller black hole branch is indeed
thermodynamically more favourable compared to the larger black hole branch.
Again the free energy profiles for different values of $\ell$ overlap with
each other, suggesting limited thermodynamic effects of this parameter.
Moreover, these results again suggest the thermodynamic stability of these
hairy black holes is qualitatively similar to the Kerr case.

Figure 10:  The variation of Hawking temperature with respect to the horizon
radius for different values of $J$. Here $\alpha=0.2$ and $\ell=0.4$ are used.
The black, red, green, blue, brown, and cyan curves correspond to $J=0$,
$0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ respectively.

Figure 11: The variation of free energy $\mathcal{G}$ with respect to the
Hawking temperature for different values of $J$. Here $\alpha=0.2$ and
$\ell=0.4$ are used. The black, red, green, blue, brown, and cyan curves
correspond to $J=0$, $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ respectively. In
all these curves, the horizon radius $r_{h}$ increases from left to right and
then up.

To make the thermodynamic analysis more complete, we have also investigated
the thermodynamic behaviour of the hairy black holes for different values of
angular momentum. These results are shown in Figures 11 and 11. Notice that
for the non-rotating case there exists only one black hole branch, the
temperature of which decreases with the horizon radius. This behaviour is
completely analogous to the Schwarzschild black hole, indicating that static
hairy black holes are thermodynamically unstable. With finite angular
momentum, a new black hole branch appears for which the temperature increases
with the horizon radii. This branch appears for small $r_{h}$, making the
rotating hairy black holes thermodynamically stable up to certain temperatures
and radii.

Figure 12:  The variation of specific heat $C$ with respect to the horizon
radius for different values of $\alpha$. Here $J=0.3$ and $\ell=0.4$ are used.
The black, red, green, blue, brown, and cyan curves correspond to $\alpha=0$,
$0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ respectively.

Figure 13: The variation of specific heat $C$ with respect to the horizon
radius for different values of $\ell$. Here $J=0.3$ and $\alpha=0.2$ are used.
The black, red, green, blue, brown, and cyan curves correspond to $\ell=0.3$,
$0.4$, $0.5$, $0.6$, $0.7$, and $0.8$ respectively.

We now discuss the local stability of the rotating hairy black holes. This
corresponds to the response of the equilibrium system under a small
fluctuation in thermodynamic variables and is quantified by the positivity of
the specific heat $C=T_{h}(\partial S_{BH}/\partial T_{h})$. In Figures 13 and
13, we have shown the behaviour of the specific heat for different values of
$\alpha$ and $\ell$. We immediately notice that the specific heat is positive
for the small black hole branch, suggesting its thermodynamically stable
nature, whereas it is negative for the large black hole branch, suggesting its
thermodynamically unstable nature.

Figure 14:  The variation of $r_{h}^{max}$ with respect to $\alpha$ for
different values of $\ell$. Here $J=0.1$ is used. The black, red, green, blue,
brown, and cyan curves correspond to $\ell=0.3$, $0.4$, $0.5$, $0.6$, $0.7$,
and $0.8$ respectively.

Figure 15: The variation of $r_{h}^{max}$ with respect to $\alpha$ for
different values of $\ell$. Here $J=0.3$ is used. The black, red, green, blue,
brown, and cyan curves correspond to $\ell=0.3$, $0.4$, $0.5$, $0.6$, $0.7$,
and $0.8$ respectively.

Figure 16:  The variation of $T_{h}^{max}$ with respect to $\alpha$ for
different values of $\ell$. Here $J=0.1$ is used. The black, red, green, blue,
brown, and cyan curves correspond to $\ell=0.3$, $0.4$, $0.5$, $0.6$, $0.7$,
and $0.8$ respectively.

Figure 17: The variation of $T_{h}^{max}$ with respect to $\alpha$ for
different values of $\ell$. Here $J=0.3$ is used. The black, red, green, blue,
brown, and cyan curves correspond to $\ell=0.3$, $0.4$, $0.5$, $0.6$, $0.7$,
and $0.8$ respectively.

It is also interesting to analyse how the horizon radius $r_{h}=r_{h}^{max}$,
at which the temperature attains a maximum value $T_{h}^{max}$, varies with
the hair parameters. This is important as $r_{h}^{max}$ specifies the maximum
horizon radius around which the stability of black hole changes, particularly,
the black holes are stable for $r_{h}<r_{h}^{max}$ whereas they are unstable
for $r_{h}>r_{h}^{max}$. The results are shown in Figures 15 and 15 for two
different angular momenta. We observe that $r_{h}^{max}$ depends non-trivially
on $\alpha$ and $\ell$, and in particular, it can increase or decrease
depending on the magnitude of these parameters. This suggests that, compared
to the Kerr case, the maximum size of the stable hairy black hole can be
smaller or larger depending on $\alpha$ and $\ell$.

One can similarly analyse the dependence of the maximum temperature
$T_{h}^{max}$ on the hairy parameters. The results are shown in Figure 17 and
17. We find that $T_{h}^{max}$ again depends non-trivially on $\alpha$ and
$\ell$. In particular, it decreases with $\alpha$ and $\ell$ for relatively
small values of $\alpha$ whereas, for higher values of $\alpha$, it can also
exhibit an opposite behaviour. Our overall analysis, therefore, suggests that
the size, stability, and thermal structure of the hairy black hole depend
significantly on the hairy parameters and can be tuned appropriately by
suitable choices of these parameters.

We end this section by making some remarks and observations about the first
law of thermodynamics like relation for the explored hairy black holes here
which we think have not been carefully discussed in the literature 444A
related discussion of the first law appeared in [91] for the static hairy
black hole.. There can actually be two different ways by which the first law
of black hole thermodynamics can be expressed in the hairy case. As mentioned
earlier the shift mass $\mathcal{M}$ is the mass that is observed by an
asymptotic observer at infinity. Therefore, one is expected to have
$\delta\mathcal{M}$ in the first law of black hole thermodynamics relation,
i.e., for the static case

$\displaystyle\delta\mathcal{M}=T_{h}\delta S_{BH}\,.$ (3.34)

We find that for the above relation to be true we must treat the shift mass
$\mathcal{M}$ as a $r_{h}$ dependent function and simultaneously must treat
$M$ as a $r_{h}$ independent function. Indeed, we have

$\displaystyle T_{h}\delta S_{BH}=\frac{1}{2}\left(1+\alpha
e^{-\frac{r_{h}}{M}}\left(1-\frac{r_{h}}{M}\right)\right)\delta r_{h}\,,$
(3.35)

and this can only be equal to $\delta\mathcal{M}$, which can be explicitly
calculated from Eq. (2.25), when the parameter $M$ is treated constant of
$r_{h}$. This suggests some kind of Legendre transformation between the
thermodynamic functions $M$ and $\mathcal{M}$ when the primary hair
$\\{\alpha,\ell\\}$ is introduced, making the former independent of $r_{h}$.
The same is true for the rotating case as well. Again treating $M$ as a
$r_{h}$ independent function, we can easily verify that the first law of
thermodynamic relation

$\displaystyle\delta\mathcal{M}=T_{h}\delta S_{BH}+\Omega_{h}\delta J\,.$
(3.36)

holds for the rotating hairy black hole case as well.

There can be another interesting scenario by which one can again construct the
first law of thermodynamic like relation for the hairy black hole, however
now, by treating the thermodynamic function $M$ as a $r_{h}$ dependent
quantity. This can be done by adding another term, corresponding to the
primary hair, in the usual first law relation

$\displaystyle\delta\mathcal{M}=T_{h}\delta S_{BH}+\Omega_{h}\delta
J+\mathcal{Q}_{\alpha}\delta\mathcal{P}_{\alpha}\,,$ (3.37)

where $\mathcal{P}_{\alpha}$ and $\mathcal{Q}_{\alpha}$ are now the charge and
the corresponding potential associated with the primary hair. Indeed, there
have been many suggestions in recent years which strongly advocated for the
modification of the standard first law in the presence of additional matter
fields. For example, in AdS spaces with an additional scalar field, not only
the standard first law gets a correction term [like in Eq. (3.37)] due to the
scalar field but also the charge and the corresponding potential associated
with the primary hair have a well defined meaning [92, 93, 94, 95]. For the
asymptotically flat black hole considered here, by treating the thermodynamic
function $M$ as a $r_{h}$ dependent function, one particular form of
$\mathcal{Q}_{\alpha}$ and $\mathcal{P}_{\alpha}$ that would satisfy Eq.
(3.37) is

$\displaystyle\mathcal{Q}_{\alpha}=\frac{\sqrt{\alpha}r_{h}^{4}e^{-\frac{r_{h}}{\mathcal{M}-\alpha\ell/2}}}{(r_{h}^{2}+a^{2})(\mathcal{M}-\alpha\ell/2)^{2}},~{}~{}~{}~{}\mathcal{P}_{\alpha}=\sqrt{\alpha}\mathcal{M}\,.$
(3.38)

From the above discussion, it therefore seems that there can be two different
formalisms [(3.36) or (3.37)] to think about the first law for the hairy black
holes. This raises many important questions such as which one of these two
cases is more suitable and accurate in the hairy black hole context, or what
are the main differences in the thermodynamic equations of states of the hairy
black hole from these formalisms. This is intriguing considering that, unlike
the AdS spaces, the additional terms and their physical meaning in the first
law due to hair are not explicitly known for the asymptotically flat spaces.
These interesting questions are certainly worthy of a separate discussion and
we therefore leave this for future work.

##  4 Conclusions

In this paper, we investigated the axisymmetric analogue of the static hairy
black holes which were recently constructed using the gravitational decoupling
method. These hairy black holes have additional hairy parameters
$\\{\alpha,\ell\\}$ in the metric and satisfy the strong energy condition (or
the dominant energy condition) outside the horizon. We first constructed the
rotating counterpart of the hairy solutions using the Newman-Janis and Azreg-
Aïnou algorithm and then studied its various properties. We find that all the
metric coefficients, except $g_{\theta\theta}$, are modified in the presence
of hair, leading to modifications in the horizon structure of the rotating
black hole. Similar to the non-hairy Kerr case, there are again two horizons
in the rotating hairy case, the smaller one increases whereas the larger one
decreases as the strength of the hair $\alpha$ increases. This also leads to
the modification in the extremal structure of the hairy black hole, in
particular, the mass of the black hole needs to always be greater than the
rotation parameter $a$ to avoid the naked singularity. Similarly, we found
that the scalar invariants remain finite everywhere outside the horizon and
that there are no additional singularities in the hairy case than those
already present in the Kerr case.

We then computed various thermodynamic variables and found corrections due to
the hair. We found that the thermodynamic structure of the hairy black hole
remains qualitatively similar to the Kerr case. In particular, there are again
thermodynamically stable small and unstable large black hole branches. We
confirmed these results from the free energy and specific heat analysis. We
further found that the horizon radius and temperature range for which the
rotating hairy black hole remains thermodynamically stable depend non-
trivially on the hairy parameters. In particular, they can be smaller/larger
compared to the Kerr case depending on the magnitude of $\alpha$ and $\ell$.
We also computed quasilocal energy and provided analytic results of it in the
slow rotating limit. Finally, we provided some insights into the first law of
thermodynamic like relation for these rotating hairy black holes and suggested
two different ways by which the first law like relation can be satisfied for
the hairy case.

There are many interesting directions to expand this work. The first one would
be to investigate the dynamical stability of these black holes against various
perturbations. It would also be interesting to verify the correct form of the
first law in the hairy case using the Ward formalism [96, 97]. It would also
be interesting to analyse the fluctuations of the thermodynamic variables and
do a comprehensive investigation of the phase transition involved in rotating
hairy black holes [98, 99]. On the application side, the obtained rotating
hairy solution can also be useful in various cosmological and astrophysical
scenarios. In particular, it would be interesting to analyse the imprints of
the hairy structure on the black hole shadow, gravitational waves, photon
orbits, quasi-normal modes, Hawking radiation etc. and distinguish them from
the Kerr case. It would also be interesting the study the effects of
cosmological constant and rotational parameter on warped curvatures near the
horizon in the lines of [100, 101]. These analyses can be further used to put
bounds on the hairy parameters. We hope to comment on these topics soon.

## Acknowledgments

The work of S.M. is supported by the Department of Science and Technology,
Government of India under the Grant Agreement number IFA17-PH207 (INSPIRE
Faculty Award). The research of I. B. is funded by the Start-Up Research Grant
from Science and Engineering Research Board, the Department of Science and
Technology, Government of India (Reg. No. SRG/2021/000418).

## Appendix A: DEC hairy solution and its rotating analogue

Another static hairy solution that instead satisfies the dominant energy
conditions (DEC)
$\\{\tilde{\rho}\geq|\tilde{p_{r}}|,\tilde{\rho}\geq|\tilde{p_{\theta}}|\\}$
was further constructed in [52]. The corresponding metric solution has the
form

$\displaystyle
e^{g(r)}=e^{-f(r)}=1-\frac{2\mathcal{M}}{r}+\frac{Q^{2}}{r^{2}}-\frac{\alpha(\mathcal{M}-\alpha\ell/2)e^{-\frac{r}{\mathcal{M}-\alpha\ell/2}}}{r}\,,$
(.1)

where again $\mathcal{M}=M+\alpha\ell/2$. The parameters $\ell$ and $Q$ denote
the charge that can be associated with the primary hair. It should be noted
that some caution is needed in the interpretation of parameter $Q$. In
particular, $Q$ does not have to be an electric charge in the usual sense. For
example, $Q$ could be the tidal charge of higher dimensional origin or some
extra source. However, when $Q$ does represent an electric charge, it means
that the electro vacuum or the energy momentum tensor of the Reissner-
Nordström black hole also contains a tensor vacuum. The effective density and
pressures are now given by

$\displaystyle\tilde{\rho}=-\tilde{p}_{r}=\Theta_{0}^{\
0}=\frac{Q^{2}}{\kappa^{2}r^{4}}-\frac{\alpha e^{-r/M}}{\kappa^{2}r^{2}}\,,$
$\displaystyle\tilde{p}_{\theta}=-\Theta_{2}^{\
2}=\frac{Q^{2}}{\kappa^{2}r^{4}}-\frac{\alpha e^{-r/M}}{2\kappa^{2}Mr}\,.$
(.2)

from which can see that DEC is readily satisfied in the region $r>2M$ i.e.
outside the horizon.

We can again straightforwardly apply the procedure mentioned in Section 3 to
find the axisymmetric analogue of (.1). In this case, the metric coefficients
in the Boyer-Lindquist coordinates takes the form

$\displaystyle
g_{tt}=\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\,,~{}~{}~{}g_{rr}=-\frac{\Sigma}{\Delta}\,,~{}~{}~{}g_{\theta\theta}=-\Sigma\,,$
$\displaystyle
g_{t\varphi}=\frac{a\sin^{2}\theta(r^{2}+a^{2}-\Delta)}{\Sigma}\,,~{}~{}~{}g_{\varphi\varphi}=-\frac{\sin^{2}\theta\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)}{\Sigma}\,,$
$\displaystyle\Delta(r)=a^{2}+r^{2}+Q^{2}-2\mathcal{M}r-\alpha
r\left(\mathcal{M}-\frac{\alpha\ell}{2}\right)e^{-r/(\mathcal{M}-\frac{\alpha\ell}{2})}\,.$
(.3)

## Appendix B: Newman-Janis algorithm and complexified null tetrads

The Newman-Janis algorithm involve finding a new null tetrads
$V^{\prime\mu}_{a}$ under the coordinate transformation (2.36), such that
transformation reduces to the old tetrad and metric when
$x^{\prime\mu}=\bar{x^{\prime}}^{\mu}$, i.e.,

$\displaystyle
V^{\prime\mu}_{a}(x^{\prime\sigma},\bar{x^{\prime}}^{\sigma})|_{\bar{x^{\prime}}^{\nu}=x^{\prime\nu}}=V^{\mu}_{a}(x^{\sigma})\,,$
(.4)

The effect of this tilde transformation is to generate a new “tilde” metric
whose components are real functions of complex variable,

$\displaystyle g_{\mu\nu}\rightarrow g^{\prime}_{\mu\nu}:x^{\prime\mu}\times
x^{\prime\nu}\longmapsto\mathbb{R}\,,$ (.5)

A particular transformation of the null tetrad vectors for the hairy metric
(2.23) that satisfies the criteria in Eqs. (.4) and (.5) and give the rotating
metric (2.53) after the transformation (2.39) is

$\displaystyle l^{\mu}\rightarrow l^{\prime\mu}=\delta^{\mu}_{r^{\prime}}\,,$
$\displaystyle n^{\mu}\rightarrow
n^{\prime\mu}=\delta^{\mu}_{u^{\prime}}-\frac{1}{2}\left(1-\mathcal{M}\left(\frac{1}{r^{\prime}}+\frac{1}{\bar{r^{\prime}}}\right)+\frac{\alpha}{4}(r^{\prime}+\bar{r^{\prime}})\left(\frac{1}{r^{\prime}}+\frac{1}{\bar{r^{\prime}}}\right)e^{\frac{-(r^{\prime}+\bar{r^{\prime}})}{2\mathcal{M}-\alpha\ell}}\right)\delta^{\mu}_{r^{\prime}}\,,$
$\displaystyle m^{\mu}\rightarrow
m^{\prime\mu}=\frac{1}{\sqrt{2}r^{\prime}}\left(\delta^{\mu}_{\theta^{\prime}}+\frac{i}{\sin{\theta^{\prime}}}\delta^{\mu}_{\phi^{\prime}}\right)\,,$
(.6)

At this point we like to emphasize that the above transformation of the null
tetrad vectors is not unique and many similar transformations which satisfy
Eqs. (.4) and (.5) can be constructed. In particular, a certain level of
arbitrariness in the method of complexification always appear in the Newman-
Janis algorithm. Indeed, most of the time, there is no unique way of
complexifying all the terms in the tetrad in the same way. This issue of
arbitrary complexification in the Newman-Janis algorithm is known for a long
time and has been discussed thoroughly in the literature. In (.6), the
complexification issue is fixed by demanding that the resultant rotational
metric using the Newman-Janis algorithm matches to the metric one gets using a
without complexification procedure used in section 3.

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