Kerr/CFT from phase space formalism

M. R. Setare 111E-mail<EMAIL_ADDRESS>, M. Koohgard 222E-mail:
<EMAIL_ADDRESS>
Department of Science,
Campus of Bijar, University of Kurdistan, Bijar, Iran

Attempts to find black hole microstates using the Hamiltonian phase space
approach have been made on the Schwarzschild spacetime. Since the
Schwarzschild spacetime is also in the larger family of the Kerr spacetimes,
and both are asymptotically flat, the Kerr black hole is a good option for the
method development. The Kerr black hole is a spinning one. We perform this
analysis on the Kerr spacetime and we obtain promising results using the
covariant phase space analysis. Although we have forced ourselves to use the
Bondi fall-off conditions, we find the gauge degrees of freedom that could be
good candidates for the black hole microstates. The charge algebra on the
boundary could be a Virasoro algebra that has a different central term than
the Schwarzschild black hole. The two dimensional theory on the black hole
boundary is conjectured to be conformally invariant.

Keywords: Kerr-spacetimes, Kerr black hole, Bondi gauge, Black hole
microstates, Fall-off conditions, Covariant phase space

##  1 Introduction

In the formation of the black hole, Hawking’s calculations show that the
presence of a thermal spectrum seems necessary, and that this spectrum should
be a unique one [1, 2]. If the Hawking radiation is essentially thermal, it is
inconsistent with both the unitarity principle and the basic principles of
quantum mechanics and leads to ”the information loss paradox” [3]. To overcome
the information paradox problem, the background metric is considered as a
classical field and this thermal spectrum receives corrections of some power
of $\frac{1}{S}$, where $S$ is the size of the system. An important source for
these corrections are the microstates which correspond to the states in the
Hilbert space. These quantum corrections are considered black hole hair, and
this is contrary to the semi-classical notion that black holes are hairless
[4].

A connection has been established the black hole physics and the critical
phenomena in Bose-Einstein condensates [5, 6, 4]. The critical point is
accompanied by the Bogoliubov modes which are small excitations around the
mean field value in the Bogoliubov approximation that are degenerate with the
ground state. The degeneracy of the modes is lifted by $\frac{1}{N}$-effects
in the quantum calculations. These gapless modes form a subspace in the
Hamiltonian phase space that is expected to have a conformal invariance. On
the other hand, lifting degenerate Bogoliubov-modes by quantum corrections can
also correspond to an anomaly in the conformal invariance.

The Kerr family solutions are the use of the concept of the critical point of
field theory in the gravity. In pure Einstein’s gravity, the Kerr family [7]
are the solutions that are asymptotically flat and they could have gapless
Bogoliubov-modes [8]. In [9], Dvali and Gomez propose a treatment to solve the
black hole information paradox. In their proposal, the black hole is
considered as a bound state of $N$ weakly interacting gravitons, and in
quantum physics, such a bound state is the critical point of a quantum phase
transition [10]. In this picture, Hawking evaporation is described as the
depletion and evaporation of the condensate of the gravitons. In the critical
point of a phase transition, the interactions of the gravitions dominate their
kinetic term, and at the point quantum effects are important. Here, similar to
what happens in the field theory, the quantum effects cannot be ignored. It
has been conjectured the Kerr black holes with extreme J spin could have two
dimensional field theories (CFTs) as their holographic dual [11, 12, 13, 14,
15, 16, 17].

We should mention another point here. The Hamiltonian phase space of the Kerr
family includes infinite number of gapless excitations, but which of these
degrees of freedom are the physical ones and can make the soft hair of the
black hole? Some of these modes are physical and can shift in the phase space.
The idea of finding the physical degrees of freedom, which has been raised,
goes back to the question that Carlip [18] posed in his calculations, which
was related to a general shape of the black hole. Carlip developed
Strominger’s work [19] in three dimensions of space-time and performed his
calculations on the physical states of the black hole for a general state.
Using the phase space formalism [20, 21, 22], Averin in [23] and [24] find the
physical degrees of freedom in the Schwarzschild black hole. The physical
microstates don’t have zero energy and they should be the soft modes. There is
a hidden conformal symmetry that appear in the near horizon region of the
phase space [12, 25, 26, 27]. According to Averin in [23], the results can be
considered as a first step in proving the Schwarzchild/CFT duality, with the
help of the phase space approach. We intend to extend this approach to the
Kerr black holes and let’s take an important step in the Kerr/CFT duality.

So in this paper we try to address the problem of black hole hair which is
defined as conserved charges in the case of Kerr spacetime. In this approach
the so-called BMS-charges defined on the null boundary of the black hole
spacetime, are responsible for ”soft-hair” or zero-energy excitations of the
background. These physical degrees of freedom can provide a statistical
mechanical interpretation for black holes by representing their microstates
semi-classically. Here especially we uses the covariant phase space method to
analyze the phase space associated to these soft hair near a Kerr black hole
solution. we first introduce the Bondi fall-off conditions [28, 29, 30] and by
briefly introducing the Kerr spacetime, we show in which coordinates the
metric can satisfy the fall-off conditions (section 2). In other words, we
will explain that among the various forms of the Kerr metric, we use the
appropriate form to satisfy the fall-off conditions. We then examine the phase
space analysis and begin by finding diffeomorphisms and non-zero excitations
of the metric (Section 3). Finding surface gauge aspects as well as examining
geometry and symmetries on the black hole boundary is our next program in
Section 3. The algebra of symmetry generators on the black hole boundary will
be found and its central terms will be discussed (section 4). We examine
whether this surface charge algebra can be used as the
$bms_{4}$-supertranslations or not. Finally in section 5, we justify that the
dual theory on the boundary has the Virasoro-algebra, and this is a good
reason that the microstates are the soft hair of the black hole.

##  2 The Bondi fall-off conditions and the Kerr metric

We seek to consider the logical form of the metric at long distances in the
Kerr spacetime as a asymptotically flat spacetime. According to the metric-
based formalism of Bondi and Sachs [28, 29] for the treatment of the
Einstein’s general relativity and the gravitational wave analysis,
asymptotically flat spacetimes have the following metrics [31]

$ds^{2}=e^{2\beta}\frac{V}{r}du^{2}-2e^{2\beta}dudr+g_{AB}(dx^{A}-U^{A}du)(dx^{B}-U^{B}du)$
(2.1)

The Bondi-Sachs coordinates $x^{a}=(u,r,x^{A})$ are based on a familly of
outgoing null hypersurfaces [32]. The above metric is obtained with the fall-
off conditions which are as follows

$g_{AB}dx^{A}dx^{B}=r^{2}\gamma_{AB}dx^{A}dx^{B}+O(r),$ (2.2)

where

$\gamma_{AB}dx^{A}dx^{B}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ (2.3)

is the conformal $2$-metric with two degrees of freedom.

The hypersurfaces $x^{0}=u=const$ are null, i.e.
$g^{ab}(\partial_{a}u)(\partial_{b}u)=0$, so that $g^{uu}=0$. Since the two
angular coordinates $x^{A}$, $(A,B,C,...=2,3)$, are constant along the null
rays, so that $g^{uA}=0$. Since we have the relation
$g^{ab}g_{bc}=\delta^{a}_{c}$ between $g^{ab}$ and $g_{ab}$, we could find two
zero components, $g_{rr}$ and $g_{rA}$, for the covariant form of the metric
which is applied in (2.1). We consider these as the gauge fixing conditions.
The $x^{1}=r$ coordinate is chosen to be in the gauge fixing condition,
$\det(g_{AB})=r^{4}\sin^{2}\theta$, and this is also because the coordinate
varies along the null rays. This condition on the metric determinants is also
considered as another part of the gauge fixing conditions. The remaining fall-
off conditions are as follows

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle O(r^{-2})$
$\displaystyle\frac{V}{r}$ $\displaystyle=$ $\displaystyle-1+O(r^{-1})$ (2.4)
$\displaystyle U^{A}$ $\displaystyle=$ $\displaystyle O(r^{-2})$

In order to be able to extend Averin’s calculations [23] to the Kerr black
holes, we need to have the Kerr metric in the Bondi coordinates. The Kerr
metric is the exact solution of the Einstein’s equations describing a
rotating, stationary and axially symmetric black hole. According to the $no-
hair$ theorem [33], all the black holes solutions of the Einstein’s equations
are characterized by three numbers: mass, charge, and angular momentum. The
Schwarzschild black hole is the only mass dependent one, but the Kerr space-
time is the spinning generalization of the Schwarzschild spherical space-time.
The explicit form of the Kerr metric in the $Boyer-Lindquist$ coordinates is
the following [7]:

$ds^{2}=-dt^{2}+\Sigma(\frac{dr^{2}}{\Delta}+d\theta^{2})+(r^{2}+a^{2})\sin^{2}\theta
d\phi^{2}+\frac{2Mr}{\Sigma}(a\sin^{2}\theta d\phi-dt)^{2}$ (2.5)

where

$\displaystyle\Delta(r)$ $\displaystyle\equiv$ $\displaystyle r^{2}-2Mr+a^{2}$
$\displaystyle\Sigma(r,\theta)$ $\displaystyle\equiv$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta$ (2.6)

From the Kerr metric (2.5), it is obvious that the Kerr black hole depends on
the two parameters: the mass parameter, $M$, and the angular momentum
parameter, $a$. Since the metric does not have an explicit dependence on the
time and the angle, the Kerr spacetime is _stationary_ and _axisymmetric_ ,
respectively. In the limit $r$ tends to infinity, this metric is reduced to
the Minkowski metric, indicating that the Kerr spacetime is an _asymptotically
flat spacetime_. In the limit angular momentum, $a$, tends to zero, this
metric is reduced to the Schwarzschild metric. The reason why the boundary of
this metric is important to us is that we want to finally show that there is a
2d CFT at the boundary.

The Boyer-Lindquist coordinates are good for many metric features, but not for
being able to use it in the Bondi-Sachs gauge and the fall-off conditions. For
this purpose, the Kerr metric is written in the Bondi coordinates as follows
[34, 35]

$ds^{2}=(-1+\frac{2M}{r})dv^{2}+2dvdr+a\frac{\cos\theta}{\sin^{2}\theta}dvd\theta+r^{2}\gamma_{AB}dx^{A}dx^{B}$
(2.7)

In the above metric, compared to the metric in [34], the following changes
have been used to satisfy the fall-off conditions and the gauge conditions,
assuming that the radius, $r$, is asymptotic.

$\displaystyle g_{v\phi}$ $\displaystyle=$
$\displaystyle-\frac{2aM\sin^{2}\theta}{r}+O(r^{-2})\to 0$ $\displaystyle
g_{v\theta}$ $\displaystyle=$
$\displaystyle\frac{a\cos\theta}{2\sin^{2}\theta}+\frac{a\cos\theta}{4}(8M+\frac{a}{sin^{3}\theta})\frac{1}{r}+O(r^{-2})\to\frac{a\cos\theta}{2\sin^{2}\theta}$
$\displaystyle g_{\theta\theta}$ $\displaystyle=$ $\displaystyle
r^{2}+\frac{a}{\sin\theta}r+O(r^{-1})\to r^{2}$ $\displaystyle g_{\phi\phi}$
$\displaystyle=$ $\displaystyle r^{2}\sin^{2}\theta-a\sin\theta r\to
r^{2}\sin^{2}\theta$ (2.8)

The inverse metric has the following form

$\displaystyle g^{vv}$ $\displaystyle=$ $\displaystyle
g^{vA}=g^{\theta\phi}=0$ $\displaystyle g^{\phi r}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle g^{vr}$ $\displaystyle=$ $\displaystyle
1-(-1+2\cos^{2}\theta)\frac{a^{2}}{r^{2}}$ $\displaystyle g^{r\theta}$
$\displaystyle=$
$\displaystyle-\frac{\cos\theta}{2\sin^{2}\theta}\frac{a}{r^{2}}$
$\displaystyle g^{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}$ $\displaystyle g^{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}\sin^{2}\theta}$ $\displaystyle g^{rr}$
$\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{1}{4r^{2}}(-8\cos^{2}\theta+4+\frac{1}{4}\frac{\cos^{2}\theta}{\sin^{4}\theta})$
(2.9)

in which the $r$ component is asymptotic. To satisfy the Bondi gauge and the
fall-off conditions, we consider the components approximately as follows

$\displaystyle g^{vr}$ $\displaystyle\to$ $\displaystyle+1$ $\displaystyle
g^{rr}$ $\displaystyle\to$ $\displaystyle 1-\frac{2M}{r}$ (2.10)

Using the Kerr metric (2.7) and its inverse (2), we calculate the connection
coefficients that appear in the appendix A. We will use the approximate form
of the metric (2.7) in the following sections. We have ignored terms that have
higher order of the inverse radius. However, the approximate form of the
metric is in its general form in the Bondi gauge (2.1). The effect of ignored
terms on the diffeomorphisms and on the metric corrections will also be of
orders of inverse radius, which can also be ignored. The approximate forms of
the diffeomorphisms and the metric corrections will be computed in the
following section.

##  3 Surface degrees of freedom of a Kerr black hole

The Bondi fall-off conditions are important in the study of the asymptotic
symmetry algebra. We can find the gauge transformations that preserve the
Bondi fall-offs and the gauge fixing conditions that mentioned in the previous
section. The generators of these transformations can be considered as the
generators of hidden symmetry that we are looking for. The authors in [31]
have found such transformations for a general case in which there is no event
horizon. They have worked in an asymptotically flat 4D spacetime and they have
shown the existence of the $bms_{4}$-symmetry as a hidden symmetry at null
infinity of the spacetime. We look for the hidden symmetry in the Kerr
spacetime in which there is the event horizon and consider the symmetry as an
enhancement in the symmetry of the spacetime. To this end, we now consider the
metric (2.7) as a reference point in the solution space and we find the
excitations $g_{ab}+h_{ab}$ that are primarily responsible for the black hole
microstates. To achieve this, we assume that there is a Hamiltonian
description of the phase space, at least at points close to the reference
point. To analysis this phase space, we use the covariant phase space method
[20, 21]. According to the black hole uniqueness theorems, the only possible
_asymptotically flat_ and _axisymmetric_ solutions of the Maxwell-Einstein
equations are the diffeomorphic solutions to the Kerr family. So it is
possible that the black hole microstates are in the form of the excitations
$h_{ab}=\mathcal{L}_{\xi}g_{ab}$ which are the Lie derivatives with respect to
the metric. These excitations involve the gauge transformations to the
reference point and are diffemorphic to the Kerr space-time. Of course, not
all of these excitations are real and physical degrees of freedom, and only
some of them could make a shift in the Hamiltonian phase space. This is
according to the Hawking, Perry and Strominger’s work [36, 37, 38, 39] on _the
soft hair of the black hole_. The degrees of freedom associated with the
microstates in the Schwarzschild black hole have been done by Averin [23, 24],
and we want to do that for the Kerr black hole.

To find the black hole microstates, the candidate excitations $h_{ab}$ should
preserve the Bondi gauge. The residual gauge transformations that preserve the
Bondi gauge fixing conditions in the form of the metric (2.7) could be
generated by vector fields satisfying the following conditions

$\mathcal{L}_{\xi}g_{rr}=0$ (3.1) $\mathcal{L}_{\xi}g_{rA}=0$ (3.2)
$\mathcal{L}_{\xi}g_{AB}g^{BA}=0$ (3.3)

By the usual definition of the Lie derivative, the first condition (3.1) is
expanded as follows

$\nabla_{r}\xi_{r}=\nabla_{r}\xi^{v}=\partial_{r}\xi^{v}=0$ (3.4)

We have used the metric compatibility condition in the first equality. The
Kerr metric components (2.7) and the Christoffel coefficients of the Appendix
A are used in the $2^{nd}$ equality in (3.4). So the first component of the
diffeomorphism that can be responsible for the residual gauge transformation
can take the following form

$\xi^{v}=X(x^{A},v).$ (3.5)

The $2^{nd}$ condition (3.2) to find the residual gauge transformation can be
expanded as follows

$g_{rv}\nabla_{A}\xi^{v}+g_{AB}\nabla_{r}\xi^{B}+g_{Av}\nabla_{r}\xi^{v}=0.$
(3.6)

By the covariant derivative expansion and the Christoffel coefficients, the
left hand side of the above equation has the following form

$\partial_{A}\xi^{v}+g_{AB}\partial_{r}\xi^{B}=0.$ (3.7)

The last condition (3.3) to find the residual gauge transformation of the
Appendix A can be expanded as follows

$g^{AB}\mathcal{L}_{\xi}g_{AB}=D_{A}\xi^{A}+\frac{2}{r}\xi^{r}=0$ (3.8)

$D_{A}$ denotes the covariant derivative on $S^{2}$ and $D^{2}$ is the
associated Laplace-operator. By differentiation with respect to $r$-component,
we find the condition (3.7) as follows

$\partial_{A}\partial_{r}\xi^{v}+2r\gamma_{AB}\partial_{r}\xi^{B}+r^{2}\gamma_{AB}\partial_{r}^{2}\xi^{B}=0$
(3.9)

Because of the $r$-independence of $\xi^{v}$ in (3.5), the first term of (3.9)
will be omitted and the general solution can take the following form

$\xi^{A}(r,v,x^{A})=X^{A}(v,x^{A})+\frac{1}{r}Z^{A}(v,x^{A}).$ (3.10)

Substituting (3.10) into (3.7), we find

$\partial_{A}\xi^{v}=\gamma_{AB}Z^{B}$ (3.11)

We substitute this result into (3.10) and we find the following form of the
$3^{rd}$ component of the diffeomorphism vector

$\xi^{A}=X^{A}+\frac{1}{r}D^{A}X$ (3.12)

where we have used $D_{A}X=\partial_{A}X$. By the results (3.5), (3.8) and
(3.12), we expect the components of the diffeomorphism vector field which
preserve the Bondi gauge fixing conditions, can take the following form

$\xi=X\partial_{v}-\frac{1}{2}(rD_{A}X^{A}+D^{2}X)\partial_{r}+(X^{A}+\frac{1}{r}D^{A}X)\partial_{A}.$
(3.13)

Here, $X=X(v,x^{A})$ and $X^{A}=X^{A}(v,x^{A})$ are an arbitrary scalar and
arbitrary vector field on the _2 sphere_ , respectively. The argument we have
followed to find the candidate form of the diffeomorphism vector is based on
the Kerr metric (2.7) and its related Christoffel coefficients in the Appendix
A. This form of the residual gauge transformation also justify the general
form of the diffeomorphisms in [31]. The authors of [31] have obtained the
general form of the residual gauge transformations in asymptotically flat 4-D
spacetime that our result (3.13) is one of the special cases of that form. In
addition, this candidate form for the diffeomorphsim is in consistent with the
form of the diffeomorphsim for the Schwarzschild black hole in [23]. Utilizing
the diffeomorphism (3.13) and its related Lie derivative
$h_{ab}=\mathcal{L}_{\xi}g_{ab}$, the non-zero excitations on the metric
components could be found as follows

$\displaystyle h_{vr}$
$\displaystyle=\partial_{v}X-\frac{1}{2}D_{B}X^{B}-\frac{1}{2}r\partial_{r}(D_{B}X^{B})$
$\displaystyle h_{Av}$
$\displaystyle=-\frac{r}{2}D_{A}D_{B}X^{B}-\frac{1}{2}D_{A}D^{2}X+VD_{A}X+r^{2}\partial_{v}X_{A}+r\partial_{v}D_{A}X$
$\displaystyle+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\partial_{v}X+\frac{M}{r^{2}}X+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}(D_{A}X^{B}+\frac{1}{r}D_{A}D^{B}X)$
$\displaystyle h_{AB}$
$\displaystyle=r^{2}(D_{A}X_{B}+D_{B}X_{A}-\gamma_{AB}D_{C}X^{C})+r(2D_{A}D_{B}X-\gamma_{AB}D^{2}X)$
$\displaystyle+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}(\delta_{B\theta}D_{A}X+\delta_{A\theta}D_{B}X)$
$\displaystyle h_{vv}$
$\displaystyle=\frac{M}{r}D_{B}X^{B}+\frac{M}{r^{2}}D^{2}X+2V\partial_{v}X-r\partial_{v}D_{B}X^{B}-D^{2}\partial_{v}X$
$\displaystyle+a\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}X^{A}\delta_{A\theta}+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}D^{A}X\delta_{A\theta}$
(3.14)

Although the diffeomorphsim form (3.13) is consistent with the case of the
Schwarzschild black hole [23], this is not the case with the metric
excitations (3).These excitations have been modified in related to the
Schwarzschild case. In general, in the phase space formalism, the set of
configurations satisfy the equation of motion forms the (not gauge-fixed)
covariant phase space $\mathcal{F}$. Working in the Bondi gauge corresponds to
fixing a gauge, which is an step to obtain the gauge-fixed solution space
$\Gamma\subseteq\mathcal{F}$. To find the physical excitations of the Kerr
black hole, we consider just the non-zero modes of the presymplectic form
related to the gauge-fixed covariant space $\Gamma$.

By examining the Hamiltonian generators of these excitations utilizing the
phase space formalism [20, 21, 22], we find out which of them are physical. In
general, the Hamiltoniant generator $H$ of a gauge transformation
$\mathcal{L}_{\xi}g_{ab}$ over a Cauchy-surface is determined by

$\delta
H[h_{ab};g_{ab}]=\int_{\Sigma}\omega[h_{ab},\mathcal{L}_{\xi}g_{ab};g_{ab}],$
(3.15)

where $\delta H$ denotes the variation of the Hamiltonian $H$ and $\omega$ is
the presymplectic current differential form. The gauge transformation
generator is composed of a bulk term and a surface contribution. The former is
related to the equations of motion, which is therefore omitted. The remaining
on-shell term of the generator (3.15) is as follows

$\delta H[h_{ab};g_{ab}]=-\frac{1}{16\pi}\oint_{\partial\Sigma}*F,$ (3.16)

where $F$ is the $2-form$ of the field strength over the spacetime. In this
case, $\partial\Sigma$ is a cross-section of the outer event horizon of the
Kerr black hole. We perform the Hamiltonian computations on the large radius
limit. The general form of the $F$ is as follows [37]

$\displaystyle F_{ab}$
$\displaystyle=\frac{1}{2}(\nabla_{a}\xi_{b}-\nabla_{b}\xi_{a})+(\nabla_{a}h^{c}_{b}-\nabla_{b}h^{c}_{a})\xi_{c}+(\nabla_{c}\xi_{a}h^{c}_{b}-\nabla_{c}\xi_{b}h^{c}_{a})$
$\displaystyle-(\nabla_{c}h^{c}_{b}\xi_{a}-\nabla_{c}h^{c}_{a}\xi_{b})-(\nabla_{a}h\xi_{b}-\nabla_{b}h\xi_{a}).$
(3.17)

Since the $\partial\Sigma$ has fixed $v$ and $r=r_{S}$ and has the topology of
the _2-sphere_ parameterized by the polar and the azimuthal coordinates, we
have

$\delta
H[h_{ab};g_{ab}]=-\frac{r^{2}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}F_{rv},$
(3.18)

where $\gamma=\det\gamma_{AB}$ and the _2-form_ $F$ has the following form

$\displaystyle F_{rv}$
$\displaystyle=\xi^{A}\big{(}\partial_{r}h_{Av}-\frac{2}{r}h_{Av}+\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r}\delta_{A\theta}h_{vr}$
$\displaystyle-\frac{a}{2r^{3}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}h^{B}_{B}\big{)}+\xi^{v}\big{(}-\frac{1}{r^{2}}D^{A}h_{Av}+\frac{1}{r^{2}}\partial_{v}h^{A}_{A}-\frac{2}{r}h_{vv}$
$\displaystyle-\frac{M}{r^{4}}h^{A}_{A}+\frac{2}{2r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\partial_{r}h_{Av}+\partial_{A}h_{rv})-\frac{V}{r^{2}}\partial_{r}h^{A}_{A}\big{)}$
$\displaystyle+\partial_{r}\xi^{v}h_{vv}+\frac{1}{r^{2}}D^{A}\xi^{v}h_{vA}-\frac{1}{2r^{2}}\partial_{v}\xi^{v}h^{A}_{A}+\xi^{r}\big{(}\frac{2}{r}h_{vr}+\frac{1}{r^{3}}h^{A}_{A}\big{)}+\frac{1}{2r^{2}}\partial_{r}\xi^{r}h^{A}_{A}.$
(3.19)

which is calculated using the non-zero metric exitations (3) and (3). Except
for the terms proportional to the angular momentum parameter $a$, the
remaining terms in (3) are similar to those form in the Schwarzschild
spacetime [37, 23]. Of course, these terms are important in terms of having
the angular momentum $a$ because the angular momentum is an important feature
of the Kerr spacetime compared to the Schwarzchild spacetime.

At this point we calculate the change of the Hamiltonian generator between
$g_{ab}$ and $g_{ab}+h_{ab}$ by substituting Eq. (3) into Eq. (3.18) and we
find that its form by the excitations $\xi=\xi(Y,Y^{A})$ (see (3.13)) and
$h_{ab}=h_{ab}(X,X^{A})$ (see (3)) is as follows:

$\displaystyle\delta H$
$\displaystyle=-\frac{r_{S}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\bigg{\\{}Y(1-D^{2})D_{B}X^{B}+D_{B}Y^{B}(D^{2}-1)X\bigg{\\}}_{r=r_{S}}$
$\displaystyle-\frac{r^{2}_{S}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\bigg{\\{}Y(-\frac{a}{2r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta})(D_{A}D_{B}X^{B})$
$\displaystyle+Y\big{[}(-\frac{a}{2r^{2}}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})(2D_{B}X^{B}\delta_{B\theta}-D_{B}X^{B})-\frac{1}{2}D^{2}\partial_{r}D_{B}X^{B}\big{]}$
$\displaystyle+D_{A}Y^{A}\big{[}(\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta})(\frac{2}{r}X^{B}\delta_{B\theta}-\frac{3}{2r^{2}}D^{B}X\delta_{B\theta})-\frac{1}{2}r\partial_{r}D_{B}X^{B}\big{]}$
$\displaystyle+Y^{A}\big{[}(\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta})(-\frac{1}{r}D_{B}X^{B}\delta_{A\theta}+\partial_{r}D_{B}X^{B}\delta_{A\theta})+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})X^{B}\delta_{B\theta}\delta_{A\theta}\big{]}\bigg{\\}}_{r=r_{S}}$
(3.20)

The first term in (3) is common with its form the Schwarzschild spacetime case
in [23] but the remaining terms are specific to the Kerr black holes geometry.

The variation $\delta H$ in (3) contain the gauge excitations $g_{ab}$ which
are the non-zero presymplectic form modes. Because the $\delta H$ in (3) is a
shift in the Hamiltonian generator, its constituents are the surface gauge
excitations. As in the Schwarzschild case in [23], all the v-dependence can be
omitted in the Hamiltonian variation in (3) in the Kerr black hole. Given the
$\delta H$ in (3), we consider the non-zero excitations as follows

$\displaystyle X$ $\displaystyle=$ $\displaystyle X(x^{A})$ $\displaystyle
X^{A}$ $\displaystyle=$ $\displaystyle X^{A}(x^{B}).$ (3.21)

where the scalar $X$ and the vector field $X^{A}$ are the physical excitations
which are on the $S^{2}$. They can be considered as the candidates for the
gauge degrees of freedom of the Kerr black hole with the metric (2.7). We call
the excitations (3) as the gauge aspects. It is needed to describe the
geometry and the physics of the candidate gauge aspects clearly.

Here, there is a point to note on the computation of the Hamiltonian variation
in (3).To calculate the variation of the Hamiltonian generator in (3), we have
used _2-Form_ (3) that is defined in terms of the diffeomorphisms (3.13) and
the metric corrections (3). Because the metric is approximate, the pre-
symplectic form (3) is also approximate. If we use the exact form of the
metric, some higher order $r$-inverse terms may be added to the integrand of
(3). Since we have chosen the gauge aspects based on the form of the integrand
terms, adding the ignored terms does not change the gauge aspects. This is
because the possible added terms are some functions of the selected gauge
aspects and their derivatives. We determine the form of the gauge aspects (3)
according to the integrand of (3). The ignored terms have no effects on the
choice of gauge aspects. The gauge aspects are important in counting the black
hole microstates. Since the ignored terms have no effect on the choice of the
gauge aspects, the effects of these terms on counting microstates and
obtaining black hole entropy is negligible.

Utilizing the statement in [31], the diffeomorphsim $\xi$ in (3.13) represents
a faithful representation of the $bms_{4}$-supertranslation. For example, we
consider an especial case of the gauge degrees as follows

$\displaystyle X$ $\displaystyle=$ $\displaystyle f(x^{A}),$ $\displaystyle
X^{A}$ $\displaystyle=$ $\displaystyle 0,$ (3.22)

where $f$ is a function on $S^{2}$. This is an especial case of
diffeomorphisms in (3.13) and can be considered as usual
$bms_{4}$-supertranslations [31]. The $bms_{4}$-supertranslations are the
degeneracy of the vacuum but they are not responsible for the physical
microstates of the black hole [40]. The authors in [41] consider some type of
the asymptotic symmetry that are not part of the $bms_{4}$-supertranslation.
They claim the latter symmetry is an enhanced symmetry at the event horizon of
the black hole and apply this claim to the Schwarzschild black hole. The
authors in [41] have argued this enhanced symmetry can be found in the general
type of the black holes. We want to extend this claim to the Kerr black hole
case (2.7) and we find the enhancement in the asymptotic symmetry of the black
hole which are the responsible for the black hole physical microstates. To
this end, we can find there is the vector field $X^{A}$ on $S^{2}$ can be
decomposed as follows [23]

$X^{A}=Y^{A}-D^{A}g,$ (3.23)

where $D_{A}Y^{A}=0$. Using the Helmholtz-Hodge decomposition (3.23), the
gauge aspects (3) can be found in the following form

$\displaystyle X$ $\displaystyle=$ $\displaystyle f$ $\displaystyle X^{A}$
$\displaystyle=$ $\displaystyle Y^{A}-D^{A}g,$ (3.24)

where $f$ and $g$ are two scalars on $S^{2}$. Substituting the following
choice of $(X,X^{A})$ into the diffeomorphisms (3.13)

$\displaystyle X$ $\displaystyle=$ $\displaystyle f,$ $\displaystyle X^{A}$
$\displaystyle=$ $\displaystyle-\frac{1}{r_{S}}D^{A}f,$ (3.25)

the diffeomorphisms at the cross-section $\partial\Sigma$ which is at the Kerr
black hole event horizon has the following form

$\xi|_{r=r_{S}}=f\partial_{v}.$ (3.26)

This can represent an enhanced asymptotic symmetry that can be called the
event horizon supertranslation which is similar to the
$bms_{4}$-supertranslation in the limit $r_{S}\to\infty$. It can be seen that
the degrees of freedom of the Kerr black hole can be considered by the
$bms_{4}$-supertranslations and the event horizon supertranslations. The
supertranslation (3) is in the form of (3) and contains a pure
$bms_{4}$-supertranslation. These are the vacuum degeneracies. The candidates
for the black hole microstates can be found, by subtracting them, as follows
to get ride of the gravitational vacuum degeneracy

$\displaystyle X$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle X^{A}$
$\displaystyle=$ $\displaystyle Y^{A}-D^{A}g.$ (3.27)

So the gauge excitations (3) contain the excitations (3) which are the
candidates for the black hole microstates. The establishment of candidates of
the Kerr black hole microstates is based on the symplectic reasoning and the
result is similar to the Schwarzschild black hole result in [23, 24]. The only
difference is that it is not possible to omit the $Y^{A}$ term in the enhanced
modes (3).

##  4 Surface charge algebra

In the section 3, we examined the surface degrees of freedom of the Kerr black
hole. In this section, we examine the surface charge algebra, and before that
we need to consider how gauge-fixing takes place in the formalism of the
covariant phase space. A gauge excitation $h_{ab}$ is not a gauge-fixed one
and can include the symplectic zero-modes. If these zero-modes are left out of
the excitation $h_{ab}$, we get a gauge-fixed excitation $\tilde{h}_{ab}$. We
find a mapping $h_{ab}\to\tilde{h}_{ab}$ that can act as an projection
operator between $h_{ab}$ and $\tilde{h}_{ab}$ and fix the gauge. We then
construct such an operator that can map $h_{ab}$ to $\tilde{h}_{ab}$ by
separating the zero modes. $\mathcal{F}$ and $\Gamma$ are the not gauge-fixed
solution space and the gauge fixed solution space of the theory, respectively,
as mentioned in the previous section. The gauge excitations are placed in the
tangent spaces of $\mathcal{F}$ and $\Gamma$. The mapping between
$h_{ab}\to\tilde{h}_{ab}$ corresponds to a map between the related tangent
spaces.

To obtain the projection operator, we consider the Hamiltonian generators
based on the surface degrees of freedom. Given the gauge excitations
$h_{ab}=\mathcal{L}_{\xi}g_{ab}$ in terms of the vector field components
$\xi^{a}$, we find the integrand ingredients of the Hamiltonian variation (3)
in terms of the $\xi^{a}$ as follows

$\displaystyle h^{A}_{A}$
$\displaystyle=(a\frac{\cos\theta}{\sin^{2}\theta})D^{A}X\delta_{A\theta}=(a\frac{\cos\theta}{\sin^{2}\theta})D^{A}\xi^{v}\delta_{A\theta}$
$\displaystyle h_{Av}$
$\displaystyle=-\frac{r}{2}D_{A}D_{B}+VD_{A}\xi^{v}+r^{2}\partial_{v}\xi_{A}-\frac{1}{2}D_{B}\xi^{B}\delta_{A\theta}$
$\displaystyle+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\partial_{v}\xi^{v}+\frac{M}{r^{2}}\xi^{v}-\frac{1}{2}D_{B}\xi^{B})+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}D_{A}\xi^{B}$
$\displaystyle\partial_{r}h_{Av}$
$\displaystyle=-\frac{1}{2}D_{A}D_{B}\xi^{B}-\frac{r}{2}\partial_{r}D_{A}D_{B}\xi^{B}-\frac{2M}{r^{2}}D_{A}\xi^{v}+2r\partial_{v}\xi_{A}+r^{2}\partial_{r}\partial_{v}\xi_{A}$
$\displaystyle-\frac{1}{2}\partial_{r}D_{B}\xi^{B}-\frac{a}{4}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}\partial_{r}D_{B}\xi^{B}+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\partial_{r}D_{A}\xi^{B}$
$\displaystyle D^{A}h_{Av}$
$\displaystyle=-\frac{r}{2}D_{B}D^{2}\xi^{B}+VD^{2}\xi^{v}+r^{2}\partial_{v}D^{A}\xi_{A}-\frac{1}{2}D^{A}D_{B}\xi^{B}\delta_{A\theta}$
$\displaystyle+\frac{a}{2}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\delta_{A\theta}(\partial_{v}\xi^{v}+\frac{M}{r^{2}}\xi^{v}-\frac{1}{2}D_{B}\xi^{B})$
$\displaystyle+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\partial_{v}D^{A}\xi^{v}+\frac{M}{r^{2}}D^{A}\xi^{v}-\frac{1}{2}D^{A}D_{B}\xi^{B})$
$\displaystyle+\frac{a}{2}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\delta_{B\theta}\delta_{A\theta}D_{A}\xi^{B}+\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}D^{2}\xi^{B}$
$\displaystyle\partial_{v}h^{A}_{A}$
$\displaystyle=a\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}D^{A}\xi^{v}\delta_{A\theta}$
$\displaystyle\partial_{r}h^{A}_{A}$ $\displaystyle=0$ (4.1)

where we have used the excitations (3), the diffeomorphism components (3.13)
and the metric (2.7).

For an arbitrary vector field $\xi=\xi(X,X^{A})$ and the gauge excitations and
their derivatives represented in (4), we find the Hamiltonian generators as
follows

$\displaystyle\delta H$
$\displaystyle=-\frac{r^{2}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\bigg{\\{}X^{A}\bigg{[}r^{2}\partial_{r}\partial_{v}\xi_{A}-\frac{1}{2}\partial_{r}D_{B}\xi^{B}-\frac{a}{4}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}\partial_{r}D_{B}\xi^{B}$
$\displaystyle+D_{B}\xi^{B}\delta_{A\theta}-\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\frac{M}{r^{2}}\xi^{v}+\frac{1}{2}D_{B}\xi^{B})$
$\displaystyle-\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}\partial_{r}\xi^{r}+\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\delta_{B\theta}\delta_{A\theta}\partial_{r}\xi^{B}$
$\displaystyle+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\delta_{B\theta}\delta_{A\theta}\xi^{B}\bigg{]}+D_{A}X^{A}\bigg{[}-\frac{1}{2}D_{B}\xi^{B}+\frac{r}{2}\partial_{r}D_{B}\xi^{B}$
$\displaystyle+\frac{2M}{r^{2}}\xi^{v}-\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\partial_{r}\xi^{B}+\frac{2V}{r}\xi^{v}+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\xi^{B}$
$\displaystyle-\partial_{v}\xi^{v}+\partial_{r}\xi^{r}+D_{B}\xi^{B}\bigg{]}+X\bigg{[}\frac{1}{2r}D_{B}D^{2}\xi^{B}+\frac{4V}{r}D^{2}\xi^{v}-\frac{2M}{r^{2}}D_{B}\xi^{B}$
$\displaystyle+\frac{a}{r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}D^{A}\xi^{v}\delta_{A\theta}-\frac{4V}{r}\partial_{v}\xi^{v}-\frac{2a}{r}\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}\xi^{A}\delta_{A\theta}$
$\displaystyle+\frac{a}{2r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(-\frac{1}{2}D_{A}D_{B}\xi^{B}-\frac{r}{2}\partial_{r}D_{A}D_{B}\xi^{B}+2r\partial_{v}\xi_{A}+r^{2}\partial_{r}\partial_{v}\xi_{A}$
$\displaystyle+\partial_{A}\partial_{v}\xi^{v}-\partial_{r}\partial_{r}\partial_{A}\xi^{r}-\partial_{A}\partial_{B}\xi^{B})-\frac{a}{r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{A\theta}(\partial_{v}D^{A}\xi^{v}-D^{A}\partial_{r}\xi^{r})$
$\displaystyle-\frac{a}{r^{2}}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})(\partial_{v}\xi^{v}-\partial_{r}\xi^{r})+\frac{1}{2}\partial_{r}D^{2}D_{B}\xi^{B}-rD^{A}\partial_{r}\partial_{v}\xi_{A}$
$\displaystyle-\frac{1}{r}D^{A}D_{B}\xi^{B}\delta_{B\theta}+\frac{1}{r}D^{2}D_{B}\xi^{B}\bigg{]}+\partial_{v}X\bigg{[}\frac{-a}{2r^{2}}\frac{\cos\theta}{\sin^{2}\theta}D^{A}\xi^{v}\delta_{A\theta}\bigg{]}\bigg{\\}}$
(4.2)

By decomposition of the vector field $\xi^{A}$ on $S^{2}$ as follows

$\ \xi^{A}=\tilde{\xi}^{A}+D^{A}h,$ (4.3)

where $h$ is a scalar on $S^{2}$ and $\tilde{\xi}^{A}$ is divergence-free, we
have (4) as follows

$\displaystyle\delta H$
$\displaystyle=-\frac{r^{2}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\bigg{(}X^{A}\big{(}D_{B}Y^{B}\delta_{A\theta}+\frac{1}{r}D^{2}Y\delta_{A\theta}+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})Y^{B}\delta_{B\theta}\big{)}$
$\displaystyle+D_{A}X^{A}\big{(}\frac{r}{2}\partial_{r}D_{B}Y^{B}+\frac{2V}{r}Y+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}Y^{B}\delta_{B\theta}+\frac{2M}{r^{2}}Y\big{)}$
$\displaystyle+X\big{(}\frac{1}{r}D^{2}D_{B}Y^{B}-\frac{1}{r}D^{A}D_{B}Y^{B}\delta_{A\theta}-\frac{2M}{r^{2}}D_{B}Y^{B}\big{)}\bigg{)}$
(4.4)

In obtaining (4) from (4), the gauge transformation $\xi=\xi(Y,Y^{A})$
considered to be an excitation of the surface degrees of freedom with gauge
aspects $Y$ and $Y^{A}$ where these are just on $S^{2}$ and they are
independent of $v$ and $r$. In addition, we have the following relations for
the $\xi$ components that are used to obtain (4)

$\displaystyle\partial_{v}\xi^{v}$
$\displaystyle=\partial_{v}Y=\partial_{v}h=\partial_{v}D_{B}\xi^{B}=0$
$\displaystyle\partial_{v}D^{A}\xi^{v}$
$\displaystyle=D^{A}\partial_{r}\partial_{v}\xi_{A}=\partial_{v}D^{2}\xi^{v}=0$
(4.5)

In calculation (4), the terms $O(r^{-2})$ have been neglected. By comparing
the coefficients of $X^{A}$, $D_{A}X^{A}$ and $X$ in the integrands of the two
equations (4) and (4), the projection operators are obtained as follows

$\displaystyle
D_{B}Y^{B}+\frac{1}{r}D^{2}Y+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})$
$\displaystyle=+D_{B}\xi^{B}-\frac{a}{2r}\frac{\cos\theta}{\sin^{2}\theta}D_{B}\xi^{B}-\frac{a}{r}\partial_{r}\xi^{r}$
$\displaystyle+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\xi^{B}\delta_{B\theta}\bigg{|}_{\partial\Sigma}$
(4.6)
$\displaystyle\frac{r}{2}\partial_{r}D_{B}Y^{B}+\frac{2V}{r}Y+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}Y^{B}\delta_{B\theta}$
$\displaystyle=-r^{2}\partial_{r}\partial_{v}h$
$\displaystyle-\frac{1}{2}D_{B}\xi^{B}+\frac{r}{2}\partial_{r}D_{B}\xi^{B}-\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\partial_{r}\xi^{B}+\frac{2V}{r}\xi^{v}$
$\displaystyle+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\xi^{B}-\partial_{v}\xi^{v}+\partial_{r}\xi^{r}+D_{B}\xi^{B}\bigg{|}_{\partial\Sigma}$
(4.7)
$\displaystyle\frac{1}{r}D^{2}D_{B}Y^{B}-\frac{1}{r}D^{A}D_{B}Y^{B}\delta_{A\theta}-\frac{2M}{r^{2}}$
$\displaystyle=\frac{1}{2r}D_{B}D^{2}\xi^{B}-\frac{2M}{r^{2}}D_{B}\xi^{B}$
$\displaystyle-\frac{4V}{r}\partial_{v}\xi^{v}-\frac{2a}{r}\frac{\cos\theta}{\sin^{2}\theta}\partial_{v}\xi^{A}\delta_{A\theta}+\frac{1}{2}\partial_{r}D^{2}D_{B}\xi^{B}-rD^{A}\partial_{r}\partial_{v}\xi_{A}$
$\displaystyle-\frac{1}{r}\partial_{v}D^{2}\xi^{v}+\frac{1}{r}D^{2}\partial_{r}\xi^{r}+\frac{1}{r}D^{2}D_{B}\xi^{B}\bigg{|}_{\partial\Sigma}$
(4.8)

In the right hand side of the above projection operators, we have arbitrary
gauge excitations $\xi$ do not satisfy the Bondi gauge but in the right hand
side of the above equations, we have the gauge aspects $(Y,Y^{A})$ that are
satisfying the Bondi gauge. So we can claim that the above operators are the
mapping between the excitations of the tangent space of
$T_{g_{ab}}\mathcal{F}$ and the excitations in the tangent space
$T_{g_{ab}}\Gamma$.

Now that we have found the projection operator, we need to calculate
$[\xi_{1},\xi_{2}]$ as well to find the gauge aspects of the Lie bracket. For
this purpose, we consider $\xi_{1}$ and $\xi_{2}$ as follows

$\displaystyle\xi_{1}(X_{1},X^{A}_{1})$ $\displaystyle=$ $\displaystyle
X_{1}\partial_{v}-\frac{1}{2}(rD_{A}X^{A}_{1}+D^{2}X_{1})\partial_{r}+(X^{A}_{1}+\frac{1}{r}D^{A}X_{1})\partial_{A}$
$\displaystyle\xi_{2}(X_{1},X^{A}_{2})$ $\displaystyle=$ $\displaystyle
X_{1}\partial_{v}-\frac{1}{2}(rD_{A}X^{A}_{2}+D^{2}X_{2})\partial_{r}+(X^{A}_{2}+\frac{1}{r}D^{A}X_{2})\partial_{A}$
(4.9)

where $X_{i}$ and $X^{A}_{i}$ are the gauge aspects that are considered as
follows

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle f_{1}$ $\displaystyle
X^{A}_{1}$ $\displaystyle=$ $\displaystyle-D^{A}g_{1}$ $\displaystyle X_{2}$
$\displaystyle=$ $\displaystyle f_{2}$ $\displaystyle X^{A}_{2}$
$\displaystyle=$ $\displaystyle-D^{A}g_{2}$ (4.10)

We are looking for gauge aspects of the Lie-bracket $[\xi_{1},\xi_{2}]$.
Therefore, substituting the gauge aspects(4) into the excitations (4), we
calculate its components as follows

$[\xi_{1},\xi_{2}]^{v}=X^{A}_{1}D_{A}X_{2}-(1\leftrightarrow 2)$ (4.11)

and

$\displaystyle[\xi_{1},\xi_{2}]^{r}$
$\displaystyle=\xi_{1}^{a}\partial_{a}\xi_{2}^{r}-(1\leftrightarrow 2)$
$\displaystyle=r(-\frac{1}{2}X^{A}_{1}D_{A}D_{B}X^{B}_{2})+(\frac{1}{4}D^{2}X_{1}D_{B}X^{B}_{2}-\frac{1}{2}X^{A}_{1}D_{A}D^{2}X_{2}-\frac{1}{2}D^{A}X_{1}D_{A}D_{B}X^{B}_{2})$
$\displaystyle+\frac{1}{r}(-\frac{1}{2}D^{A}X_{1}D_{A}D^{2}X_{2})-\frac{1}{4}\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r}\delta_{A\theta}(rD_{B}X^{B}_{1}+D^{2}X_{1})(X^{A}_{2}-\frac{1}{r}D^{A}X_{2})$
$\displaystyle-(1\leftrightarrow 2)$ (4.12)

The last component of this Lie-bracket $[\xi_{1},\xi_{2}]$ is as follows

$\displaystyle[\xi_{1},\xi_{2}]^{A}$
$\displaystyle=X_{1}^{B}D_{B}X^{A}_{2}+\frac{1}{r^{2}}(\frac{1}{2}D^{2}X_{1}D^{A}X_{2}+D^{B}X_{1}D_{B}D^{A}X_{2})$
$\displaystyle+\frac{1}{r}(\frac{1}{2}D_{B}X^{B}_{1}D^{A}X_{2}+X_{1}^{B}D_{B}D^{A}X_{2}+D^{B}X_{1}D_{B}X^{A}_{2})$
$\displaystyle-(1\leftrightarrow 2)$ (4.13)

The only difference in the components of the bracket $[\xi_{1},\xi_{2}]$,
(4.11)-(4), compared to the similar case in the Schwarzschild black hole
calculations in [23] is related to the $r$-component of the Lie bracket in (4)
compared to the other components in (4.11) and (4). This component has four
more terms in related to the Schwarzschild case in [23, 24] that are
proportional to the angular momentum. Since the Lie-bracket
$[\xi_{1},\xi_{2}]$ itself is a vector field, we replace its components
(4.11)-(4) in right hand side of the projection operators (4), (4.7) and (4.8)
and the projection operator for the Lie bracket can be found as follows

$\displaystyle
D_{B}Y^{B}+\frac{1}{r}D^{2}Y+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})Y^{B}\delta_{B\theta}$
$\displaystyle=X^{C}_{1}D_{B}D_{C}X^{B}_{2}$
$\displaystyle-\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\bigg{[}-\frac{r}{2}X^{A}_{1}\partial_{r}D_{A}D_{B}X^{B}_{2}$
$\displaystyle+\frac{1}{4}\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r^{2}}\delta_{A\theta}X^{A}_{2}(D^{2}X_{1}-r^{2}\partial_{r}D_{B}X_{1}^{B})\bigg{]}$
$\displaystyle+\frac{a}{r}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})\bigg{[}X_{1}^{C}D_{C}X_{2}^{B}+D^{C}X_{1}D_{C}X^{B}_{2}$
$\displaystyle+\frac{1}{r}(\frac{1}{2}D_{C}X_{1}^{C}D^{B}X_{2}+X_{1}^{C}D_{C}D^{B}X_{2})\bigg{]}$
$\displaystyle+(\frac{1}{r}-\frac{a}{2r}\frac{\cos\theta}{\sin^{2}\theta})\bigg{[}\frac{1}{r}(\frac{1}{2}D_{C}X_{1}^{C}D^{2}X_{2}$
$\displaystyle+D^{C}X_{1}D_{B}D_{C}X_{2}^{B})\bigg{]}$ (4.14)

This is one of the three projection operators that we neglect $O(r^{-3})$ and
$O(r^{-4})$ terms in its calculation. We have two equations for other
projection operators which are given below. Neglecting the mentioned terms has
been done in the next two operators as well.

$\displaystyle\frac{r}{2}\partial_{r}D_{B}Y^{B}+\frac{2V}{r}Y$
$\displaystyle+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}Y^{B}\delta_{B\theta}$
$\displaystyle=\frac{1}{2}X_{1}^{C}D_{C}D_{B}X_{2}^{B}+\frac{3}{2r}\bigg{(}\frac{1}{2}D_{C}X_{1}^{C}D^{2}X_{2}$
$\displaystyle+D^{C}X_{1}D_{B}D_{C}X_{2}^{B}\bigg{)}-\frac{r}{4}X_{1}^{C}\partial_{r}D_{C}D_{B}X_{2}^{B}$
$\displaystyle+\frac{1}{2r}\bigg{(}\frac{1}{2}\partial_{r}D_{C}X_{1}^{C}D^{2}X_{2}+\frac{1}{2}D_{C}X_{1}^{C}\partial_{r}D^{2}X_{2}$
$\displaystyle+D^{C}X_{1}\partial_{r}D_{B}D_{C}X_{2}^{B}\bigg{)}-\frac{1}{2}X_{1}^{C}D_{C}D_{B}X_{2}^{B}$
$\displaystyle-\frac{a}{2}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\bigg{[}X_{1}^{C}\partial_{r}D_{C}X_{2}^{B}-\frac{1}{r^{2}}(\frac{1}{2}D_{C}X_{1}^{C}D^{B}X_{2}$
$\displaystyle+X_{1}^{C}D_{C}D^{B}X_{2}+D^{C}X_{1}D_{C}X_{2}^{B})\bigg{]}$
$\displaystyle+\frac{2V}{r}\bigg{(}X_{1}^{B}D_{B}X_{2}\bigg{)}+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}\bigg{[}X_{1}^{C}D_{C}X_{2}^{B}$
$\displaystyle+\frac{1}{r}(\frac{1}{2}D_{C}X_{1}^{C}D^{B}X_{2}+X_{1}^{C}D_{C}D^{B}X_{2}+D^{C}X_{1}D_{C}X_{2}^{B})\bigg{]}$
$\displaystyle-\frac{r}{2}X_{1}^{A}\partial_{r}D_{A}D_{B}X_{2}^{B}+\frac{1}{4}\partial_{r}D^{2}X_{1}D_{B}X_{2}^{B}$
$\displaystyle+\frac{1}{4}D^{2}X_{1}\partial_{r}D_{B}X_{2}-\frac{1}{2}X_{1}^{A}\partial_{r}D_{A}D^{2}X_{2}^{B}$
$\displaystyle-\frac{1}{2}D^{A}X_{1}\partial_{r}D_{A}D_{B}X_{2}^{B}+\frac{1}{2r^{2}}D^{A}X_{1}D_{A}D^{2}X_{2}$
$\displaystyle+\frac{1}{4}\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r^{2}}\delta_{A\theta}\bigg{(}D^{2}X_{1}D_{2}^{A}-r^{2}\partial_{r}D_{B}X_{1}^{B}X_{2}^{A}\bigg{)}$
(4.15)

As expected, the second projection operator (4) has more terms than the first
operator (4). The last equation related to the projection operators is as
follows

$\displaystyle\frac{1}{r}D^{2}D_{B}Y^{B}-\frac{1}{r}D^{A}D_{B}Y^{B}\delta_{A\theta}$
$\displaystyle-\frac{2M}{r}D_{B}Y^{B}$
$\displaystyle=\frac{3}{2r}D^{2}D_{B}\big{(}X_{1}^{C}D_{C}X_{2}^{B}+D^{C}X_{1}D_{C}X_{2}^{B}$
$\displaystyle+\frac{1}{2}D_{C}X_{1}^{C}D^{B}X_{2}+X_{1}^{C}D_{C}D^{B}X_{2}\big{)}$
$\displaystyle-\frac{2M}{r^{2}}D_{B}\big{(}X_{1}^{C}D_{C}X_{2}^{B}\big{)}$
$\displaystyle-\frac{3}{4r^{2}}D^{2}D_{B}\big{(}X_{1}^{C}D_{C}X_{2}^{C}+\frac{1}{2}D_{C}X_{1}^{C}D^{B}X_{2}$
$\displaystyle+X_{1}^{C}D_{C}D^{B}X_{2}+D^{C}X_{1}D_{C}X_{2}^{B}\big{)}$
$\displaystyle+\frac{1}{r}D^{2}\bigg{[}-\frac{1}{2}X^{A}_{1}D_{A}D_{B}X_{2}^{B}-\frac{1}{2}X_{1}^{A}\partial_{r}D_{A}D_{B}X_{2}^{B}$
$\displaystyle+\frac{1}{4}\partial_{r}D^{2}X_{1}D_{B}X_{2}^{B}+\frac{1}{4}D^{2}X_{1}\partial_{r}D_{B}X_{2}^{B}$
$\displaystyle-\frac{1}{2}X_{1}^{A}\partial_{r}D_{A}D^{2}X_{2}-\frac{1}{2}D^{A}X_{1}\partial_{r}D_{A}D_{B}X_{2}^{2}\bigg{]}$
$\displaystyle-\frac{a}{4r}\frac{\cos\theta}{\sin^{2}\theta}D^{2}(\partial_{r}D_{B}X_{1}^{B}X_{2}^{A})\delta_{A\theta}-\big{(}1\leftrightarrow
2\big{)}$ (4.16)

We find many terms in the projection operators (4)- (4) are not omitted, while
many of these terms are zero in the Schwarzschild spacetime in [23, 24].

Generally, the Hamiltonian generators $H_{X}$ and $H_{Y}$ which are related to
the symplectic symmetries $X$ and $Y$ could have the following Lie algebra
[23, 24]

$\\{H_{X},H_{Y}\\}=H_{[X,Y]}+K_{X,Y},$ (4.17)

where $K_{X,Y}$ is a central extension which is a c-number.

Substituting $X=\delta_{\xi_{1}}$ and $Y=\delta_{\xi_{2}}$, the Hamilton
generators of the gauge transformations could have the following algebra,
which are accompanied by a central extension, which is a $c$-number [23]

$\\{H_{\xi_{1}},H_{\xi_{2}}\\}=H_{[\xi_{1},\xi_{2}]}+K_{\xi_{1},\xi_{2}}$
(4.18)

where $[\xi_{1},\xi_{2}]$ is the Lie-bracket of vector fields on the spacetime
manifold.

Considering the gauge aspects defined for the algebra (4.18), we get its form
as follows

$\\{H_{X_{1},X_{1}^{A}},H_{X_{2},X_{2}^{A}}\\}=H_{(Y,Y^{A})}+K_{(X_{1},X_{1}^{A}),(X_{2},X_{2}^{A})}.$
(4.19)

Given that the left hand side of the above equation is the variation in the
Hamiltonian generators as
$\big{\\{}H_{(X_{1},X_{1}^{A})},H_{(X_{2},X_{2}^{A})}\big{\\}}$, we obtain the
central term using (3) as follows:

$\displaystyle K_{(X_{1},X_{1}^{A}),(X_{2},X_{2}^{A})}$
$\displaystyle=-\frac{r^{2}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\Big{[}X_{2}^{A}(\frac{1}{r}D_{B}X_{1}^{B}+\frac{1}{r^{2}}D^{2}X_{1}-\partial_{r}D_{B}X_{1}^{B}$
$\displaystyle+\frac{a}{4}\frac{\cos\theta}{\sin^{2}\theta}\partial_{r}D_{B}X_{1}^{B}\delta_{B\theta})\delta_{A\theta}+D_{A}X_{2}^{A}(\frac{r}{2}\partial_{r}D_{B}X_{1}^{B}-\frac{1}{r}D^{2}X_{1}$
$\displaystyle+\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}X_{1}^{B}\delta_{B\theta}+\frac{2M}{r}X_{1}-\frac{1}{2}\partial_{r}D^{2}X_{1}$
$\displaystyle+\frac{5a}{4r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}D^{B}X_{1})+X_{2}(\frac{1}{r}D^{2}D_{B}X_{1}^{B}-\frac{1}{r^{2}}D^{A}D_{B}X^{B}_{1}\delta_{A\theta}$
$\displaystyle-\frac{2M}{r^{2}}D_{B}X_{1}^{B}+\frac{a}{2r^{2}}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})D_{B}X_{1}^{B}+\frac{a}{4r^{2}}\frac{\cos\theta}{\sin^{2}\theta}D^{A}D_{B}X_{1}^{B}\delta_{A\theta}$
$\displaystyle+\frac{1}{2}D^{2}\partial_{r}D_{B}X_{1}^{B})\Big{]}-H_{(Y,Y^{A})}[g_{ab}]$
(4.20)

where we have done the following replacements

$\displaystyle Y$ $\displaystyle\to$ $\displaystyle
X_{1},~{}~{}~{}~{}~{}X^{B}\to X_{2}^{B}$ $\displaystyle Y^{A}$
$\displaystyle\to$ $\displaystyle X_{1}^{A},~{}~{}~{}~{}~{}X\to X_{2}$ (4.21)

We have the freedom to put all surface charges at the reference point $g_{ab}$
equal to zero, as follows

$H_{(Y,Y^{A})}[g^{ab}]=0$ (4.22)

Utilizing the following selection for the gauge aspects into the surface
algebra,

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle f_{1}$ $\displaystyle
X_{1}^{A}$ $\displaystyle=$ $\displaystyle-D^{A}g_{1}$ $\displaystyle X_{2}$
$\displaystyle=$ $\displaystyle f_{2}$ $\displaystyle X_{2}^{A}$
$\displaystyle=$ $\displaystyle-D^{A}g_{2},$ (4.23)

the surface charge algebra for the surface degrees of freedom of a Kerr black
hole can be found as follows

$\\{H_{f_{1},g_{1}},H_{f_{2},g_{2}}\\}=H_{\hat{f},\hat{g}}+K(f_{1},g_{1};f_{2},g_{2})$
(4.24)

where the central term $K(f_{1},g_{1};f_{2},g_{2})$ of this algebra can be
found as follows

$\displaystyle K(f_{1},g_{1};f_{2},g_{2})$
$\displaystyle=+\frac{r^{2}}{16\pi}\oint_{\partial\Sigma}d^{2}x\sqrt{\gamma}\Big{[}D^{A}g_{2}\big{(}-\frac{1}{r}D^{2}g_{1}+\frac{1}{r^{2}}D^{2}f_{1}+D^{2}g_{1}$
$\displaystyle-\frac{a}{4}\frac{\cos\theta}{\sin^{2}\theta}\partial_{r}D^{2}g_{1}\big{)}\delta_{A\theta}+D^{2}g_{2}\big{(}-\frac{r}{2}\partial_{r}D^{2}g_{1}-\frac{a}{r}\frac{\cos\theta}{\sin^{2}\theta}D^{B}g_{1}\delta_{B\theta}$
$\displaystyle+\frac{2M}{r^{2}}f_{1}-\frac{1}{r}D^{2}f_{1}-\frac{1}{2}\partial_{r}D^{2}f_{1}+\frac{5a}{4r^{2}}\frac{\cos\theta}{\sin^{2}\theta}\delta_{B\theta}D^{B}f_{1}\big{)}$
$\displaystyle-
f_{2}\big{(}-\frac{1}{r}D^{2}D^{2}g_{1}+\frac{1}{r^{2}}D^{A}D^{2}g_{1}\delta_{A\theta}+\frac{2M}{r^{2}}D^{2}g_{1}$
$\displaystyle-\frac{a}{2r^{2}}\partial_{\theta}(\frac{\cos\theta}{\sin^{2}\theta})D^{2}g_{1}-\frac{a}{4r^{2}}\frac{\cos\theta}{\sin^{2}\theta}D^{A}D^{2}g_{1}\delta_{A\theta}-\frac{1}{2}D^{2}\partial_{r}D^{2}g_{1}\big{)}\Big{]}$
(4.25)

We can now consider the properties of this algebra. In the first step, we
consider the gauge aspects as follows

$f_{1}=r,~{}~{}g_{1}=0,~{}~{}f_{2}=f,~{}~{}g_{2}=g$ (4.26)

Using the above assumption (4.26) and replacing them in (4), we have the
central term as follows

$K(f_{1},g_{1};f_{2},g_{2})=0$ (4.27)

So we conclude that

$\\{H_{r_{S},0},H_{f,g}\\}=0$ (4.28)

With the above result, Averin [23] considers the $H_{r_{S},0}$ to be
corresponding to the ADM-energy subtracted of by the energy passing through
future null infinity and the portion of the event horizon between the location
of $\partial\Sigma$ and the horizon’s future end point. According to (4.28),
he considers the $H_{r_{S},0}$ as the same as the ADM-energy and then
concludes that the surface degrees of freedom are gapless excitations that
they keep the ADM-energy invariant. We can achieve the same results in the
Kerr spacetime, and we can consider the surface degrees of freedom that
provide soft black hole hair.

We have found the gauge aspects of a Kerr black hole as functions on $S^{2}$.
These surface degrees of freedom (3) justify the surface charge algebra (4.19)
with respect to the Poisson-bracket. This is an evidence for a theory living
on a lower dimension that describes part of the phase space near the Kerr
solution $g_{ab}$. This is another realization of the holographic principle
[42, 43] for a Kerr black hole spacetime.

Finding the surface degrees of freedom from (3) does not mean the microstates
as black hole soft hair have zero energy. The energy of the physical
microstates of the Kerr black hole can be given by the black hole mass and
this is not zero in the Kerr case [24]. Kerr black holes with non-zero mass
$M$ and spin $J$ have a hidden conformal symmetry [11]. This symmetry on soft
modes that appear on the low-energy theory, can not be found in a near horizon
region of spacetime. The symmetry can be found in the near horizon region of
the phase space [12, 25, 26, 27].

We have been able to find the surface degrees of freedom for the Kerr black
hole, which are the gauge aspects which can be defined as some functions on
$S^{2}$. Utilizing the Poisson bracket, the algebra of the gauge aspects can
be given by (4.19). Given the surface degrees of freedom and their algebra, we
can claim to have arrived at a lower-dimensional theory that describes part of
the phase space. This result is in consistent with the result in [11] that
suggest existence of a hidden conformal symmetry that emerges in the near
horizon of the phase space. The coordinates of this part of the phase space
are determined by gauge aspects defined on a $2D$ sphere that this is a reason
for the lower dimensions of the theory that describes the near event horizon
region.

##  5 Conclusion

We have extended a phase space formalism to find a hidden symmetry in the near
horizon region of the Kerr black hole and to describe a possible Kerr/CFT
holography that can be a good tool for description of mechanics of the black
hole. We have presented an approach to find the physical gauge degrees of
freedom that can be the responsible for the black hole microstates. These
states are the Bogoliubov gapless modes that their degeneracy can be lifted by
the quantum effects. In this conclusion section, we analyze the results in
more details.

The 2D theory on the event horizon that we have found in the last section
describes part of the phase space. The part of the phase space is near the
reference point $g_{ab}$ of the Kerr spacetime. The gauge aspects are gapless
excitations of a Kerr black hole. We guess the 2D theory on the event horizon
is a conformal field theory which is related to scale-invariance of the system
which include the gapless modes. This symmetry can be lifted by a conformal
anomaly which is seen by the appearance of central extensions in the Virasoro
algebra in the calculation of quantum effects. The suggested theory to
describe the near horizon region is a 2D conformal field theory (CFT) where
the reason for being two-dimensional goes back to the gauge excitations
dependences on 2D sphere coordinates, and the reason for being conformally
invariant is due to the scale invariance as mentioned in section 1.

A necessary condition for the dual theory to be conformally invariant is that
it has a 2D-stress tensor with the Virasoro-algebra. There, the Virasoro-
generators are the generators of the gauge transformations and the generators
from the gauge aspects satisfy a Virasoro algebra as follows

$\\{H_{n},H_{m}\\}=(m-n)H_{m+n}+K_{m,n},$ (5.1)

where the bracket $\\{,\\}$ is the Dirac bracket on the gauge fixed space and
the central term $K_{m,n}$ could have the following relation [12, 25, 26, 27]

$K_{m,n}=\delta H(\xi_{n},\mathcal{L}_{\xi_{m}}g;g),$ (5.2)

where the Hamiltonian generators utilizing the gauge aspects $(f_{n},g_{n})$
could be as follows

$H_{n}:=H_{(f_{n},g_{n})}.$ (5.3)

The authors in [27] write the diffeomorphism vector fields $\xi_{n}$ using the
Kerr metric in the conformal coordinates and are able to show that these
vectors could satisfy two copies of the Witt-algebra as follows

$\displaystyle\big{[}\xi_{m},\xi_{n}\big{]}$ $\displaystyle=$
$\displaystyle-i(m-n)\xi_{m+n}$
$\displaystyle\big{[}\bar{\xi}_{n},\bar{\xi}_{m}\big{]}$ $\displaystyle=$
$\displaystyle-i(m-n)\bar{\xi}_{m+n}$
$\displaystyle\big{[}\xi_{n},\bar{\xi_{m}}\big{]}$ $\displaystyle=$
$\displaystyle 0$ (5.4)

Since our initial metric and the metric used by the authors in [12, 25, 26,
27] are the Boyer-Lindquist metric, we can claim that such vectors can be
found in the Bondi coordinate as well to satisfy the With algebra. In the
Bondi coordinates and utilizing the fall-off conditions, we can use the
results of the central terms in [12, 25, 26, 27] as follows

$K_{m,n}=\frac{1}{12}aMm^{3}\delta_{m+n}$ (5.5)

where its dependence on the angular momentum can be seen from the
$a$-dependence terms in (4). Of course, in order to the central term to be
independent of the black hole temperature, the Wald-Zoupas counterterm must
also be added to the Hamiltonian variation, which has been done in [27].

Finding the Virasoro-algebra is a good reason why the dual theory we found in
the section 4 could be a conformally invariant one. This means that the
Virasoro-generators made of the surface degrees of freedom could justify the
algebra. In other words, the surface degrees of freedom are good candidates
for the soft hair of the black hole. Of course, it should be noted that the
part of the phase space does not cover the whole Kerr-family. The dual theory
that we have found out describes part of the phase space near the reference
point and its gapless excitations. The microstates that are candidates the
black hole soft hair must not have zero energy. To satisfy this condition, it
is needed the Virasoro zero modes $H_{0}$ and $\bar{H}_{0}$ that measure the
mass and angular momentum of the Kerr black hole [24] is non-zero. This could
be a good sign that the Kerr/CFT duality can be a good candidate to describe
the black hole soft hair.

In this way, we have a lower dimensional theory for the Kerr black hole that
governs the part of the phase space related to the physical microstates. Is
this theory a correct one? What kind of the conformal field theory is
considered to describe the Kerr black hole spacetime? Another check that one
can do to see whether the theory is a correct candidate is related to the
black hole entropy. Utilizing the central charges, the entropy can be obtained
with the help of the Cardy formula. The result obtained in [12, 25, 26, 27]
can also be used in the approach we used.

That, the part of the phase space can be responsible for the physical
microstates, is obtained from a field theoretic point of view. In a bosonic
and self-interactive field theory, when the stationary critical point of the
theory has been accompanied by some gapless modes, the degeneracy of the modes
can be lifted by the quantum effects [9]. As the field theoretic states, the
Kerr family of the stationary and asymptotically flat solutions of the
Einstein gravity are the critical field configurations and these states should
have the gapless modes. We have these modes in our analysis that are the
physical origins of the black hole entropy.

As the approach utilized in [23] and [24], we could deduce the Hamiltonian
generators of the diffeomorphisms contains the Virasoro generators of a
possible CFT that governs the black hole microstates. We can not find more
details on the CFT by this approach. As stated in [24], the approach to deduce
the existence of the Virasoro generators is very sensitive to the choice of
the phase space related to the microstates. In addition, according to the
Virasoro algebra (4.19), it is possible to count the degeneracy of the states.
By finding the degeneracy, we can compare it to the expected entropy. If we
can go this way, we can deduce the theory covers all degrees of freedom of the
Kerr black hole. So far we have been able to present a good candidate to
describe the Kerr black hole soft hair and the Kerr/CFT duality. We leave the
consistency check on the entropy counting to clarify more the 2D CFT for the
future investigations.

## Acknowledgements

We thank Artem Averin for insightful discussions and commenting on a draft of
this paper.

##  Appendix A Appendix

In this section, we present the Christoffel coefficients of the Kerr
spacetime. We use the following well-known relation

$\Gamma^{a}_{bc}=\frac{1}{2}g^{ad}(\partial_{c}g_{d}a+\partial_{b}g_{d}c-\partial_{d}g_{b}c)$
(A.1)

For $r\to\infty$ we have the following coefficients

$\displaystyle\Gamma^{v}_{vr}$ $\displaystyle=$
$\displaystyle\Gamma^{V}_{v\phi}=\Gamma^{v}_{ra}=\Gamma^{v}_{r\theta}=\Gamma^{v}_{\theta\phi}=0$
$\displaystyle\Gamma^{r}_{v\phi}$ $\displaystyle=$
$\displaystyle\Gamma^{r}_{r\phi}=\Gamma^{r}_{\theta\phi}=0$
$\displaystyle\Gamma^{\theta}_{v\phi}$ $\displaystyle=$
$\displaystyle\Gamma^{\theta}_{rr}=\Gamma^{\theta}_{r\phi}=\Gamma^{\theta}_{\theta\phi}=0$
$\displaystyle\Gamma^{\phi}_{va}$ $\displaystyle=$
$\displaystyle\Gamma^{\phi}_{rr}=\Gamma^{\phi}_{r\theta}=\Gamma^{\phi}_{\theta\phi}=\Gamma^{\phi}_{\phi\phi}=0$
$\displaystyle\Gamma^{\theta}_{vv}$ $\displaystyle=$
$\displaystyle\Gamma^{\theta}_{vr}=\Gamma^{\theta}_{\theta v}=O(r^{-4})$

which are the coefficients that are either exactly zero or are approximately
zero. The non-zero coefficients are as follows

$\displaystyle\Gamma^{r}_{rr}$ $\displaystyle=$ $\displaystyle O(r^{-3})$
$\displaystyle\Gamma^{r}_{AB}$ $\displaystyle=$ $\displaystyle rV\gamma_{AB}$
$\displaystyle\Gamma^{v}_{AB}$ $\displaystyle=$ $\displaystyle-r\gamma_{AB}$
$\displaystyle\Gamma^{\phi}_{r\phi}$ $\displaystyle=$
$\displaystyle\Gamma^{\theta}_{r\theta}=\frac{1}{r}$
$\displaystyle\Gamma^{v}_{vv}$ $\displaystyle=$
$\displaystyle\Gamma^{r}_{vv}=\frac{M}{r^{2}}$
$\displaystyle\Gamma^{v}_{\theta\theta}$ $\displaystyle=$ $\displaystyle-r$
$\displaystyle\Gamma^{\theta}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r}$
$\displaystyle\Gamma^{r}_{r\theta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\cos\theta}{\sin^{2}\theta}\frac{a}{r}$ (A.3)

In the calculation of the coefficients, we also noted that the general form of
the metric also applies to the Bondi gauge.

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