# Boundary conditions for Ashtekar-Barbero variables in the context of
asymptotically flat spacetimes which lead to supertranslations at spatial
infinity

Sepideh Bakhoda

Department of Physics, Beijing Normal University, Beijing 100875, China
<EMAIL_ADDRESS>
###### Abstract

This paper delves into the exploration of suitable boundary conditions for the
asymptotically flat scenario of general relativity presented in terms of
Ashtekar-Barbero variables. While the standard parity conditions have been
extensively studied in [1, 2], it turns out that they fail to produce non-
trivial supertranslations at spatial infinity. We propose new parity
conditions for the Ashtekar-Barbero variables that do yield non-trivial
supertranslation charges at spatial infinity. We compare our findings with
those presented in [3] and demonstrate that the new boundary conditions ensure
the finiteness of the symplectic structure. Moreover, when embarking on the
quest for appropriate parity conditions, it is essential to ensure that the
selected parities remain invariant under hypersurface deformations. Given that
working with Ashtekar-Barbero variables provides more asymptotic structure as
compared to the ADM variables, it is shown that by fixing the Lagrange
multiplier corresponding to the Gauss constraint, the invariance of certain
parity conditions can be guaranteed.

##  1 Introduction

In recent years, there has been a growing interest in the study of boundaries
[4] in general relativity (GR) [5], particularly with respect to the
asymptotic symmetries of spacetimes. Bondi, van der Burg, and Metzner [6]
discovered, and Sachs [7] later confirmed, that the asymptotic region of an
asymptotically Lorentzian flat spacetime has a richer structure than the
traditional Poincaré group, which consists of translations, rotations, and
boosts. The BMS group, a newly discovered symmetry group, is an extension of
the finite-dimensional Poincaré group augmented by infinite-dimensional
supertranslations [7]. Strictly speaking, the asymptotic region of an
asymptotically flat spacetime can be viewed as an infinite set of classical
vacua in general relativity. In this context, supertranslations describe the
transformations that connect these vacua.

The BMS symmetry was initially observed at null infinity [6, 7]. Nevertheless,
studying the BMS symmetry at spatial infinity holds significant importance as
well. There are four distinct motivations for analyzing the asymptotic
structure of gravity at spatial infinity on spacelike hypersurfaces.

First, supertranslations that are inextricably linked to the asymptotic
structure of spacetime are accountable for the emission of gravitational
radiation towards null infinity. Furthermore, it is well-known that
gravitational radiations have the potential to devastate the typical
smoothness requirements imposed at null infinity [10]. Hence, a fruitful
approach to investigating the conditions on Cauchy data that yield a
sufficiently smooth null infinity involves exploring the symmetries of the
theory in a context that separates the presence of BMS symmetry from
gravitational radiation.

Second, soft graviton theorems can be interpreted as Ward identities for the
BMS asymptotic symmetries. Ward identities are mathematical equations that
relate the symmetries of a system to its conserved charges [8]. However,
constructing the BMS charges that generate the BMS symmetries in a canonical
way is challenging. One difficulty is that, at null infinity, it is more
appropriate to consider fluxes rather than charges. Fluxes are measures of the
flow of energy and momentum into or out of a system. Unfortunately, fluxes are
not conserved when they are non-zero. Additionally, the BMS symmetries are not
generated in a canonical way, and the association of functions with these
symmetries is complex. Hypersurfaces that reach null infinity are non-Cauchy,
meaning that they do not capture the entire dynamical evolution of the system.
Only when the fluxes at null infinity vanish can a standard Hamiltonian
picture be recovered. In conclusion, constructing the BMS charges that
canonically generate the BMS symmetries is a challenging task due to the non-
conservation of fluxes at null infinity and the non-canonical generation of
the symmetries themselves.

Three, pioneer studies [9, 1, 2] of the Hamiltonian structure at spatial
infinity did not identify the BMS group as a group of physical symmetries,
leading to a contradiction with results from null infinity. Therefore, it is
important to resolve this tension in order to gain a better understanding of
the symmetries of gravity.

Finally, our main motivation for studying BMS symmetries at spatial infinity
lies in the quantum formulation of the theory, which is inherently explored on
Cauchy hypersurfaces. The BMS algebra is manifested in the quantum theory
through charges that act in the Hilbert space of states. These charges should
have an expression at spatial infinity in the ADM formulation of evolution,
which is based on foliations that approach asymptotic parallel hyperplanes,
representing inertial observers at infinity. Our ultimate objective is to
examine the quantum properties of the symmetry and its associated charges
within the framework of Loop Quantum Gravity (LQG) [19].

Since LQG is expressed in terms of the Ashtekar-Barbero variables [11, 12],
the initial step towards achieving our goal is to determine appropriate
boundary conditions for the canonical variables that allow for the emergence
of BMS charges at spatial infinity. This has already been done in terms of the
ADM variables in [3]. In their approach, they have proposed a new boundary
conditions compared to the boundary conditions in the earlier work [9]. The
boundary conditions employed in [9] at spatial infinity, with the intention of
ensuring finite angular momentum, also result in all BMS charges becoming
identically zero. This outcome, as demonstrated in [9], arises due to the
parity conditions imposed on the leading order of the metric and its conjugate
momentum as one moves towards spatial infinity. Therefore, to reconcile the
tension between the asymptotic structure at spatial infinity and the emergence
of the BMS algebra at null infinity, it seems necessary to adopt boundary
conditions at spatial infinity that differ from those of [9]. Simply
discarding the standard parity conditions is not feasible, as it leads to
logarithmic divergence of the symplectic structure, angular momentum, and
boost charges [20]. In the study by Henneaux and Troessaert [3], they
discovered alternative parity conditions for the leading terms of the metric
and its conjugate momentum. These parity conditions preserve finiteness while
allowing for a well-defined and nontrivial action of the BMS algebra. The key
to their approach is to set the leading order terms of the constraints to
zero. This is an additional restriction, but it is very mild since the leading
terms of the constraints vanish on-shell anyway, and therefore do not remove
any solutions.

In this paper, we investigate whether employing the same approach is possible
to identify suitable boundary conditions for Ashtekar-Barbero variables,
utilized in a LQG. The challenge lies in the increased asymptotic structure
that must be determined due to the freedom in selecting the internal SU(2)
frame, describing the internal orientation. However, to reproduce the ADM
results from, it is necessary to fix the internal frame at the asymptotic
boundary [1, 2]. This constraint is not inappropriate, as an SU(2) charge
should not hold any physical significance in general relativity. The optimal
parity conditions simultaneously render the asymptotic symmetry generators and
the symplectic 2-form finite while producing an integrable and finite charge.
The fall-off and parity conditions of the additional degrees of freedom must
be chosen in a manner that satisfies all these requirements concurrently.
Furthermore, a challenging aspect of this work is ensuring that all imposed
parities and fall-off conditions are preserved by the hypersurface
deformation, which is a based task to tackle.

Our paper is structured as follows:

In Section 2, we begin by reviewing some classic background information on the
ADM Hamiltonian treatment of asymptotically flat spacetimes, with particular
emphasis on the parity conditions proposed by [9] and [3]. We then proceed to
discuss the Ashtekar-Barbero variables, deriving their fall-off conditions in
asymptotically spherical coordinates. This is necessary because the new
boundary conditions are most conveniently expressed in this coordinate system.
Next, we recall the standard boundary conditions for the Ashtekar-Barbero
variables obtained in [1] and [2], as well as the strategy employed by these
papers to fix the internal frame at the asymptotic boundary.
In Section 3, we begin by stating a theorem that introduces fall-off and
parity conditions for the canonical variables and expresses and fixes some of
the leading terms of the Lagrange multipliers. We then spend the remainder of
the section proving the theorem. Specifically, we show that the symplectic
form is finite and well-defined, the boundary conditions are preserved by
hypersurface deformations, and the constraints are well-defined and
functionally differentiable for asymptotic translations and do yield to non-
trivial supertranslations at spatial infinity. Thus, we derive a finite and
integrable charge for infinite-dimensional supertranslations.
In Section 4, an analysis is carried out to assess the correlation between our
findings and the outcomes presented in [3], specifically regarding the
retrieval of the supertranslation charge within the ADM formulation.
Furthermore, a discussion is provided on the distinction observed in the
boundary terms when employing ADM variables in comparison to Ashtekar-Barbero
variables.
In the final section, we summarize our findings and provide an outlook. The
conclusive outcomes of some lengthy calculations have been included in an
appendix.

##  2 Background

This section is dedicated to establishing notation and outlining the necessary
steps for verifying the suitability of boundary conditions. Additionally, a
brief review of previous work will be provided.

In this paper, we employ the Hamiltonian formalism of GR. Consequently, it is
important to emphasize that this formulation assumes a foliation of spacetime
into spacelike hypersurfaces. Consistent with the canonical approach, we
select a Cauchy hypersurface $\Sigma$ and define cartesian coordinates
$x^{a}=(x,y,z)$ on it. The region at spatial infinity is identified as
$r\to\infty$, where $r=x^{a}x_{a}$.

Appropriate boundary conditions are those that satisfy the following
requirements collectively. These requirements will be revisited throughout the
paper. Consistent boundary conditions are those that:

  1. i)

Ensure the symplectic structure is well-defined.
A well-defined symplectic structure is crucial for working in the phase space
and calculating Poisson brackets. Therefore, it is necessary to ensure that
the proposed boundary conditions do not cause the symplectic structure to
become divergent as one approaches infinity.

  2. ii)

Remain invariant under hypersurface deformations.
To establish a consistent theory, it is essential for the boundary conditions
to remain the same across all slices. In other words, they must be invariant
as one moves from one hypersurface to the next.

  3. iii)

Enable the Hamiltonian generators of the asymptotic symmetries to be well-
defined and integrable.
The surface integrals that yield the charges associated with the asymptotic
symmetries should be finite and integrable. By integrable, we mean that the
variation can be extracted from the integral in the surface integral.
Mathematically speaking, the variation of the surface charge is a one-form in
field space obtained by performing integration by parts on the bulk generator.
This one-form must be exact.

This requirement can be explained in another way:
In gauge theories, including GR, one has to work with some constraints whose
vanishing represents equations relating the canonical variables. The
constraint surface of the phase space is defined by the vanishing of these
constraint functionals. Should these constraints meet the first-class
criterion in Dirac’s terminology [14], they are responsible for generating
gauge transformations. Consequently, it becomes necessary to calculate Poisson
brackets with these constraint functionals. In order to compute Poisson-
brackets between the constraint functionals and different functions on the
phase space, it is essential for them to be both finite and functionally
differentiable.

If, in the presence of a boundary, it occurs that the generators of gauge
transformations are not functionally differentiable, one can address this
issue by following the subsequent procedure: Compute the variation of the
constraint, which involves obtaining a surface integral. If the variation’s
volume term is well-defined, it yields the desired functional derivative of
the functional that we aim to establish as well-defined. Subtract the surface
term from the variation of the original constraint. If, after this process,
the resulting surface term is found to be exact, meaning it can be expressed
as the variation of a surface integral, then one has acquired an expression
that is functionally differentiable and, if fortunate, is already finite and
so it is well-defined.

In a totally constrained system, the calculation of the charge requires the
consideration of the variation of the sum of the smeared constraints and the
identification of appropriate surface terms that can be added to the sum to
ensure differentiability. In mathematical terms, let us consider a fully
constrained system characterized by canonical variables $(q^{a},p_{a})$ and
first-class constraints $C_{I}$ with corresponding smearing functions
$\lambda^{I}$. The variation can then be expressed as

$\delta C_{I}[\lambda^{I}]=\int d^{3}x\;(\delta_{\lambda}p_{a}\delta
q^{a}-\delta_{\lambda}q^{a}\delta p_{a})+\mathcal{B}_{\lambda}(\delta
q^{a},\delta p_{a})$ (2.1)

Here, $\mathcal{B}_{\lambda}[\delta q^{a},\delta p_{a}]$ represents the
surface term. Initially, it is crucial to ensure that
$\mathcal{B}_{\lambda}[\delta q^{a},\delta p_{a}]$ converges as the boundary
is approached. Subsequently, it must be verified if this term can be expressed
as the variation of a surface integral. If both of these criteria are met, the
charge is determined by $\mathcal{Q}_{\lambda}(q^{a},p_{a})$, defined as

$\mathcal{B}_{\lambda}(\delta q^{a},\delta
p_{a})=-\delta\mathcal{Q}_{\lambda}(q^{a},p_{a}).$ (2.2)

###  2.1 Boundary conditions for ADM variables

Within the framework of the ADM formalism, the spatial metric tensor $q_{ab}$
of $\Sigma$, as well as the lapse function $N$ and the shift vector $N^{a}$,
along with their conjugate momenta $\pi^{ab}$, $\Pi$, and $\Pi_{a}$,
respectively, serve as the canonical variables. To simplify notation, we
define $\textbf{N}:=(N,N^{a})=(N,\vec{N})$. Upon deriving the ADM action, it
becomes apparent that the action is independent of the time derivatives of $N$
and $N^{a}$. Consequently, this leads to the establishment of primary
constraints $\Pi=0$ and $\Pi_{a}=0$. Regarding the conjugate momentum of the
metric, it can be determined that $\pi^{ab}=\sqrt{q}(K^{ab}-Kq^{ab})$, where
$q$ denotes the determinant of the spatial metric $q_{ab}$ and
$K_{ab}=\frac{1}{2N}(\dot{q}_{ab}-\mathcal{L}_{\vec{N}}q_{ab})$ represents the
extrinsic curvature of $\Sigma$ with $K$ representing its trace.

Stability of the primary constraints shows that the secondary constraints are
(for a complete review of the geometrodynamics of GR the reader refer to [18])

$\displaystyle H_{a}[N^{a}]$
$\displaystyle:=-2\int_{\Sigma}d^{3}x\;N^{a}q_{ac}D_{b}P^{bc}$ (2.3)
$\displaystyle H[N]$
$\displaystyle:=-\int_{\Sigma}d^{3}x\;N\left(\frac{s}{\sqrt{q}}\left[q_{ac}q_{bd}-\frac{1}{2}q_{ab}q_{cd}\right]\pi^{ab}\pi^{cd}+\sqrt{q}R\right)$
(2.4)

where $R$ is the Ricci scalar of the spatial hypersurface $\Sigma$, and
$D_{a}$ is the Levi-civita connection associated to $q_{ab}$. By imposing
stability of these constraints under evolution, no tertiary constraints arise.
The complete introduction of the phase space of the theory is achieved through
the definition of the symplectic structure. Denoted as
$\Omega(\delta_{1},\delta_{2})$, this symplectic structure is given by

$\Omega(\delta_{1},\delta_{2})=\int_{\Sigma}d^{3}x\;(\delta_{1}q_{ab}\delta_{2}\pi^{ab}-\delta_{2}q_{ab}\delta_{1}\pi^{ab})$
(2.5)

The variation of a phase space function $\mathcal{F}[q,\pi]$ is defined as
$\delta\mathcal{F}=\Omega(\delta_{\mathcal{F}},\delta)$ where $\delta=(\delta
q_{ab},\delta\pi^{ab})$ and
$\delta_{\mathcal{F}}=(\delta_{\mathcal{F}}q_{ab},\delta_{\mathcal{F}}\pi^{ab})$
and $\delta_{\mathcal{F}}$ is simply the Hamiltonian vector field of
$\mathcal{F}$. Then the Poisson bracket of two given space time functions
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ is defined by
$\\{\mathcal{F}_{1},\mathcal{F}_{2}\\}:=\Omega(\delta_{\mathcal{F}_{1}},\delta_{\mathcal{F}_{2}})$
[2].

####  2.1.1 R-T boundary conditions

After providing this brief overview, we are now prepared to explore the
examination of the asymptotic region using the canonical formulation of GR. A
spacetime is considered asymptotically flat111Mathematically rigorous
definitions for an asymptotically flat spacetime exist, which are beyond the
scope of this paper [13]. if, outside of a compact region, the metric follows
the behavior $g_{\mu\nu}=\delta_{\mu\nu}+\frac{1}{r}h_{\mu\nu}$, where
$h_{\mu\nu}$ is a tensor on the asymptotic 2-sphere ($\partial\Sigma=S^{2}$).
In order to make use of the Hamiltonian formalism, it is essential to
comprehend the decay behaviors of the variables $q_{ab}$ and $\pi^{ab}$.
Although there is no indication of the decay behavior of the latter in the
fall-off behavior of $g_{\mu\nu}$, the former can be directly derived from it.
Demanding the symplectic structure (2.5) to be finite, one arrives at this
conclusion that $\pi^{ab}=O(r^{2+\epsilon})$ (for more detail look at [9] or
the second reference in [19]). However, it is precisely the finiteness of the
ADM momentum that necessitates the decay of $r^{-2}$. Therefore, one must
adhere to the fall-off behaviors as

$\begin{split}q_{ab}&=\delta_{ab}+\frac{1}{r}\bar{h}_{ab}(\vec{\mathbf{n}})+O(r^{-2})\\\
\pi^{ab}&=\frac{1}{r^{2}}\bar{\pi}^{ab}(\vec{\mathbf{n}})+O(r^{-3})\end{split}$
(2.6)

Here, $\bar{h}_{ab}$ and $\bar{\pi}^{ab}$ are tensor fields on the 2-sphere at
spatial infinity ($\vec{\mathbf{n}}=\frac{\vec{x}}{r}$), and now the objective
is to eliminate the divergences arising in (2.5) through an alternate
technique. This is exactly where the parity conditions come into play. A
possible way to eliminate the divergence in (2.5) is to impose the condition
that the functions $\bar{h}_{ab}$ and $\bar{\pi}^{ab}$ possess opposite
parity. In [9], the following parity conditions were proposed, which are
commonly known as the standard parity conditions. Under the antipodal map on
the asymptotic 2-sphere, $\bar{h}_{ab}$ and $\bar{\pi}^{ab}$ show the
following behavior

$\bar{h}_{ab}\left(-\vec{\mathbf{n}}\right)=\bar{h}_{ab}\left(\vec{\mathbf{n}}\right),\;\;\;\;\;\;\bar{\pi}^{ab}\left(-\vec{\mathbf{n}}\right)=-\bar{\pi}^{ab}\left(\vec{\mathbf{n}}\right)$
(2.7)

In other words, $\bar{h}_{ab}=\text{even}$ and $\bar{\pi}^{ab}=\text{odd}$.
Looking at the power expansion of the symplectic 2-form (2.5), i.e.

$\Omega(\delta_{1},\delta_{2})=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(\delta_{1}\bar{h}_{ab}\delta_{2}\bar{\pi}^{ab}-\delta_{2}\bar{h}_{ab}\delta_{1}\bar{\pi}^{ab})+\text{finite}$
(2.8)

one finds that using the parity conditions (2.7) the coefficient of the
leading logarithmic singularity is zero because the term
$\delta\bar{h}_{ab}\delta\bar{\pi}^{ab}$ is an odd function and its integral
over the sphere vanishes. Here, $d\mathbf{\sigma}$ is the standard measure on
the unit sphere. It is worth mentioning that it is not possible to interchange
the parity conditions (2.7) due to the resulting in disappearance of the ADM
energy momentum.

Now in accordance with (iii), one needs to verify if the constraints (2.3) and
(2.4) are well defined. To accomplish this, the asymptotic behaviors of the
Lagrange multipliers, i.e., $N$ and $N^{a}$, must first be determined. In the
simplest version, in [4] it is shown that the Hamiltonian and diffeomorphism
constraints are finite and functionally differentiable when the lapse and
shift have the following $r\to\infty$ asymptotic behavior

$N=S(\vec{\mathbf{n}})+O(r^{-1}),\;\;\;\;\;\;N^{a}=S^{a}(\vec{\mathbf{n}})+O(r^{-1})$
(2.9)

where $S(\vec{\mathbf{n}})$ and $S^{a}(\vec{\mathbf{n}})$ are arbitrary odd
functions on the unit sphere, i.e.

$S(-\vec{\mathbf{n}})=-S(\vec{\mathbf{n}}),\;\;\;\;\;\;S^{a}(-\vec{\mathbf{n}})=-S^{a}(\vec{\mathbf{n}})$
(2.10)

Thus, the constraints (2.3) and (2.4) with lapse and shift obeying (2.9),
(2.10) generate the gauge transformation of the theory. Note that the charge
corresponding to the so-called supertranslation $S(\vec{\mathbf{n}})$ and
$S^{a}(\vec{\mathbf{n}})$ is identically zero. We will return to this point in
section 3.4.

In the context of asymptotically flat spacetime, it is reasonable to allow for
the decay behaviors of smearing functions $N,N^{a}$ which correspond to
infinitesimal Poincaré transformations. These behaviors can be described as
follows

$\displaystyle
N=b_{a}x^{a}+a(\vec{\mathbf{n}})+O(r^{-1}),\;\;\;\;\;\;N^{a}=b^{a}_{b}x^{b}+a^{a}(\vec{\mathbf{n}})+O(r^{-1})$
(2.11)

where $b_{a}$ and $b_{ab}=-b_{ba}$ represent arbitrary constants, while
$a(\vec{\mathbf{n}})$ and $a^{a}(\vec{\mathbf{n}})$ are arbitrary functions on
the unit sphere. The constants $b_{a}$ serve as parameters for Lorentz boosts,
and the antisymmetric constants $b_{ab}=-b_{ba}$ act as parameters for spatial
rotations. Arbitrary functions $a$ and $a^{a}$ describe angle-dependent
translations whose zero modes are standard translations.

In their paper [9], Regge and Teitelboim demonstrated that the boundary
conditions (2.6) and (2.7) not only ensure the finiteness of the symplectic
structure (2.5), but also possess invariance under hypersurface deformations
with lapse and shift (2.11) provided $a,a^{a}$ (except their zero modes) are
odd functions, thereby satisfying conditions (i) and (ii). However, when
introducing the Lagrange multipliers as (2.11), it turns that the constraints
(2.3) and (2.4) are not well-defined. Following the procedure outlined in
(iii), the authors were able to identify well-defined generators. Hence, it
can be concluded that the boundary conditions (2.6) and (2.7) are appropriate
as they fulfill all the requirements (i)-(iii). The only limitation of their
work is that the resulting charge only includes those associated with Poincaré
symmetries, leaving no scope for charges associated with the non-constant
angle-dependent translations $a,a^{a}$. Therefore, there is no room for the
BMS charge. To address this issue, an alternative set of appropriate boundary
conditions must be sought. This is precisely what Henneaux and Troessaert
accomplished in their study [3]. Below, we briefly outline their approach and
encourage interested readers to consult the original paper for further
details.

####  2.1.2 H-T boundary conditions

In order to investigate the asymptotic region, it is preferable to use
spherical coordinates $(r,x^{A})$, where $x^{A}$ denotes the coordinates on
the 2-sphere. In what follows depending on the definition of the antipodal map
on the 2-sphere, two types of coordinates are utilized

  * 1.

The coordinates for which the antipodal map is $x^{A}\to-x^{A}$.

  * 2.

The traditional coordinates $x^{A}=(\theta,\varphi)$ for which the antipodal
map is $\theta\to\pi-\theta,\varphi\to\varphi+\pi$.

When working with tensorial equations, the choice of coordinates does not
affect the results. However, in terms of associating parities, it is important
to specify the coordinate system being used. It is always possible to perform
a coordinate transformation to switch between these two coordinate systems. In
a slight abuse of notation, we utilize $x^{A}$ to represent both sets of
coordinates (1) and (2). However, if any confusion arises, we will explicitly
clarify which coordinate system is being used. We will refer to them as
coordinate system (1) and coordinate system (2) respectively.

When expressed in the spherical coordinates, equation (2.6) can be written as

$\begin{split}&q_{rr}=1+\frac{1}{r}\bar{h}_{rr}+\frac{1}{r^{2}}h^{(2)}_{rr}+O(r^{-3})\\\
&q_{rA}=\bar{h}_{rA}+\frac{1}{r}h^{(2)}_{rA}+O(r^{-2})\\\
&q_{AB}=r^{2}\bar{\gamma}_{AB}+r\bar{h}_{AB}+h^{(2)}_{AB}+O(r^{-1})\\\
&\pi^{rr}=\bar{\pi}^{rr}+\frac{1}{r}\pi^{(2)rr}+O(r^{-2})\\\
&\pi^{rA}=\frac{1}{r}\bar{\pi}^{rA}+\frac{1}{r^{2}}\pi^{(2)rA}+O(r^{-3})\\\
&\pi^{AB}=\frac{1}{r^{2}}\bar{\pi}^{AB}+\frac{1}{r^{3}}\pi^{(2)AB}+O(r^{-4}),\end{split}$
(2.12)

where $\bar{\gamma}_{AB}$ is the metric on the unit 2-sphere. It should be
noted that when deriving the last three equations in (2.12), one must consider
that $\pi^{ab}$ is not a tensor field but rather a tensor density.
Furthermore, it is possible to assume, without loss of generality, that
$\bar{h}_{rA}=0$, which greatly simplifies the calculations in subsequent
sections. This assumption holds true since $\bar{h}_{rA}=0$ can always be
achieved through a coordinate transformation [3].

Expressed in spherical coordinates, equation (2.11) takes the form:

$\displaystyle N=rb+f+O(r^{-1}),$ (2.13) $\displaystyle N^{r}=W+$
$\displaystyle O(r^{-1}),\;\;N^{A}=Y^{A}+\frac{1}{r}I^{A}+O(r^{-2})$ (2.14)

Here, $b$, an arbitrary function satisfying the condition
$\bar{D}_{A}\bar{D}_{B}b+b\bar{\gamma}_{AB}=0$, serves as the boost parameter,
and $Y^{A}$ is assumed to be the rotation generator, satisfying
$\mathcal{L}_{Y}\bar{\gamma}_{AB}=0$. It should be noted that $\bar{D}_{A}$
denotes the torsion-free connection that is compatible with the metric of the
unit 2-sphere $\bar{\gamma}_{AB}$.

When attempting to compute the surface term $\mathcal{B}_{\lambda}[\delta
q^{a},\delta p_{a}]$ defined in (2.1) for the case of GR with the constraints
given by (2.3) and (2.4), and with smearing functions defined in (2.13) and
(2.14), it is found that $\mathcal{B}_{\mathbf{N}}[\delta
q_{ab},\delta\pi^{ab}]$ exhibits linear divergence, specifically

$\left(\mathcal{B}_{\mathbf{N}}[\delta
q_{ab},\delta\pi^{ab}]\right)_{\text{divergent
part}}=r\int_{S^{2}}d\sigma\;(-2Y^{A}\bar{\gamma}_{AB}\delta\bar{\pi}^{rB}-2\sqrt{\bar{\gamma}}b\delta\bar{k})$
(2.15)

where

$\bar{k}:=\bar{\gamma}^{AB}\bar{k}_{AB},\;\;\;\;\bar{k}_{AB}:=\frac{1}{2}\bar{h}_{AB}+\bar{\lambda}\bar{\gamma}_{AB},\;\;\;\;\bar{\lambda}:=\frac{1}{2}\bar{h}_{rr}$
(2.16)

In the work by Henneaux and Troessaert [3], it was observed that the divergent
term (2.15) can be eliminated with the help of the leading order terms of the
constraints, i.e. $\bar{H},\bar{H}_{A}$ where
$H=:\frac{1}{r}\bar{H}+O(r^{-2})$ and $H_{A}=:\bar{H}_{A}+O(r^{-1})$.
Specifically, the conditions $\bar{H}=0=\bar{H}_{A}$ were imposed even off-
shell, where

$\begin{split}&\bar{H}=\bar{D}_{A}\bar{D}_{B}\bar{k}^{AB}-\bar{D}_{A}\bar{D}^{A}\bar{k}=0,\;\;\;\bar{H}_{A}=\bar{\gamma}_{AB}\bar{\pi}^{rB}+\bar{\gamma}_{AB}\bar{D}_{C}\bar{\pi}^{BC}=0.\\\
\end{split}$ (2.17)

By utilizing equation (2.17), performing integration by parts, and employing
the boost property $\bar{D}_{A}\bar{D}_{B}b+b\bar{\gamma}_{AB}=0$, the
divergent term (2.15) can be eliminated. In the previous work by Regge and
Teitelboim [9], the removal of (2.15) was achieved through parity conditions.
However, in the study conducted by Henneaux and Troessaert [3], the role is
fulfilled by (2.17), enabling the relaxation of parity conditions and the
introduction of different conditions than those proposed by R-T, thus leading
to the revival of the supertranslation charge.

To present the H-T parity conditions, besides the newly defined variables
(2.16) we need to define

$\bar{p}:=2(\bar{\pi}^{rr}-\bar{\pi}^{A}_{A})$ (2.18)

Then in terms of the spherical coordinates (1), the set of parity conditions
on the boundary values proposed in [3] are

$\begin{split}\bar{\lambda}\sim\bar{\pi}^{AB}=even,\;\;\;\;\;\bar{p}\sim\bar{k}_{AB}\sim\bar{\pi}^{rA}=odd,\end{split}$
(2.19)

or in terms of the spherical coordinates (2),

$\begin{split}&\bar{\lambda}\sim\bar{\pi}^{r\varphi}\sim\bar{\pi}^{\theta\theta}\sim\bar{\pi}^{\varphi\varphi}\sim\bar{k}_{\theta\varphi}=\text{even},\\\
&\bar{p}\sim\bar{\pi}^{r\theta}\sim\bar{\pi}^{\theta\varphi}\sim\bar{k}_{\theta\theta}\sim\bar{k}_{\varphi\varphi}=\text{odd}.\end{split}$
(2.20)

Thus, (2.12) together with (2.19) are called the H-T boundary conditions [3].
It is straightforward to see that these boundary conditions remove the
divergent term of the symplectic structure, because

$\begin{split}\Omega(\delta_{1},\delta_{2})&=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{h}_{rr}\delta_{2}\bar{\pi}^{rr}+\delta_{1}\bar{h}_{AB}\delta_{2}\bar{\pi}^{AB}-(\delta_{1}\leftrightarrow\delta_{2})\right)+\text{finite}\\\
&=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{h}_{rr}\delta_{2}\bar{\pi}^{rr}+\delta_{1}\bar{h}_{AB}\delta_{2}\bar{\pi}^{AB}+\delta_{1}\bar{h}_{rr}\delta_{2}\bar{\pi}^{A}_{A}-\delta_{1}\bar{h}_{rr}\delta_{2}\bar{\pi}^{A}_{A}-(\delta_{1}\leftrightarrow\delta_{2})\right)+\text{finite}\\\
&=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{h}_{rr}\delta_{2}[\bar{\pi}^{rr}-\bar{\pi}^{A}_{A}]+\delta_{1}[\bar{h}_{AB}+\bar{h}_{rr}\bar{\gamma}_{AB}]\delta_{2}\bar{\pi}^{AB}-(\delta_{1}\leftrightarrow\delta_{2})\right)+\text{finite}\\\
&=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{\lambda}\delta_{2}\bar{p}+2\delta_{1}\bar{k}_{AB}\delta_{2}\bar{\pi}^{AB}-(\delta_{1}\leftrightarrow\delta_{2})\right)+\text{finite}\end{split}$
(2.21)

and in accordance with (2.19), the terms $\delta\bar{\lambda}\delta\bar{p}$
and $\delta\bar{k}_{AB}\delta\bar{\pi}^{AB}$ are odd functions and their
integral over the sphere are equal to zero. It should be noted that the term
$\delta\bar{h}_{rA}\delta\bar{\pi}^{rA}$ does not contribute to the divergent
term of the symplectic structure due to the assumption $\bar{h}_{rA}=0$.
Through a lengthy calculation, it can be demonstrated that the H-T boundary
conditions remain invariant under hypersurface deformations, thus meeting the
requirements (i) and (ii). As previously mentioned, the divergent part of the
surface term (2.15) is eliminated by setting the leading terms of the
constraints to zero, i.e. (2.17). Furthermore, it has been proven that the
surface term $\mathcal{B}_{\mathbf{N}}[\delta q_{ab},\delta\pi^{ab}]$ is
exact, thereby establishing the integrability of the charge [3]. Therefore,
all the requirements (i)-(iii) are satisfied by H-T boundary conditions (2.7)
and (2.19).

The non-zero supertranslation charge is hence given by

$\begin{split}\mathcal{Q}_{Supertranslation}=\int_{S^{2}}d\sigma\;(4T\sqrt{\bar{\gamma}}\bar{\lambda}+W\bar{p}),\end{split}$
(2.22)

where $T:=f+b\bar{\lambda}+b\bar{k}$ and $W$ are even and odd arbitrary
functions on the unit 2-sphere, respectively. The parities of $T$ and $W$
ensure that the H-T parity conditions (2.19) remain invariant under
hypersurface deformations. Note that the terms given in (2.22) may not
generally vanish as they are determined by integrating arbitrary even
functions.

It is noteworthy to examine why the R-T parity conditions result in a
vanishing supertranslation charge. As mentioned in the concluding paragraph of
section 2.1.1, the non-zero modes of the arbitrary functions $a$ and $a^{a}$
are odd. This can be equivalently expressed in terms of spherical coordinates,
indicating that the non-zero modes of $f$ and $W$ are odd and even,
respectively. Furthermore, when the R-T parity conditions (2.7) are translated
into spherical coordinates, it becomes evident that
$\bar{\lambda}=\text{even}$ and $\bar{p}\sim\bar{k}_{AB}=\text{odd}$, leading
to even parity for the non-zero mode of $T$. Considering all of these factors,
we can arrive at the conclusion that the arbitrary functions present in $T$
and $W$ have opposite parity to $\bar{\lambda}$ and $\bar{p}$, respectively.
Equation (2.22) indicates that these functions possess identically vanishing
surface charges, leaving no room for the BMS symmetry with the parity
conditions (2.7).

###  2.2 Standard boundary conditions for Ashtekar-Barbero variables

The Ashtekar-Barbero formalism relies on the use of tetrad variables to
represent the gravitational field. These variables consist of four covariant
fields denoted as $e^{I}_{\mu}(x)$, where $I,J,\cdots=0,1,2,3$ are flat
indices that are raised and lowered by the metric
$\eta_{IJ}=\text{diag}[s,+1,+1,+1]$. Here, $s$ represents the signature of the
metric ($s=+1$ for Euclidean and $s=-1$ for Lorentzian metric). The metric
variables can be expressed in terms of the tetrad variables using the equation
$g_{\mu\nu}=\eta_{IJ}e^{I}_{\mu}e^{J}_{\nu}$. This formulation introduces an
additional gauge invariance of $SO(3,1)$ into GR. The corresponding canonical
formalism is defined within the temporal gauge $e^{i}_{t}=0$, where
$i,j,\cdots=1,2,3$ are flat three-dimensional indices raised and lowered by
$\delta_{ij}$. In this gauge, the Lorentz group is reduced to $SU(2)$, and the
ADM configuration variables are
$q_{ab}=e^{i}_{a}e^{i}_{b},N=e^{0}_{t},e^{i}_{t}=e^{i}_{a}N^{a}$.

The Ashtekar-Barbero variables [11, 12] consist of the following connection
1-form and electric field

$\displaystyle A^{i}_{a}$ $\displaystyle:=\Gamma^{i}_{a}+\beta K^{i}_{a}$
(2.23) $\displaystyle E^{a}_{i}$ $\displaystyle:=\sqrt{q}\;e^{a}_{i}$ (2.24)

respectively, where $q:=\det(q_{ab})$, $e^{a}_{i}$ is defined by the relations
$e^{a}_{i}e^{j}_{a}=\delta^{i}_{j},\;e^{a}_{i}e^{i}_{b}=\delta^{a}_{b}$, the
one-form $K^{i}_{a}$ is defined through $K_{ab}=:K^{i}_{(a}e^{i}_{b)}$, the
parameter $\beta$ is an arbitrary complex number known as Barbero-Immirzi
parameter [12] and

$\begin{split}\Gamma^{i}_{a}:=&-\frac{1}{2}\epsilon_{ijk}E^{j}_{b}D_{a}E^{b}_{k}\\\
=&-\frac{1}{2}\epsilon^{ijk}E^{b}_{k}\left[E^{j}_{a,b}-E^{j}_{b,a}+E^{c}_{j}E^{l}_{a}E^{l}_{c,b}+E^{j}_{a}\frac{(\det(E))_{,b}}{\det(E)}\right]\end{split}$
(2.25)

is the spin connection associated with $e^{i}_{a}$ in which $E^{i}_{a}$ is the
inverse of (2.24). The phase space coordinated by the pair
$(A^{i}_{a},E^{a}_{i})$ is equipped with the symplectic structure given by

$\Omega(\delta_{1},\delta_{2})=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;(\delta_{1}A^{i}_{a}\delta_{2}E^{a}_{i}-\delta_{2}A^{i}_{a}\delta_{1}E^{a}_{i})$
(2.26)

with respect to which the Ashtekar-Barbero variables (2.23) and (2.24) form a
canonically conjugate pair. In terms of these variables, the constraints of
the theory are expressed in the form [18]

$\displaystyle\mathcal{G}_{i}[\Lambda^{i}]=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\Lambda^{i}\left(\partial_{a}E^{a}_{i}+\epsilon_{ijk}A^{j}_{a}E^{a}_{k}\right)$
(2.27) $\displaystyle
H_{a}[N^{a}]=\frac{-2s}{\beta}\int_{\Sigma}d^{3}x\;N^{a}\left(F^{i}_{ab}E^{b}_{i}-A^{i}_{a}\mathcal{G}_{i}\right)$
(2.28) $\displaystyle
H[N]=\int_{\Sigma}d^{3}x\;\tilde{N}\left[F_{ab}^{i}-(\beta^{2}-s)\epsilon_{imn}K^{m}_{a}K^{n}_{b}\right]\epsilon_{ijk}E^{a}_{j}E^{b}_{k}$
(2.29)

that are known as Gauss, diffeomorphism and Hamiltonian constraints,
respectively. Here,

$F^{i}_{ab}=\partial_{a}A^{i}_{b}-\partial_{b}A^{i}_{a}+{\epsilon^{i}}_{jk}A^{j}_{a}A^{k}_{b},$
(2.30)

and $\Lambda^{i}$ is the Lagrange multiplier corresponding to
$\mathcal{G}_{i}$ and as usual, $N^{a}$ is the shift vector and $\tilde{N}$ is
the densitized lapse function with weight $-1$. The diffeomorphism and
Hamiltonian constraints, i.e. (2.28) and (2.29), are the same as in the ADM
formulation, i.e. (2.3) and (2.4), with $q_{ab}$ and $\pi^{ab}$ expressed in
terms of the Ashtekar-Barbero variables. In addition, there is an extra
constraint, i.e. (2.27), generating the internal rotations. The canonical
Hamiltonian which will be used to compute equation of motions is a linear
combination of the constraints (2.27)-(2.29) and is expressed as

$\mathbf{H}_{\text{can}}=\int_{\Sigma}d^{3}x\;(\mathcal{G}_{i}[\Lambda^{i}]+\mathcal{H}_{a}[N^{a}]+\mathcal{H}[\tilde{N}])$
(2.31)

We are now prepared to discuss the asymptotic behaviors of the Ashtekar-
Barbero variables $(A_{a}^{i},E^{a}_{i})$ which satisfy requirements
(i)-(iii). It is important to note that simply converting the boundary
conditions imposed on the ADM-variables to the new variables does not yield a
comprehensive asymptotic theory for GR written in terms of Ashtekar-Barbero
variables. This is due to the existence of an additional internal $SU(2)$
frame, the asymptotic behavior of which must be determined while satisfying
all consistency requirements (i)-(iii). Reference [1] provides an asymptotic
analysis of the theory for $\beta=i$, which is equivalent to the R-T boundary
conditions described in section 2.1.1. Subsequently, reference [2] presents a
similar analysis for a real arbitrary Barbero-Immirzi parameter $\beta$. In
the remainder of this section, we provide a brief summary of their discussions
and highlight the findings that are necessary to explain our results in the
next section.

Given that the triad 1-form corresponds to the square-root of the 3-metric, it
is reasonable to anticipate that the electric field will exhibit a fall-off
behavior of $E^{a}_{i}=\bar{E}^{a}_{i}+\frac{\bar{f}^{a}_{i}}{r}+O(r^{-2})$.
Here, $\bar{E}^{a}_{i}$ represents the densitized triad of the asymptotic
3-metric at spatial infinity. Stated differently, the associated metric of
$\bar{E}^{a}_{i}$ is considered to be $\delta_{ab}$ appearing in (2.6). It is
important to note that $\bar{E}^{a}_{i}$ is not fixed in general, as it has
the ability to undergo rotations in the internal space, while still
maintaining $\delta_{ab}$ as its associated metric. However, ensuring a well-
defined symplectic structure requires that the electric fields asymptote to a
fixed densitized triad. Thus, we select the fixed, zeroth order asymptotic
electric field to be $\bar{E}^{a}_{i}=\delta^{a}_{i}$, where

$\delta^{a}_{i}=\begin{cases}1&\quad\text{if}\;(a,i)=(x,1),(y,2),(z,3),\\\
0&\quad\text{otherwise.}\\\ \end{cases}$

Subsequently, the boundary conditions (2.6) in terms of the Ashtekar-Barbero
variables can be expressed as

$\begin{split}&E^{a}_{i}=\delta^{a}_{i}+\frac{\bar{f}^{a}_{i}}{r}+O(r^{-2})\\\
&A_{a}^{i}=\frac{\bar{g}^{i}_{a}}{r^{2}}+O(r^{-3})\end{split}$ (2.32)

where and $\bar{f}^{a}_{i}$ and $\bar{g}_{a}^{i}$ are tensor fields defined on
the asymptotic 2-sphere admitting the definite parity conditions [1, 2]

$\begin{split}\bar{f}^{a}_{i}\left(-\vec{\mathbf{n}}\right)=f^{a}_{i}\left(\vec{\mathbf{n}}\right),\;\;\;\bar{g}^{i}_{a}\left(-\vec{\mathbf{n}}\right)=-g^{i}_{a}\left(\vec{\mathbf{n}}\right).\end{split}$
(2.33)

The well-definedness of the symplectic structure (2.26) is ensured by the
fall-off conditions (2.32) and the parity conditions (2.33). Specifically, the
symplectic structure can be expressed as

$\Omega(\delta_{1},\delta_{2})=\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(\delta_{1}\bar{f}^{a}_{i}\delta_{2}\bar{g}^{i}_{a}-\delta_{2}\bar{f}^{a}_{i}\delta_{1}\bar{g}^{i}_{a})+\text{finite}$
(2.34)

Similar to the analysis presented in section 2.1.1, the coefficient of the
leading logarithmic singularity vanishes. This is due to the parity conditions
(2.33) which render the term $\delta\bar{f}^{a}_{i}\delta\bar{g}^{i}_{a}$ an
odd function, hence its integral over the sphere evaluates to zero. Now it is
easy to see the importance of fixing the zeroth order electric field in (2.32)
in order to ensure the convergence of the integral (2.26). If we were to allow
all possible SU(2)-rotated $\bar{E}^{a}_{i}$, the convergence of this integral
could not be guaranteed. This means the requirement (i) is fulfilled.

In order to validate the requirement (iii) regarding these boundary
conditions, it is necessary to obtain well-defined forms of the constraints
(2.27)-(2.29) when the smearing functions include the Poincaré generators
(2.11). To achieve this, the appropriate decay behavior for $\Lambda^{i}$ must
first be determined. Given that the leading term of $\mathcal{G}_{i}$
presented in (2.27) is an odd function with $r^{-2}$ decay, the convergence of
$\mathcal{G}_{i}[\Lambda^{i}]$ is contingent upon the decay condition

$\begin{split}\Lambda^{i}=\frac{1}{r}\bar{\Lambda}^{i}+O(r^{-2})\end{split}$
(2.35)

where $\bar{\Lambda}^{i}$ are even functions defined on the asymptotic
$S^{2}$. It can be readily confirmed that the differentiability of
$\mathcal{G}_{i}[\Lambda^{i}]$ is also ensured by (2.35).

With regards to the diffeomorphism and Hamiltonian constraints, i.e. (2.28)
and (2.29), it is observed that even after subtracting the surface destroying
differentiability, the constraints only converge for translations and not for
boosts and rotations. This situation necessitates modification such that: 1)
the generators remain functionally differentiable and 2) the already available
well-defined generator for translations remains intact up to a pure gauge.
Upon conducting a thorough and meticulous examination, it becomes evident that
the issue at hand stems from the fact that, despite the fixation of the zeroth
order term of the electric field as $\delta^{a}_{i}$, it continues to undergo
rotation within the internal space when moving from one hypersurface to
another. Put differently, $\delta^{a}_{i}$ remains unfixed during hypersurface
deformations. For instance, it is known that under the action of the
diffeomorphism constraint, the variables change according to their Lie
derivative along the shift vector, i.e.
$\\{H_{a}[N^{a}],E^{b}_{i}\\}=\mathcal{L}_{\vec{N}}E^{b}_{i}$. By examining
the asymptotic behavior of this equation, it becomes evident that
$\\{H_{a}[N^{a}],\delta^{b}_{i}\\}=\mathcal{L}_{\vec{R}}\delta^{b}_{i}$, which
is not generally equal to zero. Here, $R$ represents the asymptotic rotations.
This problem did not occur in the ADM variables due to the fact that $\vec{R}$
acts as a Killing vector for the asymptotic 3-metric $\delta_{ab}$, i.e.
$\mathcal{L}_{\vec{R}}\delta_{ab}=0$. However, it should be noted that the
fact that $\vec{R}$ is an asymptotic Killing vector does not necessarily imply
that the asymptotic triad is also Lie-annihilated. It only means that the
asymptotic triad is rotated within the tangent space. Given that the Gauss
constraint is primarily responsible for generating internal rotations, there
is still hope of preventing these rotations by compensating for their effects
through a term proportional to the Gauss constraint equipped with suitable
Lagrange multipliers. This approach has been implemented in [1, 2]. By
subtracting a term proportional to the Gauss constraint from the
diffeomorphism and Hamiltonian constraints, not only is $\delta^{a}_{i}$
fixed, but it also renders the constraints well-defined functionals.
Specifically, it has been revealed that for real $\beta$, the final well-
defined symmetry generators are [2]

$\displaystyle\mathcal{H}_{a}[N^{a}]:=H_{a}[N^{a}]+s\mathcal{G}_{i}[\hat{\Lambda}^{i}]+\text{surface
terms}$ (2.36)
$\displaystyle\mathcal{H}[\tilde{N}]:=H[\tilde{N}]-\beta\;\mathcal{G}_{i}[\mathring{\Lambda}^{i}]+\text{surface
terms}$ (2.37)

with suitable Lagrange multipliers

$\begin{split}&\hat{\Lambda}^{i}=\Lambda^{i}-\frac{1}{2}\epsilon_{ijk}\delta^{j}_{a}\delta^{b}_{k}b^{a}_{b}\\\
&\ring{\Lambda}^{i}=\Lambda^{i}+\delta^{a}_{i}b_{a}.\end{split}$ (2.38)

Although the Gauss terms have been subtracted to eliminate the source of
divergence coming from boosts and rotations, still one needs some surface
terms subtracted to ensure the differentiability. It should be noted that, as
expected, the volume terms added to the constraints,
$s\mathcal{G}_{i}[\hat{\Lambda}^{i}]$ and
$\beta\;\mathcal{G}_{i}[\mathring{\Lambda}^{i}]$ respectively, are
proportional to the Gauss constraint, thereby preserving the invariance of the
translation generator on the constraint surface of the Gauss constraint. The
reader can find the surface terms in the original paper [2] which are derived
based on the specific boundary conditions (2.32) and (2.33). It has been
verified in [1, 2] in more details that the standard boundary conditions
(2.32) and (2.33) are preserved by hypersurfece deformations. This analysis
successfully verifies all the requirements (i)-(iii) for the standard boundary
conditions.

As the boundary conditions (2.32) and (2.33) are equivalent to R-T boundary
conditions, similar to what was discussed in section 2.1.1, there is no scope
for the supertranslation charge. In the following section, we present new
boundary conditions within the Ashtekar-Barbero phase space, without relying
on the ADM expressions, which result in non-zero supertranslation charges at
spatial infinity.

##  3 New boundary conditions for Ashtekar-Barbero variables

In order to provide a comprehensive description of our approach for
strengthening the boundary conditions, it is more practical to employ
spherical coordinates $(r,x^{A})$, wherein $x^{A}$ denotes coordinates on the
sphere. In this section, we predominantly use the second set of spherical
coordinates introduced in section 2.1.2, specifically
$a,b,\cdots\in\\{r,\theta,\varphi\\}$,$A,B,\cdots\in\\{\theta,\varphi\\}$ and
the antipodal map is defined as $\theta\to\pi-\theta,\varphi\to\varphi+\pi$.
In these coordinates, the asymptotic conditions (2.32) are expressed as

$\begin{split}E^{r}_{i}&=r^{2}\sqrt{\bar{\gamma}}\bar{\gamma}^{r}_{i}+r\sqrt{\bar{\gamma}}\bar{f}^{r}_{i}+\sqrt{\bar{\gamma}}\bar{f}^{(2)r}_{i}+O(r^{-1})\\\
E^{A}_{i}&=r\sqrt{\bar{\gamma}}\bar{\gamma}^{A}_{i}+\sqrt{\bar{\gamma}}\bar{f}^{A}_{i}+\sqrt{\bar{\gamma}}\bar{f}^{(2)A}_{i}+O(r^{-1})\\\
A^{i}_{r}&=\frac{\bar{g}^{i}_{r}}{r^{2}}+\frac{\bar{g}^{(2)i}_{r}}{r^{3}}+O(r^{-4})\\\
A^{i}_{B}&=\frac{\bar{g}^{i}_{B}}{r}+\frac{\bar{g}^{(2)i}_{B}}{r^{2}}+O(r^{-3})\end{split}$
(3.1)

where $\bar{\gamma}$ is the determinant of the unit metric $\bar{\gamma}_{AB}$
on the sphere, $\bar{\gamma}^{a}_{i}$ are asymptotic triads satisfying
$\bar{\gamma}^{a}_{i}\bar{\gamma}^{b}_{i}=\bar{\gamma}^{ab}=\text{diag}(1,\bar{\gamma}^{AB})$.
The behavior of the inverse of $E^{a}_{j}$ in the asymptotic region is given
by

$\begin{split}E_{r}^{i}&=\frac{1}{r^{2}}\frac{\bar{\gamma}_{r}^{i}}{\sqrt{\bar{\gamma}}}+\frac{1}{r^{3}}\frac{\bar{f}_{r}^{i}}{\sqrt{\bar{\gamma}}}+O(r^{-4})\\\
E_{A}^{i}&=\frac{1}{r}\frac{\bar{\gamma}_{A}^{i}}{\sqrt{\bar{\gamma}}}+\frac{1}{r^{2}}\frac{\bar{f}_{A}^{i}}{\sqrt{\bar{\gamma}}}+O(r^{-3})\end{split}$
(3.2)

Here, $\bar{\gamma}_{a}^{i}$ represents the inverse of $\bar{\gamma}^{a}_{i}$
satisfying
$\bar{\gamma}_{a}^{i}\bar{\gamma}_{b}^{i}=\bar{\gamma}_{ab}=\text{diag}(1,\bar{\gamma}_{AB})$.
It should be noted that when deriving the first two equations in (3.1) and
(3.2), one must consider that the electric field and its inverse follow the
coordinate transformation rules of tensor densities, rather than those of
tensor fields. Furthermore, from the requirement
$E^{i}_{a}E_{i}^{b}=\delta^{b}_{a}$ and the relations
$\bar{\gamma}^{a}_{i}\bar{\gamma}_{b}^{i}=\delta^{a}_{b}$ and
$\bar{\gamma}^{a}_{i}\bar{\gamma}_{a}^{j}=\delta^{j}_{i}$, we can determine
$\bar{f}_{b}^{i}$ in terms of $\bar{f}^{a}_{i}$ as

$\bar{f}_{b}^{i}=-\bar{\gamma}_{b}^{j}\bar{\gamma}^{i}_{a}\bar{f}^{a}_{j}$
(3.3)

Two properties of $\bar{\gamma}^{a}_{i}$ that will be significant for
subsequent calculations are

$\displaystyle\bar{D}_{A}\bar{\gamma}^{i}_{r}=$
$\displaystyle\bar{\gamma}^{i}_{A}$ (3.4)
$\displaystyle\bar{D}_{A}\bar{\gamma}^{i}_{B}=$
$\displaystyle-\bar{\gamma}_{AB}\bar{\gamma}^{i}_{r}$ (3.5)

Under the action of the antipodal map $\bar{\gamma}_{a}^{i}$ are

$\bar{\gamma}_{i}^{r}\sim\bar{\gamma}_{i}^{\varphi}=\text{odd}\;\;\;\;\bar{\gamma}_{i}^{\theta}=\text{even}$
(3.6)

and their lower spacetime index have the same parity because

$\bar{\gamma}^{i}_{r}=\bar{\gamma}^{r}_{i},\;\;\bar{\gamma}^{i}_{\theta}=\bar{\gamma}^{\theta}_{i},\;\;\bar{\gamma}^{i}_{\varphi}=\bar{\gamma}\bar{\gamma}^{\varphi}_{i}$
(3.7)

Moreover, since
$\\{\bar{\gamma}^{i}_{r},\bar{\gamma}^{i}_{\theta},\bar{\gamma}^{i}_{\varphi}\\}$
forms a basis for the internal space, it proves convenient to use the
expression $\bar{f}^{a}_{i}=\bar{F}^{a}_{b}\bar{\gamma}^{b}_{i}$,
$\bar{g}_{b}^{i}=\bar{G}^{a}_{b}\bar{\gamma}_{a}^{i}$ because in the end we
will associate desired parities to the components

$\displaystyle\bar{F}^{a}_{b}:=\bar{f}^{a}_{i}\bar{\gamma}^{i}_{b}$ (3.8)
$\displaystyle\bar{G}^{a}_{b}:=\bar{g}^{i}_{b}\bar{\gamma}_{i}^{a}$ (3.9)

In general, for any field $\bar{v}^{a}_{i}$ (or $\bar{v}_{a}^{i}$) on the
asymptotic 2-sphere, its components in this basis are defined as
$\bar{v}^{a}_{b}:=\bar{v}^{a}_{i}\bar{\gamma}^{i}_{b}$ (by
$\bar{v}^{b}_{a}:=\bar{v}_{a}^{i}\bar{\gamma}_{i}^{b}$).

The asymptotic behavior for lapse and shift are still assumed to be (2.13) and
(2.14). Since the smearing function for the Hamiltonian constraint (2.29) is
considered to be a scalar density of weight $-1$, we need to derive the
asymptotic behavior of $\tilde{N}$ based on that of $N$ and the relation
$\tilde{N}=\frac{N}{\sqrt{q}}$, namely,

$\tilde{N}=\frac{1}{r}\frac{b}{\sqrt{\bar{\gamma}}}+\frac{1}{r^{2}}\frac{\tilde{f}}{\sqrt{\bar{\gamma}}}+O(r^{-3})$
(3.10)

The relation between $f$ in (2.13) and $\tilde{f}$ in (3.10) is given by

$\tilde{f}=f-\frac{b}{2}(\bar{F}^{r}_{r}+\bar{F}^{A}_{A})$ (3.11)

where we have used the asymptotic expansion of the determinant of the metric

$q=\det
E^{a}_{i}=r^{4}\bar{\gamma}+r^{3}\bar{\gamma}(\bar{F}^{r}_{r}+\bar{F}^{A}_{A})+O(r^{2})$
(3.12)

From this point forward, the smearing functions we use for the diffeomorphism
and Hamiltonian constraints are (2.14) and (3.10), respectively.

Moving forward, we will assume that we are always working in a coordinate
system in which the equation
$\bar{\gamma}_{AB}\bar{F}^{A}_{r}+\bar{F}^{r}_{B}=0$ holds. This assumption is
the same as the one stated in section 2.1.2, where we set $\bar{h}_{rA}=0$
[3]. In fact, this assumption for mixed radial-angular components of metric
was imposed also in [21]. The only cost we must pay to ensure that we remain
in this coordinate when transitioning from one hypersuface to the next is to
fix $I^{A}$ in the shift vector (2.14) (for more details, refer to equations
(3.3) and (3.3)).

For future reference, it is necessary to obtain the asymptotic expansion of
the spin connection and extrinsic curvature, which are parts of the definition
of the Ashtekar-Barbero connection (2.23), as well as the curvature (2.30)
that appears in the constraints (2.28) and (2.29). These quantities can be
expanded in spherical coordinates as

$\displaystyle
K^{i}_{r}=\frac{\bar{k}^{i}_{r}}{r^{2}}+O(r^{-3}),\;\;\;\;\;\;K^{i}_{B}=\frac{\bar{k}^{i}_{B}}{r}+O(r^{-2})$
(3.13)
$\displaystyle\Gamma^{i}_{r}=\frac{\bar{\Gamma}^{i}_{r}}{r^{2}}+O(r^{-3}),\;\;\;\;\;\;\;\Gamma^{i}_{B}=\frac{\bar{k}^{i}_{B}}{r}+O(r^{-2})$
(3.14) $\displaystyle F_{rA}=\frac{1}{r^{2}}$
$\displaystyle\left(-\bar{g}^{i}_{A}-\partial_{A}\bar{g}^{i}_{r}\right)+O(r^{-3}),\;\;\;\;F_{AB}=\frac{1}{r}\left(\partial_{A}\bar{g}^{i}_{B}-\partial_{B}\bar{g}^{i}_{A}\right)+O(r^{-2})$
(3.15)

By using equations (2.25) and (3.1) and and performing a lengthy calculation,
we obtain the leading order term of the spin connection as

$\displaystyle\bar{\Gamma}^{i}_{r}=$
$\displaystyle\frac{1}{2\sqrt{\bar{\gamma}}}\left[2\epsilon^{CB}\bar{\gamma}_{AB}\bar{F}^{A}_{r}-\epsilon^{CA}\bar{D}_{A}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right]\bar{\gamma}^{i}_{C}$
$\displaystyle+\frac{1}{2\sqrt{\bar{\gamma}}}\epsilon^{BA}\left[\bar{D}_{A}(\bar{\gamma}_{CB}\bar{F}^{C}_{r}+\bar{F}^{r}_{B})+\bar{\gamma}_{CB}\bar{F}^{C}_{A}\right]\bar{\gamma}^{i}_{r}$
$\displaystyle=$
$\displaystyle\;\bar{\Gamma}^{C}_{r}\bar{\gamma}^{i}_{C}+\bar{\Gamma}^{r}_{r}\bar{\gamma}^{i}_{r}$
(3.16) $\displaystyle\bar{\Gamma}^{i}_{B}=$
$\displaystyle\frac{1}{2\sqrt{\bar{\gamma}}}\left[\epsilon^{DA}\bar{\gamma}_{AB}(\bar{F}^{C}_{C}-\bar{F}^{r}_{r})-2\epsilon^{DC}\bar{\gamma}_{AC}(\bar{D}_{B}\bar{F}^{A}_{r})\right]\bar{\gamma}^{i}_{D}$
$\displaystyle+\frac{1}{2\sqrt{\bar{\gamma}}}\left[2\epsilon^{DA}\bar{\gamma}_{BD}\bar{\gamma}_{AC}\bar{F}^{C}_{r}-\epsilon^{DA}\bar{\gamma}_{BD}\bar{D}_{A}(\bar{F}^{r}_{r}+\bar{F}^{C}_{C})+2\epsilon^{DA}\bar{\gamma}_{CD}\bar{D}_{A}\bar{F}^{C}_{B}\right]\bar{\gamma}^{i}_{r}$
$\displaystyle=$
$\displaystyle\;\bar{\Gamma}^{D}_{B}\bar{\gamma}^{i}_{D}+\bar{\Gamma}^{r}_{B}\bar{\gamma}^{i}_{r}$
(3.17)

Along with equation (2.23), these results allow us to determine the leading
terms of $K^{i}_{a}$ as

$\bar{k}^{i}_{a}=\beta^{-1}(\bar{g}^{i}_{a}-\bar{\Gamma}^{i}_{a})=\beta^{-1}(\bar{G}^{b}_{a}\bar{\gamma}_{b}^{i}-\bar{\Gamma}^{i}_{a})$
(3.18)

Finally, by replacing the expressions (3) and (3) into (3.18), we can obtain
the explicit formula for $\bar{k}^{i}_{a}$ in terms of $\bar{F}^{a}_{b}$ and
$\bar{G}^{a}_{b}$.

###  3.1 Explicit form

We are now prepared to present the new boundary conditions for the Ashtekar-
Barbero variables in the form of a theorem. The subsequent sections of this
paper will be dedicated to the thorough analysis of the implications and
substance of this theorem.

######  Theorem.

The following boundary conditions meet both requirements (i) and (ii). They
also fulfill requirement (iii) for spacetime translations and result in non-
zero supertranslation charges. The boundary conditions consist of the decay
conditions (3.1) and the parity conditions

$\displaystyle\bar{F}^{\theta}_{r}\sim\bar{F}^{\theta}_{\varphi}\sim\bar{F}^{\varphi}_{\theta}=\text{even}$
(3.19)
$\displaystyle\bar{F}^{\varphi}_{r}\sim\bar{F}^{\theta}_{\theta}\sim\bar{F}^{\varphi}_{\varphi}\sim\bar{G}^{r}_{r}=\text{odd}$
(3.20)
$\displaystyle(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})\sim(\bar{G}^{\theta}_{\theta}+\bar{G}^{r}_{r})\sim(\bar{G}^{\varphi}_{\varphi}+\bar{G}^{r}_{r})=\text{even}$
(3.21)
$\displaystyle\bar{G}^{\theta}_{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{odd}$
(3.22)
$\displaystyle\bar{G}_{\theta}^{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{odd}$
(3.23)
$\displaystyle\bar{G}^{\varphi}_{r}-\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{even}$
(3.24)
$\displaystyle\bar{G}_{\varphi}^{r}-\frac{\sqrt{\bar{\gamma}}}{2}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{even}$
(3.25)
$\displaystyle\bar{G}^{\theta}_{\varphi}+\frac{\sqrt{\bar{\gamma}}}{2}\left(\bar{F}^{r}_{r}-\bar{F}^{A}_{A}\right)=\text{odd}$
(3.26)
$\displaystyle\bar{G}_{\theta}^{\varphi}-\frac{1}{2\sqrt{\bar{\gamma}}}\left(\bar{F}^{r}_{r}-\bar{F}^{A}_{A}\right)=\text{odd}$
(3.27)

Moreover, we have assumed

$\displaystyle\bar{\gamma}_{AB}\bar{F}^{A}_{r}+\bar{F}^{r}_{B}=0$ (3.28)
$\displaystyle\bar{\gamma}_{B[E}\delta\bar{F}^{B}_{D]}=0$ (3.29)

The restrictions we have in our Lagrange multipliers are

$\displaystyle W=\text{odd},\;\;\;\;\tilde{f}=\text{even}$ (3.30)
$\displaystyle
I_{D}=\partial_{D}W-\beta\;b\left[\frac{s}{\beta}(\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r})-\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AD}\bar{\gamma}_{BC}\bar{F}^{B}_{r}\right]$
(3.31) $\displaystyle
2\bar{\Lambda}^{r}=-s\;\frac{\epsilon^{DA}}{\sqrt{\bar{\gamma}}}\bar{D}_{A}\left(b\;[\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r}]\right)-\beta\left[\bar{F}^{A}_{r}(\bar{D}_{A}b)-2b\bar{D}_{A}\bar{F}^{A}_{r}+b\;\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{C}_{A}\right]$
(3.32)
$\displaystyle\bar{\Lambda}_{\theta}+\beta\frac{b}{2}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{odd}$
(3.33)
$\displaystyle\bar{\Lambda}_{\varphi}+\beta\frac{b}{2}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=\text{even}$
(3.34)

where $\bar{\Lambda}^{a}:=\bar{\Lambda}^{i}\bar{\gamma}_{i}^{a}$.

Note that obviously the new boundary conditions differ from the standard ones.
Specifically, in terms of $\bar{F}^{a}_{b},\bar{G}^{a}_{b}$ the standard
parity conditions 2.33 is stated as

$\begin{split}&\bar{F}^{a}_{r}=\bar{f}^{a}_{i}\bar{\gamma}^{i}_{r}=\text{odd},\;\;\;\bar{F}^{a}_{\theta}=\bar{f}^{a}_{i}\bar{\gamma}^{i}_{\theta}=\text{even},\;\;\;\bar{F}^{a}_{\varphi}=\bar{f}^{a}_{i}\bar{\gamma}^{i}_{\varphi}=\text{odd},\\\
&\bar{G}^{r}_{a}=\bar{g}_{a}^{i}\bar{\gamma}_{i}^{r}=\text{even},\;\;\;\bar{G}^{\theta}_{a}=\bar{g}_{a}^{i}\bar{\gamma}_{i}^{\theta}=\text{odd},\;\;\;\bar{G}^{\varphi}_{a}=\bar{g}_{a}^{i}\bar{\gamma}_{i}^{\varphi}=\text{even}\end{split}$
(3.35)

where all the canonical variables have definite parities. Here to read the
parities we have the relations (3.6) and (3.7) in mind. Furthermore, the
parity condition $\bar{\Lambda}^{i}=\text{even}$ in the standard boundary
conditions is equivalent to
$\bar{\Lambda}^{r}\sim\bar{\Lambda}^{\varphi}=\text{even}$ and
$\bar{\Lambda}^{\theta}=\text{odd}$, while the equations (3.32)-(3.34) tells
us that all $\bar{\Lambda}^{a}$ do have both even and odd pieces in the new
setting of boundary conditions. Moreover, in the standard parity conditions
$I^{A}$ is not fixed, while here it is through the equation (3.31).

The rest of this section is dedicated to prove this theorem. First, we show
that these boundary conditions gives us a well-defined symplectic structure
(2.26). As we already saw in (2.34), since the asymptotic triads are supposed
to be fixed, the divergent part of the symplectic structure is

$\begin{split}&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(\delta_{1}\bar{f}^{a}_{i}\delta_{2}\bar{g}^{i}_{a}-\delta_{2}\bar{f}^{a}_{i}\delta_{1}\bar{g}^{i}_{a})\\\
=&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(\delta_{1}\bar{F}^{a}_{b}\delta_{2}\bar{G}^{b}_{a}-\delta_{2}\bar{F}^{a}_{b}\delta_{1}\bar{G}^{b}_{a})\\\
=&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{F}^{r}_{r}\delta_{2}\bar{G}^{r}_{r}+\delta_{1}\bar{F}^{r}_{A}\delta_{2}\bar{G}^{A}_{r}+\delta_{1}\bar{F}^{A}_{r}\delta_{2}\bar{G}^{r}_{A}+\delta_{1}\bar{F}^{A}_{B}\delta_{2}\bar{G}^{B}_{A}-(\delta_{1}\leftrightarrow\delta_{2})\right)\\\
=&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})\delta_{2}\bar{G}^{r}_{r}+\delta_{1}\bar{F}^{A}_{A}\delta_{2}\bar{G}^{r}_{r}+\delta_{1}\bar{F}^{A}_{r}\delta_{2}(\bar{G}^{r}_{A}-\bar{\gamma}_{AB}\bar{G}^{B}_{r})+\delta_{1}\bar{F}^{A}_{B}\delta_{2}(\bar{G}^{B}_{A})_{\text{sym}}-(\delta_{1}\leftrightarrow\delta_{2})\right)\\\
=&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{F}^{A}_{A}\delta_{2}\bar{G}^{r}_{r}+\delta_{1}\bar{F}^{A}_{B}\delta_{2}[(\bar{G}^{B}_{A})_{TL}+\frac{1}{2}\bar{G}^{C}_{C}\delta^{B}_{A}]-(\delta_{1}\leftrightarrow\delta_{2})\right)\\\
=&\frac{2}{\beta}\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;\left(\delta_{1}\bar{F}^{A}_{A}\delta_{2}(\bar{G}^{r}_{r}+\frac{1}{2}\bar{G}^{C}_{C})+\delta_{1}(\bar{F}^{A}_{B})_{TL}\delta_{2}(\bar{G}^{B}_{A})_{TL}-(\delta_{1}\leftrightarrow\delta_{2})\right)\\\
=&\;0\end{split}$ (3.36)

Here, in the first step we have used (3.8), (3.9) and
$\bar{\gamma}^{a}_{i}\bar{\gamma}_{b}^{i}=\delta^{a}_{b}$. In the second step
we have separated the radial and angular components. In the third step we have
added and subtracted $\delta_{1}\bar{F}^{A}_{A}\delta_{2}\bar{G}^{r}_{r}$, the
we used (3.28) to conclude that
$\delta_{1}\bar{F}^{r}_{A}\delta_{2}\bar{G}^{A}_{r}+\delta_{1}\bar{F}^{A}_{r}\delta_{2}\bar{G}^{r}_{A}=\delta_{1}\bar{F}^{A}_{r}\delta_{2}(\bar{G}^{r}_{A}-\bar{\gamma}_{AB}\bar{G}^{B}_{r})$
and we also used (3.29) to see that only the symmetric part of
$\bar{G}^{B}_{A}$ contributes in it as
$\delta_{1}\bar{F}^{A}_{B}\delta_{2}\bar{G}^{B}_{A}=\delta_{1}\bar{F}^{A}_{B}\delta_{2}(\bar{G}^{B}_{A})_{\text{sym}}$
where by $(\bar{G}^{B}_{A})_{\text{sym}}$ we mean the symmetric part of the 2
by 2 matrix $\bar{G}^{B}_{A}$. In the fourth step, first we used the parity
conditions (3.19) to see that
$\delta_{1}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})\delta_{2}\bar{G}^{r}_{r}=\text{odd}$
and also the parity conditions (3.22)-(3.25) to conclude that
$\delta_{1}\bar{F}^{A}_{r}\delta_{2}(\bar{G}^{r}_{A}-\bar{\gamma}_{AB}\bar{G}^{B}_{r})=\text{odd}$.
These two odd terms vanish because integration of an odd function over
2-sphere is equal to zero. Then we have also split the symmetric matrix
$(\bar{G}^{B}_{A})_{\text{sym}}$ into the trace piece and traceless piece
$(\bar{G}^{B}_{A})_{TL}$, i.e.
$(\bar{G}^{B}_{A})_{\text{sym}}=(\bar{G}^{B}_{A})_{TL}+\frac{1}{2}\bar{G}^{C}_{C}\delta^{B}_{A}$.
In the fifth step we have factored $\bar{F}^{A}_{A}$ out of the terms
including it and noted that only the traceless part of the symmetric matrix
$\bar{F}^{A}_{B}$ contributes in the term
$\delta_{1}\bar{F}^{A}_{B}\delta_{2}(\bar{G}^{B}_{A})_{TL}$. Finally, by using
the parity conditions (3.19)-(3.21), (3.26) and (3.27) we see that the two
remaining terms are both odd functions and so their integrations vanish.

Therefore, the parity conditions introduced in the theorem ensure convergence
of the symplectic structure and the requirement (i) is satisfied.

###  3.2 Constraints

The newly established parity conditions do ensure the finite nature of the
symplectic form, but they alone do not guarantee the cancellation of the
divergent components in the boost charges and angular momentum, in contrast to
the parity conditions presented in [9]. Therefore, these conditions must be
complemented by additional asymptotic restrictions in order to achieve the
finiteness of the charges.

As explained in section 2.1.2, the approach employed in [3] to solve this
issue involves setting the leading terms of the constraints to zero. These
supplementary conditions are relatively moderate. In this study, while working
with Ashtekar-Barbero variables, our objective is to investigate whether this
strategy leads to the elimination of divergence in surface terms. For this
purpose, in the current section, we derive the leading terms of the
constraints in terms of the Ashtekar-Barbero variables and set them to zero.
Additionally, we analyze the behavior of the constraints and their variations
at infinity.

With the given boundary conditions (3.1) and in spherical coordinates, the
constraints (2.27)-(2.29) exhibit the following decay behavior

$\displaystyle\mathcal{G}_{i}=$
$\displaystyle\;\bar{\mathcal{G}}_{i}+O(r^{-1})$ (3.37) $\displaystyle
H_{r}=\frac{1}{r}\bar{H}_{r}+O($ $\displaystyle
r^{-2}),\;\;\;H_{A}=\bar{H}_{A}+O(r^{-1})$ (3.38) $\displaystyle H=$
$\displaystyle\;r\bar{H}+O(1)$ (3.39)

The strengthened boundary conditions require the absence of leading
divergences in the constraints, thus imposing the conditions:

$\bar{\mathcal{G}}_{i}=\bar{H}_{r}=\bar{H}_{A}=\bar{H}=0$ (3.40)

As in this section we wish to thoroughly discuss aspects related to the
constraints, we will derive their actions on the canonical variables and also
examine the surface terms that affect their differentiability, which will be
addressed in section 3.4 when the asymptotic charges are discussed.

Before delving into the details, it is important to recall from section 2.2
that to ensure the finiteness of the volume part of the diffeomorphism and
Hamiltonian constraints, a term proportional to the Gauss constraint needs to
be added to their expressions (see equation (2.36) and (2.37) respectively).
In spherical coordinates, the Lagrange multipliers $\hat{\Lambda}^{i}$ and
$\mathring{\Lambda}^{i}$ are given by:

$\displaystyle\hat{\Lambda}^{i}=\Lambda^{i}+\frac{1}{2}\frac{\epsilon^{BC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AC}\left(-2Y^{A}\bar{\gamma}^{j}_{B}+(\bar{D}_{B}Y^{A})\bar{\gamma}^{j}_{r}\right)$
(3.41)
$\displaystyle\mathring{\Lambda}^{i}=\Lambda^{i}+(b\bar{\gamma}^{r}_{j}+(\partial_{A}b)\bar{\gamma}^{A}_{j}).$
(3.42)

In the subsequent discussion, we examine the behavior of each constraint
individually.

####  3.2.1 Gauss constraint

After some calculation, one can obtain the explicit expression of the
conditions $\bar{\mathcal{G}}_{i}=0$ in (3.40) as equivalent to the following
equations

$\displaystyle\bar{\mathcal{G}}^{r}=\frac{2}{\beta}\left(\sqrt{\bar{\gamma}}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})+\sqrt{\bar{\gamma}}(\bar{D}_{A}\bar{F}^{A}_{r})-\epsilon^{AB}\bar{G}^{C}_{A}\bar{\gamma}_{BC}\right)=0$
(3.43)
$\displaystyle\bar{\mathcal{G}}^{A}=\frac{2}{\beta}\left(\sqrt{\bar{\gamma}}(\bar{F}^{r}_{B}\bar{\gamma}^{AB}+\bar{F}^{A}_{r})+\sqrt{\bar{\gamma}}\bar{\gamma}^{AB}(\bar{D}_{C}\bar{F}^{C}_{B})+\epsilon^{AB}(\bar{G}^{C}_{r}\bar{\gamma}_{BC}-\bar{G}^{r}_{B})\right)$
$\displaystyle\;\;\;\;\;=\frac{2}{\beta}\left(\sqrt{\bar{\gamma}}\bar{\gamma}^{AB}(\bar{D}_{C}\bar{F}^{C}_{B})+\epsilon^{AB}(\bar{G}^{C}_{r}\bar{\gamma}_{BC}-\bar{G}^{r}_{B})\right)$
(3.44)

where
$\bar{\mathcal{G}}_{i}=\bar{\mathcal{G}}^{r}\bar{\gamma}^{i}_{r}+\bar{\mathcal{G}}^{A}\bar{\gamma}^{i}_{A}$,
and to obtain the second line of (3.44), we make use of the assumption (3.28).
It is important to ensure the convergence of the smeared constraint. In the
standard boundary conditions discussed in section 2.2, it is required to
assign definite even parity to $\bar{\Lambda}^{i}$ in order to ensure the
finiteness of (2.27). However, in this case, we do not need to restrict the
parity of $\bar{\Lambda}^{i}$ as the divergent part of (2.27) has already been
removed, as represented by the equation

$\mathcal{G}_{i}[\Lambda^{i}]=\int_{\Sigma}drd\theta
d\varphi\;\mathcal{G}_{i}\Lambda^{i}=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(\bar{\Lambda}^{i}\bar{\mathcal{G}}_{i})+\text{finite}=\text{finite}$
(3.45)

where we have used the condition $\bar{\mathcal{G}}_{i}=0$.

Now, it is necessary to verify the differentiability of the constraint. The
variation of the smeared Gauss constraint can be represented as

$\begin{split}\delta\mathcal{G}_{i}[\Lambda^{i}]=&\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\Lambda^{i}\left(\partial_{a}\delta
E^{a}_{i}+\epsilon_{ijk}(\delta
A^{j}_{a})E^{a}_{k}+\epsilon_{ijk}A^{j}_{a}(\delta E^{a}_{k})\right)\\\
=&\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\left(-\partial_{a}\Lambda^{k}+\epsilon_{ijk}A^{j}_{a}\Lambda^{i}\right)(\delta
E^{a}_{k})+\left(\epsilon_{ijk}\Lambda^{i}E^{a}_{k}\right)(\delta
A_{a}^{j})+\frac{2}{\beta}\oint dS_{a}(\Lambda^{i}\delta
E^{a}_{i})\end{split}$ (3.46)

By expressing the above variation in the form

$\delta\mathcal{G}_{i}[\Lambda^{i}]=\Omega(\delta_{\Lambda},\delta)+\mathcal{B}_{\Lambda}(\delta
A^{i}_{a},\delta E^{a}_{i})=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\delta_{\Lambda}A^{i}_{a})-(\delta
A^{i}_{a})(\delta_{\Lambda}E^{a}_{i})\right)+\mathcal{B}_{\Lambda}(\delta
A^{i}_{a},\delta E^{a}_{i})$ (3.47)

we can determine the variation of the canonical variables under the action of
the Gauss constraint, as well as the surface term that needs to be subtracted
in order to obtain a functionally differentiable generator for internal
rotations. Specifically, we find

$\displaystyle\delta_{\Lambda}A^{i}_{a}=-\partial_{a}\Lambda^{i}+\epsilon_{ijk}\Lambda^{j}A^{k}_{a}$
(3.48)
$\displaystyle\delta_{\Lambda}E^{a}_{i}=\epsilon_{ijk}\Lambda^{j}E^{a}_{k}$
(3.49) $\displaystyle\mathcal{B}_{\Lambda}(\delta A^{i}_{a},\delta
E^{a}_{i})=\frac{2}{\beta}\oint dS_{a}(\Lambda^{i}\delta E^{a}_{i})$ (3.50)

We will address the issue arising from the surface term in section 3.4.

####  3.2.2 Diffeomorphism constraint

The conditions $\bar{H}_{r}=\bar{H}_{A}=0$ stated in (3.40) can be expressed
explicitly as

$\displaystyle\bar{H}_{r}=\frac{2s}{\beta}\sqrt{\bar{\gamma}}\left(\bar{G}^{A}_{A}+2\bar{G}^{r}_{r}+\bar{D}_{A}\bar{G}^{A}_{r}\right)=0$
(3.51)
$\displaystyle\bar{H}_{A}=-\frac{2s}{\beta}\sqrt{\bar{\gamma}}\left(\bar{D}_{A}\bar{G}^{r}_{r}-\bar{G}^{B}_{r}\bar{\gamma}_{AB}+\bar{D}_{A}\bar{G}^{B}_{B}-\bar{D}_{B}\bar{G}^{B}_{A}\right)=0$
(3.52)

In order to check the convergence of the smeared constraint (2.28), we
calculate its asymptotic expansion

$\displaystyle H_{a}[N^{a}]$ $\displaystyle=\int_{\Sigma}drd\theta
d\varphi\;\left(H_{r}N^{r}+H_{A}N^{A}\right)$
$\displaystyle=\int\frac{dr}{r}\int_{S^{2}}d\mathbf{\sigma}\;(W\bar{H}_{r})+\int
dr\int_{S^{2}}d\sigma(Y^{A}\bar{H}_{A}+O(r^{-1}))+\text{finite}$ (3.53)

It is evident that even with the imposition of the conditions
$\bar{H}_{r}=\bar{H}_{A}=0$, the constraint $H_{a}[N^{a}]$ diverges due to the
logarithmic divergence arising from the second integral in (3.2.2). This issue
was already encountered in section 2.2, where applying the standard parity
conditions allowed the removal of the divergent parts including $W\bar{H}_{r}$
and $Y^{A}\bar{H}_{A}$, but the logarithmic divergence in the second integral
of (3.2.2) persisted. As mentioned previously, this problem can be resolved
without relying on any specific parity condition by working with
$\mathcal{H}_{a}[N^{a}]$ defined in (2.36) alongside the Lagrange multiplier
(3.41) instead of $H_{a}[N^{a}]$. For a more detailed explanation, the
interested reader can refer to [1, 2].

Although $\mathcal{H}_{a}[N^{a}]$ provides us with a finite generator for the
symmetries, it is still necessary to ensure its differentiability. Let us
compute the variation of $\mathcal{H}_{a}[N^{a}]$ to obtain the surface terms
mentioned in (2.36), as well as the variations of the canonical variables
under its action

$\begin{split}\delta(\mathcal{H}_{a}[N^{a}])=&\frac{-2s}{\beta}\delta\int_{\Sigma}d^{3}x\;N^{a}\left(F^{i}_{ab}E^{b}_{i}-A^{i}_{a}\mathcal{G}_{i}\right)+s\delta\mathcal{G}_{i}[\hat{\Lambda}^{i}]\\\
=&\frac{-2s}{\beta}\delta\int_{\Sigma}d^{3}x\;N^{a}\left([\partial_{a}A^{j}_{b}-\partial_{b}A^{j}_{a}]E^{b}_{i}-A^{j}_{a}\partial_{b}E^{b}_{j}\right)+s\delta\mathcal{G}_{i}[\hat{\Lambda}^{i}]\\\
=&\frac{-2s}{\beta}\int_{\Sigma}d^{3}x\;N^{a}\left(2(\partial_{[a}\delta
A^{j}_{b]})E^{b}_{i}+2(\partial_{[a}A^{j}_{b]})\delta E^{b}_{i}-(\delta
A^{j}_{a})\partial_{b}E^{b}_{j}-A^{j}_{a}\partial_{b}\delta
E^{b}_{j}\right)+s\delta\mathcal{G}_{i}[\hat{\Lambda}^{i}]\\\
=&\frac{-2s}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\mathcal{L}_{\vec{N}}A^{i}_{a})-(\delta
A_{a}^{i})(\mathcal{L}_{\vec{N}}E^{a}_{i})\right)\\\
&-\frac{2s}{\beta}\int_{S^{2}}dS_{a}\;\left(N^{a}E^{b}_{i}\delta
A^{i}_{b}-N^{b}E^{a}_{i}\delta A^{i}_{b}-N^{b}A^{i}_{b}\delta
E^{a}_{i}\right)+s\delta\mathcal{G}_{i}[\hat{\Lambda}^{i}]\\\
=&\frac{-2s}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\mathcal{L}_{\vec{N}}A^{i}_{a}-\epsilon_{ijk}\hat{\Lambda}^{j}A^{k}_{a})-(\delta
A_{a}^{i})(\mathcal{L}_{\vec{N}}E^{a}_{i}-\epsilon_{ijk}\hat{\Lambda}^{j}E^{a}_{k})\right)\\\
&-\frac{2s}{\beta}\oint_{S^{2}}dS_{a}\;\left(N^{a}E^{b}_{i}\delta
A^{i}_{b}-N^{b}E^{a}_{i}\delta A^{i}_{b}-N^{b}A^{i}_{b}\delta
E^{a}_{i}-\hat{\Lambda}^{i}\delta E^{a}_{i}\right)\end{split}$ (3.54)

By expressing the above variation in the form

$\delta\mathcal{H}_{a}[N^{a}]=\Omega(\delta_{\vec{N}},\delta)+\mathcal{B}_{\vec{N}}(\delta
A^{i}_{a},\delta E^{a}_{i})=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\delta_{\vec{N}}A^{i}_{a})-(\delta
A^{i}_{a})(\delta_{\vec{N}}E^{a}_{i})\right)+\mathcal{B}_{\vec{N}}(\delta
A^{i}_{a},\delta E^{a}_{i})$ (3.55)

We can determine the variation of the canonical variables under the action of
the diffeomorphism constraint, as well as the surface term that must be
subtracted to obtain a functionally differentiable generator for spatial
diffeomorphisms. Specifically

$\displaystyle\delta_{\vec{N}}A^{i}_{a}=-s(\mathcal{L}_{\vec{N}}A^{i}_{a}-\epsilon_{ijk}\hat{\Lambda}^{j}A^{k}_{a})$
(3.56)
$\displaystyle\delta_{\vec{N}}E^{a}_{i}=-s(\mathcal{L}_{\vec{N}}E^{a}_{i}-\epsilon_{ijk}\hat{\Lambda}^{j}E^{a}_{k})$
(3.57) $\displaystyle\mathcal{B}_{\vec{N}}(\delta A^{i}_{a},\delta
E^{a}_{i})=-\frac{2s}{\beta}\oint_{S^{2}}dS_{a}\;\left(N^{a}E^{b}_{i}\delta
A^{i}_{b}-N^{b}E^{a}_{i}\delta A^{i}_{b}-N^{b}A^{i}_{b}\delta
E^{a}_{i}-\hat{\Lambda}^{i}\delta E^{a}_{i}\right)$ (3.58)

we will treat the issue coming from the surface term in section 3.4.

####  3.2.3 Hamiltonian constraint

The explicit expression of the condition $\bar{H}=0$ in (3.40) is

$\bar{H}=-2\sqrt{\bar{\gamma}}\bar{D}_{A}\left(\epsilon^{AB}(\bar{G}^{C}_{r}\bar{\gamma}_{BC}-\bar{G}^{r}_{B})\right)=0\\\
$ (3.59)

To check convergence of the smeared constraint (2.29), we compute the
asymptotic expansion of it, namely,

$\displaystyle H[\tilde{N}]$ $\displaystyle=\int_{\Sigma}drd\theta
d\varphi\;\tilde{N}H=\int
dr\int_{S^{2}}d\sigma(b\bar{H}+O(r^{-1}))+\text{finite}$ (3.60)

Despite implementing the conditions $\bar{H}=0$, it is apparent that the
constraint $H[\tilde{N}]$ becomes divergent due to the logarithmic divergence
arising from the $O(r^{-1})$ term in equation (3.60). It is noteworthy to
mention that a similar problem was encountered in section 2.2, where the
utilization of standard parity conditions allowed for the elimination of
divergent parts, including $b\bar{H}$, while the logarithmic divergence in the
second integral of equation (3.60) persisted. As previously stated, this issue
can be resolved without relying on any specific parity condition by working
with $\mathcal{H}[N]$ as defined in (2.37) and incorporating the Lagrange
multiplier from (3.42). For a more comprehensive explanation, we recommend
referring to [1, 2].

The variation of the smeared Hamiltonian constraint is given by

$\begin{split}\delta\mathcal{H}[\tilde{N}]=&\int_{\Sigma}d^{3}x\;\tilde{N}\left(\epsilon_{ijk}\delta
F^{i}_{ab}E^{a}_{j}E^{b}_{k}+2\epsilon_{ijk}F^{i}_{ab}\delta
E^{a}_{j}E^{b}_{k}-4(\beta^{2}-s)\left[\delta
K^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}+K^{j}_{[a}K^{k}_{b]}\delta
E^{a}_{j}E^{b}_{k}\right]\right)\\\
&-\beta\delta\mathcal{G}_{i}[\mathring{\Lambda}^{i}]\\\
=&\int_{\Sigma}\tilde{N}\left(\epsilon_{ijk}[2\partial_{a}\delta
A^{i}_{b}+2\epsilon_{ilm}A^{m}_{b}\delta
A^{l}_{a}]E^{a}_{j}E^{b}_{k}+2\epsilon_{ijk}F^{i}_{ab}\delta
E^{a}_{j}E^{b}_{k}-4(\beta^{2}-s)K^{j}_{[a}K^{k}_{b]}\delta
E^{a}_{j}E^{b}_{k}\right.\\\ &\hskip
42.67912pt\left.-4\beta^{-1}(\beta^{2}-s)\delta
A^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}+4\beta^{-1}(\beta^{2}-s)\delta\Gamma^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}\right)-\beta\delta\mathcal{G}_{i}[\mathring{\Lambda}^{i}]\\\
=&\int_{\Sigma}d^{3}x\;\left(-2\epsilon_{njk}\partial_{a}(\tilde{N}E^{a}_{j}E^{e}_{k})\delta
A^{n}_{e}+2\tilde{N}\epsilon_{ijk}\epsilon_{inm}A^{m}_{b}\delta
A^{n}_{e}E^{e}_{j}E^{b}_{k}\right.\\\ &\hskip
42.67912pt+2\epsilon_{ink}\tilde{N}F^{i}_{eb}E^{b}_{k}(\delta
E^{e}_{n})-4(\beta^{2}-s)\tilde{N}K^{n}_{[e}K^{k}_{b]}E^{b}_{k}(\delta
E^{e}_{n})\\\ &\hskip
42.67912pt-2\beta^{-1}(\beta^{2}-s)\tilde{N}K^{k}_{b}E^{e}_{n}E^{b}_{k}(\delta
A^{n}_{e})+2\beta^{-1}(\beta^{2}-s)\tilde{N}K^{k}_{a}E^{a}_{n}E^{e}_{k}(\delta
A^{n}_{e})\\\ &\hskip
42.67912pt\left.+4\beta^{-1}(\beta^{2}-s)\tilde{N}\delta\Gamma^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}\right)+\oint
dS_{a}\;2\tilde{N}\epsilon_{ijk}\delta A^{i}_{b}E^{a}_{j}E^{b}_{k}\\\
=&\int_{\Sigma}d^{3}x\;(\delta
A^{n}_{e})\left(-2\epsilon_{njk}\partial_{a}(\tilde{N}E^{a}_{j}E^{e}_{k})+2\tilde{N}\epsilon_{ijk}\epsilon_{inm}A^{m}_{b}E^{e}_{j}E^{b}_{k}-4\beta^{-1}(\beta^{2}-s)\tilde{N}K^{k}_{a}E^{[e}_{n}E^{a]}_{k}-\beta\epsilon_{njk}\mathring{\Lambda}^{j}E^{e}_{k}\right)\\\
&+\int_{\Sigma}d^{3}x\;(\delta
E^{e}_{n})E^{b}_{k}\left(2\epsilon_{ink}\tilde{N}F^{i}_{eb}-4(\beta^{2}-s)\tilde{N}K^{n}_{[e}K^{k}_{b]}-\beta\epsilon_{njk}\mathring{\Lambda}^{j}A_{e}^{k}\right)\\\
&+4\beta^{-1}(\beta^{2}-s)\int_{\Sigma}d^{3}x\;\tilde{N}\delta\Gamma^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}+\oint
dS_{a}\;\left(2\tilde{N}\epsilon_{ijk}\delta
A^{i}_{b}E^{a}_{j}E^{b}_{k}-2\hat{\Lambda}^{i}\delta
E^{a}_{i}\right)\end{split}$ (3.61)

In order to obtain $\delta_{\tilde{N}}A_{a}^{i}$ and the respective surface
terms, it is necessary to express the term
$\int_{\Sigma}d^{3}x\;\tilde{N}\delta\Gamma^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}$
explicitly in terms of $\delta E^{a}_{i}$. We have performed these
calculations as shown below.

$\begin{split}\int_{\Sigma}d^{3}x\;&\tilde{N}\delta\Gamma^{j}_{[a}K^{k}_{b]}E^{a}_{j}E^{b}_{k}=\frac{1}{2}\int_{\Sigma}d^{3}x\;\left(\tilde{N}\delta\Gamma^{j}_{a}E^{a}_{j}K^{k}_{b}E^{b}_{k}-\tilde{N}\delta\Gamma^{i}_{a}E^{a}_{m}K^{m}_{d}E^{d}_{i}\right)\\\
&=\frac{1}{4}\int_{\Sigma}d^{3}x\;(\delta
E^{e}_{n})\left(\epsilon^{njk}\tilde{N}K^{i}_{a}E^{a}_{i}E^{b}_{k}\left[E^{l}_{e}E^{c}_{j}E^{l}_{c,b}+E^{j}_{e}\frac{(\det(E))_{,b}}{\det(E)}\right]+\epsilon^{ljk}\partial_{a}(\tilde{N}K^{i}_{d}E^{d}_{i}E^{b}_{k}E^{a}_{l})E^{n}_{b}E^{j}_{e}\right)\\\
&+\frac{1}{4}\int_{\Sigma}d^{3}x\;\tilde{N}\epsilon^{ijn}(\delta
E^{e}_{n})K^{m}_{d}E^{d}_{i}\left[E^{a}_{m}E^{j}_{a,e}-E^{a}_{m}E^{j}_{e,a}+E^{c}_{j}E^{m}_{c,e}+\delta^{j}_{m}\frac{(\det(E))_{,e}}{\det(E)}\right]\\\
&-\frac{1}{4}\int_{\Sigma}d^{3}x\;\tilde{N}\epsilon^{ijk}(\delta
E^{e}_{n})E^{b}_{k}K^{n}_{d}E^{d}_{i}\left[E^{c}_{j}E^{l}_{e}E^{l}_{c,b}+E^{j}_{e}\frac{(\det(E))_{,b}}{\det(E)}\right]+\frac{1}{4}\int_{\Sigma}d^{3}x\;\tilde{N}\epsilon^{ink}(\delta
E^{e}_{n})E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{m}_{e,b}\\\
&+\frac{1}{4}\int_{\Sigma}d^{3}x\;\epsilon^{ijk}(\delta
E^{e}_{n})E^{j}_{e}\left[\partial_{b}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m})E^{n}_{a}-\partial_{a}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m})E^{n}_{b}\right]\\\
&-\frac{1}{4}\int_{\Sigma}d^{3}x\;\epsilon^{ijk}(\delta
E^{e}_{n})\left[-\partial_{b}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{c}_{j})E^{n}_{c}E^{m}_{e}+\partial_{b}(\tilde{N}E^{b}_{k}K^{j}_{d}E^{d}_{i})E^{n}_{e}\right]\\\
&+\frac{1}{4}\oint
dS_{a}\epsilon^{ijk}\tilde{N}K^{n}_{e}E^{e}_{n}E^{b}_{k}E^{a}_{i}\delta
E^{j}_{b}+\frac{1}{4}\oint
dS_{b}\;\epsilon^{ijk}\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}\left[E^{a}_{m}\delta
E^{j}_{a}+E^{c}_{j}(\delta E^{m}_{c})+\delta^{j}_{m}(E^{l}_{d}\delta
E^{d}_{l})\right]\\\ &-\frac{1}{4}\oint
dS_{a}\;\epsilon^{ijk}\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m}\delta
E^{j}_{b}\end{split}$ (3.62)

where the variation of the spin connection is calculated using the relation
(2.25).

By substituting (3.62) into (3.61) and reorganizing the terms, we can express
$\delta\mathcal{H}[\tilde{N}]$ in the desired form

$\delta\mathcal{H}[\tilde{N}]=\Omega(\delta_{\tilde{N}},\delta)+\mathcal{B}_{\tilde{N}}(\delta
A^{i}_{a},\delta E^{a}_{i})=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\delta_{\tilde{N}}A^{i}_{a})-(\delta
A^{i}_{a})(\delta_{\tilde{N}}E^{a}_{i})\right)+\mathcal{B}_{\tilde{N}}(\delta
A^{i}_{a},\delta E^{a}_{i})$ (3.63)

From this expression, we can read both
$\delta_{\tilde{N}}A^{i}_{a},\;\delta_{\tilde{N}}E^{a}_{i}$, as well as the
boundary term $\mathcal{B}_{\tilde{N}}(\delta A^{i}_{a},\delta E^{a}_{i})$.
The variation of the Ashtekar-Barbero connection under the action of the
Hamiltonian constraint is

$\begin{split}\delta_{\tilde{N}}A^{n}_{e}=&\;\beta
E^{b}_{k}\left(\epsilon_{ink}\tilde{N}F^{i}_{eb}-2(\beta^{2}-s)\tilde{N}K^{n}_{[e}K^{k}_{b]}\right)\\\
&+\frac{1}{2}(\beta^{2}-s)\left(\epsilon^{njk}\tilde{N}K^{m}_{f}E^{f}_{m}E^{b}_{k}\left[E^{l}_{e}E^{c}_{j}E^{l}_{c,b}+E^{j}_{e}\frac{(\det(E))_{,b}}{\det(E)}\right]\right.\\\
&\hskip
65.44142pt+\epsilon^{ljk}\partial_{a}(\tilde{N}K^{i}_{d}E^{d}_{i}E^{b}_{k}E^{a}_{l})E^{n}_{b}E^{j}_{e}\\\
&\hskip
65.44142pt+\tilde{N}\epsilon^{ijn}K^{m}_{d}E^{d}_{i}\left[E^{a}_{m}E^{j}_{a,e}-E^{a}_{m}E^{j}_{e,a}+E^{c}_{j}E^{m}_{c,e}+\delta^{j}_{m}\frac{(\det(E))_{,e}}{\det(E)}\right]\\\
&\hskip
65.44142pt-\tilde{N}\epsilon^{ijk}E^{b}_{k}K^{n}_{d}E^{d}_{i}\left[E^{c}_{j}E^{l}_{e}E^{l}_{c,b}+E^{j}_{e}\frac{(\det(E))_{,b}}{\det(E)}\right]\\\
&\hskip
65.44142pt+\tilde{N}\epsilon^{ink}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{m}_{e,b}\\\
&\hskip
65.44142pt+\epsilon^{ijk}E^{j}_{e}\left[\partial_{b}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m})E^{n}_{a}-\partial_{a}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m})E^{n}_{b}\right]\\\
&\hskip
65.44142pt\left.-\epsilon^{ijk}\left[-\partial_{b}(\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{c}_{j})E^{n}_{c}E^{m}_{e}+\partial_{b}(\tilde{N}E^{b}_{k}K^{j}_{d}E^{d}_{i})E^{n}_{e}\right]\right)-\beta\epsilon_{njk}\mathring{\Lambda}^{j}A^{k}_{e}\end{split}$
(3.64)

The variation of the electric field is given by

$\delta_{\tilde{N}}E^{e}_{n}=\beta\epsilon_{njk}\partial_{a}(\tilde{N}E^{a}_{j}E^{e}_{k})-\beta\tilde{N}\epsilon_{ijk}\epsilon_{inm}A^{m}_{b}E^{e}_{j}E^{b}_{k}+2(\beta^{2}-s)\tilde{N}K^{k}_{a}E^{[e}_{n}E^{a]}_{k}-\beta\epsilon_{njk}\mathring{\Lambda}^{j}E^{e}_{k}$
(3.65)

The surface term associated with the Hamiltonian constraint is

$\begin{split}\mathcal{B}_{\tilde{N}}(\delta A^{i}_{a},\delta
E^{a}_{i})=&\oint dS_{a}\;2\tilde{N}\epsilon_{ijk}\delta
A^{i}_{b}E^{a}_{j}E^{b}_{k}+\beta^{-1}(\beta^{2}-s)\oint
dS_{a}\epsilon^{ijk}\tilde{N}K^{n}_{e}E^{e}_{n}E^{b}_{k}E^{a}_{i}\delta
E^{j}_{b}\\\ &+\beta^{-1}(\beta^{2}-s)\oint
dS_{b}\;\epsilon^{ijk}\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}\left[E^{a}_{m}\delta
E^{j}_{a}+E^{c}_{j}(\delta E^{m}_{c})+\delta^{j}_{m}(E^{l}_{d}\delta
E^{d}_{l})\right]\\\ &-\beta^{-1}(\beta^{2}-s)\oint
dS_{a}\;\epsilon^{ijk}\tilde{N}E^{b}_{k}K^{m}_{d}E^{d}_{i}E^{a}_{m}\delta
E^{j}_{b}-2\oint dS_{a}\;\mathring{\Lambda}^{i}\delta E^{a}_{i}\end{split}$
(3.66)

we will deal with the the surface term issue in section 3.4.

###  3.3 Preservation under hypersurface deformations

In this section we want to show that the boundary conditions proposed in the
theorem are preserved under the hypersurface deformations. To do that we use
the results we obtained in the appendix A. We begin with equation (3.28). We
should make sure that this equation is preserved when one moves from one
hypersurface to the next. Using the variations (A) and (A), we have

$\displaystyle\delta(\bar{F}^{B}_{r}\bar{\gamma}_{BD}+\bar{F}^{r}_{D})$
$\displaystyle=$
$\displaystyle\bar{\gamma}_{BD}\left(\beta\left[\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}b\;\bar{D}_{A}\bar{F}^{B}_{C}+\frac{\epsilon^{CB}}{\sqrt{\bar{\gamma}}}\bar{F}^{A}_{C}(\bar{D}_{A}b)+\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}(\bar{D}_{A}\tilde{f})+b\left(\bar{G}^{r}_{A}\bar{\gamma}^{AB}-\frac{(\beta^{2}-s)}{\beta}\bar{k}^{B}_{r}\right)\right]+\mathcal{L}_{Y}\bar{F}^{B}_{r}-\frac{\epsilon^{BC}}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{C}\right)$
$\displaystyle+\beta\left(\bar{\gamma}_{AD}\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\left[b\bar{D}_{B}\bar{F}^{r}_{r}-b(\bar{\gamma}_{BC}\bar{F}^{C}_{r}-\bar{F}^{r}_{B})+\bar{F}^{C}_{B}(\bar{D}_{C}b)+(\bar{D}_{B}\tilde{f})\right]+b\;\bar{G}^{A}_{r}\bar{\gamma}_{AD}-\frac{(\beta^{2}-s)}{\beta}b\;\bar{k}^{r}_{D}\right)$
$\displaystyle-\partial_{D}W+I_{D}+\mathcal{L}_{Y}\bar{F}^{r}_{D}+\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AD}\bar{\Lambda}_{B}$
$\displaystyle=$
$\displaystyle\beta\;b\left[\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\left(\bar{\gamma}_{BD}\;\bar{D}_{A}\bar{F}^{B}_{C}+\bar{\gamma}_{AD}\left[\bar{D}_{C}\bar{F}^{r}_{r}-2\bar{\gamma}_{BC}\bar{F}^{B}_{r}\right]\right)+\bar{G}^{r}_{D}+\bar{G}^{A}_{r}\bar{\gamma}_{AD}-\frac{(\beta^{2}-s)}{\beta}(\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r})\right]$
$\displaystyle-\partial_{D}W+I_{D}+\mathcal{L}_{Y}(\bar{\gamma}_{BD}\bar{F}^{B}_{r}+\bar{F}^{r}_{D})$
(3.67)

In order to make sure that (3.28) is preserved we have to fix $I^{A}$ by the
equation

$\displaystyle I_{D}=$
$\displaystyle\partial_{D}W-\beta\;b\left[\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\left(\bar{\gamma}_{BD}\;\bar{D}_{A}\bar{F}^{B}_{C}+\bar{\gamma}_{AD}\left[\bar{D}_{C}\bar{F}^{r}_{r}-2\bar{\gamma}_{BC}\bar{F}^{B}_{r}\right]\right)+\bar{G}^{r}_{D}+\bar{G}^{A}_{r}\bar{\gamma}_{AD}-\frac{(\beta^{2}-s)}{\beta}(\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r})\right]$
$\displaystyle=$
$\displaystyle\partial_{D}W-\beta\;b\left[\frac{s}{\beta}(\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r})-\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AD}\bar{\gamma}_{BC}\bar{F}^{B}_{r}\right]$
(3.68)

In fact inserting (3.3) in (3.3) leads to
$\delta(\bar{F}^{B}_{r}\bar{\gamma}_{BD}+\bar{F}^{r}_{D})=0$.

In order to show that the equation (3.29) is preserved we use the equations
(A) to get

$\displaystyle\bar{\gamma}_{B[E}\delta\bar{F}^{B}_{D]}=$
$\displaystyle\beta\left[\frac{1}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{B[E}\bar{\gamma}_{D]C}\left(\epsilon^{CA}b\bar{D}_{A}\bar{F}^{B}_{r}-\epsilon^{CB}\bar{F}^{A}_{r}(\bar{D}_{A}b)\right)+b\;\bar{G}^{C}_{[E}\bar{\gamma}_{D]C}\right]$
$\displaystyle-\bar{\gamma}_{B[E}\bar{D}_{D]}I^{B}+\mathcal{L}_{Y}(\bar{\gamma}_{B[E}\bar{F}^{B}_{D]})+\frac{\epsilon^{BC}}{\sqrt{\bar{\gamma}}}\bar{\Lambda}^{r}\bar{\gamma}_{B[E}\bar{\gamma}_{D]C}$
$\displaystyle=\bar{\gamma}_{B[E}\bar{\gamma}_{D]C}\left[\beta\left(\epsilon^{CA}b\bar{D}_{A}\bar{F}^{B}_{r}-\epsilon^{CB}\bar{F}^{A}_{r}(\bar{D}_{A}b)+b\bar{G}^{C}_{A}\bar{\gamma}^{AB}\right)-\bar{D}^{C}I^{B}+\frac{\epsilon^{BC}}{\sqrt{\bar{\gamma}}}\bar{\Lambda}^{r}\right]+\mathcal{L}_{Y}(\bar{\gamma}_{B[E}\bar{F}^{B}_{D]})$
(3.69)

One can solve the equation $\bar{\gamma}_{B[E}\delta\bar{F}^{B}_{D]}=0$ for
$\bar{\Lambda}^{r}$ in the following way

$\displaystyle 2\bar{\Lambda}^{r}=$
$\displaystyle\sqrt{\bar{\gamma}}\;\epsilon_{BC}(\bar{D}^{C}I^{B})-\beta\left[2\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}+b\sqrt{\bar{\gamma}}\;\epsilon_{BC}\bar{G}^{C}_{A}\bar{\gamma}^{AB}\right]$
$\displaystyle=$
$\displaystyle\sqrt{\bar{\gamma}}\;(\frac{1}{\bar{\gamma}}\epsilon^{DA}\bar{\gamma}_{DB}\bar{\gamma}_{AC})(\bar{D}^{C}I^{B})-\beta\left[2\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}+b\sqrt{\bar{\gamma}}\;(\frac{1}{\bar{\gamma}}\epsilon^{DE}\bar{\gamma}_{DB}\bar{\gamma}_{EC})\bar{G}^{C}_{A}\bar{\gamma}^{AB}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\bar{\gamma}}}\epsilon^{DA}(\bar{D}_{A}I_{D})-\beta\left[2\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}+b\;\frac{1}{\sqrt{\bar{\gamma}}}\epsilon^{AB}\bar{\gamma}_{BC}\bar{G}^{C}_{A}\right]$
$\displaystyle=$
$\displaystyle\frac{\epsilon^{DA}}{\sqrt{\bar{\gamma}}}\bar{D}_{A}\left(\partial_{D}W-\beta\;b\left[\frac{s}{\beta}(\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r})-\frac{\epsilon^{EC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{ED}\bar{\gamma}_{BC}\bar{F}^{B}_{r}\right]\right)-\beta\left[2\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}+b\;\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{C}_{A}\right]$
$\displaystyle=$
$\displaystyle-s\;\frac{\epsilon^{DA}}{\sqrt{\bar{\gamma}}}\bar{D}_{A}\left(b\;[\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r}]\right)-\beta\left[\bar{F}^{A}_{r}(\bar{D}_{A}b)-2b\bar{D}_{A}\bar{F}^{A}_{r}+b\;\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{C}_{A}\right]$
(3.70)

Plugging (3.3) into (3.3) ensures that
$\delta(\bar{\gamma}_{B[E}\bar{F}^{B}_{D]})=0$.

Using the equations (A) and (A), we compute the variation of
$\bar{F}^{r}_{r}-\bar{F}^{A}_{A}$ that is supposed to be even.

$\displaystyle\delta(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})=$
$\displaystyle\beta\left(\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}b\bar{D}_{A}\bar{F}^{r}_{B}-\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AC}b\bar{F}^{C}_{B}-b\bar{G}^{A}_{A}+\frac{(\beta^{2}-s)}{\beta}b\bar{k}^{A}_{A}\right)+2W+\bar{D}_{A}I^{A}+\mathcal{L}_{Y}\bar{F}^{r}_{r}$
$\displaystyle-\beta\left[\frac{\bar{\gamma}_{CB}}{\sqrt{\bar{\gamma}}}\left(\epsilon^{CA}b\bar{D}_{A}\bar{F}^{B}_{r}\right)+b\;(-\bar{G}^{A}_{A}-2\bar{G}^{r}_{r})+\frac{(\beta^{2}-s)}{\beta}b\;(2\bar{k}^{r}_{r}+\bar{k}^{B}_{B})\right]+2W+(\bar{D}_{A}I^{A})+\mathcal{L}_{Y}\bar{F}^{B}_{B}$
$\displaystyle=$
$\displaystyle\beta\left(2b\bar{G}^{r}_{r}-\frac{(\beta^{2}-s)}{\beta^{2}}2b\bar{G}^{r}_{r}-\frac{\epsilon^{AB}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AC}b\bar{F}^{C}_{B}\right)+\mathcal{L}_{Y}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})$
$\displaystyle=$
$\displaystyle\beta\left(2b\bar{G}^{r}_{r}-\frac{(\beta^{2}-s)}{\beta^{2}}2b\bar{G}^{r}_{r}\right)+\mathcal{L}_{Y}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})$
$\displaystyle=$
$\displaystyle\frac{2s}{\beta}b\;\bar{G}^{r}_{r}+\mathcal{L}_{Y}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})$

which is even because $b\sim\text{odd}$ and $\bar{G}^{r}_{r}\sim\text{odd}$.
This means that the first parity condition in (3.21) is preserved.

Utilizing the equation (A) and subsequent simplification, we obtain the
following expression for $\delta\bar{F}^{\theta}_{r}$

$\displaystyle\delta\bar{F}^{\theta}_{r}$
$\displaystyle=\frac{\beta}{\sqrt{\bar{\gamma}}}\left[b\bar{D}_{\theta}\bar{F}^{\theta}_{\varphi}-b\bar{D}_{\varphi}\bar{F}^{\theta}_{\theta}-\bar{F}^{\theta}_{\varphi}(\bar{D}_{\theta}b)-\bar{F}^{\varphi}_{\varphi}(\bar{D}_{\varphi}b)+\bar{D}_{\varphi}\tilde{f}+b\sqrt{\bar{\gamma}}\left(\bar{G}^{r}_{\theta}-\frac{\beta^{2}-s}{\beta}\bar{k}^{\theta}_{r}\right)\right]+\mathcal{L}_{Y}\bar{F}^{\theta}_{r}-\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\varphi}$
$\displaystyle=\beta
b\left(\bar{G}^{r}_{\theta}-\frac{\beta^{2}-s}{\beta}\bar{k}^{\theta}_{r}\right)-\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\varphi}+\text{even}$
$\displaystyle\sim\text{even}$ (3.71)

where by employing the parity conditions (3.19), (3.20), and (3.30) in the
first step, we can determine the terms with transparent even parity.
Subsequently, we make use of the parity
$\bar{k}^{\theta}_{r}=\beta^{-1}(\bar{G}^{\theta}_{r}-\bar{\Gamma}^{\theta}_{r})\sim\text{odd}$
derived from (3.22) and the equation

$\bar{\Gamma}^{\theta}_{r}=\frac{1}{2\sqrt{\bar{\gamma}}}\left(2\bar{\gamma}\bar{F}^{\varphi}_{r}-\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})\right)$
(3.72)

Furthermore, through the cancellation of the odd parts of $\beta
b\bar{G}^{r}_{\theta}$ and
$-\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\varphi}$ as shown in (3.23) and
(3.34), an even quantity remains. Thus, the parity of $\bar{F}^{\theta}_{r}$
is preserved.

A similar approach is utilized for $\delta\bar{F}^{\varphi}_{r}$. By employing
the equation (A) and simplifying, we obtain

$\displaystyle\delta\bar{F}^{\varphi}_{r}$
$\displaystyle=\frac{\beta}{\sqrt{\bar{\gamma}}}\left[b\bar{D}_{\theta}\bar{F}^{\varphi}_{\varphi}-b\bar{D}_{\varphi}\bar{F}^{\varphi}_{\theta}+\bar{F}^{\theta}_{\theta}(\bar{D}_{\theta}b)+\bar{F}^{\varphi}_{\theta}(\bar{D}_{\varphi}b)+\bar{D}_{\theta}\tilde{f}+b\sqrt{\bar{\gamma}}\left(\bar{G}^{r}_{\varphi}-\frac{\beta^{2}-s}{\beta}\bar{k}^{\varphi}_{r}\right)\right]+\mathcal{L}_{Y}\bar{F}^{\varphi}_{r}+\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\theta}$
$\displaystyle=\left(\bar{G}^{r}_{\varphi}-\frac{\beta^{2}-s}{\beta}\bar{k}^{\varphi}_{r}\right)+\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\theta}+\text{odd}$
$\displaystyle\sim\text{odd}$ (3.73)

In the first step, we used the parity conditions (3.19), (3.20), and (3.30) to
determine the terms with transparent odd parity. Next, we employed the parity
$\bar{k}^{\varphi}_{r}=\beta^{-1}(\bar{G}^{\varphi}_{r}-\bar{\Gamma}^{\varphi}_{r})\sim\text{even}$
resulting from (3.24) and the equation

$\bar{\Gamma}^{\varphi}_{r}=\frac{1}{2\sqrt{\bar{\gamma}}}\left(-2\bar{\gamma}\bar{F}^{\theta}_{r}-\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})\right)$
(3.74)

Additionally, the cancellation of the even parts of $\beta
b\bar{G}^{r}_{\theta}$ and
$-\frac{1}{\sqrt{\bar{\gamma}}}\bar{\Lambda}_{\varphi}$ is evident from (3.23)
and (3.34), resulting in an odd quantity. Consequently, the parity of
$\bar{F}^{\varphi}_{r}$ is preserved.

Furthermore, considering that the parities of $\bar{F}^{\theta}_{r}$ and
$\bar{F}^{\varphi}_{r}$ are both preserved, and due to the fact that the
equation (3.28) remains unchanged under hypersurface deformations, we can
conclude that the parities of $\bar{F}_{\theta}^{r}$ and
$\bar{F}_{\varphi}^{r}$ are also preserved. This result can also be confirmed
directly from (A).

By using the equation (A), one can derive $\delta\bar{F}^{\theta}\theta$ and
$\delta\bar{F}^{\varphi}\varphi$, which can be simplified as

$\displaystyle\delta\bar{F}^{\theta}_{\theta}$
$\displaystyle=\beta\left[\frac{1}{\sqrt{\bar{\gamma}}}b(\bar{D}_{\varphi}\bar{F}^{\theta}_{r})-b\;(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi})+\frac{(\beta^{2}-s)}{\beta^{2}}b\;(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}-\bar{\Gamma}^{\varphi}_{\varphi})\right]+W+\bar{D}_{\varphi}I^{\varphi}+\mathcal{L}_{Y}\bar{F}^{\theta}_{\theta}$
$\displaystyle\sim\text{odd}$ (3.75)
$\displaystyle\delta\bar{F}^{\varphi}_{\varphi}$
$\displaystyle=\beta\left[-\sqrt{\bar{\gamma}}b(\bar{D}_{\theta}\bar{F}^{\varphi}_{r})-b\;(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta})+\frac{(\beta^{2}-s)}{\beta^{2}}b\;(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}-\bar{\Gamma}^{\theta}_{\theta})\right]+W+\bar{D}_{\theta}I^{\theta}+\mathcal{L}_{Y}\bar{F}^{\varphi}_{\varphi}$
$\displaystyle\sim\text{odd}$ (3.76)

It is evident that both $\delta\bar{F}^{\theta}\theta$ and
$\delta\bar{F}^{\varphi}\varphi$ are odd, as can be verified by the fact that
$I^{\theta}\sim\text{even}$, $I^{\varphi}\sim\text{odd}$, and

$\bar{\Gamma}^{\varphi}_{\varphi}=\frac{1}{\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}\bar{F}^{\theta}_{r}\sim\text{even},\;\;\;\;\;\bar{\Gamma}^{\theta}_{\theta}=-\sqrt{\bar{\gamma}}\bar{D}_{\theta}\bar{F}^{\varphi}_{r}\sim\text{even}$
(3.77)

Consequently, the parities of $\bar{F}^{\theta}\theta$ and
$\bar{F}^{\varphi}\varphi$ are maintained under deformations of hypersurfaces.

The off-diagonal component of $\delta\bar{F}^{A}_{B}$ can be expressed as
follows in a simplified manner

$\displaystyle\delta\bar{F}^{\theta}_{\varphi}$
$\displaystyle=\beta\left[\sqrt{\bar{\gamma}}\left(-b\bar{D}_{\theta}\bar{F}^{\theta}_{r}+\bar{F}^{A}_{r}(\bar{D}_{A}b)\right)+b\;\bar{\gamma}\bar{G}^{\varphi}_{\theta}-\frac{(\beta^{2}-s)}{\beta}b\;\bar{k}^{\theta}_{\varphi}\right]-\bar{D}_{\varphi}I^{\theta}+\mathcal{L}_{Y}\bar{F}^{\theta}_{\varphi}+\sqrt{\bar{\gamma}}\bar{\Lambda}^{r}$
$\displaystyle=\beta
b\;\bar{\gamma}\bar{G}^{\varphi}_{\theta}+\sqrt{\bar{\gamma}}\bar{\Lambda}^{r}+\text{even}$
$\displaystyle\sim\text{even}$ (3.78)

We arrived at this result by utilizing the fact that
$\bar{k}^{\theta}_{\varphi}\sim\text{odd}$ and the observation that from
(3.32), it can be deduced that

$\bar{\Lambda}^{r}=\text{even}-\beta\frac{b}{2}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A}),$
(3.79)

Therefore, the odd components of $\beta
b\;\bar{\gamma}\bar{G}^{\varphi}_{\theta}$ and
$\sqrt{\bar{\gamma}}\bar{\Lambda}^{r}$ cancel each other out. Additionally,
given that the parity of $\bar{F}^{\theta}_{\varphi}$ remains unchanged and
equation (3.29) is unaffected by deformations in hypersurfaces, it can be
concluded that the parity of $\bar{F}^{\varphi}_{\theta}$ is likewise
preserved.

Based on the appearance of equation (A), it is not possible to ensure that it
is odd. Therefore, we proceed to apply slight adjustments in order to
transform it into a more appropriate format, facilitating the determination of
its parity. The subsequent calculation outlines the procedure employed for
this purpose.

$\displaystyle\delta\bar{G}^{r}_{r}=$
$\displaystyle-\beta\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\left(\bar{D}_{A}(b\bar{G}^{B}_{r})+b\bar{G}^{B}_{A}\right)+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\bar{D}_{A}(b\;\epsilon^{AB}\bar{\pi}^{r}_{B})+\mathcal{L}_{Y}\bar{G}^{r}_{r}+\bar{\Lambda}^{r}$
$\displaystyle=$
$\displaystyle-\beta\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{D}_{A}\left(b\left[-\frac{\epsilon^{DB}}{2\sqrt{\bar{\gamma}}}\left(\bar{\gamma}_{DE}\bar{F}^{E}_{r}-\bar{D}_{D}(\bar{F}^{r}_{r}-\bar{F^{E}_{E}})\right)+\beta\frac{1}{2\sqrt{\bar{\gamma}}}\bar{\pi}^{rB}\right]\right)-\beta
b\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{B}_{A}$
$\displaystyle+\frac{\beta^{2}-s}{2\bar{\gamma}}\bar{D}_{A}(b\;\epsilon^{AB}\bar{\pi}^{r}_{B})+\mathcal{L}_{Y}\bar{G}^{r}_{r}-\frac{s}{2\bar{\gamma}}\epsilon^{DA}\bar{D}_{A}(b\bar{\pi}^{r}_{D})+\beta\left(b(\bar{D}_{A}\bar{F}^{A}_{r})-\frac{1}{2}\bar{F}^{A}_{r}(\partial_{A}b)-\frac{1}{2}b\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{B}_{A}\right)$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\bar{\gamma}^{AD}\bar{D}_{A}\left(b\left[\bar{\gamma}_{DE}\bar{F}^{E}_{r}-\bar{D}_{D}(\bar{F}^{r}_{r}-\bar{F^{E}_{E}})\right]\right)+\mathcal{L}_{Y}\bar{G}^{r}_{r}+\beta\left(b(\bar{D}_{A}\bar{F}^{A}_{r})-\frac{1}{2}\bar{F}^{A}_{r}(\partial_{A}b)-\frac{3}{2}b\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{B}_{A}\right)$
$\displaystyle=$ $\displaystyle\frac{3}{2}\beta
b(\bar{D}_{A}\bar{F}^{A}_{r})-\frac{\beta}{2}(\bar{D}_{A}b)[\bar{D}^{A}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})]-\frac{\beta}{2}b\bar{D}_{A}\bar{D}^{A}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})-\frac{3}{2}\beta
b\frac{\epsilon^{AC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{BC}\bar{G}^{B}_{A}+\mathcal{L}_{Y}\bar{G}^{r}_{r}$
$\displaystyle=$ $\displaystyle-\frac{3}{2}\beta
b(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})-\frac{\beta}{2}(\bar{D}_{A}b)[\bar{D}^{A}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})]-\frac{\beta}{2}b\bar{D}_{A}\bar{D}^{A}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})+\mathcal{L}_{Y}\bar{G}^{r}_{r}+\frac{3}{2}\beta
b\frac{\bar{\mathcal{G}}^{r}}{\sqrt{\bar{\gamma}}}$ $\displaystyle\sim$ odd
(3.80)

The odd parity of $\delta\bar{G}^{r}_{r}$ arises due to the fact that the
first four terms in the last line of the above calculation are odd and
$\bar{\mathcal{G}}^{r}=0$ is taken into consideration. Consequently, the
parity of $\bar{G}^{r}_{r}$ remains unchanged.

The remaining parity conditions are addressed in a similar way using the
equations provided in in appendix A. The variations of the remaining variables
with definite parity conditions are presented below in a row.

$\displaystyle\delta\left(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}\right)=$
$\displaystyle\beta\sqrt{\bar{\gamma}}\left(\bar{D}_{\varphi}[b(\bar{G}^{\theta}_{r})_{\text{odd}}]-\bar{D}_{\theta}[b(\bar{G}^{\varphi}_{r})_{\text{even}}]\right)+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\left(\bar{D}_{\theta}(b[\bar{k}^{r}_{\varphi}+\bar{\gamma}\bar{k}^{\varphi}_{r}])-\bar{D}_{\varphi}(b[\bar{k}^{r}_{\theta}+\bar{k}^{\theta}_{r}])\right)$
$\displaystyle+\beta\frac{b}{\sqrt{\bar{\gamma}}}\left(-\bar{\gamma}\bar{D}_{\theta}(\bar{G}^{\varphi}_{r})_{\text{even}}+\bar{D}_{\theta}(\bar{G}^{r}_{\varphi})_{\text{even}}-\bar{D}_{\varphi}(\bar{G}^{r}_{\theta})_{\text{odd}}\right)-\beta\frac{1}{\sqrt{\bar{\gamma}}}\left((\partial_{\varphi}b)(\bar{G}^{r}_{\theta})_{\text{odd}}\right)$
$\displaystyle+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\left(\bar{D}_{\varphi}(b\bar{k}^{\theta}_{r})+b\bar{\gamma}\bar{k}^{\varphi}_{\theta}-b\bar{k}^{\theta}_{\varphi}-\bar{D}_{\theta}(b[\bar{k}^{r}_{\varphi}-\bar{\gamma}\bar{k}^{\varphi}_{r}])+\bar{D}_{\varphi}(b[\bar{k}^{r}_{\theta}-\bar{k}^{\theta}_{r}])\right)$
$\displaystyle+\mathcal{L}_{Y}\left(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}\right)-\bar{D}_{\theta}(\bar{\Lambda}^{\theta})_{\text{odd}}$
$\displaystyle\sim$ even (3.81)

$\displaystyle\delta\left(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}\right)=$
$\displaystyle\beta\sqrt{\bar{\gamma}}\left(\bar{D}_{\varphi}[b(\bar{G}^{\theta}_{r})_{\text{odd}}]-\bar{D}_{\theta}[b(\bar{G}^{\varphi}_{r})_{\text{even}}]\right)+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\left(\bar{D}_{\theta}(b[\bar{k}^{r}_{\varphi}+\bar{\gamma}\bar{k}^{\varphi}_{r}])-\bar{D}_{\varphi}(b[\bar{k}^{r}_{\theta}+\bar{k}^{\theta}_{r}])\right)$
$\displaystyle+\beta\frac{b}{\sqrt{\bar{\gamma}}}\left(\bar{\gamma}\bar{D}_{\varphi}(\bar{G}^{\theta}_{r})_{\text{odd}}-\bar{D}_{\varphi}(\bar{G}^{r}_{\theta})_{\text{odd}}+\bar{D}_{\theta}(\bar{G}^{r}_{\varphi})_{\text{even}}\right)+\beta\frac{1}{\sqrt{\bar{\gamma}}}\left((\partial_{\theta}b)(\bar{G}^{r}_{\varphi})_{\text{even}}\right).$
$\displaystyle+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\left(-2\bar{\gamma}\bar{D}_{\theta}(b\bar{k}^{\varphi}_{r})+b\bar{\gamma}\bar{k}^{\varphi}_{\theta}-b\bar{k}^{\theta}_{\varphi}-\bar{D}_{\theta}(b[\bar{k}^{r}_{\varphi}-\bar{\gamma}\bar{k}^{\varphi}_{r}])+\bar{D}_{\varphi}(b[\bar{k}^{r}_{\theta}-\bar{k}^{\theta}_{r}])\right)$
$\displaystyle+\mathcal{L}_{Y}\left(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}\right)-\bar{D}_{\varphi}(\bar{\Lambda}^{\varphi})_{\text{even}}$
$\displaystyle\sim$ even (3.82)

$\displaystyle\delta\left(\bar{G}^{\theta}_{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=\beta\frac{1}{\sqrt{\bar{\gamma}}}\left(2b\bar{\gamma}(\bar{G}^{\varphi}_{r})_{\text{even}}-b(\bar{G}^{r}_{\varphi})_{\text{even}}\right)-(\beta^{2}-s)\sqrt{\bar{\gamma}}b\;\bar{k}^{\varphi}_{r}+\mathcal{L}_{Y}\left(\bar{G}^{\theta}_{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+(\bar{\Lambda}^{\theta})_{\text{odd}}$
$\displaystyle\sim\text{odd}$ (3.83)

$\displaystyle\delta\left(\bar{G}_{\theta}^{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=-\beta\frac{1}{\sqrt{\bar{\gamma}}}\left(b(\bar{G}^{r}_{\varphi})_{\text{even}}+b\bar{D}_{\theta}(\bar{G}^{\theta}_{\varphi})_{\text{odd}}-\bar{D}_{\varphi}(b(\bar{G}^{\theta}_{\theta})_{\text{even}})+(\partial_{\theta}b)\bar{\gamma}(\bar{G}^{\varphi}_{\theta})_{\text{odd}}\right)$
$\displaystyle\hskip
14.22636pt+\frac{\beta^{2}-s}{2\sqrt{\bar{\gamma}}}\left[2b\;\bar{k}^{r}_{\varphi}+2\bar{D}_{\theta}(b\;\bar{k}^{\theta}_{\varphi})-2\beta^{-1}\bar{D}_{\varphi}\left(b[(\bar{G}^{\theta}_{\theta})_{\text{even}}+\sqrt{\bar{\gamma}}(\bar{D}_{\theta}\bar{F}^{\varphi}_{r})]\right)\right]$
$\displaystyle\hskip
14.22636pt-\frac{1}{2}\bar{D}_{\theta}\left(-s\;\frac{\epsilon^{DA}}{\sqrt{\bar{\gamma}}}\bar{D}_{A}\left(b\;[\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r}]\right)-\beta\left[\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}\right]\right)$
$\displaystyle\hskip
14.22636pt+\mathcal{L}_{Y}\left(\bar{G}^{\theta}_{r}+\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+(\bar{\Lambda}^{\theta})_{\text{odd}}$
$\displaystyle\sim\text{odd}$ (3.84)

$\displaystyle\delta\left(\bar{G}^{\varphi}_{r}-\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=-\beta\frac{b}{\sqrt{\bar{\gamma}}}\left(2(\bar{G}^{\theta}_{r})_{\text{odd}}-(\bar{G}^{r}_{\theta})_{\text{odd}}\right)+\frac{\beta^{2}-s}{\sqrt{\bar{\gamma}}}b\bar{k}^{\theta}_{r}+\mathcal{L}_{Y}\left(\bar{G}^{\varphi}_{r}-\frac{1}{2\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+(\bar{\Lambda}^{\varphi})_{\text{even}}$
$\displaystyle\sim\text{even}$ (3.85)

$\displaystyle\delta\left(\bar{G}_{\varphi}^{r}-\frac{\sqrt{\bar{\gamma}}}{2}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=-\beta\sqrt{\bar{\gamma}}\left(-b(\bar{G}^{r}_{\theta})_{\text{odd}}-b\bar{D}_{\varphi}(\bar{G}^{\varphi}_{\theta})_{\text{odd}}+\bar{D}_{\theta}(b(\bar{G}^{\varphi}_{\varphi})_{\text{even}})\right)+(\beta^{2}-s)\sqrt{\bar{\gamma}}\left(\bar{D}_{\theta}(\beta^{-1}b[(\bar{G}^{\varphi}_{\varphi})_{\text{even}}])-b\bar{k}^{r}_{\theta}-\bar{D}_{\varphi}(b\bar{k}^{\varphi}_{\theta})\right)$
$\displaystyle\hskip
14.22636pt+\frac{\beta}{\sqrt{\bar{\gamma}}}(\partial_{\varphi}b)(\bar{G}_{\varphi}^{\theta})_{\text{odd}}-\frac{1}{2}\bar{D}_{\varphi}\left(-s\;\frac{\epsilon^{DA}}{\sqrt{\bar{\gamma}}}\bar{D}_{A}\left(b\;[\bar{k}^{r}_{D}+\bar{\gamma}_{BD}\bar{k}^{B}_{r}]\right)-\beta\left[\bar{F}^{A}_{r}(\bar{D}_{A}b)-b\bar{D}_{A}\bar{F}^{A}_{r}\right]\right)$
$\displaystyle\hskip
14.22636pt+\mathcal{L}_{Y}\left(\bar{G}_{\varphi}^{r}-\frac{\sqrt{\bar{\gamma}}}{2}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+(\bar{\Lambda}_{\varphi})_{\text{even}}$
$\displaystyle\sim\text{even}$ (3.86)

$\displaystyle\delta\left(\bar{G}^{\theta}_{\varphi}+\frac{\sqrt{\bar{\gamma}}}{2}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=-\beta
b\sqrt{\bar{\gamma}}\bar{D}_{\varphi}(\bar{G}^{\varphi}_{r})_{\text{even}}+(\beta^{2}-s)\sqrt{\bar{\gamma}}\bar{D}_{\varphi}(b\bar{k}^{\varphi}_{r})-\frac{\beta}{\sqrt{\bar{\gamma}}}(\partial_{\varphi}b)(\bar{G}^{r}_{\varphi})_{\text{even}}-\bar{D}_{\varphi}(\bar{\Lambda}^{\theta})_{\text{odd}}+\mathcal{L}_{Y}\left(\bar{G}^{\theta}_{\varphi}+\frac{\sqrt{\bar{\gamma}}}{2}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle\sim\text{odd}$ (3.87)

$\displaystyle\delta\left(\bar{G}_{\theta}^{\varphi}-\frac{1}{2\sqrt{\bar{\gamma}}}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle=\beta\frac{b}{\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(\bar{G}^{\theta}_{r})_{\text{odd}}-\frac{\beta^{2}-s}{\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(b\bar{k}^{\theta}_{r})+\frac{\beta}{\sqrt{\bar{\gamma}}}(\partial_{\theta}b)(\bar{G}^{r}_{\theta})_{\text{odd}}-\bar{D}_{\theta}(\bar{\Lambda}^{\varphi})_{\text{even}}+\mathcal{L}_{Y}\left(\bar{G}_{\theta}^{\varphi}-\frac{1}{2\sqrt{\bar{\gamma}}}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)$
$\displaystyle\sim\text{odd}$ (3.88)

where to obtain the parities we have used the parity conditions in the
theorem.

This demonstrates that the newly established boundary conditions remain intact
when subjected to deformations of hypersurfaces. Hence, the requirement (ii)
is satisfied. The sole matter that still requires examination pertains to the
derived surface terms as described in section 3.2, consequently giving rise to
the asymptotic charges. This is the task that we will undertake in the
subsequent section.

###  3.4 Asymptotic charges

Thus far, we have shown that the new boundary conditions ensure their
invariance under deformations of hypersurfaces as well as the well-definedness
of the symplectic structure. We will now investigate the circumstances under
which the canonical generators of the asymptotic symmetries remain well-
defined when the parity conditions introduced in the theorem are applied. In
other words, we must first examine whether the boundary terms are finite, and
then determine if they are exact.

Our objective is to prove that the bulk portion of the generators, defined by
the smeared constraints $\int
d^{3}x(\mathcal{H}[\tilde{N}]+\mathcal{H}_{a}[N^{a}]+\mathcal{G}[\Lambda])$,
can be supplemented with appropriate surface terms that render the sum
functionally differentiable.

$\begin{split}&\delta_{(\mathbf{N},\Lambda)}\left(\int
d^{3}x(\mathcal{H}[\tilde{N}]+\mathcal{H}_{a}[N^{a}]+\mathcal{G}[\Lambda])\right)\\\
&\hskip 85.35826pt=\frac{2}{\beta}\int_{\Sigma}d^{3}x\;\left((\delta
E^{a}_{i})(\delta_{(\mathbf{N},\Lambda)}A^{i}_{a})-(\delta
A^{i}_{a})(\delta_{(\mathbf{N},\Lambda)}E^{a}_{i})\right)+\mathcal{B}_{(\mathbf{N},\Lambda)}(\delta
A^{i}_{a},\delta E^{a}_{i})\end{split}$ (3.89)

where

$\mathcal{B}_{(\mathbf{N},\Lambda)}(\delta A^{i}_{a},\delta
E^{a}_{i}):=\mathcal{B}_{\tilde{N}}(\delta A^{i}_{a},\delta
E^{a}_{i})+\mathcal{B}_{\vec{N}}(\delta A^{i}_{a},\delta
E^{a}_{i})+\mathcal{B}_{\Lambda}(\delta A^{i}_{a},\delta E^{a}_{i})$ (3.90)

and the explicit expressions of the boundary terms are given in (3.61), (3.58)
and (3.50) respectively.

By employing the asymptotic expansion and gathering all divergent and finite
terms, we derive the following expression

$\begin{split}\mathcal{B}_{(\mathbf{N},\Lambda)}&(\delta A^{i}_{a},\delta
E^{a}_{i})\\\ =&\;2r\oint
d\sigma\left[(b\epsilon^{AB}\bar{\gamma}_{BC}\delta\bar{G}^{C}_{A}-b\sqrt{\bar{\gamma}}\delta\bar{F}^{r}_{r}-\sqrt{\bar{\gamma}}(\partial_{A}b)\bar{\gamma}^{AB}\delta\bar{F}^{r}_{B})+\frac{s}{\beta}\sqrt{\bar{\gamma}}Y^{A}(\delta\bar{G}^{r}_{A}-2\epsilon^{BC}\bar{\gamma}_{AC}\delta\bar{F}^{r}_{B})\right]\\\
&+\oint
d\sigma\left[-2\epsilon^{AB}\left\\{b\bar{F}^{r}_{B}\delta\bar{G}^{r}_{A}-b\bar{F}^{r}_{r}\bar{\gamma}_{BC}\delta\bar{G}^{C}_{A}-b\bar{F}^{C}_{A}\bar{\gamma}_{BD}\delta\bar{G}^{D}_{C}-b\bar{\gamma}_{BC}\delta\bar{G}^{(2)C}_{A}-\tilde{f}\bar{\gamma}_{BC}\delta\bar{G}^{C}_{A}\right.\right.\\\
&\hskip
42.67912pt-\frac{\beta^{2}-s}{\beta}b\left.\left(\bar{k}^{D}_{B}\bar{\gamma}_{CD}\delta\bar{F}^{C}_{A}-\bar{k}^{r}_{A}\delta\bar{F}^{r}_{B}\right)\right\\}-2\sqrt{\bar{\gamma}}(b\delta\bar{F}^{(2)r}_{r}+(\partial_{A}b)\bar{\gamma}^{AB}\delta\bar{F}^{(2)r}_{B})\\\
&\hskip
42.67912pt+\frac{2s}{\beta}\sqrt{\bar{\gamma}}\left(-W\delta\bar{G}^{A}_{A}+I^{A}\delta\bar{G}^{r}_{A}+Y^{A}\delta\bar{G}^{(2)r}_{A}+Y^{A}\delta(\bar{F}^{r}_{a}\bar{G}^{a}_{A})-2Y^{A}\frac{\epsilon^{BC}}{\sqrt{\bar{\gamma}}}\bar{\gamma}_{AC}\bar{F}^{(2)r}_{B}\right)\\\
&\hskip
42.67912pt+\left.\frac{2}{\beta}\sqrt{\bar{\gamma}}\bar{\Lambda}^{a}\delta\bar{F}^{r}_{a}\right]\end{split}$
(3.91)

As previously discussed in section 2.1.2, when working with the ADM variables,
vanishing of the leading terms of the constraints were used to eliminate the
divergent portion of the surface terms. However, in this case, the divergent
part of the surface terms (i.e. the first line in equation (3.91)) cannot be
eliminated either by employing equations (3.43), (3.44), (3.51), (3.52), and
(3.59), nor by applying parity conditions. This discrepancy is unexpected
since the Ashtekar-Barbero variables formulation of General Relativity is
expected to be equivalent to the formulation utilizing the ADM variables. In
the subsequent section, we will investigate and address this fundamental
difference.

The notable aspect regarding the aforementioned surface term (3.91) is that if
we disregard asymptotic rotations and boosts (i.e. by setting $b=Y^{A}=0$),
the divergent term is eradicated, while the corresponding charge for
supertranslations remains integrable and non-zero. In other words,

$\begin{split}\mathcal{B}_{(\mathbf{N},\Lambda)}(\delta A^{i}_{a},\delta
E^{a}_{i})\left|{}_{b=Y^{A}=0}\right.&=\oint
d\sigma\left[2\epsilon^{AB}\tilde{f}\bar{\gamma}_{BC}\delta\bar{G}^{C}_{A}+\frac{2s}{\beta}\sqrt{\bar{\gamma}}\left(-W\delta\bar{G}^{A}_{A}+(\bar{D}^{A}W)\delta\bar{G}^{r}_{A}\right)\right]\\\
&=\oint
d\sigma\left[2\epsilon^{AB}\tilde{f}\bar{\gamma}_{BC}\delta\bar{G}^{C}_{A}-\frac{2s}{\beta}\sqrt{\bar{\gamma}}W\left(\delta\bar{G}^{A}_{A}+\bar{D}^{A}\delta\bar{G}^{r}_{A}\right)\right]\\\
&=\delta\left(2\oint
d\sigma\sqrt{\bar{\gamma}}\left[\tilde{f}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})+\frac{2s}{\beta}W\bar{G}^{r}_{r}\right]\right)\end{split}$
(3.92)

Hence, based on equation (2.2), the charges associated with supertranslations
can be expressed as

$\mathcal{Q}_{Supertranslation}=-2\oint
d\sigma\sqrt{\bar{\gamma}}\left[\tilde{f}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})+\frac{2s}{\beta}W\bar{G}^{r}_{r}\right]$
(3.93)

These charges do not vanish in general, as they represent integrals of non-
trivial even functions. This outcome holds significant importance since, even
in the absence of boosts and rotations in the expansion of lapse and shift,
respectively, the charges linked to supertranslations become null when
standard boundary conditions are employed (See equation (2.9) and its
accompanying explanation). Conversely, the newly proposed parity conditions in
section 3.1 yield non-zero supertranslation charges. Note that when
$\tilde{f}$ and $W$ represent arbitrary angle-dependent even and odd
functions, respectively, the non-zero supertranslation charges emerge.
Additionally, the zero modes of $\tilde{f}$ and $W$ give rise to the customary
charges associated with ordinary translations.

##  4 Comparison with Henneaux and Troessaert’s paper

In this section, we begin by transcribing the boundary conditions imposed on
the Ashtekar-Barbero variables into the ADM variables. This transcription is
performed in order to facilitate a comparison between the results obtained in
this paper and those obtained in [3]. It should be recalled, as discussed in
Section 2.1.2, that the leading terms of canonical variables are
$\bar{\lambda},\bar{h}_{rA},\bar{k}_{AB}$ (refer to (2.16) for the definition
of $\bar{\lambda}$ and $\bar{k}_{AB}$), along with leading terms of their
conjugate momenta $\bar{p},\bar{\pi}^{rA},\bar{\pi}^{AB}$ (refer to (2.18) for
the definition of $\bar{p}$). It is assumed that $\bar{h}_{rA}$ is equal to
zero, as stated in Section 2.1.2.

By utilizing the relation $q_{ab}=e^{i}_{a}e^{i}_{b}=q^{-1}E^{i}_{a}E^{i}_{b}$
and performing the asymptotic expansion of both sides, we can express the
variables $\bar{\lambda},\bar{h}_{rA},\bar{k}_{AB}$ in terms of
$\bar{F}^{a}_{b}$ as

$\displaystyle\bar{\lambda}=\frac{1}{2}\bar{h}_{rr}=\frac{1}{2}\left(\bar{F}^{a}_{a}-2\gamma^{i}_{r}\bar{f}^{i}_{r}\right)=\frac{1}{2}\left(\bar{F}^{A}_{A}-\bar{F}^{r}_{r}\right)\sim\text{even},$
(4.1)
$\displaystyle\bar{h}_{rA}=\bar{f}^{r}_{i}\gamma^{i}_{A}+\bar{\gamma}_{AB}\bar{f}^{B}_{i}\gamma^{i}_{r}=\bar{F}^{r}_{A}+\bar{\gamma}_{AB}\bar{F}^{B}_{r}=0,$
(4.2)
$\displaystyle\bar{h}_{AB}=\bar{F}^{a}_{a}\bar{\gamma}_{AB}-\gamma^{i}_{A}\bar{f}^{i}_{B}-\gamma^{i}_{B}\bar{f}^{i}_{A}$
(4.3)

Here, we have used equations (2.12), (3.2), (3.3), and (3.12). The parity of
(4.1) is inferred from (3.21) and equation (4.2) from (3.28).

Using (4.3), the components of $\bar{k}_{AB}$ can be determined as

$\displaystyle\bar{k}_{AB}=\frac{1}{2}(\bar{h}_{AB}+\bar{h}_{rr}\bar{\gamma}_{AB})\;\Rightarrow\;$
$\displaystyle\bar{k}_{\theta\theta}=\bar{f}^{\varphi}_{i}\gamma^{i}_{\varphi}=\bar{F}^{\varphi}_{\varphi}\sim\text{odd},$
(4.4)
$\displaystyle\bar{k}_{\varphi\varphi}=\bar{\gamma}\bar{f}^{\theta}_{i}\gamma^{i}_{\theta}=\bar{\gamma}\bar{F}^{\theta}_{\theta}\sim\text{odd},$
(4.5)
$\displaystyle\bar{k}_{\theta\varphi}=-\frac{1}{2}(\bar{f}^{\theta}_{i}\gamma^{i}_{\varphi}+\bar{\gamma}\bar{f}^{\varphi}_{i}\gamma^{i}_{\theta})=-\frac{1}{2}(\bar{F}^{\theta}_{\varphi}+\bar{\gamma}\bar{F}^{\varphi}_{\theta})\sim\text{even}$
(4.6)

The parities of these variables have been determined based on (3.19) and
(3.20).

Using the relation
$\pi^{ab}=2|\det(E)|^{-1}E^{a}_{k}E^{d}_{k}K^{j}_{[d}\delta^{b}_{c]}E^{c}_{j}$,
we can express the ADM momenta in terms of the Ashtekar-Barbero variables.

$\displaystyle\pi^{rr}$
$\displaystyle=2|\det(E)|^{-1}E^{r}_{k}E^{d}_{k}K^{j}_{[d}\delta^{r}_{c]}E^{c}_{j}$
$\displaystyle=|\det(E)|^{-1}E^{r}_{k}(E^{d}_{k}K^{j}_{d}E^{r}_{j}-E^{r}_{k}K^{j}_{c}E^{c}_{j})$
$\displaystyle=|\det(E)|^{-1}E^{r}_{k}(E^{A}_{k}K^{j}_{A}E^{r}_{j}-E^{r}_{k}K^{j}_{A}E^{A}_{j})$
$\displaystyle=-\sqrt{\bar{\gamma}}\;\bar{k}^{j}_{A}\gamma^{A}_{j}+O(r^{-1})$
(4.7)

$\displaystyle\pi^{rA}$
$\displaystyle=2|\det(E)|^{-1}E^{r}_{k}E^{d}_{k}K^{j}_{[d}\delta^{A}_{c]}E^{c}_{j}$
$\displaystyle=|\det(E)|^{-1}E^{r}_{k}(E^{d}_{k}K^{j}_{d}E^{A}_{j}-E^{A}_{k}K^{j}_{c}E^{c}_{j})$
$\displaystyle=|\det(E)|^{-1}E^{r}_{k}(E^{A}_{k}K^{j}_{A}E^{r}_{j}-E^{r}_{k}K^{j}_{A}E^{A}_{j})$
$\displaystyle=|\det(E)|^{-1}E^{r}_{k}(E^{r}_{k}K^{j}_{r}E^{A}_{j}+E^{B}_{k}K^{j}_{B}E^{A}_{j}-E^{A}_{k}K^{j}_{r}E^{r}_{j}-E^{A}_{k}K^{j}_{B}E^{B}_{j})$
$\displaystyle=\frac{1}{r}\sqrt{\bar{\gamma}}\;(\bar{k}^{j}_{r}\gamma^{A}_{j}+\bar{\gamma}^{AB}\bar{k}^{j}_{B}\gamma^{r}_{j})+O(r^{-2})$
(4.8)

$\displaystyle\pi^{AB}$
$\displaystyle=2|\det(E)|^{-1}E^{A}_{k}E^{d}_{k}K^{j}_{[d}\delta^{B}_{c]}E^{c}_{j}$
$\displaystyle=|\det(E)|^{-1}E^{A}_{k}(E^{d}_{k}K^{j}_{d}E^{B}_{j}-E^{B}_{k}K^{j}_{c}E^{c}_{j})$
$\displaystyle=|\det(E)|^{-1}E^{A}_{k}(E^{r}_{k}K^{j}_{r}E^{B}_{j}+E^{C}_{k}K^{j}_{C}E^{B}_{j}-E^{B}_{k}K^{j}_{r}E^{r}_{j}-E^{B}_{k}K^{j}_{C}E^{C}_{j})$
$\displaystyle=\frac{1}{r^{2}}\sqrt{\bar{\gamma}}\;(\bar{\gamma}^{AC}\bar{k}^{j}_{C}\gamma^{B}_{j}-\bar{\gamma}^{AB}\bar{k}^{j}_{r}\gamma^{r}_{j}-\bar{\gamma}^{AB}\bar{k}^{j}_{C}\gamma^{C}_{j})+O(r^{-3})$
(4.9)

From the above equations, we can easily read the leading order terms

$\displaystyle\bar{\pi}^{rr}=-\sqrt{\bar{\gamma}}\;\bar{k}^{A}_{A}$ (4.10)
$\displaystyle\bar{\pi}^{rA}=\sqrt{\bar{\gamma}}\;(\bar{k}^{A}_{r}+\bar{\gamma}^{AB}\bar{k}^{r}_{B})$
(4.11)
$\displaystyle\bar{\pi}^{AB}=\sqrt{\bar{\gamma}}\;(\bar{\gamma}^{AC}\bar{k}^{B}_{C}-\bar{\gamma}^{AB}\bar{k}^{r}_{r}-\bar{\gamma}^{AB}\bar{k}^{C}_{C})$
(4.12)

Now we need to express $\bar{p},\bar{\pi}^{rA},\bar{\pi}^{AB}$ in terms of
$\bar{F}^{a}_{b}$ and $\bar{G}^{a}_{b}$. Let’s start with $\bar{p}$, as
defined in (2.18)

$\displaystyle\bar{p}$
$\displaystyle=2(\bar{\pi}^{rr}-\bar{\pi}^{A}_{A})=2\left(-\sqrt{\bar{\gamma}}\;\bar{k}^{A}_{A}-\sqrt{\bar{\gamma}}\;(\bar{k}^{C}_{C}-2\bar{k}^{r}_{r}-2\bar{k}^{C}_{C})\right)$
$\displaystyle=4\sqrt{\bar{\gamma}}\;\bar{k}^{r}_{r}=4\sqrt{\bar{\gamma}}\beta^{-1}\;(\bar{G}^{r}_{r}-\bar{\Gamma}^{r}_{r})$
$\displaystyle=4\sqrt{\bar{\gamma}}\beta^{-1}\left[\bar{G}^{r}_{r}+\frac{1}{2}\frac{1}{\sqrt{\gamma}}\epsilon^{BA}\bar{\gamma}_{C[B}\bar{F}^{C}_{A]}\right]$
$\displaystyle=4\sqrt{\bar{\gamma}}\beta^{-1}\bar{G}^{r}_{r}\sim\text{odd}$
(4.13)

where in the last step we have used (3.29). We have found that $\bar{p}$ has
an odd parity based on the parity condition (3.20).

Next, we examine $\bar{\pi}^{r\theta}$ using equation (4.11) by substituting
in it the explicit expressions of $\bar{\Gamma}^{a}_{b}$ from (3) and (3).
Then, the expression for $\bar{\pi}^{r\theta}$ is

$\displaystyle\bar{\pi}^{r\theta}$
$\displaystyle=\sqrt{\bar{\gamma}}(\bar{k}^{\theta}_{r}+\bar{\gamma}^{\theta
B}\bar{k}^{r}_{B})=\sqrt{\bar{\gamma}}(\bar{k}^{\theta}_{r}+\bar{k}^{r}_{\theta})$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\theta}_{r}+\bar{G}_{\theta}^{r}-(\bar{\Gamma}^{\theta}_{r}+\bar{\Gamma}_{\theta}^{r})\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\theta}_{r}+\bar{G}_{\theta}^{r}-\left(\frac{1}{2\sqrt{\bar{\gamma}}}\left[2\epsilon^{\theta\varphi}\bar{\gamma}_{\varphi\varphi}\bar{F}^{\varphi}_{r}-\epsilon^{\theta\varphi}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right]\right.\right.$
$\displaystyle\hskip
108.12054pt\left.\left.+\frac{1}{2\sqrt{\bar{\gamma}}}\left[2\epsilon^{\theta\varphi}\bar{\gamma}_{\theta\theta}\bar{\gamma}_{\varphi\varphi}\bar{F}^{\varphi}_{r}-\epsilon^{\theta\varphi}\bar{\gamma}_{\theta\theta}\bar{D}_{\varphi}(\bar{F}^{r}_{r}+\bar{F}^{B}_{B})+2\epsilon^{\theta\varphi}\bar{\gamma}_{\theta\theta}\bar{D}_{\varphi}\bar{F}^{\theta}_{\theta}+2\epsilon^{\varphi\theta}\bar{\gamma}_{\varphi\varphi}\bar{D}_{\theta}\bar{F}^{\varphi}_{\theta}\right]\right)\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\theta}_{r}+\bar{G}_{\theta}^{r}+\frac{1}{\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+\text{odd}$
$\displaystyle\sim\text{odd}$ (4.14)

By using the parities proposed in the theorem, particularly (3.22) and (3.23),
we have deduced that $\bar{\pi}^{r\theta}$ has an odd parity.

Similarly, we can express $\bar{\pi}^{r\varphi}$ in terms of $\bar{F}^{a}_{b}$
and $\bar{G}^{a}_{b}$ using the equation (4.11), (3), (3) and (3.18)

$\displaystyle\bar{\pi}^{r\varphi}$
$\displaystyle=\sqrt{\bar{\gamma}}(\bar{k}^{\varphi}_{r}+\bar{\gamma}^{\varphi
B}\bar{k}^{r}_{B})=\sqrt{\bar{\gamma}}(\bar{k}^{\varphi}_{r}+\bar{\gamma}^{-1}\bar{k}^{r}_{\varphi})$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\varphi}_{r}+\bar{\gamma}^{-1}\bar{G}_{\varphi}^{r}-(\bar{\Gamma}^{\varphi}_{r}+\bar{\gamma}^{-1}\bar{\Gamma}_{\varphi}^{r})\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\varphi}_{r}+\bar{\gamma}^{-1}\bar{G}_{\varphi}^{r}-\left(\frac{1}{2\sqrt{\bar{\gamma}}}\left[2\epsilon^{\varphi\theta}\bar{\gamma}_{\theta\theta}\bar{F}^{\theta}_{r}-\epsilon^{\varphi\theta}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right]\right.\right.$
$\displaystyle\hskip
108.12054pt\left.\left.+\frac{1}{2\bar{\gamma}\sqrt{\bar{\gamma}}}\left[2\epsilon^{\varphi\theta}\bar{\gamma}_{\varphi\varphi}\bar{\gamma}_{\theta\theta}\bar{F}^{\theta}_{r}-\epsilon^{\varphi\theta}\bar{\gamma}_{\varphi\varphi}\bar{D}_{\theta}(\bar{F}^{r}_{r}+\bar{F}^{B}_{B})+2\epsilon^{\theta\varphi}\bar{\gamma}_{\theta\theta}\bar{D}_{\varphi}\bar{F}^{\theta}_{\varphi}+2\epsilon^{\varphi\theta}\bar{\gamma}_{\varphi\varphi}\bar{D}_{\theta}\bar{F}^{\varphi}_{\varphi}\right]\right)\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\varphi}_{r}+\bar{\gamma}^{-1}\bar{G}_{\varphi}^{r}-\frac{1}{\sqrt{\bar{\gamma}}}\bar{D}_{\theta}(\bar{F}^{r}_{r}-\bar{F}^{B}_{B})\right)+\text{even}$
$\displaystyle\sim\text{even}$ (4.15)

By utilizing the parities proposed in the theorem, specifically (3.24) and
(3.25), we conclude that $\bar{\pi}^{r\varphi}$ has an even parity.

To express $\bar{\pi}^{AB}$ in terms of $\bar{F}^{a}_{b}$ and
$\bar{G}^{a}_{b}$, we use the equation (4.12), (3.18), (3) and (3).

$\displaystyle\bar{\pi}^{\theta}_{\theta}$
$\displaystyle=\sqrt{\bar{\gamma}}(-\bar{k}^{r}_{r}-\bar{k}^{\varphi}_{\varphi})=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}-\bar{\Gamma}^{\varphi}_{\varphi})$
$\displaystyle=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}+\frac{1}{\sqrt{\bar{\gamma}}}\epsilon^{\varphi\theta}\bar{\gamma}_{\theta\theta}\bar{D}_{\varphi}\bar{F}^{\theta}_{r})$
$\displaystyle=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\varphi}_{\varphi}-\frac{1}{\sqrt{\bar{\gamma}}}\bar{D}_{\varphi}\bar{F}^{\theta}_{r})\sim\text{even}$
(4.16) $\displaystyle\bar{\pi}^{\varphi}_{\varphi}$
$\displaystyle=\sqrt{\bar{\gamma}}(-\bar{k}^{r}_{r}-\bar{k}^{\theta}_{\theta})=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}-\bar{\Gamma}^{\theta}_{\theta})$
$\displaystyle=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}+\frac{1}{\sqrt{\bar{\gamma}}}\epsilon^{\theta\varphi}\bar{\gamma}_{\varphi\varphi}\bar{D}_{\theta}\bar{F}^{\varphi}_{r})$
$\displaystyle=-\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{r}_{r}+\bar{G}^{\theta}_{\theta}-\frac{1}{\sqrt{\bar{\gamma}}}\bar{D}_{\theta}\bar{F}^{\varphi}_{r})\sim\text{even}$
(4.17)

where the assumption $\bar{\Gamma}^{r}_{r}=0$ (obtained from the relation
(3.29)) is used in this derivation. The parity conditions (3.19)-(3.21) are
also applied to deduce that
$\bar{\pi}^{\theta}_{\theta}\sim\bar{\pi}^{\varphi}_{\varphi}\sim\text{even}$.

Furthermore, the expression for $\bar{\pi}^{\theta}_{\varphi}$ and
$\bar{\pi}_{\theta}^{\varphi}$ are found as

$\displaystyle\bar{\pi}^{\theta}_{\varphi}$
$\displaystyle=\sqrt{\bar{\gamma}}\bar{k}^{\theta}_{\varphi}=\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}^{\theta}_{\varphi}-\bar{\Gamma}^{\theta}_{\varphi})$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\theta}_{\varphi}-\frac{1}{2\sqrt{\bar{\gamma}}}\left[\epsilon^{\theta\varphi}\bar{\gamma}_{\varphi\varphi}(\bar{F}^{C}_{C}-\bar{F}^{r}_{r})-2\epsilon^{\theta\varphi}\bar{\gamma}_{\varphi\varphi}\bar{D}_{\varphi}\bar{F}^{\varphi}_{r})\right]\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}^{\theta}_{\varphi}-\frac{\sqrt{\bar{\gamma}}}{2}\left[\bar{F}^{C}_{C}-\bar{F}^{r}_{r}-2\bar{D}_{\varphi}\bar{F}^{\varphi}_{r}\right]\right)\sim\text{odd}$
(4.18)

$\displaystyle\bar{\pi}_{\theta}^{\varphi}$
$\displaystyle=\sqrt{\bar{\gamma}}\bar{k}_{\theta}^{\varphi}=\sqrt{\bar{\gamma}}\beta^{-1}(\bar{G}_{\theta}^{\varphi}-\bar{\Gamma}_{\theta}^{\varphi})$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}_{\theta}^{\varphi}-\frac{1}{2\sqrt{\bar{\gamma}}}\left[\epsilon^{\varphi\theta}\bar{\gamma}_{\theta\theta}(\bar{F}^{C}_{C}-\bar{F}^{r}_{r})-2\epsilon^{\varphi\theta}\bar{\gamma}_{\theta\theta}\bar{D}_{\theta}\bar{F}^{\theta}_{r}\right]\right)$
$\displaystyle=\sqrt{\bar{\gamma}}\beta^{-1}\left(\bar{G}_{\theta}^{\varphi}+\frac{1}{2\sqrt{\bar{\gamma}}}\left[\bar{F}^{C}_{C}-\bar{F}^{r}_{r}-2\bar{D}_{\theta}\bar{F}^{\theta}_{r}\right]\right)\sim\text{odd}$
(4.19)

The parity conditions (3.19), (3.20), (3.26), and (3.27) are utilized to
conclude that
$\bar{\pi}^{\theta}_{\varphi}\sim\bar{\pi}_{\theta}^{\varphi}\sim\text{odd}$.

The cumulative summation of the aforementioned calculations concludes that if
the parity conditions stated in the theorem are translated to the ADM
variables, the resultant parity conditions (2.20) as introduced in [3] are
exactly obtained. The question that arises here is why, despite the
application of the same parity condition to the theory, the divergence of
surface terms is eliminated using the leading order terms of the constraints
when working with ADM variables, but not when working with Ashtekar-Barbero
variables. By examining the process of obtaining Ashtekar-Barbero variables
from ADM variables, it becomes evident that in order to establish $A^{i}_{a}$
and $E^{a}_{i}$ as conjugate variables, particularly to demonstrate that the
Poisson bracket between two $A^{i}_{a}$ is zero, it is necessary to prove

$\frac{\delta\Gamma^{j}_{a}(x)}{\delta
E^{b}_{k}(y)}-\frac{\delta\Gamma^{k}_{b}(y)}{\delta E^{a}_{j}(x)}=0$ (4.20)

which represents the integrability condition for $\Gamma^{j}_{a}$ to possess a
generating function $F$. When working with a manifold without a boundary, such
an $F$ can be easily found. In addition, when working within the framework of
asymptotically flat spacetimes, it is readily demonstrated that the standard
boundary conditions result in a well-defined $F$. Nonetheless, when one wishes
to relax the boundary conditions, matters become more complex and the well-
definedness of $F$ should be regarded as an additional requirement, in
addition to the three requirements necessary to propose appropriate boundary
conditions. If a well-defined $F$ cannot be defined under specific boundary
conditions, it indicates that the Ashtekar-Barbero variables are not
canonically transformed from the ADM variables under those boundary
conditions. This is manifested just in surface terms, and consequently, all
analyses pertaining to the bulk, such as the well-definedness of the
symplectic structure and the preservation of boundary conditions under
hypersurface deformations, remain the same in both sets of variables.

The intriguing aspect of this situation is that, despite the significant
disparity, the supertranslation charges derived in this paper using Ashtekar-
Barbero variables, denoted as equation (3.93), are equivalent to those
presented in [3], denoted as equation (2.22). To demonstrate this, we will
begin with equation (3.93) and attempt to express it in terms of ADM
variables. To elaborate, we have

$\displaystyle\mathcal{Q}_{Supertranslation}$ $\displaystyle=-2\oint
d\sigma\sqrt{\bar{\gamma}}\left[\tilde{f}(\bar{F}^{r}_{r}-\bar{F}^{A}_{A})+\frac{2s}{\beta}W\bar{G}^{r}_{r}\right]$
$\displaystyle=-2\oint
d\sigma\sqrt{\bar{\gamma}}\left[\tilde{f}\left(-2\bar{\lambda}\right)+\frac{2s}{\beta}W\left(\frac{\beta}{4\sqrt{\bar{\gamma}}}\;\bar{p}\right)\right]$
$\displaystyle=\oint
d\sigma\left[4\tilde{f}\sqrt{\bar{\gamma}}\bar{\lambda}-sW\bar{p}\right]$
(4.21)

In arriving at equation (4), we have utilized equations (4.1) and (4), which
allow us to express $\bar{F}^{r}_{r}-\bar{F}^{A}_{A}$ as $-2\bar{\lambda}$ and
$\bar{G}^{r}_{r}$ as $\frac{\beta}{4\sqrt{\bar{\gamma}}}\;\bar{p}$,
respectively. Note that, in the case of considering Lorentzian signature
($s=-1$), the charge presented in equation (4) is equivalent to that of
equation (2.22), since $\tilde{f}$ and $T$ are both arbitrary even functions.

##  5 Conclusion and outlook

In this paper, we have put forward novel boundary conditions for the Ashtekar-
Barbero variables at spatial infinity within the framework of asymptotically
flat spacetimes. These new boundary conditions are described by equation
(3.1), with the parity conditions stated in the theorem of section (3.1). We
have examined these boundary conditions without resorting to the ADM
expressions.

These boundary conditions satisfy the following consistency requirements: the
symplectic structure is well-defined and the boundary conditions are preserved
under hypersurface deformations. It turns out that by using the new parity
conditions, the generators of the asymptotic symmetries are finite only for
spacetime translations, but not for boosts and rotations. This issue also
arises when working with ADM variables, but in [3], the authors provide a
strategy to resolve it, which involves imposing faster fall-off conditions for
the constraints, namely the additional conditions (2.17). In this paper, we