# Weakly Isolated Horizons: $3+1$ decomposition and canonical formulations in
self-dual variables

Alejandro Corichi<EMAIL_ADDRESS>Centro de Ciencias Matemáticas,
Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3,
Morelia, Michoacán 58090, Mexico  Juan D. Reyes<EMAIL_ADDRESS>Facultad de
Ingeniería, Universidad Autónoma de Chihuahua, Nuevo Campus Universitario,
Chihuahua 31125, Mexico  Tatjana Vukašinac<EMAIL_ADDRESS>Facultad de
Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, Morelia,
Michoacán 58000, Mexico

###### Abstract

The notion of Isolated Horizons has played an important role in gravitational
physics, being useful from the characterization of the endpoint of black hole
mergers to (quantum) black hole entropy. In particular, the definition of
weakly isolated horizons (WIHs) as quasilocal generalizations of event
horizons is purely geometrical, and is independent of the variables used in
describing the gravitational field. Here we consider a canonical decomposition
of general relativity in terms of connection and vierbein variables starting
from a first order action. Within this approach, the information about the
existence of a (weakly) isolated horizon is obtained through a set of boundary
conditions on an internal boundary of the spacetime region under
consideration. We employ, for the self-dual action, a generalization of the
Dirac algorithm for regions with boundary. While the formalism for treating
gauge theories with boundaries is unambiguous, the choice of dynamical
variables on the boundary is not. We explore this freedom and consider
different canonical formulations for non-rotating black holes as defined by
WIHs. We show that both the notion of horizon degrees of freedom and energy
associated to the horizon is not unique, even when the descriptions might be
self-consistent. This represents a generalization of previous work on isolated
horizons both in the exploration of this freedom and in the type of horizons
considered. We comment on previous results found in the literature.

###### pacs:

04.20.Fy, 04.70.Bw, 04.20.Cv

###### Contents

  1. I Introduction
  2. II Preliminaries
    1. II.1 Canonical formalism for gauge theories defined in regions with boundary
    2. II.2 WIH boundary conditions
    3. II.3 Spinorial formulation
  3. III Canonical action in self-dual variables
  4. IV Consistency conditions and differentiability
    1. IV.1 Compatibility with 3+1 foliation and gauge fixing
    2. IV.2 Boundary terms in the variations of the constraints
  5. V Hamiltonians and boundary phase spaces
    1. V.1 Case I
      1. V.1.1 Symplectic structure and the Hamiltonian
      2. V.1.2 Gauss constraint
      3. V.1.3 Spatial diffeomorphism constraint
    2. V.2 Case II
    3. V.3 Case III
    4. V.4 Case IV
  6. VI Discussion
  7. A Potentials
  8. B Spherically symmetric non-rotating WIH

##  I Introduction

Gauge theories defined in regions with boundaries have been the subject of
extensive study from many different perspectives. To mention just a few of
them: The boundary degrees of freedom and dynamics and symmetries on the
boundary, conserved charges related to the boundary, consistency conditions on
the boundary that the theory has to satisfy in order to be well defined (see,
for example RT ; Banados1996 ; Troessaert ; HM ; BCE ; ABOL  and references
therein), a holographic principle that allows one to relate a bulk theory with
a boundary one Maldacena ; HSS ; MannMarolf , and many more Barbero ; Barbero1
; Zabzine ; MPB ; Irais . It is no overstatement to affirm that the proper
understanding of boundary contributions to a physical theory is of the
uttermost importance. The purpose of this manuscript is twofold. On the one
hand, we are interested in exploring physical systems that can be analyzed
from the perspective of a newly revised extension of the Dirac algorithm for
theories defined with a region with a boundary CV-M+P . Furthermore, as an
application of the formalism, we are interested in making contact with
research on isolated horizons and the corresponding Hamiltonian description on
them. As we shall now elaborate, these two issues become intertwined.

The particular gauge theory we shall consider is general relativity in a first
order formulation. Within general relativity, the concept of an isolated
horizon was introduced in order to give a quasi-local description of a black
hole in equilibrium PRL ; ABL ; ABLgeometry . It turns out to be very useful
to approach the quantum theory that finally leads to the calculation of the
entropy of a black hole ABCKprl ; ENPprl ; ABKquantum . From the classical
point of view, we have a theory defined by an action in terms of tetrads or
soldering forms and connections and want to account for the presence of the
horizon that is modeled as an internal boundary to the spacetime region. The
starting point in this case is a covariant action principle. The information
about the weakly isolated horizon (WIH) on the boundary is included in the
boundary conditions on the fields in the action principle. In order to arrive
at a quantum description, one needs to procure a Hamiltonian formalism.
However, the choice is not unique. For example, it is well known that there
are (at least) two Hamiltonian formalisms we can use: the better known
canonical formulation, or the covariant one. In the first one the phase space
is defined after a 3+1 decomposition of the theory and the corresponding
Legendre transformation. In the second one the covariant phase space is
defined as the space of solutions to the equations of motion; there is no
splitting of spacetime.

Both formulations have been applied to the issue at hand. Historically, the
canonical approach was used in the early works on this subject ACKclassical ;
ABF0 ; ABF , and also in Thiemann:1993zq ; ThiemannBook ; CorichiReyes  for
the asymptotically flat case, while the covariant one has received increased
attention starting with AFK  and in most papers after that, see, for example
AshtekarES ; EngleNouiPerezPranzetti ; ChatterjeeGhosh ; crv1 ; CRGV2016 ;
CRVwih2 . In both cases it was possible to obtain a consistent description
that allowed for the computation of expressions for the mass of a black hole
and to derive the zeroth and first laws of the black hole mechanics. Also, in
the early work, a black hole was modeled using an isolated horizon, whose
conditions have been since relaxed to the less restrictive weakly isolated
horizon.

Here, we will adopt the canonical approach and analyze a first order gravity
action in self-dual variables in the presence of a non-rotating WIH. Our
interest in this problem is twofold: First, we want to revisit and extend the
results of ACKclassical  and ABF , that were obtained for the case of
spherically symmetric strongly isolated horizon, with a preferred foliation.
In ACKclassical , the phase space was also restricted to configurations with
the same horizon area, that led to a Chern-Simon action and the corresponding
structure on the boundary. In ABF , the area of the horizon was not fixed,
which allowed for the derivation of the first law of black hole mechanics.
Here, we want to further relax the conditions of ABF , so that our phase space
includes configurations with non-rotating WIH as an internal boundary.

Furthermore, this manuscript can be seen as a continuation of the work on
gauge theories defined in regions with boundaries, following the ideas of CV-P
; CV-M+P . There it was shown that when the action contains a boundary
contribution with a time derivative, then a boundary contribution to the
symplectic structure arises rather naturally. This also leads to a
modification of the fundamental equations that now may contain contributions
from the boundary. In contrast to the examples studied in those works, here
the starting covariant action does not have any boundary terms that contain
time derivative of the fields, but they appear in the corresponding canonical
action. The main result in this manuscript is that one may find several,
different, descriptions of the canonical action and the corresponding degrees
of freedom at the boundary. As we will show in detail, due to WIH boundary
conditions, these boundary terms can be interpreted in different ways, for
instance as a part of the boundary symplectic structure, or as a boundary term
in the Hamiltonian. Depending on how one chooses to interpret the terms one
has a different canonical description of the system.

The paper is organized as follows. In section II we present some preliminary
material. We start by recalling the basic ideas of the canonical treatment of
gauge theories defined in regions with boundary. Then, we briefly review the
WIH boundary conditions and recall the spinorial formalism, in order to
introduce our basic self-dual variables.

In section III we derive the canonical action in self-dual variables, in the
presence of boundaries, starting from the standard covariant self-dual action.
We obtain the explicit form of a boundary term, for an arbitrary null
boundary. We also reproduce the expressions for the Hamiltonian, vector and
Gauss constraints.

In section IV we first study the compatibility with 3+1 foliations and the so
called ‘comoving gauge’, introduced in CRVwih2 , that will allow us to
explicitly write down the expressions for boundary terms on WIH. We show that
it is only the Gauss constraint that contributes to the boundary equations of
motion.

In section V we present our main results. As mentioned earlier, it turns out
that the boundary term that appears in the canonical action, can be
interpreted as a contribution to the kinetic term, and thus to the boundary
symplectic structure, or as a boundary term in the corresponding Hamiltonian.
These options correspond to different choices of the boundary phase space.
Once this election is made, we can obtain a well defined Hamiltonian, that is
unique. We will analyze each of these settings in detail. We will also show
that the boundary term in the Hamiltonian cannot always be interpreted as the
energy (mass) of the horizon. In these different situations, we also find
different gauge symmetries on the horizon. In some of the cases there is a
residual $U(1)$ gauge symmetry, generated by the Gauss constraint that is
defined in the bulk (where it is a generator of $SU(2)$ gauge symmetry), but
whose functional derivative has contribution on the horizon. We end by
commenting on the different choices that one has to make to have consistent
descriptions of the system.

In Appendix A we look into some additional details of the potentials that in
some cases represent boundary degrees of freedom. In Appendix B we reproduce
the results obtained earlier in the spherically symmetric case of strongly
isolated non-rotating horizon and the phase space with fixed horizon area
ACKclassical ; ABF . We obtain the Chern-Simons boundary symplectic structure
and show that in this case, tangential diffeomorfisms are boundary gauge
symmetries of the theory.

Throughout the manuscript, we assume the spacetime to be a four dimensional
smooth manifold $\mathcal{M}$. Furthermore, we will take $\mathcal{M}$, or at
least a portion of it $M\subseteq\mathcal{M}$, to be globally hyperbolic, with
a boundary $\partial M$, and such that it may be foliated as
$M\approx\mathbb{R}\times\Sigma$, with $\Sigma$ a spatial hypersurface with
inner boundary homeomorphic to a 2-sphere. We will use greek letters to denote
abstract (or actual component) spacetime indices running from 0 to 3.
Similarly, lowercase latin letters from the beginning of the alphabet will
correspond to abstract or component spatial indices on $\Sigma$ running from 1
to 3.

##  II Preliminaries

In this section we shall present some preliminary material in order to make
the manuscript self contained. It has three parts. In the first one, we
briefly summarize the formalism of gauge theories in the presence of a
boundary as developed in CV-M+P . In the second one, we summarize the isolated
horizon boundary conditions and its main geometrical constituents. In the last
part we introduce the basics of the spinorial formalism needed to write down
the self-dual action that we shall consider in the rest of the manuscript.
Readers familiar with either of these topics may skip the corresponding part
of this section.

###  II.1 Canonical formalism for gauge theories defined in regions with
boundary

We shall start by recalling some of the main results of CV-M+P . In that
contribution the objective was to study gauge theories that could have a
boundary term in the covariant (or canonical) action, containing time
derivatives of the fields. It was shown that the symplectic structure can
acquire a boundary contribution. In that case the standard Dirac algorithm has
to be extended in order to have a consistent description, that includes an
extended phase space with a boundary phase space, degrees of freedom and
(possibly) constraints on the boundary.

In this approach the starting point is a canonical action that is obtained
after a 3+1 decomposition of a covariant action.

To be more concrete, let us suppose that as a result of the 3+1 decomposition,
the canonical action has the form

$S[A,P]=\int\left\\{P[{\cal{L}}_{t}{A}]-H\right\\}{\mathrm{d}}t\,,$ (1)

where $P[{\cal{L}}_{t}{A}]$ is the kinetic term or symplectic potential111More
precisely, one uses $P[{\cal{L}}_{t}{A}]$ (which is a one-form on
configuration space) to define or identify the symplectic potential and
symplectic structure on phase space. For details see CV-M+P . , whose general
form in the space region with a boundary is

$P[{\cal{L}}_{t}{A}]=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,\tilde{P}^{a}_{\;BC}\,{\cal{L}}_{t}A_{a}^{\;BC}+\int_{\partial\Sigma}{\mathrm{d}}^{2}y\,\,\tilde{\pi}^{j}\,{\cal{L}}_{t}\alpha_{j}\,.$
(2)

$A_{a}^{\;BC}(x)$ are the bulk field configuration variables, and
$\tilde{P}^{a}_{\;BC}(x)$ their corresponding momenta, $a$ is a spatial index
and $(B,C)$ are internal indices.222Customary, one denotes internal indices in
gauge theories in terms of $I$, an index for the Lie algebra of the gauge
group. Here, for convenience and without loss of generality, we shall use a
pair of spinorial indices $AB$, since that shall be the structure of the
internal space we will consider. $\alpha_{j}(y)$ are to be interpreted as
boundary configuration variables, and $\tilde{\pi}^{j}(y)$ as their
corresponding momenta333We are denoting by $j$ all possible index structure
that the fields at the boundary have.. In the general case, $H$ is a candidate
for a Hamiltonian that may depend also on the boundary variables
$(\alpha_{j},\tilde{\pi}^{j})$, on top of the bulk canonical variables
restricted to the boundary. In this manuscript we will show that the boundary
term in $P[{\cal{L}}_{t}{A}]$, due to given boundary conditions, is not
unique. It can be chosen in different ways, leading to a different boundary
phase space in each case. In every one of those cases, we shall find a
corresponding well defined Hamiltonian.

The kinetic term determines the symplectic structure of the theory as CV-M+P
:

$\Omega=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{P}^{a}_{\;BC}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,\;{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;BC}+\int_{\partial\Sigma}{\mathrm{d}}^{2}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}^{j}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,\;{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\alpha_{j}\,,$
(3)

with ${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,$ and
$\wedge\\!\\!\\!\\!\\!\wedge\,$ the differential and wedge operator
respectively on (infinite dimensional) phase space. We are regarding the bulk
and boundary degrees of freedom as independent, even when one (or more) of the
boundary fields might have come from some appropriate restriction of bulk
degrees of freedom, that is imposed by boundary conditions.

The function $H$ defines a Hamiltonian vector field (HVF) $X_{H}$ through the
equation

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H(Y)=\Omega(Y,X_{H})\,.$ (4)

Given the boundary contribution to the symplectic structure $\Omega$, the bulk
part of (4) gives the equations of motion in the bulk, while the boundary term
defines the boundary components of the Hamiltonian vector field (HVF). We will
call a corresponding canonical description consistent if: a) in the case when
there are no boundary contributions to the symplectic structure, the boundary
term in ${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H$ vanishes or b) in the case
when there are boundary contributions to the symplectic structure, $X_{H}$
generates consistent evolution of both the bulk and boundary degrees of
freedom.

In gauge theories we have a certain set of (smeared) constraints $G_{I}[f]$
and in that case there is an additional set of consistency conditions:

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,G_{I}[f](X_{H})\approx 0\,.$ (5)

These conditions imply that $X_{H}$ is tangent to the constraint surface.

To write down explicitly the expressions (4) and (5) we need to write down the
generic HVF CV-M+P

$\displaystyle X$ $\displaystyle=$
$\displaystyle\int_{\Sigma}{\mathrm{d}}^{3}x\,\left[(X_{A})_{a}^{\;BC}(x)\left(\frac{\delta}{\delta
A_{a}^{\;BC}(x)}\right)+(X_{\tilde{P}})^{a}_{\;BC}(x)\left(\frac{\delta}{\delta{\tilde{P}}^{a}_{\;BC}(x)}\right)\right]$
(6) $\displaystyle+$
$\displaystyle\int_{\partial\Sigma}{\mathrm{d}}^{2}y\,\left[(X_{\alpha})_{j}(y)\left(\frac{\delta}{\delta\alpha_{j}(y)}\right)+(X_{\tilde{\pi}})^{j}(y)\left(\frac{\delta}{\delta{\tilde{\pi}}^{j}(y)}\right)\right]\,,$

whose components in the bulk are $(X_{A},X_{\tilde{P}})$ and on the boundary
$(X_{\alpha},X_{\tilde{\pi}})$.

The gradient of an eligible function $F$ will have the form,

$\displaystyle{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,F$ $\displaystyle=$
$\displaystyle\int_{\Sigma}{\mathrm{d}}^{3}\\!x\;\left(\frac{\delta F}{\delta
A_{a}^{\;BC}(x)}\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;BC}(x)+\frac{\delta
F}{\delta{\tilde{P}}^{a}_{\;BC}(x)}\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,{\tilde{P}}^{a}_{\;BC}(x)\right)$
(7)
$\displaystyle+\int_{\partial\Sigma}{\mathrm{d}}^{2}\\!y\;\left(\frac{\delta
F}{\delta\alpha_{j}(y)}\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\alpha_{j}(y)+\frac{\delta
F}{\delta{\tilde{\pi}}^{j}(y)}\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,{\tilde{\pi}}^{j}(y)\right)\,,$

with its corresponding contribution from the boundary. As usual, to find these
gradients we will identify ${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,F(Y)$ with
variations of $F$ along the direction of $Y$, i.e.
${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,F(Y)=\delta_{Y}F$. The quantities
$\frac{\delta F}{\delta\alpha_{j}(y)}$ should be thought as a generalization
of the partial derivative of the function $F$ along the coordinate
$\alpha_{j}$ and leaving the rest of the coordinates (including the bulk DOF)
constant. It is important to note that, since the function $F$ depends on all
possible variables (bulk and boundary), the boundary contribution to the
gradient might depend explicitly on the bulk variables, restricted to the
boundary, as well.

From (3) and (6) it follows that

$\Omega(Y,X)=\int_{\Sigma}{\mathrm{d}}^{3}x\,(X_{A}Y_{\tilde{P}}-Y_{A}X_{\tilde{P}})+\int_{\partial\Sigma}{\mathrm{d}}^{2}y\,(X_{\alpha}Y_{\tilde{\pi}}-Y_{\alpha}X_{\tilde{\pi}})\,,$
(8)

where the contraction over the corresponding indexes is understood.

Here we are interested in the theory of gravity in the first order formalism,
that is a diffeomorphism invariant theory. Its bulk Hamiltonian is a linear
combination of the Hamiltonian, vector and Gauss constraints, so the standard
interpretation is that the bulk part of the Hamiltonian only generates gauge
transformations (in the Hamiltonian definition of gauge, as explained in
CV-M+P ). As is well known, in the case of asymptotically flat spacetimes, in
order to obtain a well defined Hamiltonian, we need to add an asymptotic
boundary term whose numerical evaluation corresponds to the ADM mass RT ;
CorichiReyes . This functional is no longer a constraint but a true generator
of asymptotic time translations.

There is an important remark to be made about the use of the terms gauge and
gauge symmetry from the viewpoint of Hamiltonian dynamics. We are considering
a framework where the fundamental variables are to be taken as connections
with their corresponding conjugate momenta, where the Lie algebra where they
live corresponds to the Lorentz Lie algebra. Thus, there is a natural notion
of gauge in the sense of Lorentz transformations, and invariance. There is
also the Hamiltonian notion of gauge, where gauge transformation are to be
generated only by first class constraints. Sometimes in the text that follows
we shall simply refer to gauge, and we hope that the context will be enough to
differentiate between the two notions.

As a natural assumption, in ABF , the mass of an isolated horizon was defined
as the numerical, on-shell value of the horizon boundary term in the
corresponding Hamiltonian. The result was identical to the Smarr expression
for the mass of a non-rotating black hole Smarr . A priori, it is not obvious
that it would be the case, since the horizon is a null surface, in contrast to
a time-like asymptotic region where the boundary is time-like. We will show
that, depending on the choice of a boundary phase space, we can obtain a
consistent description of WIH dynamics, even in the case when the Hamiltonian
does not have a horizon boundary term. We will also show under which
assumptions we can obtain the expected expression for the mass of a black
hole.

We should note that, in a somewhat independent manner, there have been several
approaches to gravitational degrees of freedom on a null surface. Several
examples can be found in Wieland1 ; HopfmuellerFreidel ; padmadhavan ;
Flanagan . Our approach is different from those.

###  II.2 WIH boundary conditions

In this part we shall give definitions and some of the basic properties of
isolated horizons. For more details see for instance, AFK  and CRVwih2 .
Isolated horizons are modeled by special types of null hypersurfaces on
$\mathcal{M}$. A three dimensional hypersurface $\Delta\subset\mathcal{M}$ is
_null_ if its induced metric $h_{\underset{{}^{\leftarrow}}{{\mu\nu}}}$ is
degenerate of signature $(0,+,+)$ (following previous conventions, we will
denote forms pulled back to $\Delta$ using indexes with arrows under them).
This degeneracy is equivalent to the requirement that any vector $\ell^{\mu}$
normal to $\Delta$ is null: $\ell_{\mu}\ell^{\mu}=0$, and it follows that a
fundamental property of null hypersurfaces is that they are ruled by null
geodesics. Any null normal vector field $\ell^{\mu}$ on $\Delta$ satisfies the
geodesic equation

$\ell^{\mu}\nabla_{\mu}\ell^{\nu}=\kappa_{(\ell)}\ell^{\nu}\,,$

and the non-affinity parameter $\kappa_{(\ell)}$ will correspond to surface
gravity when one specializes to isolated horizons.

For null hypersurfaces, it is convenient to use the null normal $\ell^{\mu}$
at each point $p\in\Delta$ to construct or fix a _Newman-Penrose null basis_
$(k,\ell,m,\bar{m})$ on $T_{p}\mathcal{M}$ NewmanPenrose . This is done by
choosing a null direction $k^{\mu}$ transverse to $T_{p}\Delta$ on the light
cone at each $T_{p}\mathcal{M}$, normalized such that $k_{\mu}\ell^{\mu}=-1$.
The orthogonal complement of the plane spanned by $k^{\mu}$ and $\ell^{\mu}$
is a two dimensional spatial subspace of $T_{p}\Delta$, where one can choose a
null basis given by a complex vector $m^{\mu}$ and its conjugate $\bar{m}$,
such that $m_{\mu}\bar{m}^{\mu}=1$. We will call a basis constructed in this
way a null basis _adapted to_ $\Delta$.

A null basis adapted to $\Delta$ is not unique, but any two adapted null bases
are related by a Lorentz rotation preserving the direction of $\ell^{\mu}$.
The _cross sectional area two-form_
$\leftidx{{}^{2}}{\epsilon}{}:=im\wedge\bar{m}\,$, is invariantly defined on
$\Delta$, in the sense that it is the same regardless of the choice of adapted
null basis. Similarly, the (two dimensional spatial) _cross sectional metric_
$2\,m_{({\underset{{}^{\leftarrow}}{{\mu}}}}\bar{m}_{{\underset{{}^{\leftarrow}}{{\nu}}})}=h_{\underset{{}^{\leftarrow}}{{\mu\nu}}}\,$
is invariantly defined on $T_{p}\Delta$ and equal to the pullback metric.

The null vector $k^{\mu}$ allows one to define a projector to $T_{p}\Delta$,
$\Pi^{\mu}_{\;\;\nu}:=\delta^{\mu}_{\;\;\nu}+k^{\mu}\ell_{\nu}$. The second
fundamental form of $\Delta$ with respect to $\ell^{\mu}$ is defined as

$\Theta_{\mu\nu}:=\Pi^{\sigma}_{\;\;\mu}\Pi^{\rho}_{\;\;\nu}\nabla_{\sigma}\ell_{\rho}=\nabla_{\mu}\ell_{\nu}-\omega_{\mu}\ell_{\nu}+\ell_{\mu}k^{\sigma}\nabla_{\sigma}\ell_{\nu}\,,$
(9)

with

$\omega_{\mu}:=-k^{\sigma}\nabla_{\mu}\ell_{\sigma}-\ell_{\mu}k_{\sigma}k^{\rho}\nabla_{\rho}\ell^{\sigma}\,,$
(10)

defined as the _rotation 1-form_ in the context of black hole horizons.
Contracting this last definition with $\ell^{\mu}$, it follows that

$\kappa_{(\ell)}=\omega_{\mu}\ell^{\mu}.$ (11)

The second fundamental form $\Theta_{\mu\nu}$ encodes the ‘kinematics’ of the
geodesic congruence of null generators of $\Delta$ with velocities
$\ell^{\mu}$. Its trace $\theta_{(\ell)}:=g^{\mu\nu}\Theta_{\mu\nu}$ defines
the _expansion_ of the congruence and its symmetric trace-free part is the
_shear_. The _twist_ or anti-symmetric part vanishes as a consequence of the
congruence being hypersurface orthogonal.

To model equilibrium horizons resulting from gravitational collapse, one first
restricts the topology of $\Delta$ and incorporates the notion that the null
geodesic generators should be non-expanding.

A null hypersurface $\Delta\subset\mathcal{M}$ is called a _non-expanding
horizon_ (NEH) if

  1. (i)

$\Delta$ is diffeomorphic to the product $S^{2}\times\mathbb{R}$ with the
fibers of the canonical projection $P:S^{2}\times\mathbb{R}\to S^{2}$
corresponding to the null generators.

  2. (ii)

The expansion $\theta_{(\ell)}$ of any null normal $\ell^{\mu}$ to $\Delta$
vanishes.

  3. (iii)

On $\Delta$, Einstein’s equations hold and the stress-energy tensor
$T_{\mu\nu}$ of matter satisfies the _null dominant energy condition_ , i.e.
$-T^{\mu}_{\;\;\nu}\,\ell^{\nu}$ is causal and future-directed.

These conditions imply that on $\Delta$ the complete second fundamental form
$\Theta_{\mu\nu}$ vanishes AFK ; GJreview . So from (9) and
$\ell_{\underset{{}^{\leftarrow}}{{\mu}}}=0$, it follows that on a NEH

$\nabla_{{\underset{{}^{\leftarrow}}{{\mu}}}}\ell^{\nu}\overset{{}_{\Delta}}{=}\omega_{\underset{{}^{\leftarrow}}{{\mu}}}\ell^{\nu}\,,$
(12)

where we used the standard notation $\overset{{}_{\Delta}}{=}$ for equalities
valid only on $\Delta$.

Equation (12) also implies that the induced metric on $\Delta$ and the
transverse area 2-form are Lie dragged along $\ell^{\mu}$ ($\ell^{\mu}$ is a
symmetry direction for the metric so NEHs are a generalization of Killing
horizons, and the horizon area is preserved):

$\mathcal{L}_{\ell}\,h_{{\underset{{}^{\leftarrow}}{{\mu\nu}}}}\overset{{}_{\Delta}}{=}0\,\qquad\text{
and
}\qquad\mathcal{L}_{\ell}\,\leftidx{{}^{2}}{\epsilon}{}\overset{{}_{\Delta}}{=}0\,.$
(13)

In a general null hypersurface $\Delta$ each choice of $k^{\mu}$ defines a
(torsion-free) induced connection
$\widehat{\nabla}_{\underset{{}^{\leftarrow}}{{\mu}}}$ compatible with the
induced metric:
$\widehat{\nabla}_{\underset{{}^{\leftarrow}}{{\mu}}}h_{{\underset{{}^{\leftarrow}}{{\nu\rho}}}}=0$.
On a NEH this connection is unique and ‘intrinsic’ to $\Delta$ (independent of
the choice of $k^{\mu}$), but unlike spatial or time-like surfaces, on a NEH
the connection is not fully determined by the metric. It is hence the pair
$\big{(}h_{{\underset{{}^{\leftarrow}}{{\mu\nu}}}},\widehat{\nabla}_{\underset{{}^{\leftarrow}}{{\mu}}}\big{)}$
taken together that defines the _intrinsic geometry_ on $\Delta$.

Along with $\omega_{\underset{{}^{\leftarrow}}{{\mu}}}$, on a NEH there is
another relevant intrinsic one-form arising from the connection on $\Delta$,
the _transverse connection potential_ $V_{\underset{{}^{\leftarrow}}{{\mu}}}$.
Using an adapted null basis, it is defined as:

$V_{\underset{{}^{\leftarrow}}{{\mu}}}:=\bar{m}_{\nu}\widehat{\nabla}_{\underset{{}^{\leftarrow}}{{\mu}}}m^{\nu}\,.$
(14)

For restricted Lorentz transformations of the null basis, this expression is
invariantly defined except for $U(1)$-rotations for which it transforms as a
$U(1)$-connection.

The definition of NEHs is too general to capture the mechanics of black holes.
To derive the zeroth and first laws, one needs to further select an
equivalence class of normal vector fields on $\Delta$.

A _weakly isolated horizon_ $(\Delta,[\ell])$ is a non-expanding horizon where
an equivalence class of null normals $[\ell]$ satisfying

$\mathcal{L}_{{\ell}}\,\omega_{\underset{{}^{\leftarrow}}{{\mu}}}\overset{{}_{\Delta}}{=}0\qquad\text{
for all }\;\ell^{\mu}\in[\ell]$ (15)

has been singled out. Two normals $\tilde{\ell}\sim\ell$ belong to the same
equivalence class iff $\tilde{\ell}^{\mu}=c\ell^{\mu}$ for some constant $c>0$
on $\Delta$.

This symmetry condition or time-independence of a part of the induced
connection is sufficient to ensure that the surface gravity $\kappa_{(\ell)}$
is constant on a WIH, that constitutes the zeroth law of black hole mechanics.

Every NEH can be made into a WIH by appropriate (albeit infinitely many
different) choices of equivalence classes $[\ell]$. We will hence work mainly
within the class of weakly isolated horizons in this paper.

###  II.3 Spinorial formulation

Our goal is to derive the canonical Hamiltonian formulation for general WIHs
in self-dual variables starting from the covariant self-dual action.
Therefore, in this section we present a brief review of the spinorial
formalism, in order to make this manuscript more self-contained. Details can
be consulted in ABF0 ; ABF ; ACKclassical ; CRVwih2 . We follow the same
conventions used in ThiemannBook ; CRVwih2 .

For the spinorial self-dual formulation of gravity the basic variables are a
pair $(\sigma_{\mu}^{AA^{\prime}},\leftidx{{}^{+}}{\mathcal{A}}{{}_{AB}})$,
where $\sigma_{\mu}^{AA^{\prime}}$ is a soldering form for primed and unprimed
$SL(2,\mathbb{C})$ spinors and $\leftidx{{}^{+}}{\mathcal{A}}{{}_{AB}}$ is a
$SL(2,\mathbb{C})$ self-dual connection.

The spinor bundle is the vector bundle over $\mathcal{M}$ whose fiber is the
complex two dimensional vector space $\mathbb{C}^{2}$, with symplectic
structure or metric $\epsilon_{AB}$ with inverse $\epsilon^{AB}$, such that
$\epsilon_{AB}=-\epsilon_{BA}$ and
$\epsilon^{AB}\epsilon_{AC}=\delta^{B}_{C}$. ($A$, $B$, $C$ are spinor indices
running from 1 to 2.) $\epsilon_{AB}$ and $\epsilon^{AB}$ are used to raise
and lower spinor indices: $X_{B}:=X^{A}\epsilon_{AB}$ and
$X^{A}=\epsilon^{AB}X_{B}$, which implies $X^{A}Y_{A}=-X_{A}Y^{A}$. Complex
conjugation $X^{A}\mapsto\bar{X}^{A^{\prime}}$, maps spinors to the
_conjugate_ spinor space with metric $\bar{\epsilon}_{A^{\prime}B^{\prime}}$.

The _soldering vector_ $\sigma^{\mu}_{AA^{\prime}}$ provides an isomorphism
between the space of anti-hermitian spinors $X^{AA^{\prime}}$ (such that
$\bar{X}^{AA^{\prime}}=-X^{A^{\prime}A}$) and the tangent space
$T_{p}\mathcal{M}$ at each point $p$. Its inverse, the _soldering form_
$\sigma_{\mu}^{AB}$, is obtained by lowering spacetime indices with
$g_{\mu\nu}$ and raising spinor indices with $\epsilon^{AB}$ and
$\bar{\epsilon}^{A^{\prime}B^{\prime}}$. They satisfy

$\sigma_{\mu}^{AA^{\prime}}\sigma^{\nu}_{AA^{\prime}}=\delta^{\nu}_{\mu},\qquad\sigma_{\mu}^{AA^{\prime}}\sigma^{\mu}_{BB^{\prime}}=\delta^{A}_{B}\delta^{A^{\prime}}_{B^{\prime}}\qquad\text{
and }\qquad\bar{\sigma}^{\mu}_{AA^{\prime}}=-\sigma^{\mu}_{A^{\prime}A}\,.$
(16)

For convenience, one can introduce a fixed dyad basis $({\rm
o}^{A},\iota^{A})$ in spinor space $\mathbb{C}^{2}$ at each point, such that

$\iota^{A}\iota_{A}=0\,,\qquad\qquad{\rm o}^{A}{\rm
o}_{A}=0\,,\qquad\qquad\iota^{A}{\rm o}_{A}=-\iota_{A}{\rm o}^{A}=1\,,$ (17)

and similarly for the conjugate basis $(\bar{{\rm
o}}^{A^{\prime}},\bar{\iota}^{A^{\prime}})$. We assume our auxiliary
connection is such that $\bar{\partial}(\iota^{A},{\rm o}^{A})=0$, and we will
fix this internal dyad such that $(\delta\iota^{A},\delta{\rm o}^{A})=0$.

The spinors $\\{i\,{\rm o}^{A}\bar{{\rm
o}}^{A^{\prime}},i\,\iota^{A}\bar{\iota}^{A^{\prime}},i\,{\rm
o}^{A}\bar{\iota}^{A^{\prime}},i\,\iota^{A}\bar{{\rm o}}^{A^{\prime}}\\}$
define a basis for the internal space of anti-hermitian spinors
$X^{AA^{\prime}}$. One can define a Newman-Penrose null basis on the tangent
space:

$\displaystyle
l^{\mu}:=\sigma^{\mu}_{AA^{\prime}}\,l^{AA^{\prime}}:=i\sigma^{\mu}_{AA^{\prime}}{\rm
o}^{A}\bar{{\rm o}}^{A^{\prime}}\,,\qquad\qquad$ $\displaystyle
m^{\mu}:=\sigma^{\mu}_{AA^{\prime}}\,m^{AA^{\prime}}:=i\sigma^{\mu}_{AA^{\prime}}{\rm
o}^{A}\bar{\iota}^{A^{\prime}}\,,$ $\displaystyle
k^{\mu}:=\sigma^{\mu}_{AA^{\prime}}\,k^{AA^{\prime}}:=i\sigma^{\mu}_{AA^{\prime}}\iota^{A}\bar{\iota}^{A^{\prime}}\,,\qquad\qquad$
$\displaystyle\bar{m}^{\mu}:=\sigma^{\mu}_{AA^{\prime}}\,\bar{m}^{AA^{\prime}}:=i\sigma^{\mu}_{AA^{\prime}}\iota^{A}\bar{{\rm
o}}^{A^{\prime}}\,,$ (18)

The soldering form is hence expanded as

$\sigma^{\mu}_{AA^{\prime}}=-k^{\mu}l_{AA^{\prime}}-l^{\mu}k_{AA^{\prime}}+\bar{m}^{\mu}m_{AA^{\prime}}+m^{\mu}\bar{m}_{AA^{\prime}}\,.$
(19)

On $\Delta$ one can define the spin connection compatible with the soldering
form or vector, $\nabla_{\mu}\sigma^{\nu}_{AA^{\prime}}=0$, such that

$\nabla_{\mu}X^{AA^{\prime}}=\bar{\partial}_{\mu}X^{AA^{\prime}}+\leftidx{{}^{+}}{\mathcal{A}}{{}^{A}_{\;\;B}}X^{BA^{\prime}}+\leftidx{{}^{-}}{\bar{\mathcal{A}}}{{}^{A^{\prime}}_{\;\;B^{\prime}}}X^{AB^{\prime}}\,,$

where $\leftidx{{}^{+}}{\mathcal{A}}{{}^{A}_{\;\;B}}$ is the self-dual
connection potential and
$\leftidx{{}^{-}}{\bar{\mathcal{A}}}{{}^{A^{\prime}}_{\;\;B^{\prime}}}$ is the
conjugate of the anti self-dual connection potential, while $\bar{\partial}$
is a fiducial flat connection. Expanding covariant derivatives of the null
tetrad in terms of the spinor basis and using compatibility condition, one can
write down an expansion for the self-dual connection potential in terms of the
one-forms $(\mathbb{W},\mathbb{V},\mathbb{U},\mathbb{Y})$ and the dyad
$(\iota_{A},{\rm o}_{A})$:

$\leftidx{{}^{+}}{\mathcal{A}}{{}_{AB}}=-(\mathbb{W}+\mathbb{V})\,\iota_{(A}{\rm
o}_{B)}+\bar{\mathbb{U}}\,{\rm o}_{A}{\rm
o}_{B}-\mathbb{Y}\,\iota_{A}\iota_{B}\,,$ (20)

where

$\mathbb{W}_{\mu}:=-k_{\nu}\nabla_{\mu}l^{\nu}\,,\qquad\mathbb{V}_{\mu}:=\bar{m}_{\nu}\nabla_{\mu}m^{\nu}\,,\qquad\mathbb{U}_{\mu}:=m_{\nu}\nabla_{\mu}k^{\nu}\,,\qquad\mathbb{Y}_{\mu}:=m_{\nu}\nabla_{\mu}l^{\nu}\,.$
(21)

Expansion (19) is valid on the bulk as well as on $\Delta$, while the
expansion (20) is valid only on $\Delta$. Both are independent of any gauge
choice.

##  III Canonical action in self-dual variables

In this section we shall review the canonical decomposition of the standard
first order self-dual action to arrive at the canonical action, following in
detail all the boundary contributions.

Spherically symmetric isolated horizons were originally defined in the
spinorial formalism, see, for example ABF ; ACKclassical , which turned out to
be especially convenient. The canonical analysis of first order gravity
theories in self-dual spinorial variables was first performed in Samuel ;
JacobsonSmolin . The classical phase space in theories with the spherically
(strongly) isolated horizon, with fixed area, was studied for the first time
in ACKclassical . The boundary symplectic structure in this case coincides
with the one of the $U(1)$ Chern-Simons theory (such theory was then quantized
in ABCKprl ; ABKquantum ). In ABF  this analysis was extended to include the
phase space where the area of the horizon could change, leading to the
generalization of the equilibrium version of the first law of black hole
mechanics. The corresponding boundary symplectic structure in this case does
not correspond to a Chern-Simons theory.

We start from the same covariant action, and perform its 3+1 decomposition, in
a more general case, where the internal boundary is a weakly isolated horizon.
We will show that, there is not a unique election of the boundary variables.
In what follows, we will explore the consequences of these different choices.
One of them extends the results of ABF , but the other ones also lead to
consistent canonical descriptions.

To derive a canonical action and a Hamiltonian theory consistent with WIH
boundary conditions, we start with the first order covariant action for
gravity in self-dual variables, given by (see, for example, Samuel ;
JacobsonSmolin ; AshtekarLectures ; ACKclassical ; ABF )

$\displaystyle
S_{\text{SD}}(\sigma,\leftidx{{}^{+}}{\mathcal{A}}):=\int_{M}{\mathrm{d}}^{4}x\,(-\,\leftidx{{}^{4}}{\sigma})\sigma^{\mu}_{\;A}\,{}^{A^{\prime}}\sigma^{\nu}_{\;BA^{\prime}}\leftidx{{}^{+}}{\mathcal{F}}_{\mu\nu}^{\,AB}\,,$
(22)

where
$\leftidx{{}^{4}}{\sigma}=\det{(\sigma^{\mu}_{\;A}\,{}^{A^{\prime}})}=\sqrt{-g}$,
with $g=\det{(g_{\mu\nu})}$ and

$\leftidx{{}^{+}}{\mathcal{F}}_{\mu\nu}^{\,AB}=2\,\partial_{[\mu}\leftidx{{}^{+}}{\mathcal{A}}_{\nu]}^{\;AB}+\leftidx{{}^{+}}{\mathcal{A}}_{[\mu\
C}^{\ A}\leftidx{{}^{+}}{\mathcal{A}}_{\nu]}^{\,CB}\,,$ (23)

is the curvature of the self-dual connection. Without loss of generality we
are fixing the value of the coupling constant to 1. For asymtotically flat
spacetimes, we are neglecting the boundary term in the asymptotic region, that
should be added in order to make the action differentiable at infinity ABF ;
Thiemann:1993zq ; CorichiReyes . Here we are interested in the inner boundary,
WIH, and in what follows we will not take into account the contributions from
the asymptotic region, as they are well understood. For asymptotically flat
spacetimes with a WIH as inner boundary, the self-dual action (22) (with
boundary term at infinity) is well defined, i.e. finite, and differentiable in
the sense that its variations vanish at the asymptotic and inner boundaries.
This is true for arbitrary variations with the full $SL(2,\mathbb{C})$ gauge
freedom at the WIH444Due to the presence of terms of the form $\ell\cdot\delta
V$ at the horizon, in CRVwih2 , it was stated that differentiability of the
action required a partial gauge fixing and/or a counter term at $\Delta$.
There is however a subtle argument to show that these terms actually vanish..

For a 3+1 decomposition and Hamiltonian formulation of a covariant theory, one
postulates a time function $t:M\subseteq\mathcal{M}\to\mathbb{R}$ on (a
portion of) spacetime $\mathcal{M}$, whose level curves $\Sigma_{t}$ are
spatial hypersurfaces and provide a foliation of the spacetime region $M$.
Spacetime fields are split into tangential (spatial) and normal components
with respect to this foliation. For the configuration space consisting of
spacetimes admitting a weakly isolated horizon $\Delta$ as an internal
boundary, we will only consider foliations of the bulk of $M$ whose
intersection with $\Delta$ is non trivial and induces a foliation of $\Delta$
by two-spheres $\mathcal{S}_{t}=\Sigma_{t}\cap\Delta$.

Additionally, one needs to choose an _evolution_ vector field $t^{\mu}$ such
that $t^{\mu}\nabla_{\mu}t=1$ and along which spatial fields are defined to
‘evolve’. The evolution vector field will generically be time-like on the
bulk. For non-rotating horizons it will be chosen to belong to the equivalence
class $[\ell]$ on $\Delta$. That is, the vector $t^{\mu}$ becomes null as one
approaches $\Delta$.

Decomposition of the spacetime metric splits the ten independent components of
$g_{\mu\nu}$ into the six independent components of the Euclidean spatial
metric $q_{\mu\nu}$, the lapse function $N$ and the shift vector $N^{\mu}$.
Lapse and shift being respectively the normal and tangential components of the
evolution vector field $t^{\mu}=Nn^{\mu}+N^{\mu}$, where $n^{\mu}$ denotes the
future directed unit normal to the foliation and
$N=-\left|({\mathrm{d}}t)_{\mu}({\mathrm{d}}t)^{\mu}\right|^{-1/2}$. In
coordinates $(t,y^{a})$ adapted to the foliation the line element reads

$g_{\mu\nu}{\mathrm{d}}x^{\mu}{\mathrm{d}}x^{\nu}=(-N^{2}+q_{ab}N^{a}N^{b}){\mathrm{d}}t^{2}+2q_{ab}N^{b}{\mathrm{d}}t\,{\mathrm{d}}y^{a}+q_{ab}{\mathrm{d}}y^{a}\,{\mathrm{d}}y^{b}\,.$
(24)

The four-dimensional $SL(2,\mathbb{C})$ soldering form on $M$,
$\sigma^{\nu}_{\;AA^{\prime}}$, induces a three-dimensional $SU(2)$ soldering
form on $\Sigma$, $\sigma^{\mu}_{\;A}\,{}^{B}$, in the following way

$\sigma^{\mu}_{\;A}\,{}^{B}:=-i\sqrt{2}\,q^{\mu}_{\,\nu}\,\sigma^{\nu}_{\;AA^{\prime}}\,n^{A^{\prime}B}\,,$
(25)

where $q^{\mu}_{\,\nu}:=\delta^{\mu}_{\nu}+n^{\mu}n_{\nu}$ is a projector on
$\Sigma$ and $n^{A^{\prime}B}=n^{\mu}\sigma_{\mu}^{\;A^{\prime}B}$ is the
spinorial representation of $n^{\mu}$. From (25) it follows that

$\sigma^{\mu}_{\;AA^{\prime}}=-i\sqrt{2}\,\sigma^{\mu}_{\;A}\,{}^{B}\,n_{BA^{\prime}}-n^{\mu}n_{AA^{\prime}}.$
(26)

From this one gets the 3+1 decomposition (see, for example, Thiemann:1993zq )

$\displaystyle S_{\text{SD}}=\int{\mathrm{d}}t\,{\mathrm{d}}^{3}y\big{\\{}$
$\displaystyle
i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\mathcal{L}_{t}A_{a}^{\;AB}-\frac{N}{\sqrt{q}}\,\tilde{\sigma}^{a}_{\;A}\,{}^{C}\,\tilde{\sigma}^{b}_{\;CB}\,F_{ab}^{\;AB}$
$\displaystyle-i\sqrt{2}\,N^{a}\,\tilde{\sigma}^{b}_{\;AB}\,F_{ab}^{\;AB}+(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,\leftidx{{}^{A}}{D}_{a}(i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB})\big{\\}}$
$\displaystyle-\int_{M}{\mathrm{d}}^{4}x\,\partial_{\nu}$
$\displaystyle\left[(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,(i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\tilde{e}^{\nu}_{a})\right],$
(27)

with densitized soldering form

$\tilde{\sigma}^{a}_{\;AB}:=\sqrt{q}\;\sigma^{a}_{\;AB}=\sqrt{q}\;\tilde{e}^{a}_{\mu}\,\sigma^{\mu}_{\;AB},$

where $q=\det q_{ab}$, $\tilde{e}^{\mu}_{a}:=\frac{\partial x^{\mu}}{\partial
y^{a}}$ are the components of tangent vectors to $\Sigma$
($\tilde{e}^{\mu}_{a}n^{a}=0$) and
$\tilde{e}^{a}_{\mu}=g_{\mu\nu}\,q^{ab}\,\tilde{e}^{\nu}_{b}$. Furthermore,

$F_{ab}^{\;AB}:=\tilde{e}^{\mu}_{a}\,\tilde{e}^{\nu}_{b}\,\leftidx{{}^{+}}{\mathcal{F}}_{\mu\nu}^{\;AB}=\frac{\partial
x^{\mu}}{\partial y^{a}}\frac{\partial x^{\nu}}{\partial
y^{b}}\,\leftidx{{}^{+}}{\mathcal{F}}_{\mu\nu}^{\;AB},$

is the pull-back to $\Sigma$ of the self-dual curvature, which matches the
curvature of the pulled back connection

$A_{a}^{\;AB}:=\tilde{e}^{\mu}_{a}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}=\frac{\partial
x^{\mu}}{\partial y^{a}}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}\,,$

and

$\leftidx{{}^{A}}{D}_{a}\tilde{\sigma}^{a}_{\;AB}=\partial_{a}\tilde{\sigma}^{a}_{\;AB}+{A_{aA}}^{C}\tilde{\sigma}^{a}_{\;CB}+{A_{aB}}^{C}\tilde{\sigma}^{a}_{\;AC}\,.$

From Stokes’ theorem, the last term in the decomposition (27) gives a boundary
term $S_{\text{SD}}^{\partial M}$. If we use the short hand notation
$\tilde{B}^{\nu}:=-(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,(i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\tilde{e}^{\nu}_{a})$
:

$\int_{M}{\mathrm{d}}^{4}x\,\partial_{\nu}\tilde{B}^{\nu}=\oint_{\partial
M}\left(\frac{\tilde{B}^{\nu}}{\sqrt{-g}}\right){\mathrm{d}}\mathcal{S}_{\nu},$

with ${\mathrm{d}}\mathcal{S}_{\nu}$ the oriented volume form on $\partial
M=\Delta\cup\Sigma_{1}\cup\Sigma_{2}\cup\tau_{\infty}$.

On the spatial hypersurfaces $\Sigma_{1,2}$,
${\mathrm{d}}\mathcal{S}_{\nu}=n_{\nu}\,{\mathrm{d}}^{3}y$, but since
$\tilde{e}^{\nu}_{a}$ is tangent to $\Sigma$, the surface terms vanish there,
so that the boundary term has only contributions from the WIH and from the
asymptotic region.

$S_{\text{SD}}^{\partial
M}=S_{\text{SD}}^{\Delta}+S_{\text{SD}}^{\tau_{\infty}}\,.$

Here, we will neglect the last term, since its contribution can be seen in
CorichiReyes . Focusing on the inner horizon boundary $\Delta$ and using
adapted coordinates
$z^{{\underset{{}^{\leftarrow}}{{\mu}}}}=(\lambda,\Theta_{1},\Theta_{2})$,
with $\lambda$ parametrizing null geodesics $x^{\mu}(\lambda)$ such that null
normals are $\ell^{\mu}=\frac{{\mathrm{d}}x^{\mu}}{{\mathrm{d}}\lambda}$, its
volume element is Poisson

${\mathrm{d}}\mathcal{S}_{\nu}=-\ell_{\nu}\leftidx{{}^{2}}{\epsilon}\,{\mathrm{d}}\lambda\,,$
(28)

where $\leftidx{{}^{2}}{\epsilon}=\sqrt{h}\,{\mathrm{d}}^{2}\Theta$ is the
transverse area two form. Then, the WIH boundary term can be written as

$S_{\text{SD}}^{\Delta}=\int_{\Delta}\,\frac{1}{N}\,(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,(i\sqrt{2}\,{\sigma}^{a}_{\;AB}\,\tilde{e}^{\nu}_{a})\ell_{\nu}\leftidx{{}^{2}}{\epsilon}\,{\mathrm{d}}\lambda,$
(29)

where we have used $\sqrt{-g}=N\sqrt{q}$.

Action (27) has the standard canonical form:

$\displaystyle S_{\text{SD}}=\int{\mathrm{d}}t\,{\mathrm{d}}^{3}y\big{\\{}$
$\displaystyle\,i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\mathcal{L}_{t}A_{a}^{\;AB}-H[N]-C[N^{a}]+G[t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}]\,\big{\\}}+S_{\text{SD}}^{\Delta}$
(30)

with

$H[N]:=\int_{\Sigma}{\mathrm{d}}^{3}y\,\frac{N}{\sqrt{q}}\,\tilde{\sigma}^{a}_{\;A}\,{}^{C}\,\tilde{\sigma}^{b}_{\;CB}\,F_{ab}^{\;AB}$
(31)

the Hamiltonian constraint,

$C[N^{a}]:=\int_{\Sigma}{\mathrm{d}}^{3}y\,i\sqrt{2}\,N^{a}\,\tilde{\sigma}^{b}_{\;AB}\,F_{ab}^{\;AB}$
(32)

the vector constraint, and

$G[\Lambda^{AB}]:=\int_{\Sigma}{\mathrm{d}}^{3}y\,\Lambda^{AB}\,\leftidx{{}^{{}_{A}}}{D}_{a}(i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB})$
(33)

the Gauss constraint.

In the following we will analyse in detail the canonical action (30), and
especially its boundary term $S_{\text{SD}}^{\Delta}$.

##  IV Consistency conditions and differentiability

This section has two parts. First, we state compatibility conditions on the
3+1 foliations and gauge fixing that we will impose. In the second part we
explore boundary contributions to the variation of the bulk part of the
canonical action. These results will be needed in the following section.

###  IV.1 Compatibility with 3+1 foliation and gauge fixing

Amongst all possible foliations of the given spacetime region $M$ inducing a
foliation on $\Delta$ by two-spheres, the equivalence class $[\ell]$ on
$\Delta$ selects a particular subset or family of _compatible foliations_. The
curves of null geodesic generators may be parametrized by the time function as
$x^{\mu}(t)$, so we require that the corresponding velocity vectors belong to
the equivalence class:

$\ell_{t}^{\mu}:=\frac{{\mathrm{d}}x^{\mu}}{{\mathrm{d}}t}\in[\ell]\,.$ (34)

Given any other representative $\ell^{\mu}\in[\ell]$, there exists then a
constant $c_{(\ell)}$, such that

$\ell_{t}^{\mu}=c_{(\ell)}\ell^{\mu}\,.$ (35)

As discussed in CRVwih2 , a foliation selects a particular transverse null
direction $\mathbf{k}^{\mu}$ from the infinitely many choices in the null cone
at each point $p\in\Delta$. For a given representative $\ell^{\mu}$, this is
the unique null direction normal to $S_{t}=\Delta\cap\Sigma_{t}$, apart from
$\ell^{\mu}$, and such that $\mathbf{k}_{\mu}\ell^{\mu}=-1$. Our compatibility
requirement implies the 3+1 decompositions of these normals are:

$\ell^{\mu}=\frac{N}{c_{(\ell)}}\left(n^{\mu}+s^{\mu}\right)\,,$ (36)
$\mathbf{k}^{\mu}=\frac{c_{(\ell)}}{2N}\left(n^{\mu}-s^{\mu}\right)\,.$ (37)

with the lapse function $N$ assumed positive for future directed normals
$\ell^{\mu}$, and with $s^{\mu}$ the ‘outward pointing’ unit normal to $S_{t}$
in $\Sigma_{t}$.

For non-rotating WIHs, we will choose our evolution vector field as
$t^{\mu}=\ell_{t}^{\mu}$. From (36), it follows that
$t^{\mu}=N\,\left(n^{\mu}+s^{\mu}\right)$, implying that on $\Delta$ the shift
and lapse are related as $N^{\mu}=Ns^{\mu}$.

In order to have better control of the fields’ variations, we will use the
Comoving gauge CRVwih2 , that was also used in the initial treatments on
isolated horizons ABF ; ACKclassical . We will restrict the soldering forms
and corresponding $SL(2,\mathbb{C})$ gauge transformations in a way that
$l^{\mu}:=i\sigma^{\mu}_{AA^{\prime}}{\rm o}^{A}\bar{{\rm o}}^{A^{\prime}}$
always belongs to the equivalence class $[\ell]$; we called this the _Adapted
Null gauge_. In addition, we will require that the transverse null direction
$k^{\mu}:=i\sigma^{\mu}_{AA^{\prime}}\iota^{A}\bar{\iota}^{A^{\prime}}$
coincides with $\mathbf{k}^{\mu}$, reducing further the gauge to
$\mathbb{R}^{+}_{\text{global}}\rtimes U(1)$. $\mathbb{R}^{+}_{\text{global}}$
refers to the global freedom of rescaling $l^{\mu}\to c\,l^{\mu}$ and
$k^{\mu}\to c^{-1}k^{\mu}$ by a positive constant $c$, and $U(1)$ to rotations
of $m^{\mu}=i\sigma^{\mu}_{AA^{\prime}}{\rm o}^{A}\bar{\iota}^{A^{\prime}}$
and its congugate $\bar{m}^{\mu}$ in the plane they span. We called this the
_Comoving gauge_. As usual, in the 3+1 decomposition of the self-dual action,
we used the time gauge to reduce the $SL(2,\mathbb{C})$ gauge group to compact
$SU(2)$. The time gauge restricts soldering forms (and gauge transformations)
to satisfy $n^{\mu}=\sigma^{\mu}_{AA^{\prime}}n^{AA^{\prime}}$ and freezes the
rescaling freedom for $\ell$ and $\mathbf{k}$, restricting further the gauge
symmetry of the WIH to $U(1)$. We will discuss later, in the canonical
framework, the (Hamiltonian) gauge symmetry of the theory on the horizon.

Solving for $n^{\mu}$ in (36), (37) we get

$n^{\mu}=\frac{c_{(\ell)}}{2N}\ell^{\mu}+\frac{N}{c_{(\ell)}}\mathbf{k}^{\mu}\,.$
(38)

and substituting the time gauge condition
$n^{\mu}=\sigma^{\mu}_{AA^{\prime}}\,n^{AA^{\prime}}$ and (18) with the
comoving gauge gauge conditions we get

$n^{AA^{\prime}}=\frac{c_{(\ell)}}{2N}\ell^{AA^{\prime}}+\frac{N}{c_{(\ell)}}k^{AA^{\prime}}=i\left(\frac{c_{(\ell)}}{2N}{\rm
o}^{A}\bar{{\rm
o}}^{A^{\prime}}+\frac{N}{c_{(\ell)}}\iota^{A}\bar{\iota}^{A^{\prime}}\right)\,.$
(39)

In every point of $\Sigma$ one can identify $SU(2)$ transformations as
$SL(2,\mathbb{C})$ transformations that preserve $n^{AA^{\prime}}$
AshtekarLectures . Equation (39) shows explicitly how the choice of compatible
foliations (through $N$) determine the $SU(2)$ subgroup of $SL(2,\mathbb{C})$.
Without loss of generality, to simplify our subsequent expressions, we will
set the lapse function to a constant:

$\frac{N}{c_{(l)}}=\frac{1}{\sqrt{2}}\,.$ (40)

This gives

$n^{AA^{\prime}}=\frac{i}{\sqrt{2}}({\rm o}^{A}\bar{{\rm
o}}^{A^{\prime}}+\iota^{A}\bar{\iota}^{A^{\prime}})\,,$ (41)

leading to the standard positive definite Hermitian inner product on
$\mathbb{C}^{2}$.

The Comoving gauge implies an expansion for the soldering form as

$\sigma^{\mu}_{AA^{\prime}}=-\mathbf{k}^{\mu}l_{AA^{\prime}}-\ell^{\mu}k_{AA^{\prime}}+\bar{m}^{\mu}m_{AA^{\prime}}+m^{\mu}\bar{m}_{AA^{\prime}}\,,$
(42)

with $m^{\mu}$ and $\bar{m}^{\mu}$ tangent to the spheres $S_{t}$.

Substituting (19), (39) and (40) in the defining formula (25), we obtain the
gauge fixed decomposition of $\sigma^{\mu}_{\;AB}$ on $\Delta$

$\displaystyle\sigma^{\mu}_{\;AB}$
$\displaystyle=-i\sqrt{2}\,q^{\mu}_{\,\nu}\,\sigma^{\nu}_{\;AA^{\prime}}\,n^{A^{\prime}}_{\;\;\;B}$
$\displaystyle=i(\,\sqrt{2}s^{\mu}\,\iota_{(A}{\rm o}_{B)}-\bar{m}^{\mu}\,{\rm
o}_{A}{\rm o}_{B}+m^{\mu}\,\iota_{A}\iota_{B}\,).$ (43)

The decomposition of the $SU(2)$ self-dual connection, $A_{a}^{AB}$, is given
by the pullback of (20) to $\Sigma$, with $l^{\mu}=\ell^{\mu}$ and
$k^{\mu}=\mathbf{k}^{\mu}$.

###  IV.2 Boundary terms in the variations of the constraints

In the next section we will construct well defined Hamiltonians and the
corresponding HVF $X_{H}$, that also describe consistent dynamics on $\Delta$.
The first step in the construction of the WIH components of $X_{H}$ is the
analysis of boundary terms that appear in the variations of the constraints.
We need the expressions from the previous subsection in order to calculate
these variations.

The boundary contribution from variation of the Hamiltonian constraint (31) is

$\delta
H[N]|_{S_{\Delta}}=\oint_{S_{\Delta}}{\mathrm{d}}S_{a}\,\frac{2N}{\sqrt{q}}\,\tilde{\sigma}^{[a|}_{\;A}\,{}^{C}\,\tilde{\sigma}^{|b]}_{\;CB}\,\delta
A_{b}^{\;AB}\,,$ (44)

where $S_{\Delta}=\Sigma\cap\Delta$.

The boundary contribution from variation of the vector constraint (32) is

$\delta
C[N^{a}]|_{S_{\Delta}}=i\sqrt{2}\oint_{S_{\Delta}}{\mathrm{d}}S_{a}\,2Ns^{[a}\,\tilde{\sigma}^{b]}_{\;AB}\,\delta
A_{b}^{\;AB}\,.$ (45)

where we have taken into account that on $\Delta$, $N^{a}=Ns^{a}$.

Expressions (44) and (45) are $SU(2)$-gauge invariant. We can therefore
substitute expansion (20) for the connection in the comoving gauge, and the
gauge fixed expression (43) for the soldering form, to find reduced
expressions for these variations. We also need the expression for

${\mathrm{d}}S_{a}=\frac{1}{2}\epsilon_{abc}{\mathrm{d}}x^{b}\wedge{\mathrm{d}}x^{c}=\frac{s_{a}}{\sqrt{q}}\,\sqrt{h}\,{\mathrm{d}}^{2}\Theta=\frac{s_{a}}{\sqrt{q}}\,\leftidx{{}^{2}}{\epsilon}\,.$
(46)

Then, the boundary term (44) takes the form

$\delta
H[N]|_{S_{\Delta}}=\sqrt{2}\oint_{S_{\Delta}}N\,\bigl{(}\bar{m}^{b}{\rm
o}_{(A}{\rm o}_{B)}+m^{b}\iota_{A}\iota_{B}\bigr{)}\,\delta
A_{b}^{\;AB}\,\leftidx{{}^{2}}{\epsilon}=-\sqrt{2}\oint_{S_{\Delta}}N\,m^{b}\,\delta\mathbb{U}_{b}\,\leftidx{{}^{2}}{\epsilon}\,,$
(47)

where we have used that for a NEH, in Adapted Null Gauge, we have
$\mathbb{Y}_{\underset{{}^{\leftarrow}}{{\mu}}}=0$ CRVwih2 , so that
$\bar{m}^{b}\,\delta\mathbb{Y}_{b}=0$.

Similarly, boundary term (45) is of the form

$\delta
C[N^{a}]|_{S_{\Delta}}=\sqrt{2}\oint_{S_{\Delta}}N\,\bigl{(}\bar{m}^{b}{\rm
o}_{(A}{\rm o}_{B)}-m^{b}\iota_{A}\iota_{B}\bigr{)}\,\delta
A_{b}^{\;AB}\,\leftidx{{}^{2}}{\epsilon}=\sqrt{2}\oint_{S_{\Delta}}N\,m^{b}\,\delta\mathbb{U}_{b}\,\leftidx{{}^{2}}{\epsilon}\,.$
(48)

As a result, we obtain

$\delta H[N]|_{S_{\Delta}}+\delta C[N^{a}]|_{S_{\Delta}}=0\,.$ (49)

Note that this is a remarkable result (and true for general lapse $N$). While
in the bulk, the lapse $N$ and shift vector $N^{a}$ are independent,
generating “bubble time evolution”, requiring that time evolution is
compatible with all the structures at the horizon forces one to consider lapse
and shifts that work in ‘tandem’. Thus, the combined contribution to the
variation of these two terms in the canonical Hamiltonian vanishes. This also
means that if there is any boundary term one has to add to the canonical
Hamiltonian to make it differentiable (a la Regge-Teitelboim), it would be to
Gauss’ law, in stark contrast to the case in spatial infinity with
asymptotically flat boundary conditions CorichiReyes . Finally, the variation
of the Gauss constraint has indeed a boundary term

$\delta
G[t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}]|_{S_{\Delta}}=i\sqrt{2}\oint_{S_{\Delta}}{\mathrm{d}}S_{a}\,(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,\delta\tilde{\sigma}^{a}_{\;AB}=\oint_{S_{\Delta}}t\cdot(\omega+V)\,\delta\,\leftidx{{}^{2}}{\epsilon}\,,$
(50)

where we have also used that in the Adapted Null gauge
$\mathbb{W}_{\underset{{}^{\leftarrow}}{{\mu}}}=\omega_{\underset{{}^{\leftarrow}}{{\mu}}}$
and
$\mathbb{V}_{\underset{{}^{\leftarrow}}{{\mu}}}=V_{\underset{{}^{\leftarrow}}{{\mu}}}$.

In the following section we will show in detail that, due to (49), the
boundary equations of motion are obtained only from the Gauss constraint, in
contrast to the bulk equations of motion.

##  V Hamiltonians and boundary phase spaces

This is the main section of this manuscript. The objective is to explore the
boundary term in the canonical action, when considering the WIH boundary
conditions. We shall see that the remaining terms, can be interpreted as
either a contribution to the “kinetic” term, or as a boundary contribution to
the Hamiltonian. Depending on the choice, we shall have different canonical
descriptions of the dynamics of the horizon. This section has four parts, in
each of which we consider each of the four possible choices for the boundary
terms.

Let us now turn our attention to the boundary term $S_{\text{SD}}^{\Delta}$ in
the canonical action. Using the expansions (20) and (43), it can be written as

$S_{\text{SD}}^{\Delta}=\int_{\Delta}\,\frac{1}{N}\,(t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB})\,(i\sqrt{2}\,{\sigma}^{a}_{\;AB}\,\tilde{e}^{\nu}_{a})\ell_{\nu}\leftidx{{}^{2}}{\epsilon}\,{\mathrm{d}}\lambda=-\int{\mathrm{d}}t\oint_{S_{\Delta}}t\cdot(\mathbb{W}+\mathbb{V})\,\leftidx{{}^{2}}{\epsilon}$
(51)

where we have also used $\sqrt{2}\,\tilde{e}^{\nu}_{a}\ell_{\nu}=s_{a}$ and
$\lambda=c_{(l)}t$. In the Adapted Null Gauge, we can finally obtain

$S_{\text{SD}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}t\cdot(\omega+V)\,\leftidx{{}^{2}}{\epsilon}\,.$
(52)

Note that this boundary term is not invariant under the residual $U(1)$ gauge
transformations on $\Delta$, namely rotation of $m$ and $\bar{m}$ preserving
$\ell$ and $\mathbf{k}$: $\ell\to\ell$, $\mathbf{k}\to\mathbf{k}$, $m\to
e^{i\theta}m$, $\bar{m}\to e^{-i\theta}\bar{m}$, for $\theta\in\mathbb{R}$.
Under this transformation $\omega$ is invariant, but $V$ transforms as a
$U(1)$ connection: $V\to V+i{\mathrm{d}}\theta$.

We shall now show that $S_{\text{SD}}^{\Delta}$ can be seen either as a
kinetic term in the canonical action or as a boundary term in the
corresponding Hamiltonian, or a mixture of both. This statement might sound
strange since the boundary term (52) contains two contributions, both of which
contain some part of the self-dual connection. Since the action principle was
written under the assumption that the connection shall play the role of
configuration variable (as is the case in gauge theories without boundaries),
one might argue that neither term contains a time derivative of any of the
canonical variables, and thus, it cannot be a kinetic term. How can one view
either (or both) terms as kinetic? It is not difficult to imagine that one can
complete that feat by introducing new degrees of freedom at the boundary. This
is precisely what is done in electrodynamics when one introduces the magnetic
potential as a “new” degree of freedom, and the magnetic field becomes a
derived object. These degrees of freedom are not entirely new, since they are
potentials for the already existing ones. Here, the strategy is to view
$\omega$ and/or $V$ as arising from a potential. Thus, we have four
possibilities. The first one is to consider both terms as arising from a
potential. Then, we shall have two cases in which only one of the terms is
reinterpreted. An finally, the fourth case is the “original” one in which one
does not introduce a potential and simply interprets (52) as a contribution to
the Hamiltonian.

In what follows, we will explore the consistency of the Hamiltonian
formulation of the theory based on all these possibilities.

###  V.1 Case I

As we discussed before, in this part we shall explore the possibility that
both $\omega$ and $V$ arise from some potential. With this idea in mind, let
us introduce $\psi_{R}$ and $\psi_{I}$, that we should call the real and
imaginary potentials, respectively, such that

$\displaystyle\mathcal{L}_{t}{\psi_{R}}$ $\displaystyle=$ $\displaystyle
t\cdot{\mathrm{d}}\psi_{R}\overset{{}_{\Delta}}{=}t\cdot\omega\,,$ (53)
$\displaystyle\mathcal{L}_{t}{\psi_{I}}$ $\displaystyle=$ $\displaystyle
t\cdot{\mathrm{d}}\psi_{L}\overset{{}_{\Delta}}{=}t\cdot V\,.$ (54)

In this first case we will consider a complex potential of the form
$\psi:=\psi_{R}+\psi_{I}$ (recall that $V$ is already imaginary), and its
corresponding canonical momentum $\tilde{\pi}$ as our boundary degrees of
freedom. As we shall see in detail, the formalism will tell us what
$\tilde{\pi}$ corresponds to.

####  V.1.1 Symplectic structure and the Hamiltonian

The idea here is to rewrite the boundary term as

$S_{\text{SD,1}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}(\mathcal{L}_{t}\,\psi)\,\leftidx{{}^{2}}{\epsilon}\,.$
(55)

In this case, it contributes completely to the kinetic term in the canonical
action, that indicates the existence of boundary degrees of freedom on the
horizon. As a result, the canonical action takes the following form

$S_{\text{SD,1}}=\int{\mathrm{d}}t\,\big{\\{}\int_{\Sigma}{\mathrm{d}}^{3}y\,i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\mathcal{L}_{t}A_{a}^{\;AB}-\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\sqrt{h}\,\mathcal{L}_{t}\psi-
H_{\text{SD,1}}\,\big{\\}}$ (56)

where the candidate for the canonical Hamiltonian is given by

$H_{\text{SD,1}}=H[N]+C[N^{a}]-G[t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}]\,.$
(57)

Since $S_{\text{SD, 1}}$ contains time derivative of $\psi$ on $\Delta$, the
kinetic term has a contribution from the boundary. In this case we have that
the $\Gamma_{\rm Bulk}$ part of the phase space is parametrized by the
coordinates $(A_{a}^{\;AB},\tilde{P}^{a}_{\;AB})$ and $\Gamma_{\rm Bound}$ is
parametrized by $(\psi,\tilde{\pi})$, where

$\displaystyle\tilde{P}^{a}_{\;AB}$ $\displaystyle=$ $\displaystyle
i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,.$ (58) $\displaystyle\tilde{\pi}$
$\displaystyle=$ $\displaystyle-\sqrt{h}\,,$ (59)

Note that $\tilde{\pi}_{I}=\tilde{\pi}_{R}=\tilde{\pi}$, since only the
combination $\psi=\psi_{R}+\psi_{I}$ appears in the kinetic term, while
$\psi^{*}=\psi_{R}-\psi_{I}$ is not dynamical, leaving us with two canonical
degrees of freedom on the boundary $\Delta$. The corresponding symplectic
structure has a boundary term and is of the form

$\Omega_{1}=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{P}^{a}_{\;AB}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;AB}+\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\psi\,.$
(60)

In the canonical framework the equations of motions are given by

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H_{\text{SD,1}}(Y)=\Omega_{1}(Y,X_{H})\,,$
(61)

where $X_{H}$ is the corresponding Hamiltonian vector field, whose components
in the bulk are $(X_{H,A},X_{H,\tilde{P}})$ and on the boundary
$(X_{H,\psi},X_{H,\tilde{\pi}})$. The bulk part of (61) gives the equations of
motion in the bulk, while the boundary term defines the boundary components of
the Hamiltonian vector field.

Here, we are especially interested in the boundary contributions in (61).
Using the previously obtained expressions (49) and (50), it follows that

$\delta
H_{\text{SD,1}}|_{S_{\Delta}}=-\oint_{S_{\Delta}}t\cdot(\omega+V)\delta\,\leftidx{{}^{2}}{\epsilon}\,.$
(62)

We can also rewrite it in terms of en exterior derivative on the phase space,

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H_{\text{SD,1}}(Y)|_{S_{\Delta}}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\
[t\cdot(\omega+V)]\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}(Y)=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\
[t\cdot(\omega+V)]\,Y_{\tilde{\pi}}\,,$ (63)

where we have used the property
${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}(y)\left(\frac{\delta}{\delta\tilde{\pi}(y^{\prime})}\right)=\delta^{2}(y,y^{\prime})$.
On the other hand, from (60) it follows that the boundary contribution to the
symplectic structure is

$\Omega_{1}(Y,X)|_{S_{\Delta}}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,(X_{\psi}Y_{\tilde{\pi}}-Y_{\psi}X_{\tilde{\pi}})\,.$
(64)

From (63) and (64) we can read off the the boundary components of the
corresponding HVF, $X_{\text{H,1}}$,

$\displaystyle X_{H1,\tilde{\pi}}=0\ \ \ $ $\displaystyle\Rightarrow$
$\displaystyle\ \ \ \mathcal{L}_{t}\leftidx{{}^{2}}{\epsilon}=0\,,$ (65)
$\displaystyle X_{H1,\psi}=t\cdot\omega+t\cdot V\ \ \ $
$\displaystyle\Rightarrow$ $\displaystyle\ \ \
\mathcal{L}_{t}\psi=t\cdot\omega+t\cdot V\,.$ (66)

The first equation is a consequence of the WIH boundary conditions and the
second one is the definition of the potential $\psi$. In that sense, we have
obtained a consistent evolution of the boundary degrees of freedom.

Nevertheless, $H_{\text{SD,1}}$ vanishes on the constraint surface, and thus
it does not contain the information about the mass of the WIH. In the
following we shall see that, given the freedom in the election of the
symplectic structure on the boundary, we can find a Hamiltonian with a non-
vanishing boundary term. Before that, let us analyze the generators of gauge
and diffeomorphism transformations in the current setup.

####  V.1.2 Gauss constraint

Here, we want to analyse the residual gauge symmetry on a WIH. For that we
need to find out the form of the boundary components of the corresponding HVF.
Previously, we have restricted the residual internal gauge symmetry on a WIH
to $U(1)$. Now, we want to further explore this gauge reduction from the
canonical framework. We start with the Gauss constraint that is given by

$G[\Lambda^{AB}]:=\int_{\Sigma}{\mathrm{d}}^{3}y\,\Lambda^{AB}\,\leftidx{{}^{{}_{A}}}{D}_{a}\tilde{P}^{a}_{\;AB}=-\int_{\Sigma}{\mathrm{d}}^{3}y\,(\leftidx{{}^{{}_{A}}}{D}_{a}\Lambda^{AB})\,\tilde{P}^{a}_{\;AB}+\oint_{S_{\Delta}}\,{\mathrm{d}}S_{a}\,\Lambda^{AB}\,\tilde{P}^{a}_{\;AB}\,.$
(67)

The general smearing function $\Lambda^{AB}$ can be expanded as

$\Lambda^{AB}=i\Lambda\,\iota^{(A}{\rm o}^{B)}+\Omega\,{\rm o}^{A}{\rm
o}^{B}+\Theta\,\iota^{A}\iota^{B}\,,$ (68)

and it has to be traceless ${\Lambda^{A}}_{A}=0$ and anti-hermitian
$\Lambda^{AB}=-\bar{\Lambda}^{BA}$. These conditions imply that

$\Lambda^{AB}=i\Lambda\,\iota^{(A}{\rm o}^{B)}+\Omega\,{\rm o}^{A}{\rm
o}^{B}-\bar{\Omega}\,\iota^{A}\iota^{B}\,,$ (69)

with $\Lambda\in\mathbb{R}$ and $\Omega\in\mathbb{C}$.

The boundary term in (67) can be rewritten using the expansion of
$\tilde{\sigma}^{a}_{\;AB}$ in the Comoving Gauge (43) as

$\oint_{S_{\Delta}}\,{\mathrm{d}}S_{a}\,\Lambda^{AB}\,\tilde{P}^{a}_{\;AB}=-2\oint_{S_{\Delta}}\,\Lambda^{AB}\,\iota_{(A}{\rm
o}_{B)}\,\leftidx{{}^{2}}{\epsilon}=\oint_{S_{\Delta}}\,i\Lambda\,\leftidx{{}^{2}}{\epsilon}\,.$
(70)

The vector field $X_{G}$, that defines the infinitesimal gauge
transformations, is given by

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,G(Y)=\Omega_{1}(Y,X_{G})\,,$ (71)

where $Y$ is an arbitrary vector in $T\Gamma$. Again, we will only consider
the WIH contribution in (71). Taking into account (67) and (70) we obtain the
form of the boundary gauge transformation

$\delta_{g}\,\tilde{\pi}:=X_{G,\tilde{\pi}}=0$ (72)
$\delta_{g}\psi:=X_{G,\psi}=-i\Lambda\,.$ (73)

Since, $\omega={\mathrm{d}}\psi_{R}+\tilde{\omega}$ and
$V={\mathrm{d}}\psi_{I}+\tilde{V}$ (the explicit form of $\tilde{\omega}$ and
$\tilde{V}$ is given in the Appendix A), the Gauss constraint generates on a
WIH gauge transformations: $\psi_{R}\to\psi_{R}$ and
$\psi_{I}\to\psi_{I}-i\Lambda$, that corresponds to $U(1)$ transformations:
$\omega\to\omega$ and $V\to V-i\,{\mathrm{d}}\Lambda$. Note that these
transformations preserve the area of $S_{\Delta}$.

The $X_{G}$, defined by the previous expressions, is a degenerate direction of
the symplectic structure, since the pullback of the symplectic structure to
the submanifold defined by the Gauss constraint, $\Omega_{g}$, is degenerate.
Effectively,

$\Omega_{g}(Y,X_{G})=0\ \ \ \Leftrightarrow\ \ \
{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,G[\Lambda^{AB}](Y)\approx 0\,.$ (74)

The R.H.S. is always true for any vector $Y$ tangent to the Gauss constraint’s
surface. It should also be true, for any vector tangent to diffeomorfism and
Hamiltonian constraints. For that to happen all the constraints should be
first class.

####  V.1.3 Spatial diffeomorphism constraint

Let us see what are the residual diffeomorphism symmetries on a WIH, in this
case. It is well known that the generator of spatial diffeomorphisms on
$\Sigma$ is a linear combinations of the vector (32) and the Gauss (33)
constraints ThiemannBook , and is of the form

$D[N^{a}]=\int_{\Sigma}{\mathrm{d}}^{3}y\,N^{a}\,(\,\tilde{P}^{b}_{\;AB}\,F_{ab}^{\;AB}-A_{a}^{\;AB}\,\leftidx{{}^{{}_{A}}}{D}_{b}\tilde{P}^{b}_{\;AB}\,)\,.$
(75)

To analyse tangential diffeomorphisms on WIH we need to calculate $\delta
D[N^{a}]|_{S_{\Delta}}$, and the boundary components of the corresponding HVF
in the case when the shift is tangential to $S_{\Delta}$,

$N^{a}=\mu\,m^{a}+\bar{\mu}\,\bar{m}^{a}\,,$ (76)

where $\mu$ is an arbitrary functions on $S_{\Delta}$.

The corresponding HVF $X_{D}$ is determined from

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,D(Y)=\Omega(Y,X_{D})\,.$ (77)

Boundary components of $X_{D}$ determine residual diffeomorphisms on WIH. The
boundary term in $\delta D[N^{a}]$ is of the form

$\delta
D[N^{a}]|_{S_{\Delta}}=-\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\delta\bigl{[}N^{a}(\omega_{a}+V_{a})\,\tilde{\pi}\bigr{]}:=\delta\bar{D}[N^{a}]|_{S_{\Delta}}+\delta\hat{D}[N^{a}]|_{S_{\Delta}}\,,$
(78)

where, using the decompositions (131)-(132), we have introduced

$\displaystyle\delta\bar{D}[N^{a}]|_{S_{\Delta}}$ $\displaystyle:=$
$\displaystyle-\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,N^{b}\,\delta\bigl{[}(\nabla_{b}\psi)\,\tilde{\pi}\bigr{]}\,,$
(79) $\displaystyle\delta\hat{D}[N^{a}]|_{S_{\Delta}}$ $\displaystyle:=-$
$\displaystyle\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,N^{b}\,\delta[(\tilde{\omega}_{b}+\tilde{V}_{b})\,\tilde{\pi}]\,.$
(80)

As we have defined in (7) the boundary contribution in the variation of any
allowed functional should only depend on boundary degrees of freedom, in this
case it should be of the form

$\delta
F|_{S_{\Delta}}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\left(\,\frac{\delta
F}{\delta\psi}\delta\psi+\frac{\delta
F}{\delta\tilde{\pi}}\delta\tilde{\pi}\,\right)\,.$ (81)

We see that the $\delta\bar{D}[N^{a}]|_{S_{\Delta}}$ is of the form given in
(81), but $\delta\hat{D}[N^{a}]|_{S_{\Delta}}$ is not and it does not vanish
either. As a consequence $D[N^{a}]$ is an allowed funcional only in the case
when $N^{a}\overset{{}_{\Delta}}{=}0$. Then $\delta D[N^{a}]|_{S_{\Delta}}=0$,
as a result, the boundary components of $X_{D}$ vanish, that means that the
tangential diffeomorphisms are not gauge transformations on the boundary. In
the Appendix B we will show that in the special case, when we restrict to
spherically symmetric configurations and the phase space with fixed area of
$S_{\Delta}$, as in ACKclassical , $D[N^{a}]$ generates tangential
diffeomorphisms on $S_{\Delta}$.

To end this part, let us note that we can add a boundary term to $D[N^{a}]$ as

$\tilde{D}[N^{a}]=D[N^{a}]+\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\bigl{[}N^{a}(\tilde{\omega}_{a}+\tilde{V}_{a})\,\tilde{\pi}\bigr{]}\,.$
(82)

such that $\delta D[N^{a}]|_{S_{\Delta}}=\delta\bar{D}[N^{a}]|_{S_{\Delta}}$.
The boundary component of HVF corresponding to $\tilde{D}$ is obtained from

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{D}(Y)|_{S_{\Delta}}=\Omega(Y,X_{\tilde{D}})|_{S_{\Delta}}\,,$
(83)

leading to

$\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\bigl{[}-\nabla_{a}(N^{a}\tilde{\pi})\,Y_{\psi}+N^{a}(\nabla_{a}\psi)\,Y_{\tilde{\pi}}\bigr{]}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,(X_{\tilde{D},\psi}Y_{\tilde{\pi}}-Y_{\psi}X_{\tilde{D},\tilde{\pi}})\,.$
(84)

Then, it follows that

$\displaystyle X_{\tilde{D},\tilde{\pi}}=\nabla_{a}(N^{a}\tilde{\pi})\ \ \ $
$\displaystyle\Rightarrow$ $\displaystyle\ \ \
\tilde{\delta}\sqrt{h}=\nabla_{a}(N^{a}\sqrt{h})=\mathcal{L}_{\vec{N}}\sqrt{h}\,,$
$\displaystyle X_{\tilde{D},\psi}=N^{a}\nabla_{a}\psi\ \ \ $
$\displaystyle\Rightarrow$ $\displaystyle\ \ \
\tilde{\delta}\psi=N^{a}\nabla_{a}\psi=\mathcal{L}_{\vec{N}}\psi\,,$ (85)

and the ’improved’ generator indeed generate tangential diffeomorphisms on
$S_{\Delta}$. This corresponds precisely to the Regge-Teitelboim strategy of
constructing Hamiltonians to generate specific symplectomorphism on phase
space. In this case, the generator of horizon diffeomorphisms, that is a true
(non-vanishing on shell) Hamiltonian function, and not a constraint. Note that
this is very similar to the result obtained in the covariant formalism in ABL
. To summarize this choice. When we take both terms as the boundary term of
the canonical action to be dynamical, namely to contribute to the symplectic
structure, we do get a consistent description but we do not recover the
(expected) mass of the black hole as a boundary term. Let us now consider
cases in which only one of the terms are to be regarded as dynamic.

###  V.2 Case II

In this case we will consider $(\psi_{I},\tilde{\pi})$ as boundary degrees of
freedom and leave the first term in (52) in the form of a boundary term in the
Hamiltonian. This approach is an extension of the results of ABF  to WIHs. We
will show that in this case, the resulting boundary term in the Hamiltonian
shall correspond to the mass of a WIH.

Since $\kappa_{(t)}:=t\cdot\omega$ , we can rewrite the boundary term in the
canonical action (52) as

$S_{\text{SD,2}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\,(\kappa_{(t)}\,+\mathcal{L}_{t}\psi_{I})\,\leftidx{{}^{2}}{\epsilon}\,.$
(86)

The last integral over $S_{\Delta}$ in the previous expression is the
symplectic potential on the horizon, that gives rise to the symplectic
structure on the horizon.

The canonical action is now of the form

$S_{\text{SD,2}}=\int{\mathrm{d}}t\,\big{\\{}\int_{\Sigma}{\mathrm{d}}^{3}y\,i\sqrt{2}\,\tilde{\sigma}^{a}_{\;AB}\,\mathcal{L}_{\bar{t}}A_{a}^{\;AB}-\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\sqrt{h}\,\mathcal{L}_{\bar{t}}\psi_{I}-H_{\text{SD,2}}\,\big{\\}}$
(87)

where the candidate for the canonical Hamiltonian is given by

$H_{\text{SD,2}}=H_{\text{SD,1}}+\oint_{S_{\Delta}}\,\kappa_{(t)}\,\leftidx{{}^{2}}{\epsilon}\,,$
(88)

and the symplectic structure in this case is of the form

$\Omega_{2}=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{P}^{a}_{\;AB}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;AB}+\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\psi_{I}\,.$
(89)

Now, only $\psi_{I}$ and $\tilde{\pi}$ contribute to $\Omega_{2}$, and are to
be regarded as the boundary degrees of freedom.

In order to calculate the boundary contribution to (61), let us first analyze
the variation of the Hamitonian, that is given by

$\delta
H_{\text{SD,2}}|_{S_{\Delta}}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\big{[}\,-\delta\kappa_{(t)}\,\tilde{\pi}+(t\cdot
V)\,\delta\,\tilde{\pi}\,\bigr{]}\,.$ (90)

Comparing it to the variation of an allowed function, we can see that the
first term in (90) should vanish or $\delta\kappa_{(t)}$ should be a linear
combination of $\delta\psi$ and $\delta\tilde{\pi}$.

The corresponding HVF, $X_{\text{H2}}$ is obtained from

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H_{\text{SD,2}}(Y)=\Omega_{2}(Y,X_{\text{H2}})\,,$
(91)

Then, from (89) and (90), it follows that the boundary contribution to the
previous expression is given by

$\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\big{[}\,-\tilde{\pi}\,Y_{\kappa_{(t)}}+(t\cdot
V)\,Y_{\tilde{\pi}}\,\bigr{]}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,(X_{H,\psi_{I}}Y_{\tilde{\pi}}-X_{H,\tilde{\pi}}Y_{\psi_{I}})\,,$
(92)

where we have introduced the notation $Y_{\kappa_{(t)}}:=\delta\kappa_{(t)}$.
In order to have the consistent description of the evolution on the boundary
we should have $Y_{\kappa_{(t)}}=AY_{\psi_{I}}+BY_{\tilde{\pi}}$, with $A$ and
$B$ functions on $S_{\Delta}$. Then, it follows that
$X_{H,\psi_{I}}=B\sqrt{h}+t\cdot V$ and $X_{H,\tilde{\pi}}=-A$, leading to
$\mathcal{L}_{l}\psi_{I}=B\sqrt{h}+t\cdot V$ and $\mathcal{L}_{l}\sqrt{h}=A$.
We see that only when $A=B=0$, we can obtain the correct boundary condition
$\mathcal{L}_{l}\sqrt{h}\overset{{}_{\Delta}}{=}0$, and also
$\mathcal{L}_{l}\psi_{I}=t\cdot V$.

So, we are led to the conclusion that we have to eliminate the first term in
(90). That can be done by adding a counterterm to the Hamiltonian,
$H_{\text{CT}}$, such that

$\delta(H_{\text{SD,2}}+H_{\text{CT,2}})|_{S_{\Delta}}=\oint_{S_{\Delta}}\,{\mathrm{d}}^{2}\Theta\,(t\cdot
V)\,\delta\,\tilde{\pi}\,,$ (93)

that implies that

$\delta
H_{\text{CT,2}}=\oint_{S_{\Delta}}\,{\mathrm{d}}^{2}\Theta\,\delta\kappa_{(t)}\,\tilde{\pi}\,.$
(94)

In CRVwih2  and GJreview  it has been shown that
$(h_{ab},\Omega_{a},\Xi_{ab},\kappa_{(t)})$ are the free independent
parameters on a given sphere $S_{\Delta}$, completely determining the WIH
geometry. The _Háj́iček 1-form_ Hajicek
$\Omega_{\mu}:=h^{\nu}_{\;\mu}\omega_{\nu}$ is the projection of the rotation
1-form to the sphere and which also coincides with its pullback $\omega_{a}$
on $T_{p}\mathcal{S}_{\Delta}$: $\omega_{a}=\Omega_{a}$. The _transversal
deformation rate_
$\Xi_{\mu\nu}:=h^{\sigma}_{\;\;\mu}h^{\rho}_{\;\;\nu}\nabla_{\sigma}k_{\rho}$
is the analog for $k^{\mu}$ of the second fundamental form (9). $\Omega_{a}$
and $\Xi_{ab}=\nabla_{a}k_{b}$ are completely unrestricted and $\Omega_{\mu}$
and $\Xi_{\mu\nu}$ may be determined from them on all of $\Delta$ using
certain constraint equations for $\mathcal{L}_{\ell}\,\Omega_{\mu}$ and
$\mathcal{L}_{\ell}\,\Xi_{\mu\nu}$ whose exact form is given in GJreview ;
ABLgeometry .

It follows then that a general counterterm must have the form

$H_{\text{CT}}=\oint_{S_{\Delta}}F(h_{ab},\Omega_{a},\Xi_{ab},\kappa_{(t)})\,\leftidx{{}^{2}}{\epsilon}\,,$
(95)

and its variation is

$\delta H_{\text{CT}}=\oint_{S_{\Delta}}\left\\{\left(\frac{\delta F}{\delta
h_{ab}}\delta h_{ab}+\frac{\delta
F}{\delta\Omega_{a}}\delta\Omega_{a}+\frac{\delta
F}{\delta\Xi_{ab}}\delta\Xi_{ab}+\frac{\delta
F}{\delta\kappa_{(t)}}\delta\kappa_{(t)}\right)\,\leftidx{{}^{2}}{\epsilon}+F\,\delta\,\leftidx{{}^{2}}{\epsilon}\right\\}\,.$
(96)

Assuming this variation cancels the unwanted boundary term in (90), it implies
that variations $(\delta
h_{ab},\delta\Omega_{a},\delta\Xi_{ab},\delta\kappa_{(t)})$ are not fully
independent. Namely, a priori, $(h_{ab},\Omega_{a},\Xi_{ab},\kappa)$ are
independent, so are their variations. We can consider directions on phase
space such that $\delta h_{ab}=0$ (and hence
$\delta\leftidx{{}^{2}}{\epsilon}=0$) and $\delta\kappa_{(t)}=0$. This implies

$\frac{\partial F}{\partial\Omega_{a}}=\frac{\partial
F}{\partial\Xi_{ab}}=0\,.$ (97)

Now, from (94) and (96), it follows

$\delta H_{\text{CT}}=\oint_{S_{\Delta}}\left\\{\left(\frac{\delta F}{\delta
h_{ab}}\delta h_{ab}+\frac{\delta
F}{\delta\kappa_{(t)}}\delta\kappa_{(t)}\right)\,\leftidx{{}^{2}}{\epsilon}+F\,\delta\,\leftidx{{}^{2}}{\epsilon}\right\\}=-\oint_{S_{\Delta}}\,\delta\kappa_{(t)}\,\leftidx{{}^{2}}{\epsilon}\,.$
(98)

Then, taking variations in directions where $\delta\kappa_{(t)}=0$, gives the
condition

$\oint_{S_{\Delta}}\left\\{\left(\frac{\delta F}{\delta h_{ab}}\delta
h_{ab}\right)\,\leftidx{{}^{2}}{\epsilon}+F\,\delta\,\leftidx{{}^{2}}{\epsilon}\right\\}=0\,,$
(99)

that implies $F$ should only depend on the metric $h_{ab}$ through the area,
$F(a_{\Delta},\kappa_{(t)})$. Since $a_{\Delta}$ and $\kappa_{(t)}$ are
constant on $S_{\Delta}$, from (98), we obtain, for the general variations,
that

$\left(\frac{\partial
F}{\partial\kappa_{(t)}}+1\right)a_{\Delta}\delta\kappa_{(t)}+\left(a_{\Delta}\frac{\partial
F}{\partial a_{\Delta}}+F\right)\delta a_{\Delta}=0\,.$ (100)

This in turn implies that $\kappa_{(t)}$ must be a function of the area
$a_{\Delta}$. (Indeed, if $\delta\kappa_{(t)}$ and $\delta a_{\Delta}$ were
independent, we would need $\frac{\partial F}{\partial\kappa_{(t)}}+1=0$,
which implies $F(a_{\Delta},\kappa_{(t)})=-\kappa_{(t)}+C(a_{\Delta})$. But
then $a_{\Delta}\frac{\partial F}{\partial
a_{\Delta}}+F=a_{\Delta}\frac{\partial C}{\partial a_{\Delta}}+C-\kappa_{(t)}$
cannot vanish in general since $C$ is independent of $\kappa_{(t)}$.)

So in the canonical formulation, we recover the result of the covariant
Hamiltonian analysis of AFK , that the surface gravity must only be a function
of $a_{\Delta}$,

$\kappa_{(t)}=\kappa_{(t)}(a_{\Delta})\,.$ (101)

As a consequence, the counterterm is also a function of area only:

$H_{\text{CT}}=\oint_{S_{\Delta}}F(a_{\Delta})\,\leftidx{{}^{2}}{\epsilon}=F(a_{\Delta})\,a_{\Delta}\,.$
(102)

and therefore, from (98) and (102), we obtain

$\delta
H_{\text{CT}}=(\frac{{\mathrm{d}}{F}}{{\mathrm{d}}a_{\Delta}}a_{\Delta}+F)\,\delta
a_{\Delta}=-\frac{{\mathrm{d}}\kappa_{(t)}}{{\mathrm{d}}a_{\Delta}}\,a_{\Delta}\,\delta
a_{\Delta}\,,$ (103)

that leads to a differential equation for $F(a_{\Delta})$:

$a_{\Delta}\,\frac{{\mathrm{d}}F}{{\mathrm{d}}a_{\Delta}}+F=-a_{\Delta}\frac{{\mathrm{d}}\kappa_{(t)}}{{\mathrm{d}}a_{\Delta}}\,.$
(104)

It is standard to fix

$\kappa_{(t)}=\frac{1}{2R_{\Delta}}=\sqrt{\pi}\,a_{\Delta}^{-1/2}\,,$ (105)

where $a_{\Delta}=4\pi R_{\Delta}^{2}$. In this way on a static solution
$t^{\mu}$ coincides with the Killing vector field on $\Delta$. This choice
implies the choice of $t^{\mu}$, so that we can define the horizon mass as
$M_{\Delta}=E^{t}_{\Delta}$, for any point of the phase space, not only for
static spacetimes CRVwih2 .

The general solution to (104) is $F=\kappa_{(t)}+Ca_{\Delta}^{-1}$, where $C$
is an arbitrary constant, so that

$H_{\text{CT,2}}=\int_{S_{\Delta}}\,\kappa_{(t)}\,\leftidx{{}^{2}}{\epsilon}+C\,.$
(106)

Without the loss of generality, we can fix $C=0$, and obtain a well defined
Hamiltonian $H_{2}=H_{\text{SD,2}}+H_{\text{CT}}$ as

$H_{2}=\int_{\Sigma}{\mathrm{d}}^{3}y\,\big{\\{}H[N]+C[N^{a}]-G[t^{\mu}\,\leftidx{{}^{+}}{\mathcal{A}}_{\mu}^{\;AB}]\,\big{\\}}+2\oint_{S_{\Delta}}\,\kappa_{(t)}\,\leftidx{{}^{2}}{\epsilon}\,.$
(107)

The boundary term corresponds to the Smarr expression for the mass of a non-
rotating black hole Smarr .

Let us analyse now the canonical description of the evolution generated by
$H_{2}$ in the phase space. The equations of motions are given by

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H_{2}(Y)=\Omega(Y,X_{H2})\,,$ (108)

where $X_{H2}$ is the corresponding Hamiltonian vector field, whose components
in the bulk are $(X_{H2,A},X_{H2,\tilde{P}})$ and on the boundary
$(X_{H2,\psi},X_{H2,\tilde{\pi}})$. The bulk part of (108) gives the equations
of motion in the bulk, while the boundary term defines the boundary components
of the HVF. Let us see what is the boundary contribution. From the previous
discussion we have that the LHS of (108) can be obtained from (93), and, as a
result, its boundary contribution is

$\oint_{S_{\Delta}}\,{\mathrm{d}}^{2}\Theta\,(t\cdot
V)\,Y_{\tilde{\pi}}=\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,(X_{H2,\psi_{I}}Y_{\tilde{\pi}}-Y_{\psi_{I}}X_{H2,\tilde{\pi}})\,.$
(109)

It follows that

$\displaystyle X_{H2,\tilde{\pi}}=0\ \ \ $ $\displaystyle\Rightarrow$
$\displaystyle\ \ \ \mathcal{L}_{t}\leftidx{{}^{2}}{\epsilon}=0\,,$ (110)
$\displaystyle X_{H2,\psi_{I}}=t\cdot V\ \ \ $ $\displaystyle\Rightarrow$
$\displaystyle\ \ \ \mathcal{L}_{t}\psi_{I}=t\cdot V\,,$ (111)

leading to the consistent boundary equations, as in the Case I. The main
difference between the two cases is that in the first one we could obtain a
well defined Hamiltonian, but we could not obtain the expression for the mass,
while in the second one we could achieve both goals.

As to gauge transformations and tangential diffeomorphisms, all the results
from Case I still hold, the only modificacion that has to be made is
$\psi\to\psi_{I}$. Again, $U(1)$ transformation on WIH are gauge
transformations, while the tangential diffeomorphisms have to be fixed. Let us
now consider the case where we choose the other term to be dynamical.

###  V.3 Case III

In this case we shall take the opposite viewpoint to the previous case. That
is, we shall have $(\psi_{R},\tilde{\pi})$ as boundary degrees of freedom.
This is the same choice that is made within the covariant phase space approach
of, for example, AFK  and CRGV2016 . It turns out that we can obtain
consistent boundary equations of motion, but again, we need to impose
additional conditions and the boundary term in the corresponding well defined
Hamiltonian vanishes.

In this case we rewrite the boundary term in the canonical action as

$S_{\text{SD,3}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\,(\mathcal{L}_{t}\psi_{R}+t\cdot
V)\,\leftidx{{}^{2}}{\epsilon}\,.$ (112)

The symplectic structure again has a boundary term

$\Omega_{3}=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{P}^{a}_{\;AB}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;AB}+\oint_{S_{\Delta}}{\mathrm{d}}^{2}\Theta\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{\pi}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\psi_{R}\,,$
(113)

where the canonical variables on $S_{\Delta}$ are $(\psi_{R},\tilde{\pi})$.
The canonical Hamiltonian is given by:

$H_{\text{SD,3}}=H_{\text{SD,1}}+\oint_{S_{\Delta}}\,(t\cdot
V)\,\leftidx{{}^{2}}{\epsilon}\,.$ (114)

The boundary term should not contribute to the boundary equations of motion
for $\psi_{R}$ or $\tilde{\pi}$, nor to the mass of WIH, so we need to impose
an additional condition

$\oint_{S_{\Delta}}\,(t\cdot V)\,\leftidx{{}^{2}}{\epsilon}=0\,.$ (115)

Then, $H_{\text{SD,3}}=H_{\text{SD,1}}$, and again the Hamiltonian weakly
vanishes. From (62) and (115), we obtain

$\delta
H_{\text{SD,3}}|_{S_{\Delta}}=-\oint_{S_{\Delta}}\,\kappa_{(t)}\,\delta\,\leftidx{{}^{2}}{\epsilon}\,.$
(116)

Under these restrictions, we can see that from

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,H_{\text{SD,3}}(Y)=\Omega_{3}(Y,X_{\text{H3}})\,,$
(117)

we can obtain the boundary components of the HVF, that lead to a consistent
equations of motion on $\Delta$,

$\mathcal{L}_{t}\leftidx{{}^{2}}{\epsilon}=0\,,\ \ \ \
\mathcal{L}_{t}\psi_{R}=\kappa_{(t)}\,.$ (118)

In the case of the Gauss constraint, we look for a HVF $X_{G}$ that generates
the infinitesimal gauge transformations,

${\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,G(Y)=\Omega_{3}(Y,X_{G})\,,$ (119)

where $Y$ is an arbitrary vector in $T\Gamma$. In this case the components of
$X_{G}$ on the WIH are

$\delta_{g}\,\tilde{\pi}:=X_{G,\tilde{\pi}}=0$ (120)
$\delta_{g}\psi_{R}:=X_{G,\psi_{R}}=-i\Lambda\,.$ (121)

The last expression represents the change in $\psi_{R}$ that is purely
imaginary, so we must fix $\Lambda=0$ on WIH. Again, in this case the Gauss
constraint does not generate gauge transformations on WIH. Similarly, boundary
tangential diffeomorphisms have to be fixed.

###  V.4 Case IV

In this last case, we will examine the scenario where no boundary degrees of
freedom are introduced and show that it also leads to a consistent
description. In this case the Regge-Teitelboim type conditions hold and we can
obtain the expression for the mass of WIH. Nevertheless, we will show that we
need to impose some additional conditions on boundary variations and that the
gauge transformations on WIH have to be completely fixed.

In this case the boundary term in the canonical action is taken as

$S_{\text{SD,4}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}(\kappa_{(t)}+t\cdot
V)\,\leftidx{{}^{2}}{\epsilon}\,.$ (122)

There are no boundary contributions to the symplectic structure

$\Omega_{4}=\int_{\Sigma}{\mathrm{d}}^{3}x\,\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,\tilde{P}^{a}_{\;AB}\wedge\\!\\!\\!\\!\\!\>\\!\wedge\,{\mathrm{d}}\\!\\!\\!\\!{\mathrm{d}}\,A_{a}^{\;AB}\,,$
(123)

so that the variations of the allowed functionals cannot have a boundary term.

The candidate for the canonical Hamiltonian is of the form

$H_{\text{SD,4}}=H_{\text{SD,1}}+\oint_{S_{\Delta}}\,(\kappa_{(t)}+t\cdot
V)\,\leftidx{{}^{2}}{\epsilon}\,.$ (124)

and its variation has a boundary term:

$\delta
H_{\text{SD,4}}|_{S_{\Delta}}=\oint_{S_{\Delta}}\,\big{[}\,(\delta\kappa_{(t)}+\delta(t\cdot
V)\bigr{]}\,\leftidx{{}^{2}}{\epsilon}\,.$ (125)

Since $\Omega_{4}$ does not have a boundary contribution, the boundary term
(125) must vanish or be canceled by a variaton of an appropriate counterterm,
such that

$\delta(H_{\text{SD,4}}+H_{\text{CT,4}})|_{S_{\Delta}}=0\,.$ (126)

We have already seen that $H_{\text{CT,2}}$, given by (106), cancels the first
term in (125) and contributes to the correct expression for the mass of WIH.
Any additional counter term that would annulate the second term in (125),
would change the expression for the mass.

In order to have a correct expression for a mass of the WIH , we need to
impose an additional condition (115) in (124). The final Hamiltonian,
$H_{4}=H_{\text{SD,4}}+H_{\text{CT,2}}$ is the same as in (107), and, due to
(115), its variation does not have a boundary term, that is consistent with
the fact that the corresponding symplectic structure $\Omega_{4}$ does not
have a boundary term. We see that in this case, the dynamics is consistently
described by the same total Hamiltonian as in the Case II, given in (107).
Nevertheless, we have an additional condition (115) and there are no boundary
degrees of freedom on WIH.

As to gauge transformations, since there are no boundary terms in the
symplectic structure, from (67) and (69), it follows that $\Lambda=0$ on
$\Delta$. Similarly, the boundary tangential diffeomorphisms are fixed on
$S_{\Delta}$.

##  VI Discussion

The purpose of the manuscript is to put forward a possibility of an interplay
between a boundary symplectic structure and a boundary term in the Hamiltonian
in diffeomorphism invariant theories defined in regions with boundary. Here we
perform a canonical analysis of the theory of gravity in the first order
formalism, in self-dual variables, defined in a region with a weakly isolated
horizon as an internal boundary. We show that there are several choices of the
boundary phase space and the corresponding well defined Hamiltonian. In this
particular system we show that each of these choices lead to a consistent
description of boundary dynamics, though in some of them the corresponding
Hamiltonian does not have a boundary term. We have also studied the residual
gauge symmetries on the horizon. The results reported here represent the
extension of the previous studies in two important directions. First, we are
relaxing boundary conditions to WIH, in order to consider more general
horizons. Next, we are exploring the freedom available in defining boundary
phase spaces, that has not been taken previously into account. In that way we
are extending some of the results obtained earlier, for strongly isolated
horizons, that involved a particular choice of a boundary phase space.

One important issue that this results highlighted is that of the freedom
(ambiguity?) involved in the definition of the theory on the boundary. As we
have shown in detail, one can decide to introduce a potential on the boundary
for the degrees of freedom that arise due to the IH boundary conditions. This
is one possibility, but it is not a necessity. One can also take the Regge-
Teitelboim perspective and have a well defined theory without any boundary
contribution in the symplectic structure nor in the variations of phase space
functionals. We analyzed this case in detail (our case IV). We have seen that
we get a consistent description and we do recover the (Smarr expression for
the) mass of the horizon. We have also analyzed the cases in which one
introduces a potential and have arrived at consistent formulations as well,
that are not mutually equivalent. The introduction of potentials is the
prevalent choice within the IH literature. The open question that remains is
whether there is, or one can postulate, some physical principle that selects
one of these possible choices. In the known case of electromagnetism, the
introduction of a potential allows for a consistent action principle, and
opens the door for gauge fields and gauge symmetries. Here we have indeed seen
that (in one case) when a potential is introduced, one gains a $U(1)$ symmetry
on the horizon. Whether this is the preferred choice remains to be seen.

Another important issue that one should comment on pertains to the possible
quantization of the theory. In the original papers, based on spherically
symmetric strongly isolated horizon with a fixed area ABCKprl ; ACKclassical ;
ABKquantum , an important feature of the resulting theory was that there was a
degree of freedom that could be interpreted as a $U(1)$ Chern-Simons
connection. This is the degree of freedom that is normally quantized, and has
been shown to yield the quantum degrees of freedom responsible for black hole
entropy. In ENPprl ; EngleNouiPerezPranzetti  a quantization of the same type
of horizon with a $SU(2)$ Chern-Simons connection was also proposed. In the
present paper that involves, as we have already mentioned, several
generalizations on different aspects, we loose some of these features. For
instance, we only recover the $U(1)$ gauge symmetry in some of the cases (I
and II). However, we loose a direct (and obvious) interpretation of spatial
diffeomorphism as gauge (as was the case in ABCKprl ; ACKclassical ;
ABKquantum ). We have shown in Appendix B that those original results can be
extended to WIH for the non-rotating spherically symmetric case with a fixed
area. Whether the original interpretation of quantum Chern-Simons degrees of
freedom (with diffeo invariance) can be extended to generic WIH seems elusive
and remains, in our opinion, an open issue.

In this manuscript we have restricted our attention to non-rotating horizons
within the self-dual formulation. Generalizations to the rotating case and the
more general Holst action are underway.

Finally, let us comment on another of the open questions pertaining the
relationship between the canonical and covariant approaches, especially
between the corresponding boundary symplectic structures. Our results are a
cautionary tale for the methods used in ENPprl ; EngleNouiPerezPranzetti  to
read from the covariant form the canonical symplectic structure directly used
for quantization. It would be interesting to further explore whether the
freedom that we found in the canonical description is also present in the
covariant one. Investigations in this direction are ongoing.

###### Acknowledgements.

This work was in part supported by DGAPA-UNAM IN114620 and CONACyT 0177840
grants, by CIC, UMSNH, and PRODEP 511-6/17-8204.

##  Appendix A Potentials

As we have seen in the main body of the manuscript, we can introduce the
potentials $\psi_{R}$ and $\psi_{I}$ as degrees of freedom on the WIH, though
they are just ‘parts’ of the components of the self-dual connection, defined
in (53) and (54). Here, we want to further explore in what extension these
definitions determine the potentials. In order to obtain the expansions of
$\omega$ and $V$, we recall that in the comoving gauge one has CRVwih2

$\displaystyle\omega=$ $\displaystyle-\kappa_{(\ell)}\mathbf{k}+\pi
m+\bar{\pi}\bar{m}\,,$ (127) $\displaystyle V=$
$\displaystyle-(\epsilon_{\text{NP}}-\bar{\epsilon}_{\text{NP}})\mathbf{k}+(\alpha-\bar{\beta})m+(\beta-\bar{\alpha})\bar{m}\,,$
(128)

where all forms are understood to be pulled back to $\Delta$ and $\pi$, all
equations are valid on $\Delta$, $\epsilon_{\text{NP}}$, $\alpha$ and $\beta$
are Newman-Penrose spin coefficients. Now, due to (53) and (54) we have:

$\displaystyle\ell\cdot\omega=\ell\cdot{\mathrm{d}}\psi_{R}\ \ \Rightarrow\ \
{\mathrm{d}}\psi_{R}$ $\displaystyle=$
$\displaystyle-\kappa_{(\ell)}\mathbf{k}+c\,m+\bar{c}\,\bar{m}\,,$ (129)
$\displaystyle\ell\cdot V=\ell\cdot{\mathrm{d}}\psi_{I}\ \ \Rightarrow\ \
{\mathrm{d}}\psi_{I}$ $\displaystyle=$
$\displaystyle-(\epsilon_{\text{NP}}-\bar{\epsilon}_{\text{NP}})\mathbf{k}+f\,m-\bar{f}\,\bar{m}\,$
(130)

where $c$ and $f$ are arbitrary functions on $\Delta$.

Comparing previous expansions, we see that we can make a decomposition

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle{\mathrm{d}}\psi_{R}+\tilde{\omega}\,,$ (131) $\displaystyle V$
$\displaystyle=$ $\displaystyle{\mathrm{d}}\psi_{I}+\tilde{V}\,,$ (132)

where

$\displaystyle\tilde{\omega}$ $\displaystyle=$
$\displaystyle(\pi-c)m+(\bar{\pi}-\bar{c})\bar{m}\,,$ (133)
$\displaystyle\tilde{V}$ $\displaystyle=$
$\displaystyle(\alpha-\bar{\beta}-f)m-(\beta-\bar{\alpha}+\bar{f})\bar{m}\,.$
(134)

WIH boundary conditions imply restrictions on $c$ y $f$. As shown, for
example, in ChatterjeeGhosh , on WIH we have

$\displaystyle{\mathrm{d}}\omega=2(\text{Im}\,\mathbf{\Psi_{2}})\,\leftidx{{}^{2}}{\epsilon}\,,$
(135)
$\displaystyle{\mathrm{d}}V=2(\text{Re}\,\mathbf{\Psi_{2}})\,\leftidx{{}^{2}}{\epsilon}\,,$
(136)

where $\mathbf{\Psi_{2}}=C_{abcd}l^{a}m^{b}\bar{m}^{c}k^{d}$ is a complex
scalar, the Newman-Penrose component of the Weyl tensor $C_{abcd}$. For non-
rotating WIHs, $\text{Im}\,\mathbf{\Psi_{2}}=0$, so that
${\mathrm{d}}\omega=0$, and particularly ${\mathrm{d}}\tilde{\omega}=0$ and
${\mathrm{d}}\tilde{V}\neq 0$, leading to conditions for $c$ and $f$. In
contrast to the covariant phase space analysis, $\psi_{R}$ is not completely
determined by $\kappa_{(l)}$, see, for example, ABL . The same is true for
$\psi_{I}$ and $V$.

##  Appendix B Spherically symmetric non-rotating WIH

Here, we shall restrict our study to spherically symmetric WIH, with fixed
area $a_{\Delta}$ on the phase space. We shall consider all of our cases and
see for which one we can obtain the Chern-Simons symplectic structure on the
boundary. This structure has been obtained previously in AFK , for spherically
symmetric non-rotating SIH with a preferred foliation. Their result can be
extended to WIH, as shown in ChatterjeeGhosh . For spherically symmetric non-
rotating WIH, we have that $\text{Im}\,\Psi_{2}\overset{{}_{\Delta}}{=}0$,
that implies ${\mathrm{d}}\omega\overset{{}_{\Delta}}{=}0$, and also
$m\cdot\omega\overset{{}_{\Delta}}{=}0$. One of the main relations that can be
obtained in this case is ChatterjeeGhosh

${\mathrm{d}}V\overset{{}_{\Delta}}{=}-\frac{2\pi}{a_{\Delta}}\,\leftidx{{}^{2}}{\epsilon}\,.$
(137)

Let us analyse what that implies in our four cases. We will also demonstrate
that under these conditions in the first two cases boundary tangential
diffeomorphisms are gauge symmetry of the theory.

In the Case I, the boundary kinetic term is of the form

$S_{\text{SD,1}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\mathcal{L}_{t}\psi\,\leftidx{{}^{2}}{\epsilon}=-\int{\mathrm{d}}t\,\bigl{[}\oint_{S_{\Delta}}\mathcal{L}_{t}\psi_{R}\,\leftidx{{}^{2}}{\epsilon}+\oint_{S_{\Delta}}\mathcal{L}_{t}\psi_{I}\,\leftidx{{}^{2}}{\epsilon}\bigr{]}\,,$
(138)

The imaginary part can be rewritten as

$\oint_{S_{\Delta}}\mathcal{L}_{t}\psi_{I}\,\leftidx{{}^{2}}{\epsilon}=-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}(t\cdot
V)\,{\mathrm{d}}V=\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}{\mathrm{d}}(t\cdot
V)\wedge
V=-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}V\wedge\mathcal{L}_{t}V\,,$ (139)

where we have used that ${\mathrm{d}}(t\cdot
V)=\mathcal{L}_{t}V-t\cdot{\mathrm{d}}V$ and
$t\cdot{\mathrm{d}}V\sim\ell\cdot\ \leftidx{{}^{2}}{\epsilon}=0$. This is the
$U(1)$ Chern-Simons boundary kinetic term in the canonical action.

Thus, the boundary kinetic term is of the form

$S_{\text{SD,1}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\mathcal{L}_{t}\psi_{R}\,\leftidx{{}^{2}}{\epsilon}+\frac{a_{\Delta}}{2\pi}\int{\mathrm{d}}t\oint_{S_{\Delta}}\,V\wedge\mathcal{L}_{t}V\,.$
(140)

The boundary contribution to the symplectic structure is

$\Omega_{1}(\delta_{1},\delta_{2})|_{\Delta}=\oint_{S_{\Delta}}(\delta_{1}\psi_{R}\,\delta_{2}\,\leftidx{{}^{2}}{\epsilon}-\delta_{2}\psi_{R}\,\delta_{1}\,\leftidx{{}^{2}}{\epsilon})-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}\delta_{1}V\wedge\delta_{2}V\,.$
(141)

The first term in this expression vanishes, since due to spherical symmetry of
$\kappa$, $\psi_{R}$ and $\delta\psi_{R}$ are also spherically symmetric, so
that

$\oint_{S_{\Delta}}\delta_{1}\psi_{R}\,\delta_{2}\,\leftidx{{}^{2}}{\epsilon}=\delta_{1}\psi_{R}\,\oint_{S_{\Delta}}\delta_{2}\,\leftidx{{}^{2}}{\epsilon}=\delta_{1}\psi_{R}\,\delta_{2}a_{\Delta}=0\,,$
(142)

since $a_{\Delta}$ is fixed in the phase space. The second term in
$\Omega_{1}$ corresponds to the symplectic structure of the Chern-Simons
theory on $S_{\Delta}$.

Finally, we only have

$\Omega_{1}(\delta_{1},\delta_{2})|_{\Delta}=-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}\delta_{1}V\wedge\delta_{2}V\,.$
(143)

Let us analyze the dynamics on $\Delta$, starting from

$\delta
H_{\text{SD,1}}|_{\Delta}=-\oint_{S_{\Delta}}t\cdot(\omega+V)\delta\,\leftidx{{}^{2}}{\epsilon}=-\oint_{S_{\Delta}}(t\cdot\omega)\,\delta\,\leftidx{{}^{2}}{\epsilon}+\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}\mathcal{L}_{t}V\wedge\delta
V\,.$ (144)

Again, the first term vanishes

$\oint_{S_{\Delta}}(t\cdot\omega)\,\delta\,\leftidx{{}^{2}}{\epsilon}=\kappa_{(t)}\delta
a_{\Delta}=0\,.$ (145)

The boundary components of the corresponding HVF are obtained from

$\delta H_{\text{SD,1}}=\Omega_{1}(\delta,\delta_{t})|_{\Delta}$ (146)

leading to

$\delta_{t}V=\mathcal{L}_{t}V\,.$ (147)

Let us now analyze the tangential diffeomorphisms on $S_{\Delta}$. From (78)
we have

$\delta
D[N^{a}]|_{S_{\Delta}}=\oint_{S_{\Delta}}N\cdot\delta[(\omega+V)\,\leftidx{{}^{2}}{\epsilon}]=\oint_{S_{\Delta}}\delta[(N\cdot
V)\,\leftidx{{}^{2}}{\epsilon}]\,.$ (148)

For spherically symmetric WIH, $\pi=0$, so that
$\omega=-\kappa_{(\ell)}\mathbf{k}$ and $N\cdot\omega\sim m\cdot\omega=0$,
implying $N\cdot\delta\omega=0$. For fixed $a_{\Delta}$, and due to (139), we
further obtain that

$\displaystyle\delta D[N^{a}]|_{S_{\Delta}}$ $\displaystyle=$
$\displaystyle-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}[\delta(N\cdot
V)\,{\mathrm{d}}V+N\cdot V\,\delta({\mathrm{d}}V)]$ (149) $\displaystyle=$
$\displaystyle-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}[(N\cdot\delta
V)\,{\mathrm{d}}V-{\mathrm{d}}(N\cdot V)\wedge\delta V]\,.$

Now, $(N\cdot\delta V)\,{\mathrm{d}}V=N\cdot(\delta
V\wedge{\mathrm{d}}V)+\delta V\wedge N\cdot{\mathrm{d}}V$ and the first term
in this expression vanishes since $\delta V\wedge{\mathrm{d}}V$ is a 3-form on
$S_{\Delta}$, so it follows that

$\delta
D[N^{a}]|_{S_{\Delta}}=-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}\delta
V\wedge[N\cdot{\mathrm{d}}V+{\mathrm{d}}(N\cdot
V)]=-\frac{a_{\Delta}}{2\pi}\oint_{S_{\Delta}}\delta
V\wedge\mathcal{L}_{\vec{N}}V\,.$ (150)

On the other hand,

$\delta D[N^{a}]=\Omega(\delta,\delta_{\vec{N}})|_{S_{\Delta}}\,.$ (151)

Comparing (143) with (150), we see that, in this case, $D[N^{a}]$ generates
tangential diffeomorphisms on $S_{\Delta}$,

$\delta_{\vec{N}}V=\mathcal{L}_{\vec{N}}V|_{S_{\Delta}}\,.$ (152)

In the Case II, for spherically symmetric WIH with fixed $a_{\Delta}$, using
(139), we obtain

$S_{\text{SD,2}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\,(\kappa_{(t)}\,+\mathcal{L}_{t}\psi_{I})\,\leftidx{{}^{2}}{\epsilon}=\int{\mathrm{d}}t\,a_{\Delta}[-\kappa_{(t)}+\frac{1}{2\pi}\oint_{S_{\Delta}}\,V\wedge\mathcal{L}_{t}V]\,.$
(153)

As in the previous case, we obtain a consistent dynamics on the horizon,
$\delta_{t}V=\mathcal{L}_{t}V$. Again, $D[N^{a}]$ generates tangential
diffeomorphisms on $S_{\Delta}$,
$\delta_{\vec{N}}V=\mathcal{L}_{\vec{N}}V|_{S_{\Delta}}$. We see that we
recover the results of ABF .

In the case III we have

$S_{\text{SD,4}}^{\Delta}=-\int{\mathrm{d}}t\oint_{S_{\Delta}}\,(\mathcal{L}_{t}\psi_{R}+t\cdot
V)\,\leftidx{{}^{2}}{\epsilon}\,.$ (154)

As we have seen in the Case I, the first term does not contribute to the
symplectic structure and the second one has to vanish. In both cases there is
no boundary contribution to the symplectic structure.

Finally, in case IV there is no boundary contribution to the symplectic
structure.

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