9

# General gravitational charges on null hypersurfaces

Gloria Odak, Antoine Rignon-Bret and Simone Speziale
Aix Marseille Univ., Univ. de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille,
France

(November 27, 2023)

###### Abstract

We perform a detailed study of the covariance properties of the symplectic
potential of general relativity on a null hypersurface, and of the different
polarizations that can be used to study conservative as well as leaky boundary
conditions. This allows us to identify a one-parameter family of covariant
symplectic potentials. We compute the charges and fluxes for the most general
phase space with arbitrary variations. We study five symmetry groups that
arise when different restrictions on the variations are included. Requiring
stationarity as in the original Wald-Zoupas prescription selects a unique
member of the family of symplectic potentials, the one of Chandrasekaran,
Flanagan and Prabhu. The associated charges are all conserved on non-expanding
horizons, but not on flat spacetime. We show that it is possible to require a
weaker notion of stationarity which selects another symplectic potential,
again in a unique way, and whose charges are conserved on both non-expanding
horizons and flat light-cones. Furthermore, the flux of future-pointing
diffeomorphisms at leading-order around an outgoing flat light-cone is
positive and reproduces a tidal heating plus a memory term. We also study the
conformal conservative boundary conditions suggested by the alternative
polarization and identify under which conditions they define a non-ambiguous
variational principle. Our results have applications for dynamical notions of
entropy, and are useful to clarify the interplay between different boundary
conditions, charge prescriptions, and symmetry groups that can be associated
with a null boundary.

###### Contents

  1. 1 Introduction
  2. 2 Null hypersurfaces
    1. 2.1 Foliations
    2. 2.2 Affine coordinates
  3. 3 Null symplectic potential
    1. 3.1 Phase space polarizations
  4. 4 Conservative boundary conditions and the variational principle
    1. 4.1 Dirichlet boundary conditions and their ambiguity
    2. 4.2 Conformal boundary conditions
  5. 5 Leaky boundary conditions and covariant phase space
    1. 5.1 Anomalies and class-III invariance
    2. 5.2 Anomalies of the boundary Lagrangians
    3. 5.3 Anomalies of the symplectic potentials
    4. 5.4 Boundary symmetry groups
  6. 6 Charges and fluxes
    1. 6.1 Wald-Zoupas conditions on null hypersurfaces
    2. 6.2 Stationarity on flat light-cones
    3. 6.3 Larger phase spaces
    4. 6.4 Charges
  7. 7 Addenda
    1. 7.1 Second-order perturbations around flat light-cones
    2. 7.2 Wald-Zoupas prescription with field-dependent diffeomorphisms
  8. 8 Conclusions
  9. A Internal Lorentz transformations
    1. A.1 Class-I
    2. A.2 Class-III
    3. A.3 Anomalies and NP representatives
  10. B Alternative polarizations
  11. C Closure of Lie brackets
  12. D Derivation of Damour’s equation

##  1 Introduction

Chandrasekaran, Flanagan and Prabhu (CFP) characterized the symmetry group of
general relativity on generic null hypersurfaces as an extension of the BMS
group to include arbitrary diffeomorphisms and Weyl transformations of any 2d
space-like cross-section [1].111The same group that can be obtained at null
infinity relaxing the fall-off conditions in a way compatible with
renormalization of the symplectic potential [2], and was in that context
referred to as BMSW group. See also [3]. They then used the Wald-Zoupas (WZ)
procedure [4] to prescribe charges for this symmetry group and study their
flux-balance laws. The charges they obtained satisfy important properties, and
are conserved on shear-free and expansion-free hypersurfaces, or equivalently
non-expanding horizons (NEHs) in vacuum. The analysis was then specialized to
NEHs and further advanced in [5], explaining for instance how the NEH area is
the charge aspect associated with the global sector of the Weyl
transformations. The same Weyl transformations play also a prominent role in
the investigation of possible black hole soft hairs by Hawking, Perry and
Strominger [6].

The CFP construction is based on the covariant phase space with a specific
choice of symplectic potential, associated with Dirichlet boundary conditions,
and on a specific set of restrictions of the variations, corresponding to a
certain universal structure constructed along the guidelines of what has
successfully been done at future null infinity (see e.g. [7] for a review).
These restrictions clarify in particular the issue of ambiguous null boundary
terms that was raised in [8]. It was then pointed out in [9] that the CFP
charges differ from what one would obtain following a procedure à la Brown-
York by a term generated by an anomalous transformation under diffeomorphisms,
thus bringing to the forefront the extension of the covariant phase space to
include anomalies described in [10, 11] (see also [12, 13]). We show here that
the anomaly term in the charges is crucial to make them covariant, and explain
why this seemingly counterintuitive statement is actually natural.

A shortcoming of the CFP charges is that some of them are not conserved even
in the absence of radiation, specifically those whose aspect is the area,
since this grows on a flat light-cone. This limits their applicability to
study physical processes, for instance a spherical collapse would see a flat
light-cone bend into an event horizon, and this process will be poorly
described by a charge that is already varying prior to any matter infalling.
This raises the question whether a different prescription for the charges
exists that is free of this shortcoming. Indeed, the Noether charges as well
as the Wald-Zoupas charges are not guaranteed to be unique, and may depend on
a choice of polarization made in writing down the symplectic potential [14,
15, 16, 11, 17, 12, 13]. We show that there exists a different choice of
polarization that leads to charges which are conserved on flat light-cones, as
well as on shear-free and expansion-free hypersurfaces. The polarization we
use was previously considered in [18], although with a restriction that
spoiled its covariance. The application of the new charges to study dynamical
processes of BH formation was anticipated in [19], see also [20] on this. It
is the unique one that satisfies the Wald-Zoupas covariance condition and a
stationarity condition interpreted in a weaker sense than the original WZ
paper and that allows one to include both shear and expansion-free surfaces
and flat light-cones. A further useful property of this potential is that the
flux of future-pointing diffeomorphisms at leading-order around an outgoing
flat light-cone is positive, and reproduces the tidal heating term plus a
memory term.

We also show that this polarization leads to conformal boundary conditions on
a null hypersurface that provide an alternative resolution to the boundary
term ambiguities of [8] based on Dirichlet boundary conditions, and discuss
the residual ambiguities that would be present in the corner terms.

One of the restrictions on the variations considered in the CFP paper concerns
the inaffinity of the normal to the null hypersurface. This restriction plays
an important role. Relaxing it reintroduces the ambiguities of [8] and
prevents a complete implementation of the WZ procedure. Nonetheless, the
authors considered the possibility that this variation may be physically
relevant, and the question was left open whether one could construct WZ
charges in this larger phase space. We investigate this issue with a general
analysis of the covariant phase space in which all restrictions on the
variations are removed one by one, until we are left with the minimal
condition that the boundary is preserved. Removing restrictions the symmetry
group gets enlarged from the CFP group to a group which includes super-
translations of arbitrary time-dependence, to the complete hypersurface
diffeomorphisms, to an extension of the hypersurface diffeomorphism with one
additional free function which applies to the largest phase space.

Having identified the symmetry groups, we look at the symplectic potential for
all phase spaces. We perform a systematic calculation of all anomalies and
provide their interpretation. We identify a one-parameter family of symplectic
potentials that satisfy the covariance condition in the larger phase spaces,
family that includes the conformal polarization. However, none of them
satisfies the stationarity condition, neither in the original sense nor in the
weaker sense. As a consequence, the new symmetries produce fluxes which don’t
vanish in stationary solutions such as non-expanding horizons and flat light-
cones. So while it is interesting to notice that this extension of the phase
space is possible, it remains unclear to us how it should be used in physical
applications.

The stationarity condition can instead be satisfied for the whole family if
the variation of the inaffinity is non-zero but fixed to be proportional to
the variation of the expansion. The proportionality parameter labels the
members of the family, and when it vanishes the CFP choice is recovered. In
all other cases one finds symmetry vector fields with a non-trivial metric
dependence, and will be investigated elsewhere.

We complete the analysis providing the general expression for the Noether
charges for arbitrary variations and a two-parameter family of polarization
that include the covariant ones. The covariant ones are the only ones that are
invariant under arbitrary choices such as rescaling of the null normal, change
of embeddings, and change of rigging vector. We explain under which conditions
these charges satisfy the Wald-Zoupas prescription.

Our results strengthen on the one hand the value of the universal structure
defined in [1], and enrich it proposing an alternative symplectic potential
with improved stationary properties. On the other hand they open the
possibility of working with a much weaker universal structure while preserving
the Wald-Zoupas covariance criterium, if the lack of stationarity can be dealt
with.

At the technical level, a difference between our approach and [1] is that we
take a spacetime description as starting point, as opposed to the intrinsic
description based on hypersurface tensors. Our complementary approach can be
useful to provide a new angle on their analysis, and allows one to relate it
to the Newman-Penrose (NP) formalism. This gives us useful tools for the
systematic analysis of anomalies in the phase space. For instance, the
geometry of a null hypersurface depends only on the equivalence class of
normals under rescaling. But many quantities that appear in the phase space
depend also on a specific choice of normal representative, as well as on an
auxiliary rigging vector used to define a local projector on space-like cross-
sections. Identifying the quantities which are independent of these two
auxiliary and non-geometric characteristics is straightforward using internal
transformations of the NP tetrad which have long since been tabulated, and are
for instance denoted class-I and class-III in [21]. One of our technical
results is to point out the direct link between lack of invariance under these
transformations and anomalies in the covariant phase space. Another technical
difference is that we don’t use the notation for the covariant phase space à
la Wald, where field variations are interpreted as tangent vectors, but as in
as differential forms [22, 15, 11]. This makes it easier to distinguish the
Lie derivative in field space from those in spacetime, which is the basic step
to study anomalies. Some of the details we present are therefore mere
translations of the results of [1] in the formalism based on spacetime tensors
and on the exterior calculus notation for the covariant phase space. This
makes for a paper longer than initially intended, but we hope that the
dictionary it provides will be of use to navigate the literature.

As this paper was being completed we learned of a similar analysis by
Venkatesa Chandrasekaran and Eanna Flanagan which has considerable overlap
with our results and which is being submitted simultaneously to the arXiv
[23].

We use mostly-plus spacetime signature. Greek letters are spacetime indices,
and we will sometimes denote scalar products by a dot. When needed, lower case
latin letters $a,b,...$ are hypersurface indices, and upper case latin letters
$A,B,...$ are indices for 2d cross-sections of the hypersurface. In all cases,
$(,)$ denotes symmetrization, $\langle,\rangle$ trace-free symmetrization, and
$[,]$ antisymmetrization. An arrow under a $p$-form means pull-back on ${\cal
N}$, $\,\hat{=}\,$ means on-shell of the field equations, and
$\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,$ means an
equality valid at the null boundary only. We use units $16\pi G=c=1$.

##  2 Null hypersurfaces

In this Section we review basic facts of null hypersurfaces. This is useful to
fix the notation, but it will also allows us to highlight properties that are
often scattered between different literature, and to provide somewhat of a
dictionary. We will in particular review the distinction between intrinsic and
extrinsic geometries, using spacetime covariant notation and offering the
translation to hypersurface indices on the one hand and to Newman-Penrose (NP)
notation on the other hand. We will recall the notion of class-III and class-I
invariance to talk about quantities which are independent respectively of the
choice of normal and of rigging vector. We will then recall Sachs’
identification of constraint-free data and how it allows one a clear
distinction between conservative and radiative or leaky boundary conditions,
and finally some useful expressions that arise when working with the special
coordinate system provided by affine coordinates.

We consider a null hypersurface ${\cal N}$ defined by a cartesian equation
$\Phi=0$, and denote its normal

$l_{\mu}:\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,-f\partial_{\mu}\Phi,\qquad
l^{2}{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,}0,$
(2.1)

with $f>0$ as to have the vector future-pointing. The corresponding vector
$l^{\mu}$ is null and hypersurface orthogonal, hence it is also tangent to
${\cal N}$ and geodetic,

$l^{\mu}\nabla_{\mu}l^{\nu}{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,}kl^{\nu}.$ (2.2)

The hypersurface is thus naturally fibrated by null geodesics, and $k=0$ if
they are affinely parametrized. The chosen tangent vector is referred to as
generator of ${\cal N}$. We will assume that ${\cal N}$ has topology $I\times
S$, where $S=S^{2}$ and $I$ is some interval in $\mathbb{R}$. If
$I=\mathbb{R}$ in affine coordinates the hypersurface is called complete, and
in this case all null geodesics extend indefinitely in both directions. It is
called semi-complete if it extends indefinitely in one direction only, and has
a boundary in the other direction caused for instance by the formation of
caustics or crossings. We will often use adapted coordinates $(\Phi,x^{a})$,
where $x^{a}$, $a=1,2,3$ are coordinates on the leaves of the $\Phi$
foliation. The condition that ${\cal N}$ is null then induces a partial gauge-
fixing of the metric given by

$g^{\Phi\Phi}:=g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,}0.$ (2.3)

For space-like and time-like hypersurfaces, there is a canonical choice of
normal with unit norm. This makes the normal independent of the embedding of
the hypersurface, namely invariant under a change of parametrization
$\Phi\mapsto\Phi^{\prime}=\Phi F(x)$ with $F$ smooth at ${\cal N}$ that
preserve the location of the boundary. No such preferred choice exists in the
null case. The function $f$ is thus arbitrary, and one has to check on a case
by case basis whether a given quantity is independent of the embedding or not.
The geometry of the hypersurface is on the other hand only sensitive to the
equivalence class $[l=Al]$ of normals identified up to an arbitrary rescaling.
This rescaling can be obtained in two independent ways. First, changing the
choice of $f$. Second, changing the embedding, which has the effect of
multiplying $f$ by $F$.

For instance, the inaffinity $k$ appearing in (2.2) is not a geometric
property of a generic null hypersurface, since it depends on $f$ and $\Phi$.
An explicit calculation using (2.1) gives in fact

$k=\pounds_{l}\ln f-\frac{f}{2}\partial_{\Phi}g^{\Phi\Phi},$ (2.4)

written in adapted coordinates. If follows that $k$ depends on the transversal
derivative of the metric, namely it contains information about the extrinsic
geometry. We can rewrite (2.4) in a more covariant form if we parametrize an
arbitrary extension of the normal as $l=-fd\Phi+\Phi\,v$, then

$k=\pounds_{l}\ln f-\frac{1}{2}\pounds_{n}l^{2}+\frac{1}{f}l\cdot v,$ (2.5)

where $n$ is any null vector such that $n\cdot l=-1$. This expression is
slightly misleading because it may give the impression that $k$ depends on
$v$, but this dependence cancels out between the second and third term to give
back (2.4).222Independence of $k$ from the extension of $l$ can be also
checked showing that $l^{\prime}=l+\Phi v$ gives the same $k$ as $l$, and it
means that different $k$ imply different $f$ at fixed spacetime metric. From
this general expression one can also read off the special values when the
extension is null everywhere ($l^{2}=0$, $v\neq 0$), when it is hypersurface-
orthogonal everywhere ($v=0$), or both. A typical example of the first special
case is Kerr’s principal null direction, hypersurface orthogonal only at the
horizon. For the second special case, $l$ is normal to a foliation that has a
single null leaf, and for the third, $l$ is normal to a null foliation.

This discussion is valid for a generic null hypersurface. If it is a Killing
horizon in a spacetime with a global isometry, then the possibility of
selecting as generator the preferred Killing vector of asymptotic unit-norm
allows one to eliminate these arbitrariness and interpret $k$ as surface
gravity. If it is a non-expanding horizon, the arbitrariness in the normal can
be reduced to constant rescaling. Every member of the equivalence class has
$k=0$ (not to be confused with the surface gravity of the NEH, which is the
arbitrary parameter of the Weyl rescaling symmetry vector field). We recall
that the definition of a NEH coincides with a shear-free and expansion-free
null hypersurface in vacuum, but it is slightly more specific in the presence
of matter, where it requires the stronger condition
$R_{\mu\nu}l^{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,\alpha l_{\nu}$ for some function $\alpha$, as opposed to
$R_{\mu\nu}l^{\mu}l^{\nu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$ satisfied by a shear-free and expansion-free null hypersurface.
In this paper we will only deal with vacuum general relativity, hence we will
use NEH as a synonym of a shear and expansion-free hypersurface.

The null vector $n$ introduced in (2.5) is known as the rigging vector, and it
is a convenient tool to work on null hypersurfaces. It allows one the use of
covariant expressions at all times and to avoid hypersurface indices, thus
making the relation to spacetime objects transparent. It also allows one to
use the Newman-Penrose (NP) formalism and the numerous results that have been
derived in that language. To that end, we complete the pair $(l,n)$ to a
doubly-null NP tetrad $(l,n,m,\bar{m})$ on ${\cal N}$.

The downside of the rigging vector approach is its reliance on an arbitrary
choice of auxiliary vector. But it is quite easy to check which quantities are
independent of this choice. The arbitrariness is a 2-parameter family given by

$n\rightarrow n+\bar{a}m+a\bar{m}+|a|^{2}l,\qquad m\rightarrow m+al,\qquad
a\in\mathbb{C}.$ (2.6)

Quantities which are invariant under (2.6) are independent of the choice of
auxiliary rigging vector. For instance, it is easy to check that (2.5) is
invariant. The map (2.6) is an internal Lorentz transformations of the NP
tetrad that corresponds to the two translations of the ISO(2) little group
stabilizing $l$. We will refer to it as a class-I transformation (of the NP
tetrad), following [21]. In this classification, class-II transformations are
the two null translations of the ISO(2) little group stabilizing $n$. They
change $l$ and disalign it from the normal to the hypersurface, and will not
be considered in the rest of the paper. The remaining two internal
transformations are the class-III spin-boost transformations

$(l,n,m,\bar{m})\rightarrow(Al,A^{-1}n,e^{i\varphi}m,e^{-i\varphi}\bar{m}).$
(2.7)

The boost transformation acts as a rescaling of the normal by an arbitrary
real function $A$. Therefore, quantities invariant under this boost are
independent of the choices of $f$ and of the embedding used when writing
(2.1). This is not the case for the inaffinity, which transforms as $k\to
A(k+\pounds_{l}A)$.

An important result of [1] is that the equivalence class

$[l,k]=[Al,A(k+\pounds_{l}A)]$ (2.8)

can be taken to be the universal background structure in the covariant phase
space of metrics with a null hypersurface.333This is referred to as universal
boundary structure. We prefer the adjective background to emphasize that these
are quantities that will not be varied in the phase space. The paper [1] also
introduces a notion of ‘universal intrinsic structure’, based on purely
intrinsic quantities, and which we do not consider here.  That is, any two
metrics with a null hypersurface admit a coordinate system in which they have
the same (2.8). Elements in the universal structure (2.8) must thus be class-
III invariant.

These internal Lorentz transformations are thus practical tools to discern the
quantities that depend solely on the geometry of ${\cal N}$ from those that
depend on additional structures or choices. To reiterate, class-I invariance
means independence from the choice of rigging vector,444 The extended
structure of a null hypersurface plus a specific choice of normal is called
Carollian structure in some literature, and the further extension including a
specific choice of rigging vector a ‘ruled’ or rigged Carollian structure, see
e.g. [24]. and class-III invariance guarantees independence from rescaling the
normal, namely from the choice of $f$ and from reparametrizations
$\Phi\mapsto\Phi^{\prime}=\Phi F(x)$. One theme of this paper will be that
lack of class I and class III invariance translates to anomalies in the
covariant phase space.

If need be to select a specific rigging vector, there are two natural ways to
do so that are common in the literature. The first is to require it to be
parallel-transported along $l$ on ${\cal N}$, see e.g. [25]. This choice is
unique, and fixes the class-I transformation so that the NP spin-coefficient
$\pi$ vanishes.555Requiring $(m,\bar{m})$ to be also parallel-transported will
further fix the spin part of the class-III so that the NP coefficient
$\epsilon$ is real. This is the same letter used below for the volume form,
but being the first a scalar and the second a form no confusion should
hopefully arise. The second way is to require it to be adapted to a given
$2+1$ foliation of ${\cal N}$. This choice is again unique once the foliation
is given. In this case the class-I transformation is fixed setting to zero two
of the three components of the pull-back of $n$, thus making it hypersurface
orthogonal within ${\cal N}$. It follows that the 2d planes spanned by
$(m,\bar{m})$ integrate to the leaves of the foliation.

A volume form on ${\cal N}$ can be defined from the spacetime volume form
$\epsilon$ via

$\epsilon=-l\wedge\epsilon_{\cal N}.$ (2.9)

The conventional minus sign here follows from assuming $l$ outgoing, and would
be plus if incoming. The volume form $\epsilon_{\cal N}$ is class-I invariant
but not class-III invariant because it depends on $f$. This formula defines
actually an equivalence class of volume forms, related by adding any 3-form
containing $l$. A convenient representative of this equivalence class can be
chosen using the rigging vector as

$\epsilon_{\cal N}:=\mathchoice{i_{n}\epsilon\leavevmode\hbox to11.37pt{\vbox
to5.41pt{\pgfpicture\makeatletter\hbox{\hskip 11.06548pt\lower-3.80551pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{ }{}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.60551pt}\pgfsys@lineto{-10.40549pt}{-3.60551pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.40549pt}{-3.60551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{i_{n}\epsilon\leavevmode\hbox
to11.27pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
11.06548pt\lower-3.80551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.60551pt}\pgfsys@lineto{-10.40549pt}{-3.60551pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.40549pt}{-3.60551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{i_{n}\epsilon\leavevmode\hbox
to8.05pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
7.85385pt\lower-3.46107pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.26108pt}\pgfsys@lineto{-7.19386pt}{-3.26108pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-7.19386pt}{-3.26108pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{i_{n}\epsilon\leavevmode\hbox
to6.55pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
6.35301pt\lower-3.46107pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.26108pt}\pgfsys@lineto{-5.69302pt}{-3.26108pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-5.69302pt}{-3.26108pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sqrt{-g}}{f}d^{3}x,$
(2.10)

where the second equality uses adapted coordinates $(\Phi,x^{a})$, and the
arrow under the form means pull-back on ${\cal N}$. We will make this choice
from now on. Written in this way, class-I invariance may not appear as
obvious, but it follows from the pull-back and the fact that $m^{\mu}$ is
tangent to ${\cal N}$.

On ${\cal N}$, we also define the space-like area form

$\epsilon_{S}:=i_{l}\epsilon_{{\cal
N}}=i\mathchoice{m\wedge\bar{m}\leavevmode\hbox to23.07pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 22.869pt\lower-2.59996pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-22.20901pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-22.20901pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{m\wedge\bar{m}\leavevmode\hbox
to23.07pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
22.869pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-22.20901pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-22.20901pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{m\wedge\bar{m}\leavevmode\hbox
to18.43pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
18.23494pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-17.57495pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-17.57495pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{m\wedge\bar{m}\leavevmode\hbox
to15.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
15.14558pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-14.4856pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-14.4856pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\qquad
i_{l}\epsilon_{S}=0,\qquad\qquad\epsilon_{\cal N}=-n\wedge\epsilon_{S}.$
(2.11)

It is class-I invariant and defined independently of any choice of foliation
of ${\cal N}$. It satisfies

$d\epsilon_{S}=\theta\epsilon_{\cal N},$ (2.12)

where $\theta$ is the expansion of $l$, as defined below. Notice that
$\epsilon_{S}$ so defined can contain components along the null direction,
even if $n$ is adapted to a foliation and $(m,\bar{m})$ are integrable.
Choosing affine coordinates eliminates these components. From this equation
one also derives the following useful identity,

$(\pounds_{l}+\theta)X\,\epsilon_{\cal N}=d(X\epsilon_{S}).$ (2.13)

The rigging vector is also handy to introduce a local projector on 2d space-
like planes, given by

$\gamma_{\mu\nu}:=g_{\mu\nu}+2l_{(\mu}n_{\nu)}=2m_{(\mu}\bar{m}_{\nu)}.$
(2.14)

Its pull-back
$\smash{\underset{\raisebox{3.01385pt}[0.0pt][0.0pt]{\footnotesize{$\leftarrow$}}}{\gamma}{}_{\mu\nu}}$,
or $\gamma_{ab}$ in hypersurface indices, coincides with the pull-back of the
spacetime metric. This is the (degenerate) induced metric, whose null
direction is given by $l^{\mu}$ itself. The class-III invariant pair
$(\smash{\underset{\raisebox{3.01385pt}[0.0pt][0.0pt]{\footnotesize{$\leftarrow$}}}{\gamma}{}_{\mu\nu}},l^{\mu}\epsilon_{\cal
N})$ contains six independent quantities, which are the analogue of the
induced geometry in the non-degenerate case.

For the extrinsic geometry, we look at the pull-back of the gradient of the
normal vector. This quantity gives the extrinsic curvature in the case of a
space-like or time-like hypersurface. In the null case,
$W_{\mu}{}^{\nu}:=\nabla_{\underset{\leftarrow}{\mu}}l^{\nu}$ defines a a
purely hypersurface objet, satisfying
$n^{\mu}W_{\mu}{}^{\nu}=0=W_{\mu}{}^{\nu}l_{\nu}$, and
$l^{\mu}W_{\mu}{}^{\nu}=kl^{\nu}$. It is related to the Weingarten map, which
is the reason for the notation $W$. The actual map is given using hypersurface
indices as in [1, 9], but that definition is equivalent to ours in terms of
covariant 4d indices. To see the geometric content of this map, it is
convenient to use the rigging vector and decompose it as follows,

$\displaystyle W_{\mu}{}^{\nu}:=\nabla_{\underset{\leftarrow}{\mu}}l^{\nu}$
$\displaystyle\stackrel{{\scriptstyle{\cal
N}}}{{=}}\omega_{\mathchoice{\mu\leavevmode\hbox to4.62pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.41783pt\lower-3.96106pt\hbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.15248pt\lower-3.55273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}l^{\nu}+\gamma^{\nu}_{\rho}B_{\mathchoice{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.41783pt\lower-3.96106pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.15248pt\lower-3.55273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{}^{\rho}$ (2.15)
$\displaystyle=\big{(}(\bar{\alpha}+\beta)\bar{m}_{\mathchoice{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.41783pt\lower-3.96106pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.15248pt\lower-3.55273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}-\epsilon
n_{\mathchoice{\mu\leavevmode\hbox to4.62pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.41783pt\lower-3.96106pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\big{)}l^{\nu}-(\sigma\bar{m}_{\mathchoice{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.41783pt\lower-3.96106pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.15248pt\lower-3.55273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\rho
m_{\mathchoice{\mu\leavevmode\hbox to4.62pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.41783pt\lower-3.96106pt\hbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})\bar{m}^{\nu}+{\rm
cc}.$

The second line makes reference to the NP formalism (with mostly-plus
signature, we use the conventions of [26]666This formula can be found for
example in references on the NP formalism (e.g. [21]). The NP formalism
assumes an extension of $l$ which is null everywhere, but since the derivative
index is here pulled-back on ${\cal N}$, it is valid for an arbitrary
extension as well.), and the various tensors there appearing are:

$\displaystyle
B_{\mu\nu}:=\gamma_{\mu}^{\rho}\gamma_{\nu}^{\sigma}\nabla_{\rho}l_{\sigma}=\frac{1}{2}\gamma_{\mu}^{\rho}\gamma_{\nu}^{\sigma}\pounds_{l}\gamma_{\rho\sigma}{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,}\sigma_{\mu\nu}+\frac{1}{2}\gamma_{\mu\nu}\theta,$ (2.16a)
$\displaystyle\sigma_{\mu\nu}:=\gamma_{\langle\mu}^{\rho}\gamma_{\nu\rangle}^{\sigma}\nabla_{\rho}l_{\sigma}=-\bar{m}_{\mu}\bar{m}_{\nu}\sigma+cc,\qquad\
\ \theta:=2m^{(\mu}\bar{m}^{\nu)}\nabla_{\mu}l_{\nu}=-2\rho,$ (2.16b)
$\displaystyle\omega_{{\mu}}:=-\eta_{\mu}-kn_{\mu},\qquad\quad\eta_{\mu}:=\gamma_{\mu}^{\rho}n^{\sigma}\nabla_{\rho}l_{\sigma}=-(\alpha+\bar{\beta})m_{\mu}+cc,\qquad
l^{\mu}\omega_{\mu}=k=2\mathrm{Re}(\epsilon).$ (2.16c)

Here $B$ is the deformation tensor, whose antisymmetric part vanishes because
$l$ is hypersurface orthogonal at ${\cal N}$, $\sigma$ is the shear and
$\theta$ the expansion; $\omega$ is the rotational 1-form of isolated and non-
expanding horizons [26, 27], satisfying $\omega\cdot l=k$; $\eta$ is the
connection 1-form on the normal time-like planes spanned by $(l,n)$, whereas
the complementary quantity $\alpha-\bar{\beta}$ is the 2-sphere connection of
the covariant derivative $\eth$ used in NP calculus [28, 29]. $\eta$ is
sometimes called Hajicek 1-form [30], or twist, since it is related to the
non-integrability of the normal planes via

$\gamma_{\mu\nu}[n,l]^{\nu}=\eta_{\mu}-\gamma_{\mu}^{\nu}(l^{\rho}\nabla_{\rho}n_{\nu}-\partial_{\nu}\ln
f).$ (2.17)

The Weingarten map depends on a specific choice of normal and not on the
equivalence class. It is nonetheless useful to describe the geometry of the
null hypersurface. From (2.16a), we see that the shear and expansion are
entirely determined by the induced metric and a choice of $l$, so they are
part of the intrinsic geometry. The dependence on the scaling of $l$ can be
eliminated if we look at the densitized expressions $\sigma\epsilon_{\cal N}$
and $\theta\epsilon_{\cal N}$ which are class-III invariant.

Transversal derivatives of the metric enter the inaffinity $k$ and the twist
$\eta_{\mu}$. These quantities could be taken as the analogue of the extrinsic
geometry, but they are ambiguous since they depend on the choice of $l$
representative and not on the equivalence class. This dependence can be
partially removed if we consider the following shifts,

$\displaystyle\bar{k}:=k-l^{\mu}\partial_{\mu}\ln{f}=-\frac{f}{2}\partial_{\Phi}g^{\Phi\Phi},$
(2.18)
$\displaystyle\bar{\eta}_{\mu}:=\eta_{\mu}+\gamma^{\nu}_{\mu}\partial_{\nu}\ln
f=\gamma_{\mu\nu}([n,l]^{\nu}+l^{\rho}\nabla_{\rho}n^{\nu})=m_{\mu}(\bar{m}_{\nu}[n,l]^{\nu}+\pi)+{\rm
cc},$ (2.19)

where $\pi$ here is one of the NP coefficients. $\bar{\eta}_{\mu}$ and
$\bar{k}\epsilon_{\cal N}$ are invariant under changes of $f$, but not under
changes of embedding $\Phi\to\Phi F(x)$. Therefore they are still not class-
III invariant, but at least satisfy the weaker requirement of being
independent of the choice of normal representative at fixed embedding.777This
is consistent with the statement in [31] that a quantity like
$\bar{k}\epsilon_{\cal N}$ here is class-III boost invariant, because that
paper works with a fixed 2+2 foliation. If we keep the embedding fixed,
$\bar{k}\epsilon_{\cal N}$ is fully unambiguous. However $\bar{\eta}_{\mu}$ is
not, because it inherits from $\eta_{\mu}$ a dependence on the rigging vector,
hence it is still not a genuine measure of the extrinsic geometry of ${\cal
N}$. In fact, even though the Weingarten map is independent of the choice of
rigging vector, the decomposition we used on the right-hand side of (2.15)
introduces a dependence on it: only $\theta$, $k$ and (the scalar contraction)
$\sigma$ are class-I invariant, whereas $\sigma_{\mu\nu}$, $\eta_{\mu}$ and
$\omega_{\mu}$ are not. For convenience, the transformation properties of all
quantities are summarized in Table 1, with the details reported in Appendix A.

The only case in which (the pull-back of) $\bar{\eta}_{\mu}$ is class-I
invariant is on a non-expanding horizon with $k=0$. And in fact it
characterizes the shape of a non-expanding horizon via the Noether charge
construction [5]. To use it as a measure of the extrinsic geometry of a
general ${\cal N}$, one has to fix the class-I gauge freedom. If we do so
taking $n$ parallel transported by $l$ the NP spin coefficient $\pi$ vanishes
and can identify the twist $\bar{\eta}_{\mu}$ (or equivalently $\eta_{\mu}$
with a gradient normal representative) with the non-integrability of the time-
like planes, thanks to (2.19). Below we will however find it more convenient
to fix instead $n$ to be adapted to a foliation of ${\cal N}$, and we will
then show that $\bar{\eta}_{\mu}$ determines the evolving Noether charges
associated with the leaves of that foliation.

quantity | Rigging-vector | Rescaling | Boost | Spin
---|---|---|---|---
| independence | independence | weight | weight
$\sigma_{\mu\nu}$ | ✗ | ✗ | 1 | 0
$\sigma$ | ✓ | ✗ | 1 | 2
$\theta$ | ✓ | ✗ | 1 | 0
$\epsilon_{\cal N}$ | ✓ | ✗ | $-1$ | 0
$k$ | ✓ | ✗ | 1+inhom | 0
$\eta_{\mu}$ | ✗ | ✗ | 0+inhom | 0
$\alpha+\bar{\beta}$ | ✗ | ✗ | 0+inhom | -1
$\gamma_{\mu\nu}[l,n]^{\nu}$ | ✗ | ✓ | 0 | 0
Table 1:  _Behaviour under class-I and class-III transformations. Quantities
that are not invariant under ( 2.7) can be characterized in terms of their
boost and spin weights, respectively $a$ and $b$, defined by $X\to
A^{a}e^{ib\varphi}X$ (up to possible inhomogeneous terms) under (2.7). The
boost weight can also be interpreted as a conformal weight, for instance in
the case of future null infinity where the normal is the gradient of the
conformal rescaling of the metric. _

We conclude with two more remarks about the Weingarten map. First, its trace
is given by

$W:=W_{\mathchoice{\mu\leavevmode\hbox to4.62pt{\vbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to4.62pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.41783pt\lower-3.96106pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.76106pt}\pgfsys@lineto{-3.75784pt}{-3.76106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-3.75784pt}{-3.76106pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to3.35pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.15248pt\lower-3.55273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.35274pt}\pgfsys@lineto{-2.4925pt}{-3.35274pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.4925pt}{-3.35274pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mu\leavevmode\hbox
to2.51pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.30891pt\lower-3.2805pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.0805pt}\pgfsys@lineto{-1.64893pt}{-3.0805pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.64893pt}{-3.0805pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{}^{\mu}=\nabla_{\mu}l^{\mu}+\frac{1}{2}\partial_{n}l^{2}=\theta+k,$
(2.20)

and provides the boundary term for the variational principle with Dirichlet
boundary conditions on a null hypersurface [32, 8, 33, 34], the equivalent of
the Gibbons-Hawking-York term. The discrepancy between the trace of the
Weingarten map and the divergence of the normal may look unfamiliar, but it
would occur also in the time-like case if the normal $\tau$ is not of unit-
norm off the hypersurface:
$K=\nabla_{\mu}\tau^{\mu}+\frac{1}{2}\partial_{\tau}\tau^{2}$, where
$K_{\mu\nu}:=q_{\mu}^{\rho}\nabla_{\rho}\tau_{\nu}$.

Second, an alternative covariant construction of the Weingarten map can be
given in terms of the “half-projector”
$\Pi_{\mu}{}^{\nu}:=\gamma_{\mu}^{\nu}-n_{\mu}l^{\nu}$, defining
$\tilde{W}_{\mu}{}^{\nu}:=\Pi_{\mu}{}^{\rho}\nabla_{\rho}l^{\nu}$. This tensor
is rigging-vector dependent, but not its pull-back on the hypersurface. This
pull-back is the definition used in [9], and coincides with (2.15). The trace
also coincides with (2.20), namely $\tilde{W}_{\mu}{}^{\mu}=W.$

###  2.1 Foliations

The volume form $\epsilon_{\cal N}$ is not class-III invariant, and depends on
the full spacetime metric determinant $\sqrt{-g}$. On non-degenerate
hypersurfaces choosing a unit-norm normal makes the volume form depend only on
the determinant of the induced metric. The unit-norm option does not exist in
the null case, but one can achieve a similar result introducing a $2+1$
foliation of ${\cal N}$. The foliation can be arbitrary, provided that its
leaves are space-like. We take it to be defined by the level sets of some
scalar function $\lambda$, and denote $x^{a}=(\lambda,x^{A})$ the coordinates
adapted to it.

Note that if we take this choice together with the foliation defined by
$\Phi$, we obtain spacetime a coordinate system $(\Phi,\lambda,x^{A})$ adapted
to a 2+2 foliation of spacetime (see e.g. [35]). Our choice of letters for
these coordinates is meant to preserve generality of the formalism with
respect to common applications. For example, to make the link the
Schwarzschild metric in retarded Bondi coordinates we would take
$(\lambda,\Phi)=(u,r-2M)$ and ${\cal N}$ is the white hole horizon, or using
advanced time instead $(\lambda,\Phi)=(v,2M-r)$ and ${\cal N}$ is the black
hole horizon. We can also keep assuming $l^{\mu}$ future pointing namely
$g^{\Phi\lambda}<0$ without loss of generality. In the first case this leads
to $g^{ur}<0$, in the second case to $g^{vr}>0$. Or if ${\cal N}$ is a null
cone in Minkowski in a doubly-null foliation, we can identify $\Phi=u:=t-r$
and $\lambda=v:=t+r$. Since $\lambda$ is a (null) time, we will refer to
$\partial_{\lambda}$ as a time derivative, and use a dot to indicate it.

In these coordinates,

$\sqrt{-g}=-\frac{1}{g^{\Phi\lambda}}\sqrt{\gamma},$ (2.21)

where $\gamma$ is the determinant of the space-like metric $\gamma_{AB}$ on
the 2d leaves. Hence,

$\epsilon_{\cal N}=\frac{\sqrt{\gamma}}{l^{\lambda}}d\lambda d^{2}x.$ (2.22)

We see that it is a completely intrinsic quantity, but it is still not class-
III invariant and contains more information than the 2d area form $\gamma$: it
depends also on the extent of $l$ via $l^{\lambda}$. If we now choose
$f=-1/g^{\Phi\lambda}$, we obtain $l^{\lambda}=1$ and

$\epsilon_{\cal N}=\sqrt{\gamma}d\lambda d^{2}x.$ (2.23)

Notice that $\lambda$ does not need to be a parameter along the null
geodesics. In general after making these choices,

$l^{a}=(1,-b^{A}),\qquad\mathchoice{g_{ab}\leavevmode\hbox to10.89pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 10.6919pt\lower-4.5444pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-4.3444pt}\pgfsys@lineto{-10.0319pt}{-4.3444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.0319pt}{-4.3444pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to10.89pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-4.3444pt}\pgfsys@lineto{-10.0319pt}{-4.3444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.0319pt}{-4.3444pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to7.82pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
7.62094pt\lower-3.98885pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.78885pt}\pgfsys@lineto{-6.96095pt}{-3.78885pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-6.96095pt}{-3.78885pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to6.8pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
6.59525pt\lower-3.98885pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.78885pt}\pgfsys@lineto{-5.93526pt}{-3.78885pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-5.93526pt}{-3.78885pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\left(\begin{array}[]{cc}{\gamma_{AB}b^{A}b^{B}}&{\gamma_{AB}b^{B}}\\\
{}\hfil&{\gamma_{AB}}\end{array}\right),$ (2.24)

and the vector $b^{A}$ acts as a shift vector for the 2+1 foliation defined by
$\lambda$. If we partially fix the coordinate gauge requiring that $x^{A}$ are
conserved along the generators, then we are setting the shift vector to zero,
and $l^{a}=(1,0,0)$. In terms of the spacetime metric, this partial gauge-
fixing reads $g^{\Phi A}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$. We refer to it as partial Bondi gauge, as in [36, 37]. The
foliation-dependent choice $f=-1/g^{\Phi\lambda}$ was referred to as
‘canonical’ normalization in [34], for its analogy with the ADM space-like
case, since $1/g^{\Phi\lambda}$ plays the role of lapse in the $3+1$
decomposition with null slices [38].

The simplification (2.23) gives to the volume form a similar structure to the
one of non-degenerate hypersurfaces (albeit in terms of a codimension-2
determinant), and it is often used in the literature, e.g. [39]. It is valid
only in the foliation chosen, but in the partial Bondi gauge it remains valid
for any new foliation obtained by a super-translation
$\lambda^{\prime}=\lambda+T(x^{A})$. We will however not do this choice in the
following, neither for $f$ nor for the coordinates, and keep fully general $f$
and $\epsilon_{\cal N}$.

A common choice of $2+1$ foliation is the one induced by the intersections of
${\cal N}$ with a space-like foliation. In this case the cross-sections of
${\cal N}$ provide the boundary $\partial\Sigma$ of each 3d space-like leaf
$\Sigma$. Let us denote by $\tau$ the unit-norm normal to the space-like
foliation, and parametrize the scalar product as follows,

$l\cdot\tau{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,}-\frac{1}{\sqrt{2}}e^{-{\hat{\beta}}}.$ (2.25)

The overall minus sign is due to the fact that both vectors are future
pointing. The quantity ${\hat{\beta}}$ has no geometric meaning per se, since
it is not class-III invariant. It can be used to measure the change of
geometric tilt between ${\cal N}$ and $\Sigma$ only if $l$ is kept fixed. The
unit-norm normal to the cross-section within $T\Sigma$ is

$\hat{r}^{\mu}{\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,}\pm\sqrt{2}e^{\hat{\beta}}q^{\mu}_{\nu}l^{\nu},\qquad
q_{\mu\nu}:=g_{\mu\nu}+\tau_{\mu}\tau_{\nu},$ (2.26)

where the sign is plus if ${\cal N}$ is the outgoing null hypersurface from
the boundary of $\Sigma$, and minus if it is the incoming one. It can be used
to define a rigging vector adapted to $\partial\Sigma$, which is given by

$n=\frac{1}{\sqrt{2}}(e^{\hat{\beta}}\tau\mp e^{-{\hat{\beta}}}\hat{r}).$
(2.27)

Now $(l,n)$ and $(\tau,\hat{r})$ provide two possible basis for the time-like
plane normal to $\partial\Sigma$. This change of basis is used to determine
the corner terms required in the action by the variational principle.

###  2.2 Affine coordinates

The fact that a null hypersurface is ruled by null geodesics endows it with a
preferred class of foliations, in which $\lambda$ is a parameter along the
geodesics, $l^{\mu}\partial_{\mu}\lambda=1$. To use this parameter as one of
the coordinates, we fix an initial cross-section of ${\cal N}$, say at
$\lambda=0$, define angular coordinates $x^{A}$ there and then Lie-drag them
along ${\cal N}$. This defines a coordinate system with vanishing shift
vector,

$l^{a}=(1,0,0),\qquad\mathchoice{g_{ab}\leavevmode\hbox to10.89pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-4.3444pt}\pgfsys@lineto{-10.0319pt}{-4.3444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.0319pt}{-4.3444pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to10.89pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-4.3444pt}\pgfsys@lineto{-10.0319pt}{-4.3444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-10.0319pt}{-4.3444pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to7.82pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
7.62094pt\lower-3.98885pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.78885pt}\pgfsys@lineto{-6.96095pt}{-3.78885pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-6.96095pt}{-3.78885pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{g_{ab}\leavevmode\hbox
to6.8pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
6.59525pt\lower-3.98885pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.78885pt}\pgfsys@lineto{-5.93526pt}{-3.78885pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-5.93526pt}{-3.78885pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\left(\begin{array}[]{cc}{0}&{0}\\\
{}\hfil&{\gamma_{AB}}\end{array}\right).$ (2.28)

These coordinates satisfy the partial Bondi gauge. We have
$l_{\mu}=(g_{\lambda\Phi},0,0,0)$ and $g_{\Phi\lambda}=1/g^{\Phi\lambda}$,
therefore this choice of tangent vector corresponds to the ‘canonical
normalization’ for $f$. This is an example of a situation in which $f$ is
metric-dependent. We can complete this partial gauge fixing on ${\cal N}$ with
a fourth condition, for instance redefining $\Phi$ so that
$g_{\lambda\Phi}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,-1$. The metric now satisfies

$g_{\lambda\lambda}=O(\Phi),\qquad g_{\Phi\lambda}=-1+O(\Phi),\qquad
g_{\lambda A}=O(\Phi),$ (2.29)

and it is fully gauge-fixed on ${\cal N}$.

The coordinate system can be further specialized if we require the parameter
to be affine, namely

$l_{o}:=\frac{d}{d\lambda},\qquad
l_{o}^{\mu}\nabla_{\mu}l_{o}^{\nu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0.$ (2.30)

This condition fixes the first-order extension of the metric component
$g_{\lambda\lambda}$ so that
$\partial_{\Phi}g_{\lambda\lambda}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,2\partial_{\lambda}g_{\Phi\lambda}$.888 This follows from
$\Gamma^{\mu}_{\lambda\lambda}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$, which by invertibility of the metric is equivalent to
$2\partial_{\lambda}g_{\mu\lambda}-\partial_{\mu}g_{\lambda\lambda}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$. This is identically satisfied by (2.28) for $\mu=(\lambda,A)$,
and thus reduces to the single equation given in the text.  Since one can
always choose the adapted coordinate $\Phi$ such that
$g_{\Phi\lambda}=-1+O(\Phi)$, in that gauge we have
$g_{\lambda\lambda}=g^{\Phi\Phi}=O(\Phi^{2})$. 999Notice that this would be a
‘generalized’ diffeomorphism, not invertible at the hypersurface, pretty much
like going from static Schwarzschild coordinates to Eddington-Finkelstein is
singular at the horizon. At this point,

$g_{\lambda\lambda}=O(\Phi^{2}),\qquad g_{\Phi\lambda}=-1+O(\Phi),\qquad
g_{\lambda A}=O(\Phi),$ (2.31)

and the rest of the metric is arbitrary. The condition of affinity can always
be imposed via gauge-fixing, but we see that it is not a characteristic of the
hypersurface coordinates alone, since involves the first-order extension of
the metric.

In the affine coordinate system, any normal vector in the equivalence class
satisfies

$l^{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,fl_{o}^{\mu}$ (2.32)

and

$k=l^{\mu}\partial_{\mu}\ln{f}.$ (2.33)

Hence it is affine iff $f$ is chosen constant in $\lambda$, namely
$\pounds_{l}f=0$. Furthermore since
$\partial_{\Phi}g^{\Phi\Phi}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$, any extension of $l$ with $v\cdot
l\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,0$ satisfies
$\partial_{n}l^{2}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0,$ namely it is null at first-order off the hypersurface. We also
recall that in affine coordinates $\bar{k}=0$.

This coordinate system can be extended to a neighbourhood of ${\cal N}$ as
follows (see e.g. [5]). We shoot geodesics off ${\cal N}$, and Lie drag
$x^{a}$ along them. Namely, we have

$n_{o}^{\mu}=\frac{\partial}{\partial\Phi},\qquad
n_{o}^{\mu}\nabla_{\mu}n_{o}^{\nu}=k_{n_{o}}n_{o}^{\nu},\qquad\pounds_{n_{o}}x^{a}=0.$
(2.34)

We can then completely fix the bulk coordinate gauge freedom if we require
that $(i)$ $n_{o}$ is null everywhere, $(ii)$ $\Phi$ is affine (hence
$k_{n_{o}}=0$), and $(iii)$ it is the gradient of the foliation of constant
$\lambda$ on ${\cal N}$, namely
$n_{o}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,-d\lambda$. The last condition in particular means that $n_{o}$
gives a choice of rigging vector for $l_{o}$ adapted to the $\lambda$
foliation. In terms of metric components, $(i)$ fixes $g_{\Phi\Phi}=0$, then
$(ii)$ requires $\Gamma^{\mu}_{\Phi\Phi}=0$, which in turns implies
$\partial_{\Phi}g_{\Phi\mu}=0$. Finally, $(iii)$ fixes
$g_{\Phi\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,(-1,0,0,0)$. The resulting coordinates $(\lambda,\Phi,x^{A})$ are
defined in a caustic-free open neighbourhood of ${\cal N}$, in which the
metric reads

$g_{\mu\nu}=\left(\begin{array}[]{ccc}\Phi^{2}F&-1&\Phi P_{A}\\\ &0&0\\\
&&\gamma_{AB}\end{array}\right),\qquad
g^{\mu\nu}=\left(\begin{array}[]{ccc}0&-1&0\\\ &-\Phi^{2}(F-P^{2})&\Phi
P^{A}\\\ &&\gamma^{AB}\end{array}\right),$ (2.35)

where $F,P_{A}$ and $\gamma_{AB}$ are arbitrary metric coefficients. With this
gauge fixing, $\bar{\eta}_{A}=\Gamma^{\Phi}_{A\Phi}=-P_{A}/{2}$ and
$\bar{k}=0$. The latter makes it manifest that the extrinsic geometry captured
by the inaffinity, or more precisely by $\bar{k}$, describes whether the
hypersurface is described in affine coordinates or not, thus being faithful to
its name. We stress that what makes $\lambda$ an affine parameter on ${\cal
N}$ is not so much $g_{\lambda\Phi}=-1$ but $g_{\lambda\lambda}=O(\Phi^{2})$.
This coordinate system can always be reached, and if one restricts the
residual diffeomorphisms to preserve it, the whole extension of $\xi$ in the
neighbourhood is fixed. On the other hand, if one relaxes it and requires only
the minimal conditions (2.31), only the first order extension
$\hat{\xi}^{\Phi}$ is fixed, whereas $\hat{\xi}^{\lambda}$ and $\hat{\xi}^{A}$
remain arbitrary.

We can now choose an extension of $l$ such that

$l^{\mu}=fl_{o}^{\mu}\,$ (2.36)

everywhere in the chart, and not only at ${\cal N}$. This is achieved taking
$v=\Phi Fd\lambda+P_{A}dx^{A}$. This extension is not hypersurface-orthogonal
nor null nor geodesic, except at ${\cal N}$. But it satisfies $l\cdot
v\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,0$ and it is
thus null at first order around ${\cal N}$.

Summarizing, affine coordinates on ${\cal N}$ depend on extrinsic properties
of the metric, and give us (2.31) and (2.33). The normal in these coordinates
reads (2.32), is in general not null at first order, but this can be easily
achieved choosing for instance the extension (2.36).101010Another convenient
extension is $dl\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$, which implies instead $v=-df+\Phi v^{\prime}$, namely
$l=-d(f\Phi)+\Phi^{2}v^{\prime}$ is a gradient on ${\cal N}$, and $v\cdot
l\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,-\partial_{l}f$. Then choosing $\Phi^{\prime}=f\Phi$ or more in
general $f$ time independent is also enough to have $v\cdot
l\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,0$ hence
$\partial_{n}l^{2}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$. So taking
$dl\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0=\pounds_{l}f$ is another solution than (2.36) to have
$\partial_{n}l^{2}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0$.

##  3 Null symplectic potential

We start from the standard Einstein-Hilbert symplectic potential

$\theta^{\scriptscriptstyle\rm EH}=\frac{1}{3!}\theta^{\scriptscriptstyle\rm
EH}{}^{\mu}\epsilon_{\mu\nu\rho\sigma}~{}dx^{\nu}\wedge dx^{\rho}\wedge
dx^{\sigma},\qquad\theta^{\scriptscriptstyle\rm
EH}{}^{\mu}=2g^{\rho[\sigma}\delta\Gamma^{\mu]}_{\rho\sigma},$ (3.1)

and consider the most general expression for its pull-back on a null
hypersurface. This was computed for instance in [34],111111With $\omega$ here
defined with opposite sign, as to match [26, 27]. We took the time to
accurately translate notations to prove that it is indeed equivalent to the
one computed in [32], including the corner term, thus answering the question
left open in [34]. In doing so we realized that the statement in [34] that
taking an extension with $\partial_{n}l^{2}\neq 0$ produces an extra term
$\partial_{n}l^{2}n^{\mu}\delta l_{\mu}$ in $\theta^{\scriptscriptstyle\rm
EH}$ was incorrect. We thank Laurent Freidel for pointing this out to us. The
final expression generalizes the one of [8] which assumes $\delta l^{\mu}=0$,
and the one of [31] which assumes $\delta l_{\mu}=-n^{\rho}l^{\sigma}\delta
g_{\rho\sigma}l_{\mu}$, the latter implying that $n_{\mu}\delta l^{\mu}=0$. It
also generalizes the one of [10] – contrarily to what there stated –, which
assumes $\delta l_{\mu}=0$, see (B.2). We will come back to these restrictions
and their motivations below. We hope that no confusion arises because the same
letter $\theta$ appears as both the symplectic potential integrand and the
expansion of $l$. The risk should be reduced by the fact that the letter used
for the symplectic potential always comes with labels such as EH or ′.  and
reads

$\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.89444pt\lower-2.59996pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}=\big{[}(\sigma^{\mu\nu}+\frac{\theta}{2}\gamma^{\mu\nu})\delta\gamma_{\mu\nu}-2\omega_{\mu}\delta
l^{\mu}+2\delta\left(\theta+k\right)\big{]}\epsilon_{\cal
N}+d\vartheta^{\scriptscriptstyle\rm EH}.$ (3.2)

This expression holds for arbitrary variations on the null-hypersurface: the
only restriction made is to preserve the null nature of the hypersurface,
namely

$l_{\mu}\delta l^{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0.$ (3.3)

In particular, it is valid for a field-dependent $f$, so that

$\delta l_{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,\delta\ln f\,l_{\mu}$ (3.4)

doesn’t need to vanish. If $f$ is field-independent, $\delta
l_{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,0$ or
equivalently $n^{\mu}\delta
l_{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal N}}}}{{=}}\,0$.

The expression (3.2) does not depend on rescalings of $l$ nor on the choice of
auxiliary vector $n$. Independence from rescalings follows from its invariance
under the class-III spin-boost transformations (2.7), and implies independence
from changes of embeddings $\Phi\to\Phi^{\prime}(\Phi)$. Independence from $n$
follows from the invariance under class-I Lorentz transformations (2.6). See
App. A for proofs. The invariance is only a property of the full expression.
The individual quantities are not, as summarized in Table 1. This is relevant
for various considerations that we do below.

The variation of the inaffinity can be written in terms of $\delta l_{\mu}$
and $\delta l^{\mu}$:

$\displaystyle\delta k$ $\displaystyle=kn^{\mu}\delta
l_{\mu}-2n^{\mu}\nabla_{(\mu}l_{\nu)}\delta
l^{\nu}-2n^{\mu}l^{\nu}\nabla_{(\mu}\delta
l_{\nu)}+\frac{1}{2}n^{\mu}\nabla_{\mu}\delta l^{2}$
$\displaystyle=(kn^{\mu}+\frac{1}{2}n^{\nu}\nabla_{\nu}l^{\mu}-\frac{1}{2}l^{\mu}n^{\nu}\nabla_{\nu}-n^{\mu}l^{\nu}\nabla_{\nu})\delta
l_{\mu}-(n^{\nu}\nabla_{\mu}l_{\nu}+\frac{1}{2}n^{\nu}\nabla_{\nu}l_{\mu}-\frac{1}{2}l_{\mu}n^{\nu}\nabla_{\nu})\delta
l^{\mu}.$ (3.5)

Notice that presence of normal derivatives on the variations, which imply that
$\delta k$ varies even if we fix both $l_{\mu}$ and $l^{\mu}$ on the
hypersurface:

$\delta l^{\mu}=\delta l_{\mu}\stackrel{{\scriptstyle{{\cal
N}}}}{{=}}0\qquad\Rightarrow\qquad\delta
k=-\frac{1}{2}n^{\mu}\nabla_{\mu}(l_{\nu}l_{\rho}\delta g^{\nu\rho}).$ (3.6)

This vanishes if we restrict the variations to preserve affine coordinates.
Therefore $\delta k$ is an independent variation because it captures the
possibility of varying the metric between the form (2.29) and (2.31), which
are both consistent with fixing $l_{\mu}$ and $l^{\mu}$ on the hypersurface.

Finally, the corner term is [31, 34]

$\vartheta^{\scriptscriptstyle\rm EH}:=n^{\mu}\delta
l_{\mu}\epsilon_{S}-i_{\delta l}\epsilon_{\cal N}=\left(n^{\mu}\delta
l_{\mu}+n_{\mu}\delta l^{\mu}\right)\epsilon_{S}-n\wedge i_{\delta
l}\epsilon_{S}.$ (3.7)

The last term vanishes if we pull-back on a space-like cross section with $n$
adapted to it.

###  3.1 Phase space polarizations

We would like to manipulate the RHS of (3.2) so to put it in the form

$\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.89444pt\lower-2.59996pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\theta^{\prime}-\delta\ell+d\vartheta,$
(3.8)

where $\theta={\theta}^{\scriptscriptstyle\rm EH}$, and
$\theta^{\prime}=p\delta q$ for a given choice of polarization of the
(kinematical) phase space on the boundary.121212This notion of polarization
should be used only at the level of the kinematical phase space. In fact, on a
null-hypersurface the momenta do not depend on velocities, hence there are
second class constraints and the actual symplectic structure is given by a
Dirac bracket and not the initial, kinematical Poisson bracket, see e.g. [40,
38]. This form will be useful to discuss two different but related contexts:
the variational principle, and the definition of charges using covariant phase
space methods.

If we attempt to interpret (3.2) as a symplectic potential in $p\delta q$
form, we see that the $q$’s appearing are not independent, since
$\gamma_{\mu\nu}$ also determines $\theta$ through (2.16a). To put it in
diagonal form, we need two steps. The first is to observe that the variation
of the volume form has two components,

$\delta\epsilon_{\cal
N}=\left(\frac{1}{2}\gamma^{\mu\nu}\delta\gamma_{\mu\nu}+n_{\mu}\delta
l^{\mu}\right)\epsilon_{\cal N},$ (3.9)

as can be established starting from the identity

$\frac{1}{2}g^{\mu\nu}\delta
g_{\mu\nu}=\frac{1}{2}\gamma^{\mu\nu}\delta\gamma_{\mu\nu}+n_{\mu}\delta
l^{\mu}-n^{\mu}\delta l_{\mu}.$ (3.10)

Using (3.9), the pull-back (3.2) can be rewritten in the form

$\displaystyle\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.89444pt\lower-2.59996pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}=\big{[}\sigma^{\mu\nu}\delta\gamma_{\mu\nu}+\pi_{\mu}\delta
l^{\mu}+2\delta(\theta+k)\big{]}\epsilon_{\cal N}+\theta\delta\epsilon_{\cal
N}+d\vartheta^{\scriptscriptstyle\rm EH},$ (3.11)

where

$\pi_{\mu}:=-2\left(\omega_{\mu}+\frac{\theta}{2}n_{\mu}\right)=2\left(\eta_{\mu}+\left(k-\frac{\theta}{2}\right)n_{\mu}\right).$
(3.12)

The second step is to integrate by parts in field space the third term. This
leads to

$\displaystyle\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 4.89444pt\lower-2.59996pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}=\theta^{\scriptscriptstyle\rm D}-\delta\ell^{\scriptscriptstyle\rm
D}+d\vartheta^{\scriptscriptstyle\rm EH},$ (3.13)

where

$\theta^{\scriptscriptstyle\rm
D}:=\sigma^{\mu\nu}\delta\gamma_{\mu\nu}\epsilon_{\cal N}+\pi_{\mu}\delta
l^{\mu}\epsilon_{\cal N}-(\theta+2k)\delta\epsilon_{\cal N},$ (3.14)

and

$\ell^{\scriptscriptstyle\rm D}:=-2W\epsilon_{\cal
N}=-2(\theta+k)\epsilon_{\cal N}=-2k\epsilon_{\cal N}-2d\epsilon_{S}.$ (3.15)

The last equality follows from (2.12), and can be used to simplify the
boundary term reabsorbing its dependence on the expansion in the corner term,
and work with

$\ell^{\scriptscriptstyle\rm D}{}^{\prime}=-2k\epsilon_{\cal
N},\qquad\vartheta^{\scriptscriptstyle\rm
EH}{}^{\prime}=\vartheta^{\scriptscriptstyle\rm EH}+2\delta\epsilon_{S}.$
(3.16)

We can now identify $\theta^{\prime}=\theta^{\scriptscriptstyle\rm D}$, which
is diagonal form $p\delta q$ with
$\smash{q=(\smash{\underset{\raisebox{3.01385pt}[0.0pt][0.0pt]{\footnotesize{$\leftarrow$}}}{\gamma}{}_{\mu\nu}},l^{\mu})}$.
The $q$ terms only involve the intrinsic geometry, therefore the symplectic
potential is in the form of a Dirichlet polarization, whence the D label. The
diagonalization obtained involves the sum of three pairs of configuration
variables and momenta that can be characterized as spin 2, 1 and 0, as
discussed for instance in [31]. Denoting $\varepsilon:={\sqrt{-g}}/f$, we can
write the conjugate momenta as densities,

$\tilde{\pi}^{\mu\nu}:=\varepsilon\sigma^{\mu\nu},\qquad\tilde{\pi}_{\mu}:=2\varepsilon\left(\eta_{\mu}+\left(k-\frac{\theta}{2}\right)n_{\mu}\right),\qquad\tilde{\pi}:=-{\varepsilon}\left(\theta+2k\right).$
(3.17)

The first term in (3.14) can be equally written as

$\sigma^{\mu\nu}\delta\gamma_{\mu\nu}\,\epsilon_{\cal
N}=-\sigma_{\mu\nu}\delta(\gamma^{\mu\nu}\,\epsilon_{\cal
N})=-\gamma_{\mu\nu}\delta\sigma^{\mu\nu}\epsilon_{\cal N},$ (3.18)

where $\gamma^{\mu\nu}\,\epsilon_{\cal N}$ is manifestly conformal invariant.
In other words, only the trace-less part of the metric perturbations enters
here. This is the spin-2 pair.

The spin-1 pair has three components, two ‘transverse’ ones whose momentum is
the twist, and a ‘longitudinal’ one proportional to $n_{\mu}\delta l^{\mu}$.
The longitudinal variation is there to compensate the dependence of
$\gamma^{\mu\nu}\delta\gamma_{\mu\nu}$ on the choice of rigging vector. This
dependence prevents the interpretation of
$\gamma^{\mu\nu}\delta\gamma_{\mu\nu}$ as the variation of the volume form,
see (3.9), and can be removed if we restrict the variations to satisfy

$n_{\mu}\delta l^{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,0\qquad\Rightarrow\qquad\frac{1}{2}\gamma^{\mu\nu}\delta\gamma_{\mu\nu}\epsilon_{\cal
N}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,\delta\epsilon_{\cal N}.$ (3.19)

This restriction can be achieved in two ways. The first is to choose the
metric-dependence of $f$ such that $\delta
l_{\mu}\,\stackrel{{\scriptstyle{\scriptscriptstyle\rm{\cal
N}}}}{{=}}\,-n^{\rho}l^{\sigma}\delta g_{\rho\sigma}l_{\mu}$ [31], from which
(3.19) follows. The second is to choose first a foliation, and then the
‘canonical’ normalization for $f$ [34].

The split into spin pairs is appealing, but it is not canonical: the spin-2
pair is class-III invariant but not class-I invariant, and the spin-1 and
spin-0 pairs mix up under class I and class III transformations, as can be
easily seen using the formulas given in App. A. In other words, the spin pairs
are not singled out by the geometry of the hypersurface, but require
additional non-geometric choices of normal representative and of
rigging.131313They are thus singled out by a rigged Carollian structure, see
previous footnote 4.

There is however a more serious problem with $\theta^{\scriptscriptstyle\rm
D}$: it is not class-III invariant even as a whole. This can be checked
explicitly, but it can be more easily read off the fact that the standard
symplectic potential is class-III invariant, whereas the boundary Lagrangian
and corner potential are not:

$\displaystyle\delta\ell^{\scriptscriptstyle\rm
D}\to\delta\ell^{\scriptscriptstyle\rm D}-2\delta(\pounds_{l}\ln
A\,\epsilon_{\cal N}),\qquad\vartheta^{\scriptscriptstyle\rm
EH}\to\vartheta^{\scriptscriptstyle\rm EH}-2\delta\ln A\,\epsilon_{S}.$ (3.20)

Adding up and using (2.13), we find

$\theta^{\scriptscriptstyle\rm D}\to\theta^{\scriptscriptstyle\rm
D}-2(\pounds_{\delta l}\ln A-\theta\delta\ln A)\epsilon_{\cal
N}-2\pounds_{l}\ln A\,\delta\epsilon_{\cal N}.$ (3.21)

We will see below how this issue affects both the variational principle and
the covariant phase space.

Notice that insofar as the rescaling is field-independent, $\delta A=0$ and
the non-invariance of $\theta^{\scriptscriptstyle\rm D}$ stems from the
boundary Lagrangian alone. Let us first restrict attention to this case. One
way to obtain class-III invariance is to restrict the variations so that
$\delta l^{\mu}$ and $\delta\epsilon_{{\cal N}}$ vanish. The first is possible
without loss of generality by a restriction on the coordinates, whereas the
second is a strong restriction on the dynamics, and we are interested in
avoiding it.

One way to do so is to impose $\delta k=0$, as was done in the CFP paper [1].
If we do so, then

$\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}|_{\delta k=0}=\big{[}\sigma^{\mu\nu}\delta\gamma_{\mu\nu}+\pi_{\mu}\delta
l^{\mu}+2\delta\theta\big{]}\epsilon_{\cal N}+\theta\delta\epsilon_{\cal
N}+d\vartheta^{\scriptscriptstyle\rm EH}=\theta^{\scriptscriptstyle\rm
CFP}-\delta\ell^{\scriptscriptstyle\rm CFP}+d\vartheta^{\scriptscriptstyle\rm
EH},$ (3.22)

where

$\theta^{\scriptscriptstyle\rm
CFP}:=\big{(}\sigma^{\mu\nu}\delta\gamma_{\mu\nu}+\pi_{\mu}\delta
l^{\mu}\big{)}\epsilon_{\cal N}-\theta\delta\epsilon_{\cal
N},\qquad\ell^{\scriptscriptstyle\rm CFP}:=-2\theta\epsilon_{\cal N}.$ (3.23)

This choice makes the boundary Lagrangian and the symplectic potential class-
III invariant.

If we impose instead $\delta l_{\mu}=0$, we can replace $k$ with $\bar{k}$ in
(3.11) and obtain

$\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
4.89444pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-4.23445pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.4861pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}|_{\delta
l_{\mu}=0}=\theta^{\scriptscriptstyle\rm\bar{D}}-\delta\ell^{\scriptscriptstyle\rm\bar{D}}+d\vartheta^{\scriptscriptstyle\rm
EH},$ (3.24)

where

$\theta^{\scriptscriptstyle\rm\bar{D}}:=\sigma^{\mu\nu}\delta\gamma_{\mu\nu}\epsilon_{\cal
N}+(\pi_{\mu}+2\partial_{\mu}\ln f)\delta l^{\mu}\epsilon_{\cal
N}-(\theta+2\bar{k})\delta\epsilon_{\cal
N},\qquad\ell^{\scriptscriptstyle\rm\bar{D}}:=-2(\theta+\bar{k})\epsilon_{\cal
N}.$ (3.25)

This boundary Lagrangian is independent of $f$, but not of reparametrizations
of $\Phi$. This is only a partial resolution of the ambiguities, and class-III
invariance is not achieved.

Remarkably, there is a solution that does not require any restriction on the
variations, but instead performing an integration by parts in field space of
the spin-0 term:

$-(\theta+2k)\delta\epsilon_{\cal N}=-\delta[(\theta+2k)\epsilon_{\cal
N}]+\delta(\theta+2k)\epsilon_{\cal N}.$ (3.26)

This leads to

$\displaystyle\mathchoice{\theta\leavevmode\hbox to5.09pt{\vbox
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to5.09pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-4.23445pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to3.69pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-2.82611pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.82611pt}{-2.39996pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\theta\leavevmode\hbox
to2.75pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.54721pt\lower-2.59996pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundcap\pgfsys@invoke{
}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-2.39996pt}\pgfsys@lineto{-1.88722pt}{-2.39996pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-1.88722pt}{-2.39996pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\scriptscriptstyle\rm
EH}=\theta^{\scriptscriptstyle\rm Conf}-\delta\ell^{\scriptscriptstyle\rm
Conf}+d\vartheta^{\scriptscriptstyle\rm EH},$ (3.27)

where now

$\theta^{\scriptscriptstyle\rm
Conf}:=[\sigma^{\mu\nu}\delta\gamma_{\mu\nu}+\pi_{\mu}\delta
l^{\mu}+\delta(\theta+2k)]\epsilon_{\cal N},$ (3.28)

and

$\ell^{\scriptscriptstyle\rm Conf}:=-\theta\epsilon_{\cal N}=-d\epsilon_{S}.$
(3.29)

The new boundary Lagrangian and symplectic potential are class-III invariant.
This shows the type of valuable insights that can be obtained using the
freedom of changing potential via (3.8).

The decomposition (3.28) corresponds to a change of polarization in the phase
space that identifies as configuration variables the conformal class of the 2d
metric – equivalently the shear, recall (3.18) and discuss around there –, the
tangent vector, and the spin-0 momentum $\theta+2k$ instead of the volume
form. We will show that it leads to a Noether flux-balance law with no anomaly
term. This is related to that fact that this choice makes the boundary
Lagrangian unambiguous even without any restriction on the variations.
Conformal boundary conditions appear thus to be better behaved, something
argued for also in the time-like case [41]. This polarization was referred to
as $\theta^{\scriptscriptstyle\rm Y}$ in [19], by analogy with York’s
conformal boundary conditions for the time-like case. A similar decomposition
to (3.28) was considered also in [18], while looking for thermodynamical
interpretations of the symplectic potential. However they included the
additional restriction $n_{\mu}\delta l^{\mu}=0$, which spoils class-I
invariance.

There are other integrations by parts in field space that could be considered.
One could change the spin-0 sector with different numerical factors. For
instance the investigations of black hole entropy by Chandrasekaran and
Speranza (CS) in [10] motivates the choice

$\ell^{\scriptscriptstyle\rm CS}=-(k+2\theta)\epsilon_{\cal
N}=\ell^{\scriptscriptstyle\rm D}+k\epsilon_{\cal N}.$ (3.30)

The motivation from black hole entropy will be briefly explained below, but
notice that this choice is not class-III invariant. On the other hand, a
change of polarization in the spin-1 pair seems of little use, since we cannot
treat $\eta_{\mu}$ as an independent configuration variable from the spin-2
pair. This is due to the fact that the constraint equations on ${\cal N}$
relates $\eta_{\mu}$ to the radiative data which are contained in the spin-2
pair. For completeness, we report in App. B an exploration of alternative
polarizations and their boundary conditions.

In the rest of the paper we will study how changing the boundary Lagrangian as
in the examples above affects the variational principle and the construction
of gravitational charges. With the exception of (3.25), the boundary
Lagrangians that we consider have the same functional dependence, and differ
only by numerical factors. This is similar to what we had in the case of a
time-like boundary [17], and allows us to treat all cases at once writing the
boundary Lagrangian in parametric form as

$\ell^{(b,c)}=-(bk+c\theta)\epsilon_{\cal N}.$ (3.31)

A boundary Lagrangian of this family is class-III invariant for $b=0$ and any
value of $c$. The specific examples described above correspond to: