# The Wald-Zoupas prescription for asymptotic charges at null infinity in
general relativity

Alexander M. Grant<EMAIL_ADDRESS>Department of Physics, Cornell
University, Ithaca, NY 14853, USA  Department of Physics, University of
Virginia, Charlottesville, VA 22904, USA  Kartik Prabhu<EMAIL_ADDRESS>Department of Physics, University of California, Santa Barbara, CA 93106, USA
Ibrahim Shehzad<EMAIL_ADDRESS>Department of Physics, Cornell University,
Ithaca, NY 14853, USA

###### Abstract

We use the formalism developed by Wald and Zoupas to derive explicit covariant
expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs
symmetries at null infinity in asymptotically flat spacetimes in vacuum
general relativity. Our expressions hold in non-stationary regions of null
infinity, are local and covariant, conformally-invariant, and are independent
of the choice of foliation of null infinity and of the chosen extension of the
symmetries away from null infinity. We also include detailed comparisons with
other expressions for the charges and fluxes that have appeared in the
literature, including expressions in conformal Bondi-Sachs coordinates, the
Ashtekar-Streubel flux formula, the Komar formulae and the linkage and twistor
charge formulae.

###### Contents

  1. 1 Introduction
  2. 2 Asymptotic-flatness at null infinity
    1. 1 Auxiliary foliation and null normal at $\mathscr{I}$
    2. 2 News tensor
  3. 3 Universal structure and metric perturbations
    1. 1 Metric perturbations near $\mathscr{I}$
  4. 4 Asymptotic symmetries at $\mathscr{I}$: the BMS Lie algebra
    1. 1 Extensions of BMS symmetries away from $\mathscr{I}$
  5. 5 Asymptotic charges and fluxes: The Wald-Zoupas prescription
    1. 1 Wald-Zoupas flux in general relativity
    2. 2 Wald-Zoupas charge in general relativity
  6. 6 Expressions in some coordinate systems
    1. 1 Bondi-Sachs coordinates
    2. 2 Conformal Gaußian null coordinates
  7. 7 Discussion
  8. A General choices of conformal factor
  9. B The GHP formalism at $\mathscr{I}$
  10. C Comparison to other charge formulae
    1. 1 Ashtekar-Streubel flux and charge
    2. 2 Komar formulae and linkage charges
    3. 3 Twistor charge
  11. D Symmetric and tracefree tensors in two dimensions

##  1 Introduction

There has been a considerable revival of interest in asymptotic symmetries and
their associated charges in general relativity in recent years. At null
infinity in asymptotically flat spacetimes, these symmetries are elements of
an infinite-dimensional symmetry group called the Bondi-Metzner-Sachs (BMS)
group BBM ; Sachs1 . A big reason for this revival of interest has been the
conjecture by Strominger that these symmetries give rise to an infinite number
of conservation laws which constrain classical gravitational scattering in
asymptotically flat spacetimes.111This conjecture has now been partially
proven, see e.g Tro ; KP-GR-match  for details. In a series of papers,
conservation laws associated with these symmetries have also been shown to be
related to soft graviton theorems He:2014laa ; Strominger:2014pwa ;
Kapec:2015vwa ; Avery:2015gxa ; Campiglia:2015kxa ; Campoleoni:2017mbt  and
gravitational memory effects He:2014laa ; Strominger:2014pwa ;
Pasterski:2015tva ; HIW ; Mao:2017wvx ; Pate:2017fgt ; Chatterjee:2017zeb  and
have even been speculated to have a possible bearing on the black hole
information paradox Hawking:2016msc ; Strominger:2017aeh ; Hawking:2016sgy ;
Flanagan:2021ojq .

In this paper we consider the problem of deriving explicit general expressions
for the charges associated with asymptotic symmetries at null infinity in
vacuum general relativity. We list a set of criteria that these should satisfy
to be physically reasonable and, using the prescription provided by Wald and
Zoupas in WZ , provide an explicit computation of the general expressions for
these quantities.

We will consider $4$-dimensional asymptotically flat spacetimes defined in a
coordinate independent manner through Penrose’s conformal completion (see Def.
2.1). In this picture, the null infinity of the original (physical) spacetime
is represented by a smooth null surface $\mathscr{I}$ in the conformally-
completed (unphysical) spacetime. This null surface has the topology of
$\mathscr{I}\cong\mathbb{R}\times\mathbb{S}^{2}$ where $\mathbb{R}$ represents
the null directions and $\mathbb{S}^{2}$ the angular directions at infinity.
It then follows from the definition of asymptotic flatness that there is a
certain _universal_ structure associated with $\mathscr{I}$ (see Sec. 3). This
structure is universal in the sense that it is independent of which
asymptotically flat physical spacetime is being considered and thus provides a
“fixed background” at infinity which is common to all physical spacetimes. The
infinitesimal diffeomorphisms (i.e. vector fields) which preserve this
universal structure form the asymptotic symmetry algebra given by the BMS
algebra (Sec. 4).

Then, one wishes to define certain “conserved quantities” (similar to mass,
energy or angular momentum) associated with any given physical spacetime and
an asymptotic symmetry. In general dynamical spacetimes such quantities will
not be conserved due to the presence of gravitational radiation. However, one
can proceed as follows: let $S\cong\mathbb{S}^{2}$ be some cross-section (or
“cut”) of null infinity $\mathscr{I}$; the choice of $S$ represents an
“instant of time” on $\mathscr{I}$. Then, for a vector field $\xi^{a}$
representing an asymptotic symmetry and to every cross-section $S$ we define a
_charge_ $\mathcal{Q}[\xi;S]$, which is to represent the “not really conserved
quantity” at that “instant of time”. The change in this charge with “time”,
i.e. between two cross-sections, is then a _flux_
$\mathcal{F}[\xi;\Delta\mathscr{I}]$ where $\Delta\mathscr{I}$ is the region
of null infinity between those two cross-sections. However, without any
further physical criteria one could make up arbitrary expressions for such a
charge and flux formula. Even if one requires that any notion of charge
associated with time translations and rotations coincide with the mass and
angular momentum usually defined in Kerr spacetimes, it is not apparent how to
generalize the charge expressions to all BMS symmetries and to non-stationary
spacetimes.

Next we list some criteria which any physically reasonable notion of
asymptotic charges and fluxes should satisfy:  1. Any charge or flux
quantities one defines must be independent of any choice of coordinates; this
issue is already resolved since we work directly with the covariant
formulation of null infinity mentioned above, instead of fixing some
coordinates like Bondi-Sachs coordinates.  2. We want to associate the charges
and fluxes with the physical spacetime and not with the choice of conformal-
completion used to obtain the unphysical conformally-completed spacetime, or
with any additional structure used to compute these quantities. Thus, we
require that the charges and fluxes be independent of the conformal factor
used to obtain the Penrose conformal-completion. Below we will also use a
foliation of $\mathscr{I}$ to simplify our computations, and so we will also
demand that the charges and fluxes be independent of the choice of this
additional choice of foliation.  3. We also want to associate the charges and
fluxes with the BMS symmetries at $\mathscr{I}$ and so, we demand that they be
independent of any arbitrary extension of these asymptotic symmetries into the
spacetime (these extensions are considered “pure gauge” and are discussed in
Sec. 4.1).  4. The charges and fluxes are local and covariant the following
sense: the value of the charge $\mathcal{Q}[\xi;S]$ is obtained as an integral
over $S$ of a $2$-form which is constructed from the available fields and the
BMS symmetry, and finitely-many of their derivatives at $S$. The flux
$\mathcal{F}[\xi;\Delta\mathscr{I}]$ is the integral over $\Delta\mathscr{I}$
of a $3$-form constructed in a similar way from the fields and symmetry in the
region $\Delta\mathscr{I}$.  5. Finally, since we want the fluxes
$\mathcal{F}[\xi;\Delta\mathscr{I}]$ to characterize physical dynamical
processes, like gravitational radiation, we also require that the flux
associated with _any_ BMS symmetry between _any_ two cross-sections vanishes
when the physical spacetime is stationary.

A prescription for obtaining such asymptotic charges and fluxes in any local
and covariant Lagrangian theory of gravity was given by Wald and Zoupas WZ .
We will detail this procedure in Sec. 5 for our case of interest, namely
asymptotically flat spacetimes in vacuum general relativity. This case was
also considered in detail by Wald and Zoupas WZ , where it was shown, rather
indirectly, that the resulting charge expression matches the one given by Dray
and Streubel DS , which was obtained by generalizing Penrose’s formula
motivated by twistor theory (see Sec. C.3). Explicit expressions for these
charges in terms of quantities on null infinity were obtained by Flanagan and
Nichols Flanagan:2015pxa , but their analysis was done in Bondi coordinates
(where the covariance and conformal-invariance of the expressions is not
manifest) and also in stationary regions of null infinity.222A similar charge
formula was also obtained in BT  but they work with a “perturbed charge”
expression which is not integrable in phase space; see LABEL:eq:charge-bondi-2
and the subsequent discussion.

In this paper we compute the formulae for the charges and fluxes, using the
Wald-Zoupas prescription, written explicitly in terms of fields defined on
null infinity in full generality. We call these the “Wald-Zoupas (WZ) charge”
and “Wald-Zoupas (WZ) flux” respectively. To simplify our computations we will
choose an arbitrary but fixed foliation of null infinity, but show that our
formulae are independent of this choice of foliation. This computation can be
done easily for the flux, since the WZ flux expression is written entirely in
terms of quantities at null infinity; our expression for the WZ flux in terms
of a choice of foliation is given in Eq. 5.33. The computation of the charges,
however, is more complicated: unlike the flux, a portion of the expression for
the WZ charge involves the limit of the integral of a quantity that is defined
in a neighborhood of null infinity. To compute this limit, we make use of
Bondi coordinates, much like Flanagan:2015pxa ; however, we show that the
value of this limit is independent of the choices that are made in using Bondi
coordinates, such as the choice of foliation, conformal factor, and extension
of the BMS vector field off of null infinity. As such, we can convert the
value of the limit back into a covariant form, allowing us to write an
expression for the WZ charge (see Eq. 5.44) in terms of quantities defined on
null infinity. Since this procedure is somewhat involved, we then check
explicitly that the change of the charge is consistent with the flux formula
in Eq. 5.33. These covariant expressions are equal to those given by WZ , and
are therefore independent of foliation; however, picking a choice of foliation
allows us to write the charge in terms of quantities defined entirely at null
infinity. These explicit formulae can then be easily compared to the
expressions one would obtain in _any_ coordinate system of ones choosing, see
e.g. Sec. 6. These expressions can also be compared to other formulations of
BMS charges, some of which we detail in Appendix C.

The rest of this paper is organized as follows. We start by reviewing the
definition of asymptotic-flatness at null infinity in Sec. 2, and summarize
the various asymptotic fields arising at null infinity that we will use in
subsequent computations. In Sec. 3, we discuss the universal structure at null
infinity (that is, the structure common to all asymptotically-flat spacetimes
at null infinity) and the restrictions on metric perturbations which follow
from the definition of asymptotic flatness. In Sec. 4, we review the BMS
algebra and its relevant properties. In Sec. 5, we describe the Wald-Zoupas
prescription for obtaining the charges and fluxes associated with BMS
symmetries and obtain manifestly covariant expressions for these quantities
which we call the “Wald-Zoupas (WZ) charge” and the “Wald-Zoupas (WZ) flux.”
Finally, we discuss the construction of Bondi-Sachs and conformal Gaußian null
coordinate frames near null infinity in Sec. 6 and express the BMS symmetries
and the Wald-Zoupas charge in these coordinates. We end with a short
discussion of our results in Sec. 7. While in the body of the paper, we work
in a conformal frame where the Bondi condition (see Eq. 2.6) holds, we include
a summary of our main results in more general conformal frames in Appendix A.
In Appendix B we rewrite the WZ charge and flux expressions in the Geroch-
Held-Penrose (GHP) formalism at null infinity. In Appendix C, we compare the
WZ charge and flux formulae to some of the other charge and flux formulae that
have been proposed for BMS symmetries, namely, the Ashtekar-Streubel flux
formula, the Komar charge formulae and their linkage versions and Penrose’s
twistor charge formula. Appendix D collects some useful results on symmetric
tracefree tensors on a $2$-sphere.

Our notation and conventions are as follows. We follow the conventions of Wald
Wald-book  for the metric signature, Riemann tensor, and differential forms,
and use abstract index notation with Latin indices $a,b,c,\ldots$ for tensor
fields. Quantities defined on the physical spacetime are denoted by a “hat”,
while the ones on the conformally-completed, unphysical spacetime are denoted
without a “hat” (e.g., $\hat{g}_{ab}$ is the physical metric while $g_{ab}$ is
the unphysical metric). In addition, we use “$\mathrel{\mathop{\widehat{=}}}$”
to denote equality at null infinity and $\underleftarrow{}$ to denote the
pullback to null infinity. We also use both the indexed and index-free
notations for differential forms, and in the index-free notation we denote
differential forms in bold. We use “$\equiv$” to translate between indexed and
index-free notation, writing, e.g.,
$\boldsymbol{\varepsilon}_{4}\equiv\varepsilon_{abcd}$. Since “$\equiv$” is
used for this translation, we use “$\mathrel{\mathop{:}}=$” for definitions.

##  2 Asymptotic-flatness at null infinity

In this section, we recall the covariant definition of asymptotically-flat
spacetimes and define the asymptotic fields and their equations at null
infinity that will appear in our later analysis.

######  Definition 2.1 (Asymptotic flatness).

A _physical_ spacetime $(\hat{M},\hat{g}_{ab})$, which satisfies the vacuum
Einstein equation $\hat{G}_{ab}=0$, is asymptotically-flat at null infinity if
there exists an _unphysical_ spacetime $(M,g_{ab})$ with a boundary
$\mathscr{I}=\partial M$ and an embedding of $\hat{M}$ into $M$ (we use this
embedding to identify $\hat{M}$ as a submanifold of $M$), such that

  1. (1)

There exists a smooth function $\Omega$ (the _conformal factor_) on $M$
satisfying $\Omega\mathrel{\mathop{\widehat{=}}}0$ and
$\nabla_{a}\Omega\not\mathrel{\mathop{\widehat{=}}}0$ such that
$g_{ab}=\Omega^{2}\hat{g}_{ab}$ is smooth on $M$ including at $\mathscr{I}$.

  2. (2)

$\mathscr{I}$ is topologically $\mathbb{R}\times\mathbb{S}^{2}$.

  3. (3)

Defining $n_{a}:=\nabla_{a}\Omega$, the vector field $\omega^{-1}n^{a}$ is
complete on $\mathscr{I}$ for any smooth function $\omega$ on $M$ such that
$\omega>0$ on $M$ and
$\nabla_{a}(\omega^{4}n^{a})\mathrel{\mathop{\widehat{=}}}0$.

Detailed expositions on the motivations for this definition may be found in
Geroch-asymp ; Wald-book . The differentiability conditions on the unphysical
spacetime can be significantly weakened, but we restrict to the smooth case
for simplicity.

Using the conformal transformation relating the unphysical Ricci tensor
$R_{ab}$ to the physical Ricci tensor $\hat{R}_{ab}$ (see, e.g, Appendix D of
Wald-book ), the vacuum Einstein equation can be written as

$\displaystyle
S_{ab}=-2\Omega^{-1}\nabla_{(a}n_{b)}+\Omega^{-2}n^{c}n_{c}g_{ab}\,,$ (2.1)

where $S_{ab}$ is given by

$\displaystyle S_{ab}=R_{ab}-\tfrac{1}{6}Rg_{ab}.$ (2.2)

It follows from Eq. 2.1 and the smoothness of $\Omega$ and the unphysical
metric $g_{ab}$ at $\mathscr{I}$ that
$n_{a}n^{a}\mathrel{\mathop{\widehat{=}}}0$. This implies that $\mathscr{I}$
is a smooth null hypersurface in $M$ with normal $n_{a}=\nabla_{a}\Omega$ and
that the vector field $n^{a}=g^{ab}n_{b}$ is a null geodesic generator of
$\mathscr{I}$.

Further, the Bianchi identity $\nabla_{[a}R_{bc]de}=0$ on the unphysical
Riemann tensor along with Eq. 2.1 gives the following equations for the
unphysical Weyl tensor $C_{abcd}$ (see Geroch-asymp  for details)

$\displaystyle\nabla_{[a}(\Omega^{-1}C_{bc]de})=0\,.$ (2.3a)
$\displaystyle\nabla^{d}C_{abcd}=-\nabla_{[a}S_{b]c}\,.$ (2.3b)

Given a physical spacetime $(\hat{M},\hat{g}_{ab})$ there is a lot of freedom
in constructing its conformal-completion to obtain the unphysical spacetime
$(M,g_{ab})$. Firstly, there is the choice of the embedding of $\hat{M}$ into
$M$, but since this is simply used to identify points of $\hat{M}$ with those
of $M$ this freedom is innocuous and one can always fix a choice of this
embedding map. Another freedom is the choice of the conformal factor $\Omega$.
Note that for the same physical spacetime another choice of the conformal
factor $\Omega^{\prime}=\omega\Omega$, with $\omega|_{\mathscr{I}}>0$ also
satisfies Def. 2.1 with a different unphysical metric
$g^{\prime}_{ab}=\Omega^{\prime 2}\hat{g}_{ab}=\omega^{2}g_{ab}$. Since we are
interested in studying the asymptotic properties of the physical spacetime —
the conformal-completion is used only to bring the asymptotic boundary “at
infinity” of the physical spacetime to a finite boundary $\mathscr{I}$ in the
unphysical spacetime — all physical quantities (such as symmetries, charges
and fluxes) must be independent of the choice of conformal factor made.

All of our computations which follow can be done using an arbitrary conformal
factor; however it is convenient to fix some of the conformal freedom by
imposing the _Bondi condition_ as follows. On $\mathscr{I}$, let $\Phi$ be
defined by

$\displaystyle\Phi\mathrel{\mathop{:}}=\tfrac{1}{4}\nabla_{a}n^{a}\big{|}_{\mathscr{I}}\,.$
(2.4)

Under a change of conformal factor $\Omega\mapsto\omega\Omega$ and
$g_{ab}\mapsto\omega^{2}g_{ab}$ we have

$\displaystyle\Phi\mapsto\omega^{-1}(\Phi+\pounds_{n}\ln\omega)\,.$ (2.5)

Now, without loss of generality, we can choose $\omega$ to be a solution of
$\Phi+\pounds_{n}\ln\omega\mathrel{\mathop{\widehat{=}}}0$ to set
$\Phi\mathrel{\mathop{\widehat{=}}}0$ Geroch-asymp ; Wald-book .333Note that
while this condition can always be imposed on $\mathscr{I}$, it cannot be
imposed at spatial infinity where solutions, $\omega$, to the equation
$\Phi+\pounds_{n}\ln\omega\mathrel{\mathop{\widehat{=}}}0$ diverge Wald-book .
In this choice of the conformal factor, Eq. 2.1 implies the _Bondi condition_

$\displaystyle\nabla_{a}n_{b}\mathrel{\mathop{\widehat{=}}}0,$ (2.6)

and

$\displaystyle n_{a}n^{a}=O(\Omega^{2}).$ (2.7)

We will henceforth work in a conformal frame where the Bondi condition holds.
Having imposed the Bondi condition, the remaining freedom in the conformal
factor is of the form $\Omega\mapsto\omega\Omega$ where

$\displaystyle\omega|_{\mathscr{I}}>0\,,\quad\pounds_{n}\omega\mathrel{\mathop{\widehat{=}}}0.$
(2.8)

We reiterate that the choice to work in a conformal frame where the Bondi
condition holds is made purely for convenience and is not essential for the
calculations in this paper. The main results of this paper, expressed in
general conformal frames where the Bondi condition is not imposed, are
included in Appendix A. In the body of the paper, our statements about
conformal invariance will pertain to conformal transformations that satisfy
Eq. 2.8. More general conformal transformations for which
$\pounds_{n}\omega\not\mathrel{\mathop{\widehat{=}}}0$ will only be considered
in Appendices A and B.

Finally, let $q_{ab}$ denote the pullback of $g_{ab}$ to $\mathscr{I}$. This
defines a degenerate metric on $\mathscr{I}$ such that

$\displaystyle
q_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}0\,,\quad\pounds_{n}q_{ab}\mathrel{\mathop{\widehat{=}}}0,$
(2.9)

where the second condition follows from Eq. 2.6. Thus, $q_{ab}$ defines a
Riemannian metric on the space of generators of $\mathscr{I}$ which is
diffeomorphic to $\mathbb{S}^{2}$.

###  1 Auxiliary foliation and null normal at $\mathscr{I}$

For carrying out explicit computations on $\mathscr{I}$, it is convenient to
pick an additional “auxiliary” structure on $\mathscr{I}$ given by a choice of
foliation of $\mathscr{I}$ by a parameter family of _cross-sections_ which are
diffeomorphic to $\mathbb{S}^{2}$. Note that such a foliation always exists
since $\mathscr{I}\cong\mathbb{R}\times\mathbb{S}^{2}$, though there is no
unique choice for this foliation. The results in this paper can be obtained
without reference to any choice of foliation, but it is far simpler to use
some choice of foliation and then verify that the results are independent of
the choice of foliation made.

For any choice of foliation, we obtain a $1$-form $l_{a}$ on $\mathscr{I}$
which is normal to each cross-section of the foliation. Then there exists a
_unique_ vector field $l^{a}$ defined at $\mathscr{I}$ such that
$l_{a}\mathrel{\mathop{\widehat{=}}}g_{ab}l^{b}$ is the normal to the chosen
foliation and

$l^{a}l_{a}\mathrel{\mathop{\widehat{=}}}0\,,\quad
l^{a}n_{a}\mathrel{\mathop{\widehat{=}}}-1.$ (2.10)

We will call such a vector field associated with the chosen foliation as the
_auxiliary normal_. Note that the vector field $l^{a}$ can be extended
arbitrarily away from $\mathscr{I}$, and our computations will be independent
of which extension is chosen.

Using the auxiliary null normal $l^{a}$, we now define a tensor $Q_{ab}$ at
$\mathscr{I}$ by

$Q_{ab}\mathrel{\mathop{:}}=g_{ab}+2l_{(a}n_{b)}.$ (2.11)

Note that $q_{ab}=\underleftarrow{Q}{}_{ab}$, and so $Q_{ab}$ is a choice of
pushforward of the intrinsic (degenerate) metric on $\mathscr{I}$. By
definition of $Q_{ab}$ we have,

$Q_{ab}l^{b}\mathrel{\mathop{\widehat{=}}}0,\qquad
Q_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}0.$ (2.12)

We also define the symmetric trace-free part of a tensor with respect to
$Q_{ab}$ by:

$\operatorname{STF}A_{ab}\mathrel{\mathop{:}}=\left(Q_{(a}{}^{c}Q_{b)}{}^{d}-\tfrac{1}{2}Q_{ab}Q^{cd}\right)A_{cd}.$
(2.13)

Next, we define a volume form on $\mathscr{I}$ and on the cross-sections of
the chosen foliation by

$\displaystyle\varepsilon_{abc}\mathrel{\mathop{:}}=l^{d}\varepsilon_{dabc}\,,\quad\varepsilon_{ab}\mathrel{\mathop{:}}=\varepsilon_{abcd}l^{c}n^{d}\mathrel{\mathop{\widehat{=}}}\varepsilon_{abc}n^{c}.$
(2.14)

######  Remark 2.1 (Orientation conventions).

The orientation of these volume forms are such that $\mathscr{I}$ is outward-
facing as a manifold in $M$, and such that the cross-sections have a future
orientation within $\mathscr{I}$. Note that this convention for the
orientations for $\boldsymbol{\varepsilon}_{2}$ and
$\boldsymbol{\varepsilon}_{3}$ is somewhat unexpected. In Minkowski spacetime,
setting $l_{a}=-\nabla_{a}u$ and $n_{a}=\nabla_{a}\Omega$ (with $\Omega=1/r$
and $u=t-r$), we have that $\boldsymbol{\varepsilon}_{3}=-\sin\theta\,du\wedge
d\theta\wedge d\phi$ and
$\boldsymbol{\varepsilon}_{2}=-\sin\theta\,d\theta\wedge d\phi$, which is the
opposite of the “expected” sign for $\boldsymbol{\varepsilon}_{2}$. Despite
this sign difference, we still have that,

$\int\limits_{S}\boldsymbol{\varepsilon}_{2}>0;$ (2.15)

this, in fact, is the _definition_ of an orientation. In Minkowski, this can
be seen by noting that this means that the bounds of the integrals are in the
opposite order (say, the $\theta$ integral is from $\pi$ to $0$, instead of
$0$ to $\pi$).

Finally, note that, in our choice of orientation, if we consider a cross-
section $S$ as a limit of spheres $S^{\prime}$ within a spacelike hypersurface
$\Sigma$ in the unphysical spacetime, then the orientation for $S^{\prime}$
that is compatible with that of $S$ is the _inward_ -facing one within
$\Sigma$.

The _shear_ of the auxiliary normal on $\mathscr{I}$ is defined by

$\displaystyle\sigma_{ab}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\nabla_{a}l_{b}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\operatorname{STF}\pounds_{l}g_{ab}\,.$
(2.16)

The twist $\varepsilon^{ab}\nabla_{a}l_{b}$ vanishes on $\mathscr{I}$ since
$l_{a}$ is normal to cross-sections of a foliation of $\mathscr{I}$. We will
mostly not need the expansion of $l_{a}$ in our analysis (it is introduced in
Sec. 6.2 where needed).

The change in $l_{a}$ along the null generators of $\mathscr{I}$ is given by

$\tau_{a}\mathrel{\mathop{\widehat{=}}}Q_{a}{}^{b}n^{c}\nabla_{c}l_{b}.$
(2.17)

By the normalization conditions for $l^{a}$ and $n^{a}$, and the Bondi
condition we have

$\tau_{a}\mathrel{\mathop{\widehat{=}}}n^{b}\nabla_{b}l_{a}\mathrel{\mathop{\widehat{=}}}\pounds_{n}l_{a},$
(2.18)

We also can define a “sphere derivative” along the cross-sections for any
tensor field that is orthogonal to $l^{a}$ and $n^{a}$ on all indices (at
$\mathscr{I}$):

$\mathscr{D}_{a}T^{b_{1}\cdots b_{r}}{}_{c_{1}\cdots
c_{s}}\mathrel{\mathop{:}}=Q^{d}{}_{a}Q^{b_{1}}{}_{e_{1}}\cdots
Q^{b_{r}}{}_{e_{r}}Q^{f_{1}}{}_{c_{1}}\cdots
Q^{f_{s}}{}_{c_{s}}\nabla_{d}T^{e_{1}\cdots e_{r}}{}_{f_{1}\cdots f_{s}}.$
(2.19)

One can easily show that this derivative operator is compatible with both
$Q_{ab}$ and $\varepsilon_{ab}$ at $\mathscr{I}$:

$\mathscr{D}_{a}Q_{bc}\mathrel{\mathop{\widehat{=}}}0,\qquad\mathscr{D}_{a}\varepsilon_{bc}\mathrel{\mathop{\widehat{=}}}0.$
(2.20)

With our choice of $\boldsymbol{\varepsilon}_{3}$ and
$\boldsymbol{\varepsilon}_{2}$, we can derive several useful formulas from
Stokes’ theorem. Consider some portion $\Delta\mathscr{I}$ of $\mathscr{I}$
whose boundary is given by two cross-sections $S_{1}$ and $S_{2}$ in our
foliation, with $S_{2}$ to the future of $S_{1}$. Then, for any scalar
$\alpha$ we have

$\displaystyle\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}\;\pounds_{n}\alpha$
$\displaystyle=\int\limits_{S_{2}}\boldsymbol{\varepsilon}_{2}\;\alpha-\int\limits_{S_{1}}\boldsymbol{\varepsilon}_{2}~{}\alpha\,.$
(2.21)

Also, for any vector $v^{a}$ which is orthogonal to both $n_{a}$ and $l_{a}$
we have

$\displaystyle\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}~{}\nabla_{a}v^{a}\mathrel{\mathop{\widehat{=}}}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}~{}(\mathscr{D}_{a}+\tau_{a})v^{a}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}0,$ (2.22)

where the second expression is obtained using the Bondi condition along with
Eqs. 2.11 and 2.18, and the vanishing of the expression follows from the fact
that $\boldsymbol{\varepsilon}_{3}\nabla_{a}v^{a}$ is an exact $3$-form and
$l_{a}v^{a}\mathrel{\mathop{\widehat{=}}}0$. Similarly, for any cross-section
$S$ of $\mathscr{I}$ we also have

$\displaystyle\int\limits_{S}\boldsymbol{\varepsilon}_{2}\;\mathscr{D}_{a}v^{a}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}0\,.$ (2.23)

We note that Eq. 2.22 was erroneously given in Eq. 2.13 of KP-GR-match  as an
integration-by-parts identity on the sphere, instead of $\Delta\mathscr{I}$
(all instances when Eq. 2.22 was used in KP-GR-match , however, were correct,
as they were applied to $\Delta\mathscr{I}$).

The final quantities that we will need are components of the Weyl tensor at
$\mathscr{I}$. By the peeling theorem, we have
$C_{abcd}\mathrel{\mathop{\widehat{=}}}0$ at $\mathscr{I}$, and thus
$\Omega^{-1}C_{abcd}$ admits a limit to $\mathscr{I}$ (see Theorem 11 of
Geroch-asymp ). In the choice of a foliation of $\mathscr{I}$ we define the
Weyl tensor fields

$\displaystyle\mathcal{R}_{ab}$
$\displaystyle\mathrel{\mathop{:}}=(\Omega^{-1}C_{cdef})Q_{a}{}^{c}n^{d}Q_{b}{}^{e}n^{f},$
$\displaystyle\mathcal{S}_{a}$
$\displaystyle\mathrel{\mathop{:}}=(\Omega^{-1}C_{cdef})l^{c}n^{d}Q_{a}{}^{e}n^{f},$
(2.24a) $\displaystyle\mathcal{P}$
$\displaystyle\mathrel{\mathop{:}}=(\Omega^{-1}C_{cdef})l^{c}n^{d}l^{e}n^{f},$
$\displaystyle\mathcal{P}^{*}$
$\displaystyle\mathrel{\mathop{:}}=\tfrac{1}{2}(\Omega^{-1}C_{cdef})l^{c}n^{d}\varepsilon^{ef},$
(2.24b) $\displaystyle\mathcal{J}_{a}$
$\displaystyle\mathrel{\mathop{:}}=(\Omega^{-1}C_{cdef})n^{c}l^{d}Q_{a}{}^{e}l^{f},$
$\displaystyle\mathcal{I}_{ab}$
$\displaystyle\mathrel{\mathop{:}}=(\Omega^{-1}C_{cdef})Q_{a}{}^{c}l^{d}Q_{b}{}^{e}l^{f}.$
(2.24c)

Note that, due to the symmetries of the Weyl tensor, $\mathcal{R}_{ab}$ and
$\mathcal{I}_{ab}$ are symmetric and traceless. The relation of these tensors
to the Weyl scalar components in the Newman-Penrose notation is given in Eq.
B.12.

For the fields defined in Eq. 2.24, Eq. 2.3a implies the following evolution
equations along $\mathscr{I}$, which can be verified to be conformally-
invariant (Eq. B.13 in the GHP formalism):

$\displaystyle\pounds_{n}\mathcal{S}_{a}$
$\displaystyle=(\mathscr{D}^{b}+\tau^{b})\mathcal{R}_{ab},$ (2.25a)
$\displaystyle\pounds_{n}\mathcal{P}~{}$
$\displaystyle=(\mathscr{D}^{a}+2\tau^{a})\mathcal{S}_{a}-\sigma^{ab}\mathcal{R}_{ab},$
(2.25b) $\displaystyle\pounds_{n}\mathcal{P}^{*}$
$\displaystyle=-\varepsilon^{ab}(\mathscr{D}_{a}+2\tau_{a})\mathcal{S}_{b}+\varepsilon_{b}{}^{c}\sigma^{ab}\mathcal{R}_{ac},$
(2.25c) $\displaystyle\pounds_{n}\mathcal{J}_{a}$
$\displaystyle=\tfrac{1}{2}(\mathscr{D}_{b}+3\tau_{b})(Q_{a}{}^{b}\mathcal{P}-\varepsilon_{a}{}^{b}\mathcal{P}^{*})-2\sigma_{a}{}^{b}\mathcal{S}_{b},$
(2.25d) $\displaystyle\pounds_{n}\mathcal{I}_{ab}$
$\displaystyle=\operatorname{STF}(\mathscr{D}_{a}+4\tau_{a})\mathcal{J}_{b}-\tfrac{3}{2}\sigma_{ac}(Q_{b}{}^{c}\mathcal{P}-\varepsilon_{b}{}^{c}\mathcal{P}^{*}).$
(2.25e)

Next, we list the conformal transformation weights of these various
quantities. Any quantity $\alpha$ is said to have _conformal weight_ $w$ if,
under $\Omega\mapsto\omega\Omega$, it transforms as
$\alpha\mapsto\omega^{w}\alpha$. First, we clearly have that

$g_{ab}:w=2,\qquad\varepsilon_{abcd}:w=4.$ (2.26)

Next, while in the spacetime

$n_{a}\mapsto\omega n_{a}+\Omega\nabla_{a}\omega,$ (2.27)

we therefore have that $n_{a}:w=1$ on $\mathscr{I}$ itself, and so

$l_{a}:w=1,\qquad(Q_{ab},\varepsilon_{ab}):w=2,\qquad\varepsilon_{abc}:w=3.$
(2.28)

Moreover, we have that

$\sigma_{ab}:w=1,$ (2.29)

while, under a conformal transformation, $\tau_{a}$ transforms as

$\tau_{a}\mapsto\tau_{a}+\mathscr{D}_{a}\ln\omega.$ (2.30)

Finally, the fields constructed from the Weyl tensor have the following
conformal weights:

$(\mathcal{R}_{ab},\mathcal{I}_{ab}):w=-1\,,\quad(\mathcal{S}_{a},\mathcal{J}_{a}):w=-2\,,\quad(\mathcal{P},\mathcal{P}^{*}):w=-3.$
(2.31)

Finally, we remark that there is a particularly convenient choice of conformal
factor and foliation, namely one where $q_{ab}$ is given by the unit 2-sphere
metric, and $\tau^{a}\mathrel{\mathop{\widehat{=}}}0$. This is the _Bondi
frame_ , and in particular it is a restriction of the remaining conformal
freedom $\Omega\mapsto\omega\Omega$ that is more restrictive than the Bondi
condition. Even though Bondi frame can always be achieved by a conformal
transformation and a change of foliation, we do not enforce Bondi frame, since
we want to consider charges on arbitrary cross-sections of $\mathscr{I}$, and
fluxes between these cross-sections and want our expressions to be manifestly
conformally invariant (apart from the choice of Bondi condition).

###  2 News tensor

In this section, we define the _News tensor_ , which characterizes the
radiative degrees of freedom of the gravitational field at null infinity. This
tensor can be defined in terms of a foliation, or invariantly from the
universal structure that exists on $\mathscr{I}$; we review both definitions
and show that they are equal. Moreover, we review how the News tensor vanishes
in asymptotically stationary regions of $\mathscr{I}$, which motivates its
usage as a characterization of radiation.

We define the News tensor as the (projected) Lie derivative along $n^{a}$ of
the shear $\sigma_{ab}$:

$N_{ab}\mathrel{\mathop{:}}=2Q_{a}{}^{c}Q_{b}{}^{d}\pounds_{n}\sigma_{cd}=2\operatorname{STF}\pounds_{n}\sigma_{ab}\,.$
(2.32)

Note that it follows from the above definition that

$\displaystyle
N_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}g^{ab}N_{ab}\mathrel{\mathop{\widehat{=}}}Q^{ab}N_{ab}\mathrel{\mathop{\widehat{=}}}0\,.$
(2.33)

It is straightforward to verify that the News is conformally invariant.

We can also write the News $N_{ab}$ in terms of $S_{ab}$ as follows. Consider
the quantity $\operatorname{STF}\pounds_{n}\pounds_{l}g_{ab}$. Despite
involving covariant derivatives of $l_{a}$, when evaluated at $\mathscr{I}$,
this quantity independent of the choice of $l_{a}$ away from $\mathscr{I}$. By
choosing a particular extension of $l_{a}$, one can show from Eqs. 2.16 and
2.18 that

$\operatorname{STF}\pounds_{n}\pounds_{l}g_{ab}\mathrel{\mathop{\widehat{=}}}2\operatorname{STF}(\pounds_{n}\sigma_{ab}-\tau_{a}\tau_{b})\mathrel{\mathop{\widehat{=}}}N_{ab}-2\operatorname{STF}(\tau_{a}\tau_{b}),$
(2.34)

where the second equality follows from Eq. 2.32. Using
$[\pounds_{n},\pounds_{l}]=\pounds_{[n,l]}$, together with Eq. 2.18, we
therefore find that

$N_{ab}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}[\pounds_{l}\pounds_{n}g_{ab}+2(\mathscr{D}_{a}+\tau_{a})\tau_{b}].$
(2.35)

Using Eq. 2.1, we therefore find that

$N_{ab}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}[S_{ab}+2(\mathscr{D}_{a}+\tau_{a})\tau_{b}].$
(2.36)

Further, from Eqs. 2.3b and 2.36, the News is related to the Weyl tensor
components (defined in Eq. 2.24) by:

$Q_{a}{}^{c}Q_{b}{}^{d}\pounds_{n}N_{cd}\mathrel{\mathop{\widehat{=}}}2\mathcal{R}_{ab}\,,\quad\mathscr{D}^{b}N_{ab}\mathrel{\mathop{\widehat{=}}}2\mathcal{S}_{a}\,,$
(2.37)

and

$\mathcal{P}^{*}=\varepsilon^{ab}\left[\mathscr{D}_{a}(\mathscr{D}_{c}-\tau_{c})\sigma_{b}{}^{c}-\tfrac{1}{2}N_{ac}\sigma_{b}{}^{c}\right]\,.$
(2.38)

In the GHP notation these relations can be found in Eqs. B.16 and B.18.

As defined by Eq. 2.32, it is not obvious that the News tensor is independent
of our choice of foliation. However, we show below that this definition of the
News coincides with the covariant definition given by Geroch Geroch-asymp
which makes no reference to any foliation of $\mathscr{I}$. While most of the
literature exclusively uses only one of these two definitions, we will find it
convenient to use either interchangeably, and (since there does not appear to
be any proof that we could find), we will now show that these two definitions
yield the same tensor.

Consider the _Geroch News tensor_ defined in Geroch-asymp  as

$N_{ab}\mathrel{\mathop{:}}=\underleftarrow{S}_{ab}-\rho_{ab},$ (2.39)

where $\underleftarrow{S}_{ab}$ denotes the pullback of $S_{ab}$ to
$\mathscr{I}$ and $\rho_{ab}$ is the _unique_ symmetric tensor field on
$\mathscr{I}$ constructed from the intrinsic universal structure on
$\mathscr{I}$ defined in Theorem 5 of Geroch-asymp .

First note that the Geroch News tensor is conformally invariant, and satisfies
the conditions Eq. 2.33 (see Geroch-asymp ). Moreover, projections of Eq. 68
of Geroch-asymp  show that Eq. 2.37 also holds for the Geroch News tensor.
Thus, if $\lambda_{ab}$ is the difference of the Geroch News tensor and the
one defined in Eq. 2.32, then $\lambda_{ab}$ is a symmetric tensor field on
$\mathscr{I}$ which satisfies

$\lambda_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}Q^{ab}\lambda_{ab}\mathrel{\mathop{\widehat{=}}}Q_{a}{}^{c}Q_{b}{}^{d}\pounds_{n}\lambda_{cd}\mathrel{\mathop{\widehat{=}}}\mathscr{D}^{b}\lambda_{ab}\mathrel{\mathop{\widehat{=}}}0\,.$
(2.40)

Thus, $\lambda_{ab}$ is a tensor field on $S$ that is symmetric, traceless and
divergence-free, and therefore vanishes by Prop. D.2. Consequently, the News
tensor defined in Eq. 2.32 is equivalent to the covariant definition by
Geroch.

Finally, we review the key property of the News tensor, namely that it
characterizes the presence of gravitational radiation in a spacetime. This can
be seen by the following result, which is due to Geroch (see pp. 53-54 of
Geroch-asymp ): consider an asymptotically flat spacetime
$(\hat{M},\hat{g}_{ab})$, and some portion $\Delta\mathscr{I}$ of null
infinity. If $\Delta\mathscr{I}$ is asymptotically stationary, in the sense
that there exists a vector field $t^{a}$ in a neighborhood of
$\Delta\mathscr{I}$ that is a timelike Killing vector with respect to
$\hat{g}_{ab}$, then $N_{ab}\mathrel{\mathop{\widehat{=}}}0$ on
$\Delta\mathscr{I}$. Since any notion of gravitational radiation should vanish
in asymptotically stationary regions of null infinity, this motivates the News
tensor as indicating the presence of radiation. It should be noted, however,
that it is not known whether the converse of this statement is true, namely
that all regions $\Delta\mathscr{I}$ where the News tensor vanishes are
asymptotically stationary.

##  3 Universal structure and metric perturbations

In this section, we summarize the universal structure at null infinity. This
is the structure that is common to the conformal completion of _all_
spacetimes that satisfy the definition of asymptotic flatness given in Def.
2.1 and is thus independent of the specific physical spacetime under
consideration.

If $(M,g_{ab},\Omega)$ and $(M^{\prime},g^{\prime}_{ab},\Omega^{\prime})$ are
the unphysical spacetimes corresponding to _any_ two asymptotically-flat
physical spacetimes then, a priori, $M$ and $M^{\prime}$ are distinct
manifolds each with their own boundary $\mathscr{I}$ and
$\mathscr{I}^{\prime}$. However, we argue below that there exists a smooth
diffeomorphism from a neighbourhood of $\mathscr{I}$ in $M$ to a neighbourhood
of $\mathscr{I}^{\prime}$ in $M^{\prime}$ which can be used to identify these
unphysical spacetimes (in this neighbourhood) and which maps $\mathscr{I}$ to
$\mathscr{I}^{\prime}$. Since we are only interested in the the asymptotic
properties near null infinity, we can work with just one manifold $M$ and one
null boundary $\mathscr{I}$ to represent null infinity for any two (and thus,
all) asymptotically-flat spacetimes. Further, _without any loss of generality_
, this diffeomorphism can also be chosen so that $\Omega^{\prime}=\Omega$ in a
neighborhood of $\mathscr{I}$ and
$g^{\prime}_{ab}\mathrel{\mathop{\widehat{=}}}g_{ab}$.

This can be achieved by setting up a suitable, geometrically-defined
coordinate system in a neighborhood of $\mathscr{I}$ and identifying the two
unphysical spacetimes in these coordinates, as we now explain (see also the
argument on p. 22 in Geroch-asymp ). On the null infinity $\mathscr{I}$, of
any asymptotically-flat spacetime, we define a parameter $u$ along the null
generators of $\mathscr{I}$ such that
$n^{a}\nabla_{a}u\mathrel{\mathop{\widehat{=}}}1$. We then pick some cross-
section $S_{0}\cong\mathbb{S}^{2}$ with constant $u=u_{0}$. On $S_{0}$ define
a coordinate system $x^{A}$ (with $A=1,2$), and extend these coordinates to
all of $\mathscr{I}$ by parallel transport along $n^{a}$:

$\displaystyle n^{a}\nabla_{a}x^{A}\mathrel{\mathop{\widehat{=}}}0\,.$ (3.1)

This gives us a coordinate system $(u,x^{A})$ on $\mathscr{I}$. Next, since
$\Omega\mathrel{\mathop{\widehat{=}}}0$ and
$n_{a}\mathrel{\mathop{\widehat{=}}}\nabla_{a}\Omega\not\mathrel{\mathop{\widehat{=}}}0$,
we can use $\Omega$ as a coordinate transverse to $\mathscr{I}$. As discussed
above, there is considerable freedom in the choice of the conformal factor at
$\mathscr{I}$ which we need to fix to specify the choice of coordinate used.
First, as before, we pick the conformal factor so that the Bondi condition
(Eq. 2.6) is satisfied, which leaves us the freedom to change the conformal
factor on the cross-sections of $\mathscr{I}$. To fix this freedom we proceed
as follows. Consider the induced metric $q_{ab}$ on the cross-section $S_{0}$
chosen above. It follows from the _uniformization theorem_ (for instance see
Ch. 8 of Bieri-CK-ext0 )444The uniformization theorem is a _global_ result
depending on the topology of the $2$-dimensional space. Locally, all metrics
of a particular signature on a $2$-surface are conformally-equivalent, Problem
2, Ch. 3 of Wald-book . that any metric on $S_{0}$ is conformal to the unit
round metric on $\mathbb{S}^{2}$ (that is, the metric with constant Ricci
scalar equaling $2$). Thus, we can always choose the conformal factor so that
the metric $q_{ab}$ on this cross-section $S_{0}$ is also the unit round
metric of $\mathbb{S}^{2}$ and from Eq. 2.9, this holds on any cross-section.
Thus, we choose as our transverse coordinate the choice of conformal factor
$\Omega$ which satisfies the Bondi condition and makes the metric on the
cross-sections of $\mathscr{I}$ to be the unit round metric on
$\mathbb{S}^{2}$. This gives us a coordinate system $(\Omega,u,x^{A})$ at
$\mathscr{I}$ for _any_ asymptotically-flat spacetime. Since this construction
can be done for any asymptotically-flat spacetime, we can, without any loss of
generality, identify the null infinities of all asymptotically-flat spacetimes
by identifying their points in these coordinates. This shows that there exists
a diffeomorphism between neighbourhoods of null infinities of different
spacetimes such that we can identify their boundaries with one “abstract”
manifold $\mathscr{I}$ and also that we can choose the same conformal factor
$\Omega$ in this neighbourhood.

Next, we show that the unphysical metric at $\mathscr{I}$ can be chosen to be
the same for the conformal completion of any physical spacetime. Consider a
foliation of $\mathscr{I}$ by cross-sections $S_{u}$ of constant $u$. Then, by
Eq. 3.1 the null generator $n^{a}$, in these coordinates, can be written as
$n^{a}\mathrel{\mathop{\widehat{=}}}\partial/\partial u$. The $1$-form on
$\mathscr{I}$ normal to this foliation is given by
$l_{a}\mathrel{\mathop{\widehat{=}}}-\nabla_{a}u$. Using
$n_{a}=\nabla_{a}\Omega$, $l_{a}\mathrel{\mathop{\widehat{=}}}-\nabla_{a}u$
and $n^{a}l_{a}\mathrel{\mathop{\widehat{=}}}-1$ , we obtain the following
expression for the line element the unphysical metric on $\mathscr{I}$ in the
coordinate system $(\Omega,u,x^{A})$

$\displaystyle ds^{2}\mathrel{\mathop{\widehat{=}}}2d\Omega
du+s_{AB}dx^{A}dx^{B}\,,$ (3.2)

where $s_{AB}$ is the unit round metric on $\mathbb{S}^{2}$ in the chosen
coordinates $x^{A}$.555The precise choice of coordinates $x^{A}$ on the cross-
sections is irrelevant; one could pick polar coordinates $x^{A}=(\theta,\phi)$
to put the unit round sphere metric in the standard form
$s_{AB}dx^{A}dx^{B}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, but any other
coordinate system is just as good. Note that the form of the unphysical metric
Eq. 3.2 is completely independent of which physical spacetime is under
consideration, that is, _all_ asymptotically-flat spacetimes have the same
universal unphysical metric at null infinity. Different choices of the
physical spacetime are only reflected in the unphysical metric _away_ from
$\mathscr{I}$. Note that the definition of asymptotic-flatness (Def. 2.1)
includes an embedding map from the physical spacetime manifold $\hat{M}$ into
the unphysical manifold $M$. The existence of the universal structure at
$\mathscr{I}$ described above implies that one can embed _any_ physical
spacetime into an unphysical spacetime by identifying some physical spacetime
coordinates with the coordinates $(\Omega,u,x^{A})$ constructed above so that
the unphysical metric $g_{ab}$ takes the form Eq. 3.2. In the conformal-
completion formalism this is the version of the statement that
“asymptotically-flat spacetimes behave like the Minkowski spacetime to leading
order at infinity” and the difference between two physical spacetimes only
shows up at “sub-leading order”.

Note that since the manifold $\mathscr{I}$ is universal, the choice of the
foliation can also be made independently of the physical spacetime. It follows
that the auxiliary normal $l_{a}\mathrel{\mathop{\widehat{=}}}-\nabla_{a}u$ is
also universal, and (from Eq. 3.2) we also have
$l^{a}\mathrel{\mathop{\widehat{=}}}-\partial/\partial\Omega$ which is the
auxiliary null normal and is also universal.

Many of the choices made in constructing the coordinates at $\mathscr{I}$ are
irrelevant to this argument and are made just for convenience. For instance,
the choice of the unit-metric $s_{AB}$ is irrelevant. In _any_ asymptotically-
flat spacetime, we can instead choose the freedom in the conformal factor at
$\mathscr{I}$ so that the induced metric on the cross-sections
$q_{ab}=q^{(0)}_{ab}$ where $q^{(0)}_{ab}$ is _any fixed_ metric on
$\mathbb{S}^{2}$. Similarly, one could have chosen a different foliation of
$\mathscr{I}$ if one wishes. These choices simply correspond to the freedom of
choosing the embedding map from the physical spacetime into the unphysical
spacetime. Then the rest of the construction proceeds as before and
$g_{ab}|_{\mathscr{I}}$ is universal.

###  1 Metric perturbations near $\mathscr{I}$

Next we consider linearized perturbations of the physical metric and derive
the conditions on the perturbations arising from requiring asymptotic
flatness. We consider a one-parameter family of physical metrics
$\hat{g}_{ab}(\lambda)$ with $\hat{g}_{ab}=\hat{g}_{ab}(\lambda=0)$ being any
chosen background spacetime metric, and define the physical metric
perturbation by

$\displaystyle\delta\hat{g}_{ab}\mathrel{\mathop{:}}=\left.\frac{d}{d\lambda}\hat{g}_{ab}(\lambda)\right|_{\lambda=0}\,.$
(3.3)

We also use the notation $\delta$ to denote perturbations of other quantities
defined in a similar way as above. We emphasize that the $\delta$ denotes
changes of quantities when the physical metric is varied; the quantities
appearing in the universal structure described above including the unphysical
metric at $\mathscr{I}$, the conformal factor and the choice of foliation do
not vary under $\delta$. Note, this does not mean that the conformal factor
and foliation cannot be changed — our expressions for the charges and fluxes
are independent of the conformal factor and foliation — it means that these
quantities can always be held fixed when the physical metric is varied.

Taking the one-parameter family of physical metrics $\hat{g}_{ab}(\lambda)$ to
all be asymptotically-flat, we want to consider the behaviour of the metric
perturbations at null infinity. Let $g_{ab}(\lambda)$ be the one-parameter
family of unphysical metrics obtained by the conformal completion of
$\hat{g}_{ab}(\lambda)$. As discussed above, without loss of generality we can
take all the unphysical metrics $g_{ab}(\lambda)$ to be defined on the same
manifold $M$, with a common boundary $\mathscr{I}$ describing null infinity.
Further, the conformal factor $\Omega$ can also be chosen to be independent of
the parameter $\lambda$. Thus, we get

$g_{ab}(\lambda)\mathrel{\mathop{:}}=\Omega^{2}\hat{g}_{ab}(\lambda)\,,\quad\delta
g_{ab}=\Omega^{2}\delta\hat{g}_{ab}\,,$ (3.4)

where $\delta g_{ab}$ is the perturbation of the unphysical metric. Moreover,
since the unphysical metric at $\mathscr{I}$ is universal, we have that
$\delta g_{ab}\mathrel{\mathop{\widehat{=}}}0$ and thus,

$\delta g_{ab}\mathrel{\mathop{\widehat{=}}}\Omega\gamma_{ab},$ (3.5)

for some $\gamma_{ab}$ which is smooth at $\mathscr{I}$.

Since the conformal factor is chosen to satisfy the Bondi condition Eq. 2.6 in
_any_ spacetime, varying the Bondi condition we get

$\displaystyle\delta(\nabla_{a}n_{b})\mathrel{\mathop{\widehat{=}}}0\implies\gamma_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}0\,.$
(3.6)

Thus, since $\gamma_{ab}$ is smooth at $\mathscr{I}$, there exists a smooth
$\gamma_{a}$ such that

$\displaystyle\gamma_{ab}n^{b}=\Omega\gamma_{a}\,.$ (3.7)

Thus, the perturbations $\delta g_{ab}$ of the unphysical metric are given by
the tensor fields $\gamma_{ab}$ and $\gamma_{a}$ with

$\displaystyle\gamma_{ab}=\Omega^{-1}\delta
g_{ab}\,,\quad\gamma_{a}=\Omega^{-2}\delta
g_{ab}n^{b}=\Omega^{-1}\gamma_{ab}n^{b}\,.$ (3.8)

These tensor fields are constrained by the linearized vacuum Einstein
equations. A particularly important component of these equations is given by
WZ

$\nabla^{b}\gamma_{ab}-3\gamma_{a}-\nabla_{a}\gamma^{b}{}_{b}\mathrel{\mathop{\widehat{=}}}0.$
(3.9)

Further, using the auxiliary foliation of $\mathscr{I}$ we can relate the
unphysical metric perturbation characterized by $\gamma_{ab}$ to the
perturbation of the shear $\sigma_{ab}$ as follows. Varying the definition of
the shear (Eq. 2.16), noting that the foliation is kept fixed and that $\delta
g_{ab}\mathrel{\mathop{\widehat{=}}}0$, from Eq. 3.8 it is straightforward to
compute that

$\displaystyle\delta\sigma_{ab}=-\tfrac{1}{2}\operatorname{STF}\gamma_{ab}.$
(3.10)

This equation shows that, although $\gamma_{ab}$ is a subleading quantity, its
value is determined by the variation of an object that is constructed from
quantities at $\mathscr{I}$: it only depends on quantities off of
$\mathscr{I}$ through the dependence of $\sigma_{ab}$ on $\nabla_{a}$.

##  4 Asymptotic symmetries at $\mathscr{I}$: the BMS Lie algebra

Consider a smooth vector field $\xi^{a}$ in the physical spacetime $\hat{M}$,
and let $\hat{g}_{ab}(\lambda)$ be the one-parameter family of physical
metrics generated by diffeomorphims along $\xi^{a}$. The physical metric
perturbation corresponding to this family is given by

$\delta_{\xi}\hat{g}_{ab}=\pounds_{\xi}\hat{g}_{ab}.$ (4.1)

The corresponding perturbation of the unphysical metric is

$\delta_{\xi}g_{ab}=\Omega^{2}\pounds_{\xi}\hat{g}_{ab}=\pounds_{\xi}g_{ab}-2\Omega^{-1}\xi^{c}n_{c}g_{ab}.$
(4.2)

For $\xi^{a}$ to be an asymptotic symmetry the infinitesimal diffeomorphism
generated by $\xi^{a}$ must preserve the universal structure discussed in Sec.
3. We now obtain the conditions on $\xi^{a}$ for it to be an asymptotic
symmetry.

Firstly, since the unphysical spacetime $M$ is smooth up to and including
$\mathscr{I}$, $\xi^{a}$ must extend to a smooth vector field on $M$ including
at $\mathscr{I}$. Secondly, since the unphysical metric is smooth at
$\mathscr{I}$, the perturbation $\delta_{\xi}g_{ab}$ in Eq. 4.2 is also smooth
at $\mathscr{I}$. This condition implies that

$\xi^{a}n_{a}\mathrel{\mathop{\widehat{=}}}0.$ (4.3)

That is, as expected, an asymptotic symmetry $\xi^{a}$ must be tangent to
$\mathscr{I}$ and thus preserves the asymptotic boundary. For convenience we
define

$\displaystyle\alpha_{(\xi)}\mathrel{\mathop{:}}=\Omega^{-1}\xi^{a}n_{a}\,,$
(4.4)

which is smooth at $\mathscr{I}$.

Next, the $\delta_{\xi}$ preserves the universal structure, which implies that
(Eq. 3.8)

$\gamma_{ab}^{(\xi)}\mathrel{\mathop{:}}=\Omega^{-1}\delta_{\xi}g_{ab}\,,\quad\gamma_{a}^{(\xi)}\mathrel{\mathop{:}}=\Omega^{-1}\gamma_{ab}^{(\xi)}n^{b}\,,$
(4.5)

must be smooth at $\mathscr{I}$. Using Eqs. 4.2 and 4.4, the smoothness of
$\gamma_{ab}^{(\xi)}$ implies

$\pounds_{\xi}g_{ab}\mathrel{\mathop{\widehat{=}}}2\alpha_{(\xi)}g_{ab},$
(4.6)

while, the smoothness of $\gamma_{a}^{(\xi)}$ implies

$\pounds_{\xi}n^{a}\mathrel{\mathop{\widehat{=}}}-\alpha_{(\xi)}n^{a}\,,\quad\pounds_{n}\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}0.$
(4.7)

The pullback of Eq. 4.6 gives

$\pounds_{\xi}q_{ab}\mathrel{\mathop{\widehat{=}}}2\alpha_{(\xi)}q_{ab}.$
(4.8)

Intrinsically on $\mathscr{I}$, an asymptotic symmetry is, thus, given by a
vector field $\xi^{a}$ tangent to $\mathscr{I}$ which satisfies Eqs. 4.7 and
4.8 for some smooth function $\alpha_{(\xi)}$. Such vector fields are the BMS
symmetries on null infinity.

In any choice of foliation of $\mathscr{I}$ we can further characterize the
BMS symmetries as follows. Since $\xi^{a}$ is tangent to $\mathscr{I}$, we can
write

$\xi^{a}\mathrel{\mathop{\widehat{=}}}\beta n^{a}+X^{a},$ (4.9)

where

$\beta\mathrel{\mathop{:}}=-l_{a}\xi^{a},\qquad
X^{a}\mathrel{\mathop{:}}=Q^{a}{}_{b}\xi^{b},$ (4.10)

We note their conformal weights:

$\displaystyle\beta:w=1\,,\quad X^{a}:w=0\,.$ (4.11)

That is, $\beta$ is a smooth function of conformal weight $1$ and $X^{a}$ is a
smooth vector field tangent to the cross-sections of the chosen foliation on
$\mathscr{I}$. The conditions on $\beta$ and $X^{a}$ follow from Eqs. 4.7 and
4.6 as we derive next. Note that Eqs. 4.7 and 4.6 only depend on the vector
field $\xi^{a}$ at $\mathscr{I}$ and are independent of how this vector field
is extended away from $\mathscr{I}$.

The only non-trivial component of Eq. 4.6 is given by its projection tangent
to the cross-sections of the foliation which gives

$Q_{a}{}^{c}Q_{b}{}^{d}\pounds_{\xi}g_{cd}\mathrel{\mathop{\widehat{=}}}2\mathscr{D}_{(a}X_{b)}\mathrel{\mathop{\widehat{=}}}2\alpha_{(\xi)}Q_{ab},$
(4.12)

which shows both that

$\operatorname{STF}\mathscr{D}_{a}X_{b}\mathrel{\mathop{\widehat{=}}}0,$
(4.13)

and that

$\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\mathscr{D}_{a}X^{a}.$
(4.14)

Next, we consider Eq. 4.7 on $\mathscr{I}$ using Eq. 4.9 to get

$\pounds_{\xi}n^{a}=-\pounds_{n}\xi^{a}\mathrel{\mathop{\widehat{=}}}-n^{a}\pounds_{n}\beta-\pounds_{n}X^{a}\mathrel{\mathop{\widehat{=}}}-\alpha_{(\xi)}n^{a}.$
(4.15)

Projecting along the cross-sections we get

$Q^{a}{}_{b}\pounds_{n}X^{b}\mathrel{\mathop{\widehat{=}}}0,$ (4.16)

whereas $X^{a}l_{a}\mathrel{\mathop{\widehat{=}}}0$, the Bondi condition, and
Eq. 2.18 imply that

$l_{a}\pounds_{n}X^{a}\mathrel{\mathop{\widehat{=}}}-X_{a}\pounds_{n}l^{a}\mathrel{\mathop{\widehat{=}}}-X_{a}\tau^{a},\qquad
n_{a}\pounds_{n}X^{a}\mathrel{\mathop{\widehat{=}}}0,$ (4.17)

so that

$\pounds_{n}X^{a}\mathrel{\mathop{\widehat{=}}}n^{a}X_{b}\tau^{b}.$ (4.18)

Contracting Eq. 4.15 with $l_{a}$, we find that

$\pounds_{n}\beta+l_{a}\pounds_{X}n^{a}\mathrel{\mathop{\widehat{=}}}\pounds_{n}\beta-
X_{a}\tau^{a}\mathrel{\mathop{\widehat{=}}}\alpha_{(\xi)},$ (4.19)

where the second equality follows from Eq. 4.18. As such, we find that Eq.
4.19 becomes

$\pounds_{n}\beta\mathrel{\mathop{\widehat{=}}}\alpha_{(\xi)}-X_{a}\tau^{a}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(\mathscr{D}_{a}-2\tau_{a})X^{a}.$
(4.20)

In summary, in a chosen foliation of $\mathscr{I}$ any BMS symmetry can be
written as $\xi^{a}\mathrel{\mathop{\widehat{=}}}\beta n^{a}+X^{a}$ where
$X^{a}$ is tangent to the cross-sections of the foliation and the following
conditions are satisfied:

$\displaystyle\pounds_{n}\beta$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(\mathscr{D}_{a}-2\tau_{a})X^{a},$
(4.21a) $\displaystyle\pounds_{n}X^{a}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}n^{a}X_{b}\tau^{b},$ (4.21b)
$\displaystyle\operatorname{STF}\mathscr{D}_{a}X_{b}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}0,$ (4.21c)
$\displaystyle\alpha_{(\xi)}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\mathscr{D}_{a}X^{a}.$
(4.21d)

Note that, using Eqs. 4.11, 2.27, 2.30, 4.4 and 2.8, one can show that these
equations are invariant under conformal transformations that preserve Bondi
condition, that is, with $\pounds_{n}\omega\mathrel{\mathop{\widehat{=}}}0$
(for the form of these equations when Bondi condition does not hold, see Eq.
A.7).

Let $\xi^{a}_{1}\mathrel{\mathop{\widehat{=}}}\beta_{1}n^{a}+X^{a}_{1}$ and
$\xi^{a}_{2}\mathrel{\mathop{\widehat{=}}}\beta_{2}n^{a}+X^{a}_{2}$ be two BMS
symmetries (with Eq. 4.21 holding for each) then their Lie bracket can be
computed to give

$\displaystyle\xi^{a}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}[\xi_{1},\xi_{2}]^{a}\mathrel{\mathop{\widehat{=}}}\beta
n^{a}+X^{a}\quad\text{with}$ (4.22) $\displaystyle\beta$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\pounds_{X_{1}}\beta_{2}-\tfrac{1}{2}\beta_{2}\mathscr{D}_{a}X^{a}_{1}-(1\leftrightarrow
2)\,,\quad X^{a}\mathrel{\mathop{\widehat{=}}}\pounds_{X_{1}}X^{a}_{2}\,.$

It can be checked that $\xi^{a}$ is also a BMS symmetry, i.e. the $\beta$ and
$X^{a}$ in Eq. 4.22 also satisfy the conditions Eq. 4.21. Thus, the BMS
symmetries form a Lie algebra $\mathfrak{b}$.

The structure of the BMS algebra can be analyzed using Eq. 4.22. Consider a
BMS symmetry of the form $\xi^{a}_{1}\mathrel{\mathop{\widehat{=}}}f_{1}n^{a}$
where $f_{1}$ is a smooth function on $\mathscr{I}$ satisfying
$\pounds_{n}f_{1}\mathrel{\mathop{\widehat{=}}}0$ (from Eq. 4.21a). Then, from
Eq. 4.22 we see that the Lie bracket of
$\xi^{a}_{1}\mathrel{\mathop{\widehat{=}}}f_{1}n^{a}$ with any other BMS
symmetry is also of the form $\xi^{a}\mathrel{\mathop{\widehat{=}}}fn^{a}$
with $\pounds_{n}f\mathrel{\mathop{\widehat{=}}}0$, that is the set of such
vector fields is invariant under the Lie bracket. Further, the Lie bracket of
any two such symmetries vanishes. Thus, BMS symmetries of the form $fn^{a}$
form a preferred infinite-dimensional abelian subalgebra $\mathfrak{s}$ which
is a _Lie ideal_ of the BMS algebra $\mathfrak{b}$ consisting of
_supertranslations_. The quotient algebra $\mathfrak{b}/\mathfrak{s}$ can then
be parameterized by $X^{a}$. From Eq. 4.21c we see that this consists of
conformal Killing fields on the cross-sections of $\mathscr{I}$. Since the
cross-sections are diffeomorphic to $\mathbb{S}^{2}$ we get that
$\mathfrak{b}/\mathfrak{s}$ is isomorphic to the Lorentz algebra
$\mathfrak{so}(1,3)$. Note that the Lorentz algebra is only identified as a
quotient algebra — since the Lie bracket of a supertranslation and $X^{a}$ is
non-zero, there is no invariant choice of Lorentz subalgebra within the BMS
algebra $\mathfrak{b}$. Thus the BMS algebra is the semi-direct sum

$\displaystyle\mathfrak{b}\cong\mathfrak{s}\ltimes\mathfrak{so}(1,3)\,,$
(4.23)

of the Lie ideal $\mathfrak{s}$ of supertranslations with the Lorentz algebra;
the $\ltimes$ indicates the non-trivial Lie bracket between the two factors.

There is another finite-dimensional Lie ideal within the BMS algebra given by
supertranslations $fn^{a}\in\mathfrak{s}$ which satisfy the additional
condition

$\displaystyle\operatorname{STF}(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})f\mathrel{\mathop{\widehat{=}}}0\,.$
(4.24)

It can be checked that this equation is conformally-invariant. Further, the
space of solutions to this equation is $4$-dimensional and is preserved under
the Lie bracket of the BMS algebra (see Remark 4.1 below). This
$4$-dimensional Lie ideal $\mathfrak{t}$ can be viewed as the space of
_translations_. In fact, if the physical spacetime $(\hat{M},\hat{g}_{ab})$
possesses any Killing vectors $\xi^{a}$, then they can be extended to
$\mathscr{I}$; moreover, if the Killing field is of the form
$\xi^{a}\mathrel{\mathop{\widehat{=}}}fn^{a}$, then $\xi^{a}$ must be a
translation in the sense of $f$ obeying Eq. 4.24 (see AX ).

A special case of the translations are those that are given by a time-
translation in some conformal frame. A time-translation in a given conformal
frame is given by $f=1$, a definition that is motivated by the behavior of the
time-translation vector field in Minkowski (and, similarly, the time-
translation vector field in Kerr). Since $f$ transforms by $f\mapsto\omega f$
under a conformal transformation, a conformally-invariant notion of a time-
translation is given by $f>0$, since $\omega>0$.

######  Remark 4.1 (Characterization of BMS symmetries through spherical
harmonics).

Consider the Bondi frame — where the conformal factor is chosen so that the
metric on the cross-sections is the unit $2$-sphere metric and the foliation
is chosen so that $\tau_{a}\mathrel{\mathop{\widehat{=}}}0$. A general
supertranslation can then be expanded in spherical harmonics on
$\mathbb{S}^{2}$. The condition Eq. 4.24 implies that the translations are
spanned by the $\ell=0,1$ harmonics which is indeed a $4$-dimensional space.
The fact that the translations are preserved under Lie brackets can also be
shown using some spherical harmonics technology (see the appendix in
Flanagan:2019vbl ; Bonga2020 ). Note that since supertranslations have
conformal weight $1$, this spherical harmonic decomposition only holds in the
Bondi frame. Similarly, the Lorentz vector fields satisfying

$\displaystyle\operatorname{STF}\mathscr{D}_{a}X_{b}\mathrel{\mathop{\widehat{=}}}0\,,$
(4.25)

are spanned by vector spherical harmonics with $\ell=1$ (see, e.g.
Flanagan:2015pxa ) which is the $6$-dimensional space of the Lorentz algebra
$\mathfrak{so}(1,3)$.

######  Remark 4.2 (Supertranslation “ambiguity” in the Lorentz algebra).

As noted above, while the supertranslations form a subalgebra of the BMS
algebra, there is no preferred Lorentz subalgebra. Instead, the Lorentz
algebra arises as the quotient algebra $\mathfrak{b}/\mathfrak{s}$: the set of
equivalence classes $[\xi^{a}]$ of $\mathfrak{b}$, where $\xi_{1}^{a}$ and
$\xi_{2}^{a}$ are members of the same equivalence class if
$\xi_{1}^{a}-\xi_{2}^{a}\mathrel{\mathop{\widehat{=}}}fn^{a}\in\mathfrak{s}$.

For a given cross-section $S$, one can consider the algebra $\mathfrak{l}_{S}$
of BMS vector fields of the form $\xi^{a}|_{S}=X^{a}$ (that is, vector fields
tangent to $S$). This is a subalgebra of $\mathfrak{b}$, and in fact it is
straightforward to show that $\mathfrak{l}_{S}\cong\mathfrak{b}/\mathfrak{s}$.
The issue, therefore, is not that there is no Lorentz subalgebra of
$\mathfrak{b}$ but that there is an uncountably infinite number of them, one
per cross-section of $\mathscr{I}$.

The fact that the members of the quotient algebra $\mathfrak{b}/\mathfrak{s}$
are only defined up to supertranslations is known as the _supertranslation
ambiguity_. A similar situation occurs with the Poincaré algebra, which, while
possessing a Lie ideal in the form of the algebra of translations, possesses
no unique Lorentz subalgebra. For each origin in Minkowski space, there is a
Lorentz subalgebra that consists of infinitesimal rotations and boosts that
fix that point, which is analogous to the Lorentz subalgebra
$\mathfrak{l}_{S}$ associated with some cross-section $S$ of $\mathscr{I}$.

Much like how there is a Lorentz subalgebra associated with each cross-section
of $\mathscr{I}$, there is moreover a Poincaré subalgebra, given by
considering the semidirect product of this Lorentz subalgebra with the
translation subalgebra picked out by Eq. 4.24. This Poincaré subalgebra can be
shown to contain any _exact_ Killing vector field that might exist in the
physical spacetime (see Theorem 1 of AX ). Similarly, in Sec. C.3, we review a
construction (based on twistors) of a Poincaré subalgebra, once again
depending explicitly on a choice of cross-section $S$. Finally, there is no
invariant notion of a set of charges conjugate to Lorentz vector fields, which
would form an invariant notion of angular momentum. Such a set of charges only
exist for a given cross-section, so that (in particular) one cannot have an
invariant notion of the flux of angular momentum between two cross-sections.
This is similar to how, since the Poincaré algebra has no unique notion of a
Lorentz subalgebra, with each origin determining a particular Lorentz
subalgebra, the notion of angular momentum in flat space is origin-dependent.

###  1 Extensions of BMS symmetries away from $\mathscr{I}$

Up to this point, although we started this discussion with vector fields
$\xi^{a}$ defined throughout the unphysical spacetime, we have worked mostly
with the value of $\xi^{a}$ on $\mathscr{I}$ itself. The extension of the BMS
symmetries away from $\mathscr{I}$ into the spacetime is arbitrary — one could
choose some coordinate system or gauge conditions to obtain a particular
extension, but then one needs to verify that any physical quantities (like
charges and fluxes) associated with the BMS symmetries are independent of such
choices. We collect some of the relevant results on the extensions of BMS
symmetries away from $\mathscr{I}$ below.

First, we show that the extension of $\xi^{a}$ away from $\mathscr{I}$ is
determined up to $O(\Omega^{2})$:

######  Proposition 4.1 (Equivalent representatives of a BMS symmetry).

If $\xi^{a}$ and ${\xi^{\prime}}^{a}$ are vector fields in $M$ which represent
the same BMS symmetry, i.e.,
$\xi^{a}\mathrel{\mathop{\widehat{=}}}{\xi^{\prime}}^{a}\in\mathfrak{b}$ then
${\xi^{\prime}}^{a}=\xi^{a}+O(\Omega^{2})$.

###### Proof.

Since ${\xi^{\prime}}^{a}\mathrel{\mathop{\widehat{=}}}\xi^{a}$, let
${\xi^{\prime}}^{a}=\xi^{a}+\Omega Z^{a}$ from which we obtain

$\displaystyle\alpha_{(\xi^{\prime})}-\alpha_{(\xi)}=\Omega^{-1}n_{a}({\xi^{\prime}}^{a}-\xi^{a})=n_{a}Z^{a}\,.$
(4.26)

Since
$\pounds_{\xi}g_{ab}-2\alpha_{(\xi)}g_{ab}\mathrel{\mathop{\widehat{=}}}0$ for
any BMS vector field we have

$\displaystyle
0\mathrel{\mathop{\widehat{=}}}\left(\pounds_{\xi^{\prime}}g_{ab}-2\alpha_{(\xi^{\prime})}g_{ab}\right)-\left(\pounds_{\xi}g_{ab}-2\alpha_{(\xi)}g_{ab}\right)\mathrel{\mathop{\widehat{=}}}2n_{(a}Z_{b)}-2n_{c}Z^{c}g_{ab}\,.$
(4.27)

Taking the trace gives $n_{a}Z^{a}\mathrel{\mathop{\widehat{=}}}0$. Then we
have $n_{(a}Z_{b)}\mathrel{\mathop{\widehat{=}}}0$ which implies
$Z^{a}\mathrel{\mathop{\widehat{=}}}0$. Thus $Z^{a}=O(\Omega)$ and
${\xi^{\prime}}^{a}=\xi^{a}+O(\Omega^{2})$. ∎

The perturbation $\gamma_{ab}^{(\xi)}$ generated by a BMS symmetry
$\xi^{a}|_{\mathscr{I}}$ will, in general, depend on the extension of the
symmetry away from $\mathscr{I}$ and hence is _not_ well-defined for the BMS
symmetries. However, using the above lemma it can be shown that
$\operatorname{STF}\gamma_{ab}^{(\xi)}$ on $\mathscr{I}$ is, in fact,
independent of the extension of the BMS symmetry $\xi^{a}$ away from
$\mathscr{I}$, and thus is well-defined for any BMS symmetry.

######  Corollary 4.1.

If $\xi^{a}$ and ${\xi^{\prime}}^{a}$ are any two extensions of a given BMS
symmetry $\xi^{a}|_{\mathscr{I}}$ away from $\mathscr{I}$ then

$\displaystyle\operatorname{STF}\gamma_{ab}^{(\xi)}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\gamma_{ab}^{(\xi^{\prime})}.$
(4.28)

###### Proof.

From Prop. 4.1, we have ${\xi^{\prime}}^{a}=\xi^{a}+\Omega^{2}W^{a}$, from
which (using Eq. 4.4) we compute

$\displaystyle\alpha_{(\xi^{\prime})}=\Omega^{-1}n_{a}{\xi^{\prime}}^{a}=\Omega^{-1}n_{a}\xi^{a}+\Omega
n_{a}W^{a}=\alpha_{(\xi)}+\Omega n_{a}W^{a}\,.$ (4.29)

Similarly

$\displaystyle\pounds_{\xi^{\prime}}g_{ab}=\pounds_{\xi}g_{ab}+4\Omega
n_{(a}W_{b)}+O(\Omega^{2})\,.$ (4.30)

Thus, we find that

$\displaystyle\Omega\gamma_{ab}^{(\xi^{\prime})}$
$\displaystyle=\pounds_{\xi^{\prime}}g_{ab}-2\alpha_{(\xi^{\prime})}g_{ab}=\pounds_{\xi}g_{ab}-2\alpha_{(\xi)}g_{ab}+4\Omega\left(n_{(a}W_{b)}-\tfrac{1}{2}n_{c}W^{c}g_{ab}\right)+O(\Omega^{2})$
(4.31) $\displaystyle\implies\gamma_{ab}^{(\xi^{\prime})}$
$\displaystyle=\gamma_{ab}^{(\xi)}+4\left(n_{(a}W_{b)}-\tfrac{1}{2}n_{c}W^{c}g_{ab}\right)+O(\Omega)\,.$

Evaluating the $\operatorname{STF}$ of both sides on $\mathscr{I}$ of the
above equation we find the desired result. ∎

For later computations it will be useful to have an explicit expression for
$\operatorname{STF}\gamma_{ab}^{(\xi)}$ at $\mathscr{I}$ for any BMS symmetry
$\xi^{a}$ in terms of the fields defined on $\mathscr{I}$. We show below that

$\operatorname{STF}\gamma_{ab}^{(\xi)}\mathrel{\mathop{\widehat{=}}}-2\operatorname{STF}\left[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}\right].$
(4.32)

Comparing the above formula to Eq. 3.10 we find that under a BMS symmetry the
shear transforms as

$\displaystyle\delta_{\xi}\sigma_{ab}=\operatorname{STF}\left[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}\right]\,.$
(4.33)

This is the infinitesimal version of the transformation of the shear found by
Sachs Sachs1 . It is also the same as that given in, for example, Eq. 2.18b of
Flanagan:2015pxa  for the transformation of the shear $C_{AB}$ in Bondi
coordinates (which is related to $\sigma_{ab}$ by Eq. 6.10).

In the remainder of this section we detail the computations which lead to Eq.
4.32. Let $\xi^{a}|_{\mathscr{I}}=\beta n^{a}+X^{a}$ be a BMS symmetry on
$\mathscr{I}$. In the following, let $l^{a}$ be _any_ vector field in a
neighbourhood of $\mathscr{I}$ which coincides with the chosen auxiliary
normal at $\mathscr{I}$ — the result of the computation can be checked to be
independent of how the auxiliary normal is extended away from $\mathscr{I}$.

As shown above in Cor. 4.1,
$\operatorname{STF}\gamma_{ab}^{(\xi)}\big{|}_{\mathscr{I}}$ is independent of
how the BMS symmetry $\xi^{a}|_{\mathscr{I}}$ is extended away from
$\mathscr{I}$. Thus, we can choose to extend the BMS symmetry away from
$\mathscr{I}$ as follows. We first extend $\beta$ and $X^{a}$ away from
$\mathscr{I}$ to satisfy

$\pounds_{l}\beta\mathrel{\mathop{\widehat{=}}}0,\qquad\pounds_{l}X^{a}\mathrel{\mathop{\widehat{=}}}0.$
(4.34)

With this choice any extension of the BMS symmetry $\xi^{a}|_{\mathscr{I}}$
can be written as

$\xi^{a}=\beta n^{a}+X^{a}+\Omega Z^{a}.$ (4.35)

where $Z^{a}$ is some smooth vector field. Next from Prop. 4.1 we see that
$Z^{a}|_{\mathscr{I}}$ can be determined in terms of $\beta$ and $X^{a}$. To
do this, first note that, by Eq. 4.4,

$\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}-\pounds_{l}(\xi^{a}n_{a})\mathrel{\mathop{\widehat{=}}}-n_{a}\pounds_{l}\xi^{a}\mathrel{\mathop{\widehat{=}}}n_{a}Z^{a}.$
(4.36)

Next, we compute $l^{b}\pounds_{\xi}g_{ab}$, using Eq. 4.6, together with Eq.
4.34:

$l^{b}\pounds_{\xi}g_{ab}\mathrel{\mathop{\widehat{=}}}-\nabla_{a}\beta+\pounds_{X}l_{a}-Z_{a}+n_{a}l^{b}Z_{b}\mathrel{\mathop{\widehat{=}}}2\alpha_{(\xi)}l_{a}.$
(4.37)

Contracting this equation with $l^{a}$, using Eq. 4.34 and
$l^{a}\pounds_{X}l_{a}\mathrel{\mathop{\widehat{=}}}-l_{a}\pounds_{X}l^{a}\mathrel{\mathop{\widehat{=}}}0$,
we find that $l_{a}Z^{a}\mathrel{\mathop{\widehat{=}}}0$. Therefore, we can
rearrange Eq. 4.37 to solve for $Z_{a}$:

$Z_{a}\mathrel{\mathop{\widehat{=}}}-\nabla_{a}\beta-2\alpha_{(\xi)}l_{a}+\pounds_{X}l_{a}.$
(4.38)

Further, since $l_{a}$ is the normal and $X^{a}$ is tangent to the cross-
sections of the chosen foliation, we have
$Q_{a}{}^{b}\pounds_{X}l_{b}\mathrel{\mathop{\widehat{=}}}0$. So

$Z_{a}\mathrel{\mathop{\widehat{=}}}-\mathscr{D}_{a}\beta+l_{a}(\pounds_{n}\beta-2\alpha_{(\xi)}+l_{b}\pounds_{X}n^{b})\mathrel{\mathop{\widehat{=}}}-\mathscr{D}_{a}\beta-\alpha_{(\xi)}l_{a},$
(4.39)

where we have used Eqs. 4.21b and 4.21a. In summary, with the choice Eq. 4.34,
we can write any extension of a given BMS symmetry $\xi^{a}|_{\mathscr{I}}$ as

$\xi^{a}=\beta
n^{a}+X^{a}-\Omega(\mathscr{D}^{a}\beta+\alpha_{(\xi)}l^{a})+\Omega^{2}W^{a},$
(4.40)

for some smooth $W^{a}$.

With this choice of extension of the BMS symmetry, we compute
$\operatorname{STF}\gamma_{ab}^{(\xi)}\big{|}_{\mathscr{I}}$ in terms of
$\beta$ and $X^{a}$. Using the expansion in Eq. 4.40 we have

$\displaystyle\Omega\gamma_{ab}^{(\xi)}$
$\displaystyle=\pounds_{\xi}g_{ab}-2\alpha_{(\xi)}g_{ab}$ (4.41)
$\displaystyle=\beta\pounds_{n}g_{ab}+\pounds_{X}g_{ab}-2\alpha_{(\xi)}g_{ab}-2n_{(a}[\mathscr{D}_{b)}\beta+\alpha_{(\xi)}l_{b)}]$
$\displaystyle\hskip
10.00002pt-\Omega[2\nabla_{a}\nabla_{b}\beta+\alpha_{(\xi)}\pounds_{l}g_{ab}+2l_{(a}\nabla_{b)}\alpha_{(\xi)}-4n_{(a}W_{b)}]+\Omega^{2}\pounds_{W}g_{ab}$

We now want to solve this equation for
$\operatorname{STF}\gamma_{ab}^{(\xi)}\big{|}_{\mathscr{I}}$. This can be done
by taking the $\pounds_{l}$ of the above equation, evaluating on
$\mathscr{I}$, and then taking the $\operatorname{STF}$. Using
$\pounds_{l}\Omega\mathrel{\mathop{\widehat{=}}}l^{a}n_{a}=-1$ and that
$\pounds_{l}n_{a}\propto n_{a}$ at $\mathscr{I}$, a long but straightforward
computation gives

$\operatorname{STF}\gamma_{ab}^{(\xi)}\mathrel{\mathop{\widehat{=}}}-\operatorname{STF}[\beta\pounds_{l}\pounds_{n}g_{ab}+\pounds_{l}\pounds_{X}g_{ab}+2\mathscr{D}_{a}\mathscr{D}_{b}\beta-\alpha_{(\xi)}\pounds_{l}g_{ab}],$
(4.42)

where we have used Eq. 4.34. Note that by Eq. 4.34, we find that

$\operatorname{STF}\pounds_{l}\pounds_{X}g_{ab}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\pounds_{X}\pounds_{l}g_{ab}\mathrel{\mathop{\widehat{=}}}2\operatorname{STF}\pounds_{X}\sigma_{ab},$
(4.43)

where the final equality can be shown using Eqs. 2.16 and 4.21c, together with
the fact that
$X^{a}n_{a}\mathrel{\mathop{\widehat{=}}}X^{a}l_{a}\mathrel{\mathop{\widehat{=}}}0$.
Combining Eqs. 4.42 and 4.43, together with Eqs. 2.16, 2.35 and 4.21d, we find
that

$\operatorname{STF}\gamma_{ab}^{(\xi)}=-2\operatorname{STF}\left[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}\right]\,,$
(4.44)

as claimed in Eq. 4.32.

##  5 Asymptotic charges and fluxes: The Wald-Zoupas prescription

The prescription of Wald and Zoupas WZ  provides a method of determining
charges and fluxes at null infinity, and can be applied to any local and
covariant theory. However, for simplicity, we will be specializing to the case
of vacuum general relativity. Note that the addition of matter, for example,
does not significantly complicate this discussion (see for example, WZ ;
Bonga2019 ).

We start with the Einstein-Hilbert Lagrangian $4$-form $\boldsymbol{L}$:

$\boldsymbol{L}=\frac{1}{16\pi}\hat{R}~{}\hat{\boldsymbol{\varepsilon}}_{4}.$
(5.1)

The dynamical field is the physical metric $\hat{g}_{ab}$, and varying this
dynamical field we obtain

$\delta\boldsymbol{L}=\boldsymbol{E}^{ab}\delta\hat{g}_{ab}+d\boldsymbol{\theta}(\delta\hat{g}),$
(5.2)

where the $3$-form $\boldsymbol{\theta}$ is the _symplectic potential_ , and
$\boldsymbol{E}^{ab}$ is a tensor-valued $4$-form which gives the equations of
motion in the form $\boldsymbol{E}^{ab}=0$. For general relativity, we have
Burnett1990

$\displaystyle\boldsymbol{E}^{ab}$
$\displaystyle=-\frac{1}{16\pi}\hat{\boldsymbol{\varepsilon}}_{4}\hat{G}^{ab},$
(5.3a) $\displaystyle\boldsymbol{\theta}(\delta\hat{g})$
$\displaystyle\equiv-\frac{1}{8\pi}\hat{\varepsilon}_{abc}{}^{[d}\hat{g}^{e]f}\hat{\nabla}_{e}\delta\hat{g}_{df},$
(5.3b)

The _symplectic current_ is defined by taking a second, independent variation
of the symplectic potential and antisymmetrizing in the perturbations:

$\boldsymbol{\omega}(\delta_{1}\hat{g},\delta_{2}\hat{g})\mathrel{\mathop{:}}=\delta_{1}\boldsymbol{\theta}(\delta_{2}\hat{g})-\delta_{2}\boldsymbol{\theta}(\delta_{1}\hat{g}).$
(5.4)

Computing $d\boldsymbol{\omega}$ from this equation, using the fact that $d$
and $\delta$ commute, using the second variation of Eq. 5.2, and using the
fact that $\delta_{1}$ and $\delta_{2}$ commute, one finds that (for example,
Burnett1990 )

$d\boldsymbol{\omega}(\delta_{1}\hat{g},\delta_{2}\hat{g})=\delta_{1}\boldsymbol{E}^{ab}\delta_{2}\hat{g}_{ab}-\delta_{2}\boldsymbol{E}^{ab}\delta_{1}\hat{g}_{ab},$
(5.5)

which vanishes whenever the perturbations satisfy the linearized equations of
motion $\delta\boldsymbol{E}^{ab}=0$. When the dynamical fields $\hat{g}_{ab}$
satisfy the equations of motion, and $\delta\hat{g}_{ab}$ satisfy the
linearized equations of motion, one can show that (see Lee1990 ; Iyer1994 ;
Prabhu2015 )

$\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})=d\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right],$
(5.6)

for all infinitesimal diffeomorphisms generated by $\xi^{a}$, where the
$2$-form $\boldsymbol{Q}_{\xi}$ is the _Noether charge_ associated with
$\xi^{a}$. In general relativity, we have that Burnett1990 ; WZ

$\boldsymbol{\omega}(\delta_{1}\hat{g},\delta_{2}\hat{g})\equiv\frac{1}{16\pi}\hat{\varepsilon}_{dabc}\hat{P}^{defghi}\left[\delta_{2}\hat{g}_{ef}\hat{\nabla}_{g}\delta_{2}\hat{g}_{hi}-(1\leftrightarrow
2)\right],$ (5.7)

where

$\hat{P}^{abcdef}\mathrel{\mathop{:}}=\hat{g}^{ae}\hat{g}^{fb}\hat{g}^{cd}-\tfrac{1}{2}\hat{g}^{ad}\hat{g}^{be}\hat{g}^{fc}-\tfrac{1}{2}\hat{g}^{ab}\hat{g}^{cd}\hat{g}^{ef}-\tfrac{1}{2}\hat{g}^{bc}\hat{g}^{ae}\hat{g}^{fd}+\tfrac{1}{2}\hat{g}^{bc}\hat{g}^{ad}\hat{g}^{ef}.$
(5.8)

Moreover, the Noether charge is given by Iyer1994

$\boldsymbol{Q}_{\xi}\equiv-\frac{1}{16\pi}\hat{\varepsilon}_{cdab}\hat{\nabla}^{c}\xi^{d}.$
(5.9)

The symplectic current introduced above, when integrated over some
hypersurface $\Sigma$, provides a symplectic product on phase space. As
discussed in Lee1990 ; WZ , a perturbed Hamiltonian should be constructed by
considering the symplectic product of an arbitrary perturbation $\delta
g_{ab}$ with $\delta_{\xi}\hat{g}_{ab}=\pounds_{\xi}\hat{g}_{ab}$:

$\delta
H_{\xi}=\int\limits_{\Sigma}\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})=\int\limits_{\Sigma}d\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]=\int\limits_{\partial\Sigma}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right],$
(5.10)

where the second equality follows by Eq. 5.6 and $\partial\Sigma$ is the
boundary of $\Sigma$. Thus, the Hamiltonian can then naturally be thought of
as an integral over the boundary $\partial\Sigma$.

Now consider the case where the hypersurface $\Sigma$ extends as a smooth
surface to $\mathscr{I}$ in the unphysical spacetime which intersects
$\mathscr{I}$ at a cross-section $S$. We take Eq. 5.10, rewritten in terms of
the unphysical fields which are smooth at $\mathscr{I}$. In general
relativity, using the behaviour of the unphysical metric perturbations
detailed in Sec. 3.1, the symplectic current has a finite limit to
$\mathscr{I}$; as shown in WZ . However, one should not conclude from Eq. 5.6
that $\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})$
has a limit to $\mathscr{I}$; in general relativity it can be again shown that
$\boldsymbol{\theta}(\delta\hat{g})$ again has a finite limit to $\mathscr{I}$
(see WZ ), but $\delta\boldsymbol{Q}_{\xi}$ diverges in the limit to
$\mathscr{I}$. Note that any procedure to “subtract out the diverging part” is
highly non-unique. Fortunately, there is no need to resort to any such ad hoc
procedure as we can use some elementary differential geometry to proceed
directly as follows.

######  Lemma 5.1.

Let $S^{\prime}$ be some sequence of $2$-spheres in unphysical spacetime which
limits (continuously) to a chosen cross-section $S$ of $\mathscr{I}$. Then,
the limiting integral

$\displaystyle\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]$
(5.11)

defined by _first_ integrating
$\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})$ over
the sequence of $2$-spheres $S^{\prime}$ and then taking the limit as the
sequence tends to $S\subset\mathscr{I}$ exists and is independent of the
chosen sequence of $2$-spheres used in the limit.

###### Proof.

Let $S_{0}$ be some $2$-surface in the unphysical spacetime and let $\Sigma$
be a smooth $3$-surface which extends from $S_{0}$ and intersects
$\mathscr{I}$ at a cross-section $S$. Note that this surface $\Sigma$ is a
_compact_ $3$-manifold in the unphysical spacetime. The integral of
$\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})$ over $\Sigma$ is
necessarily finite since
$\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})$ is a continuous
$3$-form on $\Sigma$ (including at the “boundary” at $S$). Thus integrating
Eq. 5.6 over $\Sigma$ we obtain666We note that on the left-hand side, we have
taken $\Sigma$ to be future-oriented while on the right-hand side, the
$2$-spheres _do not_ have the usual outward-facing orientation within
$\Sigma$, but the opposite. This choice of orientation is more natural, since
(as mentioned in Remark 2.1), the limit of the orientation as $S^{\prime}\to
S$ is the future-directed orientation of $S$ within $\mathscr{I}$ as specified
by $\boldsymbol{\varepsilon}_{2}$. This choice of orientation is the opposite
of the one that is used by Wald and Zoupas (see footnotes 2, 3, and 8 of WZ ),
and so some of our equations have the opposite sign.

$\int\limits_{\Sigma}\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})=\int\limits_{S_{0}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]-\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right],$
(5.12)

where the last expression on the right-hand side means “integrate over some
$2$-sphere $S^{\prime}\subset\Sigma$, and then take the limit of this
$2$-sphere to the boundary $S$”. This limiting procedure is necessary because,
although the integral on the left-hand side of Eq. 5.12 is always finite (as
argued above), the $2$-form integrand on the right-hand side need not have a
finite limit to $\mathscr{I}$ in general. It follows that the limit in the
last expression on the right-hand side of Eq. 5.12 exists. The limit is also
independent of the choice of sequence of $S^{\prime}$ that is used in its
definition due to Eq. 5.6 and the fact that $\Sigma$ is compact and
$\boldsymbol{\omega}(\delta\hat{g},\pounds_{\xi}\hat{g})$ is continuous on
$\Sigma$. Further, by Stokes’ theorem, this expression is the same for _any_
choice of the $3$-surface $\Sigma$ whose boundary is also $S$, since
$d\boldsymbol{\omega}(\delta\hat{g},\delta_{\xi}\hat{g})=0$, by Eq. 5.5 and
that $\boldsymbol{\omega}(\delta\hat{g},\delta_{\xi}\hat{g})$ extends
continuously to $\mathscr{I}$. ∎

######  Remark 5.1 (Necessity of $\boldsymbol{\omega}$ extending continuously
to $\mathscr{I}$).

We emphasize that condition that
$\boldsymbol{\omega}(\delta\hat{g},\delta_{\xi}\hat{g})$ extend continuously
(as a $3$-form in $M$) to $\mathscr{I}$ is necessary in above argument. For
instance, if we instead only assume that
$\int\limits_{\Sigma}\boldsymbol{\omega}(\delta\hat{g},\delta_{\xi}\hat{g})$
is finite on _every_ surface $\Sigma$ (or that the pullback of
$\boldsymbol{\omega}(\delta\hat{g},\delta_{\xi}\hat{g})$ to _every_ $\Sigma$
is continuous within $\Sigma$), then $d\boldsymbol{\omega}=0$ does _not_ imply
that the limit of
$\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]$
is well-defined since it can depend on the choice of surface $\Sigma$ used to
define the limiting integral. As this point is often overlooked in the
application of Stokes’ theorem at $\mathscr{I}$ we provide a simple example on
the Euclidean plane below.

On $\mathbb{R}^{2}$, let $(x,y)$ be the usual Cartesian coordinates. Consider
the $1$-form

$\displaystyle\boldsymbol{\omega}\equiv\frac{1}{(x^{2}+y^{2})^{{{}^{3}\\!\\!/\\!_{2}}}}\left[y^{2}~{}dx-
xy~{}dy\right]\,.$ (5.13)

This $1$-form can be written as the exterior derivative of a $0$-form (i.e., a
function) as777The choice of the constant in the function $Q$ is irrelevant
for our argument.

$\displaystyle\boldsymbol{\omega}=dQ\,,\quad
Q=\frac{x}{(x^{2}+y^{2})^{{}^{1}\\!\\!/\\!_{2}}}\,.$ (5.14)

Note that $\boldsymbol{\omega}$ does not extend continuously to the origin
$(x,y)=(0,0)$, but $d\boldsymbol{\omega}=0$ everywhere else and extends
continuously to the origin. Further, it can be checked that the pullback of
this $\boldsymbol{\omega}$ to any smooth curve $\Sigma$ through the origin is
continuous at the origin _within_ this curve. However, we _cannot_ use Stokes’
theorem to conclude that $\int\limits_{\Sigma}\boldsymbol{\omega}$ is
independent of the choice of curve $\Sigma$ joining the origin to some other
point, since $\boldsymbol{\omega}$ is not continuous at the origin as a
$1$-form in $\mathbb{R}^{2}$ . A direct computation shows that for any curve
$\Sigma$ from the origin to $(1,1)$, we have

$\displaystyle\int\limits_{\Sigma}\boldsymbol{\omega}=\tfrac{1}{\sqrt{2}}-\cos\theta\,,$
(5.15)

where $\theta$ is the angle with the $x$-axis of the tangent of $\Sigma$ at
the origin. So the integral of $\boldsymbol{\omega}$ along any curve through
the origin is finite but depends on the curve $\Sigma$. Thus, we cannot define
the value of $Q$ at the origin by taking such integrals over curves $\Sigma$
since it depends on the curve used in the limiting procedure. This can be
explicitly checked from the expression of $Q$ given above.

Next we show that, in general relativity, the limiting integral of
$\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]$
is independent of the choice of extension of the BMS symmetry away from
$\mathscr{I}$.

######  Lemma 5.2.

If $\xi^{a}$ and $\xi^{\prime a}$ are equivalent representatives of a BMS
symmetry on $\mathscr{I}$ in general relativity then

$\displaystyle\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]=\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi^{\prime}}-\xi^{\prime}\cdot\boldsymbol{\theta}(\delta\hat{g})\right],$
(5.16)

for all background spacetimes, all perturbations $\delta\hat{g}_{ab}$ and all
cross-sections $S$ of $\mathscr{I}$.

###### Proof.

From Prop. 4.1, two equivalent representatives of a BMS symmetry are related
by $\xi^{\prime a}=\xi^{a}+\Omega^{2}W^{a}$ for some smooth $W^{a}$. Hence to
prove the desired result we only need to show that the above integral computed
for the vector field $\Omega^{2}W^{a}$ vanishes. In general relativity it can
be shown that the $3$-form $\boldsymbol{\theta}(\delta\hat{g})$ is finite at
$\mathscr{I}$ WZ  and thus
$(\Omega^{2}W)\cdot\boldsymbol{\theta}(\delta\hat{g})\mathrel{\mathop{\widehat{=}}}0$.
So we only need to the compute the Noether charge term which can be written as

$\displaystyle\boldsymbol{Q}_{\Omega^{2}W}\equiv-\frac{1}{16\pi}\hat{\varepsilon}_{abcd}\hat{\nabla}^{c}(\Omega^{2}W^{d})=-\frac{1}{16\pi}\varepsilon_{abcd}\nabla^{c}W^{d}\,.$
(5.17)

Note that this is manifestly finite at $\mathscr{I}$ and so we can dispense
with the limiting procedure used in the integral and evaluate the variation of
the above expression directly at $\mathscr{I}$. Using the asymptotic
conditions on the metric perturbations (Eq. 3.8), we get (we ignore the
overall signs and factors)

$\displaystyle\delta\boldsymbol{Q}_{\Omega^{2}W}\big{|}_{\mathscr{I}}\propto\varepsilon_{abcd}n^{c}\gamma^{d}{}_{e}W^{e}\,.$
(5.18)

Using the definition of the volume-forms (Eq. 2.14) and
$n_{a}l^{a}\mathrel{\mathop{\widehat{=}}}-1$, the integral over the cross-
section $S$ is then (using
$\gamma_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}\Omega\gamma_{a}$)

$\displaystyle\int\limits_{S}\delta\boldsymbol{Q}_{\Omega^{2}W}\propto\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}n^{a}\gamma_{ab}W^{b}\mathrel{\mathop{\widehat{=}}}\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\Omega\gamma_{a}W^{a}\mathrel{\mathop{\widehat{=}}}0\,.$
(5.19)

∎

The results of Lemmas 5.1 and 5.2 prove that the limiting integral of
$\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]$
is well-defined on BMS symmetries at $\mathscr{I}$. Thus, from Eq. 5.12, it
would be natural to define a charge associated with the asymptotic symmetry
$\xi^{a}$ at $S$ as a function $Q[\xi;S]$ in the phase space of the theory
such that

$\delta Q[\xi;S]\mathrel{\mathop{:}}=\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right],$
(5.20)

for all backgrounds $\hat{g}_{ab}$, all perturbations $\delta\hat{g}_{ab}$,
and all cross-sections $S$. However, in general, no such function $Q[\xi;S]$
exists, since the right-hand side is not integrable in phase space; that is,
it cannot be written as the variation of some quantity for _all_
perturbations. To see this, suppose that the charge defined in Eq. 5.20 does
exist. Then, one must have
$(\delta_{1}\delta_{2}-\delta_{2}\delta_{1})Q[\xi;S]=0$ for _all_ backgrounds
$\hat{g}_{ab}$ and _all_ perturbations $\delta_{1}\hat{g}_{ab}$, and
$\delta_{2}\hat{g}_{ab}$ (satisfying the corresponding equations of motion).
However, it follows from Eqs. 5.20 and 5.4 and the commutativity of
$\delta_{1}$ and $\delta_{2}$ that

$(\delta_{1}\delta_{2}-\delta_{2}\delta_{1})Q[\xi,S]=-\int\limits_{S}\xi\cdot\boldsymbol{\omega}(\delta_{1}\hat{g},\delta_{2}\hat{g}).$
(5.21)

Thus, a charge defined by Eq. 5.20 will exist if the right-hand side of the
above equation vanishes. This is the case if $\xi^{a}$ is tangent to $S$.
However, in general, the right-hand side is non-vanishing, and so one cannot
define any charge $Q[\xi;S]$ using Eq. 5.20.

######  Remark 5.2 (Subtracting the “non-integrable part”).

It might be tempting to simply stare at some explicit expression for the
right-hand-side of Eq. 5.20 and then subtract off the “non-integrable part” of
the expression to obtain an expression which is manifestly integrable to
define the charge at $S$. But this “procedure” is very ad hoc; for instance
suppose one manages to write the right-hand side of Eq. 5.20 as $\delta A+B$
where $\delta A$ is a manifestly integrable expression and $B$ is not.
However, one can trivially write this in the alternative form $\delta
A+B=\delta(A+C)+(B-\delta C)$ where $C$ is some tensorial expression in terms
of the available fields. Obviously, $\delta(A+C)$ is integrable while
$(B-\delta C)$ is non-integrable for any choice of $C$. Thus the procedure to
“subtract off the non-integrable part” is highly ambiguous — without any
additional criteria one cannot know which “non-integrable part” $B$ or
$B-\delta C$ should be “subtracted off”.

The obstruction to the non-integrability of Eq. 5.20 was resolved by the
rather general prescription of Wald and Zoupas WZ . Their procedure for
defining integrable charges associated with asymptotic symmetries can be
summarized as follows: let $\boldsymbol{\Theta}$ be a symplectic potential for
the pullback of the symplectic current to $\mathscr{I}$; that is,

$\underleftarrow{\boldsymbol{\omega}}(\delta_{1}\hat{g},\delta_{2}\hat{g})=\delta_{1}\boldsymbol{\Theta}(\delta_{2}\hat{g})-\delta_{2}\boldsymbol{\Theta}(\delta_{1}\hat{g}),$
(5.22)

for _all_ backgrounds and _all_ perturbations, with appropriate asymptotic
conditions and equations of motion imposed. Following WZ , we require that the
choice of $\boldsymbol{\Theta}$ satisfies the following properties:

  1. (1)

$\boldsymbol{\Theta}$ must be locally and covariantly constructed out of the
dynamical fields $g_{ab}$, the perturbations $\delta g_{ab}$, and finitely
many of their derivatives, along with any fields in the “universal background
structure” present at $\mathscr{I}$ described in Sec. 3;

  2. (2)

$\boldsymbol{\Theta}$ must be independent of any arbitrary choices made in
specifying the background structure; that is, $\boldsymbol{\Theta}$ is
conformally invariant and independent of the choice of the auxiliary normal
$l^{a}$; and

  3. (3)

if $g_{ab}$ is a stationary background solution, then
$\boldsymbol{\Theta}(\hat{g};\delta\hat{g})=0$, for _all_ (not necessarily
stationary) perturbations $\delta g_{ab}$.

The first of these criteria is motivated by the fact that, as laid out in the
introduction, the prescription used to define charges should only require the
tensor fields that exist on $\mathscr{I}$, and should not depend on additional
structure that can be used (for example, some choice of coordinates). The
second is required to ensure that the charges that are defined by this
prescription are associated with the physical spacetime and do not depend that
the particular choices that go into the conformal-completion (such as the
conformal factor) or other additional choices like the foliation of
$\mathscr{I}$. The final criterion plays an important role in showing that the
flux of these charges vanishes for stationary spacetimes as we shall see
below.

If such a symplectic potential $\boldsymbol{\Theta}$ can be found, define
$\mathcal{Q}[\xi;S]$ to be a function on the phase space at $\mathscr{I}$ by

$\delta\mathcal{Q}[\xi;S]\mathrel{\mathop{:}}=\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]+\int\limits_{S}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g}).$
(5.23)

It can easily be checked (using Eqs. 5.20, 5.21 and 5.22) that this expression
is integrable in phase space; that is,
$(\delta_{1}\delta_{2}-\delta_{2}\delta_{1})\mathcal{Q}[\xi;S]=0$. Together
with some choice of reference solution $g_{0}$ on which $\mathcal{Q}[\xi;S]=0$
for all asymptotic symmetries $\xi^{a}$ and all cross-sections $S$, Eq. 5.23
can be integrated in phase space to define the _Wald-Zoupas (WZ) charge_
$\mathcal{Q}[\xi;S]$ associated with the asymptotic BMS symmetry $\xi^{a}$ at
$S$. We note the following properties of the WZ charge defined by the above
procedure

  1. (1)

Since $\boldsymbol{\Theta}$ is locally and covariantly constructed from the
physical metric and the universal structure of $\mathscr{I}$, the charge can
similarly be written in terms of quantities that are defined at $\mathscr{I}$.
In particular the charge only depends on the BMS symmetry at $\mathscr{I}$ and
not on any choice of its extension into the spacetime. While this is clear for
the term that is an integral of $\xi\cdot\boldsymbol{\Theta}$, that this is
also true for the remaining terms, which are defined by a limit in the
unphysical spacetime, follows by Lemmas 5.1 and 5.2.

  2. (2)

Moreover, since $\boldsymbol{\Theta}$ is chosen to be conformally invariant
and independent of the choice of the auxiliary foliation of $\mathscr{I}$ and
the choice of the reference solution is to be specified without any particular
choice of its conformal completion or the choice of auxiliary foliation, the
WZ charge is also conformally invariant and independent of the choice of the
auxiliary foliation. Note, however that the WZ charge does depend on the
chosen cross-section $S$ on which it is evaluated and hence can depend on the
auxiliary normal $l_{a}$ at $S$ (but does not depend on the choice of $l_{a}$
away from $S$).

The flux of the perturbed WZ charge, through a portion $\Delta\mathscr{I}$ of
$\mathscr{I}$ whose boundary is given by two cross-sections $S_{1}$ and
$S_{2}$, is given by (see Eqs. 28 and 29 of WZ )

$\delta\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{:}}=\delta\mathcal{Q}[\xi;S_{2}]-\delta\mathcal{Q}[\xi;S_{1}]=\int\limits_{\Delta\mathscr{I}}\left[\underleftarrow{\boldsymbol{\omega}}(\delta\hat{g},\pounds_{\xi}\hat{g})+d\\{\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})\\}\right].$
(5.24)

The last term of this equation can also be written as

$d[\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})]=\pounds_{\xi}\boldsymbol{\Theta}(\delta\hat{g})=-\underleftarrow{\boldsymbol{\omega}}(\delta\hat{g},\pounds_{\xi}\hat{g})+\delta\boldsymbol{\Theta}(\pounds_{\xi}\hat{g})\,,$
(5.25)

where in the first equality, we have used the fact that $\boldsymbol{\Theta}$
is a $3$-form intrinsic to $\mathscr{I}$, and so its exterior derivative is
zero, and the second equality follows from the definition of
$\boldsymbol{\Theta}$ as a symplectic potential for
$\underleftarrow{\boldsymbol{\omega}}$ (Eq. 5.22). The flux of the perturbed
WZ charge is therefore simply given by

$\delta\mathcal{F}[\xi;\Delta\mathscr{I}]=\int\limits_{\Delta\mathscr{I}}\delta\boldsymbol{\Theta}(\pounds_{\xi}\hat{g}).$
(5.26)

To get the unperturbed charge and flux, we have to choose a reference solution
$g_{0}$ on which the charges are required to vanish. Since the symplectic
potential $\boldsymbol{\Theta}$ is required to vanish on stationary
backgrounds, we choose the reference solution $g_{0}$ to also be stationary.
For our concrete case of general relativity, we will pick $g_{0}$ to be (any
conformal completion of) Minkowski spacetime. Then, the flux of the WZ charge
is given by

$\mathcal{F}[\xi;\Delta\mathscr{I}]=\mathcal{Q}[\xi;S_{2}]-\mathcal{Q}[\xi;S_{1}]=\int\limits_{\Delta\mathscr{I}}\boldsymbol{\Theta}(\pounds_{\xi}\hat{g}).$
(5.27)

There are two important properties that follow from this expression for the
flux. The first, which the flux inherits from $\boldsymbol{\Theta}$, is that,
for any stationary background, the flux will vanish. This property captures
the fact that, if there is no radiation, there should be no flux and the
charges should be conserved quantities. The second is that, if $\xi^{a}$ is an
exact Killing vector field in the physical spacetime, then the flux will
vanish as well. This follows from the vanishing of $\pounds_{\xi}\hat{g}_{ab}$
for such vector fields. This property is reminiscent of Noether’s theorem: if
there is an exact symmetry, then there should be a quantity related to that
symmetry (in this case, the charge) that is conserved. In the next two
sections, we will find expressions for the WZ flux and the WZ charge in vacuum
general relativity. Since the charge and flux calculations will be performed
in different ways, we will check these calculations by showing that they agree
with Eq. 5.27.

Finally, we remark that the Wald-Zoupas prescription has certain ambiguities
related to the choices of the symplectic potential $\boldsymbol{\theta}$ and
the choice of $\boldsymbol{\Theta}$. However, it was argued in WZ  that these
ambiguities do not affect the final result. As the proof is quite involved, we
do not present it here, and turn our attention to computing explicit
expressions for the WZ flux and charge.

###  1 Wald-Zoupas flux in general relativity

We first consider the flux of the Wald-Zoupas charge. From Eq. 5.27, it is
apparent that determining this flux requires finding $\boldsymbol{\Theta}$.
First, a lengthy calculation starting with Eq. 5.7, then using Eq. 3.8 for the
unphysical metric perturbation $\delta g_{ab}$, along with the variation of
the vacuum Einstein equations (Eq. 2.1), shows that WZ

$\underleftarrow{\boldsymbol{\omega}}(\delta_{1}\hat{g},\delta_{2}\hat{g})\mathrel{\mathop{\widehat{=}}}-\frac{1}{32\pi}\left(\delta_{1}S_{ab}\gamma_{2}^{ab}-\delta_{2}S_{ab}\gamma_{1}^{ab}\right)\boldsymbol{\varepsilon}_{3}.$
(5.28)

Since $\gamma^{ab}n_{b}\mathrel{\mathop{\widehat{=}}}0$ by Eq. 3.8, $\delta
S_{ab}$ in this expression can be replaced with
$\delta\underleftarrow{S}_{ab}$. Moreover, $\rho_{ab}$ is universal, and so
$\delta\rho_{ab}=0$; as such, Eq. 2.39 implies that we can replace
$\delta\underleftarrow{S}_{ab}$ with $\delta N_{ab}$, yielding

$\underleftarrow{\boldsymbol{\omega}}(\delta_{1}\hat{g},\delta_{2}\hat{g})\mathrel{\mathop{\widehat{=}}}-\frac{1}{32\pi}\left(\delta_{1}N_{ab}\gamma_{2}^{ab}-\delta_{2}N_{ab}\gamma_{1}^{ab}\right)\boldsymbol{\varepsilon}_{3}\mathrel{\mathop{\widehat{=}}}\frac{1}{16\pi}\left(\delta_{1}N^{ab}\delta_{1}\sigma_{ab}-\delta_{2}N^{ab}\delta_{1}\sigma_{ab}\right)\boldsymbol{\varepsilon}_{3},$
(5.29)

where the final equality uses the fact that the News tensor is traceless and
Eq. 3.10.

A symplectic potential for $\underleftarrow{\boldsymbol{\omega}}$ is therefore
given by

$\boldsymbol{\Theta}(\delta\hat{g})\mathrel{\mathop{\widehat{=}}}-\frac{1}{32\pi}N^{ab}\gamma_{ab}\boldsymbol{\varepsilon}_{3}\mathrel{\mathop{\widehat{=}}}\frac{1}{16\pi}N^{ab}\delta\sigma_{ab}\boldsymbol{\varepsilon}_{3}.$
(5.30)

One must check that this symplectic potential satisfies the requirements
above. First, it is constructed from $g_{ab}$, $\delta g_{ab}$, their
derivatives (such as $S_{ab}$), and fields that are part of the universal
structure at $\mathscr{I}$ (such as $n^{a}$ and, less obviously, $\rho_{ab}$
WZ ). Moreover, it does not depend on the arbitrary structure that we have
provided at $\mathscr{I}$, the choice of conformal factor $\Omega$ and
auxiliary normal $l^{a}$. To see this, first note that the only piece that
depends on $l^{a}$ is
$\boldsymbol{\varepsilon}_{3}=l\cdot\boldsymbol{\varepsilon}_{4}$, but upon
taking the pullback, this dependence drops out. To see that it is conformally
invariant, use the fact that $\gamma_{ab}=\Omega\delta\hat{g}_{ab}$, and so
under a conformal transformation $\Omega\mapsto\omega\Omega$,

$\displaystyle
N^{ab}\mapsto\omega^{-4}N^{ab}\,,\quad\gamma_{ab}\mapsto\omega\gamma_{ab}\,,\quad\boldsymbol{\varepsilon}_{3}\mapsto\omega^{3}\boldsymbol{\varepsilon}_{3},$
(5.31)

and so we find that $\boldsymbol{\Theta}$ is conformally invariant. Finally,
note that this choice of $\boldsymbol{\Theta}$ vanishes on stationary
solutions $g_{ab}$, for _any_ perturbation $\delta g_{ab}$, by the argument on
pp. 53–54 of Geroch-asymp  (which shows that $N_{ab}$ vanishes for stationary
vacuum spacetimes).

Using a chosen foliation of $\mathscr{I}$, we now provide a more explicit
expression for the WZ flux in terms of fields defined on $\mathscr{I}$. Note
that, since the News tensor is traceless, we have that

$N^{ab}\gamma_{ab}^{(\xi)}\mathrel{\mathop{\widehat{=}}}N^{ab}\operatorname{STF}\gamma_{ab}^{(\xi)}.$
(5.32)

As such, from Cor. 4.1 we have that our flux is independent of the extension
of $\xi^{a}$ off of $\mathscr{I}$. Plugging in our expression for
$\operatorname{STF}\gamma_{ab}^{(\xi)}$ from Eq. 4.32, we therefore find
that888The first form of the WZ flux in Eq. 5.33 was anticipated by Geroch and
Winicour long before Wald and Zoupas (see Eq. 28 of GW )!

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}\gamma_{ab}^{(\xi)}$
(5.33)
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}].$

Note that the first expression is manifestly independent of any choice of
foliation of $\mathscr{I}$, a property it inherits from $\boldsymbol{\Theta}$.
This is not obviously true for the second, which instead is more useful for
explicit computations (such as comparing with expressions in Bondi
coordinates). Both, however, are clearly local and covariant, and can be
easily shown to be independent of the conformal factor. We reiterate that
these properties of the flux are motivated by the desire that these
expressions not be dependent on any arbitrary choices that we could make, such
as a coordinate system, conformal factor etc.

As mentioned below Eq. 5.27, the flux in Eq. 5.33 has both the property that
it vanishes for stationary backgrounds, as well as when $\xi^{a}$ is an exact
Killing vector field in the physical spacetime. The first of these properties
is evident both in the first and second expressions in Eq. 5.33, as the
integrands are both proportional to the News. That the flux vanishes when
$\xi^{a}$ is an exact Killing vector field follows immediately from the fact
that $\gamma^{(\xi)}_{ab}$ vanishes for such vector fields.

In the case where $\xi^{a}$ is a translation, namely $\xi^{a}=fn^{a}$, where
$f$ obeys Eq. 4.24, it is the case that

$\mathcal{F}[fn;\Delta\mathscr{I}]=-\frac{1}{32\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}\;fN^{ab}N_{ab}.$
(5.34)

In the case where $f$ is everywhere positive, $\xi^{a}$ corresponds to a time-
translation in some conformal frame, as mentioned below Eq. 4.24. In this
case, the flux is negative, corresponding to the _loss_ of mass/energy during
the emission of gravitational waves.

######  Remark 5.3 (BMS fluxes and Hamiltonians).

On any region $\Delta\mathscr{I}$ of null infinity we can consider the
integral of the symplectic current
$\int\limits_{\Delta\mathscr{I}}\underleftarrow{\boldsymbol{\omega}}(\delta_{1}\hat{g},\delta_{2}\hat{g})$
as defining a _symplectic form_ on the radiative phase on $\Delta\mathscr{I}$.
Then, Eq. 5.24 implies that

$\delta\mathcal{F}[\xi;\Delta\mathscr{I}]=\int\limits_{\Delta\mathscr{I}}\underleftarrow{\boldsymbol{\omega}}(\delta\hat{g},\pounds_{\xi}\hat{g})+\int\limits_{S_{2}}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})-\int\limits_{S_{1}}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g}),$
(5.35)

Note that if the boundary terms at $S_{1}$ and $S_{2}$ vanish for all
perturbations $\delta g_{ab}$ and all background solutions $g_{ab}$ then the
WZ flux $\mathcal{F}[\xi;\Delta\mathscr{I}]$ would define a _Hamiltonian
generator_ corresponding to the BMS symmetry $\xi^{a}$, as defined by WZ ;
Lee1990 . But these boundary terms do not vanish in general and the WZ flux
does not define a Hamiltonian. However, in the case where we consider all of
null infinity, instead of some finite portion and, appropriate boundary
conditions at timelike and spatial infinity, then
$\mathcal{F}[\xi;\mathscr{I}]$ will be a Hamiltonian generator on the full
radiative phase space on $\mathscr{I}$. To see this, let $u$ be a parameter
along the null generators of $\mathscr{I}$ such that
$n^{a}\nabla_{a}u\mathrel{\mathop{\widehat{=}}}1$, and impose, as
$u\to\pm\infty$, the following boundary conditions:

$N_{ab}=O(1/|u|^{1+\epsilon}),\quad\gamma_{ab}=O(1),$ (5.36)

for some $\epsilon>0$. These conditions ensure that the integral of the
symplectic current over all of $\mathscr{I}$, as given in Eq. 5.29, is finite.
Further, Eq. 4.21a implies that $\xi^{a}=O(|u|)$, and so it follows from Eqs.
5.36 and 5.30 that

$\lim_{u\to\pm\infty}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})=0.$ (5.37)

Note that the conditions Eq. 5.36 are preserved by all BMS symmetries, and
thus the total flux $\mathcal{F}[\xi;\mathscr{I}]$ defines a Hamiltonian
generator for the BMS symmetry $\xi^{a}$ on the radiative phase space on
$\mathscr{I}$.

###  2 Wald-Zoupas charge in general relativity

Having obtained the WZ flux we now wish to find an expression for the WZ
charge in terms of fields on $\mathscr{I}$. From Eq. 5.23, the WZ charge
$\mathcal{Q}[\xi;S]$ is determined by

$\displaystyle\delta\mathcal{Q}[\xi;S]=\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]+\int\limits_{S}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})\,,$
(5.38)

along with the requirement that $\mathcal{Q}[\xi;S]$ vanish on Minkowski
spacetime for all BMS symmetries $\xi^{a}$ and all cross-sections $S$. The
main difficulty in carrying out this computation directly is that the $2$-form
$\delta\boldsymbol{Q}_{\xi}$ does not have a limit to $\mathscr{I}$; as
mentioned earlier, it can be shown that $\boldsymbol{\theta}(\delta\hat{g})$
does have a limit to $\mathscr{I}$ in general relativity. To compute the
right-hand-side of Eq. 5.38, one must first choose some family of $2$-spheres
inside the spacetime, evaluate the integral and then take the limit as these
$2$-spheres tend to the chosen cross-section $S$ of $\mathscr{I}$.

To compute this term, we note that by the general arguments in Lemmas 5.1 and
5.2,

$\displaystyle\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]\,,$
(5.39)

is guaranteed to exist, is independent of the family of $2$-spheres chosen to
take the limit and is independent of how the BMS symmetry is extended into the
spacetime away from $\mathscr{I}$. Moreover, it is manifestly independent of
the choice of the conformal factor and the choice of the foliation of
$\mathscr{I}$. Thus, we can compute Eq. 5.39 in the choices given by the
conformal Bondi-Sachs coordinates in a neighbourhood of $\mathscr{I}$. The
detailed construction of these coordinates and the form of the unphysical and
physical metrics in these coordinates is given in Sec. 6.1. The conformal
Bondi-Sachs coordinates give us a family of null surfaces $\mathcal{N}_{u}$
labeled by a coordinate $u$ and a family of $2$-spheres $S^{\prime}$ labeled
by the coordinates $\Omega$ and $u$ along each null surface $\mathcal{N}_{u}$
such that as $\Omega\to 0$ the $2$-spheres limit to a cross-section $S$ of
$\mathscr{I}$. We use this family of $2$-spheres to evaluate Eq. 5.39 and then
take the limit $\Omega\to 0$ along this family.

With this setup, we take the form of the unphysical metric in Bondi-Sachs
coordinates (given by Eqs. 6.2 and 6.5), the corresponding expressions for the
unphysical metric perturbations along with the expression for $\xi^{a}$
derived in Eq. 6.19. We use this to evaluate Eq. 5.39 using Eqs. 5.3b and 5.9,
after converting the physical metric and physical metric perturbations to
their unphysical counterparts. The resulting expression for the $2$-form
$\delta\boldsymbol{Q}_{\xi}$ has a term that diverges as $\Omega\to 0$, which
is given by

$\displaystyle\boldsymbol{\varepsilon}_{2}~{}\Omega^{-1}\left(2X^{A}\delta
U^{(2)}_{A}\right)=-\boldsymbol{\varepsilon}_{2}~{}\Omega^{-1}X^{A}\mathscr{D}^{B}\delta
C_{AB}\,.$ (5.40)

The indices $A,B$ are abstract indices for tensor fields, $\mathscr{D}_{a}$ is
the covariant derivative on the chosen family of $2$-spheres, and the equality
Eq. 5.40 follows from Eq. 6.7 which is a consequence of the (linearized)
Einstein equation in the Bondi-Sachs coordinates. Now, since $X^{A}$ is a
Lorentz vector field (satisfying Eq. 6.14 in the conformal Bondi-Sachs
coordinates) and $\delta C_{AB}$ is traceless, this term vanishes when
integrated over the $2$-spheres. As a result, the limit of
$\int\limits_{S^{\prime}}\delta\boldsymbol{Q}_{\xi}$ as $S^{\prime}\to S$,
i.e. $\Omega\to 0$, is finite. Then, computing the remaining terms in Eq. 5.39
(which are manifestly finite as $\Omega\to 0$) we obtain

$\displaystyle\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]$
(5.41)
$\displaystyle=-\frac{1}{8\pi}\delta\int\limits_{S}\boldsymbol{\varepsilon}_{2}\big{[}\beta\,(\mathcal{P}+\tfrac{1}{2}\sigma_{ab}N^{ab})+X^{a}\mathcal{J}_{a}+X^{a}\sigma_{ab}\mathscr{D}_{c}\sigma^{bc}-\tfrac{1}{4}\sigma_{ab}\sigma^{ab}\mathscr{D}_{c}X^{c}\big{]}$
$\displaystyle\,\quad-\frac{1}{16\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\beta\,N^{ab}\delta\sigma_{ab}\,.$

where we have used Eq. 6.10 to convert the metric components in the conformal
Bondi-Sachs coordinates to covariant quantities defined on $\mathscr{I}$. We
can immediately see this expression is non-integrable due to the last
term.999As emphasized in Remark 5.2 above, this _should not_ be taken to mean
that the last line above is the “non-integrable part” of the expression which
should be “simply subtracted away” without specifying any additional criteria.

Using Eqs. 5.30, 6.19 and 2.14, we have

$\displaystyle\int\limits_{S}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})=\tfrac{1}{16\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\beta
N^{ab}\delta\sigma_{ab}\,.$ (5.42)

Thus we have the perturbed WZ charge

$\displaystyle\delta\mathcal{Q}[\xi;S]$ $\displaystyle=\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\delta\boldsymbol{Q}_{\xi}-\xi\cdot\boldsymbol{\theta}(\delta\hat{g})\right]+\int\limits_{S}\xi\cdot\boldsymbol{\Theta}(\delta\hat{g})$
(5.43)
$\displaystyle=-\frac{1}{8\pi}\delta\int\limits_{S}\boldsymbol{\varepsilon}_{2}\big{[}\beta\,(\mathcal{P}+\tfrac{1}{2}\sigma_{ab}N^{ab})+X^{a}\mathcal{J}_{a}+X^{a}\sigma_{ab}\mathscr{D}_{c}\sigma^{bc}-\tfrac{1}{4}\sigma_{ab}\sigma^{ab}\mathscr{D}_{c}X^{c}\big{]}\,.$

Therefore, the expression for the WZ charge is given by

$\displaystyle\mathcal{Q}[\xi;S]\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\left[\beta(\mathcal{P}+\tfrac{1}{2}\sigma^{ab}N_{ab})+X^{a}\mathcal{J}_{a}+X^{a}\sigma_{ab}(\mathscr{D}_{c}-\tau_{c})\sigma^{bc}-\tfrac{1}{4}\sigma_{ab}\sigma^{ab}(\mathscr{D}_{c}-2\tau_{c})X^{c}\right]\,.$
(5.44)

Note that in Eq. 5.44 we have added terms which depend on $\tau_{a}$; these
terms cancel amongst each other using the identity Eq. D.3. These additional
terms make each term in the integrand of conformal weight $-2$ (which can be
verified using the conformal weights given in Eqs. 4.11, 2.31, 2.29 and 2.30)
which, along with the conformal weight $+2$ of $\boldsymbol{\varepsilon}_{2}$,
makes the charge conformally-invariant. Further, since these $\tau_{a}$ terms
cancel the charge is independent of the choice of the foliation and only
depends on the chosen cross-section $S$. We will verify below that the flux of
this charge across any region $\Delta\mathscr{I}$ is given by Eq. 5.33.
Finally, since the flux vanishes in Minkowski spacetime we can compute the
charge on any cross-section of $\mathscr{I}$. Using a shear-free cross-section
we see that the charge also vanishes on _any_ cross-section of $\mathscr{I}$
in Minkowski spacetime. Thus, the charge formula in Eq. 5.44 satisfies all the
properties required by the WZ charge.

For a supertranslation symmetry with $\beta\mathrel{\mathop{\widehat{=}}}f$
satisfying $\pounds_{n}f\mathrel{\mathop{\widehat{=}}}0$ it is straightforward
to verify that Eq. 5.44 reproduces Geroch’s supermomentum Geroch-asymp ; WZ .
Further, if at some chosen cross-section $S$ we pick
$\beta|_{S}\mathrel{\mathop{\widehat{=}}}0$ then the resulting formula is
equal to the linkage charge defined by Geroch and Winicour GW ; WZ ; this was
proven in WZ  and we show this more explicitly in Sec. C.2. Now, while on any
fixed cross-section $S$ we can decompose a general BMS symmetry into a
supertranslation part and a part tangent to $S$, this decomposition is not
preserved along $\mathscr{I}$ (see also Remark 4.2). Thus, in general the WZ
charge is _not_ the sum of Geroch’s supermomentum with the Geroch-Winicour
charge. For an exact Killing vector field $\xi^{a}$ in the physical spacetime,
the WZ charge agrees with the charge that is given by the Komar formula. We
discuss in more detail Sec. C.2. To see this, note that, for exact Killing
vector fields, the linkage charge agrees with the Komar formula for Killing
vector fields which asymptotically become Lorentz vector fields at $S$, and
agrees up to a factor of two for Killing vector fields that become
translations at $S$. The WZ charge, similarly, equals the linkage charge for
Lorentz vector fields at $S$, and up to a factor of two for translations; as
such, the WZ charge and the Komar formula agree for exact Killing vector
fields. Note that the Komar formula is (up to constant factors) the integral
of the Noether charge in Eq. 5.9.

By the general arguments following Eq. 5.24, the change in the charge Eq. 5.44
between two cross-sections is given by the flux formula Eq. 5.33. However,
showing this explicitly is a non-trivial computation, which we detail in the
remainder of this section.

Let $S_{2}$ and $S_{1}$ be _any_ two cross-sections of $\mathscr{I}$, with
$S_{2}$ to the future of $S_{1}$, and let $\Delta\mathscr{I}$ be the portion
of $\mathscr{I}$ bounded by these cross-sections. Then, the change in the
charge is given by

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}\mathcal{Q}[\xi;S_{2}]-\mathcal{Q}[\xi;S_{1}]$
(5.45)
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}\pounds_{n}\left[\beta(\mathcal{P}+\tfrac{1}{2}\sigma^{ab}N_{ab})+X^{a}\mathcal{J}_{a}+X^{a}\sigma_{ab}\mathscr{D}_{c}\sigma^{bc}-\tfrac{1}{4}\sigma_{ab}\sigma^{ab}\mathscr{D}_{c}X^{c}\right]\,.$

Note that we have dropped the $\tau_{a}$ terms from the expression since they
do not contribute to the charge as explained above.

Now we simplify LABEL:eq:flux1 term-by-term starting with the first and second
terms. Using $\mathcal{J}_{a}n^{a}\mathrel{\mathop{\widehat{=}}}0$, as well as
Eqs. 2.25b, 2.25d, 2.37, 4.21a and 4.21b and the integration-by-parts formula
Eq. 2.22, we see that the first two terms in LABEL:eq:flux1 contribute

$\displaystyle-\frac{1}{8\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}\bigg{[}N^{ab}\bigg{(}\tfrac{1}{4}\beta
N_{ab}+\tfrac{1}{2}(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+(\mathscr{D}_{a}+\tau_{a})(X^{c}\sigma_{bc})+\tfrac{1}{4}\sigma_{ab}(\mathscr{D}_{c}-\tau_{c})X^{c}\bigg{)}$
(5.46)
$\displaystyle\qquad\qquad-\tfrac{1}{2}\varepsilon_{a}{}^{b}X^{a}\big{(}\mathscr{D}_{b}+3\,\tau_{b})\mathcal{P}^{\ast}\bigg{]}\,,$

to the flux. Now consider the contribution of the third term in
LABEL:eq:flux1:

$\displaystyle\pounds_{n}(X^{a}\sigma_{ab}\mathscr{D}_{c}\sigma^{bc})\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}X^{a}N_{ab}\mathscr{D}_{c}\sigma^{bc}+X^{a}\sigma_{ab}\pounds_{n}\mathscr{D}_{c}\sigma^{bc}$
(5.47)

where we have used Eq. 4.21b and Eq. 2.32. For the last term above we have

$\displaystyle\pounds_{n}\mathscr{D}_{c}\sigma^{bc}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\pounds_{n}(Q^{a}{}_{c}Q^{b}{}_{d})\nabla_{a}\sigma^{cd}+Q^{a}{}_{c}Q^{b}{}_{d}\pounds_{n}\nabla_{a}\sigma^{cd}$
(5.48)
$\displaystyle\mathrel{\mathop{\widehat{=}}}(Q^{a}{}_{c}n^{b}\tau_{d}+Q^{b}{}_{d}n^{a}\tau_{c})\nabla_{a}\sigma^{cd}+\tfrac{1}{2}\mathscr{D}_{a}N^{ab}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}(Q^{a}{}_{c}n^{b}\tau_{d})\nabla_{a}\sigma^{cd}+\tfrac{1}{2}(\mathscr{D}_{a}+\tau_{a})N^{ab}\,,$

where we have made liberal use of the Bondi condition (Eq. 2.6) along with
Eqs. 2.11 and 2.18, and commuted the $\pounds_{n}$ past the $\nabla_{a}$ using
$\sigma^{ab}n_{a}\mathrel{\mathop{\widehat{=}}}Q_{ab}n^{a}\mathrel{\mathop{\widehat{=}}}0$.
Using the above expression in Eq. 5.47 the first term vanishes as
$\sigma_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}0$ and we get

$\displaystyle\pounds_{n}(X^{a}\sigma_{ab}\mathscr{D}_{c}\sigma^{bc})\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}X^{a}N_{ab}\mathscr{D}_{c}\sigma^{bc}+\tfrac{1}{2}(\mathscr{D}_{c}+\tau_{c})(X^{a}\sigma_{ab}N^{bc})-\tfrac{1}{2}N_{bc}\mathscr{D}^{b}(X_{a}\sigma^{ac})\,.$
(5.49)

Note that the second term on the right-hand side above drops out of the flux
formula using Eq. 2.22.

Consider now the fourth term in LABEL:eq:flux1. Since
$\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\mathscr{D}_{a}X^{a}$
and $\pounds_{n}\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}0$ (Eq. 4.7),
using Eq. 2.32 we get

$\displaystyle-\tfrac{1}{4}\pounds_{n}(\sigma_{ab}\sigma^{ab}\mathscr{D}_{c}X^{c})\mathrel{\mathop{\widehat{=}}}-\tfrac{1}{4}\sigma^{ab}N_{ab}\mathscr{D}_{c}X^{c}\,.$
(5.50)

Putting together LABEL:contrib3, 5.49 and 5.50, we see that the flux is given
by

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}\bigg{[}N^{ab}\bigg{(}\tfrac{1}{4}\beta
N_{ab}+\tfrac{1}{2}(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\tfrac{1}{2}(\mathscr{D}_{a}+2\tau_{a})(X^{c}\sigma_{bc})$
(5.51)
$\displaystyle\qquad-\tfrac{1}{2}\sigma_{ab}\tau_{c}X^{c}+\tfrac{1}{2}X^{a}\mathscr{D}_{c}\sigma^{bc}\bigg{)}-\tfrac{1}{2}\varepsilon_{a}{}^{b}X^{a}(\mathscr{D}_{b}+3\tau_{b})\mathcal{P}^{*}\bigg{]}\,.$

Next we simplify the last term using the identity Eq. 2.38 for
$\mathcal{P}^{*}$. The term arising from this identity which involves
derivatives of the shear reads
$\varepsilon_{3}\,\varepsilon^{de}\varepsilon_{a}{}^{b}X^{a}(\mathscr{D}_{b}+3\tau_{b})[\mathscr{D}_{d}(\mathscr{D}_{c}-\tau_{c})\sigma_{e}{}^{c}]$.
This term can be shown to vanish upon integrating over the cross-sections as
follows. Note that this term is conformally-invariant (which can be seen using
Eq. 2.38 and the conformal weights given in Eqs. 2.28, 4.11, 2.29 and 2.31).
Therefore, we can evaluate this term in the Bondi frame where the metric on
the cross-sections is chosen to be the unit round metric and
$\tau_{a}\mathrel{\mathop{\widehat{=}}}0$. This allows us to make use of
spherical harmonics — $\sigma_{ab}$ is a symmetric and trace-free tensor and
thus is supported on $\ell\geq 2$ tensor spherical harmonics while $X^{a}$ is
supported on $\ell=1$ vector spherical harmonics (see Remark 4.1). Using the
orthogonality of the spherical harmonics, the integral of this term over the
cross-sections vanishes. The remaining term arising from Eq. 2.38 contains the
News tensor which, after an integration-by-parts using Eq. 2.22, becomes

$\displaystyle-\tfrac{1}{4}\varepsilon^{ab}\varepsilon_{e}{}^{d}N_{ac}\sigma_{b}{}^{c}(\mathscr{D}_{d}-2\tau_{d})X^{e}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}N_{ab}\sigma^{b}{}_{c}\mathscr{D}^{[a}X^{c]}-N_{ab}\sigma^{b}{}_{c}X^{[c}\tau^{a]}\,.$
(5.52)

Replacing the above for the last term in Eq. 5.51 we get

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}\bigg{[}\tfrac{1}{4}\beta
N_{ab}+\tfrac{1}{2}(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\tfrac{1}{2}(\mathscr{D}_{a}+2\tau_{a})(X^{c}\sigma_{bc})$
(5.53)
$\displaystyle\qquad\qquad-\tfrac{1}{2}\sigma_{ab}\tau_{c}X^{c}+\tfrac{1}{2}X_{a}\mathscr{D}^{c}\sigma_{bc}+\tfrac{1}{2}\sigma_{b}{}^{c}\mathscr{D}_{[a}X_{c]}-\sigma_{b}{}^{c}X_{[c}\tau_{a]}\bigg{]}\,.$

The last term on the first line and first term on the second line in the
expression above can together be written as

$\displaystyle
N^{ab}(X^{c}\sigma_{bc}\tau_{a}-\tfrac{1}{2}\sigma_{ab}\tau_{c}X^{c})\mathrel{\mathop{\widehat{=}}}N^{ab}\sigma_{b}{}^{c}(X_{c}\tau_{a}-\tfrac{1}{2}Q_{ca}X_{d}\tau^{d})\,.$
(5.54)

Moreover, it follows from Eq. D.3 that
$N_{ab}\sigma^{b}{}_{c}\mathrel{\mathop{\widehat{=}}}\text{pure trace
term}+N_{b[a}\sigma^{b}{}_{c]}$ and so Eq. 5.54 becomes
$N^{ab}\sigma_{b}{}^{c}X_{[c}\tau_{a]}$. This exactly cancels the last term in
Eq. 5.53 and therefore, the flux formula simplifies to

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}\bigg{[}$
$\displaystyle\tfrac{1}{4}\beta
N_{ab}+\tfrac{1}{2}(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta$
(5.55)
$\displaystyle+\tfrac{1}{2}\mathscr{D}_{a}(X^{c}\sigma_{bc})+\tfrac{1}{2}X_{a}\mathscr{D}^{c}\sigma_{bc}+\tfrac{1}{2}\sigma_{b}{}^{c}\mathscr{D}_{[a}X_{c]}\bigg{]}\,.$

The second line above can be simplified using Eq. D.5 to get

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}].$
(5.56)

which matches the WZ flux derived in Eq. 5.33.

##  6 Expressions in some coordinate systems

Our entire preceding analysis was completely covariant, without referring to
any particular coordinate systems. In this section, we consider two examples
of coordinate systems for the unphysical spacetime in a neighbourhood of null
infinity. These coordinates can be used to also obtain asymptotic coordinates
for the physical spacetime. We will find the asymptotic form of the metric,
both physical and unphysical, and derive the expressions for the BMS
symmetries and their charges in these coordinates.

As described in Sec. 3, one can construct a geometrically defined coordinate
system $(\Omega,u,x^{A})$ at $\mathscr{I}$ where $x^{A}$ are coordinates on
the cross-sections of $\mathscr{I}$, $u$ satisfies
$n^{a}\nabla_{a}u\mathrel{\mathop{\widehat{=}}}1$ and the conformal factor
$\Omega$ is chosen so that the Bondi condition is satisfied and so that the
induced metric on the cross-sections is the unit round metric $s_{AB}$ on
$\mathbb{S}^{2}$. Then for any asymptotically flat spacetime the line element
of the unphysical metric at $\mathscr{I}$ is

$\displaystyle ds^{2}\mathrel{\mathop{\widehat{=}}}2d\Omega
du+s_{AB}dx^{A}dx^{B}\,.$ (6.1)

These coordinates can be extended away from $\mathscr{I}$ in different ways
and these give rise to the different coordinates that are often used in the
analysis of asymptotic symmetries and their associated charges. Two commonly
used coordinates are the Bondi-Sachs coordinates and conformal Gaußian null
coordinates and we will focus on these in the remainder of this section. We
emphasize that the form of the (physical or unphysical) metric in these
coordinates follows directly from the construction of the coordinate systems
and the covariant definition of asymptotic flatness without any additional
assumptions. We also show that in any such coordinate system one can obtain
the BMS symmetries as coordinate transformations which preserve the asymptotic
form of the metric.

###  1 Bondi-Sachs coordinates

One way to extend the coordinates described above away from $\mathscr{I}$ is
as follows. Let $S_{u}$ be cross-sections of $\mathscr{I}$ with
$u=\text{constant}$, and consider a family of null surfaces $\mathcal{N}_{u}$
which intersect $\mathscr{I}$ transversely in the cross-sections $S_{u}$.
These surfaces $\mathcal{N}_{u}$ foliate a neighbourhood of $\mathscr{I}$ by
null surfaces. We first extend the coordinate $u$ away from $\mathscr{I}$ so
that it is constant along each null surface $\mathcal{N}_{u}$. Then
$l_{a}\mathrel{\mathop{:}}=-\nabla_{a}u$ is the null normal to each
$\mathcal{N}_{u}$ with $l^{a}l_{a}=0$ in addition to
$l^{a}n_{a}\mathrel{\mathop{\widehat{=}}}-1$ as above. Then we extend the
angular coordinates $x^{A}$ on each cross-section $S_{u}$ by parallel
transport i.e. $l^{a}\nabla_{a}x^{A}=0$.

Fixing the induced metric on cross sections of $\mathscr{I}$ to be the unit
round sphere metric fixes the conformal factor, $\Omega$, on $\mathscr{I}$. To
extend $\Omega$ away from $\mathscr{I}$, we use the freedom in the conformal
factor off of $\mathscr{I}$ to demand that the $2$-spheres at constant $u$ and
$\Omega$ have the same area element as the unit sphere, that is, if $h_{AB}$
is the $2$-metric on the surfaces of constant $u$ and $\Omega$ then we demand
that $\det h=\det s$ in the $x^{A}$-coordinates. This fixes $\Omega$ uniquely
away from $\mathscr{I}$. Thus we have set up the conformal Bondi-Sachs
coordinate system $(\Omega,u,\theta^{A})$ in a neighborhood of $\mathscr{I}$.

The most general form of the unphysical metric in conformal Bondi-Sachs
coordinates is given by101010Note that the function we denote by $B$ is
conventionally denoted by $\beta$, but we use a different symbol to avoid
conflict with the $n^{a}$-component of a BMS vector field.

$\displaystyle ds^{2}\equiv-We^{2B}du^{2}+2e^{2B}d\Omega
du+h_{AB}(dx^{A}-U^{A}du)(dx^{B}-U^{B}du)\,,$ (6.2)

where the metric components $g_{\Omega\Omega}$ and $g_{\Omega A}$ vanish
everywhere due to the conditions $l^{a}l_{a}=l^{a}\nabla_{a}x^{A}=0$, and $W$,
$B$, $h_{AB}$, and $U^{A}$ are smooth functions of the coordinates
$(\Omega,u,x^{A})$. Since the metric at $\mathscr{I}$ is given by Eq. 6.1 we
also have

$\displaystyle W=O(\Omega)\,,\quad B=O(\Omega)\,,\quad U^{A}=O(\Omega)\,,\quad
h_{AB}=s_{AB}+O(\Omega)\,.$ (6.3)

Further, evaluating the Bondi condition
$\nabla_{a}n_{b}\mathrel{\mathop{\widehat{=}}}0$ (Eq. 2.6) gives

$\displaystyle W=O(\Omega^{2})\,,\quad B=O(\Omega^{2})\,,\quad
U^{A}=O(\Omega^{2})\,.$ (6.4)

We therefore consider the following expansion of the metric components

$\displaystyle W=\Omega^{2}W^{(2)}-2\Omega^{3}M+O(\Omega^{4})\,,$
$\displaystyle
U^{A}=\Omega^{2}U^{(2)}{}^{A}+2\Omega^{3}L^{A}+O(\Omega^{4})\,,$ (6.5)
$\displaystyle B=\Omega^{2}B^{(2)}+O(\Omega^{3})\,,$ $\displaystyle
h_{AB}=s_{AB}+\Omega C_{AB}+\Omega^{2}d_{AB}+O(\Omega^{3})\,.$

Next, imposing $\det h=\det s$, we get

$\displaystyle s^{AB}C_{AB}=0\,,\quad
s^{AB}d_{AB}=\tfrac{1}{2}C^{AB}C_{AB}\,.$ (6.6)

We then impose the Einstein equation, Eq. 2.1, order by order in $\Omega$. At
$O(\Omega^{0})$, it gives

$\displaystyle W^{(2)}=1\,,\quad B^{(2)}=-\tfrac{1}{32}C^{AB}C_{AB}\,,\quad
U^{(2)}_{A}=-\tfrac{1}{2}\mathscr{D}^{B}C_{AB}\,,$ (6.7)

while at $O(\Omega)$ we get

$\displaystyle\partial_{u}\operatorname{STF}d_{AB}=\mathscr{D}^{A}\operatorname{STF}d_{AB}=0\,.$
(6.8)

Since $d_{AB}$ is a smooth tensor on a $2$-sphere this implies that
$\operatorname{STF}d_{AB}=0$ and thus (from Eq. 6.6)

$\displaystyle d_{AB}=\tfrac{1}{4}C^{CD}C_{CD}s_{AB}\,.$ (6.9)

The metric component $C_{AB}$ is related to the shear and the News tensor