# Calculation of asymptotic charges at the critical sets of null infinity

Mariem Magdy Ali Mohamed 111E-mail<EMAIL_ADDRESS>School of
Mathematical Sciences, Queen Mary, University of London, Mile End Road, London
E1 4NS, United Kingdom.

###### Abstract

The asymptotic structure of null and spatial infinities of asymptotically flat
spacetimes plays an essential role in discussing gravitational radiation,
gravitational memory effect, and conserved quantities in General Relativity.
Bondi, Metzner and Sachs established that the asymptotic symmetry group for
asymptotically simple spacetimes is the infinite-dimensional BMS group. Given
that null infinity is divided into two sets: past null infinity
$\mathscr{I}^{-}$ and future null infinity $\mathscr{I}^{+}$, one can identify
two independent symmetry groups: $\text{BMS}^{-}$ at $\mathscr{I}^{-}$ and
$\text{BMS}^{+}$ at $\mathscr{I}^{+}$. Associated with these symmetries are
the so-called BMS charges. A recent conjecture by Strominger suggests that the
generators of $\text{BMS}^{-}$ and $\text{BMS}^{+}$ and their associated
charges are related via an antipodal reflection map near spatial infinity. To
verify this matching, an analysis of the gravitational field near spatial
infinity is required. This task is complicated due to the singular nature of
spatial infinity for spacetimes with non-vanishing ADM mass. Different
frameworks have been introduced in the literature to address this singularity,
e.g., Friedrich's cylinder, Ashtekar-Hansen's hyperboloid and Ashtekar-
Romano's asymptote at spatial infinity. This paper reviews the role of
Friedrich's formulation of spatial infinity in the investigation of the
matching of the spin-2 charges on Minkowski spacetime and in the full GR
setting.

###### Contents

  1. 1 Introduction
  2. 2 The spin-2 asymptotic charges on Minkowski spacetime
    1. 2.1 The Minkowski spacetime in the F-gauge
    2. 2.2 NP-gauge to F-gauge
    3. 2.3 The spin-2 charges in the F-gauge
    4. 2.4 The spin-2 field equations
  3. 3 The asymptotic charges in full GR
    1. 3.1 Friedrich's regular initial value problem
    2. 3.2 The supertranslation asymptotic charges in full GR
  4. 4 Conclusions

##  1 Introduction

In classical General Relativity (GR), isolated systems are commonly described
by asymptotically flat spacetimes, with a metric approaching the Minkowski
metric far from the source. In this setting, the influential work of R.
Penrose [1, 2] offers a geometrical approach to the studies of isolated
systems. In particular, Penrose's notion of "asymptotic simplicity" identifies
spacetimes with a conformal extension similar to that of Minkowski spacetime,
implying the existence of null infinity $\mathscr{I}$, comprised of two
disjoint sets: future null infinity $\mathscr{I}^{+}$ and past null infinity
$\mathscr{I}^{-}$. The universal fields shared by asymptotically simple
spacetimes allow us to identify the infinite-dimensional asymptotic symmetry
group known as the BMS group [3], named after Bondi, Metzner and Sachs.
However, Penrose's notion of asymptotic simplicity is not concerned with the
behaviour of the gravitational field at spatial infinity, which is a crucial
ingredient in the discussion of conserved quantities in GR [4].

One of the challenging aspects of studies of the asymptotic structure at
spatial infinity is the singular conformal structure at spatial infinity
$i^{0}$ for spacetimes with non-vanishing Arnowitt-Deser-Misner (ADM) mass.
Different formulations of spatial infinity [5, 6, 7] can be used to resolve
the structure of the gravitational field in this region. Of particular
importance to this article is Friedrich's formulation of spatial infinity,
initially introduced in [7] with the goal of obtaining a regular initial value
problem at spatial infinity for the so-called conformal Einstien field
equations. This representation of spatial infinity is linked to the conformal
properties of spacetimes, and it introduces a blow-up of the spatial infinity
point $i^{0}$ to a cylinder $(-1,1)\times\mathbb{S}^{2}$ commonly known as the
cylinder at spatial infinity $\mathcal{I}$. The cylinder $\mathcal{I}$ touches
the endpoints of past and future null infinities $\mathscr{I}^{\pm}$ at the
critical sets $\mathcal{I}^{\pm}=\\{\pm 1\\}\times\mathbb{S}^{2}$. This
representation of spatial infinity is useful for relating quantities at the
critical sets $\mathcal{I}^{\pm}$ to initial data on a Cauchy hypersurface —-
see, e.g, [8, 9]. Other equally significant contributions to the studies of
the asymptotic structure at spatial infinity are Ashtekar-Hansen's and
Ashtekar-Romano's formulations of spatial infinity — see [5, 6]. While the
relation between Friedrich's and Ashtekar-Romano's formulations was
established in [10], the link between Friedrich's formulation and Ashtekar-
Hansen's remains unexplored. Ashtekar-Hansen's and Ashtekar-Romano's
formulations introduce the Spi group, denoting the infinite-dimensional
asymptotic symmetry group at spatial infinity, with a structure similar to the
BMS group at null infinity.

One physical motivation for studying symmetries is the prospect of defining
conserved quantities (also known as Noether charges or simply charges) in an
isolated system, e.g., energy, momentum and angular momentum. For fields on a
fixed background, conserved quantities are defined by considering the integral
of the local energy-momentum tensor contracted with Killing fields of the
background spacetime over a Cauchy hypersurface. Given the dual role of the
spacetime metric in GR, describing both geometrical and physical aspects of
the theory, one can only define such conserved quantities in the asymptotic
limit, where the background and physical fields can be studied separately —
see [11]. As noted in [12], there had been a clear distinction in earlier work
in defining conserved quantities associated with asymptotic symmetries at null
and spatial infinity. In particular, the Hamiltonian formulation was
prominently used in the derivation of conserved quantities at spatial infinity
[13, 14, 15] compared to null infinity. The challenge in defining "conserved
quantities" at null infinity using a standard Hamiltonian formulation is that
symplectic current can be radiated away at null infinity, and thus,
generically, there exists no Hamiltonian generating BMS transformations.
However, the general prescription in [12] allows one to define charges
associated with asymptotic symmetries even in situations where the Hamiltonian
does not exist — see also [16, 17]. This prescription was used in [18] to
derive explicit expressions for the charges associated with BMS symmetries at
null infinity. Finally, note that "conserved quantities" at null infinity are
not exactly conserved for general dynamical spacetimes. Instead, a charge
associated with a BMS symmetry will have a non-vanishing flux through null
infinity $\mathscr{I}^{\pm}$ [12].

The discussion in this article is motivated by a recent conjecture by
Strominger [19] suggesting that BMS symmetries and their associated charges
can be linked to soft theorems [20, 19, 21] and the gravitational memory
effect [22, 23, 24]. This link is based on the so-called matching problem,
i.e., the idea that the BMS groups at past and future null infinities
($\text{BMS}^{-}$ at $\mathscr{I}^{-}$ and $\text{BMS}^{+}$ at
$\mathscr{I}^{+}$) can be matched by an antipodal reflection map near spatial
infinity. The matching of these symmetries leads to a global diagonal
asymptotic symmetry group in GR, implying that the incoming flux of an
asymptotic charge at $\mathscr{I}^{-}$ would be equal to the outgoing flux of
the corresponding charge at $\mathscr{I}^{+}$. Generically, the matching of
$\text{BMS}^{+}$ and $\text{BMS}^{-}$ and their associated charges requires an
analysis of the gravitational field and the charges near spatial infinity. One
significant challenge in this analysis is the singular nature of the conformal
structure at spatial infinity $i^{0}$, further highlighting the importance of
the different representations of spatial infinity [5, 25, 6, 7] in the
discussion of the matching problem. In recent years, numerous articles
discussed the asymptotic symmetry group at spatial infinity [26, 27, 28, 29,
30, 31] and their matching with the asymptotic charges at null infinities [26,
32, 33, 34, 35, 36]. On Minkowski spacetime, the matching of supertranslation
asymptotic charges has been investigated for the spin-1 field in [26] and the
spin-2 field in [32]. For spacetimes satisfying Ashtekar-Hansen's definition
of asymptotic flatness at null and spatial infinity (see [5] or [34] for
precise definition), the matching of supertranslation asymptotic charges for
the spin-1 and the gravitational field was shown in [33, 34]. Moreover, the
matching for Lorentz charges was also investigated in [35] using Ashtekar-
Hansen's formulation of spatial infinity.

The purpose of this paper is to provide a streamlined presentation of the
calculation of asymptotic charges at the critical sets and their matching
using Friedrich's formulation of spatial infinity. The expressions of the
asymptotic charges used in this article are adapted from [34], which agrees
with the general prescription of conserved quantities given in [12]. The use
of Friedrich's formulation allows us to express the supertranslation
asymptotic charges at $\mathcal{I}^{\pm}$ in terms of initial data given on a
Cauchy hypersurface, and to show that the matching of the asymptotic charges
at $\mathcal{I}^{\pm}$ follows from certain regularity conditions on the free
initial data. The full analysis of the asymptotic charges in a full GR setting
using Friedrich's formulation will be presented elsewhere. However, the main
results can be summarised as follows:

_For the generic initial data set given in[37], the asymptotic charges (as
defined in [34]) associated with BMS supertranslation symmetries are well-
defined at $\mathcal{I}^{\pm}$ if and only if the initial data satisfy extra
regularity conditions. The regularity conditions can be imposed on the free
conformal initial data. Finally, given initial data that satisfy the
regularity conditions, the asymptotic charge $\mathcal{Q}_{l,m}$ associated
with a given harmonic $Y_{l,m}$ at $\mathcal{I}^{+}$ is related to the
corresponding asymptotic charge at $\mathcal{I}^{-}$ by:
$\mathcal{Q}_{l,m}|_{\mathcal{I}^{+}}=(-1)^{l}\mathcal{Q}_{l,m}|_{\mathcal{I}^{-}}$._

The structure of this paper is as follows: In Section 2, the calculation of
the spin-2 supertranslation asymptotic charges on Minkowski spacetime using
Friedrich's formulation is reviewed. We start by introducing Friedrich's
representation and Friedrich-gauge (F-gauge) of spatial infinity on Minkowski
spacetime in Section 2.1. Since the asymptotic charges are expressed in terms
of the so-called Newman-Penrose gauge (NP-gauge), Section 2.2 provides a brief
discussion of the transformation from the NP-gauge to the F-gauge. Finally,
Section 3 presents a brief description of the tools and techniques used to
analyse the supertranslation asymptotic charges in full GR.

### Notations and conventions

This article will use tensors and spinors separately in various calculations.
The following indices will be used:

  * •

$a,\,b,\,c,\ldots$: spacetime abstract tensorial indices.

  * •

$i,\,j,\,k,\ldots$: spatial abstract indices

  * •

$\mu,\,\nu,\dots$: spacetime coordinate indices.

  * •

$\alpha,\,\beta,\dots$: spatial coordinate indices.

  * •

$\mathcal{A},\mathcal{B},\mathcal{C},\ldots$: coordinate indices on a
2-sphere.

  * •

$A,\,B,\,C,\ldots$: abstract spinorial indices.

The components of a tensor $T_{ab}$ with respect to a tensorial frame
$\\{{\bm{e}}_{{\bm{a}}}\\}$ are defined as

$T_{{\bm{a}}{\bm{b}}}=T_{ab}{\bm{e}}_{{\bm{a}}}{}^{a}{\bm{e}}_{{\bm{b}}}{}^{b}.$

Similarly, if $\\{{\bm{o}},{\bm{\iota}}\\}$ is a spin basis defined by

$o^{A}\equiv{\bm{\epsilon}}_{{\bm{0}}}{}^{A},\qquad\iota^{A}\equiv{\bm{\epsilon}}_{{\bm{1}}}{}^{A},$

then the components of a spinor $\xi_{A}$ with respect to the spin frame
$\\{{\bm{\epsilon}}_{{\bm{A}}}\\}$ are given by

$\xi_{{\bm{A}}}=\xi_{A}{\bm{\epsilon}}_{{\bm{A}}}{}^{A}.$

The spin basis $\\{{\bm{o}},{\bm{\iota}}\\}$ satisfies $\llbracket
o,\iota\rrbracket=1$, where $\llbracket.,.\rrbracket$ is the antisymmetric
product defined by

$\llbracket\zeta,\lambda\rrbracket=\zeta_{B}\lambda^{B}=\epsilon_{AB}\zeta^{A}\lambda^{B}.$

Here, $\epsilon_{AB}$ is the antisymmetric $\epsilon$-spinor that can be
regarded as a raising/lowering object for spinor indices.

##  2 The spin-2 asymptotic charges on Minkowski spacetime

In this section, we summarise the calculation of supertranslation asymptotic
charges for the spin-2 field as presented in [38]. Given the role of
Friedrich's representation of spatial infinity in this calculation, let us
begin by introducing the F-gauge on Minkowski spacetime.

###  2.1 The Minkowski spacetime in the F-gauge

Figure 1:  (a) The domain $\tilde{\mathcal{D}}$ containing spatial infinity,
(b) The domain $\tilde{\mathcal{D}}$ on the conformal diagram of Minkowski
spacetime.

To introduce the F-gauge on Minkowski spacetime
$(\mathbb{R}^{4},\tilde{{\bm{\eta}}})$, start with the Minkowski metric
$\tilde{{\bm{\eta}}}$ in the standard Cartesian coordinates
$(\tilde{x}^{\mu})$

$\tilde{{\bm{\eta}}}=\tilde{\eta}_{\mu\nu}{\bm{d}}{\tilde{x}^{\mu}}\otimes{\bm{d}}{\tilde{x}^{\nu}},$

where $\tilde{\eta}_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. This metric can be
written in terms of the standard spherical coordinates
$(\tilde{t},\tilde{\rho},\theta^{\mathcal{A}})$ as

$\tilde{{\bm{\eta}}}={\bm{d}}{\tilde{t}}\otimes{\bm{d}}{\tilde{t}}-{\bm{d}}{\tilde{\rho}}\otimes{\bm{d}}{\tilde{\rho}}-\tilde{\rho}^{2}{\bm{\sigma}},$

where $\theta^{\mathcal{A}}$ is a choice of coordinates on $\mathbb{S}^{2}$
and ${\bm{\sigma}}$ is the standard round metric on $\mathbb{S}^{2}$. Now,
define
$\tilde{X}^{2}\equiv\tilde{\eta}_{\mu\nu}\tilde{x}^{\mu}\tilde{x}^{\nu}=\tilde{t}^{2}-\tilde{\rho}^{2}$,
where $\tilde{x}^{0}\equiv\tilde{t}$ and
$\tilde{\rho}^{2}\equiv(\tilde{x}^{1})^{2}+(\tilde{x}^{2})^{2}+(\tilde{x}^{3})^{2}$.
Then, it is clear to see that spatial infinity is contained in the domain
$\tilde{\mathcal{D}}$ (See Figure 1) defined as

$\tilde{\mathcal{D}}\equiv\\{p\in\mathbb{R}^{4}|\hskip
2.84526pt\tilde{\eta}_{\mu\nu}\tilde{x}^{\mu}(p)\tilde{x}^{\nu}(p)<0\\}.$

The conformal metric $\hat{{\bm{\eta}}}=\Xi^{2}\tilde{{\bm{\eta}}}$, with
$\Xi=\tilde{X}^{-2}$ produces a point compactification of the physical
spacetime $(\mathbb{R}^{4},\tilde{{\bm{\eta}}})$, where all the points at
infinite spatial distances in the physical spacetime
$(\mathbb{R}^{4},\tilde{{\bm{\eta}}})$ are mapped to the spatial infinity
point $i^{0}$ in $(\mathbb{R}^{4},\hat{{\bm{\eta}}})$. The metric
$\hat{{\bm{\eta}}}$ can be written explicitly as

$\hat{{\bm{\eta}}}={\bm{d}}{t}\otimes{\bm{d}}{t}-{\bm{d}}{\rho}\otimes{\bm{d}}{\rho}-\rho^{2}{\bm{\sigma}},$

where

$t=-\frac{\tilde{t}}{\tilde{t}^{2}-\tilde{\rho}^{2}},\qquad\rho=-\frac{\tilde{\rho}}{\tilde{t}^{2}-\tilde{\rho}^{2}}.$

To introduce Friedrich's blow-up of spatial infinity, define a new time
coordinate $\tau=t/\rho$ and the rescaling

${\bm{\eta}}=\frac{1}{\rho^{2}}\hat{{\bm{\eta}}},$

so that

${\bm{\eta}}={\bm{d}}{\tau}\otimes{\bm{d}}{\tau}+\frac{\tau}{\rho}({\bm{d}}{\tau}\otimes{\bm{d}}{\rho}+{\bm{d}}{\rho}\otimes{\bm{d}}{\tau})-\frac{(1-\tau^{2})}{\rho^{2}}{\bm{d}}{\rho}\otimes{\bm{d}}{\rho}-{\bm{\sigma}}.$
(1)

From this, the relation between ${\bm{\eta}}$ and $\tilde{{\bm{\eta}}}$ can be
written as

${\bm{\eta}}=\Theta^{2}\tilde{{\bm{\eta}}},\qquad\Theta=\rho(1-\tau^{2}).$ (2)

In this representation, one sees that the spacetime metric ${\bm{\eta}}$ is
singular at $\rho=0$ while the intrinsic metric on the $\rho=\text{const.}$
hypersurfaces have a well-defined limit as $\rho\to 0$ and is given by

${\bm{q}}={\bm{d}}{\tau}\otimes{\bm{d}}{\tau}-{\bm{\sigma}}.$

Figure 2: A diagram of the neighbourhood of spatial infinity in Friedrich's
representation. In this representation, the spatial infinity point $i^{0}$ is
blown up to a cylinder $\mathcal{I}$ connecting past null infinity
$\mathscr{I}^{+}$ and future null infinity $\mathscr{I}^{-}$. The critical
sets $\mathcal{I}^{\pm}$ represents the sets where $\mathcal{I}$ touches
$\mathscr{I}^{\pm}$. The set $\mathcal{I}^{0}$ represents the intersection of
the cylinder $\mathcal{I}$ with the initial hypersurface $\\{\tau=0\\}$.

Given the above, define the conformal extension $(\mathcal{M},{\bm{\eta}})$
with

$\mathcal{M}\equiv\\{p\in\mathbb{R}^{4}|-1\leq\tau(p)\leq 1,\rho(p)\geq 0\\},$

then introduce the following subsets of the conformal boundary $(\Theta=0)$ —
see Figure 2.

$\displaystyle\mathscr{I}^{\pm}\equiv\big{\\{}p\in\mathcal{M}\hskip
2.84526pt\rvert\hskip 2.84526pt\tau(p)=\pm 1\big{\\}},\qquad$ past and future
null infinity $\displaystyle\mathcal{I}\equiv\big{\\{}p\in\mathcal{M}\hskip
2.84526pt\rvert\hskip 2.84526pt|\tau(p)|<1,\hskip
2.84526pt\rho(p)=0\big{\\}},\qquad$ the cylinder at spatial infinity
$\displaystyle\mathcal{I}^{\pm}\equiv\big{\\{}p\in\mathcal{M}\hskip
2.84526pt\rvert\hskip 2.84526pt\tau(p)=\pm 1,\hskip
2.84526pt\rho(p)=0\big{\\}},\qquad$ the critical sets at of null infinity

and

$\mathcal{I}^{0}\equiv\big{\\{}p\in\mathcal{M}\hskip 2.84526pt\rvert\hskip
2.84526pt\tau(p)=0,\hskip 2.84526pt\rho(p)=0\big{\\}},$

where $\mathcal{I}^{0}$ is the intersection of $\mathcal{I}$ with the initial
hypersurface $\mathcal{S}_{*}\equiv\\{\tau=0\\}$. In subsequent discussions,
we will refer to $(\mathcal{I},{\bm{q}})$ as the cylinder at spatial infinity.
Moreover, it will be convenient to introduce a frame basis
$\\{{\bm{e}}_{{\bm{a}}}\\}$ adapted to Friedrich's cylinder at spatial
infinity on Minkowski spacetime, the so-called F-gauge frame. Start with the
Minkowski metric ${\bm{\eta}}$ given by (1). It is straightforward to see that
the metric on the hypersurfaces $\mathcal{Q}_{\tau,\varrho}$ of constant
$\rho$ and $\tau$ is the standard metric on $\mathbb{S}^{2}$. Then, introduce
the complex null frame $\\{{\bm{\partial}}_{+},{\bm{\partial}}_{-}\\}$ on
$\mathcal{Q}_{\tau,\rho}$ and propagate
$\\{{\bm{\partial}}_{+},{\bm{\partial}}_{-}\\}$ off $\mathcal{Q}_{\tau,\rho}$
by imposing

$[{\bm{\partial}}_{\tau},{\bm{\partial}}_{\pm}]=0,\qquad[{\bm{\partial}}_{\rho},{\bm{\partial}}_{\pm}]=0.$

Now, the F-gauge frame $\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ and their
dual $\\{{\bm{\omega}}^{{\bm{A}}{\bm{A}}^{\prime}}\\}$ can be defined as
follows

${\bm{e}}_{{\bm{0}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}\left((1-\tau){\bm{\partial}}_{\tau}+\rho{\bm{\partial}}_{\rho}\right),\qquad{\bm{\omega}}^{{\bm{0}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}\left({\bm{d}}{\tau}-\frac{1}{\rho}(1-\tau){\bm{d}}{\rho}\right),$
(4)
${\bm{e}}_{{\bm{1}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}\left((1+\tau){\bm{\partial}}_{\tau}-\rho{\bm{\partial}}_{\rho}\right),\qquad{\bm{\omega}}^{{\bm{1}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}\left({\bm{d}}{\tau}+\frac{1}{\rho}(1+\tau){\bm{d}}{\rho}\right),$
${\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}{\bm{\partial}}_{+},\qquad{\bm{\omega}}^{{\bm{0}}{\bm{1}}^{\prime}}=\sqrt{2}{\bm{\omega}}^{+},$
${\bm{e}}_{{\bm{1}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}{\bm{\partial}}_{-},\qquad{\bm{\omega}}^{{\bm{1}}{\bm{0}}^{\prime}}=\sqrt{2}{\bm{\omega}}^{-},$

where ${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$ is obtained from
${\bm{e}}_{{\bm{a}}}$ by contraction with the Infeld-van der Waerden symbols
$\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}$. So,
${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\equiv\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{a}}}$.
The dual frame ${\bm{\omega}}^{\pm}$ satisfy

$\langle{\bm{\omega}}^{+},{\bm{\partial}}_{+}\rangle=1,\qquad\langle{\bm{\omega}}^{-},{\bm{\partial}}_{-}\rangle=1.$

In terms of the above frame fields, the metric ${\bm{\eta}}$ can be written as

${\bm{\eta}}=\epsilon_{{\bm{A}}{\bm{B}}}\epsilon_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}{\bm{\omega}}^{{\bm{A}}{\bm{A}}^{\prime}}\otimes{\bm{\omega}}^{{\bm{B}}{\bm{B}}^{\prime}}.$

######  Remark 1.

A special class of conformal curves, known as conformal geodesics, play a
major role in Friedrich's representation of spatial infinity — see [10] for
details. For example, on Minkowski spacetime, the curves of constant $\rho$
and $\theta^{\mathcal{A}}$ describe a non-intersecting congruence of conformal
geodesics on $\mathcal{M}$, suggesting that the congruence of conformal
geodesics starting at $\rho=0$ coincide with the cylinder at spatial infinity.
One of the remarkable properties of conformal geodesics is that they specify a
canonical conformal factor $\Theta$ [39, 40], and an F-gauge metric ${\bm{g}}$
given by

${\bm{g}}=\Theta^{2}\tilde{{\bm{g}}},$ (5)

where $\tilde{{\bm{g}}}$ is the metric on a vacuum Einstein spacetime
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$. On Minkowski spacetime, the
canonical conformal factor associated with the above-mentioned congruence of
conformal geodesics on Minkowski is equivalent to $\Theta$ given in (2).

Further discussion of Friedrich's blow-up of spatial infinity and the F-gauge
on more general spacetimes will be postponed for later sections.

###  2.2 NP-gauge to F-gauge

As mentioned in the introduction, the supertranslation asymptotic charges at
$\mathscr{I}^{\pm}$ are generally expressed in terms of the NP-gauge,
comprised of certain conformal gauge conditions, certain coordinates and an
orthonormal frame field $\\{{\bm{e}}^{\bullet}_{{\bm{a}}}\\}$ satisfying
certain frame gauge conditions. A general description of these gauge
conditions was given in [8], along with a prescription of the transformation
between the NP-gauge and the F-gauge. The main observation in [8] is that the
NP-gauge frame is adapted to null infinity $\mathscr{I}^{\pm}$ while the
F-gauge frame is adapted to Cauchy hypersurfaces. Additionally, the NP
conformal gauge conditions, as described in [8], imply that a metric
${\bm{g}}^{\bullet}$ satisfying those conditions will be related to the
F-gauge metric ${\bm{g}}$ by

${\bm{g}}^{\bullet}=\theta^{2}{\bm{g}},$

with $\theta$ satisfying a linear ordinary differential equation, which can be
solved on the generators of $\mathscr{I}^{\pm}$ — see [8]. Given the above,
the NP-gauge orthonormal frame
$\\{{\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$, defined by
${\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}\equiv\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}^{\bullet}_{{\bm{a}}}$,
is related to the F-gauge orthonormal frame
$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ by

${\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}=\theta^{-1}\Lambda^{{\bm{B}}}{}_{{\bm{A}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}},\qquad\Lambda^{{\bm{B}}}{}_{{\bm{A}}}\in\text{SL}(2,\mathbb{C}).$

The general framework in [8] was used in [9] to obtain an explicit
transformation between the NP-gauge and the F-gauge on Minkowski spacetime. In
particular, the results in [9] show that $\theta=1$ and

$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{0}}}=\frac{2e^{i\omega}}{\sqrt{\rho}(1+\tau)},\qquad\Lambda^{{\bm{0}}}{}_{{\bm{1}}}=\frac{e^{-i\omega}\sqrt{\rho}(1+\tau)}{2},\qquad\Lambda^{{\bm{1}}}{}_{{\bm{1}}}=\Lambda^{{\bm{0}}}{}_{{\bm{0}}}=0,\qquad\text{on
}\mathscr{I}^{+}$
$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{0}}}=\frac{e^{-i\omega}\sqrt{\rho}(1-\tau)}{2},\qquad\Lambda^{{\bm{0}}}{}_{{\bm{1}}}=\frac{2e^{i\omega}}{\sqrt{\rho}(1-\tau)},\qquad\Lambda^{{\bm{1}}}{}_{{\bm{1}}}=\Lambda^{{\bm{0}}}{}_{{\bm{0}}}=0,\qquad\text{on
}\mathscr{I}^{-}$

where $\omega$ is an arbitrary real number that encodes the spin rotation of
the frames on $\mathbb{S}^{2}$. Given the above transformation between the NP-
gauge and the F-gauge on Minkowski spacetime, the asymptotic charges can be
evaluated on the critical sets $\mathcal{I}^{\pm}$ given solutions for the
spin-2 field equations.

###  2.3 The spin-2 charges in the F-gauge

The goal of this section is to obtain an expression of the supertranslation
asymptotic charges that can be evaluated at the critical sets
$\mathcal{I}^{\pm}$. First, introduce the NP null tetrad
$(l^{a},n^{a},m^{a},\bar{m}^{a})$ as

$l^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}},\qquad
n^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}},\qquad
m^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}},\qquad\bar{m}^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}.$
(7)

Then, let $W^{\bullet}_{abcd}$ denote a Weyl-like tensor, i.e., a tensor with
symmetries of the Weyl tensor, and define $\mathcal{W}^{\bullet}_{abcd}$ as

$\mathcal{W}^{\bullet}_{abcd}\equiv
W^{\bullet}_{abcd}+i({}^{*}W)^{\bullet}_{abcd},$

where $({}^{*}W)^{\bullet}_{abcd}$ is the left Hodge dual of
$W^{\bullet}_{abcd}$. Then, the spinorial counterpart of $W^{\bullet}_{abcd}$
can be decomposed in terms of the symmetric spin-2 spinor
$\psi^{\bullet}_{ABCD}$ as

$W^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=-\psi^{\bullet}_{ABCD}\epsilon^{\bullet}_{A^{\prime}B^{\prime}}\epsilon^{\bullet}_{C^{\prime}D^{\prime}}-\bar{\psi}^{\bullet}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon^{\bullet}_{AB}\epsilon^{\bullet}_{CD}.$
(8)

######  Remark 2.

The ∙ notation indicates that $W^{\bullet}_{abcd}$ (or
$\mathcal{W}^{\bullet}_{abcd}$), $\epsilon^{\bullet}_{AB}$ are in the NP-
gauge. Given that $\theta=1$ on Minkowski spacetime, one has
$W^{\bullet}_{abcd}=W_{abcd}$. However, this is not true for general
spacetimes, i.e., the Weyl tensor $C^{\bullet}_{abcd}$ associated with
${\bm{g}}^{\bullet}$ will be related to $C_{abcd}$ associated with ${\bm{g}}$
by $C^{\bullet}_{abcd}=\theta^{2}C_{abcd}$. Moreover, the $\epsilon$-spinor in
the NP-gauge $\epsilon^{\bullet}_{AB}$ will be related to $\epsilon_{AB}$ by
$\epsilon^{\bullet}_{AB}=\theta\epsilon_{AB}$. If one wishes to obtain a
relation between $C^{\bullet}_{{\bm{a}}{\bm{b}}{\bm{c}}{\bm{d}}}$ and
$C_{{\bm{a}}{\bm{b}}{\bm{c}}{\bm{d}}}$, where
$C^{\bullet}_{{\bm{a}}{\bm{b}}{\bm{c}}{\bm{d}}}\equiv
C^{\bullet}_{abcd}{\bm{e}}^{\bullet}_{{\bm{a}}}{}^{a}{\bm{e}}^{\bullet}_{{\bm{b}}}{}^{b}{\bm{e}}^{\bullet}_{{\bm{c}}}{}^{c}{\bm{e}}^{\bullet}_{{\bm{d}}}{}^{d}$
and $C_{{\bm{a}}{\bm{b}}{\bm{c}}{\bm{d}}}\equiv
C_{abcd}{\bm{e}}_{{\bm{a}}}{}^{a}{\bm{e}}_{{\bm{b}}}{}^{b}{\bm{e}}_{{\bm{c}}}{}^{c}{\bm{e}}_{{\bm{d}}}{}^{d}$,
then one makes use of the transformation between $C^{\bullet}_{abcd}$ and
$C_{abcd}$ as well as the transformation between the NP-gauge frame
$\\{{\bm{e}}^{\bullet}_{{\bm{a}}}\\}$ and the F-gauge frame
$\\{{\bm{e}}_{{\bm{a}}}\\}$ given by
${\bm{e}}^{\bullet}_{{\bm{a}}}=\theta^{-1}\Lambda^{{\bm{b}}}{}_{{\bm{a}}}{\bm{e}}_{{\bm{b}}}$,
where $\Lambda^{{\bm{b}}}{}_{{\bm{a}}}\in\text{O}(1,3)$.

Following the discussion in [34], the asymptotic charges associated with
smooth functions $\lambda$ on $\mathbb{S}^{2}$ can be written as

$\mathscr{Q}=\int_{\mathcal{C}}\lambda\mathcal{W}^{\bullet}_{abcd}l^{a}n^{b}m^{c}\bar{m}^{d}{\bm{d}}{S},$

where $\mathcal{C}$ denotes a cross-section of $\mathscr{I}^{\pm}$. From (7)
and (8), it can be shown that the charges $\mathscr{Q}$ can be written as

$\mathscr{Q}=-2\int_{\mathcal{C}}\lambda\bar{\psi}^{\bullet}_{2}{\bm{d}}{S},$

where
$\bar{\psi}^{\bullet}_{2}\equiv\bar{\psi}^{\bullet}_{{\bm{0}}^{\prime}{\bm{0}}^{\prime}{\bm{1}}^{\prime}{\bm{1}}^{\prime}}$.
To evaluate the charges at the critical sets, one must obtain an expression
for $\mathscr{Q}$ in terms of the F-gauge. The transformation from the NP-
gauge to the F-gauge on Minkowski spacetime, discussed in the previous
section, implies that

$\psi^{\bullet}_{2}=\psi_{2},$

where $\psi_{2}\equiv\psi_{{\bm{0}}{\bm{0}}{\bm{1}}{\bm{1}}}$. Thus, the final
expression of the charges in the F-gauge is given by

$\mathscr{Q}=-2\int_{\mathcal{C}}\lambda\bar{\psi}_{2}{\bm{d}}{S}.$ (9)

To evaluate this expression at $\mathcal{I}^{\pm}$, the next step is to obtain
a solution for $\bar{\psi}_{2}$ using the field equations.

###  2.4 The spin-2 field equations

The spinorial spin-2 field equation can be written as
$\nabla^{A}{}_{A^{\prime}}\psi_{ABCD}=0$. Applying
$-2\nabla_{E}{}^{A^{\prime}}$, the wave equation satisfied by $\psi_{ABCD}$
can be written as

$\square\psi_{ABCD}=0,$ (10)

where $\square\equiv\nabla_{AA^{\prime}}\nabla^{AA^{\prime}}$ is the
D'Alembertian operator. To analyse the solutions for this equation in a
neighbourhood of spatial infinity, assume that the components $\psi_{n}$ of
the spin-2 spinor can be expanded near $\rho=0$ in terms of spin-weighted
spherical harmonics ${}_{n}Y_{l,m}$ as

$\psi_{n}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(\tau){}_{2-n}Y_{l,m}+o_{1}(\rho),\qquad\text{for
}n=0,\ldots,4.$ (11)

where
$a_{n;l,m}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow\mathbb{C}$.

######  Remark 3.

The expansion (11) is consistent with the estimates developed in [41] that
demonstrates that for a certain non-trivial class of initial data, the
components $\psi_{n}$ can be expanded as

$\psi_{n}=\sum_{k=|2-n|}^{p-1}\frac{1}{k!}\psi_{n}^{(k)}\rho^{k}+R_{p}[\psi_{n}].$

In the above, the coefficients $\psi_{n}^{(k)}$ are explicitly known functions
of $\tau$ and the angular variables which are smooth for $\tau\in(-1,1)$ and
whose regularity at $\tau=\pm 1$ can be controlled in terms of the initial
data. The reminder satisfies $R_{p}[\psi_{n}]\in C^{m}$ for $p\geq m+6$ for
$\rho$ near 0 and $\tau\in[-1,1]$.

Using (11) and substituting in (10), one obtains second order ordinary
differential equations for the coefficients $a_{n;l,m}(\tau)$

$(1-\tau^{2})\ddot{a}_{0;l,m}+2(2-\tau)\dot{a}_{0;l,m}+l(l+1)a_{0;l,m}=0,\\\ $
(12a)
$(1-\tau^{2})\ddot{a}_{1;l,m}+2(1-\tau)\dot{a}_{1;l,m}+l(l+1)a_{1;l,m}=0,\\\ $
(12b) $(1-\tau^{2})\ddot{a}_{2;l,m}-2\tau\dot{a}_{2;l,m}+l(l+1)a_{2;l,m}=0,\\\
$ (12c)
$(1-\tau^{2})\ddot{a}_{3;l,m}-2(1+\tau)\dot{a}_{3;l,m}+l(l+1)a_{3;l,m}=0,\\\ $
(12d)
$(1-\tau^{2})\ddot{a}_{4;l,m}-2(2+\tau)\dot{a}_{4;l,m}+l(l+1)a_{4;l,m}=0.$
(12e)

Now, assume that the initial data $(\psi_{n})|_{\mathcal{S}_{*}}$ on
$\mathcal{S}_{*}\equiv\\{\tau=0\\}$ can be expanded near $\rho=0$ as

$\psi_{n}|_{\mathcal{S}_{*}}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(0){}_{2-n}Y_{l,m}+o(\rho).$

Given that the expression of the charges (9) is written in terms of
$\psi_{2}$, one only requires the solution for (12c) in order to evaluate
$\mathscr{Q}$ at $\tau=\pm 1$. Equation (12c) is a Jacobi differential
equation, with a solution that can be expressed in terms of hypergeometric
functions — see [42]. However, a simpler expression can be obtained by using
the differential equation solver of the Wolfram Language. In particular, it
can be shown that for $l\geq 0$ and $-l\leq m\leq l$, the solution $a_{2;l,m}$
is given by

$a_{2;l,m}(\tau)=A_{l,m}P_{l}(\tau)+B_{l,m}Q_{l}(\tau),$ (13)

where $P_{l}(\tau)$ is the Legendre polynomial of order $l$ and $Q_{l}(\tau)$
is the Legendre function of the second kind of order $l$. The constants
$A_{l,m}$ and $B_{l,m}$ can be expressed in terms of the initial data for
$a_{2;l,m}$. Since $Q_{l}(\tau)$ is proportional to $\ln{(1+\tau)}$ and
$\ln{(1-\tau)}$, the solution (13) will diverge logarithmically near $\tau=\pm
1$, unless $B_{l,m}=0$.

To obtain a well-defined solution for $a_{2;l,m}$ at the critical sets, the
constant $B_{l,m}$ is required to vanish. Note that $B_{l,m}$ can be written
in terms of $a_{2;l,m}(0)$ and $\dot{a}_{2;l,m}(0)$ as

$B_{l,m}=\frac{\sqrt{\pi}(l+1)}{\Gamma(-\frac{l}{2})\Gamma(\frac{l+3}{2})}a_{2;l,m}(0)+\frac{\sqrt{\pi}}{\Gamma(\frac{1}{2}-\frac{l}{2})\Gamma(1+\frac{l}{2})}\dot{a}_{2;l,m}(0),$
(14)

where $\Gamma$ denotes the Gamma function. Then, the observation that the
coefficient of $a_{2;l,m}(0)$ vanishes for even $l$ while the coefficient of
$\dot{a}_{2;l,m}(0)$ vanishes for odd l suggests the following regularity
conditions:

######  Lemma 1.

The solution (13) is well-defined at $\mathcal{I}^{\pm}$ if and only if:

  1. 1.

$a_{2;l,m}(0)=0$ for odd $l$, and

  2. 2.

$\dot{a}_{2;l,m}(0)=0$ for even $l$.

These regularity conditions can be expressed in terms of freely specifiable
data as shown in [38].

Making use of (9), (11) and (13) and by choosing initial data satisfying Lemma
1 and $\lambda=Y_{l,m}$, the charge $\mathscr{Q}_{l,m}$ associated with
$Y_{l,m}$ at $\mathcal{I}^{\pm}$ can be written as

$\mathscr{Q}_{l,m}|_{\mathcal{I}^{\pm}}=\begin{cases}2(l+1)Q_{l+1}(0)(a_{2})_{*}\qquad\text{for
even }l\geq 0,\\\
\pm\sqrt{l(l+1)}Q_{l}(0)\left((a_{1})_{*}-(a_{3})_{*}\right)\qquad\text{for
odd }l,\end{cases}$ (15)

where $(a_{n})_{*}\equiv a_{n;l,m}(0)$. The main conclusions from the above
discussion are

  1. 1.

For generic boosted initial data, the charges $\mathscr{Q}$ are not well-
defined in the limits of spatial infinity, i.e., at the critical sets
$\mathcal{I}^{\pm}$.

  2. 2.

Boosted initial data satisfying Lemma 1 allows us to obtain well-defined
expressions for $\mathscr{Q}$ at the critical sets.

  3. 3.

The antipodal matching of the charges is obtained naturally in this formalism.
In particular, we have
$\mathscr{Q}_{l,m}|_{\mathcal{I}^{+}}=(-1)^{l}\mathscr{Q}_{l,m}|_{\mathcal{I}^{-}}$.

##  3 The asymptotic charges in full GR

The process of the calculation of the asymptotic charges for the spin-2 field
at the critical sets presented in the previous section can be extended to the
full GR setting. For this, assume that
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ is a spacetime satisfying the vacuum
Einstein field equations i.e.

$\tilde{R}_{ab}=0,$ (16)

where $\tilde{R}_{ab}$ is the Ricci tensor associated with the Levi-Civita
connection $\tilde{{\bm{\nabla}}}$ of $\tilde{{\bm{g}}}$. The conformal
rescaling

${\bm{g}}=\Xi^{2}\tilde{{\bm{g}}},$ (17)

implies transformation laws for the physical fields e.g. the curvature tensor
$\tilde{R}^{a}{}_{bcd}$, the Schouten tensor $\tilde{L}_{ab}$ etc. It follows
that the vacuum Einstein field equations are not conformally invariant and
that the field equations implied by (17) cannot be analysed at the conformal
boundary $\Xi=0$ since the conformal Ricci tensor $R_{ab}$ is singular at the
points where $\Xi=0$. If $\tilde{C}^{a}{}_{bcd}$ denotes the Weyl tensor, then
the Bianchi identity can be written in terms of the Levi-Civita connection
associated with ${\bm{g}}$ as

$\nabla_{a}(\Xi^{-1}\tilde{C}^{a}{}_{bcd})=0.$ (18)

If one defines the rescaled Weyl tensor
$d^{a}{}_{bcd}\equiv\Xi^{-1}\tilde{C}^{a}{}_{bcd}$, then equation (18) can be
written as

$\nabla_{a}d^{a}{}_{bcd}=0.$ (19)

Exploiting the symmetries of the rescaled Weyl tensor implies

$\nabla_{[e}d^{a}{}_{|b|cd]}=0.$ (20)

Our calculations of the asymptotic charges rely on Friedrich's extended
conformal field equations [43, 44, 45, 39] written in terms of a Weyl
connection $\hat{{\bm{\nabla}}}$ satisfying

$\hat{\nabla}_{a}g_{bc}=-2f_{a}g_{bc},$

where $f_{a}$ is an arbitrary 1-form. The explicit form of these equations
will not be necessary for this article, interested readers can refer to
Chapter 8 in [39]. The extended conformal field equations yield differential
equations to be solved for the ${\bm{g}}$-orthonormal frame fields
$\\{{\bm{e}}_{{\bm{a}}}\\}$, the components of the Weyl connection
coefficients $\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}$, the Schouten
tensor $\hat{L}_{{\bm{a}}{\bm{b}}}$ and the rescaled Weyl tensor
$d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}$. One significant feature of the
extended conformal field equations is that they exhibit gauge freedom
indicated by the fact that there are no equations to fix the conformal factor
$\Xi$ and the Weyl connection $\hat{{\bm{\nabla}}}$. To fix this gauge
freedom, one can make use of the so-called conformal Gaussian gauge, based on
conformal geodesics, that allows us to write the field equations as a
symmetric hyperbolic system in which the evolution equations reduce to a
transport system along the conformal geodesics. Given the field equations in
this gauge, it is possible to obtain a spinorial version of these equations to
be analysed near spatial infinity.

The above discussion highlights one of the key tools of conformal methods in
GR. The following section will introduce Friedrich's regular initial value
problem for the conformal field equations.

###  3.1 Friedrich's regular initial value problem

The purpose of this section is to briefly introduce Friedrich's formulation in
full GR. As mentioned in the introduction, the aim of Friedrich's formulation
is to introduce a regular initial value problem for the conformal field
equations near spatial infinity. An extensive discussion of this framework is
provided in [7, 8]. In this framework, the spacetime
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ is assumed to be the development of
some asymptotically Euclidean and regular [11, 39] initial data
$(\tilde{\mathcal{S}},\tilde{{\bm{h}}},\tilde{{\bm{K}}})$. In particular, the
initial data $(\tilde{\mathcal{S}},\tilde{{\bm{h}}},\tilde{{\bm{K}}})$ is said
to be an asymptotically Euclidean and regular manifold if there exists a
3-dimensional smooth compact manifold
$(\mathcal{S}^{\prime},{\bm{h}}^{\prime})$ with a point
$i\in\mathcal{S}^{\prime}$, a diffeomorphism $\Phi$ from
$\mathcal{S}^{\prime}\setminus\\{i\\}$ onto $\tilde{\mathcal{S}}$ and a
conformal factor $\Omega^{\prime}$ which is analytic on $\mathcal{S}^{\prime}$
and satisfying i) $\Omega^{\prime}=0,{\bm{d}}{\Omega^{\prime}}=0$ and
$\text{Hess}(\Omega^{\prime})=-2{\bm{h}}^{\prime}$ at $i$, ii)
$\Omega^{\prime}>0$ on $\mathcal{S}^{\prime}\setminus\\{i\\}$, iii)
${\bm{h}}^{\prime}=\Omega^{\prime 2}\Phi_{*}\tilde{{\bm{h}}}$ on
$\mathcal{S}^{\prime}\setminus\\{i\\}$. To apply this, start with the initial
data satisfying the Hamiltonian and momentum constraints as introduced in
[37]:

######  Proposition 1.

For any $\xi,\zeta\in C^{2}(\mathbb{S}^{2})$, there exists a vacuum initial
data set $(\tilde{{\bm{h}}},\tilde{\bm{K}})$ such that the components of
$\tilde{{\bm{h}}}$ and $\tilde{\bm{K}}$ with respect to the standard Euclidean
coordinate chart $(x^{\alpha})$ have the following asymptotics:

$\tilde{h}_{\alpha\beta}=-\delta_{\alpha\beta}-\frac{1}{r}\left[\left(A-\frac{\xi}{2}\right)\delta_{\alpha\beta}+\xi\frac{x_{\alpha}x_{\beta}}{r^{2}}\right]+O_{2}(r^{-2}),$
(21a)
$\tilde{K}_{\alpha\beta}=\frac{1}{r^{2}}\left[-\frac{1}{2}\zeta\delta_{\alpha\beta}+\frac{1}{r}\left(-B_{\alpha}x_{\beta}-B_{\beta}x_{\alpha}+(B^{\gamma}x_{\gamma})\delta_{\alpha\beta}\right)+\zeta\frac{x_{\alpha}x_{\beta}}{r^{2}}\right]+O_{1}(r^{-3})$
(21b)

where $A$, $\\{B_{\alpha}\\}_{\alpha=1}^{3}$ are some constants and
$r=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$.

Then, define the inverse coordinates $(y^{\alpha})$ and the conformal factor
$\Omega^{\prime}$ as

$y^{\alpha}=-\frac{x^{\alpha}}{r^{2}},\qquad\Omega^{\prime}=\frac{\varrho^{2}}{\sqrt{1+A\varrho}},$

so that the components of the conformal initial data
${\bm{h}}^{\prime}=\Omega^{\prime 2}\tilde{{\bm{h}}}$ and
${\bm{K}}^{\prime}=\Omega^{\prime}\tilde{{\bm{K}}}$ can be expanded around
$\varrho=\sqrt{(y^{1})^{2}+(y^{2})^{2}+(y^{3})^{2}}=0$ as

$h^{\prime}_{\alpha\beta}=-\delta_{\alpha\beta}-\xi\varrho\left(\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}-\frac{1}{2}\delta_{\alpha\beta}\right)+O_{2}(\varrho^{2}),$
(22a)
$K^{\prime}_{\alpha\beta}=-\frac{\zeta}{2}\delta_{\alpha\beta}-\frac{1}{\varrho}\left(B_{\alpha}y_{\beta}+B_{\beta}y_{\alpha}+\frac{1}{2}(B^{\gamma}y_{\gamma})\delta_{\alpha\beta}\right)+\left(\zeta-4\frac{(B^{\gamma}y_{\gamma})}{\varrho}\right)\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}+O_{1}(\varrho),$
(22b)

The $O(\varrho)$ term in (22a) can be made to vanish by performing a
coordinate transformation from $(y^{\alpha})$ to normal coordinates
$(z^{\alpha})$ [46]. Then, the term $O(\varrho^{2})$ can be removed by
performing a further conformal transformation

$\Omega^{\prime}\to\Omega\equiv\varpi\Omega^{\prime},$

where

$\varpi\equiv e^{f},\quad\text{ with
}f=\frac{1}{2}l^{\prime}_{\alpha\beta}(i)z^{\alpha}z^{\beta}.$

Here, $l^{\prime}_{\alpha\beta}(i)$ denotes the components of the Schouten
tensor associated with ${\bm{h}}^{\prime}$ in normal coordinates
$(z^{\alpha})$ evaluated at $i(\varrho=0)$. If $h^{\prime(0)}_{\alpha\beta}$
is the metric at $i$ and $|z|^{2}\equiv
h^{\prime(0)}_{\alpha\beta}z^{\alpha}z^{\beta}$, then the components of the
conformal initial data $\bar{{\bm{h}}}=\varpi^{2}{\bm{h}}^{\prime}$ and
$\bar{{\bm{K}}}=\varpi{\bm{K}}^{\prime}$ can be written as

$\bar{h}_{\alpha\beta}=-\delta_{\alpha\beta}+O(|z|^{3}).$ (23)
$\bar{K}_{\alpha\beta}=-\frac{\zeta}{2}\delta_{\alpha\beta}-\frac{1}{2}\left(B_{\alpha}\vartheta_{\beta}+B_{\beta}\vartheta_{\alpha}+\frac{1}{2}(B^{\gamma}\vartheta_{\gamma})\delta_{\alpha\beta}\right)+\zeta\vartheta_{\alpha}\vartheta_{\beta}-4(B^{\gamma}\vartheta_{\gamma})\vartheta_{\alpha}\vartheta_{\beta}+O(|z|).$

where $\vartheta^{\alpha}=z^{\alpha}/|z|$. The initial data
$(\bar{{\bm{h}}},\bar{{\bm{K}}})$ will be referred to as the conformal normal
initial data. It can be shown, using the conformal constraint equations, that
the initial data for the components of the conformal Schouten tensor
$\bar{L}_{\alpha\beta}$ and the electric and magnetic parts of the Weyl
tensor, $\bar{d}_{\alpha\beta}$ and $\bar{d}_{\alpha\beta\gamma}$,
respectively, are singular at $|z|=0$. To introduce regular initial data, one
must introduce a further conformal rescaling as suggested in [7]

$\Omega\to\kappa^{-1}\Omega,$ (24)

with $\kappa=O(|z|)$. Let $\rho=|z|$, then the conformal factor $\Omega$ can
be expanded around $\rho=0$ as

$\Omega=\rho^{2}+\frac{1}{6}\Pi_{3}[\Omega]\rho^{3}+O(\rho^{4}),$

where $\Pi_{3}[\Omega]$ is written in terms of the angular coordinates
$\vartheta^{\alpha}$, the constant $A$, the function $\alpha$ and its
derivatives with respect to $\vartheta^{\alpha}$.

The conformal rescaling (24) introduces the conformal metric
${\bm{h}}=\kappa^{-2}\bar{{\bm{h}}}$. Then, if $\\{{\bm{e}}_{{\bm{i}}}\\}$ is
an ${\bm{h}}$-orthonormal frame, one can show

$h_{{\bm{i}}{\bm{j}}}=-\delta_{{\bm{i}}{\bm{j}}}+O(|z|^{3}),$

and

$K_{{\bm{i}}{\bm{j}}}=O(|z|),\qquad L_{{\bm{i}}{\bm{j}}}=O(|z|),\qquad
d_{{\bm{i}}{\bm{j}}}=O(1),\qquad d_{{\bm{i}}{\bm{j}}{\bm{k}}}=O(1).$

Hence, the initial data $({\bm{h}},{\bm{K}})$ for the conformal field
equations are regular at $|z|=0$.

One of the advantages of using the conformal Gaussian gauge mentioned in the
last section is that it implies a conformal factor $\Theta$ that can be
written in terms of initial data. Following [40, 7], we have

$\Theta=\kappa^{-1}\Omega\left(1-\tau^{2}\frac{\kappa^{2}}{\omega}\right),$
(25)

where $\tau$ refers to the parameter along the conformal geodesics used to
construct the conformal Gaussian system and

$\omega=\frac{2\Omega}{\sqrt{|{\bm{h}}({\bm{d}}{\Omega},{\bm{d}}{\Omega})|}}.$

######  Remark 4.

In the following, $\text{SU}(2,\mathbb{C})$ refers to the special unitary
group of degree $2$ over complex numbers. We also use $\text{SU}(\mathcal{S})$
to refer to the bundle of normalised spin frames over a manifold $\mathcal{S}$
with structure group $\text{SU}(2,\mathbb{C})$.

The basic idea of the blow-up of the point $i$ involves replacing $i$ with the
space of directions pointing out of $i$. In other words, the blow-up of $i$ is
a certain subspace of the tangent space at $i$, which is diffeomorphic to
$\mathbb{S}^{2}$. In Friedrich's formulation, rather than working with tensor
frames, the blow-up of $i$ is achieved by considering a certain subset of the
bundle of the normalised spin frames $\text{SU}(\mathcal{S}^{\prime})$ with
structure group $\text{SU}(2,\mathbb{C})$ — see [8] for details. In this
picture, the blow-up of $i$ is diffeomorphic to $\mathbb{S}^{3}$ while its
quotient by $\text{U}(1)$ is diffeomorphic to $\mathbb{S}^{2}$. The extra
dimension in this blow-up corresponds to the choice of a phase parameter given
that the choice of the spin frame is not unique. More precisely, consider a
fixed spin frame $\\{{\bm{\epsilon}}_{{\bm{A}}}{}\\}$ at $i$ and
${\bm{t}}\in\text{SU}(2,\mathbb{C})$, the transformed spin frame
${\bm{\epsilon}}_{{\bm{A}}}({\bm{t}})\equiv
t_{{\bm{A}}}{}^{{\bm{B}}}{\bm{\epsilon}}_{{\bm{B}}}{}$ can be extended to an
open ball $B_{a}(i)$ in $\mathcal{S}^{\prime}$ of radius $a$ centred at $i$ by
parallel propagation along an ${\bm{h}}$-geodesic starting at $i$. If $\rho$
is the affine parameter along the geodesic, then for a fixed ${\bm{t}}$, the
propagated spin frame can be written as
${\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})$. Next, define
$\mathcal{M}_{a,\kappa}$, a submanifold of
$\mathbb{R}\times\mathbb{R}\times\text{SU}(2,\mathbb{C})$ as

$\mathcal{M}_{a,\kappa}=\\{(\tau,\rho,{\bm{t}})\in\mathbb{R}\times\mathbb{R}\times\text{SU}(2,\mathbb{C})|\hskip
2.84526pt0\leq\rho<a,-\frac{\omega}{\kappa}\leq\tau\leq\frac{\omega}{\kappa}\\},$
(26)

with the following subsets

$\mathscr{I}^{\pm}_{a}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\hskip
2.84526pt0<\rho<a,\tau=\pm\frac{\omega}{\kappa}\\},\qquad\text{past and future
null infinity}$ (27a)
$\mathcal{I}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\hskip
2.84526pt\rho=0,-1<\tau<1\\},\qquad\text{the cylinder at spatial infinity}$
(27b)
$\mathcal{I}^{\pm}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\hskip
2.84526pt\rho=0,\tau=\pm 1\\},\qquad\text{the critical sets of null infinity}$
(27c)

and

$\mathcal{I}_{0}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\hskip
2.84526pt\rho=0,\tau=0\\}.$ (28)

To relate the structures on the fibre bundle to the spacetime manifold
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ satisfying (16), let
$(\mathcal{M},{\bm{g}})$ denote a smooth conformal extension such that i)
$\Theta>0$ and ${\bm{g}}=\Theta^{2}\tilde{{\bm{g}}}$ on $\tilde{\mathcal{M}}$,
ii) $\Theta=0$ and $d\Theta\neq 0$ on $\mathscr{I}^{\pm}_{a}$. Now let
$\mathcal{N}\subset\mathcal{M}$ denote the domain of influence of
$B_{a}(i)\setminus\\{i\\}$, then the projection map $\bar{\pi}^{\prime}$ from
$\mathcal{M}_{a,\kappa}$ to $\mathcal{N}$ can be factored as

$\mathcal{M}_{a,\kappa}\xrightarrow{\bar{\pi}^{\prime}_{1}}\mathcal{M}^{\prime}_{a,\kappa}\xrightarrow{\bar{\pi}^{\prime}_{2}}\mathcal{N},$

where
$\mathcal{M}^{\prime}_{a,\kappa}\equiv\mathcal{M}_{a,\kappa}/\text{U}(1)$ is
implied by the action of $\text{U}(1)$ on $\text{SU}(2,\mathbb{C})$. From
(26), the map $\bar{\pi}^{\prime}_{1}$ is given by the identity on the
$\mathbb{R}\times\mathbb{R}$ component, and, under the identification
$\text{SU}(2,\mathbb{C})=\mathbb{S}^{3}$, by the Hopf fibration on the
$\text{SU}(2,\mathbb{C})$ component. In other words, $\bar{\pi}^{\prime}_{1}$
maps $\mathcal{M}_{a,\kappa}$ onto
$\mathbb{R}\times\mathbb{R}\times\mathbb{S}^{2}$.

Finally, note that the spin frames ${\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})$
can be extended to the spacetime $\mathcal{M}_{a,\kappa}$ by a certain
propagation law along the conformal geodesics orthogonal to the
$\mathcal{S}_{a}$, where $\mathcal{S}_{a}$ can be thought of as the initial
hypersurface on $\mathcal{M}_{a,\kappa}$, i.e.

$\mathcal{S}_{a}=\\{(\rho,{\bm{t}})\in\mathbb{R}\times\text{SU}(2,\mathbb{C})|\hskip
2.84526pt0\leq\rho<a\\}.$

The propagated spin frames ${\bm{\epsilon}}_{{\bm{A}}}(\tau,\rho,{\bm{t}})$
are determined at any
$p\in\mathcal{M}_{a,\kappa}\setminus(\mathcal{I}\cup\mathcal{I}^{+}\cup\mathcal{I}^{-})$
up to a multiplication factor that corresponds to the action of $\text{U}(1)$
on $\text{SU}(\mathcal{M})$.

######  Remark 5.

Friedrich's formulation involves encoding the F-gauge conditions in the
initial data and the properties of the fields appearing in the conformal field
equations. For further discussions of Friedrich's formulation and the F-gauge,
readers are referred to [7, 8].

######  Remark 6.

Fields on $\mathcal{M}_{a,\kappa}$ can be decomposed in terms of complex-
valued functions
$T_{m}{}^{j}{}_{k}\mathrel{\mathop{\ordinarycolon}}\text{SU}(2,\mathbb{C})\to\mathbb{C}$,
closely related to the standard spin-weighted harmonics on $\mathbb{S}^{2}$
—see e.g. [8]. The analysis of the conformal field equations can be carried
out on $\mathcal{M}_{a,\kappa}$ and their solutions can be projected onto
$\mathbb{R}\times\mathbb{R}\times\mathbb{S}^{2}$ and used to evaluate BMS
asymptotic charges at the critical sets.

###  3.2 The supertranslation asymptotic charges in full GR

To introduce BMS asymptotic charges at $\mathscr{I}^{\pm}$, let
$d^{\bullet}_{abcd}$ denote the rescaled Weyl tensor in the NP-gauge and
introduce the spinorial counterpart
$d^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}$ which can be
decomposed as follows

$d^{\bullet}_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=-\phi^{\bullet}_{ABCD}\epsilon^{\bullet}_{A^{\prime}B^{\prime}}\epsilon^{\bullet}_{C^{\prime}D^{\prime}}-\bar{\phi}^{\bullet}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon^{\bullet}_{AB}\epsilon^{\bullet}_{CD},$
(29)

where $\phi^{\bullet}_{ABCD}$ is a symmetric valence 4 spinor. Given the
above, the asymptotic charges associated with smooth functions $f$ on
$\mathbb{S}^{2}$ can be written as

$\mathcal{Q}(f;\mathcal{C})\equiv\oint_{\mathcal{C}}\bm{\varepsilon}_{2}f(\mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})+\tfrac{1}{2}\sigma^{\bullet
ab}N^{\bullet}_{ab}),$ (30)

where $\mathcal{C}$ is some cross-section of $\mathscr{I}^{\pm}$ and
$\bm{\varepsilon}_{2}$ is its area element, $\sigma^{\bullet ab}$ is the shear
tensor, $N^{\bullet}_{ab}$ is the news tensor and

$\mathcal{P}^{\bullet}\equiv d^{\bullet}_{abcd}l^{a}n^{b}l^{c}n^{d},$ (31)
$(*\mathcal{P}^{\bullet})\equiv(*d^{\bullet})_{abcd}l^{a}n^{b}l^{c}n^{d}.$

Using (29) and (7), one gets

$\mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})=-2\bar{\phi}^{\bullet}_{2}.$
(32)

Moreover, the term involving $\sigma^{\bullet ab}N^{\bullet}_{ab}$ can be
written in terms of the NP-connection coefficients [47, 48, 49], whose
explicit form depends on whether we are considering the asymptotic charges at
$\mathscr{I}^{+}$ or $\mathscr{I}^{-}$. In particular,

$\sigma^{\bullet
ab}N^{\bullet}_{ab}=2\Delta|\sigma^{\bullet}|^{2}-|\sigma^{\bullet}|^{2}\big{(}3\mu^{\bullet}+3\bar{\mu}^{\bullet}+\gamma^{\bullet}+\bar{\gamma}^{\bullet}\big{)},\qquad\text{on
}\mathscr{I}^{+},$ (33a) $\sigma^{\bullet
ab}N^{\bullet}_{ab}=2\Delta|\lambda^{\bullet}|^{2}-|\lambda^{\bullet}|^{2}\big{(}3\rho^{\bullet}+3\bar{\rho}^{\bullet}+\epsilon^{\bullet}+\bar{\epsilon}^{\bullet}\big{)},\qquad\text{on
}\mathscr{I}^{-}.$ (33b)

Here, $\Delta\equiv n^{a}\nabla^{\bullet}_{a}$ and
$\sigma^{\bullet},\mu^{\bullet},\gamma^{\bullet},\lambda^{\bullet},\rho^{\bullet},\epsilon^{\bullet}$
are the NP-connection coefficients defined as

$\sigma^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{}^{{\bm{1}}}{}_{{\bm{0}}},\qquad\mu^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{1}}},\qquad\gamma^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{0}}},$
(34a)
$\lambda^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{1}}},\qquad\rho^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{}^{{\bm{1}}}{}_{{\bm{0}}},\qquad\epsilon^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{0}}}.$
(34b)

In the above,
$\bar{\mu}^{\bullet},\bar{\gamma}^{\bullet},\bar{\rho}^{\bullet}$ and
$\bar{\epsilon}^{\bullet}$ refer to the complex conjugates of
$\mu^{\bullet},\gamma^{\bullet},\rho^{\bullet}$ and $\epsilon^{\bullet}$,
respectively. To evaluate the expression of the charges (30) at the critical
sets $\mathcal{I}^{\pm}$, one must find a transformation between the NP-gauge
frame and the F-gauge frame in full GR. Following [8], a general
transformation between a NP-gauge spin frame
$\\{{\bm{\epsilon}}^{\bullet}_{{\bm{A}}}\\}$ and an F-gauge spin frame
$\\{{\bm{\epsilon}}_{{\bm{A}}}\\}$ is parameterised by a conformal factor
$\theta$ and an $\text{SL}(2,\mathbb{C})$ transformation matrix
$\Lambda^{{\bm{B}}}{}_{{\bm{A}}}$

${\bm{\epsilon}}^{\bullet}_{{\bm{A}}}=\theta^{-1/2}\Lambda^{{\bm{B}}}{}_{{\bm{A}}}{\bm{\epsilon}}_{{\bm{B}}},$

implying transformations for $\bar{\phi}^{\bullet}_{2}$ and the NP-connection
coefficients (34b). The expressions for these will not be presented here.

As we are interested in evaluating the expressions of the charges at
$\mathcal{I}^{\pm}$, an asymptotic solution for the conformal field equations
is analysed, given the initial data prescribed in the previous section. Given
the zero-order solution, asymptotic expansions for the conformal factor
$\theta$ and the transformation matrices $\Lambda^{{\bm{B}}}{}_{{\bm{A}}}$ are
obtained, following [8].

If $\phi_{0},\phi_{1},\phi_{2},\phi_{3},\phi_{4}$ denote the components of the
rescaled Weyl tensor in the F-gauge, then the explicit transformation from the
NP-gauge to the F-gauge implies

  1. 1.

Contributions to $\mathcal{Q}|_{\mathcal{I}^{\pm}}$ from
$\phi_{0},\phi_{1},\phi_{3},\phi_{4}$ are at most $O(\rho)$.

  2. 2.

The background term $\sigma^{\bullet ab}N^{\bullet}_{ab}$ does not contribute
to $\mathcal{Q}|_{\mathcal{I}^{\pm}}$ at zero order in $\rho$.

Hence, the asymptotic charges at $\mathcal{I}^{\pm}$ are determined by $f$ and
the zero-order solution of $\phi_{2}$, i.e.,

$\mathcal{Q}|_{\mathcal{I}^{\pm}}=\mathcal{Q}|_{\mathcal{I}^{\pm}}(f,\phi_{2}^{(0)}).$
(35)

Given that the equation for $\phi_{2}^{(0)}$ is equivalent to the equation for
$\psi_{2}$ on Minkowski spacetime, the solution for $\phi_{2}^{(0)}$ will
develop a logarithmic singularity at $\mathcal{I}^{\pm}$ unless our initial
data satisfy certain regularity conditions. The explicit form of these
regularity conditions will be presented in a later article as well as the
final expression of $\mathcal{Q}|_{\mathcal{I}^{\pm}}$. The main result is
that given initial data that satisfy our regularity conditions, one can show
that $\mathcal{Q}|_{\mathcal{I}^{\pm}}$ are fully determined by
$\Pi_{3}[\Omega]$. Moreover, if the initial data are chosen to satisfy the
regularity conditions, the asymptotic charges $\mathcal{Q}_{l,m}$ associated
with a given harmonic $Y_{l,m}$ at $\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ are
related by:

$\mathcal{Q}_{l,m}|_{\mathcal{I}^{+}}=(-1)^{l}\mathcal{Q}_{l,m}|_{\mathcal{I}^{-}}$
(36)

##  4 Conclusions

This article addresses the matching of the asymptotic charges associated with
supertranslation symmetries in the context of an initial value problem using
Friedrich's formulation of spatial infinity. The results in this paper
demonstrate that the zero-order solution of $\phi_{2}$ develops logarithmic
singularities at $\mathcal{I}^{\pm}$ given the prescribed initial data in
Section 3.1. Therefore, $\mathcal{Q}|_{\mathcal{I}^{\pm}}$ are only well-
defined if extra regularity conditions are imposed on our initial data. An
upcoming article will present the explicit form of these regularity
conditions. A significant consequence of this result is that the matching of
the BMS asymptotic charges, as defined in this article, is not feasible for
generic asymptotically flat spacetimes unless these spacetimes are the
development of initial data satisfying certain regularity conditions.

## References

  * [1] Roger Penrose.  Asymptotic properties of fields and space-times.  Phys. Rev. Lett., 10:66–68, 1963.
  * [2] Roger Penrose.  Republication of: Conformal treatment of infinity.  General Relativity and Gravitation, 43(3):901–922, Mar 2011.
  * [3] Bondi Hermann, Van der Burg M. G. J., and Metzner A. W. K.  Gravitational waves in general relativity, VII. waves from axi-symmetric isolated system.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 269(1336):21–52, August 1962.
  * [4] Robert Geroch.  Asymptotic Structure of Space-Time, pages 1–105.  Springer US, Boston, MA, 1977.
  * [5] Abhay Ashtekar and R. O. Hansen.  A unified treatment of null and spatial infinity in general relativity. i. universal structure, asymptotic symmetries, and conserved quantities at spatial infinity.  Journal of Mathematical Physics, 19(7):1542–1566, July 1978.
  * [6] A Ashtekar and J D Romano.  Spatial infinity as a boundary of spacetime.  Classical and Quantum Gravity, 9(4):1069, apr 1992.
  * [7] Helmut Friedrich.  Gravitational fields near space-like and null infinity.  Journal of Geometry and Physics, 24(2):83–163, 1998.
  * [8] Helmut Friedrich and János Kánnár.  Bondi-type systems near spacelike infinity and the calculation of the newman–penrose constants.  Journal of Mathematical Physics, 41(4):2195–2232, April 2000.
  * [9] Edgar Gasperin and Juan Antonio Valiente Kroon.  Zero rest-mass fields and the Newman-Penrose constants on flat space.  J. Math. Phys., 61(12):122503, 2020.
  * [10] Mariem Magdy Ali Mohamed and Juan A Valiente Kroon.  A comparison of Ashtekar’s and Friedrich’s formalisms of spatial infinity.  Classical and Quantum Gravity, 38(16):165015, jul 2021.
  * [11] Robert Geroch.  Structure of the gravitational field at spatial infinity.  Journal of Mathematical Physics, 13(7):956–968, 1972.
  * [12] Robert M. Wald and Andreas Zoupas.  A General definition of 'conserved quantities' in General Relativity and other theories of gravity.  Phys. Rev. D, 61:084027, 2000.
  * [13] R. Arnowitt, S. Deser, and C. W. Misner.  Dynamical Structure and Definition of Energy in General Relativity.  Physical Review, 116(5):1322–1330, December 1959.
  * [14] L. Witten.  Gravitation: An Introduction to Current Research.  Wiley, 1962.
  * [15] Tullio Regge and Claudio Teitelboim.  Role of surface integrals in the Hamiltonian formulation of General Relativity.  Annals of Physics, 88(1):286–318, 1974.
  * [16] T Dray and M Streubel.  Angular momentum at null infinity.  Classical and Quantum Gravity, 1(1):15, jan 1984.
  * [17] Hadi Godazgar, Mahdi Godazgar, and Malcolm J. Perry.  Asymptotic gravitational charges.  Phys. Rev. Lett., 125:101301, Sep 2020.
  * [18] Alexander M. Grant, Kartik Prabhu, and Ibrahim Shehzad.  The Wald–Zoupas prescription for asymptotic charges at null infinity in general relativity.  Class. Quant. Grav., 39(8):085002, 2022.
  * [19] Andrew Strominger.  On bms invariance of gravitational scattering.  Journal of High Energy Physics, 2014(7):152, Jul 2014.
  * [20] Steven Weinberg.  Infrared photons and gravitons.  Phys. Rev., 140:B516–B524, Oct 1965.
  * [21] Temple He, Vyacheslav Lysov, Prahar Mitra, and Andrew Strominger.  Bms supertranslations and weinberg's soft graviton theorem.  Journal of High Energy Physics, 2015(5):151, May 2015.
  * [22] Marc Favata.  The gravitational-wave memory effect.  Classical and Quantum Gravity, 27(8):084036, apr 2010.
  * [23] Demetrios Christodoulou.  Nonlinear nature of gravitation and gravitational-wave experiments.  Phys. Rev. Lett., 67:1486–1489, Sep 1991.
  * [24] Luc Blanchet and Thibault Damour.  Hereditary effects in gravitational radiation.  Phys. Rev. D, 46:4304–4319, Nov 1992.
  * [25] R. Beig and B. G. Schmidt.  Einstein's equations near spatial infinity.  Communications in Mathematical Physics, 87(1):65–80, Mar 1982.
  * [26] Miguel Campiglia and Rodrigo Eyheralde.  Asymptotic $U(1)$ charges at spatial infinity.  JHEP, 11:168, 2017.
  * [27] Marc Henneaux and Cédric Troessaert.  Asymptotic symmetries of electromagnetism at spatial infinity.  JHEP, 05:137, 2018.
  * [28] Marc Henneaux and Cédric Troessaert.  BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach.  JHEP, 03:147, 2018.
  * [29] Marc Henneaux and Cédric Troessaert.  Hamiltonian structure and asymptotic symmetries of the Einstein-Maxwell system at spatial infinity.  JHEP, 07:171, 2018.
  * [30] Marc Henneaux and Cédric Troessaert.  The asymptotic structure of gravity at spatial infinity in four spacetime dimensions.  Proceedings of the Steklov Institute of Mathematics, 309(1):127–149, May 2020.
  * [31] Kartik Prabhu and Ibrahim Shehzad.  Asymptotic symmetries and charges at spatial infinity in general relativity.  Class. Quant. Grav., 37(16):165008, 2020.
  * [32] Cédric Troessaert.  The BMS4 algebra at spatial infinity.  Class. Quant. Grav., 35(7):074003, 2018.
  * [33] Kartik Prabhu.  Conservation of asymptotic charges from past to future null infinity: Maxwell fields.  Journal of High Energy Physics, 2018(10):113, Oct 2018.
  * [34] Kartik Prabhu.  Conservation of asymptotic charges from past to future null infinity: supermomentum in general relativity.  Journal of High Energy Physics, 2019(3):148, Mar 2019.
  * [35] Kartik Prabhu and Ibrahim Shehzad.  Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity.  JHEP, 08:029, 2022.
  * [36] Federico Capone, Kevin Nguyen, and Enrico Parisini.  Charge and antipodal matching across spatial infinity.  SciPost Phys., 14:014, 2023.
  * [37] Lan-Hsuan Huang.  Solutions of special asymptotics to the einstein constraint equations.  Classical and Quantum Gravity, 27(24):245002, November 2010.
  * [38] Mariem Magdy Ali Mohamed and Juan A. Valiente Kroon.  Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity.  Journal of Mathematical Physics, 63(5):052502, May 2022.
  * [39] Juan A. Valiente Kroon.  Conformal Methods in General Relativity.  Cambridge University Press, 2016.
  * [40] Helmut Friedrich.  Einstein equations and conformal structure: Existence of anti-de sitter-type space-times.  Journal of Geometry and Physics, 17(2):125–184, 1995.
  * [41] Helmut Friedrich.  Spin two fields on Minkowski space near space - like and null infinity.  Class. Quant. Grav., 20:101–118, 2003.
  * [42] Eric W. Weisstein.  Jacobi differential equation.  From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiDifferentialEquation.html.
  * [43] H. Friedrich and Stephen William Hawking.  The asymptotic characteristic initial value problem for einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 378(1774):401–421, 1981.
  * [44] H. Friedrich.  On the Regular and the Asymptotic Characteristic Initial Value Problem for Einstein's Vacuum Field Equations.  Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 375(1761):169–184, 1981.
  * [45] Helmut Friedrich.  Cauchy problems for the conformal vacuum field equations in general relativity.  Communications in Mathematical Physics, 91(4):445 – 472, 1983.
  * [46] L. Brewin.  Riemann normal coordinates.  https://users.monash.edu.au/~leo/research/papers/files/lcb96-01.pdf.  [Online; accessed 16-April-2023].
  * [47] Roger Penrose and Wolfgang Rindler.  Spinors and Space-Time, volume 1 of Cambridge Monographs on Mathematical Physics.  Cambridge University Press, 1984.
  * [48] Roger Penrose and Wolfgang Rindler.  Spinors and Space-Time, volume 2 of Cambridge Monographs on Mathematical Physics.  Cambridge University Press, 1986.
  * [49] John Stewart.  Differential geometry, page 1–61.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1991.