# BMS-supertranslation charges at the critical sets of null infinity

Mariem Magdy Ali Mohamed 111E-mail address<EMAIL_ADDRESS>School of
Mathematical Sciences, Queen Mary, University of London, Mile End Road, London
E1 4NS, United Kingdom.  Kartik Prabhu 222E-mail address:
<EMAIL_ADDRESS>Raman Research Institute, Sadashivanagar, Bengaluru
560080, India.  Juan A. Valiente Kroon 333E-mail address: j.a.valiente-
<EMAIL_ADDRESS>School of Mathematical Sciences, Queen Mary, University of
London, Mile End Road, London E1 4NS, United Kingdom.

###### Abstract

For asymptotically flat spacetimes, a conjecture by Strominger states that
asymptotic BMS-supertranslations and their associated charges at past null
infinity $\mathscr{I}^{-}$ can be related to those at future null infinity
$\mathscr{I}^{+}$ via an antipodal map at spatial infinity $i^{0}$. We analyse
the validity of this conjecture using Friedrich’s formulation of spatial
infinity, which gives rise to a regular initial value problem for the
conformal field equations at spatial infinity. A central structure in this
analysis is the cylinder at spatial infinity $\mathcal{I}$ representing a
blow-up of the standard spatial infinity point $i^{0}$ to a 2-sphere. The
cylinder $\mathcal{I}$ touches past and future null infinities
$\mathscr{I}^{\pm}$ at the critical sets $\mathcal{I}^{\pm}$. We show that for
a generic class of asymptotically Euclidean and regular initial data, BMS-
supertranslation charges are not well-defined at $\mathcal{I}^{\pm}$ unless
the initial data satisfies an extra regularity condition. We also show that
given initial data that satisfy the regularity condition, BMS-supertranslation
charges at $\mathcal{I}^{\pm}$ are fully determined by the initial data and
that the relation between the charges at $\mathcal{I}^{-}$ and those at
$\mathcal{I}^{+}$ directly follows from our regularity condition.

###### Contents

  1. 1 Introduction
  2. 2 Conformal geometry tools
    1. 2.1 The metric conformal field equations
    2. 2.2 The extended conformal field equations
    3. 2.3 Conformal Geodesics
  3. 3 Friedrich’s formulation of spatial infinity
    1. 3.1 Hyperbolic reduction using the conformal Gaussian gauge
      1. 3.1.1 Spinors formulation of the extended conformal field equations
      2. 3.1.2 Scalarising the extended conformal field equations
  4. 4 The Newman-Penrose gauge
  5. 5 BMS-supertranslations charges
    1. 5.1 BMS-supertranslation charges in the F-gauge
  6. 6 The initial data and the constraint equations
    1. 6.1 The conformal constraint equations
    2. 6.2 The initial data
      1. 6.2.1 Normal coordinates and conformal normal initial data
      2. 6.2.2 The regular initial data at spatial infinity
      3. 6.2.3 Conformally flat initial data for the space spinor fields
  7. 7 Evaluating the BMS-supertranslation charges at the critical sets
    1. 7.1 Asymptotic solution of the extended conformal field equations
    2. 7.2 The explicit transformation from the NP to the F-gauge
    3. 7.3 BMS-supertranslation charges at the critical sets
  8. 8 Conclusions
  9. 9 Acknowledgements
  10. A The boundary adapted evolution and constraint equations

##  1 Introduction

Studies of isolated systems, asymptotic structures and symmetries have
received increasing interest in recent years due to their relation to black-
hole physics [1, 2, 3], the gravitational memory effect [4, 5, 6] and
developments in soft theorems [7, 8, 9]. A common approach in these studies
involves the use of conformal transformations to study the behaviour of the
gravitational field ‘at infinity’ using local differential geometry by mapping
points at infinity in one manifold to a region located at finite distances in
another manifold, the so-called conformal boundary.

The conformal approach is inspired by R. Penrose seminal work [10, 11] in
which the notion of asymptotically simple (AS) spacetimes was initially
introduced with the aim of identifying a large class of spacetimes that admit
a conformal extension similar to that of Minkowski spacetime. More precisely,
a spacetime $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ is asymptotically simple
if there exists a smooth, oriented, time-oriented and causal spacetime
$(\mathcal{M},{\bm{g}})$ and a smooth function $\Xi$ on $\mathcal{M}$ such
that (i) $\mathcal{M}$ is a manifold with boundary
$\mathscr{I}\equiv\partial\mathcal{M}$; (ii) the conformal factor $\Xi$
satisfies: $\Xi>0$ on $\mathcal{M}\setminus\mathscr{I}$ and
$\Xi=0,{\bm{d}}{\Xi}\neq 0$ on $\mathscr{I}$; (iii) the manifolds
$\tilde{\mathcal{M}}$ and $\mathcal{M}$ are related by
$\phi:\tilde{\mathcal{M}}\rightarrow\mathcal{M}$ such that
$\phi(\tilde{\mathcal{M}})=\mathcal{M}\setminus\mathscr{I}$ and
$\phi^{*}{\bm{g}}=\Xi^{2}\tilde{{\bm{g}}}$; and finally, (iv) null geodesics
of $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ starts and ends on $\mathscr{I}$.
In this context, we say that $\phi$ is a conformal transformation that maps
$\tilde{\mathcal{M}}$ to $\mathcal{M}\setminus\mathscr{I}$. The manifolds
$\tilde{\mathcal{M}}$ and $\mathcal{M}$ are referred to as the physical and
unphysical manifolds, respectively. Similarly, the metric $\tilde{{\bm{g}}}$
is the physical metric, and ${\bm{g}}$ is the unphysical metric. Generally,
the boundary $\mathscr{I}$ (null infinity) can be split into two disjoint sets
$\mathscr{I}^{+}$ (future null infinity) and $\mathscr{I}^{-}$ (past null
infinity).

The literature on the asymptotic structure of spacetimes can be divided into
two categories: studies of the asymptotic structure at null infinity or at
spatial infinity. In the null regime, it was expected that the Poincaré group
would describe the asymptotic symmetry group for AS spacetimes, given that the
background geometry of an AS spacetime is similar to that of Minkowski.
However, the work of Bondi, Metzner and Sachs [12] revealed that the
asymptotic symmetry group for AS spacetimes is given by the infinite-
dimensional BMS symmetry group, which can be written as the semi-direct
product of the Lorentz group with the infinite-dimensional group of angle-
dependent translations (supertranslations) along null infinity. Notions of
asymptotic flatness at spatial infinity also give rise to an infinite-
dimensional asymptotic symmetry group at spatial infinity, known as the Spi
group [13, 14], with a structure similar to the BMS group —see also [15, 16,
17, 18, 19].

Asymptotic symmetry groups define corresponding conserved quantities or
asymptotic charges. At spatial infinity, notions of conserved mass, momentum
and angular momentum can be derived using a Hamiltonian formulation [20, 21].
More recently, it was shown that charges associated with supertranslations at
spatial infinity are generally non-vanishing [16, 19]. At null infinity, the
challenge is that BMS charges can not be defined using a standard Hamiltonian
formulation as, generically, there exists no Hamiltonian that generates BMS
transformations at null infinity. This observation can be linked to the fact
that the symplectic current can be radiated away at null infinity, i.e., BMS
charges are not exactly conserved. In fact, BMS charges have non-vanishing
fluxes through null infinity. Nevertheless, the discussion in [22] provides a
general definition of ‘conserved quantities’ associated with asymptotic
symmetries, even in scenarios in which the Hamiltonian does not exist.

A conjecture by Strominger [8] states that a priori independent asymptotic
symmetry groups at past and future null infinities $\mathscr{I}^{\pm}$,
denoted by $\text{BMS}^{+}$ and $\text{BMS}^{-}$, respectively, can be related
via an antipodal reflection map near spatial infinity. The verification of
this conjecture, referred to as the matching problem, would imply a global
diagonal asymptotic symmetry group $\text{BMS}^{+}\times\text{BMS}^{-}$. In
other words, the incoming fluxes associated with $\text{BMS}^{+}$ would be
equal to the outgoing fluxes associated with $\text{BMS}^{-}$. The matching
problem is also a crucial ingredient in the conjectured equivalence relation
between asymptotic symmetries, the soft graviton theorem and the gravitational
memory effect [23, 24].

The strategy in validating the matching of $\text{BMS}^{+}$ and
$\text{BMS}^{-}$ and their associated charges involves expanding the fields in
suitable coordinates around null and spatial infinity. On Minkowski spacetime,
the matching of asymptotic charges at past and future null infinities has been
verified for the spin-1 and spin-2 fields [25, 26, 27]. For more general
spacetimes, the analysis is complicated due to the singular conformal
structure at spatial infinity for spacetimes with non-vanishing Arnowitt-
Deser-Misner (ADM) mass, referred to as the problem of spatial infinity —see
e.g. Chapter 20 in [28]. Another challenge is that one requires a
transformation between adapted coordinates at null and spatial infinity, which
can be explicitly computed on Minkowski spacetime but is not generally known
for general spacetimes. Nevertheless, the covariant formulation of Ashtekar
and Hansen [13] was used in [29, 30] to prove the matching of asymptotic
charges for the spin-1 and gravitational fields on spacetimes that satisfy
Ashtekar-Hansen’s notion of asymptotic flatness. Similar techniques were used
in [31] to investigate the matching of Lorentz charges for the gravitational
field on Ashtekar-Hansen asymptotically flat spacetimes —see also [32].

The purpose of this article is to verify the matching of BMS asymptotic
charges in a full GR setting using an initial value formulation of the field
equations. The argument made in [8] is that the matching of BMS-
supertranlation charges should hold for Christodoulou-Klainerman class of
spacetimes [33]. However, the Christodoulou-Klainerman class of spacetimes are
not general enough to obtain non-trivial asymptotic charges near spatial
infinity since they lead to vanishing BMS-supertranslation charges at spatial
infinity. Therefore, the aim of this article is to identify a generic class of
initial data and conditions on the initial data that implies non-trivial,
well-defined asymptotic charges at spatial infinity.

In the context of the initial value problem, Einstein’s field equations are
split into constraint equations and evolution equations; the constraint
equations are satisfied by an initial data set
$(\tilde{\mathcal{S}},\tilde{{\bm{h}}},\tilde{{\bm{K}}})$ prescribed on an
initial Cauchy hypersurface $\tilde{\mathcal{S}}$, where $\tilde{{\bm{h}}}$ is
the intrinsic metric on $\tilde{\mathcal{S}}$ and $\tilde{{\bm{K}}}$ is the
extrinsic curvature. The well-posedness of the Cauchy problem ensures that
there exists a vacuum spacetime $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$,
referred to as the development of the initial data
$(\tilde{\mathcal{S}},\tilde{{\bm{h}}},\tilde{{\bm{K}}})$, such that
$\tilde{\mathcal{S}}$ is a spacelike hypersurface in $\tilde{\mathcal{M}}$
with an intrinsic metric $\tilde{{\bm{h}}}$ induced on $\tilde{\mathcal{S}}$
by $\tilde{{\bm{g}}}$ with an associated extrinsic curvature
$\tilde{{\bm{K}}}$. It is worth noting that not every spacetime can be
globally constructed from an initial value problem. A spacetime is said to be
globally hyperbolic if it can be constructed from an initial value problem of
Einstein’s field equations. A special class of initial data relevant to this
article are the so-called asymptotically Euclidean and regular initial data
[34] defined by:

######  Definition 1 (asymptotically Euclidean and regular).

A three-dimensional Riemannian manifold
$(\tilde{\mathcal{S}},\tilde{{\bm{h}}})$ is asymptotically Euclidean and
regular if there exists a three-dimensional, orientable, compact manifold
$(\mathcal{S},{\bm{h}})$ with points $i_{k}\in\mathcal{S},k=1,\dotso,N$ with
$N$ some integer, a function $\Omega\in C^{2}$ and a diffeomorphism
$\varphi:\mathcal{S}\setminus\\{i_{1},\dotso,i_{N}\\}\rightarrow\tilde{\mathcal{S}}$
such that

  1. 1.

$\Omega(i_{k})=0,{\bm{d}}\Omega(i_{k})=0$ and $\textbf{Hess
}\Omega(i_{k})=-2{\bm{h}}(i_{k})$, for all $i_{k}\in\\{i_{1},\dotso,i_{N}\\}$,

  2. 2.

$\Omega>0$ on $\mathcal{S}\setminus\\{i_{1},\dotso,i_{N}\\}$, and

  3. 3.

${\bm{h}}=\Omega^{2}\varphi^{*}\tilde{{\bm{h}}}$ on
$\mathcal{S}\setminus\\{i_{1},\dotso,i_{N}\\}$ with ${\bm{h}}\in
C^{2}(\mathcal{S})\cap
C^{\infty}(\mathcal{S}\setminus\\{i_{1},\dotso,i_{N}\\})$.

In the above, neighbourhoods of the points $i_{k}$ can be mapped to the
asymptotic ends of $\tilde{\mathcal{S}}$ and thus, each of these points
represents spacelike infinity. Compared to the standard definition of
asymptotically Euclidean manifolds which describes the asymptotic expansion of
the intrinsic fields on $\tilde{\mathcal{S}}$ near the asymptotic ends,
Definition 1 is more geometric in nature and it imposes extra conditions on
the smoothness of the initial data, which in turn affects the asymptotic
behaviour of their evolution in time [34]. In this article, the aim is to
obtain non-trivial BMS asymptotic charges associated with the development of
some initial data that satisfy the constraint equations with prescribed
behaviour in the asymptotic region. These initial data were first considered
in [35] and are obtained by means of a gluing construction. This class of
initial data includes, as a particular case, boosted solutions to the
constraints and allows for a term with arbitrary multipolar structure in the
initial metric, which appears at the same order as the mass. As will become
evident from our analysis, this arbitrary multipolar structure will be
responsible for the existence of non-trivial BMS asymptotic charges.

It should be noted that the transformation of Einstein’s field equations from
the physical manifold $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ to the
unphysical manifold $(\mathcal{M},{\bm{g}})$ implies singular equations at the
conformal boundary $\Xi=0$. An alternative set of field equations, the so-
called metric conformal field equations, can be constructed following the
discussion in [36, 37, 38]. These equations are regular at $\Xi=0$, and they
imply solutions to Einstein’s field equations at the points where $\Xi\neq 0$
—see [28], Chapter 8. In this article, we will make use of the extended
conformal field equations (ECFEs). These equations are formulated in terms of
a Weyl connection $\hat{{\bm{\nabla}}}$ and exhibit additional gauge freedom
in contrast to the metric conformal field equations, which are formulated in
terms of the Levi-Civita connection ${\bm{\nabla}}$ associated with
${\bm{g}}$. As such, the main goal of this project will be to use the ECFEs to
evaluate BMS asymptotic charges near spatial infinity.

To address the singular conformal structure at spatial infinity, we make use
of Friedrich’s formulation of spatial infinity originally introduced in [39]
with the aim of obtaining a regular initial value problem for the conformal
field equations at spatial infinity. A central structure in this formulation
is the cylinder at spatial infinity $\mathcal{I}$ corresponding to a blow-up
of the spatial infinity point to a 2-sphere. The cylinder $\mathcal{I}$
touches the endpoints of past and future null infinities $\mathscr{I}^{\pm}$
at the critical sets $\mathcal{I}^{\pm}$. Associated with this formulation is
a particular choice of gauge, referred to as the F-gauge, in which the
coordinates and frames on an initial hypersurface are propagated along
conformal geodesics. One of the remarkable properties of conformal geodesics
is that they introduce a canonical conformal factor that depends on the proper
time along the curves and the initial data. In other words, the F-gauge is
constructed so that the location of the conformal boundary is known a priori.
Moreover, in this particular choice of gauge, the cylinder $\mathcal{I}$ is a
total characteristic of the ECFEs, i.e., the associated evolution equations
can be written as a system of transport equations on $\mathcal{I}$. A
significant advantage of Friedrich’s formulation is that it allows us to link
quantities at the critical sets $\mathcal{I}^{\pm}$ with the initial data
prescribed on an initial hypersurface. This approach was used in [40, 41] to
express the Newman-Penrose (NP) constants in terms of initial data, and it
illustrates our strategy in this work where the goal is to express BMS
asymptotic charges at $\mathcal{I}^{\pm}$ in terms of our initial data.

As mentioned earlier, one of the challenges in verifying the matching of
asymptotic charges is that a transformation between adapted coordinates at
null and spatial infinity is required. In this article, BMS asymptotic charges
are expressed in the NP-gauge, comprised of certain conformal gauge
conditions, certain coordinates and an orthonormal frame field satisfying
certain frame gauge conditions. The main difference between the F-gauge and
the NP-gauge is that the former is adapted to Cauchy hypersurfaces while the
latter is adapted to null infinity $\mathscr{I}^{\pm}$. The discussion in [40]
provides a prescription of the transformation between the NP-gauge and the
F-gauge. Given a solution to the ECFEs, an explicit transformation can be
obtained, allowing us to express the BMS asymptotic charges in terms of the
F-gauge. In turn, the BMS-asymptotic charges can be evaluated at
$\mathcal{I}^{\pm}$ given the solution to the ECFEs.

### Main result

The main results of this article can be summarised in the following:

######  Theorem.

For the generic initial data in [35], asymptotic BMS-supertranslation charges
are not well-defined at the critical sets $\mathcal{I}^{\pm}$ unless the
conformal initial data satisfy the regularity condition given in Lemma 1. If
the initial data are chosen to satisfy the extra regularity condition, the
BMS-supertranslation charges at $\mathcal{I}^{\pm}$ are fully determined by
the initial data and the matching between charges at $\mathcal{I}^{+}$ and
$\mathcal{I}^{-}$ follows directly from the regularity condition.

As it will be discussed in the main body of the article, the piece of the
freely specifiable initial data from which the value of the BMS charges at
$\mathcal{I}^{\pm}$ arise correspond to a function $\xi\in C^{2}(\mathbb{S})$.
The regularity condition in Lemma 1 ensuring that the charges are well-defined
is a statement about the parity of the function $\xi$. More precisely, all odd
parity harmonics, except for the one with $l=1$, are required to vanish. This
result provides evidence that Strominger’s antipodal matching condition is, in
fact, a regularity condition on spatial infinity. A complete proof of this
statement is, however, beyond the scope of this article.

As pointed in [42] the type of parity condition arising from our analysis (and
an analogous one for the extrinsic curvature, which is not required in the
present analysis) is required to make all Poincaré charges at spatial infinity
well-defined. This condition can be traced back to the seminal article [21].
Our analysis points out a deep connection between the regularity of null
infinity and the physical requirement that asymptotic charges are well-
defined. Again, a systematic analysis of these ideas goes beyond the scope of
this article.

### Outline of the article

In Section 2, we start by introducing some of the basic conformal tools used
throughout this article —e.g. the conformal field equations and conformal
geodesics. Friedrich’s formulation of spatial infinity and the F-gauge are
introduced in Section 3. In Section 4, we discuss the NP-gauge conditions and
the relation between the NP-gauge and the F-gauge. The expressions for BMS-
supertranslation charges are introduced in Section 5 along with their
translation to the F-gauge. In Section 6, we present the initial data utilised
in our analysis of the conformal field equations. We conclude the analysis of
this article in Section 7 by obtaining the zero-order solution of the
conformal field equations and evaluating the BMS-supertranslation charges at
the critical sets $\mathcal{I}^{\pm}$.

### Notations and conventions

In this article, Latin letters $a,\,b,\,c,\ldots$ will denote spacetime
abstract tensorial indices while $i,\,j,\,k,\ldots$ will denote spatial
abstract tensorial indices. Capital Latin letters $A,\,B,\,C,\ldots$ will
denote abstract spinorial indices.

To discuss the components of tensors with respect to a coordinate basis, the
Greek letters $\mu,\,\nu,\dots$ will be used as spacetime coordinate indices
while $\alpha,\,\beta,\ldots$ will be used as spatial coordinate indices.
Then, the components of a generic spacetime tensor $T_{ab}$ with respect to an
arbitrary coordinate system $(x^{\mu})$ will be written as
$T_{\mu\nu}=T_{ab}({\bm{\partial}}/{\bm{\partial}}x^{\mu})^{a}({\bm{\partial}}/{\bm{\partial}}x^{\nu})^{b}$.
Similarly, the components of a generic spatial tensor $l_{ij}$ with respect to
an arbitrary coordinate system $(x^{\alpha})$ will be written as
$l_{\alpha\beta}=l_{ij}({\bm{\partial}}_{\alpha})^{i}({\bm{\partial}}_{\beta})^{j}$,
where
$({\bm{\partial}}_{\alpha})^{i}\equiv({\bm{\partial}}/{\bm{\partial}}x^{\alpha})^{i}$.

To discuss the components of tensors and spinors with respect to a frame
basis, let ${\bm{a}},{\bm{b}},{\bm{c}},\ldots$ denote tensorial frame indices
and ${\bm{A}},{\bm{B}},{\bm{C}},\ldots$ denote spinorial frame indices. Then,
the components of a generic tensor $T_{ab}$ with respect to an arbitrary basis
${\bm{e}}_{a}\equiv\\{{\bm{e}}_{{\bm{a}}}\\}$ will be written as
$T_{{\bm{a}}{\bm{b}}}=T_{ab}{\bm{e}}_{{\bm{a}}}{}^{a}{}{\bm{e}}_{{\bm{b}}}{}^{b}{}$
with with ${\bm{a}},{\bm{b}}\in\\{{\bm{0}},{\bm{1}},{\bm{2}},{\bm{3}}\\}$.
Moreover, if $\\{{\bm{o}},{\bm{\iota}}\\}$ denote a spin bases satisfying
$\llbracket{\bm{o}},{\bm{\iota}}\rrbracket=1$, where $\llbracket.,.\rrbracket$
is the antisymmetric product, then the components of a generic spinor
$\zeta_{A}$ can be written as
$\zeta_{{\bm{A}}}=\zeta_{A}{\bm{\epsilon}}_{{\bm{A}}}{}^{A}{}$, where

$\displaystyle{\bm{o}}^{A}={\bm{\epsilon}}_{{\bm{0}}}{}^{A}{},\qquad{\bm{\iota}}^{A}={\bm{\epsilon}}_{{\bm{1}}}{}^{A}{},$
$\displaystyle{\bm{o}}_{A}={\bm{\epsilon}}^{{\bm{1}}}{}_{A}{},\qquad{\bm{\iota}}_{A}=-{\bm{\epsilon}}^{{\bm{0}}}{}_{A}{}.$
(1)

The antisymmetric product $\llbracket.,.\rrbracket$ of two generic spinors
$\zeta$ and $\lambda$ can be expressed as

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

where $\epsilon_{AB}$ is the $\epsilon$-spinor that can be regarded as an
index raising/lowering object for spinors. Throughout, we express spacetime
frames $\\{{\bm{e}}_{{\bm{a}}}\\}$ in spinorial notation. The spinorial
counterpart of $\\{{\bm{e}}_{{\bm{a}}}\\}$ is given by

${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}=\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{a}}}$

where $\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}$ are the Infeld-van der
Waerden symbols. Finally, the signature convention for spacetime metrics used
in this article is $(+,-,-,-)$. Throughout, we mostly follow the notation and
conventions of Penrose and Rindler [43] —see also [28].

##  2 Conformal geometry tools

The purpose of this section is to provide a brief introduction of the
conformal tools utilised throughout this article. In the following, let
$\tilde{\mathcal{M}}$ denote a four-dimensional Lorentzian manifold. The
metrics $\tilde{{\bm{g}}}$ and $\hat{{\bm{g}}}$ on $\tilde{\mathcal{M}}$ are
_conformally related_ if there exists a positive function $\Omega$ such that

$\hat{{\bm{g}}}=\Omega^{2}\tilde{{\bm{g}}},$

where $\Omega$ is known as the _conformal factor_. On the other hand, if
$\mathcal{M}$ is a four-dimensional manifold with metric ${\bm{g}}$, then one
defines a _conformal transformation_ $\phi$ as the diffeomorphic map
$\phi:\tilde{\mathcal{M}}\to\mathcal{M}$ such that

$\phi^{*}{\bm{g}}=\Xi^{2}\tilde{{\bm{g}}}.$

where $\Xi$ denotes the conformal factor which is a positive function on
$\tilde{\mathcal{M}}$ and $\phi^{*}{\bm{g}}$ is the pull-back of ${\bm{g}}$ to
$\tilde{\mathcal{M}}$. Given the above, one can define:

######  Definition 2 (conformal compactification).

Let $\mathcal{U}$ denote a compact, connected and open subset of
$\mathcal{M}$, then the diffeomorphic map
$\phi:\tilde{\mathcal{M}}\to\mathcal{U}$ defines a conformal compactification
of $\tilde{\mathcal{M}}$ if there exists a positive function $\Xi$ satisfying

  1. i.

$\Xi>0$ in $\mathcal{U}$,

  2. ii.

$\Xi=0$ on the boundary of the open set $\mathcal{U}$, denoted
$\partial\mathcal{U}$,

and if ${\bm{g}}$ is related to $\tilde{{\bm{g}}}$ by

${\bm{g}}=(\phi^{*})^{-1}(\Xi^{2}\tilde{{\bm{g}}})\qquad\text{in
}\quad\mathcal{U}.$ (2)

In this context, $\partial\mathcal{U}$ is known as the conformal boundary of
$\tilde{\mathcal{M}}$.

######  Remark 1.

Throughout, we omit $\phi^{*}$ and $(\phi^{*})^{-1}$ when discussing conformal
transformations and compactifications —e.g., eq. (2) will be written as
${\bm{g}}=\Xi^{2}\tilde{{\bm{g}}}$.

###  2.1 The metric conformal field equations

The conformal transformation given by eq. (2) implies transformation laws for
the Levi-Civita connections associated with $\tilde{{\bm{g}}}$ and ${\bm{g}}$
(denoted by $\tilde{{\bm{\nabla}}}$ and ${\bm{\nabla}}$) and other related
fields —e.g. the Riemann curvature tensors $\tilde{R}^{a}{}_{bcd}$ and
$R^{a}{}_{bcd}$, and the Ricci tensors $\tilde{R}_{ab}$ and $R_{ab}$ etc. The
derivation of these formulae is discussed in [28] but will not be necessary
for our discussion.

Now, assume that $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ satisfy the vacuum
Einstein field equations, so that

$\tilde{R}_{ab}=0.$ (3)

The transformation law of the Ricci tensor implied by eq. (2) yields a
singular expression for $R_{ab}$ at $\Xi=0$. However, the prescription in [36,
37, 38, 28] introduces a set of field equations on $\mathcal{M}$ that are
well-defined at the conformal boundary. We will refer to these equations as
the _metric conformal field equations_ , and they are given by

$\displaystyle\nabla_{a}\nabla_{b}\Xi=-\Xi L_{ab}+sg_{ab},$ (4a)
$\displaystyle\nabla_{a}s=-L_{ac}\nabla^{c}\Xi,$ (4b)
$\displaystyle\nabla_{c}L_{db}-\nabla_{d}L_{cb}=\nabla_{a}\Xi d^{a}{}_{bcd},$
(4c) $\displaystyle\nabla_{a}d^{a}{}_{bcd}=0,$ (4d) $\displaystyle 6\Xi
s-3\nabla_{c}\Xi\nabla^{c}\Xi=0,$ (4e)

where ${\bm{\nabla}}$ is the ${\bm{g}}$-Levi-Civita connection, $L_{ab}$ is
the Schouten tensor associated with ${\bm{\nabla}}$, $d^{a}{}_{bcd}$ is the
rescaled Weyl tensor, defined in terms of the Weyl tensor $C^{a}{}_{bcd}$ as

$d^{a}{}_{bcd}=\Xi^{-1}C^{a}{}_{bcd}.$ (5)

Finally, $s$ denotes Friedrich’s scalar, given by

$s\equiv\frac{1}{4}\nabla^{c}\nabla_{c}\Xi+\frac{1}{24}R\Xi,$

where $R$ is the Ricci scalar associated with ${\bm{\nabla}}$.

Note that eqs. (4) exhibit conformal gauge freedom manifested by the fact that
a solution to eq. (3) will correspond to an infinite number of solutions to
eq. (4). In our analysis, our focus will be on an equivalent set of conformal
field equations exhibiting additional gauge freedom compared to eqs. (4).

###  2.2 The extended conformal field equations

This section briefly overviews the conformal formulation of the Einstein field
equations introduced by Friedrich in [39]. The result of this formulation is a
set of equations known as the extended conformal field equations (ECFEs).

Following the discussion in [28], let $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$
denote the physical spacetime satisfying eqs. (3), and let
$(\mathcal{M},{\bm{g}})$ denote the unphysical spacetime with

$g_{ab}=\Xi^{2}\tilde{g}_{ab},$ (6)

Given a ${\bm{g}}$-orthonormal frame $\\{{\bm{e}}_{{\bm{a}}}\\}$ and a dual
frame $\\{{\bm{\omega}}^{{\bm{a}}}\\}$, one can write

$\displaystyle g_{{\bm{a}}{\bm{b}}}=\Xi^{2}\tilde{g}_{{\bm{a}}{\bm{b}}},$ (7a)
$\displaystyle g^{{\bm{a}}{\bm{b}}}=\Xi^{-2}\tilde{g}^{{\bm{a}}{\bm{b}}}.$
(7b)

To introduce the ECFEs, introduce the _Weyl connection_ as a torsion-free
connection $\hat{{\bm{\nabla}}}$ satisfying

$\hat{\nabla}_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}}=-2f_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}},$
(8)

where $\hat{\nabla}_{{\bm{a}}}={\bm{e}}_{{\bm{a}}}{}^{a}\hat{\nabla}_{a}$. The
relation between $\hat{{\bm{\nabla}}}$, ${\bm{\nabla}}$ and
$\tilde{{\bm{\nabla}}}$ is then given by

$\displaystyle\hat{\bm{\nabla}}-{\bm{\nabla}}={\bm{S}}({\bm{f}}),$ (9a)
$\displaystyle\hat{{\bm{\nabla}}}-\tilde{{\bm{\nabla}}}={\bm{S}}({\bm{\beta}}),$
(9b)

where ${\bm{\beta}}\equiv{\bm{f}}+\Xi^{-1}{\bm{d}}{\Xi}$ and
${\bm{S}}({\bm{f}})$ can be written as

${\bm{S}}({\bm{f}})=S_{ab}{}^{cd}f_{d},$

with

$S_{ab}{}^{cd}=\delta_{a}{}^{c}\delta_{b}{}^{d}+\delta_{a}{}^{d}\delta_{b}{}^{c}-g_{ab}g^{cd},$
(10)

where $\delta_{a}{}^{c}$ is the Kronecker delta. Given the above, the extended
conformal field equations can be written explicitly in terms of the zero
quantities
$\hat{\Sigma}_{{\bm{a}}{\bm{b}}},\hat{\Xi}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}},\hat{\Delta}_{{\bm{c}}{\bm{d}}{\bm{b}}}\text{
and }\hat{\Lambda}_{{\bm{b}}{\bm{c}}{\bm{d}}}$ as

$\hat{\Sigma}_{{\bm{a}}{\bm{b}}}=0,\quad\hat{\Xi}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}=0,\quad\hat{\Delta}_{{\bm{c}}{\bm{d}}{\bm{b}}}=0,\quad\hat{\Lambda}_{{\bm{b}}{\bm{c}}{\bm{d}}}=0,$
(11)

where
$(\hat{\Sigma}_{{\bm{a}}{\bm{b}}},\hat{\Xi}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}},\hat{\Delta}_{{\bm{c}}{\bm{d}}{\bm{b}}},\hat{\Lambda}_{{\bm{b}}{\bm{c}}{\bm{d}}})$
are defined as

$\displaystyle\hat{\Sigma}_{{\bm{a}}{\bm{b}}}\equiv[{\bm{e}}_{{\bm{b}}},{\bm{e}}_{{\bm{a}}}]-(\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{c}}}{}_{{\bm{b}}}-\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{c}}}{}_{{\bm{a}}}){\bm{e}}_{{\bm{c}}},$
$\displaystyle\hat{\Xi}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}\equiv\hat{P}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}-\hat{\rho}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}},$
$\displaystyle\hat{\Delta}_{{\bm{c}}{\bm{d}}{\bm{b}}}\equiv\hat{\nabla}_{{\bm{c}}}\hat{L}_{{\bm{d}}{\bm{b}}}-\hat{\nabla}_{{\bm{d}}}\hat{L}_{{\bm{c}}{\bm{b}}}-d_{{\bm{a}}}d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}},$
$\displaystyle\hat{\Lambda}_{{\bm{b}}{\bm{c}}{\bm{d}}}\equiv\hat{\nabla}_{{\bm{a}}}d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}-f_{{\bm{a}}}d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}.$

Here, $[{\bm{e}}_{{\bm{b}}},{\bm{e}}_{{\bm{a}}}]$ is the commutator defined as
$[{\bm{e}}_{{\bm{b}}},{\bm{e}}_{{\bm{a}}}]\equiv{\bm{e}}_{{\bm{b}}}({\bm{e}}_{{\bm{a}}}(f))-{\bm{e}}_{{\bm{a}}}({\bm{e}}_{{\bm{b}}}(f))$
for any function $f$ on $\mathcal{M}$, $\hat{L}_{{\bm{a}}{\bm{b}}}$ are the
components of the Schouten tensor $\hat{{\bm{L}}}$ associated with
$\hat{{\bm{\nabla}}}$, $\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}$ are
the $\hat{{\bm{\nabla}}}$-connection coefficients, and $d_{a}$ is a 1-form
related to $f_{a}$ by

$d_{a}=\Xi f_{a}+({\bm{d}}{\Xi})_{a}.$

Finally, $\hat{P}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}$ and
$\hat{\rho}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}$ are the components of the
so-called geometric and algebraic curvature with respect to
$\\{{\bm{e}}_{{\bm{a}}}\\}$, defined by

$\displaystyle\hat{P}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}\equiv{\bm{e}}_{{\bm{a}}}(\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{c}}}{}_{{\bm{d}}})-{\bm{e}}_{{\bm{b}}}(\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{c}}}{}_{{\bm{d}}})+\hat{\Gamma}_{{\bm{f}}}{}^{{\bm{c}}}{}_{{\bm{d}}}(\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{f}}}{}_{{\bm{a}}}-\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{f}}}{}_{{\bm{b}}})+\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{f}}}{}_{{\bm{d}}}\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{c}}}{}_{{\bm{f}}}-\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{f}}}{}_{{\bm{d}}}\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{c}}}{}_{{\bm{f}}},$
$\displaystyle\hat{\rho}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}\equiv\Xi
d^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}}+2S_{{\bm{d}}[{\bm{a}}}{}^{{\bm{c}}{\bm{f}}}\hat{L}_{{\bm{b}}]{\bm{f}}}.$

The ECFEs yield partial differential equations to be solved for the unknowns

$({\bm{e}}_{{\bm{a}}},\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}},\hat{L}_{{\bm{a}}{\bm{b}}},d^{{\bm{a}}}{}_{{\bm{b}}{\bm{d}}{\bm{c}}}).$

In addition to eqs. (11), introduce the zero quantities
$\delta_{{\bm{a}}},\gamma_{{\bm{a}}{\bm{b}}}$ and
$\varsigma_{{\bm{a}}{\bm{b}}}$ satisfying

$\delta_{{\bm{a}}}=0,\quad\gamma_{{\bm{a}}{\bm{b}}}=0,\quad\varsigma_{{\bm{a}}{\bm{b}}}=0,$
(14)

where

$\displaystyle\delta_{{\bm{a}}}\equiv d_{{\bm{a}}}-\Xi
f_{{\bm{a}}}-\hat{\nabla}_{{\bm{a}}}\Xi,$
$\displaystyle\gamma_{{\bm{a}}{\bm{b}}}=\hat{L}_{{\bm{a}}{\bm{b}}}-\hat{\nabla}_{{\bm{a}}}\beta_{{\bm{b}}}-\frac{1}{2}S_{{\bm{a}}{\bm{b}}}{}^{{\bm{c}}{\bm{d}}}\beta_{{\bm{c}}}\beta_{{\bm{d}}},$
$\displaystyle\varsigma_{{\bm{a}}{\bm{b}}}=\hat{L}_{[{\bm{a}}{\bm{b}}]}-\hat{\nabla}_{[{\bm{a}}}f_{{\bm{b}}]}.$

The supplementary eqs. (14) relate the solutions of the ECFEs to Einstein’s
field equations. In particular, given a solution
$({\bm{e}}_{{\bm{a}}},\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}},\hat{L}_{{\bm{a}}{\bm{b}}},d^{{\bm{a}}}{}_{{\bm{b}}{\bm{d}}{\bm{c}}})$
to the ECFEs with a choice of $\Xi$ and $d_{{\bm{a}}}$ that satisfies the
supplementary eqs. (14), then if $\Xi\neq 0$ and
$\text{det}(\eta^{{\bm{a}}{\bm{b}}}{\bm{e}}_{{\bm{a}}}\otimes{\bm{e}}_{{\bm{b}}})\neq
0$ on some open set $\mathcal{U}$ of $\mathcal{M}$, the metric

$\tilde{g}_{{\bm{a}}{\bm{b}}}=\Xi^{-2}\eta_{{\bm{a}}{\bm{b}}}{\bm{\omega}}^{{\bm{a}}}\otimes{\bm{\omega}}^{{\bm{b}}}$

is a solution to eq. (3) on $\mathcal{U}$. Here $\eta_{{\bm{a}}{\bm{b}}}$ is
used to denote the components of the Minkowski metric with respect to a
Cartesian coordinate frame field, i.e.,
$\eta_{{\bm{a}}{\bm{b}}}=\text{diag}(1,-1,-1,-1)$.

Similar to the metric conformal field equations introduced in the previous
section, the ECFEs exhibit conformal gauge freedom. To demonstrate this,
define the conformal metric $\grave{{\bm{g}}}$ such that

$\displaystyle\grave{g}_{{\bm{a}}{\bm{b}}}=\grave{\Xi}^{2}\tilde{g}_{{\bm{a}}{\bm{b}}},$
$\displaystyle\grave{g}_{{\bm{a}}{\bm{b}}}=\kappa^{2}g_{{\bm{a}}{\bm{b}}}.$

Thus, we have $\grave{\Xi}=\kappa^{-1}\Xi$. Then, introduce the Levi-Civita
connection $\grave{{\bm{\nabla}}}$ associated with $\grave{{\bm{g}}}$ and
define the Weyl connection $\check{{\bm{\nabla}}}$ as

$\check{\nabla}_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}}=-2\grave{f}_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}},$

so that the relation between $\grave{{\bm{\nabla}}}$, $\check{{\bm{\nabla}}}$
and $\hat{{\bm{\nabla}}}$ is given by

$\displaystyle\grave{{\bm{\nabla}}}-\check{{\bm{\nabla}}}={\bm{S}}(\grave{{\bm{f}}}),$
$\displaystyle\hat{{\bm{\nabla}}}-\check{{\bm{\nabla}}}={\bm{S}}({\bm{k}}+\kappa^{-1}{\bm{d}}{\kappa}),$

where ${\bm{k}}={\bm{f}}-\grave{{\bm{f}}}$. In terms of the above, the
conformal covariance of the ECFEs can now be expressed as follows: if
$({\bm{e}}_{{\bm{a}}},\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}},\hat{L}_{{\bm{a}}{\bm{b}}},d^{{\bm{a}}}{}_{{\bm{b}}{\bm{d}}{\bm{c}}})$
is a solution to eqs. (11), the collection
$(\grave{{\bm{e}}}_{{\bm{a}}},\check{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}},\check{L}_{{\bm{a}}{\bm{b}}},\grave{d}^{{\bm{a}}}{}_{{\bm{b}}{\bm{d}}{\bm{c}}})$
with

$\displaystyle\grave{{\bm{e}}}_{{\bm{a}}}=\kappa{\bm{e}}_{{\bm{a}}},$
$\displaystyle\check{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}=\kappa\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}+\delta_{{\bm{c}}}{}^{{\bm{b}}}\hat{\nabla}_{{\bm{a}}}\kappa-\kappa
S_{{\bm{a}}{\bm{c}}}{}^{{\bm{b}}{\bm{d}}}(k_{{\bm{d}}}+\kappa^{-1}\hat{\nabla}_{{\bm{d}}}\kappa),$
$\displaystyle\check{L}_{{\bm{a}}{\bm{b}}}=\kappa^{2}\hat{L}_{{\bm{a}}{\bm{b}}}-\kappa^{2}\hat{\nabla}_{{\bm{a}}}(k_{{\bm{b}}}+\kappa^{-1}\hat{\nabla}_{{\bm{b}}}\kappa)-\frac{1}{2}\kappa^{2}S_{{\bm{a}}{\bm{b}}}{}^{{\bm{c}}{\bm{d}}}(k_{{\bm{c}}}+\kappa^{-1}\hat{\nabla}_{{\bm{c}}}\kappa)(k_{{\bm{d}}}+\kappa^{-1}\hat{\nabla}_{{\bm{d}}}\kappa),$
$\displaystyle\grave{d}^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}=\kappa^{3}d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}},$

is also a solution to the ECFEs. However, note that the ECFEs exhibit
additional gauge freedom, corresponding to the freedom in the choice of
$\hat{{\bm{\nabla}}}$, compared to the metric conformal field equations.
Therefore, to obtain a solution to eqs. (11), our analysis of the ECFEs will
require a gauge choice. In Section 3.1, the final form of the ECFEs will be
obtained by introducing the so-called conformal Gaussian gauge that allows us
to fix the conformal factor $\Xi$ and $\hat{{\bm{\nabla}}}$.

######  Remark 2.

As our main focus will be on obtaining solutions to the ECFEs, rather than the
metric conformal field equations, the term conformal field equations will be
used interchangeably with ECFEs in later discussions.

#### The extended conformal field equations in spinors formulation

The analysis of the ECFEs will be carried out in terms of spinors. In the
following, let $\\{{\bm{e}}_{{\bm{a}}}\\}$ denote the ${\bm{g}}$-orthonormal
frame introduced in the previous section and let
$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ refer to the spinorial version of
the frame satisfying

${\bm{g}}({\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}},{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}})=\epsilon_{{\bm{A}}{\bm{B}}}\bar{\epsilon}_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}.$
(19)

Then, introduce the spinorial counterparts of
$\hat{P}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}},\hat{\rho}^{{\bm{c}}}{}_{{\bm{d}}{\bm{a}}{\bm{b}}},\hat{L}_{{\bm{a}}{\bm{b}}},d_{{\bm{a}}},d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}$
and $f_{{\bm{a}}}$ as

$\hat{P}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\hat{\rho}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\hat{L}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},d_{{\bm{A}}{\bm{A}}^{\prime}},d^{{\bm{A}}{\bm{A}}^{\prime}}{}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}},f_{{\bm{A}}{\bm{A}}^{\prime}}$

Since the definition of geometric and algebraic curvature is given in terms of
the connection coefficients
$\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}$, introduce the spinorial
connection coefficients
$\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}{\bm{B}}^{\prime}}{}_{{\bm{C}}{\bm{C}}^{\prime}}$
which can be decomposed as

$\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}{\bm{B}}^{\prime}}{}_{{\bm{C}}{\bm{C}}^{\prime}}=\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}\delta_{{\bm{C}}^{\prime}}{}^{{\bm{B}}^{\prime}}+\bar{\hat{\Gamma}}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}^{\prime}}{}_{{\bm{C}}^{\prime}}\delta_{{\bm{C}}}{}^{{\bm{B}}},$
(20)

where $\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}$ are
known as the spin connection coefficients. Given the above, the relation
between the Weyl connection $\hat{{\bm{\nabla}}}$ and the unphysical
connection ${\bm{\nabla}}$ given by eq. (9) implies that

$\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}=\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}+\delta_{{\bm{A}}}{}^{{\bm{B}}}f_{{\bm{C}}{\bm{A}}^{\prime}},\qquad\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{Q}}}{}_{{\bm{Q}}}=f_{{\bm{A}}{\bm{A}}^{\prime}}.$

Then,
$\hat{P}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}$
and
$\hat{\rho}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}$
can be decomposed as

$\displaystyle\hat{P}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=\hat{P}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\delta_{{\bm{D}}^{\prime}}{}^{{\bm{C}}^{\prime}}+\bar{\hat{P}}^{{\bm{C}}^{\prime}}{}_{{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\delta_{{\bm{D}}}{}^{{\bm{C}}},$
$\displaystyle\hat{\rho}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=\hat{\rho}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\delta_{{\bm{D}}^{\prime}}{}^{{\bm{C}}^{\prime}}+\bar{\hat{\rho}}^{{\bm{C}}^{\prime}}{}_{{\bm{D}}^{\prime}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\delta_{{\bm{D}}}{}^{{\bm{C}}},$

where

$\displaystyle\hat{P}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}=\hat{P}_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}+\frac{1}{2}\epsilon_{{\bm{A}}{\bm{B}}}\left(\hat{\nabla}_{{\bm{C}}{\bm{C}}^{\prime}}f_{{\bm{D}}{\bm{D}}^{\prime}}-\hat{\nabla}_{{\bm{D}}{\bm{D}}^{\prime}}f_{{\bm{C}}{\bm{C}}^{\prime}}\right),$
$\displaystyle\hat{\rho}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}=\hat{\rho}_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}+\frac{1}{2}\epsilon_{{\bm{A}}{\bm{B}}}\left(\hat{L}_{{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}-\hat{L}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}}\right).$

Given these definitions, the extended conformal field equations can be written
in terms of the zero quantities
$(\hat{\Sigma}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\hat{\Xi}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\hat{\Delta}_{{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\hat{\Lambda}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}})$
as

$\hat{\Sigma}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0,\qquad\hat{\Xi}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0,\qquad\hat{\Delta}_{{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0,\qquad\hat{\Lambda}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}=0,$
(23)

where

$\displaystyle\hat{\Sigma}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\equiv[{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}},{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}}]-\left(\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{B}}{\bm{B}}^{\prime}}-\hat{\Gamma}_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{C}}{\bm{C}}^{\prime}}{}_{{\bm{A}}{\bm{A}}^{\prime}}\right){\bm{e}}_{{\bm{C}}{\bm{C}}^{\prime}},$
$\displaystyle\hat{\Xi}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\equiv\hat{P}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}-\hat{\rho}^{{\bm{C}}}{}_{{\bm{D}}{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},$
$\displaystyle\hat{\Delta}_{{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}\equiv\hat{\nabla}_{{\bm{C}}{\bm{C}}^{\prime}}\hat{L}_{{\bm{D}}{\bm{D}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}-\hat{\nabla}_{{\bm{D}}{\bm{D}}^{\prime}}\hat{L}_{{\bm{C}}{\bm{C}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}-d_{{\bm{A}}{\bm{A}}^{\prime}}d^{{\bm{A}}{\bm{A}}^{\prime}}{}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}},$
$\displaystyle\hat{\Lambda}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}\equiv\hat{\nabla}_{{\bm{A}}{\bm{A}}^{\prime}}d^{{\bm{A}}{\bm{A}}^{\prime}}{}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}-f_{{\bm{A}}{\bm{A}}^{\prime}}d^{{\bm{A}}{\bm{A}}^{\prime}}{}_{{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}.$

One can also define the spinorial counterparts of the zero quantities
$\delta_{{\bm{a}}},\gamma_{{\bm{a}}{\bm{b}}}$ and
$\varsigma_{{\bm{a}}{\bm{b}}}$ as

$\delta_{{\bm{A}}{\bm{A}}^{\prime}},\qquad\gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},\qquad\varsigma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}},$

that can be expressed in terms of the spinorial counterparts of
$d_{{\bm{a}}},f_{{\bm{a}}},\hat{L}_{{\bm{a}}{\bm{b}}},\beta_{{\bm{a}}}$. Then,
eqs. (14) can be written as

$\delta_{{\bm{A}}{\bm{A}}^{\prime}}=0,\qquad\gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0,\qquad\varsigma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0.$
(25)

###  2.3 Conformal Geodesics

As will become evident, Friedrich’s formulation of spatial infinity relies on
a class of conformal curves known as conformal geodesics. The aim of this
section is to briefly introduce the notion of conformal geodesics utilised in
this work.

######  Definition 3 (conformal geodesics).

Let $I\subseteq\mathbb{R}$ and $\tau\in I$, then the curve $x(\tau)$ is a
conformal geodesic on the physical spacetime
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ if there exists a 1-form
${\bm{\beta}}(\tau)$ along $x(\tau)$ such that

$\displaystyle\tilde{\nabla}_{\dot{{\bm{x}}}}\dot{{\bm{x}}}=-2\langle{\bm{\beta}},\dot{{\bm{x}}}\rangle\dot{{\bm{x}}}+\tilde{{\bm{g}}}(\dot{{\bm{x}}},\dot{{\bm{x}}}){\bm{\beta}}^{\sharp},$
(26a)
$\displaystyle\tilde{\nabla}_{\dot{{\bm{x}}}}{\bm{\beta}}=\langle{\bm{\beta}},\dot{{\bm{x}}}\rangle-\frac{1}{2}\tilde{{\bm{g}}}^{\sharp}({\bm{\beta}},{\bm{\beta}})\dot{{\bm{x}}}^{\flat}+\tilde{{\bm{L}}}(\dot{{\bm{x}}},.),$
(26b)

where $\dot{{\bm{x}}}$ denotes the tangent vector and $\tilde{{\bm{L}}}$ is
the Schouten tensor associated with $\tilde{{\bm{\nabla}}}$.

In the following, let $\tilde{\mathcal{S}}$ denote a spacelike submanifold of
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ and consider the smooth initial data
on $\tilde{\mathcal{S}}$ as

$x_{\star}\in\tilde{\mathcal{S}},\qquad\dot{{\bm{x}}}_{\star}\in
T|_{x_{\star}}(\tilde{\mathcal{S}}),\qquad{\bm{\beta}}_{\star}\in
T^{*}|_{x_{\star}}(\tilde{\mathcal{S}}),$

where $T|_{x_{\star}}(\tilde{\mathcal{S}})$ and
$T^{*}|_{x_{\star}}(\tilde{\mathcal{S}})$ denote the tangent and dual tangent
space at $x_{\star}\in\tilde{\mathcal{S}}$, respectively. Given these initial
data, there exists a unique conformal geodesic $(x(\tau),{\bm{\beta}}(\tau))$
passing through each $x_{\star}\in\tilde{\mathcal{S}}$ such that

$x(0)\equiv
x_{\star},\qquad\dot{{\bm{x}}}(0)\equiv\dot{{\bm{x}}}_{\star},\qquad{\bm{\beta}}(0)\equiv{\bm{\beta}}_{\star}.$
(27)

To illustrate some of the useful properties of conformal geodesics, introduce
the Weyl connection $\hat{{\bm{\nabla}}}$ satisfying

$\hat{\nabla}_{a}\tilde{g}_{bc}=-2\beta_{a}\tilde{g}_{bc},$ (28)

where $\beta_{a}$ is the 1-form satisfying the conformal geodesics eqs. (26).
Given this definition, the relation between $\tilde{{\bm{\nabla}}}$ and
$\hat{{\bm{\nabla}}}$ is given by

$\hat{{\bm{\nabla}}}-\tilde{{\bm{\nabla}}}={\bm{S}}({\bm{\beta}}),$

where ${\bm{S}}({\bm{\beta}})$ can be written as

${\bm{S}}({\bm{\beta}})=S_{ab}{}^{cd}\beta_{d},$

with

$S_{ab}{}^{cd}=\delta_{a}{}^{c}\delta_{b}{}^{d}+\delta_{a}{}^{d}\delta_{b}{}^{c}-\tilde{g}_{ab}\tilde{g}^{cd}.$
(29)

Then, eqs. (26) implies

$\hat{{\bm{\nabla}}}_{\dot{{\bm{x}}}}\dot{{\bm{x}}}=0,\qquad\hat{{\bm{L}}}(\dot{{\bm{x}}},.)=0.$
(30)

where $\hat{{\bm{L}}}$ is the Schouten tensor associated with
$\hat{{\bm{\nabla}}}$. Furthermore, one can introduce a Weyl-propagated frame
$\\{{\bm{e}}_{{\bm{a}}}\\}$ as

$\hat{\nabla}_{\dot{{\bm{x}}}}{\bm{e}}_{{\bm{a}}}=0.$ (31)

Hence, a congruence of conformal geodesics satisfying eqs. (26) given initial
data eq. (27) singles out a Weyl connection $\hat{{\bm{\nabla}}}$ and a Weyl-
propagated frame $\\{{\bm{e}}_{{\bm{a}}}\\}$ as suggested by eq. (28) and eq.
(31).

Another essential feature of conformal geodesics is that a non-intersecting
congruence of conformal geodesics singles out a canonical conformal factor
$\Xi$ and a metric ${\bm{g}}=\Xi^{2}\tilde{{\bm{g}}}$ such that

${\bm{g}}(\dot{{\bm{x}}},\dot{{\bm{x}}})=1.$ (32)

In other words, the metric ${\bm{g}}$ is singled out by enforcing that the
parameter $\tau$ of the conformal geodesics corresponds to the
${\bm{g}}$-proper time. In particular, we have

######  Proposition 1.

_Let $(\mathcal{\tilde{M}},\tilde{{\bm{g}}})$ denote a vacuum spacetime
satisfying eq. (3) and let $(x(\tau),{\bm{\beta}}(\tau))$ denote a solution to
the conformal geodesics eqs. (26) with initial data eq. (27). Then, if
${\bm{g}}=\Xi^{2}\tilde{{\bm{g}}}$ is defined such that
${\bm{g}}(\dot{{\bm{x}}},\dot{{\bm{x}}})=1$, the conformal factor $\Xi$ can be
written as a quadratic polynomial in terms of $\tau$, i.e._

$\displaystyle\Xi(\tau)=\Xi_{\star}+\dot{\Xi}_{\star}\tau+\frac{1}{2}\ddot{\Xi}_{\star}\tau^{2}$
(33)

_with_

$\displaystyle\dot{\Xi}_{\star}=\langle{\bm{\beta}}_{\star},\dot{{\bm{x}}}_{\star}\rangle\Xi_{\star},\qquad\qquad\Xi_{\star}\ddot{\Xi}_{\star}=\frac{1}{2}\tilde{{\bm{g}}}^{\sharp}({\bm{\beta}}_{\star},{\bm{\beta}}_{\star}).$
(34)

_If $\\{{\bm{e}}_{{\bm{a}}}\\}$ is a ${\bm{g}}$-orthonormal Weyl-propagated
frame satisfying eq. (31), one can show that the components of ${\bm{\beta}}$
with respect to $\\{{\bm{e}}_{{\bm{a}}}\\}$ satisfy_

$\Xi\beta_{{\bm{0}}}=\dot{\Xi},\qquad\Xi\beta_{{\bm{i}}}=\Xi_{\star}\beta_{{\bm{i}}\star},$

_where
$\beta_{{\bm{a}}}\equiv\langle{\bm{\beta}},{\bm{e}}_{{\bm{a}}}\rangle$._

In the next section, we will use the unique properties of conformal geodesics
to introduce Friedrich’s formulation of spatial infinity.

##  3 Friedrich’s formulation of spatial infinity

As discussed in the introduction, the strategy of this work is to make use of
Friedrich’s regular initial value problem of the conformal field equations
[39] to analyse the behaviour of asymptotic charges near spatial infinity.
Following the discussion in [40], let $\tilde{\mathcal{S}}$ denote a spacelike
submanifold of $(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ which is
asymptotically Euclidean and regular (Definition 1) with one asymptotic end,
and let $\tilde{{\bm{h}}}$ denote the intrinsic metric induced by
$\tilde{{\bm{g}}}$ on $\tilde{\mathcal{S}}$. Let $\mathcal{S}$ denote the
hypersurface mentioned in Definition 1 with the asymptotic point $i$ and
denote by $SU(\mathcal{S})$ the bundle of normalised spin frames over
$\mathcal{S}$ with the structure group $SU(2,\mathbb{C})$. Let
$\\{{\bm{\epsilon}}_{{\bm{A}}}{}^{A}\\}$ with
${\bm{A}}\in\\{{\bm{0}},{\bm{1}}\\}$ denote the spin frame chosen so that the
components of the ${\bm{\epsilon}}$-spinor are given by

$\epsilon_{{\bm{0}}{\bm{1}}}=1,\quad\epsilon^{{\bm{0}}{\bm{1}}}=1.$

Also, let $\tau^{AA^{\prime}}$ denote the spinorial counterpart of the future
directed normal vector of $\tilde{\mathcal{S}}$ satisfying
$\tau_{AA^{\prime}}\tau^{AA^{\prime}}=2$. Then, $\tau^{AA^{\prime}}$ can be
written as

$\tau^{AA^{\prime}}=\epsilon_{{\bm{0}}}{}^{A}\bar{\epsilon}_{{\bm{0}}^{\prime}}{}^{A^{\prime}}+\epsilon_{{\bm{1}}}{}^{A}\bar{\epsilon}_{{\bm{1}}^{\prime}}{}^{A^{\prime}}.$
(35)

Given ${\bm{t}}=\\{t^{{\bm{A}}}{}_{{\bm{B}}}\\}\in SU(2,\mathbb{C})$, one can
show

$\displaystyle\epsilon_{{\bm{A}}{\bm{B}}}t^{{\bm{A}}}{}_{{\bm{C}}}t^{{\bm{B}}}{}_{{\bm{D}}}=\epsilon_{{\bm{C}}{\bm{D}}},$
$\displaystyle\tau_{{\bm{A}}{\bm{A}}^{\prime}}t^{{\bm{A}}}{}_{{\bm{B}}}\bar{t}^{{\bm{A}}^{\prime}}{}_{{\bm{B}}^{\prime}}=\tau_{{\bm{B}}{\bm{B}}^{\prime}}.$

Then, given a fixed spin frame $\\{{\bm{\epsilon}}_{{\bm{A}}}{}^{A}\\}$ at $i$
and ${\bm{t}}\in SU(2,\mathbb{C})$, introduce the transformed frame
${\bm{\epsilon}}_{{\bm{A}}}{}^{A}({\bm{t}})$ as

${\bm{\epsilon}}_{{\bm{A}}}{}^{A}({\bm{t}})=t_{{\bm{A}}}{}^{{\bm{B}}}{\bm{\epsilon}}_{{\bm{B}}}{}^{A}.$

To introduce spin frames in a neighbourhood of the asymptotic point
$i\in\mathcal{S}$, consider the metric ball $\mathcal{B}_{a}(i)$ in
$\mathcal{S}$ centered at $i$ and choose $a>0$ such that $\mathcal{B}_{a}(i)$
is geodesically convex and the metric ${\bm{h}}$ is smooth on
$\mathcal{B}_{a}(i)$. Then, construct the ${\bm{h}}$-geodesic starting at $i$
and let $\rho$ denote the affine parameter along the geodesic so that
$\rho(i)=0$. By fixing ${\bm{t}}$, the spin frames
${\bm{\epsilon}}_{{\bm{A}}}{}^{A}({\bm{t}})$ can be parallelly transported
along the geodesic and the frames obtained are denoted by
${\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})$, where the upper index for the
spin frames will be removed in the subsequent discussion for convenience.
Then, let $\mathcal{C}_{a}$ denote the subset of $SU(\mathcal{S})$ defined as

$\mathcal{C}_{a}=\\{{\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})\in
SU(\mathcal{S})|-a<\rho<a,{\bm{t}}\in SU(2,\mathbb{C})\\}.$ (37)

From the above, $\mathcal{C}_{a}$ is diffeomorphic to $\bar{\mathcal{S}}_{a}$
given by

$\bar{\mathcal{S}}_{a}=\\{(\rho,{\bm{t}})\in\mathbb{R}\times
SU(2,\mathbb{C})|-a<\rho<a\\},$

To relate the structures on $\mathcal{C}_{a}$ with those on $\mathcal{S}$, let
$\pi$ denote the projection from $SU(\mathcal{S})$ to $\mathcal{S}$ and then
denote by $\pi^{\prime}$ the restriction to $\mathcal{C}_{a}$ so that
$\pi^{\prime}$ is a projection map from $\mathcal{C}_{a}$ to
$\mathcal{B}_{a}(i)$. The action of $U(1)$ on $SU(\mathcal{S})$ will imply an
action of $U(1)$ on $\mathcal{C}_{a}$. The quotient under this action
$\mathcal{C}^{\prime}_{a}=\mathcal{C}_{a}/U(1)$ is diffeomorphic to
$(-a,a)\times\mathbb{S}^{2}$. Subsequently, if ${\bm{t}}\in SU(2,\mathbb{C})$
and $s^{\prime}\in U(1)$, then ${\bm{\epsilon}}_{{\bm{A}}}({\bm{t}})$ and
${\bm{\epsilon}}_{{\bm{A}}}(s^{\prime}{\bm{t}})$ can be parallelly transported
along the same geodesic and will be given by
${\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})$ and
${\bm{\epsilon}}_{{\bm{A}}}(\rho,s^{\prime}{\bm{t}})$, respectively. Since the
function $\rho$ is invariant under the action of $U(1)$, we have

$\pi^{\prime}({\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}}))=\pi^{\prime}({\bm{\epsilon}}_{{\bm{A}}}(\rho,s^{\prime}{\bm{t}})),$

and the map $\pi^{\prime}$ can be factored as

$\mathcal{C}_{a}\xrightarrow{\pi^{\prime}_{1}}\mathcal{C}^{\prime}_{a}\xrightarrow{\pi^{\prime}_{2}}\mathcal{B}_{a}(i),$

where $\pi^{\prime}_{1}$ is the Hopf fibration and $\pi^{\prime}_{2}$ is the
exponential map. Note that the set $\mathcal{C}^{\prime}_{a}$ can be split
into two components: $\mathcal{C}^{\prime}_{a}{}^{+}$ on which $\rho>0$ and
$\mathcal{C}^{\prime}_{a}{}^{-}$ on which $\rho<0$. Each of these components
can be mapped into the punctured disk $\mathcal{B}_{a}(i)\setminus\\{i\\}$
using $\pi^{\prime}_{2}$. Additionally, given these projections, the point $i$
can be replaced by the set $\pi^{\prime}_{2}{}^{-1}(i)$ which is diffeomorphic
to $\mathbb{S}^{2}$. Then if $I_{0}=\\{\rho=0\\}\subset\mathcal{C}_{a}$, we
can identify $\pi^{\prime}_{1}(I_{0})=\pi^{\prime}_{2}{}^{-1}(i)$ and
$\pi^{\prime}_{2}{}^{-1}(i)$ glues together the components
$\mathcal{C}^{\prime}_{a}{}^{+}$ and $\mathcal{C}^{\prime}_{a}{}^{-}$.

In the following, it will be assumed that
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ is the development of the initial
data $(\tilde{\mathcal{S}},\tilde{{\bm{h}}})$ satisfying Definition 1 and that
$(\mathcal{M},{\bm{g}},\Theta)$ is the smooth conformal extension of
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ such that
$\mathcal{M}=\tilde{\mathcal{M}}\cup\mathscr{I}^{+}\cup\mathscr{I}^{-}$, where
$\mathscr{I}^{\pm}$ denotes future and past null infinity, respectively and
$\Theta$ is the conformal factor satisfying i) $\Theta>0$ and
${\bm{g}}=\Theta^{2}\tilde{{\bm{g}}}$ on $\tilde{\mathcal{M}}$, ii) $\Theta=0$
and $d\Theta\neq 0$ on $\mathscr{I}^{\pm}$. Then, introduce the manifold
$\mathcal{M}_{a,\kappa}$ by

$\mathcal{M}_{a,\kappa}=\bigg{\\{}(\tau,\rho,{\bm{t}})\in\mathbb{R}\times\mathbb{R}\times
SU(2,\mathbb{C})|\>0\leq\rho<a,-\frac{\omega}{\kappa}\leq\tau\leq\frac{\omega}{\kappa}\bigg{\\}},$

where $\kappa$ is an arbitrary function such that $\kappa=\rho\kappa^{\prime}$
with $\kappa^{\prime}$ smooth and $\kappa^{\prime}(i)=1$,
$\omega\equiv\omega(\rho,{\bm{t}})$ is a smooth non-negative function such
that $\omega/\kappa\to 1$ as $\rho\to 0$. Note that a coordinate system can be
defined on $\mathcal{M}_{a,\kappa}$ by introducing a coordinate system on
$SU(2,\mathbb{C})$ together with the functions $\tau$ and $\rho$.

Similar to previous discussion, the action of $U(1)$ on $SU(2,\mathbb{C})$
implies an action of $U(1)$ on $\mathcal{M}_{a,\kappa}$. The quotient under
this action $\mathcal{M}_{a,\kappa}/U(1)$ will be denoted by
$\mathcal{M}^{\prime}_{a,\kappa}$ and the projection from
$\mathcal{M}_{a,\kappa}$ onto $\mathcal{N}\subset\mathcal{M}$ by
$\bar{\pi}^{\prime}$, where $\mathcal{N}$ is the domain of influence of
$\mathcal{B}_{a}(i)\setminus\\{i\\}$. The map $\bar{\pi}^{\prime}$ can be
factored as

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

Given the mentioned construction, define the following subsets of
$\mathcal{M}_{a,\kappa}$

$\displaystyle\mathscr{I}^{\pm}_{a}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\>0<\rho<a,\tau=\pm\frac{\omega}{\kappa}\\},$
(38a)
$\displaystyle\mathcal{I}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\>\rho=0,-1<\tau<1\\},$
(38b)
$\displaystyle\mathcal{I}^{\pm}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\>\rho=0,\tau=\pm
1\\},$ (38c)
$\displaystyle\mathcal{I}_{0}=\\{(\tau,\rho,{\bm{t}})\in\mathcal{M}_{a,\kappa}|\>\rho=0,\tau=0\\},$
(38d)

where the sets $\mathscr{I}^{\pm}_{a}$ represent past/future null infinity,
$\mathcal{I}$ is the cylinder at spacelike infinity and $\mathcal{I}^{\pm}$
are the sets at which $\mathscr{I}^{\pm}_{a}$ touches $\mathcal{I}$, known as
the _critical sets_.

######  Remark 3.

  1. i.

The subscript a in $\mathscr{I}^{\pm}_{a}$ is used to indicate that the sets
$\mathscr{I}^{\pm}_{a}$ do not represent the entirety of past/future null
infinity, rather they map to a part of null infinity close to spacelike
infinity. In subsequent discussions, we will drop the subscript for
convenience.

  2. ii.

Given that $\pi^{\prime}_{1}(I_{0})=\pi^{\prime}_{2}{}^{-1}(i)$ and that
$\pi^{\prime}_{2}{}^{-1}(i)$ is diffeomorphic to $\mathbb{S}^{2}$, it is clear
to see that $\bar{\pi}^{\prime}_{1}(\mathcal{I})$ is diffeomorphic to
$\mathbb{R}\times\mathbb{S}^{2}$, hence the use of the term _cylinder at
spatial infinity_ to refer to $\mathcal{I}$.

So far, the construction $\mathcal{C}_{a}$ has not been extended to the
spacetime $\mathcal{M}_{a,\kappa}$. To do so, assume $\mathcal{C}_{a}$ defined
as previously and define
$\mathcal{S}_{a}=\\{\rho>0\\}\subset\bar{\mathcal{S}}_{a}$, then the spin
frames ${\bm{\epsilon}}_{{\bm{A}}}(\rho,{\bm{t}})\in SU(\mathcal{S})$ will be
transported off $\mathcal{S}_{a}$ into the spacetime by a certain propagation
law along conformal geodesics which are orthogonal to $\mathcal{S}_{a}$. This
allows us to determine the spin frames
${\bm{\epsilon}}_{{\bm{A}}}(\tau,\rho,{\bm{t}})$ at points of
$\mathcal{M}_{a,\kappa}\setminus(\mathcal{I}\cup\mathcal{I}^{+}\cup\mathcal{I}^{-})$
up to multiplication by a phase parameter that corresponds to the action of
$U(1)$ on $SU(\mathcal{M})$.

######  Remark 4.

Subsequent calculations will be carried out on $\mathcal{M}_{a,\kappa}$. We
will use the same notation to refer to fields on $\mathcal{M}$ and their pull-
back to $\mathcal{M}_{a,\kappa}$ via $\bar{\pi}^{\prime}$ e.g. use $\Theta$
for $\bar{\pi}^{\prime*}\Theta$.

Following the discussion in [40], let $Z_{u_{i}}$ denote the vector fields
generated by $u_{i}$, the basis of the Lie Algebra of $SU(2,\mathbb{C})$,
given by

$u_{1}=\frac{1}{2}\begin{pmatrix}0&i\\\
i&0\end{pmatrix},\qquad\;u_{2}=\frac{1}{2}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\qquad\;u_{3}=\frac{1}{2}\begin{pmatrix}i&0\\\
0&-i\end{pmatrix}.$ (39)

Then, introduce the complex vector fields

$X_{+}=-(Z_{u_{2}}+iZ_{u_{1}}),\qquad X_{-}=-(Z_{u_{2}}-iZ_{u_{1}}),\qquad
X=-2iZ_{u_{3}},$

satisfying

$[X,X_{+}]=2X_{+},\qquad[X,X_{-}]=-2X_{-},\qquad[X_{+},X_{-}]=-X.$

Given the above, consider the smooth vector field

${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}={\bm{e}}^{0}{}_{{\bm{A}}{\bm{A}}^{\prime}}\partial_{\tau}+{\bm{e}}^{1}{}_{{\bm{A}}{\bm{A}}^{\prime}}\partial_{\rho}+{\bm{e}}^{+}{}_{{\bm{A}}{\bm{A}}^{\prime}}X_{+}-{\bm{e}}^{-}{}_{{\bm{A}}{\bm{A}}^{\prime}}X_{-},$
(40)

where ${\bm{e}}^{\mu}{}_{{\bm{A}}{\bm{A}}^{\prime}}$ denotes the components of
${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$ with respect to the local coordinate
system $(\tau,\rho,{\bm{t}})$. If ${\bm{\omega}}^{{\bm{A}}{\bm{A}}^{\prime}}$
denotes the dual frame, then
$\langle{\bm{\omega}}^{{\bm{A}}{\bm{A}}^{\prime}},{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}}\rangle=\epsilon_{{\bm{B}}}{}^{{\bm{A}}}\bar{\epsilon}_{{\bm{B}}^{\prime}}{}^{{\bm{A}}^{\prime}}$
on $\mathcal{M}_{a,\kappa}\setminus\mathcal{I}$.

Given that Friedrich’s formulation is an initial value problem formulation of
the conformal field equation, we will be interested in reintroducing the $1+3$
decomposition of the field equations in terms of so-called space-spinors —see
[44, 45, 28] for details. In the space-spinors formulation, the frame fields
${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$ can be decomposed as follows

${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}=\frac{1}{\sqrt{2}}\tau_{{\bm{A}}{\bm{A}}^{\prime}}\partial_{\tau}-\tau^{{\bm{B}}}{}_{{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{A}}{\bm{B}}},$

where $\tau_{{\bm{A}}{\bm{A}}^{\prime}}$ denotes the components of the future
directed normal of $\mathcal{S}$ and ${\bm{e}}_{{\bm{A}}{\bm{B}}}$ is defined
by

${\bm{e}}_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{A}}}{}^{{\bm{B}}^{\prime}}{\bm{e}}_{{\bm{B}}){\bm{B}}^{\prime}}={\bm{e}}^{0}_{{\bm{A}}{\bm{B}}}\partial_{\tau}+{\bm{e}}^{1}_{{\bm{A}}{\bm{B}}}\partial_{\rho}+{\bm{e}}^{+}_{{\bm{A}}{\bm{B}}}X_{+}+{\bm{e}}^{-}_{{\bm{A}}{\bm{B}}}X_{-}.$
(41)

Since the ECFEs are written in terms of a Weyl connection, let
$\hat{{\bm{\nabla}}}$ denote a Weyl connection such that

$\hat{\nabla}_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}}=-2f_{{\bm{a}}}g_{{\bm{b}}{\bm{c}}},$
(42)

where $\\{{\bm{e}}_{{\bm{a}}}\\}$ is the tensorial counterpart of the frame
fields defined by eq. (40),
$g_{{\bm{a}}{\bm{b}}}\equiv{\bm{g}}({\bm{e}}_{{\bm{a}}},{\bm{e}}_{{\bm{b}}})$,
and $f_{{\bm{a}}}$ is an arbitrary 1-form which will be fixed in later
discussions.

Now, let $\chi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ and
$\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ denote the real and imaginary parts
of $\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ of the spin connection
coefficients
$\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}\hat{\Gamma}_{{\bm{A}}){\bm{A}}^{\prime}{\bm{C}}{\bm{D}}}$
associated with $\hat{{\bm{\nabla}}}$, defined by

$\displaystyle\chi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv-\frac{1}{\sqrt{2}}(\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}+\hat{\Gamma}^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}),$
$\displaystyle\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\frac{1}{\sqrt{2}}(\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}-\hat{\Gamma}^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}),$

where $\hat{\Gamma}^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ is the Hermitian
conjugate of $\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$. Then,
$\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ can be written as

$\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=\frac{1}{\sqrt{2}}(\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}-\chi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}})=\frac{1}{\sqrt{2}}(\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}-\chi_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}})-\frac{1}{2}\epsilon_{{\bm{A}}{\bm{B}}}f_{{\bm{C}}{\bm{D}}},$
(44)

where
$f_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}f_{{\bm{A}}){\bm{A}}^{\prime}}=\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}\sigma^{{\bm{a}}}{}_{{\bm{A}}){\bm{A}}^{\prime}}f_{{\bm{a}}}$.

Given that the curvature tensor $\hat{R}^{a}{}_{bcd}$ associated with
$\hat{{\bm{\nabla}}}$ can be written in terms of the rescaled Weyl tensor
$d^{a}{}_{bcd}$ and the Schouten tensor $\hat{L}_{ab}$, it is straightforward
to see that the spinorial counterpart of $\hat{R}^{a}{}_{bcd}$ is fully
determined by the rescaled Weyl spinor
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\phi_{({\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}})}$
and the space-spinor counterpart of the Schouten tensor
$\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$, where
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ and
$\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ are defined by

$\displaystyle\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv-\frac{1}{4}\epsilon^{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}\epsilon^{{\bm{C}}^{\prime}{\bm{D}}^{\prime}}d_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
$\displaystyle\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\Theta_{{\bm{A}}{\bm{B}}({\bm{C}}{\bm{D}})}\equiv\tau_{{\bm{B}}}{}^{{\bm{A}}^{\prime}}\tau_{{\bm{D}}}{}^{{\bm{C}}^{\prime}}\hat{L}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}}.$

A further decomposition of
$\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\Theta_{{\bm{A}}{\bm{B}}({\bm{C}}{\bm{D}})}$
yields

$\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=\Theta_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}}-\frac{1}{2}\epsilon_{{\bm{A}}{\bm{B}}}\Theta_{{\bm{Q}}}{}^{{\bm{Q}}}{}_{{\bm{C}}{\bm{D}}}.$

As shown in [46], the gauge choice based on conformal geodesics implies that
the conformal factor $\Theta$ can be expressed in terms of initial data as
follows:

$\Theta=\kappa^{-1}\Omega\left(1-\tau^{2}\frac{\kappa^{2}}{\omega^{2}}\right),$
(46)

where $\omega$ is given by

$\omega=\frac{2\Omega}{\sqrt{-D_{{\bm{A}}{\bm{B}}}\Omega
D^{{\bm{A}}{\bm{B}}}\Omega}}.$ (47)

Here, $D_{AB}$ denotes the intrinsic covariant derivative on the initial
hypersurface $\mathcal{S}_{a}$. Moreover, one can show that $\Theta$ satisfies

$\displaystyle\Theta>0\qquad\;\text{on
}\mathcal{M}_{a,\kappa},\qquad\\{\Theta=0\\}=\mathscr{I}^{+}_{a}\cup\mathcal{I}^{-}\cup\mathcal{I}\cup\mathcal{I}^{+}\cup\mathscr{I}^{+}_{a},$
$\displaystyle{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}(\Theta)\neq
0,\qquad\epsilon^{{\bm{A}}{\bm{B}}}\bar{\epsilon}^{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}(\Theta){\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}}(\Theta)=0\qquad\;\text{on
}\mathscr{I}^{\pm}_{a}.$

For the rest of this article, we will refer to the frame
$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ (or equivalently
$\\{{\bm{e}}_{{\bm{a}}}\\}$) satisfying the conditions mentioned, the
coordinates $(\tau,\rho,{\bm{t}})$ and the conformal gauge defined above as
the F-gauge. In the following section, Friedrich’s approach to the conformal
field equations will be introduced, where the aim is to encode the above gauge
conditions, and their transport laws in the properties of the unknowns
appearing in the conformal field equations.

###  3.1 Hyperbolic reduction using the conformal Gaussian gauge

This section aims to introduce a hyperbolic reduction procedure of the ECFEs
using the so-called conformal Gaussian gauge based on conformal geodesics,
following the discussion of Chapter 13 in [28].

In the following, let $\tilde{\mathcal{S}}\equiv\\{\tau=0\\}$ denote a
spacelike hypersurface on the physical spacetime
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ and let

${\bm{e}}_{{\bm{a}}_{\star}}={\bm{e}}_{{\bm{a}}}|_{\tilde{\mathcal{S}}},\quad{\bm{\beta}}_{\star}={\bm{\beta}}|_{\tilde{\mathcal{S}}},$
(49)

denote a smooth initial data specified at each point
$x_{\star}\in\tilde{\mathcal{S}}$. Furthermore, assume that
$(x(\tau),{\bm{\beta}}(\tau))|_{x_{\star}}$ is the unique solution to eqs.
(26) passing through $x_{\star}\in\tilde{\mathcal{S}}$ such that

$x(0)=x_{\star},\qquad\dot{{\bm{x}}}(0)={\bm{e}}_{{\bm{0}}\star},\qquad{\bm{\beta}}(0)={\bm{\beta}}_{\star}.$

By varying $x_{\star}$, it can be shown that one obtains a smooth caustic free
congruence of conformal geodesics in a _small_ neighbourhood
$\tilde{\mathcal{U}}$ of $\tilde{\mathcal{S}}$.

######  Remark 5.

In subsequent calculations, it will be assumed that
$(x(\tau),{\bm{\beta}}(\tau))$ is a congruence of conformal geodesics
satisfying eqs. (26) with initial data eq. (49) specified on
$\tilde{\mathcal{S}}$.

The solution to the conformal geodesic equations allows us to fix the gauge
freedom in the ECFEs associated with the choice of $\hat{{\bm{\nabla}}}$. In
particular, one can introduce the Weyl connection $\hat{{\bm{\nabla}}}$ as

$\hat{{\bm{\nabla}}}-\tilde{{\bm{\nabla}}}={\bm{S}}({\bm{\beta}}).$

As mentioned in Section 2.3, the Weyl connection introduces a Weyl propagated
frame field $\\{{\bm{e}}_{{\bm{a}}}\\}$ satisfying eq. (31). Moreover, one can
obtain a canonical conformal factor $\Xi$ by imposing conditions eq. (32).
Applying $\tilde{\nabla}_{\dot{{\bm{x}}}}$ to eq. (32) and using the conformal
geodesic equations, one can show that $\Xi$ satisfies the evolution equation

$\hat{\nabla}_{\dot{{\bm{x}}}}\Xi=\Xi\langle{\bm{\beta}},\dot{{\bm{x}}}\rangle.$
(50)

Given $\\{{\bm{e}}_{{\bm{a}}}\\}$ and $\Xi$ that satisfy the above mentioned
equations and if the frame $\\{{\bm{e}}_{{\bm{a}}}\\}$ is adapted to the
conformal geodesics so that $\dot{{\bm{x}}}={\bm{e}}_{{\bm{0}}}$, then the
conformal metric ${\bm{g}}\equiv\Xi^{2}\tilde{{\bm{g}}}$ satisfies

${\bm{g}}({\bm{e}}_{{\bm{a}}},{\bm{e}}_{{\bm{b}}})=\eta_{{\bm{a}}{\bm{b}}}.$

Finally, given local coordinates $(x^{\alpha})$ on $\tilde{\mathcal{S}}$ and
setting $x^{0}=\tau$, the local coordinates on $\tilde{\mathcal{S}}$ can be
dragged along the conformal geodesics to obtain a smooth coordinate system
$(x^{\mu})\equiv(x^{0},x^{\alpha})$ on $\tilde{\mathcal{U}}$. In this gauge,
one can show that

$\dot{{\bm{x}}}={\bm{e}}_{{\bm{0}}}=\partial_{\tau},\quad\hat{\Gamma}_{{\bm{0}}}{}^{{\bm{a}}}{}_{{\bm{b}}}=0,\quad\hat{L}_{{\bm{0}}{\bm{a}}}=0\qquad\text{
on }\tilde{\mathcal{U}},$ (51)

and that the conformal factor $\Xi$ and 1-form ${\bm{d}}\equiv\Xi{\bm{\beta}}$
can be expressed in terms of the initial data as:

$\displaystyle\Xi(\tau)=\Xi|_{\tilde{\mathcal{S}}}\left(1+\langle{\bm{\beta}}_{\star},{\bm{e}}_{{\bm{0}}\star}\rangle\tau+\frac{1}{4}\tilde{{\bm{g}}}^{\sharp}({\bm{\beta}}_{\star},{\bm{\beta}}_{\star})\tau^{2}\right),$
(52a) $\displaystyle d_{{\bm{0}}}=\dot{\Xi}=\partial_{\tau}\Xi,\quad
d_{{\bm{a}}}=\Xi|_{\tilde{\mathcal{S}}}\langle{\bm{\beta}}_{\star},{\bm{e}}_{{\bm{0}}\star}\rangle\qquad\text{for
}\quad{\bm{a}}={\bm{1}},{\bm{2}},{\bm{3}}.$ (52b)

This choice of coordinates, frame fields and conformal factor will be known as
the conformal Gaussian gauge system.

To make use of this gauge in the ECFEs, let $\varphi$ denote an embedding map
$\varphi:\tilde{\mathcal{S}}\rightarrow\tilde{\mathcal{M}}$ so that
$\tilde{\mathcal{S}}$ is a submanifold of $\tilde{\mathcal{M}}$ and let $\phi$
denote the conformal transformation map
$\phi:\tilde{\mathcal{M}}\rightarrow\mathcal{M}$ so that the metric
${\bm{g}}\equiv\Xi^{2}\tilde{{\bm{g}}}$ on $\mathcal{M}$ satisfies eq. (32).
Then, the map $\phi\circ\varphi:\tilde{\mathcal{S}}\rightarrow\mathcal{M}$ is
also an embedding so that $\tilde{\mathcal{S}}$ can be considered as a
submanifold of $\mathcal{M}$. If $\tilde{{\bm{h}}}$ is the metric induced by
$\tilde{{\bm{g}}}$ on $\tilde{\mathcal{S}}$ and ${\bm{h}}$ is the metric
induced by ${\bm{g}}$ on $\tilde{\mathcal{S}}$, one has

${\bm{h}}=\Omega^{2}\tilde{{\bm{h}}},$ (53)

where $\Omega=\Xi|_{\tilde{\mathcal{S}}}$. Given the smooth coordinate system
$(x^{\mu})$ on $\tilde{\mathcal{U}}$ and if ${\bm{e}}^{\mu}{}_{{\bm{a}}}$
denote the coefficients of the frame field ${\bm{e}}_{{\bm{a}}}$ with respect
to the coordinate system $(x^{\mu})$ i.e.
${\bm{e}}^{\mu}{}_{{\bm{a}}}=\langle{\bm{d}}{x}^{\mu},{\bm{e}}_{{\bm{a}}}\rangle$,
then using the gauge conditions eq. (51), the evolution equations for
${\bm{e}}^{\mu}{}_{{\bm{a}}},\hat{\Gamma}_{{\bm{a}}}{}^{{\bm{b}}}{}_{{\bm{c}}}$
and $\hat{L}_{{\bm{a}}{\bm{b}}}$ can be written as

$\displaystyle\partial_{\tau}{\bm{e}}^{\mu}{}_{{\bm{b}}}=-\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{c}}}{}_{{\bm{0}}}{\bm{e}}^{\mu}{}_{{\bm{c}}},$
(54a)
$\displaystyle\partial_{\tau}\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{c}}}{}_{{\bm{d}}}=-\hat{\Gamma}_{{\bm{f}}}{}^{{\bm{c}}}{}_{{\bm{d}}}\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{f}}}{}_{{\bm{0}}}+\delta_{{\bm{0}}}{}^{{\bm{c}}}\hat{L}_{{\bm{b}}{\bm{d}}}+\delta_{{\bm{d}}}{}^{{\bm{c}}}\hat{L}_{{\bm{b}}{\bm{0}}}+\eta_{{\bm{0}}{\bm{d}}}\eta^{{\bm{f}}{\bm{c}}}\hat{L}_{{\bm{b}}{\bm{f}}}+\Xi
d^{{\bm{c}}}{}_{{\bm{d}}{\bm{0}}{\bm{b}}},$ (54b)
$\displaystyle\partial_{\tau}\hat{L}_{{\bm{b}}{\bm{c}}}=-\hat{\Gamma}_{{\bm{b}}}{}^{{\bm{f}}}{}_{{\bm{0}}}\hat{L}_{{\bm{f}}{\bm{c}}}+d_{{\bm{f}}}d^{{\bm{f}}}{}_{{\bm{c}}{\bm{0}}{\bm{b}}}.$
(54c)

Note that these equations contain only derivatives with respect to $\tau$, so
they can be considered as a transport system along the conformal geodesics
$(x(\tau),{\bm{\beta}}(\tau))$. If $n_{a}$ is the ${\bm{g}}$-unit normal to
$\tilde{\mathcal{S}}$, the rescaled Weyl tensor
$d^{{\bm{a}}}{}_{{\bm{b}}{\bm{c}}{\bm{d}}}$ can be decomposed in terms of the
electric part $E_{{\bm{a}}{\bm{b}}}$ and magnetic part $B_{{\bm{a}}{\bm{b}}}$
defined as

$E_{{\bm{a}}{\bm{b}}}\equiv
d_{{\bm{a}}{\bm{c}}{\bm{b}}{\bm{d}}}n^{{\bm{c}}}n^{{\bm{d}}},\qquad
B_{{\bm{a}}{\bm{b}}}=(*d)_{{\bm{a}}{\bm{c}}{\bm{b}}{\bm{d}}}n^{{\bm{c}}}n^{{\bm{d}}},$

where $(*d)_{{\bm{a}}{\bm{c}}{\bm{b}}{\bm{d}}}$ are the components of the left
Hodge dual of $d_{acbd}$. Using this decomposition, one can show that
$E_{{\bm{a}}{\bm{b}}}$ and $B_{{\bm{a}}{\bm{b}}}$ satisfy

$\displaystyle\partial_{\tau}(E_{{\bm{b}}{\bm{d}}})+D_{{\bm{a}}}B_{{\bm{c}}({\bm{b}}}\epsilon_{{\bm{d}})}{}^{{\bm{a}}{\bm{c}}}+2a_{{\bm{a}}}\epsilon^{{\bm{a}}{\bm{c}}}{}_{({\bm{b}}}B_{{\bm{d}}){\bm{c}}}-3\chi_{({\bm{b}}}{}^{{\bm{c}}}E_{{\bm{d}}){\bm{c}}}-\epsilon_{{\bm{b}}}{}^{{\bm{a}}{\bm{c}}}\epsilon_{{\bm{d}}}{}^{{\bm{e}}{\bm{f}}}E_{{\bm{a}}{\bm{c}}}\chi_{{\bm{e}}{\bm{f}}}+\chi
E_{{\bm{b}}{\bm{d}}}=0,$
$\displaystyle\partial_{\tau}(B_{{\bm{b}}{\bm{d}}})-D_{{\bm{a}}}E_{{\bm{c}}({\bm{b}}}\epsilon_{{\bm{d}})}{}^{{\bm{a}}{\bm{c}}}-2a_{{\bm{a}}}\epsilon^{{\bm{a}}{\bm{c}}}{}_{({\bm{b}}}E_{{\bm{d}}){\bm{c}}}-3\chi^{{\bm{a}}}{}_{({\bm{b}}}B_{{\bm{d}}){\bm{a}}}-\epsilon_{{\bm{b}}}{}^{{\bm{a}}{\bm{c}}}\epsilon_{{\bm{d}}}{}^{{\bm{e}}{\bm{f}}}B_{{\bm{a}}{\bm{c}}}\chi_{{\bm{e}}{\bm{f}}}+\chi
B_{{\bm{b}}{\bm{d}}}=0,$

where

$h_{{\bm{a}}{\bm{b}}}\equiv
g_{{\bm{a}}{\bm{b}}}-\tau_{{\bm{a}}}\tau_{{\bm{b}}},\quad\chi_{{\bm{a}}{\bm{b}}}\equiv
h_{{\bm{a}}}{}^{{\bm{c}}}\nabla_{{\bm{c}}}\tau_{{\bm{b}}},\quad\chi\equiv
h^{{\bm{a}}{\bm{b}}}\chi_{{\bm{a}}{\bm{b}}},\quad
a_{{\bm{a}}}\equiv\tau^{{\bm{b}}}\nabla_{{\bm{b}}}\tau_{{\bm{a}}},$

and $\tau_{{\bm{a}}}=\sqrt{2}n_{{\bm{a}}}$.

####  3.1.1 Spinors formulation of the extended conformal field equations

The ECFEs obtained from the hyperbolic reduction procedure discussed in the
previous section can be expressed in terms of spinors. In particular, if
$\tau^{AA^{\prime}}$ denotes the spinor counterpart of the vector $\tau^{a}$
and $\\{{\bm{\epsilon}}_{{\bm{A}}}{}^{A}\\}$ denotes the spin frame introduced
in Section 3 so that $\tau^{AA^{\prime}}$ is given by eq. (35), then the
evolution equations for
${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}},\hat{\Gamma}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}$,
$\hat{L}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}$ and
$d_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
can be written as

$\displaystyle\tau^{{\bm{A}}{\bm{A}}^{\prime}}\hat{\Sigma}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}}=0,\qquad\tau^{{\bm{C}}{\bm{C}}^{\prime}}\hat{\Xi}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}=0,$
(56a)
$\displaystyle\tau^{{\bm{A}}{\bm{A}}^{\prime}}\hat{\Delta}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}}=0,\qquad\tau_{({\bm{A}}}{}^{{\bm{A}}^{\prime}}\hat{\Lambda}_{|{\bm{A}}^{\prime}|{\bm{B}}{\bm{C}}{\bm{D}})}=0,$
(56b)

where we choose to omit the explicit form of these equations — see [28] for a
detailed expression of these equations.

In the following, let
$\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}},f_{{\bm{A}}{\bm{B}}}$ and
$\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ denote the components of the
space-spinor counterparts of $\hat{{\bm{\nabla}}}$-spin connection
coefficients, the 1-form ${\bm{f}}$ and the $\hat{{\bm{\nabla}}}$-Schouten
tensor, as introduced in Section 3, i.e.,

$\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}\hat{\Gamma}_{{\bm{A}}){\bm{A}}^{\prime}{\bm{C}}{\bm{D}}},\qquad\;f_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}f_{{\bm{A}}){\bm{A}}^{\prime}},\qquad\;\Theta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\tau_{{\bm{B}}}{}^{{\bm{A}}^{\prime}}\tau_{{\bm{D}}}{}^{{\bm{C}}^{\prime}}\hat{L}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}}.$

Note that $\hat{\Gamma}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ can be further
decomposed in terms of $\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ and
$\chi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ using eq. (44) while the spinorial
counterpart of the rescaled Weyl tensor
$d_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
can be decomposed in terms of the rescaled Weyl spinor
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ as follows:

$d_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}=-\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\epsilon_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}\epsilon_{{\bm{C}}^{\prime}{\bm{D}}^{\prime}}-\bar{\phi}_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}{\bm{C}}^{\prime}{\bm{D}}^{\prime}}\epsilon_{{\bm{A}}{\bm{B}}}\epsilon_{{\bm{C}}{\bm{D}}}.$
(57)

Now, define the electric and magnetic parts of the rescaled Weyl spinor
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ as

$\eta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\frac{1}{2}\left(\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}+\phi^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}},\right),\qquad\mu_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv-\frac{i}{2}\left(\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}-\phi^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\right),$

where the + sign denotes the Hermitian conjugation, so that

$\phi^{+}_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\equiv\tau_{{\bm{A}}}{}^{{\bm{A}}^{\prime}}\tau_{{\bm{B}}}{}^{{\bm{B}}^{\prime}}\tau_{{\bm{C}}}{}^{{\bm{C}}^{\prime}}\tau_{{\bm{D}}}{}^{{\bm{D}}^{\prime}}\bar{\phi}_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}{\bm{C}}^{\prime}{\bm{D}}^{\prime}}.$

In terms of the above-mentioned fields, the evolution equations for

$({\bm{e}}^{0}{}_{{\bm{A}}{\bm{B}}},{\bm{e}}^{\alpha}{}_{{\bm{A}}{\bm{B}}},\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}},f_{{\bm{A}}{\bm{B}}},\chi_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}},\Theta_{{\bm{C}}{\bm{D}}({\bm{A}}{\bm{B}})},\Theta_{{\bm{A}}{\bm{B}}{\bm{Q}}}{}^{{\bm{Q}}})$

can be written as

$\displaystyle\partial_{\tau}{\bm{e}}^{0}{}_{{\bm{A}}{\bm{B}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}{\bm{e}}^{0}{}_{{\bm{P}}{\bm{Q}}}-f_{{\bm{A}}{\bm{B}}},$
(58a)
$\displaystyle\partial_{\tau}{\bm{e}}^{\alpha}{}_{{\bm{A}}{\bm{B}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}{\bm{e}}^{\alpha}{}_{{\bm{P}}{\bm{Q}}},$
(58b)
$\displaystyle\partial_{\tau}\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}\xi_{{\bm{P}}{\bm{Q}}{\bm{C}}{\bm{D}}}+\frac{1}{\sqrt{2}}\left(\epsilon_{{\bm{A}}{\bm{C}}}\chi_{({\bm{B}}{\bm{D}}){\bm{P}}{\bm{Q}}}+\epsilon_{{\bm{B}}{\bm{D}}}\chi_{({\bm{A}}{\bm{C}}){\bm{P}}{\bm{Q}}}\right)f^{{\bm{P}}{\bm{Q}}}$
(58c)
$\displaystyle\phantom{\partial_{\tau}\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=}-\sqrt{2}\chi_{({\bm{A}}{\bm{B}})({\bm{C}}}{}^{{\bm{E}}}f_{{\bm{D}}){\bm{E}}}-\frac{1}{2}\left(\epsilon_{{\bm{A}}{\bm{C}}}\Theta_{{\bm{B}}{\bm{D}}{\bm{Q}}}{}^{{\bm{Q}}}+\epsilon_{{\bm{B}}{\bm{D}}}\Theta_{{\bm{A}}{\bm{C}}{\bm{Q}}}{}^{{\bm{Q}}}\right)-i\Xi\mu_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}},$
$\displaystyle\partial_{\tau}f_{{\bm{A}}{\bm{B}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}f_{{\bm{P}}{\bm{Q}}}+\frac{1}{\sqrt{2}}\Theta_{{\bm{A}}{\bm{B}}{\bm{Q}}}{}^{{\bm{Q}}},$
(58d)
$\displaystyle\partial_{\tau}\chi_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}\chi_{{\bm{P}}{\bm{Q}}{\bm{C}}{\bm{D}}}-\Theta_{{\bm{A}}{\bm{B}}({\bm{C}}{\bm{D}})}+\Xi\eta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}},$
(58e)
$\displaystyle\partial_{\tau}\Theta_{{\bm{C}}{\bm{D}}({\bm{A}}{\bm{B}})}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{P}}{\bm{Q}}}\Theta_{{\bm{P}}{\bm{Q}}({\bm{A}}{\bm{B}})}-\partial_{\tau}\Xi\eta_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}+i\sqrt{2}d^{{\bm{P}}}{}_{({\bm{A}}}\mu_{{\bm{B}}){\bm{C}}{\bm{D}}{\bm{P}}},$
(58f)
$\displaystyle\partial_{\tau}\Theta_{{\bm{A}}{\bm{B}}{\bm{Q}}}{}^{{\bm{Q}}}=-\chi_{({\bm{A}}{\bm{B}})}{}^{{\bm{E}}{\bm{F}}}\Theta_{{\bm{E}}{\bm{F}}{\bm{Q}}}{}^{{\bm{Q}}}+\sqrt{2}d^{{\bm{P}}{\bm{Q}}}\eta_{{\bm{A}}{\bm{B}}{\bm{P}}{\bm{Q}}}.$
(58g)

In the above,
$d_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{B}}}{}^{{\bm{A}}^{\prime}}d_{{\bm{A}}){\bm{A}}^{\prime}}$
is the space-spinor counterpart of the 1-form ${\bm{d}}$ introduced in the
previous section. Note that the conformal factor $\Xi$, its derivatives
$\partial_{\tau}\Xi$ and $d_{{\bm{A}}{\bm{B}}}$ can be expressed in terms of
the initial data as discussed earlier —see eqs. (52).

Finally, let $\mathcal{D}_{{\bm{A}}{\bm{B}}}$ and $\mathcal{P}$ denote the Sen
connection and the Fermi derivative associated with the Levi-Civita connection
${\bm{\nabla}}$, so that

$\mathcal{D}_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{A}}}{}^{{\bm{A}}^{\prime}}\nabla_{{\bm{B}}){\bm{A}}^{\prime}},\qquad\;\mathcal{P}\equiv\tau^{{\bm{A}}{\bm{A}}^{\prime}}\nabla_{{\bm{A}}{\bm{A}}^{\prime}},$

then the so-called boundary-adapted evolution system for
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ can be written as

$-2P_{{\bm{0}}{\bm{0}}{\bm{0}}{\bm{0}}}=0,\qquad\;-2P_{{\bm{0}}{\bm{0}}{\bm{0}}{\bm{1}}}-\frac{1}{2}C_{{\bm{0}}{\bm{0}}}=0,\qquad\;-2P_{{\bm{0}}{\bm{0}}{\bm{1}}{\bm{1}}}=0,$
(59a)
$-2P_{{\bm{0}}{\bm{1}}{\bm{1}}{\bm{1}}}+\frac{1}{2}C_{{\bm{1}}{\bm{1}}}=0,\qquad\;-2P_{{\bm{1}}{\bm{1}}{\bm{1}}{\bm{1}}}=0.$
(59b)

where

$P_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=-\frac{1}{2}\left(\mathcal{P}\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}-\mathcal{D}_{({\bm{D}}}{}^{{\bm{F}}}\phi_{{\bm{A}}{\bm{B}}{\bm{C}}){\bm{F}}}\right),\qquad\;C_{{\bm{A}}{\bm{B}}}=\mathcal{D}^{{\bm{E}}{\bm{F}}}\phi_{{\bm{A}}{\bm{B}}{\bm{E}}{\bm{F}}}.$

The constraint equations for $\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ can be
written as

$C_{{\bm{A}}{\bm{B}}}=0.$ (60)

Given the evolution equations for the background fields
${\bm{e}}^{0}{}_{{\bm{A}}{\bm{B}}}$, ${\bm{e}}^{\alpha}{}_{{\bm{A}}{\bm{B}}}$,
$\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$, $f_{{\bm{A}}{\bm{B}}}$,
$\chi_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}}$,
$\Theta_{{\bm{C}}{\bm{D}}({\bm{A}}{\bm{B}})}$,
$\Theta_{{\bm{A}}{\bm{B}}{\bm{Q}}}{}^{{\bm{Q}}}$ and the boundary-adapted
evolution and constraint equations for the rescaled Weyl tensor
$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$, the next step is to obtain a scalar
version of these equations that can be solved for the components of the
background fields and the rescaled Weyl tensor.

####  3.1.2 Scalarising the extended conformal field equations

To obtain a scalar version of the extended conformal field equations, we
consider the irreducible decomposition of spinors with two or four unprimed
indices.

In the following, let $\\{{\bm{\epsilon}}_{{\bm{A}}}{}^{A}\\}$ denote some
arbitrary spin frame and introduce the primary spinors
$x_{AB},y_{AB},z_{AB},\epsilon^{s}_{ABCD}$ and $h_{ABCD}$ as

$\displaystyle
x_{AB}=\sqrt{2}{\bm{\epsilon}}^{{\bm{0}}}{}_{(A}{\bm{\epsilon}}^{{\bm{1}}}{}_{B)},\qquad\;y_{AB}=-\frac{1}{\sqrt{2}}{\bm{\epsilon}}^{{\bm{1}}}{}_{(A}{\bm{\epsilon}}^{{\bm{1}}}{}_{B)},\qquad\;z_{AB}=\frac{1}{\sqrt{2}}{\bm{\epsilon}}^{{\bm{0}}}{}_{(A}{\bm{\epsilon}}^{{\bm{0}}}{}_{B)},\hskip
28.45274pt$ (61a)
$\displaystyle\epsilon^{s}_{ABCD}={\bm{\epsilon}}^{({\bm{A}}}{}_{(A}{\bm{\epsilon}}^{{\bm{B}}}{}_{B}{\bm{\epsilon}}^{{\bm{C}}}{}_{C}{\bm{\epsilon}}^{{\bm{D}})_{s}}{}_{D)},\qquad\;h_{ABCD}=-\epsilon_{A(C}\epsilon_{D)B},$
(61b)

where the indices with a bracket are symmetrised, and the lower index $s\leq
4$ is used to indicate that $s$ of the indices are set to equal ${\bm{1}}$
while the remaining are set to equal ${\bm{0}}$. The $\epsilon$-spinor can be
written in terms of $\\{{\bm{\epsilon}}_{{\bm{A}}}{}^{A}\\}$ as

$\epsilon_{AB}={\bm{\epsilon}}^{{\bm{0}}}{}_{A}{\bm{\epsilon}}^{{\bm{1}}}{}_{B}-{\bm{\epsilon}}^{{\bm{0}}}{}_{B}{\bm{\epsilon}}^{{\bm{1}}}{}_{A}.$

The spinors $\epsilon^{s}_{ABCD}$ can be written in terms of $x_{AB},y_{AB}$
and $z_{AB}$ as

$\displaystyle\epsilon^{0}_{ABCD}=\frac{2}{3}z_{AD}z_{BC}+\frac{2}{3}z_{AC}z_{BD}+\frac{2}{3}z_{AB}z_{CD},$
$\displaystyle\epsilon^{1}_{ABCD}=\frac{1}{6}x_{CD}z_{AB}+\frac{1}{6}x_{BD}z_{AC}+\frac{1}{6}x_{BC}z_{AD}+\frac{1}{6}x_{AD}z_{BC}+\frac{1}{6}x_{AC}z_{BD}+\frac{1}{6}x_{AB}z_{CD},$
$\displaystyle\epsilon^{2}_{ABCD}=\frac{1}{6}x_{AD}x_{BC}+\frac{1}{6}x_{AC}x_{BD}+\frac{1}{6}x_{AB}x_{CD},$
$\displaystyle\epsilon^{3}_{ABCD}=-\frac{1}{6}x_{CD}y_{AB}-\frac{1}{6}x_{BD}y_{AC}-\frac{1}{6}x_{BC}y_{AD}-\frac{1}{6}x_{AD}y_{BC}-\frac{1}{6}x_{AC}y_{BD}-\frac{1}{6}x_{AB}y_{CD},$
$\displaystyle\epsilon^{4}_{ABCD}=\frac{2}{3}y_{AD}y_{BC}+\frac{2}{3}y_{AC}y_{BD}+\frac{2}{3}y_{AB}y_{CD}.$

The primary spinors satisfy a number of useful identities —see appendix in
[40] for a full list. For example, the spinors $x_{AB},y_{AB}$ and $z_{AB}$
satisfy

$\displaystyle x_{AB}x^{AB}=-1,\qquad x_{AB}y^{AB}=0,\qquad x_{AB}z^{AB}=0,$
$\displaystyle y_{AB}y^{AB}=0,\qquad y_{AB}z^{AB}=-\frac{1}{2}\qquad
z_{AB}z^{AB}=0.$

An arbitrary spinor with 2 unprimed indices $l_{AB}$ can be decomposed as

$l_{AB}=l_{x}x_{AB}+l_{y}y_{AB}+l_{z}z_{AB},$

where

$l_{x}=-l_{AB}x^{AB},\qquad\;l_{y}=-2l_{AB}z^{AB},\qquad\;l_{z}=-2l_{AB}y^{AB}.$

Similarly, an arbitrary spinor with 4 unprimed indices $S_{ABCD}$ can be
written as

$\displaystyle
S_{ABCD}=S_{0}\epsilon^{0}_{ABCD}+S_{1}\epsilon^{1}_{ABCD}+S_{2}\epsilon^{2}_{ABCD}+S_{3}\epsilon^{3}_{ABCD}+S_{4}\epsilon^{4}_{ABCD}+S_{h}h_{ABCD}$
$\displaystyle\phantom{S_{ABCD}}+S_{x}(x_{BD}\epsilon_{AC}+x_{AC}\epsilon_{BD})+S_{y}(y_{BD}\epsilon_{AC}+y_{AC}\epsilon_{BD})+S_{z}(z_{BD}\epsilon_{AC}+z_{AC}\epsilon_{BD}).$

For totally symmetric spinors, $T_{ABCD}=T_{(ABCD)}$, the irreducible
decomposition can be written as

$T_{ABCD}=T_{0}\epsilon^{0}_{ABCD}+T_{1}\epsilon^{1}_{ABCD}+T_{2}\epsilon^{2}_{ABCD}+T_{3}\epsilon^{3}_{ABCD}+T_{4}\epsilon^{4}_{ABCD}.$

Given the above-mentioned decomposition, we can list the components of the
spinors appearing in the evolution eqs. (58) for the background fields as

$\displaystyle{\bm{e}}^{0}{}_{{\bm{A}}{\bm{B}}}\rightarrow\left({\bm{e}}^{0}_{x},{\bm{e}}^{0}_{y},{\bm{e}}^{0}_{z}\right),$
$\displaystyle{\bm{e}}^{\alpha}{}_{{\bm{A}}{\bm{B}}}\rightarrow\left({\bm{e}}^{1}_{x},{\bm{e}}^{1}_{y},{\bm{e}}^{1}_{z},{\bm{e}}^{2}_{x},{\bm{e}}^{2}_{y},{\bm{e}}^{2}_{z},{\bm{e}}^{3}_{x},{\bm{e}}^{3}_{y},{\bm{e}}^{3}_{z}\right),$
$\displaystyle\xi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\rightarrow\left(\xi_{0},\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{h},\xi_{x},\xi_{y},\xi_{z}\right),$
$\displaystyle f_{{\bm{A}}{\bm{B}}}\rightarrow\left(f_{x},f_{y},f_{z}\right),$
$\displaystyle\chi_{({\bm{A}}{\bm{B}}){\bm{C}}{\bm{D}}}\rightarrow\left(\chi_{0},\chi_{1},\chi_{2},\chi_{3},\chi_{4},\chi_{h},\chi_{x},\chi_{y},\chi_{z}\right),$
$\displaystyle\Theta_{{\bm{A}}{\bm{B}}({\bm{C}}{\bm{D}})}\rightarrow\left(\Theta_{0},\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4},\Theta_{h},\Theta_{x},\Theta_{y},\Theta_{z}\right),$
$\displaystyle\Theta_{{\bm{A}}{\bm{B}}}\equiv\Theta_{{\bm{A}}{\bm{B}}{\bm{Q}}}{}^{{\bm{Q}}}\rightarrow\left(\theta_{x},\theta_{y},\theta_{z}\right).$

It is possible to obtain a scalarised version of the evolution eqs. (58) by
various contractions with the primary spinors. In total, one obtains 45
equations for the 45 components listed above. In our analysis, the xAct
package [47] for Wolfram language was used to obtain an explicit form of the
scalarised equations. These will not be listed here for obvious reasons.

Given that $\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ is totally symmetric, the
components involved in the boundary-adapted evolution and constraint equations
can be listed as follows:

$\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}\rightarrow\left(\phi_{0},\phi_{1},\phi_{2},\phi_{3},\phi_{4}\right).$

In terms of the above, the evolution eqs. (59b) and the constraint eqs. (60)
for $\phi_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}$ consists of 5 and 3 equations,
respectively, to be solved for the components
$\phi_{0},\phi_{1},\phi_{2},\phi_{3}$ and $\phi_{4}$ —see Appendix A.

##  4 The Newman-Penrose gauge

As mentioned in the introduction, the BMS-asymptotic charges used in this work
are expressed in terms of the NP-gauge. Before introducing the expressions of
the charges, we will discuss the NP-gauge conditions and the general relation
between the NP-gauge frame with the F-gauge frame. As will become evident, the
main distinction between the NP-gauge and the F-gauge is that the former is
adapted to null infinity $\mathscr{I}^{\pm}$ while the latter is adapted to
Cauchy hypersurfaces.

In this section, the focus will be on the gauge conditions satisfied by the
NP-gauge at future null infinity $\mathscr{I}^{+}$. Similar conditions can be
formulated at past null infinity $\mathscr{I}^{-}$. Following the discussion
in [40], introduce the conformal metric ${\bm{g}}^{\bullet}$ related to
${\bm{g}}$ by

${\bm{g}}^{\bullet}=\theta^{2}{\bm{g}}.$ (66)

On a neighbourhood $\mathcal{U}\subset\mathcal{M}$ of $\mathscr{I}^{+}$,
introduce the smooth adapted frame
$\\{{\bm{e}}^{\circ}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ satisfying the following
conditions:

  1. i.

The frame field ${\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}$ is tangent to
and parallelly propagated along $\mathscr{I}^{+}$. Hence,

$\nabla^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}{\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}\simeq
0,$

where $\simeq$ is used to denote equality on $\mathscr{I}^{+}$.

  2. ii.

On $\mathcal{U}$, there exists a smooth function $u^{\circ}$ that induces an
affine parameter on the null generators of $\mathscr{I}^{+}$ such that

${\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}(u^{\circ})\simeq 1.$

  3. iii.

The frame ${\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}}$ is tangent to the
hypersurfaces transverse to $\mathscr{I}^{+}$ (on which
$u^{\circ}=\text{constant}$), i.e.,

${\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}}={\bm{g}}({\bm{d}}{u^{\circ}},\cdot).$

  4. iv.

The frame fields ${\bm{e}}^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}}$ and
${\bm{e}}^{\circ}_{{\bm{1}}{\bm{0}}^{\prime}}$ are tangent to the slices
$\\{u^{\circ}=\text{constant}\\}\cap\mathscr{I}^{+}$. These frame fields as
well as ${\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}}$ and
${\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}$ are parallelly propagated in
the direction of ${\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}}$.

Let $(x^{\mu})$ denote a local coordinate system on $\mathcal{U}$, then the
above gauge conditions can be expressed in terms of the spin connection
coefficients
$\Gamma^{\circ}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{B}}}{}_{{\bm{C}}}$, which
can be written in terms of the components of
${\bm{e}}^{\circ}_{{\bm{A}}{\bm{A}}^{\prime}}$ with respect to $(x^{\mu})$ as

$\Gamma^{\circ}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{C}}}=\frac{1}{2}\left({\bm{e}}^{\circ\mu}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}^{\circ\nu}{}_{{\bm{B}}{\bm{1}}^{\prime}}\nabla_{\mu}{\bm{e}}^{\circ}_{\nu{\bm{C}}{\bm{0}}^{\prime}}+{\bm{e}}^{\circ\mu}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}^{\circ\nu}{}_{{\bm{C}}{\bm{1}}^{\prime}}\nabla_{\mu}{\bm{e}}^{\circ}_{\nu{\bm{B}}{\bm{0}}^{\prime}}\right).$

Then, the gauge conditions are given by

$\displaystyle\Gamma^{\circ}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{1}}{\bm{1}}}\simeq
0,\qquad\Gamma^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}{\bm{1}}{\bm{1}}}\simeq 0,$
$\displaystyle\Gamma^{\circ}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}=\bar{\Gamma}^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{0}}^{\prime}{\bm{0}}^{\prime}},\qquad\Gamma^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}{\bm{0}}{\bm{0}}}=\bar{\Gamma}^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{0}}^{\prime}{\bm{1}}^{\prime}}+\Gamma^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{0}}{\bm{1}}},\qquad\Gamma^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}{\bm{B}}{\bm{C}}}=0\qquad\text{
on }\mathcal{U}.$

To introduce the NP-gauge frame
$\\{{\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ and the NP-gauge
conditions, consider the conformal rescaling ${\bm{g}}\to{\bm{g}}^{\bullet}$
where ${\bm{g}}^{\bullet}$ is defined by eq. (66). Given $\theta>0$ and an
arbitrary function $p>0$ which is constant on the generators of
$\mathscr{I}^{+}$, the NP-gauge frame
$\\{{\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ can be introduced as
follows: let
${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}\simeq\theta^{-2}p{\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}$
and introduce the affine parameter $u^{\bullet}$ along the generators of
$\mathscr{I}^{+}$ as follows

$u^{\bullet}(u^{\circ})=\int^{u^{\circ}}_{u^{\circ}_{\star}}\theta^{2}(u^{\prime})p^{-1}(u^{\prime})du^{\prime}+u^{\bullet}_{\star}\qquad\text{on
}\mathscr{I}^{+}.$

Hence, the frame ${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}$ is
parallelly propagated along $\mathscr{I}^{+}$ and it satisfies
${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(u^{\bullet})\simeq 0$. If
$\mathcal{C}$ denotes a cut on $\mathscr{I}^{+}$, which is diffeomorphic to
$\mathbb{S}^{2}$, then the coordinates $(\vartheta,\varphi)$ on $\mathcal{C}$
and the conformal factor $\theta$ can be fixed such that the metric on
$\mathcal{C}$ is the standard metric on $\mathbb{S}^{2}$.

In the following, assume that $u^{\circ}=u^{\circ}_{\star}$ and
$u^{\bullet}=u^{\bullet}_{\star}$ on $\mathcal{C}$ and set

${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}=p^{-1}{\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}},\qquad{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}=\theta^{-2}p{\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}},\qquad{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}=\theta^{-1}{\bm{e}}^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}}\qquad\text{on
}\mathcal{C}.$

Given that ${\bm{g}}=\Theta^{2}\tilde{{\bm{g}}}$, the transformation
${\bm{g}}\mapsto{\bm{g}}^{\bullet}$ implies
$\Theta\mapsto\Theta^{\bullet}=\theta\Theta$ and the transformation laws for
the spin connection coefficients can be written on $\mathcal{C}$ as

$\displaystyle\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}=p^{-1}\left(\Gamma^{\circ}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}-{\bm{e}}^{\circ}_{{\bm{0}}{\bm{0}}^{\prime}}(\log(\theta))\right),$
$\displaystyle\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{1}}{\bm{1}}}=p\theta^{-2}\left(\Gamma^{\circ}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{1}}{\bm{1}}}+{\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}(\log(\theta))\right).$

Thus, with a suitable choice of $d\theta$ and $p$, one can achieve the
following

$\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}=0,\qquad\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{1}}{\bm{1}}}=0,\qquad{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Theta^{\bullet})=\text{const.}\neq
0\qquad\qquad\text{ on }\mathcal{C}.$ (68)

The conformal rescaling given by eq. (66) implies a transformation of the
trace-free part of the ${\bm{g}}$-Ricci tensor, denoted by $s_{ab}$. In
particular, one has that

$s^{\bullet}_{\mu\nu}=s_{\mu\nu}-\frac{2}{\theta}\left(\nabla_{\mu}\nabla_{\nu}\theta-\frac{2}{\theta}\nabla_{\mu}\theta\nabla_{\nu}\theta-\frac{1}{4}g_{\mu\nu}(\nabla_{\lambda}\nabla^{\lambda}\theta-\frac{2}{\theta}\nabla_{\lambda}\theta\nabla^{\lambda}\theta)\right),$

where $s_{\mu\nu}$ are the components of ${\bm{s}}$ with respect to
$(x^{\mu})$. Now, define $\Phi^{\circ}_{22}$ and $\Phi^{\bullet}_{22}$ as

$\Phi^{\circ}_{22}\equiv\frac{1}{2}s_{\mu\nu}{\bm{e}}^{\circ\mu}{}_{{\bm{1}}{\bm{1}}^{\prime}}{\bm{e}}^{\circ\nu}{}_{{\bm{1}}{\bm{1}}^{\prime}},\qquad\Phi^{\bullet}_{22}\equiv\frac{1}{2}s^{\bullet}_{\mu\nu}{\bm{e}}^{\bullet\mu}{}_{{\bm{1}}{\bm{1}}^{\prime}}{\bm{e}}^{\bullet\nu}{}_{{\bm{1}}{\bm{1}}^{\prime}}.$
(69)

The condition $\Phi^{\bullet}_{22}=0$ on $\mathscr{I}^{+}$ implies a linear
ordinary differential equation (ODE) for $\theta^{-1}$ on the generators of
$\mathscr{I}^{+}$

${\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}({\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}(\theta^{-1}))+\theta^{-1}\Phi^{\circ}_{22}=0.$
(70)

The initial data for $\theta$,
${\bm{e}}^{\circ}_{{\bm{1}}{\bm{1}}^{\prime}}(\theta)$ on $\mathcal{C}$ can be
used to solve for $\theta$ and obtain

$\Phi^{\bullet}_{22}\simeq
0,\qquad\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}{\bm{1}}{\bm{1}}}\simeq 0.$

By fixing $\theta$ and ${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}$ on
$\mathscr{I}^{+}$, the frame fields
${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}$ and
${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}$ can be determined up to a
rotation. Then, the phase parameter $c$ can be determined by solving

${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(c)=-i{\bm{e}}^{\bullet\mu}{}_{{\bm{1}}{\bm{0}}^{\prime}}{\bm{e}}^{\bullet\nu}{}_{{\bm{1}}{\bm{1}}^{\prime}}\nabla^{\bullet}_{\mu}{\bm{e}}^{\bullet}_{\nu{\bm{0}}{\bm{1}}^{\prime}},$
(71)

along the generators of $\mathscr{I}^{+}$ with initial data $c=0$ on
$\mathcal{C}$. Making the replacement
${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}\to
e^{ic}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}$ allows us to obtain

$\Gamma^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}{\bm{0}}{\bm{1}}}\simeq 0.$

Given that $s^{\bullet}_{\mu\nu}$ is defined by

$\Theta^{\bullet}s^{\bullet}_{\mu\nu}=\frac{1}{2}g^{\bullet}_{\mu\nu}\nabla^{\bullet}_{\lambda}\nabla^{\bullet\lambda}\Theta^{\bullet}-2\nabla^{\bullet}_{\mu}\nabla^{\bullet}_{\nu}\Theta^{\bullet},$
(72)

contracting the above with
${\bm{e}}^{\bullet\mu}{}_{{\bm{0}}{\bm{1}}^{\prime}}{\bm{e}}^{\bullet\nu}{}_{{\bm{1}}{\bm{0}}^{\prime}}$
implies that

$\nabla^{\bullet}_{\lambda}\nabla^{\bullet\lambda}\Theta^{\bullet}\simeq 0.$

While a contraction with
${\bm{e}}^{\bullet\mu}{}_{{\bm{0}}{\bm{0}}^{\prime}}{\bm{e}}^{\bullet\nu}{}_{{\bm{1}}{\bm{1}}^{\prime}}$
and
${\bm{e}}^{\bullet\mu}{}_{{\bm{0}}{\bm{0}}^{\prime}}{\bm{e}}^{\bullet\nu}{}_{{\bm{0}}{\bm{1}}^{\prime}}$
gives

${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}({\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Theta^{\bullet}))\simeq
0,\qquad{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Theta^{\bullet})\simeq\text{const.}$

and

${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}({\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Theta^{\bullet}))=\Gamma^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}{\bm{0}}{\bm{0}}}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Theta^{\bullet}),$

respectively. Thus, one concludes

$\Gamma^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}{\bm{0}}{\bm{0}}}\simeq 0.$

The conformal rescaling given by eq. (66) implies a transformation law for the
Ricci scalars associated with ${\bm{\nabla}}^{\bullet}$ and ${\bm{\nabla}}$

$R^{\bullet}=\frac{1}{\theta^{2}}R+\frac{12}{\theta^{2}}\nabla^{\bullet}_{\alpha}\theta\nabla^{\bullet\alpha}\theta-\frac{6}{\theta}\nabla^{\bullet}_{\alpha}\nabla^{\bullet\alpha}\theta.$

By requiring that $R^{\bullet}\simeq 0$, one obtains the following linear ODE
for ${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta)$ on the generators
of $\mathscr{I}^{+}$

${\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}({\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta))-\frac{2}{\theta}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(\theta){\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta)=\mathcal{F}^{\bullet},$

where $\mathcal{F}^{\bullet}$ can be determined using the quantities already
obtained on $\mathscr{I}^{+}$. Given the initial data for
${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta)=p^{-1}\theta\Gamma^{\circ}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}$
on $\mathcal{C}$, the ODE can be integrated to obtain

$R^{\bullet}\simeq
0,\qquad\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}{\bm{0}}{\bm{0}}}\simeq 0.$

To introduce coordinates on $\mathcal{U}$, let $r^{\bullet}$ denote the affine
parameter along the null generators of the hypersurfaces
$\\{u^{\bullet}=\text{const.}\\}$ such that
${\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(r^{\bullet})=0$ and
$r^{\bullet}\simeq 0$. Then, the coordinates $(\vartheta,\varphi)$ on
$\mathcal{C}$ can be extended to $\mathscr{I}^{+}$ and the hypersurfaces
$\\{u^{\bullet}=\text{const.}\\}$ by requiring them to be constant along the
null generators of $\mathscr{I}^{+}$ and $\\{u^{\bullet}=\text{const.}\\}$
hypersurfaces, respectively. This allows us to obtain Bondi coordinates
$(u^{\bullet},r^{\bullet},\vartheta,\varphi)$ in a neighbourhood $\mathcal{U}$
of $\mathscr{I}^{+}$.

In the following, the conditions on the conformal rescaling, the frame field
and the coordinates will be referred to as the NP-gauge. We will assume that
the NP-gauge frame $\\{{\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ is a
frame field that satisfies the conditions mentioned above. The term NP-spin-
frame will be used to refer to the normalised spin frame
$\\{{\bm{\epsilon}}^{\bullet}_{{\bm{A}}}{}^{A}\\}$ implying a NP-gauge frame.
Other quantities in the NP-gauge will also be denoted by ∙.

######  Remark 6.

In subsequent calculations, the spin connection coefficients
$\Gamma^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{C}}}{}_{{\bm{D}}}$ will
be referred to as the NP connection coefficients. It will be useful to
introduce the following shorthand notation:

$\displaystyle\sigma^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{}^{{\bm{1}}}{}_{{\bm{0}}},\qquad\mu^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{1}}},\qquad\gamma^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{0}}},$
(73a)
$\displaystyle\lambda^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{1}}},\qquad\rho^{\bullet}\equiv-\Gamma^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{}^{{\bm{1}}}{}_{{\bm{0}}},\qquad\epsilon^{\bullet}\equiv\Gamma^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}{}^{{\bm{0}}}{}_{{\bm{0}}}.$
(73b)

Given the conformal relation between ${\bm{g}}^{\bullet}$ and ${\bm{g}}$ in
eq. (66), one has

$g^{\bullet}_{{\bm{a}}{\bm{b}}}=\theta^{2}g_{{\bm{a}}{\bm{b}}},$ (74)

where
$g^{\bullet}_{{\bm{a}}{\bm{b}}}\equiv{\bm{g}}^{\bullet}({\bm{e}}^{\bullet}_{{\bm{a}}},{\bm{e}}^{\bullet}_{{\bm{b}}})$
and
$g_{{\bm{a}}{\bm{b}}}\equiv{\bm{g}}({\bm{e}}_{{\bm{a}}},{\bm{e}}_{{\bm{b}}})$.
Then, the relation between the NP-gauge frame
$\\{{\bm{e}}^{\bullet}_{{\bm{a}}}\\}$ and the F-gauge frame
$\\{{\bm{e}}_{{\bm{a}}}\\}$, parameterised in terms of $\theta$ and
$\Lambda^{{\bm{a}}}{}_{{\bm{b}}}\in O(1,3)$, can be written as

${\bm{e}}^{\bullet}_{{\bm{a}}}=\theta^{-1}\Lambda^{{\bm{b}}}{}_{{\bm{a}}}{\bm{e}}_{{\bm{b}}},\qquad{\bm{e}}_{{\bm{a}}}=\theta\Lambda_{{\bm{a}}}{}^{{\bm{b}}}{\bm{e}}^{\bullet}_{{\bm{b}}}.$
(75)

Moreover, the NP-gauge spin frame
$\\{{\bm{\epsilon}}^{\bullet}_{{\bm{A}}}{}^{A}\\}$ and the F-gauge spin frame
$\\{{\bm{\epsilon}}^{\bullet}_{{\bm{A}}}{}^{A}\\}$ are related by

${\bm{\epsilon}}^{\bullet}_{{\bm{A}}}{}^{A}=\theta^{-\frac{1}{2}}\Lambda^{{\bm{B}}}{}_{{\bm{A}}}{\bm{\epsilon}}_{{\bm{B}}}{}^{A},\qquad{\bm{\epsilon}}^{\bullet{\bm{A}}}{}_{A}=\theta^{\frac{1}{2}}\Lambda_{{\bm{B}}}{}^{{\bm{A}}}{\bm{\epsilon}}^{{\bm{B}}}{}_{A}.$
(76)

where $\Lambda^{{\bm{B}}}{}_{{\bm{A}}}\in SL(2,\mathbb{C})$.

##  5 BMS-supertranslations charges

Now that we have defined the NP-gauge and its relation to the F-gauge, we can
introduce BMS-asymptotic charges expressed in terms of the NP-gauge. Following
the discussion in [30], for every $f\in C^{\infty}(\mathbb{S}^{2})$, the
associated BMS-supertranslation charge $\mathcal{Q}$ can be written as an
integral over some cut $\mathcal{C}$ of $\mathscr{I}^{\pm}$

$\mathcal{Q}(f;\mathcal{C})\equiv\oint_{\mathcal{C}}\bm{\varepsilon}_{2}f(\mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})+\tfrac{1}{2}\sigma^{\bullet
ab}N^{\bullet}_{ab}),$ (77)

where $\bm{\varepsilon}_{2}$ is the area element on $\mathcal{C}$,
$\sigma^{\bullet ab}$ is the shear tensor, $N^{\bullet}_{ab}$ is the news
tensor. $\mathcal{P}^{\bullet}$ and $(*\mathcal{P}^{\bullet})$ are defined in
terms of the rescaled Weyl tensor $d^{\bullet}_{abcd}$ and its left Hodge dual
$(*d^{\bullet})_{abcd}$ as follows:

$\displaystyle\mathcal{P}^{\bullet}\equiv
d^{\bullet}_{cdef}l^{c}n^{d}l^{e}n^{f},$
$\displaystyle(*\mathcal{P}^{\bullet})\equiv(*d^{\bullet})_{cdef}l^{c}n^{d}l^{e}n^{f},$

where the vectors $l^{a}$ and $n^{a}$ are identified as follows:

$\displaystyle
l^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}{}^{a},\qquad
n^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}{}^{a}\qquad\text{on
}\mathscr{I}^{+},$ (79a) $\displaystyle
l^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}{}^{a},\qquad
n^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}{}^{a}\qquad\text{on
}\mathscr{I}^{-}.$ (79b)

To translate eq. (77) to the F-gauge, our strategy is to obtain an expression
for $\mathcal{P}^{\bullet}$, $(*\mathcal{P}^{\bullet})$ and $\sigma^{\bullet
ab}N^{\bullet}_{ab})$ in terms of scalars for which it would be simpler to
compute the transformation to the F-gauge. Given that the spinorial
counterpart of $d^{\bullet}_{abcd}$ can be decomposed in terms of the rescaled
Weyl spinor $\phi^{\bullet}_{ABCD}$ as

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

One has

$\displaystyle\mathcal{P}^{\bullet}=d^{\bullet}_{CC^{\prime}DD^{\prime}EE^{\prime}FF^{\prime}}o^{\bullet
C}\bar{o}^{\bullet C^{\prime}}\iota^{\bullet D}\bar{\iota}^{\bullet
D^{\prime}}o^{\bullet E}\bar{o}^{\bullet E^{\prime}}\iota^{\bullet
F}\bar{\iota}^{\bullet F^{\prime}},$
$\displaystyle\phantom{\mathcal{P}^{\bullet}}=-(\phi^{\bullet}_{2}+\bar{\phi}^{\bullet}_{2}),$

where $\\{{\bm{o}}^{\bullet},{\bm{\iota}}^{\bullet}\\}$ is the spin dyad
adapted to the NP-gauge and

$\phi^{\bullet}_{2}\equiv\phi^{\bullet}_{ABCD}o^{\bullet A}\iota^{\bullet
B}o^{\bullet C}\iota^{\bullet
D},\qquad\bar{\phi}^{\bullet}_{2}\equiv\bar{\phi}^{\bullet}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\bar{o}^{\bullet
A^{\prime}}\bar{\iota}^{\bullet B^{\prime}}\bar{o}^{\bullet
C^{\prime}}\bar{\iota}^{\bullet D^{\prime}}.$

For $(*\mathcal{P}^{\bullet})$, one has

$(*\mathcal{P}^{\bullet})=(*d^{\bullet})_{CC^{\prime}DD^{\prime}EE^{\prime}FF^{\prime}}o^{\bullet
C}\bar{o}^{\bullet C^{\prime}}\iota^{\bullet D}\bar{\iota}^{\bullet
D^{\prime}}o^{\bullet E}\bar{o}^{\bullet E^{\prime}}\iota^{\bullet
F}\bar{\iota}^{\bullet F^{\prime}}.$

But, given that

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

One can show that $(*\mathcal{P}^{\bullet})$ is given by

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

Hence, one has that

$\mathcal{P}^{\bullet}-i(*\mathcal{P}^{\bullet})=-2\bar{\phi}^{\bullet}_{2}.$
(80)

For the background term involving $\sigma^{\bullet ab}N^{\bullet}_{ab}$, note
that $\sigma^{\bullet ab}$ and $N^{\bullet}_{ab}$ are defined as

$\displaystyle N^{\bullet}_{ab}\equiv
2(\mathcal{L}_{n}-\Phi)\sigma^{\bullet}_{ab},$
$\displaystyle\sigma^{\bullet}_{ab}\equiv\left(q^{\bullet}_{a}{}^{c}q^{\bullet}_{b}{}^{d}-\frac{1}{2}q^{\bullet}_{ab}q^{\bullet
cd}\right)\nabla^{\bullet}_{c}l_{d},$

where $q^{\bullet}_{ab}$ is the induced metric on
$\mathscr{I}=\mathscr{I}^{+}\cup\mathscr{I}^{-}$, and $\Phi$ is defined by
$\Phi\equiv\frac{1}{4}\nabla^{\bullet}_{a}n^{a}$ on $\mathscr{I}$ and it
satisfies the following:

$\Phi|_{i}=2.$

Note that the metric ${\bm{g}}^{\bullet}$ and the covariant derivative
${\bm{\nabla}}^{\bullet}$ can be decomposed in terms of the null tetrad
$(l^{a},n^{a},m^{a},\bar{m}^{a})$ as

$\displaystyle
g^{\bullet}_{ab}=n_{a}l_{b}+l_{a}n_{b}-\bar{m}_{a}m_{b}-m_{a}\bar{m}_{b},$
$\displaystyle\nabla^{\bullet}_{a}=n_{a}D+l_{a}\Delta-\bar{m}_{a}\delta-
m_{a}\bar{\delta},$

where $D\equiv l^{a}\nabla^{\bullet}_{a},\;\Delta\equiv
n^{a}\nabla^{\bullet}_{a},\;\delta\equiv m^{a}\nabla^{\bullet}_{a}$ and
$\bar{\delta}\equiv\bar{m}^{a}\nabla^{\bullet}_{a}$, $l^{a}$ and $n^{a}$ are
defined by eq. (79) and

$m^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{}^{a},\qquad\bar{m}^{a}\equiv{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{}^{a}\qquad\text{on
}\mathscr{I}^{\pm}.$ (83)

From the above, one can show that

$\displaystyle\sigma^{\bullet}_{ab}=\sigma^{\bullet}\bar{m}_{a}\bar{m}_{b}+\bar{\sigma}^{\bullet}m_{a}m_{b},\qquad\text{on
}\mathscr{I}^{+},$
$\displaystyle\sigma^{\bullet}_{ab}=-\lambda^{\bullet}m_{a}m_{b}-\bar{\lambda}^{\bullet}\bar{m}_{a}\bar{m}_{b},\qquad\text{on
}\mathscr{I}^{-},$

and

$\displaystyle\Phi=\frac{1}{4}(\mu^{\bullet}+\bar{\mu}^{\bullet}-\gamma^{\bullet}-\bar{\gamma}^{\bullet}),\qquad\text{on
}\mathscr{I}^{+},$
$\displaystyle\Phi=\frac{1}{4}(\epsilon^{\bullet}+\bar{\epsilon}^{\bullet}-\rho^{\bullet}-\bar{\rho}^{\bullet}),\qquad\text{on
}\mathscr{I}^{-}.$

Given that

$\mathcal{L}_{n}\sigma^{\bullet}_{ab}=\Delta\sigma^{\bullet}_{ab}+\sigma^{\bullet}_{cb}\nabla^{\bullet}_{a}n^{c}+\sigma^{\bullet}_{ac}\nabla^{\bullet}_{b}n^{c},$

a long computation yields

$\displaystyle\sigma^{\bullet
ab}N^{\bullet}_{ab}=2\Delta|\sigma^{\bullet}|^{2}-|\sigma^{\bullet}|^{2}\big{(}3\mu^{\bullet}+3\bar{\mu}^{\bullet}+\gamma^{\bullet}+\bar{\gamma}^{\bullet}\big{)},\qquad\text{on
}\mathscr{I}^{+},$ (86a) $\displaystyle\sigma^{\bullet
ab}N^{\bullet}_{ab}=2\Delta|\lambda^{\bullet}|^{2}-|\lambda^{\bullet}|^{2}\big{(}3\rho^{\bullet}+3\bar{\rho}^{\bullet}+\epsilon^{\bullet}+\bar{\epsilon}^{\bullet}\big{)},\qquad\text{on
}\mathscr{I}^{-}.$ (86b)

Using eqs. (80) and (86) and substituting in eq. (77), one obtains

$\displaystyle\mathcal{Q}(f,\mathcal{C})=\oint_{\mathcal{C}}\bm{\varepsilon}_{2}f\bigg{(}-2\bar{\phi}^{\bullet}_{2}+\Delta|\sigma^{\bullet}|^{2}-\frac{1}{2}|\sigma^{\bullet}|^{2}(3\mu^{\bullet}+3\bar{\mu}^{\bullet}+\gamma^{\bullet}+\bar{\gamma}^{\bullet})\bigg{)},\qquad\text{on
}\mathscr{I}^{+},$
$\displaystyle\mathcal{Q}(f,\mathcal{C})=\oint_{\mathcal{C}}\bm{\varepsilon}_{2}f\bigg{(}-2\bar{\phi}^{\bullet}_{2}+\Delta|\lambda^{\bullet}|^{2}-\frac{1}{2}|\lambda^{\bullet}|^{2}(3\rho^{\bullet}+3\bar{\rho}^{\bullet}+\epsilon^{\bullet}+\bar{\epsilon}^{\bullet})\bigg{)},\qquad\text{on
}\mathscr{I}^{-},$

for which it will be simpler to compute the transformation to the F-gauge.

###  5.1 BMS-supertranslation charges in the F-gauge

The next step in this analysis is to express
$\bar{\phi}^{\bullet}_{2},\;\sigma^{\bullet},\;\mu^{\bullet},\;\gamma^{\bullet},\;\lambda^{\bullet},;\rho^{\bullet}$
and $\epsilon^{\bullet}$ in terms of F-gauge quantities. Given that

$\displaystyle\sigma^{\bullet}\equiv{\bm{\epsilon}}^{\bullet}_{{\bm{0}}}{}^{A}\nabla^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{1}}}{}_{A},\qquad\mu^{\bullet}\equiv-{\bm{\epsilon}}^{\bullet}_{{\bm{1}}}{}^{A}\nabla^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{0}}}{}_{A},\qquad\gamma^{\bullet}\equiv-{\bm{\epsilon}}^{\bullet}_{{\bm{0}}}{}^{A}\nabla^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{0}}}{}_{A},$
$\displaystyle\lambda^{\bullet}\equiv-{\bm{\epsilon}}^{\bullet}_{{\bm{1}}}{}^{A}\nabla^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{0}}}{}_{A},\qquad\rho^{\bullet}\equiv{\bm{\epsilon}}^{\bullet}_{{\bm{0}}}{}^{A}\nabla^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{1}}}{}_{A},\qquad\epsilon^{\bullet}\equiv{\bm{\epsilon}}^{\bullet}_{{\bm{0}}}{}^{A}\nabla^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}{\bm{\epsilon}}^{\bullet{\bm{0}}}{}_{A},$

and using eq. (76), a long computation yields

$\displaystyle\sigma^{\bullet}=\Lambda^{{\bm{A}}}{}_{{\bm{0}}}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{1}}})-\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{1}}}\Lambda^{{\bm{C}}}{}_{{\bm{0}}}\Lambda^{{\bm{B}}}{}_{{\bm{0}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{1}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89a)
$\displaystyle\mu^{\bullet}=\theta^{-1}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(\theta)-\Lambda^{{\bm{A}}}{}_{{\bm{1}}}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{1}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{0}}})+\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{0}}}\Lambda^{{\bm{C}}}{}_{{\bm{1}}}\Lambda^{{\bm{B}}}{}_{{\bm{0}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{1}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89b)
$\displaystyle\gamma^{\bullet}=-\frac{1}{2}\theta^{-1}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(\theta)-\Lambda^{{\bm{A}}}{}_{{\bm{0}}}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{1}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{0}}})+\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{0}}}\Lambda^{{\bm{C}}}{}_{{\bm{0}}}\Lambda^{{\bm{B}}}{}_{{\bm{1}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{1}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89c)
$\displaystyle\lambda^{\bullet}=-\Lambda^{{\bm{A}}}{}_{{\bm{1}}}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{0}}})+\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{0}}}\Lambda^{{\bm{C}}}{}_{{\bm{1}}}\Lambda^{{\bm{B}}}{}_{{\bm{1}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{0}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89d)
$\displaystyle\rho^{\bullet}=\theta^{-1}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta)+\Lambda^{{\bm{A}}}{}_{{\bm{0}}}{\bm{e}}^{\bullet}_{{\bm{1}}{\bm{0}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{1}}})-\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{1}}}\Lambda^{{\bm{C}}}{}_{{\bm{0}}}\Lambda^{{\bm{B}}}{}_{{\bm{1}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{0}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89e)
$\displaystyle\epsilon^{\bullet}=\frac{1}{2}\theta^{-1}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\theta)-\Lambda^{{\bm{A}}}{}_{{\bm{0}}}{\bm{e}}^{\bullet}_{{\bm{0}}{\bm{0}}^{\prime}}(\Lambda_{{\bm{A}}}{}^{{\bm{0}}})+\theta^{-1}\Lambda_{{\bm{A}}}{}^{{\bm{0}}}\Lambda^{{\bm{C}}}{}_{{\bm{0}}}\Lambda^{{\bm{B}}}{}_{{\bm{0}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{0}}^{\prime}}\Gamma_{{\bm{B}}{\bm{B}}^{\prime}}{}^{{\bm{A}}}{}_{{\bm{C}}},$
(89f)

where the LHS in the above expressions is written in terms of F-gauge
quantities, except for ${\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}$, which
will be explicitly computed in later sections. The component
$\bar{\phi}^{\bullet}_{2}$ can be written in terms of F-gauge quantities as

$\displaystyle\bar{\phi}^{\bullet}_{2}=-\frac{3}{2}\theta^{-3}\Lambda^{{\bm{A}}}{}_{{\bm{1}}}\Lambda^{{\bm{B}}}{}_{{\bm{0}}}\Lambda^{{\bm{C}}}{}_{{\bm{1}}}\Lambda^{{\bm{D}}}{}_{{\bm{0}}}\bar{\Lambda}^{{\bm{A}}^{\prime}}{}_{{\bm{0}}^{\prime}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{1}}^{\prime}}\bar{\Lambda}^{{\bm{C}}^{\prime}}{}_{{\bm{0}}^{\prime}}\bar{\Lambda}^{{\bm{D}}^{\prime}}{}_{{\bm{1}}^{\prime}}(\bar{d}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
(90)
$\displaystyle\phantom{\bar{\phi}^{\bullet}_{2}}-\bar{d}_{{\bm{B}}{\bm{A}}^{\prime}{\bm{A}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}+\bar{d}_{{\bm{B}}{\bm{A}}^{\prime}{\bm{A}}{\bm{B}}^{\prime}{\bm{D}}{\bm{C}}^{\prime}{\bm{C}}{\bm{D}}^{\prime}}-\bar{d}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{D}}{\bm{C}}^{\prime}{\bm{C}}{\bm{D}}^{\prime}}).$

Here,
$\bar{d}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
denotes the components of the complex conjugate of
$d_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}$ in the F-gauge. The
discussion above indicates that $\mathcal{Q}$ can be written as

$\mathcal{Q}(f,\mathcal{C})\equiv\mathcal{Q}(\theta,\Lambda^{{\bm{A}}}{}_{{\bm{B}}},\Lambda_{{\bm{B}}}{}^{{\bm{A}}},\bar{\Lambda}^{{\bm{A}}^{\prime}}{}_{{\bm{B}}^{\prime}},\bar{\Lambda}_{{\bm{B}}^{\prime}}{}^{{\bm{A}}^{\prime}},{\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}},\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{C}}}{}_{{\bm{D}}},\bar{d}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}).$
(91)

Accordingly, the evaluation of $\mathcal{Q}$ at $\mathcal{I}^{\pm}$ requires a
solution for ${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$,
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{{\bm{C}}}{}_{{\bm{D}}}$ and
$\bar{d}_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{B}}^{\prime}{\bm{C}}{\bm{C}}^{\prime}{\bm{D}}{\bm{D}}^{\prime}}$
at $\mathcal{I}^{\pm}$. As will become evident, a solution of the conformal
field equations is also necessary for obtaining an asymptotic expression of
$\theta,\Lambda^{{\bm{A}}}{}_{{\bm{B}}}$ and
${\bm{e}}^{\bullet}_{{\bm{A}}{\bm{A}}^{\prime}}$.

##  6 The initial data and the constraint equations

This section discusses the initial data used in our analysis, which is
required to obtain a solution to the conformal field equations and, more
precisely, to obtain non-trivial BMS asymptotic charges at
$\mathscr{I}^{\pm}$. The conformal constraint equations, satisfied by our
initial data, will be introduced in terms of the Levi-Civita connection
associated with ${\bm{g}}$, contrary to the extended conformal field equations
discussed in Section 2.2.

###  6.1 The conformal constraint equations

To introduce the conformal constraint equations, let
$(\tilde{\mathcal{M}},\tilde{{\bm{g}}})$ denote the physical spacetime
satisfying eq. (3), and introduce the unphysical spacetime
$(\mathcal{M},{\bm{g}})$ with ${\bm{g}}$ given by eq. (6). Moreover, let
$\tilde{\mathcal{S}}$ denote a three-dimensional hypersurface on
$\tilde{\mathcal{M}}$ with an induced metric denoted by $\tilde{{\bm{h}}}$. As
mentioned earlier, the hypersurface $\tilde{\mathcal{S}}$ can be regarded as a
hypersurface on $\mathcal{M}$ given the composition map
$\phi\circ\varphi:\tilde{\mathcal{S}}\to\mathcal{M}$ where
$\phi:\tilde{\mathcal{M}}\to\mathcal{M}$ and
$\varphi:\tilde{\mathcal{S}}\to\tilde{\mathcal{M}}$. The metric ${\bm{g}}$ on
$\mathcal{M}$ also induces an intrinsic metric ${\bm{h}}$ on
$\tilde{\mathcal{S}}$ related to $\tilde{{\bm{h}}}$ by eq. (53). If
$\tilde{{\bm{n}}}$ and ${\bm{n}}$ are the $\tilde{{\bm{g}}}$ and ${\bm{g}}$
unit normals of $\tilde{\mathcal{S}}$, then

$\epsilon\equiv\tilde{{\bm{g}}}(\tilde{{\bm{n}}},\tilde{{\bm{n}}})={\bm{g}}({\bm{n}},{\bm{n}})=1,$

since $\tilde{\mathcal{S}}$ is a spacelike hypersurface. Moreover, if
${\bm{u}},{\bm{v}}$ are some arbitrary vectors on $\tilde{\mathcal{S}}$, then
the extrinsic curvatures $\tilde{{\bm{K}}}$ and ${\bm{K}}$ are defined by

$\tilde{{\bm{K}}}({\bm{u}},{\bm{v}})=\langle\tilde{{\bm{\nabla}}}_{{\bm{u}}}\tilde{{\bm{n}}},{\bm{v}}\rangle,\qquad{\bm{K}}({\bm{u}},{\bm{v}})=\langle{\bm{\nabla}}_{{\bm{u}}}{\bm{n}},{\bm{v}}\rangle.$

It can be shown that the relation between $\tilde{{\bm{K}}}$ and ${\bm{K}}$ is
given by

${\bm{K}}=\Omega(\tilde{{\bm{K}}}+\Sigma\tilde{{\bm{h}}}),$

where

$\Sigma={\bm{g}}^{\sharp}({\bm{d}}{\Xi},{\bm{n}})=\Xi^{-1}\tilde{{\bm{g}}}^{\sharp}({\bm{d}}{\Xi},\tilde{{\bm{n}}}).$

Let $\\{{\bm{e}}_{{\bm{i}}}\\}$ denote an ${\bm{h}}$-orthonormal frame. Then,
the vacuum conformal constraint equations on $\tilde{\mathcal{S}}$ are given
by

$\displaystyle D_{{\bm{i}}}D_{{\bm{j}}}\Omega+\Sigma
K_{{\bm{i}}{\bm{j}}}+\Omega L_{{\bm{i}}{\bm{j}}}-sh_{{\bm{i}}{\bm{j}}}=0,$
(92a) $\displaystyle D_{{\bm{i}}}\Sigma-
K_{{\bm{i}}}{}^{{\bm{k}}}D_{{\bm{k}}}\Omega+\Omega L_{{\bm{i}}}=0,$ (92b)
$\displaystyle D_{{\bm{i}}}s+\Sigma
L_{{\bm{i}}}+L_{{\bm{i}}{\bm{k}}}D^{{\bm{k}}}\Omega=0,$ (92c) $\displaystyle
D_{{\bm{i}}}L_{{\bm{j}}{\bm{k}}}-D_{{\bm{j}}}L_{{\bm{i}}{\bm{k}}}+\Sigma
d_{{\bm{k}}{\bm{i}}{\bm{j}}}-D^{{\bm{l}}}\Omega
d_{{\bm{l}}{\bm{k}}{\bm{i}}{\bm{j}}}+K_{{\bm{i}}{\bm{k}}}L_{{\bm{j}}}-K_{{\bm{j}}{\bm{k}}}L_{{\bm{i}}}=0,$
(92d) $\displaystyle
D_{{\bm{i}}}L_{{\bm{j}}}-D_{{\bm{j}}}L_{{\bm{i}}}+K_{{\bm{j}}}{}^{{\bm{k}}}L_{{\bm{i}}{\bm{k}}}-K_{{\bm{i}}}{}^{{\bm{k}}}L_{{\bm{j}}{\bm{k}}}-D^{{\bm{l}}}\Omega
d_{{\bm{l}}{\bm{i}}{\bm{j}}}=0,$ (92e) $\displaystyle
D^{{\bm{k}}}d_{{\bm{k}}{\bm{i}}{\bm{j}}}-K^{{\bm{k}}}{}_{{\bm{i}}}d_{{\bm{j}}{\bm{k}}}+K^{{\bm{k}}}{}_{{\bm{j}}}d_{{\bm{i}}{\bm{k}}}=0,$
(92f) $\displaystyle
D^{{\bm{k}}}d_{{\bm{k}}{\bm{j}}}-K^{{\bm{i}}{\bm{k}}}d_{{\bm{i}}{\bm{j}}{\bm{k}}}=0,$
(92g) $\displaystyle 6\Omega s-3\Sigma^{2}-3D_{{\bm{i}}}\Omega
D^{{\bm{i}}}\Omega=0.$ (92h)

In the above, ${\bm{D}}$ denotes the Levi-Civita connection associated with
${\bm{h}}$ and $D_{{\bm{i}}}\equiv{\bm{e}}_{{\bm{i}}}{}^{i}D_{i}$ while
$h_{{\bm{i}}{\bm{j}}}\equiv{\bm{h}}({\bm{e}}_{{\bm{i}}},{\bm{e}}_{{\bm{j}}})$
denotes the components of ${\bm{h}}$ with respect to
$\\{{\bm{e}}_{{\bm{i}}}\\}$. Similarly, $K_{{\bm{i}}{\bm{j}}}$ and
$l_{{\bm{i}}{\bm{j}}}$ denote the components of the extrinsic curvature and
intrinsic Schouten tensor with respect to $\\{{\bm{e}}_{{\bm{i}}}\\}$,
respectively. Furthermore, $L_{{\bm{i}}},d_{{\bm{i}}{\bm{j}}}$ and
$d_{{\bm{i}}{\bm{j}}{\bm{k}}}$ are the spatial components of the contraction
of $L_{{\bm{a}}{\bm{b}}},d_{{\bm{a}}{\bm{b}}{\bm{c}}{\bm{d}}}$ with
$n^{{\bm{a}}}$. In particular, $L_{{\bm{i}}},d_{{\bm{i}}{\bm{j}}}$ and
$d_{{\bm{i}}{\bm{j}}{\bm{k}}}$ are defined as

$L_{{\bm{i}}}\equiv
L_{{\bm{i}}{\bm{a}}}n^{{\bm{a}}},\qquad\;d_{{\bm{i}}{\bm{j}}}\equiv
d_{{\bm{i}}{\bm{a}}{\bm{j}}{\bm{b}}}n^{{\bm{a}}}n^{{\bm{b}}},\qquad\;d_{{\bm{i}}{\bm{j}}{\bm{k}}}\equiv
d_{{\bm{i}}{\bm{a}}{\bm{j}}{\bm{k}}}n^{{\bm{a}}}.$

######  Remark 7.

The spatial components of $E_{ab}$, introduced in Section 3.1 are equivalent
to $d_{{\bm{i}}{\bm{j}}}$ while the spatial components of $B_{ab}$ are related
to $d_{{\bm{i}}{\bm{j}}{\bm{k}}}$ by

$d_{{\bm{i}}{\bm{j}}{\bm{k}}}=\epsilon^{{\bm{l}}}{}_{{\bm{j}}{\bm{k}}}B_{{\bm{i}}{\bm{l}}},$

where $\epsilon^{{\bm{l}}}{}_{{\bm{j}}{\bm{k}}}$ denotes the components of the
three-dimensional volume form on $\tilde{\mathcal{S}}$.

###  6.2 The initial data

To introduce the initial data for the ECFEs, the starting point is to consider
an initial data set satisfying the Hamiltonian and momentum constraints
implied by the vacuum Einstein field eq. (3). More precisely, we will be
interested in a vacuum initial data set
$(\tilde{\mathcal{S}},\tilde{{\bm{h}}},\tilde{{\bm{K}}})$ satisfying

$\displaystyle\tilde{r}+\tilde{K}^{2}-\tilde{K}_{jl}\tilde{K}^{jl}=0,$ (93a)
$\displaystyle\tilde{D}^{j}\tilde{K}_{kj}-\tilde{D}_{k}\tilde{K}=0.$ (93b)

To obtain non-trivial BMS asymptotic charges at $\mathscr{I}^{\pm}$, let us
consider the vacuum initial data prescribed in [35]:

######  Proposition 2.

For any $\xi,\zeta\in C^{2}(\mathbb{S}^{2})$ and $q\geq 1$, there exists a
vacuum initial data set $(\tilde{{\bm{h}}},\tilde{\bm{\pi}})$ where the
components of the intrinsic metric $\tilde{{\bm{h}}}$ and the momentum tensor
$\tilde{\bm{\pi}}$ with respect to the standard Euclidean coordinate chart
$(x^{\alpha})$ have the asymptotics

$\displaystyle\tilde{h}_{\alpha\beta}=-\left(1+\frac{A}{r}\right)\delta_{\alpha\beta}-\frac{\xi}{r}\left(\frac{x_{\alpha}x_{\beta}}{r^{2}}-\frac{1}{2}\delta_{\alpha\beta}\right)+O_{2}(r^{-1-q}),$
$\displaystyle\tilde{\pi}_{\alpha\beta}=\frac{\zeta}{r^{2}}\frac{x_{\alpha}x_{\beta}}{r^{2}}+\frac{1}{r^{3}}\left(-B_{\alpha}x_{\beta}-B_{\beta}x_{\alpha}+(B^{\gamma}x_{\gamma})\delta_{\alpha\beta}\right)+O_{1}(r^{-2-q}),$

where $A$, $\\{B_{\alpha}\\}_{\alpha=1}^{3}$ are some constants, and
$r\equiv\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$. The momentum tensor
$\tilde{\bm{\pi}}$ is defined as

$\tilde{\pi}_{ij}\equiv\tilde{K}_{ij}-\tilde{K}\tilde{h}_{ij}.$ (95)

To simplify the analysis in this work, we set $q=1$ so that the components of
$\tilde{{\bm{h}}}$ and $\tilde{\bm{\pi}}$ are written as

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

where the expressions above have been rearranged to group terms of similar
orders in $r$. Given eq. (95), the components of $\tilde{{\bm{K}}}$ with
respect to $(x^{\alpha})$ can be written as

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

In order to discuss the region near spatial infinity, we introduce the inverse
coordinates $(y^{\alpha})$ related to $(x^{\alpha})$ by

$y^{\alpha}=-\frac{x^{\alpha}}{r^{2}},\qquad
y_{\alpha}=-\frac{x_{\alpha}}{r^{2}}.$

Then, the inverse coordinate transformation is given by

$x^{\alpha}=-\frac{y^{\alpha}}{\varrho^{2}},\qquad
x_{\alpha}=-\frac{y_{\alpha}}{\varrho^{2}},\qquad\varrho=\frac{1}{r},$

where $\varrho=\sqrt{(y^{1})^{2}+(y^{2})^{2}+(y^{3})^{2}}$. In terms of
$(y^{\alpha})$, the components of $\tilde{{\bm{h}}}$ and $\tilde{{\bm{K}}}$
can be written as

$\displaystyle\tilde{h}_{\alpha\beta}=-\frac{(1+A\varrho)}{\varrho^{4}}\delta_{\alpha\beta}-\frac{\xi}{\varrho^{3}}\left(\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}-\frac{\xi}{2}\delta_{\alpha\beta}\right)+O_{2}(\varrho^{-2}),$
$\displaystyle\tilde{K}_{\alpha\beta}=-\frac{\zeta}{2\varrho^{2}}\delta_{\alpha\beta}-\frac{1}{\varrho^{3}}\left(2B_{(\alpha}y_{\beta)}+\frac{1}{2}(B^{\gamma}y_{\gamma})\delta_{\alpha\beta}\right)+\frac{y_{\alpha}y_{\beta}}{\varrho^{4}}\left(\zeta-4\frac{(B^{\gamma}y_{\gamma})}{\varrho}\right)+O_{1}(\varrho^{-1}).$

In the following, let $\mathcal{S}^{\prime}$ denote a three-dimensional
compact manifold with a spatial infinity point $i$ and let $\phi$ denote the
diffeomorphic map from $\mathcal{S}^{\prime}\setminus\\{i\\}$ to
$\tilde{\mathcal{S}}$ with the conformal factor $\Omega^{\prime}$ given by

$\Omega^{\prime}=\frac{\varrho^{2}}{\sqrt{1+A\varrho}}.$

Then, the components of ${\bm{h}}^{\prime}\equiv\Omega^{\prime
2}\tilde{{\bm{h}}}$ and
${\bm{K}}^{\prime}\equiv\Omega^{\prime}\tilde{{\bm{K}}}$ with respect to
$(y^{\alpha})$ are related to $\tilde{h}_{\alpha\beta}$ and
$\tilde{K}_{\alpha\beta}$ by

$h^{\prime}_{\alpha\beta}=\Omega^{\prime 2}\tilde{h}_{\alpha\beta},\qquad
K^{\prime}_{\alpha\beta}=\Omega^{\prime}\tilde{K}_{\alpha\beta}.$

By expanding $\Omega^{\prime}$ around $\varrho=0$, the components
$h^{\prime}_{\alpha\beta}$ and $K^{\prime}_{\alpha\beta}$ can be written as

$\displaystyle h^{\prime}_{\alpha\beta}$
$\displaystyle=-\delta_{\alpha\beta}-\xi\varrho\left(\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}-\frac{1}{2}\delta_{\alpha\beta}\right)+O_{2}(\varrho^{2}),$
(98a) $\displaystyle\hskip 28.45274ptK^{\prime}_{\alpha\beta}$
$\displaystyle=-\frac{\zeta}{2}\delta_{\alpha\beta}-\frac{1}{\varrho}\left(2B_{(\alpha}y_{\beta)}+\frac{1}{2}(B^{\gamma}y_{\gamma})\delta_{\alpha\beta}\right)+\left(\zeta-4\frac{(B^{\gamma}y_{\gamma})}{\varrho}\right)\frac{y_{\alpha}y_{\beta}}{\varrho^{2}}+O_{1}(\varrho).$
(98b)

The above initial data $({\bm{h}}^{\prime},{\bm{K}}^{\prime})$ is said to be
asymptotically Euclidean and regular in the spirit of Definition 1. To analyse
the conformal constraint equations, it will be convenient to express
${\bm{h}}^{\prime}$ and ${\bm{K}}^{\prime}$ in terms of the so-called normal
coordinates and to introduce the conformal normal initial data.

####  6.2.1 Normal coordinates and conformal normal initial data

Consider the ${\bm{h}}^{\prime}$-geodesics emanating from
$i\in\mathcal{S}^{\prime}$ to nearby points in a neighbourhood
$\mathcal{U}^{\prime}\in\mathcal{S}^{\prime}$ and introduce the subset
$\mathcal{T}$ of the tangent space at $i$

$\mathcal{T}\equiv\\{{\bm{v}}\in T|_{i}(\mathcal{S}^{\prime})|\hskip
5.69054pt\gamma_{{\bm{v}}}\text{ is defined on an interval containing
}[0,1]\\},$

where $\gamma_{{\bm{v}}}\equiv\gamma_{{\bm{v}}}(t)$ is the geodesic starting
at $i$ (i.e. $\gamma_{{\bm{v}}}(0)=i$) with an initial tangent vector