# Controlled regularity at future null infinity from past asymptotic initial
data: massless fields

G. Taujanskas111Electronic address<EMAIL_ADDRESS>and J. A.
Valiente Kroon222Electronic address<EMAIL_ADDRESS>
###### Abstract

We study the relationship between asymptotic characteristic initial data at
past null infinity and the regularity of solutions at future null infinity for
the massless linear spin-$s$ field equations on Minkowski space. By
quantitatively controlling the solutions on a causal rectangle reaching the
conformal boundary, we relate the (generically singular) behaviour of the
solutions near past null infinity, future null infinity, and spatial infinity.
Our analysis uses Friedrich’s cylinder at spatial infinity together with a
careful Grönwall-type estimate that does not degenerate at the intersection of
null infinity and the cylinder (the so-called critical sets).

###### Contents

  1. 1 Introduction
  2. 2 Geometric setup
  3. 3 Estimates near $\mathscr{I}^{+}$
  4. 4 Asymptotic characteristic initial value problem
  5. 5 Estimates near $\mathscr{I}^{-}$
  6. 6 Controlling the solutions from $\mathscr{I}^{-}$ to $\mathscr{I}^{+}$
  7. 7 Concluding remarks
  8. A Geometry of $\mathrm{SU}(2)$
  9. B Spin-$s$ equations
  10. C F-expansions
  11. D Expansions near $\mathscr{I}^{-}$

##  1 Introduction

In the early 1960s, Penrose famously observed [Pen63, Pen65] that techniques
from conformal geometry could be used to study the asymptotic regime of fields
in general relativity. Since zero rest-mass fields carry no preferred length
scale, their equations of motion exhibit an essential conformal invariance,
making Penrose’s technique particularly suited in the study of the asymptotics
of massless fields. By attaching to a spacetime a conformal boundary “at
infinity”, Penrose was able to translate questions regarding the asymptotic
behaviour of physical fields into questions about the _local_ regularity of
suitably rescaled (or _unphysical_) fields, now evolving on a conformally
rescaled (unphysical) spacetime. Spacetimes for which the procedure of
attaching such a boundary was possible were termed _asymptotically simple_ ,
and since its inception the notion has been instrumental in the development of
the current understanding of the asymptotic behaviour of massless fields,
including gravity [Ger76, PR86, Fri04].

An important application of the idea of asymptotic simplicity has been to
_scattering problems_ , the study of the relationship between the asymptotics
of fields in the distant past and the distant future. Penrose’s conformal
boundary—called _null infinity_ 333Null infinity, denoted $\mathscr{I}$,
derives its name as a result of being the locus of all endpoints of
inextendible null geodesics in the spacetime. It is a co-dimension one
hypersurface which may be spacelike, timelike, or null, corresponding to the
sign of the cosmological constant $\Lambda$ being positive, negative, or zero,
respectively. When $\Lambda\geqslant 0$, $\mathscr{I}$ has a past component
and a future component, denoted $\mathscr{I}^{-}$ and $\mathscr{I}^{+}$.—fits
very naturally here, and was first understood in relation to the classical
Lax–Phillips scattering theory444Classical scattering theory has a long
history going back to at least the works of Lax and Phillips [LP64, LP67] and
Friedlander [Fri62, Fri64, Fri67] in the 1960s. For a more complete
bibliography, see for example [Tau19] and references therein. by Friedlander
in [Fri80]. In the case of zero cosmological constant, Friedlander showed that
scattering problems could be formulated as _characteristic initial value
problems_ with data at past or future null infinity, and the ideas of such
_conformal_ formulations of scattering were later taken up by Baez, Segal and
Zhou [BSZ90] and subsequently many others (see e.g. [NT22] and references
therein). In cases when the field equations are linear or the data is highly
regular, characteristic initial value problems are known to be well-posed
[Hör90, Ren90, CP12, CP13, CCW16], giving rise to a _scattering operator_
555We gloss over the issue of asymptotic completeness here, i.e. that a
scattering operator should be invertible.: a map from, say, data on
$\mathscr{I}^{-}$ to induced data on $\mathscr{I}^{+}$. A detailed
understanding of this scattering operator, however, requires an understanding
of how the structure of the data on $\mathscr{I}^{-}$ is related to the
structure of the solution near $\mathscr{I}^{+}$.

In general, this is a non-trivial task, in part due to the behaviour of the
fields near spatial infinity $i^{0}$. In Penrose’s picture [Pen65] of smoothly
compactified Minkowski space, the two components of null infinity meet at a
point, $i^{0}$, where the integral curves of the generators of
$\mathscr{I}^{\pm}$ intersect. Spatial infinity $i^{0}$ is therefore a
caustic, and, unless the data is supported away from $i^{0}$, solutions
generally evolve non-smoothly near $i^{0}$. It is now understood that generic
behaviour is at best _polyhomogeneous_ near spatial and null infinity [Fri98a,
HV20], i.e. the asymptotic expansions of fields involve polynomial and
logarithmic666It was observed in [Val04, Val04a] that there are two
conceptually distinct classes of logarithmic terms. One is a consequence of
the caustic nature of $i^{0}$, and is the focus of the present paper. The
second arises from the singularity in the Weyl tensor at $i^{0}$ on spacetimes
with non-zero ADM mass. In full general relativity these two classes are
unavoidably intertwined, but the second is of course absent on Minkowski
space. terms of the form $x^{n}\log^{m}x$. In an effort to resolve the
detailed structure of spatial infinity, in 1998 Friedrich [Fri98a]
introduced777Before that, Ashtekar and Hansen [AH78] had put forward a similar
construction of a _hyperboloid at spatial infinity_ , which turns out to be
closely related to Friedrich’s cylinder [MV21]. a different conformal
representation of $i^{0}$, now known as the _cylinder at spatial infinity_ ,
or the _F-gauge_. In general, the utility of this conformal gauge lies in the
fact that it permits the formulation of a regular initial value problem in a
neighbourhood of spatial infinity for Friedrich’s _conformal Einstein field
equations_ [Fri98]. In full generality, Friedrich’s cylinder is constructed
using conformal geodesics, and so carries a strong geometric underpinning. In
the case of Minkowski space, however, the cylinder may be constructed using an
ad hoc conformal transformation, which we recap in Section 2.2. Briefly, the
key idea is to blow up the point $i^{0}$ to a $(1+2)$-dimensional submanifold
$\mathcal{I}$ with topology $(-1,1)\times\mathbb{S}^{2}$, which then becomes a
total characteristic of the field equations. The ends
$\mathcal{I}^{\pm}=\\{\pm 1\\}\times\mathbb{S}^{2}$ of this cylinder intersect
$\mathscr{I}^{\pm}$; on these spheres—called the _critical sets_ —the field
equations lose symmetric hyperbolicity and therefore degenerate. It is this
degeneracy that is at the heart of the logarithmic divergences which we study
in this paper.

Friedrich’s cylinder has been used to study the polyhomogeneity of spin-0,
spin-1 and spin-2 fields starting from spacelike Cauchy data in [Fri98a,
Fri03, Val02, MMV22, GV22, Val07, Val09]. The case of the characteristic
initial value problem near $i^{0}$, for Friedrich’s conformal Einstein
equations, has been investigated by Paetz [Pae18], who studied conditions on
the characteristic data which guarantee smoothness at the critical sets
$\mathcal{I}^{\pm}$. The degeneracy of the spin-$s$ equations at
$\mathcal{I}^{\pm}$ arises as a loss of rank in the matrix multiplying the
time derivatives, destroying symmetric hyperbolicity. Since standard estimates
for symmetric hyperbolic systems (which rely on very general properties of the
principal part of the equations [Joh91]) now fail, new estimates are needed.
In this paper we study the linear massless spin-$s$ equations on Minkowski
space, for888Here are throughout the paper, $s\neq 0$. Although results
similar to the ones presented in this paper appear obtainable for spin-0
fields, we do not handle the case of the wave equation as it is not in the
form of a zero rest-mass free field equation
$\nabla_{A}{}^{A^{\prime}}\phi_{AB\ldots M}=0$, which manifestly exhibits
structures on which our estimates rely. $s\in\frac{1}{2}\mathbb{N}$, starting
from characteristic data on $\mathscr{I}^{-}$, and relate the structural
properties of solutions in the regions near $\mathscr{I}^{-}$, $i^{0}$, and
$\mathscr{I}^{+}$. Our starting point are the estimates of Friedrich [Fri03],
which we generalize to all non-zero spins999While the most attention in the
literature appears to have been devoted to spin-2 fields (gravity), we believe
that the validity of our results for all $s$ may find applications in related
areas. In particular, the case of Dirac fields ($s=1/2$) may be relevant for
Lorentzian index theory [BGM05, BS20]., this being an easy extension with the
relevant structure present in the equations for any $s$. Friedrich’s estimates
[Fri03], valid near $i^{0}$ and $\mathscr{I}^{+}$, exploit the specific lower
order structure of the spin-$2$ equations, and are necessarily weaker in the
sense that they allow polyhomogeneous solutions with logarithmic divergences
at $\mathcal{I}^{\pm}$. Having extended these to all spins, we then proceed to
adapt the ideas in [Fri03] to construct estimates near $\mathscr{I}^{-}$. This
is the main contribution of our paper. Specifically, we prescribe initial data
for spin-$s$ fields on a portion of past null infinity which extends all the
way to $i^{0}$, and on an outgoing null hypersurface
$\underline{\mathcal{B}}_{\varepsilon}$ emanating from $\mathscr{I}^{-}$ (see
Figure 1), and follow Luk’s strategy101010Luk’s approach, for the Einstein
equations, relies on the well-known observation that when written in a double
null gauge, these form a symmetric hyperbolic system with a hierarchical
structure. A similar hierarchical structure, for the spin-$s$ equations, is
key to obtaining the estimates in [Fri03] as well as the ones presented in
this paper. [Luk12] to construct an optimal existence domain for the
characteristic initial value problem. In this domain we prove Grönwall-type
estimates for the solution by carefully arranging the sign of a suitable
constant, which we achieve by taking sufficiently many derivatives in the
boundary-defining variable $\tau=t/r$.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$i^{0}$$\underline{\mathcal{B}}_{\varepsilon}$$\mathscr{D}$

Figure 1: The Penrose diagram of Minkowski space showing the existence domain
$\mathscr{D}$. The main technical challenge in this analysis is the degeneracy
of the spin-$s$ equations at $i^{0}$.

The necessity for weaker estimates near $i^{0}$ reflects the general fact that
solutions there acquire polyhomogeneous expansions [Fri98]. The computation of
these expansions is done by postulating an Ansatz in terms of spin-weighted
spherical harmonics, and then using the total characteristic nature of the
cylinder $\mathcal{I}$ at spatial infinity. As a consequence, the spin-$s$
equations reduce to intrinsic transport equations on $\mathcal{I}$, and, at
least formally, the terms in the expansion can be computed by solving Jacobi
differential equations up and down the cylinder. The mentioned estimates then
show that these expansions represent true solutions. The logarithmic terms in
the solutions arise as logarithmic singularities in Jacobi polynomials of the
second kind as one solves the ODEs on $\mathcal{I}$. An inspection of the
computation of the asymptotic expansions reveals that, for our Ansatz111111Our
Ansatz, given in Section C.2, roughly postulates that the $p$-th derivative of
the field contains no spherical harmonics of order higher than $p$. We make
this assumption on the basis of simplifying the presentation. In the general
case when all spherical harmonics are permitted, logarithmic terms will arise
for each spherical harmonic with order $\geqslant p$., the logarithmic terms
are, at each order in the expansion, associated to a specific “top order”
spherical harmonic. Moreover, at each order these logarithmic terms are
regularized in a very specific way by multiplication by a polynomial
expression which vanishes at $\mathcal{I}^{\pm}$. The higher the order in the
expansion, the higher the order of this smoothing polynomial—consequently, the
logarithmic divergences become milder as one looks higher in the expansion.

### Main result

The main result of this article is as follows. Given asymptotic characteristic
initial data on $\mathscr{I}^{-}$ for the massless spin-$s$ equations which
possesses a regular asymptotic expansion towards $\mathcal{I}^{-}$, there
exists a unique solution to the spin-$s$ equations in a neighbourhood of
spatial infinity which contains pieces of both past and future null infinity,
as shown in Figure 1, and moreover, one is able to control the regularity of
the solution at $\mathscr{I}^{+}$ in terms of specific properties of the data
on $\mathscr{I}^{-}$. The regularity of the solution at $\mathscr{I}^{+}$ is
determined not solely by the regularity of the data on $\mathscr{I}^{-}$, but
also by its multipolar structure at the critical set $\mathcal{I}^{-}$. In
particular, simply requiring additional regularity of the characteristic data
does not necessarily result in improved behaviour at future null infinity.
This is consistent with previous results obtained in the analysis of the
corresponding Cauchy problem [Fri03, Val02]. A detailed statement of our main
result is given in Theorem 6.1.

#### Strategy of the proof

As mentioned, in our analysis we use Friedrich’s cylinder at spatial infinity,
the F-gauge. In this gauge the causal domain depicted in Figure 1 corresponds
to the (light and dark) grey area in Figure 2. We divide this domain in two
subdomains, the _lower domain_ $\underline{\mathcal{N}}_{\epsilon}$ in light
grey, and the _upper domain_ $\mathcal{N}_{1}$ in dark grey, which are
separated by a spacelike hypersurface $\mathcal{S}_{-1+\varepsilon}$
terminating at the cylinder at spatial infinity $\mathcal{I}$. On the upper
domain $\mathcal{N}_{1}$ we look for solutions which can be written as a
formal asymptotic expansion around $\mathcal{I}$ plus a remainder term. By
using the total characteristic nature of $\mathcal{I}$, the terms in this
formal expansion are, at least in principle, explicitly computable, and the
regularity of the expansion can be controlled by fine-tuning the multipolar
structure of the initial data. For the remainder, by generalizing the
estimates of [Fri03], we ensure its control all the way up to
$\mathscr{I}^{+}$ in terms of the regularity of the Cauchy data. There is a
connection here between the regularity of the remainder and the order of the
formal expansion; the remainder becomes more regular as the order of the
expansion increases. On the lower domain
$\underline{\mathcal{N}}_{\varepsilon}$, we again look for solutions in the
form of an asymptotic expansion plus a remainder. Here, however, the
geometric/structural properties of the setting require an expansion with
respect to the boundary defining function of past null infinity,
$1+\tau=1+t/r$, rather than that of the cylinder $\mathcal{I}$. For
simplicity, we assume that this expansion takes the form of a regular Taylor
expansion in powers of $1+\tau$. In particular, we assume the data on
$\mathscr{I}^{-}$ is sufficiently regular. To control the remainder, we
construct estimates using an adaptation of the techniques in [Fri03]. This
construction proceeds in two stages: first, by making use of a bootstrap
assumption on the radiation field, we estimate the other $2s-1$ components of
the spin-$s$ field; in the second stage, we prove the bootstrap bound on the
radiation field (recall that the radiation field encodes the freely
specifiable data for the spin-$s$ equations on $\mathscr{I}^{-}$). As we
mentioned in the introduction, these estimates are obtained by arranging the
sign of a certain constant to be negative, which is achieved by taking
derivatives in $\tau$ and carefully using the lower order structure of the
equations. As in the case of the upper domain, the regularity of this
remainder depends on the order of the asymptotic expansion in a similar way.
Finally, we stitch together the solutions on the lower and upper domains by
ensuring that the lower solution has enough control at the spatial
hypersurface $\mathcal{S}_{-1+\epsilon}$ to be able to apply our estimates in
the upper domain. We thus obtain a statement controlling the solution up to
future null infinity in terms of the asymptotic characteristic initial data on
past null infinity. The existence and uniqueness of solutions in each domain
is proved using standard last slice arguments.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{I}$$\mathcal{I}^{+}$$\bullet$$\mathcal{I}^{-}$$\bullet$$\mathcal{S}_{-1+\varepsilon}$$\underline{\mathcal{B}}_{\varepsilon}$$\underline{\mathcal{N}}_{\varepsilon}$$\mathcal{N}_{1}$$\mathcal{B}_{1}$

Figure 2: A depiction of the existence domain
$\mathscr{D}=\underline{\mathcal{N}}_{\epsilon}\cup\mathcal{N}_{1}$ in the
F-gauge. This is not a Penrose diagram but rather a coordinate diagram. The
domain $\mathscr{D}$ consists of the union of two domains: the lower domain
$\underline{\mathcal{N}}_{\varepsilon}$, which arises as the development of an
asymptotic characteristic initial value problem, and which is bounded in the
future by a spacelike hypersurface $\mathcal{S}_{-1+\varepsilon}$ which
reaches the cylinder at spatial infinity $\mathcal{I}$; and the upper domain
$\mathcal{N}_{1}$, which arises as the development of a standard Cauchy
initial value problem from data on $\mathcal{S}_{-1+\varepsilon}$. The
estimates constructed in the main text will control the solution up to
$\mathscr{I}^{+}$ despite the degeneracy of the equations at
$\mathcal{I}^{\pm}$.

### Outline of the article

In Section 2 we provide a succinct discussion of Friedrich’s framework of the
cylinder at spatial infinity as well as lay out general properties of the
spin-$s$ field equations. In Section 3 we construct the estimates in the upper
domain, where, along the way, we extend Friedrich’s estimates [Fri03] to
arbitrary spins $s\in\frac{1}{2}\mathbb{N}$. In Section 4 we discuss the set-
up of the asymptotic characteristic initial value problem. Here we also
discuss the difficulties that arise when attempting to analyse the behaviour
of solutions near spatial infinity. Section 5 contains the main insight of our
paper and provides the construction of our estimates in the lower domain. In
Section 6 we combine the estimates in the lower and upper domains to establish
the control of solutions at future null infinity in terms of past asymptotic
initial data. We conclude in Section 7 with some remarks and prospective
directions for future research. In an effort to be accessible to a wider
audience, we also include four appendices containing material essential for
the analysis but whose inclusion in the main text would hinder the flow of the
reading. Much (but not all) of this material is scattered throughout the
various references noted in the introduction, and we hope that collecting it
here will make the subject more approachable. Appendix A discusses properties
of $\mathrm{SU}(2)$ which are relevant to the construction of the estimates in
the main text. Appendix B provides a detailed derivation of the spin-$s$ field
equations in the F-gauge. Appendix C gives a detailed account of the
construction of asymptotic expansions in a neighbourhood of the cylinder at
spatial infinity—these expansions are central for the construction of the
solution in the upper domain. Finally, Appendix D presents the construction of
asymptotic expansions in a neighbourhood of past null infinity—these
expansions are key to the construction of the solution in the lower domain.

### Conventions and notation

Our conventions are consistent with [Val16]. In particular, our metric
signature is $(+,-,-,-)$, and the Riemann curvature tensor associated with the
Levi-Civita connection of a metric $g_{ab}$ is defined by
$[\nabla_{a},\nabla_{b}]u^{d}=R^{d}_{\phantom{d}cab}u^{c}$. For a given spin
dyad $\\{o^{A},\iota^{A}\\}$ with $o_{A}\iota^{A}=1$, we write
$\epsilon_{\bm{A}}{}^{A}$, ${\bm{A}}\in\\{0,1\\}$, to denote
$\epsilon_{0}{}^{A}=o^{A}$ and $\epsilon_{1}{}^{A}=\iota^{A}$. Spinorial
indices are raised and lowered using the antisymmetric $\epsilon$-spinor
$\epsilon_{AB}=o_{A}\iota_{B}-\iota_{A}o_{B}$ (with inverse
$\epsilon^{AB}=o^{A}\iota^{B}-\iota^{A}o^{B}$), e.g.
$\xi_{B}=\xi^{A}\epsilon_{AB}$, using the convention that contracted indices
should be “adjacent, descending to the right”. As usual, the spacetime metric
$g_{ab}$ decomposes as
$g_{ab}=\epsilon_{AB}\bar{\epsilon}_{A^{\prime}B^{\prime}}$, where
$\bar{\epsilon}_{A^{\prime}B^{\prime}}=\overline{\epsilon_{AB}}$. The spin
dyad $\epsilon_{\bm{A}}{}^{A}$ gives rise to a tetrad of null vectors
${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}={\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{AA^{\prime}}\partial_{AA^{\prime}}=\epsilon_{\bm{A}}{}^{A}\bar{\epsilon}_{{\bm{A}}^{\prime}}{}^{A^{\prime}}\partial_{AA^{\prime}}$.
The spin connection coefficients
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\bm{B}}{}_{\bm{C}}$ are then defined as
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\bm{B}}{}_{\bm{C}}=-\epsilon_{\bm{C}}{}^{Q}\nabla_{{\bm{A}}{\bm{A}}^{\prime}}\epsilon^{\bm{B}}{}_{Q}$,
where
$\nabla_{{\bm{A}}{\bm{A}}^{\prime}}={\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}{}^{AA^{\prime}}\nabla_{AA^{\prime}}$.
In Newman–Penrose language, the spin connection coefficients
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{C}}}=\Gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{C}}{\bm{B}}}$
are exactly the spin coefficients
$\gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{C}}}$ (see [PR86], Summary of
Vol. 1). When integrating over $\mathrm{SU}(2)$, $\mu$ denotes the normalized
Haar measure on $\mathrm{SU}(2)$. Throughout the majority of the paper we will
be working on _rescaled_ Minkowski space (in the F-gauge), and we shall denote
objects (fields, connections, spin dyad) on this spacetime plainly. When
working on physical Minkowski space, we will denote objects with a tilde, e.g.
$\tilde{\phi}$.

##  2 Geometric setup

In this section we discuss the general geometric setup for our analysis in a
neighbourhood of spatial and null infinity.

###  2.1 Point representation of spatial infinity: the Penrose gauge

Let $(y^{\mu})$ be Cartesian coordinates on the Minkowski spacetime
$(\mathbb{R}^{4},\tilde{{\bm{\eta}}})$ with the standard metric
$\tilde{{\bm{\eta}}}=\tilde{\eta}_{\mu\nu}\mathbf{d}y^{\mu}\otimes\mathbf{d}y^{\nu}$,
where $\tilde{\eta}_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$. In spherical
coordinates we have that

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

where $t=y^{0}$, $r^{2}=\sum_{i=1}^{3}y_{i}^{2}$, and ${\bm{\sigma}}$ denotes
the standard metric on $\mathbb{S}^{2}$. On the region

$\mathcal{N}\equiv\\{(y^{\mu})\in\mathbb{R}^{4}\;|\;\tilde{\eta}_{\mu\nu}y^{\mu}y^{\nu}<0\\}$

which includes the asymptotic region around spatial infinity, we consider the
coordinate inversion

$y^{\mu}\mapsto x^{\mu}=-\frac{y^{\mu}}{y_{\nu}y^{\nu}},$

and formally extend the domain of validity of the coordinates $(x^{\mu})$ to
include the set
$\mathscr{I}^{\prime}=\\{x^{\mu}x_{\mu}=0\\}=\\{y^{\mu}y_{\mu}=-\infty\\}$.
The set $\mathscr{I}^{\prime}$ formally forms the part of the boundary of
$\mathcal{N}$ at infinity. Defining $\rho^{2}\equiv\sum_{i=1}^{3}x_{i}^{2}$,
the metric $\tilde{{\bm{\eta}}}$ reads

$\tilde{{\bm{\eta}}}=\frac{1}{(x^{\sigma}x_{\sigma})^{2}}\tilde{\eta}_{\mu\nu}\mathbf{d}x^{\mu}\otimes\mathbf{d}x^{\nu}=\frac{1}{\left(\rho^{2}-(x^{0})^{2}\right)^{2}}\left(\mathbf{d}\,(x^{0})\otimes\mathbf{d}\,(x^{0})-\mathbf{d}\rho\otimes\mathbf{d}\rho-\rho^{2}{\bm{\sigma}}\right),$
(1)

where on $\mathcal{N}$ the inverted coordinates $x^{0}$ and $\rho$ are given
in terms of $t$ and $r$ by

$\rho=\frac{r}{r^{2}-t^{2}},\qquad x^{0}=\frac{t}{r^{2}-t^{2}}.$

The right-hand side of (1) is a conformally rescaled Minkowski metric in the
new coordinates $(x^{\mu})$, where one notices that on $\mathcal{N}$ the
conformal factor

$\Xi=-x^{\mu}x_{\mu}=-\frac{1}{y^{\mu}y_{\mu}}=\rho^{2}-(x^{0})^{2}$

extends smoothly to $\mathscr{I}^{\prime}$. Moreover, the metric
$\eta^{\prime}_{\mu\nu}\equiv\Xi^{2}\tilde{\eta}_{\mu\nu}$ (which is just the
Minkowski metric again) does too, and now contains at its origin the point

$i^{0}\equiv\\{(x^{\mu})\in\mathbb{R}^{4}\;|\;x^{\mu}=0\\}$

called _spatial infinity_ , formally a point on the $\mathscr{I}^{\prime}$
part of the boundary of $\mathcal{N}$. The sets

$\mathscr{I}^{\pm}\equiv\mathscr{I}^{\prime}\cap\\{(x^{\mu})\in\mathbb{R}^{4}\;|\;\eta_{\mu\nu}x^{\mu}x^{\nu}=0,\;\pm
x^{0}>0\\}$

are called the future and past _null infinity_ , respectively, of Minkowski
space (near $i^{0}$), and are the hypersurfaces which null geodesics in
$\mathcal{N}$ approach asymptotically (see Figure 3).

$\mathcal{N}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$i^{0}$$\mathcal{N}$$r$$t$

Figure 3: The point representation of spatial infinity $i^{0}$ with its
neighbourhood $\mathcal{N}$. The left diagram depicts the region $\mathcal{N}$
in physical space. The right diagram depicts a conformal extension of the
region which includes spatial infinity and a portion of null infinity.

###  2.2 Cylinder representation of spatial infinity: the F-gauge

In the construction above, the endpoint $i^{0}$ of all _spacelike_ geodesics
in Minkowski space is collapsed to a point, which for our purposes turns out
to be somewhat unnatural (compare this, for example, to $\mathscr{I}^{\pm}$,
the end _surfaces_ of null geodesics). Instead, it is useful to blow up
$i^{0}$ to a cylinder as follows. Set121212Note that in terms of the physical
coordinates $t$ and $r$, the new coordinate $\tau$ is given by $\tau=t/r$.
$x^{0}=\rho\tau$, and define a new conformal factor131313In terms of physical
coordinates $t$ and $r$, this new conformal factor is simply $\Theta=1/r$.
This conformal scale in fact appears in the literature on the theory of
conformal scattering [MN04, MN12, NT22], albeit without the use of the
F-coordinates $\tau$ and $\rho$. It is the definition of these coordinates
that is at the core of the blow-up of $i^{0}$ to a cylinder.

$\Theta\equiv\frac{1}{\rho}\Xi=\rho\left(1-\tau^{2}\right).$

We then define the unphysical metric

$\displaystyle{\bm{\eta}}$ $\displaystyle\equiv\Theta^{2}\tilde{{\bm{\eta}}}$
$\displaystyle=\frac{1}{\rho^{2}}\left(\rho^{2}\mathbf{d}\tau\otimes\mathbf{d}\tau+\tau\rho\left(\mathbf{d}\tau\otimes\mathbf{d}\rho+\mathbf{d}\rho\otimes\mathbf{d}\tau\right)-(1-\tau^{2})\,\mathbf{d}\rho\otimes\mathbf{d}\rho-\rho^{2}{\bm{\sigma}}\right).$

We call the coordinates $\tau$ and $\rho$ the _F-coordinates_ on Minkowski
space. In terms of $\tau$ and $\rho$, the region $\mathcal{N}$ then reads

$\mathcal{N}=\left\\{(\tau,\rho)\times\mathbb{S}^{2}\,|\,-1<\tau<1,\;\rho>0\right\\}\simeq(-1,1)_{\tau}\times(0,\infty)_{\rho}\times\mathbb{S}^{2}$

and

$\mathscr{I}^{\pm}=\left\\{(\tau,\rho)\times\mathbb{S}^{2}\,|\,\tau=\pm
1,\;\rho>0\right\\}\simeq(0,\infty)_{\rho}\times\mathbb{S}^{2}.$

It is also convenient to introduce the sets

$\displaystyle\mathcal{I}\equiv\left\\{(\tau,\rho)\times\mathbb{S}^{2}\,|\,|\tau|<1,\;\rho=0\right\\}\simeq(-1,1)_{\tau}\times\mathbb{S}^{2},$
$\displaystyle\mathcal{I}^{0}\equiv\left\\{(\tau,\rho)\times\mathbb{S}^{2}\,|\,\tau=0,\;\rho=0\right\\}\simeq\mathbb{S}^{2},$
$\displaystyle\mathcal{I}^{\pm}\equiv\left\\{(\tau,\rho)\times\mathbb{S}^{2}\,|\,\tau=\pm
1,\;\rho=0\right\\}\simeq\mathbb{S}^{2}.$

In this representation the point
$i^{0}=\\{x^{\mu}=0\\}=\\{\rho=0,\,x^{0}=0\\}$ has therefore been blown to the
cylinder $\mathcal{I}$. Note that although $\mathcal{I}$ is at a finite
$\rho$-coordinate, it is still at infinity with respect to the metric
${\bm{\eta}}$. We call $\mathcal{I}$ the _cylinder at spatial infinity_ and
$\mathcal{I}^{\pm}$ the _critical sets_.

Let us denote the standard initial hypersurface by

$\tilde{\mathcal{S}}\equiv\\{(y^{\mu})\in\mathbb{R}^{4}\;|\;t=y^{0}=0\\}$

and set

$\mathcal{S}\equiv\tilde{\mathcal{S}}\cup\mathcal{I},\qquad\overline{\mathcal{I}}\equiv\mathcal{I}\cup\mathcal{I}^{+}\cup\mathcal{I}^{-},\qquad\overline{\mathcal{N}}\equiv\mathcal{N}\cup\mathscr{I}^{+}\cup\mathscr{I}^{-}\cup\overline{\mathcal{I}},$

and consider $\rho$, $\tau$ and the spherical coordinates ${\bm{\sigma}}$ as
coordinates on

$\overline{\mathcal{N}}\simeq[-1,1]_{\tau}\times[0,\infty)_{\rho}\times\mathbb{S}^{2}_{\bm{\sigma}}.$

In these coordinates the expressions for $\Theta$ and $\Xi=\rho\Theta$, and
the coordinate expression for the inverse metric ${\bm{\eta}}^{\sharp}$ all
extend smoothly to all points of $\overline{\mathcal{N}}$. Observe, however,
that ${\bm{\eta}}$ itself degenerates at $\rho=0$, so that ${\bm{\eta}}$ is
only smooth on $\mathcal{N}\cup\mathscr{I}^{+}\cup\mathscr{I}^{-}$. We call
this conformal completion of $\mathcal{N}$, together with the F-coordinates
$\tau$ and $\rho$, the _F-gauge_ [Fri98a, Fri98].

$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\mathcal{I}^{0}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\tau=0$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{I}$

Figure 4: The cylinder at spatial infinity in the F-gauge. On the left, a
depiction suppressing one angular dimension. On the right, a cross section.
This type of 2-dimensional diagram is a convenient way of depicting the
locations of various hypersurfaces in the F-gauge. Observe that this is a
_coordinate diagram_ and not a Penrose diagram. In particular, null geodesics
do not correspond to lines with slope of $\pm 45^{\circ}$.

###  2.3 Null frame near $\mathcal{I}^{0}$

In order to perform estimates, we will write the field equations in terms of a
null frame
$\\{{\bm{e}}_{\bm{a}}={\bm{e}}_{\bm{a}}^{\mu}{\bm{\partial}}_{\mu}\\}$. We
make use of the Infeld-van der Waerden symbols to _spinorise_ the frame
indices,
${\bm{e}}_{\bm{a}}\mapsto{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}^{\mu}{\bm{\partial}}_{\mu}$,
and introduce a frame ${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$ satisfying

${\bm{\eta}}({\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}},{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}})=\epsilon_{{\bm{A}}{\bm{B}}}\epsilon_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}.$

Specifically, we choose

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

and complex vector fields ${\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}$ and
${\bm{e}}_{{\bm{1}}{\bm{0}}^{\prime}}$ which are tangent to the spheres
$\mathbb{S}^{2}_{\tau,\rho}$ of constant $\tau$ and $\rho$. However, non-
vanishing smooth vector fields on $\mathbb{S}^{2}$ cannot be defined globally,
so we lift the dimension of the spheres by considering _all possible choices_
of such vector fields. Specifically, for each $(\tau,\rho)$ we consider the
Hopf map
$p:\mathrm{SU}(2)\simeq\mathbb{S}^{3}\to\mathbb{S}^{2}\simeq\mathrm{SU}(2)/\mathrm{U}(1)$,
which defines a principal $\mathrm{U}(1)$-bundle over $\mathbb{S}^{2}$. This
map induces a group of rotations ${\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}\mapsto
e^{i\alpha}{\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}$, $\alpha\in\mathbb{R}$, which
leaves the tangent bundle of $\mathbb{S}^{2}_{\tau,\rho}$ invariant. By
lifting each $2$-sphere along the Hopf map, we obtain a 5-dimensional manifold
$[-1,1]_{\tau}\times(0,\infty)_{\rho}\times\mathrm{SU}(2)$; note that this is
a 5-dimensional submanifold of the bundle of normalised spin frames on
$\overline{\mathcal{N}}\setminus\overline{\mathcal{I}}$.

The geometric structures on
$\overline{\mathcal{N}}\setminus\overline{\mathcal{I}}$ can be naturally
lifted to the 5-dimensional manifold introduced in the previous paragraph.
Considering $\tau$, $\rho$ and $t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)$ as
coordinates on this manifold, the lifted vector fields in (2) have the same
coordinate expressions as before. Allowing $\rho$ to take the value $0$, one
can extend all geometric structures to include this value. Thus, one considers
$(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})$ as coordinates on the extended manifold,
which we again denote by

$\overline{\mathcal{N}}\equiv[-1,1]_{\tau}\times[0,\infty)_{\rho}\times\mathrm{SU}(2)_{t^{\bm{A}}_{\phantom{{\bm{A}}}{\bm{B}}}}.$

In this setting now

$\mathcal{I}\simeq(-1,1)_{\tau}\times\mathrm{SU}(2),\quad\mathcal{I}^{0}\simeq\mathrm{SU}(2),\quad\mathcal{I}^{\pm}\simeq\mathrm{SU}(2),\quad\text{and}\quad\mathscr{I}^{\pm}\simeq(0,\infty)_{\rho}\times\mathrm{SU}(2)$

as subsets of $\overline{\mathcal{N}}$. To complete this construction, it
remains to choose a frame of complex vector fields on $\mathrm{SU}(2)$. We set

${\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}=-\frac{1}{\sqrt{2}}{\bm{X}}_{+}\quad\text{and}\quad{\bm{e}}_{{\bm{1}}{\bm{0}}^{\prime}}=-\frac{1}{\sqrt{2}}{\bm{X}}_{-},$

where ${\bm{X}}_{\pm}$ are vector fields on $\mathrm{SU}(2)$ defined in
Appendix A (briefly, they are complex linear combinations of two of the three
left-invariant vector fields on $\mathrm{SU}(2)$). The connection form on the
bundle of normalised spin frames then defines connection coefficients
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{B}}{\bm{C}}}=\Gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{C}}{\bm{B}}}$
with respect to the frame ${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}$, the only
independent non-zero values of which are

$\Gamma_{{\bm{0}}{\bm{0}}^{\prime}{\bm{0}}{\bm{1}}}=\Gamma_{{\bm{1}}{\bm{1}}^{\prime}{\bm{0}}{\bm{1}}}=-\frac{1}{2\sqrt{2}}.$

In Newman–Penrose language, these correspond to the spin coefficients
$\varepsilon$ and $\gamma$, which are gauge quantities with respect to tetrad
rescalings in the compacted spin coefficient formalism—see §4.12, [PR84].

###  2.4 Spin-$s$ equations in the F-gauge

Let

$\phi_{A_{1}\dots A_{2s}}=\phi_{(A_{1}\dots A_{2s})}$

be a totally symmetric spinor of valence $2s$, with
$s\in\frac{1}{2}\mathbb{N}$, where we take
$\mathbb{N}=\\{1,\,2,\,3,\,\ldots\\}$, i.e. we do not consider the spin zero
case (the wave equation). The massless spin-$s$ equations for $\phi$ on
$(\mathcal{N},\bm{\eta})$ then read

$\nabla^{Q}{}_{A^{\prime}}\phi_{QA_{1}\dots A_{2s-1}}=0.$ (3)

It is worth recalling that the spin-$s$ equations are conformally invariant.
That is, $\phi_{A_{1}\ldots A_{2s}}$ satisfies (3) on
$(\mathcal{N},\bm{\eta})$, where $\bm{\eta}=\Theta^{2}\tilde{\bm{\eta}}$ is
the unphysical Minkowski metric, if and only if the _physical_ spin-$s$ field

$\tilde{\phi}_{A_{1}\ldots A_{2s}}=\Theta\phi_{A_{1}\ldots A_{2s}}$

satisfies

$\tilde{\nabla}^{Q}{}_{A^{\prime}}\tilde{\phi}_{QA_{1}\ldots A_{2s-1}}=0$

on $(\mathcal{N},\tilde{\bm{\eta}})$ [PR84]. Conclusions about $\phi$ can
therefore be easily translated into conclusions about the physical field
$\tilde{\phi}$.

In the F-gauge equation (3) is equivalent to the system

$\displaystyle
A_{k}\equiv(1+\tau)\partial_{\tau}\phi_{k+1}-\rho\partial_{\rho}\phi_{k+1}-{\bm{X}}_{+}\phi_{k}+(k+1-s)\phi_{k+1}=0,$
(4a) $\displaystyle
B_{k}\equiv(1-\tau)\partial_{\tau}\phi_{k}+\rho\partial_{\rho}\phi_{k}-{\bm{X}}_{-}\phi_{k+1}+(k-s)\phi_{k}=0,$
(4b)

for $k=0,\,\ldots,\,2s-1$, where $\phi_{k}$ are the $2s$ independent
components of the spinor $\phi_{A_{1}\cdots A_{2s}}$ with respect to a spin
dyad $\\{o^{A},\iota^{A}\\}$ corresponding to the frame
$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$. The derivation of equations (4a)
and (4b) is given in Appendix B. We note here that the component
$\phi_{2s}=\phi_{A_{1}\ldots A_{2s}}\iota^{A_{1}}\ldots\iota^{A_{2s}}$ is the
_outgoing radiation field_ on $\mathscr{I}^{+}$, while
$\phi_{0}=\phi_{A_{1}\ldots A_{2s}}o^{A_{1}}\ldots o^{A_{2s}}$ is the
_incoming radiation field_ on $\mathscr{I}^{-}$.

##  3 Estimates near $\mathscr{I}^{+}$

In this section we construct estimates for the massless spin-$s$ equations
which allow us to control the solutions up to and including $\mathscr{I}^{+}$
in terms of suitable norms on initial data prescribed on a Cauchy
hypersurface. In doing so we recap the estimates of Friedrich [Fri03] and
extend them to fields of arbitrary spin $s$.

###  3.1 Geometric setup

We use the F-gauge as discussed in Sections 2.2 and 2.3. Given $t\in(-1,1]$,
$t>\tau_{\star}\in(-1,1)$, and $\rho_{\star}>0$, one defines the sets

$\displaystyle\mathcal{N}_{t}\equiv\bigg{\\{}(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})\,|\,\tau_{\star}\leqslant\tau\leqslant
t,\;0\leqslant\rho\leqslant\frac{\rho_{\star}}{1+\tau},\;t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\bigg{\\}}\subset[-1,1]_{\tau}\times[0,\infty)_{\rho}\times\mathrm{SU}(2),$
$\displaystyle\mathcal{S}_{t}\equiv\bigg{\\{}(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})\,|\,\tau=t,\;0\leqslant\rho\leqslant\frac{\rho_{\star}}{1+t},\;t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\bigg{\\}},$
$\displaystyle\mathcal{B}_{t}\equiv\bigg{\\{}(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})\,|\,\tau_{\star}\leqslant\tau\leqslant
t,\;\rho=\frac{\rho_{\star}}{1+\tau},\;t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\bigg{\\}},$
$\displaystyle\mathcal{I}_{t}\equiv\bigg{\\{}(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})\,|\,\tau_{\star}\leqslant\tau\leqslant
t,\;\rho=0,\;t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\bigg{\\}},$

and $\mathcal{S}_{\star}\equiv\mathcal{S}_{\tau_{\star}}$. A schematic
depiction of these sets is given in Figure 5.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{N}_{t}$$\mathcal{I}_{t}$$\mathcal{I}^{+}$$\bullet$$\mathcal{I}^{-}$$\bullet$$\mathcal{S}_{\star}$$\mathcal{B}_{t}$$\mathcal{S}_{t}$

Figure 5: Schematic depiction of the domain used in the construction of
estimates controlling the solutions to the spin-$s$ equations near
$\mathscr{I}^{+}$ in terms of Cauchy initial data.

###  3.2 Construction of estimates

Given non-negative integers $p$, $p^{\prime}$, $q$, $q^{\prime}$ and a
multiindex $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$, we consider the
differential operators

$D\equiv
D^{q,p,\alpha}\equiv\partial^{q}_{\tau}\partial^{p}_{\rho}{\bm{Z}}^{\alpha}\quad\text{and}\quad
D^{\prime}\equiv
D^{q^{\prime},p^{\prime},\alpha}\equiv\partial^{q^{\prime}}_{\tau}\partial^{p^{\prime}}_{\rho}{\bm{Z}}^{\alpha},$

where ${\bm{Z}}^{\alpha}$ denotes the vector fields on $\mathrm{SU}(2)$
introduced in Section A.2.2. We then have the following estimates controlling
the solutions to the spin-$s$ equations up to $\mathscr{I}^{+}$.

######  Proposition 3.1.

Let $\tau_{\star}\in(-1,1)$, $t>\tau_{\star}$, and consider the field
$\phi_{A_{1}\cdots A_{2s}}$ satisfying equations (34a) and (34b) in the region
$\mathcal{N}_{t}$. Let $(p,m)\in\mathbb{N}\times\mathbb{N}$ be such that

$p>m+s,$

and suppose that

$\sum_{k=0}^{2s}\int_{\mathcal{S}_{\star}}\sum_{q^{\prime}+p^{\prime}+|\alpha|\leqslant
m}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu<\infty.$

Then there exists a constant $C_{p,m,s}>0$ which is independent of $t$ such
that for all $0\leqslant k\leqslant 2s$

$\|\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\mathcal{N}_{t})}\equiv\int_{\mathcal{N}_{t}}\sum_{q^{\prime}+p^{\prime}+|\alpha|\leqslant
m}|D^{\prime}(\partial^{p}_{\rho}\phi_{k})|^{2}\,\mathrm{dv}\leqslant
C_{p,m,s}\sum_{k=0}^{2s}\int_{\mathcal{S}_{\star}}\sum_{q^{\prime}+p^{\prime}+|\alpha|\leqslant
m}|D^{\prime}(\partial^{p}_{\rho}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu.$
(5)

###### Proof.

Applying $D$ to the equations (34a) and (34b), multiplying by
$\overline{D\phi_{k}}$ and $\overline{D\phi_{k+1}}$ respectively, and summing
the real parts, one obtains the identity

$2\operatorname{Re}\left(\overline{D\phi_{k+1}}DA_{k}[\phi]+\overline{D\phi_{k}}DB_{k}[\phi]\right)=0.$

Commuting $D$ into the operators $A_{k}[\phi]$ and $B_{k}[\phi]$, this may be
rewritten as

$\displaystyle\begin{split}0&=\partial_{\tau}\left((1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\right)+\partial_{\rho}\left(-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\right)\\\
&\phantom{=}-{\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})-{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})\\\
&\phantom{=}-{\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})-{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})\\\
&\phantom{=}+2(k+1-s+q-p)|D\phi_{k+1}|^{2}+2(k-s-q+p)|D\phi_{k}|^{2}.\end{split}$
(6)

It is worth noting at this stage that the numerical factors in the last two
terms in (6), arising from the explicit appearance of the coordinates $\tau$
and $\rho$ in equations (34a) and (34b), are a key aspect of the calculation.
In particular, their signs may be manipulated by choosing the values of $q$
and $p$ appropriately. We now integrate eq. 6 over $\mathcal{N}_{t}$ (cf.
Figure 5) with respect to the _Euclidean_ volume element
$\mathrm{dv}=\operatorname{d}\\!\tau\wedge\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$,
where $\operatorname{d}\\!\mu$ is the Haar measure on $\mathrm{SU}(2)$. We
have

$\displaystyle 0$
$\displaystyle=\int_{\mathcal{N}_{t}}\left(\begin{array}[]{c}\partial_{\tau}\\\
\partial_{\rho}\end{array}\right)\cdot\left(\begin{array}[]{c}(1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\\\
-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\end{array}\right)\mathrm{dv}$
$\displaystyle\phantom{=}-\int_{\mathcal{N}_{t}}\Big{(}{\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})+{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})$
$\displaystyle\phantom{=}\hskip
20.00003pt+{\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})+{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})\Big{)}\,\mathrm{dv}$
$\displaystyle\phantom{=}+2(k+1-s+q-p)\int_{\mathcal{N}_{t}}|D\phi_{k+1}|^{2}\,\mathrm{dv}+2(k-s-q+p)\int_{\mathcal{N}_{t}}|D\phi_{k}|^{2}\,\mathrm{dv}.$

Using Lemma A.1, it is straightforward to check that the second integral in
the above equality will vanish when summed over $|\alpha|\leqslant m^{\prime}$
for a given $m^{\prime}$; as we will employ such a summation shortly, we
simply write $\text{angular}(\alpha)$ to denote the second integral in the
above. Next, using the Euclidean divergence theorem, one obtains

$\displaystyle 0$
$\displaystyle=\int_{\mathcal{S}_{t}}\left((1+t)|D\phi_{k+1}|^{2}+(1-t)|D\phi_{k}|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle\phantom{=}-\int_{\mathcal{S}_{\star}}\left((1+\tau_{\star})|D\phi_{k+1}|^{2}+(1-\tau_{\star})|D\phi_{k}|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle\phantom{=}+\int_{\mathcal{I}_{t}}\rho\left(|D\phi_{k}|^{2}-|D\phi_{k+1}|^{2}\right)\operatorname{d}\\!\tau\wedge\operatorname{d}\\!\mu$
$\displaystyle\phantom{=}+\int_{\mathcal{B}_{t}}\left(\begin{array}[]{c}(1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\\\
-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\end{array}\right)\cdot\nu\left(\begin{array}[]{c}\rho\\\
1+\tau\end{array}\right)\operatorname{d}\\!\mathcal{B}+\text{angular}(\alpha)$
$\displaystyle\phantom{=}+2(p-q+k-s)\int_{\mathcal{N}_{t}}|D\phi_{k}|^{2}\,\mathrm{dv}-2(p-q-k-1+s)\int_{\mathcal{N}_{t}}|D\phi_{k+1}|^{2}\,\mathrm{dv},$

where $\operatorname{d}\\!\mathcal{B}$ is the induced measure on
$\mathcal{B}_{t}$, $\nu\equiv(\rho^{2}+(1+\tau)^{2})^{-1/2}$ is a
normalisation factor for the outward normal to $\mathcal{B}_{t}$, and we note
that the integral over $\mathcal{I}_{t}$ vanishes due to the factor of $\rho$
in the integrand. Further, the integral over $\mathcal{B}_{t}$ simplifies to
just
$\int_{\mathcal{B}_{t}}2\nu\rho|D\phi_{k}|^{2}\operatorname{d}\\!\mathcal{B}$,
and in particular is manifestly non-negative. Altogether therefore

$\displaystyle\int_{\mathcal{S}_{t}}\left((1+t)|D\phi_{k+1}|^{2}+(1-t)|D\phi_{k}|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+2(p-q+k-s)\int_{\mathcal{N}_{t}}|D\phi_{k}|^{2}\,\mathrm{dv}$
$\displaystyle\leqslant\int_{\mathcal{S}_{\star}}\left((1+\tau_{\star})|D\phi_{k+1}|^{2}+(1-\tau_{\star})|D\phi_{k}|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+2(p-q-k-1+s)\int_{\mathcal{N}_{t}}|D\phi_{k+1}|^{2}\,\mathrm{dv}$
$\displaystyle-\text{angular}(\alpha).$

We now perform the relabelling

$q\longrightarrow q^{\prime},\qquad p\longrightarrow p^{\prime}+p,$

under which $D\longrightarrow D^{\prime}\partial_{\rho}^{p}$, where
$D^{\prime}=\partial_{\tau}^{q^{\prime}}\partial_{\rho}^{p^{\prime}}{\bm{Z}}^{\alpha}$,
and sum both sides over

$\mho\equiv\left\\{(q^{\prime},p^{\prime},\alpha)\,|\,q^{\prime}+p^{\prime}+|\alpha|\leqslant
m\right\\}.$

Then $\sum_{\mho}\text{angular}(\alpha)=0$, and we obtain

$\displaystyle(1+t)$
$\displaystyle\sum_{\mho}\int_{\mathcal{S}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+(1-t)\sum_{\mho}\int_{\mathcal{S}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle+2\sum_{\mho}(p^{\prime}+p-q^{\prime}+k-s)\int_{\mathcal{N}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\mathrm{dv}$
$\displaystyle\leqslant(1+\tau_{\star})\sum_{\mho}\int_{\mathcal{S}_{\star}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1}|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+(1-\tau_{\star})\sum_{\mho}\int_{\mathcal{S}_{\star}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle+2\sum_{\mho}(p^{\prime}+p-q^{\prime}-k-1+s)\int_{\mathcal{N}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\mathrm{dv}.$

We now choose $p$ and $m$ such that

$p>m+s$

and use simple uniform bounds for the numerical factors in the above,

$\displaystyle\begin{split}(1+t)&\sum_{\mho}\int_{\mathcal{S}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+(1-t)\sum_{\mho}\int_{\mathcal{S}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\\\
&+2(p-m-s+k)\sum_{\mho}\int_{\mathcal{N}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\mathrm{dv}\\\
&\leqslant(1+\tau_{\star})\sum_{\mho}\int_{\mathcal{S}_{\star}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+(1-\tau_{\star})\sum_{\mho}\int_{\mathcal{S}_{\star}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\\\
&+2(p+m+s-1)\sum_{\mho}\int_{\mathcal{N}_{t}}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\mathrm{dv}.\end{split}$
(7)

With the restriction $p>m+s$ on $p$ and $m$, the bulk integral on the left-
hand side is positive, so we deduce for $0\leqslant k\leqslant 2s-1$ that

$\displaystyle\begin{split}\int_{\mathcal{S}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu&\lesssim\int_{\mathcal{S}_{\star}}\sum_{\mho}\left(|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}+|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\\\
&\phantom{\lesssim}+2(p+m+s-1)\int_{\tau_{\star}}^{t}\left(\int_{\mathcal{S}_{\tau}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\right)\operatorname{d}\\!\tau.\end{split}$
(8)

The inequality (8) provides an estimate for all components of
$\phi_{A_{1}\cdots A_{2s}}$ except $\phi_{0}$, the incoming radiation field.
For this, we note that (7) also implies

$\displaystyle\begin{split}2(p-m-s)\int_{\mathcal{N}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{0})|^{2}\,\mathrm{dv}&\lesssim\int_{\mathcal{S}_{\star}}\sum_{\mho}\left(|D^{\prime}(\partial_{\rho}^{p}\phi_{0})|^{2}+|D^{\prime}(\partial_{\rho}^{p}\phi_{1})|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\\\
&\phantom{\lesssim}+\int_{\mathcal{N}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{1})|^{2}\,\mathrm{dv},\end{split}$
(9)

so that the norm
$\|\partial_{\rho}^{p}\phi_{0}\|^{2}_{H^{m}(\mathcal{N}_{t})}$ will be
controlled if the norm
$\|\partial_{\rho}^{p}\phi_{1}\|^{2}_{H^{m}(\mathcal{N}_{t})}$ is controlled.
Now applying Grönwall’s inequality to (8), we obtain

$\int_{\mathcal{S}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\lesssim_{p,m,s}\int_{\mathcal{S}_{\star}}\sum_{\mho}\left(|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}+|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu,$

which, upon integration, gives

$\int_{\mathcal{N}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\mathrm{dv}\lesssim_{p,m,s}\int_{\mathcal{S}_{\star}}\sum_{\mho}\left(|D^{\prime}(\partial_{\rho}^{p}\phi_{k+1})|^{2}+|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$

for $0\leqslant k\leqslant 2s-1$. This, combined with (9), then gives the
estimate for $\phi_{0}$,

$\int_{\mathcal{N}_{t}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{0})|^{2}\,\mathrm{dv}\lesssim_{p,m,s}\int_{\mathcal{S}_{\star}}\sum_{\mho}\left(|D^{\prime}(\partial_{\rho}^{p}\phi_{1})|^{2}+|D^{\prime}(\partial_{\rho}^{p}\phi_{0})|^{2}\right)\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu.$

We therefore conclude that for all $0\leqslant k\leqslant 2s$ and for $p>m+s$

$\|\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\mathcal{N}_{t})}\leqslant
C_{p,m,s}\sum_{k=0}^{2s}\int_{\mathcal{S}_{\star}}\sum_{\mho}|D^{\prime}(\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu,$
(5)

where in particular the constant $C_{p,m,s}$ is independent of $t$. ∎

######  Remark 1.

Observe that the right-hand side of (5) in principle contains large numbers of
derivatives in $\tau$. Using the spin-$s$ equations (34a) and (34b), however,
these can be eliminated entirely at the expense of producing derivatives in
$\rho$ and along $\mathrm{SU}(2)$. That is, the right-hand side may be
estimated by the initial data and its tangential derivatives on
$\mathcal{S}_{\star}$.

######  Remark 2.

As the right-hand side of (5) does not depend on $t$, one has that the left-
hand side is uniformly bounded as $t\to 1$. Therefore for any $t\in(-1,1]$, as
the boundary $\partial\mathcal{N}_{t}$ of the 5-dimensional domain
$\mathcal{N}_{t}$ is Lipschitz, the Sobolev embedding theorem gives for
$m,r\in\mathbb{N}$, $\alpha\in[0,1)$ such that $m\geqslant
r+\alpha+\frac{5}{2}$ the continuous embedding

$H^{m}(\mathcal{N}_{t})\hookrightarrow C^{r,\alpha}(\mathcal{N}_{t}).$

In particular, if $\alpha\leqslant\frac{1}{2}$ we have

$H^{r+3}(\mathcal{N}_{t})\hookrightarrow C^{r,\alpha}(\mathcal{N}_{t}).$

Note that it is important here that $\mathcal{N}_{t}$ is an open set, i.e. we
consider $\mathcal{N}_{t}$ without its boundary.

###  3.3 Asymptotic expansions near $\mathcal{I}$

Using estimate (5) and the Sobolev embedding theorem as described above, one
finds that for any $0\leqslant k\leqslant 2s$, $r\in\mathbb{N}$, and
$0\leqslant\alpha\leqslant\frac{1}{2}$,

$\partial^{p}_{\rho}\phi_{k}\in
C^{r,\alpha}(\mathcal{N}_{1})\quad\text{for}\quad p\geqslant r+s+4,$

provided the data on $\mathcal{S}_{\star}$ is sufficiently regular, as per
Proposition 3.1. This allows one to ensure the desired regularity of the
solution near $\mathscr{I}^{+}$ by providing sufficiently regular data on
$\mathcal{S}_{\star}$. Now, using the wave equations of Appendix C on
$\mathcal{I}$, one may solve for the functions

$\phi_{k}^{(p^{\prime})}(\tau,t^{\bm{A}}{}_{\bm{B}})\equiv\partial_{\rho}^{p^{\prime}}\phi_{k}|_{\mathcal{I}},$

$0\leqslant k\leqslant 2s$, on $\mathcal{I}$ for any finite order
$p^{\prime}\in\mathbb{N}\cup\\{0\\}$. The functions $\phi_{k}^{(p^{\prime})}$
are computable explicitly as a consequence of the fact that the cylinder at
spatial infinity $\mathcal{I}$ is a total characteristic of the spin-$s$
equations. Regarding these functions as functions on $\mathcal{N}_{1}$ which
are independent of $\rho$, one may integrate $\partial^{p}_{\rho}\phi_{k}$
with respect to $\rho$ to obtain for $p\geqslant r+s+4$ the expansion

$\phi_{k}=\sum_{p^{\prime}=|k-2s|}^{p-1}\frac{1}{p^{\prime}!}\phi^{(p^{\prime})}_{k}\rho^{p^{\prime}}+J^{p}(\partial_{\rho}^{p}\phi_{k})$
(10)

on $\mathcal{N}_{1}$, where $J$ denotes the operator

$J\,:\,f\mapsto
J(f)=\int_{0}^{\rho}f(\tau,\rho^{\prime},t^{\bm{A}}{}_{\bm{B}})\operatorname{d}\\!\rho^{\prime}.$

It follows then that

$J^{p}(\partial^{p}_{\rho}\phi_{k})=\phi_{k}-\sum_{p^{\prime}=0}^{p-1}\frac{1}{p^{\prime}!}\phi^{(p^{\prime})}_{k}\rho^{p^{\prime}}\in
C^{r,\alpha}(\mathcal{N}_{1})\quad\text{for}\quad p\geqslant r+s+4$ (11)

for any given $r\in\mathbb{N}$. One therefore obtains an expansion of the
solution to the spin-$s$ equations in terms of the explicitly known functions
$\phi_{k}^{(p^{\prime})}$, $0\leqslant k\leqslant 2s$, $0\leqslant
p^{\prime}\leqslant p-1$, and a remainder of prescribed smoothness. We call
such expansions F-expansions. The details of the explicit computation of the
terms in the F-expansions is given in Appendix C.

######  Remark 3.

Notice that the functions $\phi_{k}^{(p^{\prime})}$ completely control whether
the solution $\phi$ extends smoothly to $\mathscr{I}^{+}$ or acquires
logarithmic singularities there. Since the gauge in our construction is
defined in terms of smooth conformally invariant structures (conformal
geodesics and associated conformal Gaussian coordinates, see e.g. [MV21]) and
extends smoothly through $\mathscr{I}^{\pm}$, these singularities are not a
result of the choice of gauge. Rather, they are generated at all orders by the
interplay between the structure of the data on $\mathcal{S}_{\star}$ near
$\mathcal{I}$ and the nature of the evolution along $\mathcal{I}$.

######  Remark 4.

Inspecting the proof of Proposition 3.1, one notices that the bound in (11)
can in fact be improved to

$p\geqslant r+s+4-k,$

provided one is prepared to consider the regularity of the solution component-
by-component. In particular, setting $p=0=r$ shows that the components
$\phi_{k\geqslant s+4}$ are always continuous. This set of components is non-
empty for $s\geqslant 4$.

###  3.4 Existence of solutions

In this subsection we provide a brief argument showing the existence of
solutions to the spin-$s$ equations on the domain $\mathcal{N}_{1}$. The proof
is a classical _last slice argument_ which combines the local existence result
for symmetric hyperbolic systems with a contradiction argument.

As we are looking for solutions of the form (10) and the formal expansions are
explicitly known, we only need to argue the existence of the derivatives
$\partial_{\rho}^{p}\phi_{k}$. Given that the components $\phi_{k}$ satisfy a
symmetric hyperbolic system (cf. eqs. 33a and 33c), it follows then that
$\partial_{\rho}^{p}\phi_{k}$ also satisfy a symmetric hyperbolic system with
the same structural properties as eqs. 33a and 33c. Now, assume that on
$\mathcal{S}_{\star}$ one has

$(\partial_{\rho}^{p}\phi_{k})_{\star}\in H^{m}(\mathcal{S}_{\star}),\qquad
p>m+s.$

The existence theorem for symmetric hyperbolic systems (see e.g. [Kat75]) then
shows that there exists a $T>0$ such that

$\partial_{\rho}^{p}\phi_{k}(\tau,\cdot)\in H^{m}(\mathcal{S}_{\tau}),\qquad
0\leqslant\tau<T,$

and either $\partial_{\rho}^{p}\phi_{k}(\tau,\cdot)\not\in H^{m}(S_{\tau})$
for $\tau>T$, or $T$ is not maximal, the dependence on $\tau$ being
continuous. Assume now the existence of a time $\tau_{\ast}<1$ (i.e. a _last
slice_) such that

$\partial_{\rho}^{p}\phi_{k}(\tau,\cdot)\not\in
H^{m}(\mathcal{S}_{\tau})\quad\mbox{for}\quad\tau\in[\tau_{\star},1].$

It follows then that

$\int_{\tau_{\ast}}^{1}\|\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\mathcal{S}_{t})}\,\mathrm{d}t=\infty.$

As this integral is bounded by
$\|\partial_{\rho}^{p}\phi_{k}\|_{H^{m}(\mathcal{N}_{1})}$, the existence of a
last slice contradicts the fact that $\partial_{\rho}^{p}\phi_{k}\in
H^{m}(\mathcal{N}_{1})$, which is true by Proposition 3.1.

##  4 Asymptotic characteristic initial value problem

The purpose of this section is to provide a formulation of the _asymptotic
characteristic initial value problem_ for the massless spin-$s$ equations (3).
By this we understand a setting in which suitable initial data is prescribed
on a portion of past null infinity $\mathscr{I}^{-}$ and an incoming null
hypersurface $\underline{\mathcal{B}}_{\varepsilon}$ intersecting
$\mathscr{I}^{-}$ at a cut

$\displaystyle\mathcal{C}_{\star}\equiv\mathscr{I}^{-}\cap\underline{\mathcal{B}}_{\varepsilon}$
$\displaystyle\phantom{\mathcal{C}_{\star}}=\\{(\tau,\rho)\times\mathbb{S}^{2}\in\mathcal{N}\;|\;\tau=-1,\;\;\rho=\rho_{\star}\\},$

with $\rho_{\star}$ a constant.

###  4.1 Freely specifiable data on $\mathscr{I}^{-}$

Equations (4a) and (4b) are, respectively, transport equations along outgoing
and incoming null geodesics in the Minkowski spacetime, written in the
F-gauge. In particular, at $\mathscr{I}^{-}$ one has that

$(A_{k})|_{\mathscr{I}^{-}}=-\rho\partial_{\rho}(\phi_{k+1})|_{\mathscr{I}^{-}}-{\bm{X}}_{+}(\phi_{k})|_{\mathscr{I}^{-}}+(k+1-s)(\phi_{k+1})|_{\mathscr{I}^{-}}=0,\qquad
k=0,\,\ldots,\,2s-1,$ (12)

where $(\phi_{k})|_{\mathscr{I}^{-}}$ denotes the restriction of $\phi_{k}$ to
$\mathscr{I}^{-}$; (12) is a set of $2s$ transport equations along the null
generators of $\mathscr{I}^{-}$. The key observation is that only
$(\phi_{0})|_{\mathscr{I}^{-}}$ is not fixed by these transport equations.
This allows one to identify $\phi_{0}$ as the freely specifiable data on
$\mathscr{I}^{-}$. That is, given $(\phi_{0})|_{\mathscr{I}^{-}}$, one can
solve the ODEs (12) one by one, at least in a neighbourhood of
$\mathcal{C}_{\star}$, starting with the equation
$A_{0}|_{\mathscr{I}^{-}}=0$. One obtains $(\phi_{k+1})|_{\mathscr{I}^{-}}$
for $k=0,\,\ldots,\,2s-1$, provided that the initial values at the cut
$(\phi_{k+1})_{\star}\equiv(\phi_{k+1})|_{\mathcal{C}_{\star}}$ are also
known.

Next, let $\underline{\mathcal{B}}_{\varepsilon}$ be the incoming null
hypersurface whose generators are tangent to the vector
${\bm{e}}_{{\bm{0}}{\bm{0}}^{\prime}}$ and which intersects $\mathscr{I}^{-}$
at $\mathcal{C}_{\star}$. On $\underline{\mathcal{B}}_{\varepsilon}$ one has
that

$B_{k}|_{\underline{\mathcal{B}}_{\varepsilon}}={\bm{e}}_{{\bm{0}}{\bm{0}}^{\prime}}(\phi_{k})|_{\underline{\mathcal{B}}_{\varepsilon}}-{\bm{X}}_{-}(\phi_{k+1})|_{\underline{\mathcal{B}}_{\varepsilon}}+(k-s)(\phi_{k})|_{\underline{\mathcal{B}}_{\varepsilon}}=0$
(13)

for $k=0,\,\ldots,\,2s-1$. These are transport equations along the null
generators of $\underline{\mathcal{B}}_{\varepsilon}$, in which only the
component $(\phi_{2s})|_{\underline{\mathcal{B}}_{\varepsilon}}$ satisfies no
differential condition. Accordingly, it can be identified with the freely
specifiable data on $\underline{\mathcal{B}}_{\varepsilon}$. Similarly, one
can then solve the ODEs (13) in sequence (starting from $B_{2s-1}$) to obtain
$(\phi_{k})|_{\underline{\mathcal{B}}_{\varepsilon}}$ for all $0\leqslant
k\leqslant 2s$, provided the initial values
$(\phi_{k})|_{\star}\equiv(\phi_{k})|_{\mathcal{C}_{\star}}$, $0\leqslant
k\leqslant 2s-1$, at the cut $\mathcal{C}_{\star}$ are known.

We therefore have the following:

######  Lemma 4.1.

The full set of initial data for the asymptotic characteristic initial value
problem for the massless spin-$s$ equations can be computed from the reduced
data set consisting of

$\displaystyle\phi_{0}\quad\mbox{on}\quad\mathscr{I}^{-},$
$\displaystyle\phi_{2s}\quad\mbox{on}\quad\underline{\mathcal{B}}_{\varepsilon},\quad\text{and}$
$\displaystyle\phi_{1},\,\ldots,\,\phi_{2s-1}\quad\text{on}\quad\mathcal{C}_{\star}=\mathscr{I}^{-}\cap\underline{\mathcal{B}}_{\varepsilon}.$

###  4.2 Symmetric hyperbolicity and the local existence theorem

For reasons that will become clear shortly, in order to discuss the existence
of solutions to the asymptotic characteristic initial value problem in a
neighbourhood of $\mathcal{C}_{\star}$, it is convenient to use the evolution
equations in a slightly different conformal gauge than the one discussed in
Section 2.3. More precisely, we make use of the evolution equations Equations
40a and 40c as given in Section B.4 and assume that the smooth function $\mu$
is no longer identically equal to $1$. A long but straightforward calculation
shows that in this gauge the spin-$s$ equations (3) are equivalent to an
evolution system of the form

$\mathbf{A}^{\mu}\partial_{\mu}{\bm{\Phi}}=\mathbf{B}\cdot{\bm{\Phi}}$ (14)

where

$\displaystyle\mathbf{A}^{\mu}\partial_{\mu}=\left(\begin{array}[]{cccccc}\sqrt{2}\hat{{\bm{e}}}_{{\bm{0}}{\bm{0}}^{\prime}}&-\mu{\bm{X}}_{-}&0&\cdots&\cdots&0\\\
-\mu{\bm{X}}_{+}&2\partial_{\hat{\tau}}&-\mu{\bm{X}}_{-}&0&\cdots&0\\\
0&-\mu{\bm{X}}_{+}&2\partial_{\hat{\tau}}&-\mu{\bm{X}}_{-}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\
0&\cdots&0&-\mu{\bm{X}}_{+}&2\partial_{\hat{\tau}}&-\mu{\bm{X}}_{-}\\\
0&\cdots&\cdots&0&-\mu{\bm{X}}_{+}&\sqrt{2}\hat{{\bm{e}}}_{{\bm{1}}{\bm{1}}^{\prime}}\end{array}\right),$
$\displaystyle{\bm{\Phi}}=\big{(}\phi_{0},\,\ldots,\,\phi_{2s}\big{)}^{t},$

and $\mathbf{B}$ is a constant $2s\times 2s$ matrix. It can be readily
verified that the above matrix is Hermitian. Moreover,

$\mathbf{A}^{0}\equiv\mbox{diag}\big{(}1-\kappa^{\prime}\hat{\tau},\,2,\,\ldots,\,2,\,1+\kappa^{\prime}\hat{\tau}\big{)}.$

This matrix is positive definite away from $\mathcal{I}^{\pm}$. As
$\kappa^{\prime}=\mu^{\prime}\rho+\mu$ with $\mu(0)=1$, but not identically
$1$ away from $\rho=0$, it follows that

$\mathbf{A}^{0}|_{\mathcal{I}^{-}}=\mbox{diag}\big{(}2,\,2,\,\ldots,\,2,\,0\big{)}.$
(16)

An analogous expression holds on $\mathcal{I}^{+}$. Thus, the system (14) is
symmetric hyperbolic away from the critical sets $\mathcal{I}^{\pm}$, but
degenerates at $\mathcal{I}^{\pm}$.

######  Remark 5.

For the choice $\mu=1$ corresponding to the basic F-gauge discussed in Section
2.2, one has that

$\mathbf{A}^{0}|_{\mathscr{I}^{-}}=\mbox{diag}\big{(}2,\,2,\,\ldots,\,2,\,0\big{)}.$

That is, in this representation the corresponding symmetric hyperbolic system
degenerates on the _whole_ of $\mathscr{I}^{-}$, which creates problems when
attempting to solve the system (14). The use of the more general gauge of
Section B.4 shows that the degeneration of $\mathbf{A}^{0}$ on the whole of
$\mathscr{I}^{-}$ in the case $\mu=1$ is a _coordinate_ singularity. While
this more singular representation ($\mu=1$) does not lend itself naturally to
the discussion of local existence in a neighbourhood of
$\mathcal{C}_{\star}=\mathscr{I}^{-}\cap\underline{\mathcal{B}}_{\varepsilon}$,
it gives rise to a straightforward construction of asymptotic initial data.
Moreover, as we do in Sections 3 and 5, it allows us to construct robust
estimates.

Using the general theory set up in [Ren90] (see also [Val16], §12.5.3), one
can therefore obtain the following local existence theorem.

######  Proposition 4.2.

Given a smooth choice of reduced initial data as in Lemma 4.1, there exists a
neighbourhood $\mathcal{U}\subset
D^{+}(\mathscr{I}^{-}\cup\underline{\mathcal{B}}_{\varepsilon})$ of
$\mathcal{C}_{\star}$ in which the asymptotic characteristic initial value
problem for the massless spin-$s$ equation has a unique smooth solution
$\phi_{k}$ for $k=0,\ldots,2s$, where $D^{+}(S)$ denotes the future domain of
dependence of the set $S\subset\mathcal{N}$.

In fact, using a strategy similar to that in [Luk12], it is possible to ensure
the existence of solutions along a narrow causal rectangle along
$\mathscr{I}^{-}$ bounded away from $\mathcal{I}$ (see Figure 6). However, the
degeneracy in the matrix $\mathbf{A}^{0}$ at $\mathcal{I}^{-}$ as given in
(16) precludes one from obtaining a domain of existence which includes the
past critical point (see Remark 6). In order to make this domain extension
statement more precise, let us introduce some further notation. Given
$(\tau_{\bullet},\rho_{\bullet})\in[-1,1]\times[0,\infty)$, let

$\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}\equiv\big{\\{}(\tau,\rho,t^{\bm{A}}{}_{\bm{B}})\in\mathcal{N}\;|\;\tau=\tau_{\bullet},\;\rho=\rho_{\bullet},\;t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\big{\\}}.$

For a given choice of $(\tau_{\bullet},\rho_{\bullet})$, the set
$\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$ naturally projects to a 2-sphere
via the Hopf map. Suppose that $(\tau_{\bullet},\rho_{\bullet})$ are chosen so
that $\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$ lies in the chronological
future of $\mathcal{C}_{\star}=\mathcal{S}_{-1,\rho_{\star}}$. The _causal
diamond_ defined by the sets $\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$ and
$\mathcal{C}_{\star}$ is given by

$\mathscr{D}_{\tau_{\bullet},\rho_{\bullet}}=J^{+}(\mathcal{C}_{\star})\cap
J^{-}(\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}).$

A schematic representation of an example of such a causal diamond
$\mathscr{D}_{\tau_{\bullet},\rho_{\bullet}}$ is provided in Figure 6. The set
$\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$ can be thought of as the set of
points of intersection of future-directed null geodesics emanating from
$\mathcal{S}_{-1,\rho_{\circ}}\subset\mathscr{I}^{-}$ with the future-directed
null geodesics emanating from
$\mathcal{S}_{\tau_{\ast},\rho_{\ast}}\subset\underline{\mathcal{B}}_{\varepsilon}$,
where $\rho_{\circ}$, $\tau_{\ast}$, and $\rho_{\ast}$ depend on
$(\tau_{\bullet},\rho_{\bullet})$. Conversely, given points
$(-1,\rho_{\circ},t^{\bm{A}}{}_{\bm{B}})\in\mathscr{I}^{-}$ and
$(\tau_{\ast},\rho_{\ast},t^{\bm{A}}{}_{\bm{B}})\in\underline{\mathcal{B}}_{\varepsilon}$,
there exist coordinates $(\tau_{\bullet},\rho_{\bullet})$ depending on
$(\rho_{\circ},\tau_{\ast},\rho_{\ast})$ such that
$\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$ is the set of intersections of
the future-directed null geodesics emanating from
$\mathcal{S}_{-1,\rho_{\circ}}$ and $\mathcal{S}_{\tau_{\ast},\rho_{\ast}}$.
In this notation one then has the following.

######  Proposition 4.3.

Given $\rho_{\circ}\in(0,\rho_{\star})$, consider smooth initial data for the
asymptotic characteristic initial value problem for the spin-$s$ equations on
$\underline{\mathcal{B}}_{\varepsilon}\cup\mathscr{I}^{-}_{[\rho_{\circ},\rho_{\star}]}$
where

$\mathscr{I}^{-}_{[\rho_{\circ},\rho_{\star}]}\equiv\\{p\in\mathscr{I}^{-}\;|\;\rho(p)\in[\rho_{\circ},\rho_{\star}]\\}.$

Then there exists a unique smooth solution to the massless spin-$s$ field
equations on the causal diamond $\mathscr{D}_{\tau_{\bullet},\rho_{\bullet}}$.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathscr{D}_{\tau_{\bullet},\rho_{\bullet}}$$\mathcal{U}$$i^{0}$$\bullet$$\mathcal{S}_{\tau_{\bullet},\rho_{\bullet}}$$\underline{\mathcal{B}}_{\varepsilon}$

(a) The domain of existence in the Penrose gauge.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{I}$$\mathcal{I}^{-}$$\bullet$$\mathcal{I}^{+}$$\bullet$$\mathscr{D}_{\tau_{\bullet},\rho_{\bullet}}$

(b) The domain of existence in the F-gauge.

Figure 6: The domains of existence of Propositions 4.2 and 4.3. Observe that
these domains do not include spatial infinity.

######  Remark 6.

The conclusion of Proposition 4.3 ensures the existence of solutions to the
spin-$s$ equations on a causal diamond which can get arbitrarily close to
spatial infinity, but cannot actually reach it. The development of estimates
to deal with this degeneracy is the objective of the next section.

##  5 Estimates near $\mathscr{I}^{-}$

In this section we construct estimates which allow us to control the solutions
to the spin-$s$ equations in terms of asymptotic characteristic initial data
on past null infinity. At the core of these estimates is a combination of the
methods developed in [Luk12] to obtain an optimal existence result for the
characteristic initial value problem, and the strategy adopted in Section 3
for the standard initial value problem near spatial infinity. We recall once
more that the degeneracy of the equations at the critical point
$\mathcal{I}^{-}$ prevents a direct application of the results of [Luk12],
necessitating a different treatment of the region which includes spatial
infinity.

###  5.1 Estimating the radiation field

The component $\phi_{2s}$ (the _outgoing radiation field_) is the most
problematic to estimate from $\mathscr{I}^{-}$ in our setting, as the naïve
energy estimate for equation (34a) for $k=2s-1$ degenerates at past null
infinity $\mathscr{I}^{-}=\\{\tau=-1\\}$. However, we may produce estimate
analogous to those of Section 3.

Given $\rho_{\star}>0$, we consider the following hypersurfaces in
$\overline{\mathcal{N}}$:

$\displaystyle\mathscr{I}^{-}_{\rho_{\star}}$
$\displaystyle\equiv\mathscr{I}^{-}\cap\\{0\leqslant\rho\leqslant\rho_{\star}\\},$
$\displaystyle\underline{\mathcal{B}}_{\varepsilon}$
$\displaystyle\equiv\left\\{(\tau,\rho,t^{\bm{A}}{}_{{\bm{B}}})\,|\,-1\leqslant\tau\leqslant-1+\varepsilon,~{}\rho=\frac{\rho_{\star}}{1-\tau},~{}t^{\bm{A}}{}_{{\bm{B}}}\in\mathrm{SU}(2)\right\\},$
$\displaystyle\mathcal{S}_{-1+\varepsilon}$
$\displaystyle\equiv\left\\{(\tau,\rho,t^{\bm{A}}{}_{{\bm{B}}})\,|\,\tau=-1+\varepsilon,~{}0\leqslant\rho\leqslant\frac{\rho_{\star}}{2-\epsilon},~{}t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\right\\},$
$\displaystyle\mathcal{I}_{\varepsilon}$
$\displaystyle\equiv\bigg{\\{}(\tau,\rho,t^{\bm{A}}{}_{{\bm{B}}})\,|\,-1\leqslant\tau\leqslant-1+\varepsilon,~{}\rho=0,~{}t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SU}(2)\bigg{\\}}.$

Observe that the set $\underline{\mathcal{B}}$ is a short incoming null
hypersurface intersecting $\mathscr{I}^{-}$ at $\rho=\rho_{\star}$. Moreover,
let $\underline{\mathcal{N}}_{\varepsilon}$ denote the spacetime slab bounded
by the hypersurfaces $\underline{\mathcal{B}}_{\varepsilon}$,
$\mathscr{I}^{-}_{\rho_{\star}}$, $\mathcal{S}_{-1+\varepsilon}$ and
$\mathcal{I}_{\varepsilon}$ (see Figure 7). As in the previous section, given
$m\geqslant 0$ we let

$\mho\equiv\\{(q^{\prime},p^{\prime},\alpha)\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{3}\,:\,q^{\prime}+p^{\prime}+|\alpha|\leqslant
m\\}.$

$\mathscr{I}^{+}$$\mathscr{I}^{-}_{\rho_{\star}}$$\mathcal{I}_{\varepsilon}$$\mathcal{S}_{-1+\varepsilon}$$\underline{\mathcal{B}}_{\varepsilon}$$\underline{\mathcal{N}}_{\varepsilon}$

Figure 7: We perform energy estimates from
$\mathscr{I}^{-}_{\rho_{\star}}\cup\underline{\mathcal{B}}_{\varepsilon}$ to a
Cauchy surface $\mathcal{S}_{-1+\varepsilon}$ for some $\varepsilon\ll 1$.

In terms of the above we have the following estimate for the outgoing
radiation field.

######  Proposition 5.1.

Let $\varepsilon>0$. Suppose that for $m,\,q,\,p\in\mathbb{N}$ and
$(q^{\prime},p^{\prime},\alpha)\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{3}$
satisfying

$q^{\prime}+p^{\prime}+|\alpha|\leqslant m\quad\text{and}\quad m+p<s+q$

we have the bound

$\sum_{\mho}\int_{\mathscr{I}^{-}_{\rho_{\star}}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k})|^{2}\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu+\sum_{\mho}\int_{\underline{\mathcal{B}}_{\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k})|^{2}\operatorname{d}\\!\underline{\mathcal{B}}\leqslant\Omega_{\star}$
(17)

on the characteristic data for $0\leqslant k\leqslant 2s$, for some
$\Omega_{\star}>0$. Additionally, assume that the bootstrap bound

$\|\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}<\Omega_{\star}$
(18)

holds for $0\leqslant k\leqslant 2s-1$. Then there exists a constant $C>0$
such that

$\|\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{2s}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\leqslant
C\Omega_{\star}.$ (19)

######  Remark 7.

The bootstrap bound (18) is proven in Proposition 5.2 assuming only the bound
(17).

###### Proof.

As before, we write

$D\equiv
D^{q,p,\alpha}\equiv\partial_{\tau}^{q}\partial_{\rho}^{p}{\bm{Z}}^{\alpha},$

and commute $D$ into the equations (34a) and (34b) to arrive at the identity

$\displaystyle\begin{split}0&=\partial_{\tau}\left((1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\right)+\partial_{\rho}\left(-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\right)\\\
&\phantom{=}-{\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})-{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})\\\
&\phantom{=}-{\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})-{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})\\\
&\phantom{=}+2(k+1-s+q-p)|D\phi_{k+1}|^{2}+2(k-s-q+p)|D\phi_{k}|^{2},\end{split}$
(6)

where $0\leqslant k\leqslant 2s-1$. Integrating (6) over the region
$\underline{\mathcal{N}}_{\varepsilon}$ against the 5-form
$\mathrm{dv}\equiv\operatorname{d}\\!\tau\wedge\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$,
we have

$\displaystyle 0$
$\displaystyle=\int_{\underline{\mathcal{N}}_{\varepsilon}}\left(\begin{array}[]{c}\partial_{\tau}\\\
\partial_{\rho}\end{array}\right)\cdot\left(\begin{array}[]{c}(1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\\\
-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\end{array}\right)\mathrm{dv}$
$\displaystyle\phantom{=}+2(k-s-q+p)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D\phi_{k}|^{2}\,\mathrm{dv}+2(k+1-s+q-p)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D\phi_{k+1}|^{2}\,\mathrm{dv}$
$\displaystyle\phantom{=}-\int_{\underline{\mathcal{N}}_{\varepsilon}}\Big{(}{\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})+{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{+}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k+1})$
$\displaystyle\phantom{=}\underbrace{\hskip
20.00003pt+{\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})+{\bm{Z}}^{\alpha}(\partial_{\tau}^{q}\partial_{\rho}^{p}\bar{\phi}_{k}){\bm{Z}}^{\alpha}{\bm{X}}_{-}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})\Big{)}\,\mathrm{dv}}_{\text{angular}(\alpha)}.$

Using Lemma A.1, as in Section 3 we will have $\sum_{|\alpha|\leqslant
m^{\prime}}\text{angular}(\alpha)=0$ for any $m^{\prime}\in\mathbb{N}$. We
will therefore not write the angular terms out in detail in the following
computations. Using the Euclidean divergence theorem, we then find

$\displaystyle\begin{split}0&=\int_{\mathcal{S}_{-1+\varepsilon}}\varepsilon|D\phi_{k+1}|^{2}+(2-\varepsilon)|D\phi_{k}|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu-\int_{\mathscr{I}^{-}_{\rho_{*}}}2|D\phi_{k}|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\\\
&\phantom{=}-\int_{\mathcal{I}_{\varepsilon}}\rho\left(-|D\phi_{k+1}|^{2}+|D\phi_{k}|^{2}\right)\operatorname{d}\\!\tau\wedge\operatorname{d}\\!\mu\\\
&\phantom{=}+\int_{\underline{\mathcal{B}}_{\varepsilon}}\left(\begin{array}[]{c}(1+\tau)|D\phi_{k+1}|^{2}+(1-\tau)|D\phi_{k}|^{2}\\\
-\rho|D\phi_{k+1}|^{2}+\rho|D\phi_{k}|^{2}\end{array}\right)\cdot\nu\left(\begin{array}[]{c}-\rho\\\
1-\tau\end{array}\right)\operatorname{d}\\!\underline{\mathcal{B}}+\text{angular}(\alpha)\\\
&\phantom{=}+2(k-s-q+p)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D\phi_{k}|^{2}\,\mathrm{dv}+2(k+1-s+q-p)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D\phi_{k+1}|^{2}\,\mathrm{dv},\end{split}$
(20)

where $\nu\equiv(\rho^{2}+(1-\tau)^{2})^{-1/2}$ is a normalisation factor for
the outward normal to $\underline{\mathcal{B}}_{\varepsilon}$,
$\operatorname{d}\\!\underline{\mathcal{B}}$ is the induced measure on
$\underline{\mathcal{B}}_{\varepsilon}$, and we note that the integral over
$\mathcal{I}_{\varepsilon}$ vanishes due to the factor of $\rho$ in the
integrand. In a slight deviation from the computations of Section 3, here we
perform the relabeling

$q\longrightarrow q+q^{\prime},\qquad p\longrightarrow p+p^{\prime},$

so that $D\longrightarrow D^{\prime}\partial_{\tau}^{q}\partial_{\rho}^{p}$,
where
$D^{\prime}=\partial_{\tau}^{q^{\prime}}\partial_{\rho}^{p^{\prime}}{\bm{Z}}^{\alpha}$,
and sum both sides of the above equality over $\mho$. The integrals over
angular terms vanish, and noting that the integral of
$(2-\varepsilon)|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k})|^{2}$
over $\mathcal{S}_{-1+\varepsilon}$ is non-negative, we have the estimate

$\displaystyle\varepsilon\sum_{\mho}\int_{\mathcal{S}_{-1+\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle\leqslant
2\sum_{\mho}\int_{\mathscr{I}^{-}_{\rho_{*}}}|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle\phantom{=}+2\sum_{\mho}\int_{\underline{\mathcal{B}}_{\varepsilon}}\nu\rho|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\underline{\mathcal{B}}$
$\displaystyle\phantom{=}+2\sum_{\mho}(q+q^{\prime}-p^{\prime}-p+s-k)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k})|^{2}\,\mathrm{dv}$
$\displaystyle\phantom{=}+2\sum_{\mho}(-q-q^{\prime}+p^{\prime}+p+s-k-1)\int_{\underline{\mathcal{N}}_{\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1})|^{2}\,\mathrm{dv}$
$\displaystyle\leqslant
2\sum_{\mho}\int_{\mathscr{I}^{-}_{\rho_{*}}}|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu$
$\displaystyle\phantom{=}+2\sum_{\mho}\int_{\underline{\mathcal{B}}_{\varepsilon}}\nu\rho|D^{\prime}(\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k})|^{2}\,\operatorname{d}\\!\underline{\mathcal{B}}$
$\displaystyle\phantom{\leqslant}+C^{(1)}_{m,p,s,k}\|\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}+C^{(2)}_{m,p,s,k}\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})},$

where

$C^{(2)}_{m,p,s,k}\equiv 2(m-q+p+s-1-k)$

and

$\|\psi\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\equiv\sum_{\mho}\int_{\mathcal{N}_{\varepsilon}}|D^{\prime}\psi|^{2}\,\mathrm{dv}.$

Recall now that we assumed that the bootstrap bound (18) holds—that is, we
have that

$\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\leqslant\Omega_{\star}$

for $0\leqslant k\leqslant 2s-1$. Plugging in this bootstrap bound and
assumption (17) into the above estimate, we obtain

$\varepsilon\sum_{\mho}\int_{\mathcal{S}_{-1+\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\leqslant
C\left(\Omega_{\star}+C^{(2)}_{m,p,s,k}\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\right).$

The surfaces $\mathcal{S}_{-1+t}$, $t\in(0,\varepsilon)$, foliate
$\underline{\mathcal{N}}_{\varepsilon}$, so this estimate is of the form

$\varepsilon f^{\prime}_{k}(\varepsilon)\leqslant
C\left(\Omega_{\star}+C^{(2)}_{m,p,s,k}f_{k}(\varepsilon)\right),$

where
$f_{k}(\varepsilon)\equiv\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}$.
This may be rewritten as

$\frac{\operatorname{d}\\!}{\operatorname{d}\\!\varepsilon}\left(\varepsilon^{-CC^{(2)}_{m,p,s,k}}f_{k}(\varepsilon)\right)\leqslant
C\Omega_{\star}\varepsilon^{-CC^{(2)}_{m,p,s,k}-1},$ (21)

which is integrable near $\varepsilon=0$ if $C^{(2)}_{m,p,s,k}<0$. This is
satisfied for $k=2s-1$ provided

$m+p<s+q.$ (22)

Assuming (22) holds and integrating (21), we arrive at

$f_{2s-1}(\varepsilon)\leqslant\frac{\Omega_{\star}}{-C^{(2)}_{m,p,s,2s-1}}\lesssim\Omega_{\star}.$

This is the required estimate. ∎

###  5.2 The boostrap bound

To prove the bootstrap bound (18), we return to the identity (20).

######  Proposition 5.2.

Let $\varepsilon>0$. Suppose that for $m,\,q,\,p\in\mathbb{N}$ and all
$(q^{\prime},p^{\prime},\alpha)\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{3}$
satisfying

$q^{\prime}+p^{\prime}+|\alpha|\leqslant m\quad\text{and}\quad m+p<s+q$

we have the bound (17) on characteristic data for $0\leqslant k\leqslant 2s$.
Then there exists a constant $C>0$ depending on $m$, $p$, $s$ and
$\varepsilon$ such that

$\|\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{k}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\leqslant
C\Omega_{\star}$

for $0\leqslant k\leqslant 2s-1$, where
$\underline{\mathcal{N}}_{\varepsilon}$ is as in Proposition 5.1. In other
words, the bootstrap estimate (18) holds.

###### Proof.

Performing the shifts $q\to q+q^{\prime}$ and $p\to p+p^{\prime}$ as in
Proposition 5.1 and summing the identity (20) over $\mho$, we deduce (this
time dropping the manifestly positive
$\varepsilon|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1})|^{2}$
term) that

$\sum_{\mho}\int_{\mathcal{S}_{-1+\varepsilon}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu\lesssim\Omega_{*}+C^{(1)}_{m,p,s,k}\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k}\|^{2}_{H^{m}(\mathcal{N}_{\varepsilon})}+C^{(2)}_{m,p,s,k}\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k+1}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})},$
(23)

where as before

$C^{(2)}_{m,p,s,k}=2(m-q+p+s-1-k)$

and $C^{(1)}_{m,p,s,k}=2(m+q-p+s-k)$.

We begin with $k=2s-1$. Then $C^{(2)}_{m,p,s,2s-1}<0$ is guaranteed by the
assumption that $m+p<s+q$, and the estimate (23) reads

$g_{2s-1}(\varepsilon)\lesssim\Omega_{\star}+C^{(1)}_{m,p,s,2s-1}\int_{0}^{\varepsilon}g_{2s-1}(t)\operatorname{d}\\!t,$

where

$g_{k}(t)\equiv\sum_{\mho}\int_{\mathcal{S}_{-1+t}}|D^{\prime}(\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k})|^{2}\,\operatorname{d}\\!\rho\wedge\operatorname{d}\\!\mu.$

Using Grönwall’s lemma, we find

$\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{2s-1}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}=\int_{0}^{\varepsilon}g_{2s-1}(t)\operatorname{d}\\!t\lesssim\Omega_{*}e^{C_{m,p,s}\varepsilon}\lesssim_{m,p,s,\varepsilon}\Omega_{\star}.$

We can now return to (23) for $k=2s-2$. We have, using the above estimate for
$\phi_{2s-1}$,

$g_{2s-2}(\varepsilon)\lesssim\Omega_{\star}+C^{(1)}_{m,p,s,2s-2}\int_{0}^{\varepsilon}g_{2s-2}(t)\operatorname{d}\\!t,$

and so deduce that

$\|\partial_{\tau}^{q}\partial_{\rho}^{p}\phi_{2s-2}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\lesssim\Omega_{\star}.$

Proceeding inductively for $0\leqslant k\leqslant 2s-3$, we conclude that
there exists a constant $C=C(m,p,s,\varepsilon)>0$ such that

$\|\partial_{\tau}^{q}\partial^{p}_{\rho}\phi_{k}\|^{2}_{H^{m}(\underline{\mathcal{N}}_{\varepsilon})}\leqslant
C\Omega_{\star}$

for $0\leqslant k\leqslant 2s-1$. ∎

###  5.3 Asymptotic expansions near $\mathscr{I}^{-}$

The estimates obtained in Section 5 may now be used to control the solutions
to the spin-$s$ field equations in a narrow causal diamond extending along
$\mathscr{I}^{-}$ and containing a portion of $\mathcal{I}$. The strategy is
to decompose the solution into a formal expansion around $\mathscr{I}^{-}$,
the regularity of the terms of which will be known explicitly, and a remainder
whose regularity is controlled by the estimates on
$\partial_{\tau}^{q}\phi_{k}$.

It is clear from the condition $m+p<s+q$ in Proposition 5.1 that requiring
$\rho$-derivatives near $\mathscr{I}^{-}$ hinders the regularity sought on
$\mathcal{S}_{-1+\varepsilon}$. Therefore setting $p=0$ (and writing $m\to
m+3$ for convenience) in Proposition 5.1, we have $\forall\,0\leqslant
k\leqslant 2s$

$\partial_{\tau}^{q}\phi_{k}\in
H^{m+3}(\underline{\mathcal{N}}_{\varepsilon})\quad\text{if}\quad m+3<s+q.$
(24)

Sobolev embedding therefore gives

$\partial_{\tau}^{q}\phi_{k}\in
C^{m,\alpha}(\underline{\mathcal{N}}_{\varepsilon})\quad\text{for}\quad
0<\alpha\leqslant\frac{1}{2},\;\quad m+3<s+q.$

Integrating this $q$ times with respect to $\tau$, one finds

$\phi_{k}=\sum_{q^{\prime}=0}^{q-1}\frac{1}{q^{\prime}!}(\partial_{\tau}^{q^{\prime}}\phi_{k})|_{\mathscr{I}^{-}}(\tau+1)^{q^{\prime}}+I^{q}(\partial_{\tau}^{q}\phi_{k}),$
(25)

where $I^{q}(\partial_{\tau}^{q}\phi_{k})\in
C^{m,\alpha}(\underline{\mathcal{N}}_{\varepsilon})$ and $I$ denotes the
operator

$f\mapsto
I(f)=\int_{-1}^{\tau}f(\tau^{\prime},\rho,t^{\bm{A}}{}_{\bm{B}})\,\mathrm{d}\tau^{\prime}.$

######  Remark 8.

Each term (except for the remainder) in the expansion (25) can be computed
explicitly from the characteristic initial data $\phi_{k}|_{\mathscr{I}^{-}}$
on $\mathscr{I}^{-}$: see Appendix D. In particular, the regularity of these
terms is known explicitly.

###  5.4 Last slice argument

As in the case of the upper domain $\mathcal{N}_{1}$, the existence of
solutions in the lower domain $\underline{\mathcal{N}}_{\varepsilon}$ can be
shown by a last slice argument. We provide a sketch of the argument below.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{I}$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\underline{\mathcal{B}}_{\varepsilon}$$\mathcal{B}_{1}$$\hat{\mathcal{S}}_{-1+\varepsilon}$$\hat{\mathcal{S}}_{\hat{\tau}_{\ast}}$$\hat{\mathcal{S}}_{-1}$$(a)$$(b)$$(b^{\prime})$$(b^{\prime\prime})$$(c)$$\mathcal{N}_{1}$$\underline{\mathcal{N}}_{\varepsilon}$

Figure 8: Schematic depiction of the region
$\mathscr{D}=\underline{\mathcal{N}}_{\varepsilon}\cup\mathcal{N}_{1}$, now in
the $(\hat{\tau},\rho)$ coordinates. The last slice argument from
$\mathscr{I}^{-}$ is best carried out in a representation of spatial infinity
where past and future null infinities are not horizontal. In region $(a)$,
Rendall’s theorem ensures the existence of a solution in a neighbourhood of
the initial cut $\mathscr{I}^{-}\cap\underline{\mathcal{B}}_{\varepsilon}$.
This neighbourhood contains a spacelike hypersurface of constant $\hat{\tau}$.
Starting from this hypersurface, in region $(b)$ we can use the standard local
existence result for symmetric hyperbolic systems to extend the solution. As
this extension of the solution possesses Cauchy horizon in
$\underline{\mathcal{N}}_{\varepsilon}$, we solve supplementary characteristic
initial value problems in regions $(b^{\prime})$ and $(b^{\prime\prime})$
starting from data on $\mathscr{I}^{-}$ and
$\underline{\mathcal{B}}_{\varepsilon}$, and the induced data on the Cauchy
horizon. The estimates in $\underline{\mathcal{N}}_{\varepsilon}$ then ensure
that the solution can be extended up to $\hat{\mathcal{S}}_{-1+\varepsilon}$.
Once we are in the upper domain $\mathcal{N}_{1}$, we proceed in a similar
fashion. In region $(c)$ the assumption of the existence of a _last slice_ is
in contradiction with the the estimates in terms of the data on
$\hat{\mathcal{S}}_{-1+\varepsilon}$.

As for the resolution of the asymptotic characteristic initial value problem
in Section 4, here it is more convenient to consider the massless spin-$s$
equations in the more general gauge discussed in Section B.4—rather than in
the horizontal representation of Section 2.2—as in the latter the equations
are singular on the whole of $\mathscr{I}^{-}$, not only at the critical set
$\mathcal{I}^{-}$. In the more general gauge we have $x^{0}=\hat{\tau}\kappa$,
where $\kappa=\rho\mu$ and $\mu\not\equiv 1$, and the hypersurfaces of
constant $\hat{\tau}$ give rise to a foliation of the lower domain
$\underline{\mathcal{N}}_{\varepsilon}$, and indeed the region to the past of
$\underline{\mathcal{B}}_{\varepsilon}$. Since $\mathscr{I}^{-}$ is now given
by $\hat{\tau}=-\mu^{-1}$ and $\mu(0)=1$, the hypersurfaces
$\hat{\mathcal{S}}_{\hat{\tau}}$ for $\hat{\tau}<-1$ terminate at
$\mathscr{I}^{-}$, whereas the ones for $-1<\hat{\tau}<1$ terminate at
$\mathcal{I}$. In particular, the hypersurface $\hat{\mathcal{S}}_{-1}$
terminates at $\mathcal{I}^{-}$—see Figure 8. Moreover, there exists a
hypersurface $\hat{\mathcal{S}}_{\tau_{\rho_{\star}}}$ which lies entirely in
the past of $\underline{\mathcal{B}}_{\varepsilon}$, and which intersects
$\mathscr{I}^{-}$ at
$\underline{\mathcal{B}}_{\varepsilon}\cap\mathscr{I}^{-}_{\rho_{\star}}$. We
concentrate on the region $\underline{\mathcal{N}}_{\varepsilon}$, i.e. in
what follows we refer to the parts of hypersurfaces $\hat{\mathcal{S}}$ in the
future of $\underline{\mathcal{B}}_{\varepsilon}$.

As we are looking for solutions of the form (25) and the asymptotic expansion
is formally known, we only need to ensure the existence of the remainder. We
therefore consider an evolution system for the fields
$\partial_{\tau}\phi_{k}^{q}$ for $m+3<s+q$. This system can be obtained by a
repeated application of the derivative $\partial_{\tau}$ to the equations eqs.
40a and 40a. The resulting system has the same structure as the original one,
so it is a symmetric hyperbolic system for $\partial_{\tau}\phi_{k}^{q}$.

Now, for the asymptotic characteristic initial value problem for the fields
$\partial_{\tau}\phi_{k}^{q}$, Rendall’s theorem [Ren90] ensures the existence
of a solution in a neighbourhood
$\mathcal{V}\subset\underline{\mathcal{N}}_{\varepsilon}\subset
J^{+}(\underline{\mathcal{B}}_{\varepsilon}\cap\mathscr{I}^{-})$ of the
initial cut, region $(a)$ in Figure 8. Next, there exists a value
$\hat{\tau}_{\delta}<-1$ of the parameter $\hat{\tau}$ such that all leaves
$\hat{\mathcal{S}}_{\hat{\tau}}$ with
$\hat{\tau}_{\rho_{\star}}<\hat{\tau}<\hat{\tau}_{\delta}$ are contained
entirely in the region $\mathcal{V}$. On one of these hypersurfaces, say
$\hat{\mathcal{S}}_{\hat{\tau}_{\circ}}$, one can formulate a standard initial
value problem for the symmetric hyperbolic system (14) implied by the spin-$s$
field equations. With data posed on $\hat{\mathcal{S}}_{\hat{\tau}_{\circ}}$,
the standard theory of symmetric hyperbolic systems ensures the existence of a
solution in $D^{+}({\hat{\mathcal{S}}_{\hat{\tau}_{\circ}}})$, region $(b)$ in
Figure 8. The solution in region $(b)$ possesses a Cauchy horizon
$\mathcal{H}^{+}(\hat{S}_{\hat{\tau}_{\circ}})=\partial
D^{+}(\hat{\mathcal{S}}_{\hat{\tau}_{\circ}})$, depicted by the hypersurfaces
separating the region $(b)$ from regions $(b^{\prime})$ and
$(b^{\prime\prime})$. We extend the solution to $(b^{\prime})$ and
$(b^{\prime\prime})$ by posing two further characteristic initial value
problems: (i) one with data on $\mathscr{I}^{-}$ and the Cauchy horizon of
$\mathcal{S}_{\tau_{\circ}}$, and (ii) a second one with data on
$\underline{\mathcal{B}}_{\varepsilon}$ and the Cauchy horizon; the
characteristic data on the Cauchy horizon is induced by the solution in the
region $(b)$. For these characteristic initial value problems we again use
Rendall’s theorem to obtain a solution in a subset of the causal future of
$\mathscr{I}^{-}\cap\mathcal{H}^{+}(\mathcal{S}_{\tau_{\circ}})$ (region
$(b^{\prime})$) and
$\underline{\mathcal{B}}_{\varepsilon}\cap\mathcal{H}^{+}(\mathcal{S}_{\tau_{\circ}})$
(region $(b^{\prime\prime})$), respectively. In this way we obtain the
solution in a subdomain of $\underline{\mathcal{N}}_{\varepsilon}$ which
contains a hypersurface $\hat{\mathcal{S}}_{\hat{\tau}}$ with
$\hat{\tau}>\hat{\tau}_{\circ}$. This construction can then be repeated to
obtain a larger extension domain for the solution.

Now assume that there exists
$\hat{\tau}^{\prime}_{\ast}\in(\hat{\tau}_{\rho_{\star}},-1)$ such that on the
hypersurface $\hat{\mathcal{S}}_{\hat{\tau}^{\prime}_{\ast}}$ the construction
for the enlargement of the domain of existence as described in the previous
paragraph fails. It is straightforward to show (as in the case of the upper
existence domain) that this is then in contradiction with the estimates in
Proposition 5.1 and in Proposition 5.2. In this way one ensures the existence
of the solution up to the hypersurface $\hat{\mathcal{S}}_{-1}$.

Once one has reached the hypersurface $\hat{\mathcal{S}}_{-1}$, the domain
extension procedure simplifies as now one only needs one supplementary
characteristic initial value problem to complete the new slab—the one on the
intersection of the Cauchy horizon and
$\underline{\mathcal{B}}_{\varepsilon}$. Again, assuming the existence of a
hypersurface $\hat{\mathcal{S}}_{\hat{\tau}_{\ast}}$ with
$\hat{\tau}_{\ast}>-1$ beyond which it is no longer possible to formulate an
initial value problem and extend the solution (region $(c)$ in Figure 8) leads
to a contradiction with the estimates in Proposition 5.1 and Proposition 5.2.

##  6 Controlling the solutions from $\mathscr{I}^{-}$ to $\mathscr{I}^{+}$

We have obtained two sets of estimates:

  * (i)

estimates which control the solution on a spacelike hypersurface
$\mathcal{S}_{-1+\varepsilon}$ in terms of initial data on $\mathscr{I}^{-}$
(Propositions 5.1 and 5.2), and

  * (ii)

ones which control the solution up to $\mathscr{I}^{+}$ in terms of initial
data on a spacelike hypersurface, say $\mathcal{S}_{-1+\varepsilon}$
(Proposition 3.1).

Each of the estimates (i) and (ii) allow the construction of a particular type
of asymptotic expansion with its own requirements on the freely specifiable
data and associated implications for the regularity of the solutions. In this
section we show how these two sets of estimates can be stitched together to
control the solutions to the massless spin-$s$ field equation in a
neighbourhood of spatial infinity which contains parts of both
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$ (Figure 2).

######  Remark 9.

We refer to the domain in which estimates (i) hold as the lower domain, and
the domain in which estimates (ii) hold as the upper domain. The parameters
$m,\;p,\;q$ in the lower domain will be denoted by $m_{-},\;p_{-},\;q_{-}$,
while those in the upper domain will be denoted by $m_{+},\;p_{+},\;q_{+}$.

###  6.1 From $\mathscr{I}^{-}$ to $\mathcal{S}_{-1+\varepsilon}$

Proposition 5.1 controls the regularity of the solution $\phi_{k}$ near
$\mathscr{I}^{-}$ in the sense that (setting $p_{-}=0$, as remarked in Section
5.3) if the data on
$\mathscr{I}^{-}_{\rho_{\star}}\cup\underline{\mathcal{B}}_{\varepsilon}$ is
such that

$(\partial_{\tau}^{q_{-}}\phi,\,\partial_{\tau}^{q_{-}+1}\phi,\,\ldots,\,\partial_{\tau}^{q_{-}+m_{-}+3}\phi)\in
H^{m_{-}+3}\times H^{m_{-}+2}\times\dots\times L^{2}$

for

$m_{-}+3<s+q_{-},$

then the solution is $\partial_{\tau}^{q_{-}}\phi\in
H^{m_{-}+3}(\underline{\mathcal{N}}_{\varepsilon})$, and in particular for
$0<\alpha\leqslant\frac{1}{2}$

$\phi_{k}=\sum_{q^{\prime}=0}^{q_{-}-1}\frac{1}{q^{\prime}!}(\partial_{\tau}^{q^{\prime}}\phi_{k})|_{\mathscr{I}^{-}}(\tau+1)^{q^{\prime}}+C^{m_{-},\alpha}(\underline{\mathcal{N}}_{\varepsilon}).$

###  6.2 From $\mathcal{S}_{-1+\varepsilon}$ to $\mathscr{I}^{+}$

On the other hand, Proposition 3.1 controls the regularity of the solution
near $\mathscr{I}^{+}$, in the sense that if the data on the Cauchy surface
$\mathcal{S}_{-1+\varepsilon}$ is such that

$(\partial_{\rho}^{p_{+}}\phi,\,\partial_{\rho}^{p_{+}+1}\phi,\,\ldots,\,\partial_{\rho}^{p_{+}+m_{+}+3}\phi)\in
H^{m_{+}+3}\times H^{m_{+}+2}\times\dots\times L^{2}$

for

$p_{+}\geqslant m_{+}+s+4,$

then the solution is $\partial_{\rho}^{p_{+}}\phi\in
H^{m_{+}+3}(\mathcal{N}_{1})$, and in particular

$\phi_{k}=\sum_{p^{\prime}=0}^{p_{+}-1}\frac{1}{p^{\prime}!}\phi_{k}^{(p^{\prime})}\rho^{p^{\prime}}+C^{m_{+},\alpha}(\mathcal{N}_{1}).$

###  6.3 Prescribing the regularity at $\mathscr{I}^{-}$

For a desired regularity $m_{+}$ of the remainder at $\mathscr{I}^{+}$, we may
therefore put $p_{+}=m_{+}+\lceil s\rceil+4$ for the minimum number of
$\rho$-derivatives on $\mathcal{S}_{-1+\varepsilon}$ required by the estimate
of Proposition 3.1. In order to match the estimates in the two domains, in the
lower domain we therefore require the remainder to be at least $C^{p_{+}}$,
i.e. $m_{-}=p_{+}$. Combining the above bounds then gives

$q_{-}\geqslant m_{+}+8.$ (26)

Note that while in (26) the value $s$ of the spin cancels out, it still plays
a role in length of the expansions of $\phi_{k}$ via $m_{-}=p_{+}=m_{+}+\lceil
s\rceil+4$. As $s$ increases, so does the length of the expansion before the
remainder can be guaranteed to be $C^{m_{+}}$ near $\mathscr{I}^{+}$.

###  6.4 Main result

We summarise the analysis of this article in the following theorem.

######  Theorem 6.1.

Let real numbers $\rho_{\star}>0$, $\varepsilon>0$ and positive integers
$m,\;q$ such that $q\geqslant m+8$ be given, and suppose that we have
asymptotic characteristic initial data for the massless spin-$s$ equations on
$\underline{\mathcal{B}}_{\varepsilon}\cup\mathscr{I}^{-}_{\rho_{\star}}$ such
that

$(\partial_{\tau}^{q}\phi,\,\partial_{\tau}^{q+1}\phi,\,\ldots,\,\partial_{\tau}^{q+m+3}\phi)\in
H^{m+3}\times H^{m+2}\times\dots\times L^{2}.$

Then in the domain
$\mathscr{D}=\underline{\mathcal{N}}_{\varepsilon}\cup\mathcal{N}_{1}$ this
data gives rise to a unique solution to the massless spin-$s$ equations (3)
which near $\mathscr{I}^{+}$ admits the expansion

$\phi_{k}=\sum_{p^{\prime}=0}^{p-1}\frac{1}{p^{\prime}!}\phi_{k}^{(p^{\prime})}\rho^{p^{\prime}}+C^{m,\alpha}(\mathcal{N}_{1}),$

where $p=m+\lceil s\rceil+4$ and $0<\alpha\leqslant\frac{1}{2}$. The
coefficients $\phi_{k}^{(p^{\prime})}$, $0\leqslant p^{\prime}<p-1$, can be
computed explicitly in terms of Jacobi polynomials in $\tau$ and spin-weighted
spherical harmonics, and their regularity depends on the multipolar structure
of the characteristic data at $\mathcal{I}^{-}$. In particular, the
coefficients $\phi_{k}^{(p^{\prime})}$ are generically polyhomogeneous at
$\mathscr{I}^{+}$.

##  7 Concluding remarks

We have constructed estimates and asymptotic expansions for massless spin-$s$
fields on Minkowski space which control the behaviour of the solutions at
future null infinity $\mathscr{I}^{+}$ in terms of data prescribed at past
null infinity $\mathscr{I}^{-}$. In particular, we have established a
connection between the regularity of the characteristic initial data at past
null infinity and the regularity of expansions near future null infinity,
showing that the linear analogue of the logarithms found by Friedrich [Fri98a]
persists despite high regularity assumptions at past null infinity. Of course,
the regularity of the data at $\mathscr{I}^{-}$ in our setting is measured in
Sobolev spaces with respect to the F-coordinates $\tau$ and $\rho$, and we
make no claim as to the non-existence of other (perhaps comparatively large)
function spaces which may eliminate the logarithmic terms. Nonetheless,
Friedrich’s coordinates appear natural insofar as they are constructed from
conformal invariants.

It is interesting to note that for every logarithmic term near
$\mathscr{I}^{+}$, there is a corresponding logarithmic term near
$\mathscr{I}^{-}$ (see (49)), however it generally carries a different
tempering polynomial in front, cf. $(1-\tau)^{p-k+s}\log(1-\tau)$ vs.
$(1+\tau)^{p+k-s}\log(1+\tau)$. In a sense, this is the statement that
singular behaviour hidden deeper in the expansions near $\mathscr{I}^{-}$ may
resurface at higher orders near $\mathscr{I}^{+}$. It is also interesting to
note that, at the level of the Cauchy data, polyhomogeneity is _not needed_ to
produce polyhomogeneity in the solutions near $\mathscr{I}^{\pm}$. A range of
examples will be presented in a subsequent paper.

Finally, we mention that our assumptions on the asymptotic characteristic
initial data have been made on the basis of ease of presentation. Our theory
can, however, applied to a more general setting, e.g. for polyhomogeneous data
at $\mathscr{I}^{-}$.

## Appendices

##  Appendix A Geometry of $\mathrm{SU}(2)$

In this appendix we recall standard results about the geometry and
representation theory of the Lie group $\mathrm{SU}(2)$.

###  A.1 Basic properties

We make use of spinorial notation to denote elements in
$\mathrm{GL}(2,\mathbb{C})$, so that $t^{\bm{A}}{}_{\bm{B}}$,
${\bm{A}},{\bm{B}}\in\\{0,1\\}$, denotes an invertible $2\times 2$ matrix. The
subgroups $\mathrm{SL}(2,\mathbb{C})$ and $\mathrm{SU}(2,\mathbb{C})$ may then
be defined by

$\displaystyle\mathrm{SL}(2,\mathbb{C})=\left\\{t^{\bm{A}}{}_{\bm{B}}\in\mathrm{GL}(2,\mathbb{C})\;|\;\epsilon_{{\bm{A}}{\bm{B}}}t^{\bm{A}}{}_{\bm{C}}t^{\bm{B}}{}_{\bm{D}}=\epsilon_{{\bm{C}}{\bm{D}}}\right\\},$
$\displaystyle\mathrm{SU}(2,\mathbb{C})=\left\\{t^{\bm{A}}{}_{\bm{B}}\in\mathrm{SL}(2,\mathbb{C})\;|\;\tau_{{\bm{A}}{\bm{A}}^{\prime}}t^{\bm{A}}{}_{\bm{B}}\bar{t}^{{\bm{A}}^{\prime}}{}_{{\bm{B}}^{\prime}}=\tau_{{\bm{B}}{\bm{B}}^{\prime}}\right\\},$

where $\tau_{{\bm{A}}{\bm{A}}^{\prime}}$ in our frame is simply the $2\times
2$ identity matrix. It is classical that the group $\mathrm{SU}(2)$ is
diffeomorphic to the 3-sphere $\mathbb{S}^{3}$, and its elements
$t^{\bm{A}}{}_{\bm{B}}$ may be written in the explicit form

$t^{\bm{A}}{}_{\bm{B}}=\frac{1}{\sqrt{1+|\zeta|^{2}}}\left(\begin{array}[]{cc}e^{i\alpha}&ie^{-i\alpha}\zeta\\\
ie^{i\alpha}\bar{\zeta}&e^{-i\alpha}\end{array}\right),$

where $\zeta=x+iy\in\mathbb{C}$ and $\alpha\in\mathbb{R}$ coordinatize
$\mathrm{SU}(2)\simeq\mathbb{S}^{3}$. This representation makes manifest the
fact that there exist $\mathrm{U}(1)$ orbits in $\mathrm{SU}(2)$ generated by
$\alpha$, with the coordinates $(x,y)$ constant along these orbits.
Quotienting out the $\mathrm{U}(1)$ subgroup returns the $2$-sphere of the
rescaled spacetime, $\mathrm{SU}(2)/\mathrm{U}(1)\simeq\mathbb{S}^{2}$, as
described in Section 2.3.

###  A.2 The vector fields ${\bm{X}}_{\pm}$ and ${\bm{X}}$

Consider the basis

${\bm{u}}_{1}=\frac{1}{2}\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right),\qquad{\bm{u}}_{2}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),\qquad{\bm{u}}_{3}=\frac{1}{2}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),$

of the real Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ of $\mathrm{SU}(2)$,
where ${\bm{u}}_{3}$ is the generator of the $\mathrm{U}(1)$ subgroup. These
obey the commutation relations

$[{\bm{u}}_{i},\,{\bm{u}}_{j}]=\epsilon_{ijk}{\bm{u}}_{k}.$

Denote by ${\bm{Z}}_{1}$, ${\bm{Z}}_{2}$ and ${\bm{Z}}_{3}$ the left-invariant
vector fields on $\mathrm{SU}(2)$ generated by ${\bm{u}}_{1}$, ${\bm{u}}_{2}$
and ${\bm{u}}_{3}$, respectively, via the exponential map. These inherit the
commutation relations

$[{\bm{Z}}_{i},\,{\bm{Z}}_{j}]=\epsilon_{ijk}{\bm{Z}}_{k}.$

We then pass to the complexified Lie algebra
$\mathfrak{su}(2)+i\,\mathfrak{su}(2)=\mathfrak{sl}(2;\mathbb{C})$ by setting

${\bm{X}}_{+}\equiv-({\bm{Z}}_{2}+\mbox{i}{\bm{Z}}_{1}),\qquad{\bm{X}}_{-}\equiv-({\bm{Z}}_{2}-\mbox{i}{\bm{Z}}_{1}),\quad\text{and}\quad{\bm{X}}\equiv-2\mbox{i}{\bm{Z}}_{3}.$

The vector fields ${\bm{X}}_{\pm}$ and ${\bm{X}}$ then satisfy the commutation
relations

$[{\bm{X}},\,{\bm{X}}_{+}]=2{\bm{X}}_{+},\qquad[{\bm{X}},\,{\bm{X}}_{-}]=-2{\bm{X}}_{-},\quad\text{and}\quad[{\bm{X}}_{+},\,{\bm{X}}_{-}]=-{\bm{X}}.$

The vector fields ${\bm{X}}_{+}$ and ${\bm{X}}_{-}$ are complex conjugates of
each other in the sense that

$\overline{{\bm{X}}_{+}\phi}={\bm{X}}_{-}\bar{\phi}$

for any sufficiently smooth function $\phi:\mathrm{SU}(2)\to\mathbb{C}$. The
Casimir element of $\mathrm{SU}(2)$ is given by

${\bm{C}}\equiv
4({\bm{Z}}_{1}^{2}+{\bm{Z}}_{2}^{2}+{\bm{Z}}_{3}^{2})=2\\{{\bm{X}}_{+},\,{\bm{X}}_{-}\\}-{\bm{X}}^{2},$

where
$\\{{\bm{X}}_{+},\,{\bm{X}}_{-}\\}={\bm{X}}_{+}{\bm{X}}_{-}+{\bm{X}}_{-}{\bm{X}}_{+}$
is the anticommutator of ${\bm{X}}_{+}$ and ${\bm{X}}_{-}$. The
$2\\{{\bm{X}}_{+},\,{\bm{X}}_{-}\\}=4({\bm{Z}}_{1}^{2}+{\bm{Z}}_{2}^{2}$ )
part of the Casimir corresponds to the Laplacian on the $2$-sphere
$\mathbb{S}^{2}$ when acting on functions $f$ independent of $\alpha$, i.e.
ones that satisfy ${\bm{X}}f=0$.

####  A.2.1 Coordinate expressions

The vector fields ${\bm{\partial}}_{x}$, ${\bm{\partial}}_{y}$,
${\bm{\partial}}_{\alpha}$ at the identity in $\mathrm{SU}(2)$ coincide,
respectively, with the generators ${\bm{u}}_{1}$, ${\bm{u}}_{2}$ and
${\bm{u}}_{3}$ of $\mathfrak{s}\mathfrak{u}(2)$. More generally, writing

$P\equiv\frac{1}{2}(1+|\zeta|^{2}),$

one may express the vector fields ${\bm{Z}}_{i}$ in terms of
${\bm{\partial}}_{x}$, ${\bm{\partial}}_{y}$, ${\bm{\partial}}_{\alpha}$ as

$\displaystyle{\bm{Z}}_{1}=(P\cos 2\alpha){\bm{\partial}}_{x}+(P\sin
2\alpha){\bm{\partial}}_{y}+\frac{1}{2}\big{(}x\sin 2\alpha-y\cos
2\alpha\big{)}{\bm{\partial}}_{\alpha},$ $\displaystyle{\bm{Z}}_{2}=-(P\sin
2\alpha){\bm{\partial}}_{x}+(P\cos
2\alpha){\bm{\partial}}_{y}+\frac{1}{2}\big{(}y\sin 2\alpha+x\cos
2\alpha\big{)}{\bm{\partial}}_{\alpha},$
$\displaystyle{\bm{Z}}_{3}=\frac{1}{2}{\bm{\partial}}_{\alpha}.$

Then

$\displaystyle{\bm{X}}_{+}=-2\mathrm{i}Pe^{2\alpha\mathrm{i}}{\bm{\partial}}_{\zeta}-\frac{1}{2}\bar{\zeta}e^{2\alpha\mathrm{i}}{\bm{\partial}}_{\alpha},$
$\displaystyle{\bm{X}}_{-}=2\mathrm{i}Pe^{-2\alpha\mathrm{i}}{\bm{\partial}}_{\bar{\zeta}}-\frac{1}{2}\zeta
e^{-2\alpha\mathrm{i}}{\bm{\partial}}_{\alpha},$
$\displaystyle{\bm{X}}=-\mathrm{i}{\bm{\partial}}_{\alpha}.$

When acting on functions $f$ such that $\partial_{\alpha}f=0$, in these
coordinates the Casimir ${\bm{C}}$ has the form

${\bm{C}}f=2\\{{\bm{X}}_{+},\,{\bm{X}}_{-}\\}f=4(1+|\zeta|^{2})^{2}\partial_{\zeta}\partial_{\bar{\zeta}}f,$

which is precisely the Laplacian on $\mathbb{S}^{2}$ in stereographic
coordinates $\zeta=x+\mathrm{i}y$.

####  A.2.2 A technical lemma

For a given multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ of non-
negative integers $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$, set

${\bm{Z}}^{\alpha}\equiv{\bm{Z}}^{\alpha_{1}}_{1}{\bm{Z}}^{\alpha_{2}}_{2}{\bm{Z}}^{\alpha_{3}}_{3}$

with the convention that ${\bm{Z}}^{\alpha}=1$ if
$|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. We record the following
technical lemma from [Fri03], which will be useful to us when performing
estimates in the main text.

######  Lemma A.1.

For any smooth complex-valued functions $\phi$, $\psi$ on $\mathrm{SU}(2)$,
$k\in\\{1,2,3\\}$ and $m\in\mathbb{N}$, one has

$\sum_{|\alpha|=m}\big{(}{\bm{Z}}^{\alpha}{\bm{Z}}_{k}\phi{\bm{Z}}^{\alpha}\psi+{\bm{Z}}^{\alpha}\phi{\bm{Z}}^{\alpha}{\bm{Z}}_{k}\psi\big{)}=\sum_{|\alpha|=m}{\bm{Z}}_{k}\left({\bm{Z}}^{\alpha}\phi{\bm{Z}}^{\alpha}\psi\right).$

In particular, one has that

$\sum_{|\alpha|=m}\int_{\mathrm{SU}(2)}\big{(}{\bm{Z}}^{\alpha}{\bm{X}}_{\pm}\phi{\bm{Z}}^{\alpha}\psi+{\bm{Z}}^{\alpha}\phi{\bm{Z}}^{\alpha}{\bm{X}}_{\pm}\psi\big{)}\operatorname{d}\\!\mu=0.$
(27)

######  Remark 10.

Lemma A.1 is an extension of the divergence theorem on $\mathrm{SU}(2)$.
Indeed, for $m=0$ the statement of (27) reduces to

$\int_{\mathrm{SU}(2)}{\bm{X}}_{\pm}(\phi\psi)\operatorname{d}\\!\mu=0.$

This, in turn, follows from the divergence theorem and the fact that left-
invariant vector fields on unimodular Lie groups are divergence free.

###  A.3 The functions $T_{m}{}^{j}{}_{k}$

It will be useful to explicitly define the matrix elements of the unitary
irreducible representations of $\mathrm{SU}(2)$ in the conventions of [Fri86].
The irreducible representations of $\mathrm{SU}(2)$ are uniquely labelled by
the natural numbers $m\in\mathbb{N}$. For each $m\in\mathbb{N}$ there exists a
unique unitary irreducible representation $T_{m}$, and the set of irreducible
representations $\\{T_{m}\\}_{m\in\mathbb{N}}$ contains all the unitary
irreducible representations of $\mathrm{SU}(2)$. The dimension of each $T_{m}$
is $m+1$, and each $T_{m}$ is an eigenfunction of the Casimir operator
${\bm{C}}$ on $\mathrm{SU}(2)$ with eigenvalue $-m(m+2)$. The matrix elements
of these representations are given by the complex-valued functions
$T_{m}{}^{j}{}_{k}$ defined by

$\mathrm{SU}(2)\ni t^{\bm{A}}{}_{\bm{B}}\mapsto
T_{m}{}^{j}{}_{k}(t^{\bm{A}}{}_{\bm{B}})\equiv\binom{m}{j}^{1/2}\binom{m}{k}^{1/2}t^{({\bm{A}}_{1}}{}_{({\bm{B}}_{1}}\cdots
t^{{\bm{A}}_{m})_{j}}{}_{{\bm{B}}_{m})_{k}}$

for $j,\,k=0,\,\ldots,\,m$, $m=1,\,2,\,3,\ldots$, and

$T_{0}{}^{0}{}_{0}=1,$

where the notation $({\bm{A}}_{1},\,\ldots,\,{\bm{A}}_{m})_{j}$ indicates that
the ${\bm{A}}$ indices are symmetrized, and then $j$ of them are set to $1$
and the remaining $m-j$ are set to $0$. The functions are real-analytic and
the associated representation $T_{m}$ is then given by

$\mathrm{SU}(2)\ni t^{\bm{A}}{}_{\bm{B}}\mapsto
T_{m}(t^{\bm{A}}{}_{\bm{B}})=T_{m}{}^{j}{}_{k}(t^{\bm{A}}{}_{\bm{B}})\in\mathrm{SU}(m+1).$

By the Schur orthogonality relations and the Peter–Weyl Theorem, the functions

$\sqrt{m+1}\,T_{m}{}^{j}{}_{k}$

form a Hilbert basis for the Hilbert space $L^{2}(\mathrm{SU}(2),\mu)$ where
$\mu$ denotes the normalised Haar measure on $\mathrm{SU}(2)$. In particular,
any complex analytic function $\phi$ on $\mathrm{SU}(2)$ admits the expansion

$\phi(t^{\bm{A}}{}_{\bm{B}})=\sum^{\infty}_{m=0}\sum^{m}_{j=0}\sum^{m}_{k=0}\phi_{m,k,j}T_{m}{}^{j}{}_{k},$

with complex coefficients $\phi_{m,k,j}$ which decay rapidly as
$m\rightarrow\infty$. From the earlier definition, one can check that under
complex conjugation we have

$\overline{T_{m}{}^{j}{}_{k}}=(-1)^{j+k}T_{m}{}^{m-j}{}_{m-k}.$

Moreover, one has for $0\leqslant j,\,k\leqslant m$, $m=0,\,1,2,\,\ldots$,
that

$\displaystyle{\bm{X}}T_{m}{}^{j}{}_{k}=(m-2k)T_{m}{}^{j}{}_{k},$
$\displaystyle{\bm{X}}_{+}T_{m}{}^{j}{}_{k}=\beta_{m,k}T_{m}{}^{j}{}_{k-1},$
$\displaystyle{\bm{X}}_{-}T_{m}{}^{j}{}_{k}=-\beta_{m,k+1}T_{m}{}^{j}{}_{k+1},$

where

$\beta_{m,k}\equiv\sqrt{k(m-k+1)}.$

In particular, the $2$-sphere Laplacian acts on the representations
$T_{m}{}^{j}{}_{k}$ by

$\\{{\bm{X}}_{+},\,{\bm{X}}_{-}\\}T_{m}{}^{j}{}_{k}=(2k(k-m)-m)T_{m}{}^{j}{}_{k}.$

These formulae ensure that for a function $\phi$ with spin weight
$\varsigma\in\frac{1}{2}\mathbb{N}$, i.e. satisfying

${\bm{X}}\phi=2\varsigma\phi,$

the expansions above reduce to

$\phi=\sum_{m\leqslant|2\sigma|}\sum_{j=0}^{m}\phi_{m,j}T_{m}{}^{j}{}_{m/2-\sigma},$

where $m$ takes even values if $\sigma$ is an integer and odd values if
$\sigma$ is a half-integer [Fri86]. In particular, we note that for a given
$k$ ($0\leqslant k\leqslant 2s$), the component $\phi_{k}$ has spin weight
$(k-s)$; that is, ${\bm{X}}\phi_{k}=2(k-s)\phi_{k}$.

######  Remark 11.

The matrix coefficients $T_{m}{}^{j}{}_{k}$ as defined above are related to
the perhaps more widely used spin-weighted spherical harmonics ${}_{s}Y_{lm}$,
as well as Wigner’s $D$-matrices $\mathcal{D}^{j}_{m^{\prime}m}$. One has

${}_{s}Y_{lm}=(-1)^{s+m}\sqrt{\frac{2l+1}{4\pi}}T_{2l}{}^{l-m}{}_{l-s}$ (28)

for $l\in\mathbb{N}\cup\\{0\\}$, and $m,\,s\in\\{-l,-l+1,\,\ldots,\,l-1,l\\}$
[Bey+12]. Further,

$\mathcal{D}^{j}_{m^{\prime}m}\propto T_{2j}{}^{j-m}{}_{j-m^{\prime}},$

where $j\in\frac{1}{2}\mathbb{N}$ and
$m^{\prime},\,m\in\\{-j,-j+1,\,\ldots,\,j-1,j\\}$, [Wig59].

##  Appendix B Spin-$s$ equations

In this appendix we provide a derivation of the massless spin-$s$ equations in
the F-gauge. Let

$\phi_{A_{1}\dots A_{2s}}=\phi_{(A_{1}\dots A_{2s})}$

denote a totally symmetric spinor of valence $2s$, with
$s\in\frac{1}{2}\mathbb{N}$. The massless spin-$s$ equations for $\phi$ then
read

$\nabla^{Q}{}_{A^{\prime}}\phi_{QA_{1}\dots A_{2s-1}}=0.$ (29)

###  B.1 Hyperbolic reduction

The system (29) is not manifestly symmetric hyperbolic, but a _hyperbolic
reduction_ may be obtained by making use of the _space-spinor_ formalism (see
e.g. [Val16], §4) as follows. Let $\tau^{AA^{\prime}}$ denote the spinorial
counterpart of the timelike vector field $\tau^{a}$, with normalization
$\tau_{AA^{\prime}}\tau^{AA^{\prime}}=2$. We define a spin dyad

$\\{\epsilon_{\bm{A}}{}^{A}\\}=\\{o^{A},\,\iota^{A}\\}$

adapted to $\tau^{AA^{\prime}}$ by requiring that

$\tau^{AA^{\prime}}=o^{A}\bar{o}^{A^{\prime}}+\iota^{A}\bar{\iota}^{A^{\prime}}.$

It then follows that

$\tau_{AA^{\prime}}\tau^{BA^{\prime}}=\delta_{A}{}^{B}.$

Defining $\nabla_{AB}\equiv\tau_{B}{}^{A^{\prime}}\nabla_{AA^{\prime}}$, one
has the decomposition

$\nabla_{AB}=\frac{1}{2}\epsilon_{AB}\mathcal{D}+\mathcal{D}_{AB},$

where

$\mathcal{D}\equiv\tau^{AA^{\prime}}\nabla_{AA^{\prime}}\quad\text{and}\quad\mathcal{D}_{AB}\equiv\tau_{(A}{}^{A^{\prime}}\nabla_{B)A^{\prime}}$

denote, respectively, the so-called _Fermi_ and _Sen_ derivative operators.
The operators $\mathcal{D}$ and $\mathcal{D}_{AB}$ correspond, respectively,
to temporal and spatial parts of $\nabla_{AA^{\prime}}$. Using this
decomposition in equation (29), one gets

$\mathcal{D}\phi_{A_{1}\dots
A_{2s}}-2\mathcal{D}^{Q}{}_{A_{1}}\phi_{A_{2}\dots A_{2s}Q}=0.$

The above equation has two irreducible components: the totally symmetric part
and the trace. That is,

$\displaystyle\mathcal{D}\phi_{A_{1}\dots
A_{2s}}-2\mathcal{D}^{Q}{}_{(A_{1}}\phi_{{A_{2}}\dots A_{2s})Q}=0,$ (30a)
$\displaystyle\mathcal{D}^{PQ}\phi_{PQA_{1}\dots A_{2s-2}}=0.$ (30b)

Equations (30a) and (30b) will be referred to as the _evolution_ and
_constraint_ equations, respectively.

######  Remark 12.

It can be shown, through a standard propagation of constraints argument, that
if (30b) is satisfied on some initial hypersurface $\\{\tau=0\\}$, then it is
also satisfied at later times whenever (30a) holds.

######  Remark 13.

Note that the above decomposition into evolution and constraint equations
fails when $s=\frac{1}{2}$. In this case $\phi_{A}$ has only two independent
components and there are two corresponding equations,

$\mathcal{D}\phi_{A}-2\mathcal{D}^{Q}_{\phantom{Q}A}\phi_{Q}=0,$

so there are no constraints.

###  B.2 Transport equations along null geodesics

Given a spinor $\mu_{AB\dots C}$, we denote its components with respect to the
spin dyad $\\{\epsilon_{\bm{A}}{}^{A}\\}$ by
$\mu_{{\bm{A}}{\bm{B}}\dots\bm{C}}$. In particular, then

$\displaystyle\nabla_{{\bm{A}}{\bm{A}}^{\prime}}\mu_{{\bm{B}}\dots\bm{C}}$
$\displaystyle=\epsilon_{\bm{A}}{}^{A}\epsilon_{{\bm{A}}^{\prime}}{}^{A}\epsilon_{{\bm{B}}}{}^{B}\dots\epsilon_{{\bm{C}}}{}^{C}\nabla_{AA^{\prime}}\mu_{B\dots
C}$
$\displaystyle=\partial_{{\bm{A}}{\bm{A}}^{\prime}}\mu_{{\bm{B}}\dots\bm{C}}-\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\bm{Q}}{}_{\bm{B}}\mu_{{\bm{Q}}\dots\bm{C}}-\dots-\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\bm{Q}}{}_{\bm{D}}\mu_{{\bm{B}}\dots\bm{Q}},$

where

$\partial_{{\bm{A}}{\bm{A}}^{\prime}}\equiv
e_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\mu}\partial_{\mu}$ (31)

denotes the directional derivatives of the Newman–Penrose (NP) frame
$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ associated to the spin dyad
$\\{\epsilon_{\bm{A}}{}^{A}\\}$, and
$\Gamma_{{\bm{A}}{\bm{A}}^{\prime}}{}^{\bm{C}}{}_{\bm{D}}$ are the
corresponding spin connection coefficients. In order to give more explicit
expressions, it is convenient to define the following basis of symmetric
$(0,2)$-spinors with unprimed indices,

$\sigma^{0}_{AB}\equiv
o_{A}o_{B},\qquad\qquad\sigma^{1}_{AB}\equiv\iota_{(A}o_{B)},\qquad\qquad\sigma^{2}_{AB}\equiv\iota_{A}\iota_{B}.$

The spinors $\sigma^{i}_{AB}$, $i\in\\{0,1,2\\}$, satisfy the orthogonality
relations

$\displaystyle\sigma^{0}_{AB}(\sigma^{0})^{AB}=0,$
$\displaystyle\sigma^{0}_{AB}(\sigma^{1})^{AB}=0,$
$\displaystyle(\sigma^{0})^{AB}\sigma^{2}_{AB}=1,$
$\displaystyle\sigma^{1}_{AB}(\sigma^{1})^{AB}=-\frac{1}{2},$
$\displaystyle\sigma^{1}_{AB}(\sigma^{2})^{AB}=0,$
$\displaystyle\sigma^{2}_{AB}(\sigma^{2})^{AB}=0.$

For higher valence spinors, we similarly define, for half-integer $s$, the
basis

$\sigma^{0}_{A_{1}\dots A_{2s}}\equiv o_{(A_{1}}\dots
o_{A_{2s})},\quad\sigma^{1}_{A_{1}\dots
A_{2s}}\equiv\iota_{(A_{1}}o_{A_{2}}\dots
o_{A_{2s})},\quad\dots\quad\sigma^{2s}_{A_{1}\dots
A_{2s}}\equiv\iota_{(A_{1}}\dots\iota_{A_{2s})}.$

By contracting with $\tau^{{\bm{A}}{\bm{A}}^{\prime}}$, the directional
derivatives (31) can be decomposed into temporal and spatial parts as

$\partial_{{\bm{A}}{\bm{A}}^{\prime}}=\displaystyle\frac{1}{2}\tau_{{\bm{A}}{\bm{A}}^{\prime}}\partial-\tau^{\bm{Q}}{}_{{\bm{A}}^{\prime}}\partial_{{\bm{A}}{\bm{B}}},$

where we write
$\partial\equiv\tau^{{\bm{A}}{\bm{A}}^{\prime}}\partial_{{\bm{A}}{\bm{A}}^{\prime}}$
and
$\partial_{{\bm{A}}{\bm{B}}}\equiv\tau_{({\bm{A}}}{}^{{\bm{A}}^{\prime}}\partial_{{\bm{B}}){\bm{A}}^{\prime}}$.
In the particular case of the F-gauge used in the main text, one has that

$\displaystyle\partial=\sqrt{2}\partial_{\tau},$
$\displaystyle\partial_{{\bm{A}}{\bm{B}}}=\sqrt{2}\sigma^{1}_{{\bm{A}}{\bm{B}}}\left(-\tau\partial_{\tau}+\rho\partial_{\rho}\right)+\frac{1}{\sqrt{2}}\sigma^{0}_{{\bm{A}}{\bm{B}}}{\bm{X}}_{+}-\frac{1}{\sqrt{2}}\sigma^{2}_{{\bm{A}}{\bm{B}}}{\bm{X}}_{-},$
$\displaystyle\Gamma_{{\bm{A}}{\bm{A}}^{\prime}{\bm{C}}{\bm{D}}}=-\frac{1}{\sqrt{2}}\tau_{{\bm{A}}{\bm{A}}^{\prime}}\sigma^{1}_{{\bm{C}}{\bm{D}}}.$

In particular, it follows that the spatial part
$\Gamma_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=\tau_{({\bm{B}}}^{\phantom{{\bm{B}}}{\bm{B}}^{\prime}}\Gamma_{{\bm{A}}){\bm{B}}^{\prime}{\bm{C}}{\bm{D}}}$
of the spin connection coefficients vanishes. Thus, the only non-trivial part
of the connection is given by the _acceleration_
$f_{{\bm{A}}{\bm{B}}}\equiv-\tau^{{\bm{C}}{\bm{C}}^{\prime}}\Gamma_{{\bm{C}}{\bm{C}}^{\prime}{\bm{A}}{\bm{B}}}$.
In fact it is easy to see that

$f_{{\bm{A}}{\bm{B}}}=\sqrt{2}\sigma^{1}_{{\bm{A}}{\bm{B}}}.$

As a consequence of $\Gamma_{{\bm{A}}{\bm{B}}{\bm{C}}{\bm{D}}}=0$ and the fact
that the components $\sigma^{k}_{{\bm{A}}_{1}\dots\bm{A}_{2s}}$ of
$\sigma^{k}_{A_{1}\dots A_{2s}}$ are constants, it follows that

$\mathcal{D}_{{\bm{A}}{\bm{B}}}\sigma^{k}_{{\bm{A}}_{1}\dots\bm{A}_{2s}}=\partial_{{\bm{A}}{\bm{B}}}\sigma^{k}_{{\bm{A}}_{1}\dots\bm{A}_{2s}}=0,$

for all $0\leqslant k\leqslant 2s$. On the other hand, a calculation shows
that

$\mathcal{D}\sigma^{k}_{{\bm{A}}_{1}\dots\bm{A}_{2s}}=\frac{1}{\sqrt{2}}\sigma^{k}_{{\bm{A}}_{1}\dots\bm{A}_{2s}}.$

In order to write down the equations (30a) and (30b) in our gauge, we now
expand $\phi_{A_{1}\dots A_{2s}}$ in terms of the basis elements
$\sigma^{k}_{A_{1}\dots A_{2s}}$ as

$\phi_{A_{1}\dots
A_{2s}}=\sum_{k=0}^{2s}(-1)^{k}\binom{2s}{k}\phi_{2s-k}\sigma^{k}_{A_{1}\dots
A_{2s}},$ (32)

where

$\phi_{0}\equiv\phi_{A_{1}\dots A_{2s}}o^{A_{1}}\dots
o^{A_{2s}},\quad\phi_{1}\equiv\phi_{A_{1}\dots
A_{2s}}\iota^{A_{1}}o^{A_{2}}\dots
o^{A_{2s}},\quad\ldots\quad\phi_{2s}\equiv\phi_{A_{1}\dots
A_{2s}}\iota^{A_{1}}\dots\iota^{A_{2s}}$

are the components of the massless spinor $\phi_{A_{1}\dots A_{2s}}$. That is,
the component $\phi_{k}$ is obtained from $k$ contractions with $\iota^{A}$
and $2s-k$ contractions with $o^{A}$. By plugging in the expansion (32) into
the equations (30a) and (30b), one may derive the $2s+1$ scalar evolution
equations and $2s-1$ scalar constraint equations satisfied by the components
$\phi_{k}$, $0\leqslant k\leqslant 2s$. These turn out to be

$\displaystyle
E_{0}\equiv(1-\tau)\partial_{\tau}\phi_{0}+\rho\partial_{\rho}\phi_{0}-{\bm{X}}_{-}\phi_{1}-s\phi_{0}=0,$
(33a) $\displaystyle
E_{k}\equiv\partial_{\tau}\phi_{k}-\frac{1}{2}{\bm{X}}_{+}\phi_{k-1}-\frac{1}{2}{\bm{X}}_{-}\phi_{k+1}+(k-s)\phi_{k}=0,\qquad
k=1,\,\dots,\,2s-1,$ (33b) $\displaystyle
E_{2s}\equiv(1+\tau)\partial_{\tau}\phi_{2s}-\rho\partial_{\rho}\phi_{2s}-{\bm{X}}_{+}\phi_{2s-1}+s\phi_{2s}=0,$
(33c)

and

$C_{k}\equiv\tau\partial_{\tau}\phi_{k}-\rho\partial_{\rho}\phi_{k}+\frac{1}{2}{\bm{X}}_{-}\phi_{k+1}-\frac{1}{2}{\bm{X}}_{+}\phi_{k-1}=0,\qquad
k=1,\,\dots,\,2s-1.$

######  Remark 14.

Note that the equations $C_{k}=0$ are termed _constraints_ despite containing
$\partial_{\tau}$ derivatives. Indeed, the quantities $C_{k}$ are propagated
as noted in Remark 12.

The analysis in the main text makes use of certain combinations of the above
evolution and constraint equations. We set

$A_{k}\equiv
E_{k+1}+C_{k+1}\quad\text{for}~{}~{}k=0,\,\dots,\,2s-2\quad\text{and}\quad
A_{2s-1}\equiv E_{2s},$

and

$B_{0}\equiv E_{0}\quad\text{and}\quad B_{k}\equiv
E_{k}-C_{k}\quad\text{for}~{}~{}k=1,\,\dots,\,2s-1.$

Explicitly, these are

$\displaystyle
A_{k}=(1+\tau)\partial_{\tau}\phi_{k+1}-\rho\partial_{\rho}\phi_{k+1}-{\bm{X}}_{+}\phi_{k}+(k+1-s)\phi_{k+1}=0,$
(34a) $\displaystyle
B_{k}=(1-\tau)\partial_{\tau}\phi_{k}+\rho\partial_{\rho}\phi_{k}-{\bm{X}}_{-}\phi_{k+1}+(k-s)\phi_{k}=0,$
(34b)

for $k=0,\,\dots,\,2s-1$. The equations $A_{k}=0$ and $B_{k}=0$ are,
respectively, transport equations along outgoing and incoming null geodesics
in Minkowski space in the F-gauge (introduced in Section 2.2), and we shall
refer to them as the _outgoing equations_ and _incoming equations_
respectively. A crucial feature of the outgoing and incoming equations is that
they become degenerate at $\tau=-1$ and $\tau=+1$ respectively. Further, we
observe here that for a given $k$ the pair $(A_{k},B_{k})$ involves only the
components $\phi_{k}$ and $\phi_{k+1}$.

###  B.3 Wave equations

It will be useful to note that the spinor $\phi_{A_{1}\dots A_{2s}}$ satisfies
the wave equation. Applying $\nabla_{P}^{\phantom{P}A^{\prime}}$ to equation
(29) and using the decomposition

$\nabla_{PA^{\prime}}\nabla_{Q}{}^{A^{\prime}}=\tfrac{1}{2}\epsilon_{PQ}\square+\square_{PQ},$

one finds that

$\square\phi_{A_{1}\dots
A_{2s-1}{\color[rgb]{1,0,0}P}}+2\square_{P}{}^{Q}\phi_{QA_{1}\dots
A_{2s-1}}=0,$

where $\square_{AB}\equiv\nabla_{A^{\prime}(A}\nabla_{B)}{}^{A^{\prime}}$ is
the _Penrose box_ encoding the commutator of two $\nabla_{AA^{\prime}}$
derivatives. Note that if one contracts $P$ with any of the remaining free
indices in the above equation, the first term vanishes by the symmetry of
$\phi_{A_{1}\ldots A_{2s}}$, implying that the second term must also vanish.
But, using the fact that on a spinor $\kappa^{C}$, $\Box_{AB}$ acts by
$\Box_{AB}\kappa^{C}=X_{ABE}{}^{C}\kappa^{E}$ and
$\Box_{A}{}^{B}\kappa_{B}=3\Lambda\kappa_{A}=\mathrm{R}\kappa_{A}/8$, where
$X_{ABCD}=\Psi_{ABCD}+\Lambda(\varepsilon_{AC}\varepsilon_{BD}+\varepsilon_{AD}\varepsilon_{BC})$,
the second term may be expressed entirely in terms of products of $\phi$ with
the Weyl spinor $\Psi_{ABCD}$. When $s>1$, one therefore discovers the well-
known consistency condition141414There are no such restrictions in the cases
$s=1/2$ and $s=1$, i.e. for the Dirac and Maxwell fields. Moreover, while the
spin-$2$ equation is unsatisfactory on a _given_ non-conformally flat
spacetime, the Weyl spinor $\Psi_{ABCD}$ itself satisfies the spin-$2$
equation in _dynamical_ general relativity (in vacuum), when the above
consistency condition becomes $\Psi_{ABM(C}\Psi_{D)}{}^{ABM}=0$, which is
trivial by the symmetries of $\Psi_{ABCD}$. [PR84] for higher spin fields,

$\phi_{ABM(C\ldots K}\Psi_{L)}{}^{ABM}=0,$

i.e. the requirement that the background spacetime be conformally flat. Since
we are working on conformally rescaled Minkowski space, this consistency
condition is satisfied and one obtains the simple wave equation

$\square\phi_{A_{1}\dots A_{2s}}=0.$ (35)

Using the splitting

$\nabla_{AA^{\prime}}=\frac{1}{2}\tau_{AA^{\prime}}\mathcal{D}-\tau^{Q}{}_{A^{\prime}}\mathcal{D}_{AQ},$

in the F-gauge the wave operator (35) on $\phi_{A_{1}\dots A_{2s}}$ may be
written as

$\square\phi_{A_{1}\dots A_{2s}}=\frac{1}{2}\mathcal{D}^{2}\phi_{A_{1}\dots
A_{2s}}+\mathcal{D}^{AB}\mathcal{D}_{AB}\phi_{A_{1}\dots
A_{2s}}-\sqrt{2}(\sigma^{1})^{AB}\mathcal{D}_{AB}\phi_{A_{1}\dots A_{2s}}.$

Scalarising this equation and writing the derivative operators $\mathcal{D}$
and $\mathcal{D}_{AB}$ in terms of $\partial_{\tau}$ and $\partial_{AB}$, one
finds, after a calculation, that the components $\phi_{k}$ satisfy the wave
equations

$\mathcal{W}_{k}[{\bm{\phi}}]\equiv(1-\tau^{2})\partial_{\tau}^{2}\phi_{k}+2\tau\rho\partial_{\tau}\partial_{\rho}\phi_{k}-\rho^{2}\partial_{\rho}^{2}\phi_{k}-\frac{1}{2}\\{{\bm{X}}_{+},{\bm{X}}_{-}\\}\phi_{k}+2(s-k-\tau)\partial_{\tau}\phi_{k}+2(s-k)^{2}\phi_{k}=0$

for $k=0,\,\dots,\,2s$. For convenience, we also introduce the _reduced wave
operator_ acting on a scalar $\zeta$ as

$\blacksquare\zeta\equiv(1-\tau^{2})\ddot{\zeta}+2\tau\rho\dot{\zeta}^{\prime}-\rho^{2}\zeta^{\prime\prime}-2\tau\dot{\zeta}-\frac{1}{2}\\{{\bm{X}}_{+},{\bm{X}}_{-}\\}\zeta,$
(36)

where we denote $\dot{\phantom{X}}\equiv\partial_{\tau}$ and
$\phantom{X}{}^{\prime}\equiv\partial_{\rho}$. In this notation

$\mathcal{W}_{k}[{\bm{\phi}}]=\blacksquare\phi_{k}+\mathbf{L}\phi_{k},\qquad
k=0,\,\dots,\,2s,$ (37)

where $\mathbf{L}$ is a linear lower order operator such that
$[\mathbf{L},\partial_{\rho}]=0$.

######  Remark 15.

In a standard (i.e. non-characteristic) initial value problem, the system (37)
of wave equations needs to be supplemented with the initial data
$(\phi_{k}^{\star},\dot{\phi}_{k}^{\star})$ where
$\phi_{k}^{\star}\equiv\phi_{k}(\tau_{\star})$ and
$\dot{\phi}_{k}^{\star}\equiv\dot{\phi}_{k}(\tau_{\star})$ for some initial
hypersurface $\mathcal{S}_{\star}=\\{\tau=\tau_{*}\\}$,
$\tau_{\star}\in(-1,1)$. We observe that, since the system (37) arises from
the first order system (33a)–(33c), here the time derivative part
$\dot{\phi}_{k}^{\star}$ of the data is always expressible in terms of
$\phi^{\star}_{k}$.

###  B.4 A more general gauge

Certain arguments in the main text require a version of the F-gauge in which
only the critical sets $\mathcal{I}^{\pm}$ are singular sets of the evolution
equations, and $\mathscr{I}^{\pm}$ is given by a non-horizontal hypersurface
in a generalized coordinate plane $(\hat{\tau},\rho)$.

Proceeding as in Section 2.2—but instead of writing $x^{0}=\tau\rho$—we define
a new time coordinate $\hat{\tau}$ by

$x^{0}=\hat{\tau}\kappa,\quad\mbox{where}\quad\kappa=\rho\mu,$

and $\mu$ is a smooth function of $\rho$ such that $\mu(0)=1$, but
$\mu\not\equiv 1$. The specific version of the F-gauge introduced in Section
2.2 corresponds to the choice $\mu\equiv 1$. This more general choice of the
coordinate $\hat{\tau}$ leads to the conformal factor

$\hat{\Theta}=\frac{\rho}{\mu}\big{(}1-\mu^{2}\hat{\tau}^{2}\big{)}=\kappa^{-1}\Xi,$
(38)

which, in turn, gives rise to the unphysical metric

$\displaystyle\hat{{\bm{\eta}}}=\hat{\Theta}^{2}\tilde{{\bm{\eta}}},$
$\displaystyle\phantom{\hat{{\bm{\eta}}}}=\mathbf{d}\hat{\tau}\otimes\mathbf{d}\hat{\tau}+\frac{\hat{\tau}\kappa^{\prime}}{\kappa}\big{(}\mathbf{d}\hat{\tau}\otimes\mathbf{d}\rho+\mathbf{d}\rho\otimes\mathbf{d}\hat{\tau}\big{)}-\frac{(1-\hat{\tau}^{2}\kappa^{\prime
2})}{\kappa^{2}}\mathbf{d}\rho\otimes\mathbf{d}\rho-\frac{1}{\mu^{2}}{\bm{\sigma}}.$

This conformal metric is supplemented with the following choice of frame:

$\displaystyle\hat{{\bm{e}}}_{{\bm{0}}{\bm{0}}^{\prime}}=\frac{1}{\sqrt{2}}\big{(}(1-\kappa^{\prime}\hat{\tau})\big{)}{\bm{\partial}}_{\hat{\tau}}+\kappa{\bm{\partial}}_{\rho},$
$\displaystyle\hat{{\bm{e}}}_{{\bm{1}}{\bm{1}}^{\prime}}=\frac{1}{\sqrt{2}}\big{(}(1+\kappa^{\prime}\hat{\tau})\big{)}{\bm{\partial}}_{\hat{\tau}}-\kappa{\bm{\partial}}_{\rho},$
$\displaystyle\hat{{\bm{e}}}_{{\bm{0}}{\bm{1}}^{\prime}}=-\frac{1}{\sqrt{2}}\mu{\bm{X}}_{+},$
$\displaystyle\hat{{\bm{e}}}_{{\bm{1}}{\bm{0}}^{\prime}}=-\frac{1}{\sqrt{2}}\mu{\bm{X}}_{-},$

with associated non-vanishing spin connection coefficients