††institutetext: University of Vienna, Faculty of Physics
Boltzmanngasse 5, A 1090 Vienna, Austria

# Gluing variations

Piotr T. Chruściel  and Wan Cong

###### Abstract

We consider the question of gluing/embedding/extending geometric structures in
vacuum spacetimes with a cosmological constant in any spacetime dimensions
$d\geq 4$, with emphasis on characteristic data.

##  1 Introduction

In a recent series of pioneering papers, Aretakis, Czimek and Rodnianski ACR1
; ACR2 ; ACR3  presented a gluing construction for characteristic initial data
for four-dimensional vacuum Einstein equations. The purpose of this paper is
to show that _related_ gluing constructions can be done using a spacelike
gluing _à la_ Corvino Corvino . While the construction in ACR1 ; ACR2 ; ACR3
uses the structure of the four-dimensional Einstein equations in a substantial
way, our approach applies to any dimensions. As a bonus, we allow a non-
vanishing cosmological constant. The resulting spacetimes are somewhat
similar, but the intermediate steps are different.

As such, the general relativistic gluing problem can be viewed as the
following question: given two spacetimes, solutions of vacuum Einstein
equations, can one find a third one where non-trivial subsets of each of the
original spacetimes are isometrically included?

A version of this can be formulated at the level of spacelike Cauchy data:
Consider a manifold ${\Sigma}$ and two vacuum initial data sets
$({\Sigma}_{1},g_{1},K_{1})$ and $({\Sigma}_{2},g_{2},K_{2})$ defined on
overlapping subsets ${\Sigma}_{1}$ and ${\Sigma}_{2}$ of ${\Sigma}$. Can one
find a vacuum data set $(g,K)$ on ${\Sigma}$ which coincides with the original
ones away from the overlap, or away from a small neighborhood of the common
boundary? A positive answer to this has first been given by Corvino Corvino
in a restricted setting, and generalised in CorvinoSchoen ; ChDelay ; see
ChBourbaki ; CarlottoLR  for further references. The problem is well
understood for data sets which are not-too-far-away from each other in the
overlap: the gluing can be performed if the spacetimes $({\mycal
M}_{1},{\mathbf{g}}_{1})$ and $({\mycal M}_{2},{\mathbf{g}}_{2})$, obtained by
evolving $({\Sigma}_{1},g_{1},K_{1})$ and $({\Sigma}_{2},g_{2},K_{2})$, have
no Killing vectors near the overlapping region. Equivalently, the set of
Killing Initial Data (KIDs) on the overlap is trivial.

Note that a gluing of overlapping spacelike initial data leads to a gluing of
spacetimes in the following sense: the domains of dependence of
${\Sigma}_{1}\setminus{\Sigma}_{2}$ and ${\Sigma}_{2}\setminus{\Sigma}_{1}$,
within the spacetime obtained by evolving the data on ${\Sigma}$, are
isometric to the corresponding domains of dependence in the original
spacetimes $({\mycal M}_{1},{\mathbf{g}}_{1})$ and $({\mycal
M}_{2},{\mathbf{g}}_{2})$, see Figure 1(a).

An essentially identical construction applies with ${\Sigma}_{1}$ and
${\Sigma}_{2}$ lying on opposite sides of a common boundary,
$\partial{\Sigma}_{1}=\partial{\Sigma}_{2}$, see Figure 1(b).

(a) Gluing overlapping spacelike initial data sets.

(b) Gluing touching spacelike initial data sets.

Figure 1.1: Spacelike gluings. The metric is smooth and vacuum everywhere,
identical to the original one in the left (yellow) and right (green) regions,
the metric in the middle (violet) region interpolates smoothly between the
original ones.

One is then led to the question, whether something similar can be done using
null initial data.111 Classic works concerned with characteristic spacetime
gluing include BarrabesIsrael ; Israel66 ; KhanPenrose . While BarrabesIsrael
; Israel66  focus on lightlike shells, the point of our constructions is to
avoid occurrence of such shells.  For instance, consider a smooth hypersurface
${\mycal N}$ and two characteristic data sets on overlapping subsets ${\mycal
N}_{1}$ and ${\mycal N}_{2}$ of ${\mycal N}$. Suppose that the data on both
${\mycal N}_{1}\subset{\mycal M}_{1}$ and ${\mycal N}_{2}\subset{\mycal
M}_{2}$ arise by restriction from vacuum spacetimes $({\mycal
M}_{1},{\mathbf{g}}_{1})$ and $({\mycal M}_{2},{\mathbf{g}}_{2})$. Can one
find a vacuum spacetime $({\mycal M},{\mathbf{g}})$, with ${\mycal
N}\subset{\mycal M}$, so that the data on ${\mycal N}$, arising by restriction
from ${\mathbf{g}}$, coincide with the original ones away from the overlapping
region?

Here the situation is somewhat different, as a well-posed characteristic
initial-value problem requires either two _transverse_ initial-data surfaces
(not to be confused with the hypersurfaces ${\mycal N}_{1}$ and ${\mycal
N}_{2}$ considered above and in what follows, which are included in a single
smooth hypersurface ${\mycal N}$), or a light cone. In what follows we will
mainly be interested in the former case, in which case the characteristic
initial data consist of data both on each initial-data hypersurfaces and on
their intersection; see, however, Proposition 6.4 below. (In the case of a
light cone, the data on the transverse intersection are replaced by regularity
conditions at the tip.) This makes it clear that an answer in terms of
characteristic initial data on a single smooth hypersurface is not possible.
However, given $k\in\mathbb{N}$, one can complement the characteristic initial
data on ${\mycal N}_{1}$ and ${\mycal N}_{2}$ with information about $k$
derivatives of the metric in directions transverse to ${\mycal N}_{1}$ and
${\mycal N}_{2}$; such transverse derivatives can be obtained by solving
transport equations (i.e., ODE’s along the generators) on ${\mycal N}_{1}$ and
${\mycal N}_{2}$ from data on cross-sections ${\mathbf{S}}_{1}\subset{\mycal
M}_{1}$ and ${\mathbf{S}}_{2}\subset{\mycal M}_{2}$ after some gauge choices
have been made. Denoting by ${{\Psi[{\mycal N}_{1},k]}}{}$ a set of
characteristic data on ${\mycal N}_{1}$ together with transverse derivatives
up to order $k\in\mathbb{N}\cup\\{\infty\\}$, and by ${{\Psi[{\mycal
N}_{2},k]}}{}$ the set of such data on ${\mycal N}_{2}$, one can ask whether
there exist data ${{\Psi[{\mycal N},k]}}{}$ on ${\mycal N}$ which coincide
with the original data on the overlap region.

Explicit parameterisations of the data ${{\Psi[{\mycal N},k]}}{}$ are
presented in Sections 4, 5, and in Appendix A. The question of optimal
differentiability conditions of the fields parameterising ${{\Psi[{\mycal
N},k]}}{}$ is delicate, and for simplicity we will require the existence of
local coordinate systems near ${\mycal N}$ in which all the functions
parameterising ${{\Psi[{\mycal N},k]}}{}$ are smooth _on ${\mycal N}$_.

In their landmark work, Aretakis, Czimek and Rodnianski have given a positive
answer to the following variation of the gluing question, in a near-
Minkowskian setting, illustrated in Figure 1.2. Namely, supposing that
${{\Psi[{\mycal N}_{1},k]}}{}$ are close to ${{\Psi[{\mycal N}_{2},k]}}{}$ on
the overlap region, one asks:

######  Question 1.1.

Do there exist characteristic data
${{\Psi[{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}},k]}}{}$ on a null
hypersurface ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$, obtained by
slightly moving ${\mycal N}_{2}$ in $({\mycal M}_{2},{\mathbf{g}}_{2})$, and
characteristic data ${{\Psi[{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal
N}_{\,}}},k]}}{}$, on a null hypersurface
${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,}}}$ connecting ${\mycal
N}_{1}\setminus{\mycal N}_{2}$ and ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal
N}_{\,2}}}$, which coincide with ${{\Psi[{\mycal N}_{1}\setminus{\mycal
N}_{2},k]}}{}$ on ${\mycal N}_{1}\setminus{\mycal N}_{2}$ and with the data
${{\Psi[{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}},k]}}{}$?

Figure 1.2: Gluing construction of ACR1 .

The authors of ACR1 ; ACR2 ; ACR3  assume that ${\mycal N}$ has the topology
of a light cone in four-dimensional Minkowski spacetime and that $k=2$. The
overlap is taken to be far away from the tip of the light cone, so that
regularity issues at the tip are irrelevant. They show that there exists a
ten-parameter family of obstructions to carry-out the gluing. In the case
where the data ${{\Psi[{\mycal N}_{2},2]}}{}$ arise from a null hypersurface
in a Kerr space-time, they show that one can get rid of the obstructions by
adjusting the mass and angular momentum of the Kerr metric. In
CzimekRodnianski  a striking extension of the method is presented, where the
obstructions are reduced to a single one, namely a lower bound on the mass of
the Kerr extension.

In ACR3  it is also shown how to make a Corvino-type spacelike gluing using
the characteristic gluing.

The aim of this work is to show that spacelike gluings can be used to
construct spacetimes with properties similar to those resulting from the
construction of ACR1 . In the approach described here the hypersurface
${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$ of Question 1.1 is again
obtained by moving slightly ${\mycal N}_{2}$ within $({\mycal
M}_{2},{\mathbf{g}}_{2})$, but we give up the requirement that ${\mycal
N}_{1}$ and ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$ are subsets
of the same smooth null hypersurface in the final spacetime. Indeed, in our
construction the null hypersurface extending smoothly ${\mycal N}_{1}$ in the
new spacetime is obtained by moving ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal
N}_{\,2}}}$ in an auxiliary, suitably constructed, nearby vacuum metric; see
Figure 8.2 below. Our method does _not_ provide a _null gluing_ , but a
variation thereof; hence the title of this paper.

A useful tool to consider towards the above is submanifold data of order $k$,
introduced in Section 3. We provide a simple parameterisation of vacuum
characteristic data on null hypersurfaces and on submanifolds of codimension-
two in Section 4 using coordinate systems introduced by Isenberg and Moncrief
isenberg:moncrief:symmetries . A second such parameterisation is provided in
Section 5 using Bondi coordinates. In Section 6 we use a “hand-crank
construction” to show that vacuum characteristic initial data, or vacuum
spacelike data on a submanifold of codimension 2, of any order can be realised
by embedding as an _interior submanifold_ of a smooth vacuum spacetime. In
Section 7 we use a “Fledermaus construction” to show that vacuum
characteristic initial data on two transversely intersecting hypersurfaces can
likewise be realised by embedding as an _interior submanifold with corner_ of
a smooth vacuum spacetime. In Section 8 we show how to carry out a variation
of the characteristic gluing using spacelike gluing. In Appendix A we show how
the sphere data of ACR1  relate to our codimension-two data in Bondi
parameterisation. In Appendix B we show existence of a preferred, unique set
of Bondi coordinates associated with a null hypersurface ${\mycal N}$ with a
cross-section ${\mathbf{S}}$. In Appendix C we show, by quite general
considerations, that spacetime Killing vectors provide an obstruction to
carry-out certain gluing constructions.

Throughout this work “vacuum” means a solution of the vacuum Einstein
equations with a cosmological constant $\Lambda\in\mathbb{R}$ in spacetime
dimension $n+1\geq 3+1$.

##  2 The existence theorem for two null hypersurfaces intersecting
transversally

In what follows we will need an existence theorem for the characteristic
Cauchy problem, and the aim of this section is to review a version thereof.
This allows us also to introduce our notations.

Let ${\mycal N}$ be a smooth null hypersurface in an $(n+1)$-dimensional
spacetime. Introduce a coordinate system $(u,r,x^{A})$ in which ${\mycal
N}=\\{u=0\\}$ and in which $\partial_{r}$ is tangent to the null geodesics
threading ${\mycal N}$, with
${\mathbf{g}}(\partial_{r},\partial_{x^{A}})|_{{\mycal N}}=0$.

In this section, for consistency with ChPaetz  we write
${\overline{{g}}}_{\mu\nu}$ to denote the restriction of
${\mathbf{g}}_{\mu\nu}$ to ${\mycal N}$, also referred to as the _trace of
${\mathbf{g}}_{\mu\nu}$ on ${\mycal N}$_. (However, we will not do this in the
remaining sections, hoping that the domains of definition will be clear from
the context.) The trace should not be confused with pull-backs: the pull-back
to $\\{u=0\\}$ of a tensor field $A_{uu}du^{2}$ is zero, while the trace of
$A_{uu}du^{2}$ to $\\{u=0\\}$ is the tensor field $A_{uu}|_{u=0}du^{2}$
defined along $\\{u=0\\}$, which vanishes if and only if $A_{uu}|_{u=0}\equiv
0$.

On ${\mycal N}$ let

${\theta}\equiv\frac{1}{2}{\overline{{g}}}^{AB}\partial_{r}{\overline{{g}}}_{AB}$
(2.1)

be the divergence scalar, and let

$\sigma_{AB}\equiv\frac{1}{2}\partial_{r}{\overline{{g}}}_{AB}-\frac{1}{n-1}{\theta}{\overline{{g}}}_{AB}$
(2.2)

be the trace-free part of $\frac{1}{2}\partial_{r}{\overline{{g}}}_{AB}$, also
known as the _shear tensor_. The vacuum Raychaudhuri equation,

$\partial_{r}{\theta}-\kappa{\theta}+|\sigma|^{2}+\frac{{\theta}^{2}}{n-1}=0\,,$
(2.3)

where

$|\sigma|^{2}\equiv\sigma_{A}{}^{B}\sigma_{B}{}^{A}\,,\quad\sigma_{A}{}^{B}\equiv{\overline{{g}}}^{BC}\sigma_{AC}\,,$

and

$\kappa={\Gamma}{{}^{r}_{rr}}|_{u=0}$ (2.4)

(see (CCM2, , Appendix A) for a collection of explicit formulae in adapted
coordinates) provides on ${\mycal N}$ a constraint equation for the family of
$(n-1)$-dimensional metrics

$r\mapsto{\overline{{g}}}_{AB}(r,x^{C})\mathrm{d}x^{A}\mathrm{d}x^{B}\,.$

The geometric meaning of $\kappa$ is that of the connection coefficient of the
one-dimensional bundle of tangents to the null generators of ${\mycal N}$,
viewed as a bundle along each of the generators. Indeed, under a change of
coordinates $(r,x^{A})\mapsto({\check{r}}(r,x^{A}),{\check{x}}^{A}=x^{A})$ we
have

${\theta}\mapsto{\check{{\theta}}}:=\frac{1}{2}{\overline{{g}}}^{AB}\partial_{{\check{r}}}{\overline{{g}}}_{AB}=\frac{\partial
r}{\partial{\check{r}}}{\theta}\,,\quad\sigma_{AB}\mapsto{\check{\sigma}}_{AB}:=\frac{1}{2}\partial_{{\check{r}}}{\overline{{g}}}_{AB}-\frac{1}{n-1}{\check{{\theta}}}{\overline{{g}}}_{AB}=\frac{\partial
r}{\partial{\check{r}}}\sigma_{AB}\,,$ (2.5)

and the Raychaudhuri equation becomes

$\partial_{{\check{r}}}{\check{{\theta}}}-{\check{\kappa}}{\check{{\theta}}}+|{\check{\sigma}}|^{2}+\frac{{\check{{\theta}}}^{2}}{n-1}=0\,,$
(2.6)

with

$\kappa\mapsto{\check{\kappa}}=\frac{\partial
r}{\partial{\check{r}}}\kappa+\frac{\partial{\check{r}}}{\partial
r}\frac{\partial^{2}r}{\partial{\check{r}}^{2}}\,.$ (2.7)

The vanishing of $\kappa$ is equivalent to the requirement that $r$ is an
affine parameter along the generator.

We are ready now to pass to the problem at hand. Consider two smooth
hypersurfaces ${{\mycal N}}$ and ${\underline{{\mycal N}}}$ in an
$(n+1)$–dimensional manifold ${\mycal M}$, with transverse intersection at a
smooth submanifold ${\mathbf{S}}$. We choose adapted coordinates
$({u},{\underline{u}},x^{A})$ so that ${{\mycal N}}$ coincides with the set
$\\{{u}=0\\}$, while ${\underline{{\mycal N}}}$ is given by
$\\{{\underline{u}}=0\\}$. We suppose that ${{\mycal
N}}={I}\times{\mathbf{S}}$ and ${\underline{{\mycal
N}}}={\underline{I}}\times{\mathbf{S}}$, where ${I}$ and ${\underline{I}}$ are
intervals of the form $[0,{\rho})$ for some
${\rho}\in\mathbb{R}^{+}\cup\\{\infty\\}$, possibly with distinct ${\rho}$,
with the coordinate ${u}$ ranging over ${\underline{I}}$ and ${\underline{u}}$
ranging over ${I}$.

We wish for the restriction ${\overline{{g}}}_{\mu\nu}$ of the metric
functions ${\mathbf{g}}_{\mu\nu}$ to the initial surface ${{\mycal
N}}\cup{\underline{{\mycal N}}}$ to arise from a smooth Lorentzian spacetime
metric. This will be the case if ${\overline{{g}}}_{{u}{\underline{u}}}$ is
nowhere vanishing, if
${\underline{u}}\mapsto{\overline{{g}}}_{AB}({\underline{u}},\cdot)$ is a
family of Riemannian metrics on
$\\{{\underline{u}}\\}\times{\mathbf{S}}\subset{{\mycal N}}$, similarly for
${u}\mapsto{\overline{{g}}}_{AB}({u},\cdot)$ on
$\\{{u}\\}\times{\mathbf{S}}\subset{\underline{{\mycal N}}}$, and if
${\overline{{g}}}_{\mu\nu}$ is smooth on ${{\mycal N}}$ and
${\underline{{\mycal N}}}$ and continuous across $S\equiv{{\mycal
N}}\cap{\underline{{\mycal N}}}$. The hypothesis that ${{\mycal N}}$ and
${\underline{{\mycal N}}}$ are characteristic translates to

${\mathbf{g}}_{{u}{u}}\big{|}_{{\underline{{\mycal N}}}}\equiv
0\equiv{\mathbf{g}}_{{u}A}\big{|}_{{\underline{{\mycal
N}}}}\,,\qquad{\mathbf{g}}_{{\underline{u}}\,{\underline{u}}}\big{|}_{{{\mycal
N}}}\equiv 0\equiv{\mathbf{g}}_{{\underline{u}}A}\big{|}_{{{\mycal N}}}\,.$
(2.8)

We therefore impose the following conditions on ${\mathbf{S}}$:

$\displaystyle\displaystyle\lim_{{\underline{u}}\rightarrow
0}{\mathbf{g}}_{AB}|_{{{\mycal N}}}\displaystyle=\ \lim_{{u}\rightarrow
0}{\mathbf{g}}_{AB}|_{{\underline{{\mycal N}}}}\,,$ (2.9)
$\displaystyle\displaystyle\lim_{{\underline{u}}\rightarrow
0}{\mathbf{g}}_{{u}{\underline{u}}}|{{{\mycal N}}}\displaystyle=\
\lim_{{u}\rightarrow 0}{\mathbf{g}}_{{u}{\underline{u}}}|{{\underline{{\mycal
N}}}}\,,$ (2.10) $\displaystyle\displaystyle\lim_{{\underline{u}}\rightarrow
0}{\mathbf{g}}_{{u}A}|{{{\mycal N}}}=\ 0\,,\quad\lim_{{u}\rightarrow
0}{\mathbf{g}}_{{\underline{u}}A}|{{\underline{{\mycal N}}}}=\,0\,,$ (2.11)
$\displaystyle\displaystyle\lim_{{\underline{u}}\rightarrow
0}{\mathbf{g}}_{{u}{u}}|{{{\mycal N}}}\displaystyle=\
0\,,\quad\lim_{{u}\rightarrow
0}{\mathbf{g}}_{{\underline{u}}\,{\underline{u}}}|{{\underline{{\mycal
N}}}}=\,0\,.$ (2.12)

To avoid ambiguities, we will denote by $\kappa_{{\mycal N}}$ the function
$\kappa$ associated with the hypersurface ${\mycal N}$; similarly for
$\kappa_{{\underline{{\mycal N}}}}$, ${\theta}_{{{\mycal N}}}$, etc. We choose
the time-orientation by requiring that both $\partial_{u}$ and
$\partial_{\underline{u}}$ are future-oriented at ${\mathbf{S}}$. We have the
following ChPaetz :222We take this opportunity to point out two annoying
misprints in ChPaetz : in Equation (40a) there the left-hand side should be
${\mathbf{g}}|_{N_{2}}$, and in Equation (40b) the left-hand side should be
${\mathbf{g}}|_{N_{1}}$.

######  Theorem 2.1.

Let there be given functions $({\overline{{g}}}_{\mu\nu},\kappa)$ on ${{\mycal
N}}\cup{\underline{{\mycal N}}}$ such that

$\displaystyle{\mathbf{g}}_{\mu\nu}|_{{{\mycal
N}}}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}={\overline{{g}}}_{{u}{u}}(\mathrm{d}{u})^{2}+2{\overline{{g}}}_{{u}{\underline{u}}}\mathrm{d}{u}\,\mathrm{d}{\underline{u}}+2{\overline{{g}}}_{{u}A}\mathrm{d}{u}\mathrm{d}x^{A}+{\overline{{g}}}_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,,$
(2.13) $\displaystyle{\mathbf{g}}_{\mu\nu}|_{{\underline{{\mycal
N}}}}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}={\overline{{g}}}_{{\underline{u}}\,{\underline{u}}}(\mathrm{d}{\underline{u}})^{2}+2{\overline{{g}}}_{{u}{\underline{u}}}\mathrm{d}{u}\,\mathrm{d}{\underline{u}}+2{\overline{{g}}}_{{\underline{u}}A}\mathrm{d}{\underline{u}}\mathrm{d}x^{A}+{\overline{{g}}}_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,,$
(2.14)

smooth up-to-boundary on ${{\mycal N}}$ and ${\underline{{\mycal N}}}$.
Suppose that ${\overline{{g}}}_{{u}{\underline{u}}}<0$, that (2.9)-(2.12)
hold, and that

$\displaystyle\partial_{{u}}{\overline{{g}}}_{{\underline{u}}\,{\underline{u}}}|_{\mathbf{S}}=2\big{(}\partial_{{\underline{u}}}{\overline{{g}}}_{{u}{\underline{u}}}-{\overline{{g}}}_{{u}{\underline{u}}}\kappa_{{{\mycal
N}}}\big{)}|_{\mathbf{S}}\qquad\Big{(}\Longleftrightarrow\quad\Gamma^{\underline{u}}_{{\underline{u}}\,{\underline{u}}}|_{\mathbf{S}}=\kappa_{{{\mycal
N}}}|_{\mathbf{S}}\Big{)}\,,$ (2.15)
$\displaystyle\partial_{{\underline{u}}}{\overline{{g}}}_{{u}{u}}|_{\mathbf{S}}=2\big{(}\partial_{{u}}{\overline{{g}}}_{{u}{\underline{u}}}-{\overline{{g}}}_{{u}{\underline{u}}}\kappa_{{\underline{{\mycal
N}}}}\big{)}|_{\mathbf{S}}\qquad\Big{(}\Longleftrightarrow\quad\Gamma^{u}_{{u}{u}}|_{\mathbf{S}}=\kappa_{{\underline{{\mycal
N}}}}|_{\mathbf{S}}\Big{)}\,.$ (2.16)

If the Raychaudhuri equation holds both on ${{\mycal N}}$ and
${\underline{{\mycal N}}}$,

$\displaystyle-\partial_{{\underline{u}}}{\theta}_{{{\mycal
N}}}+{\kappa_{{{\mycal N}}}}{\theta}_{{{\mycal N}}}-|\sigma_{{{\mycal
N}}}|^{2}-\frac{{\theta}^{2}_{{{\mycal N}}}}{n-1}=0\ \mbox{on ${{\mycal
N}}$}\,,$ (2.17) $\displaystyle-\partial_{{u}}{\theta}_{{\underline{{\mycal
N}}}}+{\kappa_{{\underline{{\mycal N}}}}}{\theta}_{{\underline{{\mycal
N}}}}-|\sigma_{{\underline{{\mycal
N}}}}|^{2}-\frac{{\theta}^{2}_{{\underline{{\mycal N}}}}}{n-1}=0\ \mbox{on
${\underline{{\mycal N}}}$}\,,$ (2.18)

then there exists a smooth metric defined on a neighbourhood of ${\mathbf{S}}$
solving the vacuum Einstein equations to the future of ${{\mycal
N}}\cup{\underline{{\mycal N}}}$ and realising the data
$({\overline{{g}}}_{\mu\nu},\kappa)$ on ${{\mycal N}}\cup{\underline{{\mycal
N}}}$.

######  Remark 2.2.

In spacetime dimension $n+1=4$ the domain of existence of the solution
contains a one-sided neighborhood of ${{\mycal N}}\cup{\underline{{\mycal
N}}}$ by Luk  when ${\mathbf{S}}\approx S^{2}$, and by CCW  for general
topologies of ${\mathbf{S}}$; we note that ${\mathbf{S}}$ is not required to
be compact unless explicitly assumed otherwise. One expects the result to be
true in all dimensions, but this remains to be established.

######  Remark 2.3.

The maximal globally hyperbolic solution is uniquely determined by the data,
up to isometry. This can be seen by first noting that the local solution is
defined uniquely, up to isometry, by the data listed; see MarsCharacteristic2
for an extensive discussion. Choosing a spacelike Cauchy hypersurface within
the domain of existence of the coordinate solution, one can appeal to the
usual Choquet-Bruhat – Geroch uniqueness theorem for the spacelike Cauchy
problem to conclude.

######  Remark 2.4.

Rendall RendallCIVP  requires that $\kappa\equiv 0$ and that the coordinates
are solutions of the wave equation, $\Box_{g}x^{\mu}=0$. The Raychaudhuri
equations (2.17)-(2.18) are then solved using a conformal ansatz
${\overline{{g}}}_{AB}=\phi^{2}\gamma_{AB}$, with $\gamma_{AB}$ being freely
prescribable on ${{\mycal N}}\cup{\underline{{\mycal N}}}$ subject to
continuity at ${\mathbf{S}}$. These conditions determine the metric functions
${\overline{{g}}}_{\mu\nu}$ in terms of the free data uniquely up to the
choice of

$({\overline{{g}}}_{{u}{\underline{u}}},\phi,\partial_{u}\phi,\partial_{\underline{u}}\phi,\zeta_{A})\big{|}_{\mathbf{S}}\,,$

where $\zeta_{A}$ is the _torsion 1-form_ defined as follows: Assuming
${\mathbf{g}}_{{u}{\underline{u}}}|_{S}$ is negative we set
$L=\sqrt{-2{\overline{{g}}}^{{u}{\underline{u}}}}\,\partial_{\underline{u}}$
and
$\underline{{\mbox{\emph{L}}}}\,=\sqrt{-2{\overline{{g}}}^{{u}{\underline{u}}}}\,\partial_{u}$
along ${{\mycal N}}\cup{\underline{{\mycal N}}}$, then $\zeta_{A}$ on
${\mathbf{S}}$ is defined as

$\displaystyle\zeta_{A}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}{\mathbf{g}}(\nabla_{A}L,\underline{L})\big{|}_{\mathbf{S}}\equiv\frac{1}{2}{\mathbf{g}}^{{u}{\underline{u}}}(\partial_{u}{\mathbf{g}}_{{\underline{u}}A}-\partial_{\underline{u}}{\mathbf{g}}_{{u}A})\big{|}_{\mathbf{S}}\,.$
(2.19)

######  Remark 2.5.

Some comments concerning Theorem 2.1 and the Bondi form of the metric,
Equation (5.1) below, are in order. Suppose that ${\theta}_{{\mycal N}}\neq 0$
in Theorem 2.1. We can then introduce on ${{\mycal N}}$ an area coordinate $r$
(compare Appendix B), and use the Bondi parameterisation of the metric on
${{\mycal N}}$. So we take ${\underline{u}}|_{{\mycal N}}=r$, and we emphasise
that (2.14) will _not_ hold in general when ${\underline{u}}$ there is taken
to be the Bondi area coordinate. Thus the Bondi form of the metric is assumed
to hold on ${{\mycal N}}$ but not necessarily away from ${{\mycal N}}$.333One
could be tempted to think that those of the Bondi-parameterised Einstein
equations which _do not_ contain $\partial_{u}$-derivatives of the metric can
be used _as they are_ on ${{\mycal N}}$, but this is not the case: for
instance, the Raychaudhuri equation (compare (2.24)) contains a term which
involves $\partial_{u}$-derivatives of the metric and which vanishes if the
Bondi form of the metric holds in a neighborhood of ${\mycal N}$, and which
does not vanish in a general coordinate system.

Using the Bondi parameterisation on ${{\mycal N}}$ we have

$\displaystyle\displaystyle{\underline{u}}|_{{\mycal
N}}=r\,,\qquad{\theta}_{{\mycal
N}}=\frac{n-1}{r}\,,\qquad\sigma_{AB}=\frac{r^{2}}{2}\partial_{r}\gamma_{AB}|_{{\mycal
N}}\,,$ (2.20)

The constraint equation (2.17) can be solved for $\kappa_{{{\mycal N}}}$:

$\kappa_{{{\mycal
N}}}=\frac{r}{4(n-1)}{\gamma}^{AC}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}{\gamma}_{CD})\big{|}_{{\mycal
N}}\,.$ (2.21)

Equations (2.15) and (2.21) provide then a constraint on ${\mathbf{S}}$:

$\displaystyle\displaystyle\Big{[}2\partial_{r}\beta+\frac{e^{-2\beta}}{2}\overline{\partial_{u}{g}_{{\underline{u}}\,{\underline{u}}}}\Big{]}\Big{|}_{\mathbf{S}}=\kappa_{{\mycal
N}}|_{\mathbf{S}}=\frac{r}{4(n-1)}{\gamma}^{AC}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}{\gamma}_{CD})\big{|}_{\mathbf{S}}\,.$
(2.22)

Once the vacuum equations have been solved, $\kappa_{{\mycal N}}$ coincides
with $\Gamma^{\underline{u}}_{{\underline{u}}{\underline{u}}}$ restricted to
${{\mycal N}}$:

$\displaystyle\displaystyle\kappa_{{\mycal
N}}=2\partial_{r}\beta+\frac{e^{-2\beta}}{2}\overline{\partial_{u}{g}_{{\underline{u}}\,{\underline{u}}}}\,.$
(2.23)

Hence the vacuum Raychaudhuri equation (2.17) becomes

$2\partial_{r}\beta\underbrace{+\frac{e^{-2\beta}}{2}\overline{\partial_{u}{g}_{{\underline{u}}\,{\underline{u}}}}}_{\phantom{x}}=\frac{r}{4(n-1)}{\gamma}^{AC}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}{\gamma}_{CD})\Big{|}_{{\mycal
N}}\,.$ (2.24)

This is consistent with (5.5) below, since the underbraced term is zero when
${\underline{u}}$ is taken to be the area coordinate $r$ and the Bondi form of
the metric is assumed in a neighborhood of ${{\mycal N}}$.

##  3 Submanifold data

Let ${\mycal M}$ be an $(n+1)$-dimensional manifold. Recall that, for
$k\in\mathbb{N}\cup\\{\infty\\}$ and $p\in{\mycal M}$, the $k$-th jet
$j^{k}_{p}A$ at $p$ of a tensor field $A$, or of a function $A$, is the
collection of all partial derivatives of $A$ up to order $k$. We also recall
that this notion is coordinate-independent, in an obvious sense: given a
representative of $j^{k}_{p}A$ in some coordinate system, we can determine
$j^{k}_{p}A$ in any other coordinate system by standard calculus formulae.

Let ${{\mathcal{J}}}$ be a smooth submanifold of ${\mycal M}$. For
$k\in\mathbb{N}\cup\\{\infty\\}$ we define _submanifold data of order $k$_ as
the following collection of jets _defined along ${{\mathcal{J}}}$_

${{\Psi[{{\mathcal{J}}},k]}}{}:=\\{j^{k}_{p}{\mathbf{g}}_{\mu\nu},p\in{{\mathcal{J}}}\\}\,,$
(3.1)

where we further assume that the jets arise by restriction to
${{\mathcal{J}}}$ of jets of a smooth Lorentzian metric
${\mathbf{g}}_{\mu\nu}$ defined in a neighborhood of ${{\mathcal{J}}}$.
Equivalently, ${{\Psi[{{\mathcal{J}}},k]}}{}$ are those sections over
${{\mathcal{J}}}$ of the bundle of jets which are obtained by calculating the
jets of a spacetime metric defined near ${{\mathcal{J}}}$ and restricting to
${{\mathcal{J}}}$. Or in yet equivalent words: an element of
${{\Psi[{{\mathcal{J}}},k]}}{}$ is obtained by taking a Lorentzian metric
defined in a neighborhood of ${{\mathcal{J}}}$, calculating all its
derivatives up to order $k$, and restricting the resulting fields to
${{\mathcal{J}}}$.

We have assumed for simplicity that all the fields occurring in
${{\Psi[{{\mathcal{J}}},k]}}{}$ are smooth, though one could of course
consider more general situations.

It should be clear that an element of ${{\Psi[{{\mathcal{J}}},k]}}{}$ can
always be extended to a smooth metric defined in a neighborhood of
${{\mathcal{J}}}$ by using Taylor series if $k$ is finite, or by Borel
summation if $k=\infty$, with the jets of the extended field inducing the
original jets on ${{\mathcal{J}}}$. Such an extension will be called
_compatible_.

Since a field ${\mathbf{g}}_{\mu\nu}$ restricted to ${{\mathcal{J}}}$ already
carries information about its derivatives in directions tangent to
${{\mathcal{J}}}$, for $k\geq 1$ the new information in
${{\Psi[{{\mathcal{J}}},k]}}{}$ is contained in the derivatives in directions
transverse to ${{\mathcal{J}}}$.

As a special case consider a hypersurface ${\Sigma}$. An equivalent definition
of ${{\Psi[{\Sigma},k]}}{}$ is obtained by choosing a smooth vector field $X$
defined in a neighborhood of ${\Sigma}$, transverse to ${\Sigma}$, and setting

${{\Psi[{\Sigma},k]}}{}:=\Big{(}\big{(}({\mycal
L}_{X})^{i}{\mathbf{g}}_{\mu\nu}\big{)}\big{|}_{{\Sigma}}\Big{)}_{0\leq i\leq
k}\,.$ (3.2)

Given another smooth vector field $Y$ transverse to ${\Sigma}$, the collection
$\Big{(}\big{(}({\mycal
L}_{Y})^{i}{\mathbf{g}}_{\mu\nu}\big{)}\big{|}_{{\Sigma}}\Big{)}_{0\leq i\leq
k}$ can be rewritten in terms of $\Big{(}\big{(}({\mycal
L}_{X})^{i}{\mathbf{g}}_{\mu\nu}\big{)}\big{|}_{{\Sigma}}\Big{)}_{0\leq i\leq
k}$, and vice-versa.

We will say that ${{\Psi[{{\mathcal{J}}},k]}}{}$ is spacelike if only metrics
${\mathbf{g}}_{\mu\nu}$ with spacelike pull-back to ${{\mathcal{J}}}$ are
allowed in (3.1), similarly for timelike, characteristic, etc.

The submanifold data ${{\Psi[{{\mathcal{J}}},k]}}{}$ will be called _vacuum_
if the derivatives of the metric up to order $k$ satisfy the restrictions
arising from the vacuum Einstein equations, including the equations obtained
by differentiating the vacuum Einstein equations in transverse directions. The
structure of the set of vacuum jets is best understood by choosing preferred
coordinates. For example, in normal coordinates at a point $p$ the first
derivatives of the metric vanish and the second derivatives are uniquely
determined by the Riemann tensor at $p$. In these coordinates the set of
vacuum jets of second order is a subspace determined by a set of simple linear
equations.

As another example, let ${\Sigma}$ be again of codimension-one, and for $k\geq
2$ consider the set of vacuum data ${{\Psi[{\Sigma},k]}}{}$ such that
${\mathbf{g}}_{\mu\nu}$ induces a Riemannian metric on ${\Sigma}$. Then every
element of ${{\Psi[{\Sigma},k]}}{}$ can be equivalently described by the usual
vacuum general relativistic initial data fields $(g,K)$, where $g$ is a
Riemannian metric on ${\Sigma}$, $K$ represents the extrinsic curvature tensor
of ${\Sigma}$, with $(g,K)$ satisfying the (spacelike) constraint equations.
Indeed, one can then introduce e.g. harmonic coordinates for
${\mathbf{g}}_{\mu\nu}$, and algebraically determine all transverse
derivatives of ${\mathbf{g}}_{\mu\nu}$ on ${\Sigma}$ in terms of $(g,K)$ and
their tangential derivatives. In this case the introduction of the index $k$
is clearly an overkill. However, we will see examples where each $k$ adds
further information.

As a further example, consider two hypersurfaces ${{\mycal N}}$ and
${\underline{{\mycal N}}}$ in ${\mycal M}$ intersecting transversally at a
joint boundary

${\mathbf{S}}={{\mycal N}}\cap{\underline{{\mycal N}}}\,,$

and consider vacuum submanifold data ${{\Psi[{{\mycal
N}}\cup{\underline{{\mycal N}}},k]}}{}$ such that ${{\mycal N}}$ and
${\underline{{\mycal N}}}$ are characteristic for ${\mathbf{g}}_{\mu\nu}$,
with the null directions along ${{\mycal N}}$ and ${\underline{{\mycal N}}}$
being orthogonal to ${\mathbf{S}}$. Set

${{\Phi[{\mycal N},k]}}{}=\\{j^{k}_{p}g_{AB},j^{k}_{p}\kappa,p\in{\mycal
N}\\}\,,$ (3.3)

where $g_{AB}dx^{A}dx^{B}$ is the tensor field induced on ${\mycal N}$ by
${\mathbf{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$, with $\kappa$ as in (2.4)
(transforming as in (2.7)), and with the data satisfying the Raychaudhuri
equation and its transverse derivatives up to order $k$. The collection (3.3)
will be referred to as _reduced characteristic data of order $k$_ on ${\mycal
N}$. It follows from Remark 2.4 that, for $k\geq 1$, vacuum characteristic
data ${{\Psi[{{\mycal N}}\cup{\underline{{\mycal N}}},k]}}{}$ can be replaced
by a set

${{\Phi[{{\mycal N}},0]}}{}\cup{{\Phi[{\underline{{\mycal
N}}},0]}}{}\cup{{\Psi[{\mathbf{S}},1]}}{}$ (3.4)

together with the compatibility conditions and constraints described in
Theorem 2.1. This last theorem also shows that every vacuum set
${{\Psi[{{\mycal N}}\cup{\underline{{\mycal N}}},k]}}{}$ can be realised as
the boundary of a manifold-with-boundary-with-corner carrying a smooth vacuum
metric. Here the index $k$ is again an overkill for $k\geq 1$.

It is sensible to enquire about the minimum value of $k$ which makes sense in
the context of Einstein equations. A first guess would be to take $k\geq 2$,
since the equations are well posed in a classical sense when $k\geq 2$.
However, some of the equations have a constraint character, involving two
derivatives in tangential directions but only one in transverse directions. In
view of our hypothesis, that all the fields are smooth in tangential
directions, an appropriate condition appears to be $k\geq 1$.

In the next sections we will:

  1. 1.

Provide a simpler description of vacuum characteristic data ${{\Phi[{\mycal
N},k]}}{}$.

  2. 2.

Show that a class of vacuum characteristic data ${{\Phi[{\mycal N},k]}}{}$ can
be realised by a hypersurface _inside_ a smooth vacuum spacetime.

  3. 3.

Show that vacuum spacelike data ${{\Psi[{\mathbf{S}},k]}}{}$ on a submanifold
${\mathbf{S}}$ of codimension two can be realised by a submanifold _inside_ a
smooth vacuum spacetime

##  4 The Isenberg-Moncrief parameterisation

A smooth submanifold ${\mathbf{S}}$ of a smooth null hypersurface ${\mycal
N}\subset{\mycal M}$ will be called a _cross-section of ${\mycal N}$_ if every
generator of ${\mycal N}$ intersects ${\mathbf{S}}$ precisely once. The aim of
this section is to provide a simpler description of characteristic data of
order $k$ on a hypersurface ${\mycal N}$ with cross-section ${\mathbf{S}}$,
using coordinates which are often referred to as Gaussian null coordinates.
Since Gauss has never heard of these coordinates, and since they have been
introduced by Isenberg and Moncrief isenberg:moncrief:symmetries , we will
refer to them as _Isenberg-Moncrief coordinates_ ; IM for short. In these
coordinates the metric reads444Our $r$ is the same as Isenberg-Moncrief’s $t$
and our $u$ is the negative of Isenberg-Moncrief’s $x^{3}$, see
(isenberg:moncrief:symmetries, , Equation (2.8)).

${\mathbf{g}}_{\mu\nu}{\mathrm{d}x}^{\mu}{\mathrm{d}x}^{\nu}=2\left(-{\mathrm{d}{u}}+{u}\alpha{\mathrm{d}{r}}+{u}\beta_{A}{\mathrm{d}x}^{A}\right){\mathrm{d}{r}}+{g}_{AB}{\mathrm{d}x}^{A}{\mathrm{d}x}^{B}\,.$
(4.1)

We like to think of ${\mycal N}$ as the hypersurface $\\{u=0\\}$, with $r$
being a coordinate along the generators of ${\mycal N}$. The Einstein
equations for the metric (4.1), in all dimensions, can be found in (HIW, ,
Appendix A), after replacing the coordinates $({u},{r})$ of HIW  with our
$({r},-{u})$.

There are actually two choices for ${\mycal N}$ in the coordinates (4.1): as
$\\{{r}=0\\}$ or as $\\{{u}=0\\}$, since both these hypersurfaces are null for
the metric (4.1). The reader is warned that our choice here leads
unfortunately to confusions with the usual notation ${u}$ for a coordinate
whose level sets are null, as is the case for Bondi coordinates: we emphasise
that the zero-level set of ${u}$ here is null, but the remaining level sets
are not, in general. On the other hand all the level sets of $r$ are null for
a metric of the form (4.1).

One can always adjust the Isenberg-Moncrief coordinates so that $\alpha$
vanishes on ${\mycal N}$, but this might not be convenient in general and will
_not_ be assumed in this section.

We shall write $\beta^{A}:={g}^{AB}\beta_{B}$ and
$\beta\cdot\beta:={g}^{AB}\beta_{A}\beta_{B}$, where ${g}^{AB}$ is the inverse
metric to ${g}_{AB}$.

In the coordinate system as in (4.1) the key data on ${\mycal N}$ are
$({g}_{AB},\alpha)$ subject to the Raychaudhuri equation, which for the metric
(4.1) reads

$0=\left(-{1\over 2}{g}^{AB}\partial^{2}_{r}{g}_{AB}+{1\over
4}{g}^{CA}{g}^{BD}(\partial_{r}{g}_{AB})\partial_{r}{g}_{CD}+\frac{1}{2}\alpha\>{g}^{AB}\partial_{r}{g}_{AB}\right)\bigg{|}_{{\mycal
N}}$ (4.2)

(see (HIW, , Appendix A), Equation (77) with their ${r}=0$). We will refer to
the pair $({g}_{AB},\alpha)$ as the _Isenberg-Moncrief data on ${\mycal N}$_.
Note that if $\theta\equiv{1\over 2}{g}^{AB}\partial_{r}{g}_{AB}$ has no zeros
on ${\mycal N}$, the field $\alpha$ can be determined algebraically in terms
of ${g}_{AB}$ and derivatives of ${g}_{AB}$ tangential to ${\mycal N}$ using
(4.2).

Then:

  1. 1.

The equation $R_{{r}A}|_{{\mycal N}}=0$ provides a linear ODE for $\beta_{A}$
along the generators of ${\mycal N}$; as already mentioned, such ODEs will be
referred to as _transport equations along ${\mycal N}$._ Indeed, in vacuum it
holds that (HIW, , Equation (79))

$\displaystyle 0$ $\displaystyle=$ $\displaystyle R_{{r}A}$ (4.3)
$\displaystyle=$ $\displaystyle-D_{A}\alpha+{1\over
2}\partial_{r}\beta_{A}+{1\over
4}\beta_{A}{g}^{BC}\partial_{r}{g}_{BC}-D_{[A}({g}^{BC}\partial_{r}{g}_{B]C})$
$\displaystyle+\frac{{u}}{2}\times\bigg{[}\;{1\over
2}({g}^{BC}\partial_{r}{g}_{BC})\partial_{u}\beta_{A}+\partial_{r}\partial_{u}\beta_{A}+2\alpha\partial_{u}\beta_{A}$
$\displaystyle\qquad+\partial_{u}({u}\beta_{A})\cdot\left\\{-{u}^{-1}\partial_{u}({u}^{2}\beta\cdot\beta)+2\partial_{u}\alpha\right\\}$
$\displaystyle\qquad-2D_{A}(\partial_{u}\alpha)+\partial_{u}(\beta^{C}\partial_{r}{g}_{CA})+2{u}^{-1}\partial_{u}\left({u}^{2}\beta^{D}D_{[A}\beta_{D]}\right)$
$\displaystyle\qquad-{1\over
2}{g}^{BC}\partial_{u}{g}_{BC}\times\Big{\\{}({u}\beta\cdot\beta-2\alpha)\partial_{u}({u}\beta_{A})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+2D_{A}\alpha-\beta^{C}\partial_{r}{g}_{CA}-2{u}\beta^{F}D_{[A}\beta_{F]}\Big{\\}}$
$\displaystyle\qquad-2\partial_{u}(\alpha\beta_{A})-2{u}(\partial_{u}\alpha)\partial_{u}\beta_{A}-D_{B}\left\\{\beta^{B}\partial_{u}({u}\beta_{A})\right\\}$
$\displaystyle\qquad+2{g}^{CD}D_{D}D_{[A}\beta_{C]}-{g}^{BC}(\partial_{u}\beta_{B})\partial_{r}{g}_{CA}$
$\displaystyle\qquad-{g}^{BC}\partial_{u}({u}\beta_{B})\times\Big{\\{}-({u}\beta\cdot\beta-2\alpha)\partial_{u}{g}_{CA}-D_{C}\beta_{A}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\beta_{C}\partial_{u}({u}\beta_{A})+{u}\beta_{C}\beta^{E}\partial_{u}{g}_{EA}\Big{\\}}$
$\displaystyle\qquad+{g}^{BC}(\partial_{u}{g}_{CA})\times\left\\{2\beta_{B}\partial_{u}({u}\alpha)+2D_{B}\alpha+\partial_{u}\beta_{B}-2{u}\beta^{D}D_{[B}\beta_{D]}\right\\}\bigg{]}\,,\phantom{xxx}$

where $D_{A}$ is the covariant derivative operator associated with the metric
${g}_{AB}$. This equation at ${u}=0$, together with
$\beta_{A}|_{{\mathbf{S}}}$ and the IM data, determines $\beta_{A}$ uniquely
along ${\mycal N}$. Note that only the first line of the right-hand side
survives on ${\mycal N}$, but we reproduce the relevant Einstein equations
here and below in whole as the overall features of the remaining terms in the
equations are relevant for the induction argument below.

Equations (5.5)-(5.8), together with the equations obtained by differentiating
(5.5)-(5.8) transversally to ${\mycal N}$, will be referred to as _the
transport equations along ${\mycal N}$._

  2. 2.

The vacuum equation $R_{AB}|_{{\mycal N}}=-\frac{n+3}{2(n-1)}\Lambda{g}_{AB}$
provides a linear transport equation for $\partial_{u}{g}_{AB}$. Indeed, we
have (HIW, , Equation (82))

$\displaystyle R_{AB}$ $\displaystyle=$
$\displaystyle\partial_{r}\partial_{u}{g}_{AB}+\alpha\partial_{u}{g}_{AB}+{\cal
R}_{AB}-D_{(A}\beta_{B)}-{1\over 2}\beta_{A}\beta_{B}$ (4.4) $\displaystyle-$
$\displaystyle{g}^{CD}\left(\partial_{u}{g}_{D(A}\right)\partial_{r}{g}_{B)C}+{1\over
4}\left\\{({g}^{CD}\partial_{r}{g}_{CD})\partial_{u}{g}_{AB}+({g}^{CD}\partial_{u}{g}_{CD})\partial_{r}{g}_{AB}\right\\}$
$\displaystyle-$
$\displaystyle\frac{{u}}{2}\times\bigg{[}-2\alpha\partial_{u}^{2}{g}_{AB}+D_{C}(\beta^{C}\partial_{u}{g}_{AB})$
$\displaystyle\qquad\,+{1\over
2}({g}^{CD}\partial_{u}{g}_{CD})\left\\{({u}\beta\cdot\beta-2\alpha)\partial_{u}{g}_{AB}+2D_{(A}\beta_{B)}\right\\}$
$\displaystyle\qquad\,-2(\partial_{u}\alpha)\partial_{u}{g}_{AB}+{u}^{-1}\\{\partial_{u}({u}^{2}\beta\cdot\beta)\\}\partial_{u}{g}_{AB}$
$\displaystyle\qquad\,+{u}\beta\cdot\beta\partial_{u}^{2}{g}_{AB}+2\partial_{u}\\{D_{(A}\beta_{B)}\\}$
$\displaystyle\qquad\,+2\beta_{(A}\partial_{u}\beta_{B)}+{u}(\partial_{u}\beta_{A})\partial_{u}\beta_{B}+{u}\beta^{E}\beta^{F}(\partial_{u}{g}_{AE})\partial_{u}{g}_{BF}$
$\displaystyle\qquad\,-2\beta^{C}\left\\{\partial_{u}({u}\beta_{(A})\right\\}\partial_{u}{g}_{B)C}-2{g}^{CD}\left(D_{D}\beta_{(A}\right)\partial_{u}{g}_{B)C}$
$\displaystyle\qquad\,-{g}^{CD}({u}\beta\cdot\beta-2\alpha)(\partial_{u}{g}_{CA})\partial_{u}{g}_{BD}\,\bigg{]}\,,$

where ${\cal R}_{AB}$ is the Ricci tensor associated with ${g}_{AB}$. This
equation at ${u}=0$, together with $\partial_{u}{g}_{AB}|_{{\mathbf{S}}}$ and
the field $\beta_{A}|_{\mycal N}$ determined in point 1., defines now uniquely
$\partial_{u}{g}_{AB}|_{{\mycal N}}$.

  3. 3.

The equation $R_{{u}{r}}|_{{\mycal N}}=-\frac{n+3}{2(n-1)}\Lambda$, where
(HIW, , Equation (78))

$\displaystyle R_{{u}{r}}$ $\displaystyle=$
$\displaystyle-2\partial_{u}\alpha+{1\over
4}{g}^{CA}{g}^{BD}(\partial_{r}{g}_{CD})\partial_{u}{g}_{AB}-{1\over
2}{g}^{AB}\partial_{u}\partial_{r}{g}_{AB}-\frac{1}{2}\alpha\>{g}^{AB}\partial_{u}{g}_{AB}+{1\over
2}\beta\cdot\beta$ (4.5) $\displaystyle+$
$\displaystyle\frac{{u}}{2}\times\bigg{[}-2\partial_{u}^{2}\alpha-{1\over
2}{g}^{AB}\partial_{u}{g}_{AB}\cdot\left\\{2\partial_{u}\alpha-\beta^{D}\partial_{u}({u}\beta_{D})\right\\}$
$\displaystyle\qquad\,\,+\beta^{B}\partial_{u}\beta_{B}+\partial_{u}\\{\beta^{B}\partial_{u}({u}\beta_{B})\\}+{g}^{AB}D_{A}(\partial_{u}\beta_{B})\bigg{]}\,,$

determines now algebraically $\partial_{u}\alpha|_{{\mycal N}}$, as well as
any derivative of $\partial_{u}\alpha|_{{\mycal N}}$ in directions tangential
to ${\mycal N}$.

  4. 4.

The transverse derivative $\partial_{u}\beta_{A}|_{{\mycal N}}$, as well as
the derivatives of $\partial_{u}\beta_{A}|_{{\mycal N}}$ tangential to
${\mycal N}$, are now obtained from the equation $R_{{u}A}|_{{\mycal N}}=0$,
where (HIW, , Equation (81))

$\displaystyle R_{{u}A}$ $\displaystyle=$ $\displaystyle-{1\over
4}\beta_{A}{g}^{BC}\partial_{u}{g}_{BC}-\partial_{u}\beta_{A}+{1\over
2}\beta^{B}\partial_{u}{g}_{AB}-D_{[A}\left({g}^{BC}\partial_{u}{g}_{B]C}\right)$
(4.6) $\displaystyle+$
$\displaystyle\frac{{u}}{2}\times\bigg{[}\;-\partial_{u}^{2}\beta_{A}+\partial_{u}\left(\beta^{B}\partial_{u}{g}_{AB}\right)$
$\displaystyle\qquad\qquad+{1\over
2}({g}^{CD}\partial_{u}{g}_{CD})\left(-\partial_{u}\beta_{A}+\beta^{B}\partial_{u}{g}_{AB}\right)\bigg{]}\,.$

One can now consider the equations obtained by successively applying
$\partial_{u}^{i}$, $i\geq 1$, to (4.4)-(4.6) to determine all transverse
derivatives of ${g}_{AB}$ on ${\mycal N}$ by first order linear transport
equations, whose solutions are determined uniquely by their initial or final
data, and all transverse derivatives of $\alpha$ and $\beta_{A}$ on ${\mycal
N}$ from linear algebraic equations.

We note that the constraint (2.16) is trivially satisfied, while (2.15) reads

$\alpha|_{\mathbf{S}}=\kappa_{\mycal N}|_{\mathbf{S}}\,.$ (4.7)

We have therefore proved:

######  Proposition 4.1.

In the Isenberg-Moncrief coordinate system the reduced vacuum characteristic
data ${{\Phi[{\mycal N},k]}}{}$ of (3.3) can be replaced by the following
collection of fields:

${{\Phi_{\mathrm{IM}}[{\mycal
N},k]}}{}:=\\{(\partial_{u}^{j}{g}_{AB},\alpha,\beta_{A})_{0\leq j\leq k}\
\mbox{on}\ {\mathbf{S}}\ \mbox{and}\ ({g}_{AB},\alpha)\ \mbox{on}\ {\mycal
N}\\}\,,$ (4.8)

with $({g}_{AB},\alpha)$ subject to the constraint equation (4.2). Moreover
the fields $(\alpha,\beta_{A})$ are not needed in (4.8) if $k=0$, as they
appear multiplied by ${u}$ in the metric.

We will refer to this set of data as _Isenberg-Moncrief vacuum characteristic
data of order $k$._

The above also shows:

######  Proposition 4.2.

Let ${\mathbf{S}}$ be of codimension-two and $k\in\mathbb{N}\cup\\{\infty\\}$.
Then:

  1. 1.

Spacelike vacuum data ${{\Psi[{\mathbf{S}},k]}}{}$ of (3.1) can be reduced to
the following collection of fields on ${\mathbf{S}}$:

${{\Psi_{\mathrm{IM}}[{\mathbf{S}},k]}}{}:=(\partial_{r}^{i}\partial_{u}^{j}{g}_{AB},\alpha,\beta_{A})_{0\leq
i+j\leq k}\,,$ (4.9)

with $(\alpha,\beta_{A})$ not needed if $k=0$.

  2. 2.

Let ${\mathbf{S}}_{1}$ and ${\mathbf{S}}_{2}$ be two cross-sections of a null
hypersurface ${\mycal N}$ with vacuum data ${{\Psi[{\mathbf{S}}_{1},k]}}{}$
and ${{\Psi[{\mathbf{S}}_{2},k]}}{}$ induced by vacuum characteristic data
${{\Psi[{\mycal N},k]}}{}$. Then ${{\Psi[{\mathbf{S}}_{2},k]}}{}$ can be
determined uniquely in terms of ${{\Psi[{\mathbf{S}}_{1},k]}}{}$ and
${{\Psi[{\mycal N},0]}}{}$ by solving linear transport equations along the
generators of ${\mycal N}$, or by solving linear algebraic equations on
${\mycal N}$; and vice-versa.

###### Proof.

1\. This can be seen directly by repeating the proof of Proposition 4.1.
Alternatively, let ${\mathbf{g}}_{\mu\nu}$ be any Lorentzian metric near
${\mathbf{S}}$ compatible with ${{\Psi[{\mathbf{S}},k]}}{}$. Denote by
${{\mycal N}}$ either of the null hypersurfaces orthogonal to ${\mathbf{S}}$,
and by ${\underline{{\mycal N}}}$ the other one. Use the data induced by
${\mathbf{g}}_{\mu\nu}$ to solve the vacuum Einstein equations to the future
of ${{\mycal N}}\cup{\underline{{\mycal N}}}$. Introduce Isenberg-Moncrief
coordinates so that ${{\mycal N}}=\\{{u}=0\\}$. The result follows now from
Proposition 4.1.

2\. It suffices to prove the result in Isenberg-Moncrief coordinates. In these
we have just shown that one can reduce ${{\Psi[{\mathbf{S}}_{1},k]}}{}$ to
${{\Psi_{\mathrm{IM}}[{\mathbf{S}}_{1},k]}}{}$ and
${{\Psi[{\mathbf{S}}_{2},k]}}{}$ to
${{\Psi_{\mathrm{IM}}[{\mathbf{S}}_{2},k]}}{}$, with
${{\Psi_{\mathrm{IM}}[{\mathbf{S}}_{2},k]}}{}$ determined uniquely in terms of
${{\Psi_{\mathrm{IM}}[{\mathbf{S}}_{2},k]}}{}$ and ${{\Psi[{\mycal N},0]}}{}$
by solving linear transport equations along the generators of ${\mycal N}$ and
by solving linear algebraic equations on ${\mycal N}$, as desired.

##  5 The Bondi parameterisation

A parameterisation of the metric which has often been used in the literature,
in spacetime-dimension equal to four, is that of Bondi et al. (cf., e.g.,
Sachs ; BBM ; Frittelli:2004pk ; Frittelli:1999yr ; MaedlerWinicour ),

$\displaystyle{\mathbf{g}}$ $\displaystyle\equiv$
$\displaystyle{\mathbf{g}}_{\alpha\beta}\mathrm{d}x^{\alpha}\mathrm{d}x^{\beta}$
(5.1) $\displaystyle=$
$\displaystyle-\frac{V}{r}e^{2\beta}\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\mathrm{d}r+r^{2}{\gamma}_{AB}\Big{(}\mathrm{d}x^{A}-U^{A}\mathrm{d}u\Big{)}\Big{(}\mathrm{d}x^{B}-U^{B}\mathrm{d}u\Big{)}\,,$

together with the conditions

$\partial_{r}\det{{\gamma}_{AB}}=\partial_{u}\det{{\gamma}_{AB}}=0\,.$ (5.2)

A coordinate system satisfying (5.2) exists on ${\mycal N}$ with ${\mycal
N}=\\{u=0\\}$ if and only if the expansion scalar of ${\mycal N}$ has no
zeros. We show in Appendix B, in all spacetime dimensions $n+1\geq 3$, that
given a cross-section ${\mathbf{S}}$ of a smooth null hypersurface ${\mycal
N}$ with ${\theta}>0$, there exists a unique Bondi coordinate system near
${\mycal N}$ in which

$u|_{{\mycal N}}=0$, $r|_{{\mathbf{S}}}=1$, ${\theta}|_{{\mathbf{S}}}=n-1$,
$V|_{{\mathbf{S}}}=-\frac{2{\underline{\theta}}}{n-1}$ and
$\beta|_{{\mathbf{S}}}=0={\mycal U}^{A}|_{{\mathbf{S}}}$.  (5.3)

Here ${\theta}$ is the null mean curvature of ${\mathbf{S}}$ associated with
$\partial_{r}$, and ${\underline{\theta}}$ is the null mean curvature of
${\mathbf{S}}$ associated with the null vectors, say $\underline{\check{L}}$,
orthogonal to ${\mathbf{S}}$ and transverse to $\\{u=0\\}$ at ${\mathbf{S}}$,
normalised so that
${\mathbf{g}}(\underline{\check{L}},\partial_{r})|_{{\mathbf{S}}}=-1$.

In the notation of (5.1), cross-section data can be defined as follows. Let
${\mathbf{S}}$ be a submanifold of a null hypersurface ${\mycal N}$ transverse
to its generators. Let $k\in\mathbb{N}\cup\\{\infty\\}$ be the number of
_transverse_ derivatives of the metric that we want to control at
${\mathbf{S}}$. Using the Bondi parameterisation of the metric, we define the
_Bondi cross-section data of order $k$_ as the collection of fields

${{\Psi_{\mathrm{Bo}}[{\mathbf{S}},k]}}{}:=(\partial_{r}^{i}\partial_{u}^{j}\gamma_{AB}|_{{\mathbf{S}}},\,\partial_{u}^{j}\beta|_{{\mathbf{S}}},\,\partial_{u}^{j}U^{A}|_{{\mathbf{S}}},\,\partial_{r}U^{A}|_{{\mathbf{S}}},\,V|_{{\mathbf{S}}})_{0\leq
i+j\leq k}\,.$ (5.4)

As already pointed out, for simplicity we assume that all the fields in (5.4)
are smooth. We note that a finite sufficiently large degree of
differentiability, typically different for distinct fields, would suffice for
most purposes; this can be determined by chasing the number of derivatives
needed in the relevant equations.

In Appendix A we show how the sphere data of ACR1  relate to the Bondi ones.

We will refer to the Bondi field $\gamma_{AB}|_{\mycal N}$ as _free Bondi
data_ on ${\mycal N}$.

Note that the data ${{\Psi_{\mathrm{Bo}}[{\mathbf{S}},k]}}{}$ contain more
fields than the Isenberg-Moncrief data
${{\Psi_{\mathrm{IM}}[{\mathbf{S}},k]}}{}$ of (4.9). This is related to the
residual freedom remaining in Bondi coordinates; compare Proposition B.3,
Appendix B.

To continue, suppose that we are given Bondi cross-section data of order $k$
and a field $\gamma_{AB}$ on ${\mycal N}$. We assume that the field
$\gamma_{AB}$ and its $\partial^{j}_{r}$-derivatives with $0\leq j\leq k$,
when restricted to ${\mathbf{S}}$, coincide with the Bondi cross-section
fields $\partial^{j}_{r}\gamma_{AB}|_{{\mathbf{S}}}$; such data sets will be
said to be compatible. Then (see MaedlerWinicour  in spacetime-dimension
four):

  1. 1.

Using the equation

$0=\frac{r}{2(n-1)}G_{rr}=\partial_{r}\beta-\frac{r}{8(n-1)}{\gamma}^{AC}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}{\gamma}_{CD})$
(5.5)

we can determine $\partial_{r}\beta|_{{\mathbf{S}}}$. Further, integrating
(5.5), the value $\beta|_{{\mathbf{S}}}$ and the field $\gamma_{AB}$ on
${\mycal N}$ can be used to determine $\partial_{r}\beta$ on ${\mycal N}$, and
hence all radial derivatives $\partial_{r}^{i}\beta|_{{\mathbf{S}}}$.

  2. 2.

The fields $U^{A}|_{{\mathbf{S}}}$ and $\partial_{r}U^{A}|_{{\mathbf{S}}}$ are
used to obtain $U^{A}|_{{\mycal N}}$ by integrating

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 2r^{n-1}G_{rA}$
$\displaystyle=$
$\displaystyle\partial_{r}\left[r^{n+1}e^{-2\beta}{\gamma}_{AB}(\partial_{r}U^{B})\right]-2r^{2(n-1)}\partial_{r}\Big{(}\frac{1}{r^{n-1}}{D}_{A}\beta\Big{)}+r^{n-1}{\gamma}^{EF}{D}_{E}(\partial_{r}{\gamma}_{AF})\,.$

Further, the radial derivatives $\partial_{r}^{i}U^{A}|_{\mathbf{S}}$ can be
algebraically determined in terms of
$\partial_{r}^{i}\gamma_{AB}|_{\mathbf{S}}$.

We note that $U_{A}$ can be set to zero on ${\mathbf{S}}$ by a refinement of
the coordinates, but this restriction is not convenient when interpolating
between cross-section data, and will therefore not be assumed.

  3. 3.

The function $V|_{{\mathbf{S}}}$ is needed to integrate the equation

$\displaystyle-2\Lambda r^{2}$ $\displaystyle=$ $\displaystyle
r^{2}e^{-2\beta}(2G_{ur}+2U^{A}G_{rA}-V/r\,G_{rr})$ (5.7) $\displaystyle=$
$\displaystyle
R[{\gamma}]-2{\gamma}^{AB}\Big{[}{D}_{A}{D}_{B}\beta+({D}_{A}\beta)({D}_{B}\beta)\Big{]}+\frac{e^{-2\beta}}{r^{2(n-2)}}{D}_{A}\Big{[}\partial_{r}(r^{2(n-1)}U^{A})\Big{]}$
$\displaystyle-\frac{1}{2}r^{4}e^{-4\beta}{\gamma}_{AB}(\partial_{r}U^{A})(\partial_{r}U^{B})-\frac{(n-1)}{r^{n-3}}e^{-2\beta}\partial_{r}(r^{n-3}V)\,,$

obtaining thus $V|_{{\mycal N}}$ by integration. Further, all radial
derivatives $\partial^{i}_{r}V|_{{\mathbf{S}}}$ can be determined
algebraically in terms of the already-known fields by inductively
$r$-differentiating this equation.

  4. 4.

The field $\partial_{u}\gamma_{AB}|_{{\mathbf{S}}}$ is needed to integrate

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
r^{(n-5)/2}\mathop{\mathrm{TS}}[G_{AB}]$ (5.8) $\displaystyle=$
$\displaystyle\partial_{r}\Big{[}r^{(n-1)/2}\partial_{u}{\gamma}_{AB}-\frac{1}{2}r^{(n-3)/2}V\partial_{r}{\gamma}_{AB}-\frac{n-1}{4}r^{(n-5)/2}V{\gamma}_{AB}\Big{]}$
$\displaystyle+\frac{n-1}{4}\partial_{r}(r^{(n-5)/2}V){\gamma}_{AB}+r^{(n-5)/2}\mathop{\mathrm{TS}}\bigg{[}e^{2\beta}r^{2}{\mathring{R}}_{AB}-2e^{\beta}{D}_{A}{D}_{B}e^{\beta}$
$\displaystyle+r^{3-n}{\gamma}_{CA}{D}_{B}[\partial_{r}(r^{n-1}U^{C})]-\frac{1}{2}r^{4}e^{-2\beta}{\gamma}_{AC}{\gamma}_{BD}(\partial_{r}U^{C})(\partial_{r}U^{D})$
$\displaystyle+\frac{r^{2}}{2}(\partial_{r}{\gamma}_{AB})({D}_{C}U^{C})+r^{2}U^{C}{D}_{C}(\partial_{r}{\gamma}_{AB})$
$+\displaystyle\frac{1}{2}rV\gamma^{CD}\partial_{r}\gamma_{AC}\partial_{r}\gamma_{BD}-\frac{1}{2}r^{2}\gamma^{CD}(\partial_{r}\gamma_{BD}\partial_{u}\gamma_{AC}+\partial_{u}\gamma_{BD}\partial_{r}\gamma_{AC})$
$\displaystyle-r^{2}(\partial_{r}{\gamma}_{AC}){\gamma}_{BE}({D}^{C}U^{E}-{D}^{E}U^{C})\bigg{]}\,,$

where the symbol $\mathop{\mathrm{TS}}$ denotes the traceless-symmetric part
of a tensor; note that the TS part of each of the terms in the box is zero
when $n=3$. One thus determines $\partial_{u}\gamma_{AB}|_{{\mycal N}}$.

  5. 5.

The $u$-derivative of $\beta$ on ${\mycal N}$ can be calculated by integrating
the equation obtained by $u$-differentiating (5.5), after expressing the
right-hand side in terms of the fields determined so far:

$\partial_{r}\partial_{u}\beta=\frac{r}{8}\Big{(}{\gamma}^{AC}\partial_{u}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}{\gamma}_{CD})+{\gamma}^{AC}{\gamma}^{BD}(\partial_{r}{\gamma}_{AB})(\partial_{r}\partial_{u}{\gamma}_{CD})\Big{)}\,.$
(5.9)

For this we also need the initial value $\partial_{u}\beta|_{{\mathbf{S}}}$.

  6. 6.

The equation

$\displaystyle-2e^{2\beta}G_{uA}=0\,,$ (5.10)

reads

$\displaystyle\partial_{r}\Big{[}e^{4\beta}\partial_{u}\Big{(}e^{-4\beta}r^{2}\gamma_{AB}U^{B}\Big{)}\Big{]}+\partial_{r}(r^{2}\gamma_{AB})\partial_{u}U^{B}$
(5.11) $\displaystyle=$
$\displaystyle\partial_{r}\Big{[}e^{2\beta}\partial_{r}\Big{(}r\gamma_{AB}U^{B}Ve^{2\beta}\Big{)}-2rV\partial_{r}(\gamma_{AB}U^{B})+r^{2}U^{B}\partial_{u}\gamma_{AB}\Big{]}+2e^{2\beta}\partial_{A}\partial_{u}\beta$
$\displaystyle+\Big{(}U^{B}\partial_{r}\big{(}r^{2}\gamma_{AB}\big{)}+1/2\gamma_{AB}r^{2}\partial_{r}U^{B}\Big{)}e^{4\beta}\partial_{u}e^{4\beta}-2\Lambda
r^{2}e^{2\beta}\gamma_{AB}U^{B}+F_{A}\,,\phantom{xxxxxx}$

with

$F_{A}=F_{A}\big{(}{n},r,{\mathbf{g}}_{\mu\nu},\partial_{i}{\mathbf{g}}_{\mu\nu},\partial_{u}\gamma_{AB},\partial_{A}\partial_{B}{\mathbf{g}}_{\mu\nu},\partial_{r}\partial_{B}{\mathbf{g}}_{\mu\nu},\partial_{A}\partial_{u}\gamma_{AB}\big{)}\,,$
(5.12)

where $\\{\partial_{i}\\}=\\{\partial_{r},\partial_{A}\\}$. This equation
allows us to determine algebraically
$\partial_{r}\partial_{u}U^{A}|_{\mathbf{S}}$ in terms of the already known
fields. The field $\partial_{r}\partial_{u}U^{A}|_{\mathbf{S}}$ is needed to
determine $\partial_{u}U^{A}|_{\mycal N}$ by integrating in $r$ the
$u$-derivative of Equation (LABEL:22XII22.3). The explicit form of $F_{A}$ is
not very enlightening and is too long to be usefully displayed here.

  7. 7.

We can determine algebraically $\partial_{u}V$ either on ${\mathbf{S}}$ or on
${\mycal N}$ from the Einstein equation $(G_{uu}+\Lambda g_{uu})|_{\mycal
N}=0$:

$G_{uu}=\frac{n-1}{2r^{2}}\partial_{u}V+...\,,$ (5.13)

where $...$ is an explicit expression in all fields already known on ${\mycal
N}$, and which is too long to be usefully displayed here.

One can inductively repeat the procedure above using the equations obtained by
differentiating Einstein equations with respect to $u$. One thus obtains a
hierarchical system of ODEs in $r$, or algebraic equations for the transverse
derivatives of $\partial_{r}U^{A}$ and $V$, for

$(\beta,\,U^{A},\,V,\,\partial_{u}\gamma_{AB},\,\partial_{u}\beta,\,\partial_{u}U^{A},\,\partial_{u}V,\,\partial_{u}^{2}\gamma_{AB},\ldots,\,\partial_{u}^{k}\beta,\,\partial_{u}^{k}U^{A},\,\partial^{k}_{u}V)\,,$
(5.14)

which can be integrated or algebraically solved in the order indicated in
(5.14).

It might be of some interest to note that there are no obstructions to
integrate the transport equations globally on $I\times{\mathbf{S}}$. Here one
should keep in mind that (5.5) is the Raychaudhuri equation written in terms
of the Bondi fields, and the blow-up of solutions of this equation is at the
origin of various incompleteness theorems in general relativity.

Summarising, we have:

######  Proposition 5.1.

In the Bondi coordinate system (the existence of which requires $\theta>0$ on
${\mycal N}$) the reduced vacuum characteristic data ${{\Phi[{\mycal
N},k]}}{}$ of (3.3) can be replaced by the following collection of fields:

${{\Phi_{\mathrm{Bo}}[{\mycal
N},k]}}{}:=\\{(\partial_{u}^{j}\gamma_{AB}|_{{\mathbf{S}}},\,\partial_{u}^{j}\beta|_{{\mathbf{S}}},\,\partial_{u}^{j}U^{A}|_{{\mathbf{S}}},\,\partial_{r}U^{A}|_{{\mathbf{S}}},\,V|_{{\mathbf{S}}})_{0\leq
j\leq k}\ \mbox{on}\ {\mathbf{S}}\ \mbox{and}\ \gamma_{AB}\ \mbox{on}\ {\mycal
N}\\}\,.$ (5.15)

Equivalently, Einstein equations and smooth cross-section Bondi data (5.4) of
order $k\in\mathbb{N}\cup\\{\infty\\}$, together with smooth compatible Bondi
free data $\gamma_{AB}$ on a null hypersurface ${\mycal N}$, determine
uniquely the metric functions

$\partial_{u}^{i}{\mathbf{g}}_{\mu\nu}|_{{\mycal N}}$

in Bondi gauge, for all $0\leq i\leq k$, through linear transport equations
along the generators, or through linear algebraic equations.

Furthermore, in vacuum, the Bondi data
${{\Psi_{\mathrm{Bo}}[{\mathbf{S}},k]}}{}$ of (5.4) suffice to determine
${{\Psi[{\mathbf{S}},k]}}{}$ in Bondi coordinates.

##  6 The “hand-crank construction”

As already pointed out, characteristic initial data on a single null
hypersurface do not lead to a well posed Cauchy problem, i.e., a setup which
guarantees both existence and uniqueness of associated solutions of the field
equations. In this section we present a construction which provides existence
of solutions of the vacuum Einstein equations realising the data. A similar
idea can be found in Kehle:2022uvc .

The proof below applies when the solution, invoked below, of the Cauchy
problem for the vacuum Einstein equations with data on two transverse
characteristic hypersurfaces, exists in a future neighborhood of the
hypersurfaces. However the existing theorems only guarantee this in spacetime
dimension four (see Remark 2.2), which is at the origin of the restriction to
this dimension in our next result:

######  Proposition 6.1.

Given a smooth vacuum characteristic initial data set ${{\Psi[{\mycal
N},k]}}{}$, $k\in\mathbb{N}\cup\\{\infty\\}$, on a three dimensional manifold
of the form

${\mycal N}=[r_{1},r_{2}]\times{\mathbf{S}}\,,\qquad r_{1}<r_{2}\,,$

there exist (many) smooth solutions of the vacuum Einstein equations $({\mycal
M},{\mathbf{g}})$ so that ${{\Psi[{\mycal N},k]}}{}$ is obtained by
restriction to a characteristic hypersurface within ${\mycal M}$.

###### Proof.

If $k<\infty$ we can extend ${{\Psi[{\mycal N},k]}}{}$ to ${{\Psi[{\mycal
N},\infty]}}{}$ in any way. Therefore it suffices to assume that $k=\infty$.

Let us write ${\mathbf{S}}_{c}$ for ${\mycal N}\cap\\{r=c\\}$. Let
${{\Psi[{\mathbf{S}}_{r},k]}}{}$ be the cross-section data induced by
${{\Psi[{\mycal N},k]}}{}$ on ${\mathbf{S}}_{r}$. Let ${{{\underline{{\mycal
N}}}}_{\,r_{1}}}=[0,1]\times{\mathbf{S}}$ be a hypersurface on which we
prescribe any smooth Isenberg-Moncrief data $({g}_{AB},\alpha)$ compatible
with ${{\Psi[{\mathbf{S}}_{r_{1}},k]}}{}$; thus $({{{\underline{{\mycal
N}}}}_{\,r_{1}}},{g}_{AB},\alpha)$ are characteristic initial data on the
hypersurface ${{{\underline{{\mycal N}}}}_{\,r_{1}}}$ meeting ${\mycal N}$
transversally at $r=r_{1}$ towards the _future_ of ${\mathbf{S}}_{r_{1}}$, see
Figure 6.1. By definition of compatibility, all derivatives of
$({g}_{\mu\nu},\alpha)$ in directions tangential to ${{{\underline{{\mycal
N}}}}_{\,r_{1}}}$ and transverse to ${\mycal N}$ have to match at
${\mathbf{S}}_{r_{1}}$ with those in ${{\Psi[{\mathbf{S}}_{r_{1}},k]}}{}$,
which can be achieved by Borel summation.

Similarly let ${{{\underline{{\mycal N}}}}_{\,r_{2}}}=[0,1]\times{\mathbf{S}}$
be a hypersurface, meeting ${\mycal N}$ transversally at
${\mathbf{S}}_{r_{2}}$ towards the _past_ , on which we give smooth
characteristic data compatible with ${{\Psi[{\mathbf{S}}_{r_{2}},k]}}{}$.

Figure 6.1: The “hand-crank construction”.

We can solve the characteristic Cauchy problem to the future with data on the
transversally intersecting hypersurfaces ${\mycal N}$ and
${{{\underline{{\mycal N}}}}_{\,r_{1}}}$, resulting in a smooth vacuum metric,
say ${\mathbf{g}}_{+}$, defined on $J^{+}({{{\underline{{\mycal
N}}}}_{\,r_{1}}}\cup{\mycal N})$. The Einstein equations guarantee that the
transport and the algebraic equations in Isenberg-Moncrief coordinates
described in Section 4 hold on ${\mycal N}$. The initial data for these
transport equations at ${\mathbf{S}}_{r_{1}}$ are given by
${{\Psi[{\mathbf{S}}_{r_{1}},k]}}{}$. Uniqueness of the solutions of the
transport equations shows that the transverse derivatives of the metric
coincide with the ones listed in ${{\Psi[{\mycal N},k]}}{}$.

One can likewise solve the characteristic Cauchy problem to the past with data
on the transversally intersecting hypersurfaces ${\mycal N}$ and
${{{\underline{{\mycal N}}}}_{\,r_{2}}}$, resulting in a vacuum metric, say
${\mathbf{g}}_{-}$, defined on $J^{-}({\mycal N}\cup{{{\underline{{\mycal
N}}}}_{\,r_{2}}})$. Uniqueness of the solutions of the transport equations
shows that the transverse derivatives of the metric coincide with the ones
listed in ${{\Psi[{\mycal N},k]}}{}$.

By construction both piecewise-smooth spacetime metrics agree, together with
all transverse derivatives, on ${\mycal N}{}$, and define thus a smooth metric
on the union of the original domains of definition. $\Box$

As a corollary of the construction of Proposition 6.1 we have:

######  Corollary 6.2.

Let $k\in\mathbb{N}\cup\\{\infty\\}$, $k\geq 2$, and let
${\mathbf{S}}\subset{\mycal M}$ be of codimension two. For every spacelike
vacuum data set ${{\Psi[{\mathbf{S}},k]}}{}$ there exists a smooth vacuum
Lorentzian metric defined near ${\mathbf{S}}$ and inducing the data.

###### Proof.

Let ${\mycal N}=[-1,1]\times{\mathbf{S}}$, where we identify ${\mathbf{S}}$
with $\\{0\\}\times{\mathbf{S}}$. Let ${{\Psi[{\mycal N},k]}}{}$ be any
characteristic data on ${\mycal N}$ compatible with
${{\Psi[{\mathbf{S}},k]}}{}$ at $\\{0\\}\times{\mathbf{S}}$. If $k<\infty$, we
complement the data with higher-derivatives to $k=\infty$ in any way. Hence it
suffices again to assume that $k=\infty$.

In spacetime dimension four we can apply Proposition 6.1 to ${{\Psi[{\mycal
N},k]}}{}$, establishing thus the corollary.

In higher dimension Cauchy-stability allows us conclude again, as follows.
There exists $k_{0}>0$ so that Cauchy-stability for the characteristic initial
value problem holds in the $C^{k_{0}}({{\mycal N}}\cup{\underline{{\mycal
N}}})$-topology on the data. For $r_{1}\leq 0$ let $({{{\underline{{\mycal
N}}}}_{\,r_{1}}},{g}_{AB},\alpha)$ be as in the proof of Proposition 6.1. We
can choose the data to vary continuously in $C^{k_{0}}$ as $r_{1}$ varies.
Cauchy stability guarantees that there exists $\epsilon_{1}>0$ so that the
solution of the Cauchy problem with the data ${{\Psi[{\mycal N},\infty]}}{}$
on ${\mycal N}$ and the data ${{\Psi[{{{\underline{{\mycal
N}}}}_{\,-\epsilon_{1}}},\infty]}}{}$ on ${{{\underline{{\mycal
N}}}}_{\,-\epsilon_{1}}}$ contains a future neighborhood of ${\mathbf{S}}$. A
similar continuity argument applies to the data on the hypersurfaces
${{{\underline{{\mycal N}}}}_{\,r_{2}}}$ with $\epsilon_{2}\geq r_{2}>0$. One
thus obtains a smooth vacuum spacetime metric as in Figure 6.1 with
$r_{1}=-\epsilon_{1}$ and $r_{2}=\epsilon_{2}$. $\Box$

A construction in the same spirit allows us to extend vacuum Cauchy data
defined on a manifold with boundary $\Sigma$ beyond the boundary, to a larger
initial data manifold $\check{\Sigma}$, where the boundary of $\Sigma$ becomes
an interior hypersurface, while satisfying the vacuum general relativistic
constraint equations:

######  Proposition 6.3.

Let ${\Sigma}$ be a manifold with boundary $\partial{\Sigma}$ carrying smooth-
up-to-boundary vacuum initial data $(g,K)$. There exists a manifold
$\check{\Sigma}$ with vacuum data $(\check{g},\check{K})$ and an isometric
embedding of $({\Sigma},g)$ into $(\check{\Sigma},\check{g})$ such that
$\check{K}$ coincides with $K$ on ${\Sigma}$.

###### Proof.

The Cauchy data $({\Sigma},g,K)$ induce spacelike vacuum data
${{\Psi[\partial{\Sigma},\infty]}}{}$ on $\partial{\Sigma}$. The maximal
globally hyperbolic development of $({\Sigma},g,K)$ induces smooth
characteristic data, compatible with ${{\Psi[\partial{\Sigma},\infty]}}{}$, on
a null boundary emanating from $\partial{\Sigma}$ in the direction of the null
normal pointing towards ${\Sigma}$, denoted by $\partial{\mycal
D}^{+}({\Sigma})$ in Figure 6.2.

Figure 6.2: Extending ${\Sigma}$ to $\check{\Sigma}$.

Choose any characteristic data ${{\Psi[{\mycal N},\infty]}}{}$, compatible
with ${{\Psi[\partial{\Sigma},\infty]}}{}$, on a hypersurface ${\mycal N}$
intersecting $\partial{\mycal D}^{+}({\Sigma})$ transversally at
$\partial{\Sigma}$; ${\mycal N}$ provides a smooth continuation, to the
future, of the hypersurface $\partial{\mycal D}^{-}({\Sigma})$ of Figure 6.2.
The construction of the proof of Proposition 6.1 provides a vacuum metric, say
${\mathbf{g}}$, defined in a neighborhood ${\mycal O}$ of ${\mycal N}$. The
spacelike hypersurface ${\Sigma}$ can be extended smoothly within ${\mycal O}$
across $\partial{\Sigma}$ to a spacelike hypersurface $\check{\Sigma}$. The
data induced on $\check{\Sigma}$ by ${\mathbf{g}}$ provide the desired
extension $(\check{\Sigma},\check{g},\check{K})$. $\Box$

We can also find vacuum metrics which extend vacuum metrics on solid light
cones. For this we need to truncate the cone at finite distance by a spacelike
acausal hypersurface ${\Sigma}$. Consider, then, smooth characteristic vacuum
data on a light cone $\mycal C_{p}$ with vertex at $p$. By ChruscielSigma
there exists a neighborhood ${\mycal O}$ of $p$ and a smooth vacuum metric
${\mathbf{g}}$ defined on ${\mycal O}\cap J^{+}(p)$ which realises the data.
(In space-time dimension four the set ${\mycal O}$ constitutes a full future
neighborhood of $\mycal C_{p}$.) Let ${\Sigma}\subset{\mycal O}$ be any smooth
spacelike hypersurface included in $J^{+}(p)$ with smooth compact boundary on
$\mycal C_{p}$:

${\mathbf{S}}:=\partial{\Sigma}={\Sigma}\cap\mycal C_{p}\,.$

We denote by $\mycal C_{p}^{{\mathbf{S}}}$ the cone $\mycal C_{p}$ truncated
at ${\mathbf{S}}$,

$\mycal C_{p}^{{\mathbf{S}}}:=\mycal C_{p}\cap J^{-}({\mathbf{S}})\,,$

see Figure 6.3.

Figure 6.3: Truncating the light cone of $p$ at $\partial{\Sigma}$.

######  Proposition 6.4.

For any smooth characteristic vacuum data on a truncated light cone $\mycal
C_{p}^{{\mathbf{S}}}$ as above, there exists a smooth vacuum metric defined in
a neighborhood of $\mycal C_{p}^{{\mathbf{S}}}$ realising the data.

###### Proof.

By Proposition 6.3 we can extend ${\Sigma}$ beyond its boundary as in Figure
6.4.

Figure 6.4: Extending a vacuum metric on a truncated future cone $J^{+}(p)\cap
J^{-}({\Sigma})$ to a neighborhood thereof.

Solving backwards in time the Cauchy problem with the extended data one
obtains a vacuum metric defined in a neighborhood of $\mycal
C_{p}^{{\mathbf{S}}}$ which extends smoothly the metric inside the light cone.

######  Corollary 6.5.

Let $k\in\mathbb{N}\cup\\{\infty\\}$. For any vacuum data
${{\Psi[\\{p\\},k]}}{}$ at a point $p$, there exists a vacuum metric defined
in a neighborhood of $p$ realising the data.

###### Proof.

By definition, there exists a smooth Lorentzian metric
$\mathring{{\mathbf{g}}}$ inducing ${{\Psi[\\{p\\},k]}}{}$; we emphasise that
we do not assume that $\mathring{{\mathbf{g}}}$ is vacuum. Denote by
$J^{+}(p;\mathring{{\mathbf{g}}})$ the causal future of $p$ in
$\mathring{{\mathbf{g}}}$, and by $\mycal C_{p}$ the light cone of
$\mathring{{\mathbf{g}}}$ emanating from $p$. Let
$\mathring{g}_{AB}dx^{A}dx^{B}$ be the tensor field of signature
$(0,+,\ldots,+)$ obtained by restricting $\mathring{{\mathbf{g}}}$ on $\mycal
C_{p}$. By ChruscielSigma  there exists a neighborhood ${\mycal O}$ of $p$ and
a smooth vacuum metric ${\mathbf{g}}$ defined on

${\mycal O}\cap J^{+}(p;\mathring{{\mathbf{g}}})={\mycal O}\cap
J^{+}(p;{\mathbf{g}})\,,$

with $\mycal C_{p}$ being the light cone of ${\mathbf{g}}$, and with
${\mathbf{g}}$ inducing on $\mycal C_{p}$ the same degenerate tensor as
$\mathring{{\mathbf{g}}}$. It follows that the data induced at $p$ by
${\mathbf{g}}$ coincide with those induced by $\mathring{{\mathbf{g}}}$, i.e.
${{\Psi[\\{p\\},k]}}{}$. The result follows by Proposition 6.4.

##  7 The “Fledermaus construction”

We have shown in Section 6 how to find a vacuum metric which realises vacuum
characteristic data ${{\Psi[{\mycal N},k]}}{}$ on a hypersurface as an
_interior submanifold_. Here we describe a construction which realises vacuum
characteristic data on two transverse vacuum characteristic hypersurfaces
${{\mycal N}}\cup{\underline{{\mycal N}}}$ as an _interior submanifold with
corner_ in a vacuum spacetime. This should be contrasted with Theorem 2.1,
which realises the data as the _boundary_ of a spacetime with boundary-with-
corner. Not unexpectedly, the resulting metric is only uniquely defined to the
future of ${{\mycal N}}\cup{\underline{{\mycal N}}}$.

Indeed, in this section we use a “Fledermaus construction” to show:

######  Proposition 7.1.

In spacetime dimension four, consider smooth vacuum initial data on two
hypersurfaces

${{\mycal N}}_{[0,1]}\approx[0,1]\times{\mathbf{S}}\ \mbox{and}\
{\underline{{\mycal N}}}_{[0,1]}\approx[0,1]\times{\mathbf{S}}$

meeting transversally at a compact submanifold ${\mathbf{S}}$. There exists a
smooth solution of vacuum Einstein equations ${\mathbf{g}}$ which is defined
in a neighbourhood of ${{\mycal N}}_{[0,1]}\cup{\underline{{\mycal
N}}}_{[0,1]}$ and which realises the data.

######  Remark 7.2.

The already-given arguments involving Cauchy stability show that the result
remains true in higher dimensions with the neighborhood of ${{\mycal
N}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}$ replaced by a neighborhood of
${\mathbf{S}}$.

######  Remark 7.3.

The metric constructed in Proposition 7.1 is uniquely determined by the
characteristic data on the hypersurface
$\,\,\,\,\,{\widehat{\\!\\!\\!\\!\\!{{\mycal
N}}_{\,}}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}\cup{{\mycal
N}}_{[0,1]}\cup\underline{\,\,\,\,\,{\widehat{\\!\\!\\!\\!\\!{{\mycal
N}}_{\,}}}}_{[0,1]}$ of Figure 7.1.

Figure 7.1: The “Fledermaus construction” to embed characteristic initial data
on two transverse hypersurfaces in a vacuum spacetime.

###### Proof.

Let us denote by ${\mathbf{g}}_{T}$, where “$T$” stands for “top”, the smooth
solution of the vacuum Einstein equations obtained by solving the
characteristic Cauchy problem to the future of ${{\mycal
N}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}$ with the given data. The
solution induces a set of spacelike vacuum data ${{\Psi[{\mathbf{S}},k]}}{}$
with $k=\infty$ on ${\mathbf{S}}$ and sets of characteristic vacuum data
${{\Psi[{{\mycal N}}_{[0,1]},k]}}{}$ and ${{\Psi[{\underline{{\mycal
N}}}_{[0,1]},k]}}{}$ on ${{\mycal N}}_{[0,1]}$ and ${\underline{{\mycal
N}}}_{[0,1]}$, again with $k=\infty$.

We view ${{\mycal N}}_{[0,1]}$ as a subset of a smooth hypersurface

${\mycal N}=[-1,1]\times{\mathbf{S}}\,,$

and we view ${\mathbf{S}}$ as the subset $\\{r=0\\}$ of ${{\mycal N}}$. We
denote by ${\mathbf{S}}_{r}$ the crossection $\\{r\\}\times{\mathbf{S}}$. For
$r\geq 0$ the metric ${\mathbf{g}}_{T}$ induces smooth spacelike vacuum data
${{\Psi[{\underline{{\mathbf{S}}}}_{r},k]}}{}$ with $k=\infty$ on
${\mathbf{S}}_{r}$.

Similarly we view ${\underline{{\mycal N}}}_{[0,1]}$ as a subset of a smooth
hypersurface

${\underline{{\mycal N}}}=[-1,1]\times{\mathbf{S}}\,,$

with crossections $\\{r\\}\times{\mathbf{S}}\subset{\underline{{\mycal N}}}$
denoted by $\underline{{\mathbf{S}}}_{r}$, and with induced vacuum data
${{\Psi[{\underline{{\mathbf{S}}}}_{r},k]}}{}$ for $r>0$.

Let

$\,\,\,\,\,{\widehat{\\!\\!\\!\\!\\!{{\mycal
N}}_{\,}}}_{[0,1]}:=[0,1]\times{\underline{{\mathbf{S}}}}_{1}$

be a null hypersurface meeting ${\underline{{\mycal N}}}$ transversally at
${\underline{{\mathbf{S}}}}_{1}$ towards the past; see Figure 7.1. We choose
any Isenberg-Moncrief fields $({g}_{AB},\alpha)$ on
$\,\,\,\,\,{\widehat{\\!\\!\\!\\!\\!{{\mycal N}}_{\,}}}_{[0,1]}$ compatible
with ${{\Psi[{\underline{{\mathbf{S}}}}_{1},k]}}{}$ and we solve the
characteristic Cauchy problem to the past with data on
$\,\,\,\,\,{\widehat{\\!\\!\\!\\!\\!{{\mycal
N}}_{\,}}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}$. One thus obtains a
vacuum metric, say ${\mathbf{g}}_{L}$, where $L$ stands for “left”, on the
left wing of the Fledermaus . Uniqueness of solutions of transport equations
for the transverse derivatives of the metric along ${\underline{{\mycal
N}}}_{[0,1]}$ implies that the metric ${\mathbf{g}}_{L}$ extends smoothly
${\mathbf{g}}_{T}$ across ${\underline{{\mycal N}}}_{[0,1]}$. The intersection
of the domain of existence of ${\mathbf{g}}_{L}$ with ${{\mycal N}}_{[-1,0]}$
contains the hypersurface

${{\mycal N}}_{[-\epsilon,0]}:=[-\epsilon,0]\times{\mathbf{S}}\subset{{\mycal
N}}\,,$

which will be made-use of shortly.

A similar construction provides a smooth vacuum metric ${\mathbf{g}}_{R}$ on
the right wing of the Fledermaus, extending smoothly ${\mathbf{g}}_{T}$ across
${\underline{{\mycal N}}}_{[-1,0]}$, with domain of existence containing a
hypersurface

${\underline{{\mycal
N}}}_{[-\epsilon,0]}:=[-\epsilon,0]\times{\mathbf{S}}\subset{\underline{{\mycal
N}}}\,.$

The three metrics ${\mathbf{g}}_{L}$, ${\mathbf{g}}_{T}$ and
${\mathbf{g}}_{R}$ match smoothly at ${{\mycal
N}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}$, in particular also at
${\mathbf{S}}$.

Let ${\mathbf{g}}_{B}$ be obtained by solving the characteristic Cauchy
problem to the past with data on ${{\mycal
N}}_{[-\epsilon,0]}\cup{\underline{{\mycal N}}}_{[-\epsilon,0]}$. From what
has been said so far it should be clear that the four metrics
${\mathbf{g}}_{L}$, ${\mathbf{g}}_{T}$, ${\mathbf{g}}_{R}$ and
${\mathbf{g}}_{B}$ match smoothly wherever more than one is defined, and
provide the desired smooth vacuum metric ${\mathbf{g}}$ defined on a
neighborhood of ${{\mycal N}}_{[0,1]}\cup{\underline{{\mycal N}}}_{[0,1]}$.

##  8 Null hypersurfaces and spacelike gluing

It is well known Moncrief75 ; Corvino  that the existence of Killing vectors
near a spacelike Cauchy surface provides an obstruction to the Corvino-Schoen
approach to spacelike gluing; see, however, CzimekRodnianski . More precisely,
there is an obstruction to the gluing construction based on the implicit
function theorem involving the adjoint of the linearised constraint operator.
We show in Appendix C that an identical obstruction arises in the
characteristic gluing.

In fact, the obstruction arising from Killing vectors is a local one: We shall
say that there are _no local Killing vectors near ${\mathbf{S}}$_ if the
Killing vector equation has only trivial solutions on all sufficiently small
neighborhoods of ${\mathbf{S}}$. Formally: every neighborhood of
${\mathbf{S}}$ contains another neighborhood of ${\mathbf{S}}$ on which only
trivial solutions of the Killing equations exist.

It might be of some interest to note that, given a submanifold
${{\mathcal{J}}}$ of ${\mycal M}$ of any dimension and type, the notion of
_absence of local Killing vectors_ can be defined in terms of submanifold data
${{\Psi[{{\mathcal{J}}},k]}}{}$ of order $k\geq 1$, For this, note that the
Killing equations at ${{\mathcal{J}}}$,

$(\nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu})|_{{\mathcal{J}}}=0\,,$ (8.1)

and their derivatives in both transverse and tangential directions to
${{\mathcal{J}}}$, up to order $k$, evaluated at ${{\mathcal{J}}}$ (e.g.,
(8.1) together with

$\nabla_{\mu}\nabla_{\nu}X_{\rho}|_{{\mathcal{J}}}=R_{\sigma\mu\nu\rho}X^{\sigma}$

when $k=2$), can be viewed as an overdetermined set of equations for the jets
of order $k$ over ${{\mathcal{J}}}$ of a vector field $X$. We will say that
_there are no local Killing vectors at ${{\mathcal{J}}}$_ if there exists
$k\geq 1$ such that these equations have only the trivial solution. Standard
arguments show that the absence of local Killing vectors at ${{\mathcal{J}}}$
implies that every metric near ${{\mathcal{J}}}$ compatible with
${{\Psi[{{\mathcal{J}}},k]}}{}$ will have no local Killing vectors near
${{\mathcal{J}}}$.

We have:

######  Theorem 8.1.

Consider two smooth vacuum metrics ${\mathbf{g}}_{1}$ and ${\mathbf{g}}_{2}$
on ${\mycal M}$ and let ${\mycal N}\subset{\mycal M}$ be a hypersurface in
${\mycal M}$ which is null both for ${\mathbf{g}}_{1}$ and ${\mathbf{g}}_{2}$.
Let ${\mathbf{S}}\subset{\mycal N}$ be a compact cross-section of ${\mycal N}$
and suppose that there are no local Killing vectors near ${\mathbf{S}}$ for
${\mathbf{g}}_{1}$. If ${\mathbf{g}}_{1}$ and ${\mathbf{g}}_{2}$ are
sufficiently close to each other near ${\mathbf{S}}$ in $C^{5}$-topology, then
there exists a smooth vacuum metric ${\mathbf{g}}$ on ${\mycal M}$ and a null
hypersurface ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$ in ${\mycal
M}$ with spacelike boundary $\partial{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal
N}_{\,2}}}$, with ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$ near to
${\mycal N}\setminus J^{-}({\mathbf{S}})$ (cf. Figure 8.2), so that

  1. 1.

${\mathbf{g}}$ coincides with ${\mathbf{g}}_{1}$ on $\overline{J^{+}({\mycal
N})\setminus J^{+}({\mathbf{S}})}$, and

  2. 2.

${\mathbf{g}}$ coincides with ${\mathbf{g}}_{2}$ on
$\overline{J^{-}({\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}})\setminus
J^{-}(\partial{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}})}$.

In particular ${\mathbf{g}}$ induces the original vacuum data ${{\Psi[{\mycal
N}\cap J^{-}({\mathbf{S}}),k]}}{}$ induced by ${\mathbf{g}}_{1}$ on ${\mycal
N}\cap J^{-}({\mathbf{S}})$ for any $k\in\mathbb{N}$, with the data
${{\Psi[{\mycal N}\cap J^{+}({\mathbf{S}}),k]}}{}$ induced by ${\mathbf{g}}$
on ${\mycal N}\cap J^{+}({\mathbf{S}})$ being close to the data induced by
${\mathbf{g}}_{2}$ there, and with the data
${{\Psi[{\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,}}},k]}}{}$ induced by
${\mathbf{g}}$ on ${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,}}}$
coinciding with the data induced by ${\mathbf{g}}_{2}$ there.

######  Remark 8.2.

In Theorem 8.1 it suffices to assume that the metric ${\mathbf{g}}_{1}$ is
defined to the future of ${\mycal N}$ and ${\mathbf{g}}_{2}$ is defined to the
past of ${\mycal N}$, as in Figure 8.1.

######  Remark 8.3.

In the spacelike gluing the obstruction arising from Killing vectors can be
circumvented by gluing to a family of initial data which carries a set of
compensating parameters; the gluing construction chooses a member of the
family. There is an obvious version of Theorem 8.1 in such a situation,
whenever a family of metrics with compensating parameters is available;
compare Section 9 below. In particular one has a similar result for gluing a
vacuum metric ${\mathbf{g}}_{1}$ with a member ${\mathbf{g}}_{2}$ of the Kerr,
Kerr-de Sitter, or Kerr Anti-de Sitter family. Note that the condition of
nearness to a member of the Kerr-(A)dS family is more severe, as compared to
the $\Lambda=0$ case, in the following sense: nearness to Kerr can be achieved
by receding in spacelike directions for a large class of asymptotically
Minkowskian initial data sets, while no such construction is known when
$\Lambda\neq 0$. Compare ChDelayAH ; CortierKdS ; HintzdSBH .

###### Proof.

We can choose spacelike hypersurfaces ${\Sigma}_{1}$ and ${\Sigma}_{2}$ as in
Figure 8.1 so that the vacuum Cauchy data $({\Sigma}_{1},g_{1},K_{1})$ and
$({\Sigma}_{2},g_{2},K_{2})$, induced by the respective spacetime metrics
${\mathbf{g}}_{1}$ and ${\mathbf{g}}_{2}$, are near to each other in a
neighborhood of ${\mathbf{S}}$ in a $C^{5}\oplus C^{4}$ topology. By (ChDelay,
, Section 8.6) the data $({\Sigma}_{1},g_{1},K_{1})$ and
$({\Sigma}_{2},g_{2},K_{2})$ can be smoothly glued together to a smooth vacuum
data set

Figure 8.1: Before the gluing of metrics with nearby characteristic data along
a null hypersurface ${\mycal N}$.

$({\Sigma}_{1}\cup{\Sigma}_{2},g,K)$, so that $(g,K)$ coincides with the
original Cauchy data except for a small neighborhood ${\mycal
O}\subset{\Sigma}_{2}$ of ${\mathbf{S}}$. Solving the Cauchy problem with
these data one obtains the desired spacetime, see Figure 8.2. The hypersurface
${\,\,\,\,\widetilde{\\!\\!\\!\\!{\mycal N}_{\,2}}}$ is taken to be
$\partial{\mycal D}^{+}({\,\widetilde{\\!\,{\Sigma}}_{\,2}})$, where
${\,\widetilde{\\!\,{\Sigma}}_{\,2}}={\Sigma}_{2}\setminus{\mycal O}$.

Figure 8.2: After the gluing, zoom to the gluing region; $\tilde{\Sigma}_{2}$
is ${\Sigma}_{2}\setminus{\mycal O}$.

$\Box$

##  9 Gluing cross-section data to Kerr data

We turn our attention now to the question addressed in ACR1 , of gluing two
sets of cross-section data, one of them arising from the Kerr family.

Thus, in spacetime dimension four, let $({\mycal M},{\mathbf{g}})$ be the
Schwarzschild metric with non-zero mass parameter. We consider a null
hypersurface ${\mycal N}$ in $({\mycal M},{\mathbf{g}})$ with two disjoint
cross-sections ${\mathbf{S}}_{1}$ and ${\mathbf{S}}_{2}$, say
${\mathbf{S}}_{2}\subset J^{+}({\mathbf{S}}_{1})$. On ${\mathbf{S}}_{1}$ we
are given spacelike vacuum data ${{\Psi[{\mathbf{S}}_{1},k]}}{}$ distinct from
but close to the data induced by ${\mathbf{g}}$. On ${\mathbf{S}}_{2}$ we
consider the family of data ${{\Psi[{\mathbf{S}}_{2},k]}}{}$ arising by
restriction from all Kerr metrics. The goal is to find null hypersurface data
which interpolate between ${{\Psi[{\mathbf{S}}_{1},k]}}{}$ and a sufficiently
small perturbation of one of the ${{\Psi[{\mathbf{S}}_{2},k]}}{}$’s in a way
such that we can carry out the spacetime gluing of Theorem 8.1. The result
will be a spacetime metric which coincides with a Kerr metric in the right-
wedge ${\mycal D}({\,\widetilde{\\!\,{\Sigma}}_{\,2}})$ of Figure 8.2. The
difficulty is to arrange smallness of the perturbation of a large number of
transverse derivatives of the metric at ${\mathbf{S}}_{2}$.

Now, there exists $k_{0}<\infty$ such that characteristic data ${{\Psi[{\mycal
N},k_{0}]}}{}$ which are $\epsilon$-close to the data induced by
${\mathbf{g}}$ will lead, through the construction of Proposition 6.1, to a
metric ${\mathbf{g}}_{1}$ which is $\epsilon$-close in $C^{5}$-topology to the
metric ${\mathbf{g}}$. The dimension-dependent number $k_{0}$ can be
determined in principle by chasing losses of differentiability through all the
steps of the construction.

So let $\epsilon$ be a measure of the deviation of the data
${{\Psi[{\mathbf{S}}_{1},k_{0}]}}{}$ from those induced by ${\mathbf{g}}$.

A brute-force gluing proceeds as follows: We choose a member of the Kerr
family such that ${{\Psi[{\mathbf{S}}_{2},k_{0}]}}{}$ is $\epsilon$-close to
the data induced by ${\mathbf{g}}$, and has the same linearly conserved radial
charges as ${{\Psi[{\mathbf{S}}_{1},k_{0}]}}{}$. We find any smoothly
interpolating free data on $J^{+}({\mathbf{S}}_{1})\cap
J^{-}({\mathbf{S}}_{2})\subset{\mycal N}$ which deviate from the Schwarzschild
data by $O(\epsilon)$. We use these data in the source terms of the transport
and algebraic equations of Section 5, including their transverse derivatives,
to obtain a solution of these equations on $J^{+}({\mathbf{S}}_{1})\cap
J^{-}({\mathbf{S}}_{2})$ which matches ${{\Psi[{\mathbf{S}}_{2},k_{0}]}}{}$ up
to error terms of order $\epsilon$. If $\epsilon$ is sufficiently small,
Theorem 8.1 applies.

The spacelike-gluing version of the more sophisticated scheme of ACR1 , which
appears to be critical for some applications such as CzimekRodnianski ,
proceeds as follows:

It has been shown in ChCong1  how to find linearised Bondi free data $h_{AB}$
so that the metric $\gamma_{AB}+h_{AB}$ interpolates between
${{\Psi[{\mathbf{S}}_{1},k_{0}]}}{}$ and one of the data sets
${{\Psi[{\mathbf{S}}_{2},k_{0}]}}{}$ at a linearised level. One can then use
these data in the source terms of the transport and algebraic equations of
Section 5 to obtain hypersurface data on $J^{+}({\mathbf{S}}_{1})\cap
J^{-}({\mathbf{S}}_{2})$ which match ${{\Psi[{\mathbf{S}}_{2},k_{0}]}}{}$ to
order $\epsilon^{2}$. The gluing then follows again from Theorem 8.1. We note
that the improvement from $\epsilon$ to $\epsilon^{2}$ is critical for some
applications, such as CzimekRodnianski ,

The same arguments apply to metrics which are near to a four-dimensional
Birmingham-Kottler metric with higher genus at infinity and with nonzero mass,
where the mass parameter has to be adjusted to do the gluing, see (ChCong1, ,
Table 1.1).

We believe that the same scheme can be used to interpolate between data with
$\Lambda\in\mathbb{R}$ near Birmingham-Kottler data in any spacetime
dimensions $n+1\geq 4$, we plan to return to this in the near future. In
spacetime dimension four with $\Lambda\neq 0$ the linearised analysis of
ChCong1  applies and, in the spherical case, the same argument will lead to
the desired conclusion after checking that the Kerr-(A)dS metrics provide the
required family of compensating metrics.

##  Appendix A ACR sphere data

Here we calculate the sphere data of ACR1  of order two in terms of Bondi
section data. Given a cross-section ${\mathbf{S}}$ of $\mathcal{N}$, by which
we mean a submanifold of $\mathcal{N}$ intersecting all the generators of
$\mathcal{N}$ transversally, the field $\partial_{r}\equiv L$ is the field of
null normals both to ${\mathbf{S}}$ and $\mathcal{N}$, while $\underline{L}$
is the field of null normals to ${\mathbf{S}}$ transverse to $\mathcal{N}$. In
typical applications both $L$ and $\underline{L}$ are chosen to be future-
directed, but the choice is irrelevant for the problem at hand.

In ACR1  the cross-section ${\mathbf{S}}$ is chosen to be a sphere and the
space-time dimension is four: both assumptions are essential for the analysis
there, compare ChCong1 .

In Bondi coordinates we can choose

$L=\partial_{r}\,,\quad\underline{L}=\partial_{u}+U^{A}\partial_{A}-\frac{V}{2r}\partial_{r}\,.$
(A.1)

This gives

$\Omega:=\sqrt{-\tfrac{1}{2}g(L,\underline{L})}=\frac{e^{\beta}}{\sqrt{2}}\,.$
(A.2)

The sphere data of ACR1  further involve the fields

$\hat{L}=L/\Omega$ and $\underline{\hat{L}}=\underline{L}/\Omega$.

First, the Ricci coefficients are defined as, for $X$ and $Y$
${\mathbf{S}}$-tangent vector fields,

$\displaystyle\chi(X,Y)$ $\displaystyle=g(\nabla_{X}\hat{L},\,Y)\,,$
$\displaystyle\underline{\chi}$
$\displaystyle=g(\nabla_{X}\hat{\underline{L}},\,Y)\,,$
$\displaystyle\zeta(X)$
$\displaystyle=\frac{1}{2}g(\nabla_{X}\hat{L},\,\hat{\underline{L}})\,,$
$\displaystyle\underline{\zeta}(X)$
$\displaystyle=\frac{1}{2}g(\nabla_{X}\hat{\underline{L}},\,\hat{L})\,,$
$\displaystyle\eta$ $\displaystyle=\zeta+d\log\Omega\,,$
$\displaystyle\underline{\eta}$ $\displaystyle=-\zeta+d\log\Omega\,,$
$\displaystyle\omega$ $\displaystyle=D\log\Omega\,,$
$\displaystyle\underline{\omega}$ $\displaystyle=\underline{D}\log\Omega\,,$
(A.3)

where $D$ and $\underline{D}$ respectively denote the projection of the Lie
derivative along $L$ and $\underline{L}$ onto the tangent space of
${\mathbf{S}}$. The null curvature components involved in the sphere data are

$\alpha(X,Y)=R(X,\hat{L},Y,\hat{L})\,,\quad\underline{\alpha}(X,Y)=R(X,\hat{\underline{L}},Y,\hat{\underline{L}}).$
(A.4)

The $C^{2}$-sphere data of ACR1  is the collection of fields

$(\Omega,\,\cancel{g},\,\Omega\mbox{tr}\chi,\,\hat{\chi},\,\Omega\mbox{tr}\underline{\chi}\,,\hat{\underline{\chi}}\,,\eta\,,\omega,\,D\omega,\,\underline{\omega},\,\underline{D}\underline{\omega},\,\alpha,\underline{\alpha})\,,$
(A.5)

where tr denotes the trace with respect to the metric $\cancel{g}$ on
${\mathbf{S}}$ and the hat $\hat{\cdot}$ above a tensor denotes the traceless
part. Leting $\cancel{\nabla}$ denote the covariant derivative of the metric
$\cancel{g}$, In Bondi coordinates the fields (A.5) read

$\displaystyle\Omega=$
$\displaystyle\frac{e^{\beta}}{\sqrt{2}}\,,\quad\cancel{g}=r^{2}\gamma\,,$
$\displaystyle\chi_{AB}=$
$\displaystyle\frac{1}{\Omega}(r\gamma_{AB}+\tfrac{1}{2}r^{2}\partial_{r}\gamma_{AB})\equiv\frac{1}{2\Omega}\partial_{r}(r^{2}\gamma_{AB})\,,$
$\displaystyle\underline{\chi}_{AB}=$
$\displaystyle-\frac{2\gamma_{AB}V-4\cancel{\nabla}_{(A}U_{B)}+r(V\partial_{r}\gamma_{AB}-2r\partial_{u}\gamma_{AB})}{2\sqrt{2}e^{\beta}}\,,$
(A.6) $\displaystyle\eta_{A}=$
$\displaystyle\partial_{A}\beta-\frac{\gamma_{AB}r^{2}\partial_{{r}{}}U^{B}}{2e^{2\beta}}\,,\quad\omega=\partial_{r}\beta\,,\quad
D\omega=\partial^{2}_{r}\beta\,,\quad\underline{\omega}=U^{A}\partial_{A}\beta-\frac{V\partial_{{r}{}}\beta}{2r}+\partial_{{u}{}}\beta\,,$
(A.7) $\displaystyle\underline{D}\underline{\omega}=$
$\displaystyle-\frac{U^{A}V\partial_{A}\partial_{r}\beta}{r}+2U^{A}\partial_{A}\partial_{u}\beta+U^{A}\partial_{A}U^{B}\partial_{B}\beta+U^{A}U^{B}\partial_{B}\partial_{A}\beta-\frac{V^{2}\partial_{r}\beta}{4r^{3}}-\frac{U^{A}\partial_{A}V\partial_{r}\beta}{2r}$
$\displaystyle-\frac{V\partial_{A}\beta\partial_{r}U^{A}}{2r}+\frac{V\partial_{r}\beta\partial_{r}V}{4r^{2}}+\frac{V^{2}\partial_{r}^{2}\beta}{4r^{2}}-\frac{V\partial_{r}\partial_{u}\beta}{r}+\partial_{A}\beta\partial_{u}U^{A}-\frac{\partial_{r}\beta\partial_{u}V}{2r}+\partial_{u}^{2}\beta\,,$
(A.8) $\displaystyle\alpha_{AC}=$
$\displaystyle\frac{4\gamma_{AC}r\partial_{r}\beta}{e^{2\beta}}-\frac{2r\partial_{r}\gamma_{AC}}{e^{2\beta}}+\frac{2r^{2}\partial_{r}\beta\partial_{r}\gamma_{AC}}{e^{2\beta}}+\frac{\gamma^{BD}r^{2}\partial_{r}\gamma_{AB}\partial_{r}\gamma_{CD}}{2e^{2\beta}}-\frac{r^{2}\partial_{r}^{2}\gamma_{AC}}{e^{2\beta}}\,,$
(A.9) $\displaystyle\underline{\alpha}_{AC}=$ $\displaystyle
e^{-2\beta}\bigg{(}r^{-1}V\cancel{\nabla}_{(A}\partial_{r}U_{B)}-2\cancel{\nabla}_{(A}\partial_{u}U_{B)}+\frac{e^{2\beta}\cancel{\nabla}_{B}\cancel{\nabla}_{A}V}{r}-2U^{C}\cancel{\nabla}_{C}\cancel{\nabla}_{(A}U_{B)}$
$\displaystyle-\frac{V^{2}\partial_{r}^{2}\gamma_{AB}}{4}+rV\partial_{r}\partial_{u}\gamma_{AB}-r^{2}\partial_{u}^{2}\gamma_{AB}\bigg{)}$
$\displaystyle+\tilde{\underline{\alpha}}_{AC}[r,r^{-1},\gamma_{BD},\gamma^{BD},\partial\gamma_{BD},e^{-2\beta},\partial\beta,U_{B},\partial
U_{B},V,\partial V,\partial_{u}\gamma_{BD},\partial_{u}\beta]\,,$ (A.10)

with a polynomial function $\tilde{\underline{\alpha}}_{AC}$ of the arguments
indicated; the explicit formula is not very enlightening and too long to be
usefully displayed. We use “$\partial$” in the arguments of
$\tilde{\underline{\alpha}}_{AC}$ to denote $\partial_{r}$ and $\partial_{A}$
derivatives, with $\partial_{u}$ derivatives indicated explicitly there.

##  Appendix B Bondi coordinates anchored at ${\mathbf{S}}$

The construction of Bondi coordinates in four spacetime dimensions starting
from ${\mycal I}$ is well known GerochWinicour81 , and generalises immediately
to higher dimensions. We indicate here how to adapt the construction to our
setting, to make clear the freedom involved.

Let ${\mathbf{S}}$ be a cross-section of a smooth, null, connected
hypersurface ${\mycal N}$ in an $(n+1)$-dimensional spacetime $({\mycal
M},g)$. Let ${\underline{{\mycal N}}}$ be a null hypersurface such that
${\mycal N}\cap{\underline{{\mycal N}}}={\mathbf{S}}$, with transverse
intersection. (Thus ${\mathbf{S}}$ is spacelike, with both $T{\mycal N}$ and
$T{\underline{{\mycal N}}}$ orthogonal to $T{\mathbf{S}}$.) Let
$\tilde{x}^{A}$ denote local coordinates on ${\mathbf{S}}$. We consider,
first, Isenberg-Moncrief VinceJimcompactCauchyCMP  coordinates
$(\tilde{u},{\tilde{\underline{u}}},\tilde{x}^{C})$ around

${\mathbf{S}}=\\{\tilde{u}=0={\tilde{\underline{u}}}\\}\,,$

and with the metric taking the form

${\mathbf{g}}=\big{(}{\tilde{\underline{u}}}\,{\alpha}\,{\mathrm{d}}\tilde{u}-2{\mathrm{d}}{\tilde{\underline{u}}}+2\,{\tilde{\underline{u}}}\,{\beta}_{A}{\mathrm{d}}\tilde{x}^{A}\big{)}{\mathrm{d}}\tilde{u}+g_{AB}{\mathrm{d}}\tilde{x}^{A}{\mathrm{d}}\tilde{x}^{B}\,,$
(B.1)

for some fields ${\alpha}$ and ${\beta}_{A}$. Here we have denoted by
$(\tilde{u},{\tilde{\underline{u}}},\tilde{x}^{A})$ the coordinates
$({u},{r},x^{A})$ of (4.1), to avoid confusion with the $(u,r,x^{A})$ Bondi
coordinates that we are about to construct.

Note that while the $\tilde{x}^{A}$’s are local coordinates, the coordinate
functions $\tilde{u}$ and ${\tilde{\underline{u}}}$ are defined globally in a
neighborhood of ${\mathbf{S}}$. The level sets of the coordinate $\tilde{u}$
are null hypersurfaces and we denote ${\mycal N}:=\\{\tilde{u}=0\\}$ in this
Appendix;555We caution that this does not agree with the notation in Section
4, where ${\mycal N}$ was chosen to be $\\{{\tilde{\underline{u}}}=0\\}$. we
will be constructing Bondi coordinates $(u,r,x^{A})$ such that ${\mycal
N}=\\{u=0\\}$. The hypersurface ${\underline{{\mycal
N}}}:=\\{{\tilde{\underline{u}}}=0\\}$ is also null, but not necessarily so
the hypersurfaces ${\tilde{\underline{u}}}\neq 0$. The sign of
${\mathbf{g}}_{\tilde{u}{\tilde{\underline{u}}}}$ has been determined by our
signature $(-,+,\ldots,+)$ together with the requirement that
$\partial_{\tilde{u}}$ and $\partial_{\tilde{\underline{u}}}$ are consistently
time-oriented at ${\mathbf{S}}$, say future oriented.

In the Isenberg-Moncrief construction one can take the integral curves of
$\ell:=\partial_{\tilde{\underline{u}}}|_{\mycal N}$ to be affinely-
parameterised future-directed null geodesics (in which case ${\alpha}$
vanishes on ${\mycal N}$), then the coordinate system above is uniquely
defined up to the choice of this last parameterisation. In order to get rid of
this freedom, consider the divergence ${\tilde{\theta}}$ of ${\mycal N}$
defined in (2.1), where we decorate ${\theta}$ with a tilde to emphasise its
dependence upon the coordinate ${\tilde{\underline{u}}}$. Under the rescaling
$\ell\mapsto f(\tilde{x}^{A})\ell$ we have

${\tilde{\theta}}=\frac{1}{2}g^{AB}\partial_{{\tilde{\underline{u}}}}g_{AB}\mapsto\frac{f}{2}g^{AB}\partial_{{\tilde{\underline{u}}}}g_{AB}=f{\tilde{\theta}}\,.$
(B.2)

Assuming that _${\tilde{\theta}}$ has no zeros on ${\mathbf{S}}$_, we can
choose a unique function $f>0$ so that, after the above rescaling has been
done, the new function ${\tilde{\theta}}$ satisfies

${\tilde{\theta}}|_{\mathbf{S}}=\pm(n-1)\,,$ (B.3)

thus preserving the future-directed character of $\ell$, or choose a unique
$f$ so that

${\tilde{\theta}}|_{\mathbf{S}}=(n-1)$ (B.4)

if the time-orientation of $\ell$ is ignored.

The field

${\mu}:=\sqrt{\det g_{AB}}|_{{\mathbf{S}}}$

defines a scalar density on ${\mathbf{S}}$. We extend ${\mu}$ to
${\underline{{\mycal N}}}$ by requiring ${\mycal
L}_{\partial_{\tilde{u}}}{\mu}|_{{\underline{{\mycal N}}}}=0$, and then we
extend it away from ${\underline{{\mycal N}}}$ by requiring ${\mycal
L}_{\partial_{{\tilde{\underline{u}}}}}{\mu}=0$. Still denoting by $\mu$ the
field so extended, since $\partial_{\tilde{\underline{u}}}$ and
$\partial_{\tilde{u}}$ commute we find that ${\mycal
L}_{\partial_{{\tilde{\underline{u}}}}}{\mycal
L}_{\partial_{\tilde{u}}}{\mu}=0$, which further implies

${\mycal L}_{\partial_{{\tilde{\underline{u}}}}}{\mu}={\mycal
L}_{\partial_{\tilde{u}}}{\mu}=0$

throughout the domain of definition of the coordinates.

We define a function $r=r(\tilde{u},{\tilde{\underline{u}}},\tilde{x}^{A})$ by
the formula

$r^{1-n}:=\frac{{\mu}}{\sqrt{\det g_{AB}}}\,.$ (B.5)

Note that

$r|_{{\mathbf{S}}}=1\,.$ (B.6)

We wish to replace the coordinates
$(\tilde{u},{\tilde{\underline{u}}},\tilde{x}^{A})$ by

$(u=\tilde{u},r,x^{A}=\tilde{x}^{A})\,.$

Using monotonicity and the implicit function theorem, we see that this is
possible on the set where

$0\neq\partial_{{\tilde{\underline{u}}}}r=\frac{r}{2(n-1)}g^{AB}\partial_{{\tilde{\underline{u}}}}g_{AB}\,.$
(B.7)

This is directly related to the divergence ${\tilde{\theta}}$ of ${\mycal N}$:

$\partial_{{\tilde{\underline{u}}}}r\Big{|}_{\mycal
N}=\frac{r{\tilde{\theta}}}{(n-1)}\,.$ (B.8)

We thus obtain a well behaved coordinate system $(r,x^{A})$ on ${\mycal N}$,
and $(u,r,x^{A})$ near ${\mycal N}$, unless $r$ becomes zero, which happens
e.g. at the vertex of a light cone, or unless ${\tilde{\theta}}$ acquires a
zero. Hence we restrict ourselves to the subset of ${\mycal N}$ where $r>0$
and $|{\tilde{\theta}}|>0$.

Similarly to (B.7) we have

$\frac{\partial
r}{\partial\tilde{u}}=\frac{r}{2(n-1)}g^{AB}\partial_{\tilde{u}}g_{AB}\equiv\frac{r{\underline{\tilde{\theta}}}}{(n-1)}\,,$
(B.9)

where ${\underline{\tilde{\theta}}}$ is the expansion of the level sets of
${\tilde{\underline{u}}}$.

We note that the vector field $\partial_{\tilde{u}}$ is uniquely determined on
${\mathbf{S}}$ by the requirement that $\partial_{\tilde{u}}$ is orthogonal to
$T{\mathbf{S}}$ and satisfies
${\mathbf{g}}(\partial_{\tilde{u}},\partial_{{\tilde{\underline{u}}}})|_{\mathbf{S}}=-1$.
Hence ${\underline{\tilde{\theta}}}|_{{\mathbf{S}}}$ is uniquely determined by
the requirement (B.3) and by ${\mathbf{S}}$, without the need to introduce the
null transverse hypersurface ${\underline{{\mycal N}}}$. The function
${\underline{\tilde{\theta}}}|_{{\mathbf{S}}}$ is sometimes called the _null
mean curvature of ${\mathbf{S}}$ along $\partial_{\tilde{u}}$_.

The change of coordinates
$(\tilde{u},{\tilde{\underline{u}}},\tilde{x}^{A})\mapsto(u,r,x^{A})$ brings
the metric to the form

${\mathbf{g}}=\,{\tilde{\underline{u}}}\,{\alpha}\,\mathrm{d}u^{2}-2\mathrm{d}u\,(\partial_{u}{\tilde{\underline{u}}}\,\mathrm{d}u+\partial_{r}{\tilde{\underline{u}}}\,\mathrm{d}r+\partial_{A}{\tilde{\underline{u}}}\,\mathrm{d}x^{A})+2\,{\tilde{\underline{u}}}\,{\beta}_{A}\mathrm{d}x^{A}\mathrm{d}u+g_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,,$
(B.10)

which can be rewritten using the Bondi parameterisation

$\displaystyle{\mathbf{g}}$ $\displaystyle=$
$\displaystyle-\frac{V}{r}e^{2\beta}\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\,\mathrm{d}r+r^{2}{\gamma}_{AB}\Big{(}\mathrm{d}x^{A}-U^{A}\mathrm{d}u\Big{)}\Big{(}\mathrm{d}x^{B}-U^{B}\mathrm{d}u\Big{)}\,,$
(B.11)

where it is assumed that $\partial_{r}$ and $\partial_{u}$ are consistently
time-oriented at ${\mathbf{S}}$.

From (B.4) and (B.7) we find

$\partial_{{\tilde{\underline{u}}}}r\Big{|}_{\mathbf{S}}=\pm 1$ (B.12)

if and only if (B.3) holds. Assuming the associated parameterisation of the
generators of ${\mycal N}$ we obtain

$\frac{\partial(r,u,x^{A})}{\partial({\tilde{\underline{u}}},\tilde{u},\tilde{x}^{B})}\Big{|}_{\mathbf{S}}=\left(\begin{array}[]{ccc}\pm
1&\partial_{\tilde{u}}r&0\\\ 0&1&0\\\ 0&0&\delta^{A}_{B}\\\
\end{array}\right)\,.$ (B.13)

Hence

$\frac{\partial({\tilde{\underline{u}}},\tilde{u},\tilde{x}^{B})}{\partial(r,u,x^{A})}\Big{|}_{\mathbf{S}}=\left(\begin{array}[]{ccc}\pm
1&\mp\partial_{\tilde{u}}r&0\\\ 0&1&0\\\ 0&0&\delta^{B}_{A}\\\
\end{array}\right)\,.$ (B.14)

This leads to the following form of (B.10) at ${\mathbf{S}}$

${\mathbf{g}}|_{\mathbf{S}}=\mp
2\mathrm{d}u\,(-\partial_{\tilde{u}}r\,\mathrm{d}u+\mathrm{d}r)+g_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,,$
(B.15)

which together with (B.9), after changing $u$ to $-u$ if necessary, shows that
at ${\mathbf{S}}$ it holds

$U_{A}|_{{\mathbf{S}}}:=-r^{-2}{\mathbf{g}}(\partial_{u},\partial_{A})\big{|}_{{\mathbf{S}}}\equiv{\gamma}_{AB}U^{B}\big{|}_{{\mathbf{S}}}=0\,,\quad\beta|_{{\mathbf{S}}}=0\,,\quad
V|_{{\mathbf{S}}}=-\frac{2{\underline{\tilde{\theta}}}}{n-1}\,.$ (B.16)

Summarising, we have proved:

######  Proposition B.1.

Let ${\mycal N}$ be a null hypersurface with $\theta>0$ and suppose that
${\mycal N}$ contains a smooth submanifold ${\mathbf{S}}$ which meets every
generator of ${\mycal N}$ transversally and precisely once. There exists a
unique coordinate system near ${\mycal N}$ in which the metric takes the Bondi
form (B.11) with ${\mycal N}=\\{u=0\\}$ and in which

$r|_{\mathbf{S}}=1\,,\qquad\beta|_{\mathbf{S}}=0\,.$ (B.17)

In this coordinate system we have

$U^{A}|_{\mathbf{S}}=0\,,\qquad{\tilde{\theta}}|_{\mathbf{S}}=(n-1)\,,\qquad
V|_{{\mathbf{S}}}=-\frac{2{\underline{\tilde{\theta}}}}{n-1}\,,$ (B.18)

where ${\tilde{\theta}}$ is the expansion of $\\{u=0\\}$ with respect to the
affinely-normalised geodesic null vector field
$\partial_{{\tilde{\underline{u}}}}$, and ${\underline{\tilde{\theta}}}$ is
that of the level sets of ${\tilde{\underline{u}}}$ with respect to the
Isenberg-Moncrief geodesic vector field $\partial_{\tilde{u}}$.

The coordinates of Proposition B.1 will be referred to as _Bondi coordinates
adapted to ${\mycal N}$ anchored at ${\mathbf{S}}$._

######  Remark B.2.

The proof of Proposition B.1 applies word-for-word in spacetimes in which
${\mycal N}$ is a smooth boundary, or in spacetimes with a boundary consisting
of two null hypersurfaces ${{\mycal N}}$ and ${\underline{{\mycal N}}}$
intersecting at ${\mathbf{S}}$, in which cases the coordinates are only
defined on one side of ${\mycal N}$.

######  Remark B.3.

The Isenberg-Moncrief construction can be carried-out using any
parameterisation of the generators of ${\mycal N}$, not necessarily affine.
Likewise our construction above applies for any parameterisation of the
generators, which shows that there is a lot of residual coordinate freedom in
the Bondi form of the metric near a null hypersurface. This explains in
particular why the Bondi hypersurface data of Section 5 have more freedom than
the Isenberg-Moncrief hypersurface data of Section 4.

More generally, whether or not (B.3) holds we have

$\frac{\partial(r,u,x^{A})}{\partial({\tilde{\underline{u}}},\tilde{u},\tilde{x}^{B})}=\left(\begin{array}[]{ccc}\partial_{{\tilde{\underline{u}}}}r&\partial_{\tilde{u}}r&\frac{\partial
r}{\partial\tilde{x}^{B}}\\\ 0&1&0\\\ 0&0&\delta^{A}_{B}\\\
\end{array}\right)\,,$ (B.19)

so that

$\frac{\partial({\tilde{\underline{u}}},\tilde{u},\tilde{x}^{B})}{\partial(r,u,x^{A})}=\left(\begin{array}[]{ccc}\big{(}\partial_{{\tilde{\underline{u}}}}r\big{)}^{-1}&\
-\big{(}\partial_{{\tilde{\underline{u}}}}r\big{)}^{-1}\partial_{\tilde{u}}r&\
\
-\big{(}\partial_{{\tilde{\underline{u}}}}r\big{)}^{-1}\partial_{\tilde{x}^{A}}r\\\
0&1&0\\\ 0&0&\delta^{B}_{A}\\\ \end{array}\right)\,,$ (B.20)

and (B.10) reads

$\displaystyle{\mathbf{g}}$ $\displaystyle=$
$\displaystyle\,{\tilde{\underline{u}}}\,{\alpha}\,\mathrm{d}u^{2}-\frac{2\Big{[}\mathrm{d}r-\partial_{\tilde{u}}r\,\mathrm{d}u-\partial_{\tilde{x}^{A}}r\,\mathrm{d}x^{A}\Big{]}}{\partial_{{\tilde{\underline{u}}}}r}\mathrm{d}u+2\,{\tilde{\underline{u}}}\,{\beta}_{A}\mathrm{d}x^{A}\mathrm{d}u+g_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}$
(B.21) $\displaystyle=$
$\displaystyle\Big{(}\,{\tilde{\underline{u}}}\,{\alpha}+2\frac{\partial_{\tilde{u}}r}{\partial_{{\tilde{\underline{u}}}}r}\Big{)}\mathrm{d}u^{2}-\frac{2\mathrm{d}r\,\mathrm{d}u}{\partial_{{\tilde{\underline{u}}}}r}+2\Big{(}{\tilde{\underline{u}}}\,{\beta}_{A}+\frac{\partial_{\tilde{x}^{A}}r}{\partial_{{\tilde{\underline{u}}}}r}\Big{)}\mathrm{d}u\,\mathrm{d}x^{A}+g_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,.$

This, together with (B.8)-(B.9), after changing $u$ to $-u$ if necessary,
shows that

$\displaystyle\beta|_{{\mycal N}}$
$\displaystyle=\frac{1}{2}\ln\frac{n-1}{r{\tilde{\theta}}}\,,\quad
U_{A}|_{{\mycal
N}}=-\frac{{\tilde{\underline{u}}}}{r^{2}}{\beta}_{A}-\frac{(n-1)}{r^{3}{\tilde{\theta}}}\partial_{\tilde{x}^{A}}r\,,$
$\displaystyle V|_{{\mycal N}}$
$\displaystyle=-\frac{2r^{2}{\underline{\tilde{\theta}}}}{n-1}-\frac{{\tilde{\underline{u}}}{\alpha}r^{2}{\tilde{\theta}}}{n-1}+\frac{r^{2}{\tilde{\underline{u}}}^{2}g^{AB}{\beta}_{A}{\beta}_{B}\,{\tilde{\theta}}}{n-1}+\frac{n-1}{{\tilde{\theta}}}g^{AB}\partial_{\tilde{x}^{A}}r\,\partial_{\tilde{x}^{B}}r\,.$
(B.22)

In particular we see that $r={\underline{u}}$ in Theorem 2.1 is possible if
and only if

$\partial_{\tilde{x}^{A}}r\big{|}_{{\mathbf{S}}}=0\,,\quad{\underline{\tilde{\theta}}}\big{|}_{{\mathbf{S}}}=0\,.$
(B.23)

##  Appendix C A variational identity

The aim of this Appendix is to show that the restriction of spacetime Killing
vectors to a hypersurface lies in the kernel of the adjoint of the
linearisation of the vacuum constraints operator. This shows in particular
that spacetime Killing vectors provide obstructions to characteristic gluing
based on the implicit function theorem.

Let ${\mathbf{g}}$ be a solution of the vacuum Einstein equations with a
cosmological constant ${\Lambda}\in\mathbb{R}$. Let ${\Sigma}$ be a
hypersurface of any causal type, possibly with boundary, and let
$\lambda\mapsto{\mathbf{g}}(\lambda)$ be a family of Lorentzian metrics along
${\Sigma}$ depending differentiably on a parameter $\lambda$ such that
${\mathbf{g}}\equiv{\mathbf{g}}(0)$. The variational operator $\delta$ is
defined by evaluating $\frac{\partial}{\partial\lambda}$ at $\lambda=0$. We
set

$\displaystyle\mycal
E^{{\alpha\beta}}:=-\frac{\sqrt{|{\mathbf{g}}|}}{16\pi}\left({G}^{{\alpha\beta}}+{\Lambda}{\mathbf{g}}^{{\alpha\beta}}\right)\,,$
(C.1)
$\displaystyle{\pi}^{{\alpha\beta}}:=\frac{1}{16\pi}\sqrt{|{\mathbf{g}}|}\
{\mathbf{g}}^{{\alpha\beta}}\,,$ (C.2) $\displaystyle
A^{\lambda}_{{\alpha\beta}}:={\Gamma}^{\lambda}_{{\alpha\beta}}-{\delta}^{\lambda}_{(\alpha}{\Gamma}^{\kappa}_{\beta)\kappa}\,,$
(C.3)

where $G_{\alpha\beta}$ is the Einstein tensor and ${\Lambda}$ the
cosmological constant. The following variational identity has been proved in
CJKKerrdS ,666The reader might notice that CJKKerrdS  uses a non-standard
convention on the sign of the cosmological constant, opposite to the one here.
see Equation (2.27) there:

$\displaystyle({\mycal
L}_{X}A^{\lambda}_{{\alpha\beta}})\delta{\pi}^{{\alpha\beta}}-({\mycal
L}_{X}{\pi}^{{\alpha\beta}})\delta
A^{\lambda}_{{\alpha\beta}}=-2X^{\mu}\delta{\cal
E}^{\lambda}{}_{{\mu}}+X^{\lambda}{\cal
E}^{{\alpha\beta}}{\delta{\mathbf{g}}}_{{\alpha\beta}}$ (C.4)
$\displaystyle+\frac{1}{16\pi}\partial_{\mu}\bigg{\\{}\delta\big{[}\sqrt{|{\mathbf{g}}|}(\nabla^{\mu}X^{\lambda}-\nabla^{\lambda}X^{\mu})\big{]}-\sqrt{|{\mathbf{g}}|}(\nabla^{\mu}\delta
X^{\lambda}-\nabla^{\lambda}\delta X^{\mu})\bigg{\\}}$
$\displaystyle+\partial_{\mu}\left\\{X^{\lambda}{\pi}^{{\alpha\beta}}\delta
A^{{\mu}}_{{\alpha\beta}}-X^{\mu}{\pi}^{{\alpha\beta}}\delta
A^{\lambda}_{{\alpha\beta}}\right\\}\,.$

Integrating (C.4) over ${\Sigma}$, assuming that the integral converges, and
that the metrics ${\mathbf{g}}(\lambda)$ coincide with ${\mathbf{g}}$ near the
boundary of $\partial{\Sigma}$, if any, one finds

$\displaystyle\int_{{\Sigma}}\big{(}({\mycal
L}_{X}A^{\lambda}_{{\alpha\beta}})\delta{\pi}^{{\alpha\beta}}-({\mycal
L}_{X}{\pi}^{{\alpha\beta}})\delta
A^{\lambda}_{{\alpha\beta}}\big{)}dS_{\lambda}=-2\int_{{\Sigma}}X^{\mu}\delta{\cal
E}^{\lambda}{}_{{\mu}}dS_{\lambda}\,.$ (C.5)

Let $N_{\lambda}$ be any field of conormals to ${\Sigma}$, thus $\delta{\cal
E}^{\lambda}{}_{{\mu}}$ enters this identity only through the components
$\delta{\cal E}^{\lambda}{}_{{\mu}}N_{\lambda}$. The equations ${\cal
E}^{\lambda}{}_{{\mu}}N_{\lambda}=0$ are the constraint equations on
${\Sigma}$, and (C.5) expresses the well-known fact that, for spacelike
${\Sigma}$’s, the constraint equations provide an action principle for the
Einstein equations. This remains true for characteristic hypersurfaces in view
of (C.5), but is perhaps somewhat less known; compare KorbiczTafel ;
CJKbhthermo .

The operator $\delta{\cal E}^{\lambda}{}_{{\mu}}N_{\lambda}$ is the
linearisation of the constraint equations on ${\Sigma}$ acting on linearised
gravitational initial data on ${\Sigma}$. Integration by parts reexpresses the
right-hand side as the adjoint operator of the linearised constraint equations
acting on $X$.

Now, Killing vectors in space-time annihilite the left-hand side of (C.5). It
follows that Killing vectors of the spacetime metric are in the kernel of this
operator, in the following sense: _if $X$ is a vector field satisfying the
Killing equations and their first derivatives on ${\mycal N}$_, then

$\displaystyle\int_{{\Sigma}}X^{\mu}\delta{\cal
E}^{\lambda}{}_{{\mu}}dS_{\lambda}=0$ (C.6)

for all variations ${\delta{\mathbf{g}}}_{\mu\nu}$ as described.

It is known, for spacelike ${\Sigma}$’s, that spacetime Killing vectors
exhaust the kernel Moncrief75 : the left-hand side of (C.6) vanishes for all
variations as above _if and only if $X$ is a vector field satisfying the
Killing equations and their first derivatives on ${\mycal N}$_. Our gluing
results in this paper suggest strongly that this remains true for
characteristic hypersurfaces, but this remains to be seen; compare ChPaetzKIDs
.

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