# High-frequency solutions to the Einstein vacuum equations: local existence
in generalised wave gauge

Arthur Touati Centre de Mathématiques Laurent Schwartz (UMR 7640), Ecole
polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France
<EMAIL_ADDRESS>
###### Abstract

We construct a family $(g_{\lambda})_{\lambda\in(0,1]}$ of high-frequency
metrics solutions to the Einstein vacuum equations in 3+1 dimensions without
any symmetry assumptions. In the limit $\lambda\longrightarrow 0$ this family
approaches a fixed background $g_{0}$ solution of the Einstein-null dust
system, illustrating the backreaction phenomenon. We introduce a precise
second order high-frequency ansatz and identify a generalised wave gauge as
well as polarization conditions at each order. The validity of our ansatz is
ensured by a weak polarized null condition satisfied by the quadratic non-
linearity in the Ricci tensor. The Einstein vacuum equations for $g_{\lambda}$
are recast as a hierarchy of transport and wave equations, and their coupling
induces a loss of derivatives. In order to solve it, we take advantage of the
null structure associated to $g_{0}$ as well as a Fourier cut-off adapted to
our ansatz. The construction of high-frequency initial data solving the
constraint equations is the content of our companion paper [Tou22b].

###### Contents

  1. 1 Introduction
    1. 1.1 Presentation of the result
    2. 1.2 High-frequency gravitational waves
      1. 1.2.1 Gravitational waves in linearized gravity
      2. 1.2.2 Gravitational waves and geometric optics
    3. 1.3 The Burnett conjecture
      1. 1.3.1 The direct conjecture
      2. 1.3.2 The indirect conjecture
    4. 1.4 Acknowledgment
    5. 1.5 Outline of the article
  2. 2 Statement of the results
    1. 2.1 Preliminaries
    2. 2.2 The background spacetime
    3. 2.3 Initial data
    4. 2.4 Local existence
  3. 3 Strategy of proof
    1. 3.1 The high-frequency ansatz
      1. 3.1.1 The quasi-linearity
      2. 3.1.2 The gauge part
      3. 3.1.3 The weak polarized null condition
      4. 3.1.4 Propagation of the polarization conditions
    2. 3.2 Solving the hierarchy of equations
      1. 3.2.1 The background transport equations
      2. 3.2.2 The quasi-linear coupling and the loss of derivatives
      3. 3.2.3 Regaining the missing derivative
      4. 3.2.4 The order of precision of the ansatz
    3. 3.3 Outline of the proof
  4. 4 The high-frequency ansatz
    1. 4.1 High-frequency expansion of the metric
    2. 4.2 High-frequency expansion of the Ricci tensor
      1. 4.2.1 The wave part
      2. 4.2.2 The gauge part
      3. 4.2.3 The quadratic non-linearity
  5. 5 The hierarchy of equations
    1. 5.1 A guide to the hierarchy
    2. 5.2 The background system
    3. 5.3 The reduced system
    4. 5.4 Generalised wave gauge and polarization conditions
      1. 5.4.1 Generalised wave gauge condition
      2. 5.4.2 Polarization conditions
    5. 5.5 Definition of $g^{(3)}$
    6. 5.6 Initial data for the spacetime metric
      1. 5.6.1 Initial data for the background system
      2. 5.6.2 Initial data for the reduced system
      3. 5.6.3 Initial estimates and properties
  6. 6 Solving the background system
    1. 6.1 Background transport estimates
    2. 6.2 Solving for $F^{(1)}$, $F^{(2,1)}$ and $F^{(2,2)}$
    3. 6.3 Algebraic properties of $F^{(1)}$
      1. 6.3.1 Propagation of the first polarization
      2. 6.3.2 Propagation of the energy
  7. 7 Solving the reduced system
    1. 7.1 Bootstrap assumptions
    2. 7.2 First consequences
    3. 7.3 Estimates for $\mathfrak{F}$
    4. 7.4 Estimates for $\tilde{\Box}_{g}\mathfrak{F}$
      1. 7.4.1 Estimates for $\Box_{g_{0}}\mathfrak{F}$
      2. 7.4.2 Estimates for $(g-g_{0})\partial^{2}\mathfrak{F}$
    5. 7.5 Estimates for $\mathfrak{h}$
    6. 7.6 Conclusion of the bootstrap
  8. 8 Solving the Einstein vacuum equations
    1. 8.1 The reduced Ricci tensor
      1. 8.1.1 The zeroth order
      2. 8.1.2 The first order
      3. 8.1.3 The higher orders
    2. 8.2 The Einstein tensor and the contracted Bianchi identities
      1. 8.2.1 The Einstein tensor
      2. 8.2.2 The contracted Bianchi identities
      3. 8.2.3 Divergence of $G\left(\Upsilon\right)$
    3. 8.3 Propagation of the second polarization
    4. 8.4 Propagation of the generalised wave gauge and conclusion
  9. A The commutator $[\Box_{g_{0}},L_{0}]$ and proof of Lemma 3.1
  10. B The spectral projections
    1. B.1 Littlewood-Paley theory
    2. B.2 Definition and first properties of $\Pi_{\leq}$ and $\Pi_{\geq}$
    3. B.3 Proof of Lemma 3.2

##  1 Introduction

We are interesting in special solutions of the Einstein vacuum equations

$\displaystyle R_{\mu\nu}(g)=0$ (1.1)

where $R_{\mu\nu}(g)$ is the Ricci tensor of the Lorentzian metric $g$ defined
on a manifold $\mathcal{M}$. More precisely, we construct a sequence of
metrics $(g_{\lambda})_{\lambda}$ solutions of (1.1) converging when
$\lambda\longrightarrow 0$ in a particular sense to a background metric
$g_{0}$ solution of the Einstein equations coupled to a null dust (equations
(2.1)-(2.2)-(2.3) below). As is standard when solving the Einstein equations,
we need to choose a gauge. In this article, we work in generalised wave gauge,
i.e we rewrite (1.1) as a system of quasi-linear wave equations for the metric
coefficients. The initial data for this system are given by our companion
paper [Tou22b] and in particular they solve the constraint equations on
$\mathbb{R}^{3}$ (see Theorem 2.1 below for the main result of this article).

###  1.1 Presentation of the result

The main feature of the sequence $(g_{\lambda})_{\lambda}$ constructed is its
high-frequency character. Concretely, the metric $g_{\lambda}$ is defined by a
high-frequency expansion ($\lambda$ being a small wavelength) with a non-
oscillating remainder in the spirit of geometric optics and the work of
Choquet-Bruhat in [CB69]. The system (1.1) is then recast as a hierarchy of
equations for each term of the ansatz: transport equations along the
background rays for the oscillating terms and a wave equation for the
remainder. Our result constitutes the first rigorous justification of the
geometric optics approximation for the Einstein vacuum equations in $3+1$ (see
[HL18b] for a similar construction in $2+1$) and we highlight now its main
challenges.

The quasi-linear aspect of (1.1) implies a loss of derivatives in the coupling
between the wave equation for the remainder and one of the transport equation.
Regaining this derivatives is the main analytical challenge of our proof. It
heavily relies on the expansion defining $g_{\lambda}$ and on the use of the
background null structure, i.e the foliation associated to a solution of the
eikonal equation for the background metric $g_{0}$.

The quadratic terms in (1.1) cause the creation of harmonics, which we need to
describe precisely in order to define our high-frequency ansatz. We rely on
the weak polarized null structure of these terms. This allows us to recover a
linear behaviour for the propagation of the waves along the rays, despite the
non-linear nature of (1.1). This is known as transparency in the geometric
optics literature, see [Mé09].

Due to their gauge invariance, the Einstein vacuum are not hyperbolic. In the
case of high-frequency waves considered here, this implies the presence of
polarization conditions. They play a role in the control of the semi-linear
and quasi-linear terms presented above but they need to be propagated by the
evolution. As opposed to [CB69], we propagate the polarization conditions of
the second order perturbation in the high-frequency ansatz relying on the
contracted Bianchi identities rather than on the transport equations along the
rays.

A rough version of our main result can then be stated as follows.

######  Theorem 1.1 (Rough version of Theorem 2.2).

There exists a family of metrics $(g_{\lambda})_{\lambda}$ of the form

$\displaystyle g_{\lambda}=g_{0}+\lambda
g^{(1)}\left(\frac{u_{0}}{\lambda}\right)+O\left(\lambda^{2}\right)$ (1.2)

solving (1.1) in generalised wave gauge on $[0,1]\times\mathbb{R}^{3}$ where
$g_{0}$ is a given background metric, $u_{0}$ a solution of
$g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})=0$ and $g^{(1)}$ is a periodic
tensor in the variable $\frac{u_{0}}{\lambda}$.

The high-frequency character of the perturbation in (1.2) manifests itself
when we consider derivatives of the metric. Indeed, one loses one power of
$\lambda$ for each derivatives, which implies that the $H^{s}$ norms of the
coefficients of $g_{\lambda}$ explodes in the limit $\lambda\longrightarrow 0$
for $s>1$ and in particular for $s=2$. The exponent $s=2$ is particularly
interesting in the context of general relativity since from the seminal work
of Klainerman, Rodnianski and Szeftel in [KRS15], it is known that the time of
existence of a generic solution to (1.1) is controlled by the initial $H^{2}$
norm. Therefore, Theorem 1.1 does not follow from standard arguments but
rather requires a specific construction adapted to the high-frequency ansatz
to ensure that the time of existence is uniform in $\lambda$.

High-frequency spacetimes satisfying (1.1) arose first in the physics
literature in the work of Isaacson in [Isa68a]-[Isa68b] where he addressed the
mathematical understanding of strong gravitational fields and in particular of
the gravitational waves resulting from them. We will explain the whys and
wherefores of this question in Section 1.2.

Theorem 1.1 also states that a generic small solution to the Einstein-null
dust system can arise as an appropriate limit of a sequence of solutions to
the Einstein vacuum equations. Thus, this result also shed new light on the
Burnett conjecture in 3+1 dimension. This aspect is discussed in Section 1.3.

###  1.2 High-frequency gravitational waves

High-frequency ansatz as (1.2) are one way of describing the propagation of
gravitational waves in a non-linear context. We start by giving a quick
overview of gravitational waves in the linearized gravity setting, following
Chapter 35 of [MTW73].

####  1.2.1 Gravitational waves in linearized gravity

Considering a metric of the form $g=m+h$ with $m$ the Minkowski metric and
neglecting all non-linear terms in the perturbation $h$ and its derivatives in
the Ricci tensor of $g$, one obtain the linearized gravity equations

$\displaystyle\Box\bar{h}_{\mu\nu}$ $\displaystyle=0,$ (1.3)
$\displaystyle\partial^{\alpha}\bar{h}_{\alpha\beta}$ $\displaystyle=0,$ (1.4)

with
$\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}m^{\alpha\beta}h_{\alpha\beta}m_{\mu\nu}$
and where the second equation is the Lorentz gauge condition. Plane waves
solutions to (1.3)-(1.4) describe the propagation of gravitational waves and
are of the form

$\displaystyle\bar{h}_{\mu\nu}$
$\displaystyle=A_{\mu\nu}\cos\left(k_{\alpha}x^{\alpha}\right)$

with $k_{\alpha}k^{\alpha}=0$ and $k^{\alpha}A_{\alpha\beta}=0$. One can
impose extra gauge conditions and obtain the TT-gauge, where
$\bar{h}_{\mu\nu}$ is traceless and transverse, i.e $\mathrm{tr}\bar{h}=0$ and
$h_{t\mu}=0$. Such a plane wave possesses exactly two degrees of freedom
corresponding to two polarizations.

Though being historically crucial, this framework fail to describe
gravitational wave in a more physical context. First, the restriction to small
perturbations $h$ can’t, by definition, describe strong gravitational fields.
Moreover, one cannot describe the energy of a gravitational wave since any
quadratic expression in the perturbation $h$ is neglected. This has the
following consequence : the background spacetime on which the gravitational
wave propagates doesn’t ”see” the wave. Therefore, a mathematical framework is
needed to go beyond the linearized gravity setting and to describe the global
impact of the wave on the background spacetime.

####  1.2.2 Gravitational waves and geometric optics

In [BH64] and later in [Isa68a]-[Isa68b], the authors consider averaging
schemes allowing the self-interaction of a gravitational wave to affect the
background. A rigorous version of these constructions, where the average are
replaced by weak limits, has been given by Choquet-Bruhat in [CB69] (see also
Chapter 11 of [CB09]), a seminal article we describe now.

The article [CB69] constitutes the first application of the geometric optics
approach to the Einstein vacuum equations. The author considers WKB ansatz for
the metric of the spacetime, i.e introduce a small parameter $\lambda>0$ and
construct a metric $g_{\lambda}$ of the form

$\displaystyle g_{\lambda}$ $\displaystyle=g_{0}+\lambda
g^{(1)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}g^{(2)}\left(\frac{u_{0}}{\lambda}\right)$

satisfying $R_{\mu\nu}(g_{\lambda})=O\left(\lambda\right)$. As in this
article, the notation $g^{(1)}\left(\frac{u_{0}}{\lambda}\right)$ means here
that $g^{(i)}$ are symmetric 2-tensor depending on a extra phase factor
$\frac{u_{0}}{\lambda}$. Since the Ricci tensor involves at most two
derivatives of the metric, we have formally

$\displaystyle
R_{\mu\nu}(g_{\lambda})=\frac{1}{\lambda}R_{\mu\nu}^{(-1)}+R_{\mu\nu}^{(0)}+O\left(\lambda\right).$

Therefore, the goal of Choquet-Bruhat’s construction is to obtain conditions
on $g_{0}$, $g^{(1)}$ and $g^{(2)}$ so that
$R_{\mu\nu}^{(-1)}=R_{\mu\nu}^{(0)}=0$. These conditions are:

  * •

to ensure $R_{\mu\nu}^{(-1)}=0$, one must have
$g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})=0$ and

$\displaystyle
g_{0}^{\mu\nu}\left(\partial_{\mu}u_{0}g^{(1)}_{\alpha\nu}-\frac{1}{2}\partial_{\alpha}u_{0}g^{(1)}_{\mu\nu}\right)=0,$
(1.5)

  * •

to ensure $R_{\mu\nu}^{(0)}=0$, one must have

$\displaystyle
2\partial^{\alpha}u_{0}\mathbf{D}_{\alpha}g^{(1)}_{\mu\nu}+(\mathbf{D}^{\alpha}\mathbf{D}_{\alpha}u_{0})g^{(1)}_{\mu\nu}$
$\displaystyle=0$ (1.6)

and $R_{\mu\nu}(g_{0})=\tau\partial_{\mu}u_{0}\partial_{\nu}u_{0}$ with
$\tau>0$ depending quadratically on $g^{(1)}$.

These conditions imply that the perturbation $g^{(1)}$ is transported along
the rays of the null phase $u_{0}$, as expected in the geometric optics
approach. The polarization condition (1.5) on $g^{(1)}$ are propagated by the
transport equation and they imply that the physically significant part of
$g^{(1)}$ has two degrees of freedom and is transverse and traceless, as in
the TT-gauge of linearized gravity. Note that $g^{(2)}$ also satisfies
polarization conditions that we neglect in this introduction.

It is an a priori magic property of the Einstein vacuum equations that
gravitational waves do not suffer self-distortion, which translates as the
linearity of the transport equation (1.6). This is due to particular structure
of the semi-linear terms in the Ricci tensor and will be explained in Section
3.1.3.

These semi-linear terms are also at the origin of the non-zero RHS in the
equation satisfied by the background metric. This shows how one can associate
a stress-energy tensor to the gravitational wave described by $g^{(1)}$ and
how it appears as a source term for the background curvature. This is also a
first example of the backreaction phenomenon in general relativity, see
Section 1.3 for further discussions.

For these reasons, the framework of [CB69] goes indeed beyond the linearized
gravity setting. However, Choquet-Bruhat does not solve the full Einstein
vacuum equations in this article and her computations are formal. The main
motivation behind the present article is precisely to fill that gap and our
main result, Theorem 2.2 below, provides the first construction of high-
frequency gravitational waves solution to the full Einstein vacuum equations
(1.1), and therefore the first justification of the geometric approximation in
general relativity.

######  Remark 1.1.

WKB ansatz have also been considered in the linearized gravity setting in
[AJOR21] in order to describe new geometric effect such as the spin Hall
effect.

###  1.3 The Burnett conjecture

In [CB69], we can already see a family of almost vacuum spacetimes approaching
$g_{0}$, that is a non-vacuum spacetime solution. In the case of ansatz (1.2)
considered in this article, the uniform limit in $L^{\infty}$ when
$\lambda\longrightarrow 0$ of $g_{\lambda}$ is $g_{0}$ and the weak limit in
$L^{2}$ of $\partial g_{\lambda}$ is $\partial g_{0}$. Since
$R_{\mu\nu}(g_{\lambda})=0$ and $R_{\mu\nu}(g_{0})\neq 0$, this is a concrete
example of backreaction in the Einstein equations, i.e when a matter model is
derived as a limit of vacuum spacetimes. The general question of backreaction
in general relativity was first raised by Burnett in [Bur89]. In this article,
he considered a family of metrics $(h_{\lambda})_{\lambda}$ converging to a
metric $h_{0}$ in the same way as the family constructed in Theorem 1.1, that
is strong convergence in $L^{\infty}$ and weak convergence in $H^{1}$.
Moreover, the metrics $h_{\lambda}$ are all assumed to solve the Einstein
vacuum equations, that is $G_{\mu\nu}(h_{\lambda})=0$ (where we denote by $G$
the Einstein tensor). Burnett then asks the following: what is
$G_{\mu\nu}(h_{0})$ ? In other words, what effective stress-energy tensor can
be produced via weak limits of vacuum spacetimes ?

####  1.3.1 The direct conjecture

In [Bur89] Burnett formulates what we call the direct Burnett conjecture: the
limit metric $h_{0}$ satisfies the Einstein-massless Vlasov system, that is

$\displaystyle G_{\mu\nu}(h_{0})=\int_{P}fp_{\mu}p_{\nu}\mathrm{d}\mu$ (1.7)

where $P$ is the null cotangent bundle, $\mathrm{d}\mu$ is the volume measure
on $P$ and where the density $f(x,p)$ satisfies the Liouville-Vlasov equation

$\displaystyle
p^{\alpha}\partial_{x^{\alpha}}f-\Gamma^{\alpha}_{\mu\nu}p^{\mu}p^{\nu}\partial_{p^{\alpha}}f=0.$

The Einstein-massless Vlasov system describes the coupling between gravitation
and kinetic theory, and we refer to the Chapter 10 of [CB09] for more details
on this system or related ones. This article does not tackle the direct
Burnett conjecture so we only mention the most recent results.

In [HL19], Huneau and Luk proved the direct Burnett conjecture assuming
$\mathbb{U}(1)$ symmetry, that is when the spacetime is invariant by
translation in a spatial direction. Under this symmetry, the 3+1 Einstein
vacuum equations reduce to the 2+1 Einstein-wave map system (see [Mon86]). The
proof in [HL19] is based on microlocal defect measures and the solution
$h_{0}$ is a measure-valued solution of the Einstein-massless Vlasov system.
It has been improved in [GdC21], where the authors require less regularity for
the sequence $(h_{\lambda})_{\lambda}$. While reaching the full direct Burnett
conjecture, the proof in [HL19] relies on the special cancellations occuring
in $\mathbb{U}(1)$ symmetry. A proof without symmetry has recently been given
in [LR20], but since it also tackles the indirect Burnett conjecture we
postpone to next section the discussion of this article.

####  1.3.2 The indirect conjecture

In [Bur89] Burnett also formulates what we call the indirect Burnett
conjecture: for all solutions $h_{0}$ of the Einstein-massless Vlasov system,
we can construct a sequence $(h_{\lambda})_{\lambda}$ of vacuum spacetimes
converging to $h_{0}$.

In the forthcoming [HL], Huneau and Luk prove the indirect Burnett conjecture
still under the $\mathbb{U}(1)$ symmetry assumption. Their strategy is to
approach first a solution of the Einstein-massless Vlasov system by null dusts
and then approach the null dusts by vacuum solutions (see also [HL18b] for
this last step). They choose to work in elliptic gauge to obtain more
regularity for the metric coefficients, which solve semi-linear elliptic
equations. The key well-posedness result in this gauge is obtained in [Tou21].
Moreover, the elliptic gauge is only available with two spatial dimensions, so
the $\mathbb{U}(1)$ symmetry is a fundamental assumption in [HL].

While allowing oscillations in an arbitrarily large number of null directions,
the results of [HL18b] and [HL] heavily relies on the chosen symmetry of the
spacetime. We now discuss Luk and Rodnianski’s [LR20], which is the first
proof of Burnett conjecture, both the direct and indirect one, without
assuming symmetry for the spacetime. Their result is based on a low-regularity
theorem for the characteristic initial value problem obtained in [LR17] and
uses the double null foliation setting first introduced in [KN03]. In this
setting, the spacetime is foliated by ingoing and outgoing null cones which
are level sets of two optical functions $u$ and $\underline{u}$ and which
intersect on 2-spheres. The vacuum spacetimes they construct are angularly
regular, meaning that they have very low regularity in the
$\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial\underline{u}}$
directions (such derivatives of metric components are only in $L^{2}$) while
being very regular in the directions tangential to the 2-spheres. This
property constraints backreaction to occur only along the two null directions.
Therefore, the stress-energy tensor obtained in the limit describes a family
of two null dusts.

The results obtained in [LR20] are the first ones tackling the Burnett
conjecture without symmetry assumptions in 3+1, and also shed new lights on
the Einstein-null dusts system itself as well as the trapped surface formation
issue. Nevertheless, it assumes special regularity linked to the double null
foliation which forbids any superposition of more than two waves. On the
opposite, Theorem 1.1 constructs high-frequency vacuum 3+1 spacetimes without
symmetry in generalised wave gauge. Local existence in this gauge does not
particularise any directions and we think that our result will allow
superposition of an arbitrarily large number of null dusts.

As a final word on the Burnett conjecture, let us mention how the direct and
indirect conjecture can be linked, while being two very different mathematical
problems. The direct conjecture deals with given sequences of solutions to the
Einstein vacuum equations and tries to characterize their limits’ Ricci
tensor. The singular nature of these vacuum solutions implies that the own
existence of the sequence is not trivial. Therefore, by answering the indirect
conjecture one actually constructs concrete examples of vacuum solutions
exhibiting the singular behaviour required by any theorem dealing with the
direct conjecture. In that sense, Theorem 1.1 paves the way for a proof of the
direct Burnett conjecture in generalised wave gauge.

###  1.4 Acknowledgment

The author would like to thank his advisor Cécile Huneau for all the helpful
suggestions and encouragement.

###  1.5 Outline of the article

The remainder of this article is structured as follows.

  * •

In Section 2, we introduce the usual analytical material, define our class of
initial data and state the main result on the existence of the sequence
$(g_{\lambda})_{\lambda}$, i.e Theorem 2.2.

  * •

In Section 3, we present the strategy of the proof of Theorem 2.2.

  * •

Sections 4 to 8 are devoted to the proof of Theorem 2.2.

##  2 Statement of the results

###  2.1 Preliminaries

We will work on the manifold
$\mathcal{M}\vcentcolon=[0,1]\times\mathbb{R}^{3}$. For $t\in[0,1]$, we denote
by $\Sigma_{t}$ the spacelike hypersurface $\\{t\\}\times\mathbb{R}^{3}$. The
greek indices go from 0 to 3 and will always refer to the usual global
coordinates system $(t,x^{1},x^{2},x^{3})$ on $\mathcal{M}$, while latin
indices go from 1 to 3 and correspond to the three spatial directions.

Let $T$ and $S$ be two symmetric 2-tensors on $\mathcal{M}$ and let $g$ be a
Lorentzian metric on $\mathcal{M}$. We define a scalar product by

$\displaystyle|T\cdot
S|_{g}=g^{\alpha\beta}g^{\mu\nu}T_{\alpha\mu}S_{\beta\nu}$

with associated norm $|T|^{2}_{g}=|T\cdot T|_{g}$. The trace of $T$ with
respect to $g$ is defined by
$\mathrm{tr}_{g}T=g^{\alpha\beta}T_{\alpha\beta}$.

If $f$ is a scalar function on $\mathcal{M}$, we denote by $\nabla f$ a
spatial derivative of $f$ and by $\partial f$ a time derivative
$\partial_{t}f$ or $\nabla f$. The wave operator associated to a Lorentzian
metric is defined by

$\displaystyle\Box_{g}f$
$\displaystyle=g^{\alpha\beta}\left(\partial_{\alpha}\partial_{\beta}f-\Gamma^{\mu}_{\alpha\beta}\partial_{\mu}f\right)$

where $\Gamma^{\mu}_{\alpha\beta}$ are the Christoffel symbols. The principal
part of this operator is denoted $\tilde{\Box}_{g}$, i.e
$\tilde{\Box}_{g}f=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}f$.

On each slice $\Sigma_{t}$ we consider the usual functions spaces $L^{p}$ and
$W^{k,p}$, by which we always mean $L^{p}(\Sigma_{t})$ and
$W^{k,p}(\Sigma_{t})$ for some $t$ depending on the context. We will also use
the weighted Sobolev spaces defined below.

######  Definition 2.1 (Weighted Sobolev spaces).

For $1\leq p<+\infty$, $\delta\in\mathbb{R}$ and $k\in\mathbb{N}$ we define
the space $W^{k,p}_{\delta}$ as the completion of $C^{\infty}_{c}$ for the
norm

$\displaystyle\left\|u\right\|_{W^{k,p}_{\delta}}$
$\displaystyle=\sum_{0\leq|\alpha|\leq k}\left\|\langle
x\rangle^{\delta+|\alpha|}\nabla^{\alpha}u\right\|_{L^{p}},$

where the $L^{p}$ norm is defined with the Euclidean volume element. We extend
this definition to tensors of any type by summing over all components in the
Euclidean coordinates. Some special cases are
$H^{k}_{\delta}\vcentcolon=W^{k,2}_{\delta}$ and
$L^{p}_{\delta}\vcentcolon=W^{0,p}_{\delta}$.

We also define the following $L^{\infty}$-based spaces:

######  Definition 2.2.

For $k\in\mathbb{N}$ and $\delta\in\mathbb{R}$ we define $C^{k}_{\delta}$ as
the completion of $C^{\infty}_{c}$ for the norm

$\displaystyle\left\|u\right\|_{C^{k}_{\delta}}$
$\displaystyle=\sum_{0\leq|\alpha|\leq k}\left\|\langle
x\rangle^{\delta+|\alpha|}\nabla^{\alpha}u\right\|_{L^{\infty}},$

Let us recall some usual facts about these spaces (see [CB09] for the proofs).

######  Proposition 2.1.

Let $s,s^{\prime},s_{1},s_{2},m\in\mathbb{N}$,
$\delta,\delta^{\prime},\delta_{1},\delta_{2},\beta\in\mathbb{R}$ and $1\leq
p<+\infty$.

  1. 1.

If $s\leq\min(s_{1},s_{2})$, $s<s_{1}+s_{2}-\frac{3}{p}$ and
$\delta<\delta_{1}+\delta_{2}+\frac{3}{p}$ we have the continuous embedding

$\displaystyle W^{s_{1},p}_{\delta_{1}}\times W^{s_{2},p}_{\delta_{2}}\subset
W^{s,p}_{\delta}.$

  2. 2.

If $m<s-\frac{3}{p}$ and $\beta\leq\delta+\frac{3}{p}$ we have the continuous
embedding

$\displaystyle W^{s,p}_{\delta}\subset C^{m}_{\beta}.$

There won’t be any issue of regularity in the time variable so if $f$ is a
scalar function defined on $\mathcal{M}$, then $\left\|f\right\|_{X}\leq C$
without any precision on $t$ will always mean

$\displaystyle\sup_{t\in[0,1]}\left\|f\right\|_{X(\Sigma_{t})}\leq C$

where $X$ is one of the function spaces defined above.

###  2.2 The background spacetime

The background quantities are defined on $\mathcal{M}$ and are composed of a
metric $g_{0}$, an optical function $u_{0}$ and a density $F_{0}$. They are
solution of the Einstein null dust system on $\mathcal{M}$:

$\displaystyle R_{\mu\nu}(g_{0})$
$\displaystyle=F_{0}^{2}\partial_{\mu}u_{0}\partial_{\nu}u_{0},$ (2.1)
$\displaystyle g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})$ $\displaystyle=0,$
(2.2) $\displaystyle-2L_{0}F_{0}+(\Box_{g_{0}}u_{0})F_{0}$ $\displaystyle=0,$
(2.3)

where $R_{\mu\nu}(g_{0})$ is the Ricci tensor associated to $g_{0}$ and
$L_{0}$ is the spacetime gradient of $u_{0}$, i.e

$\displaystyle
L_{0}=-g_{0}^{\alpha\beta}\partial_{\alpha}u_{0}\partial_{\beta}.$ (2.4)

By (2.2), $L_{0}$ is null and geodesic, i.e

$\displaystyle
g_{0}(L_{0},L_{0})=0\quad\text{and}\quad\mathbf{D}_{L_{0}}L_{0}=0,$

where $\mathbf{D}$ denotes throughout this article the covariant derivative
associated to $g_{0}$. We also assume that $g_{0}$ satisfy in the standard
coordinates $(t,x^{1},x^{2},x^{3})$ the wave condition

$\displaystyle g_{0}^{\mu\nu}\Gamma(g_{0})^{\alpha}_{\mu\nu}=0.$ (2.5)

Note that this condition implies that the wave operator associated to $g_{0}$
reduces to its principal part, i.e $\Box_{g_{0}}=\tilde{\Box}_{g_{0}}$. A
standard computation shows that under the wave condition (2.5) the Einstein
type equation (2.1) rewrites as

$\displaystyle\tilde{\Box}_{g_{0}}(g_{0})_{\alpha\beta}$
$\displaystyle=P_{\alpha\beta}(g_{0})(\partial g_{0},\partial
g_{0})-2F^{2}_{0}\partial_{\alpha}u_{0}\partial_{\beta}u_{0}.$ (2.6)

where $P_{\alpha\beta}(g_{0})(\partial g_{0},\partial g_{0})$ is a quadratic
non-linearity (see (3.2) for its exact expression). We make the following
assumptions on the background quantities $(g_{0},u_{0},F_{0})$. In what
follows, $N\geq 10$, $\delta>-\frac{3}{2}$ and $\varepsilon>0$ is our
smallness threshold.

  * •

Assumptions on the metric $g_{0}$. There exists a constant $\varepsilon>0$
such that for all $t\in[0,1]$

$\displaystyle\left\|g_{0}-m\right\|_{H^{N+1}_{\delta}}+\left\|\partial_{t}g_{0}\right\|_{H^{N}_{\delta+1}}+\left\|\partial_{t}^{2}g_{0}\right\|_{H^{N-1}_{\delta+2}}\leq\varepsilon$
(2.7)

where $m$ is the Minkowski metric on $\mathcal{M}$. For convenience, we also
assume that $\partial_{t}$ is the unit normal to $\Sigma_{0}$ of $g_{0}$, and
that the second fundamental form of $\Sigma_{0}$ is traceless. The former
assumption simplifies the construction of the initial data for the high-
frequency metric (see Section 5.6) and the latter is only use in [Tou22b]
where we solve the constraint equations. The estimates (2.7) means that we
work in the asymptotically flat setting.

  * •

Assumptions on the optical function $u_{0}$. We assume that there exists a
constant non-zero vector field
$\mathfrak{z}=(\mathfrak{z}_{1},\mathfrak{z}_{2},\mathfrak{z}_{3})$ such that

$\displaystyle\left\|\nabla
u_{0}-\mathfrak{z}\right\|_{H^{N}_{\delta+1}}\leq\varepsilon$ (2.8)

where $\nabla u_{0}=(\partial_{1}u_{0},\partial_{2}u_{0},\partial_{3}u_{0})$
is the spatial Euclidean gradient of $u_{0}$. This implies that the level sets
of $u_{0}$ restricted to any $\Sigma_{t}$ are asymptotically planes in
$\mathbb{R}^{3}$ and that if $\varepsilon$ is small enough there exists $c>0$
such that

$\displaystyle\inf_{x\in\mathbb{R}^{3}}|\nabla u_{0}|(x)>c.$ (2.9)

This in turn implies that $u_{0}$ has no critical point. Finally, we assume
that $L_{0}$ defined by (2.4) is futur-directed and since $\partial_{t}$ is
assumed to be the unit normal of $g_{0}$ to $\Sigma_{0}$ and $u_{0}$ solves
(2.2), this implies

$\displaystyle\partial_{t}u_{0}=|\nabla u_{0}|_{\bar{g}_{0}}$ (2.10)

on $\Sigma_{0}$.

  * •

Assumptions on the density $F_{0}$. We assume that $F_{0}$ is initially
compactly supported, i.e there exists $R>0$ such that
$F_{0}\restriction{\Sigma_{0}}$ is supported in $\\{|x|\leq R\\}$. Since
$L_{0}$ is null and geodesic, (2.3) shows that

$\displaystyle\mathrm{supp}\left(F_{0}\right)\subset J_{0}^{+}\left(\\{|x|\leq
R\\}\right)$

where $J_{0}^{+}$ denotes the causal future associated to $g_{0}$. We can
assume for simplicity that

$\displaystyle J_{0}^{+}\left(\\{|x|\leq
R\\}\right)\subset\left\\{(t,x)\in\mathcal{M}~{}\middle|~{}|x|\leq
C_{\mathrm{supp}}R\right\\}$

for some constant $C_{\mathrm{supp}}>0$. Finally we assume the following
estimate:

$\displaystyle\left\|F_{0}\right\|_{H^{N}}\leq\varepsilon.$ (2.11)

In this article we don’t prove the existence of the background solution
$(g_{0},F_{0},u_{0})$ on $\mathcal{M}$ since it follows from standard
arguments. We will denote by $C_{0}$ any numerical constant depending on the
background estimates stated here, that is depending on $\delta$,$N$, $R$ or
$\mathfrak{z}$.

In addition to the usual coordinates on $[0,1]\times\mathbb{R}^{3}$ we will
use a frame linked to the optical function $u_{0}$, i.e the null frame, which
is defined as follows. We denote by $\mathcal{H}_{u}$ the level sets of
$u_{0}$, i.e

$\displaystyle\mathcal{H}_{u}$
$\displaystyle=\left\\{(t,x)\in\mathcal{M}~{}\middle|~{}u_{0}(t,x)=u\right\\}.$

Since $u_{0}$ is a solution of the eikonal equation (2.2), each
$\mathcal{H}_{u}$ is a null hypersurface generated by the geodesic vector
field $L_{0}$. Thanks to (2.9) they induce a foliation of the spacetime. For
$t\in[0,1]$ and $u$ in the image of $u_{0}$ we define the following 2-surfaces

$\displaystyle P_{t,u}=\Sigma_{t}\cap\mathcal{H}_{u}.$

Thanks to (2.8), each $P_{t,u}$ have the topology of a plane in
$\mathbb{R}^{3}$. We denote by $\bar{g}_{0}$ the induced metric on
$\Sigma_{t}$ and $\mathring{g}_{0}$ the induced metric on $P_{t,u}$. We
consider a null vector field $\underline{L}_{0}$ such that
$g_{0}(\underline{L}_{0},L_{0})=-2$ and an orthonormal frame $(e_{1},e_{2})$
of $TP_{t,u}$ for $\mathring{g}_{0}$. This defines the background null frame
$(L_{0},\underline{L}_{0},e_{1},e_{2})$, and such a choice is always possible,
at least locally. We use the latin upper case letters as indices for the frame
$(e_{1},e_{2})$ on $TP_{t,u}$. We consider the subset

$\displaystyle\mathcal{T}_{0}=\\{L_{0},e_{1},e_{2}\\}$ (2.12)

of the background null frame. The use of such a frame originates in the
seminal work of Christodoulou and Klainerman on the stability of Minkowski
spacetime (see [CK93]) and a concise presentation can be found in [Sze18].

Further geometric objects related to the background null frame will be needed
for the proof of Theorem 2.2, linked to the commutator $[L_{0},\Box_{g_{0}}]$.
We refer to Lemma 3.1 for the main result on this commutator and to Appendix A
for its proof.

###  2.3 Initial data

We reproduce here the main result of [Tou22b] where we construct high-
frequency solutions $(\bar{g}_{\lambda},K_{\lambda})$ of the constraint
equations on $\mathbb{R}^{3}$, that is

$\displaystyle
R(\bar{g}_{\lambda})+(\mathrm{tr}_{\bar{g}_{\lambda}}K_{\lambda})^{2}-|K_{\lambda}|^{2}_{\bar{g}_{\lambda}}$
$\displaystyle=0,$ (2.13)
$\displaystyle-\mathrm{div}_{\bar{g}_{\lambda}}K_{\lambda}+\mathrm{d}\mathrm{tr}_{\bar{g}_{\lambda}}K_{\lambda}$
$\displaystyle=0.$ (2.14)

######  Theorem 2.1.

Let $(g_{0},u_{0},F_{0})$ be the solution of the Einstein-null dust system
described in Section 2.2, and let $\varepsilon>0$ be the smallness threshold.
There exists $\varepsilon_{0}=\varepsilon_{0}(\delta,R)>0$ such that if
$0<\varepsilon\leq\varepsilon_{0}$, there exists for all $\lambda\in(0,1]$ a
solution $(\bar{g}_{\lambda},K_{\lambda})$ solution of the constraint
equations (2.13)-(2.14) on $\mathbb{R}^{3}$ of the form

$\displaystyle\bar{g}_{\lambda}$
$\displaystyle=\bar{g}_{0}+\lambda\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}+\lambda^{2}\left(\sin\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(2,1)}+\cos\left(\frac{2u_{0}}{\lambda}\right)\bar{F}^{(2,2)}\right)+\lambda^{2}\bar{\mathfrak{h}}_{\lambda},$
(2.15) $\displaystyle K_{\lambda}$ $\displaystyle=K^{(0)}_{\lambda}+\lambda
K^{(1)}_{\lambda}+\lambda^{2}K^{(\geq 2)}_{\lambda},$ (2.16)

with

$\displaystyle K^{(0)}_{\lambda}$
$\displaystyle=K_{0}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)},$ (2.17)
$\displaystyle\left(K^{(1)}_{\lambda}\right)_{ij}$
$\displaystyle=-\frac{1}{2}\cos\left(\frac{u_{0}}{\lambda}\right)\left(-N_{0}\bar{F}^{(1)}_{ij}+(\partial_{t}+N_{0})^{\rho}\Gamma(g_{0})_{\rho(i}^{k}\bar{F}^{(1)}_{j)k}+\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}(\Box_{g_{0}}u_{0})\bar{F}^{(1)}_{ij}\right)$ (2.18)
$\displaystyle\quad-\frac{1}{2}\left|\nabla
u_{0}\right|_{\bar{g}_{0}}\left(\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(2,1)}_{ij}-2\sin\left(\frac{2u_{0}}{\lambda}\right)\bar{F}^{(2,2)}_{ij}\right),$

where $K_{0}$ is the second fundamental form of $\Sigma_{0}$ for $g_{0}$.
Moreover:

  * (i)

the tensors $\bar{F}^{(1)}$, $\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$ are
supported in $\\{|x|\leq R\\}$ and there exists
$C_{\mathrm{cons}}=C_{\mathrm{cons}}(\delta,R)>0$ such that

$\displaystyle\left\|\bar{F}^{(1)}\right\|_{H^{N}}+\left\|\bar{F}^{(2,1)}\right\|_{H^{N-1}}+\left\|\bar{F}^{(2,2)}\right\|_{H^{N-1}}$
$\displaystyle\leq C_{\mathrm{cons}}\varepsilon,$ (2.19)

  * (ii)

the tensor $\bar{F}^{(1)}$ is $\bar{g}_{0}$-traceless, tangential to $P_{0,u}$
and satisfies

$\displaystyle\left|\bar{F}^{(1)}\right|^{2}_{\bar{g}_{0}}=8F_{0}^{2},$ (2.20)

  * (iii)

the tensors $\bar{\mathfrak{h}}_{\lambda}$ and $K^{(\geq 2)}_{\lambda}$ belong
to the spaces $H^{5}_{\delta}$ and $H^{4}_{\delta+1}$ respectively and satisfy

$\displaystyle\max_{r\in\llbracket
0,4\rrbracket}\lambda^{r}\left\|\nabla^{r+1}\bar{\mathfrak{h}}_{\lambda}\right\|_{L^{2}_{\delta+r+1}}$
$\displaystyle\leq C_{\mathrm{cons}}\varepsilon,$ (2.21)
$\displaystyle\max_{r\in\llbracket
0,4\rrbracket}\lambda^{r}\left\|\nabla^{r}K_{\lambda}^{(\geq
2)}\right\|_{L^{2}_{\delta+r+1}}$ $\displaystyle\leq
C_{\mathrm{cons}}\varepsilon.$ (2.22)

###  2.4 Local existence

The following theorem is our main result.

######  Theorem 2.2.

Let $(g_{0},u_{0},F_{0})$ be the solution of the Einstein-null dust system
described in Section 2.2, and let $\varepsilon>0$ be the smallness threshold.
There exists $\lambda_{0}\in\mathbb{R}$ and
$\varepsilon_{0}=\varepsilon_{0}(\delta,R)>0$ such that if
$0<\varepsilon\leq\varepsilon_{0}$, there exists for all
$\lambda\in(0,\lambda_{0}]$ a solution $g_{\lambda}$ of the form

$\displaystyle g_{\lambda}=g_{0}+\lambda
g^{(1)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}g^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}\tilde{\mathfrak{h}}_{\lambda}$
(2.23)

to (1.1) on $[0,1]\times\mathbb{R}^{3}$ in generalised wave gauge. Moreover:

  1. (i)

the tensors $g^{(i)}$ for $i=1,2$ are supported in $J_{0}^{+}\left(\\{|x|\leq
R\\}\right)$ and are periodic and smooth functions of the argument
$\frac{u_{0}}{\lambda}$,

  2. (ii)

there exists $C=C(\delta,R)>0$ such that the following estimates hold

$\displaystyle\max_{r\in\llbracket
0,10\rrbracket}\lambda^{r}\left\|\nabla^{r}g^{(1)}\right\|_{L^{2}}+\max_{r\in\llbracket
0,6\rrbracket}\lambda^{r}\left\|\nabla^{r}g^{(2)}\right\|_{L^{2}}+\max_{r\in\llbracket
1,5\rrbracket}\lambda^{r}\left\|\nabla^{r+1}\tilde{\mathfrak{h}}_{\lambda}\right\|_{L^{2}_{\delta+r}}$
$\displaystyle\leq C\varepsilon,$ (2.24)

  3. (iii)

the tensor
$g^{(1)}_{\mu\nu}=\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}_{\mu\nu}$
satisfy the following polarization and energy conditions

$\displaystyle
g_{0}^{\mu\nu}\left(\partial_{\mu}u_{0}F^{(1)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}u_{0}F^{(1)}_{\mu\nu}\right)$
$\displaystyle=0,$ (2.25)
$\displaystyle\frac{1}{8}\left|F^{(1)}\right|^{2}_{g_{0}}-\frac{1}{16}\left(\mathrm{tr}_{g_{0}}F^{(1)}\right)^{2}$
$\displaystyle=F_{0}^{2}.$ (2.26)

Before we present the strategy behind the proof of Theorem 2.2 in Section 3,
let us make some remarks on this result.

  * •

The notation $g^{(i)}\left(\frac{u_{0}}{\lambda}\right)$ is used in (2.23) to
emphasize the oscillating character of $g^{(i)}$, but these tensors also
depends on the spacetime variables and this notation actually stands for the
following functions on the spacetime

$\displaystyle
g^{(i)}_{\alpha\beta}\left(\frac{u_{0}}{\lambda}\right)(t,x)=g^{(i)}_{\alpha\beta}\left(t,x,\frac{u_{0}(t,x)}{\lambda}\right)$

where $(t,x)\in[0,1]\times\mathbb{R}^{3}$.

  * •

The exact expression of the ansatz for $g_{\lambda}$ is given in Section 4.1.
Compared to (2.23), we will then describe precisely the oscillating behaviour
of each $g^{(i)}$ for $i=1,2$, i.e give a finite frequency decomposition. Note
that in Theorem 2.2 we already gave the frequency decomposition of $g^{(1)}$,
that is $g^{(1)}=\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}$.

  * •

As explained in depth in the introduction, our main goal is to consider high-
frequency limits, that is $\lambda\longrightarrow 0$. Since the smallness
threshold $\varepsilon_{0}$ is independent from $\lambda$, our construction is
valid in the high-frequency limit and the estimates

$\displaystyle\left\|g_{\lambda}-g_{0}\right\|_{L^{\infty}}$
$\displaystyle\lesssim\lambda,$
$\displaystyle\left\|\partial\left(g_{\lambda}-g_{0}\right)\right\|_{L^{2}}$
$\displaystyle\lesssim 1$

hold even for small $\lambda$, as well as the time of existence equal to 1.
From this perspective, the restriction $\lambda\in(0,\lambda_{0}]$ with
$\lambda_{0}$ potentially very small is meaningless and only plays a role in
the improvement of one of our bootstrap assumptions (see Proposition 7.1).

  * •

The tensor $g^{(2)}$ also satisfies polarization conditions similar to (2.25)
but with complicated RHS depending on $g^{(1)}$. See Section 5.4.2 for more
details.

  * •

The generalised wave condition defining our gauge has a complicated expression
and we choose not to present it in Theorem 2.2. In particular, it depends on
the exact ansatz for $g_{\lambda}$ and the coupling between the transport
equations for the waves and the wave equation for the remainder. See Section
5.4.1 for its exact expression.

  * •

The oscillating terms in (2.23) depend on the phase $\frac{u_{0}}{\lambda}$,
where $u_{0}$ is the solution of the background eikonal equation, i.e (2.2).
We emphasize the fact that we don’t solve the eikonal equation for the metric
$g_{\lambda}$, as in [HL18a] but as opposed to [LR20]. This would require the
construction of a null foliation in 3+1 dimensions and hence would complicate
our proof.

##  3 Strategy of proof

In this section, we sketch the proof of Theorem 2.2. The challenge we face is
twofold: first we need to define the high-frequency ansatz for $g_{\lambda}$
in a fully non-linear context, and then we need to solve the hierarchy of
equations obtained which in particular amounts to solve an apparent loss of
derivatives.

###  3.1 The high-frequency ansatz

In any coordinates system, the Ricci tensor of a Lorentzian metric $g$ reads

$\displaystyle
2R_{\alpha\beta}(g)=-\tilde{\Box}_{g}g_{\alpha\beta}+H^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}H^{\rho}+P_{\alpha\beta}(g)(\partial
g,\partial g).$ (3.1)

Let us detail the different terms in this expression.

  * •

$\tilde{\Box}_{g}$ stands for the principal part of the wave operator, i.e
$g^{\mu\nu}\partial_{\mu}\partial_{\nu}$. Because of this term, the Einstein
vacuum equations (1.1) are quasi-linear. Moreover, if we think of (1.1) as a
system of second order partial differential operators for the metric
coefficients, the $-\tilde{\Box}_{g}g_{\alpha\beta}$ term represents the
diagonal part of the Ricci operator.

  * •

The terms involving $H^{\rho}=g^{\mu\nu}\Gamma^{\rho}_{\mu\nu}$ are the gauge
part of the Ricci operator and they involve second order derivatives of the
metric coefficients. They represent the non-diagonal part of the Ricci
operator and working in generalised wave gauge precisely means that we
prescribe the value of $H^{\rho}$ in order to solve the loss of hyperbolicity
it causes.

  * •

The quadratic non-linearity $P_{\alpha\beta}(g)(\partial g,\partial g)$ is
given in coordinates by

$\displaystyle P_{\alpha\beta}(g)(\partial g,\partial g)$
$\displaystyle=g^{\mu\rho}g^{\nu\sigma}\partial_{(\alpha}g_{\rho\sigma}\partial_{\mu}g_{\beta)\nu}-\frac{1}{2}g^{\mu\rho}g^{\nu\sigma}\partial_{\alpha}g_{\rho\sigma}\partial_{\beta}g_{\mu\nu}$
(3.2)
$\displaystyle\quad-g^{\mu\rho}g^{\nu\sigma}\partial_{\rho}g_{\alpha\nu}\partial_{\sigma}g_{\beta\mu}+g^{\mu\rho}g^{\nu\sigma}\partial_{\rho}g_{\sigma\alpha}\partial_{\mu}g_{\nu\beta}.$

It enjoys what we call the weak polarized null condition defined in Section
3.1.3.

While being highly non-linear, the Ricci tensor and the three parts defining
it all enjoy some special structure at the heart of our construction. The
precise ansatz we choose is

$\displaystyle g_{\lambda}=g_{0}+\lambda
g^{(1)}+\lambda^{2}\left(g^{(2)}+\mathfrak{h}_{\lambda}\right)+\lambda^{3}g^{(3)}$
(3.3)

The remainder $\mathfrak{h}_{\lambda}$ is a symmetric 2-tensor and each
$g^{(i)}_{\alpha\beta}$ is a finite linear combination of terms of the form
$\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)S_{\alpha\beta}$ for $S$ a
symmetric 2-tensor and $\mathrm{T}$ a trigonometric function, that is

$\displaystyle\mathrm{T}\in\left\\{s\longmapsto\cos(ks)~{}\middle|~{}k\in\mathbb{N}\right\\}\cup\left\\{s\longmapsto\sin(ks)~{}\middle|~{}k\in\mathbb{N}\right\\}.$
(3.4)

Without loss of generality, we can assume that $g^{(1)}$ has the form
$g^{(1)}_{\alpha\beta}=\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}_{\alpha\beta}$,
as stated in Theorem 2.2. The Ricci tensor being non-linear, we must
incorporate the creation of harmonics into $g_{\lambda}$. Even though terms
like $P_{\alpha\beta}(g)(\partial g,\partial g)$ are of the form
$g^{-1}g^{-1}\partial g\partial g$ and thus corresponds to an interaction of
four oscillating terms, the derivatives imply that this term corresponds
actually to an interaction of only two oscillating terms, at least at leading
order. Therefore, we expect $g^{(2)}$ to oscillate with frequency $\theta$ and
$2\theta$. More precisely, by analogy with our toy model in [Tou22a] $g^{(2)}$
oscillates like $\sin(\theta)$ and $\cos(2\theta)$.

The Ricci tensor involves second derivatives of the metric, and
$R_{\mu\nu}(g_{\lambda})$ admits a priori a formal expansion of the form

$\displaystyle R_{\mu\nu}(g_{\lambda})$
$\displaystyle=\frac{1}{\lambda}R^{(-1)}_{\mu\nu}+R^{(0)}_{\mu\nu}+\lambda
R^{(1)}_{\mu\nu}+\cdots.$

In the sequel, when considering the formal expansion of the Ricci tensor in
powers of $\lambda$, we refer to all the terms multiplied by a certain power
$\lambda^{k}$ as the $\lambda^{k}$ level. Moreover, ”solving the $\lambda^{k}$
level” will mean obtaining $R^{(k)}_{\mu\nu}=0$.

In the rest of this section and for the sake of clarity we drop the index
$\lambda$ and only write $g$ and $\mathfrak{h}$ instead of $g_{\lambda}$ and
$\mathfrak{h}_{\lambda}$.

####  3.1.1 The quasi-linearity

We start by studying the quasi-linear part of the Ricci tensor, also called
the wave part in the sequel, that is $-\tilde{\Box}_{g}g_{\alpha\beta}$. We
simply compute

$\displaystyle\tilde{\Box}_{g}\left(\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)f\right)$
$\displaystyle=\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)\tilde{\Box}_{g}f+\frac{1}{\lambda}\mathrm{T}^{\prime}\left(\frac{u_{0}}{\lambda}\right)\left[2g^{\mu\nu}\partial_{\mu}u_{0}\partial_{\nu}f+(\tilde{\Box}_{g}u_{0})f\right]$
(3.5)
$\displaystyle\quad+\frac{1}{\lambda^{2}}\mathrm{T}^{\prime\prime}\left(\frac{u_{0}}{\lambda}\right)g^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})f$

for any scalar function $f$ and $\mathrm{T}$ a trigonometric function. Since
$g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})=0$ we have schematically

$\displaystyle g^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})=\lambda
g^{(1)}_{L_{0}L_{0}}+\lambda^{2}g^{(2)}_{L_{0}L_{0}}+O\left(\lambda^{3}\right).$
(3.6)

Therefore, the contribution to $R_{\mu\nu}(g_{\lambda})$ of each
$\tilde{\Box}_{g}\lambda^{i}g^{(i)}$ reads schematically

$\displaystyle\lambda^{0}\left(L_{0}\partial_{\theta}g^{(1)}+g^{(1)}_{L_{0}L_{0}}\partial_{\theta}^{2}g^{(1)}\right)+\lambda^{1}\left(L_{0}\partial_{\theta}g^{(2)}+g^{(1)}_{L_{0}L_{0}}\partial_{\theta}^{2}g^{(2)}+g^{(2)}_{L_{0}L_{0}}\partial_{\theta}^{2}g^{(1)}\right)$
(3.7)

where we neglect for now $\mathfrak{h}_{\lambda}$. We recover here the main
result of the geometric optics approximation: the waves $g^{(i)}$ solve
transport equations along the rays of the optical function $u_{0}$ (recall
(2.4)). These equations will absorb all the terms oscillating as $g^{(i)}$. To
make things clearer, we define the notion of admissible frequencies:

######  Definition 3.1.

Let $k=0,1$. The admissible frequencies at the $\lambda^{k}$ level are
$\ell\theta$ for $\ell\in\llbracket 1,k+1\rrbracket$. A non-admissible
frequency at a certain level is said to be forbidden at this level.

Looking at (3.7) and at the oscillating behaviour of each $g^{(i)}$ described
above, we see that the transport equations for the $g^{(i)}$ will precisely
absorb the admissible frequencies.

The quasi-linearity of the Ricci tensor manifests itself by the presence of
the eikonal terms $g^{(i)}_{L_{0}L_{0}}$. More precisely, if $k=0,1$, it
produces at the $\lambda^{k}$ level the terms
$g^{(i)}_{L_{0}L_{0}}\partial_{\theta}^{2}g^{(k+2-i)}$, which in turn produces
$(k+2)\theta$ oscillations if one consider only the top frequency of each
$g^{(i)}$ ($\theta$ for $g^{(1)}$ and $2\theta$ for $g^{(2)}$). Looking at
Definition 3.1, these are forbidden frequencies and we need to use the
structures of the gauge part and the quadratic part of the Ricci tensor to
remove them, see the end of Section 3.1.3.

The remainder $\mathfrak{h}_{\lambda}$ in (3.3) being non-oscillating, it
simply produces $\tilde{\Box}_{g}\mathfrak{h}_{\lambda}$ at the $\lambda^{2}$
level. Therefore, $\mathfrak{h}_{\lambda}$ will solve a wave equation
absorbing any frequency appearing at the $\lambda^{2}$ level, which explains
why there is no need to write down this level in (3.7), and why $g^{(3)}$
won’t be obtained by solving a transport equation.

####  3.1.2 The gauge part

In the last section, we saw how the second order derivatives of the wave part
do not produce a loss of two powers of $\lambda$ since the eikonal term
$g^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})$ is of order $\lambda$. However,
second order derivatives of the metric also appear in the gauge part of the
Ricci tensor, that is the one involving first order derivatives of $H^{\rho}$
whose expression is given by

$\displaystyle
H^{\rho}=g^{\mu\nu}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}g_{\mu\nu}\right).$

Let us compute the action of this term on the waves $g^{(i)}$ as we did in the
previous section for the wave part:

$\displaystyle
g^{\mu\nu}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}g_{\mu\nu}\right)$
$\displaystyle=\frac{1}{\lambda}g_{0}^{\mu\nu}g_{0}^{\rho\sigma}\left(\partial_{\mu}u_{0}\partial_{\theta}g^{(1)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}u_{0}\partial_{\theta}g^{(1)}_{\mu\nu}\right)+O\left(1\right).$

The leading term in the previous expression will be of great importance in our
construction, it corresponds to the polarization tensor already identified in
[CB69].

######  Definition 3.2.

Let $S$ a symmetric 2-tensor, we define its polarization tensor by

$\displaystyle\mathrm{Pol}_{\sigma}(S)=g_{0}^{\mu\nu}\left(\partial_{\mu}u_{0}S_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}u_{0}S_{\mu\nu}\right).$
(3.8)

Its components in the background null frame (see Section 2.2) are

$\displaystyle\mathrm{Pol}_{L_{0}}(S)$ $\displaystyle=-S_{L_{0}L_{0}},$ (3.9)
$\displaystyle\mathrm{Pol}_{A}(S)$ $\displaystyle=-S_{AL_{0}},$ (3.10)
$\displaystyle\mathrm{Pol}_{\underline{L}_{0}}(S)$
$\displaystyle=-\delta^{AB}S_{AB}.$ (3.11)

Since the main term in the gauge part of the Ricci tensor is
$g_{\rho(\alpha}\partial_{\beta)}H^{\rho}$, the main contributions of the
waves $g^{(i)}$ to $R_{\mu\nu}(g_{\lambda})$ read schematically

$\displaystyle\frac{1}{\lambda}$
$\displaystyle\partial_{(\alpha}u_{0}\mathrm{Pol}_{\beta)}\left(\partial_{\theta}^{2}g^{(1)}\right)+\lambda^{0}\partial_{(\alpha}u_{0}\mathrm{Pol}_{\beta)}\left(\partial_{\theta}^{2}g^{(2)}\right)+\lambda^{1}\partial_{(\alpha}u_{0}\mathrm{Pol}_{\beta)}\left(\partial_{\theta}^{2}g^{(3)}\right)$
(3.12)

Since the quadratic non-linearity only involves first order derivatives,
(3.12) already implies that solving the $\lambda^{-1}$ level of the Ricci
tensor implies $\mathrm{Pol}\left(g^{(1)}\right)=0$. Together with (3.9) this
implies that

$\displaystyle g^{(1)}_{L_{0}L_{0}}=0$ (3.13)

which remove some of the quasi-linear forbidden frequencies in (3.7). The
identity (3.13) is called the exceptionality condition by Choquet-Bruhat in
[CB69] and is a special feature of the Einstein vacuum equations. It also
implies that $g^{(1)}_{L_{0}A}=0$, and we will actually prove that
$g^{(1)}_{L_{0}\alpha}=0$.

Let us now try to understand the role of $\mathrm{Pol}\left(g^{(i)}\right)$
for $i\geq 2$. We introduce the following notions to emphasize the structure
at stake:

######  Definition 3.3.

Let $S$ be a symmetric 2-tensor and recall (2.12).

  1. 1.

We say that $S$ is non-tangential if $S_{XY}=0$ for all
$X,Y\in\mathcal{T}_{0}$. In coordinates this is equivalent to the existence of
a 1-form $Q$ such that $S_{\alpha\beta}=\partial_{(\alpha}u_{0}Q_{\beta)}$.

  2. 2.

We say that $S$ is fully non-tangential if the only non-zero null components
of $S$ is $S_{\underline{L}_{0}\underline{L}_{0}}$. In coordinates this is
equivalent to the existence of a scalar function $f$ such that
$S_{\alpha\beta}=f\partial_{\alpha}u_{0}\partial_{\beta}u_{0}$.

We see in (3.12) that the polarization of the waves $g^{(i)}$ only manifest
themselves as non-tangential terms in the Ricci tensor, at one level lower
than the transport equations that $g^{(i)}$ satisfy. In terms of oscillations,
they add the forbidden frequency $(k+2)\theta$ at the $\lambda^{k}$ level, for
$k=0,1$. Therefore, the gauge part in the Ricci tensor can remove some
forbidden frequencies with polarization conditions of the form

$\displaystyle\mathrm{Pol}\left(g^{(i)}\right)=\text{lower order terms}$

if and only if the problematic terms are non-tangential in the Ricci tensor.
This holds for the forbidden frequencies produced by the quadratic non-
linearity (see next section) but doesn’t hold for the quasi-linear forbidden
frequencies in (3.7).

The computation (3.12) also shows the role of $g^{(3)}$. As explained above,
it won’t solve a transport equation but will only be useful to absorb
forbidden frequency at the $\lambda^{1}$ level.

####  3.1.3 The weak polarized null condition

The semi-linear terms $P_{\alpha\beta}$ (short for
$P_{\alpha\beta}(g)(\partial g,\partial g)$) are crucial to our construction
since they are the source of backreaction. Indeed we have schematically

$\displaystyle P_{\alpha\beta}$ $\displaystyle=(\partial
g_{0})^{2}+\left(\partial u_{0}\partial_{\theta}g^{(1)}\right)^{2}+\partial
g_{0}\partial u_{0}\partial_{\theta}g^{(1)}+O\left(\lambda\right).$ (3.14)

The term $(\partial g_{0})^{2}$ is part of the background equation, while
$\partial g_{0}\partial u_{0}\partial_{\theta}g^{(1)}$ corresponds to an
admissible frequency and thus can be absorbed by the transport equation for
$g^{(1)}$. The middle term in (3.14) is the most important since it contains a
non-oscillating term and an harmonic $2\theta$, that is a forbidden frequency
at the $\lambda^{0}$ level. The non-oscillating term produces the backreaction
as it is absorbed by the background equation, which thus becomes an Einstein-
type equation with a RHS (see (2.1)). Without any additional structure, the
harmonic $2\theta$ would be absorbed by the transport equation for $g^{(1)}$,
which would therefore be non-linear. However, $P_{\alpha\beta}$ does enjoy
additional structure, which we call the weak polarized null condition and
present now.

The history of null conditions goes back to the work of Klainerman in [Kla86]
on the global existence of solutions for semi-linear quadratic wave equations
in 3+1 dimensions.

In order to define the weak polarized null condition, we consider the general
quadratic form $Q_{\alpha\beta}(T,S)$ acting on symmetric 2-tensors defined by

$\displaystyle Q_{\alpha\beta}(T,S)$
$\displaystyle=g_{0}^{\mu\rho}g_{0}^{\nu\sigma}\partial_{(\alpha}u_{0}T_{\rho\sigma}\partial_{\mu}u_{0}S_{\beta)\nu}-\frac{1}{2}g_{0}^{\mu\rho}g_{0}^{\nu\sigma}\partial_{\alpha}u_{0}T_{\rho\sigma}\partial_{\beta}u_{0}S_{\mu\nu}$
$\displaystyle\quad-
g_{0}^{\mu\rho}g_{0}^{\nu\sigma}\partial_{\rho}u_{0}T_{\alpha\nu}\partial_{\sigma}u_{0}S_{\beta\mu}+g_{0}^{\mu\rho}g_{0}^{\nu\sigma}\partial_{\rho}u_{0}T_{\sigma\alpha}\partial_{\mu}u_{0}S_{\nu\beta}$

It corresponds to the leading term in $\lambda$ when the derivatives $\partial
g\partial g$ in $P_{\alpha\beta}$ both hit oscillating parts of the waves
$g^{(i)}$ and where $T$ and $S$ denote here the amplitude of the waves. The
discussion of Section 3.1.2 shows that each wave $g^{(i)}$ will satisfy
polarization conditions through their polarization tensor
$\mathrm{Pol}\left(g^{(i)}\right)$. We can show that

$\displaystyle Q_{\alpha\beta}(T,T)$
$\displaystyle=\mathcal{E}\left(T,T\right)\partial_{\alpha}u_{0}\partial_{\beta}u_{0}+g_{0}^{\nu\sigma}\mathrm{Pol}_{\sigma}(T)\partial_{(\alpha}u_{0}T_{\beta)\nu}-\mathrm{Pol}_{\alpha}(T)\mathrm{Pol}_{\beta}(T)$
(3.15)

where we define the following energy

$\mathcal{E}\left(T,S\right)=\frac{1}{4}\left(\mathrm{tr}_{g_{0}}T\right)\left(\mathrm{tr}_{g_{0}}S\right)-\frac{1}{2}\left|T\cdot
S\right|_{g_{0}}.$ (3.16)

Let us explain our classification of null conditions.

  * •

The standard null condition introduced by Klainerman in [Chr86] and [Kla86]
would imply that $Q_{\alpha\beta}(T,T)=0$, since $L_{0}$ is a null vector
field.

  * •

In [CB00], Choquet-Bruhat introduces the polarized null condition, which would
imply

$\displaystyle\mathrm{Pol}(T)=0\Longrightarrow Q_{\alpha\beta}(T,T)=0.$

  * •

We now define the weak polarized null condition by

$\displaystyle\mathrm{Pol}(T)=0\Longrightarrow Q_{\alpha\beta}(T,T)\text{ is
fully non-tangential}.$

As the computation (3.15) shows, the quadratic non-linearity in the Ricci
tensor does satisfy the weak polarized null condition. Note the analogy with
the weak null condition of Lindblad and Rodnianski, since the term
$\mathcal{E}\left(T,T\right)\partial_{\alpha}u_{0}\partial_{\beta}u_{0}$
precisely corresponds to their problematic term
$P(\partial_{\mu}h,\partial_{\nu}h)$, see Proposition 3.1 in [LR10]. This
explains our choice of designation.

The weak polarized null condition is the key structure allowing us to deal
with the forbidden frequencies caused by the self-interaction of $g^{(1)}$.
Indeed, the self-interaction of $g^{(1)}$ produces only fully non-tangential
term since $\mathrm{Pol}\left(g^{(1)}\right)=0$. As explained above, this
interaction produces the $2\theta$ forbidden frequency at the $\lambda^{0}$
level (see (3.14)), but thanks to the fully non-tangential structure it can be
absorbed by the polarization tensor of $g^{(2)}$ which thus satisfies

$\displaystyle\mathrm{Pol}_{\beta}\left(g^{(2)}\right)$
$\displaystyle=\frac{1}{2}\mathcal{E}\left(\partial_{\theta}g^{(1)},\partial_{\theta}g^{(1)}\right)\partial_{\beta}u_{0}.$
(3.17)

In particular this implies

$\displaystyle\mathrm{Pol}_{\mathcal{T}_{0}}\left(g^{(2)}\right)=0.$ (3.18)

This last property (together with (3.9)) implies that $g^{(2)}_{L_{0}L_{0}}=0$
(if we only consider the top frequency) which remove the $3\theta$ forbidden
frequency at the $\lambda^{1}$ level coming from the quasi-linearity (see
(3.7)).

We summarize the discussion of Sections 3.1.1-3.1.2-3.1.3 since it describes
the key mechanism of our result. Let $k=-1,0,1$. The admissible frequencies
are absorbed by the transport equation of $g^{(k+1)}$ (for $k\geq 0$). The
quasi-linear terms produce forbidden frequencies which rewrite as polarization
terms while the semi-linear terms produce forbidden frequencies with the fully
non-tangential structure. This allows the polarization tensors of $g^{(k+2)}$
originating from the non-diagonal part of the Ricci tensor to absorb the
latter. For this last step, the non-tangential structure would be enough but
the fully non-tangential structure implies in addition that the quasi-linear
forbidden frequencies actually vanish. In particular, this shows how a non-
linear theory such as general relativity still implies propagation without
deformation since the transport equation for $g^{(1)}$ is linear. This is
called the transparency phenomenon in the geometric optics literature, see
[Mé09].

####  3.1.4 Propagation of the polarization conditions

As explained above, we need the waves $g^{(1)}$ and $g^{(2)}$ to satisfy both
transport equations along the rays and polarization conditions. In [CB69],
Choquet-Bruhat proves that the polarization condition for $g^{(1)}$, that is
$\mathrm{Pol}\left(g^{(1)}\right)=0$, is propagated by the transport equation
$g^{(1)}$ satisfies, which schematically writes $L_{0}g^{(1)}=0$. She shows
that if $\mathrm{Pol}\left(g^{(1)}\right)=0$ holds on $\Sigma_{0}$, and if
$g^{(1)}$ solves $L_{0}g^{(1)}=0$, then $\mathrm{Pol}\left(g^{(1)}\right)=0$
holds on the whole spacetime. This is proved by commuting $L_{0}$ and
$\mathrm{Pol}$ which gives an equation of the form

$\displaystyle L_{0}\mathrm{Pol}\left(g^{(1)}\right)=0.$

Unfortunately, the transport equations and polarization conditions satisfied
by $g^{(2)}$ (see Sections 5.2 and 5.4.2) are much more complicated than the
one $g^{(1)}$ satisfies and the previous strategy seems unfeasible. Instead of
deriving the equations for $\mathrm{Pol}\left(g^{(2)}\right)$ directly from
the equations that $g^{(2)}$ satisfies, we treat the polarization conditions
for $g^{(2)}$ as gauge conditions. This means that we first solve the
hierarchy of equations as if they were satisfied, and then derive the desired
transport equations from the contracted Bianchi identities, which states that
the Einstein tensor of a Lorentzian metric is divergence free. This shows how
we recover the polarization conditions on the whole spacetime, since our
choice of initial data will be such that they hold initially (see Corollary
5.1).

Therefore, we are able to obtain both the transport equations and the
polarization conditions for all the waves in $g$, proving that the latter is a
solution of (1.1). It remains to explain how we solve the hierarchy of
equations and the challenges its resolution pose from an analytical point of
view.

###  3.2 Solving the hierarchy of equations

In this section, we explain how we solve the hierarchy of equations derived
from putting the ansatz (3.3) into the Einstein vacuum equations (1.1). As
explained above, the hierarchy of equations is made of transport equations for
$g^{(1)}$ and $g^{(2)}$ and a wave equation for $\mathfrak{h}$.

####  3.2.1 The background transport equations

As explained above, the transport equation for $g^{(1)}$ is linear and reads
schematically

$\displaystyle L_{0}\partial_{\theta}g^{(1)}$ $\displaystyle=0.$ (3.19)

If we only focus on the $2\theta$ oscillation in $g^{(2)}$, the second
transport equation reads schematically

$\displaystyle L_{0}\partial_{\theta}g^{(2)}$
$\displaystyle=\partial^{2}g^{(1)}$ (3.20)

where in the RHS of (3.20) we only consider the worst term from the
perspective of regularity. The system (3.19)-(3.20) is triangular and solving
it causes no problem. In the sequel, the exact form of this system will be
called the background system because the regularity of its solution only
depends on the background spacetime and does not require any bootstrap.

######  Remark 3.1.

The threshold for the background regularity, i.e $N\geq 10$ (see Section 2.2),
is chosen so that all these background quantities and their derivatives can be
ultimately bounded in $L^{\infty}$. Let us explain briefly why the value $10$
fills this requirement.

If the background metric coefficients are in $H^{N+1}$ (we neglect the weights
at spacelike infinity in this discussion), then (3.19) and (3.20) imply that
$g^{(1)}\in H^{N}$ and $g^{(2)}\in H^{N-2}$. Second derivatives of $g^{(2)}$
will appear as a source term in the wave equation for the remainder
$\mathfrak{h}$. In order to prove local well-posedness for this equation we
will differenciate it 4 times (see (3.29) below) and our worse term is thus
$\partial^{6}g^{(2)}$. We want to bound it in $L^{\infty}$ so the usual
Sobolev embedding $H^{2}\xhookrightarrow{}L^{\infty}$ implies that we need
$N-2\geq 8$.

####  3.2.2 The quasi-linear coupling and the loss of derivatives

Let us now look at what we call the reduced system, that is the system solved
by the $\theta$ oscillation in $g^{(2)}$ and $\mathfrak{h}$. For a semi-linear
equation like the one considered in [Tou22a], this system would also be
perfectly triangular. However, due to the quasi-linear terms, our reduced
system is not triangular.

As announced, we focus only on the $\theta$ oscillation in $g^{(2)}$ and write
$g^{(2)}_{\alpha\beta}=\sin\left(\frac{u_{0}}{\lambda}\right)\mathfrak{F}_{\alpha\beta}$.
Considering now the term proportionnal to $\mathfrak{h}$ in the inverse of $g$
that we neglected in the eikonal term (3.6), the term
$\tilde{\Box}\left(\lambda g^{(1)}\right)$ produces a term proportional to
$\lambda\cos\left(\frac{u_{0}}{\lambda}\right)\mathfrak{h}_{L_{0}L_{0}}F^{(1)}$.
Moreover, the wave part of the Ricci also contains terms of the form

$\displaystyle\lambda\cos\left(\frac{u_{0}}{\lambda}\right)L_{0}\mathfrak{F}+\lambda^{2}\left(\Box_{g}\mathfrak{h}+\sin\left(\frac{u_{0}}{\lambda}\right)\Box_{g}\mathfrak{F}\right).$

Therefore, in order to solve the $\lambda^{1}$ and $\lambda^{2}$ levels we
need to solve the following coupled system

$\displaystyle L_{0}\mathfrak{F}$ $\displaystyle=\mathfrak{h},$ (3.21)
$\displaystyle\Box_{g}\mathfrak{h}$
$\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)\Box_{g}\mathfrak{F},$
(3.22)

where we only write the terms relevant to the present discussion. Performing
energy estimates for (3.21) implies schematically that
$\partial\mathfrak{F}\sim\partial\mathfrak{h}$ while energy estimates for
(3.22) give $\partial\mathfrak{h}\sim\partial^{2}\mathfrak{F}$. Therefore,
there is an a priori loss of derivatives in the coupling between (3.21) and
(3.22). Regaining this derivatives is the main analytical challenge of our
work.

####  3.2.3 Regaining the missing derivative

The main idea is to show that $\Box_{g}\mathfrak{F}$ is better than
$\partial^{2}\mathfrak{F}$, i.e better than any second order derivatives of
$\mathfrak{F}$. To show this, we would like to derive a transport operator for
$\Box_{g}\mathfrak{F}$ from (3.21). However we can’t hope for any structural
property of $[L_{0},\Box_{g}]$ since $L_{0}$ is a null vector for $g_{0}$ and
not $g$. One option would be to construct the full null foliation associated
to $g$, that is consider $[L,\Box_{g}]$ for $L$ the spacetime gradient of an
optical function $u$ solving $g^{-1}(\mathrm{d}u,\mathrm{d}u)=0$. This would
require to consider the null structure equations and the Bianchi equations in
3+1 as part as our bootstrap argument. To avoid this and to stick to the
background null foliation, we benefit from the expansion defining $g-g_{0}$
and write schematically

$\displaystyle\Box_{g}\mathfrak{F}$
$\displaystyle=\Box_{g_{0}}\mathfrak{F}+\lambda\partial^{2}\mathfrak{F}.$
(3.23)

We improve the first term in (3.23) by considering the commutator
$[L_{0},\Box_{g_{0}}]$ for which we can now hope for structural properties.
Formally we have from (3.21)

$\displaystyle L_{0}\left(\Box_{g_{0}}\mathfrak{F}\right)$
$\displaystyle=[L_{0},\Box_{g_{0}}]\mathfrak{F}+\Box_{g_{0}}\mathfrak{h}.$
(3.24)

We need to know what are the second derivatives appearing in the commutator
$[L_{0},\Box_{g_{0}}]$. This is the content of the following key lemma.

######  Lemma 3.1.

Let $f$ be compactly supported, we have

$\displaystyle\left\|[L_{0},\Box_{g_{0}}]f\right\|_{L^{2}}\leq
C(C_{0})\left(\left\|\partial
L_{0}f\right\|_{L^{2}}+\left\|\Box_{g_{0}}f\right\|_{L^{2}}+\left\|\partial
f\right\|_{L^{2}}\right).$ (3.25)

Moreover, if $r\geq 1$ we have

$\displaystyle\left\|\nabla^{r}[L_{0},\Box_{g_{0}}]f\right\|_{L^{2}}\leq
C(C_{0})\left(\left\|\nabla^{r}\partial
L_{0}f\right\|_{L^{2}}+\left\|\nabla^{r}\Box_{g_{0}}f\right\|_{L^{2}}+\left\|\partial
f\right\|_{H^{r}}+\left\|\partial^{2}f\right\|_{H^{r-1}}\right).$ (3.26)

The proof of this lemma is postponed to Appendix A and is based on the null
structure induced by the optical function $u_{0}$. In the case of equation
(3.24), the term $\Box_{g_{0}}\mathfrak{F}$ from (3.25) is treated with
Gronwall’s inequality and the term $\partial L_{0}\mathfrak{F}$ rewrites as
$\partial\mathfrak{h}$ thanks to (3.21). By using again the expansion for $g$,
we can use equation (3.22) for the term $\Box_{g_{0}}\mathfrak{h}$ in (3.24).
Therefore, we are able to show that $\Box_{g_{0}}\mathfrak{F}$ is at the level
of $\partial\mathfrak{h}$, which is consistent with (3.22). This is done in
details in Section 7.4.1.

It remains to improve the second term in (3.23). As our notation suggests it,
we don’t use any structure of the second order derivatives involved but rather
benefit from the small factor $\lambda$ in front. The idea is to cancel one
derivative with this factor, which rigorously involves a spatial Fourier
projector $\Pi_{\leq}$, that is a Fourier multiplier supported in
$\left\\{|\xi|\leq\frac{1}{\lambda}\right\\}$. If we also define
$\Pi_{\geq}=\mathrm{Id}-\Pi_{\leq}$, we have indeed Bernstein-like estimates
in $L^{2}$:

$\displaystyle\lambda\partial\Pi_{\leq}(f)\sim
f\quad\text{and}\quad\frac{1}{\lambda}\Pi_{\geq}(f)\sim\partial f.$

The proof of these estimates and the exact definition of the projectors
$\Pi_{\leq}$ and $\Pi_{\geq}$ are postponed to Appendix B.3. In order to
benefit from these projectors, we need to introduce them into the system
(3.21)-(3.22). Instead of this system, we thus solve

$\displaystyle L_{0}\mathfrak{F}$ $\displaystyle=\Pi_{\leq}(\mathfrak{h}),$
(3.27) $\displaystyle\Box_{g}\mathfrak{h}$
$\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)\Box_{g}\mathfrak{F}+\frac{1}{\lambda}\Pi_{\geq}(\mathfrak{h}).$
(3.28)

Using the two Bernstein-like estimates, equation (3.27) now implies that
$\lambda\partial^{2}\mathfrak{F}\sim\partial\mathfrak{h}$ while the extra term
in (3.28) is absorbed by the energy estimate for the wave operator. This
modification of the system is only possible because (3.21)-(3.22) are obtained
as part of the high-frequency expansion of the Ricci tensor. This allows us to
artificially remove $\Pi_{\geq}(\mathfrak{h})$ from the transport equation for
$\mathfrak{F}$ by simply rewriting the $\lambda\mathfrak{h}$ term as

$\displaystyle\lambda\Pi_{\leq}(\mathfrak{h})+\lambda^{2}\times\frac{1}{\lambda}\Pi_{\geq}(\mathfrak{h}).$

This shows how one can use (3.23) to gain a derivative, solve (3.21)-(3.22)
and obtain

$\displaystyle\left\|\partial\nabla^{r}\mathfrak{h}\right\|_{L^{2}}+\left\|\partial\nabla^{r}\mathfrak{F}\right\|_{L^{2}}\lesssim\frac{1}{\lambda^{r}}$
(3.29)

for $r\in\llbracket 0,4\rrbracket$ (we omitted the weight at spacelike
infinity for $\mathfrak{h}$). Note that since we modified the equation for
$\mathfrak{F}_{\lambda}$, we need the following commutator estimate to deal
with the first term in (3.23):

######  Lemma 3.2.

Let $u$ and $v$ be two tempered distribution. We have

$\displaystyle\left\|\left[u,\Pi_{\leq}\right]\nabla
v\right\|_{L^{2}}\lesssim\left(\left\|\nabla
u\right\|_{L^{\infty}}+\lambda\left\|u\right\|_{H^{\frac{7}{2}}}\right)\left\|v\right\|_{L^{2}}.$
(3.30)

This lemma will be used to estimate $\left[\Pi_{\leq},\Box_{g_{0}}\right]$.
Its proof is postponed to Appendix B.3 and relies on the Littlewood-Paley
decomposition, and we introduce in Section B.1 the material needed.

####  3.2.4 The order of precision of the ansatz

Finally, let us comment on the order of precision of the high-frequency ansatz
defining $g_{\lambda}$. As (3.3) shows, the order of precision is 2, since the
non-oscillating remainder $\mathfrak{h}_{\lambda}$ appears first at the order
$\lambda^{2}$ in $g_{\lambda}$. This is similar to the construction of high-
frequency ansatz in [HL18b] in $\mathbb{U}(1)$ symmetry, but different from
the results of [Tou22a], where we are able to construct high-frequency ansatz
of arbitrary precision for a toy model.

As in [Tou22a], the value of this order of precision is linked to the method
of proof we use to solve the equations at stake. To make this clearer, let us
assume that this value, denoted by $K\in\mathbb{Z}$, is not fixed yet.

The transport equations only involve the transport operator $L_{0}$ which is
part of the background and therefore their resolution is independent from
$\lambda$. However, we use energy estimates associated to $g_{\lambda}$ to
solve (3.28). As shown in [Sog95] (see also Lemma 7.9 in this article) this
energy estimates reads schematically

$\displaystyle\left\|\partial
u\right\|_{L^{2}}(t)\lesssim\exp\left(\int_{0}^{t}\left\|\partial
g_{\lambda}\right\|_{L^{\infty}}\right)\left(\left\|\partial
u\right\|_{L^{2}}(0)+\int_{0}^{t}\left\|\tilde{\Box}_{g_{\lambda}}u\right\|_{L^{2}}\right).$
(3.31)

Therefore, in order to construct a solution to (3.28) with a time of existence
independent of $\lambda$, we need $\left\|\partial
g_{\lambda}\right\|_{L^{\infty}}$ to be bounded uniformly in the high-
frequency limit $\lambda\longrightarrow 0$, since it appears as coefficient in
(3.31). If we focus on the term $\lambda^{K}\mathfrak{h}_{\lambda}$ in
$g_{\lambda}$, the usual Sobolev embedding $H^{2}\xhookrightarrow{}L^{\infty}$
gives

$\displaystyle\left\|\lambda^{K}\partial\mathfrak{h}_{\lambda}\right\|_{L^{\infty}}\lesssim\lambda^{K}\left\|\partial\mathfrak{h}_{\lambda}\right\|_{H^{2}}\lesssim\lambda^{K-2}$

where we used the fact that one loses one power of $\lambda$ for each
derivatives of $\partial\mathfrak{h}_{\lambda}$ (see (3.29)). See Lemma 7.1
for a more rigorous proof. We see why we need $K$ to be at least equal to 2.
The present article then shows that $K=2$ is sufficient to construct local
high-frequency solutions to (1.1) on $[0,1]\times\mathbb{R}^{3}$.

An interesting problem would be to construct similar solutions but of greater
order of precision, which could pave the way for the study of the long time
dynamics of high-frequency waves. Indeed, the main result of [Tou22a] shows
with a toy model how the vector field method requires $K\geq 4$.

###  3.3 Outline of the proof

We give here an outline of the article’s remainder which is concerned with the
proof of Theorem 2.2 and follows the strategy depicted in the previous
sections.

  * •

In Section 4, we give the final expression of the ansatz for $g_{\lambda}$ and
expand its Ricci tensor.

  * •

In Section 5, we state the hierarchy of equations as well as the polarization
and generalised wave gauge conditions. The hierarchy is composed of the
background system and the reduced system. We also define the initial data.

  * •

In Section 6, we solve the background system.

  * •

In Section 7, we solve the reduced system.

  * •

In Section 8, we show that the metric $g_{\lambda}$ is actually a solution of
(1.1) by propagating the polarization and generalised wave gauge conditions.

##  4 The high-frequency ansatz

In this section, we present the ansatz for the spacetime metric $g_{\lambda}$
and compute its Ricci tensor.

###  4.1 High-frequency expansion of the metric

The metric we consider is given by

$\displaystyle g_{\lambda}=g_{0}+\lambda
g^{(1)}+\lambda^{2}\left(g^{(2)}+\mathfrak{h}_{\lambda}\right)+\lambda^{3}g^{(3)}$
(4.1)

where

$\displaystyle g^{(1)}$
$\displaystyle=F^{(1)}\cos\left(\frac{u_{0}}{\lambda}\right),$ $\displaystyle
g^{(2)}$
$\displaystyle=\mathfrak{F}_{\lambda}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,1)}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,2)}\cos\left(\frac{2u_{0}}{\lambda}\right).$

The symmetric 2-tensors $F^{(1)}$ and $F^{(2,i)}$ for $i=1,2$ are called the
background perturbations. They satisfy the background system, defined in the
Section 5.2. The symmetric 2-tensors $\mathfrak{F}_{\lambda}$ and
$\mathfrak{h}_{\lambda}$ satisfy the reduced system, defined in Section 5.3.
We use a different font to emphasize their special role. The exact definition
of $g^{(3)}$ will be given in Section 5.5, let us just say for now that it is
a polynomial in terms of the background perturbations and their derivatives
and is independent $\mathfrak{h}_{\lambda}$ but does depends on
$\mathfrak{F}_{\lambda}$ without derivatives. The background perturbations
tensors don’t depend on $\lambda$, unlike the tensors
$\mathfrak{F}_{\lambda}$, $\mathfrak{h}_{\lambda}$ and the metric
$g_{\lambda}$ itself. However, for the sake of clarity we drop from now on
this notation and only write $\mathfrak{F}$, $\mathfrak{h}$ and $g$.

######  Remark 4.1.

The presence of $\mathfrak{F}$ and $F^{(2,1)}$ in $g^{(2)}$ seems redundant
and is not necessary. However it seems useful to the author to distinguish
between the part of the $\theta$ oscillation in $g^{(2)}$ coupled to
$\mathfrak{h}$ (see Sections 3.2.2 for a informal discussion) and the part
whose only purpose is to absorb terms coming from $g^{(1)}$ such as
$\Box_{g_{0}}F^{(1)}$ and which can thus be treated as a background quantity.

We can formally compute the high-frequency expansion of the inverse of $g$. In
coordinates it is given by

$\displaystyle g^{\mu\nu}$
$\displaystyle=g_{0}^{\mu\nu}+\lambda(g^{\mu\nu})^{(1)}+\lambda^{2}(g^{\mu\nu})^{(2)}+\lambda^{3}H^{\mu\nu}$
(4.2)

where we define

$\displaystyle(g^{\mu\nu})^{(1)}$
$\displaystyle=-(F^{(1)})^{\mu\nu}\cos\left(\frac{u_{0}}{\lambda}\right),$
(4.3) $\displaystyle(g^{\mu\nu})^{(2)}$
$\displaystyle=g_{0}^{\mu\rho}(F^{(1)})^{\nu\sigma}F^{(1)}_{\rho\sigma}\cos^{2}\left(\frac{u_{0}}{\lambda}\right)-\mathfrak{F}^{\mu\nu}\sin\left(\frac{u_{0}}{\lambda}\right)-\mathfrak{h}^{\mu\nu}$
(4.4)
$\displaystyle\quad-(F^{(2,1)})^{\mu\nu}\sin\left(\frac{u_{0}}{\lambda}\right)-(F^{(2,2)})^{\mu\nu}\cos\left(\frac{2u_{0}}{\lambda}\right)$

On the RHS of these expressions, all the inverse are taken with respect of the
background metric $g_{0}$. For example we have
$(F^{(1)})^{\mu\nu}=g_{0}^{\mu\rho}g_{0}^{\nu\sigma}F^{(1)}_{\rho\sigma}$. The
quantity $H^{\mu\nu}$ is defined so that (4.2) holds with $(g^{\mu\nu})^{(i)}$
for $i=1,2$ being defined by (4.3)-(4.4), and one can verify by hand that
$H^{\mu\nu}=O\left(1\right)$.

######  Remark 4.2.

Note that we denote by $g^{-k}$ any product of $k$ coefficients of the inverse
metric $g^{-1}$. This also applies to the background inverse metric
$g_{0}^{-1}$.

###  4.2 High-frequency expansion of the Ricci tensor

In this section, we compute the Ricci tensor of the metric $g$ given by (4.1).
In order to lighten the computations, we assume that
$\mathrm{Pol}\left(F^{(1)}\right)=0$, where $\mathrm{Pol}$ is the polarization
operator defined in (3.8). We also assume
$F^{(1)}_{L_{0}\underline{L}_{0}}=0$, which altogether gives
$F^{(1)}_{L_{0}\alpha}=0$ and $\mathrm{tr}_{g_{0}}F^{(1)}=0$. The fact that
$F^{(1)}$ satisfy these conditions will be proved in Section 6.3.1.

The expansion of the Ricci tensor is based on (3.1), i.e its decomposition in
generalised wave coordinates, and we first decompose the wave part
$\tilde{\Box}_{g}g_{\alpha\beta}$, then the gauge part $H^{\rho}$ and finally
the quadratic non-linearity $P_{\alpha\beta}$. We recall the notion of
admissible/forbidden frequencies (see Definition 3.1): at the $\lambda^{k}$
level, only the frequencies $\ell\theta$ for $\ell\in\llbracket
1,k+1\rrbracket$ are admissible, with $k=0,1$.

The exact expression of the expansions are complicated and we proceed as
follows to present them in the clearest way possible. We start by giving the
formal expansions in powers of $\lambda$ (see (4.5), (4.17) and (4.27)), and
then give the oscillating expansion of each coefficients in Lemmas 4.1, 4.2
and 4.3 for the wave part, Lemmas 4.4, 4.5 and 4.6 for the gauge part and
Lemmas 4.7 and 4.8 for the quadratic non-linearity.

####  4.2.1 The wave part

We start by the expansion of the wave part of the Ricci tensor of $g$.
Formally we have

$\displaystyle\tilde{\Box}_{g}g_{\alpha\beta}$
$\displaystyle=W^{(0)}_{\alpha\beta}+\lambda
W^{(1)}_{\alpha\beta}+\lambda^{2}W^{(\geq 2)}_{\alpha\beta}.$ (4.5)

######  Lemma 4.1 (Expression of $W^{(0)}$).

We have

$\displaystyle W^{(0)}_{\alpha\beta}$
$\displaystyle=\tilde{\Box}_{g_{0}}(g_{0})_{\alpha\beta}-\sin\left(\frac{u_{0}}{\lambda}\right)\left[-2L_{0}F^{(1)}_{\alpha\beta}+(\tilde{\Box}_{g_{0}}u_{0})F^{(1)}_{\alpha\beta}\right].$
(4.6)

###### Proof.

The main ingredient is the computation (3.5). We can use the expansion of the
inverse given in (4.2)-(4.4) and $F^{(1)}_{L_{0}\alpha}=0$ to expand first the
eikonal term $g^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})$:

$\displaystyle g^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})$
$\displaystyle=g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})-\lambda
F^{(1)}_{L_{0}L_{0}}\cos\left(\frac{u_{0}}{\lambda}\right)$
$\displaystyle\quad-\lambda^{2}\left(\mathfrak{F}_{L_{0}L_{0}}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,1)}_{L_{0}L_{0}}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,2)}_{L_{0}L_{0}}\cos\left(\frac{2u_{0}}{\lambda}\right)+\mathfrak{h}_{L_{0}L_{0}}\right)+O\left(\lambda^{3}\right)$
$\displaystyle=-\lambda^{2}\left(\mathfrak{F}_{L_{0}L_{0}}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,1)}_{L_{0}L_{0}}\sin\left(\frac{u_{0}}{\lambda}\right)+F^{(2,2)}_{L_{0}L_{0}}\cos\left(\frac{2u_{0}}{\lambda}\right)+\mathfrak{h}_{L_{0}L_{0}}\right)+O\left(\lambda^{3}\right)$
(4.7)

where we also used (2.2). The absence of a $\lambda^{0}$ term in (4.7)
explains why there is no $\lambda^{-1}$ term in the expansion of
$\tilde{\Box}_{g}g_{\alpha\beta}$ given by (4.5). Similarly we have

$\displaystyle g^{\mu\nu}\partial_{\mu}u_{0}\partial_{\nu}f$
$\displaystyle=-L_{0}f+O\left(\lambda^{2}\right)$ (4.8)

and

$\displaystyle\tilde{\Box}_{g}f$
$\displaystyle=\tilde{\Box}_{g_{0}}f-\lambda\cos\left(\frac{u_{0}}{\lambda}\right)(F^{(1)})^{\mu\nu}\partial_{\mu}\partial_{\nu}f+O\left(\lambda^{2}\right)$
(4.9)

for $f$ a scalar function. Using the general formula (3.5) and the expansions
(4.7), (4.8) and (4.9) we can expand $\tilde{\Box}_{g}g_{\alpha\beta}$ and
obtain the different terms in (4.5). In particular the $\lambda^{0}$ terms in
$\tilde{\Box}_{g}g_{\alpha\beta}$, i.e $W^{(0)}_{\alpha\beta}$ come from
$\partial^{2}g_{0}$ and $\partial u_{0}\partial_{\theta}\partial g^{(1)}$. ∎

######  Remark 4.3.

Note that the absence of the forbidden frequency $2\theta$ is due to
$F^{(1)}_{L_{0}\alpha}=0$.

######  Lemma 4.2 (Expression of $W^{(1)}$).

We have

$\displaystyle W^{(1)}_{\alpha\beta}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)W^{(1,1)}_{\alpha\beta}+\sin\left(\frac{2u_{0}}{\lambda}\right)W^{(1,2)}_{\alpha\beta}$
(4.10)
$\displaystyle\quad+\frac{1}{2}\sin\left(\frac{2u_{0}}{\lambda}\right)\mathfrak{F}_{L_{0}L_{0}}F^{(1)}_{\alpha\beta}+\cos\left(\frac{u_{0}}{\lambda}\right)\cos\left(\frac{2u_{0}}{\lambda}\right)F^{(2,2)}_{L_{0}L_{0}}F^{(1)}_{\alpha\beta}$

where $W^{(1,1)}$ and $W^{(1,2)}$ are given in (4.11) and (4.12).

###### Proof.

The $\lambda$ terms in $\tilde{\Box}_{g}g_{\alpha\beta}$ come from
$\partial^{2}g^{(1)}$ and $\partial u_{0}\partial_{\theta}\partial g^{(2)}$,
which explains the oscillating decomposition of $W^{(1)}$ in (4.10). More
precisely, the terms $W^{(1,i)}$ are given by

$\displaystyle W^{(1,1)}_{\alpha\beta}$
$\displaystyle=-2L_{0}F^{(2,1)}_{\alpha\beta}+(\tilde{\Box}_{g_{0}}u_{0})F^{(2,1)}_{\alpha\beta}+\tilde{W}^{(1,1)}_{\alpha\beta}$
(4.11)
$\displaystyle\quad-2L_{0}\mathfrak{F}_{\alpha\beta}+(\tilde{\Box}_{g_{0}}u_{0})\mathfrak{F}_{\alpha\beta}+\mathfrak{h}_{L_{0}L_{0}}F^{(1)}_{\alpha\beta},$
$\displaystyle W^{(1,2)}_{\alpha\beta}$
$\displaystyle=-2\left(-2L_{0}F^{(2,2)}_{\alpha\beta}+(\tilde{\Box}_{g_{0}}u_{0})F^{(2,2)}_{\alpha\beta}+\tilde{W}^{(1,2)}_{\alpha\beta}-\frac{1}{4}F^{(2,1)}_{L_{0}L_{0}}F^{(1)}_{\alpha\beta}\right),$
(4.12)

with $\tilde{W}^{(1,i)}$ being some irrelevant lower order terms formally
admitting the following expression

$\displaystyle\tilde{W}^{(1,i)}=g_{0}^{-2}F^{(1)}\partial^{2}g_{0}+g_{0}^{-2}\partial^{2}F^{(1)}+g_{0}^{-2}(F^{(1)})^{2}\partial^{2}u_{0}.$
(4.13)

∎

######  Remark 4.4.

Note the appearance of the eikonal terms $\mathfrak{F}_{L_{0}L_{0}}$ and
$F^{(2,2)}_{L_{0}L_{0}}$ in $W^{(1)}$, which will require a special treatment.
The exact expression in front of $F^{(2,2)}_{L_{0}L_{0}}$ in (4.10) is
irrelevant, the only important remark to be made is that it involves the
$3\theta$ frequency, which is forbidden at the $\lambda$ level.

######  Lemma 4.3 (Expression of $W^{(\geq 2)}$).

We have

$\displaystyle W^{(\geq 2)}_{\alpha\beta}$
$\displaystyle=\tilde{\Box}_{g}\mathfrak{h}_{\alpha\beta}+\sin\left(\frac{u_{0}}{\lambda}\right)\tilde{\Box}_{g}\mathfrak{F}_{\alpha\beta}+\lambda\tilde{\Box}_{g}g^{(3)}_{\alpha\beta}+\tilde{W}^{(\geq
2)}_{\alpha\beta}$ (4.14)

where $\tilde{W}^{(\geq 2)}_{\alpha\beta}$ contains only lower order terms and
is given in (4.15).

###### Proof.

Since the $\lambda^{2}$ terms in the Ricci tensor will be absorbed by the wave
equation for $\mathfrak{h}$, there is no need for a decomposition in
frequencies and (4.14) simply follows from considering first all the second
derivatives of $\mathfrak{F}$, $\mathfrak{h}$ and $g^{(3)}$ and then putting
all the lower order quantities into $\tilde{W}^{(\geq 2)}_{\alpha\beta}$. This
gives the following schematic expression:

$\displaystyle\tilde{W}^{(\geq 2)}$ $\displaystyle=(g^{-1})^{(\geq
3)}(\partial u_{0})^{2}F^{(1)}+(g^{-1})^{(\geq
2)}\left(\partial^{2}g_{0}+\partial^{2}u_{0}+\partial u_{0}\partial
F^{(1)}+(\partial u_{0})^{2}F^{(2,i)}\right)$ (4.15)
$\displaystyle\quad+(g^{-1})^{(\geq
1)}\left(\partial^{2}F^{(1)}+\partial^{2}u_{0}F^{(2,i)}+\partial u_{0}\partial
F^{(2,i)}\right)+g^{-1}\partial^{2}F^{(2,i)}.$

∎

As explained in Section 3.1.1, the forbidden frequency $3\theta$ at the
$\lambda^{1}$ level comes from the eikonal term in (3.5). Moreover, it is
caused by the ”top frequency term”, i.e $F^{(2,2)}_{L_{0}L_{0}}$. See the end
of Section 5.4.2 to see how we deal with these quasi-linear forbidden
frequencies.

####  4.2.2 The gauge part

In Section 3.1.2, we explained how the main part in the gauge term $H^{\rho}$
are the polarization tensors $\mathrm{Pol}$ of each wave in (4.1). These
tensors will play an important role in the consistency of our ansatz by
absorbing forbidden frequencies. The gauge term is also responsible for a
potential loss of hyperbolicity. Indeed, the two non-oscillating terms in
(4.1), i.e $g_{0}$ and $\mathfrak{h}$ will solve wave equations, but the gauge
term $H^{\rho}$ contains first order derivatives of these tensors which leads
to the loss of hyperbolicity announced (since the Ricci tensor contains
derivatives of $H^{\rho}$). The generalised wave gauge choice is precisely
made to cancel these problematic terms. We already assume the usual wave gauge
condition for the background in (2.5), and for $\mathfrak{h}_{\lambda}$ the
problematic terms are given by

$\displaystyle\Upsilon^{\rho}$
$\displaystyle=g^{\mu\nu}g^{\rho\sigma}\left(\partial_{\mu}\mathfrak{h}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}\mathfrak{h}_{\mu\nu}+\sin\left(\frac{u_{0}}{\lambda}\right)\left(\partial_{\mu}\mathfrak{F}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}\mathfrak{F}_{\mu\nu}\right)+\lambda\left(\partial_{\mu}g^{(3)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}g^{(3)}_{\mu\nu}\right)\right)$
(4.16) $\displaystyle\quad-
g_{0}^{\rho\sigma}\mathfrak{h}^{\mu\nu}\left(\partial_{\mu}(g_{0})_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}(g_{0})_{\mu\nu}-\sin\left(\frac{u_{0}}{\lambda}\right)\left(\partial_{\mu}u_{0}F^{(1)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}u_{0}F^{(1)}_{\mu\nu}\right)\right)$

Concretely, $\Upsilon^{\rho}$ contains all derivatives of $\mathfrak{h}$ and
$\mathfrak{F}$. Though it doesn’t satisfy a wave equation, we also put
$\partial\mathfrak{F}$ in (4.16), its presence will be justified by the
analysis of the hierarchy of equations, which will show that
$\partial\mathfrak{F}$ are at the level of $\partial\mathfrak{h}$. Note that
in the first line, $\partial g^{(3)}$ is the non-frequential derivative of
$g^{(3)}$, i.e we don’t differentiate the oscillating parts of $g^{(3)}$.
These $\partial g^{(3)}$ terms are here since they involve
$\partial\mathfrak{F}$. The presence of the last line in (4.16) is linked to
$g^{(3)}$ and will be explained at the end of this section.

We now introduce the formal high-frequency expansion of $H^{\rho}$:

$\displaystyle H^{\rho}$
$\displaystyle=\lambda\left((H^{(1)})^{\rho}+\cos\left(\frac{u_{0}}{\lambda}\right)g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(\mathfrak{F}\right)\right)+\lambda^{2}\left((H^{(2)})^{\rho}+\Upsilon^{\rho}\right)+\lambda^{3}(H^{(\geq
3)})^{\rho}.$ (4.17)

Note that there is no $\lambda^{0}$ term thanks to (2.5) and
$\mathrm{Pol}\left(F^{(1)}\right)=0$. For clarity, we denote by $\mathring{H}$
the part of $H$ containing only allowed terms from the wave equation
perspective, that is

$\displaystyle\mathring{H}^{\rho}=H^{\rho}-\lambda^{2}\Upsilon^{\rho}.$ (4.18)

######  Lemma 4.4 (Expression of $H^{(1)}$).

We have

$\displaystyle(H^{(1)})^{\rho}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\left(g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(F^{(2,1)}\right)+(\tilde{H}^{(1,1)})^{\rho}\right)$
(4.19)
$\displaystyle\quad+\sin\left(\frac{2u_{0}}{\lambda}\right)\left(-2g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(F^{(2,2)}\right)+(\tilde{H}^{(1,2)})^{\rho}\right)$

where $\tilde{H}^{(1,1)}$ and $\tilde{H}^{(1,2)}$ are given in (4.20) and
(4.21).

###### Proof.

We decompose $(H^{(1)})^{\rho}$ into frequencies and highlight its dependence
on $g^{(2)}$. This gives the expression (4.19). We only need a formal
expression of $\tilde{H}^{(1,1)}$:

$\displaystyle\tilde{H}^{(1,1)}=g_{0}^{-2}\left(\partial
F^{(1)}+g_{0}^{-1}F^{(1)}\partial g_{0}\right).$ (4.20)

However, we need the precise expression of $\tilde{H}^{(1,2)}$, wich comes
from $g_{0}^{-1}(g^{-1})^{(1)}\partial u_{0}\partial_{\theta}g^{(1)}$ and is
given by

$\displaystyle(\tilde{H}^{(1,2)})^{\rho}$
$\displaystyle=-\frac{1}{4}g_{0}^{\rho\sigma}\partial_{\sigma}u_{0}\left|F^{(1)}\right|^{2}_{g_{0}}.$
(4.21)

∎

######  Remark 4.5.

Note that $\mathfrak{F}$ doesn’t appear in $H^{(1)}$ (even though it plays a
similar role as $F^{(2,i)}$) since its polarization contribution is already in
(4.17). As Lemma 8.1 will show, this contribution actually vanishes.

######  Lemma 4.5 (Expression of $H^{(2)}$).

We have

$\displaystyle(H^{(2)})^{\rho}$
$\displaystyle=g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(\partial_{\theta}g^{(3)}\right)+(\tilde{H}^{(2)})^{\rho}$
(4.22)

where $\tilde{H}^{(2)}$ is given in (4.23).

###### Proof.

Since $H^{(2)}$ will be absorbed at the $\lambda^{1}$ level by $g^{(3)}$, we
don’t need to specify its decomposition in frequency. The expression (4.22)
only means that we highlight the dependency of $H^{(2)}$ on $g^{(3)}$, meaning
that the quantity $(\tilde{H}^{(2)})^{\rho}$ is defined so that (4.22) holds,
and does not depends on $g^{(3)}$. Moreover, thanks to our choice of
$\Upsilon^{\rho}$ (see (4.16)), $(\tilde{H}^{(2)})^{\rho}$ also does not
depends on $\mathfrak{h}$. Indeed, $(\tilde{H}^{(2)})^{\rho}$ could depend on
$\mathfrak{h}$ through two types of terms.

  * •

The $\partial\mathfrak{h}$ appearing in $g_{0}^{-1}(\partial g)^{(2)}$, which
corresponds to the first line in (4.16). The latter also ensures that
$(\tilde{H}^{(2)})^{\rho}$ does not depend on $\partial\mathfrak{F}$.

  * •

The $\mathfrak{h}$ appearing in $(g^{-1})^{(2)}(\partial g)^{(0)}$, which
corresponds to the last line in (4.16).

The term $(\tilde{H}^{(2)})^{\rho}$ does however depend in a polynomial
fashion on the quantities $g_{0}$, $F^{(1)}$ and $F^{(2,i)}$ and their first
derivatives. Therefore, we can write schematically

$\displaystyle\tilde{H}^{(2)}$
$\displaystyle=g_{0}^{-3}\left((F^{(1)})^{2}+F^{(2,i)}+\mathfrak{F}\right)(\partial
g_{0}+\partial u_{0}F^{(1)})$ (4.23)
$\displaystyle\quad+g_{0}^{-3}F^{(1)}(\partial F^{(1)}+\partial
u_{0}F^{(2,i)}+\partial u_{0}\mathfrak{F})+g_{0}^{-2}\partial F^{(2,i)}$

where we used a schematic version of (4.4), that is

$\displaystyle(g^{-1})^{(2)}=g_{0}^{-2}\left((F^{(1)})^{2}+F^{(2,i)}+\mathfrak{h}+\mathfrak{F}\right).$

∎

######  Lemma 4.6 (Expression of $H^{(\geq 3)}$).

We have

$\displaystyle H^{(\geq 3)}$ $\displaystyle=(g^{-2})^{(\geq 3)}\left(\partial
g_{0}+\partial u_{0}F^{(1)}\right)$ (4.24) $\displaystyle\quad+(g^{-2})^{(\geq
2)}\left(\partial F^{(1)}+\partial u_{0}F^{(2,i)}+\partial
u_{0}\mathfrak{F}\right)+(g^{-2})^{(\geq 1)}\partial F^{(2,i)}.$

###### Proof.

Since $H^{(\geq 3)}$ will only appear at the RHS of the wave equation for the
remainder $\mathfrak{h}$, its only important property is its independancy from
$\partial\mathfrak{h}$ and $\partial\mathfrak{F}$. This holds thanks to
(4.16), where in particular we put the $\partial g^{(3)}$ in $\Upsilon^{\rho}$
since $g^{(3)}$ contains linear terms in $\mathfrak{F}$ (see Section 5.5
below). It does depend on $g_{0}$, $F^{(1)}$, $F^{(2,i)}$, and their first
derivatives, and on $\mathfrak{F}$, $\mathfrak{h}$ and $g^{(3)}$. The terms
depending on $g^{(3)}$ in $H^{(\geq 3)}$ are still $O\left(1\right)$ with
respect to $\lambda$, since we lose one $\lambda$ power when the derivatives
hit the oscillating parts of $g^{(3)}$, producing the
$\partial_{\theta}g^{(3)}$ already included in $H^{(2)}$. This justifies the
schematic expression (4.24). ∎

The gauge term $H^{\rho}$ is differenciated in the Ricci tensor, and we now
give the first two orders of the high-frequency expansion of these terms. We
have

$\displaystyle\left(H^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}H^{\rho}\right)^{(0)}=(g_{0})_{\rho(\alpha}\partial_{\beta)}u_{0}\partial_{\theta}(H^{(1)})^{\rho}-\sin\left(\frac{u_{0}}{\lambda}\right)\partial_{(\alpha}u_{0}\mathrm{Pol}_{\beta)}(\mathfrak{F}),$
(4.25)
$\displaystyle\left(H^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}H^{\rho}\right)^{(1)}$
$\displaystyle\quad=\left((H^{(1)})^{\rho}+\cos\left(\frac{u_{0}}{\lambda}\right)g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(\mathfrak{F}\right)\right)\left(\partial_{\rho}(g_{0})_{\alpha\beta}-\sin\left(\frac{u_{0}}{\lambda}\right)\partial_{\rho}u_{0}F^{(1)}_{\alpha\beta}\right)$
(4.26)
$\displaystyle\quad\quad+(g_{0})_{\rho(\alpha}\left(\partial_{\beta)}(H^{(1)})^{\rho}+\cos\left(\frac{u_{0}}{\lambda}\right)\partial_{\beta)}\left(g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(\mathfrak{F}\right)\right)\right)+(g_{0})_{\rho(\alpha}\partial_{\beta)}u_{0}\partial_{\theta}\left((H^{(2)})^{\rho}+\Upsilon^{\rho}\right)$
$\displaystyle\quad\quad+\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}_{\rho(\alpha}\partial_{\beta)}u_{0}\left(\partial_{\theta}(H^{(1)})^{\rho}-\sin\left(\frac{u_{0}}{\lambda}\right)g_{0}^{\rho\sigma}\mathrm{Pol}_{\sigma}\left(\mathfrak{F}\right)\right),$

where by $\partial_{\beta)}(H^{(1)})^{\rho}$ in (4.26) we denote the
derivative of $(H^{(1)})^{\rho}$ without differenciating its oscillating parts
(i.e the $\cos\left(\frac{u_{0}}{\lambda}\right)$ and
$\sin\left(\frac{2u_{0}}{\lambda}\right)$ in (4.19)).

####  4.2.3 The quadratic non-linearity

In this section we expand $P_{\alpha\beta}(g)(\partial g,\partial g)$ (see
(3.2) for an exact expression). Formally we have

$P_{\alpha\beta}(g)(\partial g,\partial g)=P^{(0)}_{\alpha\beta}+\lambda
P^{(1)}_{\alpha\beta}+\lambda^{2}P^{(\geq 2)}_{\alpha\beta}.$ (4.27)

The main structural remarks on $P_{\alpha\beta}$ were already made in Section
3.1.3, where we defined the weak polarized null condition. We recall the main
consequences of this structure: the self-interaction of a wave $T$ produces
fully non-tangential terms and $\mathrm{Pol}(T)$ terms (see (3.15)).

######  Lemma 4.7 (Expression of $P^{(0)}$).

We have

$\displaystyle P^{(0)}_{\alpha\beta}$
$\displaystyle=P^{(0)}_{\alpha\beta}\left[g^{(1)}\right]+P^{(0)}_{\alpha\beta}\left[g_{0}\right]$
(4.28)

where

$\displaystyle P^{(0)}_{\alpha\beta}\left[g_{0}\right]$
$\displaystyle=P_{\alpha\beta}(g_{0})(\partial g_{0},\partial g_{0}),$ (4.29)
$\displaystyle P^{(0)}_{\alpha\beta}\left[g^{(1)}\right]$
$\displaystyle=\frac{1}{2}\mathcal{E}\left(F^{(1)},F^{(1)}\right)\partial_{\alpha}u_{0}\partial_{\beta}u_{0}+\cos\left(\frac{2u_{0}}{\lambda}\right)\partial_{(\alpha}u_{0}\hat{P}_{\beta)}^{(0,2)}\left[g^{(1)}\right]$
(4.30)
$\displaystyle\quad-\sin\left(\frac{u_{0}}{\lambda}\right)\left(-2L_{0}^{\rho}\Gamma(g_{0})^{\nu}_{(\alpha\rho}F^{(1)}_{\beta)\nu}+\partial_{(\alpha}u_{0}\hat{P}^{(0,1)}_{\beta)}\left[g^{(1)}\right]\right),$

and where $\hat{P}_{\beta}^{(0,2)}\left[g^{(1)}\right]$ and
$\hat{P}^{(0,1)}_{\beta}\left[g^{(1)}\right]$ are given in (4.31) and (4.32).

###### Proof.

The $\lambda^{0}$ terms in $P_{\alpha\beta}(g)(\partial g,\partial g)$ are of
the form $g_{0}^{-2}(\partial g)^{(0)}(\partial g)^{(0)}$, with $(\partial
g)^{(0)}$ being itself of the form $\partial g_{0}+\partial
u_{0}\partial_{\theta}g^{(1)}$. We decompose first $P^{(0)}$ in terms
depending on $g^{(1)}$ which gives (4.28).The background dependent terms are
simply given by $g_{0}^{-1}g_{0}^{-1}\partial g_{0}\partial g_{0}$, that is
the quadratic non-linearity appearing on the RHS of (2.6), hence (4.29). The
terms depending on $g^{(1)}$ come from the self-interaction of $g^{(1)}$ and
from the interaction between $g^{(1)}$ and the background. Since
$\mathrm{Pol}\left(F^{(1)}\right)=0$ we obtain (4.30) where

$\hat{P}^{(0,1)}_{\beta}\left[g^{(1)}\right]=g_{0}^{-2}F^{(1)}\partial g_{0}.$
(4.31)

and

$\displaystyle\hat{P}_{\beta}^{(0,2)}\left[g^{(1)}\right]$
$\displaystyle=-\frac{1}{4}\mathcal{E}\left(F^{(1)},F^{(1)}\right)\partial_{\beta}u_{0}.$
(4.32)

∎

######  Remark 4.6.

The forbidden frequency in (4.28) has the non-tangential structure, allowing
the polarization tensors of $g^{(2)}$ to absorb it.

######  Lemma 4.8 (Expression of $P^{(1)}$).

We have

$\displaystyle P^{(1)}_{\alpha\beta}$
$\displaystyle=P^{(1)}_{\alpha\beta}\left[g^{(2)}\right]+P^{(1)}_{\alpha\beta}\left[g^{(\leq
1)}\right]$ (4.33)

where

$\displaystyle P^{(1)}_{\alpha\beta}\left[g^{(\leq 1)}\right]$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)P^{(1,1)}_{\alpha\beta}+\sin\left(\frac{2u_{0}}{\lambda}\right)P^{(1,2)}_{\alpha\beta}+\cos\left(\frac{3u_{0}}{\lambda}\right)P^{(1,3)}_{\alpha\beta}$
(4.34) $\displaystyle P^{(1)}_{\alpha\beta}\left[g^{(2)}\right]$
$\displaystyle=-2L_{0}^{\rho}\Gamma(g_{0})^{\nu}_{(\alpha\rho}\left(\cos\left(\frac{u_{0}}{\lambda}\right)\mathfrak{F}_{\nu\beta)}+\cos\left(\frac{u_{0}}{\lambda}\right)F^{(2,1)}_{\nu\beta)}-2\sin\left(\frac{2u_{0}}{\lambda}\right)F^{(2,2)}_{\nu\beta)}\right)$
(4.35)
$\displaystyle\quad+\partial_{(\alpha}u_{0}\hat{P}^{(1)}_{\beta)}\left[g^{(2)}\right]$

and where $P^{(1,1)}_{\alpha\beta}$, $P^{(1,2)}_{\alpha\beta}$ and
$P^{(1,3)}_{\alpha\beta}$ are given in (4.37) and (4.38), and
$\hat{P}^{(1)}_{\beta}\left[g^{(2)}\right]$ is given in (4.40) (see also
(4.41)).

###### Proof.

The $\lambda$ terms in $P_{\alpha\beta}(g)(\partial g,\partial g)$ are of the
form

$\displaystyle g_{0}^{-1}(g^{-1})^{(1)}\left((\partial g_{0})^{2}+\partial
g_{0}\partial u_{0}\partial_{\theta}g^{(1)}+\left(\partial
u_{0}\partial_{\theta}g^{(1)}\right)^{2}\right)$ (4.36)
$\displaystyle\quad+g_{0}^{-2}\partial g_{0}\left(\partial g^{(1)}+\partial
u_{0}\partial_{\theta}g^{(2)}\right)+g_{0}^{-2}\partial
u_{0}\partial_{\theta}g^{(1)}\left(\partial g^{(1)}+\partial
u_{0}\partial_{\theta}g^{(2)}\right).$

We decompose $P^{(1)}$ into a part depending on $g^{(2)}$ and a part depending
only on the background and on $g^{(1)}$, hence (4.33). Next, we can decompose
the two terms in (4.33) in frequencies following (4.36). For
$P^{(1)}_{\alpha\beta}\left[g^{(\leq 1)}\right]$, this gives (4.34). Since the
frequencies $\theta$ and $2\theta$ are admissible at the $\lambda$ level, a
formal expression of $P^{(1,1)}$ and $P^{(1,2)}$ is enough:

$\displaystyle P^{(1,k)}$
$\displaystyle=\left(g_{0}^{-1}(g^{-1})^{(1)}\partial g_{0}+g_{0}^{-2}\partial
g^{(1)}\right)\left(\partial g_{0}+\partial
u_{0}\partial_{\theta}g^{(1)}\right)$ (4.37)

for $k\in\llbracket 1,2\rrbracket$. The frequency $3\theta$ is forbidden at
the $\lambda$ level, therefore we need the exact expression of $P^{(1,3)}$. It
comes from the term $g_{0}^{-1}(g^{-1})^{(1)}\partial
u_{0}\partial_{\theta}g^{(1)}\partial u_{0}\partial_{\theta}g^{(1)}$ and is
given by

$\displaystyle P^{(1,3)}_{\alpha\beta}$
$\displaystyle=\partial_{(\alpha}u_{0}\hat{P}_{\beta)}^{(1,3)}$ (4.38)

with

$\displaystyle\hat{P}_{\beta}^{(1,3)}$
$\displaystyle=-\frac{1}{8}\partial_{\beta}u_{0}g_{0}^{\mu\rho}(F^{(1)})^{\nu\sigma}F^{(1)}_{\rho\sigma}F^{(1)}_{\mu\nu}.$
(4.39)

Let us now look at the part of $P^{(1)}$ depending on $g^{(2)}$, denoted in
(4.33) by $P^{(1)}_{\alpha\beta}\left[g^{(2)}\right]$. We first decompose it
into terms coming from the interaction between $g^{(2)}$ and the background,
i.e terms of the form $g_{0}^{-2}\partial g_{0}\partial
u_{0}\partial_{\theta}g^{(2)}$, and terms coming from the interaction between
$g^{(2)}$ and $g^{(1)}$, i.e terms of the form $g_{0}^{-2}(\partial
u_{0})^{2}\partial_{\theta}g^{(1)}\partial_{\theta}g^{(2)}$. This gives (4.35)
with

$\displaystyle\hat{P}^{(1)}_{\beta}\left[g^{(2)}\right]$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\hat{P}^{(1,1)}_{\beta}\left[g^{(2)}\right]+\sin\left(\frac{2u_{0}}{\lambda}\right)\hat{P}^{(1,2)}_{\beta}\left[g^{(2)}\right]+\cos\left(\frac{3u_{0}}{\lambda}\right)\hat{P}^{(1,3)}_{\beta}\left[g^{(2)}\right].$
(4.40)

We don’t need a precise expression for
$\hat{P}^{(1,i)}_{\beta}\left[g^{(2)}\right]$ for $i=1,2,3$ and only write
down the following schematic expression

$\displaystyle\hat{P}^{(1,i)}_{\beta}\left[g^{(2)}\right]$
$\displaystyle=g_{0}^{-2}\left(\partial g_{0}+\partial
u_{0}F^{(1)}\right)\left(\mathfrak{F}+F^{(2,k)}\right).$ (4.41)

∎

Since $P^{(\geq 2)}$ will be absorbed by the wave equation for the remainder
$\mathfrak{h}$, we don’t need to know its frequency decomposition. As all the
terms coming from $P_{\alpha\beta}(g)(\partial g,\partial g)$ it is of the
form $g^{-2}\partial g\partial g$.

##  5 The hierarchy of equations

In this section, we present the equations that the different terms in (4.1)
solve. We divide them into two groups of equations, the background system
presented in Section 5.2 and the reduced system presented in Section 5.3. Note
that $g^{(3)}$ is defined in Section 5.5 but doesn’t solve a differential
equation. The background and reduced system are choosen so that (4.1) satisfy
the Einstein vacuum equations, this will be proved in Section 8.

Before we present the equations, we define the following transport operator:

$\displaystyle\mathscr{L}_{0}\vcentcolon=-2\mathbf{D}_{L_{0}}+\Box_{g_{0}}u_{0}$
(5.1)

where we recall that $\mathbf{D}$ is the covariant derivative associated to
$g_{0}$. It acts on tensor fields of all type, including scalar functions.

###  5.1 A guide to the hierarchy

The exact expression of the equations satisfied by the different terms in
(4.1) are quite complicated and in this section we break down the role these
equations play in cancelling the different parts of the Ricci tensor. First
note that the assumptions that $\mathrm{Pol}\left(F^{(1)}\right)=0$ (which has
been made in Section 4.2) implies that there is no $\lambda^{-1}$ terms in the
Ricci tensor of $g$. Therefore, we have the following decomposition

$\displaystyle R_{\alpha\beta}(g)=R^{(0)}_{\alpha\beta}+\lambda
R^{(1)}_{\alpha\beta}+\lambda^{2}R^{(\geq 2)}_{\alpha\beta}.x$

The $\lambda^{0}$ terms in $R_{\alpha\beta}(g)$ are given by

$\displaystyle
2R_{\alpha\beta}^{(0)}=-W^{(0)}_{\alpha\beta}+P^{(0)}_{\alpha\beta}+\left(H^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}H^{\rho}\right)^{(0)},$
(5.2)

where these different terms are given by Lemmas 4.1, 4.7 and (4.25). In order
to obtain $R^{(0)}_{\alpha\beta}=0$, we proceed as follows:

  * •

the non-oscillating terms in (5.2) are absorbed by the background equation
(2.6) for $g_{0}$,

  * •

the terms in $\sin\left(\frac{u_{0}}{\lambda}\right)$ (which corresponds to an
admissible frequency) are absorbed by a transport equation for $F^{(1)}$ (see
(5.4) below), except for the terms with the non-tangential structure which are
absorbed by the polarization tensor of $F^{(2,1)}$,

  * •

the terms in $\cos\left(\frac{2u_{0}}{\lambda}\right)$ (which corresponds to a
forbidden frequency and all have the non-tangential structure thanks to the
weak polarized null condition) are absorbed by the polarization tensor of
$F^{(2,2)}$.

The polarization conditions that $F^{(2,1)}$ and $F^{(2,2)}$ needs to solve
are precisely given in Section 5.4.2 below.

The $\lambda^{1}$ terms in $R_{\alpha\beta}(g)$ are given by

$\displaystyle 2R^{(1)}_{\alpha\beta}$
$\displaystyle=-W^{(1)}_{\alpha\beta}+P^{(1)}_{\alpha\beta}+\left(H^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}H^{\rho}\right)^{(1)},$
(5.3)

where these different terms are given by Lemmas 4.2, 4.8 and (4.26). In order
to obtain $R^{(1)}_{\alpha\beta}=0$, we proceed as follows. First we look at
the terms independent from $\mathfrak{h}$ or $\mathfrak{F}$:

  * •

the terms in $\cos\left(\frac{u_{0}}{\lambda}\right)$ and
$\sin\left(\frac{2u_{0}}{\lambda}\right)$ (which corresponds to admissible
frequencies) are absorbed by transport equations for $F^{(2,1)}$ and
$F^{(2,2)}$ (see (5.5) and (5.6) below), except for the terms with a non-
tangential structure which are absorbed by the polarization tensor of
$g^{(3)}$,

  * •

the terms in $\cos\left(\frac{3u_{0}}{\lambda}\right)$ (which corresponds to a
forbidden frequency) with the non-tangential structure are absorbed by the
polarization tensor of $g^{(3)}$,

  * •

the term $\cos\left(\frac{3u_{0}}{\lambda}\right)$ without the non-tangential
structure (see (4.10)) rewrites itself as a polarization term (see (5.17))
which eventually vanish at the end of the proof.

The polarization condition that $g^{(3)}$ needs to solve is precisely given in
Section 5.5. The main terms depending on $\mathfrak{h}$ and $\mathfrak{F}$ are
all multiplied by $\cos\left(\frac{u_{0}}{\lambda}\right)$ and are absorbed by
the transport equation for $\mathfrak{F}$ (see (5.7) below).

In order to solve the Einstein vacuum equations, it remains to obtain
$R^{(\geq 2)}_{\alpha\beta}=0$. This is done by putting all the remaining
terms on the RHS of a wave equation for $\mathfrak{h}$ (see (5.8) below)
except for the $\Upsilon^{\rho}$ terms coming from the $\lambda^{2}$ terms in
(4.17).

######  Remark 5.1.

Following this procedure, we don’t actually solve the full Einstein vacuum
equations. It will remain some polarization conditions and generalised wave
gauge terms which are proved to vanish in Section 8.

###  5.2 The background system

We present now the system solved by the background perturbations, called the
background system. The equation for $F^{(1)}$ is

$\displaystyle\mathscr{L}_{0}F^{(1)}_{\alpha\beta}$ $\displaystyle=0.$ (5.4)

The equations for $g^{(2)}$ are

$\displaystyle\mathscr{L}_{0}F^{(2,1)}_{\alpha\beta}$
$\displaystyle=-\tilde{W}^{(1,1)}_{\alpha\beta}+P^{(1,1)}_{\alpha\beta}-\mathbf{D}_{(\alpha}\hat{P}^{(0,1)}_{\beta)}\left[g^{(1)}\right],$
(5.5) $\displaystyle\mathscr{L}_{0}F^{(2,2)}_{\alpha\beta}$
$\displaystyle=-\tilde{W}^{(1,2)}_{\alpha\beta}-\frac{1}{2}P^{(1,2)}_{\alpha\beta}+\frac{1}{4}\left(F^{(2,1)}_{L_{0}L_{0}}+L_{0}^{\sigma}\hat{P}^{(0,1)}_{\sigma}\left[g^{(1)}\right]\right)F^{(1)}_{\alpha\beta}+\frac{1}{4}\mathbf{D}_{(\alpha}\hat{P}^{(0,2)}_{\beta)}\left[g^{(1)}\right]$
(5.6)

The background system is a coupled transport-wave system which admits a
triangular structure. It will be solved in Section 6 and all the estimates and
support properties of the background perturbations are given in Theorem 6.1.
The purpose of the equations (5.4)-(5.6) is given in Section 5.1. Their exact
expression follows from the computations performed in Section 4.2.

######  Remark 5.2.

Note that the terms

$\displaystyle-\mathbf{D}_{(\alpha}\hat{P}^{(0,1)}_{\beta)}\left[g^{(1)}\right]$

and

$\displaystyle\frac{1}{4}L_{0}^{\sigma}\hat{P}^{(0,1)}_{\sigma}\left[g^{(1)}\right]F^{(1)}_{\alpha\beta}+\frac{1}{4}\mathbf{D}_{(\alpha}\hat{P}^{(0,2)}_{\beta)}\left[g^{(1)}\right]$

in (5.5) and (5.6) are obtained from the gauge terms (4.26) after having
replaced the polarization tensors $\mathrm{Pol}\left(F^{(2,i)}\right)$ coming
from $H^{(1)}$ by the non-tangential terms they absorbed in
$R^{(0)}_{\alpha\beta}$.

Since (5.4), (5.5) and (5.6) are first order transport equation, the initial
data for the background system consist in

$\displaystyle
F^{(1)}_{\alpha\beta}\restriction{\Sigma_{0}}\quad\text{and}\quad
F^{(2,i)}_{\alpha\beta}\restriction{\Sigma_{0}}.$

They are given in Section 5.6.1.

###  5.3 The reduced system

We present now the system solved by the tensors $\mathfrak{F}$ and
$\mathfrak{h}$, called the reduced system. As explained in Section 3.2.3, we
introduce the Fourier cut-offs $\Pi_{\leq}$ and $\Pi_{\geq}$:

######  Definition 5.1.

Let $\chi_{1}:\mathbb{R}^{3}\longrightarrow[0,1]$ be a smooth function
supported in

$\left\\{\xi\in\mathbb{R}^{3}~{}\middle|~{}|\xi|\leq 2\right\\}$

and such that $\chi_{1}(\xi)=1$ if $|\xi|\leq 1$. For $\lambda>0$ we define
$\chi_{\lambda}(\xi)=\chi_{1}(\lambda\xi)$, thus $\chi_{\lambda}$ is supported
in
$\left\\{\xi\in\mathbb{R}^{3}~{}\middle|~{}|\xi|\leq\frac{2}{\lambda}\right\\}$
and $\chi_{\lambda}(\xi)=1$ if $|\xi|\leq\frac{1}{\lambda}$. We define
$\Pi_{\leq}$ by

$\displaystyle\Pi_{\leq}(f)=\mathcal{F}^{-1}\left(\chi_{\lambda}\mathcal{F}(f)\right)$

for $f\in L^{2}$. We also define

$\displaystyle\Pi_{\geq}$ $\displaystyle=\mathrm{Id}-\Pi_{\leq}.$

The equations for $\mathfrak{F}$ and $\mathfrak{h}$ are then

$\displaystyle\mathscr{L}_{0}\mathfrak{F}_{\alpha\beta}$
$\displaystyle=-\Pi_{\leq}\left(\mathfrak{h}_{L_{0}L_{0}}\right)F^{(1)}_{\alpha\beta},$
(5.7) $\displaystyle\tilde{\Box}_{g}\mathfrak{h}_{\alpha\beta}$
$\displaystyle=-\sin\left(\frac{u_{0}}{\lambda}\right)\tilde{\Box}_{g}\mathfrak{F}_{\alpha\beta}-\lambda\tilde{\Box}_{g}g^{(3)}_{\alpha\beta}-\frac{1}{\lambda}\cos\left(\frac{u_{0}}{\lambda}\right)\Pi_{\geq}\left(\mathfrak{h}_{L_{0}L_{0}}\right)F^{(1)}_{\alpha\beta}+\mathcal{R}(g)$
(5.8)

where we set

$\displaystyle\mathcal{R}(g)$ $\displaystyle=-\tilde{W}^{(\geq
2)}_{\alpha\beta}+\left(\mathring{H}^{\rho}\partial_{\rho}g_{\alpha\beta}+g_{\rho(\alpha}\partial_{\beta)}\mathring{H}^{\rho}\right)^{(\geq
2)}+P^{(\geq 2)}_{\alpha\beta}.$ (5.9)

with $\mathring{H}$ defined in (4.18).

Equations (5.7)-(5.8) form a coupled transport-wave system and its purpose is
to solve the tangential part of the remaining orders of the Ricci tensor of
$g$, i.e some part of its order 1 and the orders superior or equal to 2, as
explained in Section 5.1. The spectral projectors $\Pi_{\leq}$ and
$\Pi_{\geq}$ are used to avoid the loss of derivatives described in the
introduction, see Section 3.2.3.

Since (5.7) is a first order transport equation and (5.8) is a second order
wave equation the initial data for the reduced system consist in

$\displaystyle\mathfrak{F}_{\alpha\beta}\restriction{\Sigma_{0}},\quad\mathfrak{h}_{\alpha\beta}\restriction{\Sigma_{0}}\quad\text{and}\quad\partial_{t}\mathfrak{h}_{\alpha\beta}\restriction{\Sigma_{0}}.$

They are given in Section 5.6.2.

###  5.4 Generalised wave gauge and polarization conditions

As it is usual when solving the Einstein equations, we impose gauge
conditions. Since we solve a wave equation for $\mathfrak{h}$ we need a
generalised wave gauge. Moreover, as the study of first order high-frequency
waves by Choquet-Bruhat shows, the oscillating parts of the ansatz need to
satisfy polarization conditions, i.e their polarization tensor (see (3.8) for
its definition) is prescribed.

####  5.4.1 Generalised wave gauge condition

The tensor $\mathfrak{h}$ solves the wave equation (5.8) and we associate to
it a generalised wave gauge condition. In Section 4.2.2 we already defined
$\Upsilon^{\rho}$, see (4.16).

######  Definition 5.2.

The generalised wave gauge considered in Theorem 2.2 is defined by

$\displaystyle\Upsilon^{\rho}=0.$ (5.10)

######  Remark 5.3.

For later use, we define $\tilde{\Upsilon}^{\rho}$ as the terms in
$\Upsilon^{\rho}$ independent from $\partial\mathfrak{h}$:

$\displaystyle\tilde{\Upsilon}^{\rho}$
$\displaystyle=g^{\mu\nu}g^{\rho\sigma}\left(\sin\left(\frac{u_{0}}{\lambda}\right)\left(\partial_{\mu}\mathfrak{F}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}\mathfrak{F}_{\mu\nu}\right)+\lambda\left(\partial_{\mu}g^{(3)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}g^{(3)}_{\mu\nu}\right)\right)$
(5.11) $\displaystyle\quad-
g_{0}^{\rho\sigma}\mathfrak{h}^{\mu\nu}\left(\partial_{\mu}(g_{0})_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}(g_{0})_{\mu\nu}-\sin\left(\frac{u_{0}}{\lambda}\right)\left(\partial_{\mu}u_{0}F^{(1)}_{\sigma\nu}-\frac{1}{2}\partial_{\sigma}u_{0}F^{(1)}_{\mu\nu}\right)\right).$

####  5.4.2 Polarization conditions

The polarization conditions prescribe the polarization tensor of $F^{(2,i)}$
in terms of $F^{(1)}$ and the background metric.

######  Definition 5.3.

We define the following tensors:

$\displaystyle V^{(2,1)}_{\beta}$
$\displaystyle=\mathrm{Pol}_{\beta}\left(F^{(2,1)}\right)+(g_{0})_{\rho\beta}(\tilde{H}^{(1,1)})^{\rho}+\hat{P}^{(0,1)}_{\beta}\left[g^{(1)}\right],$
(5.12) $\displaystyle V^{(2,2)}_{\beta}$
$\displaystyle=\mathrm{Pol}_{\beta}\left(F^{(2,2)}\right)-\frac{1}{2}(g_{0})_{\rho\beta}(\tilde{H}^{(1,2)})^{\rho}-\frac{1}{4}\hat{P}_{\beta}^{(0,2)}\left[g^{(1)}\right].$
(5.13)

The polarization conditions are

$\displaystyle V^{(2,j)}_{\beta}=0$ (5.14)

together with

$\displaystyle\mathrm{Pol}_{\beta}\left(F^{(1)}\right)=0.$ (5.15)

We don’t introduce the notation $V^{(1)}$ since the proof of (5.15) and (5.14)
are different. They are both based on the same idea: computing transport
equations satisfied by either $\mathrm{Pol}_{\beta}\left(F^{(1)}\right)$ or
$V^{(2,j)}$ and use the fact that these quantities vanish on $\Sigma_{0}$ (see
Corollary 5.1 below). However, the method used to derive these transport
equations differs, see Section 3.1.4 for more details.

The exact expression of $V^{(2,1)}$ is irrelevant. However, (4.21) and (4.32)
imply

$\displaystyle
L_{0}^{\beta}\left(-\frac{1}{2}(g_{0})_{\rho\beta}(\tilde{H}^{(1,2)})^{\rho}-\frac{1}{4}\hat{P}_{\beta}^{(0,2)}\left[g^{(1)}\right]\right)$
$\displaystyle=0$ (5.16)

which together with (3.9) and (5.13) implies

$\displaystyle F^{(2,2)}_{L_{0}L_{0}}=-V^{(2,2)}_{L_{0}}.$ (5.17)

This shows how we deal with the forbidden frequency coming from the quasi-
linearity in the wave part of the Ricci tensor. The frequency $3\theta$ at the
$\lambda$ level (see (4.10)) is proportional to $F^{(2,2)}_{L_{0}L_{0}}$ which
thanks to (5.17) rewrite as $V$ terms, i.e it vanishes if we are able to prove
that the polarization conditions (5.14) hold.

###  5.5 Definition of $g^{(3)}$

The tensor $g^{(3)}$ is introduced in the ansatz (4.1) in order to absorb the
non-tangential part of the $\lambda^{1}$ level in the Ricci tensor of $g$.
Unlike the other terms in (4.1) it doesn’t solve any differential equation but
only a polarization condition. More precisely we want $g^{(3)}$ to satisfy

$\displaystyle\mathrm{Pol}_{\beta}\left(\partial_{\theta}^{2}g^{(3)}\right)$
$\displaystyle=-(g_{0})_{\rho\beta}\partial_{\theta}(\tilde{H}^{(2)})^{\rho}-\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}_{\rho\beta}\partial_{\theta}(H^{(1)})^{\rho}$
(5.18)
$\displaystyle\quad-\hat{P}^{(1)}_{\beta}\left[g^{(2)}\right]-\cos\left(\frac{3u_{0}}{\lambda}\right)\hat{P}^{(1,3)}_{\beta}$

A necessary and sufficient condition for (5.18) to be satisfied is that its
RHS is purely oscillating, this holds thanks to (4.19) and (4.40). Indeed,
(4.40) implies that $\hat{P}^{(1)}_{\beta}\left[g^{(2)}\right]$ is purely
oscillating while (4.19) implies that the product
$\cos\left(\frac{u_{0}}{\lambda}\right)\partial_{\theta}(H^{(1)})^{\rho}$ is
also purely oscillating.

We are left with the question of solving an equation of the form

$\mathrm{Pol}_{\beta}(T)=\Omega_{\beta}$ (5.19)

with $\Omega_{\beta}$ a 1-form for $T$ a symmetric 2-tensor. According to
(3.9), (3.10) and (3.11) we just need to set

$\displaystyle T_{L_{0}L_{0}}$ $\displaystyle=-\Omega_{L_{0}}$ $\displaystyle
T_{L_{0}A}$ $\displaystyle=-\Omega_{A}$ $\displaystyle T_{11}$
$\displaystyle=-\Omega_{\underline{L}_{0}}$

with all the other coefficients of $T$ being zero. This implies that we can
solve (5.19) for $T$ satisfying $|T|\lesssim|\Omega|$ with an implicit
constant depending only on the background spacetime.

As it is clear from the RHS of (5.18), $g^{(3)}$ depends on the background
quantities $g_{0}$, $F^{(1)}$, $F^{(2,i)}$ and their first derivatives and on
the non-background field $\mathfrak{F}$ through the terms $\tilde{H}^{(2)}$
and $\hat{P}^{(1)}_{\beta}\left[g^{(2)}\right]$ (see (4.23) and (4.41)).
However, it does not depend on $\mathfrak{h}$. We summarize this discussion in
the next lemma.

######  Lemma 5.1.

Given $F^{(1)}$, $F^{(2,i)}$ and $\mathfrak{F}$, there exists a symmetric
2-tensor $g^{(3)}$ solution of (5.18) of the form

$\displaystyle g^{(3)}$
$\displaystyle=\sum_{\mathrm{T}\in\mathcal{A}}\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)\left(g^{(3,\mathrm{T})}(\mathfrak{F})+g^{(3,\mathrm{T},\mathrm{BG})}\right)$

where $\mathcal{A}$ is a finite subset of trigonometric functions (see (3.4)),
$g^{(3,\mathrm{T})}(\mathfrak{F})$ satisfy schematically

$\displaystyle g^{(3,\mathrm{T})}(\mathfrak{F})=g_{0}g_{0}^{-3}\left(\partial
g_{0}+\partial u_{0}F^{(1)}\right)\mathfrak{F}$ (5.20)

and where $g^{(3,\mathrm{T},\mathrm{BG})}$ is a polynomial in the background
quantities $g_{0}$, $F^{(1)}$, $F^{(2,i)}$ and their first derivatives. In the
rest of this article, we will use the notation

$\displaystyle g^{(3)}(\mathfrak{F})$
$\displaystyle=\sum_{\mathrm{T}\in\mathcal{A}}\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)g^{(3,\mathrm{T})}(\mathfrak{F}),$
$\displaystyle g^{(3,\mathrm{BG})}$
$\displaystyle=\sum_{\mathrm{T}\in\mathcal{A}}\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)g^{(3,\mathrm{T},\mathrm{BG})}.$

Note that since $g^{(3)}$ only absorb oscillating terms, it has the same
support as $F^{(1)}$, $F^{(2,i)}$ or $\mathfrak{F}$, which will be proved to
be compact in this article.

###  5.6 Initial data for the spacetime metric

In this section, we define the initial data on $\Sigma_{0}$ for the different
tensors appearing in the high-frequency ansatz (4.1). They are based on the
solution of the constraint equations presented in Theorem 2.1, which is the
main result of [Tou22b].

####  5.6.1 Initial data for the background system

We start with the tensors $F^{(1)}$, $F^{(2,1)}$ and $F^{(2,2)}$. For the
spatial components, we simply set

$\displaystyle F^{(1)}_{ij}=\bar{F}^{(1)}_{ij},\quad
F^{(2,1)}_{ij}=\bar{F}^{(2,1)}_{ij},\quad
F^{(2,2)}_{ij}=\bar{F}^{(2,2)}_{ij}.$ (5.21)

For the time components of $F^{(1)}$, we set

$\displaystyle F^{(1)}_{0\alpha}\restriction{\Sigma_{0}}=0.$ (5.22)

For the time components of $F^{(2,i)}$, we need to ensure that $F^{(2,i)}$
satisfy on $\Sigma_{0}$ the polarization conditions (5.14). For clarity, we
define the 1-forms $\Omega^{(i)}$ for $i=1,2$ so that

$\displaystyle
V^{(2,i)}_{\beta}=\mathrm{Pol}_{\beta}\left(F^{(2,i)}\right)-\Omega^{(i)}_{\beta}.$
(5.23)

Note from (5.12), (5.13), (4.20), (4.21), (4.31) and (4.32) that
$\Omega^{(i)}$ involve only $g_{0}$ and its derivatives or $F^{(1)}$ and its
first derivatives $\partial F^{(1)}$. For $\nabla F^{(1)}$, the initial data
on $\Sigma_{0}$ are obtained by differenciating directly
$F^{(1)}_{\alpha\beta}$ defined in (5.21) and (5.22) and for
$\partial_{t}F^{(1)}_{\alpha\beta}$ they are obtained by rewriting the
equation we want $F^{(1)}$ to satisfy, that is (5.4) which rewrites as

$\displaystyle\partial_{t}F^{(1)}_{\alpha\beta}$
$\displaystyle=-N_{0}F^{(1)}_{\alpha\beta}+(\partial_{t}+N_{0})^{\rho}\Gamma(g_{0})^{\mu}_{\rho(\alpha}F^{(1)}_{\mu\beta)}+\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}(\Box_{g_{0}}u_{0})F^{(1)}_{\alpha\beta}$ (5.24)

where we used the fact that on $\Sigma_{0}$ we have $L_{0}=|\nabla
u_{0}|_{\bar{g}_{0}}\left(\partial_{t}+N_{0}\right)$. At this stage, the RHS
of (5.24) is fully defined on $\Sigma_{0}$, which implies that this is also
the case for $\Omega^{(i)}$. It remains to define the initial data for
$F^{(2,i)}_{0\alpha}$ for $i=1,2$ by

$\displaystyle F^{(2,i)}_{0A}$
$\displaystyle=-\bar{F}^{(2,i)}_{AN_{0}}-\frac{1}{|\nabla
u_{0}|_{\bar{g}_{0}}}\Omega^{(i)}_{A},$ (5.25) $\displaystyle
F^{(2,i)}_{0N_{0}}$
$\displaystyle=-\frac{1}{2}\bar{F}^{(2,i)}_{N_{0}N_{0}}-\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}^{2}}\Omega^{(i)}_{L_{0}},$ (5.26) $\displaystyle
F^{(2,i)}_{00}$ $\displaystyle=0.$ (5.27)

Note that these definitions only ensure

$\displaystyle V_{\mathcal{T}_{0}}^{(2,i)}\restriction{\Sigma_{0}}=0$ (5.28)

thanks to (3.9) and (3.10). The fact that $V^{(2,i)}_{\underline{L}_{0}}$ also
vanishes on $\Sigma_{0}$ is the content of Lemma 8.9.

####  5.6.2 Initial data for the reduced system

We define here the initial data for the solution of the reduced system. For
the tensor $\mathfrak{F}$ we simply set

$\displaystyle\mathfrak{F}_{\alpha\beta}\restriction{\Sigma_{0}}$
$\displaystyle=0.$ (5.29)

Before we look at $\mathfrak{h}$, note that with (5.29) we can apply Lemma 5.1
and fully define $g^{(3)}$ on $\Sigma_{0}$. Indeed it is independent from
$\mathfrak{h}$ and depends only on $F^{(1)}$, $F^{(2,i)}$ and their
derivatives which can be computed on $\Sigma_{0}$ by looking at the transport
equations we want them to satisfy, that is (5.4), (5.5) and (5.6).

Since $\mathfrak{h}$ solves a second order equation we need to prescribe
$\mathfrak{h}$ and $\partial_{t}\mathfrak{h}$ on $\Sigma_{0}$. We start with
the spatial components of $\mathfrak{h}$, defined so that the induced metric
on $\Sigma_{0}$ of the spacetime metric $g$ defined by (4.1) is precisely the
Riemannian metric $\bar{g}$ obtained in Theorem 2.1. This imposes

$\displaystyle\mathfrak{h}_{ij}=\bar{\mathfrak{h}}_{ij}-\lambda g^{(3)}_{ij}.$
(5.30)

where $\bar{h}$ is given in (2.15). For the time components of $\mathfrak{h}$,
we choose them so that they compensate the one of $g^{(3)}$, that is

$\displaystyle\mathfrak{h}_{0\alpha}$ $\displaystyle=-\lambda
g^{(3)}_{0\alpha}.$ (5.31)

Note that we can now obtain the initial data for the derivatives of
$\mathfrak{F}$. Indeed, (5.29) obviously gives us that
$\nabla\mathfrak{F}_{\alpha\beta}\restriction{\Sigma_{0}}=0$ and since we want
$\mathfrak{F}$ to solve (5.7), the time derivative of $\mathfrak{F}$ is
initially given by

$\displaystyle\partial_{t}\mathfrak{F}_{\alpha\beta}$
$\displaystyle=\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}\Pi_{\leq}\left(\mathfrak{h}_{L_{0}L_{0}}\right)F^{(1)}_{\alpha\beta}$
(5.32)

where we also used (5.29) to simplify (5.7).

We now want to define the initial data for
$\partial_{t}\mathfrak{h}_{\alpha\beta}$. The spatial components, i.e
$\partial_{t}\mathfrak{h}_{ij}$, are chosen so that the second fundamental
form of $\Sigma_{0}$ as an hypersurface in $(\mathcal{M},g)$ is given by $K$
obtained in Theorem 2.1. The second fundamental form is defined by
$-\frac{1}{2}\mathcal{L}_{T}g$ where $T$ is the unit normal to $\Sigma_{0}$
for $g$ and $\mathcal{L}$ is the Lie derivative. A priori, the unit normal $T$
depends on $\lambda$. This shows that in order to obtain the dependence of
$\partial_{t}\mathfrak{h}_{ij}$ on $\lambda$, we need to know more about $T$.

######  Lemma 5.2.

The future unit normal $T$ to $\Sigma_{0}$ for the metric $g$ given by (4.1)
satisfies

$\displaystyle T=\partial_{t}+Z$ (5.33)

with the vector field $Z=Z^{t}\partial_{t}+\bar{Z}^{i}\partial_{i}$ given by

$\displaystyle Z^{t}$
$\displaystyle=\left(1+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}\right)^{-\frac{1}{2}}-1,$
(5.34) $\displaystyle\bar{Z}^{i}$
$\displaystyle=-\lambda^{2}\bar{g}^{ij}g^{(2)}_{0j}\left(1+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}\right)^{-\frac{1}{2}}.$
(5.35)

###### Proof.

Since the unit normal to a spacelike hypersurface for a given Lorentzian
metric is unique, we just need to prove that we can construct the vector field
$Z$ such that $T$ defined by (5.33) satisfies

$\displaystyle g(T,T)=-1\quad\text{and}\quad g(T,\partial_{i})=0.$

Note that thanks to (5.22), (5.27), (5.29) and (5.31) and the fact that
$\partial_{t}$ is the unit normal to $\Sigma_{0}$ for $g_{0}$ we have
$g_{00}=-1$ and $g_{0i}=\lambda^{2}g^{(2)}_{0i}$. This gives

$\displaystyle g(T,\partial_{i})$
$\displaystyle=\lambda^{2}g^{(2)}_{0i}\left(1+Z^{t}\right)+\bar{g}_{ij}\bar{Z}^{j}$

so we impose

$\displaystyle\bar{Z}^{i}$
$\displaystyle=-\lambda^{2}\bar{g}^{ij}g^{(2)}_{0j}\left(1+Z^{t}\right).$
(5.36)

Moreover, we have

$\displaystyle g(T,T)$
$\displaystyle=-1+2g_{0\alpha}Z^{\alpha}+g_{\alpha\beta}Z^{\alpha}Z^{\beta}$

so we impose $2g_{0\alpha}Z^{\alpha}+g_{\alpha\beta}Z^{\alpha}Z^{\beta}=0$.
After expanding the quadratic term and inserting (5.36) this condition
rewrites as a quadratic equation for $Z^{t}$

$\displaystyle\left(1+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}\right)\left(Z^{t}\right)^{2}+2\left(1+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}\right)Z^{t}+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}=0.$

The two roots of this equation are

$\displaystyle\pm\left(1+\lambda^{4}\bar{g}^{ij}g^{(2)}_{0i}g^{(2)}_{0j}\right)^{-\frac{1}{2}}-1.$

Since we want $T$ to be future directed, we choose the root where $\pm=+$, i.e
(5.34). Inserting this into (5.36) gives (5.35). ∎

We now compute the expression of the second fundamental form of $\Sigma_{0}$
as an hypersurface in $(\mathcal{M},g)$ with $g$ being given by (4.1).

######  Lemma 5.3.

If $g$ is given by (4.1) and if $\partial_{t}F^{(1)}_{\alpha\beta}$ is given
by (5.24) then on $\Sigma_{0}$ the following holds

$\displaystyle-\frac{1}{2}\mathcal{L}_{T}g_{ij}$
$\displaystyle=K^{(0)}_{\lambda}+\lambda
K^{(1)}_{\lambda}+\lambda^{2}\left(-\frac{1}{2}\mathcal{L}_{T}\mathfrak{h}+\tilde{K}_{evol}^{(\geq
2)}\right)$

where $K^{(0)}_{\lambda}$ and $K^{(1)}_{\lambda}$ are given by (2.17) and
(2.18). Moreover, we have schematically

$\displaystyle\lambda^{2}\tilde{K}_{evol}^{(\geq 2)}$
$\displaystyle=Z^{\alpha}\partial g_{0}+\lambda Z^{\alpha}\partial
g^{(1)}+\lambda^{2}\left(\partial\mathfrak{F}+\partial
F^{(2,i)}+Z^{\alpha}\partial g^{(2)}\right)+\lambda^{3}Tg^{(3)}.$ (5.37)

###### Proof.

Recalling that $g$ is defined by (4.1) we have

$\displaystyle-\frac{1}{2}\mathcal{L}_{T}g$
$\displaystyle=-\frac{1}{2}\mathcal{L}_{T}g_{0}-\frac{\lambda}{2}\mathcal{L}_{T}g^{(1)}-\frac{\lambda^{2}}{2}\mathcal{L}_{T}g^{(2)}-\frac{\lambda^{2}}{2}\mathcal{L}_{T}\mathfrak{h}-\frac{\lambda^{3}}{2}\mathcal{L}_{T}g^{(3)}.$

We compute or schematically estimate each term in this expression, using the
expansion of $T$ given in Lemma 5.2. The latter gives

$\displaystyle-\frac{1}{2}\mathcal{L}_{T}g_{0}=K_{0}+O\left(Z^{\alpha}\partial
g_{0}\right)$

where $Z^{\alpha}$ denotes either $Z^{t}$ or $\bar{Z}^{i}$ and where we used
the fact that $\partial_{t}$ is the unit normal to $\Sigma_{0}$ for $g_{0}$.
Similarly, using $\partial_{t}u_{0}=|\nabla u_{0}|_{\bar{g}_{0}}$ on
$\Sigma_{0}$ we have

$\displaystyle-\frac{\lambda}{2}\mathcal{L}_{T}g^{(1)}_{ij}$
$\displaystyle=\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{ij}-\frac{\lambda}{2}\cos\left(\frac{u_{0}}{\lambda}\right)\partial_{t}F^{(1)}_{ij}+O\left(\lambda
Z^{\alpha}\partial g^{(1)}\right)$
$\displaystyle=\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{ij}$
$\displaystyle\quad-\frac{\lambda}{2}\cos\left(\frac{u_{0}}{\lambda}\right)\left(-N_{0}\bar{F}^{(1)}_{ij}+(\partial_{t}+N_{0})^{\rho}\Gamma(g_{0})^{k}_{\rho(i}\bar{F}^{(1)}_{kj)}+\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}(\Box_{g_{0}}u_{0})\bar{F}^{(1)}_{ij}\right)$
$\displaystyle\quad+O\left(\lambda Z^{\alpha}\partial g^{(1)}\right)$

where we used (5.24) and (5.22) to compute $\partial_{t}F^{(1)}_{ij}$ and
(5.21) to compute $F^{(1)}_{ij}$. Using in addition (5.29) we now obtain

$\displaystyle-\frac{\lambda^{2}}{2}\mathcal{L}_{T}g^{(2)}_{ij}$
$\displaystyle=-\frac{\lambda}{2}\cos\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(2,1)}_{ij}+\lambda\sin\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(2,2)}_{ij}$
$\displaystyle\quad+O\left(\lambda^{2}\left(\partial\mathfrak{F}+\partial
F^{(2,i)}+Z^{\alpha}\partial g^{(2)}\right)\right).$

This concludes the proof of the lemma, by recalling the expression of
$K^{(i)}_{\lambda}$ for $i=1,2$ given in (2.17) and (2.18). ∎