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\DOI{10.5802/crmeca.288}
\datereceived{2024-11-22}
\daterevised{2025-02-03}
\dateaccepted{2025-02-06}
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\begin{document}

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\begin{noXML}

\CDRsetmeta{articletype}{review}


\title{Burnett's conjecture in general relativity}

\alttitle{La conjecture de Burnett en relativit\'e g\'en\'erale}

\author{\firstname{Arthur} \lastname{Touati}\CDRorcid{0000-0002-0359-1665}}
\address{CNRS \& IMB, Bordeaux University, France}
\email{arthur.touati@math.u-bordeaux.fr}

\begin{abstract} 
We present the literature on Burnett's conjecture in general
relativity, which relate weak limits of vacuum solutions to
relativistic kinetic theory. A special care is put on relating these
works with early Choquet-Bruhat's results on high-frequency
gravitational waves and geometric optics.
\end{abstract}

\begin{altabstract} 
Nous passons en revue la litt\'erature sur la conjecture de Burnett en
relativit\'e g\'en\'erale, qui relie les limites faibles des solutions
du vide \`a la th\'eorie cin\'etique relativiste. Une attention
particuli\`ere est port\'ee sur le lien entre ces travaux et les
premiers r\'esultats de Choquet-Bruhat concernant les ondes
gravitationnelles haute fr\'equence et l'optique g\'eom\'etrique.
\end{altabstract}

\keywords{\kwd{Einstein equations}
\kwd{Backreaction}
\kwd{Relativistic kinetic theory}
\kwd{Compensated compactness}
\kwd{Geometric optics}
\kwd{High-frequency gravitational waves}}

\altkeywords{\kwd{\'Equations d'Einstein}
\kwd{R\'etroaction} 
\kwd{Th\'eorie cin\'etique relativiste} 
\kwd{Compacit\'e compens\'ee} 
\kwd{Optique g\'eom\'etrique} 
\kwd{Ondes gravitationnelles haute fr\'equence}}

\maketitle

\end{noXML}

\section{Introduction}\label{sec:introduction}

The author of the present text is more than honored to celebrate Yvonne
Choquet-Bruhat's  legacy in the field of mathematical general
relativity, a field that, one could say, she launched in her seminal
article of 1952~\cite{ChoquetBruhat1952}.
There, she built a local existence theory for the Einstein equations of
general relativity. Of course, she did not stop there and went on to
achieve an incredibly diverse career in general relativity, analysis
and mathematical physics over more than 60 years. Her
book~\cite{ChoquetBruhat2009} is the concrete proof of that, and
fifteen years after its publication, this scientific testament still
contains hidden treasures and research directions only waiting to be
explored (see Section~\ref{section conclusion} for an example linked to
this survey's topic). The goal of this survey is to present the
mathematical literature on the so-called Burnett conjecture in general
relativity. At this stage, we state the conjecture in the roughest way
possible: weak limits of solutions to the Einstein vacuum equations can
be identified with solutions to the massless Einstein--Vlasov system.
Our first introductory task is to explicit the nature of the link
between this conjecture and Choquet-Bruhat's work.

At first glance they seem rather unrelated, since the 785 pages
of~\cite{ChoquetBruhat2009} don't contain any reference to Burnett
himself, his conjecture or his original 1989
article~\cite{Burnett1989}. However, and this may appease the reader
rightfully questioning the presence of this text in this volume,
Burnett's article~\cite{Burnett1989} does contain a reference to
Choquet-Bruhat's article~\cite{ChoquetBruhat1969} from 1969. In this
article, Choquet-Bruhat, following an approach pioneered by some
physicists~\cite{Bril1964,Isaacson1968,Isaacson1968a,MacCallum1973},
has put on a solid mathematical footing an approximate geometric optics
construction for these equations. Since an oscillating behaviour in the
limit where the frequency tends to infinity is the simplest example of
weak convergence in the mathematical sense, the approximate solutions
of~\cite{ChoquetBruhat1969} fit into Burnett's abstract framework and
provide a precursory and neat illustration of the so-called
backreaction phenomenon, i.e.\ the effect of small scale inhomogeneities
on a background. This connects
Choquet-Bruhat's~\cite{ChoquetBruhat1969} and
Burnett's~\cite{Burnett1989} through the description of a common
physics. But the link between these two works has recently been
strengthened. Indeed, after being almost completely silent for decades,
the mathematical activity around Burnett's conjecture has been revived
by the first rigorous result obtained by Green and Wald in
2011~\cite{Green2011}. Since then, Choquet-Bruhat's early approximate
construction from~\cite{ChoquetBruhat1969} has been revisited and its
rigorous justification has proved to be an efficient way of attacking
Burnett's conjecture.

Because of these two links, an early one from physics and a modern one
from mathematics, the narrative of Burnett's conjecture will start in
Section~\ref{section article CB} by a description
of~\cite{ChoquetBruhat1969}. Another, more prosaic, explanation behind
this choice lies in the author's personal interests and his will to
honor in this volume a relatively unknown, at least from the
mathematical community, aspect of Choquet-Bruhat's work (corresponding
to the eleventh chapter of~\cite{ChoquetBruhat2009}). We also refer the
reader to her autobiography~\cite{ChoquetBruhat2016}, where her work on
high-frequency gravitational waves is mentioned several times.

Needless to say that the Burnett conjecture and its literature are also
connected to many more areas of the mathematical theory of general
relativity and of the partial differential equations' world. To
highlight the richness of these connections is also one of the
objectives of this text, which should be viewed as a complement to
other reviews on Burnett's conjecture like the
recent~\cite{Huneau2024a}. 

\section{Choquet-Bruhat's article from 1969} \label{section article CB}

As early as~\cite{Einstein1916}, physicists have tried to linearize the
equations of general relativity around simple exact solutions of the
Einstein vacuum equations, such as the Minkowski spacetime. This has
led early general relativity physicists to think that spacetime itself
could be a potential medium for a new kind of waves, called
gravitational waves. We refer to Part VIII of~\cite{Misner1973} for a
rich discussion of how gravitational waves propagate, how they are
generated and how one can detect them. Chapter 35 of~\cite{Misner1973}
is particularly relevant for the present discussion.

The obvious drawback of any linearization procedure is that it assumes
that the perturbation is small, as well as all its derivatives, thus
forbidding the description of more interesting physical situations
where the gravitational field is strong. Moreover, the linearization
procedure rules out the possibility of describing the ``energy'' of the
gravitational waves themselves, since such a quantity, though
complicated to define in general relativity, must be of quadratic
nature. This has led physicists to imagine various averaging schemes
(see for
instance~\cite{Bril1964,Isaacson1968,Isaacson1968a,MacCallum1973})
unfortunately lacking a rigorous mathematical framework. Thanks to her
(at the time) recent experience in nonlinear geometric optics
(see~\cite{ChoquetBruhat1967,ChoquetBruhat1969a}\footnote{This paper 
appears to be related to two other versions by the same or similar
title, one earlier and one contemporaneous with the present entry. They
are, respectively:~\cite{16} and ~\cite{17}.}), Yvonne
Choquet-Bruhat was the first to rigorously applied the WKB expansion
method to the Einstein vacuum equations themselves in the
article~\cite{ChoquetBruhat1969}. We will discuss in more details the
general strategy of geometric optics in Section~\ref{section GO}, but
for now we summarize the approach and main outcomes of this pioneer
article.

In~\cite{ChoquetBruhat1969}, Choquet-Bruhat considers a family of
metrics of the form
{\begin{equation}\label{ansatzCB}
\g_\la = \g_0 + \la \g^{(1)} \pth{ \frac{\ffi}{\la}} + 
\la^2 \g^{(2)}\pth{\frac{\ffi}{\la}},
\end{equation}}\unskip
where $\g_0$ is another metric, the phase $\ffi$ is a scalar function
and where by $\g^{(i)} \pth{ \sfrac{\ffi}{\la}}$ we actually mean
$\g^{(i)} \pth{x, \sfrac{\ffi(x)}{\la}}$. Note that we adapt her
original notations in order to ease the comparison with more modern
results on Burnett's conjecture (in particular those of
Section~\ref{section reverse}). We assume moreover that each $\g^{(i)}$
is periodic with respect to the second variable. The metric $\g_\la$
should be thought as describing a vacuum spacetime in which a
gravitational wave of wavelength $\la$ is propagating. Crucially, the
geometric optics expansion \eqref{ansatzCB} allows the perturbation to
be small but its derivatives to be large in the high-frequency limit
$\la\to 0$. The goal is to find conditions on $\ffi$, $\g_0$,
$\g^{(1)}$ and $\g^{(2)}$ such that $\g_\la$ is an approximate solution
of the Einstein vacuum equations at order 1, that is
$R_{\mu\nu}(\g_\la)=\GO{\la}$. For that, we compute the Ricci tensor of
the metric $\g_\la$ defined by~\eqref{ansatzCB}, and because of the
curvature being second order with respect to the metric, it is of the
form
{\begin{equation}\label{ricciCB}
R_{\mu\nu}(\g_\la) = \frac{1}{\la} R^{(-1)}_{\mu\nu} + 
R^{(0)}_{\mu\nu} + \GO{\la},
\end{equation}}\unskip
where $R^{(-1)}_{\mu\nu}$ depends on $\g_0$ and $\g^{(1)}$ and
$R^{(0)}_{\mu\nu}$ depends on $\g_0$, $\g^{(1)}$ and $\g^{(2)}$. In
order to formulate the main result of~\cite{ChoquetBruhat1969}, we
introduce the polarization tensor of a tensor $T$ with respect to a
phase $\ffi$:
{\begin{equation}\label{def pola}
\Pol{T}{\ffi}_\a  : = \g_0^{\mu\nu}\pth{ \dr_\mu 
\ffi T_{\nu\a} - \half \dr_\a \ffi T_{\mu\nu}}.
\end{equation}}\unskip

\begin{theo}[{\cite{ChoquetBruhat1969}}]\label{theo CB}
Consider $\g_\la$ defined by \eqref{ansatzCB} and $R^{(-1)}_{\mu\nu}$
and $R^{(0)}_{\mu\nu}$ as in \eqref{ricciCB}.
\begin{enumerate}[(i)]
\item $R^{(-1)}_{\mu\nu}=0$ if and only if $\g_0^{-1}(\d\ffi,\d\ffi)=0$
and $\g^{(1)}$ satisfies the polarization condition
{\begin{equation}\label{pola condition CB}
\Pol{\g^{(1)}}{\ffi}  = 0.
\end{equation}}\unskip
\item If $R^{(-1)}_{\mu\nu}=0$, then $R^{(0)}_{\mu\nu}=0$ if and only
if $\g_0$ satisfies
{\begin{equation}\label{backreaction CB}
R_{\mu\nu}(\g_0)  = \left\langle \tfrac{1}{4} \g_0^{\rho\si}
\g_0^{\a\b} \pth{ \dr_\th \g^{(1)}_{\rho\a} \dr_\th 
\g^{(1)}_{\si\b} - \half \dr_\th \g^{(1)}_{\rho\si} 
\dr_\th \g^{(1)}_{\a\b} } \right\rangle_\th \dr_\mu \ffi \dr_\nu \ffi,
\end{equation}}\unskip
$\g^{(1)}$ satisfies the transport equation
{\begin{equation}\label{transport CB}
2\dr^\a \ffi \mathbf{D}_\a \g^{(1)} + (\Box_\g\ffi) \g^{(1)}  = 0,
\end{equation}}\unskip
and $\g^{(2)}$ satisfies a polarization condition of the form
$\Pol{\g^{(2)}}{\ffi} = \g^{(1)}\star \g^{(1)}$, where $\star$ denotes
some tensor contractions.
\end{enumerate}
\end{theo}

In \eqref{backreaction CB}, $\dr_\th$ and $\langle\cdot \rangle_\th$
denote respectively a derivative and the average with respect to the
oscillating variable. Note that in Theorem~\ref{theo CB}, the case
$\g_0^{-1}(\d\ffi,\d\ffi)\neq 0$ is excluded for physical reasons,
since in this case a high-frequency change of variables can remove
$\g^{(1)}$. Though Theorem~\ref{theo CB} only produces approximate
solutions to the Einstein vacuum equations, it provides a more rigorous
averaging scheme than the
results~\cite{Bril1964,Isaacson1968,Isaacson1968a,MacCallum1973}. It
shows how one can describe high-frequency gravitational waves in the
nonlinear regime while still retaining some surprising linear features.
For instance, as in the linearized gravity setting, the wave propagate
along a null direction and the polarization condition~\ref{pola
condition CB} is actually equivalent to the TT-gauge conditions of
linearized gravity which imply in particular that a gravitational wave
has two degrees of freedom. Moreover, the (surprisingly linear)
transport equation~\eqref{transport CB} is shown to propagate the
polarization condition~\eqref{pola condition CB}, so that the wave
retains its form! Physically, this distinguishes greatly gravitational
waves from other kind of waves, say in fluids, since it shows that the
former don't self-steepen while the latter usually can accumulate and
form shocks.

In addition to exhibiting this exceptional linear structure,
Theorem~\ref{theo CB} also shows the nonlinear effects caused by the
wave on the background spacetime $\g_0$. Indeed, even though $\g_\la$
solves (approximately) the Einstein vacuum equations, this is not the
case of $\g_0$ which ``sees'' the presence of the wave propagating, via
the non-zero stress energy tensor in the right hand side of
\eqref{backreaction CB} (note that this expression, here derived
rigorously, matches those found with other averaging schemes). In this
sense, theorem~\ref{theo CB} clearly prefigures Burnett's conjecture,
which aims at characterizing completely the backreaction phenomenon
(see Section~\ref{section burnett} below).

We also mention subsequent works in the framework
of~\cite{ChoquetBruhat1969}, most of them being by Choquet-Bruhat
herself. All these works show how important to her this direction of
research was, from early to later stages of her career. High-frequency
waves in relativistic fluids are considered
in~\cite{ChoquetBruhat1967,ChoquetBruhat1969a,Anile1990}. The coupling
between gravitational waves and electromagnetic or fluid waves are the
objects of~\cite{ChoquetBruhat1971a} and~\cite{ChoquetBruhat1996}
respectively. The case of a charged scalar field is considered
in~\cite{ChoquetBruhat1977} and the one of the Yang--Mills equations
in~\cite{ChoquetBruhat1983}. She even considers high-frequency
gravitational waves with Gauss--Bonnet corrections or with stringy
effects in~\cite{ChoquetBruhat1988} and~\cite{ChoquetBruhat1989}
respectively. As we will see, the results on Burnett's conjecture
discussed in Section~\ref{section reverse} can also be thought as
extensions of the pioneer work of Yvonne Choquet-Bruhat on
high-frequency gravitational waves.


\section{Statement of Burnett's conjecture}\label{section burnett}

In this Section, we first introduce the massless Einstein--Vlasov
system, state and discuss conceptualy Burnett's conjecture, and present
the first rigorous result on the conjecture.

\subsection{Relativistic kinetic theory}
 
Relativistic kinetic theory is the name given to the physical
description of a system of particles living in a spacetime described by
general relativity, see the tenth chapter of~\cite{ChoquetBruhat1969}
or~\cite{Andreasson2011}. While in classical kinetic theory the
particles in consideration are most often gas or liquid particles, in
general relativity, this perspective is adopted to describe ``particles''
at very different scales, such as stars, galaxies or even clusters of
galaxies. Nevertheless, the framework of relativistic kinetic theory is
identical to its classical counterpart: particles are described by a
density function defined in phase space and solving a PDE encoding the
interactions between the particles, and integrating this density with
respect to the momentum variable provides macroscopic information on
the system of particles.

If $(\mathcal{M},\g)$ is a Lorentzian manifold, we first look for an
equivalent of the classical phase space encoding both position in space
and time and velocity or momentum. Since tangent vectors to curves are
represented by vector fields in differential geometry, the phase space
in relativistic kinetic theory will be a particular subset of the
tangent bundle $T\M$, the latter consisting of pairs $(x,p)$ where
$x\in\M$ and $p\in T_x\M$. More precisely, for $m\geq 0$ we define the
mass shell at a point $x$ by
{\begin{equation*}
\P_{m,x} : = \enstq{p\in T_x\M }\g_{|_x}(p,p)=-m^2 \quad
\text{and}\quad  p \text{ is future oriented} .
\end{equation*}}\unskip
If we set $\P_m :=\bigsqcup_{x\in\M}\P_{m,x}$, then a density $f$
describing particles of (rest) mass $m$ is simply a function $f: \P_m
\longrightarrow \mathbb{R}_+$, and $f(x,p)$ is the ``number'' of
particles at position $x\in\M$ with momentum $p\in T_x\M$. To lighten
the notations, in what follows we don't write down the measure on
$\P_{m,x}$ induced by the metric $\g$.


We will assume here the absence of collisions between the particles in
the system, so that the particles interact with each other only through
their interaction with other fields. If moreover we neglect the charges
of the particles, the only reasonable such field is the gravitational
one and such a matter model is sometimes referred to as
self-gravitating. This means that each particle is following a geodesic
on $\M$, which is equivalent to the following Vlasov-type equation for
the distribution function $f$: 
{\begin{equation}\label{vlasov}
p^\a \dr_\a f - \Ga^\a_{\mu\nu}p^\mu p^\nu \dr_{p^\a}f  = 0.
\end{equation}}\unskip
If $(\M,\g)$ is given, the behaviour of the solutions to this equation
is of tremendous physical importance. However, as always in general
relativity, we can also describe the effect of a matter model on the
Lorentzian metric $\g$ itself by plugging its stress energy tensor into
the right hand side of the Einstein equations. In the case of kinetic
theory, the stress energy tensor is simply the second moment of the
density $f$, which leads us to the so-called Einstein--Vlasov system
with rest mass $m$:
{\begin{equation}\label{Einstein Vlasov}
\left\{
\begin{array}{rcl}
\displaystyle R_{\mu\nu}(\g) - \dhalf R(\g)\g_{\mu\nu} 
&  =& \displaystyle\int_{\P_{m,x}}f p_\mu p_\nu,\\ 
\displaystyle p^\a \dr_\a f - \Ga^\a_{\mu\nu}p^\mu p^\nu \dr_{p^\a}f & =& 0.
\end{array}
\right.
\end{equation}}\unskip
In the context of \eqref{Einstein Vlasov}, the Vlasov equation for $f$
can also be interpreted as the condition for the stress energy tensor
to have zero divergence. In the case $m=0$, i.e.~when the particles in
consideration have zero rest mass, the system \eqref{Einstein Vlasov}
is called the massless Einstein--Vlasov system. Note that in this case,
the mass shell $\P_0$ is simply the collection of null cones for the
metric $\g$ over each point of the manifold.

The literature on the system \eqref{Einstein Vlasov} is very rich and
is nowadays concerned with global existence results, stationary
solutions etc. Note that an important step was the proof of
well-posedness in wave coordinates in~\cite{ChoquetBruhat1971}.
Finally, as is standard in general relativity, we expect the limit
$c\to +\infty$ to correspond to the classical Newtonian gravity. This
is indeed true for relativistic kinetic theory, since it is proved
in~\cite{Rendall1994} that the famous Vlasov--Poisson system is the
Newtonian limit of \eqref{Einstein Vlasov}.

\subsection{First discussion of the conjecture}\label{section first discussion}

We are now ready to state Burnett's conjecture from~\cite{Burnett1989}.
For that, fix a manifold $\M$ and consider sequences of metrics
$(\g_\la)_{\la\in(0,1]}$ solving the Einstein vacuum equations
{\begin{equation}\label{EVEbis}
R_{\mu\nu}(\g_\la)  = 0, 
\end{equation}}\unskip
and such that there exists another metric $\g_0$ such that in some
coordinates system $(x^\a)_{\a=0,1,2,3}$ on $\M$ we have
{\begin{equation}\label{burnett convergence}
(\g_\la)_{\a\b} - (\g_0)_{\a\b} \longrightarrow 0 
\quad\text{uniformly}, \quad \dr_\mu\pth{(\g_\la)_{\a\b} - 
(\g_0)_{\a\b} } \longrightarrow 0 \quad\text{weakly},
\end{equation}}\unskip
when the parameter $\la$ tends to $0$. Note that instead of the second
assumption in \eqref{burnett convergence}, one can ask the sequence of
first order derivatives $\pth{\dr_\mu(\g_\la)_{\a\b}}_{\la\in(0,1]}$ to
be bounded.

\begin{conjecture}[Burnett's conjecture]\label{conj}
One conjectures that:
\begin{itemize}
\item If $(\g_\la)_{\la\in(0,1]}$ and $\g_0$ are as above, then $\g_0$
is a solution to the massless Einstein--Vlasov system \eqref{Einstein
Vlasov} for a suitable density.
\item Conversely, if $\g_0$ is a solution to the massless
Einstein--Vlasov system \eqref{Einstein Vlasov}, then there exists a
sequence $(\g_\la)_{\la\in(0,1]}$ as above and with limit $\g_0$.
\end{itemize}
\end{conjecture}

The above conjecture aims at completely describing backreaction for the
Einstein vacuum equations, i.e.\ the effect of small scales
inhomogeneities onto a background spacetime. These inhomogeneities are
described by the lack of strong convergence of the first derivatives
$\dr_\mu\pth{(\g_\la)_{\a\b} - (\g_0)_{\a\b} }$. In his original
article, Burnett refers to this weak convergence as being the trace of
some high-frequency oscillations in the metric itself. However, this
regime also includes concentration effects, and only two references in
the sequel of this text will allow for concentration effects, while all
the others will be concerned with a true high-frequency regime, meaning
where \eqref{burnett convergence} is replaced by the stronger
{\begin{equation*}
\left| \dr^k\pth{(\g_\la)_{\a\b} - (\g_0)_{\a\b} } \right| 
\leq \la^{1-k}, \quad k=0,\ldots,K,
\end{equation*}}\unskip
for some $K\geq 2$. 

The core idea behind Conjecture~\ref{conj} is the following. If first
order derivatives of $\g_\la$ converge only weakly, then products
$\dr\g_\la\dr\g_\la$ of these derivatives don't necessarily converge to
$\dr\g_0\dr\g_0$, and since the Ricci tensor contains such semilinear
terms, passing to the limit in \eqref{EVEbis} does not lead {a
priori} to $R_{\mu\nu}(\g_0)=0$ but rather to
{\begin{equation*}
R_{\mu\nu}(\g_0)-\half R(\g_0)(\g_0)_{\mu\nu}  =: 
T_{\mu\nu}^{\mathrm{effective}},
\end{equation*}}\unskip
where $T_{\mu\nu}^{\mathrm{effective}}$ can be called the effective
stress energy tensor of the inhomogeneities. The first part of
Conjecture~\ref{conj} can be thus rephrased as: there must exist some
density function $f$ defined on $\P_0$ (defined with the metric $\g_0$)
such that
{\begin{equation}\label{Teff}
T_{\mu\nu}^{\mathrm{effective}}  = \int_{\P_{0,x}}f p_\mu p_\nu.
\end{equation}}\unskip
The effective stress energy tensor $T_{\mu\nu}^{\mathrm{effective}}$ is
thus caused by the lack of convergence of the semilinear terms
$\dr\g_\la\dr\g_\la$. This shows that in the particular case (obviously
included in Conjecture~\ref{conj}) where the convergence of
$\dr_\mu\pth{(\g_\la)_{\a\b} - (\g_0)_{\a\b}}$ is actually strong, then
we can pass to the limit in \eqref{EVEbis} and $\g_0$ thus solves the
Einstein vacuum equations.


As explained in the introduction of~\cite{Burnett1989}, the fact that
backreaction is necessarily of kinetic nature can be first observed on
plane wave solutions to \eqref{EVEbis}, where $\g_\la$ is explicitly
given by
{\begin{equation*}
 - \d u\d v + B_\la(u)^2 \pth{ {\mathrm{e}}^{\la\a(u)\cos
 \pth{\frac{u}{\la}} } \d x^2 + {\mathrm{e}}^{- \la\a(u)
 \cos\pth{\sfrac{u}{\la}}} \d y^2},
\end{equation*}}\unskip
where $\a$ is a given function of $u$ and where $B_\la$ needs to solve
an ODE. One can show that $\g_\la$ converge in the sense \eqref{burnett
convergence} to $\g_0$ satisfying $R_{\mu\nu}(\g_0)=\sfrac12 \a^2 \dr_\mu
u\dr_\nu u$. Formally summing such plane waves gives rise to the
integrated expression in \eqref{Teff}. However, and though Burnett does
not mention it, the same kind of ``discrete'' effective stress energy
tensor is obtained in the approximate geometric optics construction
of~\cite{ChoquetBruhat1969} described in Section~\ref{section article
CB} (recall \eqref{backreaction CB}). Burnett's explicit example has
the benefit of being a family of exact solutions to \eqref{EVEbis},
while Choquet-Bruhat's approximate solutions have the benefit of living
outside any symmetry class. Moreover, as Section~\ref{section reverse}
will show, Choquet-Bruhat's strategy can be extended to produce exact
solutions. However, Burnett's and Choquet-Bruhat's concrete examples of
backreaction have one thing in common: the background kinetic spacetime
metric $\g_0$ is a particular measure-valued solution of the massless
Einstein--Vlasov system. More precisely, by considering several
solutions of the eikonal equations $(u_\A)_{\A\in\AA}$ for $\AA$ a
finite set, we can consider the density function
$f(x,p)=\sum_{\A\in\AA}F^2_\A(x)\otimes \de_{p=\pth{\d u_\A}^{\text{\hashSymb}}}$,
where $\de_{p=\pth{\d u_\A}^{\text{\hashSymb}}}$ denotes the Dirac measure at $\pth{\d
u_\A}^{\text{\hashSymb}}$ in each tangent space and where the $F_\A$'s are scalar
functions. In this case the massless Einstein--Vlasov becomes the
so-called Einstein-null dusts system:
{\begin{equation}\label{null dusts}
\left\{
\begin{array}{rcl}
R_{\mu\nu}(\g_0) & =&\displaystyle \sum_{\A\in\AA}F_\A^2 \dr_\mu u_\A \dr_\nu u_\A,\\ 
\g_0^{-1}(\d u_\A,\d u_\A) & =& 0, \quad \text{for all } \A\in\AA,\\
2\g_0^{\a\b}\dr_\b u_\A \dr_\a F_\A + \pth{\Box_{\g_0}u_\A}F_\A & =& 0, 
\quad \text{for all }\A\in\AA,
\end{array}
\right.
\end{equation}}\unskip
The system \eqref{null dusts} can be viewed as a ``discrete'' version of
the massless Einstein--Vlasov system \eqref{Einstein Vlasov}, but it
also corresponds to the Einstein equations coupled to a perfect null
fluid with zero pressure. Other examples of weak limits of vacuum
spacetimes have been considered with two-dimensional symmetry, such as
the $\mathbb{T}^2$ symmetry in~\cite{Lefloch2020}.


\subsection{What does it say about the structure of the Einstein vacuum
equations?}\label{section what}

Even before discussing actual results on Conjecture~\ref{conj}, we wish
to reflect here on its significance. We start by noticing that, as it
is apparent from its statement, Conjecture~\ref{conj} is a double
conjecture. Its two parts are of very different mathematical nature.

 
The first part, referred to as the \textit{direct Burnett conjecture},
aims at describing the generic behaviour of sequences of solutions to
the Einstein vacuum equations. In particular, it shows that this set is
not closed for the topology of \eqref{burnett convergence}. Since this
backreaction is due to a lack of strong convergence, results on the
direct Burnett conjecture fall into the scope of homogenization theory
(we refer to~\cite{Tartar2009} for an introduction to this field, which
can be roughly defined as the study of averaging processes in
disordered media), and the direct Burnett conjecture is an instance of
compensated compactness. On the other hand, the second part of
Conjecture~\ref{conj}, referred to as the \textit{reverse Burnett
conjecture}, aims at constructing solutions to the Einstein vacuum
equations that converge in the sense of \eqref{burnett convergence} to
a given kinetic target. If it holds together with the direct part, then
the set of massless kinetic spacetimes completely characterizes the
closure of the set of solutions to the Einstein vacuum equations for
the topology of \eqref{burnett convergence}. Therefore, if Burnett's
conjecture is true, the Einstein vacuum equations stand in between two
kinds of PDEs:
\begin{itemize}
\item PDEs for which the set of solutions is weakly closed, such as
semilinear wave systems with the classical null condition
of~\cite{Christodoulou1986,Klainerman1986}. In this case, compensated
compactness can be seen as a consequence of the famous div-curl lemma
(see~\cite{Murat1978,Tartar1979}). Such PDEs are said to display
\textit{rigidity}.
\item PDEs for which weak limits of solutions are somehow arbitrary,
such as incompressible fluid equations. In the case of incompressible
Euler, weak limits of solutions are shown to solve the Euler--Reynolds
system (see~\cite{DeLellis2013} or the survey~\cite{DeLellis2022}),
which plays there a similar role as the massless Einstein--Vlasov. As
opposed to the latter, the Euler--Reynolds system is underdetermined,
thus leading to the display of \textit{flexibility} (see the resolution
of Onsager's conjecture in~\cite{Isett2018} for a consequence of this
flexibility).
\end{itemize}
Therefore, one could say that the Einstein vacuum equations display
some \textit{flexible rigidity} or \textit{rigid flexibility} (we let
the reader decide which one is more accurate).

Not only does Burnett's conjecture distinguish the Einstein vacuum
equations from incompressible fluid equations, in the sense that the
failure of convergence satisfies a transport equation, it also reveals
a hidden linear behaviour, despite the equations being very much
nonlinear (already observed by Choquet-Bruhat
in~\cite{ChoquetBruhat1969}, see the discussion after Theorem~\ref{theo
CB}). Indeed, the fact that backreaction can be described by the
massless Einstein--Vlasov system, i.e.\ a self-gravitating matter model,
shows that small scale inhomogeneities don't interact directly but only
through their impact on the background metric. In other words, the
linearity of the Vlasov equation with respect to the density, or the
fact that no collision operator shows up in the right hand side, is a
second feature of the Einstein vacuum equations. Depending on the
context, these surprising nonlinear cancellations can be related to
compensated compactness again (for instance in~\cite{Huneau2024a} it is
described as a secondary form of compensated compactness), to the
classical null condition and Shatah's normal form approach
(see~\cite{Shatah1985}) in semilinear waves or to transparency in
geometric optics (see~\cite{Joly2000}). The deep link between all these
approaches is discussed in depth in the Bourbaki
seminar~\cite{Lannes2013}. 

 
However, it is very well-known from the works on stability of Minkowski
(see~\cite{Christodoulou1993,Lindblad2010}) that the Einstein vacuum
equations don't satisfy the null condition in wave coordinates. It is
complicated to directly compare the structure of the Einstein vacuum
equations in different gauges, but if Burnett's conjecture were to be
true in full generality, it would show that they don't satisfy the null
condition in any gauge or setting, since otherwise backreaction would
not occur and vacuum spacetimes would not approach anything else but
other vacuum spacetimes. Burnett's conjecture thus hints at a very fine
structure: semilinear terms in the Einstein vacuum equations fail to
satisfy the null condition (otherwise there would be no backreaction)
but the failure of the null condition has a unique structure still
leading to some cancellations (otherwise the Vlasov equation would be
nonlinear). 

 
In~\cite{ChoquetBruhat2000}, Choquet-Bruhat introduced gauge-invariant
definitions of the null condition and of a relaxation of it, but
perhaps the most famous relaxed version of the null condition in wave
coordinates is the weak null condition introduced
in~\cite{Lindblad2003}. One could then wonder if the aforementioned
unique structure of the failure of the null condition is precisely
Lindblad and Rodnianski's weak null condition. This is in fact not the
case and if true, Burnett's conjecture thus demonstrates that the
Einstein vacuum equations possess an even finer structure, strictly in
between the null condition and the weak null condition. This last fact
actually brings Burnett's conjecture closer to the bounded $L^2$
curvature theorem proved in~\cite{Klainerman2015}, which (to the best
of the author's knowledge) is the only other instance where the
Einstein vacuum equations behave better than a system satisfying the
weak null condition. The link between Burnett's conjecture and
low-regularity solutions in general relativity is actually even
stronger, but we postpone this discussion to Section~\ref{section
strategy}. 

\subsection{First rigorous result in the context of cosmology}

Before discussing in the rest of this survey the ``PDE'' results on
Conjecture~\ref{conj}, we start with a result coming from the physics
community. Obtained by Green and Wald in~\cite{Green2011}, this is the
first mathematically rigorous result on Conjecture~\ref{conj} since
Burnett's original article~\cite{Burnett1989}. 

Their motivation comes from cosmology, where a wealth of literature has
been produced on the question of the effect of small scale
inhomogeneities in the universe we live in. In particular, backreaction
from these inhomogeneities onto the cosmological background has been
believed to be a potential explanation of the presence of dark matter
in our universe, in replacement of the standard explanation via the
presence of (so far undetected) weakly-interacting ``dark particles'' or
modifications of Einstein's gravity, see for
instance~\cite{Buchert2008,Kolb2006}. However, if small scale
inhomogeneities are described by sequences of metrics with the
convergence regime \eqref{burnett convergence}, then Burnett's
Conjecture~\ref{conj} contradicts this belief, since the effect of
backreaction is characterized by a massless Vlasov field. This
motivated the work~\cite{Green2011}, where the authors get the
following theorem:

\begin{theo}[\cite{Green2011}]\label{theo GW}
If $(\g_n)_{n\in\mathbb{N}}$ is a sequence of solutions to
{\begin{equation*}
R_{\mu\nu}(\g_n) - \half R(\g_n)(\g_n)_{\mu\nu} + \La (\g_n)_{\mu\nu}
= (T_n)_{\mu\nu},
\end{equation*}}\unskip
where each $T_n$ is trace free and satisfy the weak energy condition
with respect to $\g_n$, and if $(\g_n)_{n\in\mathbb{N}}$ converges to
some $\g_0$ as in \eqref{burnett convergence}, then 
{\begin{equation*}
R_{\mu\nu}(\g_0) - \half R(\g_0)(\g_0)_{\mu\nu} + \La (\g_0)_{\mu\nu}
= (T_0)_{\mu\nu},
\end{equation*}}\unskip
where $T_0$ is trace free and satisfy the weak energy condition with
respect to $\g_0$.
\end{theo}

The conclusion of Theorem~\ref{theo GW} completely rules out the
possibility of dark matter being described by backreaction under the
convergence regime \eqref{burnett convergence} (see
also~\cite{Green2013} where the authors provide interesting explicit
examples of such regime). Note also that the conclusion of
Theorem~\ref{theo GW} are in accordance with Burnett's conjecture,
since the stress energy tensor of a massless Vlasov field is indeed
traceless and satisfies the weak energy condition. Moreover,
Theorem~\ref{theo GW} fails to describe precisely the structure of the
effective stress energy tensor, or the fact that the Vlasov transport
equation is satisfied. However, some features distinguish it from the
``PDE'' results described in Sections~\ref{section U(1) GWC}
and~\ref{section double null} below: it allows for both cosmological
constant and sequences of non-vacuum spacetimes, it does not require
any symmetry nor gauge choice, it allows for both concentration and
oscillation. For all these reasons, Theorem~\ref{theo GW} reaches a
high level of generality compared to the PDE results.

In the physics community, the article~\cite{Green2011} has led to
strong debates on the nature of backreaction, see for
instance~\cite{Buchert2015,Green2015}. It also provoked an important
increase in the number of citations of Burnett's original
article~\cite{Burnett1989}, and it raised the interest of
mathematicians and analysts working on the Einstein equations.

\subsection{Low-regularity and two different strategies}\label{section
strategy}

In Sections~\ref{section U(1) GWC} and~\ref{section double null} below,
we will present the
results~\cite{Guerra2021,Huneau2018a,Huneau2024b,Huneau2024c,Huneau2024,Luk2020,Touati2023a,Touati2024} 
on Burnett's Conjecture~\ref{conj}. In order to motivate the splitting
of the results between these two sections and their inner structures,
we need to address a crucial feature of Burnett's conjecture, already
hinted at in Section~\ref{section what}. To put it plainly, Burnett's
conjecture is intrinsically a low-regularity phenomenon from the point
of view of the Einstein vacuum equations. Indeed, the most generic
well-posedness result for the Einstein vacuum equations is the
celebrated bounded $L^2$ curvature theorem, which can be formulated as
follows:

\begin{theo}[\cite{Klainerman2015}]\label{theo H2}
The time of existence, with respect to a maximal foliation, of a
classical solution to the Einstein vacuum equations depends only on the
$L^2$ norm of the curvature and second fundamental form of the initial
data set and on a lower bound of its volume radius.
\end{theo}

In other words, the Einstein vacuum equations are well-posed in $H^s$
if $s\geq 2$. Theorem~\ref{theo H2} is expected to be sharp, meaning
that the \textit{if} in the previous sentence is supposedly an
\textit{if and only if}\footnote{The presence of symmetries can lower
the regularity threshold, see for instance Christodoulou's BV theory in
spherical symmetry~\cite{Christodoulou1993bis}.}. However, in order to
produce a nontrivial backreaction, a sequence $(\g_\la)_{\la\in(0,1]}$
of solutions to the Einstein vacuum equations must necessarily be
unbounded in any $H^s$ for $s>1$, since otherwise the convergence of
$\dr_\mu\pth{(\g_\la)_{\a\b} - (\g_0)_{\a\b}}$ would be strong. This
shows that Burnett's conjecture is a low-regularity phenomenon and that
the mere existence of sequences $(\g_\la)_{\la\in(0,1]}$ of solutions
converging to some $\g_0$ on a time interval uniform in $\la$ and being
unbounded in $H^s$ for $s>1$ is not at all guaranteed by the general
theory (said differently, the best we can do {a priori} is a
time of existence of order $\la$). Since such sequences are the
starting point of any theorems on the direct conjecture, they might be
empty theorems. This leaves us with two possible strategies:

\begin{itemize}
\item First, if in a given setting (either a symmetry class or a
particular gauge choice), no improvement of Theorem~\ref{theo H2} is
available, then one needs to construct concrete examples of such
sequences unbounded in $H^s$ for $s>1$ bypassing the general
obstruction, which is equivalent to proving the reverse conjecture!
This motivates discussing first the reverse conjecture in the two
frameworks fitting this category, and then the corresponding results on
the direct conjecture (which are then ensured to be non-empty).

\item Second, if in a given setting it is possible to somewhat improve
Theorem~\ref{theo H2}, in the sense that the time of existence of a
solution in this setting can be shown to depend only on the $H^1$ norm
of the data in \emph{some} directions, then the existence of the
unbounded sequences would be ensured. As we will see, in this case
proving the full Conjecture~\ref{conj} actually reduces to proving its
direct part.
\end{itemize}
Section~\ref{section U(1) GWC} will present the results fitting into
the first category,
i.e.~\cite{Guerra2021,Huneau2018a,Huneau2024b,Huneau2024c,Huneau2024,Touati2023a,Touati2024}, 
in which Burnett's conjecture is studied in $\mathbb{U}(1)$ symmetry
and generalized wave coordinates. Section~\ref{section double null}
will present the only result fitting into the second catagory,
i.e.~\cite{Luk2020}, which is concerned with angularly regular
spacetimes in double null gauge.

\section{The $\mathbb{U}(1)$ symmetry and generalized wave coordinates}\label{section U(1) GWC}

In this section, we discuss first the
results~\cite{Huneau2018a,Huneau2024,Touati2023a,Touati2024}, which
attack the reverse conjecture via multiphase geometric optics
constructions, and then the
results~\cite{Guerra2021,Huneau2024b,Huneau2024c}, which prove the
direct conjecture by means of microlocal defect measures.

\subsection{The reverse conjecture}\label{section reverse}

As explained in Section~\ref{section strategy}, without any improvement
of the general theory, proving the reverse conjecture amounts to the
construction of particular solutions defined by special ansatz. As
mentioned in Section~\ref{section first discussion}, oscillation and
concentration are the two different phenomena that, if present at the
level of the first order derivatives of the metric, can cause
boundedness in $H^1$ and unboundedness in $H^s$ for $s>1$ as required
by the Burnett regime \eqref{burnett convergence}. If one chooses
oscillation, then one actually studies the geometric optics
approximation for the Einstein vacuum equations already considered by
Choquet-Bruhat in~\cite{ChoquetBruhat1969}.

\subsubsection{Geometric optics}\label{section GO}

Broadly speaking, the field of geometric optics is concerned with the
propagation of waves, and in particular with the description of rays
out of it (see~\cite{Metivier2009,Rauch2012} for a more in depth
introduction to this field). Concretely, the goal is to construct
high-frequency solutions to nonlinear hyperbolic systems. These
solutions are defined via multiscale asymptotic expansions of the form
{\begin{equation}\label{ansatz GO} 
u^\e \sim \e^p\sum_{n\geq 0} \e^n
U_n\pth{ t,x,\frac{\ffi}{\e}},
\end{equation}}\unskip
where $\e$ is a small
parameter and where the $U_n$ are periodic with respect to their third
variable. This ansatz is very much in the spirit of WKB expansions
(after Wentzel, Kramers and Brillouin) from the 1920's used to recover
classical mechanics from quantum mechanics. In the case considered
in~\cite{Lax1957,Garding1964} of a linear hyperbolic system, the
oscillating profiles $U_n$ can be shown to solve a hierarchy of
transport equations along the ``rays'' defined by the phase $\ffi$, which
needs to solve the eikonal equation associated to the differential
operator. The first to have considered nonlinear systems is actually
Yvonne Choquet-Bruhat herself, after having followed Leray's (``mon maitre'', 
as she calls him in~\cite{ChoquetBruhat2016}) lectures at
Princeton, see~\cite{ChoquetBruhat1967,ChoquetBruhat1969a}. Of course,
the article~\cite{ChoquetBruhat1969} is another instance of a geometric
optics being applied to a nonlinear system, but with the subtlety that
for the Einstein vacuum equations true hyperbolicity comes at a cost of
a gauge choice. See also~\cite{Jeanne2002,Salvi2024} for examples of
geometric optics constructions on gauge-invariant semilinear systems.
Given a nonlinear system, the basic questions are: can we construct
formal solutions of the form \eqref{ansatz GO}? are these approximate
solutions stable, i.e.\ can we solve for a remainder in the expansion
and transform them into exact solutions? In general, this is a very
difficult problem strongly depending on the system and on the strength
of the wave.

The parameter $p\geq 0$ in \eqref{ansatz GO} describes the strength of
the wave. If $p$ is too large, no nonlinear effects will be observed in
the transport hierarchy for the profiles. If one decreases the value of
$p$, one can reach a particular value where nonlinear interactions
terms enter the transport equation for the first profile, thus making
it nonlinear. This is the regime of weakly nonlinear geometric optics,
see~\cite{Hunter1983,Joly1993} for classical results. For quadratic
systems such as the Einstein equations, the critical value is $p=1$. We
remark crucially that~\cite{ChoquetBruhat1969} considers precisely the
weakly nonlinear regime for the Einstein vacuum equations, and that if
$p=1$ then an ansatz of the form~\eqref{ansatz GO} precisely enters
Burnett's convergence regime \eqref{burnett convergence}. In
conclusion, we see that proving the reverse Burnett conjecture via
oscillations is equivalent to proving the stability of the geometric
optics approximation for the Einstein vacuum equations in the weakly
nonlinear regime.

\subsubsection{The $\mathbb{U}(1)$ symmetry}\label{section U(1) reverse}

The first setting where the reverse Burnett conjecture has been studied
is the one of $\mathbb{U}(1)$ symmetry, that is when the spacetime has
a spacelike Killing vector field. The reduction of the $3+1$ Einstein
vacuum equations in this case can be deduced from the general
Kaluza--Klein theory (found for instance in Appendix VII
of~\cite{ChoquetBruhat2009}). A general metric on a $3+1$ manifold with
$\dr_{x^3}$ Killing is of the form
{\begin{equation*}
\g  = {\mathrm{e}}^{-2\ffi} g + {\mathrm{e}}^{2\ffi}\pth{ \d x^3 + A_\a\d x^\a },
\end{equation*}}\unskip
where $g$ is a $2+1$ Lorentzian metric, $\ffi$ is a scalar function and
$A$ a 1-form. One can prove that the Einstein vacuum equations for $\g$
as above are equivalent to the following Einstein-wave map system in
$2+1$ dimension
{\begin{equation}\label{system U(1) vacuum}
\left\{
\begin{array}{rcl}
\Box_g \ffi & =& - \half {\mathrm{e}}^{-4\ffi} g^{-1}(\d \om,\d \om), \\
\Box_g \om &=& 4 g^{-1}(\d \om,\d\ffi), \\ 
R_{\mu\nu}(g) & =& 2\dr_\mu\ffi\dr_\nu\ffi + \half {\mathrm{e}}^{-4\ffi}\dr_\mu\om\dr_\nu\om,
\end{array}
\right.
\end{equation}}\unskip
where $\om$ is related to $A$. This system has been studied for
instance in~\cite{ChoquetBruhat1994,Choquetbruhat2001}, motivated by
its connection with cosmology, and the constraint equations for
\eqref{system U(1) vacuum} are solved in~\cite{Huneau2016a}. Several
choices of gauge are possible for \eqref{system U(1) vacuum}, for
instance generalized wave coordinates in~\cite{Huneau2018b} or null
geodesic in~\cite{Alexakis2022}, but it is the elliptic gauge which
proves to be useful for Burnett's conjecture (or other low-regularity
problems such as~\cite{Luk2021,Luk2023}). It is defined in the
following way: after having decomposed $g$ with lapse and shift, we ask
its spacelike part to be conformal to the Euclidean metric (thanks to
the uniformization theorem of surfaces) and each time slice to be
maximal, i.e.\ with zero mean curvature. We now state the main result
of~\cite{Huneau2018a} by Huneau and Luk:

\begin{theo}[\cite{Huneau2018a}]\label{theo U(1) reverse}
If $\pth{g_0,\ffi_0,\om_0,F_\A,u_\A}$, with $\A$ belonging to a finite
set, is a generic solution on $[0,1]\times \RRR^2$ of the Einstein-null
dusts system in $\mathbb{U}(1)$ symmetry in the elliptic gauge and
close to Minkowski, then there exists a sequence
$(g_\la,\ffi_\la,\om_\la)_{\la\in(0,1]}$ of solutions of \eqref{system
U(1) vacuum} in the elliptic gauge on $[0,1]\times \RRR^2$ and such
that $(g_\la,\dr g_\la, \ffi_\la,\om_\la) \longrightarrow (g_0,\dr g_0,
\ffi_0,\om_0)$ uniformly and $(\dr\ffi_\la,\dr\om_\la)
\longrightarrow(\dr\ffi_0,\dr\om_0) $ weakly when $\la$ tends to 0.
\end{theo}

The Einstein-null dusts system in $\mathbb{U}(1)$ symmetry mentioned in
Theorem~\ref{theo U(1) reverse} is the equivalent of \eqref{system U(1)
vacuum} with the addition of the stress energy tensor of the null dusts
system, see \eqref{null dusts}. The existence of the background
solution $\pth{g_0,\ffi_0,\om_0,F_\A,u_\A}$ follows from a
low-regularity existence result in elliptic gauge
from~\cite{Huneau2018}. If we denote by $\mathfrak{g}$ either the
lapse, the shift or the conformal factor, and by $\psi$ either $\ffi$
or $\om$, the system \eqref{system U(1) vacuum} in elliptic gauge can
be rewritten in the following schematic form
{\begin{equation}\label{system U(1) vacuum elliptic}
\left\{
\begin{array}{rcl}
\Box_g \psi & =& g^{-1}(\d \psi,\d \psi),\\ 
\De \mathfrak{g} & =& (\dr\psi)^2 + (\nabla \mathfrak{g})^2.
\end{array}
\right.
\end{equation}}\unskip
The sequence $(\mathfrak{g}_\la,\psi_\la)_{\la\in(0,1]}$ is defined by
a high-frequency ansatz, a simplified version of which being
{\begin{equation}\label{ansatz U(1)}
\begin{array}{rcl}
\psi_\la & =&\displaystyle \psi_0 + \la \sum_\A \psi^{(1)}_\A\pth{\frac{u_\A}{\la}} + \la^2 
\pth{\sum_\A \psi^{(2)}_\A\pth{\frac{u_\A}{\la}} + \sum_{\A\neq\B,\pm}
\psi^{(2,\pm)}_{\A\B}\pth{\frac{u_\A\pm u_\B}{\la}}} + \text{remainder},\\ 
\mathfrak{g}_\la & =&\displaystyle \mathfrak{g}_0 + \la^2 \pth{\sum_\A 
\mathfrak{g}^{(2)}_\A\pth{\frac{u_\A}{\la}} + \sum_{\A\neq\B,\pm}
\mathfrak{g}^{(2,\pm)}_{\A\B}\pth{\frac{u_\A\pm u_\B}{\la}}} + \text{remainder},
\end{array}
\end{equation}}\unskip
where, for instance, by $f\pth{\sfrac{u_\A}{\la}}$ we mean that $f$ is a
quantity depending in a periodic manner on $\sfrac{u_\A}{\la}$ (as the
profiles $U_n$ in \eqref{ansatz GO}). By plugging \eqref{ansatz U(1)}
in \eqref{system U(1) vacuum elliptic}, we derive transport equations
for the profiles $\psi^{(i)}$ and ``algebraic'' equations for the
profiles $\mathfrak{g}^{(i)}$ (the solvability of the latter requires a
coherence assumption on the $u_\A$'s, as it is common in multiphase
geometric optics). Solving for the remainders in \eqref{ansatz U(1)}
over the required time scale is the most technical part of the proof
and relies crucially on the structure of \eqref{system U(1) vacuum
elliptic}. From the present perspective, this structure can be
summarized as follows: the dynamics and the geometry are completely
decoupled into a quasilinear hyperbolic system with the null condition
and a semilinear elliptic system without the null condition. This
observation explains why there is no $\la$ term in $\mathfrak{g}_\la$,
or equivalently why $\dr\mathfrak{g}_\la$ converges strongly, as
opposed to $\dr\psi_\la$. It also sheds light on the discussion of
Section~\ref{section what}: the lack of null condition is compensated
here by an elliptic operator, very much in the spirit of the Coulomb
gauge used in~\cite{Klainerman2015}. To conclude the discussion of
Theorem~\ref{theo U(1) reverse}, let us clarify how dust-like
backreaction can arise. If say
$\psi^{(1)}_\A\pth{\sfrac{u_\A}{\la}}=\cos\pth{\sfrac{u_\A}{\la}}\tilde{\psi}^{(1)}_\A$, 
then the quadratic term $\dr_\mu\psi_\la\dr_\nu\psi_\la$ in
\eqref{system U(1) vacuum} or \eqref{system U(1) vacuum elliptic}
becomes
{\begin{eqnarray*}
\dr_\mu\psi_\la\dr_\nu\psi_\la & =& \dr_\mu\psi_0\dr_\nu\psi_0 + 
\dhalf \sum_{\A,\B} \pth{\tilde{\psi}^{(1)}_\A}^2  \dr_\mu u_\A \dr_\nu u_\A \\
&&\quad -\, \dhalf  \sum_\A \sin\pth{\frac{u_\A}{\la}} \tilde{\psi}^{(1)}_\A 
\dr_{\mathop{(}\mu}\psi_0  \dr_{\nu\mathop{)}} u_\A - \dhalf \sum_{\A,\B} 
\pth{\tilde{\psi}^{(1)}_\A}^2   
\cos\pth{\frac{2u_\A}{\la}}\dr_\mu u_\A \dr_\nu u_\A\\
&&\quad +\, \sum_{\A\neq\B} \sin\pth{\frac{u_\A}{\la}} 
\sin\pth{\frac{u_\B}{\la}} \tilde{\psi}^{(1)}_\A\tilde{\psi}^{(1)}_\B 
\dr_\mu u_\A \dr_\nu u_\B + \GO{\la}.
\end{eqnarray*}}\unskip
Thanks to the coherence assumption, the last two lines converge weakly
to 0 when $\la$ tends to 0, showing how a new resonant term appears in
the high-frequency limit.

In the forthcoming~\cite{Huneau2024}, Huneau and Luk prove that it is
possible to rigorously send the number of dusts to infinity. Relying on
an improved existence result in this setting~\cite{Touati2022c}, they
perform a double limit argument, i.e.\ construct ansatz of the form
\eqref{ansatz U(1)} with uniform estimates with respect to both $\la$
and the number of dusts. In particular, this allows to target general
solutions to the massless Einstein--Vlasov system, by approximating any
measure which is absolutely continuous with respect to the Lebesgue
measure by sums of Dirac measures.

\subsubsection{Generalized wave coordinates}\label{GWC reverse}

In this section, we discuss results on the reverse Burnett conjecture
for the Einstein vacuum equations in generalized wave coordinates. To
define them, we give the expression of the Ricci tensor in any
coordinates system:
{\begin{equation}\label{ricci}
2R_{\a\b}(\g)  = - \tilde{\Box}_\g \g_{\a\b} + \g_{\rho\mathop{(}\a}\dr_{\b\mathop{)}}
H^\rho + H^\rho\dr_\rho\g_{\a\b} + P_{\a\b}(\g)(\dr\g,\dr\g), 
\end{equation}}\unskip
where $\tilde{\Box}_\g : = \g^{\mu\nu}\dr_\mu\dr_\nu$, $H^\rho : =
\g^{\mu\nu}\Ga^\rho_{\mu\nu}$ and $P_{\a\b}(\g)(\dr\g,\dr\g)$ contains
every quadratic semilinear terms. Constructing coordinates such that $H
= 0$ while simultaneously solving the reduced equations $
\tilde{\Box}_\g \g_{\a\b}=P_{\a\b}(\g)(\dr\g,\dr\g)$ has been shown to
be possible in the pioneering~\cite{ChoquetBruhat1952}. Since $H^\rho =
- \Box_\g x^\rho$, such coordinates are called wave coordinates. They
are in fact a special instance of the so-called generalized wave
coordinates, where instead of asking $H=0$ we allow $H$ to be equal to
a non-zero quantity depending on $\g$ and not on $\dr\g$. In the
literature, this extra freedom has been used for instance to help
determine some spacelike or timelike asymptotic behaviour in stability
proofs such as~\cite{Fang2021,Fang2022,Hintz2018,Huneau2018b} or even
in numerical simulations~\cite{Pretorius2005}. In the context of
Burnett's conjecture, the role of $H$ is somewhat different. First, the
main result of~\cite{Touati2024} by the author of this review:

\begin{theo}[\cite{Touati2024}]\label{theo GWC reverse}
If $\pth{\g_0,F_\A,u_\A}$, with $\A$ belonging to a finite set, is a
generic solution on $[0,1]\times \RRR^3$ of \eqref{null dusts} in wave
coordinates and close to Minkowski, then there exists a sequence
$(\g_\la)_{\la\in(0,1]}$ of solutions of the Einstein vacuum equations
in generalized wave coordinates on $[0,1]\times \RRR^3$ and such that
$\g_\la \longrightarrow \g_0$ uniformly and $\dr\g_\la \longrightarrow
\dr\g_0$ weakly when $\la$ tends to 0.
\end{theo}

The existence of the background solution $\pth{\g_0,F_\A,u_\A}$ is
proved in~\cite{ChoquetBruhat2006}, and the case of a single null dust
is considered in~\cite{Touati2023a,Touati2023b}. Theorem~\ref{theo GWC
reverse} is the strict equivalent of Theorem~\ref{theo U(1) reverse}
without any symmetry assumption and the multiphase geometric optics
strategy is also followed as $\g_\la$ is of the form
{\begin{equation}\label{ansatz GWC}
\g_\la  = \g_0 + \la \sum_\A \g^{(1)}_\A\pth{\frac{u_\A}{\la}} + 
\la^2 \pth{\sum_\A \g^{(2)}_\A\pth{\frac{u_\A}{\la}} + 
\sum_{\A\neq\B,\pm}\g^{(2,\pm)}_{\A\B}\pth{\frac{u_\A\pm u_\B}{\la}}} + \text{remainder}.
\end{equation}}\unskip
Note that if we forget about the remainder and consider only one dust,
the ansatz \eqref{ansatz GWC} is precisely the one considered in the
approximate construction~\cite{ChoquetBruhat1969}. By plugging
\eqref{ansatz GWC} into \eqref{ricci} and asking for
$R_{\a\b}(\g_\la)=0$, we derive a system of transport equations and
polarization conditions for the profiles $\g^{(i)}$, the former coming
from the truely hyperbolic terms in \eqref{ansatz GWC} and the latter
from the gauge terms $\dr H$. Contrary to the elliptic gauge in
$\mathbb{U}(1)$ symmetry and because of the absence of decoupling, the
transport equations and polarization conditions must be shown to be
compatible at any order in $\la$. The polarization conditions are of
the schematic form
{\begin{equation}\label{pola condition GWC}
\Pol{\g^{(1)}_\A}{u_\A}  = 0, 
\quad \Pol{\g^{(2)}_\A}{u_\A}  = 
\left|\g^{(1)}_\A \right|^2_{\g_0}\d u_\A\otimes \d u_\A, \quad 
\g^{(2,\pm)}_{\A\B}- \Pol{\g^{(2,\pm)}_{\A\B}}{u_\A\pm u_\B}  = \g^{(1)}_\A \star \g^{(1)}_\B,
\end{equation}}\unskip
where the $\star$ denotes some tensor contractions and $\mathbf{P}$ has
been defined in \eqref{def pola}. The first condition in \eqref{pola
condition GWC} is the equivalent of Choquet-Bruhat's \eqref{pola
condition CB} and the non-trivial RHS's of the last two conditions are
due to the lack of null condition of the semilinear terms in
\eqref{ricci}. The solvability of the last two conditions in
\eqref{pola condition GWC} is a structural miracle, in some sense
equivalent to the ellipticity of the equations for $\mathfrak{g}$ in
\eqref{system U(1) vacuum elliptic}. However, as opposed
to~\cite{Huneau2018a}, here the metrics $\g_\la$ don't satisfy the same
gauge conditions as their limit $\g_0$, precisely because of the last
two conditions in \eqref{pola condition GWC}, showing that $H$ plays
the role of a bin in which we hide undesired terms. Note that the first
condition in \eqref{pola condition GWC} still implies that 
{\begin{equation}\label{convergence gauge}
H(\g_\la) \longrightarrow H(\g_0), \quad \text{uniformly}, 
\quad \text{and} \quad \dr H(\g_\la) \longrightarrow \dr H(\g_0)\quad 
\text{weakly},
\end{equation}}\unskip
when $\la$ tends to 0.

As explained in Section~\ref{section strategy}, the existence of an
exact solution of the form \eqref{ansatz GWC} up to time 1 does not
follow from any general result. This can be directly seen on the wave
equation satisfied by the remainder in \eqref{ansatz GWC}, which, due
to the quasilinearity of the wave operator in \eqref{ricci} is coupled
to the transport equation for $\g^{(2)}$, ultimately leading to the
apparent loss of one derivative (in consistence with Burnett's regime
living one derivative lower than what Theorem~\ref{theo H2} provides).
Regaining this derivative makes crucial use of the high-frequency
ansatz \eqref{ansatz GWC}.

\subsection{The direct conjecture}

In this section, we review the results
in~\cite{Guerra2021,Huneau2024b,Huneau2024c} on the direct conjecture
in $\mathbb{U}(1)$ symmetry or in generalized wave coordinates. These
results will assume the existence of a sequence of vacuum metrics
$(\g_\la)_{\la\in(0,1]}$ converging in some sense, actually in a
stronger compared to \eqref{burnett convergence}, to a given metric
$\g_0$. Their goal is then to compute the Einstein tensor of $\g_0$ and
show that it satisfies a transport equation. The main tool are
microlocal defect measures and the structural miracle is compensated
compactness.

\subsubsection{Microlocal defect measures}

Microlocal defect measures (sometimes also called H-measures) have been
introduced to detect the lack of strong convergence of a sequence in
$L^2$. We refer to~\cite{Burq1997} for a nice introduction to these
measures and to their semi-classical counterparts. Introduced
in~\cite{Gerard1991,Tartar1990}, microlocal defect measures are
associated to sequences $(u_k)_{k\in\mathbb{N}}$ of functions in
$L^2(\mathbb{R}^{d+1},\mathbb{C})$ and converging weakly to 0, and in
these references it is shown that there exists a non-negative
complex-valued Radon measure $\mu$ (the so-called microlocal defect
measure) such that
{\begin{equation}\label{MDM}
\lim_{k\to +\infty}  \ps{A u_k,u_k}_{L^2(\mathbb{R}^{d+1},
\mathbb{C})} = \int_{S^*\mathbb{R}^{d+1}}  a(x,\xi)\d\mu, 
\end{equation}}\unskip
for all order 0 pseudo-differential operator $A$ with principal symbol
$a$ homogeneous of degree 0, and where $S^*\mathbb{R}^{d+1}$ is the
cosphere bundle of $\mathbb{R}^{d+1}$. For instance, if
$u_k(x)=k^{\sfrac{d+1}{2}}\chi(k(x-x_0))$ (with $\chi$ compactly
supported) then $\d\mu$ is of the form $\de_{x_0}\otimes \d\nu$, and if
$u_k(x)=\ffi(x)\cos(x\cdot\om)$ then $\d\mu=|\ffi|^2\d x\otimes
\de_\om$. These two simple examples illustrate respectively lack of
strong convergence by concentration and oscillation and the second one
reminds us of the backreaction's dust nature obtained via geometric
optics (see Section~\ref{section reverse}). Microlocal defect measures
satisfy several important properties, the first of them being
localization: if $P$ is an order $m$ differential operator with
principal symbol $p$, and if $(P(u_k))_{k\in\mathbb{N}}$ is relatively
compact in $H^{-m}_{\mathrm{loc}}$ then $p\d\mu=0$. This shows that
$\d\mu$ is supported (in phase space) on the zero set of the symbol
$p$. In the context of Burnett's conjecture, $P$ will always be a wave
operator associated to a Lorentzian metric, and the zero set of $p$ in
this case is precisely $\mathcal{P}_0$, the zero mass shell! The second
key property of microlocal defect measures in the context of wave
equations is that they propagate. This has been first proved in the
linear case in~\cite{Francfort1992}. As a toy model, consider a
sequence $u_k$ such that $\dr u_k$ and $\Box_{\mathbf{m}} u_k$ (where
$\mathbf{m}$ is the Minkowski metric) converge respectively weakly and
strongly to 0 in $L^2$, it can be shown that the microlocal defect
measure associated to derivatives of $u_k$ satisfy
{\begin{equation}\label{poisson}
\left\{ \mathbf{m}^{\a\b}\xi_\a\xi_\b, a \right\}\frac{\d\nu}{|\xi|^2} =0
\end{equation}}\unskip
for all $a$ homogeneous of degree 1, and where $\{\cdot,\cdot\}$
denotes the Poisson bracket in the $(x,\xi)$ variables. 
The relation \eqref{poisson} is a weak formulation of the Vlasov
equation~\eqref{vlasov}.

Note that microlocal defect measures are only sensible to the direction
$\sfrac{\xi}{|\xi|}$ of the lack of strong convergence, and in this
regard semi-classical defect measures are more precise. Both types of
defect measures are powerful tools with applications in the theory of
homogenization but also to the control of wave equations, see for
instance~\cite{Bardos1992,Burq1997a}. However, as mentioned earlier in
this text, the phenomenon at the heart of Burnett's conjecture is
compensated compactness, the most famous example of which is the
div-curl lemma (see~\cite{Murat1978,Tartar1979}). Generalized
in~\cite{Gerard1991}, it stipulates that one can pass to the limit in
particular quadratic expressions. For the application to Burnett's
conjecture, one would like a similar statement for cubic interactions.
Prior to the results discussed in the sections below, the only known
example of trilinear compensated compactness was the
result~\cite{Joly1995}, where it is shown that $u^{(1)}_k u^{(2)}_k
u^{(3)}_k$ converges weakly to $u^{(1)}_\infty u^{(2)}_\infty
u^{(3)}_\infty$ if $u^{(i)}_k$ converges weakly to $u^{(i)}_\infty$ in
$L^2$ and if there exists vector fields $X^{(i)}$ satisfying a
particular coherence condition and such that $X^{(i)}u^{(i)}_k$ is
bounded in $L^2$. 

As geometric optics is the perfect tool to prove Burnett's reverse
conjecture, it turns out that microlocal defect measures are the
perfect tools to prove Burnett's direct conjecture, as we will now
show.

\subsubsection{The $\mathbb{U}(1)$ symmetry}

In this section, we describe the work~\cite{Huneau2024b} and the
subsequent~\cite{Guerra2021}, which consider spacetimes enjoying
$\mathbb{U}(1)$ symmetry in the elliptic gauge, a framework already
described in Section~\ref{section U(1) reverse}. To state the main
theorems of these works, we first define the kinetic spacetimes under
consideration. If $\d\nu$ is a non-negative Radon measure on
$S^*\mathbb{R}^{2+1}$, we say that $(g,\ffi,\om,\d\nu)$ is a
radially-averaged measure solution to the massless Einstein--Vlasov
system in $\mathbb{U}(1)$ symmetry if the first two equations in
\eqref{system U(1) vacuum} hold, if for all vector field $X$ smooth and
compactly supported we have
{\begin{equation}\label{weak1}
\int_{\mathbb{R}^{2+1}}R_{\mu\nu}(g)X^\mu X^\mu  = 
\int_{\mathbb{R}^{2+1}}\pth{ 2(X\ffi)^2 + \dhalf {\mathrm{e}}^{-4\ffi}(X\om)^2 } + 
\int_{S^*\mathbb{R}^{2+1}}g(\xi,X)^2\frac{\d\nu}{|\xi|^2},
\end{equation}}\unskip
if for all $f$ compactly supported on $\mathcal{M}$ we have
{\begin{equation}\label{weak2}
\int_{S^*\mathbb{R}^{2+1}}f g^{\a\b}\xi_\a\xi_\b \frac{\d\nu}{|\xi|^2}  = 0,
\end{equation}}\unskip
and if for all $a$ defined on $T^*\mathbb{R}^{2+1}$ and homogeneous of
degree 1 we have
{\begin{equation}\label{weak3}
\int_{S^*\mathbb{R}^{2+1}} \pth{ g^{\a\b}\xi_\a \dr_{\b} a - 
\dhalf \dr_\mu g^{\a\b} \xi_\a\xi_\b \dr_{\xi_\mu}a } 
\frac{\d\nu}{|\xi|^2}  = 0.
\end{equation}}\unskip
Note that, when put together, Equations~\eqref{weak1}--\eqref{weak3}
are a weak formulation of the massless Einstein--Vlasov system in
$\mathbb{U}(1)$ symmetry. The following is the main result
of~\cite{Huneau2024b} by Huneau and Luk and of~\cite{Guerra2021} by
Guerra and Teixeira da Costa:

\begin{theo}[{\cite{Huneau2024b,Guerra2021}}]\label{theo U(1) direct}
Let $(g_n,\ffi_n,\om_n)_{n\in\mathbb{N}}$ a sequence of solutions to
\eqref{system U(1) vacuum} in the elliptic gauge such that there exists
$(g_0,\ffi_0,\om_0)$ satisfying the elliptic gauge conditions and such
that $(g_n,\ffi_n,\om_n)$ converges to $(g_0,\ffi_0,\om_0)$ uniformly
and weakly in $W^{1,p}$ for some $p>2$, then there exists $\d\nu$ a
non-negative Radon measure on $S^*\mathbb{R}^{2+1}$ such that
$(g_0,\ffi_0,\om_0,\d\nu)$ satisfy \eqref{weak1} and \eqref{weak2}. If
moreover there exists a sequence $(\la_n)_{n\in\mathbb{N}}$ converging
to 0 such that for all compact $K\subset \mathbb{R}^{2+1}$ we have 
{\begin{equation}\label{estimee HF}
\sum_{k=0}^{q} \la_n^{k-1} \l \dr^k\pth{ (g_n,\ffi_n,\om_n) - 
(g_0,\ffi_0,\om_0) } \r_{L^\infty(K)} \leq C_K,
\end{equation}}\unskip
with $q=4$ in~\cite{Huneau2024b} and $q=2$ in~\cite{Guerra2021}, then
$(g_0,\ffi_0,\om_0,\d\nu)$ satisfy \eqref{weak3}.
\end{theo}

Note that the existence of sequences
$(g_n,\ffi_n,\om_n)_{n\in\mathbb{N}}$ as in Theorem~\ref{theo U(1)
direct} and moreover unbounded in $W^{s,p}$ if $s>1$ (so that the
measure $\d\nu$ obtained has a chance to be non-zero) is a consequence
of~\cite{Huneau2018a}. As it can be seen in Theorem~\ref{theo U(1)
direct}, the existence of the measure $\d\nu$ such that
$(g_0,\ffi_0,\om_0,\d\nu)$ satisfy (weakly) the Einstein equations with
a massless kinetic stress energy tensor only follows from the Burnett
convergence regime \eqref{burnett convergence} (allowing both
concentration and oscillation), while the fact that $\d\nu$ solves
(weakly) a Vlasov equation requires the stronger high-frequency
assumption \eqref{estimee HF}. We now discuss these two parts of
Theorem~\ref{theo U(1) direct} successively, using again our concise
notations $(\mathfrak{g},\psi)$ from Section~\ref{section U(1)
reverse}.

The existence of the measure $\d\nu$ follows from the general existence
results~\cite{Gerard1991,Tartar1990} applied to the sequence $\pth{ \dr
(\psi_n-\psi_0)}_{n\in\mathbb{N}}$, which by assumption converge to 0
only weakly. We wish now to pass to the limit in the system
\eqref{system U(1) vacuum elliptic} that $(\mathfrak{g}_n,\psi_n)$
satisfy. For the wave map system, this is possible thanks to standard
bilinear compensated compactness applied to the null form
$g^{-1}(\d\psi_n,\d\psi_n)$. For the Ricci equation, the elliptic gauge
allows to improve weak convergence to strong convergence for
$\nabla\mathfrak{g}_n$, so that we can deal with
$(\nabla\mathfrak{g})^2$, and thanks to the equivalent of \eqref{MDM}
here, the term $(\dr\psi_n)^2$ converges weakly to
$(\dr\psi_0)^2+\xi^2\sfrac{\d\nu}{|\xi|^2}$. The support property of
$\d\nu$ follows again from bilinear compactness used together with the
localization property of microlocal defect measure and the wave maps
system for $\psi_n-\psi_0$. In conclusion, we have proved the first
part of Theorem~\ref{theo U(1) direct}.

The most difficult part of~\cite{Huneau2024b,Guerra2021} is to deduce
the weak Vlasov equation~\eqref{weak3} from the additional
high-frequency assumption~\eqref{estimee HF}. We repeat that a similar
statement was known for linear wave equations, while here a quasilinear
hyperbolic-elliptic system is considered and trilinear compensated
compactness needs to be uncover. The derivation relies fully on the
wave map structure of the equations for $\psi$, both for the semilinear
terms satisfying the null condition and for the coupling between the
two components of the wave map, and on the coupling to the elliptic
equations for the metric in order to handle quasilinear terms. 

\subsubsection{Generalized wave coordinates} 

We describe now the work~\cite{Huneau2024c}, which proves Burnett's
direct conjecture in generalized wave gauge. This framework has already
been discussed in Section~\ref{GWC reverse} above. In consistence with
Theorem~\ref{theo GWC reverse} and in particular \eqref{convergence
gauge},~\cite{Huneau2024c} considers sequences
$(\g_n)_{n\in\mathbb{N}}$ of solutions to the Einstein vacuum equations
satisfying two sets of assumptions \eqref{assum1} and \eqref{assum2}
below, with $\g_0$ a given metric
{\begin{eqnarray}
\left\| \g_n - \g_0 \right\|_\infty + \left\| H(\g_n) - H(\g_0) 
\right\|_\infty \leq \la_n, \quad \left\| \dr\g_n \right\|_\infty &\leq& C, \label{assum1}\Seqnsplit
\la \left\| \dr^2\g_n \right\|_\infty  + \left\| \dr H(\g_n) \right\|_\infty &\leq& C, \label{assum2}
\end{eqnarray}}\unskip
where $C>0$ and $(\la_n)_{n\in\mathbb{N}}$ is a given sequence
converging to 0. The following result from~\cite{Huneau2024c} is due to
Huneau and Luk:

\begin{theo}[\cite{Huneau2024c}]\label{theo GWC direct}
Let $(\g_n)_{n\in\mathbb{N}}$ be a sequence of solutions to the
Einstein vacuum equations and $\g_0$ a metric. If \eqref{assum1} holds,
then there exists a non-negative Radon measure $\d\mu$ on
$T^*\mathbb{R}^{3+1}$ supported on the zero mass shell of $\g_0$ and
such that
{\begin{equation*}
\int_{\mathbb{R}^{3+1}}\psi R_{\mu\nu}(\g_0) = 
\int_{S^*\mathbb{R}^{3+1}}\psi \xi_\mu\xi_\nu\d\mu
\end{equation*}}\unskip
for all function $\psi$ smooth and compactly supported. If moreover
\eqref{assum2} holds, then for any $a$ homogeneous of degree 1 we have
$\{ \g_0^{\a\b}\xi_\a\xi_\b, a\} \d\mu =0$.
\end{theo}

Theorem~\ref{theo GWC direct} is the equivalent of Theorem~\ref{theo
U(1) direct} in generalized wave coordinates, which in particular
removes any symmetry assumption. Its conclusions are the same, and it
splits similarly into two parts. First, getting the existence of
$\d\mu$ and the Ricci equation for $(\g_0,\d\mu)$ requires only
\eqref{assum1} and follows again from~\cite{Gerard1991,Tartar1990}.
More precisely we set $\d\mu := \g_0^{\a\rho}\g_0^{\b\si}\pth{
\sfrac{1}{4}\d\mu_{\rho\b\a\si} - \sfrac12 \d\mu_{\rho\a\b\si}}$ where each
$\d\mu_{\rho\b\a\si}$ is associated to $\dr(\g_n-\g_0)$ in the
following way
{\begin{equation*}
\lim_{n\to+\infty}\ps{\dr_\ga (\g_n-\g_0)_{\a\b}, A\dr_\de 
(\g_n-\g_0)_{\rho\si}}_{L^2} = \int_{S^*\mathbb{R}^{3+1}}a 
\xi_\ga \xi_{\de}\d\mu_{\a\b\rho\si}.
\end{equation*}}\unskip
The measure $\d\mu$ is tuned to match the exact expression of the
semilinear terms in the Ricci tensor that don't satisfy the null
condition. Thanks to the div-curl lemma applied to the null form
$\g^{-1}(\d\ffi,\d\psi)$ and to estimates from~\cite{Ionescu2022}
applied to the null forms
$\dr_\mu\ffi\dr_\nu\psi-\dr_\mu\psi\dr_\nu\ffi$ (which are new compared
to the $\mathbb{U}(1)$ symmetry setting), one shows that the semilinear
terms without the null condition are the only one contributing when
passing to the limit in $R_{\mu\nu}(\g_n)=0$. The proof of the
propagation property of $\d\mu$, i.e.\ the fact that under the stronger
assumption \eqref{assum2} we have $\{ \g_0^{\a\b}\xi_\a\xi_\b,
a\} \d\mu =0$, requires to compute the weak limit of terms of the
form $\ps{\dr (\g_n-\g_0), A \pth{\dr (\g_n-\g_0),\dr (\g_n-\g_0) }
}_{L^2}$. One of the new issue compared to the $\mathbb{U}(1)$ symmetry
are the semilinear terms not satisfying the null condition, and
trilinear compensated compactness for these terms can be shown using
the generalized wave coordinates condition $\| \dr H(\g_n)
\|_\infty\leq C$. This shows again that the failure of null
condition can be tamed by gauge terms in the Ricci tensor.

\section{Angularly regular spacetimes}\label{section double null}

As explained in Section~\ref{section strategy}, the
article~\cite{Luk2020} deserves its own section because it is very
different in spirit from the articles discussed in Section~\ref{section
U(1) GWC} above. In particular, it provides an analytical framework
where both parts of Burnett's conjecture can be studied, and in fact
the reverse part will be deduced from the direct part.

This framework is the one of angularly regular spacetimes in double
null gauge. Requiring no symmetry assumption, the double null gauge has
proved very flexible over the years, see for instance the important
works~\cite{Christodoulou2009,Dafermos2019,Klainerman2003}. If $S$ is a
compact 2-surface and $u_*,\u_*>0$, the Lorentzian manifold
$\pth{[0,u_*]\times[0,\u_*]\times S ,\g}$ is said to be in double null
gauge if 
{\begin{equation}\label{g double null}
\g = -4\Om^2 \d u \d \u + \ga_{AB}\pth{ \d\th^A - b^A \d u}\pth{ \d\th^B - b^B \d u},
\end{equation}}\unskip
where $\Om>0$ is a function, $\ga$ is a Riemannian metric on $S$,
$(\th^1,\th^2)$ are coordinates on $S$ and $b$ is vector field tangent
to $S$. The local existence of spacetimes in double null gauges is
usually formulated with a characteristic initial value problem, where
characteristic data are put on two intersecting null hypersurfaces
$H_0=\{0\}\times[0,\u_*]\times S$ and
$\underline{H}_0=[0,u_*]\times\{0\}\times S$, see~\cite{Luk2012}.
In~\cite{Luk2017}, motivated by the study of impulsive gravitational
waves (see also~\cite{Luk2015}), Luk and Rodnianski prove a
low-regularity local existence result in double null gauge where the
$\dr_u$ and $\dr_{\u}$ derivatives of the metric coefficients in
\eqref{g double null} are allowed to be only in $L^2$, while the
$\dr_{\th^A}$ derivatives must be more regular than $H^1$. Compared to
Theorem~\ref{theo H2}, the spacetimes produced by~\cite{Luk2017} are
more regular in the angular directions $\th^A$ but require one
derivative less in the null directions $u$ and $\u$, the higher
regularity compensating for the lower regularity. The details of the
proof of the main result in~\cite{Luk2017} are out of the scope of this
text, let us just say that the structure of the Einstein vacuum
equations is exploited at its fullest and that ellipticity in the
angular directions plays again a significant role.

Since in the framework of~\cite{Luk2017} derivatives in the null
directions $u$ and $\u$ are allowed to be only in $L^2$, the Burnett
convergence regime \eqref{burnett convergence} (and thus Burnett's
conjecture itself) is in some sense no longer a low-regularity
phenomenon, if we restrict ourself to the situation where derivatives
in the null directions are the only ones lacking strong convergence.
This allows for a very soft proof of both sides of the conjecture with
the only restriction that the kinetic spacetimes under consideration
must be two null dusts propagating in the $u$ and $\u$ directions. More
precisely, there target kinetic spacetimes are angularly regular weak
solutions to the Einstein-null dusts system of the form
$(\g,\d\nu,\d\underline{\nu})$ where $\d\nu$ and $\d\underline{\nu}$
are two non-negative Radon measures such that
{\begin{equation}\label{weak}
\left\{
\begin{array}{rcl}
\displaystyle \int_{\mathcal{M}} \pth{ \mathrm{div} X \mathrm{div} Y - D_\mu X^\nu D_\nu Y^\mu }
\d \mathrm{Vol}_\g & =&\displaystyle \int_{\mathcal{M}} (X u)(Yu)
\d\nu + \int_{\mathcal{M}} (X\u)(Y\u)\d\underline{\nu},\vspace*{3pt}\\ 
\displaystyle\int_{\mathcal{M}} \g^{-1}(\d u, \d \ffi) \d\nu & =& 
\displaystyle\int_{\mathcal{M}} \g^{-1}(\d\u,\d\ffi)\d\underline{\nu} =0,
\end{array}
\right.
\end{equation}}\unskip
for all smooth $X$, $Y$ and $\ffi$. Note that the equations in
\eqref{weak} are respectively the weak formulation of the first and
second equations in \eqref{Einstein Vlasov} in the case of two null
dusts. We gather in the following statement rough versions of the main
results of~\cite{Luk2020} by Luk and Rodnianski:

\begin{theo}[\cite{Luk2020}]\label{theo luk}
The following hold:
\begin{enumerate}[(i)]
\item Any sequence of characteristic angularly regular vacuum initial
data satisfying uniform bounds gives rise to a sequence of angularly
regular vacuum spacetimes in double null gauge (with a common region of
existence). Up to a subsequence, this sequence converges in the sense
\eqref{burnett convergence} to an angularly regular weak solution to
\eqref{weak}.
\item Any angularly regular weak solution of \eqref{weak} in double
null gauge can be approximated in the sense \eqref{burnett convergence}
by a sequence of angularly regular vacuum spacetimes in double null
gauge.
\item Any characteristic angularly regular initial data for
\eqref{weak} gives rise to a unique angularly regular weak solution of
\eqref{weak}.
\end{enumerate}
\end{theo}

The three points of Theorem~\ref{theo luk} address respectively
Burnett's direct conjecture, Burnett's reverse conjecture and the local
theory for angularly regular measure-valued null dusts. As we will see,
points (ii) and (iii) are consequences of point (i), so we first
discuss point (i). The existence part of point (i) is a direct
consequence of the local existence result of~\cite{Luk2017}, as does
the existence of the limit, with the help of standard compactness
arguments using the uniform bounds provided by~\cite{Luk2017}. The fact
that the limit is an angularly regular weak solution to \eqref{weak}
follows from a careful analysis of each quadratic products
$\Ga^{(1)}\Ga^{(2)}$ in the structure equations of the double null
gauge, where the $\Ga^{(i)}$'s denote Christoffel symbols in the null
frame. It turns out that the only such products for which\footnote{Here
w-lim denotes the weak limit.}
{\begin{equation*}
\text{w-lim}\pth{\Ga^{(1)}\Ga^{(2)}} \neq \pth{ \text{w-lim}\Ga^{(1)}}\pth{\text{w-lim}\Ga^{(1)}}
\end{equation*}}\unskip
are $| \hat{\chi}|^2$ and $| \underline{\hat{\chi}}|^2$ in the
Raychaudhuri equations, where $\hat{\chi}$ and $\underline{\hat{\chi}}
$ are the shears of the null hypersurfaces. These products, which fully
characterize the lack of null condition for the Einstein vacuum
equations in this setting, have the additional property to have a sign
and to satisfy elliptic equations (the so-called Codazzi equations),
two facts that should not surprise the reader anymore. The Radon
measures representing the two dusts are then roughly defined by
{\begin{equation*}
\d\nu  : = \text{w*-lim}| \hat{\chi} |^2 - 
| \text{w-lim}\hat{\chi} |^2, \quad  \d\underline{\nu} 
: = \text{w*-lim}| \underline{\hat{\chi}} |^2 - 
| \text{w-lim} \underline{\hat{\chi}}  |^2.
\end{equation*}}\unskip
Once point (i) of Theorem~\ref{theo luk} is proved, points (ii) and
(iii) are reduced to initial data statements. Indeed, given a solution
of \eqref{weak}, if we are able to explicitly (say with oscillations)
approach its data by vacuum data, then point (i) and a uniqueness
low-regularity statement implies point (ii). Point (iii) follows from
similar arguments, and is the first local theory for measure-valued
null dusts, in particular allowing the construction of null dust shell
solutions (see~\cite{Hawking1990,Penrose1973} for their significance).

Appart from the fact that the kinetic spacetimes considered in the
framework of angularly regular spacetimes in double null gauge are
restricted to only two null dusts, Theorem~\ref{theo luk} is much more
powerful then the statements in Section~\ref{section U(1) GWC} and we
would like to discuss further some aspects of it. In addition to
providing a unified treatment of both parts of Burnett's conjecture,
Theorem~\ref{theo luk} completely drops geometric optics as a tool to
construct examples. As a consequence, the target kinetic spacetime can
be very rough, while the justification of geometric optics expansions
in Section~\ref{section reverse} relies strongly on the smoothness of
the target. Moreover, the target kinetic spacetime can be far from
Minkowski, while some smallness is required in Theorems~\ref{theo U(1)
reverse} and~\ref{theo GWC reverse}. Lastly, as opposed to
Theorems~\ref{theo U(1) direct} and~\ref{theo GWC direct} who needed
oscillations in order to get the Vlasov equation, lack of strong
convergence by concentration is allowed in Theorem~\ref{theo luk}.

\section{Conclusion}\label{section conclusion}

In this section we highlight interesting future directions of research
on Burnett's conjecture. The review~\cite{Huneau2024a} already offers a
very rich list of such directions, and we start by briefly presenting
some of them. 

Even though they clearly give Burnett's conjecture some validity, the
various results presented in Sections~\ref{section U(1) GWC}
and~\ref{section double null} have serious limitations. Most of these
results concern kinetic solutions close to Minkowski spacetime, with
exception Theorem~\ref{theo luk} but which is restricted to two null
dusts, and dropping the smallness assumption in $\mathbb{U}(1)$
symmetry or generalized wave coordinates would be very interesting. All
of these results are local in time, and one could imagine considering
as target kinetic spacetimes a global perturbation of Minkowski as
constructed in~\cite{Bigorgne2021,Taylor2017} and obtaining a
global-in-time Burnett approximation result. Opposite to this scenario,
the approximation by vacuum spacetimes of the static spherically
symmetric self-gravitating solutions of~\cite{Andreasson2017} doesn't
seem to be true globally in time, in view of the final state
conjecture, but perhaps it is for large times. More generally, given
any ``exotic'' solution to the massless Einstein--Vlasov with particular
properties, the question of its approximation by vacuum in the spirit
of Burnett's conjecture is an interesting one.

By transforming the early~\cite{ChoquetBruhat1969} into more analytical
theorems, the results of Section~\ref{section reverse} bring
mathematical general relativity closer to the vast geometric optics
literature. However they represent only the very first step and many
more phenomena remain to be explored, such as caustics, diffractive
effects etc. Because of the beautiful transparency properties of the
Einstein vacuum equations, one could also try to go beyond Burnett's
regime and construct stronger oscillations by concretely lowering the
value of $p$ in \eqref{ansatz GO}. This would correspond to the
strongly nonlinear geometric optics regime, used for instance to prove
ill-posedness of some supercritical wave equations by
Lebeau~\cite{Lebeau2001}. In this spirit, the approximate constructions
of large amplitudes gravitational waves in~\cite{Ali1999,Ali2007} are
definitely of interest, also since they seem to be stronger short
pulses than what exists in the mathematical general relativity
literature
(see~\cite{Christodoulou2009,Klainerman2012,Luk2017a,Alexakis2022}).

We conclude this text by coming back to Yvonne Choquet-Bruhat, in honor
of which this text has been conceived. In the 90's, she explored with
her collaborators alternative formulations of the Einstein vacuum
equations, and in particular higher order formulations built with first
or second order derivatives of the Ricci tensor. For instance
in~\cite{Abrahams1996} they give a fourth-order system (with respect to
the metric), equivalent to the Einstein vacuum equations, and show that
without any choice of gauge this system still retains a very weak form
of hyperbolicity~\cite{Leray1967}. In Remark 6 of the
review~\cite{Choquet-Bruhat:2001ygq}, Choquet-Bruhat even hints that
this fourth-order system would enjoy better structural semilinear
properties than the Einstein equations, in link with the relaxed null
condition introduced in~\cite{ChoquetBruhat2000}. She repeats this
remark in Chapter VIII Section 4 of~\cite{ChoquetBruhat2009}, sending
the reader to Chapter XI where high-frequency waves are studied.
However, no mention to this exotic formulation is made in Chapter XI,
let alone to its application to high-frequency waves, the nature of
which thus remains a mystery.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}

The author would like to thank Thibault Damour, C\'ecile Huneau and
Jonathan Luk for illuminating discussions over the years.

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