\makeatletter
\@ifundefined{HCode}                  
{\PassOptionsToPackage{russian,english}{babel}
\documentclass[screen,CRMECA,Unicode,biblatex,published]{cedram}
\addbibresource{crmeca20240836.bib}
\newenvironment{noXML}{}{}%
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#2}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\xtbody{\noalign{\relax}}
\def\tabnote#1{\vskip4pt\parbox{.51\linewidth}{#1}}
\def\xxtabnote#1{\vskip4pt\parbox{.65\linewidth}{#1}}
\def\mathbi#1{\boldsymbol{#1}}
\def\xmathop#1{\mathop{\text{#1}}}
\def\rmDelta{\Delta}
\usepackage{etoolbox}
\usepackage{upgreek}
\usepackage{graphicx}
\usepackage{amsfonts}
\usepackage{amssymb} 
\usepackage{bm} 
\usepackage{slashed}
\csdef{Seqnsplit}{\\}
\def\botline{\\\hline}
\newcounter{runlevel}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20240836}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}} 
\def\rmmu{\upmu}
\def\mn{\phantom{$-$}}
\def\refinput#1{}
\def\back#1{}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newenvironment{inftab}{}{}
\def\Custref#1#2{\hyperlink{#2}{#1}}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\makeatother
\DOI{10.5802/crmeca.284}
\datereceived{2024-10-11}
\daterevised{2025-01-13}
\dateaccepted{2025-01-13}
\ItHasTeXPublished
\newtheorem{defn}{Definition}[section]
}
{
\documentclass[crmeca]{article}
\usepackage[T1]{fontenc} 
\usepackage[T2A,T1]{fontenc}
\usepackage{upgreek}
\usepackage{slashed}
\def\bibliographystyle#1{}
\def\bibliography#1{}
\let\refinput\input
\def\CDRdoi{10.5802/crmeca.284}
\def\newline{\unskip\break}
\def\sfrac#1#2{{#1}/{#2}}
\def\xxtabnote#1{\tabnote{#1}}
\def\xmathop#1{\mathop{{#1}}}
\def\xtbody{\tbody}
\def\hypertarget#1#2{\label{#1}}
\newtheorem{theo}{Theorem}
\newtheorem{defn}{Definition}[section]
\newtheorem{definition}[theo]{Definition}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{conj}[theo]{Conjecture}
\newtheorem{question}[theo]{Question}
}

\usepackage{moreverb}

\dateposted{2025-02-06}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{Weak cosmic censorship, trapped surfaces, and naked
singularities for the Einstein vacuum equations}

\alttitle{Censure cosmique faible, surfaces pi\'eg\'ees et
singularit\'es nues pour les \'equations d'Einstein dans le vide}

\author{\firstname{Yakov} \lastname{Shlapentokh-Rothman}\CDRorcid{0000-0001-7782-8752}}
\address{Department of Mathematics, University of Toronto, 40
St. George Street, Toronto, ON, Canada}
\address{Department of Mathematical and Computational Sciences, 
University of Toronto Mississauga, 3359 Mississauga Road, Mississauga,
ON, Canada}
\email{yakovsr@math.toronto.edu}

\begin{abstract}
The weak cosmic censorship conjecture posits that, generically, all
singularities in General Relativity arising from regular asymptotically
flat initial data should have a complete future null infinity. While
this conjecture remains wide open, it has inspired many mathematical
works concerning topics such as trapped surface formation and the
construction of naked singularities. In this article we will review
some of these works and attempt to emphasize their interconnectedness.
\end{abstract}

\begin{altabstract}
La conjecture de censure cosmique faible postule que, de mani\`ere
g\'en\'erale, toutes les singularit\'es en relativit\'e g\'en\'erale
provenant de donn\'ees initiales r\'eguli\`eres asymptotiquement plates
devraient avoir un infini nul futur complet. Bien que cette conjecture
reste largement ouverte, elle a inspir\'e de nombreux travaux
math\'ematiques portant sur des sujets tels que la formation de
surfaces pi\'eg\'ees et la construction de singularit\'es nues. Dans
cet article, nous passerons en revue certains de ces travaux et
tenterons de mettre en \'evidence leurs interconnexions.
interconnectedness.
\end{altabstract}

\keywords{\kwd{Weak cosmic censorship}\kwd{Trapped surfaces}\kwd{Naked singularities}}

\altkeywords{\kwd{Censure cosmique faible}\kwd{Surfaces pi\'eg\'ees}\kwd{Singularit\'es nues}}

\thanks{NSERC discovery grants (RGPIN-2021-02562 and DGECR-2021-00093)}

\maketitle

\end{noXML}

\makeatletter
\renewcommand\theequation{\thesection.\arabic{equation}}
\@addtoreset{equation}{section}
\makeatletter

\section{Introduction}

We are interested here in General Relativity as a part of classical
physics, and thus the Cauchy problem will be paramount for us. We now
quickly recall the relevant notions. We say that the pair of a
$3$-dimensional Riemannian manifold $(\Sigma,h)$ and a symmetric
$(0,2)$-tensor $k$ on $\Sigma$ satisfy the \textit{constraint
equations} if
{\[
R(h) + (\mathrm{tr}_hk)^2 - |k|^2_h  = 0, \quad \mathrm{div}_hk - \nabla 
\mathrm{tr}_hk = 0.
\]}\unskip
In this case we say that $(\Sigma,h,k)$ forms an \textit{initial data
set}. The fundamental theorem concerning the Cauchy problem is the
following celebrated result of Choquet-Bruhat and Geroch:

\begin{theo}\cite{init,geroch}\label{cauchy} 
For every sufficiently regular initial data set $(\Sigma,h,k)$, there
exists a unique maximal globally hyperbolic spacetime $(\mathcal{M},g)$
solving the Einstein vacuum equations
{\[
\mathrm{Ric}(g) = 0,
\]}\unskip
so that $\Sigma$ may be embedded in $(\mathcal{M},g)$ with first and
second fundamental form $h$ and $k$ respectively.
\end{theo}

The proof of Theorem~\ref{cauchy} also yields that continuous
dependence on the data holds in a suitable sense. We will assume in
this article that the reader is familiar with the basics of Lorentzian
geometry (such as the term ``globally hyperbolic development'' above in
Theorem~\ref{cauchy}). See~\cite{wald} for an introduction to
Lorentzian geometry and causality theory.

As with any evolutionary PDE from mathematical physics, once a locally
well-posed initial value problem has been established, the next natural
question is, supposing we have  sufficiently ``nice'' initial data, do
we obtain a corresponding ``global and singularity free''  solution?
Note, however, that unlike the case for the more traditional PDEs of
mathematical physics, in view of the geometric nature of the Einstein
equations, it is not immediately clear what ``global and singularity
free'' should mean! Nevertheless, some necessary conditions are
immediate; for example, on physical grounds it is clear that we should
require that global existence at least include causal geodesic
completeness. As we shall discuss in the next section, suitably
understood, the famous Schwarzschild solution already illustrates the
ubiquitousness of the failure of global existence for the Einstein
vacuum equation. 

\subsection{The Schwarzschild spacetime and geodesic incompleteness}

The first discovered non-trivial exact solution for the Einstein
equations was the Schwarzschild spacetime~\cite{schwarzschild1916}. For
every $M > 0$, the \textit{exterior region} of the Schwarzschild
spacetime of mass $M$ may be covered by so-called ``Schwarzschild''
coordinates $(t,r,\theta,\phi) \in \mathbb{R} \times (2M,\infty) \times
\mathbb{S}^2$:
{\[
g_{\mathcal{S}} = -
\left(1-\dfrac{2M}{r}\right)\,\mathrm{d}t^2 + 
\left(1-\dfrac{2M}{r}\right)^{-1}\,\mathrm{d}r^2 + r^2
(\mathrm{d}\theta^2 + \sin^2\theta \,\mathrm{d}\phi^2).
\]}\unskip
It took over 30 years for the global geometry of the Schwarzschild
spacetime to be properly understood. We do not have the space here to
review this in full detail  (see the detailed exposition in Section 2
of~\cite{ClayLectureNotes}), but it will be useful to recall some of
the most salient points. The maximal extension of the Schwarzschild
spacetime is a globally hyperbolic spacetime with a two-ended Cauchy
hypersurface $\Sigma \approx \mathbb{R} \times \mathbb{S}^2$ and may be
globally covered by a double-null coordinate system where
{\[
g_{\mathcal{S}} = -\Omega^2 (U,V)\,\mathrm{d}U\,\mathrm{d}V +  r^2
(U,V) (\mathrm{d}\theta^2 + \sin^2\theta \,\mathrm{d}\phi^2),
\]}\unskip
for suitable functions $\Omega(U,V)$ and $r(U,V)$. Furthermore, we may
rescale the $U$ and $V$ coordinates so that their ranges lie in a
bounded set. In Figure~\ref{SchwPen} we then depict the range of the
$U$ and $V$ coordinates with the corresponding Penrose diagram. We
emphasize that each point in the diagram corresponds to a sphere in the
actual spacetime. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{SchwPen}Penrose diagram of maximally extended
Schwarzschild.}
\end{figure}

The original Schwarzschild coordinates correspond to the rightmost
diamond in the Penrose diagram. (The leftmost diamond corresponds to an
isometric copy of the rightmost copy. We will mostly ignore this
leftmost diamond in what follows.) Introducing a new coordinate $v = t
+ r + 2M\log (r-2M)$, we obtain in $(v,r,\theta,\phi)$ coordinates
{\[
g_{\mathcal{S}} = -\left(1-\dfrac{2M}{r}\right)\,\mathrm{d}v^2 + 2
\,\mathrm{d}v\,\mathrm{d}r + r^2 (\mathrm{d}\theta^2 +  \sin^2\theta
\,\mathrm{d}\phi^2).
\]}\unskip
We may then let these coordinates run over the range $(v,r,\theta,\phi)
\in \mathbb{R} \times (0,\infty) \times \mathbb{S}^2$, and these
coordinates cover the union of the original exterior region with the
region $\mathcal{B}$ from the Penrose diagram. Time-orienting by
$\partial_v$, one now easily checks that for each
$(c_0,\theta_0,\phi_0) \in \mathbb{R} \times \mathbb{S}^2$, after
reparameterization, the curve
{\[
s \mapsto (v,r,\theta,\phi) = (c_0,-s,\theta_0,\phi_0),\quad s < 0,
\]}\unskip
represents a future oriented incomplete null geodesic. Furthermore, as
$r\to 0$ along the geodesic, one may compute that the Kretschmann
scalar $R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$
blows-up. It immediately follows that there is no way to acquire future
null geodesic completeness by extending the Schwarzschild spacetime in
a $C^2$ fashion\footnote{In fact, it is not even possible to extend
maximal Schwarzschild in a $C^0$ fashion!
See~\cite{SbierskiCzero,JanSchw2}.}, and thus that the Schwarzschild
spacetime represents an example of singularity formation for the Cauchy
problem!

Having established the existence of a singular spacetime, one
immediately asks if the presence of this incompleteness is a generic
feature of solutions? Could it be the case that this incompleteness is
a special feature of highly symmetric spacetimes? Penrose's notion of a
trapped surface, which we will discuss in the next section, quickly
yields insight into these questions.
 
\subsection{Trapped surfaces}

Penrose's concept of a trapped surface will play a central role in this
article.

\begin{defn}\cite{PenTrap} 
\upshape
Let $\mathcal{S}$ be a closed oriented spacelike surface in a $3+1$
dimensional time-oriented Lorentzian manifold. At every point $p \in
\mathcal{S}$ one may pick two linearly independent future oriented null
and normal vectors $L$ and $\underline{L}$. We say that $\mathcal{S}$
is trapped if both of the corresponding mean curvatures are negative.
If one mean curvature is negative and the other vanishes, we say that
$\mathcal{S}$ is marginally trapped. We say that $\mathcal{S}$ is
anti-trapped if both of the corresponding mean curvatures are positive,
and if one mean curvatures is positive and the other vanishes, we say
that $\mathcal{S}$ is marginally anti-trapped. 
\end{defn}
 
Note that by the first variation formula for the area form, a surface
$\mathcal{S}$ is trapped if the area form locally contracts under any
flow of null vector congruences. This is physically interpreted as a
sign that one is in a regime of strong gravity. A straightforward
calculation shows  that a sphere of radius $r$ in the Schwarzschild
spacetime is trapped if and only if $t > 0$ and $r < 2M$. This region
of trapped spheres exactly corresponds to the region $\mathcal{B}$ in
Figure~\ref{SchwPen}. 

The usefulness of the trapped surface notion follows from two
fundamental facts. The first is that being a trapped surface is a
manifestly stable notion. Trapped surfaces cannot disappear from small
perturbations of a metric, and, more importantly, in view of Cauchy
stability, any sufficiently small perturbation of the Cauchy data of a
spacetime with trapped surfaces will still lead to a spacetime which
possesses trapped surfaces. In particular, any sufficiently small
perturbation of Schwarzschild's Cauchy data will still lead to a
spacetime with a trapped surface. The second fact underlying the
usefulness of the trapped surface notion is the following famous
theorem due to Penrose.

\begin{theo}\cite{PenTrap} 
Suppose that $(\mathcal{M},g)$ is a globally hyperbolic spacetime with
a non-compact Cauchy surface and satisfies
{\[
\mathrm{Ric}(V,V) \geq 0,\quad \forall V \mbox{ with }g(V,V) = 0.
\]}\unskip
Then, if $(\mathcal{M},g)$ possesses a trapped surface, it must be
future causally geodesically incomplete. 
\end{theo}

We immediately conclude that the geodesic incompleteness of
Schwarzschild is a stable property to perturbations of the Cauchy data!
We are thus forced to take seriously the failure of global existence in
General Relativity.

\subsection{Completeness of future null infinity and the weak cosmic
censorship conjecture}

While the Schwarzschild spacetime poses serious conceptual questions
concerning its singularity as $r\to 0$, the spacetime geometry also
suggests a ``way out''. On the Penrose diagram in Figure~\ref{SchwPen},
we have denoted part of the boundary by $\mathcal{I}^+$. This denotes
``future null infinity'' and corresponds to future end points of the
null hypersurfaces $t = r + 2M\log(r-2M) + c$ for any choice of $c \in
\mathbb{R}$. Physically, one imagines that a far away observer
propagates along a causal curve which is close to $\mathcal{I}^+$. The
following proposition is straightforward to establish.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}The hypersurface $\mathcal{C}$ and null
geodesic $\gamma$ in Schwarzschild from Proposition~\ref{SchwNotNaked}.}
\end{figure}

\begin{prop}\label{SchwNotNaked}
For $r_0 \gg 1$, let $\mathcal{C}$ denote the outgoing null
hypersurface $\{t = r + 2M\log(r-2M)\} \cap \{r \geq r_0\}$. For each
$\tilde{r} \geq r_0$ and $(\tilde{\theta},\tilde{\phi}) \in
\mathbb{S}^2$, we may consider the ingoing null geodesic
$\gamma_{\tilde{r},\tilde{\theta},\tilde{\phi}}$ uniquely defined by
the requirement that 
$\gamma_{\tilde{r},\tilde{\theta},\tilde{\phi}}(0) =
\mathcal{C}|_{(r,\theta,\phi) =
(\tilde{r},\tilde{\theta},\tilde{\phi})}$ and
$\dot{\gamma}_{\tilde{r},\tilde{\theta},\tilde{\phi}}(0) = \partial_t
-(1-({2M}/{r}))\partial_r$. We then define $A(\tilde{r})$ to be the
affine length of $\gamma$ (it is immediate that $A$ does not depend on
$(\tilde{\theta},\tilde{\phi})$). We then have that 
{\[
\lim_{\tilde{r} \to \infty}A(\tilde{r}) = \infty.
\]}\unskip
\end{prop} 
	
See Figure~\ref{fig2} for the Penrose diagram corresponding to
Proposition~\ref{SchwNotNaked}.
Proposition~\ref{SchwNotNaked} has the physical interpretation that far
away observers ``at'' future null infinity $\mathcal{I}^+$ live forever
and may make observations without ever needing to interact with the
Schwarzschild singularity. This good property of future null infinity
is referred to as \textit{completeness}. Our next goal is formulate a
general definition of this property, but we need a few preliminaries.

\begin{definition}
\upshape
Let $(\Sigma,h,k)$ be an initial data set. We say that $(\Sigma,h,k)$
is strongly asymptotically flat with $n$-ends if there exists a compact
set $K$ so $\Sigma\setminus K$ is diffeomorphic to $n$ copies of
$\mathbb{R}^3$ minus a ball, and there exists coordinates $\{x^a\}$ on
each of these so that
{\[
h_{ab} = \left(1+\dfrac{2M}{r}\right)\delta_{ab} + 
o_4 (r^{-3/2}),\quad k_{ab} = o_3 (r^{-5/2}),
\]}\unskip
where the $o_i\left(\cdot\right)$ means that up to $i$ derivatives may
be applied and that each derivative yields an extra power of decay in
$r$. 
\end{definition}

It is a consequence of the stability of Minkowski space~\cite{CK,KN},
that associated to each end of a strongly asymptotically flat initial
data set, the corresponding maximal development will posses an outgoing
null hypersurface $\mathcal{C}$ with future complete null generators
and whose geometry asymptotes  to a null cone from Minkowski space.
(The details of the asymptotic geometry will not be important for us
here.)

We now are ready to give a definition of a complete future null
infinity. We note that there are various definitions existing in the
literature. Our definition is in the spirit of the definition given
in~\cite{Chrmil}. A key point in both is that we do not formally define
future null infinity as a boundary of a conformal compactification of
spacetime. This avoids various technical regularity issues concerning
such compactifications which are largely irrelevant for the physical
content of the definition. 

\begin{defn}\label{compnullinf}
\upshape
Let $(\Sigma,h,k)$ be a strongly asymptotically flat initial data set
and $(\mathcal{M},g)$ be the corresponding maximal development.
Associated to each asymptotically flat end, choose an outgoing null
hypersurface $\mathcal{C}$ in $\mathcal{M}$ with future complete null
generators whose geometry approaches that of a standard Minkowski cone.
Let $L$ denote a geodesic null normal along $\mathcal{C}$. Then let
$\underline{L}$ be a choice of a transversal null vector field along
$\mathcal{C}$ which is parallel transported by $L$. For every point $p
\in \mathcal{C}$, we let $A(p)$ denote the maximal affine length  of
the null geodesic starting at $p$ with initial tangent $\underline{L}$,
and we let $p(s)$ denote the outgoing null geodesic which starts at $p$
and has initial tangent vector equal to $L$. We then say that
$\left(\mathcal{M},g\right)$ has a \textit{complete future null
infinity} if, for every $p \in \mathcal{C}$, $\lim_{s\to \infty}A(p(s))
= \infty$.
\end{defn}

We note that we have chosen to work with strongly asymptotically
initial data sets so that we may invoke~\cite{CK,KN}. However, newer
proofs of the stability of Minkowski space which allow weaker versions
of asymptotic flatness (cf.~\cite{shenslowmink}) in principle allow for
weaker asymptotic flatness assumptions in Definition~\ref{compnullinf}.
The corresponding reformulations of Definition~\ref{compnullinf} are
straightforward to state. 

Finally, we are ready for the weak cosmic censorship conjecture.

\begin{conj}[Weak cosmic censorship]\label{wccConj} 
Generic strongly asymptotically flat initial data sets possess a
complete future null infinity.
\end{conj}

Spacetimes with an incomplete future null infinity are called
\textit{naked singularities}\footnote{In this review we will restrict
attention to naked singularities which arise in evolution from complete
and sufficiently regular initial data. This requirement rules out
solutions such as negative mass Schwarzschild or the super extremal
Kerr black hole.}. The earliest version of Conjecture~\ref{wccConj}
originates in~\cite{Penroseergo}, which, in particular, introduced the
notion of a ``cosmic censor'' who prevents the appearance of naked
singularities. However, the addition of the word ``generic'' into the
conjecture only followed the discovery of naked singularity solutions
for the spherically symmetric Einstein-scalar field
system~\cite{ChristNaked}. See the discussion in
Section~\ref{nakedsingsecChrist} below.

We close this section by noting some important ambiguities underlying
Conjecture~\ref{wccConj}. The most apparent of these is the presence of
the word ``generic;'' a successful resolution of
Conjecture~\ref{wccConj} will have to make this precise. A more subtle
ambiguity is the fact that the maximal development of an initial data
set depends on the functional framework used to solve the Einstein
equations. In particular, as is the case famously with the strong
cosmic censorship conjecture (see~\cite{DafLuk1}), different regularity
requirements on initial data or on what constitutes an admissible
development of data could lead to different outcomes for
Conjecture~\ref{wccConj}. See the discussions below in
Sections~\ref{nakedsingsecChrist}--\ref{stablenakedJay}.

\subsection{Outline of rest of the article}

Though the focus on this article is on the Einstein vacuum equations,
much of our intuition about the vacuum equations derives from an
influential sequence of works carried out by Christodoulou in the 1990s
concerning the spherically symmetric Einstein-scalar field system. In
Section~\ref{sphsymmscalfield}, we will review Christodoulou's work; in
particular we will discuss Christodoulou's examples of naked
singularities, the study of the instabilities of naked singularities,
and the central role of trapped surface formation. Finally we will
briefly discuss recent work of Jaydeep Singh which yields new insights
on the naked singularities of Christodouloou.

In Section~\ref{trappedvacuum} we will start with a discussion of
Christodoulou's short-pulse method and corresponding breakthrough work
on trapped surface formation for the Einstein vacuum equations. We will
then survey various works which lead to significant extensions of
Christodoulou's original work.

Finally, in Section~\ref{nakedvacsec} we will discuss the recent
construction of naked singularity solutions for the Einstein vacuum
equations. We will explain various notions of self-similarity and give
a high level overview of the proof of existence of the naked
singularities.
 
\section{The spherically symmetric Einstein-scalar field system}
\label{sphsymmscalfield}

Before trying to tackle the weak cosmic censorship conjecture for the
Einstein vacuum equations, it is natural to work first with easier
``model problems''. For many PDEs of mathematical physics, such a model
problem, which is consistent with asymptotic flatness, may be obtained
by restricting considerations to spherically symmetric solutions.
Unfortunately, for the Einstein equations, Birkhoff's theorem implies
that any solution to the Einstein vacuum equations which is spherically
symmetric must be static and isometric to a Schwarzschild spacetime.
Arguably, the simplest way to remove this rigidity while retaining many
fundamental features of the Einstein vacuum equations is to consider
the spherically symmetric Einstein-scalar field system.

The unknowns for the spherically symmetric Einstein (real) scalar-field
system are a $1+1$ dimensional Lorentzian manifold with boundary
$(\mathcal{Q},g)$ which arises from quotienting out the original
spacetime by the spherical symmetry, an area radius function $r :
\mathcal{Q} \to [0,\infty)$, and a scalar field $\phi : \mathcal{Q} \to
\mathbb{R}$. The equations are
{\begin{equation}\label{esf}
\begin{array}{@{}rcl@{}}
r\nabla_{\alpha}\nabla_{\beta} r &=& \tfrac{1}{2} g_{\alpha\beta}
(1-|\nabla r|^2_g) - r^2 (\partial_{\alpha}\phi\partial_{\beta}\phi -
\tfrac{1}{2}g_{\alpha\beta}|\nabla\phi|_g^2),  \\ 
\nabla^{\alpha} (r^2\nabla_{\alpha}\phi) &=& 0 \\ 
K(g) &=& r^{-2} (1-|\nabla r|^2_g) + |\nabla\phi|_g^2,
\end{array}
\end{equation}}\unskip
where $K(g)$ denotes the Gauss curvature of $g$. It is straightforward
to give analogues of Definition~\ref{compnullinf} and
Conjecture~\ref{wccConj} for this system.

In the Sections~\ref{secBV}--\ref{unstablenakedChrist} we review the 
results from an important sequence of papers due to
Christodoulou~\cite{ChristSphTrap,ChristBV,ChristNaked,Christodoulou4}.
In our discussion of these papers we will follow closely the review
article~\cite{Chrmil}.

Finally, in Section~\ref{stablenakedJay} we discuss the recent
work~\cite{jaystablenaked} concerning the behavior of Christodoulou's
naked singularities under relatively smooth perturbations.
 
\subsection{Scale-invariance and BV solutions}\label{secBV}

We set $\Gamma \doteq \partial \mathcal{Q}$ to be the set of fixed
points of the $SO(3)$ action. Using that every contractible $1+1$
dimensional Lorentzian manifold is conformally equivalent to a suitable
subset of $\mathbb{R}^{1+1}$, it is straightforward to show that the
metric $g$ may be written as
{\begin{equation}\label{metricformsph}
g = -\Omega^2 \,\mathrm{d}u\,\mathrm{d}v,
\end{equation}}\unskip
for some function $\Omega(u,v)$ and where the null
coordinates $u$ and $v$ satisfy $v = u$ along $\Gamma$. We note that
the double-null form~(\ref{metricformsph}) is preserved by any
coordinate change $u\mapsto f(u)$ and $v \mapsto h(v)$ for $f',h' > 0$.
The system~(\ref{esf}) then becomes the following system of $1+1$
dimensional PDEs for the functions $\Omega(u,v)$, $r(u,v)$, and
$\phi(u,v)$:
{\begin{equation}\label{esf2}
\begin{array}{@{}rcl@{}}
r\dfrac{\partial^2r}{\partial u\partial v} + 
\dfrac{\partial r}{\partial u}
\dfrac{\partial r}{\partial v} &=& -
\dfrac{1}{4}\Omega^2, \\ 
\dfrac{\partial^2r}{\partial u^2} - 2\Omega^{-1}
\dfrac{\partial r}{\partial u}\dfrac{\partial \Omega}{\partial u} &=&
-r\left(\dfrac{\partial \phi}{\partial u}\right)^2, \\ 
\dfrac{\partial^2r}{\partial v^2} - 2\Omega^{-1}
\dfrac{\partial r}{\partial v}
\dfrac{\partial \Omega}{\partial v} &=& 
-r\left(\dfrac{\partial \phi}{\partial v}\right)^2, \\ 
r^2\dfrac{\partial^2\log\Omega}{\partial u\partial v} -
\dfrac{1}{4}\Omega^2 &=& -r^2
\dfrac{\partial\phi}{\partial u}
\dfrac{\partial\phi}{\partial v} + 
\dfrac{\partial r}{\partial u}
\dfrac{\partial r}{\partial v}, \\ 
r\dfrac{\partial^2\phi}{\partial u\partial v} + 
\dfrac{\partial r}{\partial u}
\dfrac{\partial \phi}{\partial v} + 
\dfrac{\partial r}{\partial v}
\dfrac{\partial \phi}{\partial u} &=& 0.
\end{array}
\end{equation}}\unskip
        
Closely related to the rescaling gauge freedom in the double-null
coordinate system~(\ref{metricformsph}) is the fact that there is a
natural group action of $\mathbb{R}_+$ on solutions to the
system~(\ref{esf2}) defined by
{\begin{equation}\label{rescalingmap}
\{r(u,v),\Omega(u,v),\phi(u,v)\} \mapsto 
\{a^{-1}r(au,av),\Omega(au,av),\phi(au,av)\},\quad a \in (0,\infty).
\end{equation}}\unskip
         
Following Christoudoulou~\cite{ChristBV}, we refer to solutions which
are fixed points of the map~(\ref{rescalingmap}) as being
\textit{scale-invariant}. In~\cite{ChristBV} Christodoulou classifies
scale-invariant solutions possessing a past complete incoming null
curve emanating from $(u,v) = (0,0)$. The derivative $\partial_v\phi$
will have a jump discontinuity along $\{v = 0\}$ and the spacetimes
fall into three classes depending on the value of this jump
{\[
A \doteq \tfrac{1}{2}\left(\lim_{v\to^+ 0}r\partial_v\phi -
\lim_{v\to^- 0}r\partial_v\phi\right).
\]}\unskip

\begin{enumerate}[(1)]
\item When $A^2 < 1$, the spacetimes is global. The scalar field $\phi$
is constant to the past of $\{v = 0\}$ and future of $\{u = 0\}$, the
transversal null derivatives of $\phi$ jump along $\{u = 0\}$ and $\{v
= 0\}$, and in the region $\{v \geq 0\} \cap \{u \leq 0\}$ we have 
{\[
\phi = \dfrac{1}{2}\log\left|\dfrac{u-(1+A)v}{u-(1-A)v}\right|.
\]}\unskip

\item When $A^2 = 1$, the spacetime exists up to $\{ u = 0\}$ where $r$
vanishes and the scalar field $\phi$ blows-up.

\item When $A^2 > 1$, the, spacetime exists up to the curve $\{ u =
-(A-1)v\}$ and is preceded by an \textit{apparent horizon} along $\{u =
-(A^2-1)v\}$. The spheres in between the apparent horizon and the final
singular curve are all trapped. 
\end{enumerate}
Though these solutions are low-regularity in that the derivative of
$\phi$ may jump, the scale-invariant spacetimes may be considered to be
physically reasonable in the following sense. They admit an
interpretation as an incoming spherical impulsive gravitational wave.
When the amplitude of the wave is small, the wave bounces off the
center and flies back out to infinity. Once the amplitude of the wave
is large enough, upon collapse to the center we have singularity
formation. However, generically this singularity is preceded by a
trapped region which is consistent with the singularity begin hidden
from infinity (cf.~the region of trapped surfaces in the Schwarzschild
spacetime and also the discussion in Section~\ref{tappedscalar} below).

Christodoulou was thus inspired in~\cite{ChristBV} to build a
functional framework which includes the scale-invariant solutions, has
norms which are invariant under the map~(\ref{rescalingmap}), and
allows one to establish a local well-posedness result for the
spherically symmetric Einstein-scalar field system. The class of
spacetimes considered are referred to as \textit{spacetimes of bounded
variation}. In view of our space constraints in this article we will
not describe in detail the norms in this space, but we note that the
scalar field $\phi$ is required to be an absolutely continuous
function, $r({\partial \phi}/{\partial v})$ is required to be a
function of bounded variation on each hypersurface $\{u = 
\mathrm{const}\}$, and $r({\partial \phi}/{\partial u})$ is required to
be a function of bounded variation on each hypersurfae $\{v =
\mathrm{const}\}$. In addition to having scale-invariant norms defining
the space, solutions of bounded variation are intimately tied to the
scale-invariant solutions discussed in the paragraph above in that for
any solution of bounded variation, one may consider the rescaling
map~(\ref{rescalingmap}) as $a\to 0$ and obtain an exactly scale
invariant solution in the limit.

Finally, focusing again only on the scalar field, we note that the
work~\cite{ChristBV} shows that a sufficient condition for global
existence as a solution of bounded variation is, in addition to natural
normalization assumptions on the geometry, for the scalar field $\phi$
to satisfy the following along an initial outgoing null hypersurface
$\{u = \mathrm{const}\}$:
{\begin{equation}\label{globalinBVcond}
\int_0^{\infty}|\partial_v^2(r\phi)|^2\,\mathrm{d}v \ll 1.
\end{equation}}\unskip
Finally, for later use, we use the term \textit{absolutely continuous
spacetimes} to refer to the situation when the scalar field satisfies
that $\partial_v(r\phi)$ and $\partial_u (r\phi)$ are absolutely
continuous functions along constant $u$ and $v$ hypersurfaces
respectively. 

\subsection{Trapped surface formation}\label{tappedscalar}

We have seen in our earlier discussions that the Schwarzschild
spacetime possesses a complete future null infinity and that all of the
spheres immediately preceding the $\{r = 0\}$ singularity are trapped.
This turns out not to be a coincidence and, in fact, it was shown
in~\cite{ChristSphTrap} that for the spherically symmetric
Einstein-scalar field system, the existence of a single trapped surface
guarantees the completeness of future null infinity. (Similar
statements can in fact be made for a large class of spherically
symmetric Einstein matter systems~\cite{dafermostrapped}.) It is thus
in principle quite useful to have various criteria which imply the
eventual formation of a trapped surface.

Before proceeding, we define an important quantity, the \textit{Hawking
mass} $m$ via the formula
{\[
1-\dfrac{2m}{r} = |\nabla r|^2.
\]}\unskip
The Hawking mass of a sphere is always non-negative and can be
considered a quasi-local measure of the energy/mass contained inside
the sphere. It is useful to keep in mind that that the ratio ${m}/{r}$
is scale-invariant in that it is left invariant under the
map~(\ref{rescalingmap}), and that we always have $\partial_vm \geq 0$.

Christodoulou's key result concerning trapped surface formation for the
spherically symmetric Einstein-scalar field system is the following.

\begin{theo}\cite{ChristSphTrap}\label{christsphtrapptheo} 
Let $\mathcal{S}_{u_0,v_0}$ denote the sphere lying at the intersection
$\{u = u_0\} \cap \{v=v_0\}$. Consider two spheres
$\mathcal{S}_{0,v_0}$ and $\mathcal{S}_{0,v_1}$ lying along $\{u = 0\}$
with $v_1 > v_0 > 0$. We define the following two quantities:
{\begin{equation}\label{sizefortrapped}
\delta \doteq \dfrac{r(\mathcal{S}_{0,v_1})}{r(\mathcal{S}_{0,v_0})} - 1, \quad 
\eta \doteq \dfrac{2(m(\mathcal{S}_{0,v_1}) - m(\mathcal{S}_{0,v_0}))}{r(\mathcal{S}_{0,v_1})}.
\end{equation}}\unskip
Assume that the axis $\Gamma$ is regular up to the point where it
intersects $\{ v = v_0\}$. Then there exists a constant $c_1,c_2 > 0$
so that $0 < \delta < c_1$ and $\eta > c_2 \delta \log (\delta^{-1})$
implies that there exists $\tilde{u} > 0$ with $v_0 > \tilde{u}$ and so
that the sphere $\mathcal{S}_{\tilde{u},v_1}$ is a trapped sphere.
\end{theo}

In Figure~\ref{illustrationtrappedsph} we provide the corresponding
Penrose diagram.

\begin{figure}
\includegraphics{fig03}
\caption{\label{illustrationtrappedsph}Penrose diagram for
Theorem~\ref{christsphtrapptheo}.}
\end{figure}

The quantities $\delta$ and $\eta$ defined in~(\ref{sizefortrapped})
are both invariant under the map~(\ref{rescalingmap}), and the
condition that $\eta > c\delta \log (\delta^{-1})$ is sharp up to the
log factor. More specifically, let $\delta > 0$ be a small constant.
Then it is straightforward to construct smooth initial data for the
Einstein-scalar field system along $\{u = 0\}$ so that $r|_{u=0} = v$
and so that scalar field is smooth, satisfies~(\ref{globalinBVcond}),
is compactly supported in $v \in (1,1+\delta)$, and also satisfies
{\begin{equation}\label{somelargeness}
\int_1^{1+\delta} |\partial_v (r\phi)|^2\,\mathrm{d}v \gtrsim \delta,
\end{equation}}\unskip
where the implied constant in~(\ref{somelargeness}) depends on the
implied constant in~(\ref{globalinBVcond}). In view of Christodoulou's
theory of BV solutions (see Section~\ref{secBV} above) such data will
give rise to a solution which is free of trapped surfaces. Furthermore,
we will have that
{\[
r(\mathcal{S}_{0,1+\delta}) = 1 + \delta, \quad 
{r(\mathcal{S}_{0,1})}= 1,
\]}\unskip
and as consequence of the $\partial_v$-propagation equation for the
Hawking mass\footnote{This equation is
{\[
\partial_vm = \dfrac{1}{2} \left(\dfrac{\partial r}{\partial v}\right)^{-1}
\left(1-\dfrac{2m}{r}\right)r^2 (\partial_v\phi)^2.
\]}\unskip
} that
{\[
m(\mathcal{S}_{0,1}) = 0,\quad m (\mathcal{S}_{0,1+\delta}) \gtrsim \delta.
\]}\unskip
Thus, for such solutions we will have both the absence of trapped
surfaces in evolution and that $\eta \gtrsim \delta$ (for $\eta$ and
$\delta$ as in~(\ref{sizefortrapped})).
        
Finally, we note that it is a remarkable fact that the quantities
considered in~(\ref{sizefortrapped}) only involve the behavior of the
solution tangentially along $\{u = 0\}$ and that, other than the
regularity of the axis $\Gamma$, no a priori assumptions are made about
the solution on any transversal hypersurface $\{v = \mathrm{const}\}$!

\subsection{Naked singularities}\label{nakedsingsecChrist}

In the work~\cite{ChristNaked} Christodoulou constructed solutions to
the spherically symmetric Einstein-scalar field system which correspond
to naked singularities. The first step in the construction of these
solutions is to introduce a generalization of the scale-invariant
solutions that we discussed in Section~\ref{secBV}. More specifically,
we observe that the map~(\ref{rescalingmap}) in fact sits inside a
$1$-parameter family of $\mathbb{R}_+$ actions parameterized by $k \in
\mathbb{R}$ and defined by    
{\begin{equation}\label{rescalingmapwithk}
\{r(u,v),\Omega(u,v),\phi(u,v)\} \mapsto
\{a^{-1}r(au,av),\Omega(au,av),\phi(au,av)+k\log(a)\},\quad
a \in (0,\infty).
\end{equation}}\unskip
A spacetime which is invariant under the map~(\ref{rescalingmapwithk})
is called $k$-\textit{self similar}. Since it can be useful to not work
in the double-null coordinates underlying~(\ref{rescalingmapwithk}), we
observe that we can phrase $k$-self similarity in a coordinate
invariant way: a spacetime is $k$-self similar if there exists a
scaling vector field $\mathcal{K}$ so that
{\begin{equation*}%\label{kselfsimform}
\mathcal{L}_{\mathcal{K}}g = 2g, \quad 
\mathcal{L}_{\mathcal{K}}r = r, \quad 
\mathcal{L}_{\mathcal{K}}\phi = -k.
\end{equation*}}\unskip

When $k \neq 0$ it is no longer possible to write down the self-similar
solutions explicitly; however, Christodoulou showed that it is still
possible to reduce the $k$-self similar spherically symmetric
Einstein-scalar system to a two dimensional autonomous system.
In~\cite{ChristNaked} Christodoulou carried out a systematic phase
plane analysis of this system, and he found that when $0 < k^2 < 1/3$
and after a suitable truncation of the scalar field in the large $r$
region, the $k$-self similar spacetimes have a Penrose diagram as
depicted in Figure~\ref{illustrationsphenaked}, are asymptotically
flat, and moreover have an incomplete future null infinity. This
showed, for the first time, that one needs the word ``generic'' in
Conjecture~\ref{wccConj}. It will be useful for the discussion later in
Section~\ref{unstablenakedChrist} to observe that along the past light
cone of the singularity, the ratio ${m}/{r}$ is constant and non-zero.

\begin{figure}
\includegraphics{fig04}
\caption{\label{illustrationsphenaked}Penrose diagram for the naked
singularities of~\cite{ChristNaked} with a depicted incoming null
geodesic $\gamma$ of finite affine length.}
\end{figure}

We make a brief note on global gauges for the $k$-self similar
spacetimes: if one has a $k$-self similar spacetime in the double-null
coordinate system of the map~(\ref{rescalingmap}), then one may easily
check that $\mathcal{K} = u\partial_u + v\partial_v$. However, it turns
out that for $k$-self similar spacetimes, it is \underline{not}
possible to have such a coordinate system which covers the past light
cone of the singularity, a hypersurface which is of fundamental
importance to the analysis. Christodoulou in fact works in
\textit{Bondi coordinates} $(u,r)$ where $u$ is an eikonal function.
Another option is to work in a rescaled double-null coordinate system
where the map~(\ref{rescalingmapwithk}) and hence $\mathcal{K}$ do not
take their usual form (see Appendix A of~\cite{jaybackwardnaked}).

These naked singularities do not have smooth data along the initial
outgoing hypersurface $\mathcal{C}$; however, the lack of regularity
there only occurs at the intersection of the past cone of the
singularity and $\mathcal{C}$. At this point the tangential null
derivative of the scalar field $\phi$ is H\"{o}lder continuous. This is
more regular than initial data for a solution of bounded variation and
we may thus consider the initial data to be ``regular''. Moreover,
Christodoulou shows that the singular point is such it is impossible to
extend the solution to the singular point as a solution of bounded
variation. The Cauchy horizon $\mathcal{CH}$ emanating from the
singularity is regular in that the derivative of the scalar field
extends to $\mathcal{CH}$ in a H\"{o}lder continuous function; in fact,
it is possible to extend the spacetime as a $k$-self similar solution
across $\mathcal{CH}$. (Thus these spacetimes are relevant for the
strong cosmic censorship conjecture.)

We will discuss the instability of these spacetimes in
Section~\ref{unstablenakedChrist} below. Before addressing the issue of
instability, a natural question that arises is whether it is possible
to construct naked singularity spacetimes which do not possess any
exact self-similarity. Exactly such spacetimes were constructed using a
perturbative scattering approach in~\cite{jaybackwardnaked}. 

Though we do not have the space here to discuss these in detail, it is
also interesting to construct naked singularity solutions for other
spherically symmetric Einstein matter systems, for example,
see~\cite{dustnaked,eineulernaked,bizonwassermanwavemap}. Finally, we
note that it is an open problem to rigorously construct smooth (or even
$C^2$!) initial data for the spherically symmetric Einstein-scalar
field system which leads to a naked singularity. With regards to this,
it is worth observing that the physics literature around
\textit{critical collapse} (see~\cite{chop,critsurv}) does suggest that
naked singularities arising from smooth initial data should exist.
         
\subsection{Instability of naked singularities}
\label{unstablenakedChrist}

In the work~\cite{Christodoulou4} Christodoulou studied the instability
of a general class of naked singularities for the spherically symmetric
Einstein-scalar field system. Before discussing this result in its full
generality, it is conceptually useful to focus on the instability of
the specific (truncated) $k$-self similar naked singularities discussed
in Section~\ref{nakedsingsecChrist} above. 

Let $\underline{\mathcal{C}}$ denote the past light cone of the
singularity. The scaling vector field $\mathcal{K}$ is tangent to
$\underline{\mathcal{C}}$, and we may introduce a coordinate $u < 0$
along $\underline{\mathcal{C}}$ so that
$\mathcal{K}|_{\underline{\mathcal{C}}} = u\partial_u$ and so that
$\underline{\mathcal{C}} \cap \mathcal{C}$ corresponds to $u = -1$. The
singularity then corresponds to the limit $u \to 0$ along
$\underline{\mathcal{C}}$. Let $\mathfrak{p} =
(\partial_vr)^{-1}\partial_v\phi$ be a suitably normalized transversal
derivative of the scalar field $\phi$ along $\underline{\mathcal{C}}$,
so that in view of the underlying self-similarity, we have
$|\mathfrak{p}| \sim u^{-1}$. Along $\underline{\mathcal{C}}$ the wave
equation may be considered a transport equation for $\mathfrak{p}$:
{\begin{equation}\label{whereblueshiftcomesfrom}
\partial_u\mathfrak{p} + \dfrac{1+k^2}{u}\mathfrak{p} = F(u),
\end{equation}}\unskip
where the $1+k^2$ and $F(u)\sim u^{-2}$ depend only on the behavior of
the geometry and scalar field \underline{tangential} to
$\underline{\mathcal{C}}$. Even though $|\mathfrak{p}| \sim u^{-1}$, it
is straightforward to see that a solution $\check{\mathfrak{p}}$ of
$\mathfrak{p}$'s equation with generic initial data at $\{u = -1\}$ 
will have $|\check{\mathfrak{p}}| \sim |u|^{-1-k^2}$, which is a faster
blow-up rate than the self-similar rate. Heuristically, we expect that
if there exists a perturbation of the initial data so that the
corresponding $\mathfrak{p}$ is more singular as $u\to 0$ than the
self-similar rate, then we will have an instability for the naked
singularity background. Optimistically, we could then hope to apply
Theorem~\ref{christsphtrapptheo} to obtain the formation of a trapped
surface and then deduce the existence of a complete future null
infinity. We note that one may connect this instability in the
evolution equation for transversal null derivatives of $\phi$ along
$\underline{\mathcal{C}}$ to an analogous growth of the  energy of null
geodesics traveling along $\underline{\mathcal{C}}$; this leads us to
refer to the instability as a \textit{blue-shift instability}. 

This instability can indeed be triggered by considering a perturbation
of the scalar field where the solution $\phi$ is kept constant in the
causal past of the singularity, so as to preserve the coefficients and
right hand side of the Equation~(\ref{whereblueshiftcomesfrom}), and
then introducing a perturbation of $\mathfrak{p}$ exactly along
$\underline{\mathcal{C}}\cap \mathcal{C}$. Since the transversal
derivative of $\phi$ thus must have a jump discontinuity, this leads to
the perturbation being only in the class of solutions of bounded
variation. However, it turns out that by exploiting the gap between
$1+k^2$ and $1$, one can also excite a similar instability with a
perturbation which again keeps $\phi$ constant in the causal past of
singularity and arranging for the perturbed $\mathfrak{p}$ to be
H\"{o}lder continuous with a small $k$-dependent exponent $\gamma > 0$.
See \cite{jaystablenaked} for more details.

We now turn to the much more general instability results
of~\cite{Christodoulou4}.

\begin{theo}\cite{Christodoulou4}\label{unstablenaked} 
We parameterize initial data for the spherically symmetric
Einstein-scalar field system by a choice of $\alpha = \partial_v
(r\phi)$ along an initial outgoing cone $\mathcal{C}$, where
$\partial_vr = 1/2$. Suppose that for some $\alpha = \alpha_0$, a
function of bounded variation, the corresponding spacetime has an
incomplete future null infinity and that, along the past light cone of
the singularity, $({m}/{r}) \not\to 0$ as $r\to 0$. Then there exists
$f_1$ which is a function of bounded variation and $f_2$ which is an
absolutely continuous function so that whenever $(c_1,c_2) \in
\mathbb{R}^2\setminus\{(0,0)\}$ the spacetime arising from the initial
data $\alpha_0 + c_1f_1 + c_2f_2$ has a complete future null infinity. 
\end{theo}

We note that the proof also applies to the case of a \textit{locally
naked singularities}. As in the instability of the specific naked
singularities discussed above, a key role is played by a blue-shift
instability along the past light cone of the singularity. However, in
this more general scenario, we do not have precise information about
the geometry available.  Remarkably, using only the information on
${m}/{r}$ as a starting point, Christodoulou is able to show that a
blue-shift instability may be triggered by a generic (in the sense of
Theorem~\ref{unstablenaked}) perturbation, and eventually leads to the
conditions of Theorem~\ref{christsphtrapptheo} being satisfied.
Finally, we note that~\cite{Christodoulou4} justifies the assumption on
${m}/{r}$ along the past light cone of the singularity by a connection
to continuation criteria for solutions of bounded variation
from~\cite{ChristBV} (see also the corresponding discussion
in~\cite{Chrmil}).

These instability proofs have been revisited with techniques developed
originally for use with the Einstein vacuum equations (outside of
symmetry): such techniques also allow for the study of certain
instabilities generated by non-spherically symmetric and ansiostropic
gravitational perturbations
(see~\cite{robustinstabnaked,instabsphnakedsinggrav,anperturbnaked}).
We note in particular the recent work~\cite{anperturbnaked} where An
revisits the instability for the specific $k$-self similar naked
singularitities of Christodoulou. By considering anistropic
perturbations An is able to construct infinitely many more unstable
directions. For these perturbations he is furthermore able to construct
a corresponding anisotropic apparent horizon and to consider
perturbations which vanish at the past light cone of the singularity
when measured using suitable scale-invariant norms. This final fact is
related to the discussion earlier in the section that the blue-shift
instability can be triggered by scalar field perturbations which have a
H\"{o}lder continuous first derivative as opposed to the derivative
merely being absolutely continuous. 

\subsection{Epilogue: the importance of the functional framework}
\label{stablenakedJay}

The naked singularities considered in Theorem~\ref{unstablenaked} arose
from data at least as regular as  a solution of bounded variation and
the perturbations considered could be as regular as a solution with
$\partial_v (r\phi)$ being an absolutely continuous function. It is
natural to ask what happens if we consider naked singularities arising
from more regular initial data but then only allow perturbations which
are also more regular. In fact, this question is already interesting if
we restrict to the $k$-self similar naked singularities constructed by
Christodoulou; for these solutions $\partial_v (r\phi)$ is H\"{o}lder
continuous.

In a very interesting recent work~\cite{jaystablenaked}, Singh has
initiated the study of these questions by considering the dynamics of
the wave equation $\Box_g\phi = 0$ (with no symmetry restriction) on
the $k$-self similar naked singularity backgrounds. If the initial data
for $(\partial_vr)^{-1}\partial_v\phi$ lies in a more regular
H\"{o}lder space than the regularity of the background scalar field,
then Singh showed that the blue-shift instability is \underline{not}
present, and the corresponding solution to $\Box_g\phi = 0$ obeys
bounds which are less singular than the scalar field of the background.
For initial data which lies in the same H\"{o}lder space as the
background scalar field, the solution obeys bounds which are analogous
to the background scalar field. In so far as solutions to the wave
equation $\Box_g\phi = 0$ are a proxy for perturbations to the
Einstein-scalar field system and non-spherically symmetric
gravitational perturbations, these results are consistent with the
$k$-self similar singularities being nonlinearly stable to sufficiently
regular perturbations! We conclude that the validity of the weak cosmic
censorship conjecture may depend sensitively on the precise functional
framework which is used both to define admissible spacetimes and
``generic'' perturbations. A similar connection between the validity of
the strong cosmic censorship conjecture and the specific functional
framework used has been an active area of research, for example,
see~\cite{DafLuk1,QNMSCC,roughdatacosm,roughsmooth}.

\section{Trapped surface formation in vacuum}\label{trappedvacuum}

We have seen in Section~\ref{sphsymmscalfield} that trapped formation
plays an important role in issues related to weak cosmic censorship. In
moving from the spherically symmetric Einstein-scalar field system to
the vacuum Einstein equations with no symmetry restrictions, the
following simple sounding questions already turn out to be highly
non-trivial: 

\begin{question}\label{trappedquestion}
Is it possible for a trapped surface to form in evolution from 
trapped, marginally trapped, anti-trapped, and marginally anti-trapped
surface free initial data with one asymptotically flat end? What about
the case of  scattering initial data posed on past null infinity
$\mathcal{I}^-$?
\end{question} 

Note that every Cauchy hypersurface in a Schwarzschild or a Kerr black
hole spacetime is both two-ended and possesses either a trapped,
anti-trapped, or marginally trapped surface. In contrast, the natural
setting for the study of gravitational collapse is in the evolution of
Cauchy data with one asymptotically flat end which is far from
possessing any trapped or anti-trapped surface.

Question~\ref{trappedquestion} was answered positively by Christodoulou
in the monumental work~\cite{christtrappedvac}. Moreover, what is even
more significant than the knowledge of a positive answer to
Question~\ref{trappedquestion}, is that the
work~\cite{christtrappedvac} introduced an entirely new paradigm for
understanding solutions to the Einstein vacuum equations for
(relatively) long times in certain large data regimes and with
\underline{no symmetry}. In the following sections we will start with a
brief description of~\cite{christtrappedvac} and then we will proceed
to describe various follow-up works which have introduced powerful new
ideas and lead to many extensions of the original result of
Christodoulou. The end result is that we currently have a quite
sophisticated understanding of trapped surface formation  even if we
still do not quite have a theorem for the Einstein vacuum equations
which is as strong as Theorem~\ref{christsphtrapptheo}. (It should be
noted, however, that it is far from clear that such an analogue can
exist without symmetry assumptions.)

Christodoulou's work~\cite{christtrappedvac} and essentially all of the
follow-up works on trapped surface formation have worked with the
Einstein vacuum equations expressed in a double-null gauge (note,
however, the interesting recent exception~\cite{chenklainerman}). Many
of these works have furthermore fundamentally exploited the
characteristic initial value problem. We assume familiarity with these
in the following sections and have included brief reviews in
Sections~\ref{eqnsdblenull}--\ref{charreview} of the Appendix.
 
\subsection{The short pulse method of Christodoulou}\label{shortpulsesec}

In this section we will review the short pulse method introduced in the
work~\cite{christtrappedvac}. For a more thorough review of the
original work~\cite{christtrappedvac} and a broader discussion of the
relevant context, in addition to the original
text~\cite{christtrappedvac}, we recommend the
article~\cite{dafermossurveytrapped}.

A key idea in~\cite{christtrappedvac} was to consider a certain class
of characteristic initial data for the Einstein vacuum equations which
allows for some ``largeness'' to be pumped into the system in a
controlled way. We now give a brief overview of this strategy. We
consider a characteristic initial value problem with data posed along
two transversely intersecting null hypersurfaces $\{u = u_0\}$ and $\{v
= 0\}$ for $u_0 < 0$ and $|u_0| \gg 1$. In order to eventually obtain
scattering data on $\mathcal{I}^-$, one should keep in mind that we
will eventually take take the limit $u_0 \to -\infty$. Along the
incoming hypersurface $\{v = 0\}$ Christodoulou placed trivial
Minkowskian data. Along $\{u = u_0\}$, we take $\Omega = 1$. The key
starting point for the short pulse method is to choose the remaining
characteristic data along $\{ u = u_0\}$ so that, in orthonormal frames
on $\mathbb{S}^2$,
{\begin{equation}\label{shortpulseans}
\hat{\chi}_{AB} (v,\theta^C) = |u_0|^{-1}\delta^{-1/2}f_{AB}
\left(\dfrac{v}{\delta},\theta^C\right),
\end{equation}}\unskip
where $0 < \delta \ll 1$ may be taken arbitrarily small, and
$f_{AB}(x,\theta^A)$ is a smooth symmetric $(0,2)$-tensor compactly
supported in $(x,\theta^A) \in (0,1) \times \mathbb{S}^2$ and
satisfying, for each $\theta^C \in \mathbb{S}^2$,
{\begin{equation}\label{lowerboundf}
\int_0^1 (f_{AB}(x,\theta^C))^2\,\mathrm{d}x = c_0,\quad 
\Vert f_{AB}\Vert _{C^k((1,2)\times \mathbb{S}^2)} \lesssim 1,
\end{equation}}\unskip
where $c_0 \sim 1$ is a suitable $\delta$-independent constant and $k$
is a suitably large $\delta$-independent integer. The gravitational
radiation field along $\mathcal{I}^-$ is given by $\lim_{u\to
-\infty}\hat{\chi}_{AB} (v,\theta^C)|u|$, and, in the limit as $u_0 \to
-\infty$, Equations~(\ref{shortpulseans}) and~(\ref{lowerboundf}) end
up corresponding to a lower bound for the incoming energy per unit
solid angle in each direction in the advanced time interval
$[0,\delta]$.

There are now two main interrelated issues to understand. First, what
is the explicit mechanism we expect to lead to the formation of the
trapped surface? Second, in view of the fact that~(\ref{shortpulseans})
manifestly puts us in a large data regime, why can we expect to
construct enough of the spacetime to show that this mechanism takes
place? 

The first step in the analysis is, still at the level of the initial
null hypersurface $\{u = u_0\}$, to obtain suitable estimates for all
of the Ricci coefficients and null curvature components. Given the
complexity of the equations of the double-null gauge (see
Section~\ref{eqnsdblenull}) this at first appears like a task of
daunting complexity. However, Christodoulou exploits a remarkable
inductive structure in the Einstein equations, which enables one to
sequentially estimate all of the required terms in terms by solving
transport and elliptic equations only in terms of quantities already
controlled. A key point in this analysis is that
while~(\ref{shortpulseans}) is large in $L^{\infty}_v$, it is small in
any $L^p_v$ with $p \in [1,2)$, and thus solutions to transport
equations in this process often gain smallness in $\delta$. We list
here the results of this analysis for the null curvature components:
{\begin{equation}\label{curvh1}
|\alpha| \lesssim \delta^{-3/2}|u_0|^{-1},\quad 
|\beta| \lesssim \delta^{-1/2}|u_0|^{-2},\quad 
|\rho| + |\sigma| \lesssim |u_0|^{-3},
\end{equation}}\unskip
{\vspace*{-1pc}}
{\begin{equation}\label{curvh2}
|\underline{\beta}| \lesssim \delta |u_0|^{-4},\quad 
|\underline{\alpha}| \lesssim \delta^{3/2}|u_0|^{-5}.
\end{equation}}\unskip
We note that the $u_0$ weights here reflect a hierarchy consistent with
\textit{peeling} (see the discussion in~\cite{dafermossurveytrapped}).
It is also worth remarking that the powers of $\delta$ one sees
in~(\ref{curvh1}) and~(\ref{curvh2}) deviate from what one would expect
purely based on linear considerations. 

The bulk of the technical work in~\cite{christtrappedvac} is concerned
with showing that these hierarchies of estimates understood for the
characteristic initial data can, with a few modifications, be
propagated into the spacetime in a region $\{(u,v,\theta^A) \in
[u_0,\check{u}] \times [0,\delta] \times \mathbb{S}^2\}$. Here
$\check{u} < 0$ may be taken arbitrarily small as long $\delta$ is then
taken sufficiently small depending on $\check{u}$. In order to prove
this ``semi-global'' existence result, in addition to transport and
elliptic equation estimates, Christodoulou introduces an intricate
scheme of energy estimates using weighted vector fields in conjunction
with the Bel--Robinson tensor. While we will not discuss these estimates
in more detail here, it is worth emphasizing that it is a remarkable
fact that the structure behind the initial data hierarchy can be
exploited in evolution all the way from $\{u = u_0\}$ to $\{u =
\check{u}\}$!

Now that we have discussed the spacetime construction, we can explain
at a schematic level how the formation of the trapped surface occurs.
First we must guarantee that there is no trapped surface in the initial
data. In view of~(\ref{shortpulseans}) and~(\ref{lowerboundf}), we have
that $\Vert \hat{\chi}\Vert ^2_{L^{\infty}_{\mathbb{S}^2}L_{u=u_0}^2}
\lesssim c_0|u_0|^{-2}$. It then follows by integrating Raychaudhuri's
Equation~(\ref{4trchi}), the relation $\mathcal{L}_{e_4}\slashed{g} =
\chi$, the fact that $\Omega = 1$ along $\{u = u_0\}$, and the
triviality of the data along $\{v = 0\}$, that we have, along $\{u =
u_0\}$
{\[
\left|\mathrm{tr}\chi - \dfrac{2}{|u_0|}\right| \lesssim
\dfrac{c_0}{|u_0|^2}.
\]}\unskip
In particular, it is clear that $\mathrm{tr}\chi > 0$ and we are far
from having a trapped surface along $\{u = u_0\}$. Next we would like
to show that a trapped surface emerges at $(u,v) = (\check{u},\delta)$.
Our plan will be again to eventually integrate the Raychaudhuri
Equation~(\ref{4trchi}); however, first we will show $\hat{\chi}$
undergoes an amplification while evolving from $u = u_0$ to $u =
\check{u}$. In order to see we use the $\nabla_3$ propagation equation
for $\hat{\chi}$~(\ref{3hatchi}) as well as the following upper bounds
which are a consequence of the estimates in the aforementioned
semi-global existence result:
{\[
\left|\mathrm{tr}\underline{\chi} - \dfrac{2}{u}\right|
\lesssim \delta |u|^{-2},\quad |\Omega\underline{\omega}| 
\lesssim \delta |u|^{-3},\quad |\eta| + |u|\delta^{1/2}
|\slashed{\nabla}\eta| \lesssim \delta^{1/2}|u|^{-2},
\]}\unskip
{\vspace*{-1pc}}
{\[
|\hat{\underline{\chi}}| \lesssim \delta^{1/2}|u|^{-2},\quad 
|\log\Omega| \lesssim \delta |u|^{-3}.
\]}\unskip
In particular we may write~(\ref{3hatchi}) as 
{\[
\nabla_3\hat{\chi} - (|u|^{-1}+ O(\delta |u|^{-2}))\hat{\chi} = 
O(|u|^{-3}).
\]}\unskip
Keeping in mind that $\delta$ may be assumed small depending on
$\check{u}$, integrating from $\{u = u_0\}$ to $\{u = \check{u}\}$ then
yields, in an orthonormal frame
{\begin{equation*}%\label{whathatchiis}
\hat{\chi}_{AB} (v,\theta^C)|_{u=\check{u}} = 
|\check{u}|^{-1}\delta^{-1/2}f_{AB} 
\left(\dfrac{v}{\delta},\theta^C\right) + O (|\check{u}|^{-2}).
\end{equation*}}\unskip
Now we turn again to the Raychaudhuri Equation~(\ref{4trchi}) which
implies
{\begin{equation}\label{tointray}
\partial_v (\Omega^{-1}\mathrm{tr}\chi) \leq -|\hat{\chi}|^2.
\end{equation}}\unskip
Since we have Minkowskian data along $\{v = 0\}$, we
integrate~(\ref{tointray}) from $v = 0$ along $\{u = \check{u}\}$ and
obtain
{\[
\Omega^{-1}\mathrm{tr}\chi|_{(u,v) = (\check{u},\delta)} \leq 
\dfrac{2}{|\check{u}|} - \dfrac{c_0}{2|\check{u}|^2} \Rightarrow 
\mathrm{tr}\chi < 0. 
\]}\unskip
In view of the smallness of $\check{u}$ we have the formation of a
trapped surface.

The initial data for Christodoulou's short pulse spacetimes are given
either on null hypersurfaces, or, once the limit $u_0 \to -\infty$ is
taken, on past null infinity. It is natural to also ask for a version
of this result with asymptotically flat initial data given along a
spacelike hypersurface free of trapped surfaces. Exactly this
construction was achieved in~\cite{spaceliketrapped}. The proof used
both the geometric information obtained from Christodoulou's estimates
as well as gluing techniques for the constraint equations developed by
Corvino--Schoen~\cite{corvsch}. We close the section by noting the
recent work~\cite{nullgluetrap} which provides a new proof of the
formation of trapped surfaces from trapped surface free Cauchy data in
the context of $C^2$ spacetimes. The proof is quite short compared to
other such results and employs a short pulse ansatz and null gluing
techniques. An interesting aspect of the proof is that the only
estimates for the Einstein equations in evolution needed are those
following from Cauchy stability (in particular it bypasses any need to
understand a semi-global existence problem).

In Figure~\ref{illustrationshortpulse} we have drawn the Penrose
diagram for Christodoulou's short pulse spacetime.

\begin{figure}
\includegraphics{fig05}
\caption{\label{illustrationshortpulse}Penrose diagram for short pulse
spacetime.}
\end{figure}

\subsection{A relaxed hierarchy and anisotropic trapped surface
formation}

Soon after Christodoulou's original work~\cite{christtrappedvac},
Klainerman--Rodnianski, focusing on the case of fixed $u_0$, introduced
a new strategy for the semi-global existence result which leads to
significant simplifications and opens the door to considering trapped
surface formation by anisotropic initial
data~\cite{KlRodTrap,KlRodScar}. The basic idea can be explain as
follows (we take $u_0 = -1$ in what follows). Christodoulou's choice of
data for $\hat{\chi}$~(\ref{shortpulseans}) leads to the following
upper bounds for any pair of integers $n$ and $m$ with $n+m \leq k$
(here $k$ is as in~(\ref{lowerboundf}))
{\begin{equation}\label{stronghier}
\delta^n\Vert \nabla_4^n\slashed{\nabla}^m
\hat{\chi}\Vert _{L^2_{\{u=-1\}}} \lesssim 1.
\end{equation}}\unskip
Klainerman--Rodnianski instead consider a wider class of initial data,
where one only requires the following weaker bound along $u = -1$:
{\begin{equation}\label{weakhier}
\delta^{n+m/2}\Vert \nabla_4^n\slashed{\nabla}^m
\hat{\chi}\Vert _{L^2_{\{u=-1\}}} \lesssim 1.
\end{equation}}\unskip

There are two main motivations for considering data of the
form~(\ref{weakhier}). First, the upper bound~(\ref{weakhier}) is
consistent with considering an initial $\hat{\chi}$ along $\{u = -1\}$
of the form
{\[
\hat{\chi}_{AB} (v,\theta^C) = \delta^{-1/2}f_{AB}
\left(\dfrac{v}{\delta},\dfrac{\theta^C}{\delta^{1/2}}\right).
\]}\unskip
This allows one to consider initial data which is highly localized in a
particular angular region of the sphere. This is of interest in order
to study various anisotropic phenomenon. 

Before explaining the second motivation, we need to review the notion
of \textit{signature} for the Einstein vacuum equations. For any Ricci
coefficient or curvature component $\psi$, we define
{\[
s(\psi) = N_4(\psi) + \tfrac{1}{2}N_A(\psi) - 1,
\]}\unskip
where $N_4$ is the number of times $L$ appears in the definition of the
component $\psi$ and $N_A$ is the number of angular frames:
{\[
s(\alpha) = 2, \quad s(\beta) = 3/2, \quad s(\rho,\sigma) = 1,\quad 
s(\underline{\beta}) = 1/2, \quad s(\underline{\alpha}) = 0,
\]}\unskip
{\vspace*{-1pc}}
{\[
s(\chi,\omega) = 1,\quad s(\zeta,\eta,\underline{\eta}) = 1/2,\quad
s(\underline{\chi},\underline{\omega}) = 0,
\]}\unskip
{\vspace*{-1pc}}
{\[
s(\nabla _4\phi) = 1 + s (\phi), \quad
s(\slashed{\nabla}\phi) = 1/2+ s (\phi),\quad
s(\nabla _3\phi) = s (\phi).
\]}\unskip
We also set
{\[
s(\phi_1\cdot\phi_2) = s(\phi_1) + s(\phi_2),
\]}\unskip
where $\cdot$ stands for any type of contraction. A key point behind
the usefulness of the signature concept is that the equations of the
double-null gauge preserve signature in that all terms in any given
equation have the same signature. Signature is furthermore preserved
upon commutation with $\nabla_4$, $\nabla_3$, and $\slashed{\nabla}$.
The weaker bound~(\ref{weakhier}) turns out to mesh nicely with a
hierarchy of $\delta$-bounds which is based on signature, and  this
ultimately allows for many simplifications in the corresponding
semi-global existence result. We close by noting that this relaxed
hierarchy turns to not quite be strong enough to see the amplification
of $\hat{\chi}$ discussed at the end of Section~\ref{shortpulsesec},
however, by interpolating the relaxed bounds with some other relatively
straightforward bounds based on~(\ref{stronghier}), the
work~\cite{KlRodTrap} is also able to recover the trapper surface
formation result of Christodoulou. The work~\cite{anthesis} extended
these methods to also allow for data on past null infinity as in
Christodoulou's original work.

While the works~\cite{KlRodTrap,KlRodScar} allowed for highly
anisotropic data~(\ref{weakhier}) they were not quite able to prove
that trapped surfaces arise from such data (though they did show the
formation of so-called pre-scarred surfaces where $\mathrm{tr}\chi <0$
on an arbitrarily large portion of a sphere). In the
work~\cite{trappedaniso} Klainerman, Luk, and Rodnianski were able to
establish a fully anisotropic trapped surface formation result. The
assumptions of the main theorem can be taken to be the same as
Christodoulou's work~\cite{christtrappedvac} except with one
significant change, namely that instead of the left hand side
of~(\ref{lowerboundf}) holding for every point in $\mathbb{S}^2$, we
only now require that
{\begin{equation}\label{assumaniso}
\sup _{\theta^A \in \mathbb{S}^2} \int _0^1
(f_{AB}(x,\theta^A))^2\,\mathrm{d}x \sim 1.
\end{equation}}\unskip
The key idea in the work~\cite{trappedaniso} was to look for a trapped
surface lying along $\{v = \delta\}$ but with respect to a foliation by
a new function $\tilde{u}$ as opposed to $u$. At a technical level, the
work~\cite{trappedaniso} shows that if one perturbs the null coordinate
$u$ along $\{v = \delta\}$ with the addition of a function
$R\left(\theta^A\right)$ defined along $\mathbb{S}^2$, then the
outgoing second fundamental forms for the new spheres have a
transformation formula which involves an elliptic operator applied to
the function $R$. It is then shown that the
assumption~(\ref{assumaniso}) is sufficient to find an $R$ satisfying a
corresponding elliptic differential inequality leading to a trapped
surface in the new foliation.

\subsection{Renormalization: impulsive waves and applications to
trapped surface formation}

A connection can be drawn between the semi-global existence result
underlying Christodoulou's trapped surface formation result and
low-regularity well-posedness for the Einstein vacuum equations.
Namely, since the ansatz~(\ref{shortpulseans}) corresponds to the data
for the metric being large in $H^s$, when $s > 1$, bounded when $s =
1$, and small when $s < 1$, one is lead to ask, in general, what type
of initial Sobolev regularity is required for well-posedness of the
initial value problem. As it turns out, a large literature exists
around low-regularity well-posedness results for the Einstein equations
which, in view of space constraints, we will not attempt to survey
here. We will just point out that the culmination of this line of
research was the remarkable collection of
works~\cite{L2curv,szefstrich,L2param3,L2param12,L2param4} which,
roughly speaking, establish that the Einstein vacuum equations are
locally well-posed when the initial spacetime curvature lies in $L^2$
along a Cauchy hypersurface. Moreover, in view of forthcoming work of
Luk--Moschidis (see Theorem 4.6 of~\cite{icmluk}), this well-posedness
result is sharp when regularity is measured in standard Sobolev spaces.

The works~\cite{impulsivefirst,impulsive} by Luk and Rodnianski
initiated a new paradigm for considering certain classes of
low-regularity solutions to the Einstein vacuum equations. These works
were motivated by the desire to establish a local well-posedness result
for the propagation (and possible interaction) of \textit{impulsive
gravitational waves} (see~\cite{penimp,kahnpenrose}). Certain
components of the curvature tensors of such spacetimes possess
$\delta$-function singularities which propagate along null
hypersurfaces; for one or two propagating impulsive waves in a suitable
double-null foliation, the singular curvature components correspond to
$\alpha$ and $\underline{\alpha}$. One key insight
from~\cite{impulsivefirst,impulsive}, which has played a crucial role
in many later developments, is that $\alpha$ and $\underline{\alpha}$
can be removed from the Bianchi equations for curvature in such a way
that the remaining system still allows for energy estimates (see the
equations of Section~\ref{renormalbianchi}). The upshot of this theory
is that one has local well-posedness for characteristic initial value
problems when, schematically put, the shears $\hat{\chi}$ and
$\hat{\underline{\chi}}$ satisfy that 
{\begin{equation*}%\label{thenormimpulse}
\sum_{i=0}^3\Vert \slashed{\nabla}^i\hat{\chi}\Vert _{L^2_{u=u_0}} +
\sum_{i=0}^3\Vert \slashed{\nabla}^i\hat{\underline{\chi}}\Vert _{L^2_{v=v_0}} 
\doteq \mathcal{A} (\hat{\chi},\underline{\chi}) < \infty.
\end{equation*}}\unskip

Remarkably, Christodoulou's initial data~(\ref{shortpulseans}) exactly
satisfies that $\mathcal{A} (\hat{\chi},\underline{\chi}) \lesssim 1$,
and so from the ``renormalized'' point of view, the semi-global
existence result now becomes a question of local existence. Using this,
the work~\cite{impulsive} is able to attain another proof both of
Christodoulou's semi-global existence result and the corresponding
trapped surface formation. Furthermore, this new proof allowed, for the
first time, for non-trivial data along the incoming null hypersuface
$\{v = 0\}$. 

One natural question left open by all of these various results on
trapped surface formation, is whether anything can be said about the
short pulse spacetimes as the parameter $\delta \to 0$. This question
has been addressed in~\cite{rodlukburn} where the authors employ their
renormalization techniques to understand the connection between certain
types of high-frequency limits of the Einstein equations and null dust
solutions\footnote{These questions are motivated by Burnett's
conjecture~\cite{burnett}; see the survey article~\cite{lukcecreview}.}.
In particular, they show that one may take the $\delta \to 0$ (weak)
limit of the short pulse spacetimes and obtain a solution to the
Einstein-null dust system which also exhibits the formation of a
trapped surface.

\subsection{Scale-invariant trapped surface formation}

Christodoulou's trapped surface criteria in
Theorem~\ref{christsphtrapptheo} is scale invariant in that it is
invariant under the rescaling map~(\ref{rescalingmap}). It is natural
to ask if we can also have a scale invariant criteria for trapped
surface formation in vacuum, though first we must formulate the
question more precisely. In any given set of local coordinates
$\{x^{\alpha}\}$, the analogue of the rescaling
map~(\ref{rescalingmap}) is simply the map $x^{\alpha} \mapsto \lambda
x^{\alpha}$ for $\lambda > 0$. The Sobolev space $\dot{H}^{3/2}$ is the
unique homogeneous Sobolev space which is left invariant under this
rescaling. Thus, we ask if one can find a trapped surface formation
criteria which only requires largeness of the initial data for the
metric in $H^s$ for some $s \geq 3/2$. 

The ansatz~(\ref{shortpulseans}) leads to initial data which large in
$H^{1+\epsilon}$, for any $\epsilon > 0$. Thus a new type of data is
required. This new class of data was provided in the work~\cite{anluk}
by An and Luk which succeeded in establishing the desired
scale-invariant trapped surface formation result. While the
work~\cite{anluk} in fact includes a large family of data which
interpolates between Christodoulou like short pulse data and
scale-invariant data, here we will just describe an example of the
scale-invariant like data considered. Let $A \gg 1$ and $B \gg 1$ and
then choose $0 < \delta \ll 1 $,  depending on $A$ and $B$. We then
take the initial data along $\{v = 0\}$ to be trivial, and suppose that
$\hat{\chi}$ along $\{u = -1\}$ satisfies the following upper and lower
bounds (see the discussion in the introduction
to~\cite{anemergyappar}):
{\begin{equation}\label{scaleinvdata}
\sum_{i=0}^5\Vert \slashed{\nabla}^i\hat{\chi}
\Vert _{L^{\infty}_vL^2(\mathbb{S}^2_{-1,v})} \lesssim A, \quad 
\mathrm{inf}_{\omega \in \mathbb{S}^2}\delta^{-1}
\int_0^{\delta} |\hat{\chi}(v,\omega)|^2\,\mathrm{d}v \geq 4A^2.
\end{equation}}\unskip
Then the solution exists in a characteristic rectangle $(u,v,\theta^A)
\in [-1,-\delta BA ] \times [0,\delta] \times \mathbb{S}^2$, and the
sphere at $(u,v) = (-\delta BA,\delta)$ is trapped. 

To see the connection to $\hat{\chi}$ being small in $H^{1/2-s}$ for 
$s > 0$ (and hence the characteristic data being small in $H^{3/2-s}$),
it is useful to first imagine that $\hat{\chi}$ is a scalar function
$\tilde{\chi}(v,\theta^A)$, in which case, we could set $\tilde{\chi}
(v,\theta^A) = CAf({v}/{\delta})$ for any smooth, non-zero, compactly
supported function $f(x) : [0,1] \to \mathbb{R}$ and suitable constant
$C$ (independent of $\delta$). This would satisfy~(\ref{scaleinvdata}),
and we then have, for any $s \in [-1/2,1/2]$ 
{\[
\Vert \tilde{\chi}\Vert _{H^{1/2 - s}} \lesssim 
\Vert \tilde{\chi}\Vert _{L^2}^{1/2+s}
\Vert \tilde{\chi}\Vert _{H^1}^{1/2-s} \lesssim
(A\delta^{1/2})^{1/2+s} (\delta^{-1/2}A)^{1/2 - s} \lesssim 
A \delta^s.
\]}\unskip
To actually construct true scale-invariant characteristic initial data
for $\hat{\chi}$, one has to work a bit harder since topological
obstructions prevent the existence of a nowhere vanishing trace-free
$(0,2)$-tensor on $\mathbb{S}^2$. Nevertheless, it remains possible,
see Appendix C of~\cite{anemergyappar} for an explicit such
construction.

We close the section by noting the sequence of works
\cite{anemergyappar,anscaletrappedsig,andynaapparent,anisoapparent} 
where An and collaborators have studied the emergence of apparent
horizons and further refined the various trapped surface formation
results discussed above. We note in particular the
work~\cite{anscaletrappedsig} which by use of signature considerations
and a rescaling argument is able to provide new simpler proofs of the
scale-invariant trapped surface formation results from~\cite{anluk} and
also to connect the analysis to the peeling properties of the Einstein
equations. 
    
\section{Self-similarity and naked singularities in vacuum}
\label{nakedvacsec}

In view of Christodoulou's construction~\cite{ChristNaked} of a naked
singularity for the spherically symmetric Einstein-scalar field system,
it is natural to try to construct naked singularity solutions for the
Einstein vacuum equations. Unfortunately, there is no direct way to
import Christodoulou's construction to the Einstein vacuum equations in
view of the following:
\begin{enumerate}[(1)]
\item Birkhoff's theorem states that all spherically symmetric
solutions to the Einstein vacuum equations are isometric to
Schwarzschild. Thus, we cannot expect to simply turn off the scalar
field in Christodoulou's example.

\item Christodoulou's analysis relied heavily on the fact that the
spherically symmetric Einstein-scalar field system reduced, upon the
imposition of self-similarity, to a two dimension autonomous system.
There is no such reduction for the Einstein vacuum equations which is
also consistent with asymptotic flatness.
\end{enumerate}

Nevertheless, we have the following theorem:

\begin{theo}\cite{nakedext,nakedint}\label{nakedvaccum} 
There exists naked singularity solutions for the Einstein vacuum
equations.
\end{theo}

In Figure~\ref{illustrationsvacnaked} we have depicted a Penrose
diagram of the naked singularity solution. (Note that other than the
use of different labels for the hypersurfaces, the diagram is the same
as Figure~\ref{illustrationsphenaked}.) In the following sections we
will describe in more details various properties of these solutions and
discuss some of the main ideas which go into their construction.

\begin{figure}
\includegraphics{fig06}
\caption{\label{illustrationsvacnaked}Penrose diagram for vacuum naked
singularity.}
\end{figure}

\subsection{Qualitative properties and comparison with the solutions of
Christodoulou}

In this section we will compare the naked singularity solutions of
Christodoulou with the solutions produced by Theorem~\ref{nakedvaccum}.
When drawing analogies between solutions to the Einstein vacuum
equations and solutions to the spherically symmetric Einstein-scalar
field system, it is useful to identify the following:
{\[
\partial _u\phi \leftrightarrow \underline{\hat{\chi}}, \quad 
\partial _v\phi \leftrightarrow \hat{\chi}.
\]}\unskip
We will see that both classes of naked singularities share many
qualitative properties.

As with Christodoulou's naked singularities, the vacuum naked
singularities arise from asymptotically flat data along a null
hypersurface. We recall that for Christodoulou's solutions, the initial
characteristic data for the scalar field satisfies that
$\left(\partial_vr\right)^{-1}\partial_v\phi$ is smooth except when it
crosses the past light cone of the singular point where it is
H\"{o}lder continuous. The initial characteristic data for the
solutions of Theorem~\ref{nakedvaccum} can be considered analogous in
that, with respect to a suitable $v$-coordinate and for any $N \in
\mathbb{Z}$ with $N \gg 1$, Theorem~\ref{nakedvaccum} produces 
solutions so that the initial $\hat{\chi}$ is $C^N$ except where it
crosses the past light cone of the singularity. Across the past light
cone of the singularity, we have that $\hat{\chi}$ is H\"{o}lder
continuous.  The loss of regularity for $\hat{\chi}$ is only with
respect to $\partial_v$; it in fact is
$C^N_{\mathbb{S}^2}C^{1,\gamma}_v$.

We cannot appeal to Christodoulou's well-posedness result in BV in
order to justify that this class of initial data is ``sufficiently
regular''. Instead we start by observing that $\hat{\chi}$ being
H\"{o}lder continuous is more than what it is needed for the local
theory developed by~\cite{impulsivefirst,impulsive}. The analysis
from~\cite{impulsivefirst,impulsive} did not attempt to cover the
solution near the ``axis'' but, for this type of data with a H\"{o}lder
continuous $\hat{\chi}$, one expects such a well-posedness result to
hold.

We now turn to a discussion of the main singularity of the spacetime.
We have already mentioned that Christodoulou's naked singularities
cannot be extended to the singular point in such a way as to remain a
solution of bounded variation. One way to see this is to observe that
along the past light cone of singularity $\mathcal{C}$ we have
{\begin{equation*}%\label{blowupnotbv}
\int _{\mathcal{C}} |\partial _u\phi|\,\mathrm{d}u = \infty.
\end{equation*}}\unskip
For the naked singularities of Theorem~\ref{nakedvaccum} we have the
following analogous fact. There exists (at least one) null geodesic
$\gamma$ converging to the singularity so that
{\[
\int _{\gamma} |\hat{\chi}|\,\mathrm{d}s = \infty.
\]}\unskip
In fact, one can also check that there exists Jacobi fields which
blow-up along $\gamma$ and that the singularity is a \textit{strong
singularity} in the sense of Tipler~\cite{tipler}.

\subsection{Self-similarity for the Einstein vacuum equations}

A fundamental role in the naked singularity construction is played by
various types of \textit{self-similarity}. In
Section~\ref{FGsecselfsim} we discuss an important precursor, the
self-similar spacetimes of Fefferman--Graham. Then in
Section~\ref{twistsecselfsim} we will discuss a new type of
self-similar spacetimes and discuss why one could expect them to be
useful for the construction of naked singularities. 

\subsubsection{The self-similarity of Fefferman--Graham}
\label{FGsecselfsim}

Fefferman--Graham~\cite{FG1,FG2} were the first to systematically study
a class of self-similar solutions to the Einstein vacuum equations
without any symmetry assumptions. Their class of spacetimes was the
following:

\begin{definition}\label{FGdefselfsim} 
\upshape
We say that a spacetime $(\mathcal{M},g)$ is Fefferman--Graham
self-similar (FGSS) if $(\mathcal{M},g)$ may be equipped with a
double-null foliation~(\ref{dblenullform}) which covers $u \in
(-\infty,0)$ and at least one of $({v}/{-u}) \in [0,c)$ or $({v}/{-u})
\in (-c,0]$ for some small constant $c$, and so that $K = u\partial_u +
v\partial_v$ satisfies
{\begin{equation}\label{seflseflseflsimilar}
\mathcal{L}_Kg = 2g.
\end{equation}}\unskip
\end{definition}

Using~(\ref{seflseflseflsimilar}), a straightforward computation shows
that the metric components $\Omega$, $b$, and $\slashed{g}$ of a FGSS
spacetime must satisfy in coordinate frames that 
{\begin{equation}\label{formulaselfsimmetriccomponents}
\begin{array}{@{}c@{}}
\Omega (u,v,\theta^A) = \tilde{\Omega} 
\left(\dfrac{v}{u},\theta^A\right), \quad 
b^A (u,v,\theta^B) = u^{-1}\tilde{b}^A 
\left(\dfrac{v}{u},\theta^B\right), \\ 
\slashed{g}_{AB} (u,v,\theta^C) = 
u^2\tilde{\slashed{g}}_{AB}\left(\dfrac{v}{u},\theta^C\right),
\end{array}
\end{equation}}\unskip
for some $\tilde{\Omega}$, $\tilde{b}$, and $\tilde{\slashed{g}}$.

Fefferman--Graham did not actually work in the double-null gauge, but
it is possible to translate their study of self-similar solutions to
the double-null gauge. As a preliminary step in their program to
understand local conformal invariants, Fefferman--Graham studied formal
power series expansions for FGSS spacetimes. If we specialize their
results concerning these formal expansions to spacetime dimension $4$
and recast them in the language of the double null-gauge , we can
informally\footnote{See~\cite{scaleinvariant,twistedselfsimilar} for
precise statements.} phrase their result as follows: 

\begin{theo}
For every smooth metric $\slashed{g}_0$ on $\mathbb{S}^2$ and smooth
$\slashed{g}_0$-trace-free symmetric $(0,2)$-tensor $k_0$ on
$\mathbb{S}^2$, there exists a smooth FGSS metric $g$  defined in the
double-null gauge~(\ref{dblenullform}) for $\{0 \leq ({v}/{-u}) < 
1/2\} \cap \{ u \in (-\infty,0)\}$ so that $g$ has a unique formal
Taylor series expansion in ${v}/{-u}$, $g \approx \sum_{j=1}^{\infty}
g_j$, and 
\begin{enumerate}[(1)]
\item The induced metric $\slashed{g}$ on $\mathbb{S}^2_{u,v}$
satisfies $\slashed{g}|_{v=0} = u^2\slashed{g}_0$ in the coordinate
frame, the quantities $b$ and $\log\Omega$ associated to $g$ vanish
along $\{v = 0\}$, and the restriction of $\hat{\chi}_{AB}$ to $\{v =
0\}$ satisfies $\hat{\chi}_{AB} = u\left(k_0\right)_{AB}$ in the
coordinate frame.

\item\label{ricciboundfg} For every sufficiently large positive integer
$n$ we have $\mathrm{Ric} (\sum_{j=1}^ng^{(j)}) (X,Y) = O
(|{v}/{u}|^n)$, for $X,Y \in
\{\partial_v,\partial_u,u^{-1}\partial_{\theta^A}\}$.
\end{enumerate}
\end{theo}

In Figure~\ref{fgpict} we depict the region of spacetime formally
covered by these expansions:

\begin{figure}
\includegraphics{fig07}
\caption{\label{fgpict}Region covered the Fefferman--Graham formal
expansions.}
\end{figure}

We note that if $\slashed{g}_0$ and $k_0$ are analytic, then the formal
expansions produced by the above theorem will actually converge
(see~\cite{FG2}). If we leave the analytic category, then we can appeal
to the work~\cite{scaleinvariant} to show the existence of a true
solution in the region to the future of $\{v = 0\}$ provided
$\slashed{g}$ and $k_0$ lie in suitable Sobolev spaces.

In particular, in view of~(\ref{formulaselfsimmetriccomponents}) it is
plausible that a FGSS spacetime cannot be extended in a regular fashion
to $u = 0$ along curves of constant ${v}/{u}$. Thus  it may appear a
reasonable strategy to use a FGSS spacetime to try to build a naked
singularity\footnote{We note that even if one did not know about the
work of Fefferman--Graham, in view of the long tradition of the use of
self-similarity to produce singular solutions to PDEs in mathematical
physics, it would be reasonable to try to search for self-similar
solutions of the Einstein equations as part of the pursuit of naked
singularities.}. However, the formal theory of Fefferman--Graham and its
extensions in~\cite{scaleinvariant} only concern the description of the
spacetime \underline{locally} near $\{v = 0\}$, while the construction
of a naked singularity involves a \underline{global} analysis, that is,
we need our spacetime to contain a hypersurface corresponding to
complete and asymptotically flat initial data. The global analysis
turns out to be problematic because there is no known way to extend a
FGSS spacetime to the past of $\{v = 0\}$ in such a way which is
compatible with the existence of a past complete Cauchy hypersurface
other than by gluing a \textit{flat} spacetime to the past of $\{v =
0\}$. Unfortunately, such a flat extension to the past of $\{v = 0\}$
prevents the resulting spacetime from being considered (part of) a
naked singularity for (at least) the following two reasons: 
\begin{enumerate}[(1)]
\item If the spacetime is flat to the past of $\{v = 0\}$, then it is a
consequence of the formal theory of Fefferman--Graham that  in order
for the spacetime to not be globally flat, we must allow $\hat{\chi}$
to have a jump discontinuity along $\{v = 0\}$.  This lowers the
regularity of whatever we pick to be our Cauchy hypersurface, the
resulting spacetime represents an impulsive gravitational wave with an
ingoing spherical front, and it is difficult to argue that the
``singularity'' at $\{u = 0\}$ represents a loss of regularity.

\item If the spacetime is flat to the past of $\{v = 0\}$ then any
causal curve entering the ``singularity'' at $\{u=0\}$ will only pass
through a flat spacetime and thus not actually encounter any truly
singular behavior.
\end{enumerate}

\subsubsection{Twisted self-similarity}\label{twistsecselfsim}

One of the key ideas in our work~\cite{nakedext} was the introduction
of a generalization of FGSS spacetimes which is consistent with the
possibility of non-flat, past complete extensions to the past of $\{v =
0\}$. We will give a brief introduction in this section to this new
type of self-similarity.  We direct the reader to the
work~\cite{twistedselfsimilar} for a systematic treatment. 

The rigidity of FGSS spacetimes that we have mentioned in the previous
section has its source in the following three closely related facts
about the geometry of the null hypersurface $\{v = 0\}$ in the FGSS
spacetimes:
\begin{enumerate}[(1)]
\item The induced (indefinite) metric along $\{v = 0\}$ is
$u^2\tilde{\slashed{g}}_{AB} (\theta^C)\,\mathrm{d}\theta^A\,\mathrm{d}
\theta^B$. In view of the uniformization theorem, we can write
$\tilde{\slashed{g}}_{AB} =
\mathrm{e}^{2\varphi}\mathring{\slashed{g}}_{AB}$ where
$\mathring{\slashed{g}}_{AB}$ denotes the round metric on the sphere
and $\varphi$ is a function on $\mathbb{S}^2$. Then, defining a new
coordinate $\tilde{u} \doteq u \mathrm{e}^{\varphi}$, in the
$(\tilde{u},\theta^A)$ coordinates, the induced metric along $\{v =
0\}$ becomes $\tilde{u}^2
\mathring{\slashed{g}}_{AB}\,\mathrm{d}\theta^A\,\mathrm{d}\theta^B$. 
This is identical to the induced metric along $\{v = 0\}$ in Minkowski
space!

\item A computation shows that all components of the curvature tensor
which are tangential to $\{v = 0\}$ must vanish.

\item\label{parallelorbit} The orbits of the restriction of the
conformal Killing field $K = u\partial_u + v\partial_v$ to $\{v = 0\}$
coincide with the orbits of the null generators of $\{v = 0\}$.
Equivalently, the shift vector vanishes on $\{v=0\}$.
\end{enumerate}

In~\cite{nakedext} we modified Definition~\ref{FGdefselfsim} by only
requiring that the double-null coordinate system was valid for
$({v}/{-u}) \in (0,c)$ instead of $[0,c)$. We then required that the
spacetime extends to $\hat{v} = 0$ in a new coordinate system
$(\hat{v},u,\theta^A)$ where 
{\begin{equation*}%\label{thisishowhatvdef}
\hat{v} \doteq v^{1-2\kappa}
\end{equation*}}\unskip
for some $0 < \kappa \ll 1$. We refer to such a self-similar spacetime
as a $\kappa$-self-similar ($\kappa$SS) spacetime. Of fundamental
important for the construction is the ingoing Raychaudhuri
Equation~(\ref{3trunchi}) along the null hypersurface $\{\hat{v} =
0\}$. If $\lim_{v\to 0}\Omega^2 ({v}/{-u})^{2\kappa} = 1$, then this
becomes the following equation along $\{\hat{v} = 0\}$:
{\begin{equation}\label{selfsimilartwistyayv0constraint}
\slashed{\mathrm{div}}b - \mathcal{L}_b
\slashed{\mathrm{div}}b - 
\tfrac{1}{2} 
(\slashed{\mathrm{div}}b)^2 = 
\tfrac{1}{4}
|\slashed{\nabla}\hat{\otimes}b|^2-4\kappa.
\end{equation}}\unskip
The presence of the free constant $-4\kappa$ turns out to allow us to
find many solutions to~(\ref{selfsimilartwistyayv0constraint}) with a
non-trivial $\slashed{\nabla}\hat{\otimes}b$. If we can construct
solutions with this new type of self-similarity and non-trivial
$\slashed{\nabla}\hat{\otimes}b$, then the vector field $K$ will still
extend to $\{\hat{v} = 0\}$, but now the orbits of $K$ will
\textit{not} coincide with null generators of the null hypersurface
$\{\hat{v} = 0\}$, and the intrinsic geometry of the $\{\hat{v} = 0\}$
hypersurface will not agree with the intrinsic geometry of a standard
null hypersurface in Minkowski space.

Under a suitable smallness assumption on $b|_{v=0}$, one may establish
an analogue of the formal expansions of Fefferman--Graham for
$\kappa$SS spacetimes. While a systematic discussion of these formal
expansions is outside the scope of the survey article we will observe
here some of the consequences of this analysis:
\begin{enumerate}[(1)]
\item The best regularity in the $\hat{v}$-variable we can expect for a
$\kappa$SS spacetime is $C^{1,\gamma}$ for $0 < \gamma \ll 1$. This is
compensated, however, by additional regularity in the $u$ and angular
variables as long as we make certain specific (non-trivial) choices of
the shift vector $b$; for any $N \geq 1$, if the shift vector takes a
certain specific form along $\{v = 0\}$ (to leading order in
$\epsilon$) we expect that our spacetime is at least in the class
$C^{1,\gamma}_{\hat{v}}C^N_{u,\theta^A}$.

\item A key role is played by developing an existence theory and a
priori estimates for equations  which are of the form $\psi +
\mathcal{L}_X\psi + W \psi = H$, where $\psi$ is an unknown tensor on
$\mathbb{S}^2$ and $(X,W,H)$ are given tensors satisfying a suitable
smallness condition (cf.~(\ref{selfsimilartwistyayv0constraint})). It
is also important to study the evolutionary analogue of these
equations: $(-v)\mathcal{L}_{\partial_v}\psi + \psi + \mathcal{L}_X\psi
+ W \psi = H$, where now all of the tensors additionally depend on $v
\in (0,c)$ for some $c > 0$. 

\item As in the choice of $(\slashed{g}_0,k_0)$ in the
Fefferman--Graham expansions, there is a certain set of ``free data''
which parametrizes the expansions for $\kappa$SS solutions.
\end{enumerate}

Finally, we note that one may in fact consider versions of the above
modification of FGSS where we allow for the form of the regular
coordinate $\hat{v}$ to be much more general than $\hat{v} \doteq
v^{1-2\kappa}$. We refer to this class of more general spacetimes as
``twisted self-similar'' (TSS) spacetimes,  and we systematically
studied them in the work~\cite{twistedselfsimilar} (some specific
instances of TSS spacetimes are also important in~\cite{nakedint}, see
Section~\ref{secint} below). 

\subsection{The exterior}

In this section we will discuss construction of the \textit{exterior}
region of the naked singularity which was carried out
in~\cite{nakedext}. This is the region to the future of the past light
cone of the singularity and is labeled $\mathcal{M}_{\mathrm{ext}}$ in
Figure~\ref{illustrationsvacnaked}.

Let $0 < \epsilon \ll \delta \ll 1$. We start by constructing our
desired spacetime in the region $\{({v}/{-u}) \in [0,\delta]\} \cap
\left\{u \in (0,-1]\right\}$ as depictied in Figure~\ref{firstregion}.
While our spacetime will not be an exact $\kappa$SS spacetime, it will
be approximately $\kappa$SS for $\kappa \sim \epsilon^2$ as $v\to 0$.
As with true $\kappa$SS spacetimes, we will have a regular coordinate
system $(\hat{v},u,\theta^A)$ for $\hat{v} \doteq v^{1-2\kappa}$ where
the spacetime extends to $\{\hat{v} = 0\}$. Tangentially along
$\{\hat{v} = 0\}$ the spacetime will be exactly $\kappa$SS in that
{\[
(v^{\kappa}\Omega)|_{\hat{v} = 0} (u,\theta^A) = (-u)^{\kappa}, \quad 
b^A|_{\hat{v} = 0} (u,\theta^B) = u^{-1}\tilde{b}^A (\theta^B),\quad
\slashed{g}_{AB}|_{\hat{v}= 0} (u,\theta^C) =
u^2\tilde{\slashed{g}}_{AB}(\theta^C),
\]}\unskip
for suitable $\tilde{b}$ and $\tilde{\slashed{g}}$ so that, in
particular, the fundamental constraint
Equation~(\ref{selfsimilartwistyayv0constraint}) holds. We also
require, further, that along $\left\{\hat{v} = 0\right\}$, $|b| \sim
\epsilon$, and that $b$ takes a certain precise form to leading order
as $\epsilon \to 0$.

\begin{figure}
\includegraphics{fig08}
\caption{\label{firstregion}Depiction of first constructed region of
exterior of naked singularity.}
\end{figure}

Our method for constructing this spacetime is to solve a characteristic
initial value problem with data along $\{\hat{v} = 0\}$  as described
in the paragraph above, and data along $\{u =-1\}$ as described  below.
Then, the construction of the spacetime essentially boils down to
suitable a priori estimates. We do not have space in this survey
article to discuss all of these, in the following we will only discus
some of the ideas involved in the energy estimates for curvature.  

In order to keep the presentation here simpler, in our discussion of
the outgoing characteristic initial data along $\left\{u=-1\right\}$,
we will only discuss the choice of $\Omega^{-1}\hat{\chi}$. Our a
priori estimates will rely on a renormalization scheme which in turn
relies on our spacetime being self-similar to leading order as
$({v}/{-u}) \to 0$ (see the discussion below). In particular, it turns
out to be necessary that $\Omega^{-1}\hat{\chi}|_{u=-1} =
\vartheta|_{u=-1} + O\left(v^a\right)$ for suitable $a \gtrsim 1$ and
where $\vartheta$ must satisfy
{\begin{equation}\label{freedatachihat}
v\mathcal{L}_{\partial_v}\vartheta_{AB} + 
\mathcal{L}_{b|_{v=0}}\vartheta_{AB} -
(\tfrac{1}{2}\slashed{\mathrm{div}}b|_{v=0} + 2\kappa)\vartheta_{AB}
- ((\slashed{\nabla}\hat{\otimes}b)|_{v=0})^C_{\ \ \left(A\right.}\vartheta_{\left.B\right)C} =
H_{AB}, 
\end{equation}}\unskip
where $H_{AB}$ is some tensor independent of $v$, which is determined
in terms of $b|_{v=0}$ and satisfies $|H| \sim \epsilon$. We also have
that $\kappa \sim \epsilon^2$ and that the operator
{\begin{equation}\label{theopthatisthere}
\vartheta \mapsto \mathcal{L}_{b|_{v=0}}\vartheta_{AB} - 
\tfrac{1}{2}\slashed{\mathrm{div}}b|_{v=0}\vartheta_{AB} - 
((\slashed{\nabla}\hat{\otimes}b)|_{v=0})^C_{\ \ \left(A\right.}\vartheta_{\left.B\right)C},
\end{equation}}\unskip
is anti-symmetric with respect to $L^2(\slashed{g})$.

Some of the key phenomenon ultimately associated with the solutions
$\vartheta$ to~(\ref{freedatachihat}) can already be understood if we
examine the much simpler scalar equation:
{\begin{equation}\label{simple}
v\mathcal{L}_{\partial_v}f - \epsilon^2 f = \epsilon .
\end{equation}}\unskip
One solution to~(\ref{simple}) is
{\[
f = -\dfrac{1-v^{\epsilon^2}}{\epsilon}.
\]}\unskip
Even though we will have that $f \sim \epsilon^{-1}$ when $v = 0$, we
also have the bound $|f| \lesssim \epsilon |\log(v)|$. For $\vartheta$
we also obtain similar estimates; however, a significant additional
difficulty, which is not present in the scalar Equation~(\ref{simple}),
is that we also need to control angular derivatives of $\vartheta$, and
naive commutation of the Equation~(\ref{freedatachihat}) destroys the
underlying anti-symmetric structure. Thus a careful commutation scheme
must be designed.  

We now briefly discuss the commutation scheme. The first key fact is
that we consider characteristic data which is almost axisymmetric.
Thus, the commutation errors with the axisymmetric vector field
$\partial_{\phi}$ will come with a small constant. The next realization
is that one can commute with the operator~(\ref{theopthatisthere}). The
vector field $b$ turns out to be linearly independent of
$\partial_{\phi}$ whenever it does not vanish and it thus turns out
these commutators suffice to control all derivatives of the solution
except at the points where $\partial_{\phi}$ or $b$ vanish. (We note,
however, that in view of the fact that $|b| \lesssim \epsilon$ the
estimates we obtain for higher derivatives have a very bad dependence
on $\epsilon^{-1}$.) At the points where $\partial_{\phi}$ or $b$
vanish we simply commute with $\slashed{\nabla}$ and the potentially
problematic terms that show up turn out to have a good sign.

A key idea to obtain the necessary a priori estimates is to subtract
off the expected leading order self-similar behavior of certain
curvature and Ricci components before carrying energy, transport, or
elliptic estimates. For example, the leading order self-similar
behavior of $\alpha$ is  generated by a self-similar extension of the
$\vartheta$ we discussed above. We obtain a new renormalized unknowns
$\tilde{\alpha}$ by subtracting off a self-similar extension of
$\vartheta$. In particular, with the renormalized unknowns we may use
weights are (mildly) singular as $({v}/{-u}) \to 0$. We note that one
may anticipate that the error terms generated by this renormalization
procedure are small because we know that we can produce formal power
series expansions for $\kappa$SS solutions which are valid as
$({v}/{-u})\to 0$. It is furthermore crucial for this scheme, that even
though $\vartheta$ is size $\epsilon^{-1}$ when $v = 0$, it quickly
becomes size $\epsilon$ once $\log(v) \sim 1$. One particular
consequence of these estimates will be that along the hypersurface
$({v}/{-u}) = \delta$ all curvature and Ricci components will be small in
that (letting $\Psi$ denote a curvature component and $\psi$ a Ricci
coefficient)
{\[
\left|\Psi|_{\frac{v}{-u} = \delta}\right| \lesssim \epsilon |u|^{-2}, \quad 
\left|\psi|_{\frac{v}{-u} = \delta}\right| \lesssim \epsilon |u|^{-1}.
\]}\unskip

The next region we study is the region where ${v}/{-u}$ is bounded away
from $0$ and $\infty$. The relevant Penrose diagram is depicted in
Figure~\ref{figreg2}. The region covered by the dashed lines
corresponds to the previous section, and the light-gray filled
in-region is what we will discuss in this section. In this region the
estimates are considerably simpler than in the previous region; for
example, the key curvature bounds essentially follow from a
Gr\"{o}nwell estimate with self-similar weights. 

\begin{figure}
\includegraphics{fig09}
\caption{\label{figreg2}Penrose diagram for second constructed region.}
\end{figure}

Finally, we come to the construction of the solution up to the
hypersurface $\{u=0\}$. This corresponds to the light gray region in
Figure~\ref{figreg3} (and the dashed region corresponds to the part we
have already constructed).

\begin{figure}
\includegraphics{fig10}
\caption{Penrose diagram for the third constructed
region.\label{figreg3}}
\end{figure}

As a consequences of the of the analysis of the previous region we will
already have suitable estimates for all of the double-null quantities
along $\{{v}/{-u} = \delta^{-1}\}$. We then extend data to an
asymptotically flat null hypersurface $\mathcal{W}$. As usual, the key
to then constructing the spacetime to the future of $\{{v}/{-u} =
\delta^{-1}\}$ and $\mathcal{W}$ is to establish suitable a priori
estimates. (After constructing the spacetime to the future of
$\mathcal{W}$ we can rescale the $u$-coordinate so that $\mathcal{W}$
corresponds to $\{u = -1\}$.) 

This region is, in many ways, dual to the region near $\{v = 0\}$ that
we discussed above. We expect the geometry near $\{u=0\}$ to be modeled
on a twisted self-similar solution (see~\cite{twistedselfsimilar}); in
particular, we expect to $\Omega\hat{\underline{\chi}}$ to be, at best,
H\"{o}lder continuous at $\{u = 0\}$ and for $\underline{\alpha}$ to
not lie in $L^2$. In the region near $\{v = 0\}$ there were similar
difficulties for $\hat{\chi}$ and $\alpha$ and we solved these via a
renormalization scheme which explicitly subtracted off from
$\hat{\chi}$ and $\alpha$ a self-similar term generated by the free
data $\vartheta$ which captured the singular and large parts of
$\hat{\chi}$ and $\alpha$. However, in this region, because we are not
posing initial data along $\{u = 0\}$, it is not possible to carry out
such an explicit renormalization procedure. Instead, we simply
eliminate $\underline{\alpha}$ from our Bianchi system using the
renormalized Bianchi equations derived
in~\cite{impulsivefirst,impulsive}. 

\subsection{The interior}\label{secint}

In this section we will discuss construction of the \textit{interior}
region of the naked singularity which was carried out
in~\cite{nakedint}. This is the region to the future of the past light
cone of the singularity and is labeled $\mathcal{M}_{\mathrm{in}t}$ in
Figure~\ref{illustrationsvacnaked}. Our plan will first be to construct
the solution in a double-null gauge which holds in a region
{\begin{equation}\label{regionregion}
(u,v,\theta^A) \in \left\{ u \in (-\infty,0)\mbox{ and } \dfrac{v}{-u}
\in (-1,0)\right\},
\end{equation}}\unskip
and where the self-similar vector field takes the form $u\partial_u +
v\partial_v$. We do not expect this coordinate system to be regular at
$\{v = 0\}$ (which will correspond to the past light cone of the
singularity) or along the ``axis'' $\left\{u = v\right\}$. In order to
carry out this construction we start by identifying a subset of the
null-structure equations and use these to derive various equations for
the metric components $\Omega$, $b$, and $\slashed{g}$. (It is outside
the scope of this survey article to describe these equations in
specific detail.) Using self-similarity we reduce these to a set of
equations along the null hypersurface $\{u = -1\}$, and then show the
existence of solutions along this hypersurface with suitable boundary
conditions at $\{v = -1\}$ and $\{v=0\}$. After using self-similarity
to expand our metric off of $\{u = -1\}$ to the entire
region~(\ref{regionregion}), the result will be a spacetime where a
certain subset of the Einstein vacuum equations hold. We then run a
``propagation of constraints'' argument to show that the spacetime is
actually Ricci flat.  Finally, we then construct regular coordinate
systems both at $\{v = 0\}$ and at the axis $\{v = u\}$ and show that
the interior spacetime may then be glued to our previously constructed
exterior spacetime.

Many of the equations we solve will be roughly of the following
form\footnote{These are relevant both for the original existence
argument and for the ``propagation of constraints'' argument. Also,
though we suppress this feature for our survey article, many of the
equations considered additionally involve various projections on
suitable spaces of spherical harmonics.}:
{\begin{equation}\label{thisiswhatasecondmodlookslike}
(-v)\partial _v^2\psi + b^A\slashed{\nabla}_A\partial _v\psi + 
\dfrac{(-v)A_1}{v+1} \partial _v\psi + A_2 \partial _v\psi +  \Omega^2
(\slashed{\Delta}+A_3(v+1)^{-2})\psi = H,\quad \psi|_{v=0} = h.
\end{equation}}\unskip
Here $\psi(v,\theta^A) : (-1,0) \times \mathbb{S}^2 \to
\mathbb{R}$ is the unknown, the $A_i$ are suitable constants, $H$ and
$h$ can be thought to be given tensors or represent nonlinear terms,
$\left(\Omega,b,\slashed{g}\right)$ will be assumed to satisfy various
bootstrap estimates, and $\slashed{\Delta}$ is the Laplacian associated
to~$\slashed{g}$.

A key difficulty is that because the shift vector $b$ does not vanish
identically when $v = 0$, our equation is of a mixed
elliptic--hyperbolic type. A relatively simple model equation for the
elliptic--hyperbolic aspects of the
problem~(\ref{thisiswhatasecondmodlookslike}) is
{\begin{equation*}%\label{modelsimpler}
(-v)\partial_v^2\psi + \epsilon \partial_{\phi}\partial_v\psi + 
\partial_{\phi}^2\psi = 0,\quad 
\psi|_{v=-1} = 0,\quad 
\psi|_{v=0} = h,
\end{equation*}}\unskip
where our unknown is $\psi(v,\phi) : (-1,0) \times (-\uppi,\uppi) \to
\mathbb{R}$, we have $0 < \epsilon \ll 1$, and $h : (-\uppi,\uppi) \to
\mathbb{R}$ is given. We consider a suitable family of smoothing
operators $\left\{\Pi_{\delta}\right\}_{\delta > 0}$ on $\mathbb{S}^2$
which converge to the identity as $\delta \to 0$, and then, for each $0
< \delta \ll 1$, introduce the regularized equation
{\begin{equation*}%\label{modelsimplerregularizeityay}
(-v)\partial_v^2\psi_{\delta} + \epsilon \Pi_{\delta} 
(\partial_{\phi}\partial_v\psi_{\delta}) + 
\partial_{\phi}^2\psi_{\delta} = 0,\quad 
\psi_{\delta}|_{v=-1} = 0,\quad 
\psi_{\delta}|_{v=-\delta} = h,
\end{equation*}}\unskip
for an unknown $\psi_{\delta} : (-1,-\delta) \times (-\uppi,\uppi) \to
\mathbb{R}$. This equation is elliptic\footnote{Note that the principle
symbol of the regularized equation is $(-v)\partial_v^2 +
\partial_{\phi}^2$ and that $-v$ is always positive in $v \in
(-1,-\delta)$.} using $-(\epsilon \partial_{\phi}\psi_{\delta} +
\psi_{\delta})$ as a multiplier, one may establish suitable a priori
estimates which hold uniformly as $\delta \to 0$. Then it is relatively
straightforward to establish the existence of a solution
$\psi_{\delta}$ and to obtain the solution $\psi$ as an appropriate
limit $\psi_{\delta} \to_{\delta\to 0} \psi$. Though we will not
discuss it in more detail, we note in passing that another important
difficulty in the analysis of equations of the
form~(\ref{thisiswhatasecondmodlookslike}) is that they are singular
near $v = -1$, which corresponds to the breakdown of the double-null
foliation at the axis. 

In order to construct a good coordinate system near the axis $\{u =v\}$
we introduce self-similar wave coordinates 
$(\xi_t,\xi_x,\xi_y,\xi_z)$, in the region $\{({v}/{-u}) \in
(-1,-\delta)\}$ for suitable $\delta > 0$, which satisfy
{\begin{equation}\label{howtodefinetheseselfsimilarcoordiantes}
\Box_g\xi_{\cdot} = 0, \quad K \xi_{\cdot} = \xi_{\cdot},
\end{equation}}\unskip
as well as suitable boundary conditions. If we work sufficiently close
to the axis, then $K$ will be timelike in the region we are defining
$(\xi_t,\xi_x,\xi_y,\xi_z)$ and the
system~(\ref{howtodefinetheseselfsimilarcoordiantes}) reduces to an
elliptic equation. Since, as is well-known, the Einstein vacuum
equations become quasilinear wave equations in harmonic coordinates, we
are eventually able to exploit various elliptic estimates to show that
$g$ is smooth in these harmonic coordinates. 

Near $\{v = 0\}$ we cannot expect our self-similar wave coordinates to
provide a good coordinate system.  The solution we construct in the
interior region will not directly correspond to the $\kappa$SS
solutions discussed in Section~\ref{twistsecselfsim}; instead it will
be of the more general TSS solutions discussed
in~\cite{twistedselfsimilar}. The key good coordinate system turns out
to be ``lapse renormalized coordinates'' $(\hat{v},u,\theta^A)$ where
we define $\hat{v}$ by
{\begin{equation*}\label{anewhatvyay}
\hat{v} (v,u,\theta^A) \doteq \int _0^v\Omega^2 
(\tilde{v},u,\theta^A) \,\mathrm{d}\tilde{v}.
\end{equation*}}\unskip
One shows that these coordinates allows for the exterior metric and
interior metric to be glued together along $\{\hat{v} = 0\}$ to produce
a $C^{1,\gamma}$ metric (with additional regularity in the $\theta^A$
and $u$ direction). The resulting spacetime is the desired naked
singularity solution.

\subsection{A few open questions}

We close the section by quickly listing some open questions and further
directions about naked singularities for the Einstein vacuum questions:
\begin{enumerate}[(1)]
\item It will be interesting to study both the stability and
instability problems of the solutions produced by
Theorem~\ref{nakedvaccum}. Do there exists similar phenomenon to those
discovered in Singh's work~\cite{jaystablenaked}?

\item Can one construct naked singularities arising from smooth, or
even $C^2$ initial data?

\item In higher dimensions, naked singularities are expected to be much
more ubiquitous~\cite{lehpretnakedsing,AnNaked}. It would be very
interesting to establish existence of asymptotically flat higher
dimensional naked singularities and to understand to what extent their
presence is generic. 
\end{enumerate}

\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*{Dedication}

The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Funding}

The author acknowledges support from an Alfred P. Sloan Fellowship in
Mathematics and from NSERC discovery grants RGPIN-2021-02562 and
DGECR-2021-00093.

\CDRGrant[NSERC]{RGPIN-2021-02562}
\CDRGrant[NSERC]{DGECR-2021-00093}

\back{}

\appendix{}

\def\appendixlabel{Appendix \Alph{section}}

\section{Equations of the double-null gauge}\label{eqnsdblenull}

In this appendix we collect the equations of the double-null gauge for
ease of reference and to fix the notation. For a proper introduction to
the double-null gauge we refer the reader to the
works~\cite{christtrappedvac,KN}. We will say that $(\mathcal{M},g)$ is
in the double-null gauge with the shift in the $u$-direction if
{\begin{equation}\label{dblenullform}
g = -2\Omega^2 
(\mathrm{d}u \otimes \,\mathrm{d}v + 
\mathrm{d}v\otimes \,\mathrm{d}u) + \slashed{g}_{AB}
(\mathrm{d}\theta^A - b^A\,\mathrm{d}u)\otimes
(\mathrm{d}\theta^B - b^B\,\mathrm{d}u),
\end{equation}}\unskip
holds, and that $(\mathcal{M},g)$ is in the double-null gauge with the
shift in the $v$-direction if~(\ref{dblenullform}) holds with the roles
of $u$ and $v$ interchanged. In the following, unless indicated
otherwise, we assume that the shift in the $u$-direction.

We define the null vector fields $e_4$ and $e_3$ by
{\[
e_4 \doteq \Omega^{-1}\partial _v,\quad 
e_3 \doteq \Omega^{-1} (\partial _u + b).
\]}\unskip
The $\mathbb{S}^2_{u,v}$-projected derivatives in the $e_4$ and $e_3$
direction are denoted by $\nabla_4$ and $\nabla_3$. The induced
covariant derivative along each $\mathbb{S}^2_{u,v}$ is denoted by
$\slashed{\nabla}$, $*$ refers to the Hodge star operator on
$(\mathbb{S}^2,\slashed{g})$, slashed operators are defined with
$\slashed{g}$, and we use $K$ to denote the Gaussian curvature of
$\slashed{g}$. We will use $\mathcal{L}$ to denote the Lie derivative.

The Ricci coefficients
$(\omega,\underline{\omega},\zeta_A,\eta_A,\underline{\eta}_A,\chi_{AB},\underline{\chi}_{AB})$
satisfy
{\begin{equation}\label{basicdefnull}
\begin{array}{@{}c@{}}
\omega = -\tfrac{1}{2}\nabla_4\log\Omega, \quad 
\underline{\omega} = -\tfrac{1}{2}\nabla_3\log\Omega, \quad 
\mathcal{L}_{\partial_v}b^A = -4\Omega^2\zeta^A, \\ 
\eta_A = \zeta_A + \slashed{\nabla}_A\log\Omega, \quad 
\underline{\eta}_A = -\zeta_A + \slashed{\nabla}_A\log\Omega, \\
\mathcal{L}_{e_4}\slashed{g}_{AB} = \chi_{AB}, \quad 
\mathcal{L}_{e_3}\slashed{g}_{AB} = \underline{\chi}_{AB}.
\end{array}
\end{equation}}\unskip
We split $\chi_{AB}$ and $\underline{\chi}_{AB}$ into their trace and
trace-free parts via
{\[
\chi_{AB} = \hat{\chi}_{AB} + 
\tfrac{1}{2}\mathrm{tr}\chi\slashed{g}_{AB}, \quad 
\underline{\chi}_{AB} = \hat{\underline{\chi}}_{AB} + 
\tfrac{1}{2}\mathrm{tr}\underline{\chi}\slashed{g}_{AB}.
\]}\unskip

The null curvature components
$ (\alpha_{AB},\beta_A,\rho,\sigma,\underline{\beta}_A,\underline{\alpha}_{AB})$ a
re related to the Ricci coefficients via the various null-structure
equations:
{\begin{equation}\label{4trchi}
\nabla_4\mathrm{tr}\chi + \tfrac{1}{2} (\mathrm{tr}\chi)^2 =
-|\hat{\chi}|^2 - 2\omega\mathrm{tr}\chi,
\end{equation}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{4hatchi}
\nabla_4\hat{\chi} + \mathrm{tr}\chi \hat{\chi} = -\alpha - 2\omega
\hat{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation}\label{3trunchi}
\nabla_3\mathrm{tr}\underline{\chi} + \tfrac{1}{2}
(\mathrm{tr}\underline{\chi})^2 = -|\underline{\hat{\chi}}|^2 -
2\underline{\omega}\mathrm{tr}\underline{\chi},
\end{equation}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{3hatunchi}
\nabla_3\hat{\underline{\chi}} + \mathrm{tr}\underline{\chi} 
\hat{\underline{\chi}} = -\underline{\alpha} - 2\underline{\omega} 
\hat{\underline{\chi}},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation}\label{3hatchi}
\nabla_3\hat{\chi} + \tfrac{1}{2}\mathrm{tr}\underline{\chi} \hat{\chi}
= 2\underline{\omega}\hat{\chi} + \slashed{\nabla}\hat{\otimes}\eta +
\eta\hat{\otimes}\eta - \tfrac{1}{2}\mathrm{tr}\chi
\hat{\underline{\chi}},
\end{equation}}\unskip
{\vspace*{-1pc}}
{\begin{equation}\label{3trchi}
\nabla_3\mathrm{tr}\chi + \tfrac{1}{2}\mathrm{tr}\chi
\mathrm{tr}\underline{\chi} = 2\rho +
2\underline{\omega}\mathrm{tr}\chi +2 \slashed{\mathrm{div}}\eta +
2|\eta|^2 - \hat{\chi}\cdot \hat{\underline{\chi}},
\end{equation}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{4hatunchi}
\nabla_4\hat{\underline{\chi}} + \tfrac{1}{2}\mathrm{tr}\chi
\hat{\underline{\chi}} = 2\omega\hat{\underline{\chi}} +
\slashed{\nabla}\hat{\otimes}\underline{\eta} +
\underline{\eta}\hat{\otimes}\underline{\eta} -
\tfrac{1}{2}\mathrm{tr}\underline{\chi} \hat{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{4trunchi}
\nabla_4\mathrm{tr}\underline{\chi} + \tfrac{1}{2}\mathrm{tr}\chi
\mathrm{tr}\underline{\chi} = 2\rho +
2\omega\mathrm{tr}\underline{\chi} +2
\slashed{\mathrm{div}}\underline{\eta} + 2|\underline{\eta}|^2 -
\hat{\chi}\cdot \hat{\underline{\chi}},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{4eta}
\nabla_4\eta = -\chi\cdot(\eta-\underline{\eta}) - \beta,
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{3uneta}
\nabla_3\underline{\eta} = -\underline{\chi}\cdot 
(\underline{\eta}-\eta) + \underline{\beta},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{curleta}
\slashed{\mathrm{curl}}\eta = \sigma +
\tfrac{1}{2}\hat{\underline{\chi}}\wedge \hat{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{curlueta}
\slashed{\mathrm{curl}}\underline{\eta} = -\sigma -
\tfrac{1}{2}\hat{\underline{\chi}}\wedge \hat{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{4uomega}
\nabla_4\underline{\omega} = \tfrac{1}{2}\rho +
2\underline{\omega}\omega + \tfrac{1}{2}|\eta|^2 -
\eta\cdot\underline{\eta},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{3omega}
\nabla_3\omega = \tfrac{1}{2}\rho + 2\underline{\omega}\omega +
\tfrac{1}{2}|\underline{\eta}|^2 - \eta\cdot\underline{\eta},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{Keqn}
K = -\rho + \tfrac{1}{2}\hat{\chi}\cdot\hat{\underline{\chi}} -
\tfrac{1}{4}\mathrm{tr}\chi \mathrm{tr}\underline{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{cod}
\slashed{\mathrm{div}}\hat{\chi} -
\tfrac{1}{2}\slashed{\nabla}\mathrm{tr}\chi = -\beta +
\tfrac{1}{2}\mathrm{tr}\chi \zeta - \zeta \cdot \hat{\chi},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{ucod}
\slashed{\mathrm{div}}\hat{\underline{\chi}} -
\tfrac{1}{2}\slashed{\nabla}\mathrm{tr}\underline{\chi} =
\underline{\beta} - \tfrac{1}{2}\mathrm{tr}\underline{\chi} \zeta +
\zeta \cdot \hat{\underline{\chi}},
\end{equation*}}\unskip

Finally we have the Bianchi equations for the curvature components:
{\begin{equation*}%\label{alphabeta}
\nabla_3\alpha + \tfrac{1}{2}\mathrm{tr}\underline{\chi} \alpha =
\slashed{\nabla}\hat{\otimes}\beta + 4\underline{\omega}\alpha -
3(\hat{\chi}\rho + {}^*\hat{\chi}\sigma) + (\zeta +
4\eta)\hat{\otimes}\beta,
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{betaalpha}
\nabla_4\beta + 2\mathrm{tr}\chi \beta = \slashed{\mathrm{div}}\alpha -
2\omega \beta + (2\zeta+\underline{\eta})\cdot \alpha,
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{betarhosigma}
\nabla_3\beta + \mathrm{tr}\underline{\chi}\beta = \slashed{\nabla}\rho
+ 2\underline{\omega} \beta + {}^*\slashed{\nabla}\sigma +
2\hat{\chi}\cdot\underline{\beta} + 3(\eta \rho + {}^*\eta
\sigma),
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{sigmabeta}
\nabla_4\sigma + \tfrac{3}{2}\mathrm{tr}\chi \sigma =
-\slashed{\mathrm{div}}{}^*\beta +
\tfrac{1}{2}\hat{\underline{\chi}}\wedge \alpha - \zeta \wedge \beta -
2\underline{\eta}\wedge \beta,
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{rhobeta}
\nabla_4\rho + \tfrac{3}{2}\mathrm{tr}\chi \rho =
\slashed{\mathrm{div}}\beta - \tfrac{1}{2}\hat{\underline{\chi}}\cdot
\alpha +\zeta\cdot\beta -2\eta\cdot\beta
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{sigmaubeta}
\nabla_3\sigma + \tfrac{3}{2}\mathrm{tr}\underline{\chi} \sigma =
-\slashed{\mathrm{div}}{}^*\underline{\beta} -
\tfrac{1}{2}\hat{\chi}\wedge \underline{\alpha} + \zeta \wedge
\underline{\beta} - 2\eta\wedge \underline{\beta},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{rhoubeta}
\nabla_3\rho + \tfrac{3}{2}\mathrm{tr}\underline{\chi} \rho =
-\slashed{\mathrm{div}}\underline{\beta} - \tfrac{1}{2}\hat{\chi}\cdot
\underline{\alpha} +\zeta\cdot\underline{\beta}
+2\underline{\eta}\cdot\underline{\beta},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{ubetarhosigma}
\nabla_4\underline{\beta} + \mathrm{tr}\chi\underline{\beta} =
-\slashed{\nabla}\rho + 2\omega \underline{\beta} +
{}^*\slashed{\nabla}\sigma + 2\hat{\underline{\chi}}\cdot\beta -
3(\underline{\eta} \rho - {}^*\underline{\eta} \sigma),
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{ubetaualpha}
\nabla_3\underline{\beta} + 2\mathrm{tr}\underline{\chi}
\underline{\beta} = -\slashed{\mathrm{div}}\underline{\alpha} -
2\underline{\omega} \underline{\beta} - (-2\zeta+\eta)\cdot
\underline{\alpha},
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{ualphabeta}
\nabla_4\underline{\alpha} + \tfrac{1}{2}\mathrm{tr}\chi
\underline{\alpha} = -\slashed{\nabla}\hat{\otimes}\underline{\beta} +
4\omega\underline{\alpha} - 3(\hat{\underline{\chi}}\rho -
{}^*\hat{\underline{\chi}}\sigma) - (-\zeta +
4\underline{\eta})\hat{\otimes}\underline{\beta}.
\end{equation*}}\unskip

\section{The renormalized Bianchi equations
from~\cite{impulsivefirst,impulsive}}\label{renormalbianchi}

We list here the renormalized Bianchi Equations
from~\cite{impulsivefirst,impulsive}. We first define
{\[
\check{\rho} \doteq \rho -
\tfrac{1}{2}\hat{\chi}\cdot\hat{\underline{\chi}}, \quad 
\check{\sigma} \doteq \sigma -
\tfrac{1}{2}\hat{\chi}\wedge\hat{\underline{\chi}}.
\]}\unskip
Then we have 
{\begin{eqnarray*}%\label{betacheckrhochecksigma}
\nabla_3\beta + \mathrm{tr}\underline{\chi}\beta &=& 
\slashed{\nabla}\check{\rho} + {}^*\slashed{\nabla}\check{\sigma} + 
2\underline{\omega}\beta + 2\hat{\chi}\cdot\underline{\beta} + 
3(\eta \check{\rho} + {}^*\eta\check{\sigma}) \\ 
&& +\, \tfrac{1}{2} (\slashed{\nabla}(\hat{\chi}\cdot
\hat{\underline{\chi}}) + {}^*\slashed{\nabla}
(\hat{\chi}\wedge \hat{\underline{\chi}})) + 
\tfrac{3}{2}(\eta\hat{\chi}\cdot\hat{\underline{\chi}} +
{}^*\eta \hat{\chi}\wedge \hat{\underline{\chi}}),
\end{eqnarray*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{checksigmabeta}
\nabla_4\check{\sigma} + \tfrac{3}{2}\mathrm{tr}\chi \check{\sigma} =
-\slashed{\mathrm{div}}{}^*\beta - \zeta\wedge \beta -
2\underline{\eta}\wedge \beta +
\tfrac{1}{2}\hat{\chi}\wedge\left(\slashed{\nabla}\hat{\otimes}\underline{\eta}\right) - \tfrac{1}{2}\hat{\chi}\wedge 
(\underline{\eta}\hat{\otimes}\underline{\eta}),
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{checkrhobeta}
\nabla_4\check{\rho} + \tfrac{3}{2}\mathrm{tr}\chi \check{\rho} =
\slashed{\mathrm{div}}\beta + \zeta\cdot\beta +
2\underline{\eta}\cdot\beta -
\tfrac{1}{2}\hat{\chi}\cdot (\slashed{\nabla}\hat{\otimes}
\underline{\eta}) - \tfrac{1}{2}\hat{\chi}\cdot
(\underline{\eta}\hat{\otimes}\underline{\eta}) + 
\tfrac{1}{4}\mathrm{tr}\underline{\chi}|\hat{\chi}|^2.
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{checksigmaubeta}
\nabla_3\check{\sigma} +
\tfrac{3}{2}\mathrm{tr}\underline{\chi}\check{\sigma} =
\slashed{\mathrm{div}}{}^*\underline{\beta} + \zeta \wedge
\underline{\beta} - 2\eta \wedge \underline{\beta} +
\tfrac{1}{2}\hat{\underline{\chi}}\wedge
(\slashed{\nabla}\hat{\otimes}\eta) + \tfrac{1}{2}\hat{\underline{\chi}}
(\eta\hat{\otimes}\eta),
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{equation*}%\label{checkrhoubeta}
\nabla_3\check{\rho} + \tfrac{3}{2}\mathrm{tr}\underline{\chi}
\check{\rho} = -\slashed{\mathrm{div}}\underline{\beta} + \zeta \cdot
\underline{\beta} - 2\eta \cdot \underline{\beta} -
\tfrac{1}{2}\hat{\underline{\chi}}\cdot
(\slashed{\nabla}\hat{\otimes}\eta) - \tfrac{1}{2}
\hat{\underline{\chi}}\cdot (\eta\hat{\otimes}\eta) + 
\tfrac{1}{2}\mathrm{tr}\chi |\hat{\underline{\chi}}|^2,
\end{equation*}}\unskip
{\vspace*{-1pc}}
{\begin{eqnarray*}%\label{ubetacheckrhochecksigma}
\nabla_4\underline{\beta} + \mathrm{tr}\chi \underline{\beta} &=&
-\slashed{\nabla}\check{\rho} + {}^*\slashed{\nabla}\check{\sigma} + 
2\omega \underline{\beta} + 2\hat{\underline{\chi}}\cdot\beta - 
3(\underline{\eta}\check{\rho} - {}^*\underline{\eta} \check{\sigma}) \\
&& -\,\tfrac{1}{2}(\slashed{\nabla}(\hat{\underline{\chi}}\cdot
\hat{\chi}) + {}^*\slashed{\nabla} (\hat{\underline{\chi}}\wedge 
\hat{\chi})) - \tfrac{3}{2} (\underline{\eta}\hat{\underline{\chi}}
\wedge \hat{\chi} + {}^*\underline{\eta} \hat{\underline{\chi}}
\wedge \hat{\chi}).
\end{eqnarray*}}\unskip

\section{Characteristic initial value problem}\label{charreview}

In this section we will quickly review the well-posedness of the
characteristic initial value problem for the Einstein equations. Choose
real numbers $u_0$, $u_1$, $v_0$, and $v_1$ with $u_0 < u_1$ and $v_0 <
v_1$. We say a triple of $\left(\Omega^{(\mathrm{in})},\left(b^A\right)^{(\mathrm{in})},\slashed{g}^{(\mathrm{in})}_{AB}\right)$
defined along $[u_0,u_1] \times \mathbb{S}^2$, a pair
$\left(\Omega^{(\mathrm{out})},\slashed{g}^{(\mathrm{out})}_{AB}\right)$ defined
along $[v_0,v_1] \times \mathbb{S}^2$, and a $1$-form $\tilde{\zeta}_A$
on $\mathbb{S}^2$ form a characteristic initial data set if
\begin{enumerate}[(1)]
\item We have $\left(\Omega^{(\mathrm{in})},\slashed{g}^{(\mathrm{in})}
\right)|_{u=u_0} =
\left(\Omega^{(\mathrm{out})},\slashed{g}^{(\mathrm{out})}\right)|_{v =
v_0}$.

\item We consider $\left(\Omega^{(\mathrm{in})},
\left(b^A\right)^{(\mathrm{in})},\slashed{g}
^{(\mathrm{in})}_{AB}\right)$ as being defined along a hypersurface
$\{v = v_0\}$ and $\left(\Omega^{(\mathrm{out})},
\slashed{g}^{(\mathrm{out})}_{AB}\right)$ as being defined along a
hypersurface $\{u = u_0\}$. After defining $\underline{\omega}$,
$\omega$, $\chi$, and $\underline{\chi}$ using~(\ref{basicdefnull}), we
require that the Raychaudhuri Equations~(\ref{4trchi})
and~(\ref{3trchi}) hold along $\{u = u_0\}$ and $\{v = v_0\}$
respectively. 
\end{enumerate}

We can now state a local existence theorem.

\begin{theo}\cite{rendallchar,charLuk}
Let $\left(\Omega^{(\mathrm{in})},\left(b^A\right)^{(\mathrm{in})},
\slashed{g}^{(\mathrm{in})}_{AB}\right)$, $\left(\Omega^{(\mathrm{out})},
\slashed{g}^{(\mathrm{out})}_{AB}\right)$, and  $\tilde{\zeta}_A$ form
an initial data set of sufficiently high regularity. Then there exists
a constant $c > 0$ and a unique solution $\left(\mathcal{M},g\right)$
to the Einstein vacuum equations expressed in a double-null gauge and
defined in a region $\left(u,v,\theta^A\right) \in \left([u_0,u_1]
\times [v_0,v_0+c] \times \mathbb{S}^2\right) \cup \left([u_0,u_0+c]
\times [v_0,v_1] \times \mathbb{S}^2\right)$ so that 
{\[
\left(\Omega^{(\mathrm{in})},\left(b^A\right)^{(\mathrm{in})},
\slashed{g}^{(\mathrm{in})}_{AB}\right) = 
\left(\Omega,b^A,\slashed{g}_{AB}\right)|_{v = v_0}, \quad 
\left(\Omega^{(\mathrm{out})},\slashed{g}^{(\mathrm{out})}_{AB}\right) = 
\left(\Omega,\slashed{g}_{AB}\right)|_{u=u_0}
\]}\unskip
{\vspace*{-1pc}}
{\[
\tilde{\zeta}_A = \zeta_A|_{(u,v) = (u_0,v_0)}.
\]}\unskip
\end{theo}

In Figure~\ref{charinitprobfigure} we have drawn a Penrose diagram
depicting the range of the solution.

\begin{figure}
\includegraphics{fig11}
\caption{\label{charinitprobfigure}Penrose diagram for spacetime
arising from characteristic initial value problem.}
\end{figure}

\printbibliography
\refinput{crmeca20240836-reference.tex}

\end{document}
