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\begin{document}\mbox{}
\vspace{0.25in}
\begin{center}
\huge{Conformal scattering and the Goursat problem~: general ideas and
what perspectives for black holes?}
\vspace{0.25in}
\large{Seminar, Newton Institute, Cambridge, 6$^\mathrm{th}$ september
2005, 15:30.}
\end{center}
\section{Introduction}
This talk is about time-dependent scattering in general
relativity. Dietrich H\"afner has amply described the topic in his
talk on friday. So let me just summarize the essential ideas.
%
\begin{itemize}
\item Fixed background space-time, Lorentzian metric, globally
hyperbolic ${\cal M} = \R_t \times \Sigma$, $\Sigma$
$3$-manifold with asymptotic ends.
\item Study the behaviour in the asymptotic ends of solutions of
covariant field equations on $\cal M$. Does so usually by
comparison with simplified dynamics (ideally asymptotic
profiles, i.e. the flow of a null congruence) in each asymptotic
region~: construction of wave operators and completeness. The
terminology ``time-dependent'' means that the construction is
based on the dynamics (the evolution) and not on stationary
solutions of the form $e^{i\omega t} \phi (x)$.
\end{itemize}
To this day, most of the works in this domain are constructions outside
black holes~: for Schwarzschild or other spherical black holes
(approximately chronologically) J. Dimock \cite{Di1985}, J. Dimock and
B. Kay \cite{DiKa1986a, DiKa1986b, DiKa1987}, A. Bachelot
\cite{Ba1991, Ba1994, Ba1997, Ba1999, Ba2000}, A. Bachelot and
A. Motet-Bachelot \cite{BaMo1993}, J.-P. Nicolas \cite{Ni1995b},
W.M. Jin \cite{Ji1998}, F. Melnyk \cite{Me2003, Me2004}, T. Daud\'e
(PhD thesis, 2004)~; for Kerr or Kerr-Newmann (approximately
chronologically) D. H\"afner \cite{Ha2003}, D. H\"afner and
J.-P. Nicolas \cite{HaNi2004}, T. Daud\'e (PhD thesis, 2004).
As I just mentionned, time-dependent scattering in relativity deals
with covariant field equations, i.e. equations that derive their
structure entirely from that of the metric. {\bf The fundamental
information for studying the scattering properties of such equations
is therefore the asymptotics of the metric in the asymptotic ends of
$\Sigma$.} All time-dependent scattering theories cited above are
constructed following the same essential procedure~:
\begin{enumerate}
\item Encoding of the fundamental information in an analytic
expression of the metric via a choice of coordinate system and
(for non scalar equations) of local frame. This step can already
be quite critical (Schwarzschild, good choice of radial
variable~; Kerr, example of our paper with Dietrich, we chose a
tetrad that allows a spherically symmetric comparison without
long range terms, which accounts for the fact that the effects
of the rotation are short range).
\item The information is then extracted via spectral techniques to
obtain the scattering theory itself. These techniques are
unstable with respect to time dependence of the coefficients of
the equation~: even for so-called time-dependent method, the
time dependence allowed for the coefficients is very simple and
means essentially that the methods can only be applied in
stationary situations (or at least locally stationary, such as
for the Kerr metric). This is a purely technical limitation, due
to the methods used, not to the nature of the problem studied.
\end{enumerate}
\section{A geometrical alternative~: ``conformal scattering''}
A different approach to scattering theory is to use the geometrical
information of the asymptotics of the metric more directly to infer
the asymptotic properties of the field. A natural tool for doing this
is Penrose's conformal compactification. It allows us to interpret the
complete scattering theory as the well-posedness of a Goursat problem
on null infinity.
Not a new idea~: Goursat problem on null infinity studied by
R. Penrose in 1963 \cite{Pe63}. First used in the framework of
time dependent scattering by F.G. Friedlander \cite{Fri1980, Fri2001},
for the wave equation on static spacetimes with regular conformal
structure at spacelike infinity. Idea used also by J.C. Baez,
I.E. Segal and Z.F. Zhou \cite{BaSeZho1990} for a non linear wave
equation on flat spacetime. So this is all for static spacetimes.
\begin{itemize}
\item Technique based on conformal compactifications, hence carries
limitations of its own (these depend also of our understanding of the
geometries studied and of the technique itself, there is therefore
some hope that they will become less important)~:
\begin{enumerate}
\item spacetimes whose asymptotic structure is well described by a
conformal compactification and whose conformal metric is regular
enough at infinity (essentially, in the present state of things,
asymptotically simple space-times with enough regularity at null
and timelike infinity)~;
\item conformally invariant equations or at least conformally
``controllable''.
\end{enumerate}
%
\item But the technique is totally indifferent to time
dependence. Generic time dependence can be allowed.
\end{itemize}
A first result has been published, in collaboration with L. Mason
\cite{MaNi2004}. Work on asymptotically simple space-times with
regular null and timelike infinities. Construction of
a scattering operator via conformal techniques~: do a picture with a
``vertical slice'' of the compactified spacetime, a Cauchy
hypersurface $\Sigma_0$, data in ${\cal C}^\infty_0 (\Sigma_0 )$,
trace operators $T^\pm$ on $\scri^\pm$, natural candidate for
scattering operator is $S:= T^+ \left( T^- \right)^{-1}$. The whole
work is therefore to show that $T^\pm$ are isomorphisms, i.e. to solve
the Goursat problem. Note that the trace operators associate to the
data only a part of the trace of the solution, called scattering data,
that comes out naturally in the energy estimates needed to solve
the Goursat problem. The construction is done for~:
%
\begin{itemize}
\item {\bf massless Dirac fields~:} no restriction, conformally
invariant closed $3$-form~;
\item {\bf Maxwell fields~:} closed $3$-form but not conformally
invariant, at least not fully, the normal vector field is not
rescaled, not applicable for energy estimates involving
$\scri$~; estimates are easy away from $i_0$ because we have a
symmetric hyperbolic system on a regular globally hyperbolic
space-time~; near $i_0$ we need an exact Killing vector field to
have exactly conserved quantities, otherwise, the singularity is
a problem~; our construction is valid only for the space-times
of Chrusciel-Delay and Corvino-Schoen, the exact Killing vector
field near $i_0$ is useful for obtaining the energy estimates~;
\item {\bf massless scalar fields~:} the construction is not quite
complete, the Goursat problem is solved, it is easier than for
Dirac and Maxwell, but for the energy estimates, we have a
problem with the scalar curvature, it is said how to obtain
them, but this is not done explicitely and again we need the
exact symmetry near $i_0$.
\end{itemize}
%
The construction for the wave equation is now complete. Extensions to
nonlinear cases is in progress.
The resolution of the Goursat problem is of course the fundamental
construction in this work. It follows the ideas of a work by
L. H\"ormander \cite{Ho1990} in which he describes a technique
based on energy estimates. His method is well adapted to
scattering in that it gives convergences in minimum regularity
spaces~; in addition, we slightly modify his approximation of
solutions, this allows us to show the equivalence with an analytic
scattering theory, defined in terms of classical wave operators whose
comparison dynamics are asymptotic profiles.
In order to work with completely generic asymptotically simple
space-times, without explicit symmetry near $i_0$, one possibility is
to handle the Goursat problem with very little regularity (at least at
one point) since the metric is not much better than Lipschitz near
$i_0$. Also for the space-times of Christodoulou and Klainerman, that
are the most general solutions (in terms of regularity) of the
Einstein vacuum equations that are asymptotically simple (almost, not
quite regular enough at $\scri$), we need weaker regularity for the
Goursat problem.
\section{H\"ormander's work (J.F.A. 1990)}
What we describe is a slightly simplified version of H\"ormander's
work for the fully characteristic Cauchy problem. His work in fact
treats the Cauchy problem, the Goursat problem and anything in between
where the initial data surface is allowed to be locally spacelike or
null. However, we are interested here in the Goursat problem.
\subsection{Geometrical framework}
\begin{itemize}
\item $X$ (space) a ${\cal C}^\infty$ compact manifold without
boundary, $\dim_\R X =n \geq 1$.
\item $\tilde{X} = \R_t \times X$ (spacetime).
\item $g(t) \in {\cal C}^\infty \left( \tilde{X} \, ;~ T^*X \odot T^* X
\right)$ a time-dependent Riemannian metric on $X$.
\item Hypersurface $\Sigma$ on which the initial data are fixed, defined as a
graph
%
\begin{equation} \label{Sigma}
\Sigma = \left\{ \left( \phi (x) , x \right) \, ;~ x \in X \right\}
\end{equation}
%
where $\phi \, :~ X \rightarrow \R$ is Lipschitz-continuous (which authorizes
singularities like the vertex of a cone). $\Sigma$ is assumed to be
characteristic (or null)~: this means that at each point where $\phi$
is differentiable (hence for almost every $x\in X$, since $\phi$ is
Lipschitz-continuous and therefore differentiable almost everywhere),
the normal (or even simpler the co-normal) to $\Sigma$ is null for the
Lorentzian metric
%
\[ \tilde{g} = \d t^2 - g \, ,\]
%
i.e.
%
\[ g^{\alpha \beta} (\phi (x),x) \partial_\alpha \phi (x) \partial_\beta \phi
(x) = 1 \, ,~\mathrm{for~a.e.~} x\in X \, .\]
\end{itemize}
\subsection{Analytical framework}
\begin{itemize}
\item We choose a smooth measure $\d \nu$ on $X$, for example that
associated with the metric $g(0)$, that, in a local system of
coordinates, is written $\d \nu = \gamma \d x$, where $\gamma$
is a smooth density.
%
\item Wave equation~:
%
\begin{equation} \label{Weq}
\square u + L_1 u = 0 \, ,
\end{equation}
%
where
%
\[ \square := \partial_t^2 -\gamma^{-1} \partial_\alpha \left( \gamma
g^{\alpha \beta} \partial_\beta \right) \]
%
is the simplified d'Alembertian and
%
\[ L_1 := b^0 \partial_t + b^\alpha \partial_\alpha + c\, ,~ b^0\, ,~b^\alpha \,
,~c~\in {\cal C}^\infty \left( \tilde{X} \right) \, .\]
%
\item On $X$ we have natural Sobolev spaces $H^s (X)$, $s\in \R$, defined via
local charts. On $\tilde{X}$ we shall only use local Sobolev spaces
$H^s_\mathrm{loc} (\tilde{X})$.
%
\item $\Sigma$ is a Lipschitz hypersurface, we can therefore define on $\Sigma$
the spaces $H^s (\Sigma )$ for $-1 \leq s\leq 1$ that are canonically isomorphic
to the corresponding $H^s (X)$ via parametrization (\ref{Sigma}). We have a
natural norm on $H^1 (\Sigma )$~: two possible expressions
%
\begin{description}
\item[(i)] $u \in H^1 (\Sigma )$, lift on $\Sigma$ of $v \in H^1 (X)$,
%
\[ \| u \|_{H^1 (\Sigma )}^2 = \int_X \left( |v(x)|^2 + g^{\alpha \beta}
(\varphi (x),x) \partial_\alpha v(x) \partial_\beta v(x) \right) \d \nu (x) \,
;\]
%
\item[(ii)] $u \in H^1 (\Sigma )$, trace on $\Sigma $ of $\psi \in
H^{3/2}_\mathrm{loc} (\tilde{X})$,
%
\[ \| u \|_{H^1 (\Sigma )}^2 = \int_\Sigma \left( |\psi|^2 + g^{\alpha \beta}
\left( \partial_\alpha \psi + \partial_\alpha \varphi \partial_t \psi \right)
\left( \partial_\beta \psi + \partial_\beta \varphi \partial_t \psi \right)
\right) \d \nu_\Sigma \, ,\]
%
where $\d \nu_\Sigma$ is the lift on $\Sigma$ via (\ref{Sigma}) of the measure
$\d \nu$.
\end{description}
\item Energy norm on $X_t:= \{ t \} \times X$~:
%
\begin{eqnarray*}
E(t,u) &=& \| u \|^2_{H^1 (X_t)} + \| \partial_t u \|^2_{L^2 (X_t )} \\
&=& \int_{X_t} \left( |u|^2 + g^{\alpha \beta} \partial_\alpha u
\partial_\beta u + |\partial_t u|^2 \right) \d \nu \, .
\end{eqnarray*}
%
\item Cauchy problem for (\ref{Weq})~: well known to be well-posed in
the energy space, i.e. for data $u_{|_{t=0}} \in H^1 (\Sigma )$
and $\partial_t u_{|_{t=0}} \in L^2 (\Sigma )$. With equivalence
locally uniform in time between the energy norms at any two
given times
%
\begin{equation} \label{EnEst1}
E(t,u) \leq e^{C_1 (T,g,L_1 ) |t-s|} E(s,u) \, ,~\forall t,s \in [-T,T] \, .
\end{equation}
%
We also have regularity properties of the solutions~: if $u_0$, $u_1$ $\in {\cal
C}^\infty (X)$, then the associated solution $u$ belongs to ${\cal C}^\infty
(\tilde{X})$.
%
\item We denote by $\cal E$ the space of solutions of (\ref{Weq}) in
%
\[ {\cal F} := {\cal C}^0 \left( \R_t \, ;~ H^1 (X) \right) \cap {\cal
C}^1 \left( \R_t \, ;~ L^2 (X) \right) \, .\]
%
Thanks to energy estimate (\ref{EnEst1}), we see that
we can choose on $\cal E$ the $H^1 \times L^2$ norm of the
initial data at $t=0$ and that makes it a Banach space
isomorphic to $H^1 (X) \times L^2 (X)$.
%
\item Energy norm on $\Sigma$~:
%
\[ E_\Sigma (u) = \| u_{|_\Sigma} \|^2_{H^1 (\Sigma )} \, , \]
%
all information conerning the time derivative of $u$ restricted to
$\Sigma$ is apparently lost.
\end{itemize}
\subsection{H\"ormander's theorem}
We consider the operator $T_\Sigma$ that to solutions $u\in {\cal E}
\cap {\cal C}^\infty (\tilde{X})$ associates $u_{|_\Sigma}$.
%
\begin{theorem}[H\"ormander 1990]
The operator $T_\Sigma$ extends in a unique manner as a linear continuous map,
still denoted $T_\Sigma$, from $\cal E$ to $H^1 (\Sigma )$. Moreover,
$T_\Sigma$ is an isomorphism.
\end{theorem}
\subsection{structure of the proof}
\begin{itemize}
\item Two fundamental energy estimates~: let $T > \max \{ |\varphi
(x)| \, , ~x\in X\}$, there exists $C_2 (T , g , L_1) >0$,
continuous in $T$, the Lipschitz norms of $g$ and $g^{-1}$ and
the $L^\infty$ norms of the corfficients of $L_1$ on $]-T,T[
\times X$, such that, for all $u \in {\cal E} \cap {\cal
C}^\infty (\tilde{X})$,
%
\begin{eqnarray}
\left\| T_\Sigma u \right\|^2_{H^1 (\Sigma )} = E_\Sigma (u) &\leq &
C_2 (T,g,L_1) \| u\|^2_{\cal E} = E(0,u) \, , \label{EnEst2} \\
E(0,u) &\leq & C_2 (T,g,L_1) E_\Sigma (u) \, .\label{EnEst3}
\end{eqnarray}
%
These are standard energy estimates proved for smooth solutions by
integration by parts (Stokes's formula really).
Estimate (\ref{EnEst2}) allows to extend $T_\Sigma$ in a unique manner
as a linear continuous map from $\cal E$ to $H^1 (\Sigma )$. Estimate
(\ref{EnEst3}) not only tells us that $T_\Sigma$ is one-to-one, it
also gives us a reciprocal estimate, hence, for proving the
surjectivity of $T_\Sigma$, we only need to define an inverse to
$T_\Sigma$ on a dense subspace of $H^1 (\Sigma )$.
%
\item Surjectivity of $T_\Sigma$~: we construct a solution to the
Goursat problem for some data $v$ that is the lift on
$\Sigma$ of a smooth function (still denoted $v$) on $X$
(this is our dense subspace of data). This is done in three
steps (note that we will need some data $w$ for the time
derivative of $u$ on $\Sigma$ in each approximation, provided we
always keep the same $w$, the choice is totally irrelevant, we
simply take $w=0$)~:
%
\begin{enumerate}
\item $\Sigma$ smooth and spacelike, the result is clear (usual Cauchy
problem)~;
\item $\Sigma$ Lipschitz and spacelike, we approach $\Sigma$ by smooth
spacelike surfaces and use compactness arguments.
%
\item $\Sigma$ Lipschitz and null~: we approach this
situation by the previous one by slowing down the propagation of
the equation~; we consider, for $0<\lambda <1$,
%
\begin{equation} \label{WeqSlow}
\square_\lambda u+ L_1 u=0 \, ,~ \square_\lambda = \partial_t^2 -
\lambda \gamma \partial_\alpha \left( \gamma g^{\alpha \beta}
\partial_\beta \right) \, .
\end{equation}
%
$\Sigma$ is uniformly spacelike for (\ref{WeqSlow}). Then use weak
convergence and compactness.
\end{enumerate}
\end{itemize}
\subsection{Our slightly different version more adapted to scattering}
In our scattering constructions, we adopt H\"ormander's technique, but
we prove existence in a slightly different way. We describe the
construction for future infinity.
We consider on our physical spacetime a global time function $t$ and
the foliation by the level hypersurfaces $\{ \Sigma_t \}_{t\in \R}$
of $t$. Say that $\Sigma_0$ is the surface on which we fix the initial
data for the Cauchy problem. The gradient of $t$ will serve to
identify the points on the different slices. We consider a compact
domain $K\subset \Sigma$, large enough so that outside $\R^+_t \times
K$ we can define a congruence of outgoing null geodesics, let us
denote $\cal C$ such a congruence.
Now consider some scattering data for our equation
$\hat{\phi}_{\scri^+} \in {\cal C}^\infty_0 (\scri^+ )$ (for the wave
equation, it will simply be the trace of $u$, for Dirac or Maxwell the
trace of one component of the spinor field. We project this data onto
$\Sigma_t$ along $\cal C$ and cut off inside $K$ (with a smooth
cut-off if necessary). We complete the data for the Cauchy problem
with identically zero functions. We propagate backwards down to
$\Sigma_0$ using the full dynamics. We show that as $t \rightarrow
+\infty$, the data we obtain on $\Sigma_0$ tends to the initial data
for a solution whose image by $T^+$ is precisely
$\hat{\phi}_{\scri^+}$ (strong limit in the energy space) , i.e. we
have constructed an inverse to $T^+$ on a dense domain. Moreover we
have constructed it as a classical wave operator~: free dynamics (null
geodesic flow), followed by cut-off, followed by backwards physical
dynamics.
Hence, we recover a complete scattering theory defined in terms of
classical wave operators, in a generically non stationary framework
where usual analytic scattering techniques are not applicable.
\subsection{H\"ormander's remark}
At the end of his paper, H\"ormander remarks that although all the
results are expressed and proved for a smooth metric and smooth
coefficients of $L_1$, the bounds in the estimates only depend on the
Lipschitz norm of the metric and the $L^\infty$ norm of the
coefficients of $L_1$ and so this is the proper generality of the
theorem.
The problem is that all parts of the
proof rely heavily of regular solutions that allow to perform
integrations by parts to prove energy estimates. It in fact suffices
to have $H^2_\mathrm{loc}$ solutions to be able to make the whole
proof run in the same manner. However, for the regularity setting
proposed by H\"ormander, we do not have access to such
solutions.
This remark is not proved in H\"ormander paper and to my knowledge it
has not been checked ever since.
\section{Extension to lower regularity}
This is a recently submitted work \cite{Ni2005}
\subsection{Cauchy problem for a Lipschitz metric}
\begin{theorem} \label{Cauchy}
We assume that $g$ is Lipschitz-continuous on $\tilde{X}$ and the
coefficients of $L_1$ are locally $L^\infty$ on $\tilde{X}$. Then, for
any $u_0 \in H^1 (X)$, $u_1 \in L^2 (X)$, there exists a unique
%
\[ u \in {\cal F}:= L^\infty_\mathrm{loc} (\R_t \, ;~ H^1 (X) ) \cap
{\cal C}^1 (\R_t \, ;~ L^2 (X) ) \]
%
solution of (\ref{Weq}) associated with the data $u_0$, $u_1$ at, say,
$t=0$. Moreover, $u$ satisfies estimate (\ref{EnEst1}) for almost all
$s,t$ and it turns out that $u \in {\cal F}$ (i.e. the solutions are
in fact continuous in time with values in $H^1 (X)$). Hence, we still
denote by $\cal E$ the space of solutions in $\tilde{\cal F}$.
\end{theorem}
%
\begin{corollary}
Under the same regularity hypotheses as theorem \ref{Cauchy}, we
consider the equation
%
\begin{equation} \label{WeqHomog}
\partial_t^2 u - g^{\alpha \beta} \partial_\alpha \partial_\beta u = 0
\, ,
\end{equation}
%
this is equation (\ref{Weq}) with
%
\[ L_1 = \gamma^{-1} \partial_\alpha \left( \gamma g^{\alpha \beta}
\right) \partial_\beta \, ,\]
%
hence the Cauchy problem for (\ref{WeqHomog}) is well-posed in the
same function space as for (\ref{Weq}). Moreover, if the initial data
%
\[ (u_0,u_1) \in H^2(X) \times H^1(X) \, ,\]
%
the associated solution $u$ of (\ref{WeqHomog}) belongs to
%
\[ \bigcap_{l=0}^2 {\cal C}^l \left( \R_t \, ;~H^{2-l} (X) \right) \,
.\]
\end{corollary}
The proof of the corollary is trivial, we just commute derivatives
into the equation.
\vspace{0.1in}
{\bf Ideas of the proof of theorem \ref{Cauchy}.} \begin{enumerate}
\item {\bf Uniqueness.} Obtained by regularizing purely in space. We
obtain energy estimate (\ref{EnEst1}) (for all $t,s$ such that
$u(t),u(s) \in H^1 (X)$).
\item {\bf Existence.} We regularize the metric and coefficients of
$L_1$. The rest makes use of weak convergence techniques and
compactness arguments. We see that $u(t) \in H^1 (X)$ for all
$t$.
\item {\bf Continuity in time.} Weak continuity in time plus energy
estimate (\ref{EnEst1}).
\end{enumerate}
\subsection{Goursat problem for a weekly regular metric}
The trace operator $T_\Sigma$ associates to a solution $u \in {\cal
E}$ its trace on $\Sigma$, this is well defined in ${\cal L} ({\cal
E} \, ;~ L^2 (\Sigma ))$ by standard trace theorems, we need energy
estimates to show it is valued in $H^1$ and one-to-one and then we
also need to prove surjectivity.
\begin{enumerate}
\item {\bf Estimates (\ref{EnEst2}) and (\ref{EnEst3}).} Idea is to
regularize the solution so that the approximating functions
satisfy analogues of (\ref{EnEst2}) and (\ref{EnEst3}), and
converge strongly in $H^1 (\Sigma )$ and $H^1 (X)$~; will
guarantee that the energy estimates remain valid in the limit.
We write equation (\ref{Weq}) as (\ref{WeqHomog}) plus first
order perturbation $\tilde{L}_1$. Then regularize the first
order perturbation and the initial data. We get solutions $u_k$
in $H^2_\mathrm{loc} (\tilde{X})$ thanks to the corollary
above. They all satisfy estimates of the type (\ref{EnEst2}),
(\ref{EnEst3}), that can simply be proved by integration by
parts, with constants uniform in $k$. All the equations have the
same principal part, this allows to get for $u_k -u_l$ estimates
of the type (\ref{EnEst2}), (\ref{EnEst3}) with a source term
$(L_1^k - L^l_1)u_l$. We simply need to read the regularity of
the coefficients of $\tilde{L}_1$ that guarantees that this term
tends to zero in say $L^1_\mathrm{loc} (\R_t \, ;
~L^2(X))$. We see that we need
%
\begin{eqnarray}
g &\in & L^\infty_\mathrm{loc} (\R_t \, ;~ {\cal C}^1 (X)) \cap
W^{1,\infty}_\mathrm{loc} (\R_t \, ;~ {\cal C}^0 (X)) \, ,
\label{Reg1} \\
b^0 \, ,~ b^\alpha &\in & L^\infty_\mathrm{loc} (\R_t \, ;~ {\cal C}^0
(X)) \, , \label{Reg2} \\
c &\in & L^\infty_\mathrm{loc} (\tilde{X} ) \, . \label{Reg3}
\end{eqnarray}
%
\item {\bf Surjectivity.} Regularize metric, coefficients of $L_1$ and
slow down propagation, plus compactness.
\end{enumerate}
\begin{theorem} \label{Goursat}
Under the regularity assumptions (\ref{Reg1})-(\ref{Reg3}), the
operator $T_\Sigma$ is an isomorphism from $\cal E$ onto
$H^1(\Sigma)$.
\end{theorem}
\section{Perspectives for black hole space-times}
The conformal geometry of black hole space-times is quite different
from that of asymptotically simple space-times~:
%
\begin{itemize}
\item horizon~: this means that there will be four trace operators,
one for each asymptotic region for the future and the past~;
\item singularity at timelike infinity~: this is where the whole
structure collapses.
\end{itemize}
If the conformal scattering constructions are to be extended to black
hole space-times, it means that we must understand the singularity of
the conformal metric at timelike infinity. This is in itself a whole
research project (judging from the time people have spent on spacelike
infinity). But it seems just as important to understand. The recent
work of M. Dafermos and I. Rodnianski gives strong hopes to construct the
scattering theory for the wave equation on spherically symmetric
blach-hole space-times in a conformal manner using their results. This
seems an interesting path that could be pursued for other
equations. Entensions to other geometries seems however very difficult
for the moment.
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\end{document}