Black hole formation from a complete regular past
\sc Mihalis Dafermos\footnote{supported in part by the NSF grant DMS-0302748}
Department of Mathematics
MIT
An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole in the future and yet past-causally geodesically complete, in particular, containing no white holes. We give such an example in this paper--in fact a family of such examples, stable in a suitable sense--for the case of a self-gravitating scalar field.
bibliog.tex
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to appear in Math. Proc. Camb. Phil. Soc.

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\emph{The global nonlinear stability of the Minkowski space}
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\bibitem{cs:wm} Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh
\emph{On the regularity of spherically symmetric wave maps} 
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\bibitem{md:bb} Mihalis Dafermos \emph{Stability and Instability
of the Reissner-Nordstr\"om Cauchy Horizon and the Problem of Uniqueness
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\end{thebibliography}

note.tex
\documentclass{article}
\usepackage{latexsym}
\usepackage{graphics}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{assumption}{Assumption}
\hyphenation{Chri-sto-dou-lou}
\newcommand{\tl}{\frac{\theta}{\lambda}}
\newcommand{\zn}{\frac{\zeta}{\nu}}
\begin{document}
\author{\sc Mihalis Dafermos\footnote{supported in part by
the NSF grant DMS-0302748} \\ Department of Mathematics, MIT \\
dafermos@math.mit.edu}
\title{Black hole formation from a complete regular past}
\maketitle
\begin{abstract}
An open problem in general relativity has been to construct
an asymptotically flat solution to a reasonable
Einstein-matter system containing a black hole in the future
and yet past-causally geodesically complete, in particular, containing no white 
holes. We give such an example in this paper--in fact a family of
such examples, stable in a suitable sense--for 
the case of a self-gravitating scalar field.
\end{abstract}
\section{Introduction}
The problem of gravitational collapse is typically formulated as
the study of the future maximal evolution of complete asymptotically
flat Cauchy data for an appropriate Einstein-matter system. 
The question of identifying physically admissible initial data,
however, is best characterized by properties of their \emph{past} 
evolution. In particular, it seems reasonable to restrict
to Cauchy data whose past evolution is regular.
Unfortunately, however, with the exception of certain classical results
for dust~\cite{os:?},
current theorems on the evolution of asymptotically flat Cauchy data 
ensuring a regular past~\cite{ck:book, chr:sgsf,rr:ge} 
also ensure a regular future; this is due to
 the fact that these theorems depend
on smallness in function-space norms that
do not distinguish past from future.\footnote{One can compare
with the cosmological case, where choice of gauge based on
the expansion of the universe can be
used to prove future completeness theorems
in cases where the past is known to be singular. See for 
instance~\cite{arr:ohs,
gr:fgc}.}
As a consequence, even a single example of a solution to a reasonable 
Einstein-matter system
with a regular asymptotically flat past and a singular future has thus
 far been lacking.

In the present paper, we shall prove
\begin{theorem}
There exist past-causally 
geodesically complete solutions of the coupled
Einstein-scalar field equations, 
which to the future terminate in a $C^0$-singularity hidden inside a 
black hole. These solutions are maximal developments of complete
asymptotically flat Cauchy data. 
\end{theorem}
In fact, a large class of such solutions will be proven to exist.
The solutions are spherically symmetric and
their conformal diagram\footnote{This is defined to be the image
of a conformal representation of the quotient manifold $Q=M/SO(3)$
in a bounded domain of $2$-dimensional Minkowski space.} is as follows:
\[
\includegraphics{summorfo.eps}
\]
Moreover,
the above picture will be stable to a certain class of perturbations.

Although the solutions we construct will admit complete asymptotically
flat Cauchy surfaces, they will not be constructed
\emph{a priori} as developments of such data. Rather, the examples
of this paper will be constructed by
pasting together the solutions to three distinct characteristic
initial value problems. In particular,
the issue of formulating a criterion
on Cauchy data ensuring a singular future yet regular
past is sidesteped in this paper. 

Replacing the Cauchy problem with several distinct
characteristic initial value problems allows
one to completely separate the problem of the future from the past.
The future is determined by data prescribed on a future-outgoing cone (see
$A$ in the diagram below),
whereas the past is determined by data on a past-outgoing cone (see $B$).
To complete the spacetime, however, one has to construct (and understand)
the solution ``between'' the two cones (see $\Gamma$):
\[
\includegraphics{Goursatlar.eps}
\]
Since in spherical symmetry, the equations reduce to a quasilinear
system in $2$-dimensions, the problem depicted by
$\Gamma$ is also a well-posed characteristic initial value problem.
Conceptually, this 
is perhaps easiest understood by replacing the two-dimensional metric
on $Q$ by its negative, thus interchanging the notions of space and 
time.\footnote{Estimates in the direction depicted by $\Gamma$ have many
applications in the study of $2$-dimensional hyperbolic problems. See
\cite{md:tp, dr:dr, cs:wm, chr:mt}.}


The data for the initial value
problem $B$ can be taken to be trivial, but alternatively,
from Christodoulou~\cite{chr:sgsf, chr:bv}, one can 
prescribe an arbitrary scalar field sufficiently small in an appropriate
norm, and this
will also guarantee a past complete development. On the other hand,
Christodoulou has given in~\cite{chr:fbh}
a criterion on initial data ensuring that
for the initial value problem $A$, a black hole forms 
in the future. The important feature, from
the point of view of this paper, is that this criterion can be applied
to data which are not trapped initially, and for which the mass aspect
function $\mu$ and the scalar field multiplied by the area radius 
$r\phi$ can be arbitrarily small in the supremum norm.

Theorem 1 thus reduces to completely understanding the evolution
of problem $\Gamma$. It should be noted that the global features of
this problem are quite different from the more standard characteristic initial
value problems $A$ and $B$; in particular, 
there are no \emph{a priori} global energy
bounds. The main analytical result of this paper is
that $\Gamma$ leads to a ``complete'' wedge in an asymptotically flat
spacetime, given sufficiently small initial data. The requisite
smallness, however, is asymmetric in the two initial characteristics;
on the future outgoing characteristic, only pointwise 
smallness of $\mu$ and $r\phi$
are required. In particular, the requirements for the data for $\Gamma$
are compatible with the data necessary in $A$ to produce a black hole. 

For completeness, in the next section,
the Einstein-scalar field equations will be presented under
spherical symmetry. The initial value problem $\Gamma$ is studied in
Section 3. From this, the theorem of this paper is easily obtained
in Section 4 along the considerations outlined above.
\vskip1pc
\noindent{\bf Acknowledgement:} The importance of identifying asymptotically
flat initial data whose future development is singular but
whose past development is regular has been stressed by Sergio Dain and Alan
Rendall. I thank them for several very useful discussions.

\section{The Einstein-scalar field equations under spherical
symmetry}
For a detailed discussion of the role of the
Einstein-scalar field system in the study of
the problem of gravitational collapse, the reader may refer to
\cite{md:si}.
In rationalized units, the equations take the form
\[
R_{\mu\nu}-\frac12Rg_{\mu\nu}=2T_{\mu\nu},
\]
\[
g^{\mu\nu}\phi_{;\mu\nu}=0,
\]
\[
T_{\mu\nu}=\phi_{;\mu}\phi_{;\nu}
-\frac12g_{\mu\nu}\phi^{;\alpha}\phi_{;\alpha},
\]
where $g_{\mu\nu}$ is a Lorentzian metric defined on a $4$-dimensional
manifold $M$ and $\phi$ is a scalar function (the scalar field).

Recall that spherical symmetry is the assumption that
$SO(3)$ acts by isometry on the spacetime and preserves $\phi$.
Under this assumption,
the equations reduce to a second order system for 
functions $(r,g_{ij},\phi)$ 
defined on the space of group orbits $Q$ of
the $SO(3)$ action. Here $r$ is the so-called \emph{area radius} function,
i.e.~$r$ evaluated at a point $p$ of $Q$ retrieves up 
to a constant the square root of
the area of the group orbit corresponding to $p$ in $M$. This group orbit
is necessarily a spacelike sphere. The tensor $g_{ij}$ is a 
$1+1$-dimensional Lorentzian metric on $Q$, which is induced from
the metric on $M$. The functions
$r$ and $g_{ij}$ together determine $g_{\mu\nu}$ as follows:
\[
g_{\mu\nu}dx^\mu dx^\nu=g_{ij}dy^idy^j+r^2\gamma,
\]
where $\gamma$ is the standard metric on the unit $2$-sphere.
Since the scalar field is constant on group orbits, it descends
to a function on $Q$, and will still be denoted by $\phi$.

One can prescribe null coordinates $u$ and $v$ on $Q$ so that its metric
becomes $-\Omega^2dudv$. These coordinates then represent
the characteristic directions of the equations. 
To exploit the method of characteristics, it
is more convenient to write the equations as a first order system.
Moreover, $\Omega$ can be replaced as an unknown in the equations
by the Hawking mass 
\[
m=\frac{r}2(1-|\nabla r|^2).
\]
Introducing
the first order derivatives $\partial_ur=\nu$, $\partial_vr=\lambda$,
$r\partial_u\phi=\zeta$, $r\partial_v\phi=\theta$, and also
the so-called mass aspect function $\mu=\frac{2m}r$, we obtain the
system
\begin{equation}
\label{ruqu}
\partial_u{r}=\nu,
\end{equation}
\begin{equation}
\label{rvqu}
\partial_v{r}=\lambda,
\end{equation}
\begin{equation}
\label{lqu}
\partial_u\lambda=\lambda\left(\frac{2\nu}{1-\mu}\frac{m}{r^2}
\right),
\end{equation}
\begin{equation}
\label{nqu}
\partial_v\nu=\nu\left(\frac{2\lambda}{1-\mu}\frac{m}{r^2}
\right), 
\end{equation}
\begin{equation}
\label{puqu}
\partial_u m=\frac{1}{2}(1-\mu)\left(\zn\right)^2\nu,
\end{equation}
\begin{equation}
\label{pvqu}
\partial_v m=\frac{1}{2}(1-\mu)\left(\tl\right)^2\lambda,
\end{equation}
\begin{equation}
\label{sign1}
\partial_u\theta=-\frac{\zeta\lambda}r,
\end{equation}
\begin{equation}
\label{sign2}
\partial_v\zeta=-\frac{\theta\nu}r.
\end{equation}

From the above we also derive
\begin{equation}
\label{avafora}
\partial_v(\phi\nu+\zeta)=\frac{\phi}{r^2}\frac{2\lambda\nu}{1-\mu}
				m.
\end{equation}
\begin{equation}
\label{avafora2}
\partial_u(\phi\lambda+\theta)=\frac{\phi}{r^2}\frac{2\lambda\nu}{1-\mu}
				m.
\end{equation}
We shall see that the $r^{-2}$ factor on the right hand side above
allows us to show that the quantities $\phi\lambda+\zeta$,
$\phi\lambda+\theta$
have better decay in $r$ than $\phi$, $\theta$, or $\zeta$; this 
fact plays an important role in our argument. 

We also easily obtain from the above the equations
\begin{equation}
\label{fdb1}
\partial_u\frac{\lambda}{1-\mu}=
\frac1{r}\left(\zn\right)^2\nu\frac{\lambda}{1-\mu}
\end{equation}
and
\begin{equation}
\label{fdb2}
\partial_v\frac{\nu}{1-\mu}=
\frac{1}r\left(\tl\right)^2\lambda\frac{\nu}{1-\mu}.
\end{equation}


\section{The characteristic initial value problem $\Gamma$}
In this section, we will study
a characteristic initial value problem posed
in the direction indicated by $\Gamma$ in the Introduction.
This will be the main new analytic element in the
proof of Theorem 1 in
the next section.

For the time being, fix constants $R>0$, $C>0$, $M>0$, and 
$m_0\ge0$.
The initial characteristic segments will be $u=-R$ and $v=R$.
At $(-R,R)$, we prescribe 
\[
r(-R,R)=R, 
\]
\[
m(-R,R)=m_0.
\]
On $u=-R$, we prescribe $\mathcal{C}^2$ 
functions of $v$ $r(-R,v):[R,\infty)\to{\bf R}$, 
$\phi(-R,v):[R,\infty)\to{\bf R}$ so as to satisfy
\begin{equation}
\label{vd1}
\partial_vr=\lambda=1,
\end{equation}
\begin{equation}
\label{vd2}
|r\phi|\le C,
\end{equation}
\begin{equation}
\label{vd3}
m\le M.
\end{equation}
On $v=R$, we prescribe $\mathcal{C}^2$ functions 
$r(u,R):(-\infty,-R]\to{\bf R}$, 
and $\phi(u,R):(-\infty,-R]\to{\bf R}$, so as to satisfy
\begin{equation}
\label{ud1}
\partial_ur=\nu=-1, 
\end{equation}
\begin{equation}
\label{ud2}
|\partial_u(r\phi)|=|\zeta+\nu\phi|\le C|u|^{-2},
\end{equation}
\begin{equation}
\label{ud3}
m\le M.
\end{equation}

We have the following
\begin{proposition}
\label{LWP}
Consider the initial value problem described above.
There exists a unique subset ${\bf D}\subset(-\infty,-R]\times[R,\infty)$, 
open in the subspace topology of the latter set,
and a unique set of sufficiently regular functions $r$, $\phi$, and $m$
defined on ${\bf D}$, such that
\begin{enumerate}
\item
$-R\times[R,\infty)\cup(-\infty,-R]\times R\subset {\bf D}$ and
$r$, $\phi$, $m$ match with the data prescribed.
\item
The functions $r$, $\phi$, $m$ satisfy $(\ref{ruqu})$--$(\ref{sign2})$.
\item
If $(u,v)\in{\bf D}$, then $(\tilde{u},\tilde{v})\in{\bf D}$,
for all $\tilde{u}\ge u$, $\tilde{v}\le v$.
\item
Given any other $\tilde{\bf D}$, and sufficiently regular
$\tilde{r}$, $\tilde{m}$, $\tilde{\phi}$,
satisfying properties 1, 2, and 3, it follows that
$\tilde{\bf D}\subset {\bf D}$,
and $\tilde{r}=r$, $\tilde{m}=m$, and $\tilde{\phi}=\phi$ on 
$\tilde{\bf D}$.
\end{enumerate}
Moreover, if the initial data are $C^\infty$, then 
$r$, $\phi$, $m$ are $C^\infty$ functions on ${\bf D}$.
\end{proposition}

\noindent\emph{Proof.} The proof is by standard techniques and
is omitted. $\Box$

We shall call ${\bf D}$ the maximal development of 
our initial value
problem. Time orienting the negative of the induced metric 
$-g_{ij}$ of our solution in the
obvious way, property 2 above says that ${\bf D}$ is a past set
with respect to $-g$. In what follows, we will 
denote causal relations with respect to the 
negative metric under the
above time orientation by $J^+_{-g}$, $J^{-}_{-g}$,
etc.

\begin{proposition}
For the intial value problem described
above, $\nu<0$ and $\lambda>0$ throughout ${\bf D}$.
\end{proposition}

\noindent\emph{Proof.} 
By continuity, the above inequalities indeed
 hold in a neighborhood of initial data.
Thus, assuming the proposition is false, there exists a
$(u,v)$ such that either $\nu(u,v)=0$ or $\lambda(u,v)=0$, but
for which $\nu<0$, $\lambda<0$ in $J^{-}_{-g}(u,v)$:
\[
\includegraphics{prwto.eps}
\]
Let us first assume that $\lambda(u,v)=0$.
Integrating equation $(\ref{fdb1})$ on $[u,-R]\times\tilde{v}$
for each $\tilde{v}\in[R,v]$, we 
obtain that 
\begin{eqnarray*}
\frac\lambda{1-\mu}(u,\tilde{v})	&=&		\frac\lambda{1-\mu}(-R,\tilde{v})
										e^{\int_{-R}^u\frac{1}{r}
										\frac{\zeta^2}
										\nu d\tilde{u}}\\
							&=&		\frac\lambda{1-\mu}(-R,\tilde{v})
										e^{-\int_{u}^{-R}\frac{1}{r}
										\frac{\zeta^2}
										\nu d\tilde{u}}\\
							&\ge&	\frac\lambda{1-\mu}(-R,\tilde{v}),
\end{eqnarray*}
since $\nu\le0$ in $\overline{J^{-}_{-g}(u,v)}$.
It then follows from the above that we have $1-\mu\ge0$ in
$\overline{J^{-}_{-g}(u,v)}$.

Integrating now $(\ref{lqu})$ on $[u,-R]\times v$, we obtain
\begin{eqnarray*}
\lambda(u,v)	&=&		\lambda(-R,v)e^{\int_{-R}^u{\frac{2\nu}{1-\mu}
					\frac{m}{r^2}d\tilde{u}}}\\
		&\ge&	\lambda(-R,v)\\
		&\ge&	1.
\end{eqnarray*}
But this contradicts the assumption $\lambda(u,v)=0$.
In an entirely similar fashion, integrating $(\ref{fdb2})$,
the assumption $\nu(u,v)=0$ leads
to a contradiction. The proposition is thus proven. $\Box$

\begin{proposition}
In ${\bf D}$, 
\begin{equation}
\label{rkaiR}
r\ge R,
\end{equation}
\[
m\ge m_0,
\]
\begin{equation}
\label{nu1}
\nu\le-1,
\end{equation}
\begin{equation}
\label{la1}
\lambda\ge1,
\end{equation}
\begin{equation}
\label{1-mu1}
0<1-\mu\le1.
\end{equation}
\end{proposition}
 
\noindent\emph{Proof.} The proof is immediate from the previous proposition
and the signs in the equations $(\ref{ruqu})$--$(\ref{pvqu})$.
$\Box$

Let $\partial{\bf D}$ denote the boundary of ${\bf D}$ in the topology
of $(-\infty,-R]\times[R,\infty)$. By monotonicity, $r$ and $m$
extend to continuous functions on ${\bf D}\cup\partial{\bf D}$ valued
in the extended real numbers, and causality relations can still be
applied. We have the following extension proposition:

\begin{proposition}
\label{epektasn}
If $p\in{\bf D}\cup\partial{\bf D}$, $J^-_{-g}(p)\subset{\bf D}$,
and $r(p)<\infty$, $m(p)<\infty$, then $p\in{\bf D}$.
\end{proposition}

\noindent\emph{Proof} omitted. $\Box$ 

The achronal (with respect to $-g$) structure of the boundary
$\partial{\bf D}$ immediately yields the following 

\begin{corollary}
If $\partial{\bf D}\ne\emptyset$ then there exists a 
$p\in\partial{\bf D}$ such that either $r(p)=\infty$ or $m(p)=\infty$.
\end{corollary}

We now proceed to the main result of this section:

\begin{proposition}
\label{kurioprop}
Fix $\alpha>1$. Consider an initial value problem of the type described 
before, where\footnote{We have here not attempted to give a scale-invariant
condition.}
\[
R>\max\left\{\left(\frac{4\alpha}{\alpha-1}\right)^{2},
\left(\frac{1+16\alpha^4M}{\alpha}\right)^2,
\frac{8\alpha^2M}{\log \alpha},\frac{6\alpha^2C^2}{\alpha-1},\frac{4\alpha M}
{1-\alpha}\right\}.
\]
Then
\[
{\bf D}=(-\infty,-R]\times[R,\infty),
\]
and the following estimates
hold:
\begin{equation}
\label{E1}
|r\phi|\le \alpha C,
\end{equation}
\begin{equation}
\label{E2}
|\nu\phi+\zeta|\le (1+\alpha^4M)C|u|^{-2},
\end{equation}
\begin{equation}
\label{E3}
-1\ge\nu\ge-\alpha
\end{equation}
\begin{equation}
\label{E4}
1\le\lambda\le\alpha
\end{equation}
\begin{equation}
\label{E5}
1-\mu\ge (2\alpha)^{-1}
\end{equation}
Moreover, if in addition 
\begin{equation}
\label{AA1}
|\lambda\phi+\theta|(R,v)\le\tilde{C}v^{-2},
\end{equation}
then 
\begin{equation}
\label{E6}
|\lambda\phi+\theta|(R,v)\le(\tilde{C}+\alpha^4MC)v^{-2}
\end{equation}
holds as well.
\end{proposition}
\noindent\emph{Proof.} 
Consider the region $\mathcal{R}$ 
defined to be the set of all $p$ such that
the inequalities:
\begin{equation}
\label{BS1}
|r\phi|<2\alpha C
\end{equation}
\begin{equation}
\label{BS2}
|\nu\phi+\zeta|<2\alpha C|u|^{-\frac32}
\end{equation}
\begin{equation}
\label{BS3}
\nu>-2\alpha
\end{equation}
\begin{equation}
\label{BS4}
\lambda<2\alpha
\end{equation}
\begin{equation}
\label{BS5}
1-\mu>(2\alpha)^{-1}
\end{equation}
\begin{equation}
\label{BS6}
m<2\alpha M
\end{equation}
hold for all $q\in J^{-}_{-g}(p)$. With the exception of
$(\ref{BS2})$ and 
$(\ref{BS5})$, the above inequalities hold on the intial data
segments regardless of the value of $R$. Inequality $(\ref{BS2})$ holds since $R>1$, and $(\ref{BS5})$ holds
since
\[
R>\frac{4M}{1-\alpha^{-1}}.
\]
By Proposition \ref{LWP}, $\mathcal{R}$ is non-empty, 
and contains a 
neighborhood of the initial segments; moreover, it is easily seen 
to be open
in the topology of ${\bf D}$.
We shall show first that in $\overline{\mathcal{R}}$, 
$(\ref{BS1})$--$(\ref{BS6})$ hold
with $\alpha$ replacing $2\alpha$. 

Let $(u,v)$ thus be in $\overline{\mathcal{R}}$. It follows by continuity that
$(\ref{BS1})$--$(\ref{BS6})$ hold at $(u,v)\cup J^{-}_{-g}(u,v)$,
where, however, the $<$ sign is replaced by the $\le$ sign.

We estimate $r\phi(u,v)$ as follows. Integrating the equation
\[
\partial_u(r\phi)=\nu\phi+\zeta,\]
we obtain from $(\ref{vd2})$ and $(\ref{BS2})$ 
\begin{eqnarray*}
|r\phi(u,v))|	&\le&	|r\phi(-R,v)|+\int_{u}^{-R}{|\nu\phi+\zeta|d\tilde{u}}\\
	&\le&	C+4C\alpha R^{-\frac12}\\
	& = &	C(1+4R^{-\frac12}\alpha)\\
	& < &	\alpha C
\end{eqnarray*}
since $R>\left[4\alpha(\alpha-1)^{-1}\right]^2$.

On the other hand, integrating $(\ref{avafora})$ and applying
$(\ref{ud2})$, $(\ref{BS1})$, $(\ref{BS3})$, $(\ref{BS5})$, $(\ref{BS6})$, 
we estimate
\begin{eqnarray*}
|\nu\phi+\zeta|(u,v)	&\le&	|\nu\phi+\zeta|(u,R)+
				\int_R^v{\frac{|\phi|}{r^2}
				\frac{2\lambda|\nu|}{1-\mu}m
				d\tilde{v}}\\
		&\le&	C|u|^{-2}+\int_R^{r(u,v)}{\frac{|\phi|}{r^2}
				\frac{2|\nu|}{1-\mu}m
				dr}\\
		&\le&	C|u|^{-2}+32C\alpha^4M
				\int_R^{r(u,v)}{\frac{dr}{r^3}}\\
		&\le&	C|u|^{-2}+16C\alpha^4Mr(u,R)^{-2}\\
		& = &	[C+16C\alpha^4M]|u|^{-2}\\
		&\le&	CR^{-\frac12}[1+16\alpha^4M]
				|u|^{-\frac32}\\
		& < &	\alpha C|u|^{-\frac32},
\end{eqnarray*}
where for the last inequality we use that
$R>\alpha^{-2}(1+16\alpha^4M)^2$.

To estimate $\nu(u,v)$, we integrate $(\ref{nqu})$, applying $(\ref{ud1})$,
$(\ref{BS3})$, $(\ref{BS5})$, $(\ref{BS6})$, to obtain
\begin{eqnarray*}
\nu(u,v)	&\ge&	\nu(u,R)e^{\int_R^v{\frac{2}{r^2}
			\frac{\lambda|\nu|}{1-\mu}md\tilde{v}}}\\
	&\ge&	-e^{8\alpha^2M\int_R^{r(u,v)}{\frac{dr}{r^2}}}\\
	&\ge&	-e^{8\alpha^2MR^{-1}}\\
	& > &	-\alpha
\end{eqnarray*}
since $R> 8\alpha^2 M(\log\alpha)^{-1}$.
Similarly, from $(\ref{lqu})$ we estimate $\lambda(u,v)$,
applying $(\ref{vd1})$, $(\ref{BS4})$, $(\ref{BS5})$, $(\ref{BS6})$: 
\begin{eqnarray*}
\lambda(u,v)	&\le&	\lambda(-R,v)e^{\int_{u}^{-R}{\frac{2}{r^2}
			\frac{\lambda|\nu|}{1-\mu}md\tilde{u}}}\\
	&\le&	e^{8\alpha^2M\int_{r(u,v)}^{-R}{\frac{dr}{r^2}}}\\
	&\le&	e^{8\alpha^2MR^{-1}}\\
	& < &	\alpha.
\end{eqnarray*}

For $m$, from $(\ref{puqu})$ we compute
\begin{eqnarray*}
m(u,v)	& = &	m(-R,v)+\int_{-R}^u{\frac12\frac{1-\mu}
			\nu\zeta^2d\tilde{u}}\\
	&\le&	M+\int_{-R}^u{\frac12\frac{1-\mu}\nu\left[
			|\zeta+\nu\phi|+|\nu\phi|\right]^2d\tilde{u}}\\
	&\le&	M+\int_{-R}^u{\frac{1-\mu}\nu\left[
			(\zeta+\nu\phi)^2+\nu^2\phi^2\right]d\tilde{u}}\\
	&\le&	M+4\alpha^2C^2\int_{-R}^u{|u|^{-3}du}	
			+4\alpha^2 C^2\int_{-R}^{r(u,v)}{\frac{dr}{r^2}}\\
	&\le&	M+2\alpha^2C^2R^{-2}+2\alpha^2 C^2R^{-1}\\
	& < &	\alpha M,
\end{eqnarray*}
where for the last inequality, we use $R>\max\left\{
6\alpha^2C^2(\alpha-1)^{-1},1\right\}$.

Finally from this, we obtain
$1-\mu=1-\frac{2m}r\ge1-\frac{4\alpha M}R > \alpha^{-1}$
since $R>4\alpha M(1-\alpha^{-1})^{-1}$.

Since 
\[
J^{-}_{-g}(\overline{\mathcal{R}})\subset \overline{J^{-}_{-g}(\mathcal{R})}=
\overline{\mathcal{R}}
\]
we have just shown that for all $(u,v)\in \overline{\mathcal{R}}$,
the inequalities $(\ref{BS1})$--$(\ref{BS6})$ hold
throughout $J^{-}_{-g}(u,v)$ with $\alpha$ replacing $2\alpha$.
Thus $(u,v)\in \mathcal{R}$, so 
\[
\mathcal{R}=\overline{\mathcal{R}}.
\]
Since ${\bf D}$ is connected, and $\mathcal{R}$ is open,
it follows that 
\[
\mathcal{R}={\bf D}.
\]

Suppose now $(u,v)\in\partial{\bf D}$. By continuity, we
have $m(u,v)\le \alpha M<\infty$, and 
\begin{eqnarray*}
r(u,v)	&\le&	R+\int_{-R}^u{\nu(\tilde{u},v)d\tilde{u}}\\
	&\le&	R+(-R-u)\alpha\\
	& < &	\infty.
\end{eqnarray*}
Thus, by the Corollary to Proposition \ref{epektasn}, 
it follows that $\partial{\bf D}=\emptyset$, and thus
\[
{\bf D}=\mathcal{R}=(-\infty,-R]\times[R,\infty).
\]

To complete the proof of the proposition,
it remains to show $(\ref{E2})$, and, under the additional assumption 
$(\ref{AA1})$, $(\ref{E6})$.
Integrating $(\ref{avafora})$ and applying
$(\ref{ud2})$, $(\ref{E1})$, $(\ref{E3})$, $(\ref{E5})$, $(\ref{E6})$, 
we obtain
\begin{eqnarray*}
|\nu\phi+\zeta|(u,v)	&\le&	|\nu\phi+\zeta|(u,R)+
				\int_R^v{\frac{|\phi|}{r^2}
				\frac{2\lambda|\nu|}{1-\mu}m
				d\tilde{v}}\\
		&\le&	C|u|^{-2}+\int_R^{r(u,v)}{\frac{|\phi|}{r^2}
				\frac{2|\nu|}{1-\mu}m
				dr}\\
		&\le&	C|u|^{-2}+2C\alpha^4M
				\int_R^{r(u,v)}{\frac{dr}{r^3}}\\
		&\le&	C|u|^{-2}+C\alpha^4Mr(u,R)^{-2}\\
		& = &	(1+\alpha^4M)C|u|^{-2}.
\end{eqnarray*}
This gives $(\ref{E2})$. On the other hand, assuming $(\ref{AA1})$
and integrating $(\ref{avafora2})$, we obtain
\begin{eqnarray*}
|\lambda\phi+\theta|(u,v)	&\le&	|\lambda\phi+\theta|(-R,v)+
				\int_{u}^{-R}{\frac{|\phi|}{r^2}
				\frac{2\lambda|\nu|}{1-\mu}m
				d\tilde{u}}\\
		&\le&	\tilde{C}v^{-2}+\int_R^{r(u,v)}{\frac{|\phi|}{r^2}
				\frac{2\lambda}{1-\mu}m
				dr}\\
		&\le&	\tilde{C}v^{-2}+2C\alpha^4M
				\int_R^{r(u,v)}{\frac{dr}{r^3}}\\
		&\le&	\tilde{C}v^{-2}+C\alpha^4Mr(-R,v)^{-2}\\
		& = &	(\tilde{C}+C\alpha^4M)v^{-2}.
\end{eqnarray*}
This completes the proof. $\Box$
 
\section{Proof of the Theorem}

In this section, we shall combine Proposition \ref{kurioprop}
with previous
results of Christodoulou to give the proof of Theorem 1.

Fix $\alpha$, $C$ and $M$, and let $R$ be as in Proposition \ref{kurioprop}.
On the ray $v=R$, $u\le0$, prescribe
\[
m(0,R)=0,
\]
\[
r=-u,
\]
and an arbitrary $\mathcal{C}^2$ function $\phi(u)$
satisfying
\begin{equation}
\label{aptovXri}
\int_{-\infty}^0{|\partial^2_u(u\phi)|du}<\epsilon_*,
\end{equation}
\begin{equation}
\label{akoma1}
|\partial_u(u\phi)|(u,R)<C|u|^{-2},
\end{equation}
for $u\ge R$,
and
\[
|R\phi(-R,R)|<C,
\]
\begin{equation}
\label{akoma2}
\int_{-\infty}^0\frac12(\partial_u\phi)^2u^2<M.
\end{equation}
If $\epsilon_*>0$ is sufficiently small, then, by Theorem 6.1 of~\cite{chr:bv}, 
$(\ref{aptovXri})$ implies that this initial value problem has
a past-causally geodesically complete asymptotically flat development:
\[
\includegraphics{parel9ov.eps}
\]
When written as a first order system on $Q$, the boundary conditions
imposed on the axis will be $r=0$, $-u+v=R$. 
Moreover, if $\phi$ is $\mathcal{C}^\infty$, then this development
is also $\mathcal{C}^\infty$.

From the above spacetime, consider the ray $u=-R$, and extend it
off the shaded region to all positive values of $v$:
\[
\includegraphics{mellov.eps}
\]
Where $v\ge R$, prescribe 
\begin{equation}
\label{kaivourgio}
r=v
\end{equation}
and
denote by $C_0=|R\phi(-R,R)|$.

Now, choose some $r_1$ satisfying $R<r_1<R+1$, and, setting $m(-R,R)=m_0$,
choose some $m_1$ satisfying $m_0<m_1<M$.
Define the function 
\[
I(V)=\min\left\{\left(\frac{C-C_0}2\right)^2\frac{1}{(V-r_1)^2},
	\frac{M-m_1}{2R^2(V-r_1)}\right\}.
\]

Fixing for the time being a $V>r_1$,
from the definition of $I$, it is clear by a partition of unity
argument that one can construct a large class of functions 
$\phi:-R\times[0,\infty)\to{\bf R}$, coinciding with
the $\phi$ induced from the solution already
obtained in $-R\times[0,R]$, and 
such that
\begin{enumerate}
\item
The function $\phi$, when viewed as a function of $r$ on $[0,\infty)$,
is regular, for instance $\mathcal{C}^\infty$. 
\item
The inequality
\begin{equation}
\label{buyuk}
(\partial_v\phi)^2(u,v)> I(V),
\end{equation}
holds for
$v\in[r_1,V]$
\item
The inequalities 
\begin{equation}
\label{sugxrovws1}
|r\phi|(-R,v)< C,
\end{equation}
\begin{equation}
\label{sugxrovws2}
m(-R,v)< M,
\end{equation}
hold
for all $v\ge R$,
where $m$ is defined by integrating $(\ref{pvqu})$.
\end{enumerate}
We will denote the class of functions satisfying properties
1, 2, and 3 above as $\mathcal{F}_V$.

Define the quantities
\[
\eta(V)=\min_{\phi\in\mathcal{F}_V}V^{-1}(m(-R,V)-m_1)
\]
and
\[
\delta(V)=Vr_1^{-1}-1.
\] 
It follows that
\begin{eqnarray*}
\eta(V)	& = &	\min_{\phi\in\mathcal{F}_V}V^{-1}\int_{r_1}^V
			{\frac12\theta^2
			\frac{1-\mu}\lambda dv}\\
	&\ge&	\min_{\phi\in\mathcal{F}_V}
			R^2V^{-1}\int_{r_1}^V{\frac12
			(\partial_v\phi)^2dv}\\
	&\ge&	\frac12R^2V^{-1}(V-r_1)I(V)\\
	&\ge&	\frac12R^2(R+1)^{-1}(V-r_1)I(V).
\end{eqnarray*}
Thus for small enough $V-r_1$ we have
\begin{equation}
\label{biryildiz}
\eta(V)\ge\frac{M-m_1}{4(R+1)}.
\end{equation}

Recall the function $E(y)$ defined in~\cite{chr:fbh} by
\[
E(y)=\frac{y}{(1+y)^2}\left[\log\left(\frac{1}{2y}\right)+5-y\right].
\]
Note that $E(y)\to0$ as $y\to0$. Choose $V-r_1$ small enough so that
\begin{equation}
\label{ikiyildiz}
E(\delta)<\frac{M-m_1}{4(R+1)}
\end{equation}
Inequalities
$(\ref{biryildiz})$ and 
$(\ref{ikiyildiz})$ together imply
\begin{equation}
\label{suv9nkn}
\eta>E(\delta).
\end{equation}
From Theorem 5.1 of~\cite{chr:fbh}, property $(\ref{suv9nkn})$ ensures
that initial data in
$\mathcal{F}_V$ has an incomplete future
development; moreover, its conformal structure is as depicted below:
\[
\includegraphics{pkm.eps}
\]
(See~\cite{chr:fbh} for details.)
On the other hand, $(\ref{akoma1})$, $(\ref{akoma2})$,
$(\ref{sugxrovws1})$, and $(\ref{sugxrovws2})$
imply that the rays $u=-R$, and $v=R$,
and the functions $r$, $\phi$, and $m$ defined on them satisfy the
conditions of Proposition \ref{kurioprop}. 
Applying this proposition one obtains a spacetime as depicted in the conformal
diagram of the Introduction. 
It is easy to see that this spacetime admits an asymptotically
flat Cauchy hypersurface. $\Box$

The strict inequalities in $(\ref{akoma1})$, $(\ref{akoma2})$,
$(\ref{sugxrovws1})$, and $(\ref{sugxrovws2})$ immediately
indicate one notion according to which our solutions are stable in the 
spherically symmetric category. Alternatively,
considering a Cauchy surface $\mathcal{S}$ intersecting $(-R,R)$,
it follows that given any solution of the kind we have constructed,
sufficiently small\footnote{For instance, in the $\mathcal{C}^\infty$ topology,
if we consider only $\mathcal{C}^\infty$ solutions.}
 perturbations supported on $\mathcal{S}$ in $r<R$ lead to a Cauchy
development with a similar conformal diagram. For on 
the backwards characteristic, $(\ref{aptovXri})$, $(\ref{akoma1})$,
and $(\ref{akoma2})$ hold on account of Cauchy stability (this takes
care of $r\le R$) and the domain of dependence theorem (for $r\ge R$),
thus guaranteeing the past completeness. On the other hand,
Cauchy stability implies that $\eta>E(\delta)$ for these purturbations,
as this refers to a compact set of the development. Thus, the formation
of a black-hole in the future is also a stable feature, in this sense.

Another point is worth mentioning. As the formation of a black hole
depends only on the solution on $-R\times[r_1,V]$, we can replace
the part of the solution for $v\ge V$
with the solution of yet another characteristic
initial value problem, where initial data are given on
$v=V$ and on past null infinity for $v\ge V$. In particular, one
can provide solutions with the conformal diagram indicated in
the Introduction, for which there is no incoming radiation near
spacelike infinity, i.e.~for which $m$ is constant along 
past null infinity in a neighborhood of spacelike infinity.
On the other hand, by the results of~\cite{chr:mt},
if past null infinity is complete and $m$ is constant throughout,
then either $m=0$, in which case the solution is Minkowski space,
or else it contains a white hole.
\newpage
\input{bibliog}
\end{document}







As discussed in the Introduction, the method of this paper will apply, 
not only to the coupled system outlined in the previous section,
but also to solutions of the linear wave equation on a \emph{fixed}
Schwarzschild or Reissner-Nordstr\"om background, in fact, on
a wide class of fixed static black-hole backgrounds. 
This uncoupled problem can be formulated in such a way
so as to satisfy equations very similar to those considered in the
previous section. To see this, consider a static spherically
symmetric background whose quotient manifold has the conformal geometry of a characteristic rectangle.
Define the quantity $\varpi$ arbitrarily at the lower left endpoint
of this characteristic rectangle, and then extend the definition of
$\varpi$
to the two sides intersecting at that endpoint by imposing the equations
$(\ref{puqu})$ and $(\ref{pvqu})$ on the constant $v$ and $u$ sides, respectively. Finally, extend the definition of $\varpi$ throughout the rectangle by imposing
for instance $(\ref{puqu})$.
By applying the divergence theorem 
to $T_{\mu\nu}X^{\mu}$, where
$T_{\mu\nu}$ is the energy-momentum tensor of the scalar field and $X^\mu$ is the timelike Killing field, it follows
that $(\ref{pvqu})$ holds as well
throughout the rectangle.

With the above definition of $\varpi$, one
sees easily that a Reissner-Nordstr\"om solution with 
parameters $\tilde\varpi$, 
$e$, together with a solution $\phi$ of the wave 
equation, are completely described by
the equations $(\ref{ruqu})$, $(\ref{rvqu})$, $(\ref{puqu})$, $(\ref{pvqu})$, $(\ref{sign1})$, $(\ref{sign2})$, and
\[
\partial_v\nu=\partial_u\lambda=\lambda\left(-\frac{2\nu}{1-\mu}\frac{1}{r^2}
\left(\frac{e^2}{r}-\tilde\varpi\right)\right),
\]
where, as before, $1-\mu=1-\frac{2m}r=1-\frac{2\tilde\varpi}r+\frac{e^2}r$.
The equations
\begin{equation}
\label{devallazei1}
\partial_u\frac\lambda{1-\mu}=0,
\end{equation}
\begin{equation}
\label{devallazei2}
\partial_v\frac\nu{1-\mu}=0,
\end{equation}
are also clearly satisfied. Equation
$(\ref{zex})$ is replaced by
\[
\partial_v\left(\zn\right)=
-\frac{\theta}r+\zn\left[\frac\lambda{1-\mu}\frac2{r^2}\left(
\frac{e^2}r-\tilde\varpi\right)\right],
\]
and 
$(\ref{avafora})$ by
\[
\partial_u(\phi\lambda+\theta)=-\frac{\phi}{r^2}\frac{2\lambda\nu}{1-\mu}
				\left(\frac{e^2}r-\hat\varpi\right).
\]

In order to avoid overly cumbersome references
to separate equations in the uncoupled case, we 
shall adopt the following
convention: Any time the quanitity $\varpi$ arises in the
expression $\frac{e^2}r-\varpi$, then in the uncoupled case
$\varpi$ is to be replaced by $\tilde\varpi$.
With this convention and
the above redefinition of the basic equations, the uncoupled case
can be understood in essentially the same notation as the coupled case.
Moreover, the uncoupled case is genuinely ``simpler'', as
it will turn out that
the right hand side of $(\ref{fdb1})$ and $(\ref{fdb2})$,
together with the difference of $\varpi$ and $\tilde\varpi$ in the expression
$\frac{e^2}r-\varpi$, are always
treated as ``error terms''. 
For the benefit of the reader primarily interested in this
latter case, however,
we will make explicit note of the simplifications in what follows.
This essentially means that certain arguments in
Section 5 can be ignored.

\section{Assumptions}
We have stated the theorems in the Introduction for complete compactly supported
spacelike initial data, in order to make explicit the relevance of the present paper to 
the problem of gravitational collapse
and to typical formulations of cosmic censorship. However,
the natural setting for the method of this paper is the evolution of characteristic initial data
with suitably decaying initial conditions, where it is assumed \emph{a priori} that an event horizon forms
in the future.
A complete set of assumptions for our problem is summarized by the following six conditions:
\begin{enumerate}
\item[a.]
We have a solution to equations $(\ref{ruqu})$--$(\ref{sign2})$
in an open $2$-dimensional spacetime $U$ which conformally contains
the spacetime ${\bf D}$ depicted below\footnote{The authors make no apologies for
describing conformal relations with respect to diagrams and without recoures to words, 
as, in the context of spherical
symmetry, these diagrams are completely descriptive.}:
\[
\includegraphics{basiko4.eps}
\]
Here, the segment labelled ``event horizon'' is understood to
be part\footnote{i.e.~local
regularity assumptions \emph{will} apply here} of ${\bf D}$, 
while the segment labelled null infinity
is not. 
\item[b.]
Along outgoing null rays ``terminating''
on null infinity, $r\to\infty$. 
\item[c.]
Along the event horizon, 
$\sup{r}=r_+<\infty$.
\item[d.]
On the initial outgoing null ray,
$r$ is monotonically increasing, $1-\mu>0$, $0<\sup m<\infty$,
and\footnote{Note that $\tl$ is
a coordinate invariant quantity.}
\begin{equation}
\label{byil}
r\left|\frac\theta\lambda\right|\le C,
\end{equation}
\begin{equation}
\label{pioduvato}
r^3\left|\phi+\frac\theta\lambda\right|\le C.
\end{equation}
\item[e.]
On the initial ingoing null ray, $r$ is monotonically
decreasing and is strictly greater than $0$.
\item[f.]
$r_+\ne e$.
\end{enumerate}

In the case where $e\ne0$,  
the evolution of complete spacelike initial data is
related to the above conditions by the following proposition, which
can be easily proven:

\begin{proposition}
Consider complete spacelike spherically symmetric initial data with
asymptotically flat ends\footnote{Such data
necessarily will have two asymptotically flat ends.} for 
the Einstein-Maxwell-scalar field equations where
$e\ne0$,
and assume that $\phi$ and $\nabla\phi$ are compactly supported
on the initial hypersurface. Then 
the initial hypersurface contains a trapped or marginally trapped surface.
Moreover, the maximal development of the data contains
a domain of outer communications satisfying conditions (a)-(e) above. If, in addition,
$r>e$ on the initial hypersurface, then condition (f) is also satisfied.
\end{proposition}

On the other hand, in the case $e=0$,
conditions (a)-(f) follow for the evolution of ``generic''
non-dispersing, compactly supported initial data from the results of Christodoulou:

\begin{theorem} \cite{chr:ins,chr:mt}
Consider complete spacelike asymptotically flat
spherically symmetric initial data 
in the BV class defined in~\cite{chr:bv} for
the Einstein-scalar field equations, and assume
$\phi$ and $\nabla\phi$ are compactly
supported on the initial hypersurface. Then, as long as
the data lie outside a certain set of codimension one
in the BV class, the maximal development contains a domain
of outer communications possessing a complete future null infinity.
Moreover, if the final Bondi mass is zero, this development is
future causally geodesically complete
and approaches Minkowski
space, while if the final Bondi mass is nonzero, conditions (a)-(f) hold.
\end{theorem}

In the original heuristic analysis of Price, besides
data of compact support, data which are ``initially static''
are also considered, that is to say, data which on the initial
outgoing null ray coincide with an asymptototically flat
static solution. This motivates the following variant
of condition (d):

\begin{enumerate}
\item[d$'$.]
On the initial outgoing null ray, the 
inequality $(\ref{pioduvato})$ in (d) is replaced by the weaker assumption
\begin{equation}
\label{ligoduvato}
r^2\left|\phi+\frac\theta\lambda\right|\le C.
\end{equation}
\end{enumerate}

Our final conclusions under condition (d$'$) will be slightly weaker than under
condition (d), in complete agreement with the heuristics of Price.


Finally, to describe the assumptions for the uncoupled case, 
it suffices to replace condition (a) with
\begin{enumerate}
\item[a$'$.]
We have a solution to , etc.
\end{enumerate}

Unless otherwise noted, in what follows, all statements will apply 
for both (a) and (a$'$), and both (d) and (d$'$).

\section{Outline of the argument}
The argument of this paper is rather involved, even in skeleton form. 
The main ingredients, however, when viewed separately, are quite transparent.
They can be thought of as relatively simple analytical manifestations or consequences of the following
robust\footnote{i.e.~features well-known from the Schwarzschild solution,
but not special to it} geometrical features of black hole spacetimes
\begin{enumerate}
\item[(A)] 
a well-defined notion of infinity,
\item[(B)]
the global conformal geometry; in particular, the existence of an affine
complete null hypersurface (the event horizon) which does not terminate at null infinity
and whose past is the entire domain of outer communications,
\item[(C)]
the celebrated red-shift effect.
\end{enumerate}
In this section, we will 
first make some general remarks on how these features can be connected to one
another in the analysis of the system.  Then we will attempt
to give a more complete outline of how these features are actually put together in our
proof of decay. The reader may want to refer back to this sketch when 
reading the subsequent sections.

We first discuss (A). 
This feature is of course the least exotic of the above, as it is familiar
from Minkowksi space and small data dispersing solutions. For us, (A), together with
the special form of energy conservation that is valid for our system, will allow us to obtain 
\emph{a priori} uniform decay in $r$ for the scalar field.  It is this 
$r$-decay that we hope to translate into $v$-decay along the event horizon, 
and $u$-decay along null infinity. 

The analytic significance of (B) is that 
the event horizon carries a positive finite flux, in view of energy conservation. 
Since the event horizon
is affine complete, every so often there must be large
regions where the flux is quantitatively small. In this paper, this smallness
will act as a catalyst for turning decay in $r$ into decay in $v$.

An efficient use of the event horizon flux in the above role is only possible
when one is able to transport information from the event
horizon without much loss to other regions of spacetime, in particular, to
regions where $r$ is large, and thus, decay in $r$ has some content. In other words, one needs
to be able to derive good estimates in the
direction indicated in the figure below.
\[
\includegraphics{kateu93.eps} 
\]
One might expect that the smallness of the flux on the segment
of the event horizon, which we hope to preserve,
is completely dwarfed by initial data on the conjugate null segment
depicted above, on which we know nothing. The remarkable fact is that
this is not the case. Herein lies the signicance of the red-shift effect (C).

Feature (C) is tied to the very name of black holes,
as it is the red-shift effect that makes the black holes
of gravitational collapse look ``black'', rather than like frozen stars.
To measure quantitatively this effect, one considers the ratio of the proper time between
the reception of two signals,
as measured by an observer travelling to timelike infinity,
to the proper time between the emission of the two signals by
an observer entering the black hole. As the emitter approaches
the black hole, and the proper time between his emissions tends to zero,
this ratio goes to $\infty$. In the context of the scalar field, 
this shifts the frequency of the radiation on the incoming null
ray to the red. This will allow us to 
essentially ``forget'' about this part of the initial data when
estimating as in the previous diagram.

Following up on the above general remarks, we now
outline our proof of decay.
We define a $v$-coordinate by setting
\[
\frac\lambda{1-\mu}=1
\]
on an outgoing null ray. 
The first step will be to show
\begin{equation}
\label{unifbounds}
|r\phi|+|r\theta|+|r^2(\phi\lambda+\theta)|+\left|\zn\right|\le C.
\end{equation}
For large $r$, these are the uniform decay bounds which were discussed
in the context of (A) above. For $r\sim r_+$, on the other hand,
showing these bounds requires making use 
of the red-shift effect in integrating $(\ref{zex})$. 
From $(\ref{unifbounds})$ and energy conservation,
we will be able to obtain
\[
1\ge\frac\lambda{1-\mu}\ge c
\]
in the domain of outer communications, for some $c>0$. (In the
case of (a$'$), this holds trivially from $(\ref{devallazei1})$ 
with $c=1$.)
The above inequality suggests
that the $v$-coordinate defined above 
is indeed appropriate for in the coupled case
the power-law decay on the event horizon predicted
by Price's heuristics for the linearized theory, since
rescaling so that $\frac\lambda{1-\mu}=1$ on the event horizon,
as in the linear theory, would then not change the rate of decay.

>From these basic estimates,
our argument will proceed in steps:
\begin{enumerate}
\item
Energy conservation (encoded in the non-linear equations by
the renormalized Hawking mass $\varpi$) implies that the total
scalar field energy flux through the event horizon is finite. 
The pigeon-hole principle then implies that there is an infinite
sequence of segments on the event horizon
\[
\includegraphics{peril.eps}
\]
having the following properties:
\begin{enumerate}
\item[a.]
If $v$ denotes the $v$-value of the right endpoint
of a segment, then the distance to the next interval
is bounded above and below by constants $\alpha v$
and $\beta v$, with $\alpha>\beta>0$.
\item[b.]
The length of each segment is bounded above and below by constants
$\alpha v^p$, $\beta v^p$, where $1>p>0$. Thus the lengths \emph{increase}
in $v$.
\item[c.]
Both, the scalar field's energy flux through each segment, and $\theta$ at the
right endpoint of each segment decay as $v^{-s}$, for some $s>0$.
\end{enumerate}
\item
Starting out from the event horizon and estimating in the spacelike
direction, the above decay estimates for the energy flux and $\theta$
induce a pointwise estimate
\begin{equation}
\label{ekpln3n}
|\theta|\le Cv^{-s}r^\alpha,
\end{equation}
for some $\alpha>0$,
on a sequence of constant-$v$ segments depicted below as dashed lines:
\[
\includegraphics{peril2.eps}
\]
This estimate relies on the red-shift effect discussed previously.
These dashed constant-$v$ segments depicted above can be chosen 
so that at the lower right endpoints
\begin{equation}
\label{lre}
r\approx v^q,
\end{equation}
where $q>0$.  This can be effected by carefully keeping
track of the balance between the size
of the original event-horizon segments
in conjunction with the 
size of the region near the event horizon where the ``red-shift'' effect
is sufficiently strong.
\item
Combining the $r^{-1}$ decay estimate $(\ref{unifbounds})$ 
for $\theta$ with $(\ref{ekpln3n})$, in view of $(\ref{lre})$,
we deduce that $\theta$
decays as $v^{-s'}$ on elongated versions
of the dashed segments, where now the right endpoint can be
taken so that 
\[
r\approx v,
\]
for some
$0<s'\le s$. 
These elongated segments are depicted as the solid constant $v$-segments
below:
\[
\includegraphics{peril3.eps}
\]
Furthermore, since the remaining right segments of the shaded rectangles lie
in the region $r\sim v$, we have that
$\phi$ and $\phi\lambda+\theta$ also decay as negative
powers of $v$ on those segments.
Recalling that in each rectangle,
all $v$ values
are comparable, it follows that to show decay in $v$ in the union of all shaded rectangles
it would suffice to estimate in the interior of
each rectangle in terms of sup bounds on the boundary. 
This will actually be accomplished in stages, considering various  parts of the
rectangles separetely.
In the end, the
decay for $|\phi|$ and $|\theta|$ thus obtained will be $r^{-1}v^{-s'}$.
\end{enumerate}

Besides a rate of decay for $\phi$ and $\theta$, this argument
yields a sharper estimate for the energy
flux on the event horizon. The whole argument can then be iterated
to obtain progressively improved values for $s'$. 
The proof of decay will thus be formulated as an induction.
The limit to the order of decay
which can be shown in each shaded rectangle above is given by the rate of decay in $v$
of $\phi\lambda+\theta$ on a sufficiently late outgoing null ray, and this is
what gives rise to the
power $3$ in Price's law for compactly supported initial data.

The preliminary estimates on $\frac{\lambda}{1-\mu}$, $r\phi$,
$r\theta$, and $r^2(\phi\lambda+\theta)$ will be derived in the next section. Section 6
will then derive the characteristic rectangle estimatate which will be used in
step 2 of the above outline. Section 7 will
derive the estimates required for step 3. 
In Section 8, the estimates of
the previous two sections will be incorporated into an induction argument,
to retrieve a preliminary decay result, in particular decay of $v^{-\frac32}$ for $\phi$
on the event horizon. Proving sharper decay will require minor modifications to
the induction scheme. In Section 9,
it will be shown that the decay for $\phi$ on the event horizon
can be pushed forward to $v^{-3+\epsilon}$, under condition (d), and to
$v^{-2+\epsilon}$, under condition (d$'$). Also, in that section, appropriate decay
rates along null infinity will be derived, not discussed in this outline.

\section{Decay in $r$}
We show in this section a series of 
global bounds which will yield decay in $r$ for $\phi$,
$\theta$, and $\lambda\phi+\theta$. These bounds follow more or
less from energy conservation alone. 

We define a $v$ coordinate so that 
\begin{equation}
\label{INITv}
\frac{\lambda}{1-\mu}=1
\end{equation}
on the initial outgoing null ray
(refer to the figure in Section 3).
We thus have, on the initial outgoing null ray, $v^{-1}$ decay for
$|\theta|$, and $v^{-2}$ decay for $|\theta+\phi\lambda|$.


 
To get started we will define a $u$ coordinate by the condition $\nu=-1$
on the initial incoming null ray. It can be shown easily from 
our basic assumptions that this $(u,v)$
coordinate system provides a regular chart covering 
all of ${\bf D}$,\footnote{Note that by considerations arising from
the Raychadhuri equation, $\nu(0,0)<0$
implies that $\nu(U,0)<0$ for any coordinate system regular at $(U,0)$.
See the proof of the next Proposition.}
specifically, ${\bf D}=[0,U]\times[0,\infty)$, for
some $\infty>U>0$.

Our system of equations possesses rich monotonicity properties, many of
which can be derived solely from the positivity of energy and the assumption of
spherical symmetry. The reader may wish to consult the nice discussion in~\cite{chr:sgrf}
for general matter. The proof of the next Proposition, which
we will provide for completeness, can in fact be easily
derived from the methods of~\cite{chr:sgrf}.

\begin{proposition}
The following inequalities hold in ${\bf D}$, except possibly on the event horizon:
\begin{equation}
\label{vumov}
\partial_ur=\nu<0,
\end{equation}
\begin{equation}
\label{lamov}
\partial_vr=\lambda>0, 
\end{equation}
\begin{equation}
\label{xwrisovoma}
1-\mu>0,
\end{equation}
\begin{equation}
\label{II}
\partial_u\frac{\lambda}{1-\mu}\le0,
\end{equation}
\begin{equation}
\label{IIb}
\partial_v\frac{\nu}{1-\mu}\le0,
\end{equation}
\begin{equation}
\label{pumov}
\partial_u\varpi\le0,
\end{equation}
\begin{equation}
\label{pvmov}
\partial_v\varpi\ge0.
\end{equation}
By continuity, we have in addition $\lambda\ge0$, $1-\mu\ge0$, $(\ref{pumov})$, $(\ref{pvmov})$
on the event horizon.
\end{proposition}

\noindent\emph{Proof.} Whether in the case of condition (a),
or (a$'$), our spacetime clearly satisfies the
positive energy condition. It follows from the
Raychadhuri equation (see~\cite{he:lssst})
that given a null ray in the quotient 
manifold,
and an affine parallel vector field $L$ on that ray,
if $\nabla_Lr(p)\le 0$, then $\nabla_L\nabla_L r\le0$
to the future of $p$, and consequently also $\nabla_Lr\le0$.
Moreover, if $\nabla_Lr(p)<0$, then clearly also $\nabla_Lr<0$
along the ray in the future of $p$.

On the initial outgoing ray, we see explicitly that $\nu<0$,
by integrating $(\ref{nqu})$. Given now any ingoing null rays\,
and defining $L$ to be a future-pointing parallel affine null vector field along that
ray, it follows that
$\nabla_L r<0$ initially, and thus by the previous paragraph
$\nabla_L r<0$ along the whole ray. It follows that
$(\ref{vumov})$ holds.

Similarly, given any outgoing null ray in the past
of the event horizon, and defining again
$L$ to be a future-pointing parallel null vector field
along that ray, if $\nabla_L r(p)\le0$, then $\nabla_L r\le0$
in the future of $p$. But then $r$ cannot tend to $\infty$
along this ray, contradicting condition (b). Thus in fact
we have $\nabla_L r>0$ in the past of the event horizon. This implies $(\ref{lamov})$.
 
In the case of condition (a), in view of $(\ref{fdb1})$, 
inequality $(\ref{vumov})$ immediately implies $(\ref{II})$,
and similarly $(\ref{lamov})$ implies $(\ref{IIb})$.
On the other hand,
on any segment $[0,u]\times{v}$, where $u<U$,
$r^{-1}\frac{\zeta^2}{\nu}$ is necessarily bounded by compactness.
Thus
\[
\frac{\lambda}{1-\mu}(u,v)=\frac{\lambda}{1-\mu}(0,v)
e^{\int_0^u{r^{-1}\frac{\zeta^2}{\nu}(u',v)du'}}
\]
is strictly positive.
Together with $(\ref{lamov})$, this yields $(\ref{xwrisovoma})$.

In the case of condition (a$'$), $(\ref{II})$ and $(\ref{IIb})$
are trivialities, and moreover, $\frac{\lambda}{1-\mu}=1$ identically,
and thus in particular, this quantitiy is strictly positive,
and so, by $(\ref{lamov})$, we again have $(\ref{xwrisovoma})$.

In both cases (a) and (a$'$), 
the remaining inequalities follow immediately from
$(\ref{puqu})$, $(\ref{pvqu})$, $(\ref{fdb2})$. $\Box$

Denoting by $\varpi_1$, $r_1$ the values of $\varpi$, $r$,
respectively, at the point of intersection of the ingoing null
ray and the event horizon, and $M$ the value of $\varpi$ at the intersection
of the outgoing null ray and null infinity, we have, by $(\ref{vumov})$, $(\ref{lamov})$,
$(\ref{pumov})$, $(\ref{pvmov})$, that
\begin{equation}
\label{rlower}
r\ge r_1,
\end{equation}
\begin{equation}
\label{pi-pavwkatw}
\varpi_1\le\varpi\le M.
\end{equation}
By conditions (d) and (e), it follows that both $r_1>0$,
and, in the case of condition (a), $M>0$. 
In the case of condition (a'), we are free to redefine $\varpi$
by adding a constant, and so,
we can arrange that $M=\tilde{\varpi}>0$.
We note then that $(\ref{pi-pavwkatw})$ holds also for
$\tilde\varpi$. 

Note that the signs of $\nu$, $\lambda$ imply in particular that
constant-$r$ curves not crossing the event horizon are timelike. 
It is clear from $(\ref{pi-pavwkatw})$, that for large enough $R$, 
\begin{equation}
\label{pekcokmuhim}
1-\mu(u,v)\ge c
\end{equation}
for $r(u,v)\ge R$, where $c$ is some positive constant such that
$c\to1$ as $R\to\infty$.

\begin{proposition} In $r\ge R$, we have the bounds 
\label{mprop}
\begin{equation}
\label{makrua1}
|\phi|\le \frac{\hat{C}}r,
\end{equation}
\begin{equation}
\label{makrua2}
|\phi\lambda+\theta|\le \frac{\hat{C}}{r^2},
\end{equation}
\begin{equation}
\label{makrua3}
\hat{c}\le\frac{\lambda}{1-\mu}\le1,
\end{equation}
for some constants $\hat{c},\hat{C}>0$, where
$\hat{c}\to1$ as $R\to\infty$.
\end{proposition}

\noindent\emph{Proof.} 
We will prove the bounds above in several steps. First we note that
\[
\frac12\int{\theta^2\frac{1-\mu}{\lambda}dv}\le M-\varpi_1,
\]
and
\[
\frac12\int{\zeta^2\frac{1-\mu}{(-\nu)}du}\le M-\varpi_1,
\]
where the integrals can be taken over any constant $u$ and
$v$ line segments respectively.

As noted earlier, in the case of condition (a$'$), 
$(\ref{makrua3})$ holds with $\hat{c}=1$. In the case
of condition (a),
we estimate
\begin{eqnarray*}
\frac{\lambda}{1-\mu}(u,v)	& = &	\frac{\lambda}{1-\mu}(0,v)e^
								{\int_0^u{\frac1r\zeta^2\frac{1-\mu}\nu
								\frac{1}{1-\mu}(u^*,v)du^*}}\\
					&\ge&	e^{c^{-1}R^{-1}
								\int_0^u{\zeta^2\frac{1-\mu}\nu du^*}}\\
					&\ge&	\hat{c},
\end{eqnarray*}	
where $\hat{c}=\exp\left[2c^{-1}R^{-1}(M-\varpi_1)\right]$.
Thus, in view also of $(\ref{INITv})$ and $(\ref{II})$, we 
have established $(\ref{makrua3})$.

Note that a bound
\[
|\phi|(0,v)\le\frac{\tilde{C}_1}r
\]
holds on the initial outgoing null ray by $(\ref{byil})$, $(\ref{ligoduvato})$, and the triangle inequality.
Integrating the equation
\[
\partial_u\phi=\frac{\zeta}r,
\]
from the initial outgoing null ray
yields
\begin{eqnarray*}
|\phi(u,v)|	&\le&	\frac{\tilde{C}_1}{r(0,v)}+
					\sqrt{\int_0^u\zeta^2\frac{1-\mu}{-\nu}
					(u^*,v)du^*}
					\sqrt{\int_0^u\frac{-\nu}
					{1-\mu}\frac{1}{r^2}(u^*,v)du}\\
		&\le&	\frac{\tilde{C}_2}{\sqrt{r}},
\end{eqnarray*}
where the last inequality follows in view of the bound $(\ref{pekcokmuhim})$.

Integrating now
$(\ref{avafora})$
from the initial outgoing null segment, in view of $(\ref{makrua3})$, 
\begin{equation}
\label{all1ast}
\left|\frac{e^2}r-\varpi\right|\le e^2r_1^{-1}+M+|\varpi_1|,
\end{equation}
and
$(\ref{ligoduvato})$,
we obtain
\[
|\phi\lambda+\theta|(u,v)\le \tilde{C}_3r^{-\frac32}.
\]
(In the case of condition (a$'$), the estimate
$(\ref{all1ast})$ is of course to be understood with 
the convention of Section 3. It holds because of
the remark following $(\ref{pi-pavwkatw})$. In what follows,
we shall make use of our convention without further comment.)

For $R\ge\sup_{\tilde{u}}{r(\tilde{u},0)}$, if $r(u,v)\ge R$, then
there exists a unique $v_*\le v$ such that $r(u,v_*)=R$. 
We can thus integrate the equation
\[
\partial_v(r\phi)=\phi\lambda+\theta
\]
from $(u,v_*)$ to $(u,v)$
 to obtain
\begin{eqnarray*}
|r\phi|(u,v)	&\le&	\frac{\tilde{C}_2R}{\sqrt R}+\left|\int_{v_*}^v{\frac{\tilde{C}_3}{r\sqrt r}dv}\right|\\
		&\le&	\tilde{C}_2\sqrt{R}+\frac{2\tilde{C}_3}{\sqrt{R}}
\end{eqnarray*}
and thus
\begin{equation}
\label{kaluterob}
|\phi|(u,v)\le\frac {\tilde{C}_4}r.
\end{equation}
Integrating again $(\ref{avafora})$
using the bound $(\ref{kaluterob})$ we obtain finally
\[
|\lambda\phi+\theta|(u,v)\le\frac{\tilde{C}_5}{r^2}.
\]
Thus $(\ref{makrua1})$ and $(\ref{makrua2})$ hold with $\hat{C}=\max\{\tilde{C}_4,\tilde{C}_5\}$.
$\Box$

In the region $r\ge R$, by $(\ref{pekcokmuhim})$ and the bound
$(\ref{all1ast})$, we have a
uniform bound
\begin{equation}
\label{eb}
\left|\int{\frac\lambda{1-\mu}\frac2{r^2}\left(\frac {e^2}r-\varpi\right)
dv}\right|\le 2R^{-1}c^{-1}[e^2r_1^{-1}+M+|\varpi_1|],
\end{equation}
where the integral can be taken on any constant-$u$ interval.
On the other hand, in the region $r\le R$, we have that $v\le f(u)$,
where $f(u)<\infty$ for $u<U$. Thus, 
\begin{equation}
\label{ab}
\exp{\int_0^\infty{\frac\lambda{1-\mu}\frac2{r^2}\left(\frac {e^2}r-\varpi\right)
dv}}=\tilde{f}(u),
\end{equation}
is a bounded function of $u$, and,
by similar considerations, can also be seen to be differentiable.
Define now a new null coordinate $u^*$ by
\[
u^*(u)=\int_0^u{\tilde{f}(u)du}.
\]
This clearly defines a regular change of coordinates for $u<U$.
Moreover $\nu^*=\partial_{u^*}r\to-1$ as $v\to\infty$,
and this defines a continuous extension of $\nu^*$ to null infinity. We will
say that $u^*$ is retarded time normalized at null infinity.

In what follows, we shall drop the $*$ and refer to our new coordinate as $u$,
and $\nu^*$, etc., as $\nu$.
Note that the bound $(\ref{eb})$ 
implies for this new coordinate that in the region $r\ge R$ we have
\begin{equation}
\label{ebgtv}
|\nu|\le\exp\{2R^{-1}c^{-1}[e^2r_1^{-1}+M+|\varpi_1|]\}.
\end{equation}


\begin{proposition} We have $u\to\infty$ along null infinity
as timelike infinity is approached, i.e.~null infinity is
complete.
\end{proposition}

\noindent\emph{Proof.}
Integrating $(\ref{ruqu})$ with $(\ref{ebgtv})$ we have
\[
r(u,v)\ge r(0,v)-\exp\{2R^{-1}c^{-1}[e^2r_1^{-1}+M+|\varpi_1|]\}u,
\]
for $(u,v)$ in the region $r\ge R$.
By the monotonicity properties of $r$, for $R>r_+$, $r(u,v)=R$ is
a timelike curve terminating at timelike infinity. In particular, there exist
points on this curve with arbitrarily large $v$. But since 
$r(0,v)\to\infty$ as $v\to\infty$, by assumption, it follows
that for these points $u\to\infty$. $\Box$.

The event horizon cannot be covered in our new $u$-coordinate system,
but it can be thought of, by a mild abuse of notation, as $u=\infty$.
Of course, to deal with points on the event horizon, the correct thing would be to
switch back to our old coordinates which were regular there. 
We will avoid this inconvenience, however, by restricting to functions
invariant of choice of $u$ coordinate when dealing in regions containing
the event horizon. Thus, expressions like $\zn(\infty,0)$,
$\int_0^\infty\lambda(\infty,v)dv$ should cause no concern to the reader,
who can easily reinterpret them in terms of the old coordinate if he prefers.

\begin{proposition} 
\label{ZBD}
In $r\le R$, we have a bound
\begin{equation}
\label{2A}
\left|\zn\right|\le \hat{C},
\end{equation}
for some $\hat{C}>0$.
\end{proposition}

\noindent\emph{Proof.}
We note that on any
constant-$v$ segment, $\left|\zn\right|$ is bounded up to
and including the event horizon, as this is a coordinate 
invariant quantity, with constant depending on $v$.
Thus, the difficulty involves estimates for
sufficiently large $v$.
In particular, the main difficulty in the proof
of this proposition involves is showing that the estimate
$(\ref{3A})$ holds. In the case of condition (a$'$), however,
$(\ref{3A})$, with the convention of Section 3 of course, can be
seen to hold immediately, as a consequence of the relations
$1-\mu=0$ and $r_+\ne e$, on the event horizon, and monotonicity. Thus, the reader interested only in the case (a$'$)
should skip to the paragraph after $(\ref{3A})$. 

We assume in the meantime condition (a).
We first wish to exlude the case where there exists a $v'$ such that
\begin{equation}
\label{excludeit}
1-\mu(\infty,v)\ge c',
\end{equation}
for all $v\ge v'$ and some \emph{positive} constant $c'$.

 
Assuming for the time being $(\ref{excludeit})$, 
we can choose 
sufficiently small intervals $[r',r'']$, $[\varpi',\varpi'']$,
containing $r_+$, $\varpi_+$, respectively,
so that for $r^*\in[r',r'']$, $\varpi^*\in[\varpi',\varpi'']$,
we have $1-\frac{2\varpi^*}{r^*}+\frac{e^2}{(r^*)^2}\ge \frac {c'}2$.

Choose $v'$ such that in addition to $(\ref{excludeit})$, we have
$r(\infty,v')\ge r'$, and $\varpi(\infty,v')\ge \varpi'$,
and consider the region 
\[
{\bf R}=\{v\ge v'\}\cap\{\varpi\in[\varpi',\varpi'']\}\cap\{r\in[r',r'']\}.
\]
It is clear from inequalities $(\ref{vumov})$, $(\ref{lamov})$, $(\ref{pumov})$, and $(\ref{pvmov})$
that if $(u,v)\in{\bf R}$, then so is $u\times[v',v]$, as well
as $[u,\infty]\times v$,
and moreover that 
\begin{equation}
\label{excludeit2}
1-\mu(u,v)\ge\frac{c'}2
\end{equation}
holds in this region.

Integrating $(\ref{zex})$
from $v=v'$, we obtain:
\begin{eqnarray*}
\left|\zn(u,v)\right|	&\le&	\left[\sqrt{\int_{v'}^v{\frac{1-\mu}\lambda\theta^2dv^*}}
							\sqrt{\int_{v'}^v{\frac{\lambda}{1-\mu}
							\frac1{r^2}dv^*}}\right.\\
				&   & \hbox{\ \ }\left.+\left|\zn(u,v')\right|\right]
							e^{\int_{v'}^v{\left[\frac\lambda{1-\mu}
							\frac2{r^2}\left|
							\frac{e^2}r-\varpi\right|\right]dv^*}}\\
				&\le&	\left[\sqrt{2(M-\varpi_1)}\sqrt{\int_{v'}^v{\frac{\lambda}{1-\mu}
							\frac1{r^2}dv^*}}
							+\left|\zn(u,v')\right|
							\right]\\
				&   &\hbox{\ \ }\cdot e^{\int_{v'}^v{\left[\frac\lambda{1-\mu}
							\frac2{r^2}\left|
							\frac{e^2}r-\varpi\right|\right]dv^*}}
\end{eqnarray*}
%				&\le&	\left[\sqrt{2(M-\varpi_1)}\sqrt{\int_{v'}^v{\frac1{r^2}
%							\frac{\lambda}{1-\mu}(\infty,v^*)
%							e^{\int_u^\infty\frac1r\left
%							(\zn\right)^2
%							(-\nu)du^*}dv^*}}\right.\\
%				&   & \hbox{\ \ }\left.+\left|\zn(u,v')\right|
%							\right]e^{\int_{v'}^v{C\frac{\lambda}{1-\mu}(\infty,v^*)
%							e^{\int_u^\infty\frac 
%							1r\left(\zn\right)^2
%							(-\nu)du^*}dv^*}}.
%\end{eqnarray*}
For $(u,v)\in{\bf R}$, it follows from $(\ref{excludeit2})$ and the bounds on $\varpi$ and $r$ that
\begin{equation}
\label{stobfR}
\left|\zn\right|\le\tilde{C},
\end{equation}
where
\begin{eqnarray*}
\tilde{C}&=&\left[\sqrt{4c'^{-1}(M-\varpi_1)(r'^{-1}-r''^{-1})}+\sup_u\left|\zn(u,v')\right|\right]\cdot\\
	&&\hbox{\ \ }\cdot
		\exp\left[2c^{-1}(r'^{-1}-r''^{-1})(e^2r'^{-1}+M+|\varpi_1|)\right].
\end{eqnarray*}
It is clear from the properties of ${\bf R}$ discussed above
that $\partial{\bf R}\cap\{u<\infty\}\cap\{v>v'\}$ must be a causal curve
which, by condition (b) of our assumptions, must be non-empty and terminate at timelike infinity.
We can choose a sequence of points 
\[
p_i=(u_i,v_i)\in\partial{\bf R}\cap\{u<\infty\}\cap\{v>v'\}
\]
such that $(u_i,v_i)\to(\infty,\infty)$.
and 
either
\begin{equation}
\label{3ava1}
r(u_i,v_i)=r''
\end{equation}
or
\begin{equation}
\label{3ava2}
\varpi(u_i,v_i)=\varpi''.
\end{equation}

Assume first $(\ref{3ava1})$. We will show this leads to a contradiction.
Refer to the figure below:
\[
\includegraphics{avtifasn.eps}
\]
Defining
\[
I_i=[u_i,\infty]\times v_i,
\]
equation $(\ref{3ava1})$ yields
\begin{equation}
\label{aptnmia0}
\int_{I_i}{\nu du}\to r_+-r'',
\end{equation}
as $i\to\infty$.
Since $u\times[v',v_i]\subset{\bf R}$ for $(u,v_i)\in I_i$,
we have the bound
\[
\int_{v'}^{v_i}{\frac{\lambda}{1-\mu}\frac{2}{r^2}\left(\frac{e^2}r-\varpi\right)(u,v^*)dv^*}<
2c^{-1}(r'^{-1}-r''^{-1})(e^2r'^{-1}+M+|\varpi_1|)
\]
on $I_i$.
Integrating $(\ref{nqu})$ in $v$, and then integrating in $u$,
we obtain that
\begin{equation}
\label{aptnvalln0}
\int_{I_i}{(-\nu) du}\le \exp\left[2c^{-1}(r'^{-1}-r''^{-1})(e^2r'^{-1}+M+|\varpi_1|)\right]
\int_{\tilde{I}_i}{(-\nu) du},
\end{equation}
where $\tilde{I}_i$ is $I_i$ translated in the $v$ direction to the 
null segment $v=v'$.

Since the right hand side of $(\ref{aptnvalln0})$ tends to $0$, as $i\to\infty$,
this contradicts $(\ref{aptnmia0})$. Thus, $(\ref{3ava1})$ is impossible.

We must have then $(\ref{3ava2})$.
But this gives
\begin{eqnarray*}
\varpi''-\varpi_+			&\le&	\varpi(u_i,v_i)-\varpi(\infty,v_i)\\
					&=&	\int_{u_i}^\infty\frac12\left(\zn\right)^2(-\nu)(1-\mu)(u,v_i)du\\
					&\le&	\frac12\tilde{C}^2(1+2|\varpi_1|r'^{-1}+e^2r'^{-2})
							\int_{I_i}{(-\nu) du}.
\end{eqnarray*}
Since, by $(\ref{aptnvalln0})$, the right hand side of the above tends to $0$, whereas
the left hand side is a strictly positive number independent of $i$, we immediately
obtain a contradiction.

We have thus excluded $(\ref{excludeit})$ and can assume
in what follows that
there is a sequence
of points $p_i$ on the event horizon for which $1-\mu(p_i)\to0$. But then by the monotonicity
properties of $\varpi$, it follows that in fact $1-\mu(\infty,v)\to0$.
In particular:
\[
\varpi_+=\frac{r_+}2+\frac{e^2}{2r_+}.
\]
This means we have either
\begin{equation}
\label{duvato}
r_+=\varpi_++\sqrt{\varpi_+^2-e^2}
\end{equation}
or 
\begin{equation}
\label{aduvato}
r_+=\varpi_+-\sqrt{\varpi_+^2-e^2}.
\end{equation}

The possibilities $(\ref{duvato})$ and $(\ref{aduvato})$
coincide if and only if $e=\varpi_+$. This would give, however, $r_+=e$, which contradicts condition
(f) in our assumptions. Thus $(\ref{duvato})$ and $(\ref{aduvato})$ are 
necessarily distinct. Note that if $e=0$ we certainly have $(\ref{duvato})$. 
We show now that in general,
$(\ref{aduvato})$ leads to a contradiction.\footnote{If we modify our
initial assumptions by requiring in addition $m\ge0$ on the initial
hypersurface, then $(\ref{duvato})$ follows immediately from
the inequality $\partial_v m\ge0$ in the domain of outer communications.}

In the case ($\ref{aduvato})$, it is clear that we can choose $0<r_2<r_+$, $0<\varpi_2<\varpi_+$ close enough to 
$r_+$, $\varpi_+$, respectively, so that
\[
\frac{e^2}{r_2}-\varpi_2>0,
\]
and such that $r(\infty,v')\ge r_2$, $\varpi(\infty,v')\ge \varpi_2$ for some $v'<\infty$. In what
follows, restrict
attention to $[0,\infty]\times[v',\infty)$.
Fix an $\epsilon>0$. By the inequalities
$1-\mu\ge0$, $r\ge r_2$, it follows that for $r\le r_++\epsilon$, we have 
$\varpi\le\varpi_++\tilde{\epsilon}$, 
where
\[
\tilde{\epsilon}=\epsilon\left[\varpi_++\frac{2e^2(r_++\epsilon)^2\epsilon}{r_+^2r_2^2}\right].
\]
If $\epsilon$ is chosen sufficiently small, we thus have
\begin{equation}
\label{ivo}
\frac{e^2}r-\varpi>\tilde{c},
\end{equation}
in $r\le r_++\epsilon$ for some $\tilde{c}>0$.

Now choose a sequence of points $p_i$ tending to timelike infinity
on the timelike curve $r=r_++\epsilon$, and consider the constant $v$ segment $I_i$
joining these points to the event horizon. (Refer to the previous
figure.) Again, we must have
\begin{equation}
\label{aptnmia}
\int_{I_i}{\nu du}\to-\epsilon.
\end{equation}
On the other hand, integrating equation
$(\ref{nqu})$,
noting that its right hand side is positive in view of $(\ref{ivo})$,
we obtain that 
\begin{equation}
\label{aptnvalln}
\int_{I_i}{(-\nu) du}\le\int_{\tilde{I}_i}{(-\nu) du},
\end{equation}
where $\tilde{I}_i$ is as before $I_i$ translated in the $v$ direction to the $v=v'$
null segment.
Because the right hand side of $(\ref{aptnvalln})$ tends to $0$,
$(\ref{aptnvalln})$ contradicts $(\ref{aptnmia})$. Thus $(\ref{aduvato})$ is impossible.

We are thus in the case $(\ref{duvato})$. Again choosing
$r_3<r_+$, $0<\varpi_3<\varpi_+$ sufficiently close to $r_+$, $\varpi_+$,
respectively,
we can arrange so that
\[
\frac{e^2}{r_3}-\varpi_3=-\tilde{c}'<0,
\]
and $r(\infty,v')\ge r_3$, $\varpi(\infty,v')\ge\varpi_3$ for some $v'$.
Restricting to $[0,\infty]\times[v',\infty)$ it follows from
$(\ref{vumov})$, $(\ref{lamov})$, $(\ref{pumov})$, and $(\ref{pvmov})$
that
\begin{equation}
\label{3A}
\frac{e^2}r-\varpi\le\frac{e^2}{r_3}-\varpi_3=-\tilde{c}'.
\end{equation}

What follows applies to both cases (a) and (a$'$).
Integrating again $(\ref{zex})$,
we obtain
\begin{eqnarray*}
\left|\zn(u,v)\right|	&\le&	\left|\int_{v'}^v{\frac\theta{r} e^{\int_{v^*}^v
							{\frac{\lambda}{1-\mu}
							\frac{2}{r^2}
							\left(\frac{e^2}r-\varpi\right)}}dv^*}
							\right|
							+\left|\zn\right|(u,v')
							e^{\int_0^v{\frac{\lambda}
							{1-\mu}\frac{2}{r^2}\left(\frac{e^2}r
							-\varpi\right)dv^*}}\\
				&\le&	r_1^{-1}\sqrt{\int_{v'}^v{\frac{1-\mu}\lambda\theta^2}}
							\sqrt{\int_{v'}^v{\frac{\lambda}{1-\mu}
							e^{2\int_{v^*}^v{\frac{\lambda}{1-\mu}
							\frac{2}{r^2}\left(\frac{e^2}r-
							\varpi\right)}}}}
							\\
				&	&	\hbox{\ \ }+\left|\zn\right|(u,v')
							e^{\int_0^v{\frac{\lambda}{1-\mu}
							\frac{2}{r^2}\left(\frac{e^2}r-\varpi\right)
							dv^*}}\\
				&\le& 	r_1^{-1}\sqrt{2(M-\varpi_3)}\sqrt{\int_{v'}^v{\frac{\lambda}{1-\mu} 
							e^{-4\tilde{c}'R^{-2}\int_{v^*}^v
							{\frac{\lambda}{1-\mu}}}}}+\sup_u\left|\zn\right|(u,v')\\
				&\le&	\hat{C},
\end{eqnarray*}
where
\[
\hat{C}=\sqrt{2(M-\varpi_3)}\left[2r_1R\sqrt{\tilde{c}}\right]^{-1}+\sup_u\left|\zn\right|(u,v')
\]
This completes the proof. $\Box$		

Finally, we have the following two propositions:
\begin{proposition}
\label{decaystor}
The inequalities  $(\ref{makrua1})$, $(\ref{makrua2})$,
$(\ref{makrua3})$, and $(\ref{2A})$ hold throughout ${\bf D}$,
for some constants $\hat{C},\hat{c}>0$.
\end{proposition}

\noindent\emph{Proof.} The bounds $(\ref{makrua1})$, $(\ref{makrua2})$,
and $(\ref{makrua3})$ applied to the region $r\le R$ follow immediately by integrating
the equations:
\[
\partial_u\phi=\zn\frac{\nu}r,
\]
\[
\partial_u\theta=-\zn\frac\nu{r}\frac\lambda{1-\mu}(1-\mu),
\]
and $(\ref{fdb1})$ (or $(\ref{devallazei1})$ 
in the case of condition (a$'$))
to the future from the curve $r=R$, in view of the
previous two propositions and the bounds $r\ge r_1$, $1-\mu\le1+2|\varpi_1|r_1^{-1}+e^2r_1^{-2}$.
The bound $(\ref{2A})$ applied to $r\ge R$ follows by integrating $(\ref{zex})$ to the right
from $r=R$, in view of the
bounds $(\ref{all1ast})$, $(\ref{pekcokmuhim})$.
$\Box$

\begin{proposition}
\label{taG}
Fix $r_0>r_+$. There exist constants $c$ and $G$, depending on $r_0$, such that
the inequalities 
\begin{equation}
\label{1-mPK}
c\le1-\mu\le1
\end{equation}
\begin{equation}
\label{upo1}
0<G\le\lambda\le 1,
\end{equation}
\begin{equation}
\label{upo2}
-1\le\nu\le-G
\end{equation}
hold in the region $\{r\ge r_0\}\cap\{v\ge v'\}$, where
$v'$ is as defined before $(\ref{3A})$.
As $r_0\to\infty$ then $G$ and $c$ can be chosen arbitrarily close to $1$.
\end{proposition}

\noindent\emph{Proof.} 
The right inequality of $(\ref{1-mPK})$ arises from
\begin{eqnarray*}
1-\mu	& = &	1-\frac{2m}r\\
	& = &	1-\frac{e^2}{r^2}+\frac2r\left(\frac{e^2}r-\varpi\right)\\
	&\le&	1-\frac{e^2}{r^2}-\frac{2\tilde{c'}}r\\
	& < &	1,
\end{eqnarray*}
where for the third inequality we use $(\ref{3A})$.
We note that the last statement of the Proposition follows then immediately from
Proposition \ref{mprop} and integration of $(\ref{nqu})$.


To show the left inequality of $(\ref{1-mPK})$, as well as $(\ref{upo1})$ and $(\ref{upo2})$,
we fix an $\epsilon>0$, and choose
$r_+<r''<r_0$ sufficiently close to $r_+$ so that
\begin{equation}
\label{idiotnta}
1-\frac{2\varpi_*}r+\frac{e^2}{r_*}\le\frac\epsilon{2\hat{C}^2},
\end{equation}
\begin{equation}
\label{idiotnta2}
r_*-\left(\varpi_*-\sqrt{\varpi_*^2-e^2}\right)>\epsilon
\end{equation}
for any
\[
r_*\in[r_+-(r''-r_+),r''], \varpi_*\in[\varpi_+-(r''-r_+)4^{-1}\hat{C}^2,\varpi_+-(r''-r_+)4^{-1}\hat{C}^2].
\]
Let $\tilde{v}$ be such that $r(\infty,\tilde{v})\ge r_+-(r''-r_+)$,
$\varpi(\infty,\tilde{v})\ge \varpi_+-(r''-r_+)4^{-1}\hat{C}^2$,
and consider the region
\[
{\bf X}=\{v\ge\tilde{v}\}\cap\{r\le r''\}.
\]
Integrating $(\ref{puqu})$ from the event horizon,
in view of the right inequality of $(\ref{1-mPK})$, and Proposition , it follows
that for $(u,v)\in{\bf X}$, 
\[
\varpi_+-(r''-r_+)4^{-1}\hat{C}^2\le\varpi(u,v)\le\varpi_+-(r''-r_+)4^{-1}\hat{C}^2
\]
and thus $(\ref{idiotnta})$ holds.
Integrating $(\ref{puqu})$ again, but using $(\ref{idiotnta})$ we obtain 
\begin{equation}
\label{giaparakatw}
|\varpi(u,v)-\varpi_+|\le\frac\epsilon2(r''-r_+).
\end{equation}

Consider the quantity $\varpi+\sqrt{\varpi^2-e^2}-r(u,v)$.
for $(u,v)$ on the curve $r=r''$.
We have
\begin{eqnarray}
\nonumber
\label{ammeta}
\varpi+\sqrt{\varpi^2-e^2}-r	& = &	\varpi-\varpi_++\sqrt{\varpi^2-e^2}-\sqrt{\varpi_+^2-e^2}-r+r_+\\
				&\le&	K|\varpi-\varpi_+|-(r''-r_+)\\
				&\le&	\left[K\frac\epsilon2-1\right](r''-r_+)
\end{eqnarray}
where for the last inequality we have used $(\ref{giaparakatw})$.
Thus for $r(u,v)=r''$, we have
\begin{eqnarray*}
1-\mu(u,v)	& = &	\frac{1}{r^2}\left[r-\left(\varpi+\sqrt{\varpi^2-e^2}\right)\right]
					\left[r-\left(\varpi-\sqrt{\varpi^2-e^2}\right)\right]\\
		&\ge&	r''^{-2}\left[1-K\frac\epsilon2\right](r''-r_+)\epsilon
\end{eqnarray*}
where for the last inequality we have used $(\ref{idiotnta2})$ and $(\ref{ammeta})$.
Choosing $\epsilon>0$ sufficiently small, the above gives a bound
\begin{equation}
\label{kalo1-m}
1-\mu(u,v)\ge\epsilon'>0
\end{equation}
for all $(u,v)$ on $r=r''$.

>From the bound $(\ref{kalo1-m})$ together with the lower bound on $\frac\lambda{1-\mu}$ from Proposition
\ref{decaystor},
we obtain, on the curve $r=r''$, a bound
\[
\lambda=\frac{\lambda}{1-\mu}\cdot{1-\mu}\ge\hat{c}\epsilon'.
\]
But from the equation $(\ref{lqu})$ and $(\ref{3A})$, it follows that $\partial_u\lambda\le0$.
Thus, we have
$(\ref{upo1})$ in $r\ge r''$ as long as $G\le\hat{c}\epsilon'$.
>From this, we immediately derive
\[
1-\mu=\frac{1-\mu}{\lambda}\cdot\lambda\ge\hat{c}\epsilon'
\]
for $r\ge r''$,
which yields the left inequality of $(\ref{1-mPK})$.
Finally, $(\ref{upo2})$ follows after integration of
$(\ref{nqu})$ from $v=\infty$, in view of $(\ref{1-mPK})$.


\section{The $v^p$ rectangles}
The following propositions will refer to a rectangle whose dimensions
are related to a constant $D$. In Section 8, the propositions of this section
will be applied to a sequence of rectangles, whose top right endpoints $(\infty,v_i)$
will approach the point at infinity on the event horizon, and for which $D\sim v_i^p$,
for some $0<p<1$. 

In the remainder of this paper, we shall always assume that $v\ge v'$ for some
$v'$, in particular, so that $(\ref{3A})$ holds.
When we say in what follows that a constant $C_3$ depends ``only'' on 
given constants $C_1$ and $C_2$,
this is meant \emph{in addition} to the constants $C$ of bounds $(\ref{byil})$ and $(\ref{ligoduvato})$,
$\hat{C}$ and $\hat{c}$ of Proposition~\ref{decaystor}, $\tilde{c}'$ of $(\ref{3A})$, and
$r_3$, $M$, and $\varpi_3>0$. 



\begin{proposition}
\label{aptovoriz}
Let $D$ be any constant $D\ge1$. Consider a characteristic rectangle as depicted:
\[
\includegraphics{claim.eps}
\] 
Assume the $v$-lengths of all $3$ differently-shaded
regions are each contained in the interval $[LD,L'D]$ for constants
$0<L<1<L'$, and that $r=r_0$ is
a timelike curve bounding the lightest-shaded region.
Assume, moreover, that
\begin{equation}
\label{stageg}
\int{\theta^2dv}\le AD^{-\rho}
\end{equation} 
on
the top left edge, for some constant $A>0$.  
Finally, assume that in the shaded regions we have
\[
\triangle r\le\epsilon
\]
for some $\epsilon$.
(Here $\triangle f=\sup{|f(p_1)-f(p_2)|}$ over all points
$p_1$, $p_2$ in the shaded regions.)
Then, 
for $\epsilon$ small enough,
\begin{equation}
\label{apot1}
|\zeta|\le \tilde{C}D^{-\frac\rho2}
\end{equation}
in the two lighter-shaded regions,
and
\begin{equation}
\label{apot2}
\triangle \varpi\le \tilde{C}D^{-\rho},
\end{equation}
where $\tilde{C}$ is a constant depending only on
$r_0$, $\epsilon$, $A$, $L$, and $L'$.
\end{proposition}
\noindent\emph{Proof.} We first derive estimates  for
points $Q$ in the left-most, darkest-shaded region.

For points $Q$, we are content
with showing that $\int_b^Q{\theta^2dv}\le A'D^{-\rho}$ for some constant $A'$.
We know that $\int{|\nu|}\le\epsilon$ on the left segment
of the lightest-shaded region. Denote the constant $v$-value of
this segment by $\tilde{v}$, and let $Q=(u,v)$.
>From $(\ref{nqu})$, we obtain 
\begin{eqnarray*}
\int_u^\infty{|\nu(u^*,v)|du^*}	& = &	\int_u^\infty{|\nu|(u^*,\tilde{v})e^
						{\int_v^{\tilde{v}}{
						\frac{\lambda}{1-\mu}\frac2{r^2}
						\left(\frac{e^2}r-\varpi\right)}du^*}}\\
				&\le& 	\int_u^\infty{e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}|\nu|(u^*,\tilde{v})du^*}\\
		  		&\le&	e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}\epsilon.
\end{eqnarray*}
Since we know that $\left|\zn\right|$ is bounded by Proposition \ref{decaystor}, and
in addition $1-\mu\le1$, it follows
that
\begin{eqnarray*}
\varpi(Q)-\varpi(q)	&\le&	\frac12\int_Q^q{\left(\zn\right)^2(1-\mu)(-\nu) du}\\
			&\le& 	\hat{C}^22^{-1}
					\epsilon e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}.
\end{eqnarray*}
The same estimate holds of course for $\varpi(b)-\varpi(a)$.
But then 
\begin{eqnarray*}
\frac12\int_b^Q{\theta^2dv}	&\le&	\varpi(Q)-\varpi(b)\\
				& = &	(\varpi(Q)-\varpi(q))+(\varpi(q)-\varpi(a))
							+(\varpi(a)-\varpi(b))\\
				&\le&	(\varpi(Q)-\varpi(q))+(\varpi(q)-\varpi(a))\\
				&\le&	\hat{C}^22^{-1}
						\epsilon e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}
						+(2\hat{c})^{-1}D^{-\rho}\\
				&\le&	A'D^{-\rho},
\end{eqnarray*}
as desired.

For points $P$ in one of the other two shaded regions,
we will derive a pointwise bound for $\zn$.
Integrating the equation:
\begin{equation}
\label{zne3is}
\partial_v\left(\zn\right)=
-\frac{\theta}r+\zn\left[\frac\lambda{1-\mu}\frac2{r^2}\left(
\frac{e^2}r-\varpi\right)\right]
\end{equation}
we obtain
\[
\zn(P)=
\int_c^P{-\frac{\theta}r(u,v)e^{\int_v^P
\frac\lambda{1-\mu}\frac2{r^2}\left(
\frac{e^2}r-\varpi\right)}dv}
+\zn(c)e^{\int_c^P
\frac\lambda{1-\mu}\frac2{r^2}\left(
\frac{e^2}r-\varpi\right)}.
\]
Thus,
\begin{eqnarray*}
\left|\zn\right|(P)	&\le&	r_1^{-1}\sqrt{\int_c^P\theta^2}
						\sqrt{\int_c^P{e^{2\int_v^P
						\frac\lambda{1-\mu}\frac2{r^2}\left(
						\frac{e^2}r-\varpi\right)}}}
						+\hat{C}e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}\\
			&\le&	r_1^{-1}\sqrt{\int_c^P\theta^2}\sqrt{\int_c^P
						{e^{-{4\tilde{c}'\hat{c}r_0^{-2}}
						(v_P-v)}dv}}+\hat{C}e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}\\
			&\le&	r_1^{-1}\sqrt{\int_c^P\theta^2}\sqrt{({4\tilde{c}'\hat{c}r_0^{-2}})^{-1}
						[1-e^{-{4\tilde{c}'\hat{c}r_0^{-2}}LD}}]
						+\hat{C}e^{-2\tilde{c}'\hat{c}r_0^{-2}LD}\\
			&\le&	r_0\left[2r_1\sqrt{\tilde{c}'\hat{c}}\right]^{-1}	
						\sqrt{\int_c^P\theta^2}+\hat{C}e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}.
\end{eqnarray*}
The notation $v_P$ above represents the $v$ value of the point $P$.
On the other hand we have that
\begin{eqnarray*}
\frac12\int_c^P\theta^2	&\le&	\varpi(P)-\varpi(c)\\
				&\le&	(\varpi(P)-\varpi(p))+(\varpi(p)-\varpi(a))\\
				&\le&	\int_P^p{\left(\zn\right)^2(1-\mu)(-\nu) du}
							+A(2\hat{c})^{-1}D^{-\rho}\\
				&\le&	\epsilon\left[\sup_{I}\left
						(\zn\right)^2\right]
							+A(2\hat{c})^{-1}D^{-\rho},
\end{eqnarray*}
where $I$ here denotes the constant $v$ segment joing $p$ and $P$.
Putting these two estimates together yields
\begin{eqnarray*}
\left|\zn\right|(P)	&\le&	r_0\left[2r_1\sqrt{\tilde{c}'\hat{c}}\right]^{-1}\sqrt{\epsilon\sup_{I}
						\left(\zn\right)^2
						+A(2\tilde{c})^{-1}D^{-\rho}}\\
			&   &	\hbox{\ \ }+\hat{C}
						e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}\\
			&\le&	r_0\left[2r_1\sqrt{\tilde{c}'\hat{c}}
						\right]^{-1}\left[\sqrt{\epsilon}\sup_{I}\left|\zn\right|
						+\sqrt{A(2\tilde{c})^{-1}}D^{-\frac\rho2}\right]\\
			&   & 	\hbox{\ \ }+\hat{C}e^{-{2\tilde{c}'\hat{c}r_0^{-2}}LD}\\
			&\le&	\tilde{\epsilon}\sup_{I}\left|\zn\right|
						+C'D^{-\frac\rho2},
\end{eqnarray*}
where
\begin{equation}
\label{tildeorism}
\tilde{\epsilon}=\sqrt{\epsilon}\left[r_0\left(2r_1\sqrt{\tilde{c}'\hat{c}}\right)^{-1}\right].
\end{equation}
Now clearly, the above estimate also applies to
$\sup_{I}\left|\zn\right|$, i.e.~we have
in addition
\[
\sup_{I}\left|\zn\right|
\le
\tilde{\epsilon}\sup_{I}\left|\zn\right|
+C'D^{-\frac\rho2}
\]
and thus,
\begin{equation}
\label{divei}
\sup_{I}\left|\zn\right|
\le\frac{C'D^{-\frac\rho2}}{1-\tilde{\epsilon}}.
\end{equation}
This gives in particular $(\ref{apot1})$ with $\tilde{C}=2C'$, as long
as $\epsilon$ is chosen small enough so that the right hand side of $(\ref{tildeorism})$ is
less than $\frac12$. 
The bound $(\ref{apot2})$ follows immediately by integrating $(\ref{puqu})$ using
$(\ref{divei})$, together with the bound $(\ref{stageg})$.
$\Box$

In the proof of the next proposition, we recall the constant $G$, $G=G(r_0)$,
defined in Proposition \ref{taG}.
\begin{proposition}
\label{metobeta}
Let $R\ge r_0>r_+$, and suppose
that on the portion of the curve $r=R$ enclosed in a characteristic rectangle ${\bf X}$
depicted as the larger rectangle below:
\[
\includegraphics{claim2.eps}
\]
we have the bound
\[
|\zeta|\le \bar{A},
\]
as well as the bound
\[
\triangle_R \varpi\le \bar{A}^2R,
\]
for some positive constant $\bar{A}$,
where 
\[
\triangle_R \varpi=\sup_{p_i\in\{r\le R\}\cap{\bf X}}{|\varpi(p_1)-\varpi(p_2)|}.
\]
Then, there is a constant $\tilde{\beta}>1$, depending
only on $r_0$, such that in 
\[
\{R\le r\le\tilde\beta R\}\cap{\bf X}
\]
we have the bound
\[
|\zeta|\le 3\bar{A},
\]
and
\[
\triangle_{\tilde{\beta}R} \varpi\le 9 \bar{A}^2\tilde{\beta}R.
\]
\end{proposition}

\noindent\emph{Proof.} Let $P$ be any point in $R\le r\le \tilde\beta R$.
Integrating the equation
\[
\partial_v\zeta=-\frac{\theta}r\nu
\]
from $c$
 we obtain
\begin{eqnarray*}
|\zeta|(P)	&\le&	\int_c^P{\left|\frac{\theta\nu}{r}\right|}+
					|\zeta(c)|\\
		&\le&	\sqrt{G^{-1}}\sqrt{\int_c^P{\theta^2dv}}
					\sqrt{\int_c^P{\frac{\lambda}{r^2}dv}}
					+\bar{A}\\
		&\le&	\sqrt{G^{-1}}\sqrt{\int_c^P{\theta^2dv}}\sqrt{
					\frac{\tilde{\beta}-1}{\tilde{\beta}R}}
					+\bar{A}.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\frac12\int_c^P{\theta^2dv}	&\le&	\varpi(P)-\varpi(c)\\
					& = &	(\varpi(P)-\varpi(p))+
								(\varpi(p)-\varpi(c))\\
					&\le& 	\int_P^p{\left(\zn\right)^2
								(1-\mu)(-\nu) du}
								+R \bar{A}^2\\
					&\le&	\left[\sup_I
								{\left(\zn\right)^2}\right]
								\int_P^p{|\nu| du}+R\bar{A}^2\\
					&\le&	(\tilde{\beta}-1)R\left[\sup_I
								{\left|\zn\right|}\right]^2+
								R\bar{A}^2,
\end{eqnarray*}
where $I$ here denotes the constant-$v$ interval from $p$ to $P$.
Combining this with the previous, we obtain
\begin{eqnarray}
\nonumber
\label{pavekt}
|\zeta|(P)	&\le&	\sqrt{G^{-1}}\sqrt{2(\tilde{\beta}-1)
					R\left[\sup_I
					{\left|\zn\right|}\right]^2+2R\bar{C}^2}
					\sqrt{\frac{\tilde{\beta}-1}{\tilde{\beta}R}}
					+\bar{A}\\
		&\le&	\frac12\sup_{I}{|\zeta|}+
					\frac{\bar{A}}2
					+\bar{A},
\end{eqnarray}
where to obtain the first two terms on the right in the latter inequality,
we have chosen $\tilde{\beta}>1$ such that
\begin{equation}
\label{evab}
\tilde{\beta}-1\le\frac{G}{8}.
\end{equation}
Since the estimate $(\ref{pavekt})$ is clearly also valid when one
replaces $|\zeta|$ with $\sup_{I}{|\zeta|}$,
we obtain:
\[
\sup_{I}{|\zeta|}\le
\frac12\sup_{I}{|\zeta|}
+\frac{\bar{A}}{2}
+\bar{A},
\]
and
thus
\[
\sup_{I}{|\zeta|}
\le
3\bar{A}.
\]

For $\tilde{p}$ in $R\le r\le\tilde{\beta}R$, we have by
the above
\begin{eqnarray*}
\frac12\int_{\tilde{p}}^{p}{\left(\zn\right)^2(-\nu)(1-\mu)du}&\le&
		\frac12G_1^{-2}9 {\bar{A}}^2R(\tilde{\beta}-1)\\
	&\le&	8\tilde{\beta}\bar{A}^2R,
\end{eqnarray*}
where the second estimate follows provided that $\tilde{\beta}$ is chosen to satisfy,
in addition to $(\ref{evab})$, the inequality
\[
\tilde{\beta}-1\le\frac{16G^2}{9}.
\]
Thus we have
\begin{eqnarray*}
\triangle_{\tilde{\beta} R}\varpi	&\le&	\triangle_R\varpi+8\tilde{\beta}\bar{A}^2R\\
					&\le&	\tilde{C}^2R+
								8\tilde{\beta}\bar{A}^2R\\
							&\le&	9\bar{A}^2\tilde{\beta}R,
\end{eqnarray*}
and this completes the proof of the Proposition. $\Box$

\begin{proposition}
\label{sautatelos}
Under the assumptions of Proposition \ref{aptovoriz}, where in addition we require $r_0>r_+$,
we have
\[
\left|\zn\right|\le\hat{C}'r^\alpha D^{-\frac \rho2},
\]
everywhere in the rectangle, outside the darkest-shaded region,
for some constant $\hat{C}'>0$ depending only on $r_0$, $\epsilon$, $A$, $L_1$ and $L_2$,
where
\[
\alpha=\frac{\log3}{\log{\tilde{\beta}}}.
\]
\end{proposition}

\noindent\emph{Proof.} The assumptions of Proposition \ref{metobeta} hold on $r=r_0$,
from the result of Proposition \ref{aptovoriz}, where 
\[
\bar{A} =\max\left\{\tilde{C}D^{-\frac\rho2},\sqrt{\tilde{C}r_0^{-1}}D^{-\frac\rho2}\right\}.
\] 
Denote $r_k=\tilde{\beta}^kr_0$.
By iterating the result of Proposition \ref{metobeta}, one obtains
for points with $r_{k-1}\le r(u,v)\le r_k$, a bound 
\[
|\zeta (u,v)|\le3^k\max\left\{\tilde{C},\sqrt{\tilde{C}r_0^{-1}}\right\}
D^{-\frac \rho2}.
\]
From
\[
k=\frac{\log r_k-\log r_0}{\log \tilde\beta},
\]
we have
\begin{eqnarray*}
\left|\zn(u,v)\right|	&\le&	G^{-1}3^{-\log r_0(\log\tilde{\beta})^{-1}}
					\max\left\{\tilde{C},\sqrt{\tilde{C}r_0^{-1}}\right\}3^{(\log\tilde\beta)^{-1}
					\log r_k}D^{-\frac \rho2}\\
			&\le&	G^{-1}3^{-\log r_0(\log\tilde{\beta})^{-1}}
					\max\left\{\tilde{C},\sqrt{\tilde{C}r_0^{-1}}\right\}
					e^{\log r_k^{(\log 3)(\log\tilde\beta)^{-1}}}
					D^{-\frac \rho2}\\
			&\le&	G^{-1}3^{-\log r_0(\log\tilde{\beta})^{-1}}
					\max\left\{\tilde{C},\sqrt{\tilde{C}r_0^{-1}}\right\}r_k^{\frac{\log 3}
					{\log\tilde\beta}}D^{-\frac \rho2}\\
			&\le&	\hat{C}'r^{\frac{\log 3}
					{\log\tilde\beta}}D^{-\frac \rho2}.
\end{eqnarray*}
This completes the proof.
$\Box$?



\section{Spreading decay}
In the previous section we have shown how to relate the flux on a segment of the event horizon
to a pointwise estimate in the interior of a rectangle. In this section we will also 
show that certain bounds on certain boundary segments of a characteristic rectangle 
yield estimates in their interior. The estimates here will be of a very different character.
In the next section, the estimates of this and the previous section will be combined
in the context of an induction argument to obtain decay for $\phi$ in $v$.

\begin{proposition}
\label{sprpro}
Let $R>r_+$, and suppose we have a characteristic
rectangle $[u_1,u_2]\times[v_1,v_2]$, such that $r(u_2,v_1)=R$:
\[
\includegraphics{sprop1.eps}
\]
Let $A$ and $B$ be two chosen constants and suppose we have a bound 
\begin{equation}
\label{3ast}
|\theta|(u,v_1)\le A
\end{equation}
for all $u\in[u_1,u_2]$, and 
bounds
\begin{equation}
\label{kaiv1}
|\phi\lambda+\theta|(u_1,v)\le B
\end{equation}
\begin{equation}
\label{kaiv2}
|\phi|(u_1,v)\le A
\end{equation}
for $v_1\le v\le v_2$,
and 
\begin{equation}
\label{kaiv3}
|\phi|(u,v_2)\le A,
\end{equation}
for all $u_1\le u\le u_2$.
It follows that there exist constants $\kappa_1,\kappa_2>0$, depending
only on $R$,
such that
\begin{equation}
\label{phifragma}
|\phi|(u,v)\le\kappa_1[\max\{A,B\}],
\end{equation}
\begin{equation}
\label{plfragma}
|\phi\lambda+\theta|(u,v)\le B+\kappa_1r^{-1}(u,v)\max\{A,B\},
\end{equation}
\begin{equation}
\label{thetafragma}
|\theta|(u,v)\le\kappa_1[\max\{A,B\}]
\end{equation}
for all points $(u,v)$
with $u_1\le u\le u_2$, $v_1\le v\le v_2$,
\end{proposition}
and 
\begin{equation}
\label{TOP-flux}
\varpi(u_2,v_2)-\varpi(u_2,v_1)
\le\kappa_2\left\{[\max\{A,B\}]^2+B^2r(u_2,v_2)\right\}
\end{equation}
\noindent\emph{Proof.}
We first show that the bound
\begin{equation}
\label{prwtaauto}
|\phi|\le\kappa'\max\{A,B\}
\end{equation}
holds throughout the rectangle for some sufficiently large
constant $\kappa'>1$. 
For this, define the number $u_*\ge u_{1}$ given by
\[
u_*=\sup_{u_1\le u\le u_2}
\left\{|\phi(u',v)|<\kappa'\max\{A,B\}
{\rm\ for\ all\ }v\in[v_1,v_2],u'\in[u_1,u].
\right\}
\]
(Note that if $\kappa'>1$ then the set referred to on the right is non-empty,
and $u_*$ is in fact necessarily strictly greater that $u_{1}$ by continuity of $\phi$.)

If $u_*$=$u_2$ then clearly $(\ref{prwtaauto})$ holds throughout the rectangle,
and we are done.
Otherwise, $u_*<u_2$, and there exists $v_*\in[v_1,v_{2}]$ such that
\[
|\phi(u_*,v_*)|=\kappa'\max\{A,B\}.
\]

Consider first the possibility that $v_*\in(v_1,v_2)$:
\[
\includegraphics{dev0.eps}
\]
Since $|\phi(u_*,v)|\le\kappa'\max\{A,B\}$ for all $v\in[v_1,v_2]$,
it follows that
\[
\partial_v\phi(u_*,v_*)=0.
\]
Integrating now
\begin{equation}
\label{toallo}
\partial_u(\phi\lambda+\theta)=-\phi\frac2{r^2}\frac{\lambda\nu}{1-\mu}
\left(\frac{e^2}r-\varpi\right)
\end{equation}
on the dotted line segment depicted, where $p=(u_*,v_*)$,
we obtain that
\begin{eqnarray}
\nonumber
\label{palitob}
|(\phi\lambda+\theta)(p)|	&\le&	B+\frac{2}{r(p)}[e^2r_1^{-1}+M]
					\kappa'\max\{A,B\}\\
				&\le&	\frac12 \kappa'\max\{A,B\}
\end{eqnarray}
where here we are using the fact that $r\ge R$ and $R$ and $\kappa'$ have
been chosen sufficiently
large. 		 	
But now we note that at $p$, $\theta=r^{-1}\partial_v\phi=0$ so
\begin{eqnarray*}
|\phi|(p)	& = &	\lambda^{-1}|\phi\lambda+\theta|\\
		&\le&	\frac12G^{-1}\kappa'\max\{A,B\}\\
		& < &	 \kappa'\max\{A,B\},
\end{eqnarray*}
where again, for the last inequality, we use that $R$ has been chosen sufficiently large.
But this is a contradiction.

Since $v_*=v_2$ would contradict $(\ref{kaiv3})$ for $\kappa'>1$,
we clearly must have $v^*=v_1$.
Integrating $(\ref{toallo})$
as above, 
we again obtain the bound $(\ref{palitob})$ for $|\phi\lambda+\theta|(u_*,v_i)$. Thus,
we have
\begin{eqnarray*}
|\phi(u_*,v_i)|	&\le&	\lambda^{-1}|\phi\lambda+\theta|(u_*,v_i)+
					\lambda^{-1}|\theta|(u_*,v_i)\\
		&\le&	G^{-1}\frac12 \kappa'\max\{A,B\}
					+G^{-1}A\\
		& < &	 \kappa'\max\{A,B\},
\end{eqnarray*}
which is a contradiction. Note that to derive the second inequality above we use
$(\ref{3ast})$ and to derive the third we require $\kappa'$ to be sufficiently large.
Thus we have established $(\ref{prwtaauto})$.

Integrating $(\ref{toallo})$ from $u=u_1$, we obtain
\begin{equation}
\label{varimiz}
|\phi\lambda+\theta|(u,v)	\le	B+r^{-1}2[e^2r_1^{-1}+M]\kappa'\max\{A,B\}.
\end{equation}
>From $(\ref{varimiz})$, $(\ref{plfragma})$ follows for sufficiently large $\kappa_1$.

Integrating the equation
\begin{equation}
\label{e3is}
\partial_v(r\phi)=\phi\lambda+\theta
\end{equation}
along $u_2\times[v_1,v_2]$, from left to right,
we obtain, for $v\in[v_i,v_{i+1}]$,
\begin{eqnarray}
\label{prwtoprob}
\nonumber
|r\phi(u_2,v)|	&\le&	R\kappa_1\max\{A,B\}+BG_1^{-1}(r-R)\\
		&   &	\hbox{\ \ }+
					(\log(r/R))G_1^{-1}\kappa'\max\{A,B\}[2M]\\
		&\le&	 \kappa''\max\{A,B\}(\log{r})+Br
\end{eqnarray}
and
thus
\begin{equation}
\label{privtosuv}
|\phi(u_2,v)|\le  \kappa''\left[\max\{A,B\}r^{-1}\log r+B\right],
\end{equation}
and this yields $(\ref{phifragma})$ for sufficiently large $\kappa_1$.
In view now of $(\ref{varimiz})$ and $(\ref{privtosuv})$ it follows
by the triangle inequality and $\lambda\le G^{-1}$ that
\begin{equation}
\label{suvepagetai}
|\theta(p)|\le \kappa'''\left[\max\{A,B\}r^{-1}\log r+B\right],
\end{equation}
and this yields $(\ref{thetafragma})$ for sufficiently large $\kappa_1$.


>From $(\ref{suvepagetai})$ and $(\ref{pvqu})$ we obtain finally the ``energy'' estimate
\begin{eqnarray*}
\varpi(u_2,v)-\varpi(u_2,v_1)	&\le&	\frac12\hat{c}^{-1}\int_{v_i}^v{\theta^2(u_2,v^*)dv^*}\\
				&\le&	\frac12\hat{c}^{-1}G_1^{-1}\kappa'''
						\left[\left[\max\{A,B\}\right]^2\right.\\
				&   &	\hbox{\ \ }\left.\cdot
						\int_{v_1}^v{r^{-2}\log^2 r(u_1,v^*)\lambda dv^*}
						+B^2r(u_2,v)\right]\\
				&\le&	\kappa_2\left[\left[\max\{A,B\}\right]^2+B^2r(u_2,v)\right],
\end{eqnarray*}
for $\kappa_2$ suitably large. This yields $(\ref{TOP-flux})$.
$\Box$

\begin{proposition}
\label{piokovtaekei}
Let $r_*>r_0$ be given and consider a characteristic rectangle $[u_1,u_2]\times[v_1,v_2]$
\[
\includegraphics{sprop2.eps}
\]
and assume
that for $u\in[u_1,u_2]\cap\{r(u,v_2)\ge r_*\}$
\begin{equation}
\label{alavaf}
|\phi|(u,v_2)\le A,
\end{equation}
and
\begin{equation}
\label{ekt-ev}
\varpi(u_1,v_2)-\varpi(u_2,v_1)\le A^2.
\end{equation}
It follows that there exists a constant $\kappa$ depending only on $r_*$,
such that for $(u,v)\in[u_1,u_2]\times[v_1,v_2]\cap\{r\ge r_*\}$
\begin{equation}
\label{allottet}
|\phi|(u,v)\le\kappa A.
\end{equation}
Assuming in addition that 
\begin{equation}
\label{prosupo}
\left|\zn\right|(u,v_1)\le A,
\end{equation}
we obtain that
\begin{equation}
\label{pros9apo1}
\left|\zn\right|(u,v)\le\kappa A,
\end{equation}
\begin{equation}
\label{pros9apo2}
|\phi|(u,v)\le\kappa A,
\end{equation}
in $[u_1,u_2]\times[v_1,v_2]\cap\{r\le r_*\}$.
\end{proposition}

\noindent\emph{Proof.}
It follows from $(\ref{pumov})$, $(\ref{pvmov})$, and $(\ref{alavaf})$
that for any two points $p_1$ and $p_2$ in the rectangle,
we have $|\varpi(p_1)-\varpi(p_2)|\le A^2$,
and thus in particular
\begin{equation}
\label{EnEs}
\frac12\int{\frac{1-\mu}{\lambda}\theta^2 dv}\le A^2,
\end{equation}
\[
\frac12\int{\left(\zn\right)^2(1-\mu)(-\nu) du}\le A^2,
\]
where
where the integrations can be carried out over any
constant $u$ and $v$ segments, respectively, contained in the
rectangle.

Integrating the equation
\[
\partial_v\phi=\frac\theta{r}
\]
from $v=v_{2}$, i.e.~from right to left,
using Schwarz inequality, and the bound $(\ref{alavaf})$,
we obtain that for $(u,v)\in\{r\ge r_*\}$
\begin{eqnarray*}
|\phi(u,v)|	&\le&	|\phi(u,v_2)|+\sqrt{\int_v^{v_2}\theta^2\frac{1-\mu}{\lambda}dv}
					\sqrt{\int_v^{v_2}\frac{\lambda}{1-\mu}\frac{1}{r^2}dv}\\
		&\le&	A+
					\sqrt{2}Ac^{-1}
					\sqrt{\int_v^{v_2}\frac{\lambda}{r^2}}\\
		&\le&	\kappa A,
\end{eqnarray*}
and this yields $(\ref{allottet})$. 

Now consider the additional assumption $(\ref{prosupo})$. For $r\le r_*$,
we will use the ``red-shift'' technique. Integrating $(\ref{zex})$, we obtain
\begin{eqnarray*}
\left|\zn\right|(u,v)	&\le&	\sqrt{\int_{v_1}^{v}\theta^2\frac{1-\mu}{\lambda}}\sqrt
					{\int_{v_1}^v{\frac\lambda{1-\mu}
					\frac{1}{r^2}e^{-2\hat{c}\tilde{c}'r_*^{-2}(v-v')}dv'}}\\
			&   &	\hbox{\ \ }+\left|\zn\right|(u,v_1)\\
			&\le&	\kappa'\sqrt{A^2}+A\\
			&\le&	\kappa A
\end{eqnarray*}	
for sufficiently large $\kappa>1$. Thus we have $(\ref{pros9apo1})$. Integrating the equation
$\partial_u\phi=\zn\frac{\nu}r$ from $r=R$,
we immediately obtain $(\ref{pros9apo2})$.
$\Box$

\begin{proposition}
\label{arketamegaln}
Suppose $R$ is a sufficiently large constant, and let $r(u_1,v_1)=R$:
\[
\includegraphics{sprop3.eps}
\]
Consider the region 
\[
{\bf X}=[u_1,u_2]\times[v_1,v_2]\cap\{r\ge R\},
\]
and suppose that 
\[
|\phi|\le A
\]
on ${\bf X}\cap\{r=R\}$, and 
\[
|\phi|(u,v_2)\le A
\]
\[
|\phi\lambda+\theta|(u_1,v)\le B
\]
for $v_1\le v\le v_2$, and $u_1\le u\le u_2$. 
Then there exists a constant $\kappa>1$ depending only on
$R$ such that
\begin{equation}
\label{evakap}
|\phi|(u,v)\le\kappa\left[\max\{A,B\}r^{-1}R+B\right],
\end{equation}
\begin{equation}
\label{evakap2}
|\phi\lambda+\theta|(u,v)\le\kappa\left[B+r^{-2}\log r\max\{A,B\}\right],
\end{equation}
for all $(u,v)\in{\bf X}$.
\end{proposition}

\noindent\emph{Proof.}
We first note that by integrating the equation $\partial_v(r\phi)=\phi\lambda+\theta$
along $u_1\times[v_1,v]$, we obtain
\[
|r\phi(u_1,v)|\le RA+G^{-1}(r(u_1,v)-R)B,
\]
and thus
\[
|\phi(u_1,v)|\le \kappa'B+A
\]
for $\kappa'$ sufficiently large.
Applying the argument of Proposition \ref{sprpro}
we immediately obtain
\begin{equation}
\label{GT}
|\phi(u,v)|\le \kappa''\max\{A,B\}
\end{equation}
for all $(u,v)\in{\bf X}$.

Integrating now $(\ref{toallo})$ with the bound $(\ref{GT})$, 
we obtain
\[
|\phi\lambda+\theta|\le B+\kappa''\max\{A,B\}r^{-1}[2M].
\]
Integrating  $(\ref{e3is})$ with the above bound, from $r=R$, we obtain
\[
|r\phi|\le RA+B(r-R)+G^{-1}\kappa''\max\{A,B\}\log (r/R)[2M],
\]
so 
\[
|\phi|\le B+\kappa'''\max\{A,B\}r^{-1}\log r.
\]
Once again integrating $(\ref{toallo})$, we obtain
\[
|\phi\lambda+\theta|\le B+Br^{-1}+G^{-1}\kappa'''r^{-2}\log (r/R)\max\{A,B\}[2M].
\]
This yields $(\ref{evakap2})$ for sufficiently large $\kappa$.
Integrating yet again $(\ref{e3is})$ yields
\[
|r\phi|\le AR+G^{-1}B(r-R)+G^{-1}B\log r+\kappa''''\max\{A,B\}.
\]
Dividing by $r$, we finally obtain,
\[
|\phi|\le \kappa[B+r^{-1}R\max\{A,B\}],
\]
for large enough $\kappa$.
This completes the proof. $\Box$


\section{Decay by induction}
All the tools necessary to extract $v$-decay for $\phi$ and $\theta$ are in place.
What remains is to put everything together. As discussed in Section 4,
the key here is the global geometry of the event horizon. 
The boundedness of the energy flux of $\phi$ through the
event horizon, together with pigeon-hole principle, will allow us to erect
a sequence of rectangles, with top left edge on the event horizon, such that
the assumptions of Section 6 hold with $D=v_i^p$, where $v_i$ is the $v$-coordinate
of the top right endpoint, and such that 
\begin{equation}
\label{sim}
v_i-v_{i-1}\sim v_i.
\end{equation}
The estimates derived from applying the results of Section 6
will then be applied to
the propositions of the
previous section to show that decay spreads
in particular to the whole event horizon.

As explained in the Section 4, the argument in this section in fact proceeds by induction,
as the above process must be repeated many times. At each step, the rate of decay is improved
by a small amount. There is a limit, however, to how much the decay can be improved,
and this limit ultimately derives from the initial $r$ decay of $\phi\lambda+\theta$.
In this section, however, we will in fact meet an earlier obstruction, which will force
us to restrict to $v^{-\frac32}$ decay on the event horizon. This obstruction is
related to our choice $(\ref{sim})$. Going beyond this decay rate will require a 
modification of our argument, which we defer to the next section.

The result of this section is the following
\begin{theorem}
\label{3/2}
There exist constants $C_{3/2}>0$, $\delta_{3/2}>0$, such that
in the region $r\le\delta_{3/2} v$, we have
\begin{equation}
\label{+}
|\phi|\le C_{3/2}\left[v^{-\frac32}r^{-1}+v^{-2}\right],
\end{equation}
\begin{equation}
\label{+++}
|\theta|\le C_{3/2}\left[v^{-\frac32}r^{-1}+v^{-2}\right],
\end{equation}
and in addition: 
\[
\int_v^\infty\theta^2(\infty,v^*)dv^*\le C_{3/2}v^{-3}.
\]
\end{theorem}

\noindent\emph{Proof.}
A series of real numbers $w_k$ will be defined inductively.
Given $w_k$, define the statement $S_k$ as follows:
\begin{enumerate}
\item[($S_k$)]
There exist
constants $\bar{C}_k, \delta_k>0$, such that
for $r\le\delta_k v$,
\begin{equation}
\label{22y}
|\phi|\le \bar{C}_k\left[v^{-w_k}r^{-1}+ v^{-2}\right],
\end{equation}
\begin{equation}
\label{33y}
|\theta|\le \bar{C}_k\left[v^{-w_k}r^{-1}+v^{-2}\right],
\end{equation}
\begin{equation}
\label{11y}
\int_{v}^{\infty}\theta^2(\infty,v^*)dv^*\le\bar{C}_kv^{-2w_k}.
\end{equation}
\end{enumerate}
The 
inductive step we shall prove is
\begin{proposition}
\label{inductstep}
$S_k$ implies $S_{k+1}$, where $w_{k+1}$
is defined by
\begin{equation}
\label{indef}
w_{k+1}=\min\left\{w_k+p,\frac32\right\}.
\end{equation}
\end{proposition}
The real number $0<p<1$ will be chosen later.
Assume for now Proposition \ref{inductstep} is true.
Setting $w_0=0$, Proposition \ref{decaystor} implies that
$S_0$ is true. It then
follows by induction
that $S_k$ is true for all $k\ge0$ where $w_{k}$ is defined inductively
by $(\ref{indef})$. 
Since $w_k=\frac32$ for $k=[\frac{3}{2p}]+1$,
the theoreom follows. $\Box$
\vskip1pc
\noindent\emph{Proof of Proposition \ref{inductstep}.}
Fix constants $h_{k+1}>1$, $\hat{v}_{0,k+1}\ge v'$.
Define $\xi_{i,k+1}$ to be the points on the
event horizon $\xi_{i,k+1}=(\infty,h^i_{k+1}\hat{v}_{0,k+1})$.
We have
\begin{lemma}
Assume $S_k$ holds for some $w_k>0$.
There exist a sequence of
points $p_{i, k+1}=(\infty,v_{i,k+1})$ on the event horizon, 
and constants $1<\alpha_{k+1}<\beta_{k+1}<\infty$,
such that
\[
\alpha_{k+1} v_{i,k+1}\le v_{i+1,k+1}\le \beta_{k+1} v_{i,k+1},
\]
and constants $c$, $\tilde{C}$ such that
\[
\int_{\tilde{p}_{i,k+1}}^{p_{i,k+1}}{\theta^2dv}
\le \tilde{C}v_{i,k+1}^{-2w_k-1+p},
\]
and 
\[
|\theta(p_{i,k+1})|\le \tilde{C}v_{i,k+1}^{-w_k-p}
\]
for points $\tilde{p}_{i,k+1}=(\infty,\tilde{v}_{i,k+1})$ satisfying
\[
cv_{i,k+1}^p\le v_{i,k+1}-\tilde{v}_{i,k+1}
\le 2cv_{i,k+1}^p
\]
\[
\includegraphics{snmeia.eps}
\]
\end{lemma}

\noindent\emph{Proof.} 
Denote $\hat{v}_{i,k+1}=h^i_{k+1}\hat{v}_{0,k+1}$.
Bisect the interval from $\xi_{i,k+1}$ to $\xi_{i+1,k+1}$, and partition
the second half into $\hat{v}_{i,k+1}^{1-p}$ subintervals of 
length approximately $\hat{v}_{i,k+1}^p$ each. By $(\ref{11y})$,
\[
\int_{\xi_{i,k+1}}^{\xi_{i+1,k+1}}{\theta^2dv}\le \bar{C}_k\hat{v}_{i,k+1}^{-2w_k},
\]
it follows that for each $i$, on one of these intervals,
call it $I_{i,k+1}$ depicted darker below: 
\[
\includegraphics{terata.eps}
\]
we have
\[
\int_{I_{i,k+1}}{\theta^2dv}\le \tilde{C}\hat{v}_{i,k+1}^{-2w_k-1+p},
\]
for some constant $\tilde{C}$.
Now bisect $I_{i,k+1}$ and call the second half $J_{i,k+1}$.
>From $(\ref{22y})$ and the equation $\partial_v\phi=r^{-1}\theta$,
we obtain that on $J_{i,k+1}$,
\[
2\hat{C}_kr_0^{-1}
\hat{v}_{i,k+1}^{-w_k}\ge\left|\int_{J_{i,k+1}}{\frac{\theta}rdv}\right|.
\]
On the other hand, $J_{i,k+1}$ has length approximately $\hat{v}_{i,k+1}^p$,
so there must be a point, call it $p_{i,k+1}$:
\[
\includegraphics{terata2.eps}
\]
where $|\theta|(p_{i,k+1})\le \tilde{C}v_{i,k+1}^{-w_k}v_{i,k+1}^{-p}$ for some
constant $\tilde{C}$.
Denote by $\tilde{p}_{i,k+1}$ the left endpoint of $J_{i,k+1}$.
It is clear now that these two points have the requisite properties.
$\Box$

Fix now $r_0>r_+$ and $v_0$ so that $r_0-r(v_0)\le\epsilon$ where
$\epsilon$ is sufficiently small as in Proposition \ref{aptovoriz}.
 
For each $i$, trisect the interval
from $\tilde{p}_{i,k+1}$ to $p_{i,k+1}$, and from the point closer
to $p_{i,k+1}$, consider the constant $v$ null segment.
Denoting by $q$ the point on this segment where $r=r_0$, consider the
unique null rectangle with endpoints on $\tilde{p}_{i,k+1}$ and
$p_{i,k+1}$ which in addition passes through $q$:
\[
\includegraphics{terata3.eps}
\]
These characteristic rectangles clearly satisfy the assumptions of
Proposition \ref{sautatelos} with $D=v_{i,k+1}^p$ and $\rho=2w_k+1-p$.
Applying then Proposition \ref{sautatelos}, it follows
that along the top right edge of the rectangles,
we have
\begin{equation}
\label{privtovbela}
\left|\zn\right|(u,v_{i,k+1})\le \hat{C}'r^\alpha v_{i,k+1}^{-w_k-\frac12+\frac p2}.
\end{equation}
Integrating the equation
\[
\partial_u\theta=-\zn\frac{\nu}r\lambda
\]
from the point $p_{i,k+1}$
we obtain that
\begin{equation}
\label{belas}
|\theta|(u,v_{i,k+1})\le\alpha^{-1}\hat{C}'r^\alpha v_{i,k+1}^{-w_k-\frac12+\frac p2}
+\tilde{C}v_{i,k+1}^{-w_k-p},
\end{equation}
for all points $(u,v_{i,k+1})$ on the top right edge of the $i$'th rectangle.
By our lower bound on $|\nu|$, $|\lambda|$, this means that there exists
a positive constant $\tilde{\delta}>0$ such that
for all $(u,v_{i,k+1})$ with $r(u,v_{i,k+1})\le\tilde{\delta}v_{i,k+1}^p$, $(\ref{belas})$ holds.

We have now the following
\begin{lemma}
\label{deutlemm}
Let $w,x,y,z,\tilde{\delta}, p$ be positive real numbers, and let $v\ge1$.
Suppose for all $(u,v)$ such that $r\le\tilde{\delta}v^p$, we have
\begin{equation}
\label{111}
|\theta|(u,v)\le \tilde{C}[r^{\alpha}v^{-y}+v^{-z}],
\end{equation}
and for some $(u_0,v_0)$ we have
\begin{equation}
\label{222}
|\theta|(u_0,v_0)\le \tilde{C}[r^{-1}v_0^{-w}+v_0^{-x}].
\end{equation}
Then it follows that there exists a constant $\tilde{C}''$ independent of $u_0$ and $v_0$ such that
\begin{equation}
\label{333}
|\theta|(u_0,v_0)\le \tilde{C}''v_0^{-\tilde{w}},
\end{equation}
where
\[
\tilde{w}=\min\left\{\frac{y}{\alpha+1}+w\left[1-\frac{1}{\alpha+1}\right],w+p,z,x\right\}.
\]
\end{lemma}

\noindent\emph{Proof.}
If 
\[
r^\alpha v_0^{-y}\ge v_0^{-w}r^{-1}
\]
then 
\[
r\ge v_0^{\frac{1}{1+\alpha}\left(y-w\right)},
\]
and thus, by
$(\ref{222})$, we have
\begin{eqnarray}
\label{al1}
\nonumber
|\theta|(u_0,v_0)	&\le&	\tilde{C}'v_0^{-w}r^{-1}+\tilde{C}'v_0^{-x}\\
\nonumber	
		&\le&	\tilde{C}'v_0^{-w-\frac{1}{1+\alpha}
					\left(y-w\right)}+\tilde{C}'v_0^{-x}\\
		& = &	\tilde{C}''v_0^{-\min\left\{\frac{y}{(\alpha+1)}+
					w\left[1-\frac1{\alpha+1}\right],x\right\}},
\end{eqnarray}
while
if 
\[
r^\alpha v_0^{-y}\le v_0^{-w}r^{-1},
\]
then
\[
r\le v_0^{\frac{1}{1+\alpha}\left(y-w\right)},
\]
and thus by
$(\ref{111})$ and $(\ref{222})$, we have
\begin{eqnarray}
\label{al2}
\nonumber
|\theta|(u_0,v_0)	&\le& 	
\tilde{C}'r^\alpha v_0^{-y}+\tilde{C}'\tilde{\delta}^{-p}v_0^{-w-p}+
					\tilde{C}'v_0^{-z}+\tilde{C}'v_0^{-x}\\
\nonumber
		&\le&	\tilde{C}'v_0^{-y+\frac{\alpha}{\alpha+1}
					\left(y-w\right)}+\tilde{C}'\tilde{\delta}^{-p}v_0^{-w-p}+
						\tilde{C}'v_0^{-z}+\tilde{C}'v_0^{-x}\\
		& = &	\tilde{C}''v_0^{-\min\left\{\left[\frac{y}{(1+\alpha)}
					+w\left[1-\frac{1}{1+\alpha}\right]\right],
					w+p,x,z\right\}}.
\end{eqnarray}
Inequalities
$(\ref{al1})$ and $(\ref{al2})$ together yield the result of the lemma. $\Box$

>From the curve $r=R$ for sufficiently large $R$, and the $v$-values $v_{i,k+1}$,
which we shall denote from now on simply as $v_i$, we construct the infinite sequence of
rectangles ${\bf X}_i$ depicted below by solid
segments:
\[
\includegraphics{moreclaim2.eps}
\]
Denote $(u_i,v_i)$ the unique point of intersection of
$v=v_i$ and $r=R$.
The rectangle ${\bf X}_i$ is then bounded by segments 
where $v=v_i$ and $v=v_{i+1}$ and $u=u_{i-1}$ and $u=u_i$.\footnote{The 
quantity
$u_0$ can be defined
arbitrarily.} Two important properties of the rectangles should here be noted: 
Firstly, it is clear that $r\ge R$ on $\bigcup_i{\bf X}_i$. Secondly, from Proposition $\ref{taG}$
it easily follows that
there exist positive constants $L_1<1<L_2$, depending only on $R$, $\beta$, and $\gamma$,
such that 
\begin{equation}
\label{rsavv}
L_1v_i\le r\le L_2v_i
\end{equation}
along the bottom and top right segments and in addition, such that for $(u,v)\in{\bf X}_i$,
\begin{equation}
\label{BY}
L_1v\le v_i\le L_2v.
\end{equation}
  
Choose $p$ such that
\[
p<\frac{1}{2(\alpha+1)+1},
\]
and apply Lemma \ref{deutlemm}, with $w=w_k$, $z=w_k+p$, $y=w_k+\frac12-\frac p2$,
and any large enough $x$.
It follows that on $[u_{i-1},u_i]\times v_i$, we have
\[
|\theta|\le \tilde{C}''v_i^{-w_{k+1}}.
\]
On the other hand, for sufficiently small $h_{k+1}$ and sufficiently large $\hat{v}_{0,k+1}$,
$\cup_i{\bf X}_i$ is contained in the region $r\le\delta_k v$,
and thus, it follows from
$S_k$, $(\ref{rsavv})$, and $(\ref{BY})$ that
on $[u_{i-1},u_i]\times v_{i+1}$,
\begin{equation}
\label{As}
|\phi|\le \tilde{C}[v_i^{-w_k-1}+v_i^{-2}],
\end{equation}
while on 
$u_{i-1}\times[v_i,v_{i+1}]$, we have
\begin{equation}
\label{Bs}
|\phi|\le \tilde{C}[v_i^{-w_k-1}+v_i^{-2}],
\end{equation}
\begin{equation}
\label{kiautomnto3exvame}
|\phi\lambda+\theta|\le\tilde{C}[v_i^{-2}],
\end{equation}
for some constant $\tilde{C}$, where for the latter inequality we use Proposition \ref{decaystor}.
Since $v_i\sim v_{i+1}$, $w_{k+1}<2$, and $w_k+1>w_{k+1}$, 
it follows that we can apply 
Proposition \ref{sprpro} on ${\bf X}_i$ 
with $A=2\tilde{C}v_i^{-w_{k+1}}$, $B=\tilde{C}v_i^{-2}$,
to obtain
\[
|\phi|\le\tilde{C}'v_i^{-w_{k+1}},
\]
\[
|\theta|\le\tilde{C}'v_i^{-w_{k+1}},
\]
\begin{equation}
\label{Cs}
|\lambda\phi+\theta|\le\tilde{C}''\left[v_i^{-2}+r^{-1}v_i^{-w_{k+1}}\right]
\end{equation}
in ${\bf X}_i$,
and 
\begin{equation}
\label{Ds}
\varpi(u_i,v_{i+1})-\varpi(u_i,v_i)\le\tilde{C}'[v_i^{-2w_{k+1}}+r(u_i,v_{i+1})v_i^{-4}].
\end{equation}
Since $w_{k+1}\le\frac32$ and $r(u_i,v_{i+1})\sim v_{i+1}$, we have in fact, 
after suitably redefining the constant $\tilde{C}''$, the bound
\begin{equation}
\label{kiedwflux}
\varpi(u_i,v_{i+1})-\varpi(u_i,v_i)\le\tilde{C}''v_i^{-2w_{k+1}}.
\end{equation}

Let ${\bf Y}_i$ denote the rectangle $[u_i,\infty]\times [v_i,v_{i+1}]$.
It follows from $(\ref{privtovbela})$ and $(\ref{kiedwflux})$ that the
assumptions of Proposition \ref{piokovtaekei} hold for ${\bf Y}_i$
with 
\[
A=\max\left\{\hat{C}'R^\alpha,\sqrt{\tilde{C}'}\right\}v_i^{-w_{k+1}}.
\]
Thus
\begin{equation}
\label{kaistoRetsi}
|\phi|\le \tilde{C}''v_i^{-w_{k+1}},
\end{equation}
\begin{equation}
\label{gialigopiokatw}
\left|\zn\right|\le \tilde{C}''v_i^{-w_{k+1}},
\end{equation}
in ${\bf Y}_i$, and also
\begin{equation}
\label{*}
\int_{v_i}^{v_{i+1}}\theta^2(\infty,v)dv\le 2\tilde{C}^2\hat{c}^{-1}v_i^{-2w_{k+1}}.
\end{equation}
Inequality $(\ref{*})$ immediately implies $(\ref{11y})_{k+1}$ for sufficiently
large $\hat{C}_{k+1}$, by summation of geometric series.  

Finally, consider the rectangles ${\bf Z}_i=[u_i,u_{i+2}]\times[v_i,v_{i+2}]$. 
By $(\ref{kiautomnto3exvame})$ and $(\ref{kaistoRetsi})$
it follows that the ${\bf Z}_i$ satisfy the assumptions of Proposition \ref{arketamegaln},
with $A=\tilde{C}''v_i^{-w_{k+1}}$, and $B=\tilde{C}v_i^{-2}$.
Thus, we have
\begin{equation}
\label{pragmkovte1}
|\phi|\le\tilde{C}'''[r^{-1} v_i^{-w_{k+1}}+v_i^{-2}],
\end{equation}
\[
|\lambda\phi+\theta|\le\tilde{C}'''v_i^{-2}
\]
in ${\bf Z}_i\cap\{r\ge R\}$. The above two inequalities immediately yield
\begin{equation}
\label{pragmkovte2}
|\theta|\le G^{-1}\tilde{C}'''[r^{-1}v_i^{-w_{k+1}}+v_i^{-2}]
\end{equation}
by the triangle inequality and the bound $\lambda^{-1}\le G$.

Finally, we note that by $(\ref{kaistoRetsi})$, 
$(\ref{gialigopiokatw})$ and the equation $(\ref{sign1})$,
equations $(\ref{pragmkovte1})$ and $(\ref{pragmkovte2})$ hold on
\[
{\bf H}=\bigcup_i\left({\bf X}_i\cup{\bf Y}_i\right).
\]
But by $(\ref{rsavv})$, there exists a $\delta_{k+1}>0$ such
that ${\bf H}$ covers the region $r\le\delta_{k+1}v$. 
Finally, by
$(\ref{BY})$ we obtain from $(\ref{pragmkovte1})$ and $(\ref{pragmkovte2})$, applied
in ${\bf H}$, the bounds $(\ref{22y})_{k+1}$ and $(\ref{33y})_{k+1}$.
This completes the proof of the proposition. $\Box$






\section{More decay}
So far we have only obtained decay away from null infinity. It turns out, however,
that we can easily extend our results to obtain estimates on the part of the domain
of outer communications not covered by Theorem \ref{3/2}. We have the following
\begin{theorem}
\label{nulinf1}
There exists a constant $\tilde{C}>0$ such that
in $r\ge \delta_{3/2}v$, we have the bounds
\begin{equation}
\label{**}
|\theta|\le \tilde{C}v^{-1}u^{-1},
\end{equation}
\[
|\zeta|\le \tilde{C}u^{-\frac32},
\]
\begin{equation}
\label{ekei;}
|r\phi|\le \tilde{C}u^{-1}.
\end{equation}
\end{theorem}

\noindent\emph{Proof.}
Fix $0<\delta'<1$ and consider the curve $\Gamma$ defined by $u=\delta' v$. For $\delta'$ sufficiently close
to $1$, $\Gamma$ is clearly contained in $r\le\delta_{3/2}v$. This follows immediately by noting that
$r(0,v)v^{-1}\to1$ as $v\to\infty$ on the one hand,
while on the other, if $r(u,v)\ge R$, then
$r(0,v)-r(u,v)\ge G^{-1}u$, and $G^{-1}\to 1$ as $R\to\infty$.
Choosing now such a sufficiently large $\delta'<1$, it is clear
that on $\Gamma$, $r\ge\kappa(1-\delta')v$, for some $\kappa>0$.
Thus, it follows from Theorem \ref{3/2} that $(\ref{ekei;})$ holds
on $\Gamma$ where $\tilde{C}$ is replaced by a constant we shall call $\tilde{C}'$.

If $(u,v)$ is such that $u\le \delta' v$, then we can
integrate $(\ref{e3is})$ along $u\times[\delta'^{-1}u,v]$ to obtain
\begin{eqnarray}
\nonumber
\label{a*}
|r\phi|(u,v)		&\le& 	\tilde{C}'u^{-1}+\int_{\delta'^{-1}u}^v{\hat{C}r^{-2}}\\
			&\le&	\tilde{C}'u^{-1}+G^{-1}\hat{C}r^{-1}(u,\delta'^{-1}u),
\end{eqnarray}
which yields $(\ref{ekei;})$
for $\tilde{C}$ sufficiently large. 
We then obtain
\begin{eqnarray}
\label{gtt}
\nonumber
|\theta|(u,v)	&\le&	G^{-1}|\phi\lambda+\theta|(u,v)+r^{-1}|r\phi|(u,v)\\
		&\le&	\tilde{C}'[v^{-2}+v^{-1}u^{-1}],
\end{eqnarray}
and this yields $(\ref{**})$,
again for sufficiently large $\tilde{C}$.
Finally, we note that in the proof of Theorem \ref{3/2}, we have in fact shown that
for $r(u,v_*)=r_0$, we have a bound,
\[
|\zeta|(u,v_*)\le\tilde{C}'v_*^{-\frac32},
\]
for some $\tilde{C}'$.
Integrating $(\ref{sign2})$ using $(\ref{+++})$ we obtain the bound
\begin{eqnarray*}
|\zeta|(u,\delta'^{-1}u)	&\le&	\tilde{C}'v_*^{-\frac32}+\int_{v_*}^{\delta'^{-1}u}\frac{|\theta\nu|}r\\
				&\le&	\tilde{C}'v_*^{-\frac32}+C_{3/2}G^{-1}\left[r^{-1}v_*^{-\frac32}+
						\log (r(u,\delta'^{-1}u)/r_0)\right]v_*^{-2}\\
				&\le&	\tilde{C}''v_*^{-\frac32}.
\end{eqnarray*}
Now let $v\ge\delta'^{-1}u$.  Integrating from $\Gamma$ using
our bound $(\ref{gtt})$ for $\theta$ we obtain
\begin{eqnarray*}
|\zeta|(u,v)		&\le&	\tilde{C}''u^{-\frac32}+\int_{\delta'^{-1}u}^{v}\frac{|\theta\nu|}r\\
			&\le&	\tilde{C}''u^{-\frac32}+\tilde{C}'''u^{-1}r^{-1}(u,\delta'^{-1}u)\\
			&\le&	\tilde{C}u^{-\frac32}
\end{eqnarray*}
for $\tilde{C}$ sufficiently large.
This completes the proof.
$\Box$

In the case where condition (d) $(\ref{pioduvato})$ holds, we can use the above theorem
to obtain a better decay estimate in $r$ for the quantity $\phi\lambda+\theta$:
\begin{proposition}
Assume condition (d). It follows that there exists a constant $\hat{C}'$ such that
\begin{equation}
\label{prwtnkalu}
|\phi\lambda+\theta|\le \hat{C}'r^{-3}\log r.
\end{equation}
\end{proposition}

\noindent\emph{Proof.}
The inequality $(\ref{prwtnkalu})$ holds on the initial outgoing null curve.
Integrating $(\ref{toallo})$ using $(\ref{ekei;})$,
we obtain immediately $(\ref{prwtnkalu})$ in the region
$r\ge\delta_{3/2} v$. Using $(\ref{+})$, 
the estimate extends to $r\le\delta_{3/2}v$. $\Box$

The above proposition easily leads to the following:

\begin{theorem}
\label{5/2}
Assume condition (d). 
It follows that there exist constants $C_{5/2}>0$, $\delta_{5/2}>0$ such that
in the region $r\le\delta_{5/2} v$, we have
\[
|\phi|\le C_{5/2}\left[v^{-\frac52}r^{-1}+v^{-3}\right],
\]
\[
|\theta|\le C_{5/2}\left[v^{-\frac52}r^{-1}+v^{-3}\right],
\]
and in addition
\[
\int_v^\infty\theta^2(\infty,v^*)dv^*\le C_{5/2}v^{-5}.
\]
\end{theorem}

\noindent\emph{Proof.}
We first show the above where $\frac 52$ is replaced by $s$ for any
$\frac32<s<\frac52$, $5$ is replaced by $2s$, and $3$ is replaced by $s+\frac12$.
For this, one procedes exactly as in the proof of Theorem \ref{3/2}, except that we now
define 
\[
w_{k+1}=\min\left\{w_k+p,s\right\},
\]
for $p>0$ defined as before.
Inequality $(\ref{prwtnkalu})$ implies $|\phi\lambda+\theta|\le C(s)r^{-s-\frac12}$,
and thus the $v_i^{-2}$ in $(\ref{As})$, $(\ref{Bs})$, and
$(\ref{Cs})$ can be replaced by $v_i^{-s-\frac 12}$, and in $(\ref{Ds})$,
the $v_i^{-4}$ can be replaced by $v_i^{-2s-1}$. Finally, since $w_k\le \frac52$,
$(\ref{kiedwflux})$ holds as well,
and the rest follows as before.

Given now this, one improves Proposition \ref{nulinf1} to obtain in particular
$r\phi\le\tilde{C}[u^{-s}+u^{-2}]$ for some constant $\tilde{C}$. 
>From this one improves $(\ref{prwtnkalu})$ to obtain
the uniform decay in $r$
\begin{equation}
\label{deutkalu}
|\phi\lambda+\theta|\le \hat{C}'r^{-3}.
\end{equation}
for sufficiently large $\hat{C}'$. But now one can repeat the argument
above replacing $s$ with $\frac52$. 
$\Box$


The $v^{-2}$ bound on $|\lambda\phi+\theta|$ in the case of (d$'$),
or the $v^{-3}$ bound in the case of (d) can clearly not be improved
for we estimate this quantity from initial data by integrating in $u$, and
these power-law tails are present initially. In the context of our above method,
this prevents
an improvement in $w_k$ beyond $\frac32$ and $\frac52$ respectively
on account of the second term of $(\ref{TOP-flux})$ from Proposition \ref{sprpro}. 
Thus, to go beyond $\frac32$ in the case of (d$'$),
and $\frac52$ in the case of (d), we must apply the propositions
of Section 7 to rectangles whose dimensions are smaller, i.e.~so that $r(u_i,v_{i+1})\sim v_{i+1}^s$
with $s<1$. We will finally obtain
\begin{theorem}
Let $\epsilon>0$, and define $\omega=3$ in the case of (d) and $\omega=2$ in the case of (d$'$).
Then there exist constants $C_{\omega-\epsilon}>0$, $\delta_{\omega-\epsilon}>0$ such that
for $r\le\delta_{w-\epsilon}v$, we have
\[
|\phi|\le C_{\omega-\epsilon}\left[v^{-(\omega-\epsilon)}r^{-1}+v^{-\omega}\right],
\]
\[
|\theta|\le C_{\omega-\epsilon}\left[v^{-(\omega-\epsilon)}r^{-1}+v^{-\omega}\right],
\]
\[
|\phi\lambda+\theta|\le C_{\omega-\epsilon}\left[v^{-(\omega-\epsilon)}r^{-1}+v^{-\omega}\right].
\]
\end{theorem}
 
\noindent\emph{Proof.}
Consider the statement  $S'_\tau$
\begin{enumerate}
\item[$(S'_\tau)$]
There exist constants
constants $\bar{C}_\tau, \delta_\tau>0$, such that
for $r\le\delta_\tau v$,
\begin{equation}
\label{22y''}
|\phi|\le \bar{C}_\tau\left[v^{-(\omega-\tau)}r^{-1}+v^{-\omega}\right],
\end{equation}
\begin{equation}
\label{33y''}
|\theta|\le \bar{C}_\tau\left[v^{-(\omega-\tau)}r^{-1}+v^{-\omega}\right],
\end{equation}
\[
|\lambda\phi+\theta|\le\bar{C}_\tau\left[v^{-(\omega-\tau)}r^{-1}+v^{-\omega}\right],
\]
\[
\int_{v}^{v+v^\tau}\theta^2(\infty,v^*)dv^*\le\bar{C}_\tau v^{-2(\omega-\tau)}.
\]
\end{enumerate}

We will prove the following
\begin{proposition}
\label{meto'}
$S'_\tau$ implies $S'_{\frac\tau2}$.
\end{proposition}

Assuming for the moment the above Proposition, and given any $\epsilon>0$, there
exists an $n$ such that $\omega-\frac1{2^n}>\omega-\epsilon$. Since $S'_\frac12$ is true by
Theorems \ref{3/2} and \ref{5/2}, then by 
induction, $S'_{\frac1{2^n}}$ is true. But $S'_{\frac1{2^n}}$ immediately implies
the theorem. $\Box$.
\vskip1pc
\noindent\emph{Proof of Proposition \ref{meto'}.}
This we shall also prove by induction!
Consider the statement $S''_k$ defined by
\begin{enumerate}
\item[$(S''_k)$]
There exist
constants $\bar{C}_k, \delta_k>0$, such that
for $r\le\delta_k v$,
\begin{equation}
\label{22y'''}
|\phi|\le \bar{C}_k[v^{-w_k}r^{-1}+v^{-\omega}],
\end{equation}
\begin{equation}
\label{33y'''}
|\theta|\le \bar{C}_k[v^{-w_k}r^{-1}+v^{-\omega}],
\end{equation}
\begin{equation}
\label{44y'''}
|\phi\lambda+\theta|\le \bar{C}_k[v^{-w_k}r^{-1}+v^{-\omega}],
\end{equation}
\begin{equation}
\label{11y'''}
\int_{v}^{v+v^\tau}\theta^2(\infty,v^*)dv^*\le\bar{C}_kv^{-2w_k}.
\end{equation}
\end{enumerate}
We will prove
\begin{proposition}
\label{telprop;}
$S''_k$ implies $S''_{k+1}$
where 
\[
w_{k+1}=\min\left\{w_k+p, \omega-\frac\tau2\right\},
\]
and $p>0$ is to be defined below.
\end{proposition}
Assuming the result of Proposition \ref{telprop;}, Proposition \ref{meto'} follows
immediately by induction. For setting $w_0=\omega - \tau$, $S''_0$ is true, as it is implied by $S'_\tau$,
and thus by induction $S''_k$ is true for all $k\ge 0$. There clearly exists a $k$
such that $w_k=\omega-\frac\tau2$, and for such a $k$, $S''_k$ immediately gives $S'_{\frac\tau2}$.
$\Box$
\vskip1pc
\noindent\emph{Proof of Proposition \ref{telprop;}.}
We fix $h_{k+1}$, $\hat{v}_{0,k+1}$, and define 
\[
\hat{v}_{i+1,k+1}=\hat{v}_{i,k+1}+h_{k+1}\hat{v}_{i,k+1}^\tau.
\]
We choose $p<\tau$,
and we obtain in analogy with the proof of Theorem \ref{3/2}
a sequence of intervals $\tilde{J}_{i,k+1}$ on the event horizon,
with right endpoint $(\infty,v_{i,k+1})$,
such that $v_{i,k+1}\sim \hat{v}_{i,k+1}$ and
\[
\int_{\tilde{J}_{i,k+1}}\theta^2\le \tilde{C}v_{i,k+1}^{-2w_k-\tau+p},
\]
\[
|\theta|(\infty, v_{i,k+1})\le \tilde{C}v_{i,k+1}^{-w_k+\frac p2-\frac \tau2}.
\]
Applying Proposition \ref{sautatelos}, we obtain that there exists $\delta>0$ such that,
for $r\le \delta v_i^p$, we have
\[
|\theta(u,v_i)|\le \tilde{C}'\left[v_i^{-w_k-p}+r^\alpha v_i^{-w_k+\frac p2-\frac\tau2}\right],
\]
for some $\tilde{C}'$. Again, we have here dropped the $k+1$ subscript from $v_{i,k+1}$.
On the other hand, 
from our assumption $S''_k$, we have that for $r(u,v_i)\le \delta_k v$
\[
|\theta(u,v_i)|\le \bar{C}_k\left[v_i^{-w_k}r^{-1}+v_i^{-\omega}\right].
\]
Choose $p$ such that
\[
p<\frac{\tau}{2\alpha+3}
\]
and apply Lemma \ref{deutlemm} 
with $w=w_k$, $x=\omega$, $y=w_k+\frac\tau2-\frac p2$, $z=w_k+p$. It follows that for
$r\le \delta v_i$, if $\delta$ is chosen sufficiently small, we have
\[
|\theta(u,v_i)|\le \tilde{C}'v_i^{-w_{k+1}},
\]
for some $\tilde{C}'$.
We then construct rectangles ${\bf X}_i$, ${\bf Y}_i$ from the sequence $v_i$ as before, 
arranging by appropriate choice
of $h_{k+1}$ and $\hat{v}_{0,k+1}$ for $\bigcup_i{\bf X}_i$ to be contained in the region $r\le \delta_{k}v$. 
On the other hand, $\bigcup_i\left({\bf X}_i\cup{\bf Y}_i\right)$ covers the region
$r\le \delta_{k+1}v^\tau$ for small enough $\delta_{k+1}<\delta_k$. Since 
$w_k\ge \omega -\tau$, in the region
$\delta_{k+1}v^\tau\le r\le \delta_{k+1}v$, $(\ref{22y'''})_k$, $(\ref{33y'''})_k$, $(\ref{44y'''})_k$
immediately yield $(\ref{22y'''})_{k+1}$, $(\ref{33y'''})_{k+1}$, and $(\ref{44y'''})_{k+1}$,
since in that region, the second term of these inequalities always dominates.
Thus, to show the proposition, it suffices to restrict to $\bigcup_i\left({\bf X}_i\cup{\bf Y}_i\right)$.
The assumptions
of Proposition \ref{aptovoriz} clearly apply to ${\bf X}_i$ with $A=\tilde{C}'v_i^{-w_{k+1}}$, 
$B=\tilde{C}'v_i^{-\omega}$. 
Since $2\omega-\tau\ge 2w_{k+1}$, it follows that the assumptions
of Proposition \ref{metobeta} then apply to the rectangles ${\bf Y}_i$ with $A=\tilde{C}'v_i^{-w_{k+1}}$.
In particular this yields $(\ref{11y'''})_{k+1}$. Finally, 
the assumptions of Proposition \ref{sautatelos} apply to the rectangles ${\bf Z}_i$ with $A=\tilde{C}'v_i^{-w_{k+1}}$
and $B=\tilde{C}'v_i^{-\omega}$. All together, the bounds
obtained yield $(\ref{22y'''})_{k+1}$, $(\ref{33y'''})_{k+1}$, and $(\ref{44y'''})_{k+1}$ in
$\bigcup_i\left({\bf X}_i\cup{\bf Y}_i\right)$ and the proof of the
proposition is complete.
$\Box$.

>From the above theorem, we can deduce also decay in a ``neighborhood'' of null infinity.
\begin{theorem}
Let $\epsilon>0$ and define $\omega$ as in the previous theorem. Then
there exists a constant $\tilde{C}_\epsilon>0$ such that
in $r\ge \delta_{\omega-\epsilon}v$, we have the bounds
\[
|\theta|\le \tilde{C}_\epsilon v^{-1}u^{-2+\epsilon}
\]
\[
|\zeta|\le \tilde{C}_\epsilon u^{-3+\epsilon}
\]
\[
|r\phi|\le \tilde{C}_\epsilon u^{-2}.
\]
\end{theorem}

\noindent\emph{Proof.} The proof is similar to Theorem \ref{nulinf1}. $\Box$

\section{Sharpness}
In view of Theorem 2 of \cite{md:cbh}, we have, in addition to
Theorem \ref{int-the} of the Introduction, the following complementary  
result:
\begin{theorem} Under the assumptions of Corollary \ref{cor-1}, if $(\ref{kritiknp})$ does
not hold, and if there exists a $V>0$ such that, for $v\ge V$, 
\begin{equation}
\label{po9oume}
|\theta|(\infty,v)\ge cv^{-p}, 
\end{equation}
for some $p<9$, $c>0$,
then the maximal extension of initial data is \emph{inextendible}
as a $C^1$ metric. 
\end{theorem}

Numerical studies seem to indicate that the inequality $(\ref{po9oume})$ holds
with $p=3$ for generic compactly supported data. This remains, however,
an open problem.

On the other hand, it seems that the red-shift techniques of this paper should easily
yield the following
\begin{theorem}
\label{conj2} 
Given any $\epsilon>0$, and $\omega>1$, there exist characteristic asymptotically
flat initial data, on
which $|\phi|(0,v)\le Cr^{-1}$, $|\theta+\phi\lambda|(0,v)\le Cr^{-\omega}$, and for 
which 
\[
|\phi(\infty,v)|\le Cv^{-\omega-\epsilon},
\]
for some constant $C>0$.
\end{theorem}

A proof of the above theorem would procede by prescribing initial data
on the event horizon and a conjugate constant-$v$ ray, and solving this
characteristic initial value problem. Note that this problem is well-posed
in spherical symmetry, as the spacelike and timelike directions can be
interchanged.

Theorem \ref{conj2} would say that the decay rate of Theorem \ref{5/2} is sharp (modulo an
$\epsilon$) in the context of the characteristic initial value problem
formulated in Section 3. At the same time, it would
indicate that any argument guaranteeing $(\ref{po9oume})$ will have
to come to terms with whatever is special about ``generic''
compactly supported data. 
\newpage
\input{bibliog}
\end{document}




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