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\hyphenation{Ro-bin-son}
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\begin{document}
\large

\begin{titlepage}
 {\LARGE{\bf
 \centerline {The cosmic no-hair conjecture and black-hole formation:}
 \centerline {an exact model with gravitational radiation}}}
 \vspace{15mm}
 \centerline {Ji\v r\'\i\  Bi\v c\' ak}
 \centerline {Department of Theoretical Physics, Faculty of}
 \centerline {Mathematics and Physics, Charles University,}
 \centerline {V Hole\v sovi\v ck\'ach 2, 180 00 Prague 8, Czech Republic }
 \centerline {bicak@aci.cvut.cz}
 \centerline {and}
 \centerline {Department of Physics, University of Utah}
 \centerline {Salt Lake City, UT 84112}

 \vspace{10mm}
 \centerline {Ji\v r\' \i\  Podolsk\' y}
 \centerline {Department of Theoretical Physics, Faculty of}
 \centerline {Mathematics and Physics, Charles University,}
 \centerline {V Hole\v sovi\v ck\'ach 2, 180 00 Prague 8, Czech Republic }
 \centerline {podolsky@aci.cvut.cz}
\end{titlepage}


\newpage
\begin{abstract}
Robinson-Trautman radiative space-times with a positive
cosmological constant are studied by analytical methods. They are
shown to approach the Schwarzschild-de Sitter solution at large
retarded times. Their global structure is analyzed, and it is
demonstrated that they represent explicit models exhibiting the
cosmic no-hair conjecture and black-hole formation under the
presence of gravitational waves.

\vspace{3mm}
\noindent
PACS number(s): 04.30.-w, 04.20.Jb, 98.80.Hw
\end{abstract}



\newpage
\section {Introduction}


   Cosmological no-hair conjectures have been discussed in
both relativistic and Newtonian contexts for more than a decade
(see, e.g., \cite{Maeda}, \cite{Barrow}
for reviews and a number of references). The
most common version claims that all expanding universes with a
positive cosmological constant approach the de Sitter space-time
locally. If an inflationary phase (described by the de Sitter
universe) is really an ``attractor'' of cosmological space-times
with positive $\Lambda$, the whole inflationary scenario becomes
a more natural indication of the present isotropy and homogeneity
of the universe.

   The cosmic no-hair conjecture has been studied under various
additional assumptions. For example, it was rigorously demonstrated
within the Bianchi cosmologies \cite{Wald}. Counter examples,
however, were also found. A simple counter example is
given by the Schwarzschild-de Sitter space-time: the metric does
not evolve into the de Sitter universe everywhere. Nevertheless,
already Gibbons and  Hawking \cite{Gibb} speculated that under the
presence of a black hole, a space-time with positive cosmological
constant would settle down to one of the Kerr-Newman-de Sitter
solutions.

   Just recently, the cosmic no-hair conjecture was also
investigated under the presence of gravitational waves. Maeda,
Nakamura and their collaborators \cite{Nakao} have analyzed the
dynamical evolution of axisymmetric gravitational waves in the
asymptotically de Sitter space-time by using numerical
computations. Their numerical work indicates that either the
space-time evolves into the de Sitter space-time with small
perturbations, or that a Schwarzschild-de Sitter-like space-time
arises.

   The purpose of the present work is to study the cosmic no-hair
conjecture under the presence of gravitational waves by using
analytical methods. Simultaneously, we shall discuss the
formation of a Schwarzschild-de Sitter black hole in a vacuum
radiative space-time with a positive cosmological constant. The
model space-times used in our discussion are represented by the
Robinson-Trautman metrics with $\Lambda>0$. They will also be
referred to as the ``cosmological'' Robinson-Trautman space-times.
(Very recently, black-hole space-times with $\Lambda>0$ have been
discussed extensively in various contexts - see, e.g.,
\cite{Brill} and references therein.)

   It is well known that the Robinson-Trautman metrics are the
general vacuum solutions which admit a geodesic, shearfree, and
twistfree null congruence of diverging rays \cite{RT1}
\cite{RT2}. The best candidates for describing gravitational
radiation from isolated sources are the Robinson-Trautman metrics
of the Petrov type II, which are asymptotically flat and --- in
contrast to ``traditional'' belief --- contain no nodal
singularities although, in general, a naked singularity is present
at the center. These space-times have attracted an increased
attention in the last decade, in particular in the works by
Luk\'acs et al. \cite{Luk}, Schmidt \cite{Schm}, Rendall
\cite{Ren}, Singleton \cite{Sin}, and, most recently, by
Chru\'sciel \cite{Chru1},\cite{Chru2} and Chru\'sciel and Singleton
\cite{ChruSin}. (We refer the reader to the last three papers for
further references.)  In these studies the Robinson-Trautman
space-times were shown to exist globally for all positive
``times'', and to converge asymptotically to a Schwarzschild
metric. Interestingly, the extension of these space-times across
the  ``Schwarzschild-like'' event horizon can only be made with a
finite degree of smoothness. All these rigorous studies are based
on the derivation and analysis of an asymptotic expansion
describing the long-time behavior of the solutions of the
nonlinear, parabolic Robinson-Trautman equation.

   The Robinson-Trautman metrics can easily be generalized to
solve the vacuum Einstein equations with a non-vanishing
$\Lambda$ \cite{KSMH}. The results proving the global existence
and convergence of the solutions of the Robinson-Trautman
equation can be taken over from the previous studies since
$\Lambda$ does not explicitly enter this equation.

   In the  next section we shall review the recent mathematical
studies of the Robinson -Trautman equation. We try to give an
intuitive (``physical'') summary of the basic results obtained by
using rigorous mathematical concepts and techniques. Section 3 is
devoted to the analysis of the Schwarzschild-de Sitter solution.
Its global structure is first described in terms of
Kruskal-type coordinates. Then synchronous (Lema\^ \i
tre-type) coordinates based on free particles moving radially
outwards towards infinity are introduced, and the properties of
the Schwarzschild-de Sitter space-time are discussed in the
context of the cosmic no-hair conjecture. The global structure
of the Robinson-Trautman  space-times of the Petrov type II with
$\Lambda>0$ is analyzed in Section 4. It is shown that these
space-times approach the Schwarzschild-de Sitter space-time in a
certain limit. They admit a smooth future (space-like) infinity
but, in general, a smooth past infinity does not exist (or, vice
versa). The continuation of the metric across the
``Schwarzschild-de Sitter-like'' black-hole horizon can be made
with a higher degree of smoothness than in the corresponding
cases with $\Lambda=0$.

   Finally, in Section 5 the asymptotic properties of the
Robinson-Trautman space-times with $\Lambda>0$ are studied at
future infinity and the validity of the cosmic no-hair conjecture
is explicitly demonstrated. The results are briefly summarized
and open problems indicated in Section 6.

   Some of the following results were presented in a preliminary
form in \cite{BiPo}.








\section {The global structure of the Robinson-Trautman
          space-times with $\Lambda=0$}

   In the standard coordinates, the Robinson-Trautman metrics have the
form (see \cite{RT1}, \cite{RT2}, \cite{KSMH})
\begin{equation}
ds^2=-\Phi du^2 - 2dudr + 2r^2P^{-2}d\zeta d\bar\zeta\ , \label{E2.2}
\end{equation}
where $P=P(u,\zeta,\bar\zeta)$, $\zeta$ is a complex spatial
(stereographic) coordinate, $r\in[0,\infty)$ is the affine
parameter along the rays $u=const$, $\zeta=const$, and
\begin{equation}
\Phi=\Delta\ln P -2r(\ln P)_{,u} - \frac{2m}{r}\ .        \label{E2.3}
\end{equation}
Here $\Delta=2P^2\partial^2/\partial\zeta\partial\bar\zeta$
and $m$ is a constant related to the Bondi mass of the system
($m=const$ can always be achieved by a coordinate transformation
 \cite{KSMH}). The function $P$ satisfies the fourth-order
Robinson-Trautman equation
\begin{equation}
(\ln P)_{,u}=-\frac{1}{12m}\Delta\Delta(\ln P)\ .          \label{E2.4}
\end{equation}

The Robinson-Trautman equation can be formulated (see, e.g.,
\cite{Chru2}, \cite{Chru1}, \cite{ChruSin})
by introducing a smooth metric $g^{0}_{ab}(x^c)$ on a
two-dimensional, compact, orientable Riemannian manifold ${\cal M}^2$
and a $u$-dependent family of 2-metrics
\begin{equation}
g_{ab}=\left[ f(u,x^c)\right]^{-2}\, g^{0}_{ab}\ ,          \label{E2.5}
\end{equation}
which, with respect to the local stereographic coordinate $\zeta$,
take the form $2P^{-2} d\zeta d\bar\zeta$. In general, the last
term in (\ref{E2.2}) can be written as $r^2f^{-2}g^{^0}_{ab}dx^a dx^b$,
and the function $f$, as a consequence of the vacuum equations, satisfies
\begin{equation}
{\partial f\over\partial u}= - \frac{f}{24m}\Delta_g R\ ,    \label{E2.6}
\end{equation}
where $R$ is the curvature scalar and $\Delta_g$ the Laplacian of
the metric $g_{ab}$. Using $R_0$ and $\Delta_0$ to denote the curvature
scalar and the Laplacian of $g^{0}_{ab}$, one has
\begin{equation}
R=f^2(R_0+2\Delta_0\ln f),\ \ \ \Delta_g=f^2\Delta_0 \ .      \label{E2.7}
\end{equation}
In general, ${\cal M}^2$ can have the topology of a 2-torus or be a higher
genus surface; however, the corresponding space-times are then
not asymptotically flat at null infinity \cite{Chru2}. Equation (\ref{E2.6})
is known in the mathematical literature as Calabi's equation \cite{Calabi} in
dimension 2. It is defined on any manifold, but here we shall primarily
concentrate on the physical case ${\cal M}^2=S^2$. Choosing standard
coordinates on the sphere,
\begin{equation}
\zeta=\sqrt2 e^{i\varphi}\tan{\theta\over2} \ ,      \label{E2.8}
\end{equation}
we obtain
\begin{eqnarray}
&&P_0=1+\frac{1}{2}\zeta\bar\zeta=
 \left(\cos^2{\theta\over2}\right)^{-1} \ , \nonumber\\
&&2P_0^{-2}d\zeta d\bar\zeta=d\theta^2+\sin^2\theta d^2\varphi \ ,
                                                      \label{E2.9}\\
&&\Delta_0\ln P_0=1\ \ ,\ \  R_0=+2\ .\nonumber
\end{eqnarray}
The metric (\ref{E2.2}) with $P=P_0$ is just the Schwarzschild metric.
Writing then (cf. (\ref{E2.5}))
\begin{equation}
P=f\,P_0 \ ,                                         \label{E2.10}
\end{equation}
we find the `original' Robinson-Trautman equation (\ref{E2.4}) to go
over into Calabi's equation (\ref{E2.6}) with $R$, $\Delta_g$ given by
(\ref{E2.7}) (and $R_0=2$).

   As mentioned in the Introduction, the most general, rigorous
and detailed analysis of the existence and behavior of the
solutions of the Robinson-Trautman equations was recently given
by Chru\'sciel \cite{Chru1}, \cite{Chru2} and by Chru\'sciel and Singleton
\cite{ChruSin} (cf. also \cite{Schm}, \cite{Tod}, \cite{Sin}). We now
very briefly summarize their main theorems and propositions;
these are formulated and proved by employing various rigorous
mathematical concepts and techniques, but we shall describe them
in a simplified, intuitive way.

   The main result of \cite{Chru1} (see Proposition 5.1 therein) shows that
when $f_0\equiv f(u=u_0,x^a)$ is an arbitrary, sufficiently smooth
initial-value function for $f$, then $f$ satisfying (\ref{E2.6}),
(\ref{E2.7}) exists for all times $u\ge u_0$, and there always exists
a constant $C$ such that
\begin{equation}
|f(u,x^a)-f_{Schw}|\le C\,e^{-2u/m} \ ,         \label{E2.11}
\end{equation}
where $f_{Schw}$ corresponds to a --- perhaps boosted --- Schwarzschild solution.
(The constant $C$  depends on the initial data $f_0$.)
Therefore, as $u\rightarrow+\infty$, {\it Robinson-Trautman
metrics approach exponentially fast a boosted Schwarzschild metric}.
Performing this boost, we can without loss of generality assume that
\begin{equation}
f_{Schw}=1 \ .                                  \label{E2.12}
\end{equation}
(How the boosts can be handled is described in
Appendix B in \cite{Chru1}.)

   In a subsequent paper \cite{Chru2} Chru\'sciel found an asymptotic
expansion of $f(u,x^a)$ for large $u$ to have the form
\begin{eqnarray}
f=1+f_1e^{-2u/m}+f_2e^{-4u/m}+\cdots+f_{14}e^{-28u/m} \nonumber\\
   +f_{log}ue^{-30u/m}+f_{15}e^{-30u/m}+\cdots \ ,   \label{E2.13}
\end{eqnarray}
where $f_1, f_2,\cdots,f_{log},\cdots$ are smooth functions on
$S^2$. Some of these functions may vanish, but Chru\'sciel and
Singleton \cite{ChruSin} prove that there exist Robinson-Trautman
space-times for which $f_{log}$ is non-vanishing.

This implies a rather surprising fact that, in general, the
Robinson-Trautman metrics cannot be smoothly extended through the null
hypersurface ${\cal H}^+$ given by $u=+\infty$. There exist an
infinite number of non-isometric Robinson-Trautman extensions
through ${\cal H}^+$ which are obtained by gluing a given
Robinson-Trautman space-time to any other Robinson-Trautman space-time
with the same mass, as shown in Fig.1. In particular, we may join
the radiative metrics to the Schwarzschild metric so that the
Robinson-Trautman space-time ``settles down''
to the Schwarzschild space-time including the interior of the
Schwarzschild black hole. The extension through the
horizon is $C^5$, in general. There exist extensions which are
$C^{117}$ through ${\cal H}^+$ --- an example can be obtained by
gluing a copy of the Robinson-Trautman space-time to itself,
as one does in the Kruskal diagram for two copies of the Schwarzschild
space-time. (The extensions will even be smoother in special
cases in which some of the coefficients of the terms $\sim
u^j e^{-ku/m}$ vanish.)

In order to see the smoothness across ${\cal H}^+$, one
introduces an advanced time coordinate $v$ by
$v=u+2r+4m\ln(r/2m-1)$,  and Kruskal-type coordinates
$\hat u, \hat v$ by (see, e.g., \cite{Tod})
\begin{eqnarray}
&&\hat u= -\exp(-u/4m) \ , \nonumber\\             \label{E2.14}
&&\hat v=\quad\exp( v/4m) \ .
\end{eqnarray}
(Such coordinates in the context of the Robinson-Trautman metrics
were first used by Tod \cite{Tod} and taken over by Chru\'sciel
\cite{Chru2}. Here we modified $v$ used in \cite{Tod},
\cite{Chru2} by a constant additive factor so as to make the
dimensions in $v$ and in the final metric (\ref{E2.15}) correct.)
The hypersurface $u=+\infty$ now becomes a boundary given by
$\hat u=0$. The original metric (\ref{E2.2}) becomes
\begin{equation}
ds^2=-\frac{32m^3}{r}\exp(-r/2m) d\hat u d\hat v - 16m^2\hat\Phi
d\hat u^2 + 2r^2P^{-2}d\zeta d\bar\zeta\ ,      \label{E2.15}
\end{equation}
where
\begin{equation}
\hat\Phi=e^{u/2m}\left(\frac{1}{2}R-1+
    \frac{r}{12m}\Delta_g R\right) ,            \label{E2.16}
\end{equation}
with $R$ and $\Delta_g$ being given by (\ref{E2.7}). In terms of
$\hat u$, the expansion (\ref{E2.13}) becomes
\begin{eqnarray}
f&=&1+f_1\hat u^8+f_2\hat u^{16}+\cdots+f_{14}\hat u^{112} \nonumber\\
 && -4mf_{log}(\ln|\hat u|)(\hat u)^{120}+f_{15}\hat u^{120}+\cdots
 \ .
                                                \label{E2.17}
\end{eqnarray}
Due to the presence of the $\ln|\hat u|$ terms, the function $f$ is
not smooth at the $\hat u=0$ --- indeed it is $C^{119}$ if
$f_{log}\not=0$. The full metric (\ref{E2.15}) is $C^{117}$, since
$\hat\Phi$ entering $g_{00}$ contains the additional factor
$e^{u/2m}\approx 1/\hat u^2$.

Although from a physical point of view a $C^5$ function may
appear as good as a smooth function, the difference between the
corresponding metrics is, in principle, observable. The resulting
space-times, despite the fact that they have a
``Schwarzschild-like'' event horizon at $r=2m$, should better be
called Robinson-Trautman black holes.

Before we turn to the cosmological Robinson-Trautman metrics we
shall discuss some properties of the Schwarzschild-de Sitter
solution.









\section {The Schwarzschild-de Sitter metric and
          synchronous coordinates}

Although the solution generalizing the Schwarzschild metric to
the case of a non-vanishing cosmological constant was found in
1918 \cite{Kott}, its global structure has been analyzed in detail only
recently \cite{Baz}.

Gibbons and Hawking \cite{Gibb} and Lake and Roeder \cite{Lake}
give a brief  discussion of its
conformal diagram; however, only Ba\. za\' nski and Ferrari
\cite{Baz} performed its ``Kruskalization'' in detail. They
considered the general case of a Schwarzschild black hole in a
de Sitter universe, thus omitting ``extreme'' black holes and
naked singularities. In the following, we shall also restrict
ourselves to such general black-hole cases.

In the standard Schwarzschild-type coordinates the metric reads
\begin{equation}
ds^2=-\left(1-\frac{2m}{r}-\frac{\Lambda}{3}r^2\right)dt^2+
      \left(1-\frac{2m}{r}-\frac{\Lambda}{3}r^2\right)^{-1}dr^2+
      r^2(d\theta^2+\sin^2\theta d\varphi^2) \ ,   \label{E3.1}
\end{equation}
where $m>0$ is the mass of the central hole  and $\Lambda>0$
is the ``repulsive'' cosmological constant. If $\Lambda=0$ we get
the Schwarzschild metric; for $m=0$ we get the de~Sitter metric in
static coordinates. Neither of these coordinate systems
covers the whole manifold. The Killing vector
$\partial/\partial t$ is timelike in a certain region of $r$
provided that the condition
\begin{equation}
0\le 9\Lambda m^2<1                              \label{E3.2}
\end{equation}
is satisfied. Hereafter we shall assume this, if not stated
otherwise. As shown in \cite{Baz}, the metric (\ref{E3.1}) can be written
in the form
\begin{equation}
ds^2=-\Phi(r)dt^2+\Phi^{-1}(r)dr^2+r^2(d\theta^2+\sin^2\theta d\varphi^2) \ ,
                                                  \label{E3.3}
\end{equation}
where
\begin{equation}
\Phi(r)= {{(r-r_+)(r_{++}-r)(r+r_++r_{++})}\over
         {r(r_+^2+r_+r_{++}+r_{++}^2)}}\ .
                                                 \label{E3.4}
\end{equation}
The parameters  $r_+$, $r_{++}$, $0<r_+<r_{++}$, are two real
positive roots of the equation $g_{00}=0$; the third root $r_-$
is negative. These roots can be parametrized as follows (see,
e.g., \cite{Jak}):
\begin{eqnarray}
r_+&=& \frac{2}{\sqrt\Lambda}
       \cos\left({\frac{\alpha}{3}+\frac{4\pi}{3}}\right)\ , \nonumber\\
r_{++}&=&\frac{2}{\sqrt\Lambda}
       \cos\frac{\alpha}{3}\ ,                      \label{E3.4b} \\
r_-   &=&\frac{2}{\sqrt\Lambda}
       \cos\left({\frac{\alpha}{3}+\frac{2\pi}{3}}\right)\ , \nonumber
\end{eqnarray}
where $\cos\alpha=-3m\sqrt\Lambda$. Here $r=r_+$ describes the
black-hole horizon, and $r=r_{++}$ is the cosmological horizon.
Only $r=0$ is a real singularity. For $r_+<r<r_{++}$, the solution
(\ref{E3.1}) is static.

In \cite{Baz}, Kruskal-type coordinate systems are found in which the
metric (\ref{E3.3}) is regular in the neighborhood of either the
black-hole horizon or the cosmological horizon. Here we shall
first convert the metric (\ref{E3.1}) into a Robinson-Trautman
form.

Define the usual ``tortoise-type'' coordinate $r^*$ by
\begin{equation}
r^*=\int\frac{dr}{\Phi(r)}=
\delta_+\ln\frac{|r-r_+|}{r+r_++r_{++}}-
\delta_{++}\ln\frac{|r_{++}-r|}{r+r_++r_{++}}+
\delta_+\left[\ln\left(\frac{r_{++}}{r_+}\right)-\frac{1}{2}\right]\ ,
                                                  \label{E3.5}
\end{equation}
where
\begin{equation}
\delta_{+}=\frac{r_+}{1-\Lambda r_+^2}\ \ ,\ \
\delta_{++}=-\frac{r_{++}}{1-\Lambda r_{++}^2}\ .
                                                  \label{E3.6}
\end{equation}
In contrast to \cite{Baz}, we have added a constant term to
$r^*$; this makes the limit $\Lambda\rightarrow 0$ more
transparent: $r_{++}\rightarrow\infty$,
$\frac{\Lambda}{3}r^2_{++}\rightarrow 1$,
$\delta_{++}\rightarrow \frac{1}{2}r_{++}$,
$\delta_+\rightarrow r_+$.
Now, by introducing the retarded time coordinate
\begin{equation}
u=t-r^*\ ,                                         \label{E3.7}
\end{equation}
instead of $t$, we convert the metric (\ref{E3.1}), resp. (\ref{E3.3}), into
\begin{eqnarray}
ds^2&=&-\left(1-\frac{2m}{r}-\frac{\Lambda}{3}r^2\right)du^2-
  2dudr+r^2(d\theta^2+\sin^2\theta d\varphi^2)   \nonumber \\
&=&-\Phi du^2 - 2dudr + r^2(d\theta^2+\sin^2\theta d\varphi^2)\ .\label{E3.8}
\end{eqnarray}
This is exactly the Robinson-Trautman form (\ref{E2.2}). (It is also the
generalization of the Schwarzschild metric in the outgoing
Eddington-Finkelstein coordinates.) It is not difficult to
show that for fixed $m$ but increasing $\Lambda$
from $\Lambda=0$ to $\Lambda=1/9m^2$, the parameters $r_+$ and
$\delta_+$ monotonically increase within the intervals
\begin{equation}
2m\le r_+\le 3m\ \ ,\ \ 2m\le\delta_+\le +\infty \ ,   \label{E3.9}
\end{equation}
whereas $r_{++}$ decreases:
\begin{equation}
  +\infty>r_{++}\ge 3m \ .                         \label{E3.10}
\end{equation}
Further, it is easy to see that $\delta_{++}>0$; it decreases from
$+\infty$ for $\Lambda=0$ to a minimum positive value, and then
increases back to $+\infty$ for $\Lambda\rightarrow1/9m^2$. At
the black-hole horizon, we have $t\rightarrow+\infty$,
$r^*\rightarrow-\infty$, and $u\rightarrow+\infty$; at the
cosmological horizon, $t\rightarrow+\infty$,
$r^*\rightarrow+\infty$, and $u$ remains finite.
On the other hand, the advanced time coordinate,
\begin{equation}
v=t+r^*\ ,                                         \label{E3.11}
\end{equation}
remains finite at the black-hole horizon, while
$v\rightarrow+\infty$ at the cosmological horizon.

In order to cover the black-hole horizon by regular coordinates,
let us introduce Kruskal-type null coordinates
\begin{eqnarray}
&&\hat u=-\exp(-u/2\delta_+)\ , \nonumber\\             \label{E3.12}
&&\hat v=\quad\exp( v/2\delta_+) \ .
\end{eqnarray}
The metric (\ref{E3.8}) now reads
\begin{equation}
ds^2=-\frac{4\Lambda\delta_+^2 r_+ e^{\frac{1}{2}}}{3r_{++}r}
  (r_{++}-r)^{1+\delta_{++}/\delta_+}
  (r+r_++r_{++})^{2-\delta_{++}/\delta_+} d\hat u d\hat v
  +r^2(d\theta^2+\sin^2\theta d\varphi^2) \ .     \label{E3.13}
\end{equation}
Letting $\Lambda\rightarrow 0$, we obtain back the metric
(\ref{E2.15}) with $\hat\Phi=0$. Clearly, at the black-hole
horizon with $r=r_+$, $\hat u=0$, $\hat v$ finite, the
metric (\ref{E3.13}) is regular. Analogously, we can find
Kruskal-type null coordinates covering the cosmological
horizon in which the metric is again regular; since, however,
we shall not need them in the following,
we refer to \cite{Baz}. The conformal diagram of the
Schwarzschild-de Sitter space-time is given in Fig.2. Notice that
both future and past infinities are spacelike as a consequence of
$\Lambda>0$. (In the following section, this point is discussed
for a general cosmological Robinson-Trautman space-time.)

Next, we shall introduce the synchronous (Lema\^\i tre-type)
coordinates in the Schwarzschild - de Sitter space-time. These
coordinates have not been
discussed so far in literature. We connect new
coordinates with free particles moving radially {\it outward}
across the past black-hole horizon and across the cosmological
horizon towards infinity since such coordinates are
suitable for the discussion of the cosmic no-hair conjecture.
(The worldline of such a particle is illustrated in Fig.2.)

Introduce the function
\begin{equation}
h(r)=\left(\frac{2m}{r}+\frac{\Lambda}{3}r^2\right)^{\frac{1}{2}}
                                           \ ,     \label{E3.14}
\end{equation}
and the coordinates $\tau$, $R$ by the relations
\begin{eqnarray}
d\tau&=&\quad dt-\frac{h}{1-h^2} dr\ ,                    \label{E3.15}\\
dR   &=&-dt+\frac{1}{h(1-h^2)}dr\ ,                \label{E3.16}
\end{eqnarray}
or, inversely,
\begin{eqnarray}
dr &=& h(dR+d\tau)\ ,                                \label{E3.17}\\
dt &=& \frac{1}{1-h^2}(d\tau+h^2dR)\ .          \label{E3.18}
\end{eqnarray}
Hence, we obtain $r(\tau,R)$ by integrating (\ref{E3.17}),
\begin{equation}
R+\tau=\int\left({\frac{2m}{r}+\frac{\Lambda}{3}r^2}\right)
       ^{-\frac{1}{2}}\,dr\ .                        \label{E3.19}
\end{equation}
The integral can be calculated explicitly to yield
\begin{equation}
R+\tau=\frac{1}{\sqrt{3\Lambda}}\ln{[2\sqrt{Y(Y+1)}+2Y+1]}+const
                                           \ ,     \label{E3.20}
\end{equation}
where
\begin{equation}
Y=\frac{\Lambda}{6m}r^3                    \ .     \label{E3.21}
\end{equation}
The relation (\ref{E3.20}) can be inverted (we put the additive
constant equal zero),
\begin{equation}
Y=\frac{(1-Z^3)^2}{4Z^3}                   \ ,     \label{E3.22}
\end{equation}
where
\begin{equation}
Z=e^{-H(R+\tau)}                           \ ,     \label{E3.23}
\end{equation}
and we introduced the ``Hubble parameter''
\begin{equation}
H=\sqrt{\frac{\Lambda}{3}}                 \ .     \label{E3.24}
\end{equation}
In the coordinates $\tau$, $R$ the metric (\ref{E3.1}) reads
\begin{equation}
ds^2=-d\tau^2+\left({\frac{2m}{r}+H^2r^2}\right)dR^2+
      r^2(d\theta^2+\sin^2\theta d\varphi^2) \ ,   \label{E3.25}
\end{equation}
with $r$ being given in term of $R$ and $\tau$ by
(\ref{E3.21})-(\ref{E3.23}). The metric is regular at the
horizons $r_+$, $r_{++}$. The particles with $R=const$,
$\theta=const$, $\varphi=const$, satisfy the equations (see
(\ref{E3.15}), (\ref{E3.16}))
\begin{eqnarray}
\frac{dr}{dt}&=&\left({\frac{2m}{r}+H^2r^2}\right)^\frac{1}{2}
    \left({1-\frac{2m}{r}-H^2r^2}\right) \ ,                   \nonumber\\
\frac{dr}{d\tau}&=&\left({\frac{2m}{r}+H^2r^2}\right)^\frac{1}{2}
                                       \ ,          \label{E3.26}\\
\frac{dt}{d\tau} &=&\left({1-\frac{2m}{r}-H^2r^2}\right)^{-1} \ .   \nonumber
\end{eqnarray}
Hence, these particles, with $\tau$ being their proper time, move
from the singularity ($r=0$, $R+\tau=0$),
across the past black-hole horizon ($r=r_+$,
$t=-\infty$), through the static region ($r_+<r<r_{++}$,
$dt/d\tau>0$), and then continue across the cosmological horizon
($r=r_{++}$, $t=+\infty$) towards infinity ($r=\infty$,
$\tau\rightarrow+\infty$) (cf. Fig.2). The equation for
$dr/d\tau$ also shows immediately that the particles cross both
horizons in a finite proper time and reach $r=\infty$ only with
$\tau\rightarrow\infty$. Their local velocity, as measured by
local static observers at $r=const$, $\theta=const$, $\varphi=const$
in the static region $r\in (r_+,r_{++})$, is given by
$v_{loc}=dl_{loc}/d\tau_{loc}=(2m/r+H^2r^2)^{\frac{1}{2}}$. At
the horizons $v_{loc}=1$ since the local ``static'' observers
themselves ``move'' with the velocity of light there.

In order to bring the metric (\ref{E3.25}) into the form in
which the de Sitter metric in ``standard'' coordinates arises
explicitly, we relabel the radial comoving coordinate $R$ as a
new coordinate $\chi$ given by
\begin{equation}
\chi=\left(\frac{m}{2H^2}\right)^\frac{1}{3} e^{HR} \ .   \label{E3.27}
\end{equation}
The metric (\ref{E3.25})  then reads as follows:
\begin{equation}
ds^2=-d\tau^2+e^{2H\tau}(1-Z^3)^\frac{4}{3}
    \left[{\left(\frac{1+Z^3}{1-Z^3}\right)^2 d\chi^2+
      \chi^2(d\theta^2+\sin^2\theta d\varphi^2)}\right] \ ,    \label{E3.28}
\end{equation}
where $Z$ is given by (\ref{E3.23}), or
\begin{equation}
Z=\left(\frac{m}{2H^2}\right)^\frac{1}{3} \chi^{-1} e^{-H\tau}\ . \label{E3.29}
\end{equation}
This is the exact form of the Schwarzschild-de Sitter metric in
the outgoing comoving coordinates. Hence it describes a white hole
in the de Sitter universe (cf. Fig.2).
The past central singularity at $r=0$ is given by $Z=1$. As $r$
increases, $Z$ decreases; at $r\rightarrow\infty$, or
$R+\tau\rightarrow\infty$, we find $Z\rightarrow0$, and the metric
(\ref{E3.28}) turns into
\begin{equation}
ds^2=-d\tau^2+e^{2H\tau} \left[{ d\chi^2+
      \chi^2(d\theta^2+\sin^2\theta d\varphi^2)}\right] \ .    \label{E3.30}
\end{equation}
This is the de Sitter metric written in the standard synchronous
Friedmann-Robertson-Walker form with the exponentially growing
(``inflationary'') expansion factor. Keeping the leading order
terms in the expansion of the Schwarzschild-de Sitter metric
(\ref{E3.28}) for $\tau\rightarrow\infty$, we obtain
\begin{eqnarray}
ds^2&=&-d\tau^2+e^{2H\tau} \left[{ d\chi^2+
      \chi^2(d\theta^2+\sin^2\theta d\varphi^2)}\right]  \label{E3.31N}\\
    &&+\frac{2m}{3H^2}\chi^{-3}e^{-H\tau} \left[{ 2d\chi^2-
      \chi^2(d\theta^2+\sin^2\theta d\varphi^2)}\right]
      +{\cal O}\left(e^{-4H\tau}\right) \ . \nonumber
\end{eqnarray}
%***
Observe that the ``traces'' of the central black hole,
characterized by the terms in which the mass $m$ enters, completely
disappear as $\tau\rightarrow\infty$ in full agreement with
the cosmic no-hair conjecture. Of course, a completely different
picture is seen by observers falling into the hole --- those,
however, are not described by our ``outgoing'' comoving
coordinates.

Before concluding this section, let us give the asymptotic
relation between the Robinson-Trautman-type coordinates $\{u, r,
\theta, \varphi\}$ and the inflationary-type coordinates
$\{\tau, \chi, \theta, \varphi\}$ for large $r$ and large $\tau$
in the present case of the Schwarzschild-de~Sitter space-time. It
is easy to see that the exact relation (\ref{E3.20}) for large
$r$ and large $\tau$ implies
\begin{equation}
r\sim \left(\frac{m}{2H^2}\right)^\frac{1}{3}e^{H(R+\tau)}\ , \label{E3.32}
\end{equation}
i.e., regarding (\ref{E3.27}), we obtain asymptotically at
$r\rightarrow\infty$, $\tau\rightarrow\infty$:
\begin{equation}
r=\chi e^{H\tau}+{\cal O}(1)+{\cal O}\left( e^{-H\tau}\right)\ .\label{E3.33}
\end{equation}
Next, for large $r$ we can integrate (\ref{E3.15}),
\begin{equation}
H\tau=Ht+\ln Hr+const   \ ;                                  \label{E3.34}
\end{equation}
and by realizing that $r^*\rightarrow const$ at $r\rightarrow\infty$ (see
(\ref{E3.5})), we can choose a constant in the last relation so
that (cf. (\ref{E3.7}))
\begin{equation}
e^{Hu}\sim e^{H\tau}(Hr)^{-1}+\cdots=\frac{1}{H\chi}+
{\cal O}\left(e^{-H\tau}\right)   \ .                        \label{E3.35}
\end{equation}
Finally, let us note that the de Sitter metric (\ref{E3.30})
preserves its form at large $\tau$ up to  terms $\sim
e^{-2H\tau}$ under the transformation
$(\tau,\chi)\rightarrow(\tau',\chi')$ such that
\begin{eqnarray}
e^{H\tau}&=&(H\chi')^2e^{H\tau'}-e^{-H\tau'}+\cdots\ ,   \\ \nonumber
H\chi&=&\frac{1}{H\chi'}+\frac{1}{(H\chi')^3}e^{-2H\tau'}+\cdots\ ,\label{E3.36}
\end{eqnarray}
which implies $\chi e^{H\tau}=\chi' e^{H\tau'}+{\cal
O}\left(e^{-3H\tau'}\right)$. Therefore, omitting primes, we can
write down the asymptotic relations at ${\cal I}^+$
$(r\rightarrow\infty,\>\tau\rightarrow\infty)$ between the
coordinate systems $(u,r)$ and $(\tau,\chi)$ in the form in which
the leading terms are both linear in $\chi$:
\begin{eqnarray}
r     &=&\chi e^{H\tau}+{\cal O}(1)\ ,   \\ \nonumber
e^{Hu}&=&H\chi+{\cal O}\left(e^{-H\tau}\right)\ .           \label{E3.37}
\end{eqnarray}
These relations obtained in the special case of the
Schwarzschild-de Sitter space-time will motivate our ansatz for
an asymptotic transformation for general Robinson-Trautman
space-times in Section 5.







\section {The global structure of the Robinson-Trautman
          space-times with $\Lambda\not=0$}

When a Robinson-Trautman space-time with $\Lambda=0$ is known in
the standard form  (\ref{E2.2}), it is straightforward to
generalize it to the case of a non-vanishing $\Lambda$
(cf. \cite{KSMH}). The metric still keeps the form (\ref{E2.2}) with
$P$ satisfying the Robinson-Trautman equation (\ref{E2.4}). The
only place where $\Lambda$ enters is the function $\Phi$. The
cosmological Robinson-Trautman metric reads
\begin{equation}
ds^2=-\Phi_\Lambda du^2 - 2dudr + 2r^2P^{-2}d\zeta d\bar\zeta\ ,\label{E4.1}
\end{equation}
where
\begin{equation}
\Phi_\Lambda=\Delta\ln P -2r(\ln P)_{,u} -
             \frac{2m}{r}-\frac{\Lambda}{3}r^2\ .  \label{E4.1*}
\end{equation}
We may still write $P=f\,P_0$, as in Eq.(\ref{E2.10}), where
$P_0$ satisfies (\ref{E2.9}) and $f$ satisfies (\ref{E2.6}),
(\ref{E2.7}). Since $\Lambda$ does not enter the evolution
equation for $f$, we may take over the results for $\Lambda=0$
described in Section 2.  The Robinson-Trautman metric will now
approach the Schwarzschild-de Sitter metric in the form
(\ref{E3.8}) as $u\rightarrow\infty$ (see Fig.3). The approach
of $f$ to its Schwarzschild-de Sitter form is again characterized
by the inequality (\ref{E2.11}) and by the asymptotic expansion
(\ref{E2.13}). We shall also assume an analog of (\ref{E2.12}),
i.e.,
\begin{equation}
f_{Schw-de Si}=1 \ .                                  \label{E4.2}
\end{equation}

The presence of a positive cosmological constant does not affect
the smoothness of future infinity in these space-times; however,
the future null infinity becomes spacelike in contrast to the
cases with $\Lambda=0$ (cf. Fig.3). The smoothness of
${\cal I}^+$ is easily seen if one introduces an inverse radial
coordinate $l=r^{-1}$ and uses $\Omega=l$ as a conformal factor.
One finds
\begin{equation}
\Omega^2 ds^2=2dudl-l^2 \Phi_\Lambda du^2 +
                           2P^{-2}d\zeta d\bar\zeta\ , \label{E4.2*}
\end{equation}
where
\begin{equation}
\Phi_\Lambda=\Delta\ln P -2l^{-1}(\ln P)_{,u} -
             2ml-\frac{\Lambda}{3}l^{-2}\ .           \label{E4.2**}
\end{equation}
It is easy to see that $l=0$ is a regular hypersurface for smooth
$P(u,\zeta,\bar\zeta)$; and it is spacelike due to the presence
of the last term in $\Phi_\Lambda$. As in the Robinson-Trautman
space-times with $\Lambda=0$, {\it past} null infinity will not
exist, in general, because the Robinson-Trautman (parabolic)
equation does not admit smooth solutions both into the future and
into the past (cf. e.g \cite{Chru2}).

The presence of the cosmological constant, however, may have a
considerable effect on the smoothness of the extensions through
the null hypersurface ${\cal H}^+$ given by $u=+\infty$. The
transformation to the Kruskal-type null coordinates is now given
by (\ref{E3.12}). Hence, instead of (\ref{E2.17}), we find the
expansion of $f$ in terms of $\hat u$ to read
\begin{eqnarray}
f&=&1+f_1(\hat u)^{4\delta_+/m}+f_2(\hat u)^{8\delta_+/m}+\cdots+
 f_{14}(\hat u)^{56\delta_+/m} \nonumber\\
 && -2\delta_+ f_{log}(\ln|\hat u|)(\hat u)^{60\delta_+/m}+
 f_{15}(\hat u)^{60\delta_+/m}+\cdots \ .
                                                \label{E4.3}
\end{eqnarray}
The full metric now takes the form
\begin{eqnarray}
ds^2&=&-\frac{4\Lambda\delta_+^2 r_+ e^{\frac{1}{2}}}{3r_{++}r}
  (r_{++}-r)^{1+\delta_{++}/\delta_+}(r+r_++r_{++})^{2-\delta_{++}/\delta_+}
  d\hat u d\hat v  \nonumber \\
 &&-4\delta_+^2\hat\Phi_\Lambda\,d\hat u^2
 +2r^2P^{-2}d\zeta d\bar\zeta \ ,              \label{E4.4}
\end{eqnarray}
where
\begin{equation}
\hat\Phi_\Lambda=e^{u/\delta_+}\left(\frac{1}{2}R-1+
    \frac{r}{12m}\Delta_g R\right) ,            \label{E4.5}
\end{equation}
and $R$, $R_0$, $P$, $P_0$, $\Delta_g$ are given by Eqs.
(\ref{E2.7})-(\ref{E2.10}), with $f$ being of the form (\ref{E4.3})
above. The constant parameters $r_+$, $r_{++}$, $\delta_+$, $\delta_{++}$
are the same as in Section 3.

It is interesting to see how the presence of $\Lambda$ influences
the smoothness of the extension of the Robinson-Trautman metrics across
the null hypersurface ${\cal H}^+$ (``horizon'') given by
$u=\infty$, or $\hat u=0$.  In particular, we may join the
radiative Robinson-Trautman metrics to the Schwarzschild-de
Sitter metric so that a Robinson-Trautman space-time with
$\Lambda\not=0$ ``settles down'' to the Schwarzschild-de Sitter
space-time including the interior of the Schwarzschild-de Sitter
black hole (see Fig.3). As summarized in Section 2, such an
extension will, in general, be $C^{117}$ in the case of vanishing
$\Lambda$. With $\Lambda\not=0$, much higher smoothness can be
obtained. Indeed, since $\delta_{+}=r_+/(1-\Lambda r_+^2)$
(see (\ref{E3.6})), the horizon ${\cal H}^+$ can be made
``arbitrarily smooth'' by letting $\Lambda$ approach
arbitrarily close its extremal value,
$\Lambda\rightarrow 1/9m^2$ ($r_+\rightarrow 3m$), which
corresponds to an almost extreme
Schwarzschild-de Sitter black hole. Then
$\delta_+$  becomes arbitrarily large and the terms $\sim\hat
u^{k(\delta_+/m)}$, $k=4,8,\cdots$, appearing in (\ref{E4.3}),
will guarantee arbitrarily high smoothness of the function $f$ at
$\hat u=0$; this in turn determines the smoothness of the functions
$R$, $P$ and thus of the full metric (\ref{E4.4}) at $\hat u=0$.
(Notice that $\hat\Phi_\Lambda$ entering $g_{00}$ contains, in
addition, the factor $e^{u/\delta_+}\approx1/\hat u^2$ which,
however, decreases the smoothness of the metric by 2 only.)

On the other hand, there are cases when the presence of $\Lambda$
can decrease the smoothness of the metric (\ref{E4.4}) across
$\hat u=0$. If $4\delta_+/m$ is not an integer, the
second term in the expansion (\ref{E4.3}),
$f_1(\hat u)^{4\delta_+/m}$, is already not smooth. Since $4\delta_+/m\ge8$
(see (\ref{E3.9})), the function $f$ is then at least $C^8$ and the
full metric is at least $C^6$. For those values of $\Lambda$ which
imply $4\delta_+/m$ equals an integer, the smoothness is
always better than for $\Lambda=0$. In general, denoting by $n$
the largest integer smaller than $4\delta_+/m$, the metric (\ref{E4.4})
is at least $C^n$ at $\hat u=0$. As mentioned above,
$n\rightarrow\infty$ with $4\delta_+/m\rightarrow\infty$.

We also expect to find analogs of the past black-hole horizon
and the cosmological horizon of the Schwarzschild-de Sitter
metric in a Robinson-Trautman solution with $\Lambda>0$ (see
Fig.3). Possible analogs can be considered in a way similar to
Tod's treatment \cite{Tod} for the case $\Lambda=0$.






\section {The cosmic no-hair conjecture}

After having examined the behavior of the cosmological
Robinson-Trautman metrics at the null hypersurface
$u\rightarrow+\infty$, we shall now turn to their properties
close to ${\cal I}^+$, i.e., near $r\rightarrow\infty$, $u$ finite (cf.
Fig.2). If the cosmic no-hair conjecture is to be verified in
these space-times, we should be able to demonstrate that they all
approach the de Sitter space-time locally. For this purpose, it
would be sufficient to convert the metrics (\ref{E4.1}), with
$\Phi_\Lambda$ given by (\ref{E4.1*}), into the Starobinsky
asymptotic form \cite{Starob}. As will be indicated below, such
an asymptotic form automatically guarantees that the cosmic no-hair
conjecture is satisfied.

Starobinsky's form represents the asymptotic behaviour of an
inhomogeneous cosmological model with $\Lambda>0$ in synchronous
coordinates at large $\tau$ ($\tau$ being the proper time
measured by the observers at rest in these synchronous
coordinates). In Section 3, the Starobinsky form of the pure
Schwarzschild-de Sitter metric was obtained in Eq.(\ref{E3.31N}).
We now wish  to find a similar form under the presence of
gravitational waves described by a general Robinson-Trautman
metric (\ref{E4.1}).

In order to convert the metric (\ref{E4.1}) into the desired
form, we first make an ansatz which is inspired by the asymptotic form
of the transformation (\ref{E3.36}) between the
Robinson-Trautman-type coordinates and the synchronous
(``inflationary-type'') coordinates for the Schwarzschild-de
Sitter metric.

Assume that the required transformation is of the asymptotic form
\begin{eqnarray}
r     &=&\chi e^{H\tau}
       -H^{-2}(f_{\infty,u}/f_\infty)+\sum_{n=1}^\infty
       \psi_n  e^{-nH\tau}\ ,   \nonumber \\
e^{Hu}&=&H\chi-e^{-H\tau}\ ,                  \label{E5.1}
\end{eqnarray}
where $H=\sqrt{\Lambda/3}$, $\psi_n$ are functions of
$\chi,\zeta,\bar\zeta$ to be chosen appropriately, and
$f_\infty=f|_{\tau\rightarrow\infty}=f(u\rightarrow
H^{-1}\ln|H\chi|,\zeta,\bar\zeta)$ is determined by the function
$f(u,\zeta,\bar\zeta)$ entering the metric (\ref{E4.1}).
(Whenever the gravitational waves are present, $f\not=1$; only
$f_{Schw-de Si}=1$ (cf. (\ref{E4.2})).) Under transformation
(\ref{E5.1}), the metric (\ref{E4.1}) becomes
\begin{eqnarray}
ds^2=&-d\tau^2+e^{2H\tau} \left[ d\chi^2+
      (\chi/f_\infty P_0)^2\, 2d\zeta d\bar\zeta\right]+
      h_{ij}^{(0)}\,dx^i dx^j    \nonumber \\
 &\ \ +e^{-H\tau}h_{\mu j}^{(1)}dx^\mu dx^j+\sum_{m=2}^\infty e^{-mH\tau}
      h_{\mu\nu}^{(m)}\, dx^\mu dx^\nu \ .          \label{E5.2}
\end{eqnarray}
Here $f_\infty$ depends only on the spatial coordinates
$\{x^i, i=1,2,3\}=\{\chi,\zeta,\bar\zeta\}$; all $h_{\mu\nu}$'s
appearing in (\ref{E5.2}) also depend only on $x^i$, but
$\{x^\mu\}=\{\tau,x^i\}$ so that the last two terms in (\ref{E5.2})
involve $d\tau$ as well. Nevertheless, the non-diagonal
components of the metric $h_{0j}^{(1)}, h_{0j}^{(2)},\ldots$ can
be transformed away by transformations of the following form:
\begin{eqnarray}
\chi^{(m-1)}  &=& \chi^{(m)}   +a_m\,e^{-(m+2)H\tau}\ ,  \nonumber \\
\zeta^{(m-1)} &=& \zeta^{(m)}\,+b_m\,e^{-(m+2)H\tau}\ ,  \label{E5.3}
\end{eqnarray}
where $m=1,2,3,\ldots$,
$\{\chi^{(0)},\zeta^{(0)},\bar\zeta^{(0)}\}=
\{\chi,\zeta,\bar\zeta\}$, and the coefficients $a_m$, $b_m$
depend on $\{\chi^{(m)},\zeta^{(m)},\bar\zeta^{(m)}\}$. After
choosing $a$'s and $b$'s so that $h_{0j}^{(m)}=0$, the metric (\ref{E5.2})
reads
\begin{eqnarray}
ds^2=&-d\tau^2+e^{2H\tau} \left[
     d\tilde\chi^2+{(\tilde\chi/f_\infty)}^2
     {(1+\frac{1}{2}\tilde\zeta \bar{\tilde\zeta})}^{-2}\,
     2d\tilde\zeta d\bar{\tilde\zeta}  \right]  \nonumber \\
    &+\sum_{m=0}^\infty e^{-mH\tau} \tilde h_{ij}^{(m)}\,
     d\tilde x^i d\tilde x^j
     +\sum_{m=2}^\infty e^{-mH\tau} \tilde h_{00}^{(m)}
     \, d\tau^2 \ .                              \label{E5.4}
\end{eqnarray}
Furthermore, the metric coefficients $\tilde h_{00}^{(m)}(\tilde
x^{(i)})$ can be transformed away by performing a transformation
of the form (\ref{E5.1}) in which $\psi_n$ are chosen
appropriately.

To summarize, a transformation of the form
\begin{eqnarray}
r     &=& \chi e^{H\tau}-H^{-2}(f_{\infty,u}/f_\infty)+\sum_{n=1}^\infty
  A_n\,e^{-nH\tau}\ ,   \nonumber \\
e^{Hu}&=&H\chi-e^{-H\tau}\quad+\   \sum_{n=3}^\infty
  B_n\,e^{-nH\tau}\ ,           \label{E5.5} \\
\zeta &=& \eta\quad+\qquad\qquad\ \  \sum_{n=3}^\infty
  C_n\,e^{-nH\tau}\ ,   \nonumber
\end{eqnarray}
in which $A_n$, $B_n$, $C_n$ are suitable functions of
$\chi,\eta,\bar\eta$ brings the metric (\ref{E4.1})
into the following asymptotic form:
\begin{eqnarray}
ds^2=&-d\tau^2+e^{2H\tau} \left[ d\chi^2+
      f_\infty^{-2} \chi^2\,(d\theta^2+\sin^2\theta d\varphi^2)
      \right]              \nonumber \\
&+\sum_{m=0}^\infty e^{-mH\tau}  h_{ij}^{(m)}\, dx^i dx^j\ ,  \label{E5.6}
\end{eqnarray}
where we have reintroduced the standard angular coordinates
$\theta$, $\phi$ ($\eta=\sqrt2 e^{i\varphi}\tan{(\theta/2)}$ --- cf.
Eq.(\ref{E2.8})), $f_\infty=f|_{\tau\rightarrow\infty} = f(u=
H^{-1}\ln|H\chi|,\theta,\phi)$, and $h_{ij}^{(m)}$ depend on
$\{x^i\} = \{\chi,\theta,\phi\}$ only. This asymptotic form of the
metric shows explicitly that for $\tau\rightarrow\infty$ the
space-time metric does {\it not} approach the de Sitter metric
globally. In contrast to the case of the pure Schwarzschild-de
Sitter black hole where outside of the hole all its ``traces''
disappear as $\tau\rightarrow\infty$  (cf. Eq.(\ref{E3.30})),
the gravitational waves now leave ``an imprint'' on the
${\cal I}^+$ which is
demonstrated by the presence of the function $f_\infty$ in the
``angular'' part of the metric (\ref{E5.6}). Only as
$u\rightarrow\infty$, $\chi\rightarrow\infty$,
$r\rightarrow\infty$, so that $f_\infty\rightarrow 1$ (see the
point $P$ in Fig.3), do the Robinson-Trautman metrics
approach the de Sitter metric since  the terms in (\ref{E5.6})
with the factor $e^{2H\tau}$ dominate those proportional to
$e^{-mH\tau}$.

In fact, the cosmic no-hair conjecture claims that a universe with
a positive cosmological constant approaches exponentially fast the
de Sitter space-time only locally in general. That this is so in
our case can easily be demonstrated as follows (cf. \cite{GiBo}).

Write the metric (\ref{E5.6}) in the form
\begin{equation}
ds^2=-d\tau^2+e^{2H\tau}\, a_{ij}(x^s)\, dx^i dx^j
     + {\cal O}(1)\ . \label{E5.7}
\end{equation}
Consider a worldline $x^i=x^i_{(0)}=const$. It is a geodesic (in
Fig.3 indicated by ${\cal O}_1$). Make a linear transformation
$x^i\rightarrow \bar x^i$, $x^i-x^i_{(0)}=A^i_k \bar x^k$ such
that $\bar a_{ij}(0)=\delta_{ij}$. Now introduce new coordinates
$y^i$, $T$ by
\begin{eqnarray}
\bar x^i &=&y^i\,e^{-HT}/(1-H^2R^2)^{\frac{1}{2}}\ , \nonumber \\
e^{H\tau}&=&e^{HT}(1-H^2R^2)^{\frac{1}{2}}\ ,    \label{E5.8}
\end{eqnarray}
where $R^2=(y^1)^2+(y^2)^2+(y^3)^2$. Putting
$y^1=R\sin\tilde\theta\cos\tilde\varphi$,
$y^2=R\sin\tilde\theta\sin\tilde\varphi$,
$y^3=R\cos\tilde\theta$, we find that (\ref{E5.7}) takes the form
\begin{eqnarray}
ds^2&=&-\left(1-\frac{\Lambda}{3}R^2\right)\,dT^2+
       \left(1-\frac{\Lambda}{3}R^2\right)^{-1}\,dR^2  \nonumber \\
&&+R^2\,(d\tilde\theta^2+\sin^2\tilde\theta d\tilde\varphi^2)
 +\sum_{m=1}^\infty e^{-mHT}\tilde a_{\mu\nu}^{(m)}\,
 dy^\mu dy^\nu\ ,                                 \label{E5.9}
\end{eqnarray}
where $\tilde a^{(m)}_{\mu\nu}$ are functions of of $\{\tilde y^i\} =
\{R,\tilde\theta,\tilde\varphi\}$, and $\tilde y^0=T$. The
transformation (\ref{E5.8})  cannot be applied globally --- it is
meaningful only for $1-H^2R^2>0$. A geodesic observer
(say ${\cal O}_1$ in Fig.3) moving with $y^i=0$ will see, inside his past
light cone, the space-time approach the de Sitter space-time
exponentially fast as $T\rightarrow\infty$. Thus for a freely
falling observer the observable universe becomes quite bald. This
is what the cosmic no-hair conjecture claims.






\section {Concluding remarks}

Starting from the {\it exact} model radiative space-times with a
positive cosmological constant, we demonstrated the validity of
the cosmic no-hair conjecture by purely analytic means. We have
also shown that these cosmological Robinson-Trautman solutions
settle down to the Schwarzschild-de Sitter solution at large
retarded times.  The interior of a Schwarzschild-de Sitter black
hole can be joined to an ``external'' cosmological
Robinson-Trautman space-time across the horizon with a higher
degree of smoothness than in the corresponding case with
$\Lambda=0$. As far as we are aware, these models represent the
only exact analytic demonstration of the cosmic no-hair
conjecture under the presence of gravitational waves. They also
appear to be the only exact examples of black-hole formation in
non-spherical space-times which are not asymptotically flat.
Hopefully, these models may serve as tests of various
approximation methods, and as test beds in numerical studies of
more realistic situations.

Finally,  let us note that the ideas discussed here can also be
applied to the Robinson-Trautman space-times containing an
incoherent homogeneous radiation field (null fluid). For
$\Lambda=0$, such space-times were studied by Bi\v c\' ak and
Perj\'es \cite{BiPe} who showed that they approach the Vaidya
metric asymptotically. With $\Lambda\not=0$ the radiating Vaidya
metric in the de Sitter universe should arise.






\section {Acknowledgments}

We want to thank Piotr Chru\' sciel for several very enlightening
discussions of his work on the Robinson-Trautman space-times, and
Joe Romano for reading the manuscript. J.B. is grateful to Karel
Kucha\v r for the warm hospitality in Salt Lake City where this
work was finished.

We also acknowledge the support of the U.S.-Czech Science and
Technology Grant No 92067, and grants GACR-0503 and
GAUK-318 of the Czech republic and Charles University.


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\newpage
{\LARGE Figure Captions}

\bigskip
\bigskip
\noindent
Figure 1. Starting with arbitrary, smooth initial data for the
Robinson-Trautman equation at $u=u_0$, the Robinson-Trautman
metrics converge exponentially fast to a Schwarzschild metric as
$u\rightarrow\infty$. However, the vacuum extension beyond the
null hypersurface ${\cal H}^+(u=+\infty)$ that includes,
for example, the interior of the vacuum Schwarzschild black hole,
can only be done with a finite degree of smoothness. (See the
text for more details.)

\bigskip
\bigskip
\noindent
Figure 2. The conformal diagram for the Schwarzschild-de Sitter
space-time. The worldline $R=const$ ($\theta=const$,
$\varphi=const$) represents the history of a free particle moving
radially outward across the past black-hole horizon and across
the cosmological horizon towards future
(spacelike) infinity. For an observer moving with such a
particle, all ``traces'' of the hole disappear as he approaches
${\cal I}^+$; only the de Sitter geometry persists.

\bigskip
\bigskip
\noindent
Figure 3. Starting with arbitrary, smooth initial data at $u=u_0$,
the Robinson-Trautman metrics with positive cosmological
constant converge exponentially fast to a Schwarzschild-de
Sitter metric as $u\rightarrow\infty$. In the neighborhood of $P$
at future infinity ${\cal I}^+$, the metric is approaching the de
Sitter metric exponentially fast. Although traces of
gravitational waves will persist in other regions of future
infinity, all geodesic observers (such as ${\cal O}_1$) will
observe the metric to approach the de Sitter metric exponentially
fast within their past light cone. Observer ${\cal O}_2$ falls
into the black hole. The metric at the horizon ${\cal
H}^+$ has only a finite degree of smoothness, however, although this can
be higher than in the case with $\Lambda=0$.


\end{document}