\documentstyle[eqsecnum,prd,twocolumn,aps]{revtex}
\begin{document}
% corrected version of the paper DE6328 by Podolsky and Griffiths
\draft
\title{Impulsive gravitational waves generated by null particles\\
in de~Sitter and anti-de~Sitter backgrounds}

\def \d {{\rm d}}

\author{Ji\v{r}\'{\i} Podolsk\'y}
\address{Department of Theoretical Physics,\\
Faculty of Mathematics and Physics, Charles University,\\
V Hole\v{s}ovi\v{c}k\'ach 2, 18000 Prague 8, Czech Republic.}

\author{Jerry B. Griffiths}
\address{Department of Mathematical Sciences,\\
Loughborough University,\\
Loughborough, Leicestershire LE11 3TU, U.K.}


\date{\today}
\maketitle
\begin{abstract}
By boosting the metric of the Schwarzschild--de~Sitter and
Schwarzschild--anti-de~Sitter space-times in the
ultrarelativistic limit as the mass parameter vanishes,
 impulsive gravitational waves are obtained in
backgrounds with a nonzero cosmological constant. We determine the geometry
of these impulsive wave surfaces using various coordinate charts. In the
de~Sitter background they are spheres containing two null particles located
at the poles. In the anti-de~Sitter background they are hyperboloidal
surfaces of constant negative curvature containing a single null particle on
the axis of symmetry. The global structure and conformal diagrams for the
complete manifolds are also determined

\end{abstract}
% insert suggested PACS numbers in braces on next line
\pacs{04.20.Jb, 04.30.Nk}

\narrowtext

\section{Introduction}
In a now classic paper, Aichelburg and Sexl \cite{R1} obtained an exact solution of
Einstein's equations for a null particle in a Minkowski background by
boosting the Schwarzschild metric to the ultrarelativistic limit as the mass
becomes arbitrarily small. The resulting space-time is an impulsive {\it
pp}-wave. Generalisations of this solution have been obtained by a number of
authors \cite{R2}--\cite{R5} by similarly boosting the Kerr or Kerr--Newman solutions.


A similar approach has been adopted by Hotta and Tanaka \cite{R6} in boosting the
Schwarz\-schild--de~Sitter metric to a similar limit. Effectively, they
consider this as a perturbation of the de~Sitter metric and boost the
particle to the ultrarelativistic limit as its mass vanishes in an
appropriate way. This approach also results in impulsive gravitational waves.
However, the background in this case is not flat and the wavefronts are not
simple planes in Minkowski space. Rather they are surfaces of constant
curvature in the de~Sitter background. The calculations can easily be adapted
to describe impulsive gravitational waves in an anti-de~Sitter background.
However, the global properties of the solution in this case differ
significantly.


It is the purpose of the present paper to clarify certain aspects of these
solutions and to investigate the geometrical properties of the impulsive wave
surfaces as they would be seen by different families of observers.


\section{Boosting the Schwarzschild--de~Sitter solution}

Let us start with the Schwarzschild--de~Sitter solution in the familiar form
 \begin{eqnarray}
 \d s^2 =\left(1-{r^2\over a^2}-{2m\over r}\right)\d t^2
&&-\left(1-{r^2\over a^2}-{2m\over r}\right)^{-1}\d r^2 \nonumber\\
&&-r^2(\d\theta^2+\sin^2\theta\,\d\phi^2)
\end{eqnarray}
 in which the cosmological constant is given by $\Lambda=3/a^2$
and $m$ is the mass parameter.  In the limit when $m=0$, this is simply the
de~Sitter solution which, as is well known, can be represented as a
four-dimensional hyperboloid
 \begin{equation}
 {Z_0}^2 -{Z_1}^2 -{Z_2}^2 -{Z_3}^2 -{Z_4}^2 =-a^2
\end{equation}
 embedded in a five-dimensional Minkowski space-time
 \begin{equation}
 \d s^2= \d{Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2 -\d{Z_4}^2 .
\end{equation}
 For $m=0$, this can be achieved by putting
  \begin{eqnarray}
 Z_0&&=\sqrt{a^2-r^2}\,\sinh(t/a) \nonumber\\
 Z_4&&=\pm\sqrt{a^2-r^2}\,\cosh(t/a) \nonumber\\
 Z_1&&=r\cos\theta \\
 Z_2&&=r\sin\theta\cos\phi \nonumber\\
 Z_3&&=r\sin\theta\sin\phi . \nonumber
 \end{eqnarray}
 From this it can immediately be observed that the origin $r=0$ corresponds
to the two world lines
 \begin{equation}
 (Z_0, Z_1, Z_2, Z_3, Z_4) =\big(a\sinh(t/a), 0, 0, 0, \pm a\cosh(t/a)\big)
 \end{equation}
 along the hyperboloid, as indicated in figure~1 in which the coordinates
$Z_2$ and $Z_3$ are suppressed. It may seem surprising that the
Schwarzschild--de~Sitter solution (2.1) does not just have a single source in
the limit when the mass is small. However, it is known that the
coordinates of (2.1) do not cover the complete space-time whose analytic
extension must include both the black hole and white hole parts. This fact can
be deduced from its conformal structure (see eg. Bi\v{c}\'ak and Podolsk\'y,
\cite{R7}).


\bigskip\centerline{ Fig. 1}\bigskip


Following the approach of Aichelburg and Sexl \cite{R1}, we may now boost the
singularity representing the point mass in the
Schwarzschild--de~Sitter solution to the limit in which its speed approaches that of light
and its mass is scaled to zero in an appropriate way.


As observed by Hotta and Tanaka \cite{R6}, in this case it is first appropriate to
take the linear approximation of (2.1) for small $m$ in the form
 \begin{eqnarray}
 \d s^2 &&=\left(1-{r^2\over a^2}-{2m\over r}\right)\!\d t^2 \nonumber\\
&&\qquad-\left(1-{r^2\over a^2}\right)^{-1}\!
\left[1+\left(1-{r^2\over a^2}\right)^{-1}\!{2m\over r}\right]\!\d r^2
\nonumber\\
&&\qquad-r^2(\d\theta^2+\sin^2\theta\,\d\phi^2).
 \end{eqnarray}
 This can be considered as a first order perturbation of the de~Sitter line
element. Using the coordinates $Z_i$ of (2.4), the first order perturbation of
the line element (2.3) can be written as
 \begin{eqnarray}
 \d s_1^2 &&=-{2ma^2\over r} \Bigg[
\left( {Z_4\d Z_0-Z_0\d Z_4\over{Z_4}^2-{Z_0}^2} \right)^2 \nonumber\\
 &&\qquad\qquad +{a^2\over r^2}
\left( {Z_0\d Z_0-Z_4\d Z_4\over{Z_4}^2-{Z_0}^2} \right)^2
\Bigg],
\end{eqnarray}
 where $r=(a^2+{Z_0}^2-{Z_4}^2)^{1/2}$.  A boost can then be
performed in any direction orthogonal to $Z_4$ (a boost in the $Z_4$ direction
simply corresponds to a time shift).
Taking the boost in the $Z_1$ direction, we can put
 \begin{eqnarray}
 &&Z_0\to{Z_0+vZ_1\over\sqrt{1-v^2}}, \qquad
Z_1\to{vZ_0+Z_1\over\sqrt{1-v^2}}, \nonumber\\
 &&Z_2\to Z_2, \qquad Z_3\to Z_3, \qquad Z_4\to Z_4,
 \end{eqnarray}
 where $v$ is the boost parameter. It can then be seen that, in the limit as
$v\to1$, the linear perturbation term vanishes everywhere except on the
hypersurface $Z_0+Z_1=0$. Then, by also scaling the mass according to
 $$ m=p\sqrt{1-v^2}, $$
 where $p$ is a constant, it can be seen that the only non-vanishing term in
(2.7) is that which becomes a multiple of $(\d Z_0+\d Z_1)^2$. The line
element thus takes the form
 \begin{eqnarray}
 \d s^2 &&=\d{Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2 -\d{Z_4}^2 \nonumber\\
&&\qquad -2pa^2H(Z_4)\,\delta(Z_0+Z_1)\,(\d Z_0+\d Z_1)^2 ,
 \end{eqnarray}
 where, putting
 $$ x^2={(Z_0+vZ_1)^2\over1-v^2}, $$
 \begin{eqnarray}
 H(Z_4)&&\,\delta(Z_0+Z_1)  \\
       &&=\lim_{v\to1}{1\over\sqrt{1-v^2}}
{(a^2-{Z_4}^2){Z_4}^2+(a^2+{Z_4}^2)x^2 \over
\left(a^2-{Z_4}^2+x^2\right)^{3/2}
\left({Z_4}^2-x^2\right)^2} . \nonumber
 \end{eqnarray}
 In evaluating this limit, we follow Hotta and Tanaka \cite{R6}, in making use of
the identity
 \begin{eqnarray}
 \lim_{v\to1}{1\over\sqrt{1-v^2}} &&f\left({(Z_0+vZ_1)^2\over1-v^2}\right)
\nonumber\\ &&=\delta(Z_0+Z_1) \int_{-\infty}^\infty f(x^2)\d x,
\end{eqnarray}
which can easily be proven in the theory of distributions.
 However, in this case, the integrand involved contains singularities at
$x=\pm Z_4$. Nevertheless, it is possible to remove the effect of these by
first making a singular coordinate transformation such that
 \begin{eqnarray}
 \d Z_0&&-\d Z_1 \to \d Z_0-\d Z_1 \nonumber\\
 && -\lim_{v\to1} {4pa^3\over\sqrt{1-v^2}}
\left(a^2-{(Z_0+Z_1)^2\over1-v^2}\right)^{-2}(\d Z_0+\d Z_1). \nonumber
 \end{eqnarray}
 Then, using (2.11), it can be shown that the function $H(Z_4)$ in the line
element (2.9) takes the form
 \begin{equation}
 H(Z_4) =-{2\over a^2} \left[
  2-{Z_4\over a}\ln\left|{a+Z_4\over a-Z_4}\right| \right].
\end{equation}


It can thus be seen that the resulting space-time differs from the de~Sitter
space-time only by the inclusion of an impulsive wave located on the null
hypersurface given by
 \begin{equation}
 Z_0+Z_1=0, \qquad {Z_2}^2 +{Z_3}^2 +{Z_4}^2 =a^2.
 \end{equation}
It may be observed that the shock wave is a 2-sphere in
the 5-dimensional Minkowski space at any time $Z_0$.


In addition, as shown in \cite{R6}, the singularities that
were originally located at $r=0$ have now been boosted to become two
null particles described by delta functions in the energy-momentum
tensor on the wave surface. These have the worldlines
 \begin{equation}
  Z_0+Z_1=0, \qquad Z_2=Z_3=0, \qquad Z_4=\pm a.
 \end{equation}
 These are the two null straight lines on the four-dimensional hyperboloid as
indicated in figure~2.

\bigskip
\bigskip\centerline{ Fig. 2}\bigskip


\section{Global coordinates ($\Lambda>0$)}

Let us now investigate the geometry of the shock wave in the de~Sitter
background. To do this, we set up some appropriate coordinate system. At each
event, we introduce some ``time'' and ``space'' labels. We can then ask the
question: what are the space coordinates of the shock wave at a given
``synchronized'' time? Of course, although the geometry is unique, the
coordinates can be chosen arbitrarily, so that the coordinate shape is not
unique. In practice, however, ``comoving coordinates'' can always be
introduced, so the coordinate shape of the shock wave will correspond to its
shape as seen by the associated family of observers.


In fact there are a number of privileged families of observers in a de~Sitter
space. These correspond to those in which the spatial sections at any
synchronous time have constant curvature. We will consider the three cases of
positive, zero and negative curvature. Of these the most natural choice of
coordinates are those associated with spatial sections of constant positive
curvature. These will be considered in the present section, with the other
cases considered in section~4. Further properties of the coordinates used for
the de~Sitter spacetime and the relations between them have been described by
Eriksen and Gr\o n \cite{R8}, although in a slightly different notation.



The most natural set of coordinates covering the four-dimensional hyperboloid
(2.2) are the ``global coordinates'' given by
 \begin{eqnarray}
 Z_0&&=a\sinh(\tau/a) \nonumber\\
Z_1&&=a\cosh(\tau/a)\cos\chi \nonumber\\
Z_4&&=a\cosh(\tau/a)\sin\chi\cos\theta \\
Z_2&&=a\cosh(\tau/a)\sin\chi\sin\theta\cos\phi \nonumber\\
Z_3&&=a\cosh(\tau/a)\sin\chi\sin\theta\sin\phi \nonumber
 \end{eqnarray}
in which the coordinates $\theta$ and $\phi$ are different from
those of section~2. Indeed the same symbols are used for a number of
different coordinate parametrisations below. With this, the line element
becomes
 \begin{eqnarray}
 \d s^2=\d \tau^2 -&&a^2\cosh^2(\tau/a)\\
   &&\times\big(\d\chi^2
+\sin^2\chi(\d\theta^2+\sin^2\theta\d\phi^2)\big). \nonumber
 \end{eqnarray}
 In this case, the privileged family of timelike observers have worldlines on
which $\chi$, $\theta$ and $\phi$ are constant. For this family, the 3-spaces
$\tau=$ const., which represent the universe at any time, have the geometry
of 3-spheres $S^3$ which appear to contract to a minimum size at $\tau=0$ and
then re-expand.


\bigskip\centerline{ Fig. 3}\bigskip


Let us now return to consider the shock wave, which is given by (2.13) with
the null particles having the worldlines (2.14). Using (3.1), we see that the
shock wave is located at $\cos\chi=-\tanh(\tau/a)$. Thus, the hyperspherical
coordinate $\chi$ monotonically increases from $\chi=0$ as $\tau\to -\infty$
to $\chi=\pi/2$ at $\tau=0$ and $\chi=\pi$ as $\tau\to +\infty$. This has a
natural physical interpretation shown in figure~3. As the synchronous
(proper) time $\tau$ of comoving observers at fixed points $P$ given by
constant $\chi_0$, $\theta_0$ and $\phi_0$ increases, the impulse propagates
from the ``North pole'' ($\chi=0$) to the ``South pole'' ($\chi=\pi$) of the
closed space $S^3$. Moreover, the radius of $S^3$ is given by
$R=a\cosh(\tau/a)$; i.e. as $\tau$ increases, for $\tau<0$, the space
contracts to a minimum radius $a$ (at $\tau=0$) and then re-expands for
$\tau>0$.


This is visualized in figure~3 where the coordinate $\phi$ was suppressed so
that the evolution of the universe is represented by a sequence of 2-spheres
(with coordinates $\chi$ and $\theta$) of which only hemispheres
are shown. At any time, the impulse is here given by a (semi-)circle
which is located exactly in the circular intersection of the
corresponding sphere with a cylinder of radius $a$. Indeed, the distance of
the impulse from the axis through the poles is given by
$R\sin\chi=a\cosh(\tau/a)\sqrt{1-\tanh^2(\tau/a)}\equiv a$. The null
particles which are the singular sources of the wave are located at
$\theta=0$ and $\theta=\pi$, as follows immediately from (2.14) and (3.1).



In order to investigate the global conformal structure of the space-time,
it is useful to introduce the conformal time $\eta_p$ defined by
 \begin{equation}
 \eta_p=2\arctan(e^{\tau/a}).
\end{equation}
 Then $\sinh(\tau/a) = -\cot\eta_p$,
$\cosh(\tau/a) = 1/\sin\eta_p$ and the line element (3.2) becomes
 \begin{equation}
 \d s^2={a^2\over\sin^2\eta_p}\big(\d\eta_p^2 -\d\chi^2
-\sin^2\chi(\d\theta^2+\sin^2\theta\d\phi^2)\big).
\end{equation}
 In this case, the appropriate conformal diagram which is shown
in figure~4 covers the coordinate ranges from $\eta_p=0$ ($\tau=-\infty$)
to $\eta_p=\pi$ ($\tau=\infty$), and from $\chi=0$ to $\chi=\pi$.



\bigskip\centerline{ Fig. 4}\bigskip



It may then be observed that the condition $Z_0+Z_1=0$
corresponds to the single root $\chi=\eta_p$, and hence that
 $$ \delta(Z_0+Z_1)={1\over a}\,\delta(\chi-\eta_p). $$
The impulsive wave in the conformal diagram in figure~4 is simply
located along the line $\chi=\eta_p$. Moreover,
since the function $H(Z_4)$ only need be evaluated for
$\chi=\eta_p$,
it can be seen that the impulsive addition to the de~Sitter line element
(3.4) is given by
 \begin{eqnarray}
 \d s_1^2 =4pa&&\left[2
-\cos\theta\ln\left|{1+\cos\theta\over1-\cos\theta}\right|\right]
 \nonumber\\
&&\times\delta(\eta_p-\chi)(\d\eta_p-\d\chi)^2.
 \end{eqnarray}
        From this, it can immediately be seen that the two null particles are
located at the points $\theta=0$ and $\theta=\pi$, which correspond to the
opposite poles of the spherical wave surface.




\section{Other coordinate charts ($\Lambda>0$)}


Although the coordinates introduced in the previous section are the most
natural in the sense that they cover the entire de~Sitter space-time
and provide a simple interpretation of the shock geometry, it is
also appropriate to consider some alternative coordinate systems as they can
be interpreted in terms of different families of observers.


Taking the case in which spatial sections at synchronous times have zero
curvature, we can introduce the conformally flat coordinates
 \begin{eqnarray}
 Z_0&&={1\over2\eta}\big(a^2-\eta^2+(x-a)^2+y^2+z^2\big) \nonumber\\
Z_1&&={a\over\eta}(x-a) \nonumber\\
Z_2&&={a\over\eta}y \\
Z_3&&={a\over\eta}z \nonumber\\
Z_4&&={1\over2\eta}\big(a^2+\eta^2-(x-a)^2-y^2-z^2\big) . \nonumber
 \end{eqnarray}
 With this, the line element becomes
 \begin{equation}
 \d s^2={a^2\over\eta^2}\big(\d\eta^2 -\d x^2 -\d y^2 -\d z^2\big).
\end{equation}


When coming to describe the geometry of the impulsive wave in these
coordinates, it is convenient to put
 $$ x=\rho\cos\theta, \qquad y=\rho\sin\theta\cos\phi, \qquad
z=\rho\sin\theta\sin\phi , $$
 where $\rho\in[0,\infty)$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi)$,
 so that
 $$ \rho^2=x^2+y^2+z^2 $$
 and then we have
 \begin{equation}
 Z_0+Z_1=-{1\over2\eta}(\eta^2-\rho^2)
 \end{equation}
 and
 $$ Z_4={1\over2\eta}(\eta^2-\rho^2+2ax). $$
 Thus, on the impulsive wave on which $Z_0+Z_1=0$, we have $\eta=\pm\rho$,
and
 $$ Z_4 ={ax\over\eta} =a{\rho\over\eta}\cos\theta. $$
 It may be noted that, in order to include the maximum analytic extension of
the space-time, it is necessary to permit $\eta$ to cover the range
$(-\infty,\infty)$. In addition, from (4.3) it follows that
 $$ \delta(Z_0+Z_1) =\delta(\eta-\rho) +\delta(\eta+\rho) $$
 which indicates that the impulsive wave has the two components:
 \begin{eqnarray}
{\rm when} \quad\ \eta=\rho, \ \qquad &&Z_4=a\cos\theta \nonumber\\
{\rm when} \quad \eta=-\rho, \qquad &&Z_4=-a\cos\theta. \nonumber
 \end{eqnarray}
 Combining these, we see that the impulsive addition to the de~Sitter line
element (4.2) is given by
 \begin{eqnarray}
 \d s_1^2&& =4p\left[2
-\cos\theta\ln\left|{1+\cos\theta\over1-\cos\theta}\right|\right] \nonumber\\
&&\times\left[\delta(\eta-\rho)(\d\eta-\d\rho)^2
+\delta(\eta+\rho)(\d\eta+\d\rho)^2 \right] .
 \end{eqnarray}
This looks like two impulses. However, both components are required for the
conformal picture to be geodesically complete: since $\rho\ge 0$,
the term $\delta(\eta-\rho)$ is nonvanishing only for $\eta\ge0$
while $\delta(\eta+\rho)$ applies only for $\eta\le 0$.
Again it can be seen that the
two null source particles are located at the poles $\theta=0$ and
$\theta=\pi$ on the spherical wave surface.


The conformal diagram of the section $y=z=0$ is shown in figure~5. In this,
the whole de~Sitter universe is covered by two regions in which $\eta\ge0$
and $\eta\le0$. It is also convenient to use $x=\rho\cos\theta$ where
$\theta=0$ or $\pi$. The diagram thus illustrates the paths of the two null
particles propagating in opposite directions. The impulsive waves $\rho=\eta$
and $\rho=-\eta$ can also be seen to represent either a contracting 2-sphere
in an exponentially expanding universe or an expanding 2-sphere in a
contracting universe.


\bigskip\centerline{ Fig. 5}\bigskip


This point can also be clarified by introducing the proper time $\tau$
defined by
 $$ \eta=\pm ae^{-\tau/a} , $$
 so that the line element (4.2) of de~Sitter space-time becomes
 \begin{equation}
 \d s^2 =\d\tau^2 -e^{2\tau/a}\big(\d\rho^2
+\rho^2(\d\theta^2+\sin^2\theta\d\phi^2)\big) .
 \end{equation}
 It can thus be seen that the region in which $\eta>0$ corresponds to an
exponentially expanding ``inflationary'' universe. The region included in the
conformal diagram in which $\eta<0$ corresponds to a time reversed
exponentially contracting universe.


It may also be observed that the ``radius'' of the spherical impulsive wave
is given by
 $$ \rho =|\eta| =ae^{-\tau/a}. $$
 Thus, in the {\it expanding universe} with the 3-dimensional
spaces $\tau=$ const. being flat, the impulsive wave is an exponentially
{\it contracting} 2-sphere. Conversely (changing $\tau\to-\tau$),
it can also be considered as an expanding spherical wave
in an exponentially contracting universe.


Let us finally consider how the impulsive wave would appear to the family of
observers which, at any synchronous time, are located in a space of
a negative curvature. In this case, the four-dimensional hyperboloid (2.2) can
be parametrised by
 \begin{eqnarray}
 Z_0&&=a\sinh(\tau/a)\cosh\psi \nonumber\\
Z_1&&=a\cosh(\tau/a) \nonumber\\
Z_4&&=a\sinh(\tau/a)\sinh\psi\cos\theta \\
Z_2&&=a\sinh(\tau/a)\sinh\psi\sin\theta\cos\phi \nonumber\\
Z_3&&=a\sinh(\tau/a)\sinh\psi\sin\theta\sin\phi \nonumber
 \end{eqnarray}
 and the de~Sitter line element takes the form
 \begin{eqnarray}
 \d s^2=\d \tau^2 - &&a^2\sinh^2(\tau/a) \\
  && \times\big(\d\psi^2
 +\sinh^2\psi(\d\theta^2+\sin^2\theta\d\phi^2)\big). \nonumber
 \end{eqnarray}
 However, it may be noted that these coordinates only cover part of the
hyperboloid.


Again it is convenient to introduce a conformal time defined here by
 \begin{equation}
 \eta_n =\cases{ 2\coth^{-1}e^{\tau/a} \hskip 15pt &for $\tau\ge0$ \cr
 \noalign{\medskip}
               2\tanh^{-1}e^{\tau/a} \hskip 15pt &for $\tau\le0$. \cr }
 \end{equation}
 It may be noted that, as $\tau$ increases from $-\infty$ to 0 and then to
$\infty$, $\eta_n$ increases from 0 to $\infty$ and then decreases back to 0.
Using this, the hyperboloid is parametrised by (4.6) where
$\sinh(\tau/a) = \pm1/\sinh\eta_n$,
$\cosh(\tau/a) = \cosh\eta_n/\sinh\eta_n$ and the line element (4.7) becomes
 \begin{equation}
 \d s^2={a^2\over\sinh^2\eta_n}\big(\d\eta_n^2 -\d\psi^2
-\sinh^2\psi(\d\theta^2+\sin^2\theta\d\phi^2)\big).
\end{equation}
 Since
 $$ Z_0+Z_1={a\over\sinh\eta_n}(\cosh\eta_n\pm\cosh\psi), $$
 we have
 $$ \delta(Z_0+Z_1) ={1\over a}\big[ \delta(\eta_n-\psi)
+\delta(\eta_n+\psi)\big]. $$
 Thus, on the impulsive wave, $\psi=\pm\eta_n$ and it can again be shown that
the impulsive addition to the de~Sitter line element (4.9) is given by
 \begin{eqnarray}
 \d s_1^2&& =4pa\left[2
-\cos\theta\ln\left|{1+\cos\theta\over1-\cos\theta}\right|\right]\\
&&\times\big[\delta(\eta_n-\psi)(\d\eta_n-\d\psi)^2
  +\delta(\eta_n+\psi)(\d\eta_n+\d\psi)^2 \big] . \nonumber
 \end{eqnarray}
 This again looks like two impulses. However, the first term is nonvanishing
only for $\psi>0$, while the second term is nonvanishing only for $\psi<0$.
Both components are required for the conformal picture to be geodesically
complete. This is described in figure~6.


\bigskip\centerline{ Fig. 6}\bigskip


The spherical pulse is now propagating in a 3-space $\tau=$ const.
which has a negative curvature and contracts for $\tau<0$ according
to $a|\sinh(\tau/a))|$. The impulse is located at
$|\psi|=\eta_n=2\tanh^{-1} e^{\tau/a}$, i.e. it is expanding from
$\psi=0$ to $|\psi|=\infty$.





\section{A null particle in an anti-de~Sitter background}

For a vacuum universe with a negative cosmological constant ($\Lambda<0$),
the Schwarzschild--anti-de~Sitter solution can be written in the form (2.1)
but with $a^2$ replaced by $-a^2$. It is then possible to proceed
as in section~2 to boost the singular point $r=0$ in the limit as
$m\to0$ in an appropriate way.
This again produces an impulsive wave, but this time in an
anti-de~Sitter background.

Before proceeding, however, it is first appropriate to recall that the
anti-de~Sitter solution can be represented as the four-dimensional
hyperboloid
 \begin{equation}
 {Z_0}^2 -{Z_1}^2 -{Z_2}^2 -{Z_3}^2 +{Z_4}^2 =a^2
\end{equation}
 embedded in a five-dimensional Minkowski space-time
 \begin{equation}
 \d s^2= \d{Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2 +\d{Z_4}^2
\end{equation}
 with two timelike dimensions, $Z_0$ and $Z_4$. The natural parametrisation
of this is given
by
  \begin{eqnarray}
 Z_4&&=\sqrt{a^2+r^2}\,\cos(t/a) \nonumber\\
 Z_0&&=\sqrt{a^2+r^2}\,\sin(t/a) \nonumber\\
 Z_1&&=r\cos\theta \\
 Z_2&&=r\sin\theta\cos\phi \nonumber\\
 Z_3&&=r\sin\theta\sin\phi . \nonumber
 \end{eqnarray}
 From this it can be seen that a source at the origin $r=0$ is located on the
line
 \begin{equation}
 (Z_0, Z_1, Z_2, Z_3, Z_4) =\big(a\sin(t/a), 0, 0, 0, a\cos(t/a)\big)
 \end{equation}
 which is a closed timelike geodesic which returns to the same event after
$\Delta t=2\pi a$. This is illustrated in figure~7 in which the spacelike
coordinates $Z_2$ and $Z_3$ are suppressed. It can thus be seen that the
$m\to0$ limit of the Schwarzschild--anti-de~Sitter solution has a single
source following a closed timelike line.

\bigskip\centerline{ Fig. 7}\bigskip

In this case, the first order perturbation of the anti-de~Sitter line
element, equivalent to (2.7), can be written as
 \begin{eqnarray}
 \d s_1^2 &&=-{2ma^2\over r} \Bigg[
\left( {Z_4\d Z_0-Z_0\d Z_4\over{Z_4}^2+{Z_0}^2} \right)^2 \nonumber\\
 &&\qquad\qquad +{a^2\over r^2}
\left( {Z_0\d Z_0+Z_4\d Z_4\over{Z_4}^2+{Z_0}^2} \right)^2 \Bigg] ,
\end{eqnarray}
 where $r=({Z_0}^2+{Z_4}^2-a^2)^{1/2}$. A boost can then be performed in the
$Z_1$ direction using (2.8). Again it can be seen that, in the limit as
$v\to1$, this linear perturbation term vanishes everywhere except on the
hypersurface $Z_0+Z_1=0$. By putting $m=p\sqrt{1-v^2}$, the line element then
takes the form
 \begin{eqnarray}
 \d s^2 &&=\d{Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2 +\d{Z_4}^2 \nonumber\\
&&\qquad -2pa^2H(Z_4)\,\delta(Z_0+Z_1)\,(\d Z_0+\d Z_1)^2 ,
 \end{eqnarray}
 where using (2.11) with $x^2=(Z_0+vZ_1)^2/(1-v^2)$,
 \begin{equation}
 H(Z_4) =\int_{-\infty}^{\infty}
{({Z_4}^2-a^2){Z_4}^2+(a^2+{Z_4}^2)x^2 \over
\left({Z_4}^2-a^2+x^2\right)^{3/2}
\left({Z_4}^2+x^2\right)^2}\,\d x.
\end{equation}
Since in this case the integrand is nonsingular we obtain
immediately
 \begin{equation}
 H(Z_4) =-{2\over a^2} \left[
2-{Z_4\over a}\ln\left|{a+Z_4\over a-Z_4}\right| \right].
\end{equation}


It can be seen that in this case the resulting space-time differs from the
anti-de~Sitter space-time by the inclusion of an impulsive wave located on
the null hypersurface given by
 \begin{equation}
 Z_0+Z_1=0, \qquad {Z_4}^2 -{Z_2}^2 -{Z_3}^2 =a^2.
\end{equation}
 Unlike the case in a de~Sitter background, this is a now an evolving
2-hyperboloid in a 5-dimensional Minkowski space with two time dimensions.


Also, the source is now a null particle having the worldline
 $$ Z_0+Z_1=0, \qquad Z_2=Z_3=0, \qquad Z_4=\pm a. $$
 This is represented by the two null straight lines on the four-dimensional
hyperboloid as indicated in figure~8. However, since time is here effectively
wrapped around the hyperboloid, this represents a null particle which
propagates to infinity and then comes back again in an endless cycle.

\bigskip\centerline{ Fig. 8}\bigskip


\section{A global coordinate chart ($\Lambda<0$)}


In an anti-de~Sitter space-time, it is appropriate initially to consider the
natural global coordinate system obtained by putting $r=a\sinh R$ in (5.3)
by which
 \begin{eqnarray}
 Z_4&&=a\cosh R\cos(t/a) \nonumber\\
 Z_0&&=a\cosh R\sin(t/a) \nonumber\\
 Z_1&&=a\sinh R\cos\theta \\
 Z_2&&=a\sinh R\sin\theta\cos\phi \nonumber\\
 Z_3&&=a\sinh R\sin\theta\sin\phi \nonumber
 \end{eqnarray}
 and the line element becomes
 \begin{equation}
 \d s^2=\cosh^2\!R\,\d t^2 -a^2(\d R^2
+\sinh^2\!R(\d\theta^2+\sin^2\theta\d\phi^2)).
\end{equation}
 In this case, the privileged family of timelike observers have worldlines on
which $R$, $\theta$ and $\phi$ are constant, their proper times
are given by $\tau=t\cosh R$. The timelike slices $t=$ const.
are three dimensional spaces of constant negative curvature.


We may now consider the impulsive wave on which $Z_0+Z_1=0$. It can be seen
that this is given by
 \begin{equation}
 \tanh R\cos\theta=-\sin(t/a).
 \end{equation}
 It may be noticed that this does not give such a convenient coordinate
condition as does the equivalent case in a de~Sitter background.
Nevertheless, we can show that
 $$ \delta(Z_0+Z_1)={\delta\big(t+a\sin^{-1}(\tanh R\cos\theta)\big)
\over\sqrt{1+\sinh^2R\sin^2\theta}} $$
 and, evaluating $H(Z_4)$ using (6.1), we obtain that the impulsive addition
to the anti-de~Sitter line element (6.2) is given by
 \begin{eqnarray}
 \d s_1^2 &&=4p\left[ {2\over\cosh R\cos(t/a)} -\ln \left| {1+\cosh
R\cos(t/a)\over1-\cosh R\cos(t/a)} \right| \right] \nonumber\\
  \times&&\delta\big(t+a\sin^{-1}(\tanh R\cos\theta)\big) \\
  \times&&\big(a{\cos\theta\over\cosh R} \d R -a\sinh R\sin\theta\d\theta
  +\cosh R\cos(t/a)\d t \big)^2, \nonumber
 \end{eqnarray}
which looks somewhat complicated.


We would now like to visualize this wave as it would be observed by the
natural family of observers in the 3-dimensional space $(R,\theta,\phi)$. For
this, we consider a section $t=$ const. through the space-time. It is then
convenient to introduce the coordinate $\tilde Z_0=a\cosh R$ which, together
with the coordinates $Z_1$, $Z_2$ and $Z_3$ span a 4-dimensions
pseudo-Euclidean space
 \begin{equation}
 \d{s_4}^2 =\d{\tilde Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2 .
 \end{equation}
The hypersurface of the impulsive wave can now be
visualised at any time as the intersection of the plane $Z_1=-\sin(t/a)\tilde
Z_0$ with half of a rotational hyperboloid
 \begin{equation}
 \tilde Z_0^2 -Z_1^2 -(\sqrt{Z_2^2+Z_3^2})^2 =a^2
 \end{equation}
 in the pseudo-Euclidean half space $(\tilde Z_0, Z_1,\sqrt{Z_2^2+Z_3^2})$
representing the 3-dimensional space of constant negative
curvature with the metric
$ \d s^2_3=-a^2(\d R^2+\sinh^2\!R(\d\theta^2+\sin^2\theta\d\phi^2))$.
This is illustrated in figure~9.

\bigskip\centerline{ Fig. 9}\bigskip

The source of the impulsive wave is a single null particle located on the
wave at the point $\sqrt{Z_2^2+Z_3^2}=0$, $Z_1=-a\tan(t/a)$. This indicates
that the null particle moves from $Z_1=\infty$ to $Z_1=-\infty$ as $t$ goes
from $-{\pi\over2}a$ to ${\pi\over2}a$.


It is then possible to visualize the wavefront at any time by projecting the
intersection of the plane and the hyperboloid onto the space of $Z_1$ and
$\sqrt{Z_2^2+Z_3^2}$ as illustrated in figure~9, and then rotating it about
the $Z_1$ axis. In this way, the $Z_2$ and $Z_3$ coordinates are reintroduced
by putting
 $$ Z_2=\sqrt{Z_2^2+Z_3^2}\cos\phi, \qquad Z_3=\sqrt{Z_2^2+Z_3^2}\sin\phi. $$
 The impulsive wave is thus represented by the sequence of hyperboloidal
surfaces illustrated in figure~10. While the angle $\phi$ is a ``polar''
angle associated with the axial symmetry of the surfaces, the angle $\theta$
is measured from the (positive-oriented) axis of symmetry. It can be shown
that each hyperboloidal surface representing the impulsive wave front has an
asymptotic angle $\theta_a$ given by $\theta_a=t/a+\pi/2$. Therefore, as
$t/a$ grows from $-\pi/2$ to $\pi/2$ (and the source null particle moves from
$Z_1=\infty$ to $Z_1=-\infty$) the wave surfaces open from $\theta_a=0$ to
$\theta_a=\pi$.

\bigskip\centerline{ Fig. 10}\bigskip

Since the time coordinate $t$ is periodic, it can be seen that the impulsive
wave propagates to $Z_1=-\infty$ and then back to $Z_1=\infty$ in an endless
cycle.


In order to investigate the conformal structure of the space-time in these
coordinates, we introduce the conformal time $\chi$ defined by
 \begin{equation}
 \chi=2\arctan(e^R)-\pi/2.
\end{equation}
 In this case we have $\cosh R=\sec\chi$, $\sinh R=\tan\chi$
 and line element becomes
 \begin{equation}
 \d s^2={a^2\over\cos^2\!\chi}\left({1\over a^2}\d t^2 -\d\chi^2
-\sin^2\!\chi(\d\theta^2+\sin^2\theta\d\phi^2)\right).
 \end{equation}
 In this case, the impulsive wave on which $Z_0+Z_1=0$ can be seen to occur
when $\sin\chi\cos\theta=-\sin(t/a)$ in agreement with (6.3).
The location of the wave in the conformal diagram of the
anti-de~Sitter space-time for $\theta=0$ is shown in figure~11.

\bigskip\centerline{ Fig. 11}\bigskip


\section{Other coordinate charts ($\Lambda<0$)}


Let us now consider how the impulsive wave would appear to the alternative
family of observers which, at any synchronous time, are located in a space of
constant negative curvature. In this case, the four-dimensional hyperboloid
(5.1) can be parametrised by
 \begin{eqnarray}
 Z_0&&=a\sin(\tau/a) \nonumber\\
Z_1&&=a\cos(\tau/a)\sinh\psi\cos\theta \nonumber\\
Z_2&&=a\cos(\tau/a)\sinh\psi\sin\theta\cos\phi \\
Z_3&&=a\cos(\tau/a)\sinh\psi\sin\theta\sin\phi \nonumber\\
Z_4&&=a\cos(\tau/a)\cosh\psi \nonumber
 \end{eqnarray}
 and the anti-de~Sitter line element takes the form
 \begin{eqnarray}
 \d s^2=\d \tau^2 - && a^2\cos^2(\tau/a) \\
  &&\times \big(\d\psi^2
+\sinh^2\psi(\d\theta^2+\sin^2\theta\d\phi^2)\big). \nonumber
 \end{eqnarray}
The time-slices $\tau=$ const. are 3-dimensional spaces of constant negative
curvature, but which do not have a constant ``size'' as occurs for the global
coordinates (6.2) in the previous section. Their curvature change according
to $a\cos(\tau/a)$. However, it may be noted that these coordinates only
cover part of the hyperboloid.


Again it is convenient to introduce a conformal time defined by
 \begin{equation}
 \eta_n =\tanh^{-1}\big(\sin(\tau/a)\big) .
 \end{equation}
 It may be noted that, as $\tau/a$ increases from $-\pi/2$ to 0 and then to
$\pi/2$, $\eta_n$ increases from $-\infty$ to $\infty$. With these
parameters, the hyperboloid is now given by (7.1) with
$\sin(\tau/a)=\tanh\eta_n$, $\cos(\tau/a)={\rm sech}\,\eta_n$
 and line element (7.2) becomes
 \begin{equation}
 \d s^2={a^2\over\cosh^2\eta_n}\big(\d\eta_n^2 -\d\psi^2
-\sinh^2\psi(\d\theta^2+\sin^2\theta\d\phi^2)\big).
\end{equation}


Proceeding as before, we can consider the impulsive wave on which
$Z_0+Z_1=0$. However, this again does not give such a convenient coordinate
condition as does the equivalent case in a de~Sitter background. Calculations
can again be performed to yield an expression similar to (6.4). However, this
does not provide any further insights. The conformal diagram with
the impulse for $\theta=0$ is given in figure~12.



\bigskip\centerline{ Fig. 12}\bigskip


It is finally appropriate here to consider a conformally flat coordinate
system. Accordingly, we can put
 \begin{eqnarray}
 Z_0&&={1\over2x}\big(a^2-\eta^2+x^2+y^2+(z-a)^2\big) \nonumber\\
Z_1&&={a\over x}(z-a) \nonumber\\
Z_2&&={1\over2x}\big(a^2+\eta^2-x^2-y^2-(z-a)^2\big) \\
Z_3&&={a\over x}y \nonumber\\
Z_4&&={a\over x}\eta .\nonumber
 \end{eqnarray}
 With this, the line element becomes
 \begin{equation}
 \d s^2={a^2\over x^2}\big(\d\eta^2 -\d x^2 -\d y^2 -\d z^2\big).
\end{equation}
 It is also convenient to put
 \begin{equation}
 x=\rho\cos\theta, \quad y=\rho\sin\theta\cos\phi, \quad
z=\rho\sin\theta\sin\phi
 \end{equation}
 so that the line element becomes
 \begin{equation}
 \d s^2={a^2\over \rho^2\cos^2\theta}\big(\d\eta^2 -\d\rho^2
-\rho^2(\d\theta^2 +\sin^2\theta\d\phi^2)\big).
\end{equation}
 It can also be shown that the form of the impulse is given by
 \begin{equation}
 \delta(Z_0+Z_1) = {|x|\over\rho}
 \big[\delta(\eta-\rho) +\delta(\eta+\rho)\big]
 \end{equation}
and the impulsive addition to the anti-de~Sitter line
element is given by
 \begin{eqnarray}
 \d s_1^2 &&={4p\over|\cos\theta|}\left[2
-{1\over\cos\theta}\ln\left|{1+\cos\theta\over1-\cos\theta}\right|\right] \\
&&\quad \times\left[\delta(\eta-\rho)(\d\eta-\d\rho)^2
+\delta(\eta+\rho)(\d\eta+\d\rho)^2 \right]. \nonumber
 \end{eqnarray}
This is clearly the simplest coordinate form of the shock geometry (5.6) in
an anti--de~Sitter background. Again this is only one impulse for $\eta>0$
and $\eta<0$. Also, the impulsive wave is given by $\eta=\pm\rho$.
The metric is regular for $\theta=\pi/2$ so that it is singular
only for $\theta=0$ and $\theta=\pi$.
The conformal diagram is given in figure~13.

\bigskip\centerline{ Fig. 13}\bigskip

 For any value of $\eta$, the surfaces $\rho=$ const. according to (7.7) look
like spheres. However, {\it they are not spheres}: they are surfaces that are
conformal to spheres. Actually,  they are hyperboloidal. Indeed, performing
the transformation $\xi=\tanh^{-1}(\sin\theta)$ we can write the section
$\eta=\pm\rho$ of the line element (7.8) as
$\d\sigma^2=-a^2(\d\xi^2+\sinh^2\xi\d\phi^2)$. This represents the
hyperboloidal 2-space of constant {\it negative} curvature $-a$.


We may thus conclude that the conformally flat coordinate system (7.8) is
well adapted for describing the impulsive wave (5.6) in anti-de~Sitter
space-time. Conversely, the coordinates (6.2) and (7.2) are not suitable due
to the fact that their corresponding 2-surfaces {\it are spheres} having the
line element $\d\sigma^2=-C^2(\d\theta^2+\sin^2\theta\d\phi^2)$, $C=$ const.
This naturally explains the complicated forms of the impulsive additions to
the anti-de~Sitter line element (6.4) in these coordinates. By contrast, we
have previously observed that all three coordinate systems (3.2), (4.2) and
(4.7) that we have introduced in the $\Lambda>0$ case are suitable for
describing {\it spherical} impulsive gravitational waves in the de~Sitter
universe since they also have
$\d\sigma^2=-C^2(\d\theta^2+\sin^2\theta\d\phi^2)$.


\section{Conclusions}

Following the method of Hotta and Tanaka \cite{R6}, we have
boosted the Schwarzschild--de~Sitter and the Schwarzschild--anti-de~Sitter
metrics to the ultrarelatistic limit and have thus constructed exact
solutions for impulsive waves in backgrounds with a nonzero cosmological
constant. In a de~Sitter background, a spherical
gravitational wave is generated by two null particles propagating in opposite
directions.  We have used three privileged families of observers in which the
constant time surfaces have constant positive, negative or zero curvature.
We have shown that each of these coordinate systems are suitable for describing
the geometry of the impulsive wave. In the closed form of the de~Sitter
universe which contracts to a minimum size and then re-expands,
the impulse propagates from the ``North pole'' to the ``South
pole''. In an expanding flat and open de~Sitter universe it is a contracting
sphere; alternatively in a contracting universe the sphere is expanding.


By contrast, the impulsive wave generated in an anti-de~Sitter background is
hyperboloidal. It is generated by a single null particle. We have also shown
that it is only the conformally flat coordinates that form a suitable natural
parameterisation of this solution.


It may also be noted that a family of impulsive spherical gravitational waves
have been constructed by Hogan \cite{R9} which appear to propagate either in a
de~Sitter or an anti-de~Sitter background. However, these are not related to
the solutions given above.


\acknowledgments

JP is grateful for hospitality at the Mathematical Sciences Department at
Loughborough University and for support from the grant GACR-202/96/0206 of
the Czech Republic and the grant GAUK-230/96 of Charles University while
this work was undertaken.




\begin{references}

\bibitem{R1} P.C. Aichelburg and R.U. Sexl,  Gen. Relativ. Gravit.
      {\bf 2}, 303 (1971).

\bibitem{R2} V. Ferrari and P. Pendenza,  Gen. Relativ. Gravit.
      {\bf 22}, 1105 (1990).

\bibitem{R3} H. Balasin and H. Nachbagauer, Class. Quantum Grav.
      {\bf 12}, 707 (1995).

\bibitem{R4} H. Balasin and H. Nachbagauer,  Class. Quantum Grav.
      {\bf 13}, 731 (1996).

\bibitem{R5} C.O. Loust\'o and N. S\'anchez, Nucl. Phys.
      {\bf B383}, 377 (1992).

\bibitem{R6} M. Hotta and M. Tanaka,  Class. Quantum Grav.
      {\bf 10}, 307 (1993).

\bibitem{R7} J. Bi\v{c}\'ak and J. Podolsk\'y, \prd {\bf 52}, 887 (1995).


\bibitem{R8} E. Eriksen and \O. Gr\o n, Int. J. Mod. Phys. D
      {\bf 4}, 115 (1995).

\bibitem{R9} P.A. Hogan,  \pl {\bf 171A}, 21 (1992).

\end{references}

\begin{figure}
\caption{The de~Sitter space-time represented as a four-dimensional
hyperboloid embedded in a five-dimensional Minkowski space-time. The
coordinates $Z_2$ and $Z_3$ have been suppressed, and the timelike world
lines are indicated of the two test particles which correspond to the weak
limit of the source of the Schwarzschild--de~Sitter space-time.}
\label{fig1}
\end{figure}

\begin{figure}
\caption{The de~Sitter space-time represented as a four-dimensional
hyperboloid embedded in a five-dimensional Minkowski space-time with the
coordinates $Z_2$ and $Z_3$ suppressed. The null world lines are indicated of
the two particles which correspond to the source of the
Schwarzschild--de~Sitter space-time boosted to the limit in which its
velocity is that of light and its mass vanishes.}
\label{fig2}
\end{figure}

\begin{figure}
\caption{In natural coordinates the de~Sitter space-time is a closed
universe with the spatial geometry of a 3-sphere whose radius is given by
$R=a\cosh(\tau/a)$. Suppressing the coordinate $\phi$, its evolution can be
visualized as a sequence of 2-spheres which are indicated here by
hemispheres which contract to a minimum size and then re-expand. At any
time, the impulsive wave is a 2-sphere represented here by a semi-circle
located at the circular intersection of a cylinder of radius $a$ with the
corresponding sphere representing the universe. This propagates from the
``North pole'' $\chi=0$ as $\tau\to-\infty$ across the equator
$\chi=\pi/2$ at $\tau=0$ to the ``South pole'' $\chi=\pi$ as $\tau\to
\infty$ while the whole universe contracts to a minimum size $R=a$ at
$\tau=0$ and then re-expands. }
\label{fig3}
\end{figure}

\begin{figure}
\caption{The conformal diagram of the de~Sitter space-time with
natural global coordinates $\eta_p, \chi$. Each point of the
diagram represents a 2-sphere parametrized by coordinates $\theta,
\phi$. The lines $\chi=0$ and $\chi=\pi$ are identified. In
contrast to Minkowski space-time, de~Sitter has a space-like
future and past infinity for null and time-like lines. The
impulsive wave is located along $\chi=\eta_p$ with two singular
points sources at $\theta=0$ and $\theta=\pi$.}
\label{fig4}
\end{figure}

\begin{figure}
\caption{The conformal diagram of the section $y=z=0$ of the de~Sitter
space-time with coordinates $\eta, \rho=|x|$. The wave is located
along $\eta=\pm\rho$ and represents either an exponentially
contracting 2-sphere in an exponentially expanding flat
inflationary universe or an expanding 2-sphere in a contracting
space.}
\label{fig5}
\end{figure}

\begin{figure}
\caption{The conformal diagram of the de~Sitter space-time with
coordinates $\tau, \psi$. For the corresponding comoving
observers the spherical impulsive wave expands in a contracting
space of negative curvature. }
\label{fig6}
\end{figure}

\begin{figure}
\caption{The anti-de~Sitter space-time represented as a four-dimensional
hyperboloid embedded in a five-dimensional space-time with two timelike
dimensions $Z_0$ and $Z_4$. Here the coordinates $Z_2$ and $Z_3$ have been
suppressed, and the closed world line of the particle which corresponds to
the weak limit of the source of the Schwarzschild--anti-de~Sitter space-time
is indicated.}
\label{fig7}
\end{figure}

\begin{figure}
\caption{The anti-de~Sitter space-time represented as a four-dimensional
hyperboloid embedded in a five-dimensional Minkowski space-time with
two timelike dimensions $Z_0$ and $Z_4$. The null world line is indicated of
the particle which correspond to the source of the
Schwarzschild--anti-de~Sitter space-time boosted to the limit in which its
velocity is that of light and its mass vanishes. }
\label{fig8}
\end{figure}

\begin{figure}
\caption{The geometry of the impulsive wave in anti-de~Sitter
space-time as observed by natural family of observers at any time
is given by the intersection of the plane $Z_1=-\sin(t/a)\tilde
Z_0$ with half of a rotational hyperboloid in the
pseudo-Euclidean space so that the wavefronts are the spaces of
constant negative curvature.}
\label{fig9}
\end{figure}

\begin{figure}
\caption{By projecting the intersections representing the wave
fronts from figure~9 onto the plane of $Z_1$, ${Z_2^2+Z_3^2}$
and rotating them about the $Z_1$ axis we get the visualization
of the impulsive wave in the anti-de~Sitter space-time as a
sequence of hyperboloidal surfaces. The source of the wave is a
null particle moving along the axis of symmetry which at any time
is located in the corresponding surface.}
\label{fig10}
\end{figure}

\begin{figure}
\caption{The conformal diagram of the anti-de~Sitter space-time
with natural global coordinate chart $t, \chi$. Each point
represents a 2-sphere parametrized by $\theta, \phi$, antipodal
points on the spheres correspond to $\pm\chi$. Null and
space-like infinity form a time-like surface $\chi=\pi/2$,
$(R=\infty)$. The impulsive wave for $\theta=0$ is also indicated. }
\label{fig11}
\end{figure}

\begin{figure}
\caption{The synchronous coordinates $\tau, \psi, \theta, \phi$
cover only a part of the whole anti-de~Sitter space-time. The
time-like geodesics $\psi, \theta, \phi$ constants all converge.
In this diagram drawn for $\theta=0$ the impulsive wave
propagates along the indicated null lines. }
\label{fig12}
\end{figure}

\begin{figure}
\caption{The conformal diagram of the anti-de~Sitter space-time
and conformally flat coordinates $\eta, \rho$ which cover the
whole universe. Here we consider the section $Z_2=Z_3=0$ so that
$\phi=\pi/2$ and $\sin\theta=(\rho^2-\eta^2)/(2a\rho)$. The
impulsive wave is given simply by $\eta=\pm\rho$. }
\label{fig13}
\end{figure}



\end{document}