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\newcommand{\keywords}{gravitational waves, impulsive waves, {\sl pp}-waves}
\newcommand{\PACS}{04.20.Jb, 04.30.Nk}
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{J. B. Griffiths et al., Impulsive gravitational waves}
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\title{Exact solutions for impulsive gravitational waves}
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\author{J. B. Griffiths$^{1}$, and J. Podolsk\'y$^{2}$}
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  {$^{1}$Department of Mathematical Sciences, Loughborough University,
Loughborough,
\\ \hspace*{0.5mm} Leicestershire LE11 3TU, U.K. \\
  $^{2}$Institute of Theoretical Physics, Charles University,
V Hole\v{s}ovi\v{c}k\'ach 2, 18000 Prague 8,
\\ \hspace*{0.5mm} Czech Republic}
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\newcommand{\email}{\tt J.B.Griffiths@Lboro.ac.uk}
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\begin{abstract}
All impulsive waves (with no tails) in vacuum backgrounds of constant
curvature (i.e. Minkowski, de~Sitter and anti-de~Sitter) are classified
according to whether they are expanding or non-expanding. It is
demonstrated that the expanding waves may, in some sense, be regarded as
impulsive limits of Robinson--Trautman type~N solutions.
\end{abstract}
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\section{Introduction}

Many exact solutions of Einstein's equations are known which describe pure
impulsive gravitational waves in Minkowski and (anti-) de~Sitter
backgrounds. These belong to two distinct families in which the waves are
either expanding or non-expanding. In each case, solutions can be
constructed by three distinct methods. The non-expanding case is briefly
reviewed. Using an analogous approach, it is argued that the expanding cases
may be regarded as impulsive limits of Robinson--Trautman type~N solutions.


\section{Non-expanding impulsive waves with ${\bf\Lambda=0}$}
\label{Nonex.Lam0}


We first use Penrose' \cite{Pen72} geometrical method for constructing
impulsive gravitational waves in which space-time is cut along a null
hypersurface and the two pieces re-attached with a suitable warp. Consider
Minkowski space in the form
\begin{equation}
 {\rm d}s^2= 2{\rm d}u\,{\rm d}v -2{\rm d}\zeta{\rm d}\bar\zeta,
\label{M1}
\end{equation}
 and perform the coordinate transformation $u=U$, $v=V-H+UH_{Z}H_{\bar Z}$,
$\zeta=Z-UH_{\bar Z}$, where $H=H(Z,\bar Z)$ is an arbitrary real function,
to obtain
\begin{equation}
 {\rm d}s^2= 2{\rm d}U\,{\rm d}V
-2\left|{\rm d}Z
-U(H_{Z\bar Z}{\rm d}Z+H_{\bar Z\bar Z}{\rm d}\bar Z)\right|^2.
\label{M2}
\end{equation}
 Following Penrose's ``cut and paste'' method, we now take (\ref{M1}) with
$u=U$, $v=V$ and $\zeta=Z$ for $U<0$ and combine this with (\ref{M2}) for
$U>0$. The resulting line element
\begin{equation}
 {\rm d}s^2= 2{\rm d}U\,{\rm d}V
-2\left|{\rm d}Z
-U\Theta(U)(H_{Z\bar Z}{\rm d}Z+H_{\bar Z\bar Z}{\rm d}\bar Z)\right|^2,
\label{RosImp}
\end{equation}
 where $\Theta(u)$ is the Heaviside step function, is continuous across the
null hypersurface $U=0$. However, the discontinuity in the derivatives of
the metric produces impulsive components in the curvature tensor.


Let us now consider the combination of the transformations above as applied
to the two regions of space-time. This can be expressed in the explicit form
\begin{equation}
 u=U, \qquad
v=V-H\,\Theta(U)+U\,\Theta(U)\,H_{Z}H_{\bar Z},  \qquad
\zeta=Z-U\,\Theta(U)\,H_{\bar Z}.
\label{trans1}
\end{equation}
 This is obviously discontinuous at $u=0$ in such a way that \
$(u=0,v,\zeta,\bar\zeta)_{M^-}=\break
(u=0,v+H(\zeta,\bar\zeta),\zeta,\bar\zeta)_{M^+}$.
 This is the Penrose junction condition for re-attaching the two halves of
the space-time $M^-(u<0)$ and $M^+(u>0)$ with a ``warp''.


Significantly, the transformation (\ref{trans1}), relates the continuous form
of the impulsive wave metric (\ref{RosImp}), not to the initial metric
(\ref{M1}), but to the following metric which also includes an impulsive
component explicitly located on the wavefront $u=0$
\begin{equation}
 {\rm d}s^2= 2{\rm d}u\,{\rm d}v -2{\rm d}\zeta\,{\rm
d}\bar\zeta +2H(\zeta,\bar\zeta)\,\delta(u)\,{\rm d}u^2.
\label{BrinkImp}
\end{equation}
 It may be observed that this general solution may also be obtained as the
impulsive limit of the well known class of {\sl pp}-waves expressed in
standard Brink\-mann form. In fact, the solutions (\ref{BrinkImp}) also
arise \cite{Podol98} as impulsive limits of the more general class of Kundt
waves. Using a natural tetrad, the nonzero components of the curvature
tensor are given by $\Psi_4=H_{\zeta\zeta}\delta(u)$ and
\hbox{$\Phi_{22}=H_{\zeta\bar\zeta}\delta(u)$}.


The problems associated with the discontinuity in the transformation
(\ref{trans1}) have been investigated by Kunzinger and Steinbauer
\cite{KunSte99}. By considering the impulsive solution as limits of
sandwich waves in the two coordinate systems they have given a precise
meaning to the ``physical equivalence'' of the two metrics (\ref{RosImp})
and (\ref{BrinkImp}). They have also shown that the transformation can be
defined rigorously within Colombeau's theory of generalized functions.


Particular cases of this family of solutions can also be obtained by
boosting some well known space-times to the ultra\-relativistic limit.
Interesting examples have been obtained by boosting the Schwarz\-schild
space-time \cite{AicSex71}, the Kerr solution \cite{FerPen90, BalNac95,
BalNac96}, the Kerr--Newman family \cite{LouSan92}, and the Weyl solutions
\cite{PodGri98b}. In addition, since the main field equation is linear
\cite{Bonnor69}, separate solutions can always be superposed, either on
separate wave surfaces $u=$~const. thus forming a sequence of plane-fronted
impulsive waves, or by superposing different sources on the same wave
surface $u=0$.


\section{Non-expanding impulsive waves with ${\bf\Lambda\ne0}$}
\label{Nonex.Lamne0}

Non-expanding impulsive waves in a de~Sitter (or anti-de~Sitter) background
can be obtained in exactly analogous ways to those described above. They can
be derived using Penrose's ``cut and paste'' method \cite{PodGri99b}, or as
the impulsive limit of Kundt waves with non-zero cosmological constant
\cite{Podol98}. In addition, they can be derived by boosting sources
\cite{HotTan93, PodGri97} by elements of the (anti-) de~Sitter group for
solutions with $\Lambda\ne0$.


However, the global properties of the wavefront is different in these cases
\cite{PodGri97, PodGri98a}. When $\Lambda>0$, the impulsive wave is located
on the spherical horizon of the de~Sitter universe. In an anti-de~Sitter
background, the space-time contains closed timelike lines and the wavefront
is hyperboloidal and propagates from one side of the universe to the other
and is then reflected back in an endless cycle.


\section{Expanding impulsive waves with ${\bf\Lambda=0}$}
\label{Exp.Lam0}

We now consider the expanding impulsive waves that can be obtained using
Penrose's method of cutting Minkowski space-time on a null cone and
re-attaching with ``warp''.  This is exactly analogous to the case for
non-expanding impulsive waves. However, there is one important difference.
Although these solutions may also contain sources on the wave-front,
additional sources necessarily appear in this case due to the global
structure of the space-times. For the non-expanding case, the Minkowski
space on either side of the impulse is the entire space-time cut along a
null hypersurface. For spherical waves, on the other hand, points on one
side of the null cone do not match to the entire null cone on the other
side. This incompleteness can be interpreted in terms of deficit angles,
cosmic strings, or other forms of topological structure in the Minkowski
space on at least one side of the wave surface.


Applying the transformation $u\to u+v\xi\bar\xi$, $\zeta=v\xi$, the
line element (\ref{M1}) becomes
\begin{equation}
 {\rm d}s^2 =2{\rm d}u\,{\rm d}v-2v^2{\rm d}\xi\,{\rm d}\bar\xi,
\label{M3}
\end{equation}
 in which the hypersurfaces $u=$~const. are a family of null cones. To this
we apply the additional transformation $u=U|h'|/{\cal A}$, $v=V{\cal
A}/|h'|$,
$\xi=h(Z)+Uh'\bar\beta /2V{\cal A}$, where $h(Z)$ is an arbitrary
holomorphic function, $h'=h_{,Z}$, $\beta=h''/h'$ and
${\cal A}=1-U\beta\bar\beta/4V$. This takes (\ref{M3}) to the form
\begin{equation}
 {\rm d} s^2=2{\rm d} U\,{\rm d} V
-2\left|V{\rm d} Z-U\bar H\,{\rm d}\bar Z\right|^2,
\label{M4}
\end{equation}
 where $H={1\over2}\beta'-{1\over4}\beta^2=
{1\over2}(h'''/h')-{3\over4}(h''/h')^2$. Using the Penrose method, we take
(\ref{M3}) for $u<0$ and join this to  (\ref{M4}) for $U>0$ to produce a
spherical impulsive wave on the null cone $u=0=U$ given by the continuous
metric (\cite{NutPen92}--\cite{Hogan94})
\begin{equation}
 {\rm d}s^2 =2{\rm d}U\,{\rm d}V
-2\left|V{\rm d} Z-U\Theta(U)\bar H\,{\rm d}\bar Z\right|^2.
\label{ContImp}
\end{equation}


As in the nonexpanding case, it is possible to combine the two
transformations either side of the wavefront (for details see
\cite{PodGri99a}). Putting $F(\xi)=h(Z)-Z$ at $U=0$, this relates the
continuous form of the line element (\ref{ContImp}), not to (\ref{M3}), but
to the form
\begin{eqnarray}
 &&{\rm d}s^2 =2{\rm d}u\,{\rm d}v-2v^2{\rm d}\xi\,{\rm d}\bar\xi
+2v^2\bar F\delta(u){\rm d}\xi\,{\rm d}u
+2v^2F\delta(u){\rm d}\bar\xi\,{\rm d}u \nonumber\\
 &&\qquad\qquad\qquad\qquad -\left[
2v^2F\bar F\delta^2(u) -v(F_\xi+\bar F_{\bar\xi})\delta(u) \right]
\,{\rm d}u^2,
\end{eqnarray}
 which includes an impulsive component explicitly located on the spherical
wavefront $u=0$. This can immediately be seen to be identical to the
impulsive limit of the simplest type~N Robinson--Trautman solution
$RTN({\Lambda=0},{\epsilon=0})$ which, expressed using
Garc\'{\i}a--Pleba\'nski coordinates, has the form
\begin{equation}
 {\rm d}s^2 =2{\rm d}u\,{\rm d}v-2v^2{\rm d}\xi\,{\rm d}\bar\xi
 +2v^2\bar f{\rm d}\xi\,{\rm d}u +2v^2f{\rm d}\bar\xi\,{\rm d}u
-\left[ 2v^2f\bar f
-v(f_\xi+\bar f_{\bar\xi}) \right]{\rm d}u^2,
\end{equation}
 where, $f=f(\xi,u)$ is an arbitrary complex function holomorphic in~$\xi$,
and the impulsive limit is obtained by putting $f(\xi,u)=F(\xi)\delta(u)$.


It must be observed that this impulsive limit includes the square of the
Dirac $\delta$-function, and so this form should not be used in practice.
Nevertheless, it explicitly demonstrates that the spherical impulsive waves
obtained using the ``cut and paste'' method are equivalent to the impulsive
limit of Robinson--Trautman type~N solutions. Further analysis of the
distributional features of this solution is obviously required.


It may also be observed that Bi\v{c}\'ak and Schmidt \cite{BicSch89}
obtained a particular case of this class of solutions as a limiting case of
a solution with boost-rotation symmetry. In this limit, two null particles
recede from a common point generating an impulsive spherical gravitational
wave and there are either cosmic strings attaching each particle to
infinity, or there is an expanding cosmic string along the axis of symmetry
separating the two particles. This solution was given independently in
\cite{GlePul89} and discussed in \cite{Bicak90}.


\section{Expanding impulsive waves with ${\bf\Lambda\ne0}$}
\label{Exp.Lamne0}

As in the previous sections, it can be shown that there are again three
possible approaches to the construction of expanding impulsive waves in
de~Sitter and anti-de~Sitter backgrounds. (1) General solutions can be
obtained using Penrose's ``cut and paste'' method with a suitable ``warp'',
cutting (anti-) de~Sitter space-time on a null cone. This was performed in
\cite{Hogan92}. (2) We can also take the impulsive limit of the type~N
Robinson--Trautman solutions with cosmological constant \cite{PodGri99a}.
(3) Particular cases may be obtained as limits of space-times with
boost-rotation symmetry and non-zero cosmological constant. However, these
have not yet been considered.


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%%%%%%%%%%%%%%%%%%%%%% Acknowledgements %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vspace*{0.25cm} \baselineskip=10pt{\small \noindent
This work was supported by a visiting fellowship from the Royal Society and
by the grant GACR-202/99/0261 of the Czech Republic. }
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%%%%%%%%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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