%******  Smooth sandwich gravitational waves  *** Oct 20, 1998 ***
%******  by J. Podolsky
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 {\large{\bf
 \centerline { SMOOTH SANDWICH GRAVITATIONAL WAVES}}}
 \vspace{14mm}
 \centerline { J. Podolsk\' y
   \footnote{  E-mail:  podolsky@mbox.troja.mff.cuni.cz}}
 \vspace{5mm}

 {\it
 \centerline {  Department of Theoretical Physics,
           Faculty of Mathematics and Physics,}

 \centerline {   Charles University, V Hole\v sovi\v ck\'ach 2,
     180 00 Prague 8, Czech Republic}}
 \vspace{12mm}




\begin{abstract}
Gravitational waves which are  smooth and contain two
asymptotically flat regions are constructed from
the homogeneous {\it pp} wave vacuum solution.
Motion of free test particles is calculated explicitly and the
limit to an impulsive wave is also considered.


\vspace{3mm}
\noindent
PACS numbers: 04.30.-w, 04.20.Jb, 98.80.Hw
\end{abstract}
\vspace{14mm}


The widely known class of {\it pp} waves \cite{KSMH} is characterized by
the existence of a quadruple Debever-Penrose null vector field which
is covariantly constant. In vacuum  the metric can be written as
\begin{equation}
\D s^2=2\,\D\zeta \D\bar\zeta-2\,\D u\D v-(f+\bar f)\,\D u^2\ , \label{E1}
\end{equation}
where  $f(u,\zeta)$ is an arbitrary function of $u$,
analytic in $\zeta$. The simplest case for which the metric
describes  gravitational waves arises when $f$ is of the form
\begin{equation}
f(u,\zeta)=d(u)\zeta^2\ , \label{E2}
\end{equation}
where $d(u)$ is an {\it arbitrary} function of $u$;
such solutions are called homogeneous {\it pp} waves
(or ``plane'' gravitational waves).
Performing the transformation \cite{Penrose}
\begin{equation}
\zeta=\frac{1}{\sqrt{2}}\,(Px+iQy)      \ ,\qquad
 v   =\frac{1}{2}\,(t+z+PP'x^2+QQ'y^2)  \ ,\qquad
 u   =t-z                               \ ,\label{E3}
\end{equation}
where real functions $P(u)\equiv P(t-z)$, $Q(u)\equiv Q(t-z)$
are solutions of differential equations
\begin{equation}
P''+d(u)\,P=0\ ,\qquad Q''-d(u)\,Q=0\ , \label{E4}
\end{equation}
(here prime denotes the derivative with respect to $u$)
the metric can be written in the (Rosen) form
\begin{equation}
\D s^2 = - \D t^2 + P^2 \D x^2 + Q^2 \D y^2 + \D z^2\ , \label{E5}
\end{equation}
which is more suitable for physical interpretation.
Considering privileged free test particles
standing at fixed $x$, $y$ and $z$, their {\it relative} motion in the
$x$-direction is given by the function $P(u)$ while it is given
by $Q(u)$ in the $y$-direction, namely
$\Delta l_x=P\Delta x$ and $\Delta l_y=Q\Delta y$.
The motions are unaffected in the
$z$-direction which demonstrate transversality of the  waves.

Assuming ``profile''  functions $d(u)$ non-vanishing  on some
finite interval of $u$ only, sandwich gravitational waves were constructed
in \cite{BPR}-\cite{PV} and elsewhere. Here we
consider the  function $d(u)$ of the form
\begin{equation}
d(u)=\frac{a}{\cosh^2(bu)}   \ ,
\label{E6}
\end{equation}
where $a$ and $b$  are arbitrary real positive constants.
Since all derivatives of $d(u)$ are continuous the corresponding
gravitational waves are {\it smooth}, contrary to ``standard'' sandwich
waves explicitly presented in literature. On the other hand, the
space-times  (\ref{E1}) given by  Eqs. (\ref{E2}) and  (\ref{E6}) do not contain
flat regions in front of the wave and behind it. They
are curved everywhere, becoming flat only {\it asymptotically} as
$u\to\pm\infty$. Therefore, we should call them
``smooth asymptotic sandwich waves''.



In order to find the form (\ref{E5}) of the metric exhibiting
naturally the particle motions we have to solve the equations (\ref{E4}).
Introducing a substitution
\begin{equation}
\xi=-\tanh(bu)   \ ,
\label{E7}
\end{equation}
the equations take the form
\begin{equation}
(1-\xi^2)\frac{\D^2R}{\D\xi^2}-2\xi\frac{\D R}{\D\xi}\pm\frac{a}{b^2}R =0 \ ,
\label{E8}
\end{equation}
where the upper sign is applied for $R=P$ and the lower sign for $R=Q$.
Clearly, a general solution can be written as a linear
combination of the Legendre functions of the first kind $P_\alpha(\xi)$
and the second kind $Q_\alpha(\xi)$ where
$\alpha=(\sqrt{1 \pm 4a/b^2}-1)/2$. It is natural to impose the
conditions  $R(u\to-\infty)\to 1$ and
$R'(u\to-\infty)\to 0$ so that the metric (\ref{E5}) is
written in explicit Minkowski form in the asymptotic region where
$u\to-\infty$. These conditions are satisfied by particular solutions of
the form
\begin{equation}
P(u)=P_\mu(\xi(u))    \ ,\qquad
Q(u)=P_\nu(\xi(u))\ ,    \label{E9}
\end{equation}
where
\begin{equation}
\mu=(\sqrt{1 + 4a/b^2}-1)/2 \ ,\qquad
\nu=(\sqrt{1 - 4a/b^2}-1)/2 \ .\label{E10}
\end{equation}
Typical behavior of the functions $P, Q$ is shown in Fig. 1. It can
be observed that in both asymptotic regions $u\to\pm\infty$
the particles move uniformly. This can be shown
analyticly using the definition
$P_\alpha(\xi)\equiv F(-\alpha,\alpha+1;1;(1-\xi)/2)$ where $F$
is the hypergeometric function. Clearly, $P\to1, Q\to1$ as
$u\to-\infty$. For $u\to+\infty$ (corresponding to $\xi\to-1$)
we use the identity 15.3.10. in \cite{Abram} according to which
\begin{eqnarray}
F\Big(-\alpha,\alpha+1;1;\frac{1-\xi}{2}\Big) &=&-\frac{\sin \pi\alpha}{\pi}
   \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(n!)^2} \label{E11}\\
&&\hskip-30mm\times\Big[2\psi(n+1)-\psi(n-\alpha)-\psi(\alpha+n+1)-
     \ln\Big(\frac{1+\xi}{2}\Big)\Big]\Big(\frac{1+\xi}{2}\Big)^n
    \ , \nonumber
\end{eqnarray}
where $\psi$ is the digamma function. Using also the identity
$\psi(-\alpha)=\psi(1-\alpha)+1/\alpha$ and the limit
 $\ln[(1+\xi)/2]\to-2bu$ for  $u\to+\infty$ we
conclude that
\begin{equation}
P(u\to+\infty)=C_1+C_2u    \ ,\qquad
Q(u\to+\infty)=D_1+D_2u    \ ,    \label{E12}
\end{equation}
where
\begin{eqnarray}
&&C_1=\frac{\sin \pi\mu}{\pi}
 \Big[\frac{1}{\mu} +\psi(1+\mu)+\psi(1-\mu)-2\psi(1)\Big]    \ ,\quad
C_2=-\frac{2b}{\pi}\sin \pi\mu\ , \nonumber\\
&&D_1=\frac{\sin \pi\nu}{\pi}
 \Big[\frac{1}{\nu} +\psi(1+\nu)+\psi(1-\nu)-2\psi(1)\Big]    \ ,\quad
D_2=-\frac{2b}{\pi}\sin \pi\nu\ .
    \label{E13}
\end{eqnarray}
For particular values of $\mu$ corresponding to $\mu=m=0,1,2,\cdots$,
the constant $C_2$ vanishes so that $P(u\to+\infty)=C_1$. These
cases coincide with solutions $P(u)$ given by Legendre
{\it polynomials} $P_m(\xi)$ and we easily get
$C_1=P_m(\xi=-1)=(-1)^m$. Explicit particular solutions of this
type arise when $a/b^2=m(m+1)$ so that
$P(u)=-\tanh(\sqrt{a/2}\,u)$ for $m=1$,
$P(u)=[3\tanh^2(\sqrt{a/6}\,u)-1]/2$ for $m=2$, etc.
Note also that there are no analogous solutions for $Q(u)$ since
(\ref{E10}) admits $\nu\in\langle -1,0\rangle$ only. Therefore,
the only non-negative integer (for which the constant $D_2$ vanishes) is
$\nu=0$ representing a trivial case $d(u)=0$.
 For all other values of $\nu$ the constant $D_2$ is positive so
that $Q(u)\to+\infty$ as $u\to+\infty$.

We can also use the above results to construct impulsive gravitational
waves. The sequence of smooth functions (\ref{E6}) approach the
Dirac $\delta$-function (in a distributional sense) as
$a\to\infty$ if the second parameter is $b=2a$ (so that the
normalization condition $\int_{-\infty}^{+\infty} d(u)\,\D u=1$
holds for arbitrary $a$). Considering this limit, indicated also
in Fig.~1,
$\alpha=(\sqrt{1 \pm 1/a}-1)/2\to 0$ and using Eqs.~(\ref{E13}) we get
\begin{equation}
P(u)=1-u\,\Theta(u)\ ,\qquad
Q(u)=1+u\,\Theta(u)\ ,\label{E14}
\end{equation}
where $\Theta$ is the Heaviside step function ($\Theta=0$ for
$u<0$, $\Theta=1$ for $u>0$).
Therefore, particle motion in the impulsive gravitational wave is
exactly the same as in the limit of ``standard'' sandwich waves
\cite{Penrose}. This is just an explicit demonstration of recent
general result \cite{Stein} according to which the distributional
limit of a geodesic motion in impulsive {\it pp} waves is
independent of the specific form of (smooth) regularization.
In fact, the sequence $d_\varepsilon(u)$ of smooth asymptotic sandwich
waves given by Eq. (\ref{E6})  with $b=2a=1/\varepsilon$ is a
simple explicit example of a regularization sequence
within the largest ``reasonble'' class of smooth regularizations
of the Dirac $\delta$-distribution introduced in \cite{KS}.

In general, sandwich {\it pp} waves may serve as (local) exact toy
models of gravitational radiation far away from a  finite source.
However, the class of explicit solutions discussed in the paper
is more interesting from purely theoretical point of view
due to its specific feature that the corresponding waves have
smooth profiles.


\vspace{8mm}




\centerline{\bf  Acknowledgments}
\vspace{3mm}

\noindent
I acknowledge the support of grants No. GACR-202/96/0206 and
No. GAUK-230/1996 from the Czech Republic and Charles University.
\vspace{3mm}

\begin{thebibliography}{99}
\setlength{\itemsep}{-1mm}

\bibitem{KSMH} D.~Kramer, H.~Stephani, M.A.H.~MacCallum, and
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\bibitem{Penrose}  R.~Penrose: in
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\bibitem{BPR} H.~Bondi,  F.A.E.~Pirani, and  I.~Robinson:
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\bibitem{BP}  H.~Bondi and F.A.E.~Pirani:
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\bibitem{Rindler}   W.~Rindler:
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\bibitem{PV}  J.~Podolsk\'y and K.~Vesel\' y:
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\bibitem{Abram} M.~Abramowitz and  I.A.~Stegun:
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\bibitem{Stein}  R.~Steinbauer:
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\bibitem{KS}  M.~Kunzinger and R.~Steinbauer:
    preprint gr-qc/9806009 (1998).

\end{thebibliography}



\newpage

{\LARGE Figure Caption}

\vspace{3mm}
\noindent
Fig. 1. Typical behavior of the functions $P(u)$ and $Q(u)$
determining relative motion of free test particles (initially at
rest) in $x$ and $y$-directions, respectively, caused by smooth
sandwich gravitational waves for different values of
$a=\frac{1}{48},\frac{2}{48},\cdots, \frac{12}{48}$, with $b=2a$. The
values $a=\frac{1}{8},\frac{1}{24},\frac{1}{48}$
correspond to $\mu=m=1, 2, 3$, respectively, for which $P\to(-1)^m$
as $u\to+\infty$.



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