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\newcommand{\keywords}{{\it pp\,}-waves, chaotic motion, impulsive
 gravitational waves}
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{J. Podolsk\'y, Chaos in {\sl pp\,}-wave spacetimes}
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\title{Chaos in {\it pp\,}-wave spacetimes \\
and its smearing in the impulsive limit }
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\author{J.\ Podolsk\'y}
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  {Institute of Theoretical Physics, Charles University,
V Hole\v{s}ovi\v{c}k\'ach 2, 180~00 Prague 8, Czech
Republic}

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\begin{abstract}
In recent works \cite{PVcha1, PVcha2} we demonstrated by
a fractal method that geodesic motion in non-homogeneous vacuum
{\it pp\,}-wave spacetimes with a constant wave-profile is chaotic.
These results were generalized to non-homogeneous sandwich
{\it pp\,}-waves \cite{PVcha3}. It was shown that chaos smears
as the duration of these gravitational  waves is reduced:
the outcome channels to infinity become fuzzy, and the fractal
structure is cut at lower and lower levels.   Here we calculate
the box-counting dimension of the basin boundaries. In the
impulsive limit its value approaches an integer which indicates
that the fractal boundaries become regular. Indeed, in the limit
of impulsive waves, the motion is fully non-chaotic.
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\section{Chaotic motion in {\it pp\,}-waves with a constant profile}

The geodesic equations for the widely known class of vacuum
{\it pp\,}-waves \cite{KSMH}
\begin{equation}
d s^2=2\,d\zeta d\bar\zeta-2\,d ud v
-[f(\zeta,u)+\bar f(\bar\zeta,u)]\,d u^2\ , \label{E1}
\end{equation}
reduce to
\begin{equation}
\zeta'' + \textstyle{\frac{1}{2}}\bar f_{,\bar\zeta} = 0\ ,\label{E2}
\end{equation}
$v(u)=\int\,[\,\zeta' {\bar\zeta}'
  -\frac{1}{2}(f+\bar f)-
   \frac{1}{2}(\epsilon/U^2)\,]\,du+\tilde V$,
$u(\tau)=U\tau+\tilde U$,
where prime denotes $d/d u$, $\tau$ is a normalized affine parameter,
$U, \tilde U, \tilde V$ are constants, and $\epsilon=-1, 0, +1$
for timelike, null or spacelike geodesics, respectively. It is
thus crucial to solve (\ref{E2}) for $\zeta(u)$.
Introducing real coordinates $x$ and $y$ by $\zeta=x+\hbox{i}y$,
Eq. (\ref{E2}) follows from the Hamiltonian
$H=\frac{1}{2}\left(p_x^2+p_y^2 \right)+V(x, y, u)$,
where $V(x,y,u)=\frac{1}{2}\, {\cal R}e\,f$.
For non-homogeneous {\it pp\,}-waves given by
\begin{equation}
f=\textstyle{\frac{2}{n}}\,h(u)\,\zeta^n\ ,\quad n=3, 4, \cdots\ , \label{E7a}
\end{equation}
the corresponding potential $V$ is polynomial. Motion in these Hamiltonian
systems with $h(u)=C=$ const. is chaotic since the Smale horseshoe map can
be embedded along specific homoclinic orbits. In \cite{PVcha1, PVcha2} the
structure of chaotic motion was investigated by a fractal method \cite{Ott}.
Geodesics starting at $x(0)=\cos\phi_0$, $y(0)=\sin\phi_0$
were evolved. It was observed that they
escaped to infinity only along  one of the $n$  distinct outcome
channels. Moreover, the  basin boundaries which separated
initial conditions leading to different outcomes were shown to be
fractal, exhibiting chaos, i.e. an extremely sensitive dependence on the
choice of initial data.






\section{Smearing of chaos in the impulsive limit of sandwich {\it pp\,}-waves}



In \cite{PVcha3} we investigated motion in sandwich waves
(\ref{E7a}) with a `square' profile
$h(u) =a^{-1}[\Theta(u)-\Theta(u-a)]$,
where $a$ is a positive constant. Test particles which are
initilly at rest start moving at $u=0$. At $u=a$ the sandwich
wave ends and the potential barriers disappear.
For $u>a$, the particles move uniformly along straight
lines in the flat space and their trajectories are not
constrained to the former channels. Thus, the  fractal structure
of basin boundaries indicating chaos is `cut' at some level related
to $a$: there is not enough time for particles to bounce  arbitrarily
many times between the potential walls. As $a\to0$, the number of
bounces tends to zero so that the fractal structure is completely
erased. The narrower the sandwich wave, the `less chaotic' the motion.
This effect can be called a `smearing of chaos'.
Its detailed formal description can be found in  \cite{PVcha3}.
In the impulsive limit $a\to0$ (i.e. for
$f=\frac{2}{n}\delta(u)\zeta^n$), the motion is non-chaotic.
Indeed, the geodesics in these impulsive {\it pp\,}-wave spacetimes
can explicitly be written
\begin{eqnarray}
&& x(u) = \rho_0\cos\phi_0-C\rho_0^{n-1}\cos[(n-1)\phi_0]\,u \
,\nonumber\\
&& y(u) = \rho_0\sin\phi_0\ +C\rho_0^{n-1}\sin[(n-1)\phi_0]\,u
 \ ,\label{E14}
\end{eqnarray}
where $\rho_0$, $\phi_0$ parametrize the initial position of the particle.
At any time $u>0$ the initial circle $\rho=\rho_0$ is deformed into
a hypotrochoid, a curve generated by a point attached to a small
circle rolling around the inside of a large fixed circle,
 see \cite{PVcha3}.


The smearing of chaos can rigorously by characterized by the
dependence of the box-counting dimension $d$ of the basin
boundaries \cite{Ott} on $a$. This can numerically be calculated
from the relation $P(\varepsilon)\sim\varepsilon^{1-d}$, where
$P(\varepsilon)$ is the fraction of initial conditions which are
uncertain as to which escape channel is chosen when there is an
initial error $\varepsilon$ (i.e. comparing geodesics starting
from $\phi_0$ and $\phi_0\pm\varepsilon$). From the
dependence of $\ln P(\varepsilon)$ on $\ln\varepsilon$ for
$\varepsilon\to 0$, we obtain the following results (for $n=3$):
$d=0.2$ for $a=100$, $d=0.3$ for $a=60$, $d=0.6$ for $a=20$, $d=0.9$
for $a=10$. In fact, the box-counting dimension approaches
an integer ($d\to1$ as $a\to0$) which means  that the
basin boundaries completely lose their fractal structure
in the impulsive limit.



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\vspace*{0.25cm} \baselineskip=10pt{\small \noindent
The support of grant no GACR-202/99/0261
from the Czech Republic is acknowledged. }

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%%%%%%%%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{thebibliography}{9}

\bibitem{PVcha1} J. Podolsk\'y and K. Vesel\'y,
    Phys. Rev.~D {\bf 58} (1998) 081501

\bibitem{PVcha2} J. Podolsk\'y and K. Vesel\'y,
    Class. Quantum Grav. {\bf 15} (1998) 3505

\bibitem{PVcha3} J. Podolsk\'y and K. Vesel\'y,
    Class. Quantum Grav. {\bf 16} (1999) 3599

\bibitem{KSMH}  D. Kramer, H. Stephani, M.~A.~H. MacCallum, and
    E. Herlt, {\it Exact Solutions of Einstein's Field
    Equations}, Cambridge University Press, Cambridge 1980

\bibitem{Ott} E. Ott, {\it Chaos in Dynamical Systems},
    Cambridge University Press, Cambridge 1993

\end{thebibliography}

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