Chaos in self-gravitating many-body systems: Lyapunov time dependence of
  $N$ and the influence of general relativity
  In self-gravitating $N$-body systems, small perturbations introduced at the
start, or infinitesimal errors that are produced by the numerical integrator or
are due to limited precision in the computer, grow exponentially with time. For
Newton's gravity, we confirm earlier results that for relatively homogeneous
systems, this rate of growth per crossing time increases with $N$ up to $N \sim
30$, but that for larger systems, the growth rate has a weaker scaling with
$N$. For concentrated systems, however, the rate of exponential growth
continues to scale with $N$. In relativistic self-gravitating systems, the rate
of growth is almost independent of $N$. This effect, however, is only
noticeable when the system's mean velocity approaches the speed of light to
within three orders of magnitude. The chaotic behavior of systems with more
than a dozen bodies for the usually adopted approximation of only solving the
pairwise interactions in the Einstein-Infeld-Hoffmann equation of motion is
qualitatively different than when the interaction terms (or cross terms) are
taken into account. This result provides a strong motivation for follow-up
studies on the microscopic effect of general relativity on orbital chaos, and
on the influence of higher-order cross-terms in the Taylor-series expansion of
the Einstein-Infeld-Hoffmann equations of motion.
