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In self-gravitating $N$-body systems, small perturbations introduced at thestart, or infinitesimal errors that are produced by the numerical integrator orare due to limited precision in the computer, grow exponentially with time. ForNewton's gravity, we confirm earlier results that for relatively homogeneoussystems, this rate of growth per crossing time increases with $N$ up to $N \sim30$, but that for larger systems, the growth rate has a weaker scaling with$N$. For concentrated systems, however, the rate of exponential growthcontinues to scale with $N$. In relativistic self-gravitating systems, the rateof growth is almost independent of $N$. This effect, however, is onlynoticeable when the system's mean velocity approaches the speed of light towithin three orders of magnitude. The chaotic behavior of systems with morethan a dozen bodies for the usually adopted approximation of only solving thepairwise interactions in the Einstein-Infeld-Hoffmann equation of motion isqualitatively different than when the interaction terms (or cross terms) aretaken into account. This result provides a strong motivation for follow-upstudies on the microscopic effect of general relativity on orbital chaos, andon the influence of higher-order cross-terms in the Taylor-series expansion ofthe Einstein-Infeld-Hoffmann equations of motion.

Chaos in self-gravitating many-body systems: Lyapunov time dependence of $N$ and the influence of general relativity