Accuracy of the non-relativistic approximation to relativistic probability densities for a low-speed weak-gravity system

The Newtonian and general-relativistic position and velocity probability densities, which are calculated from the same initial Gaussian ensemble of trajectories using the same system parameters, are compared for a low-speed weak-gravity bouncing ball system. The Newtonian approximation to the general-relativistic probability densities does not always break down rapidly if the trajectories in the ensembles are chaotic -- the rapid breakdown occurs only if the initial position and velocity standard deviations are sufficiently small. This result is in contrast to the previously studied single-trajectory case where the Newtonian approximation to a general-relativistic trajectory will always break down rapidly if the two trajectories are chaotic. Similar rapid breakdown of the Newtonian approximation to the general-relativistic probability densities should also occur for other low-speed weak-gravity chaotic systems since it is due to sensitivity to the small difference between the two dynamical theories at low speed and weak gravity. For the bouncing ball system, the breakdown of the Newtonian approximation is transient because the Newtonian and general-relativistic probability densities eventually converge to invariant densities which are close in agreement.