Abstract
The statistical predictions of Newtonian and special-relativistic mechanics, which are calculated from an initially Gaussian ensemble of trajectories, are compared for a low-speed scattering system. The comparisons are focused on the mean dwell time, transmission and reflection coefficients, and the position and momentum means and standard deviations. We find that the statistical predictions of the two theories do not always agree as conventionally expected. The predictions are close if the scattering is non-chaotic but they are radically different if the scattering is chaotic and the initial ensemble is well localized in phase space. Our result indicates that for low-speed chaotic scattering, special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to obtain empirically-correct statistical predictions from an initially well-localized Gaussian ensemble.
Highlights
The standard practice in dynamics is to use Newtonian mechanics to study the motion of low-speed particles, instead of using the specialrelativistic theory
This practice is rooted in the conventional belief [1,2,3] that the dynamics predicted by special-relativistic mechanics for a low-speed system is always well-approximated by the dynamics predicted by Newtonian mechanics from the same parameters and initial conditions
We have shown that the Newtonian and special-relativistic statistical predictions for the mean dwell time, transmission and reflection coefficients, and the position and momentum means and standard deviations, which are calculated from an initially Gaussian ensemble of trajectories, for a low-speed scattering system are radically different if the scattering is chaotic and the initial ensemble is well localized in phase space
Summary
The standard practice in dynamics is to use Newtonian mechanics to study the motion of low-speed (i.e., much smaller than the speed of light) particles, instead of using the specialrelativistic theory. This practice is rooted in the conventional belief [1,2,3] that the dynamics predicted by special-relativistic mechanics for a low-speed system is always well-approximated by the dynamics predicted by Newtonian mechanics from the same parameters and initial conditions. The breakdown of agreement between the Newtonian and specialrelativistic trajectories is, much faster in the chaotic case compared to the non-chaotic, since the difference between the two trajectories grows exponentially in the former case but linearly in the latter case. For the scattering system in [13], the rapid breakdown of agreement was found to be due to a sufficiently-long exponential growth of the difference between the two trajectories in the scattering region when the scattering is chaotic
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