Statistical Predictions for the Dynamics of a Low-Speed System: Newtonian versus Special-Relativistic Mechanics

The Newtonian and special-relativistic statistical predictions for the mean, standard deviation and probability density function of the position and momentum are compared for the periodically-delta-kicked particle at low speed. Contrary to expectation, we find that the statistical predictions, which are calculated from the same parameters and initial Gaussian ensemble of trajectories, do not always agree if the initial ensemble is sufficiently well-localized in phase space. Moreover, the breakdown of agreement is very fast if the trajectories in the ensemble are chaotic, but very slow if the trajectories in the ensemble are non-chaotic. The breakdown of agreement implies that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly calculate the statistical predictions for the dynamics of a low-speed system

Comparison of Newtonian and Special-Relativistic Trajectories with the General-Relativistic Trajectory for a Low-Speed Weak-Gravity System

We show, contrary to expectation, that the trajectory predicted by general-relativistic mechanics for a low-speed weak-gravity system is not always well-approximated by the trajectories predicted by special-relativistic and Newtonian mechanics for the same parameters and initial conditions. If the system is dissipative, the breakdown of agreement occurs for chaotic trajectories only. If the system is non-dissipative, the breakdown of agreement occurs for chaotic trajectories and non-chaotic trajectories. The agreement breaks down slowly for non-chaotic trajectories but rapidly for chaotic trajectories. When the predictions are different, general-relativistic mechanics must therefore be used, instead of special-relativistic mechanics (Newtonian mechanics), to correctly study the dynamics of a weak-gravity system (a low-speed weak-gravity system)

Comparison of classical dynamical predictions for low-speed, weak-gravity, and low-speed weak-gravity chaotic systems.

There are three fundamental classical theories which can be used to study the motion of dynamical systems: Newtonian mechanics (NM), special-relativistic mechanics (SRM) and general-relativistic mechanics (GRM). It is conventionally believed that (i) the predictions of SRM are well approximated by those of NM in the low-speed limit, (ii) the predictions of GRM are well approximated by those of SRM in the weak-gravity limit, and (iii) the predictions of GRM are well approximated by those of NM in the low-speed weak-gravity limit. In my research project, numerically-accurate predictions of the theories were compared in the three limits for chaotic dynamical systems to check the validity of the conventional beliefs. The results of this study have overturned the conventional beliefs: for each limit, I showed that the two predictions can rapidly disagree completely. This new conceptual understanding of the relationships between the predictions of the theories for low-speed, weak-gravity, and low-speed weak-gravity chaotic dynamical systems implies that physicists and engineers must replace the theories they have conventionally been using to study these systems with the more general theories. In particular, NM must be replaced by SRM for low-speed systems, SRM must be replaced by GRM for weak-gravity systems, and NM must be replaced by GRM for low-speed weak-gravity systems. These paradigm shifts could potentially lead to new understanding and discoveries in these systems

Testing general relativity using a bouncing ball

In a recent article (Liang and Lan, (2011)), we showed that the trajectories predicted by general-relativistic and Newtonian mechanics from the same parameters and initial conditions for a low-speed weak-gravity bouncing ball system will rapidly disagree completely if the trajectories are chaotic. Here, we determine how accurate the parameters and initial conditions of the system must be known so that the two different calculated chaotic trajectories are sufficiently accurate for an empirical test

Newtonian versus special-relativistic statistical predictions for low-speed scattering

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Comparison of probability densities for the first example.

<p>Newtonian (shaded grey) and special-relativistic (bold line) position (top plot) and momentum (bottom plot) probability densities for the first example at kick 17.</p

Comparison of trajectories for the second example.

<p>Comparison of the Newtonian (squares), special-relativistic (diamonds) and general-relativistic (triangles) positions (top plot) and velocities (bottom plot) for the chaotic second example.</p

Comparison of momentum standard deviations for the first example.

<p>Newtonian (squares) and special-relativistic (diamonds) momentum standard deviations for the first example: first 15 kicks (top plot), kick 15 to 30 (bottom plot). The Newtonian and special-relativistic momentum standard deviations in the bottom plot are completely different from each other - they appear to be close from kick 25 onwards because the natural log of the standard deviations is plotted.</p