Newtonian and special-relativistic predictions for the trajectories of a low-speed scattering system

Newtonian and special-relativistic predictions, based on the same model parameters and initial conditions for the trajectory of a low-speed scattering system are compared. When the scattering is chaotic, the two predictions for the trajectory can rapidly diverge completely, not only quantitatively but also qualitatively, due to an exponentially growing separation taking place in the scattering region. In contrast, when the scattering is nonchaotic, the breakdown of agreement between predictions takes a very long time, since the difference between the predictions grows only linearly. More importantly, in the case of low-speed chaotic scattering, the rapid loss of agreement between the Newtonian and special-relativistic trajectory predictions implies that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly describe the scattering dynamics.

Figure 1

Scattering potential (solid line) of Eq. (1) and the corresponding force (dotted line) for $V_0$ $=$8 and $\beta$ $=$4.

Figure 2

Newtonian and special-relativistic predictions for the nonchaotic scattering case: position (top) and momentum (bottom). Newtonian and special-relativistic values are plotted with squares and diamonds, respectively.

Figure 3

Newtonian and special-relativistic predictions for the first example of the chaotic scattering case: position (top) and momentum (bottom). Newtonian and special-relativistic values are plotted with squares and diamonds, respectively.

Figure 4

Magnitude of the difference between the Newtonian and special-relativistic predictions for the nonchaotic scattering case of Fig. 2: position difference (top) and momentum difference (bottom).

Figure 5

Natural log of the magnitude of the difference between the Newtonian and special-relativistic predictions versus kick for the first example of the chaotic scattering case of Fig. 3: position difference (top) and momentum difference (bottom).

Figure 6

Newtonian and special-relativistic predictions for the second example of the chaotic scattering case: position (top) and momentum (bottom). Newtonian and special-relativistic values are plotted with squares and diamonds, respectively.

Figure 7

Natural log of the magnitude of the difference between the Newtonian and special-relativistic predictions versus kick for the second example of the chaotic scattering case of Fig. 6: position difference (top) and momentum difference (bottom).