Time dilation in relativistic two-particle interactions

TL;DR

This paper investigates relativistic two-particle orbits, revealing that while Lorentz transformations fail for observables, Einstein's time dilation accurately predicts orbital periods across interaction strengths.

Contribution

It demonstrates the limitations of Lorentz transformations in interacting systems and confirms the validity of Einstein's time dilation for orbital periods.

Findings

Lorentz transformations do not correctly relate particle observables in interacting systems.

Einstein's time dilation accurately predicts orbital periods regardless of interaction strength.

Kinematic Lorentz formulas are generally inaccurate in relativistic two-particle interactions.

Abstract

We study the orbits of two interacting particles described by a fully relativistic classical mechanical Hamiltonian. We use two sets of initial conditions. In the first set (dynamics 1) the system's center of mass is at rest. In the second set (dynamics 2) the center of mass evolves with velocity V. If dynamics 1 is observed from a reference frame moving with velocity -V, the principle of relativity requires that all observables must be identical to those of dynamics 2 seen from the lab frame. Our numerical simulations demonstrate that kinematic Lorentz space-time transformations fail to transform particle observables between the two frames. This is explained as a result of the inevitable interaction-dependence of the boost generator in the instant form of relativistic dynamics. In spite of general inaccuracies of Lorentz formulas, the orbital periods are correctly predicted by the Einstein's time dilation factor for all interaction strengths.