Lorentz transformation and vector field flows

TL;DR

This paper explores how Lorentz transformations induce vector field flows in parameter space, deriving the exact Thomas rotation angle and visualizing parameter evolution through phase portraits, revealing fixed points and invariants.

Contribution

It provides a novel analysis of Lorentz transformation parameter flows, including an exact Thomas rotation angle and a visual phase portrait approach.

Findings

Exact finite Thomas rotation angle derived

Phase portraits visualize parameter evolution

Analytic invariant correlates parameter changes

Abstract

The parameter changes resulting from a combination of Lorentz transformation are shown to form vector field flows. The exact, finite Thomas rotation angle is determined and interpreted intuitively. Using phase portraits, the parameters evolution can be clearly visualized. In addition to identifying the fixed points, we obtain an analytic invariant, which correlates the evolution of parameters.