On the relation of Thomas rotation and angular velocity of reference frames

TL;DR

This paper critically examines the methods for calculating Thomas rotation in relativistic physics, clarifies the applicability of a key principle, and establishes properties of rotating reference frames.

Contribution

It clarifies the correct application of the Rindler principle for Thomas rotation and analyzes the limitations of previous approaches.

Findings

Identifies the conditions under which the Rindler principle applies.

Shows that applying the principle to different frames yields inconsistent results.

Establishes general properties of rotating reference frames.

Abstract

In the extensive literature dealing with the relativistic phenomenon of Thomas rotation several methods have been developed for calculating the Thomas rotation angle of a gyroscope along a circular world line. One of the most appealing concepts, introduced in \cite{rindler}, is to consider a rotating reference frame co-moving with the gyroscope, and relate the precession of the gyroscope to the angular velocity of the reference frame. A recent paper \cite{herrera}, however, applies this principle to three different co-moving rotating reference frames and arrives at three different Thomas rotation angles. The reason for this apparent paradox is that the principle of \cite{rindler} is used for a situation to which it does not apply. In this paper we rigorously examine the theoretical background and limitations of applicability of the principle of \cite{rindler}. Along the way we also establish some general properties of {\it rotating reference frames}, which may be of independent interest.