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

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 Rindler and Perlick (Gen Rel Grav 22:1067, 1990), 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 (Herrera and di Prisco in Found Phys Lett 15:373, 2002), 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 Rindler and Perlick (Gen Rel Grav 22:1067, 1990) 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 Rindler and Perlick (Gen Rel Grav 22:1067, 1990). Along the way we also establish some general properties of rotating reference frames, which may be of independent interest.