A simple criterion for nonrotating reference frames (Q1878381)

A reference frame of a non-rotating observer in the sense of A. G. Walker is a frame in which the acceleration of a free test particle in the neighborhood of the observer is independent of its velocity. Such a frame is e.g. given by an orthonormal tetrad which is spanned by the unit tangent vector to a trajectory in spacetime \({\mathbf{x}} = {\mathbf{x}}(t)\) and the unit vectors in the directions of the space axes if each of its vectors satisfies the Fermi-Walker equations. Given the assumption that the metric is diagonal (assumption (2a)) and stationary (assumption (2b)) on the spacetime path \({\mathbf{x}}(t)\), the authors reveal sufficient and necessary criteria for obtaining such a nonrotating reference frame in the sense of A. G. Walker, leading to their main theorem in section 2. They not only provide a detailed proof of this theorem, but also apply their main result to the van Stockum metric case of which the solution represents a rotating dust cylinder of infinite extent along the axis of symmetry but of finite radius. By using suitable coordinate transformations (a rotation and a subsequently following change to Cartesian coordinates) they show that the above assumptions (2a) and (2b) hold in the van Stockum case, so that their main theorem can be applied. Moreover, this application directly implies a known result of van Stockum which originally had been proven by using direct calculations within the framework of the Fermi-Walker equations and a technical limiting argument.