Relativistic Sagnac effect and Ehrenfest paradox

There seems to exist a dilemma in the literature as to the correct relativistic formula for the Sagnac phase-shift. The paper addresses this issue in the light of a novel, kinematically equivalent linear Sagnac-type thought experiment, which provides a vantage point from which the effect of rotation in the usual Sagnac effect can be analyzed. The question is shown to be related to the so-called rotating disc problem known as the Ehrenfest paradox. The relativistic formula for the Sagnac phase-shift seems to depend on the way the paradox is resolved. Kinematic resolution of the Ehrenfest paradox proposed by some authors predicts the usually quoted formula for the Sagnac delay but the resolution itself is shown to be based upon some implicit assumptions regarding the behaviour of solid bodies under acceleration. In order to have a greater insight into the problem, a second version of the thought experiment involving linear motion of a "special type'' of a nonrigid frame of reference is discussed. It is shown by analogy that the usually quoted special relativistic formula for the Sagnac delay follows, provided the material of the disc matches the "special type.''