Consistent Coordinate Transformation for Relativistic Circular Motion and Speeds of Light

Abstract

A world system is composed of the world lines of the rest observers in the system. We present a relativistic coordinate transformation, termed the transformation under constant light speed with the same angle (TCL-SA), between a rotating world system and the isotropic system. In TCL-SA, the constancy of the two-way speed of light holds, and the angles of rotation before and after the transformation are the same. Additionally a transformation for inertial world systems is derived from it through the limit operation of circular motion to linear motion. The generalized Sagnac effect involves linear motion, as well as circular motion. We deal with the generalized effect via TCL-SA and via the framework of Mansouri and Sexl (MS), analyzing the speeds of light. Their analysis results correspond to each other and are in agreement with the experimental results. Within the framework of special and general relativity (SGR), traditionally the Sagnac effect has been dealt with by using the Galilean transformation (GT) in cylindrical coordinates together with the invariant line element. Applying the same traditional methods to an inertial frame in place of the rotating one, we show that the speed of light with respect to proper time is anisotropic in the inertial frame, even if the Lorentz transformation, instead of GT, is employed. The local speeds of light obtained via the traditional methods within SGR correspond to those derived from TCL-SA and from the MS framework.