Relativistic approach for circular motion and the Sagnac effect

초록

The Sagnac effect could not have been exactly explained through rigorous theoretical derivations under special relativity or general relativity. The conundrum has been solved fully in the relativistic context. To this end, the Lorentz transformation, which makes transformations between the unprimed and the primed coordinates, is reformulated without any postulates except the space-time isotropy of inertial frames. In the reformulation, a complex Euclidean space (CES) is employed which is an extension of the Euclidean space from the real number to the complex number. The time and space axes of each coordinate system in CES are represented as being perpendicular, and the Lorentz transformation has the form of rotation. These characteristics of the formulation pave the way for the relativistic approach to circular motion. In CES, the orientation of the coordinate systems provides important information on angles of rotation that the relativistic approach exploits. The relativistic transformation from the inertial to the rotating frames is formulated such that it has the same form as that from the rotating to the inertial. The difference between the travel times of two light beams traversing a circumference in the Sagnac experiment can be exactly found by the circular approach presented, which shows that the nonrelativistic and relativistic analysis results are the same within a first order approximation. The circular approach can also provide exact coordinate transformations between the inertial and the rotating frames, which enables us to examine the speeds of light on a rotating plate. The primed observation systems, which consist of the world lines of rotating observers, have the same light speeds as the unprimed corresponding ones unless the metric tensor for the light speed constancy is employed. © 2016 Physics Essays Publication.

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