Derivation of the General Case Sagnac Result using Non-time-orthogonal Analysis

TL;DR

This paper derives a comprehensive formula for the Sagnac effect in complex geometries using non-time-orthogonal tensor analysis, revealing insights beyond traditional methods.

Contribution

It introduces a novel non-time-orthogonal tensor approach to derive the general Sagnac effect for arbitrary enclosed areas and geometries.

Findings

Derived the general Sagnac fringe shift formula for non-circular areas

Extended the analysis to cases where the area does not pass through the rotation axis

Showed limitations of conventional local Lorentz frame approaches

Abstract

The experimentally determined Sagnac fringe shift dependency on angular velocity and enclosed area is derived from the rotating reference frame using non-time-orthogonal tensor analysis. The relationship for the most general case, in which the area enclosed is not circular and does not have the axis of rotation passing through its center, is determined. It is submitted that this quantitative result, along with a related thought experiment, can not be found using the conventional approach of local co-moving Lorentz frames.