Geometry of rotating disk and the Sagnac effect

TL;DR

This paper demonstrates that the geometry of a rotating disk remains Euclidean under Lorentz transformations and explains the Sagnac effect as a difference in light transit times, challenging the notion of non-Euclidean geometry in rotating frames.

Contribution

It shows that Lorentz transformations preserve Euclidean geometry on a rotating disk and clarifies the physical origin of the Sagnac effect from an inertial observer’s perspective.

Findings

Lorentz transformations leave the Euclidean metric invariant on a rotating disk.

Experiments without tidal or Coriolis forces cannot distinguish rotation.

The Sagnac effect results from different light transit times in opposite directions.

Abstract

In this paper we demonstrate that subsequent application of Lorentz transformations to the cylindrical coordinates on a rotating disk leaves the Euclidean metric invariant. Therefore, the geometry on rotating disk is the Euclidean geometry, and any experiment which do not involve tidal forces or Coriolis forces cannot identify either the disk rotates or not. We also show that, from the point of view of external inertial observer, the difference in the transit times for the light running along a circle of radius R in the opposite directions (with respect to the rotation) is 2w/c^2 S, where S is the area the circle, and w is the angle velocity.