[Paper Review] Geometry of rotating disk and the Sagnac effect

В. И. Башков, M. Malakhaltsev|ArXiv.org|Nov 18, 2000

Tribology and Lubrication Engineering2 references3 citations

TL;DR

This paper argues that the geometry on a rotating disk is Euclidean when properly accounting for local Lorentz transformations in rotating frames, contradicting the conventional view of non-Euclidean geometry. By applying infinitesimal Lorentz transformations to cylindrical coordinates and preserving the spacetime metric $ds^2 = c^2dt^2 - dr^2 - r^2d heta^2 - dz^2$, the authors show that only experiments involving inertial forces (e.g., Coriolis or centrifugal forces) can detect rotation. The key result is a derivation of the Sagnac effect time difference $\Delta t = 2\frac{\omega}{c^2}S$ using this geometric framework.

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.

Motivation & Objective

• To re-express the geometry of a rotating disk using successive Lorentz transformations applied to inertial coordinates.
• To challenge the widely held view that rotating disks exhibit non-Euclidean geometry by showing the metric remains invariant under local transformations.
• To derive the Sagnac effect time difference from first principles using a geometric approach that preserves the Minkowski metric.
• To clarify that only dynamical effects (Coriolis, centrifugal forces) can detect rotation, not kinematic geometry alone.

Proposed method

• Apply infinitesimal Galilean transformation to map inertial coordinates (r, θ, z, t) to rotating frame (r′, θ′, z′, t′) with θ′ = θ - ωt.
• Apply Lorentz-type transformations to the rotating frame, including spatial contraction along the tangential direction and local time correction.
• Use the infinitesimal transformation rules: $dr'' = dr$, $r''d heta'' = (rd heta - ωrdt)/\sqrt{1 - \omega^2r^2/c^2}$, and $dt'' = (dt - \omega r^2 d heta / c^2)/\sqrt{1 - \omega^2r^2/c^2}$.
• Show that the spacetime interval $ds^2 = c^2dt^2 - dr^2 - r^2d\theta^2$ is preserved in the transformed coordinates, confirming invariance.
• Extract the spatial metric $dl''^2 = dr''^2 + r''^2 d\theta''^2$, proving Euclidean geometry on the disk.
• Derive the Sagnac time difference by integrating the differential time shift along a circular path of radius R, yielding $\Delta t = 2\omega S / c^2$.

Experimental results

Research questions

• RQ1 Does the geometry of a rotating disk remain Euclidean when properly accounting for local Lorentz transformations?
• RQ2 Can the Sagnac effect be derived from a geometric framework that preserves the Minkowski metric in rotating coordinates?
• RQ3 Why do some authors claim non-Euclidean geometry on rotating disks, and what is the flaw in their reasoning?
• RQ4 What role do inertial forces (Coriolis, centrifugal) play in detecting rotation, and why are they necessary?
• RQ5 Is the time difference in light propagation around a rotating ring independent of the choice of coordinates, and how is it derived?

Key findings

• The spacetime metric $ds^2 = c^2dt^2 - dr^2 - r^2d\theta^2$ remains invariant under successive Lorentz transformations applied to a rotating disk, confirming the geometry is Minkowskian in the transformed coordinates.
• The spatial metric on the rotating disk is $dl''^2 = dr''^2 + r''^2 d\theta''^2$, proving the geometry is Euclidean, contrary to claims of non-Euclidean structure.
• The Sagnac effect time difference is derived as $\Delta t = 2\frac{\omega}{c^2}S$, where $S = \pi R^2$ is the area of the circular path, consistent with experimental observations.
• Light transit times in opposite directions on the rotating disk are equal in the rotating frame ($dt''_+ = dt''_-$), meaning rotation cannot be detected kinematically without dynamical effects.
• The transformation from inertial to rotating coordinates is not globally integrable, indicating that $r''$, $\theta''$, $t''$ are local coordinates, not global ones.
• The discrepancy with prior works arises because they neglected the time transformation in the rotating frame, leading to an incorrect non-Euclidean spatial metric.