The Dynamics of Relativistic Length Contraction and the Ehrenfest Paradox

TL;DR

This paper explores the dynamic aspects of length contraction in special relativity, analyzing the Ehrenfest paradox and revealing how internal forces and geometry change during acceleration and rotation.

Contribution

It introduces a dynamical approach to length contraction, connecting internal forces with geometric transformations, and explains the transition from Euclidean to hyperbolic geometry in rotating disks.

Findings

Length contraction is influenced by internal forces during acceleration.

Rotating disks exhibit hyperbolic geometry due to non-Euclidean circumferential deformation.

Rest mass of spinning disks includes potential energy from geometric deformation.

Abstract

Relativistic kinematics is usually considered only as a manifestation of pseudo-Euclidean (Lorentzian) geometry of space-time. However, as it is explicitly stated in General Relativity, the geometry itself depends on dynamics, specifically, on the energy-momentum tensor. We discuss a few examples, which illustrate the dynamical aspect of the length-contraction effect within the framework of Special Relativity. We show some pitfalls associated with direct application of the length contraction formula in cases when an extended object is accelerated. Our analysis reveals intimate connections between length contraction and the dynamics of internal forces within the accelerated system. The developed approach is used to analyze the correlation between two congruent disks - one stationary and one rotating (the Ehrenfest paradox). Specifically, we consider the transition of a disk from the state of rest to a spinning state under the applied forces. It reveals the underlying physical mechanism in the corresponding transition from Euclidean geometry of stationary disk to Lobachevsky's (hyperbolic) geometry of the spinning disk in the process of its rotational boost. A conclusion is made that the rest mass of a spinning disk or ring of a fixed radius must contain an additional term representing the potential energy of non-Euclidean circumferential deformation of its material. Possible experimentally observable manifestations of Lobachevsky's geometry of rotating systems are discussed.