Deformation of a Rotating Dust Disk within the Framework of Special Relativity

This paper derives the geometry of a rotating dust disk of infinitesimal mass within the framework of special relativity. The Ehrenfest paradox (1909) demonstrates that a Born-rigid disk cannot undergo relativistic rotation without deformation. This article determines the shape of the deformed disk in the absence of centrifugal forces (due to its negligible mass) by applying Lorentz contraction to the circumference while maintaining a constant radial distance. The vertical displacement z(x) is obtained by integrating the differential geodesic length under the condition that it remains equal to the initial parameter prior to rotation. It is shown that the approximate formula aligns well with the exact solution up to velocities of approximately 0.85c, making it applicable to a wide range of problems involving relativistic rotating systems.