The Identity Error Common to Rotating Disk Treatments

The Identity Error Common to Rotating Disk Treatments Max Karson November 29, 2025 Abstract Ehrenfest’s 1909 reductio ad absurdum correctly shows that “Born rigid” and “uniformly rotating” are mutually exclusive. His argument, however, presupposed a fixed radial coordinate (r=r′) and applied the inertial Lorentz contraction formula to the curved rim. Later analyses of non-rigid disks, notably Grøn’s, retained these geometric premises, thereby canonizing a coordinate error as physical fact. We demonstrate operationally that a laboratory radius maps to a curved null path in the co-rotating frame, and the laboratory circumference maps to an open spacetime helix. Treating these distinct geometries as one object yields the familiar stress formula, but at the cost of assigning material properties to a geometry that no rotating observer operationally constructs. 1 Ehrenfest’s Methodological Premise Ehrenfest [1] disproved rigid rotation by assuming the radius is unchanged while the circumference contracts, obtaining C′= 2πR′. The premise r=r′tacitly imports the laboratory radius into the rotating description and ignores that spatial distances are defined by observer-dependent light paths. 2 Canonization of the Premise Subsequent elastic-disk treatments, e.g. Grøn [2], adopted both Ehrenfest’s conclusion and his invalid rotating coordinates: (i) the rim is a closed circle that contracts tangentially; (ii) the radius is a straight segment that remains invariant. This imported identity anchors later stress calculations. 3 Radial Misidentification A radius is fixed by a two-way light signal. In the laboratory frame the path is straight; in rotating coordinates the same null geodesic is curved due to relativistic aberration. Thus, the coordinate transformation r=r′is physically incorrect. 4 Circumferential Misidentification The standard elastic–disk treatment begins by applying the inertial contraction rule L′=L/γ to the laboratory circumference, while declaring the radius unchanged. This transformation embeds two distinct errors: 1 1. Closed-loop preservation. Contraction is applied tangentially but the rim is still treated as a spatially closed circle in the rotating description. A rotating observer, however, cannot extend simultaneity consistently around the loop; a full circuit returns to the starting point with a proper-time offset ∆τrot. The laboratory circle therefore corresponds to an open helix on the rim world-tube, not to a shorter closed circle. 2. Misuse of the contraction formula. The relation L′=L/γ is valid only for endpoints linked by a straight world-line segment in an inertial frame. A rotating rim element has no such global inertial rest frame; its world-line is curved. Contraction, Sagnac time shift, and relativistic aberration are different facets of the same coordinate transformation, but the contraction rule cannot be inverted to define a single “proper circumference” for a non-inertial observer. 5 Conclusion The incompatibility of rigidity and rotation has been settled since Ehrenfest’s 1909 analysis. Nevertheless, later treatments of rotating elastic disks canonized Ehrenfest’s geometric assumptions into the non-rigid case. Operationally, the laboratory and rotating observers slice the rim’s world-tube into geometrically distinct objects, making the coordinate equality r=r′physically invalid. The stresses in a rotating elastic disk cannot be interpreted as a comparison between “lab” and “comoving proper” circumferences, because the rotating frame is non-inertial and admits no unique proper circumference. AI Disclosure The author used AI language models to assist with drafting, derivations, and algebraic checks. References [1] P. Ehrenfest, “Gleichf¨ormige Rotation starrer K¨orper und Relativit¨atstheorie,” Physikalische Zeitschrift 10, 918 (1909). [2] Ø. Grøn, “Relativistic description of a rotating disk,” American Journal of Physics 43, 869 (1975). 2