The Rotating Disk Paradox Is a Transformation Error

The Rotating Disk Paradox Is a Transformation Error Max Karson November 20, 2025 Abstract Standard treatments of the Ehrenfest paradox rely on the coordinate transformation postulate rrot =rlab. We demonstrate that this identity is physically invalid. Because spatial coordinates are defined by light paths, the geometric object defined as a ”disk” in the lab frame maps to a non-circular geometry in the rotating frame. The paradox is not a physical phenomenon but a transformation error arising from an invalid assumption that the object’s identity persists across frames. The coordinate error. The ”rotating disk paradox” [1] is typically derived by assuming that the radial coordinate in the rotating frame is identical to that in the stationary lab frame (r=r′). Standard treatments (e.g., Grøn [2]) explicitly stipulate this identity. This assumption treats the disk as a Newtonian object, rather than a geometric measurement derived from light paths. In relativity, a spatial dimension is defined by simultaneous measurement. Since simultaneity and light propagation paths differ between inertial and rotating frames, the set of events that defines a straight radius in one frame does not define a straight radius in the other. The identity r=r′is therefore false by definition. Proof via mutually exclusive geometries. Consider a ”disk” defined in the lab frame: a set of points simultaneously equidistant from a center, with straight lines through the center connecting antipodal points. For an observer on this rotating object’s rim, a photon directed radially inward will miss the lab-defined antipode because it rotates away during transit. Operationally, the rotating observer measures a chord, not a diameter. Thus, the object defined as a disk by the lab is defined as an asymmetric, non-circular shape by the rotating observer. Conclusion. The paradox arises because the transformation r=r′forces the rotating observer to occupy the lab’s geometry while retaining their own time. This is impossible. There is no single ”disk” that exists in both frames; there are two distinct geometric definitions that share no common shape. 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). 1 [2] Ø. Grøn, “Relativistic description of a rotating disk,” American Journal of Physics 43, 869 (1975). 2