The Radius of a Rotating Disk Must Also Contract

The Radius of a Rotating Disk Must Also Contract Max Karson December 9, 2025 Abstract Ehrenfest’s paradox asserts the incompatibility of rigid rotation with special relativity, presuming that the radius of a rotating disk remains equal to its non-rotating counterpart (r=r′). Subsequent analyses, notably Grøn (1975), canonized this coordinate identity, implicitly treating the rotating disk as a hybrid geometry: a non-Euclidean rim coupled rigidly to a Euclidean radius. We demonstrate that this identification ignores relativistic operational measurement via two-way photon signals, which show that a rotating disk’s radius is not a static Euclidean line but a materially longer spiral path. Proper calculation confirms that the tangential component of this spiral undergoes Lorentz contraction in the laboratory frame. The standard assertion of an invariant radius is thus an identity error arising from the misapplication of coordinate geometry to a physically distinct operational reality. 1 Ehrenfest’s Premise: r=r′ Ehrenfest [1] disproved rigid rotation by assuming the non-rotating radius remains unchanged after rotation while the circumference contracts, yielding the contradiction C′= 2πR′. Subsequent rotating-disk treatments, e.g. Grøn [2] or standard texts [3], adopted this logic and used the coordinate identity r=r′to identify the non-rotating and rotating radii. This deductive identification ignores the relativistic requirement that spatial distances be defined operationally through observer-dependent light paths. 2 Radial Misidentification The standard assertion r=r′effectively defines the radius as the spatial distance between the center and the rim at a single instant of laboratory time (dt = 0). This definition is deductive rather than operational, relying on Ehrenfest’s assumption that the rotating radius is not lengthcontracted in the lab frame. That assumption is a category error, treating length contraction as an instantaneous state rather than a path integral defined by signal exchange. This leads to a logical circle in which the radius is assumed to be perpendicular to the motion in order to prove that it remains perpendicular. Operationally, any “object” is defined by light paths. For a disk at rest, the lab’s measurement photon traces a straight radius across the material. For a rotating disk, the lab’s measurement photon sweeps over a different chain of atoms: the set of intersection events forms a material spiral connecting center and rim, not a fixed Euclidean line. (In a full two–way Einstein measurement, the inbound and outbound legs select two distinct spirals; we return to this in Sec. 4.) The proper rotating radius must therefore be defined by this operational connectivity. Because signal propagation is not instantaneous, the measurement interval is non-zero (dt>0). During this 1 interval, the disk rotates, so the material path connecting the center to the rim is a spiral in lab coordinates. In the laboratory frame, a photon travels a radial distance dr in time dt =dr/c. During this interval, the disk material at radius rrotates tangentially by a distance dx =v dt =β(r)dr, where β(r) = v(r)/c =ωr/c. Proper length of spiral segment. Transforming to the instantaneous rest frame of the material element: •The radial component dr is perpendicular to the element’s motion and is unchanged. •The tangential component is Lorentz–contracted in the lab; its proper value is γ(r)dx = γ(r)β(r)dr. The differential proper length is therefore dLp=pdr2+ (γβ dr)2=drp1+γ2β2.(1) Using the identity 1 + γ2β2=γ2, this simplifies to dLp=γ(r)dr . (2) The total proper length of the material radius is the integral of the local Lorentz factor from the center to the rim: Lp=ZR 0 dr q1−ω2r2 c2 =c ωarcsinωR c> R . (3) 3 One-Way Lab Radii as Contracted Material Spirals Conversely, each proper segment dLpof this material chain contributes only its radial projection dr =dLp/γ(r) (the local Lorentz–contraction relation) to what the laboratory counts as a one–way radius between center and rim. Summing these projected contributions from center to rim yields the shorter coordinate span Rfor each leg of the two–way measurement. In this sense, each laboratory “radius” of length Ris the Lorentz–contracted radial projection of a longer material spiral of proper length Lp. Identifying these distinct quantities as equal (r=r′) conflates the contracted coordinate radius with the materially defined, operational radii. 4 No Unique Center in Rotating Coordinates Because any measurement photon traces two distinct spirals on its outbound and inbound legs, the rotating observer finds that a center-bound photon must be emitted in the anti-rotational direction to compensate for relativistic aberration, yet returns from a rotationally forward direction. Consequently, the rotating observer cannot define a single disk center. 2 5 Conclusion The incompatibility of rigidity and rotation has been settled since Ehrenfest’s 1909 analysis. Nevertheless, subsequent treatments preserved the paradox’s underlying category error by enforcing radial rigidity via the coordinate identity r=r′. This identity implicitly models the rotating disk as a hybrid object: a non-Euclidean rim attached to a Euclidean radius. This hybrid geometry is operationally incoherent. Defined correctly via observer-dependent, material light-paths, the radius of a rotating disk is not a straight Euclidean line but a spiral with proper length Lp> R. Consequently, the laboratory observer measures a length-contracted radius, just as they measure a length-contracted circumference. The standard assertion that the radius remains unchanged by rotation is therefore an identity error equating the geometric radius of empty laboratory space with the materially defined radius of the object. 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). [3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, Oxford, 1975), Chap. 3. 3