The Radius of a Rotating Disk Is Not Invariant

The Radius of a Rotating Disk Is Not Invariant Max Karson Independent Researcher maxk[email protected] December 19, 2025 Abstract Ehrenfest’s paradox asserts the incompatibility of rigid rotation with special relativity. Standard resolutions assume the radius of a rotating disk remains equal to its non-rotating counterpart (r=r′), while the circumference undergoes Lorentz contraction. We show that this assumed invariance of the radius is incorrect, arising from a fundamental category error: it mistakes the coordinate distance between independent atoms for the proper length of a causally connected material object. By analyzing the operational light paths required to define a radius, we show that the physical radius is not a static Euclidean line but a material spiral. Proper calculation confirms the radius itself undergoes Lorentz contraction in the laboratory frame, resolving the apparent incompatibility between radial and tangential geometry. The radius of a rotating disk is therefore not invariant, contradicting the standard interpretation. 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 analyses, from Grøn [2] and standard texts like Landau [3], to the modern treatments collected by Rizzi and Ruggiero [4], have largely canonized the logic that r=r′. This asserts that because the radius is perpendicular to the velocity vector, it suffers no contraction. However, this assertion promotes a coordinate label identity to a metric identity of proper lengths. It ignores the operational reality that ”length” in special relativity must be defined by causal signal exchange, not by geometric projection. When we demand that the radius be defined by the same causal connectivity used for the circumference, the orthogonality argument fails. 2 Radial Misidentification The standard metric assertion dσ′=dr (implied by 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 length-contracted in the laboratory frame. 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. We must instead define the material radius operationally via light paths. To visualize this, consider the case of a single rotating rod. A photon sent from the center to the tip does not trace a straight Euclidean line, but a longer curved trajectory. 1 Rotation is, by definition, an extension of the light path across the material. Since the photon must traverse a distance greater than the coordinate radius (Lp> R), the proper length of the rod is longer than the geometric space it occupies. Perhaps unintuitively, this implies that the radius undergoes length contraction despite the motion being purely tangential. This occurs because the operational radius (the material spiral) possesses a tangential component aligned with the velocity vector. By ignoring this spiral nature and selecting a geometric line at dt = 0, standard theory misidentifies a succession of unconnected atoms as the material radius. A disk is a collection of such causal radii; if the individual rod contracts, the disk must also. Proper length of the spiral Operationally, the laboratory frame 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(r)dt =β(r)dr, where v(r)=ωr. 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 Laboratory Radii as Contracted Material Spirals 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 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 2 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 inconsistent. 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. Declarations 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. [4] G. Rizzi and M. L. Ruggiero (eds.), Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames, (Kluwer Academic Publishers, Dordrecht, 2004). 3