Radial Accommodation in the Ehrenfest Problem A Conditional Layered-Ring Model with an Effective Covariant Phase-Stiffness Realization

Abstract

The Ehrenfest problem is traditionally formulated by considering an ideally Born-rigid disk accelerated from rest into uniform rotation. Tangential material elements are associated with local Lorentz contraction in an inertial laboratory frame, while radial elements are instantaneously perpendicular to the tangential motion and therefore do not undergo direct longitudinal Lorentz contraction. The further conclusion that the material radius must remain unchanged is, however, an additional mechanical assumption rather than a direct consequence of the Lorentz transformation. A conditional layered-ring construction is presented. Each concentric material ring is allowed to change its laboratory radius while preserving the sum of its local comoving circumferential lengths. Under this closure condition, the radial map is r(ρ) = ρ p 1 + ω 2 ρ 2 /c 2 , where ρ is the initial material radius and ω is the final angular velocity. The construction preserves C lab = 2πr on laboratory simultaneity slices while satisfying C proper = γC lab = 2πρ. The kinematic construction is supplemented by an effective radial-support model. Reducing a covariant spatial phase-gradient sector on a uniformly wound thin ring produces the proper circumference energy U wind (C) = K s 2 C "  2πN C  2 − q 2 0 # 2 . Its minimum fixes the summed local comoving circumference to C 0 = 2π|N| q 0 . Near that minimum, the reduced theory yields the effective stiffness κ eff = 4K s q 4 0 C 0 . In the strong-locking limit, the stationary thin ring satisfies C proper = C 0 , and the radial map follows exactly. A phenomenological fixed-ω radial Lagrangian gives a positive finite-stiffness radius correction and a local radial stability condition within that reduced model. A separate covariant strong-locking embedding retains an independent temporal phase variable and imposes the zero-temporal-frequency constraint u µ ∇ µ ϕ = 0. It reproduces the same radial map and gives the stationary constrained angular momentum J(ω) = M 0 ρ 2 ω (1 + ω 2 ρ 2 /c 2 ) 3/2 + N p χ , where p χ is the conserved phase charge. The previously used one-term expression is recovered in the sector p χ = 0. The finite-stiffness Hamiltonian completion and passage through the point dJ/dω = 0 remain open. The construction therefore supplies an explicit fixed-ω radial support model and a constrained strong-locking time-dependent embedding for a thin ring. It does not establish that all physical matter possesses the proposed phase sector

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