Ehrenfest disk

Compare laboratory circumferences with measurements made using locally co-rotating rulers.

About this tool

Enter laboratory radius R and rim speed β=v(R)/c; valid values update the comparison immediately. The exact radial probe ρ=r/R and its slider select from the center (0) to the rim (1), changing the local measurement while keeping the comparison curves. Options contains examples and More measurements, including the rim comparison. Reset restores R=1000 m, β=0.6 and ρ=1. Your exact input texts, including unfinished drafts, are kept; each visit computes a fresh comparison. Invalid R/β removes stale results; an invalid probe hides only its markers and readings. All examples use R=1000 m. The laboratory disk keeps its prescribed radius. The tangential arrow indicates counterclockwise motion; its length is schematic.

What the measurements mean

Laboratory rulers give C_L(r) = 2πr. For rulers moving tangentially with the disk, their lengths appear shortened in the laboratory, so more local ruler lengths fit around that same laboratory circle. Summing their locally measured lengths gives C_M(r) = 2πrγ(r), which is larger when the local speed is nonzero. Radial rulers are perpendicular to the motion, and the radial length from the center to the rim remains R.

The local spatial measurement rule is dl² = dr² + γ(r)²r²dφ² at constant z. Integrating its tangential term around a circle gives C_M. This local co-rotating measurement does not define a globally Einstein-synchronized snapshot of the rotating system. It does not embed a warped material disk in laboratory space; the underlying Minkowski spacetime remains flat.

Formulas and example

With c = 299792458 m/s, ω = βc/R, v(r) = βcρ and γ(r) = 1/√[1 − (βρ)²]. The extra circumference is ΔC = C_L(γ − 1). The curves show C_L/(2πR) = ρ and C_M/(2πR) = ργ(r). The tangential ruler ratio L_lab/L₀ is 1/γ.

At rim β = 0.6, γ = 1.25: the sum of local co-rotating lengths is 25% greater than the laboratory circumference. For R = 1000 m, C_L ≈ 6283.185307 m and C_M ≈ 7853.981634 m. At rest γ = 1 and both curves coincide. At the center both circumferences, speed and excess are exactly zero; C_M/(2r) is undefined there, while for r > 0 it equals πγ.

Scope and controls

This is stationary rotation at a specified laboratory radius, with R from 0.01 to 1000000 m and rim β from 0 to 0.99. The upper value 0.99 is this tool’s display domain, not a universal physical limit. Input values are never silently clamped.

The model does not describe spinning up a material disk, elastic deformation, stresses or material limits. It does not claim a Born-rigid acceleration from rest. There are no gaps in the disk and no contraction of its prescribed laboratory radius. Longer circumferences here arise from a different spatial measurement procedure.

Sources