Ehrenfest Rotating Disk Paradox

The Ehrenfest Paradox

In 1909, Paul Ehrenfest posed a fascinating puzzle that revealed deep connections between special relativity and general relativity. Imagine a disk at rest with radius R₀ and circumference C₀ = 2πR₀. Now spin the disk. What happens to these measurements?

The Paradox

Consider the disk from the lab frame (stationary observer):

• Circumference: Points on the rim move tangentially at velocity v = ωR₀ (where ω is angular velocity). Due to length contraction, the tangential distances between points are contracted by γ = 1/√(1-β²) where β = v/c. Therefore, the circumference measured in the lab frame is C = C₀/γ < 2πR₀.
• Radius: The radius is perpendicular to the motion, so it experiences no length contraction. Therefore, R = R₀.
• The Problem: This gives us C/R = (C₀/γ)/R₀ = 2π/γ < 2π! The ratio is less than 2π, which seems to violate Euclidean geometry.

The Resolution

The paradox reveals a profound truth: the rotating reference frame is non-Euclidean. Space itself appears curved from the rotating frame's perspective, even though we're only dealing with special relativity (flat spacetime)!

Why C/R < 2π Is Not a Contradiction

Euclidean geometry only applies in inertial (non-accelerating) frames. The rotating frame is accelerating (centripetal acceleration), so we should not expect Euclidean geometry to hold. The rotating observers experience space as having negative curvature (hyperbolic geometry).

Born Rigidity

A related question: can a rigid disk even be spun up? The answer depends on "Born rigidity" - the condition that proper distances between neighboring material points remain constant. For a disk, you can't spin up a Born-rigid disk instantaneously; you must apply forces gradually, and the disk will experience stresses. In practice, the material would deform or break at relativistic speeds.

Connection to General Relativity

The Ehrenfest paradox was historically important because it showed Einstein that acceleration and rotation induce geometric effects similar to gravity. This insight contributed to his development of general relativity, where spacetime curvature is fundamental. The rotating disk is equivalent (via the equivalence principle) to a gravitational field, providing a bridge between special and general relativity.

What You're Seeing

• The disk: Shown from the lab frame. The disk rotates, and points on the circumference experience length contraction tangentially.
• Measurement points: Equally spaced points (in the lab frame) around the circumference. The spacing between adjacent points is contracted by a factor of γ(r), which depends on the radius r from the center.
• Effective circumference: The contracted circumference C = C₀/γ, where γ is evaluated at the rim.
• Geometry ratio: C/R < 2π indicates non-Euclidean geometry in the rotating frame.

Things to Try

1. Slow Rotation: At low ω, β is small, so γ ≈ 1. The ratio C/R is very close to 2π - the geometry is nearly Euclidean.
2. Increase Angular Velocity: As ω increases, the rim velocity β = ωR approaches c. Watch how C/R decreases, revealing stronger non-Euclidean effects. The circumference contracts while the radius stays the same.
3. Approach the Speed of Light: Set ω very high (near 0.95). The rim approaches the speed of light, and γ becomes very large. The ratio C/R can drop well below 2π, showing extreme departure from flat geometry.
4. Compare Different Radii: Notice that points closer to the center move slower (v = ωr), so they experience less contraction. The geometry is radially dependent - the disk's space is inhomogeneous.
5. Toggle Measurements: Turn measurements on/off to see the contracted spacing more clearly. The spacing between adjacent points shrinks tangentially but not radially.

Key Formulas

• Rim velocity: v = ωR₀
• Beta at rim: β = v/c = ωR₀/c
• Lorentz factor: γ = 1/√(1-β²)
• Contracted circumference: C = C₀/γ = 2πR₀/γ
• Radius: R = R₀ (no contraction perpendicular to motion)
• Geometry ratio: C/R = 2π/γ < 2π

Historical Context

Paul Ehrenfest introduced this paradox in 1909, just four years after Einstein published special relativity. The paradox sparked extensive debate and helped clarify the limitations of special relativity (which assumes inertial frames) and the need for general relativity (which handles accelerating frames and curved spacetime). Einstein himself wrestled with the rotating disk problem while developing GR, and it influenced his thinking about the equivalence principle and the geometric nature of gravity.

Today, the Ehrenfest paradox is understood as a straightforward consequence of relativistic kinematics: rotating frames are non-inertial, and non-inertial frames exhibit effective spacetime curvature. It remains a beautiful pedagogical example of how special relativity connects to general relativity.