Ehrenwheel

Ehrenwheel spins her discus, and its rim ripples as it turns.

The Physics

Spin a disk. Seen from the lab, each rim element moves along its own length and is Lorentz-contracted, while each radial element moves sideways and is not. So a body cannot be spun up from rest and stay Born-rigid. On a disk already turning steadily, co-rotating rulers laid along the rim are contracted as seen from the lab, so more of them fit: the rim measures more than 2πR while the radius still measures R. Geometry on the turning disk is not Euclidean.

C′ = 2πR / √(1 − ω²R²/c²) > 2πR (co-rotating rulers)

• C′: rim circumference measured with co-rotating rulers (m)
• R: disk radius (m)
• ω: angular speed of the disk (rad/s)
• c: speed of light, 299 792 458 m/s (exact)

Paul Ehrenfest posed the puzzle in Physikalische Zeitschrift 10, 918 (1909).

Honest caveat: How to interpret the spinning disk's geometry was argued over for decades. The "more rulers fit" result is now standard, but finer points of interpretation are still discussed.

On the shirt

Front: Ehrenwheel, the pixel-art character, with the title. Back: the rim and radial rulers, the circumference formula, the Born-rigidity result and Ehrenfest's 1909 paper. Printed on a unisex tee.