Ehrenwheel: Disk That Can't Stay Flat
Relativity & Gravity arc, page 31 of 193. Back to The Pixelated Physics Universe
Who they are
Ehrenwheel spins her discus, and its rim ripples as it turns.
Ehrenwheel is a character of the Relativity & Gravity arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
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. Paul Ehrenfest posed the puzzle in Physikalische Zeitschrift 10, 918 (1909).
$$ \begin{gathered} C' = \frac{2\pi R}{\sqrt{1-\omega^{2}R^{2}/c^{2}}} > 2\pi R \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $C'$ | rim circumference measured with rulers turning with the disk | m |
| $R$ | disk radius | m |
| $\omega$ | angular speed of the disk | rad/s |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
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.
Sources
- Ehrenfest (1909) Gleichförmige Rotation starrer Körper und Relativitätstheorie, Phys. Z. 10, 918 (Wikisource transcription)
- Grøn (1975) Relativistic description of a rotating disk, Am. J. Phys. 43, 869
- Grøn (2004) Space Geometry in Rotating Reference Frames: A Historical Appraisal, in Relativity in Rotating Frames, Springer, 285–333
Crossovers
Ehrenwheel meets Tethra (Bell's Spaceship Paradox). Tethra's thread snaps because the rockets' own frame sees their gap grow. Ehrenwheel's spinning disk has the same problem around its rim: each rim element is Lorentz-contracted while radial lengths are not, so a rigid flat disk cannot be spun up.
Ehrenwheel meets The Accelerant (Unruh Effect). The Accelerant and Ehrenwheel both live in accelerated frames, where flat-spacetime rulers and clocks stop agreeing in simple ways. Ehrenwheel's spinning rim cannot be flat and rigid; the Accelerant's horizon gives the vacuum a temperature. Neither needs gravity, only acceleration.
Wear the lore
Concept keywords: Disk That Can't Stay Flat, Ehrenwheel, Relativity & Gravity, physics lore, Pixelated Physics