Tethra: Bell's Spaceship Paradox
Relativity & Gravity arc, page 30 of 193. Back to The Pixelated Physics Universe
Who they are
Same burn. Same speed. Tethra's twin rockets fly in perfect step, and the thread between them still snaps.
Tethra 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
Two ships start at rest a distance L apart and accelerate identically. In the launch frame their gap stays L the whole time, but a thread moving with them is Lorentz-contracted: to span a gap of L at speed v it would need a proper length of γL. Its proper length is only L, so it is stretched and breaks. Equivalently, once both coast at speed v, in the ships' momentary rest frame their separation has grown to γL while the thread's rest length is still L. E. Dewan and M. Beran published the puzzle in 1959; J. S. Bell made it famous in "How to teach special relativity" (1976).
$$ \begin{gathered} \text{launch frame: } \Delta x = L \\ \text{ships' frame (both coasting at } v\text{): gap} = \gamma L, \qquad \gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $L$ | initial separation, and the launch-frame separation at all times | m |
| $\Delta x$ | separation measured in the launch frame | m |
| $v$ | final common speed | m/s |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $\gamma$ | Lorentz factor | 1 |
Honest caveat: The γL gap applies after both ships reach the same final speed; during acceleration they share no single rest frame.
Sources
- Dewan & Beran (1959) Note on Stress Effects due to Relativistic Contraction, Am. J. Phys. 27, 517
- Bell (1976) How to teach special relativity, reprinted in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 2004)
- Franklin (2010) Lorentz contraction, Bell's spaceships and rigid body motion in special relativity, Eur. J. Phys. 31, 291
Crossovers
Tethra meets Twinne & Twynne (Twin Paradox). Twinne and Twynne show that clocks disagree. Tethra shows that lengths do too: two identically accelerating rockets keep their launch-frame gap, so in their own frame the thread between them is stretched until it snaps.
Tethra meets Ehrenwheel (Disk That Can't Stay Flat). 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.
Wear the lore
Concept keywords: Bell's Spaceship Paradox, Tethra, Relativity & Gravity, physics lore, Pixelated Physics