Bell's spaceship paradox

Bell's spaceship paradox, named for the best known presentation by John S. Bell, but discussed since the 1920s , is an apparent paradox in relativity.

Thought experiment

On a space platform two spaceships, their noses in positive x-direction, and separated by a distance $l$ are ready to start. Their autopilots are programmed to perform the same acceleration programme, for example a constant eigen-acceleration of $\alpha$ for a duration $T$. They are linked by a rope of length $l$, which should considered to weak to transport any forces and inflexible.

Will the rope break?

Special relativity's prediction

According to special relativity it will break. Seen from platform, the the rope will undergo Lorentz contraction but the distance between spaceships will stay the same. Seen from the spaceship, the length of the rope will stay the same, but the distance between the spaceships will enlarge, as during acceleration, at any point of time in the co-moving frame, the front spaceship accelerates more.

The corollary of this is, that an accelerated rigid rod accelerates more at the back than at the front. This example is often discussed in modern textbooks of relativiry, also Nikolic recently gave a most complete paper on it.

Controversies

This result was called into question in debates re-emerging from time to time. 1962 the American Journal of Physics accepted a paper by P. J. Nawrocki, which opposed previous correct treatments of the subject. Takuya Matsuda and Atsuya Kinoshita report that a controvery in Japanese physics journal arose after their publication of the paradox and is standard solution. As recently as 2004, the physicist J.H.Field wrote a paper stating the opposite result, but it wasn't accepted for publication.

Literature

• J. S. Bell, How to teach special relativity, Progress in Scientific Culture, Vol 1, No 2, 1976
• J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge University Press, 1987, ISBN 0521523389 (includes the above treatise)
• P. J: Nawrocki, Am. J. Phys. 30, 771 (1962)
• H. Nikolic, Relativistic contraction of an accelerated rod, Am. J. Phys. 67, 1007 (1999), preprint online as physics/9810017
• J. H. Field, On the Real and Apparent Positions of Moving Objects in Special Relativity: The Rockets-and-String and Pole-and-Barn Paradoxes Revisited and a New Paradox, preprint online as physics/0403094

External links

• Bell's Spaceship Paradox USENET Relativity FAQ
• A Paradox of Two Space Ships in Special Relativity Takuya Matsuda and Atsuya Kinoshita