Bell’s SpaceShips Paradox is interesting because it highlights the issue of whether Lorentzian length contraction only applies to physical objects travelling at very high speeds, or whether it applies to the abstract concept of length in a fast moving frame generally.
Introduction
Bell’s spaceship paradox is a thought experiment in Special Relativity, first described by E. Dewan and M. Beran in 1959 (Dewan & Beran, 1959). It became more widely known after John Stewart Bell elaborated the idea further in 1976.
Two spaceships initially at rest in an inertial frame and one in front of the other are connected by a weak inelastic string. At exactly the same time (in the observer’s frame) the two spaceships accelerate at exactly the same constant rate and in the same direction. They have an identical ever-increasing velocity as viewed in the observer’s frame. As the spaceships gain a significant speed v they contract by the usual Lorentzian gamma factor 𝛾 =[1 − (v⁄𝑐)2]-1/2.
The question is … does the string break or not? It is a simple question but one which has generated a lot of discussion. The Wikipedia article titled Bell’s Spaceship Paradox has over thirty references and there are many online presentations about it. Strangely enough the predictions and explanations are not all in agreement.
On the one hand it seems that the separation between the rockets must remain constant. On the other hand, it seems that this separation should Lorentz contract. That’s the paradox.
Some argue that the whole ensemble shrinks by the same factor. So the picture above just scales down and nothing breaks. Some argue that if the acceleration stops the resultant situation is just a Lorentz transformation of the initial situation, so the state of the string should be the same.
Others argue that the string must break. But all agree that any and all inertial observers of the spaceships and string must be able to agree on whether the string breaks or not. And all agree that both the spaceships contract as viewed from the rest frame.
Sometimes the argument seems to depend on where the string is fastened. Is it nose to tail or midpoint to midpoint? The drawing above shows the string running from tail to tail, for reasons that will be explained later.
Sometimes the disagreement seems to hinge on whether only physical objects contract. If both rockets and also the string contracts but the separation between the spaceships remains the same then the string must break.
Some presentations switch to the viewpoint of observers travelling on the spaceships. This becomes quite complicated because such viewpoints are taking place in an accelerating reference frame, and this requires the use of Rindler coordinates or even the complicated mathematics of General Relativity.
Some presentations introduce arguments related to relativistic effects on the electro-magnetic forces holding the string together.
It is a bit of a concern that this simple physical thought experiment causes so much trouble.
Explanation in Q Theory
The explanation and resolution offered within Q theory is elegantly simple. The spaceships move faster and faster in the Q field. As they do so they contract by the usual gamma factor. But they contract back towards the tail, which is where the thrust is coming from. It is an actual physical compression able to be photographed against suitable static measuring rods by inertial observers.
The distance from tail to tail stays the same. However, the string is also a physical object. It contracts by the usual gamma factor and so, being weak and inelastic, it breaks.
Furthermore it would break where it was attached to the tail of the spaceship in front. This is because the string is being pulled along by the spaceship in front. It is being accelerated by tension. It becomes harder and harder to pull along. If it were elastic it might stretch backwards. But the string has been assumed to be inelastic and so it breaks. It breaks where the tension is greatest, which is most likely to be where it is tied to the tail of the spaceship in front.
(The question of how material objects respond to relativistic effects is one that interested Lorentz, Ehrenfest, Born and many others (see Born rigidity) but is not is not needed for this simple Q theory resolution of the Spaceships Paradox.)
Summary
It is disconcerting how much trouble the Spaceships Paradox seems to create. However, the explanation in Q theory is simple. Both spaceships contract towards the source of the force accelerating them, the separation measured tail to tail stays the same but the string contracts and hence it breaks.
References
Van de Vusse, Sjoerd B.A., 2024, Some ideas and experiments for issues affecting modern physics, https://hereticalphysics.com.au
Dewan, Edmond M.; Beran, Michael J. (March 20, 1959). Note on stress effects due to relativistic contraction. American Journal of Physics. 27 (7): 517–518. Bibcode:1959AmJPh..27.517D. doi:10.1119/1.1996214.
Author contact: SBAvan@utas.edu.au
Author’s location: Hobart, Australia
