Bell's spaceship paradox

*Above*: In S the distance between the spaceships stays the same, while the string contracts. *Below*: In S′ the distance between the spaceships increases, while the string length stays the same.

Bell's spaceship paradox is a thought experiment in special relativity. It was first described by E. Dewan and M. Beran in 1959^[1] but became more widely known after John Stewart Bell elaborated the idea further in 1976.^[2] A delicate thread hangs between two spaceships headed in the same direction. They start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times as viewed from S. Therefore, they are all subject to the same Lorentz contraction, so the entire assembly seems to be equally contracted in the S frame with respect to the length at the start. At first sight, it might appear that the thread will not break during acceleration.

This argument, however, is incorrect as shown by Dewan and Beran, and later Bell.^[1]^[2] The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start, because in S, it is effectively defined to remain the same, due to the equal and simultaneous acceleration of both spaceships in S. It also turns out that the rest length between the two has increased in the frames in which they are momentarily at rest (S′), because the accelerations of the spaceships are not simultaneous here due to relativity of simultaneity. The thread, on the other hand, being a physical object held together by electrostatic forces, maintains the same rest length. Thus, in frame S, it must be Lorentz contracted, which result can also be derived when the electromagnetic fields of bodies in motion are considered. So, calculations made in both frames show that the thread will break; in S′ due to the non-simultaneous acceleration and the increasing distance between the spaceships, and in S due to length contraction of the thread.

In the following, the rest length^[3] or proper length^[4] of an object is its length measured in the object's rest frame. (This length corresponds to the proper distance between two events in the special case, when these events are measured simultaneously at the endpoints in the object's rest frame.^[4])

^ ^a ^b 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.
^ ^a ^b J. S. Bell: How to teach special relativity, Progress in Scientific culture 1(2) (1976), pp. 1–13. Reprinted in J. S. Bell: Speakable and unspeakable in quantum mechanics (Cambridge University Press, 1987), chapter 9, pp. 67–80.
^ Cite error: The named reference franklin was invoked but never defined (see the help page).
^ ^a ^b Moses Fayngold (2009). Special Relativity and How it Works. John Wiley & Sons. p. 407. ISBN 978-3527406074. Note that the proper distance between two events is generally not the same as the proper length of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length l(0). If you are in the rest frame K0 of the rod, and you want to measure its length, you can do it by first marking its end-points. And it is not necessary that you mark them simultaneously in K0. You can mark one end now (at a moment t1) and the other end later (at a moment t2) in K0, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K0, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K0.{{cite book}}: CS1 maint: location missing publisher (link)

[dewanberan-1] 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.

[Bell76-2] J. S. Bell: How to teach special relativity, Progress in Scientific culture 1(2) (1976), pp. 1–13. Reprinted in J. S. Bell: Speakable and unspeakable in quantum mechanics (Cambridge University Press, 1987), chapter 9, pp. 67–80.

[franklin-3] Cite error: The named reference franklin was invoked but never defined (see the help page).

[fayngold-4] Moses Fayngold (2009). Special Relativity and How it Works. John Wiley & Sons. p. 407. ISBN 978-3527406074. Note that the proper distance between two events is generally not the same as the proper length of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length l(0). If you are in the rest frame K0 of the rod, and you want to measure its length, you can do it by first marking its end-points. And it is not necessary that you mark them simultaneously in K0. You can mark one end now (at a moment t1) and the other end later (at a moment t2) in K0, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K0, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K0.{{cite book}}: CS1 maint: location missing publisher (link)

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Bell's spaceship paradox

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