Bell's spaceship paradox
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 initially at rest in the inertial frame S. They start accelerating in the same direction simultaneously and equally, as measured in 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.
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.
They concluded: Dewan and Beran also discussed the result from the viewpoint of inertial frames momentarily comoving with the first rocket, by applying a Lorentz transformation: They concluded: Then they discussed the objection, that there should be no difference between a) the distance between two ends of a connected rod, and b) the distance between two unconnected objects which move with the same velocity with respect to an inertial frame.
Dewan and Beran removed those objections by arguing: In Bell's version of the thought experiment, three spaceships A, B and C are initially at rest in a common inertial reference frame, B and C being equidistant to A.
According to Bell, this implies that B and C (as seen in A's rest frame) "will have at every moment the same velocity, and so remain displaced one from the other by a fixed distance."
Now, if a fragile thread is tied between B and C, it's not long enough anymore due to length contractions, thus it will break.
He concluded that "the artificial prevention of the natural contraction imposes intolerable stress".
[2] Bell reported that he encountered much skepticism from "a distinguished experimentalist" when he presented the paradox.
To attempt to resolve the dispute, an informal and non-systematic survey of opinion at CERN was held.
Bell goes on to add, In general, it was concluded by Dewan & Beran and Bell, that relativistic stresses arise when all parts of an object are accelerated the same way with respect to an inertial frame, and that length contraction has real physical consequences.
The distorted electromagnetic intermolecular fields cause moving objects to contract, or to become stressed if hindered from doing so.
They agreed with the result that the string will break due to unequal accelerations in the rocket frames, which causes the rest length between them to increase (see the Minkowski diagram in the analysis section).
However, they denied the idea that those stresses are caused by length contraction in S. This is because, in their opinion, length contraction has no "physical reality", but is merely the result of a Lorentz transformation, i.e. a rotation in four-dimensional space which by itself can never cause any stress at all.
Thus the occurrence of such stresses in all reference frames including S and the breaking of the string is supposed to be the effect of relativistic acceleration alone.
[3][6] Paul Nawrocki (1962) gives three arguments why the string should not break,[7] while Edmond Dewan (1963) showed in a reply that his original analysis still remains valid.
[8] Many years later and after Bell's book, Matsuda and Kinoshita reported receiving much criticism after publishing an article on their independently rediscovered version of the paradox in a Japanese journal.
Matsuda and Kinoshita do not cite specific papers, however, stating only that these objections were written in Japanese.
[9] However, in most publications it is agreed that the string will break, with some reformulations, modifications and different scenarios, such as by Evett & Wangsness (1960),[10] Dewan (1963),[8] Romain (1963),[11] Evett (1972),[12] Gershtein & Logunov (1998),[13] Tartaglia & Ruggiero (2003),[14] Cornwell (2005),[15] Flores (2005),[16] Semay (2006),[17] Styer (2007),[18] Freund (2008),[19] Redzic (2008),[20] Peregoudov (2009),[21] Redžić (2009),[22] Gu (2009),[23] Petkov (2009),[6] Franklin (2009),[3] Miller (2010),[24] Fernflores (2011),[25] Kassner (2012),[26] Natario (2014),[27] Lewis, Barnes & Sticka (2018),[28] Bokor (2018).
[32][33] Similarly, in the case of Bell's spaceship paradox the relation between the initial rest length
For instance, if the acceleration is finished the ships will constantly remain at the same location in the final rest frame S′, so it's only necessary to compute the distance between the x-coordinates transformed from S to S′.
Due to this B will always have higher velocity than A up until the moment A is finished accelerating too, and both of them are at rest with respect to S′.
in S′, which is contracted in S by the same factor, so it stays the same in S:[6][14][18] Summarizing: While the rest distance between the ships increases to
in S′, the relativity principle requires that the string (whose physical constitution is unaltered) maintains its rest length
This leads to hyperbolic motion, in which the observer continuously changes momentary inertial frames[34] where
