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Bell's spaceship paradox is a thought experiment in special relativity. It was first designed by E. Dewan and M. Beran in 1959 and became more widely known when J. S. Bell included a modified version.〔 in chapter 9 of his popular 1976 book on quantum mechanics.〕 A delicate string or thread hangs between two spaceships. Both spaceships now start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times in 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. Therefore at first sight it might appear that the thread will not break during acceleration. This argument, however, is incorrect as shown by Dewan & Beran and Bell.〔〔 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, in S due to length contraction of the thread. In the following, the rest length〔 or ''proper length''〔 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.〔: p. 407: "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."〕) == Dewan and Beran == Dewan and Beran stated the thought experiment by writing: :"Consider two identically constructed rockets at rest in an inertial frame S. Let them face the same direction and be situated one behind the other. If we suppose that at a prearranged time both rockets are simultaneously (with respect to S) fired up, then their velocities with respect to S are always equal throughout the remainder of the experiment (even though they are functions of time). This means, by definition, ''that with respect to S'' the distance between the two rockets does not change even when they speed up to relativistic velocities."〔 Then this setup is repeated again, but this time the back of the first rocket is connected with the front of the second rocket by a silk thread. They concluded: :"According to the special theory the thread must contract with respect to S because it has a velocity with respect to S. However, since the rockets maintain a constant distance apart with respect to S, the thread (which we have assumed to be taut at the start) cannot contract: therefore a stress must form until for high enough velocities the thread finally reaches its elastic limit and breaks."〔 Dewan and Beran also discussed the result from the viewpoint of inertial frames momentarily comoving with the first rocket, by applying a Lorentz transformation: :"Since , (..) each frame used here has a different synchronization scheme because of the factor. It can be shown that as increases, the front rocket will not only appear to be a larger distance from the back rocket with respect to an instantaneous inertial frame, but also to have started at an earlier time."〔 They concluded: :"One may conclude that whenever a body is constrained to move in such a way that all parts of it have the same acceleration with respect to an inertial frame (or, alternatively, in such a way that with respect to an inertial frame its dimensions are fixed, and there is no rotation), then such a body must in general experience relativistic stresses."〔 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: * Since the rockets are constructed exactly the same way, and starting at the same moment in S with the same acceleration, they must have the same velocity all of the time in S. Thus they are traveling the same distances in S, so their mutual distance cannot change in this frame. Otherwise, if the distance were to contract in S, then this would imply different velocities of the rockets in this frame as well, which contradicts the initial assumption of equal construction and acceleration. * They also argued that there indeed is a difference between a) and b): Case a) is the ordinary case of length contraction, based on the concept of the rod's rest length l0 in S0, which always stays the same as long as the rod can be seen as rigid. Under those circumstances, the rod is contracted in S. But the distance cannot be seen as rigid in case b) because it is increasing due to unequal accelerations in S0, and the rockets would have to exchange information with each other and adjust their velocities in order to compensate for this – all of those complications don't arise in case a). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bell's spaceship paradox」の詳細全文を読む スポンサード リンク
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