# T_NAKÄ[uO

## ux̉FDphbNXvx󂵂Ă݂悤iQj

<<   쐬 F 2016/11/29 00:01   >>

 wikipediaBell's spaceship paradox̖̑܂B [Dewan and Beran]̕łB [Dewan and BeranFf[Aƃx]===================================== 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." f[Aƃx͎vlȉ̂悤ɏF un S ɐÎ~Ă2̓Pbgl悤BɌāAЕ̌Ɉʒu悤ɂBiSɂāj\ߌ߂ꂽɗ̃Pbgɔ˂Ɖ肷ƁAiԂ̊֐łƂłjʂSɑ΂鑬x͏ɓB́A SɊւđΘ_Ixɑx𑝉Ă2̃PbgԂ͕̋ωȂƂӖBv 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." ̌A̐ݒxJԂA͍ŏ̃Pbǧ㕔2Ԗڂ̃Pbg̑OɌłȂĂƂA ނ͎̂悤Ɍ_ÂF uꑊΘ_ɂ΁ASɊւđxL̂ŎSɊւĎkȂ΂ȂȂBAPbgԋSɑ΂ĈɈێ̂ŁAiJnɂْ͋ĂƑz肵Ăj͎k邱ƂłȂBāA͂AeEɒB\ȑxƍŏIIɎ͐؂Av 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 $\110dpi {t}'= (t-vx/c^{2})/\sqrt{1-v^{2}/c^{2}}$, each frame used here has a different synchronization scheme because of the factor $\110dpi vx/c^{2}$. It can be shown that as $\110dpi v$ 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." f[Aƃx͂܂[cϊKpāAPڂ̃PbguԓIɐÎ~銵n̊ϓ_̌ʂɂċc_F u$\110dpi {t}'= (t-vx/c^{2})/\sqrt{1-v^{2}/c^{2}}$ AŎgpeńA $\110dpi vx/c^{2}$ q̂߂ɏꏊɂĈقȂ铯@\LB$\110dpi v$ ɂāAOPbǵAuԓIȊnɊւČPbg傫ȋɌ邾łȂA葁ɔ˂ƂƂłBv 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." ނ͌_ÂF ûnɑ΂ēxL悤ɍSƂ͂łi邢́Anɑ΂Ă̐@Œ肳悤 A]ȂꍇjAʓIɑΘ_I͂oȂ΂ȂȂBv 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: ESince 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. EThey 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 $\110dpi S_{0}$ , 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). ɁAނ́Aajڑꂽ_̗[Ԃ̋@ƁAbjnɑ΂ēxœ2̖ڑ̊Ԃ̋Ƃ̊ԁ@ɍȂ΂ȂȂƂ_ɂċc_B f[Aƃx͎̂悤Ɏ咣Ăً̈c\Ă菜B EPbg͂܂悤ɍ\ĂAxS̓uԂn܂AɓxBāAނS̓ړĂ̂ŁA݋͂̊nł͕ωȂBłȂASŎk΂̌nł̃Pbg̑xقȂ邱ƂӖA͓\Ɖ̍ŏ̉ƖB Eނ͂܂@ajƁ@bj̊ԂɎۂɈႢƎ咣BP[X aj $\110dpi S_{0}$ ɂ_̐Î~ $\110dpi l_{0}$ ̊TOɊÂk̒ʏ̏ꍇłA_̂̍ƌȂƂłB̂悤ȏ󋵉ł́A_SŎkBA $\110dpi S_{0}$ ̉xsψȂ߂ɑĂ̂ bj̏ꍇ͍ƌȂƂłAPbg͂₤߂ɏđx𒲐KvB-ׂ̎͂ aj̏ꍇɂ͔ȂB ========================================================================== Ō̂Ƃ낪ǂ󂵂̂H܂B킯łȂ񂩈ӖʂȂȂĂ܂B ܂̂̕ŗǂƂ܂傤B ͂̕ӂŁBB

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