# T_NAKÄ[uO

## ux̉FDphbNXvx󂵂Ă݂悤iUj

<<   쐬 F 2016/12/12 00:01   >>

 wikipediaBell's spaceship paradox̖̑܂B [Analysis][Accelerating ships][Born rigidity]̕łB [Accelerating shipsFFD]====================================== Similarly, in the case of Bell's spaceship paradox the relation between the initial rest length L between the ships (identical to the moving length in S after acceleration) and the new rest length L' in S after acceleration, is: lɁAx̉FDphbNX̏ꍇA S 'nɂVÎ~ L'ƁADԂ̏Î~ Li SnɂړƓjƂ @${L}'=&space;\gamma&space;L$ This length increase can be calculated in different ways. 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 $\110dpi x$-coordinates transformed from S to S. If $\110dpi x_{A}$ and $\110dpi x_{B} = x_{A} + L$ are the ships' positions in S, the positions in their new rest frame Sare: ̒̑́A܂܂ȕ@ŌvZłB Ⴆ΁AIƁAD͍ŏIÎ~n S '̓ʒuɏɗ܂̂ŁAS S'ɕϊꂽ $\110dpi x$ WԂ̋vZ΂悢B $\110dpi x_{A}$ $\110dpi x_{B} = x_{A} + L$ S ̑D̈ʒułꍇA̐VÎ~n S ' ̈ʒû͎ƂłB @${x}'_{A}= \gamma (x_{A}-vt)\; ,\; {x}'_{B}= \gamma (x_{A}+L-vt)$ @${L}'={x}'_{B}- {x}'_{A}=\gamma (x_{A}+L-vt)- \gamma (x_{A}-vt)= \gamma L$ Another method was shown by Dewan (1963) who demonstrated the importance of relativity of simultaneity. The perspective of frame S is described, in which both ships will be at rest after the acceleration is finished. The ships are accelerating simultaneously at $\110dpi t_ {A} = t_ {B}$ in S (assuming acceleration in infinitesimal small time), though B is accelerating and stopping in S before A due to relativity of simultaneity, with the time difference: ̕@́Ȃΐ̏dv؂Dewani1963jɂĎꂽB IẢFDx~n S 'ł̌ЉĂB S 'ł B ͓̑ΐ̂߂ A ̑ỎĒ~ĂƂԍ邪 AS ł́iԂ̉肵āj$\110dpi t_ {A} = t_ {B}$ œɉB @$\Delta {t}'={t}'_{A}- {t}'_{B}=\gamma \left (t_{A}-\frac{vx_{A}}{c^{2}} \right)- \gamma \left (t_{B}-\frac{vx_{B}}{c^{2}} \right )= \gamma (t_{A}-t_{B})+\gamma \frac{v(x_{B}-x_{A})}{c^{2}}$ @@$= \frac{\gamma vL}{c^{2}}$ Since the ships are moving with the same velocity in S before acceleration, the initial rest length $\110dpi L$ in S is shortened in S by $\110dpi {L}'_{old}= L / \gamma$ due to length contraction. This distance starts to increase after B came to stop, because A is now moving away from B with constant velocity during $\110dpi \Delta {t}'$ until A stops as well. Dewan arrived at the relation (in different notation): FD͉O S 'œxœĂ̂ŁA[ck̂߂ S' ̍ŏ̐Î~ $\110dpi L$ $\110dpi {L}'_{old}= L / \gamma$ ɂȂB ̋́AB@~ɑn߂BȂȂAA ͎g~܂ł̏\Ȏ $\110dpi \Delta {t}'$ Ɉ葬xł B 痣邩łB Dewan͂̊֌Wiʂ̋L@Łjɂǂ蒅F @${L}'= {L}'_{old}+v\Delta {t}'= \frac{L}{\gamma }+\frac{\gamma v^{2}L}{c^{2}}= \gamma L\left ( \frac{1}{\gamma ^{2}} +\frac{v^{2}}{c^{2}}\right )= \gamma L\left (1-\frac{v^{2}}{c^{2}} +\frac{v^{2}}{c^{2}}\right )$ @@$= \gamma L$ It was also noted by several authors that the constant length in S and the increased length in S is consistent with the length contraction formula $\110dpi L = {L}'/ \gamma$ , because the initial rest length $\110dpi L$ is increased by $\110dpi \gamma$ in S, which is contracted in S by the same factor, so it stays the same in S: lɂĎwEĂ邪AS ̈蒷 S ' ̑̓[ck@$\110dpi L = {L}'/ \gamma$ ƈvB́A $\110dpi L$ S ' $\110dpi \gamma$ ɂđ邽߂łA S œqŏkĂ邽߁AS ł͓܂܂łB @$L_{contr}= \frac{{L}'}{\gamma }= \frac{\gamma L}{\gamma }= L$ Summarizing: While the rest distance between the ships increases to $\110dpi \gamma L$ in S, the relativity principle requires that the string (whose physical constitution is unaltered) maintains its rest length $\110dpi L$ in its new rest system S. Therefore, it breaks in S due to the increasing distance between the ships. As explained above, the same is also obtained by only considering the start frame S using length contraction of the string (or the contraction of its moving molecular fields) while the distance between the ships stays the same due to equal acceleration. vFDԂ̐Î~ S ' $\110dpi \gamma L$ ܂őŁAΐ_̌ł́AVÎ~n@S ' ŁiI\ύXĂȂjÎ~ $\110dpi L$ ێBāADԂ̋Ȃ邽߂ S 'Ŏ͐؂B q悤ɁA̒̎ki܂͂̈ړ镪q̎kjpĊJnn S l邱ƂɂĂƂ邪Ax̂߂ɑDԋ͓łB [}] Minkowski diagram: The world lines (navy blue curves) of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A and B, the observers stop accelerating. The dotted line is a "line of simultaneity" for either observer after acceleration stops. ~RtXL[}F2l̊ϑAB̐EiF̋ȐjBɓ̉xŉB A 'B'ł́Aώ@҂͉~߂B _́A~̂̊ώ@҂ɂƂĂu̐vłB Loedel diagram: Length $\110dpi {L}'$ between the ships in S after acceleration is longer than the previous length $\110dpi {L}'_{old}$ in S, and longer than the unchanged length {\displaystyle L} L in S. The dashed lines indicate the broken string in S and S. Loedel }i~RtXL[Όŕ\킵́jF S '̑DԂ̒ $\110dpi {L}'$ ́AS ̑O̒ $\110dpi {L}'_{old}$ 蒷A j S S 'ɂ؂ꂽB ========================================================================== [Born rigidityF{]=============================================== The mathematical treatment of this paradox is similar to the treatment of Born rigid motion. However, rather than ask about the separation of spaceships with the same acceleration in an inertial frame, the problem of Born rigid motion asks, "What acceleration profile is required by the second spaceship so that the distance between the spaceships remains constant in their proper frame?". In order for the two spaceships, initially at rest in an inertial frame, to maintain a constant proper distance, the lead spaceship must have a lower proper acceleration. ̃phbNX̐wI̓{̂̉^̑ΏƓłB AЂƂ̊n̓xFD̕ɂčl@ƂA{̉^̖́AuFDԂ̋K؂Ȍnňɕۂ悤ɁA2FDvvtB[͂ǂ̂悤Ȃ̂HvƂ̂łB 2̉FDŏ͊nɐÎ~āA̓Kێ邽߂ɂ́AsFD͓K؂ȉxႭȂ΂ȂȂB ========================================================================== Bell's spaceship paradox̑Ŝǂł݂āA_̖_Ƌc_ɂėǂ܂Ƃ܂ĂƊ܂B wikipedia Ƃgł͎dȂ̂łA̘b̃cbR~󂭁Au؂ȂvƎvłl[邩HƂƂƖȂƎv܂B AǂohbNXH͕Ǝv܂B

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