The Pole-Barn Paradox

杆仓悖论

The pole-barn paradox is a famous variation on the twin paradox which must be addressed with the ideas of simultaneity in relativity. The fact that two events are simultaneous in one frame of reference does not imply that they are simultaneous as seen by an observer moving at a relativistic speed with respect to that frame.

杆-车库悖论是双生子悖论的一个著名变种，必须用相对论中的同时性概念来解决。在一个参考系中两个事件是同时发生的，并不意味着在另一个以相对论速度运动相对于该参考系的观察者看来它们也是同时发生的。

To calculate the times for the two frames of reference, consider the pole entering the barn and set t=t'=0 at that instant and x=x'=0 to establish the coordinate system. In the events described below, x' and t' refer to the pole frame while x and t refer to the barn frame. The relativity factor γ = 2.29 and the Lorentz transformation is used to transform quantities from one frame to the other.

为计算两个参考系的时间，考虑杆进入 barn 的情况，并在该时刻将 t=t'=0 和 x=x'=0 以建立坐标系。在下面描述的事件中，x' 和 t' 指杆的参考系，而 x 和 t 指 barn 的参考系。相对论因子 γ = 2.29，使用洛伦兹变换将量从一个参考系转换到另一个参考系。

Barn frame of reference

巴恩参考系

The barn is considered to the be the reference frame, and x and t are used for positions and times.

Front of pole enters: t = 0
杆子前端进入：t = 0
Back of pole enters: t = 8.73m/0.9c = 32.29 ns
杆尾端进入：t = 8.73m/0.9c = 32.29 ns
Front of pole leaves: t = 10m/0.9c = 37.04 ns
Back of pole leaves: t = 32.35ns + 37.04 ns = 69.38 ns
杆后端离开：t = 32.35ns + 37.04 ns = 69.38 ns

杆前端离开：t = 10m/0.9c = 37.04 ns

The back of the pole enters the barn before the front of the pole leaves, so a 1 ns gate could be closed on both ends, containing the entire pole.

杆子的后端进入谷仓之前，前端已经离开谷仓，因此一个1 ns的逻辑门可以在两端同时关闭，将整个杆子包含在内。

Pole frame of reference

极坐标参考系

Front of pole enters: t' = 0
杆前端进入：t' = 0
Front of pole leaves barn: t' = 4.37m/0.9c = 16.14 ns
杆前端离开谷仓的时间：t' = 4.37m/0.9c = 16.14 ns
Back of pole enters: t = 20m/0.9c = 74.07 ns
杆尾端进入：t = 20m/0.9c = 74.07 ns
Back of pole leaves: t = 16.14 ns + 74.07 ns = 90.21 ns
杆尾端的时刻：t = 16.14 ns + 74.07 ns = 90.21 ns
Front gate closes at t = 32.35 ns, but t'= γ(t-vx/c²) = 2.29(32.29 ns) = 74.07 ns
前逻辑门在 t = 32.35 ns 时关闭，但 t' = γ(t - vx/c²) = 2.29(32.29 ns) = 74.07 ns
Back gate closes at t=32.35 ns, but at x=10m. It is simultaneous in the barn frame, but not in the pole grame. The time for back gate closing in the pole frame is t'=γ(t-vx/c²) = 2.29(32.35 - (0.9c)(10 m)/c²) = 5.38 ns.
后门在t=32.35 ns时关闭，但位于x=10m处。在杆框中这是同时发生的，但在杆框中不是。在杆框中后门关闭的时间为t'=γ(t - vx/c²) = 2.29(32.35 - (0.9c)(10 m)/c²) = 5.38 ns。

From the pole point of view, the front gate closes just as the back of the pole enters. The surprising result is that the back gate is seen to close earlier from the pole framework, before the front of the pole reaches it. The gate closings are not simultaneous, and they permit the pole to pass through without hitting either gate.

从杆的视角来看，前门在杆的后端进入时刚好关闭。令人惊讶的是，在杆的框架中，后门被看到在前门关闭之前就关闭了，而前门的前端尚未到达该位置。门的关闭并非同时发生，这使得杆能够通过而不碰到任何一扇门。

barn 被视为参考系，x 和 t 用于表示位置和时间。