The Bug-Rivet Paradox

The bug-rivet paradox is a variation on the twin paradox and is similar to the pole-barn paradox in that the ideas of simultaneity in relativity must be addressed. 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.

To calculate the times for the two frames of reference, consider the rivet entering the hole 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 rivet frame while x and t refer to the bug frame. The relativity factor γ = 2.29 and the Lorentz transformation is used to transform quantities from one frame to the other.

The rivet is considered to the be the reference frame, and x' and t' are used for positions and times.

Transforming the times to the bug's frame of reference:

End of rivet enters hole: t = 0 铆钉进入孔底的时间：t = 0 Rivet head hits wall: t = γ(t'+vx'/c2) = 2.29(29.63 ps + (-0.9c)(0.8 cm)/c2) = 12.9 ps 铆钉头撞墙：t = γ(t' + vx'/c²) = 2.29(29.63 ps + (-0.9c)(0.8 cm)/c²) = 12.9 ps Time for rivet to reach end of hole: t = 2.29(16.14 ps) = 37.04 ps 铆钉到达孔底所需时间：t = 2.29(16.14 ps) = 37.04 ps

The bug disagrees with this analysis and finds the time for the rivet head to hit the wall is earlier than the time for the rivet end to reach the bottom of the hole. The paradox is not resolved. 这只虫子不同意这一分析，认为rivet头击中墙的时间早于rivet尾端到达孔底的时间。这个悖论并未得到解决。

Transforming the times measured in the bug's frame of reference to the rivet frame:

Rivet head impact time t' = γ(12.9 ps) = 29.63 ps 铆钉头冲击时间 t' = γ(12.9 ps) = 29.63 ps Rivet end in bug frame is simultaneously at x=-.35 cm, but it is not simultaneous in the rivet frame. Transforming gives t' =γ(12.9 ps - .9c(.0035)/c2) = 5.63 ps. 在铆钉框架中，铆钉末端同时位于x=-0.35 cm处，但在虫洞框架中并非同时。变换后得到t' = γ(12.9 ps - 0.9c(0.0035)/c²) = 5.63 ps。 If you try to find a rivet frame time when the rivet end hits the bottom of the hole, x = -1 cm, t' =γ(37.04 ps - .9c(.01)/c2) = 16.14 ps. 如果你试图找到一个铆钉框架的时间，当铆钉末端击中孔底时，x = -1 cm，t' = γ(37.04 ps - 0.9c(0.01)/c² ) = 16.14 ps.

Transforming times from the bug frame to the rivet frame gives a time for the end to reach -0.35 cm before the rivet head hits, and even suggests that it reaches the bottom of the hole before the rivet head hits. The paradox is not resolved. 将虫洞参考系中的时间转换为铆钉参考系，得到一个结果：钉头撞击前，钉尾到达孔底的时间为-0.35 cm，甚至暗示钉尾在钉头撞击前已到达孔底。这个悖论仍未得到解决。