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 = 2.29 and the Lorentz transformation is used to transform quantities from one frame to the other. Rivet frame of referenceThe rivet is considered to the be the reference frame, and x' and t' are used for positions and times. End of rivet enters hole: t' = 0

Rivet head hits wall: t' = 0.8 cm/0.9c = 29.63 ps

Rivet end hits length-contracted bottom of hole: t' = 1 cm/0.9cg = 16.14 ps

The end of the rivet hits the bottom of the hole before the head of the rivet hits the wall. So it looks like the bug is squashed.

Transforming the times to the bug's frame of reference: End of rivet enters hole: t = 0

Rivet head hits wall: t = (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

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.

Bug frame of referenceFront of rivet enters hole: t = 0

Head of rivet hits wall: t = 0.8 cm/0.9c = 12.91 ps

All this is nonsense from the bug's point of view. The rivet head hits the wall when the rivet end is just 0.35 cm down in the hole! The rivet doesn't get close to the bug.

Transforming the times measured in the bug's frame of reference to the rivet frame: Rivet head impact time t' = (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' =(12.9 ps - .9c(.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' =(37.04 ps - .9c(.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.

The Pole-Barn Paradox

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The Pole-Barn Paradox 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.

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 = 2.29 and the Lorentz transformation is used to transform quantities from one frame to the other. Barn frame of referenceThe 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

Back of pole enters: 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

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.

Pole frame of referenceFront of pole enters: t' = 0

Front of pole leaves barn: t' = 4.37m/0.9c = 16.14 ns

Back of pole enters: t = 20m/0.9c = 74.07 ns

Back of pole leaves: t = 16.14 ns + 74.07 ns = 90.21 ns

Front gate closes at t = 32.35 ns, but t'= (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'=(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.

The Bug-Rivet Paradox

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