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Old September 25th 11, 08:07 PM posted to sci.astro,sci.math
Pentcho Valev
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Default WHAT IF FASTER-THAN-LIGHT TRAVEL IS POSSIBLE

The solution to the *train in a tunnel* problem is on p. 57 in David
Morin's text:

http://www.people.fas.harvard.edu/~djmorin/chap11.pdf
p. 57: "Yes, the bomb explodes. This is clear in the frame of the
train... (...) We can, however, also look at things in the frame of
the tunnel... (...) Therefore, the deactivation device gets triggered
before the front of the train passes the far end of the tunnel, so you
might think that the bomb does not explode. We appear to have a
paradox. The resolution to this paradox is that the deactivation
device cannot instantaneously tell the bomb to deactivate itself. It
takes a finite time for the signal to travel the length of the train
from the sensor to the bomb. And it turns out that this transmission
time makes it impossible for the deactivation signal to get to the
bomb before the bomb gets to the far end of the tunnel, no matter how
fast the train is moving. Let's show this. The signal has the best
chance of winning this "race" if it has speed c, so let's assume this
is the case..."

It can be rigorously proved that, if the deactivation signal's speed
is higher than c (e.g. neutrinos faster than light are used), the
signal does get to the bomb before the bomb gets to the far end of the
tunnel. So special relativity predicts that the bomb explodes in the
frame of the train and does not explode in the frame of the tunnel.

Pentcho Valev wrote:

My claim:

"Insofar as the fate of special relativity is concerned, the neutrinos-
faster-than-light hysteria is a red herring par excellence."

was not quite correct. It can be proved, although the proof is
somewhat complicated, that if a signal is faster than light, special
relativity predicts that an event (bomb explosion) occurs according to
one observer and does not occur according to the other:

http://www.people.fas.harvard.edu/~djmorin/chap11.pdf
pp. 41-42: "11.6. Train in a tunnel. A train and a tunnel both have
proper lengths L. The train moves toward the tunnel at speed v. A bomb
is located at the front of the train. The bomb is designed to explode
when the front of the train passes the far end of the tunnel. A
deactivation sensor is located at the back of the train. When the back
of the train passes the near end of the tunnel, the sensor tells the
bomb to disarm itself. Does the bomb explode?"

A much easier and more convincing proof of the contradictory nature of
special relativity is supplied by the so-called bug-rivet paradox:

http://hyperphysics.phy-astr.gsu.edu.../bugrivet.html
"The bug-rivet paradox is a variation on the twin paradox and is
similar to the pole-barn paradox.....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.....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....The paradox is not resolved."

Pentcho Valev