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We can PROVE it is false since it produces absurd couples of propositions
of the type (p, not-p). As you know, length contraction is a corollary of
the axiom so let us consider the following textbook example:
"Two bombs lie on a train platform, a distance L apart. As a train passes by
at constant speed,the bombs explode simultaneously (in the platform frame)
and leave marks on the train. Due to the length contraction of the train,
the marks on the train will be a distance gamma*L apart when viewed in the
train's frame (since this distance is what is length-contracted down to the
given distance L in the platform frame)."
Millions of professors have taught this example and yet nobody has found it
suitable to introduce the following modifications. The bombs are replaced
with two barriers which are simultaneously (in the platform frame) stretched
across the railway. The barriers are strong enough to be able to stop the
train. Also, the length of the train is L', a value limited by L<L'<gamma*L.
Is the barrier mechanism capable of "catching" the train? The observer in
the platform frame will first say "yes" since, in this frame, the moving
train is shorter than L. Then the same observer will say "no" since the
train cannot remain length-contracted after joining the platform frame.
Finally, millions of relativists will say "never mind" since for them
relativity is a cult, not a theory that should be verified.
Of course, if for some reason you don't like this textbook example, we
could analyse others.