[net.physics] Relativistic Paradox problem

stekas@hou2g.UUCP (J.STEKAS) (02/13/84)

Ok, the no paper/no pencil problem was too easy.

Consider the following paradox.  Suppose arrows of length L are fired
through a tube of slightly shorter length.  If the arrows are fired at
a high enough velocity, they will be sufficiently Lorentz contracted
to be completely contained within the tube.  Therefore, one should be
able to take a photo of the tube while the arrow was completely inside
it, and none of the arrow showing.

                      ________________________________

                        >--------ARROW------------>

                      ________________________________

But using the same arguement, form the arrows point of view it is the
the TUBE which is Lorentz contracted.  Therefore, any picture of the arrow
passing through the tube should show it sticking out the end.

                           ______________________

                       >---------ARROW-------------->

                           ______________________


What would the picture show?   If this isn't a paradox, how does one reconcile
the two points of view?

                     
                                                          Jim

ntt@dciem.UUCP (Mark Brader) (02/15/84)

This is an old chestnut.  To recap the problem, we are asked to consider an
arrow shot at high speed through a tube of similar length.  In the frame
of reference of the tube, the arrow is moving and is contracted, and fits
entirely within the tube, but in the frame of reference of the arrow, the
tube is contracted and the arrow does not fit entirely within it.  So if we
take a photograph when the center of the arrow is at the center of the tube,
what does it show?

It is convenient that the problem referred to taking a photograph, because
the answer is:  It depends on the velocity of the camera!  If it is moving
along with the arrow, it will show the arrow sticking out both ends; if it is
in the tube frame of reference, the arrow will be shown within the tube.

In relativity, events (an event is a time and place) may be connected in a
"spacelike" or "timelike" way.  If a particle travelling from one event to
the other would have to go faster than light, the events are connected in a
spacelike way, and it is guaranteed that there will exist frames of reference
where one event happens first and frames of reference where the other event
happens first.  (Otherwise, they are connected in a timelike way, and everyone
agrees on which one happens first.)

So in this problem it is not possible to state whether the front of the arrow
leaves the tube before or after the rear of the arrow enters it, because the
two events are connected in a spacelike way.  What you observe depends on
your own velocity.

Mark Brader (v = 336 m/s (?) (allowing only for the rotation of the earth))