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))