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Unformatted text preview: Physics 325 Lecture 21 More on Minkowski Diagrams The Minkoski diagrams make some of the strange things weve been investigating very apparent. Consider for example (Kogut sec. 3.2) the world lines of two clocks at rest in frame S: Clocks 1 (on the left) and 2 (on the right) are at rest in the frame S. Their world lines take them from points along the x-axis at ct=0 to some later time d=ct away. The picture has been drawn intentionally such that the x axis intersects clock 1 at t=0 and clock 2 at t=d . This means that both clocks lie on the x axis in S at t=0 . At this time, clock 1 reads t=0 , but clock 2 reads t=d/c . If the separation between the clocks in S is A (the proper distance between the clocks), then tan d = = A . so that d = A and the time on clock 2 is 2 v d t c c c = = = A A This is the same result that we got before: The two clocks are at rest and synchronized in S (they are at constant t in the Minkowski diagram), but separated by A . When viewed from S, the clocks are moving to the left at velocity v . When clock 1 reads t=t=0, clock 2 reads 2 t v c = A . x x ct 1 tan = d 1 2 x x ct 1 tan = d x x ct 1 tan = d 1 2 We can turn this around to view two clocks at rest in S on the same Minkowski diagram. axis, so it looks like this: In S, the two clocks are synchronized to t=0 when they are on the x axis. From this picture we can see that when they are compared at a constant time t in S, clock 2 is behind clock 1. This is apparent because at t=0, clock 1 is at t Their world lines will be parallel to the t = , but clock 2 is at t d c = . Again, geometry tells us precisely how much clock 2 is behind clock 1. We have ( ) sin tan sin 2 d = = = + A Since d ct = we get 2 v d t c c c = = = A A just like before. Of course we expect these results since the slopes of the axes are determined by the Lorentz transformations. Nevertheless, the Minkowski diagrams are a powerful tool for visualizing these transformations....
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This note was uploaded on 10/06/2011 for the course PHYS 225 taught by Professor Makins during the Spring '10 term at University of Illinois, Urbana Champaign.
- Spring '10