Special Relativity 1

A sharp projection is extended from the side of the

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Unformatted text preview: ) z ￿ x = L1,0 (β ) ct￿ + L1,1 (β ) x￿ + L1,2 (β ) y ￿ + L1,3 (β ) z ￿ y = L2,0 (β ) ct￿ + L2,1 (β ) x￿ + L2,2 (β ) y ￿ + L2,3 (β ) z ￿ z = L3,0 (β ) ct￿ + L3,1 (β ) x￿ + L3,2 (β ) y ￿ + L3,3 (β ) z ￿ So. . . we need to find 16 matrix elements5 . Simplification . . . or not. Consider the following experiment: orient the axes so that the rocket travels in the +x direction in the lab. A sharp projection is extended from the side of the rocket in such a way that it scrapes a very long wall that runs parallel to the rocket’s motion through the lab. If that projection is mounted at y ￿ = a in the rocket, where will the resulting mark appear in the lab? Remember - the frames were coincident at t = t￿ = 0, and the rocket frame is moving with a velocity along the +x direction in the lab. Relative motion does not affect the transverse spacial coordinates of an event. If the rocket is moving in the x direction, y = y ￿ and z = z ￿ . With the exception of the identity sub-matrix in the transform, the rest of the elements relating to transverse spacial components go to zero! The transform will look like: ￿ ct L 0 ,0 ( β ) L 0 ,1 ( β ) 0 0 ct x L1,0 (β ) L1,1 (β ) 0 0 x￿ = y 0 0 1 0 y ￿ z 0 0 01 z￿ or. . . ct = L0,0 (β ) ct￿ + L0,1 (β ) x￿ x = L1,0 (β ) ct￿ + L1,1 (β ) x￿ y = y￿ z = z￿ 5 Note the conventional use of 0 to denote the temporal index. 4 Since there is not a lot happening in the transverse dimensions, they are frequently ignored. It is common practice to omit the transverse coordinates when writing a transform. So - for relative motion along the x direction in the lab. . . ￿￿ ￿ ￿ ￿ ￿￿ ct L 0 ,0 ( β ) L 0 ,1 ( β ) ct = x L 1 ,0 ( β ) L 1 ,1 ( β ) x￿ or. . . ct = L0,0 (β ) ct￿ + L0,1 (β ) x￿ x = L1,0 (β ) ct￿ + L1,1 (β ) x￿ with the understanding that. . . y = y￿ z = z￿ Galilean Transformation Let’s derive the (2 × 2) relativistic transformation for Galileo’s world....
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This note was uploaded on 12/03/2013 for the course PHYSICS 105A taught by Professor Corbin during the Winter '13 term at UCLA.

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