Special Relativity 1

This forces l 1 1 1 finally we can describe o in the

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Unformatted text preview: What are the constraints he would have imposed? • i) Time is the same for all observers. • ii) Space is the same for all observers. • iii) The origin of the rocket frame (O￿ ) was co-located with the origin of the lab frame (O) at t = t￿ = 0 and moves through the lab frame with a velocity ￿ = β cx. v ˆ Since the clocks were synchronized at t = t￿ = 0, the first constraint says t = t￿ (always), and. . . L 0 ,0 ( β ) = 1 and L 0 ,1 ( β ) = 0 Since the origins were co-located at t = t￿ = 0, the second constraint tells us that x = x￿ at t = t￿ = 0. This forces L 1 ,1 ( β ) = 1 Finally, we can describe O￿ in the rocket frame. . . ￿ ￿￿ ct ￿ r= 0 The third constraint tells us that in the lab, the description of O￿ looks like 5 r= ￿ ct β ct ￿ From which we get. . . L 1 ,0 ( β ) = β Put it all together, and the Galilean transform looks like L( β ) = ￿ 1 β 0 1 ￿ An Instructive Example S S’ v =β c u’ Lab Rocket Suppose a mouse leaves the origin of the rocket-ship at t￿ = 0 and walks forward with a velocity (in the rocket) u ￿ x￿ . The position of the mouse (in the rocket) ˆ would be given by x￿ = u￿ t￿ and the spacetime coordinates of the mouse (in the rocket) would look like: ￿ ￿￿ ct r￿ = u ￿ t￿ What will the mouse’s motion look like in the lab? Since the origins coincide at t = t￿ = 0, the mouse will appear to leave the origin in the lab at t = 0 and proceed forward with a velocity (in the lab) u x. The position of the mouse (in ˆ the lab) at any instant would be given by x = u t and the spacetime coordinates of the mouse (in the lab) would look like: ￿￿ ct r= ut Let’s relate these two descriptions by the appropriate Galilean transform. . . ￿￿ ￿ ￿￿ ￿ ￿ ct 10 ct = ut β1 u ￿ t￿ ct = ct￿ 6 ut = (β c + u￿ ) t￿ Divide the second equation by the first. . . u = u￿ + β c which is just another way of saying. . . Vmouse,lab = Vmouse,rocket + Vrocket,lab We recover the classical relative velocity equation! The G...
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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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