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Unformatted text preview: 1.3 Lorentz Transformation Since Galilean transformations are inconsistent with Einstein’s postulate of the speed of light, we must modify them. Consider our two inertial frames S and S ′ , and let the axes be parallel, with S ′ moving w.r.t. S with a speed u in the x direction. The times t and t ′ are zero when the origins coincide. For events with y ′ = x ′ 2 = 0, these should also have y = x 2 = 0 independent of the rest of the variables. Similarly for z = x 3 , and we get x ′ 2 = x 2 and x ′ 3 = x 3 (24) These are the same as the Galilean transformation. Along the x direction, we derive the most general linear transformation, x ′ 1 = ax 1 + bt (25) At the origin of S ′ , where x ′ 1 = 0, we expect x 1 = ut . Substitute this initial condition in. 0 = aut + bt ⇒ b =- au (26) and our general linear combination reduces to x ′ 1 = a ( x 1- ut ) (27) By symmetry, we also have x 1 = a ( x ′ 1 + ut ′ ) (28) Now let’s apply the second postulate of Special Relativity. If a pulse of light is sent fromNow let’s apply the second postulate of Special Relativity....
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- Spring '10
- Special Relativity, Spacetime, ruler, system S, Lorentz Transformation, yB