Relativity and the Wave Equation. (a) Consider the

Chapter 37, Problem 37.7

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Relativity and the Wave Equation. (a) Consider the Galilean transformation along the x@direction: x = x - vt and t = t. In frame S the wave equation for electromagnetic waves in a vacuum is 0 2 E1x, t2 0x2 - 1 c2 0 2 E1x, t2 0t 2 = 0 where E represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame S is found to be a1 - v2 c2 b 0 2 E1x, t2 0x=2 + 2v c2 0 2 E1x, t2 0x= 0t = - 1 c2 0 2 E1x, t2 0t =2 = 0 This has a different form than the wave equation in S. Hence the Galilean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint: Express the derivatives 0>0x and 0>0t in terms of 0>0x and 0>0t by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame S the wave equation has the same form as in frame S: 0 2 E1x, t2 0x 2 - 1 c2 0 2 E1x, t2 0t 2 = 0 Explain why this shows that the speed of light in vacuum is c in both frames S and S.