(Lorentz invariance of the wave equation) Thinking of the
Chapter 9, Problem 4(choose chapter or problem)
(Lorentz invariance of the wave equation) Thinking of the coordinates of space-time as 4-vectors (x, y, z, t), let be the diagonal matrix with the diagonal entries 1, 1, 1, 1. Another matrix L is called a Lorentz transformation if L has an inverse and L1 = tL, where tL is the transpose. (a) If L and M are Lorentz, show that LM and L1 also are. (b) Show that L is Lorentz if and only if m(Lv)=m(v) for all 4vectors v=(x, y,z,t), where m(v)= x2 + y2 +z2 t2 is called the Lorentz metric. (c) If u(x, y,z,t) is any function and L is Lorentz, let U(x, y,z,t)= u(L(x, y,z,t)). Show that Uxx+Uyy+Uzz Utt =uxx+uyy+uzz utt. (d) ExplainthemeaningofaLorentztransformationinmoregeometrical terms. (Hint: Consider the level sets of m(v).)
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