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Conjugate Speed Transformations |
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Define the “speed” of an interval with components dx and dt to be dx/dt, regardless of whether the interval is spacelike or timelike. We’ve seen in another note that if a Lorentz transformation with parameter v maps the speed u to the speed u′, then it also maps c2/u to c2/u′. Thus a Lorentz transformation gives a consistent treatment of both the phase and the group velocities. |
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However, in both cases the parameter of the transformation is v, rather than c2/v, despite the formal three-way symmetry |
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(The symmetry isn’t perfect, though, in the sense that, with the noted indices, two of the velocities must have one sign and one must have the other.) The question arises as to how the Lorentz transformation with parameter v would be affected if v is replaced with c2/v. Letting L[v] denote the Lorentz transformation matrix with parameter v, which is to say |
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we find that |
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where the sign of ±i is positive for v < -1 and for 0 < v < 1, and otherwise it is negative. Thus in geometrical units with c=1 we have |
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This essentially just swaps x and t, and negates their squares. So, if the application of L[v] to the vector [x,t]T yields [x′,t′]T, then the application of L[v−1] to [x,t]T yields [±it′,±ix′]T. We note the identities |
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Naturally we also have the relation L[v]L[−v] = I for any v. The individual coefficient matrices also satisfy the relations |
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so in a sense these matrices are the square roots of negations. |
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