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Oct. 8, 2022 Note: Equation ((6)) should use sin(A') and sin(B') if A' and B' are measured with respect to the normal i.e. y axis. Oct. 10, 2002 Equation ((10)) is really c/ v(relative) = t moving frame / t lab frame. We use c=1 in this note. Fermat’s principle (in general) is linked to a stationary path for: Integral n ds where n is the index of refraction and ds is a three space interval. In (1) it is argued that for special relativity ds becomes dT where T is proper time t sqrt(1-vv) and L= msqrt(1-vv) - V(x) with c=1. For a free particle then Integral m sqrt(1-vv) dT should be stationary and rest mass m is linked to n. In (2) we argued that for a mirror moving in the negative y direction with constant v one could map n, the index of refraction to 1/ v(relative) for both the incident and reflected rays. In (3) we argued that v(relative) = Integral dt ‘(moving frame) / Integral dt (lab frame) ((1)). This was shown explicitly using Lorentz transformations, but here we argue that one does need to know the explicit form of the Lorentz transformation to link the moving mirror to special relativity. Ultimately we find that a generalized index of refraction is linked to ((1)) with ds remaining as a spatial distance and not becoming dT with m taking the place of the index of refraction for the moving mirror problem. In this approach, however, we are no longer varying x as in Fermat’s principle. Instead we use the hypothetical velocity approach which is equivalent to having the x velocity projection in the moving frame remain constant for the moving and reflected rays i.e. incident angle equals reflected in the moving frame with c the same in the moving and lab frame. |
