lew@ihlpa.UUCP (Lew Mammel, Jr.) (03/13/85)
Given the Lorentz Transformation: ( x', ct' ) = gamma * ( x - beta * ct, ct - beta * x ) there are two ways to derive the red shift: The easy way: ( p, E/c ) = ( p, p ) transform as (x, ct) so ( p', p' ) = gamma * (1-beta) * ( p, p ) The hard way: Let event 1 be the coincidence of the origins of two reference frames with a wave front. ( x, ct ) = ( x', ct' ) = ( 0, 0 ) At time t=0 the next wavefront is at x = -lambda in the "stationary" frame. It has to catch up to the moving origin at a relative speed ( in the stationary frame! ) of ( c - v ). So event 2, the arrival of the second wavefront at the moving origin, has coordinates: ( x, ct ) = ( v * lambda/( c - v ), c * lambda/( c - v ) ) ( x', ct' ) = ( 0, c * gamma * ( 1 - beta^2 ) * lambda/( c - v ) ) = ( 0, lambda/( gamma * ( 1 - beta ) ) ) Note that ct' = lambda', since t' is the time between arrival of two wavefronts moving at speed c. So that's it. Lew Mammel, Jr. ihnp4!lhpa!lew