Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84 exptools; site ihlpa.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!ihnp4!ihlpa!lew
From: lew@ihlpa.UUCP (Lew Mammel, Jr.)
Newsgroups: net.physics
Subject: derivation of redshift (for lightsail problem)
Message-ID: <148@ihlpa.UUCP>
Date: Wed, 13-Mar-85 14:12:15 EST
Article-I.D.: ihlpa.148
Posted: Wed Mar 13 14:12:15 1985
Date-Received: Thu, 14-Mar-85 05:26:54 EST
Distribution: net
Organization: AT&T Bell Laboratories
Lines: 32

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