Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · lorentz-transformation-derivation
Problem
Two equivalent inertial frames K and K′ are such that K′ moves in the positive x direction with speed v as seen from K. The spatial coordinate axes in K′ are parallel to those in K and the two origins are coincident at times t=t′=0.
(a) Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x,y,z,t) and (x′,y′,z′,t′) is the linear transformation,
x′=f(v2)x−vf(v2)t;t′=g(v2)t−vh(v2)x;y′=y;z′=z
and the inverse,
x=f(v2)x′+vf(v2)t′;t=g(v2)t′+vh(v2)x′;y=y′;z=z′
where f, g, and h are functions of v2, the structures of the x′ and x equations are determined by the definition of the inertial frames in relative motion, and the signs in the inverse equation are a reflection of the reversal of roles of the two frames.
(b) Show that consistency of the initial transformation and its inverse require
f=gandf2−v2fh=1
(c) If a physical entity has speed u′ parallel to the x′ axis in K′, show that its speed u parallel to the x axis in K is
u=1+vu′(h/f)u′+v
Using the second postulate 2' (universal limiting speed C), show that h=f/C2 is required and that the Lorentz transformation of the coordinates results. The universal limiting speed C is to be determined from experiment.