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 its inverse,
x=f(v2)x'+vf(v2)t'; t=g(v2)t'+vh(v2)x'; y=y'; z'=z'
The Attempt at a Solution
Now, I know that homogeneity implies that the transformation must be linear in x and t and that the isotropy of space implies that the coefficients can only be functions of the magnitude of the velocity (not the direction) at most. Therefore, I am stuck at the following:
x'=f(v2)x-vf0(v2)t; t'=g(v2)t-vh(v2)x; y'=y; z'=z
However, I am having trouble proving that f = f0. The solution I am looking at says that this follows from the homogeneity of space-time but I am having trouble using that fact to prove it.