- #1

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## Homework Statement

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(v

^{2})x-vf(v

^{2})t; t'=g(v

^{2})t-vh(v

^{2})x; y'=y; z'=z

and its inverse,

x=f(v

^{2})x'+vf(v

^{2})t'; t=g(v

^{2})t'+vh(v

^{2})x'; y=y'; z'=z'

## Homework Equations

## 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(v

^{2})x-vf

_{0}(v

^{2})t; t'=g(v

^{2})t-vh(v

^{2})x; y'=y; z'=z

However, I am having trouble proving that f = f

_{0}. 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.