Deriving the Lorentz Transformation from the Homogeneity of Spacetime

[hgandh]

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(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'

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(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.

 

[strangerep]

Hmm. Which source/solution are you looking at? (Can you post a reference or link?)

In my derivation of this stuff, the result that $f = f_0$ emerges from the properties of any 1-parameter Lie group (which is what you're deriving here). I.e., 2 successive transformations (in the same direction) with different parameters $v,v'$ must commute. This imposes some constraints on the various functions.

 

[hgandh]

Hmm. Which source/solution are you looking at? (Can you post a reference or link?)

In my derivation of this stuff, the result that $f = f_0$ emerges from the properties of any 1-parameter Lie group (which is what you're deriving here). I.e., 2 successive transformations (in the same direction) with different parameters $v,v'$ must commute. This imposes some constraints on the various functions.

http://faculty.uml.edu/cbaird/all_homework_solutions/Jackson_11_1_Homework_Solution.pdf

 

[strangerep]

Hmm. (Sigh.) Well, I will say that I think there are some unnecessary fudges in that solution. So I'm not sure how I can usefully help you. I could (possibly) show you (a version of) my derivation, but it would be rather different from the solution you've been given.