Deriving the Lorentz Transformation from the Homogeneity of Spacetime

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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(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.
 

Answers and Replies

  • #2
strangerep
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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.
 
  • #3
27
2
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
 
  • #4
strangerep
Science Advisor
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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.
 

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