Wikipedia:Reference desk/Archives/Mathematics/2012 June 8
{{#ifeq:{{PAGENAME}}|Special:Undelete| |{{#if:|
Lorentz transformation
The article on the Lorentz transformation says that "[i]f space [and time are] homogeneous, then the Lorentz transformation must be a linear transformation." It's intuitively clear that this is true, but despite going through a dozen pages of Google I could not find a (reasonably) rigorous proof of this simple statement. Can anyone here help? 65.92.7.168 (talk) 04:11, 8 June 2012 (UTC)
:Are you looking for this for your own satisfaction, or to add to the article. If it's the latter then I wouldn't worry. We only really need to reference information that is either controversial, or is likely to be challenged. It seems like a perfectly sensible claim to me, and I doubt that anyone might disagree. — Fly by Night (talk) 16:35, 8 June 2012 (UTC)
::Personal satisfaction. 65.92.7.168 (talk) 17:13, 8 June 2012 (UTC)
:::The Lorentz transform in local form are valid, even if space-time is not homogeneous. You can even have fields that spontaneously break Lorentz invariance (they could get a vacuum expectation value which will make the vacuum non-invariant under Lorentz transforms) and yet, Nature would still be Lorentz invariant. Count Iblis (talk) 20:02, 8 June 2012 (UTC)
::::While that seems interesting, I'm afraid it went over my head. That said, there should still exist a proof that spacetime homogeneity implies linear Lorentz transformation, even if the converse is false (which I gathered you were arguing). 65.92.7.168 (talk) 20:45, 8 June 2012 (UTC)
::My inexpert browsing suggests that the statement "If space is homogeneous, then the Lorentz transformation must be a linear transformation." in itself is not valid; it is based on too many unstated assumptions. In particular, it would seem that three things are required: homogeneity, isotropy (both of, one assumes, spacetime) as well as linear synchronization. For a challenge, see [http://www.everythingimportant.org/relativity/generalized.htm]. For a listing of the assumptions, see [http://onlinelibrary.wiley.com/doi/10.1002/pamm.200310502/pdf]. I'm not saying these are authoritative, but since the statement adds little, I think the OP's observation is quite pertinent, and the statement should be removed from the article. — Quondum☏ 06:40, 9 June 2012 (UTC)