Deriving Lorentz transformations

samalkhaiat

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However I reaffirm my only objection to your demonstration, i.e. that straight lines transforming into straight lines requires that the transformation be linear.(this is the only argument you should attack)
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How about you give me a counter-example?
Okay, I give you a transition functions [itex]F_{21}: x \to \bar{x}[/itex] between two Minkowski charts. It ia assumed that [itex]F_{21}[/itex] is a [itex]\mathscr{C}^{3}[/itex] homeomorphism, i.e., continuosly thirce differentiable (smooth and regular). Furthermore, we demand that [itex]F_{21}[/itex] (or its inverse) maps straight (world) lines onto straight lines. Now, you show me one such homeomorphism that does not correspond to linear tranformation?
 
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How about you give me a counter-example?
Okay, I give you a transition functions F21:x→x¯F_{21}: x \to \bar{x} between two Minkowski charts. It ia assumed that F21F_{21} is a C3\mathscr{C}^{3} homeomorphism, i.e., continuosly thirce differentiable (smooth and regular). Furthermore, we demand that F21F_{21} (or its inverse) maps straight (world) lines onto straight lines. Now, you show me one such homeomorphsm that does not correspond to linear tranformation?
Well, the discussion, as I understood it, is not at the mathematical level you are using. I' m sorry if I misunderstood that. At the level I took it there is no place for homeomorphisims, diffeomorphisms, bijective maps, etc. May be I will look like an ignorant fool to you but I'm not the only one, for instance, "Einstein Gravity in a Nutshell" by Zee was written at the mathematical level I'm using in this discussion. Please don't take this the wrong way, I respect the mathematics and I'm not saying it in a demeaning way, on the contrary, I think it's way over my head.
Having clarified this point, what I meant is that many authors working at the mathematical level to which I am referring (Einstein included), attribute the linearity either to the homogeneity and isotropy of space and time or to thecondition that straight lines must be transformed in straight lines, without further explanation, which I find inappropriate even for this level of rigor. At this level of rigor, a fractional linear transformation can work as a counter example, of course even I Know that this is not an homeormorphism for R4
 
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the discussion, as I understood it, is not at the mathematical level you are using.
This thread is at an "I" level, which means undergraduate level. However, the subthread you are participating in is probably on the borderline between "I" and "A" (which is graduate level). That's probably unavoidable given the nature of the topic; to talk about "derivation" of something you have to have enough rigor to be able to precisely specify the starting point and the conclusion.
 
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his thread is at an "I" level, which means undergraduate level. However, the subthread you are participating in is probably on the borderline between "I" and "A" (which is graduate level). That's probably unavoidable given the nature of the topic; to talk about "derivation" of something you have to have enough rigor to be able to precisely specify the starting point and the conclusion.
This thread started with an innocent question and I believe by now it went too far, however I must confess I really enjoyed it and learned a lot from it.
 

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