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facenian
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Erland said:The 2D Lorentz transformation can be derived from the following mathematical assumptions, which all have physical motivations.
In that respect I think that strangerep and Fredrik gave the answers
Erland said:The 2D Lorentz transformation can be derived from the following mathematical assumptions, which all have physical motivations.
The main diificulty in going to 4D is my point 8, about isotropy. How to formultate this mathematically in a sufficiently simple manner?facenian said:I think it's a good warm up to attack the real important case in 4D.
The term “linear” in that thread is not the same as linearity in the algebraic sense: [itex]f(ax + by) = af(x) + bf(y)[/itex]. Linearity there means polynomial of degree one in the coordinates, i.e., solutions to the system of 2-order PDE’sfacenian said:Very interesting post. However I would modify the demonstration because it has one flaw. The problem I see is the assertion that since straight lines must transform into straight lines, the transformation must be linear. I think this is a common mistake(for an example see, for instance, "The Special Theory of Relativity" by Aharoni)
This is incorrect. The conformal group does not even act on Minkowski space [itex]M^{4}[/itex]. It acts on the (conformally) compactified version of Minkowski space [itex]\bar{M}^{4}\cong S^{3}\times S^{1} / Z_{2}[/itex].The principle of Relativity only implies the conformal group and this means
An argument like the one given in Landau and Lifshitz Volume 2 now proves [itex]\alpha=1[/itex]
The theorem you are talking about is the followingFinally it can be shown(see, for instance, "Gravitation and Cosmology" by S. Weingberg) that the only transformations that leave [itex]ds^2[/itex] invariant are linear tranformations.
Well, here is a sketch to get you started...Erland said:The main diificulty in going to 4D is my point 8, about isotropy. How to formultate this mathematically in a sufficiently simple manner?
Yes, and this is precisely the kind of linearity I was reffering to. 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)samalkhaiat said:The term “linear” in that thread is not the same as linearity in the algebraic sense: f(ax+by)=af(x)+bf(y) f(ax + by) = af(x) + bf(y). Linearity there means polynomial of degree one in the coordinates, i.e., solutions to the system of 2-order PDE’s
∂ 2 F σ ∂x μ ∂x ν =0.
You are right, I should have said that the light principle(LP) only implies that the transformation is conformalsamalkhaiat said:This is incorrect. The conformal group does not even act on Minkowski space M 4 M^{4}. It acts on the (conformally) compactified version of Minkowski space M ¯ 4 ≅S 3 ×S 1 /Z 2 \bar{M}^{4}\cong S^{3}\times S^{1} / Z_{2}.
As for the rest of your comments ,I was only sketching a demonstration and citing known authors because some other people may read the post. Yes I read the post and I noticed you used a similar agument.samalkhaiat said:Did you read the whole post? I used argument similar to Landau’s argument to show α=1 \alpha = 1.
How about you give me a counter-example?facenian said: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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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.samalkhaiat said: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?
facenian said:the discussion, as I understood it, is not at the mathematical level you are using.
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.PeterDonis said: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.