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It seems that there is a considerable number of ways of deriving the Lorentz transformations. Does anyone know how many ways are there?
I doubt there is a finite number of ways.Does anyone know how many ways are there?
I like the so-called "1-postulate" group-theoretic method. I.e., start with the Relativity Principle ("RP"), spatial isotropy plus physical continuity and regularity. (I.e., no a-priori light principle.) From this (smaller-than-usual) set of assumptions, one can derive Lorentz transformations (though it takes quite a lot of work).Or at least the most illuminating [of deriving Lorentz transformations]...
You get even more: The only two spacetimes (up to redefinitions of units) are the Galilei-Newton and the Minkowski spacetimes. It's not the quickest approach but one learns a lot about the underlying group theory. See, e.g.,I like the so-called "1-postulate" group-theoretic method. I.e., start with the Relativity Principle ("RP"), spatial isotropy plus physical continuity and regularity. (I.e., no a-priori light principle.) From this (smaller-than-usual) set of assumptions, one can derive Lorentz transformations (though it takes quite a lot of work).
Afaict, all other derivations are just a modified version of the above, obtained by assuming something extra to create a shortcut to the end result.
Yes, I know. But I didn't want to overdo my post #5 unless the OP wants more.You get even more:
Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story.The only two spacetimes (up to redefinitions of units) are the Galilei-Newton and the Minkowski spacetimes.
The applicable references require some nontrivial additional explanation. I'm a bit busy right now, but I'll try to send you a PM in the next few days.Interesting! Do you have a reference? To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.
For de Sitter spacetime, at least, doesn't it have the same ten-parameter group of symmetries as Minkowski spacetime?To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.
The translation generators are noncommutative, of the form $$[P_\mu \,,\, P_\nu] ~=~ \Lambda \, J_{\mu\nu} ~,$$ where ##J_{\mu\nu}## are the usual Lorentz generators and ##\Lambda## is a constant with dimensions of inverse length squared. The limit ##\Lambda\to 0## contracts the algebra back to Poincare.For de Sitter spacetime, at least, doesn't it have the same ten-parameter group of symmetries as Minkowski spacetime?
Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story.
Start from Maxwell's equations and find the set of linear transformations of space and time that leave Maxwell invariant. You get the Poincare transformations, which contain the Lorentz transformations.Or at least the most illuminating
Correct.Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?
Indeed, this is just an example from the theory of homogeneous spaces.I think it should be possible to construct Minkowski space from the Poincare group, [...]
why you say that? are there non-linear transformations that leave Maxwell invariant?Start from Maxwell's equations and find the set of linear transformations of space and time that leave Maxwell invariant. You get the Poincare transformations, which contain the Lorentz transformations.
Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.why you say that?
oh yea, I knew that already. But then is'nt it unecessary to say "linear transformations"? For they are the only possible transformations.Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.
The English language is not a precision instrument representing all concepts with minimal redundancy.... but in this case the redundancy is helpful because it tells you something about how to choose your ansatz.But then is'nt it unecessary to say "linear transformations"?
Yes -- the conformal transformations. They have the Poincare group as a subgroup. If you search PF for "conformal" articles written by @samalkhaiat, you'll find a good tutorial on this subject.are there non-linear transformations that leave Maxwell invariant?
That depends how you define "straight" lines. If one defines them via a condition of zero acceleration, then fractional-linear ("FL") transformations are the most general.Dale said:[...] Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.
This one: https://www.physicsforums.com/threads/conformal-group-poincare-group.420204/?If you search PF for "conformal" articles written by @samalkhaiat, you'll find a good tutorial on this subject.
In response to this I was searching for information about fractional linear transforms. All of them that I saw were mappings from the complex plane to the complex plane. I didn’t see anything on FL transforms as a mapping from R4 to R4. I am not sure how they are even applicable here.If one defines them via a condition of zero acceleration, then fractional-linear ("FL") transformations are the most general.
Can you state what you have to change from the assumptions in the Gorini paper to achieve this? I thought the derivations in that paper (@vanhees71 provided it earlier in this thread) were quite rigorous. Or were there hidden assumptions in the derivation?Yes, I know. But I didn't want to overdo my post #5 unless the OP wants more.
Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story.
No, I was thinking of this one.