Derivation of the Lorentz transformations

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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?
 

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  • #2
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Or at least the most illuminating
 
  • #3
PAllen
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Using LED lighting will be the most efficiently illuminating. :-p
 
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  • #5
strangerep
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Or at least the most illuminating [of deriving Lorentz transformations]...
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.
 
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  • #6
vanhees71
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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.
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.,

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math.
Phys. 10, 1518 (1969)
http://dx.doi.org/10.1063/1.1665000.
 
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strangerep
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You get even more:
Yes, I know. But I didn't want to overdo my post #5 unless the OP wants more.

The only two spacetimes (up to redefinitions of units) are the Galilei-Newton and the Minkowski spacetimes.
Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story. :oldwink:
 
  • #8
vanhees71
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Interesting! Do you have a reference? To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.
 
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strangerep
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Interesting! Do you have a reference? To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.
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.
 
  • #10
PeterDonis
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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?
 
  • #11
strangerep
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For de Sitter spacetime, at least, doesn't it have the same ten-parameter group of symmetries as Minkowski spacetime?
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.
 
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  • #12
vanhees71
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I think it should be possible to construct Minkowski space from the Poincare group, i.e., it should turn out to be an affine pseudo-Euclidean space with the fundamental form of signature ##(1,-1,-1,-1)## or, equivalently, ##(-1,1,1,1)##. Obviously de Sitter spacetime is not an affine space though it's a homogeneous (admitting translations) and isotropic (admitting "pseudo-Rotations") pseudo-Riemannian space.
 
  • #13
haushofer
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Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story. :oldwink:
Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?
 
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  • #14
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Or at least the most illuminating
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.
 
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  • #15
strangerep
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Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?
Correct.
 
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  • #16
strangerep
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I think it should be possible to construct Minkowski space from the Poincare group, [...]
Indeed, this is just an example from the theory of homogeneous spaces.
 
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  • #17
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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.
why you say that? are there non-linear transformations that leave Maxwell invariant?

do you mean in the sense that I can write down the wave equations in either cartesian or spherical coordinates, for example?
 
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why you say that?
Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.

*Actually that is slightly too strong, they could be affine and still map straight lines to straight lines, but linear is easier.
 
  • #19
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Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.
oh yea, I knew that already. But then is'nt it unecessary to say "linear transformations"? For they are the only possible transformations.
 
  • #20
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But then is'nt it unecessary to say "linear transformations"?
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.
 
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  • #21
strangerep
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are there non-linear transformations that leave Maxwell invariant?
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.

Dale said:
[...] Because the transformations between inertial frames need* to be linear so as to map straight lines to straight lines.
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.

The intersection of FL and Conformal transformations leaves us with Poincare.
 
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  • #23
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If one defines them via a condition of zero acceleration, then fractional-linear ("FL") transformations are the most general.
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.
 
  • #24
PAllen
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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. :oldwink:
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?
 

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