Reverse engineering relativity?

[selfAdjoint]

Is it possible to start with the Lorentz transformations - I mean the whole Poincare group SO(1,3) - and derive that there exists an invariant speed? Is this a known derivation? I suppose we could assume a "flat" manifold with metric Diag(1,-1,-1,-1,).

 

[marlon]

Is it possible to start with the Lorentz transformations - I mean the whole Poincare group SO(1,3) - and derive that there exists an invariant speed? Is this a known derivation? I suppose we could assume a "flat" manifold with metric Diag(1,-1,-1,-1,).

I am just wondering, wouldn't it be one heck of a coincidence that you start with "such" transformations ? I don't see the point in doing so, to be honest. However, isn't this quite easy to prove ? Just apply those transformations to two reference frames...

marlon

 

[marlon]

Besides, velocity being absolute is an inherent property of the Lorentz transformations, right ? They are constructed starting from the universal constant c and the homogenity of space...So if you start from the L transformations, you already have that property

marlon

 

[dextercioby]

It would be interesting SA.But it would be cheating.You cannot teach that.Students might ask you:"Why SO(3,1)"??What would you say ?What would be the reasoning behind not follwing the Einstein approach,the aximatical one?


It sort of reminds of the "deriving Schrödinger equation" threads.

Daniel.

 

[robphy]

Is it possible to start with the Lorentz transformations - I mean the whole Poincare group SO(1,3) - and derive that there exists an invariant speed? Is this a known derivation? I suppose we could assume a "flat" manifold with metric Diag(1,-1,-1,-1,).

Suggestion: Find the eigenvectors. (They should lie on a cone through the origin.)
It's probably easiest to leave off the translations.
To see that this is plausible, try the usual Lorentz Transformations in 2D Minkowski.

Edit: Furthermore, one should be able to show that if the eigenvalue corresponding to an eigenvector is not 1 or -1, then that eigenvector must have zero norm [with respect to the metric preserved by the transformation].

 

[selfAdjoint]

Suggestion: Find the eigenvectors. (They should lie on a cone through the origin.)
It's probably easiest to leave off the translations.
To see that this is plausible, try the usual Lorentz Transformations in 2D Minkowski.

Edit: Furthermore, one should be able to show that if the eigenvalue corresponding to an eigenvector is not 1 or -1, then that eigenvector must have zero norm [with respect to the metric preserved by the transformation].

Thanks robphy, that's what I was looking for. Here for the rest of you is my secret motivation. Ingo Kirsch of Harvard posted a paper, http://www.arxiv.org/abs/hep-th/0503024, in which he starts from diffeomorphism invariance on a non metric space and by repeated symmetry breaking gets down to SO(1,3) acting on the tangent space of a Lorenzian (1,3)-manifold. And supposing you came at it all from that direction, what could you find out and how would you go about it?

 

[quantumdude]

I don't see the point in doing so, to be honest.

I do! There are a whole gaggle of amateur aether theorists on the net who accept the Lorentz transformations, and yet do not accept the postulates of SR. They insist that there is no "if and only if" relationship between the two. By this insistence they maintain that their point of view is upheld by experiment every bit as much as the SR point of view. But if it can be proven rigorously that the postulates of SR and the derived results share a biconditional relationship, it would go a long way towards ending that debate.