Derivation of the Lorentz transformations

  • #26
strangerep
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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.
Yes, the literature on this is quite sparse, and mostly poor. :oldfrown:

I am not sure how they are even applicable here.
In Fock & Kemmer, [Ref: FK64, Appendix A] there's a derivation of the most general transformations that map solutions of the free EoMs among themselves.

Stepanov [Ref: Step99, Appendix 1] gives a simplified derivation in 1+1D (although the main body of that paper is rather poor, IMHO).

These transformations are also known as "Fock-Lorentz" transformations (which coincidenally has the same initials "FL"). But you can mostly ignore the Wikipedia page on that subject, since it gives an impression that FL transformations necessarily involve a varying (local) speed of light, which is a false claim.

Kerner [Ref: Ker76] also attempted some work on this, but he didn't get very far and (in subsequent publications) develops an increasingly aggressive/desperate tone. He progresses to de Sitter, but doesn't get very far beyond that.

Manida [Ref: Man99], also derives Fock-Lorentz transformations, duplicating some of Kerner's early work (though apparently without citing him). But his attempts to develop this into a cosmoglogy are (imho) fruitless, with shortcomings reminiscent of Milne's work.

References:

FK64: V. Fock, N. Kemmer (translator), The theory of space, time and gravitation.
2nd revised edition. Pergamon Press, Oxford, London, New York, Paris (1964).

Step99: S. S. Stepanov,
Fundamental Physical Constants & the Principle of Parametric Incompleteness,
arXiv:physics/9909009.

Ker76: E. H. Kerner,
An extension of the concept of inertial frame and of Lorentz transformation,
Proc. Nat. Acad. Sci. USA, Vol. 73, No. 5, pp. 1418-1421, May 1976 .

Man99: S. N. Manida,
Fock-Lorentz transformations and time-varying speed of light,
Available as: arXiv:gr-qc/9905046 .
(Ignore the 2nd part of the title: he's not talking about a varying local speed of light.)
 
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  • #27
strangerep
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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?
On p1519 of that paper (Berzi & Gorini, 1969), section II, they interpret "homogeneity" to mean that the transformations must not affect "the relation between 2 observers", and from this they derive that the transformations must be linear. That rules out de Sitter -- for which "homogeneity" needs a more general meaning, i.e., that an inertial observer "here" perceives essentially the same laws of physics as an inertial observer "there". Iow, there is no preferred point in spacetime. This leads eventually to a de Sitter space of constant curvature.
 
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  • #28
samalkhaiat
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Indeed, this is just an example from the theory of homogeneous spaces.
Indeed, given Minkowski spacetime [itex]M^{(1,3)}[/itex], one can show that the Poincare group [itex]\Pi (1,3)[/itex] is its maximal symmetry group. Conversely, given [itex]\Pi (1,3)[/itex], one can show (using the theory of induced representations) that [tex]M^{(1,3)} \cong \frac{\Pi (1,3)}{SO^{\uparrow} (1,3)} .[/tex] That is Minkowski space-time is diffeomorphic to (or identified with) the space of orbits that the Lorentz group [itex]SO^{\uparrow}(1,3)[/itex] sweeps out in the Poincare group. In fact the powerful methods of induced representations make it possible to derive the physical notions of spacetimes, fields and transformations.
 
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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?
I mentioned linear because with that assumption it is quite easy to deduce the Lorentz transformations. Non-linear was just opening a can of worm I wished to avoid due to my ignorance.

I was only thinking of Cartesian coordinates, but you can write down the wave equation in any set of coordinates.
 
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