Derivation of the Lorentz transformations

[kent davidge]

It seems that there is a considerable number of ways of deriving the Lorentz transformations. Does anyone know how many ways are there?

 

[kent davidge]

Or at least the most illuminating

 

[PAllen]

Using LED lighting will be the most efficiently illuminating.

 

[Dale]

Does anyone know how many ways are there?

I doubt there is a finite number of ways.

 

[strangerep]

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.

 

[vanhees71]

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.

 

[strangerep]

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.

 

[vanhees71]

Interesting! Do you have a reference? To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.

 

[strangerep]

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.

 

[PeterDonis]

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?

 

[strangerep]

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.

 

[vanhees71]

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.

 

[haushofer]

Actually, one can also get de Sitter and a time-asymmetric Poincare. But that's another, even longer, story.

Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?

 

[Michael Price]

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.

 

[strangerep]

Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?

Correct.

 

[strangerep]

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.

 

[kent davidge]

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?

 

[Dale]

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.

 

[kent davidge]

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.

 

[Nugatory]

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.

 

[strangerep]

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.

[...] 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.

 

[Ibix]

If you search PF for "conformal" articles written by @samalkhaiat, you'll find a good tutorial on this subject.

This one: https://www.physicsforums.com/threads/conformal-group-poincare-group.420204/?

 

[Dale]

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.

 

[PAllen]

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.

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?

 

[strangerep]

This one: https://www.physicsforums.com/threads/conformal-group-poincare-group.420204/?

No, I was thinking of this one.

 

[strangerep]

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.

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.)

 

[strangerep]

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.

 

[samalkhaiat]

Indeed, this is just an example from the theory of homogeneous spaces.

Indeed, given Minkowski spacetime M^{(1,3)}, one can show that the Poincare group \Pi (1,3) is its maximal symmetry group. Conversely, given \Pi (1,3), one can show (using the theory of induced representations) that M^{(1,3)} \cong \frac{\Pi (1,3)}{SO^{\uparrow} (1,3)} . That is Minkowski space-time is diffeomorphic to (or identified with) the space of orbits that the Lorentz group SO^{\uparrow}(1,3) 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.

 

[Michael Price]

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.