- #1
kent davidge
- 933
- 56
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.kent davidge said: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).kent davidge said: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.,strangerep said: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.vanhees71 said: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.vanhees71 said:Interesting! Do you have a reference? To get other spacetimes than Galilei-Newton and Minkowski, you have to relax obviously some (symmetry) constraints.
vanhees71 said: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.PeterDonis said: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?strangerep said: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.kent davidge said:Or at least the most illuminating
Correct.haushofer said: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.vanhees71 said: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?Michael Price said: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.kent davidge said: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.Dale said: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.kent davidge said: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.kent davidge said: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/?strangerep said: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.strangerep said: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?strangerep said: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.Ibix said:
Yes, the literature on this is quite sparse, and mostly poor.Dale said: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.
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.I am not sure how they are even applicable here.
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.PAllen said: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?
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.strangerep said:Indeed, this is just an example from the theory of homogeneous spaces.
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.kent davidge said: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?
The Lorentz transformations are a set of equations that describe how time and space coordinates change between two reference frames that are moving relative to each other at a constant velocity. They were developed by Dutch physicist Hendrik Lorentz in the late 19th century and later refined by Albert Einstein in his theory of special relativity.
The Lorentz transformations were developed to explain the results of the Michelson-Morley experiment, which showed that the speed of light is constant and does not depend on the motion of the observer. This contradicted the classical laws of physics and led to the development of the theory of special relativity.
The Lorentz transformations can be derived using mathematical equations and principles from both classical mechanics and special relativity. They involve the concepts of time dilation, length contraction, and the relativity of simultaneity, and are based on the postulate that the laws of physics are the same in all inertial reference frames.
The Lorentz transformations have several important implications in physics. They show that the laws of physics are the same for all observers moving at a constant velocity, and that the speed of light is the same in all reference frames. They also lead to the concept of spacetime, where time and space are not separate entities but are intertwined.
Yes, the Lorentz transformations are still widely used in modern physics, particularly in the fields of special relativity and particle physics. They are also important in practical applications, such as GPS technology, which relies on the principles of special relativity to function accurately.