# Derivation of the Lorentz transformations

• I
• kent davidge
In summary: It is necessary to say "linear transformations" because you can't have a nonlinear transformation between two inertial frames, regardless of how they are related. But that's a different question.
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?

Or at least the most illuminating

Using LED lighting will be the most efficiently illuminating.

vanhees71 and kent davidge
kent davidge said:
Does anyone know how many ways are there?
I doubt there is a finite number of ways.

kent davidge said:
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 and Ibix
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.
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.

vanhees71 said:
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.

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:
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.

vanhees71 said:
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?

PeterDonis said:
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.

strangerep said:
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?

vanhees71
kent davidge said:
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.

vanhees71 and Dale
haushofer said:
Then one should also be able to obtain the non-relativistic equivalent of de Sitter, namely Newton-Hooke, no?
Correct.

vanhees71
vanhees71 said:
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.

samalkhaiat and vanhees71
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.
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?

Last edited:
kent davidge said:
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.

Dale said:
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.

kent davidge said:
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.

vanhees71
kent davidge said:
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.

vanhees71
strangerep said:
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.

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.
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?

Ibix said:
No, I was thinking of this one.

vanhees71 and Ibix
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.
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.)

Last edited:
Michael Price
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?
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
strangerep said:
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.

strangerep
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?
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.

kent davidge

## 1. What are the Lorentz transformations?

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.

## 2. Why were the Lorentz transformations developed?

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.

## 3. How are the Lorentz transformations derived?

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.

## 4. What are the implications of the Lorentz transformations?

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.

## 5. Are the Lorentz transformations still relevant today?

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.

• Special and General Relativity
Replies
20
Views
2K
• Special and General Relativity
Replies
33
Views
2K
• Special and General Relativity
Replies
17
Views
3K
• Special and General Relativity
Replies
101
Views
4K
• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
54
Views
2K
• Special and General Relativity
Replies
18
Views
2K
• Special and General Relativity
Replies
7
Views
2K
• Special and General Relativity
Replies
6
Views
1K