# Lorentz transformations: 1+1 spacetime only

• I
• robwilson

#### robwilson

[Mentors' note: This thead was forked from another thread - hence the reference to "these replies" in the first post]

I am wondering why all these replies only discuss Lorentz transformations in 1+1 spacetime dimensions. That is the easy bit. The problems in understanding arise in 2+1 dimensions, and even more so in 3+1 dimensions. I understand Lorentz transformations in 1+1 dimensions, where we are talking about 2 independent observers. I struggle in 2+1 dimensions, where we have three independent observers. In 3+1 dimensions, with four independent observers, I cannot make any sense of the Lorentz group. And I am a group theorist.

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Yes, of course, I understand all that. As I said, I am a group theorist, so there is nothing you can tell me about coordinate systems for 4-dimensional spacetime that I don't already know. But no physicist has ever been able, or willing, to give me a real physical interpretation of the Lorentz group that makes sense.

But no physicist has ever been able, or willing, to give me a real physical interpretation of the Lorentz group that makes sense.
Perhaps you should set out or reference an interpretation that doesn't make sense to you and explain what you see as its deficiencies.

• robphy
Yes, of course, I understand all that. As I said, I am a group theorist, so there is nothing you can tell me about coordinate systems for 4-dimensional spacetime that I don't already know. But no physicist has ever been able, or willing, to give me a real physical interpretation of the Lorentz group that makes sense.
It's all the possibilities for inertial reference frames: boosts, rotations; and, if you add translations you get the Poincare Group.

If I'm moving inertially relative to you, then my coordinates will be a Lorentz transformation of your coordinates. This can be defined by two physical parameters:

My velocity relative to you; and then I have the choice of orientation of my coordinate system.

I will, but not here, because it will get me banned. No-one has answered my original question, in any case. The group-theoretical problem is that from the one-dimensional Lorentz transformations it is impossible to infer what the two-dimensional group is. Therefore there is a physical assumption going into the process somewhere. It is a physical assumption about the nature of spacetime, and I'd like to know explicitly what it is. I have my own ideas, but this forum is not the place for me to expound on them.

Yes, of course, I understand all that. As I said, I am a group theorist, so there is nothing you can tell me about coordinate systems for 4-dimensional spacetime that I don't already know. But no physicist has ever been able, or willing, to give me a real physical interpretation of the Lorentz group that makes sense.
It's in fact the proper orthochronous Poincare group, which determines not only Minkowski space-time as a physical space-time model but also a general framework to build the dynamical theories that describe the physical world. Together with the concept of locality, i.e., the field description of physics, which is the most natural description within a relativistic theory (note how problematic the concept of point particles in relativistic physics indeed is, not having a fully consistent dynamical theory for interacting relativistic point particles after about 120 years since Einstein's discovery of special relativity), to formulate dynamical models compatible with the space-time description based on this symmetry group.

Of course, which (quantum) field theories successfully describe the real world cannot be deduced from group-theoretical or mathematical considerations alone but you always need observations and experiments to get the right idea which models make sense and have a chance to stand experimental tests.

For a good introduction, particularly for a mathematician who already knows group theory very well but also for physicists who want a gentle but non-trivializing introduction to Lie-group-theoretical methods, see

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles, Springer, Wien (2001).

• PeroK
It's in fact the proper orthochronous Poincare group, which determines not only Minkowski space-time as a physical space-time model but also a general framework to build the dynamical theories that describe the physical world. Together with the concept of locality, i.e., the field description of physics, which is the most natural description within a relativistic theory (note how problematic the concept of point particles in relativistic physics indeed is, not having a fully consistent dynamical theory for interacting relativistic point particles after about 120 years since Einstein's discovery of special relativity), to formulate dynamical models compatible with the space-time description based on this symmetry group.

Of course, which (quantum) field theories successfully describe the real world cannot be deduced from group-theoretical or mathematical considerations alone but you always need observations and experiments to get the right idea which models make sense and have a chance to stand experimental tests.

For a good introduction, particularly for a mathematician who already knows group theory very well but also for physicists who want a gentle but non-trivializing introduction to Lie-group-theoretical methods, see

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles, Springer, Wien (2001).
Thank you for this reference and for your serious response. I think you understand, as I do, that the group, whatever it is, determines the model of spacetime, and what field theories are possible, and so on. And of course I agree that group theory cannot determine these things, which can only be determined by experiment. That is why I, as a group theorist, presented with data, look for the right group to describe the data. Since I know a lot of groups very well indeed, I have a large menu to choose from, and I can look at the problem in some depth. And over the years I have collected a lot of data from the literature. And the more I look at it, the more it doesn't seem to fit. For 45 years, since I first learned the theory of special relativity, I never once questioned the appropriateness of the Lorentz group. But it doesn't seem to fit the data. In 1+1 dimensions it works, as all the textbooks prove. In 2+1 dimensions it can be made to work, with a bit of persuasion. In 3+1 dimensions, I don't see it.

• PeroK
What do you mean by that "it doesn't seem to fit the data"? To the contrary the more we test special and general relativity with ever more precise measurements (and the measurment of time is among the most accurate measurements possible today having all the quantum-optics technology at hand) the more the relativistic space-time model is confirmed. There is not a single high-precision measurement contradicting the assumption of (local) Poincare symmetry.

Yes, that is correct. But all the tests are carried out in 2+1 spacetime.

Yes, that is correct. But all the tests are carried out in 2+1 spacetime.
I think it's safe to say that no experiment has ever been conducted in 2+1 spacetime!

• • Doc Al, vela and vanhees71
I think it's safe to say that no experiment has ever been conducted in 2+1 spacetime!
You make a joke, which is fair enough. If you address the substance of my remarks, it would be more useful.

I am wondering why all these replies only discuss Lorentz transformations in 1+1 spacetime dimensions. That is the easy bit.
It's simple pragmatism on the part of the physicists, who often are interested in the math only to the extent that it is needed to effectively model the universe around them. As you say, the 1+1 case is easy - and that plus an assumption of isotropy, a coordinate rotation, and a wave of the hands is enough to work with problems in the locally flat space of special relativity.

And I am a group theorist.

Do you mean that you are a professional mathematician who specializes in group theory?
Or something else?

It's simple pragmatism on the part of the physicists, who often are interested in the math only to the extent that it is needed to effectively model the universe around them. As you say, the 1+1 case is easy - and that plus an assumption of isotropy, a coordinate rotation, and a wave of the hands is enough to work with problems in the locally flat space of special relativity.
If we take the transformation of the Dirac electron as an example, with different representations of the Lorentz Group for the four-momentum and the spinor components, then there is a lot more than hand-waving going on.

Outside of elementary texts on SR, the 3+1 spacetime model is the default, and in the case of the electron spin can hardly be avoided.

• vanhees71
If we take the transformation of the Dirac electron as an example, with different representations of the Lorentz Group for the four-momentum and the spinor components, then there is a lot more than hand-waving going on.
I think the Dirac equation maybe takes us beyond the original question (although splitting the thread may have changed the question - the post you're quoting had originally been in the B-level thread and considering only classical particles). But yes, you make a good point if we're going to dig deeper.

But all the tests are carried out in 2+1 spacetime.
I must confess that I don't understand this statement. Everything we do - including all tests - is carried out in the 3+1 spacetime in which we live, so I expect you meant something else here?

• vanhees71
I think the Dirac equation maybe takes us beyond the original question (although splitting the thread may have changed the question - the post you're quoting had originally been in the B-level thread and considering only classical particles). But yes, you make a good point if we're going to dig deeper.
I think the gist of this thread is that the Lorentz Group only works in 1D (in some sense) and that physicists generally have never taken the trouble to look at the 3D group properly.

Your comment was somewhat unfortunate in that respect, I'm sorry to say!

• vanhees71
Do you mean that you are a professional mathematician who specializes in group theory?
Or something else?
Yes, that is what I mean.

I must confess that I don't understand this statement. Everything we do - including all tests - is carried out in the 3+1 spacetime in which we live, so I expect you meant something else here?
What I meant was that the third dimension of space is not of the essence in the experiments. It may be that there are some experiments on SR that do essentially use the three dimensions, but if so I am not aware of them. My impression is that most experiments use only one or two space dimensions in an essential way. Some that do use three dimensions, which includes the neutrino oscillation experiments, and the kaon oscillation experiments, are in my opinion not well explained by the standard model.

The group-theoretical problem is that from the one-dimensional Lorentz transformations it is impossible to infer what the two-dimensional group is. Therefore there is a physical assumption going into the process somewhere. It is a physical assumption about the nature of spacetime, and I'd like to know explicitly what it is. I have my own ideas, but this forum is not the place for me to expound on them.

At times, it seems that aspects that are seen in Special Relativity
are actually seen elsewhere.
(That is, it may not really be a "special relativity" problem that "special relativity" has to resolve.)

Sometimes, it may be helpful (as a "toy problem") to consider what happens in those other cases.
Specifically,
• How does 2-dimensional Euclidean geometry generalize to 3-dimensional Euclidean geometry?
...possibly, first keeping the first direction invariant as one considers extensions to the "second dimension"

• How does the 1+1 Galilean geometry of nonrelativistic physics along the x-direction
generalize to 2+1 and 3+1 Galilean geometry to the xy-plane and to 3-d space?
These (Euclidean, Galilean, and Minkowskian) are examples of the Cayley-Klein affine geometries
https://en.wikipedia.org/wiki/Cayley–Klein_metric
which are related by an underlying projective geometry.

Possibly useful reference:
https://www.springer.com/gp/book/9783642172854
Perspectives on Projective Geometry
A Guided Tour Through Real and Complex Geometry
by
Jürgen Richter-Gebert

• vanhees71
The most general Lorentz transformation can be represented as
$$\left( \begin{array}{c|c} \gamma & -\gamma \textbf{v}^T / c\\ \hline -\gamma \textbf{v} / c & \textbf{I} + (\gamma - 1) \textbf{v} \textbf{v}^T / (\textbf{v}^T\textbf{v}) \\ \end{array} \right) \left( \begin{array}{c|c} 1 & \textbf{0}^T \\ \hline \textbf{0} & \textbf{U} \\ \end{array} \right)$$ where
• ##\textbf{v}## is the velocity of one observer relative to another, as a 3×1 column vector
• ##\gamma## is the usual Lorentz factor ##1 / \sqrt{1 - \textbf{v}^T\textbf{v} / c^2} ##
• ##\textbf{U}## is any orthogonal 3×3 matrix, representing a rotation of one observer's spatial axes relative to the other
(You can verify it by considering the special case ##\textbf{v} = (v, 0, 0)^T##.)

That transformation represents a transformation between a pair of observers' coordinates. The whole group represents all possible transformations between pairs of observers, an infinite number of observers in total. I don't understand why you talk about 3 observers in 2+1 spacetime or 4 observers in 3+1 spacetime.

• PeroK and etotheipi
The most general Lorentz transformation can be represented as
$$\left( \begin{array}{c|c} \gamma & -\gamma \textbf{v}^T / c\\ \hline -\gamma \textbf{v} / c & \textbf{I} + (\gamma - 1) \textbf{v} \textbf{v}^T / (\textbf{v}^T\textbf{v}) \\ \end{array} \right) \left( \begin{array}{c|c} 1 & \textbf{0}^T \\ \hline \textbf{0} & \textbf{U} \\ \end{array} \right)$$ where
• ##\textbf{v}## is the velocity of one observer relative to another, as a 3×1 column vector
• ##\gamma## is the usual Lorentz factor ##1 / \sqrt{1 - \textbf{v}^T\textbf{v} / c^2} ##
• ##\textbf{U}## is any orthogonal 3×3 matrix, representing a rotation of one observer's spatial axes relative to the other
(You can verify it by considering the special case ##\textbf{v} = (v, 0, 0)^T##.)

That transformation represents a transformation between a pair of observers' coordinates. The whole group represents all possible transformations between pairs of observers, an infinite number of observers in total. I don't understand why you talk about 3 observers in 2+1 spacetime or 4 observers in 3+1 spacetime.
I'm sorry, I'm not expressing myself well. Observer 1 and observer 2 understand each other via Lorentz transformations. Observer 1 and observer 3 likewise. The problem is for observer 1 to understand how observer 2 and observer 3 see each other, without talking to both of them.

If we take the transformation of the Dirac electron as an example, with different representations of the Lorentz Group for the four-momentum and the spinor components, then there is a lot more than hand-waving going on.

Outside of elementary texts on SR, the 3+1 spacetime model is the default, and in the case of the electron spin can hardly be avoided.
Of course, it is the electron spin that really points up the problem, although I didn't really want to get into all that. The action of the Lorentz group on the Dirac electron includes some variables that cannot be observed (e.g. spin direction), and does not include some variables that can be observed (e.g. generation). Plus the incompatibility of QM with GR is a serious mathematical problem, rather than a physical problem. But to sort out the mathematics, it is necessary to look at the experimental data with an uncluttered mind.

• Killtech
At times, it seems that aspects that are seen in Special Relativity
are actually seen elsewhere.
(That is, it may not really be a "special relativity" problem that "special relativity" has to resolve.)

Sometimes, it may be helpful (as a "toy problem") to consider what happens in those other cases.
Specifically,
• How does 2-dimensional Euclidean geometry generalize to 3-dimensional Euclidean geometry?
...possibly, first keeping the first direction invariant as one considers extensions to the "second dimension"

• How does the 1+1 Galilean geometry of nonrelativistic physics along the x-direction
generalize to 2+1 and 3+1 Galilean geometry to the xy-plane and to 3-d space?
These (Euclidean, Galilean, and Minkowskian) are examples of the Cayley-Klein affine geometries
https://en.wikipedia.org/wiki/Cayley–Klein_metric
which are related by an underlying projective geometry.

Possibly useful reference:
https://www.springer.com/gp/book/9783642172854
Perspectives on Projective Geometry
A Guided Tour Through Real and Complex Geometry
by
Jürgen Richter-Gebert
Yes, all this is true. The question is exactly as you say, how does the geometry generalise from 1 space dimension to 3 space dimensions. The examples you give are not the only possibilities.

I'm sorry, I'm not expressing myself well. Observer 1 and observer 2 understand each other via Lorentz transformations. Observer 1 and observer 3 likewise. The problem is for observer 1 to understand how observer 2 and observer 3 see each other, without talking to both of them.
If we denote the Lorentz transformation between observer ##n## and observer ##m## by ##\Lambda_{nm}## then the essential results are that
\begin{align*} \Lambda_{11} &= I \\ \Lambda_{12} \Lambda_{23} &= \Lambda_{13} \\ \Lambda_{12}^{-1} &= \Lambda_{21} \end{align*}
So is your question really, what is the physical justification for this?

• vanhees71
Yes, all this is true. The question is exactly as you say, how does the geometry generalise from 1 space dimension to 3 space dimensions. The examples you give are not the only possibilities.
True. Issues like homogeneity and isotropy would restrict the possibilities.

" How does 2-dimensional Euclidean geometry generalize to 3-dimensional Euclidean geometry? ".

If we denote the Lorentz transformation between observer n and observer m by Λnm then the essential results are that
\begin{align*} \Lambda_{11} &= I \ \Lambda_{12} \Lambda_{23} &= \Lambda_{13} \ \Lambda_{12}^{-1} &= \Lambda_{21} \end{align*}
So is your question really, what is the physical justification for this?
It took me a while to put in the line breaks into your formulae, but then I have no problem with the first and third, it is the second one which needs justification. This is exactly the issue that I cannot resolve.

It took me a while to put in the line breaks into your formulae, but then I have no problem with the first and third, it is the second one which needs justification. This is exactly the issue that I cannot resolve.
Apologies, there's a bug in the forum software that sometimes corrupts the LaTeX during composition and preview, I've now fixed my previous post.

Surely the second equation is an inevitable consequence of Einstein's first postulate? If you accept that, then are you looking for experimental justification of the first postulate?

True. Issues like homogeneity and isotropy would restrict the possibilities.

" How does 2-dimensional Euclidean geometry generalize to 3-dimensional Euclidean geometry? ".
This is a highly nontrivial question. If you did 2-dimensional Euclidean geometry at school, as I did, taught not quite from Euclid himself, but not far off, you will know that 3-dimensional Euclidean geometry was not taught. It is far too hard, and the generalisations are far from obvious.

Apologies, there's a bug in the forum software that sometimes corrupts the LaTeX during composition and preview, I've now fixed my previous post.

Surely the second equation is an inevitable consequence of Einstein's first postulate? If you accept that, then are you looking for experimental justification of the first postulate?
Well, correct me if I'm wrong, but Einstein's first postulate seems to assume that if frame 1 and frame 2 are inertial with respect to each other, and frame 1 and frame 3 are inertial with respect to each other, then frame 2 and frame 3 are inertial with respect to each other. That isn't a mathematical theorem, so it must be a physical assumption.

• dextercioby
I think possibly the content of my remark is not so much that it is a physical assumption rather than a mathematical theorem, but that it is not actually a reasonable physical assumption, if you think about it. That doesn't necessarily mean it isn't true, but it does cast some doubt, I think.

If we denote the Lorentz transformation between observer ##n## and observer ##m## by ##\Lambda_{nm}## then the essential results are that
\begin{align*} \Lambda_{11} &= I \\ \Lambda_{12} \Lambda_{23} &= \Lambda_{13} \\ \Lambda_{12}^{-1} &= \Lambda_{21} \end{align*}
So is your question really, what is the physical justification for this?
Oops, I realize there's a mistake here. I had forgotten to include the phenomenon of Thomas–Wigner rotation, so the second equation should really be $$\Lambda_{12} \Lambda_{23} =\left( \begin{array}{c|c} 1 & \textbf{0}^T \\ \hline \textbf{0} & \textbf{U} \\ \end{array} \right) \Lambda_{13}$$ for some 3×3 orthogonal matrix ##\textbf{U}## which complicates the argument.

Well, correct me if I'm wrong, but Einstein's first postulate seems to assume that if frame 1 and frame 2 are inertial with respect to each other, and frame 1 and frame 3 are inertial with respect to each other, then frame 2 and frame 3 are inertial with respect to each other. That isn't a mathematical theorem, so it must be a physical assumption.
It's a simple theorem if we use the Lorentz group in the definition of "inertial with respect to each other". Physical assumptions only get you so far. At some point you have declare the mathematical assumptions that are the basis of the model. Then it's a question of whether what you can crank out of the theory matches experiment.

It's clear that the historical postulates of SR are airy fairy - but, then, once you have your theory you can dispense with the historical baggage (if you so wish) and just say: flat spacetime is a 3+1 manifold with the Minkowski metric and be done with it.

Just recently I came across a pseudo-science link that said something like "if the speed of light is found not to be invariant then SR falls". Which is nonsense. SR is not dependent upon precise physical assumptions.

In answer to your question, the mathematical framework chosen for SR has the property that inertial relationships are transitive (if I can put it like that). If that's not justfified, then we'll eventually experimentally know about it.

It's a simple theorem if we use the Lorentz group in the definition of "inertial with respect to each other". Physical assumptions only get you so far. At some point you have declare the mathematical assumptions that are the basis of the model. Then it's a question of whether what you can crank out of the theory matches experiment.

It's clear that the historical postulates of SR are airy fairy - but, then, once you have your theory you can dispense with the historical baggage (if you so wish) and just say: flat spacetime is a 3+1 manifold with the Minkowski metric and be done with it.

Just recently I came across a pseudo-science link that said something like "if the speed of light is found not to be invariant then SR falls". Which is nonsense. SR is not dependent upon precise physical assumptions.

In answer to your question, the mathematical framework chosen for SR has the property that inertial relationships are transitive (if I can put it like that). If that's not justfified, then we'll eventually experimentally know about it.
Yes, every word of this is correct. But you use the word "if" in the first sentence, and it is precisely this assumption that I am questioning. What "if not"?

Yes, every word of this is correct. But you use the word "if" in the first sentence, and it is precisely this assumption that I am questioning. What "if not"?
And, by the way, we do know experimentally that this assumption is not justified.