I Lorentz transformations: 1+1 spacetime only

[robwilson]

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.

 

[DrGreg]

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.

 

[PeroK]

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.

 

[robwilson]

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"?

 

[robwilson]

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.

 

[Herman Trivilino]

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.

The first postulate, better known as the Principle of Relativity, is the assertion that all inertial frames are equivalent. I don't understand why you're saying things like frames 1 and 2 are inertial with respect to each other. You can establish, for example, that frame 1 is inertial. There is no need to compare it to frame 2 to establish this. Moreover, if frame 2 is inertial then of course both frames are inertial. There is no need to add the qualifier "with respect to each other".

So there are two experimental issues here. First, do inertial frames exist, and second, are they all equivalent to each other. The validity of these statements is established through experiment.

 

[PeroK]

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"?

Then you formulate an alternative theory; establish where it makes different predictions and test experimentally.

E.g. if (flat) spacetime is not isotropic and homogeneous, then we'd need a way to identify in what way it is not. That said, the expanding universe is a case in point. Globally we have (approximately) spatial flatness but a time-dependent scale factor.

 

[robwilson]

The first postulate, better known as the Principle of Relativity, is the assertion that all inertial frames are equivalent. I don't understand why you're saying things like frames 1 and 2 are inertial with respect to each other. You can establish, for example, that frame 1 is inertial. There is no need to compare it to frame 2 to establish this. Moreover, if frame 2 is inertial then of course both frames are inertial. There is no need to add the qualifier "with respect to each other".

So there are two experimental issues here. First, do inertial frames exist, and second, are they all equivalent to each other. The validity of these statements is established through experiment.

Then you formulate an alternative theory; establish where it makes different predictions and test experimentally.

E.g. if (flat) spacetime is not isotropic and homogeneous, then we'd need a way to identify in what way it is not. That said, the expanding universe is a case in point. Globally we have (approximately) spatial flatness but a time-dependent scale factor.

Indeed. But don't ask me about my alternative theory here!

 

[robwilson]

The first postulate, better known as the Principle of Relativity, is the assertion that all inertial frames are equivalent. I don't understand why you're saying things like frames 1 and 2 are inertial with respect to each other. You can establish, for example, that frame 1 is inertial. There is no need to compare it to frame 2 to establish this. Moreover, if frame 2 is inertial then of course both frames are inertial. There is no need to add the qualifier "with respect to each other".

So there are two experimental issues here. First, do inertial frames exist, and second, are they all equivalent to each other. The validity of these statements is established through experiment.

I am sceptical. I have never heard a definition of "inertial frame" that isn't circular. Without a definition, an experimental test is meaningless.

 

[PeroK]

I have never heard a definition of "inertial frame" that isn't circular. Without a definition, an experimental test is meaningless.

The first thing is probably true. It's the same with Newton's Laws: $\vec F = m\vec a$ defines both force and mass, essentially. Physics is like that. It can't be bootstrapped from nothing.

The second point is that in physics you can (and must) tie experiment to theory without complete philosophical watertightness.

 

[robwilson]

The first thing is probably true. It's the same with Newton's Laws: $\vec F = m\vec a$ defines both force and mass, essentially. Physics is like that. It can't be bootstrapped from nothing.

The second point is that in physics you can (and must) tie experiment to theory without complete philosophical watertightness.

True, of course. We've probably gone about as far down this discussion as it is reasonable to go. I wanted to see if I could get some good physical reason for assuming the Lorentz group is the correct symmetry group for spacetime, if indeed spacetime is a meaningful concept at all, and on my terms the answer is no, I couldn't. Other people may be convinced, and it may be correct, but for me, I feel it may be useful to try other groups instead, and treating spacetime as emergent rather than fundamental. If these ideas are wrong, experiment will prove it, and nothing is wasted except my time, which is an investment I'm prepared to make.

 

[robphy]

True, of course. We've probably gone about as far down this discussion as it is reasonable to go. I wanted to see if I could get some good physical reason for assuming the Lorentz group is the correct symmetry group for spacetime, if indeed spacetime is a meaningful concept at all, and on my terms the answer is no, I couldn't. Other people may be convinced, and it may be correct, but for me, I feel it may be useful to try other groups instead, and treating spacetime as emergent rather than fundamental. If these ideas are wrong, experiment will prove it, and nothing is wasted except my time, which is an investment I'm prepared to make.

In many approaches to quantum gravity,
there is the issue of what "spacetime" really is and what are its properties.
So, it is a non-trivial problem.

In my opinion, a first step toward the answer is clarifying the [possibly tentative] starting points,
then seeing what follows from them... and seeing what more is needed to advance.

With the many symmetries of Minkowski spacetime,
there are possibly many starting points
(from an old thread)
https://www.physicsforums.com/threa...challenge-for-experts-only.83373/#post-694535

..but this already assumes a lot... like a continuum.

Maybe spacetime is fundamentally discrete at the microscopic scale...
and maybe the causal ordering is what is fundamental
and this is what implies the group structure
https://aip.scitation.org/doi/10.1063/1.1704140
Causality Implies the Lorentz Group
Journal of Mathematical Physics 5, 490 (1964); https://doi.org/10.1063/1.1704140
E. C. Zeeman

https://en.wikipedia.org/wiki/Alfred_Robb (with his "after" relation) was probably among the first to consider causal ordering as fundamental.

Alternatively, maybe causality isn't fundamental (but only seems so macroscopically).
Some other structure may be more fundamental.

[A related question I've had is the dimensionality of space[time].]
Maybe the metric isn't even pseudo-riemannian. ( See my comment here mentioning the Ehlers-Pirani-Schild (EPS) approach : https://physics.stackexchange.com/a/511370/148184 )
Yes, many questions.

But the point is... one has to declare one's starting points.
(e.g. homogeneity?, isotropy?, etc...)finally, a link to another old post, offering possible starting points in the literature
https://www.physicsforums.com/threads/the-foundations-of-relativity-ii.106296/post-878413

 

[robwilson]

In many approaches to quantum gravity,
there is the issue of what "spacetime" really is and what are its properties.
So, it is a non-trivial problem.[A related question I've had is the dimensionality of space[time].]
Maybe the metric isn't even pseudo-riemannian. ( See my comment here mentioning the Ehlers-Pirani-Schild (EPS) approach : https://physics.stackexchange.com/a/511370/148184 )
Yes, many questions.

But the point is... one has to declare one's starting points.
(e.g. homogeneity?, isotropy?, etc...)finally, a link to another old post, offering possible starting points in the literature
https://www.physicsforums.com/threads/the-foundations-of-relativity-ii.106296/post-878413

Yes, many questions. I just want a group that describes the symmetries that experiment reveals. I'm not happy with the groups that are in the current theories, and I've tried many alternatives, which I am not happy with either. Questioning the Lorentz group and the structure of spacetime is more or less a last resort - beyond this there is nowhere else to go.

 

[strangerep]

I wanted to see if I could get some good physical reason for assuming the Lorentz group is the correct symmetry group for spacetime, if indeed spacetime is a meaningful concept at all, and on my terms the answer is no, I couldn't.

I arrive late to this thread, and maybe I shouldn't get involved, but it's a Sunday afternoon here and I'm a bit bored. So,...

Starting from the relativity principle, we ask "what is the largest group of coordinate transformations that maps inertial (non-accelerating) worldlines among themselves". I.e., coordinate transformations $(t,x^i) \to (t',x'^i)$ such that $$\frac{d^2x^i}{dt^2} ~=~ 0 ~~~ \leftrightarrow~~~ \frac{d^2x'^i}{dt'^2} ~=~ 0 ~.$$ Hidden in this is already an assumption of spatial isotropy and a technical assumption about continuity and differentiability of the admissible coordinate systems (which I gloss over for now).

The largest such group is the 24-parameter fractional linear transformations ("FLT" hereafter). Although it's possible to work with this group, I'll sketch the part of the argument relevant to ordinary Lorentz transformations by restricting the FLT group down to the affine group. It's obvious that ordinary translations in space and in time form subgroups, so by using this freedom I'll restrict to the case where coordinate origin is preserved by the transformation. I.e., by an ordinary translation, the primed origin can be made to coincide with the unprimed origin. Then, it's also obvious that spatial rotations are a subgroup here which preserves coordinate origins.

Now we can investigate velocity-changing transformations. Pick an arbitrary direction in space (anchored at the origin), and rename the spatial coordinates so that one spatial coordinate lies along this arbitrary direction -- I'll call it $x$. By a spatial rotation of the primed coordinates we can make its spatial axes coincide with the unprimed coordinates.
(This also assumes we've set up our spatial axes to be Euclidean-orthogonal.)

Now we ask for the most general coordinate transformation (using the few degrees of freedom remaining) that maps $$\frac{dx}{dt} ~=~ 0 ~~~ \to ~~~ \frac{dx'^i}{dt'} ~=~ v ~,$$where $v \ne 0$. We further restrict such transformations by requiring that they form a 1-parameter Lie group (with parameter $v$), where $v=0$ corresponds to the identity, and is well-defined on a open set in $v$-space containing $v=0$.

Introducing such a Lie group requirement here is physically motivated by using (again) the relativity principle, i.e., that no inertial reference frame is in any way physically distinguished.
There's also an implicit assumption that if observer A is moving with velocity $v_1$ relative to observer B, who is moving with velocity $v_2$ relative to observer C (all reference frames having been set up as described above, i.e., all origins and all spatial axes coincide),
then there should exist a well-defined velocity $v_3$ with which C is moving relative to A.

Cranking the mathematical handle on the above yields the Lorentz transformations along spatial direction $x$. We can repeat the whole procedure along directions $y$ and $z$ separately, then determine the full Lie algebra among the generators of those transformations (and discover that we much include the spatial rotations for the algebra to close). Then we can add the translation generators and discover the Poincare algebra.

You asked for a "physical" reason for assuming the Lorentz group is the correct symmetry group for spacetime. The answer is that we do NOT "assume" it. Rather, it is derived via analysis of the most general symmetry group that maps non-accelerating worldlines amongst themselves.

All my explanation above was in terms of abstract coordinates. Although most people then "promote" these coordinates to a "physically real" spacetime entity, but there is (imho) no need to do so. The basic (intrinsic) physical properties (mass, spin, etc) of elementary particles can be found from the unitary irreducible representations of the Poincare group, and the possible interactions between particles (and exchange of these properties) is also strongly constrained by the Poincare group.

[...]if indeed spacetime is a meaningful concept at all,[...]

It is meaningful only in the sense of being isomorphic to a homogeneous space for the Poincare group. The symmetry group governing inertial reference frames is what really matters.

<Phew!> ... I sure do hope this has actually helped...

 

[A.T.]

I have never heard a definition of "inertial frame" that isn't circular.

What is actually wrong with circular definitions? I see the problem with circular reasoning, but I never understood why circular definitions are bad.

 

[PeterDonis]

Starting from the relativity principle, we ask "what is the largest group of coordinate transformations that maps inertial (non-accelerating) worldlines among themselves".

You seem to be making an additional implicit assumption, namely that worldlines with zero coordinate acceleration have zero proper acceleration. Or are you including this in what you are calling the hidden assumption of spatial isotropy? (A worldline with nonzero proper acceleration picks out a preferred direction in space, the direction of the proper acceleration.)

 

[robwilson]

It's obvious that ordinary translations in space and in time form subgroups, so by using this freedom I'll restrict to the case where coordinate origin is preserved by the transformation. I.e., by an ordinary translation, the primed origin can be made to coincide with the unprimed origin. Then, it's also obvious that spatial rotations are a subgroup here which preserves coordinate origins.

(This also assumes we've set up our spatial axes to be Euclidean-orthogonal.)

There's also an implicit assumption that if observer A is moving with velocity $v_1$ relative to observer B, who is moving with velocity $v_2$ relative to observer C (all reference frames having been set up as described above, i.e., all origins and all spatial axes coincide),
then there should exist a well-defined velocity $v_3$ with which C is moving relative to A.

You asked for a "physical" reason for assuming the Lorentz group is the correct symmetry group for spacetime. The answer is that we do NOT "assume" it. Rather, it is derived via analysis of the most general symmetry group that maps non-accelerating worldlines amongst themselves.

The symmetry group governing inertial reference frames is what really matters.

<Phew!> ... I sure do hope this has actually helped...

Thank you for this detailed and helpful explanation. I've selected a few places where you use words like "obvious" or "assumption" for analysis. I feel there are gaps in the argument in one or two of these places, such as the one pointed out by PeterDonis. In particular, the conclusion that the Lorentz group maps non-accelerating worldlines among themselves does not seem to be justified by the analysis: unless I am much mistaken, the Lorentz group includes rotating worldlines, which are accelerating.

 

[PeroK]

... unless I am much mistaken, the Lorentz group includes rotating worldlines, which are accelerating.

I'd say you are very much mistaken. It's the difference between a fixed rotated set of axes and a rotating (changing with time) set of axes.

 

[mitochan]

@robwilson , in order to get your point, could you tell me do you have same question on Galilean transformation or you have no problem on it ?

 

[robwilson]

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.

Thanks for this elucidation, and especially for the very helpful reference to the wikipedia article on Thomas-Wigner rotation. I note in particular the prominent sentence near the beginning: "There are still ongoing discussions about the correct form of the equations for the Thomas rotation in different reference systems, with contradictory results."
This is really my point: that if you assume the Lorentz group is a symmetry group of the system, then you get a particular prediction for this rotation, which presumably can be tested. The article does not suggest that it ever has been tested, however. Other groups will give other predictions. Is it just too difficult to test these predictions? Or is it a theoretical problem, because rotating frames are non-inertial, so that one needs general relativity as well?

 

[robwilson]

@robwlinson, in order to get your point, could you tell me do you have same question on Galilean transformation or you have no problem on it ?

I have no problem with Galilean transformations. The question is, what group do you need if you want the speed of light to be finite and the same for all observers. The Lorentz group is a good approximation, but is it good enough?

 

[robwilson]

I'd say you are very much mistaken. It's the difference between a fixed rotated set of axes and a rotating (changing with time) set of axes.

A distinction without a difference, I'm afraid. If observers 2 and 3 are traveling at constant velocities with respect to observer 1, in different directions and at different speeds, then observers 2 and 3 are rotating around each other according to observer 1. So yes, you can transform to a different frame in which observers 2 and 3 agree that they are not rotating, but the transformation between observer 1's frame and that frame is a rotating transformation not a rotated transformation.

 

[PeroK]

A distinction without a difference, I'm afraid. If observers 2 and 3 are traveling at constant velocities with respect to observer 1, in different directions and at different speeds, then observers 2 and 3 are rotating around each other according to observer 1. So yes, you can transform to a different frame in which observers 2 and 3 agree that they are not rotating, but the transformation between observer 1's frame and that frame is a rotating transformation not a rotated transformation.

In that example: according to observer 2, observer 3 will have a constant velocity. You can calculate that velocity using the Lorentz Transformations.

How do you come to the conclusion that it would be otherwise?

 

[strangerep]

You seem to be making an additional implicit assumption, namely that worldlines with zero coordinate acceleration have zero proper acceleration.

OK. Of course I must start with inertial observers, i.e., who all experience zero proper acceleration. I omitted a description of how each constructs their own coordinate system such that the other observers are not accelerating wrt those coordinates. (I figured I could just appeal to the various textbook discussions involving rods and clocks, etc, for such construction.)

I feel there are gaps in the argument in one or two of these places, such as the one pointed out by PeterDonis. In particular, the conclusion that the Lorentz group maps non-accelerating worldlines among themselves does not seem to be justified by the analysis [...]

That's not a "conclusion" -- one of the input assumptions is that we are dealing only with inertial observers, i.e., observers who experience zero acceleration according to their respective accelerometers (which is another way of saying "zero proper acceleration").

 

[robwilson]

In that example: according to observer 2, observer 3 will have a constant velocity. You can calculate that velocity using the Lorentz Transformations.

How do you come to the conclusion that it would be otherwise?

As usual, you are assuming the Lorentz group is the correct group in which to calculate. I don't find this group well-motivated, and I am not convinced that its conclusions are necessarily exactly correct (although clearly they are very nearly correct in a very wide range of experimental circumstances that have been extensively investigated).
However, I think I am beginning to see that the difference between my assumptions and the standard assumptions must involve gravity, so that if we ignore gravity, then the Lorentz group probably works just fine.

 

[PeroK]

As usual, you are assuming the Lorentz group is the correct group in which to calculate. I don't find this group well-motivated, and I am not convinced that its conclusions are necessarily exactly correct (although clearly they are very nearly correct in a very wide range of experimental circumstances that have been extensively investigated).
However, I think I am beginning to see that the difference between my assumptions and the standard assumptions must involve gravity, so that if we ignore gravity, then the Lorentz group probably works just fine.

If the starting point is the Lorentz Transformation in 1D, then the transformations in 2D and 3D follow. There can be no ambiguity about the 2D and 3D transformations.

There is no gravity in SR. SR is flat spacetime. If you have gravity, then that is a different ballgame, as there are no global inertial reference frames in curved spacetime. Instead, locally (over a "small enough" region of spacetime) you have approximately SR. This is one form of the equivalence principle. The Lorentz group only applies locally, therefore. For example, experiments at CERN may assume approximately flat spacetime as they are confined to a region of spacetime where the Earth's gravity is approximately constant - and we have SR locally, hence no need to invoke GR.

 

[robwilson]

If the starting point is the Lorentz Transformation in 1D, then the transformations in 2D and 3D follow. There can be no ambiguity about the 2D and 3D transformations.

There is no gravity in SR. SR is flat spacetime. If you have gravity, then that is a different ballgame, as there are no global inertial reference frames in curved spacetime. Instead, locally (over a "small enough" region of spacetime) you have approximately SR. This is one form of the equivalence principle. The Lorentz group only applies locally, therefore. For example, experiments at CERN may assume approximately flat spacetime as they are confined to a region of spacetime where the Earth's gravity is approximately constant - and we have SR locally, hence no need to invoke GR.

Exactly. A whole load of simplifying assumptions. I am not interested in a simplified approximate local answer. I am interested in a mathematically consistent globally correct answer. SR and GR taken together do not provide that.

 

[vanhees71]

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!

This is indeed utter nonsense. The standard theories use (1+3)D space-time models. All attempts to find "extra dimensions" so far were in vain. Of course you can build the Poincare group out of its subgroups and so this group is presented in standard textbooks. In fact most you already know from Newtonian mechanics (translation and rotation symmetry). All you have to change are the boosts, and that's why most textbooks start with the Lorentz boost in 1 direction, building a one-parameter subgroup of the Lorentz group. From that you can build the entire proper orthochronous Lorentz group by these boosts and rotations. As a group theorist the OP should know that.

 

[vanhees71]

Exactly. A whole load of simplifying assumptions. I am not interested in a simplified approximate local answer. I am interested in a mathematically consistent globally correct answer. SR and GR taken together do not provide that.

All fundamental models based on relativity are local, and all observations are indeed based on local measurements. It's not simplified and not approximate, it's the most comprehensive model we have in physics today.

 

[robwilson]

This is indeed utter nonsense. The standard theories use (1+3)D space-time models. All attempts to find "extra dimensions" so far were in vain. Of course you can build the Poincare group out of its subgroups and so this group is presented in standard textbooks. In fact most you already know from Newtonian mechanics (translation and rotation symmetry). All you have to change are the boosts, and that's why most textbooks start with the Lorentz boost in 1 direction, building a one-parameter subgroup of the Lorentz group. From that you can build the entire proper orthochronous Lorentz group by these boosts and rotations. As a group theorist the OP should know that.

I do know that. I also know that there are other groups that can also be built from one-parameter subgroups of Lorentz transformations. The local structure of a group does not determine the global structure.