Second postulate of SR quiz question

[PAllen]

Thanks for the hint, but what's the merit of should such an overcomplication? There's nothing in Maxwell's theory, which can be regarded as the most accurate theory ever discovered (when you take into account even QED as its quantized refinement, it's even the most accurate theory ever discovered), hinting at a direction dependence (anisotropy?) for the phase velocity of em. waves.

I agree. That is why I refer to the view that harps on the inability to disprove such conspiratorial anisotropy as pedantic. I prefer to define isotropy as a feature of physical law as the property that:

- assuming it leads to simplification
- only conspiratorial anisotropy is consistent with experiment

Then I can just say physical law is isotropic to the best of our experiments, per this definition.

 

[vanhees71]

Well, it's of course another question to check empirically whether Maxwell's equations hold true. In principle this should be done, because so unlikely it seems that they are flawed somehow, you never know, and high-precision measurements often lead to important groundbreaking discoveries in science. One example is the high-precision measurement of the black-body spectrum in the Physikalisch Technische Reichsanstalt in Berlin around 1900 over a wide range of frequencies, which lead Planck to find the correct radiation law and then derive the theoretical consequences, leading finally to the discovery of quantum mechanics in 1925.

 

[loislane]

Isn't the reason that discussions about isotropy, synchronization and simultaneity usually go in circles that Minkowski space being an afine space is just homogeneous but simply doesn't define angles(you need more structure for that) and therefore can't determine rotational invariance? It is precisely the affine structure what allows Minkowski space to accommodate an indefinite signature bilinear form, and equivalently but more physically the relativity of simultaneity determined by the indefinite signature that allows the planes of simultaneity not to be fixed as would happen in a Euclidean geometry with isotropy.
In this context the requirement of isotropy for physical laws and the empirical evidence of it achieved by experiments seem to be at odds with the structure of SR, that requires the ambiguity about isotropy and one-way vs two-way speeds, and the conventional synchronizations view to be kept since it is encoded in its affine mathematical formulation.

 

[PeterDonis]

Minkowski space being an afine space is just homogeneous but simply doesn't define angles (you need more structure for that) and therefore can't determine rotational invariance?

Minkowski spacetime is not just an affine space; it has a metric defined on it. The metric is not positive definite (some purists might call it a "pseudo-metric" because of that), but it's sufficient to define rotational invariance (there is a three-parameter group of Killing vector fields corresponding to spatial rotations).

 

[loislane]

Minkowski spacetime is not just an affine space; it has a metric defined on it. The metric is not positive definite (some purists might call it a "pseudo-metric" because of that), but it's sufficient to define rotational invariance (there is a three-parameter group of Killing vector fields corresponding to spatial rotations).

Minkowski space is a real affine space with an indefinite bilinear form that determines certain symmetries in the vector space associated when fixing a point as the origin, namely the Lorentz group, a 6-parameter group of symmetry, 3 for the boosts and 3 for rotational invariance in three dimensions restricted to the tangent vector spaces at the points in affine space, not for affine space itself where there is no origin fixed.
The isotropy I was talking about is the one corresponding to rotations in 4 dimensions-O(4)-, this symmetry is not present in the affine Minkowski space, if it were it would be Euclidean Riemannian space in 4 dimensions which it isn't and it wouldn't have arbitrary simultaneity planes.
The kind of isotropy of light that PAllen and others were discussing, that includes time(synchronization, simultaneity), cannot be assumed or postulated because of the lack of this symmetry(of course the possibility remains to choose orthogonal coordinates at every point and in that sense a coordinate isotropy, this is equivalent to the possibility of choosing the Einstein synchronization in the rest frame), even if all experiments like Michelson-Morley, Kennedy-Thorndike... in their modern versions apparently show isotropy beyond a reasonable doubt.

 

[loislane]

Maybe we have a different view of the same facts. Adding or not adding the simultaneity convention does not introduce internal inconsistency. It simply follows from and is consistent with the implied assumption that the reference system of choice is in rest - so that other reference systems are not in rest. In other words, when assuming (or pretending) that your system of choice is in rest, you make your own chosen space-time homogeneous so that other space-times become inhomogeneous (according to you; it's the inverse according to others).

[addendum:]That is not very different from momentum in Newton's mechanics: The momentum of a particle that is co-moving with your system of choice is taken as zero by you, while it is taken as non-zero in other systems. Disagreement by convention is not the same as contradiction.

I am actually saying that it is impossible to show internal inconsistency.

I don't know if he proposed that footnote himself or if an editor proposed it and he agreed without thinking of a better way to clarify it. The way he formulated it in the original text implies, when taken at face value, that the Newtonian equations hold perfectly in the new theory, which is incorrect; they only hold to first approximation in the new theory. IMHO he should have phrased it as follows in the original text:
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good according to Newtonian mechanics" (which is a bit exhausting), or
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics are believed to hold good" (which is simpler but may still be misunderstood).

Once more, SR relates to the reference systems of classical mechanics exactly: the Lorentz transformations are exactly valid if we can ignore the effects of gravitation.

I can agree with this way of looking at it, which is a possible interpretation of the added footnote but it is also compatible with the logic in my question about the formulation of the second postulate (or both postulates). As written it would seem to refer to the Newtonian equations holding perfectly also in SR, not just to first order in v/c.
Newton mechanics was set in the context of the Euclidean geometry symmetries, where the inertial frames(cartesian coordinates and linear time parameter) hold exactly.

And that (the first postulate) is not what the second postulate is about. According to Newtonian mechanics, if we assume that light is made up of particles, then the laws of optics are included in the first postulate. The problem at the time, which was solved by Lorentz and Einstein, was how to combine Newton's mechanics with Maxwell's electrodynamics. The first postulate is an essential feature of Newton's mechanics, but was at odds with Maxwell's electrodynamics when assuming that Newton's laws are exactly valid. The second postulate is an essential feature of Maxwell's electrodynamics, but it appeared to be at odds with the first postulate.

Well I was choosing the interpretation of the SR first postulate that includes bot mechanics and EM laws which is not exactly the same as the galilean principle of relativity, if so then the second postulate just specifies something postulated in the new SR principle of relativity, but not in the galilean one. If Einstein was using the interpretation including just the laws of mechanics then you are right.

Or, as Einstein phrased it, "[the second postulate] is only apparently irreconcilable with the [first postulate]".

We are back to the ambiguity of the paper(that is more obvious from our privileged hindsight pov),one would have to know exactly to what physics laws the first postulate is referring to, if it refers to just the laws of mechanics the postulates are apparently irreconcilable and precisely the way in which Einstein reconciled them is the relativity of simultaneity as shown in the mathematical Minkowskian representation(as discussed in my previous post), so that is probably the right interpratation of his first postulate.
Otherwise I can't see the apparent contradiction and the second postulate is just a specific implementation of the first postulate in optics.

 

[harrylin]

[.] Well I was choosing the interpretation of the SR first postulate that includes bot mechanics and EM laws which is not exactly the same as the galilean principle of relativity, [..] If Einstein was using the interpretation including just the laws of mechanics then you are right.

The relativity principle does not state or include any specific law of physics - instead it prescribes a requirement for the laws of physics. Another formulation of the PoR is that it is impossible to detect absolute inertial motion - or as Einstein phrased it, the phenomena possesses no properties corresponding to the idea of absolute rest. Classical mechanics and SR both use "Galilean" frames and apply the same relativity principle to those frames. However, by the time of MMX it was assumed that optical phenomena could not obey the PoR - see next.

We are back to the ambiguity of the paper(that is more obvious from our privileged hindsight pov),one would have to know exactly to what physics laws the first postulate is referring to, if it refers to just the laws of mechanics the postulates are apparently irreconcilable and precisely the way in which Einstein reconciled them is the relativity of simultaneity as shown in the mathematical Minkowskian representation(as discussed in my previous post), so that is probably the right interpratation of his first postulate.
Otherwise I can't see the apparent contradiction and the second postulate is just a specific implementation of the first postulate in optics.

The apparent contradiction was already explained in the intro of Einstein's 1905 paper, but perhaps it was better explained by Michelson and Morley, as they first redid Fizeau's experiment which gave, as I already cited, a firm basis for the second postulate. Based on that result, they next performed their famous experiment (often indicated as "MMX") which they expected to give a positive result and therewith also prove beyond doubt that the PoR is not valid for optics. But instead, that latter experiment supported the first postulate. If it is not clear why these two experimental outcomes were in apparent contradiction with each other, then perhaps Wikipedia clarifies it well enough: https://en.wikipedia.org/wiki/Michelson–Morley_experiment#Most_famous_.22failed.22_experiment

 

[loislane]

The relativity principle does not state or include any specific law of physics - instead it prescribes a requirement for the laws of physics.

That's why we are discussing this point, yes.

The apparent contradiction was already explained in the intro of Einstein's 1905 paper

AFAICS it was mentioned in the intro, which is not exactly the same as explained. But this is of very little importance as I already agreed that the logic leads to interpreting what Einstein meant in his first postulate your way . That doesn't mean that it is the only interpretation, other posters have manifested their preference for the meaning that makes the second postulate just a consequence of the first.

 

[PAllen]

The kind of isotropy of light that PAllen and others were discussing, that includes time(synchronization, simultaneity), cannot be assumed or postulated because of the lack of this symmetry(of course the possibility remains to choose orthogonal coordinates at every point and in that sense a coordinate isotropy, this is equivalent to the possibility of choosing the Einstein synchronization in the rest frame), even if all experiments like Michelson-Morley, Kennedy-Thorndike... in their modern versions apparently show isotropy beyond a reasonable doubt.

Isotropy in physics is always taken to be spatial. When I was formulating an assumption of isotropy (for SR) I explicitly said there exists a group (Poincare group of all global inertial coordinates) of global coordinates such that observed physical laws expressed ins such coordinates display manifest isotropy, with the latter meaning spatial because that is always assumed.

 

[vanhees71]

There is nothing ambiguous in Einstein's paper in the important first part on kinematics. At least, I don't see anything that's ambiguous there, but very clearly derived from the two postulates, which are reconciled by the synchronization procedure described by Einstein as a convention, based on the two postulates, particularly on the 2nd one. I think, you overcomplicate things. Is it, perhaps, the English translation, which makes the paper look ambiguous? Of course, you can lament about a lot in the further parts of the paper, particularly the introduction of various relativistic masses in the mechanical part of the paper. Nowadays we struggle still with remnants of this. Fortunately nobody uses Einstein's transverse and longitudinal masses anymore, but sometimes one uses the quantity $m \gamma$ as a relativistic mass, which is bad enough ;-).

 

[loislane]

Isotropy in physics is always taken to be spatial. When I was formulating an assumption of isotropy (for SR) I explicitly said there exists a group (Poincare group of all global inertial coordinates) of global coordinates such that observed physical laws expressed ins such coordinates display manifest isotropy, with the latter meaning spatial because that is always assumed.

Then I don't understand what your discussion is about, rotational invariance in 3 dimensions is clearly a symmetry when fixing an origin(it clearly cannot be "not assumed"), but it is independent of anything related to synchronization or simultaneity conventions, or the one-way versus 2-way speed of light debate, that is related to the boosts part of the Lorentz group and the absence of a 4 dimensional spatial isotropy.

 

[PeterDonis]

Minkowski space is a real affine space with an indefinite bilinear form that determines certain symmetries in the vector space associated when fixing a point as the origin, namely the Lorentz group, a 6-parameter group of symmetry, 3 for the boosts and 3 for rotational invariance in three dimensions restricted to the tangent vector spaces at the points in affine space, not for affine space itself where there is no origin fixed.

Let me try to restate what I think you're saying in different terms. In order to give meaning to the term "isotropy", we need to pick a point to serve as the origin, so that we can pick out the particular group of transformations that count as "rotations" about that origin. But Minkowski spacetime does not have any particular point which is preferred as the origin; any point will do. So the term "isotropy" has no meaning if we just look at Minkowski spacetime itself; it only has meaning once we pick a particular point as the origin, and then it means "isotropy about this chosen point", not "isotropy of spacetime in general".

I agree with this as far as it goes; but I think what most people implicitly mean when they say that Minkowski spacetime is isotropic is that you can pick any point you like as the origin and you will have isotropy (in the sense of the 3-parameter group of spacelike Killing vector fields being present) about that point. Technically this is sloppy phrasing, but I think physicists are often sloppy in that way, at least from the viewpoint of mathematicians.

 

[harrylin]

other posters have manifested their preference for the meaning that makes the second postulate just a consequence of the first.

That misunderstanding was debunked in posts #22 and #67: There is no necessity to deviate from Galilean relativity on the basis of the relativity postulate alone, and there is no apparent contradiction between a consequence of something and that something.

 

[loislane]

Let me try to restate what I think you're saying in different terms. In order to give meaning to the term "isotropy", we need to pick a point to serve as the origin, so that we can pick out the particular group of transformations that count as "rotations" about that origin. But Minkowski spacetime does not have any particular point which is preferred as the origin; any point will do. So the term "isotropy" has no meaning if we just look at Minkowski spacetime itself; it only has meaning once we pick a particular point as the origin, and then it means "isotropy about this chosen point", not "isotropy of spacetime in general".

I agree with this as far as it goes; but I think what most people implicitly mean when they say that Minkowski spacetime is isotropic is that you can pick any point you like as the origin and you will have isotropy (in the sense of the 3-parameter group of spacelike Killing vector fields being present) about that point. Technically this is sloppy phrasing, but I think physicists are often sloppy in that way, at least from the viewpoint of mathematicians.

Yes, that's right.
Then my point about PAllen comments about "isotropy" and its relation with convention of synchronization for diferent frames, and how this determines the relations between one-way and two way speed of light is purely geometric.
The absence of a 4 dimensional-O(4)- rotational symmetry in Minkowski space that comes from not having a positive definite euclidean inner product leaves one degree of freedom in the affine space that is used in the arbitrary choice of simultaneity hyperplane-O(1.3) symmetry- for a certain time axis(i.e. tilting is allowed), that is what boosts are, while allowing also obviously the orthogonal coordinate choice. This specific choice is what an Einstein synchronization amounts to, and what I think PAllen refers to by "isotropy" convention that one is free to assume or not.

 

[loislane]

That misunderstanding was debunked in posts #22 and #67: There is no necessity to deviate from Galilean relativity on the basis of the relativity postulate alone, and there is no apparent contradiction between a consequence of something and that something.

I've stepped on a LET supporter' toe maybe?

 

[PAllen]

Then I don't understand what your discussion is about, rotational invariance in 3 dimensions is clearly a symmetry when fixing an origin(it clearly cannot be "not assumed"), but it is independent of anything related to synchronization or simultaneity conventions, or the one-way versus 2-way speed of light debate, that is related to the boosts part of the Lorentz group and the absence of a 4 dimensional spatial isotropy.

The argument about inability to prove isotropy of (one way) light speed is, in effect, there exists a different group of coordinates which lead to identical measurements of observables, where not only one way light speed is anisotropic, but (necessarily to match experiment) equations of EM and mechanics are also anisotropic in just the right way. This is taken to mean isotropy can't be proven. My argument is that the important physical point is existence of the large group that manifests isotropy for all laws. One could easily envision some universe where you have to establish coordinates different ways for different laws to show isotropy of that law. The ability to find such a large group that shows isotropy for all laws is, by my argument, the definition of what it means for the universe to show isotropy as fundamental symmetry. Then, IMO, the existence of coordinate schemes that don't manifest isotropy becomes irrelevant to the question of whether isotropy is an intrinsic symmetry of our universe.

 

[vanhees71]

Let me try to restate what I think you're saying in different terms. In order to give meaning to the term "isotropy", we need to pick a point to serve as the origin, so that we can pick out the particular group of transformations that count as "rotations" about that origin. But Minkowski spacetime does not have any particular point which is preferred as the origin; any point will do. So the term "isotropy" has no meaning if we just look at Minkowski spacetime itself; it only has meaning once we pick a particular point as the origin, and then it means "isotropy about this chosen point", not "isotropy of spacetime in general".

I agree with this as far as it goes; but I think what most people implicitly mean when they say that Minkowski spacetime is isotropic is that you can pick any point you like as the origin and you will have isotropy (in the sense of the 3-parameter group of spacelike Killing vector fields being present) about that point. Technically this is sloppy phrasing, but I think physicists are often sloppy in that way, at least from the viewpoint of mathematicians.

An affine space is in any case translation invariant. So if it is isotropic wrt. one point, it must be isotropic around any point. Minkowski space admits not an SO(4) group. So it is not isotropic in the strict sense.

 

[PeterDonis]

An affine space is in any case translation invariant. So if it is isotropic wrt. one point, it must be isotropic around any point.

Yes, agreed. This is consistent with what I said.

Minkowski space admits not an SO(4) group.

No, but once you've picked an origin, it does admit an SO(3) symmetry group about that origin (and, as you agree, any origin will do). That's what "isotropy" is usually taken to mean.

 

[vanhees71]

Sure, that's the isotropy of space for one (and thus all) inertial observers, which is tacitly assumed in Einstein's 1905 paper.

 

[loislane]

No, but once you've picked an origin, it does admit an SO(3) symmetry group about that origin (and, as you agree, any origin will do). That's what "isotropy" is usually taken to mean.

If by what physicists usually understand by isotropy is meant the concept in classical Newtonian physics, then I disagree, in classical mechanics the geometry the physics is based on is euclidean and in three dimensional euclidean geometry the SO(3) symmetry has a different mathematical sense from that in four dimensional Minkowskian geometry. In the euclidean case the rotational invariance is part of the isometry group, and isometries are distance-preserving maps between metric spaces. While the SO(3) symmetry of the vector spaces associated to Minkowski space has nothing to do with metric spaces, it is not a metric space property like the isotropy of classical mechanics.

 

[loislane]

The argument about inability to prove isotropy of (one way) light speed is, in effect, there exists a different group of coordinates which lead to identical measurements of observables, where not only one way light speed is anisotropic, but (necessarily to match experiment) equations of EM and mechanics are also anisotropic in just the right way. This is taken to mean isotropy can't be proven. My argument is that the important physical point is existence of the large group that manifests isotropy for all laws. One could easily envision some universe where you have to establish coordinates different ways for different laws to show isotropy of that law. The ability to find such a large group that shows isotropy for all laws is, by my argument, the definition of what it means for the universe to show isotropy as fundamental symmetry. Then, IMO, the existence of coordinate schemes that don't manifest isotropy becomes irrelevant to the question of whether isotropy is an intrinsic symmetry of our universe.

You would need to make explicit what symmetry group you are referring to now when you say isotropy and also what large group you mean: SO(4), SO(1,3)? It is hard for me to understand your post without that information.

 

[PeterDonis]

If by what physicists usually understand by isotropy is meant the concept in classical Newtonian physics, then I disagree

But as you note, the classical concept is not applicable in Minkowski spacetime, so clearly in the context of SR "isotropy" must mean something different.

In the euclidean case the rotational invariance is part of the isometry group, and isometries are distance-preserving maps between metric spaces.

Agreed.

While the SO(3) symmetry of the vector spaces associated to Minkowski space has nothing to do with metric spaces

But there is a connection between the two. If we pick a point of Minkowski spacetime as the origin, and we pick a particular spacelike hypersurface of simultaneity through that origin, then that hypersurface is a metric space with the geometry of Euclidean 3-space, including an SO(3) isometry group about the chosen origin. And this SO(3) isometry group is the same one we get when we use the properties of Minkowski spacetime as an affine space to determine the symmetries of the vector space at the chosen origin. (More precisely, the restriction, if that's the right term, of the symmetries of that vector space to the chosen spacelike hypersurface.)

 

[loislane]

But there is a connection between the two. If we pick a point of Minkowski spacetime as the origin, and we pick a particular spacelike hypersurface of simultaneity through that origin, then that hypersurface is a metric space with the geometry of Euclidean 3-space, including an SO(3) isometry group about the chosen origin. And this SO(3) isometry group is the same one we get when we use the properties of Minkowski spacetime as an affine space to determine the symmetries of the vector space at the chosen origin. (More precisely, the restriction, if that's the right term, of the symmetries of that vector space to the chosen spacelike hypersurface.)

Of course there is a certain connection but it is ultimately conventional, the first conditional (picking an origin) simply reflects and undoes the generalization from a vector space to an affine space.
The second conditional you use to stablish the connection between SO(3) in both geometries is the requirement of an arbitrary choice of a particular plane of simultaneity in the second case, it reflects the generalization from the euclidean metric geometry to the affine non-metric geometry. This is a particular choice of local coordinates at a particular point and thus conventional.

 

[harrylin]

I've stepped on a LET supporter' toe maybe?

I think that you stepped on no toe; did I step on one?

 

[vanhees71]

SO(3) is of course a subgroup of SO(1,3), and this is due to an important additional assumption on the structure of relativistic space-time, namely that for each inertial observer space is a Euclidean 3-dimensional affine space as in Newtonian mechanics. The new thing is that this is indeed restricted to inertial observers, because accelerated observers find a non-Euclidean space in special relativity.

 

[PeterDonis]

accelerated observers find a non-Euclidean space in special relativity.

There are some caveats to this statement that I think are worth mentioning.

First, there is one particular family of accelerated observers for whom "space" is still Euclidean: Rindler observers. These have proper acceleration all in the same (fixed) direction, and with magnitude that varies in just the right way to keep the radar distance between them constant. In the accelerated coordinates in which these observers are at rest, the metric is

$$
ds^2 = - a^2 x^2 dt^2 + dx^2 + dy^2 + dz^2
$$

where the observers' proper acceleration is in the $x$ direction, and the observer at $x = 1$ has proper acceleration of magnitude $a$. It is evident that spacelike slices of constant coordinate time $t$ are Euclidean for this metric.

Second, when we look at a family of accelerated observers for whom "space" is non-Euclidean, we have to be careful defining what "space" means. For example, consider the family of Langevin observers, who are all moving in circular trajectories about a common origin, with the same angular velocity $\omega$. In the accelerated coordinates in which these observers are at rest (we use cylindrical coordinates here to make things look as simple as possible), the metric is

$$
ds^2 = - \left( 1 - \omega^2 r^2 \right) dt^2 + 2 \omega r^2 dt d\phi + dz^2 + dr^2 + r^2 d\phi^2
$$

Note that this metric is only valid for $0 < r < 1 / \omega$; at larger values of $r$, there are no Langevin observers (if there were, they would be moving around their circles faster than light).

If we look at a spacelike slice of constant coordinate time $t$ in this metric, we find something unexpected: it is Euclidean! The metric of such a slice is simply $dz^2 + dr^2 + r^2 d\phi^2$, which is the metric of Euclidean 3-space in cylindrical coordinates. Why, then, is it always said that "space" is not Euclidean for such observers?

The answer is that, although the observers are at rest (constant spatial coordinates $z$, $r$, $\phi$) in this chart, the spacelike slices of constant coordinate time $t$ are not simultaneous spaces for those observers. That is, the set of events all sharing a given coordinate time $t$ are not all simultaneous (by the Einstein definition of simultaneity) for the observers. In fact, the set of events which are simultaneous, by the Einstein definition of simultaneity, to a given event on a given observer's worldline do not even form a well-defined spacelike hypersurface at all. So we can't even use that obvious definition of "space" for such observers.

In fact, the "space" that is non-Euclidean for these observers is a different kind of mathematical object: it is the 3-dimensional abstract space obtained by taking the quotient space of the 4-dimensional spacetime by the set of worldlines of the Langevin observers. In other words, we take each worldline and treat it as a "point", and look at the structure of the 3-dimensional space of such "points". We find that this "space" has a non-Euclidean metric, and that this metric gives a good description of the distances the observers would measure between themselves and neighboring observers. But this "space" does not correspond to any spacelike slice that can be taken out of the 4-dimensional spacetime.

Further discussion here:

https://en.wikipedia.org/wiki/Born_coordinates

 

[loislane]

SO(3) is of course a subgroup of SO(1,3), and this is due to an important additional assumption on the structure of relativistic space-time, namely that for each inertial observer space is a Euclidean 3-dimensional affine space as in Newtonian mechanics. The new thing is that this is indeed restricted to inertial observers, because accelerated observers find a non-Euclidean space in special relativity.

There are some caveats to this statement that I think are worth mentioning.

First, there is one particular family of accelerated observers for whom "space" is still Euclidean: Rindler observers. These have proper acceleration all in the same (fixed) direction, and with magnitude that varies in just the right way to keep the radar distance between them constant. In the accelerated coordinates in which these observers are at rest, the metric is

$$
ds^2 = - a^2 x^2 dt^2 + dx^2 + dy^2 + dz^2
$$

where the observers' proper acceleration is in the $x$ direction, and the observer at $x = 1$ has proper acceleration of magnitude $a$. It is evident that spacelike slices of constant coordinate time $t$ are Euclidean for this metric.

Second, when we look at a family of accelerated observers for whom "space" is non-Euclidean, we have to be careful defining what "space" means. For example, consider the family of Langevin observers, who are all moving in circular trajectories about a common origin, with the same angular velocity $\omega$. In the accelerated coordinates in which these observers are at rest (we use cylindrical coordinates here to make things look as simple as possible), the metric is

$$
ds^2 = - \left( 1 - \omega^2 r^2 \right) dt^2 + 2 \omega r^2 dt d\phi + dz^2 + dr^2 + r^2 d\phi^2
$$

Note that this metric is only valid for $0 < r < 1 / \omega$; at larger values of $r$, there are no Langevin observers (if there were, they would be moving around their circles faster than light).

If we look at a spacelike slice of constant coordinate time $t$ in this metric, we find something unexpected: it is Euclidean! The metric of such a slice is simply $dz^2 + dr^2 + r^2 d\phi^2$, which is the metric of Euclidean 3-space in cylindrical coordinates. Why, then, is it always said that "space" is not Euclidean for such observers?

The answer is that, although the observers are at rest (constant spatial coordinates $z$, $r$, $\phi$) in this chart, the spacelike slices of constant coordinate time $t$ are not simultaneous spaces for those observers. That is, the set of events all sharing a given coordinate time $t$ are not all simultaneous (by the Einstein definition of simultaneity) for the observers. In fact, the set of events which are simultaneous, by the Einstein definition of simultaneity, to a given event on a given observer's worldline do not even form a well-defined spacelike hypersurface at all. So we can't even use that obvious definition of "space" for such observers.

In fact, the "space" that is non-Euclidean for these observers is a different kind of mathematical object: it is the 3-dimensional abstract space obtained by taking the quotient space of the 4-dimensional spacetime by the set of worldlines of the Langevin observers. In other words, we take each worldline and treat it as a "point", and look at the structure of the 3-dimensional space of such "points". We find that this "space" has a non-Euclidean metric, and that this metric gives a good description of the distances the observers would measure between themselves and neighboring observers. But this "space" does not correspond to any spacelike slice that can be taken out of the 4-dimensional spacetime.

Further discussion here:

https://en.wikipedia.org/wiki/Born_coordinates

At the risk of making a pedantic point I must say that geometrically it makes little sense to talk about the spaces either in inertial or noninertial coordinates in terms of euclidean or non-euclidean unless is done in a purely metaphorical sense. The choice of observers or coordinates can never affect the intrinsic geometry of a space or a subspace of lower dimensions.
The Lorentz group of symmetry in the tangent space indeed has SO(3) as a subgroup, but it has other three dimensional subgroups that have nothing to do with euclidean geometry(like the group of isometries of the hyperbolic plane). What's important here is that these symmetries have in the context of points in an affine space nothing to do with euclidean or non-euclidean metric geometries.

 

[PeterDonis]

The choice of observers or coordinates can never affect the intrinsic geometry of a space or a subspace of lower dimensions.

Agreed. But it can affect which subspace you pick out as worthy of interest. When we talk about the spatial geometry of inertial coordinates being Euclidean, we mean that the 3-dimensional subspaces of Minkowski spacetime that are picked out as "spacelike hypersurfaces of constant coordinate time" in inertial coordinates have Euclidean geometry (if you pick a particular point as the spatial origin, so you have a metric space).

My point in the long post you quoted was that, for at least some non-inertial coordinates (Born coordinates in the case I described), saying that "space is non-Euclidean" for observers at rest in the coordinates isn't a statement about a 3-dimensional subspace of the spacetime at all. It's a statement about a quotient space.

 

[loislane]

Agreed. But it can affect which subspace you pick out as worthy of interest. When we talk about the spatial geometry of inertial coordinates being Euclidean, we mean that the 3-dimensional subspaces of Minkowski spacetime that are picked out as "spacelike hypersurfaces of constant coordinate time" in inertial coordinates have Euclidean geometry

When you pick inertial coordinates with standard synchronization you may describe the x, y, z coordinates as defining a cartesian space that one can identify with a Euclidean space.

(if you pick a particular point as the spatial origin, so you have a metric space).

No, you have a vector space, metric spaces are not related to fixing an origin but with determining distances, indefinite bilinear forms cannot determine distances.

My point in the long post you quoted was that, for at least some non-inertial coordinates (Born coordinates in the case I described), saying that "space is non-Euclidean" for observers at rest in the coordinates isn't a statement about a 3-dimensional subspace of the spacetime at all. It's a statement about a quotient space.

SR postulates don't hold for noninertial coordinates and again the possibility of assigning Euclidean or non-euclidean spatial relations from the choice of different curvilinear coordinates is trivial, but it doesn't say much about the underlying spaces, and it is my understanding that it cannot determine anything physical either. They are just labels.

 

[vanhees71]

True, it's a very bad habit of physicists to talk about "metrics" when dealing with the fundamental forms of pseudo-Riemannian (or pseudo-Euclidean) manifolds, but that's the jargon used. Usually people look a bit confused, when you start talking about pseudo-metrics or something like this, but strictly speaking it would be good to change this habit, particularly for beginners in the field!