Second postulate of SR quiz question

[PAllen]

Not if the simultaneity convention can be realized by a physical procedure, as Einstein simultaneity can. That convention does not assume isotropy; the fact that, when an inertial observer adopts this convention, he finds that his coordinates have spatial isotropy, is a physical fact about his state of motion and the physical procedure he uses to realize the simultaneity convention (i.e., the procedure used for Einstein clock synchronization).

Not sure what you are getting at - Einstein simultaneity convention explicitly assumes isotropy of one way speed of light (or defines it that way). [The factor of 1/2, and that you do the same in all directions, are explicit assumptions of isotropy. They make no sense without such an assumption.] On the other hand, if you assume anisotropy of the Edward's frame variety, you use a derive from this assumption that a different synchronization should be used, and you then 'confirm' that both one way light speed and the laws of mechanics are anisotropic - and consistent with experiment if this regime is carried out properly.

What I think is non-trivial, and makes it pedantic to really argue that there is nothing to isotropy, is that if you define coordinates to achieve isotropy for anyone phenomenon, you find it holds for all others. But the pedant can certainly say you have not ruled out anisotropy of just the right kind, coordinated in just the right way for all phenomena.

My point remains that I think the physical definition of isotropy should be that all laws display isotropy in the same group of frames/coordinates. If we find this, we say the universe is isotropic - by definition. Then we can say SR combined with EM and mechanics is isotropic - in the sense we have defined isotropic.

 

[PeterDonis]

Einstein simultaneity convention explicitly assumes isotropy of one way speed of light (or defines it that way).

This may be a matter of language more than anything else, but to me, if you specify a particular procedure for determining which events are simultaneous, you aren't assuming anything about isotropy; if that procedure leads to isotropy, that's something that you've discovered about the procedure, not something you put into it to start with.

if you assume anisotropy of the Edward's frame variety, you use a derive from this assumption that a different synchronization should be used

But you could also simply adopt the different synchronization method, and then discover that it leads to anisotropy. In other words, I'm viewing different synchronization procedures as procedures, not assumptions. You adopt the procedure, and then you discover what kind of symmetry (or asymmetry) it leads to.

 

[PAllen]

This may be a matter of language more than anything else, but to me, if you specify a particular procedure for determining which events are simultaneous, you aren't assuming anything about isotropy; if that procedure leads to isotropy, that's something that you've discovered about the procedure, not something you put into it to start with.
But you could also simply adopt the different synchronization method, and then discover that it leads to anisotropy. In other words, I'm viewing different synchronization procedures as procedures, not assumptions. You adopt the procedure, and then you discover what kind of symmetry (or asymmetry) it leads to.

I think it is a question of direction of inference, which is a choice. You can assume isotropy/anisotropy, derive a procedure implied by this, and find that isotropy/anisotropy. Or you can assume a procedure, for whatever reason and find whether it leads to isotropy/anisotropy. To my mind, there is little doubt, based on the history of classical mechanics and EM, that Einstein chose his synchronization as the one implied by assuming isotropy of one way light speed.

Again, the non-trivial feature is that defining coordinates to achieve isotropy of one phenomenon, also leads to isotropy of unrelated phenomena (e.g. mechanics). Further, Edward's frames, which preserver two way light speed isotropy, lead to anisotropic mechanics and EM. That is, even assuming anisotropy, you have to assume just the right form, and the description of all phenomena are affected in tandem.

 

[vanhees71]

In general I would tend to agree, but the OP has already said that ambiguity is inevitable because there is no one canonical version of the formulation of SR, so basically we're each answering our own version of the question.

Well, it's just a great example, why I usually abhorr such questionaires and polls. They do this even as a "scientific method" in the non-hard sciences like sociology to get opinions of "representative ensembles" of people. I usually ignore such nonsense, because it's not scientific at all, because you already filter the cohort by only having people who respond to such internet polls. When it comes to the "prediction" of, e.g., results of political elections, overwhelming evidence tells me that this way of getting representative opinions is flawed. I don't know any example, where the predicted outcome of the election was accurate. Often it's not even accurate in just a qualitative sense. Also the media coverage of polls usually omits to quote the uncertainties (at least statistical ones should be mandatory, but also some educated guess of the systematic ones should be considered).

Well, that's off-topic, but anyway...

 

[vanhees71]

I think it is a question of direction of inference, which is a choice. You can assume isotropy/anisotropy, derive a procedure implied by this, and find that isotropy/anisotropy. Or you can assume a procedure, for whatever reason and find whether it leads to isotropy/anisotropy. To my mind, there is little doubt, based on the history of classical mechanics and EM, that Einstein chose his synchronization as the one implied by assuming isotropy of one way light speed.

Again, the non-trivial feature is that defining coordinates to achieve isotropy of one phenomenon, also leads to isotropy of unrelated phenomena (e.g. mechanics). Further, Edward's frames, which preserver two way light speed isotropy, lead to anisotropic mechanics and EM. That is, even assuming anisotropy, you have to assume just the right form, and the description of all phenomena are affected in tandem.

I don't know what Edward's frame is. It is however clear that if you use Minkowski space the physics doesn't change with choosing a different frame of reference. Choosing a non-inertial frame usually leads to non-Euclidean spatial "time slices", i.e., the 3D space-like hypersurfaces of simultaneity wrt. the chosen (usually local) frame (or a congruence of timelike curves in a region os spacetime). That's all. The question, whether it's the right spacetime model is unaffected by this.

Also it's clear that there are many ways to look at special relativity. Einstein's original way is, in my opinion, a masterpiece in several aspects, because it uses the most simple assumptions with the "two postulates" thinkable. Although starting from electromagnetism and the "asymmetries", which are only due to the "contemporary interpretation but not in the phenomena", he uses just the most simple aspect to add to the assumptions Galilei-Newton spacetime is based on, namely the existence of a universal speed of propagation of electromagnetic waves.

I still have to think about the mirror thing. It's still not clear to me, whether the gedanken experiment with a mirror to measure a distance with light signals (i.e., wave packets using some kind of group velocity) is accurate. I doubt it (at least for real mirrors with a finite conductivity). I guess, I'll have to do some "numerical experiments" for that, although the setup is very easy for the most simple case (em. plane-wave packet perpendicular on a conducting half-space) in terms of Fourier representations. The speed of light in vacuo as it appears in Maxwell's equations is the phase velocity and usually the postulate is tested as such, e.g., with Michelson-Morley experiments and variations thereof. The most recent one is amazing in its accuracy (it's open access, so it should be downloadable for everybody, but I give the arXiv link anyway):

M. Nagel et al, Direct terrestrial test of Lorentz symmetry in electrodynamics to $10^{−18}$, Nature Communications 6, 8174 (2015)
http://dx.doi.org/doi:10.1038/ncomms9174
http://arxiv.org/abs/1412.6954

 

[stevendaryl]

Edited:

I suppose I should have asked:

(1) Let's define the speed of light to be just the conversion factor between space and time, ie. we assume Minkowski spacetime and measurement outcomes as spacetime events.

(2a) Do we run into any paradoxes if we allow the one-way signal speed to be greater than the conversion factor between space and time, or do those only arise if our two-way signal speed is greater than the conversion factor between space and time?

(2b) Does the requirement that spacelike observables commute impose a restriction on the one-way signal speed or the two-way signal speed?

I think the only "paradoxes" arise from closed, time-like loops. Because FTL in one frame is back-in-time in another, FTL normally leads to CTL, but that isn't necessarily true if the FTL is only available at specific spots, and in specific directions. So no, I don't think that FTL by itself is paradoxical.

 

[harrylin]

Exactly, that's why I said in the last part of the quote that any internal contradiction in this respect cannot be addressed from the theory as not only the semantic ambivalence of the postulates but mainly the relativity of simultaneity act as a safeguard against demonstration of internal inconsistency. The downside of the convention thing is that it is more of a philosophical stance than math or physics.

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.

But then why did Einstein make the correction(it was added after the first publication of the paper) that he was referring to inertial frames where the Newtonian mechanics equations hold good to the first approximation only? The inertial frames of classical mechanics must have held exactly in the Newtonian theory, don't they? So they must be slightly different within SR, as used in postulating Einstein relativity principle.

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).

The second postulate is just a specific example of how not only the laws of mechanics are included in the first postulate but also those of optics and electrodynamics, but if the inertial frames are now taken as valid just to the first approximation, it seems odd that the constancy of c in the second postulate doesn't refer just to v/c in the relative motion.

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

 

[PAllen]

I don't know what Edward's frame is. It is however clear that if you use Minkowski space the physics doesn't change with choosing a different frame of reference. Choosing a non-inertial frame usually leads to non-Euclidean spatial "time slices", i.e., the 3D space-like hypersurfaces of simultaneity wrt. the chosen (usually local) frame (or a congruence of timelike curves in a region os spacetime). That's all. The question, whether it's the right spacetime model is unaffected by this.

Also it's clear that there are many ways to look at special relativity. Einstein's original way is, in my opinion, a masterpiece in several aspects, because it uses the most simple assumptions with the "two postulates" thinkable. Although starting from electromagnetism and the "asymmetries", which are only due to the "contemporary interpretation but not in the phenomena", he uses just the most simple aspect to add to the assumptions Galilei-Newton spacetime is based on, namely the existence of a universal speed of propagation of electromagnetic waves.

Edwards frames is a term sometimes used for schemes like those described here, of which Edwards was an early writer:

https://en.wikipedia.org/wiki/One-w...ansformations_with_anisotropic_one-way_speeds

 

[vanhees71]

I don't know, about what you debate here. The two postulates in Einstein's famous work of 1905 read (translation mine):

(1) The laws, according to which the states of physical systems change with time, are independent from to which two reference frames that are in uniform translationary movement relative to each other these state changes are related.

(2) Each light ray moves in the "resting" coordinate system with a certain speed $V$, independently from whether it is emitted from a source at rest or in motion.

Before Einstein had defined the synchronization of clocks via the one-way speed of light. I don't see that he refers anywhere to the approximate validity of the non-relativistic spacetime model.

 

[vanhees71]

Edwards frames is a term sometimes used for schemes like those described here, of which Edwards was an early writer:

https://en.wikipedia.org/wiki/One-w...ansformations_with_anisotropic_one-way_speeds

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.

 

[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!

 

[vanhees71]

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

Thanks for the clarification, but this statement is a bit misleading. You cannot simply set $\mathrm{d} t$ to 0 to infer what the observer considers the geometry of his "space" (time slice). The local geometry is rather defined by a metric of a 3D submanifold, where ("infinitesimal") distances are defined via the two-way speed of light (this is most clearly explained in Landau-Lifshitz vol. 2).

So the rotating observer sends a light signal towards an infinitesimal distant point, where it is reflected and measures the time his signal needs to come back to him. This time determines the distance (modulo a factor $c$, which I set to 1). The corresponding times for the light signal to travel forth and back is determined by the null-geodesics condition for the light ray:
$$g_{\mu \nu} \mathrm{d} q^{\mu} \mathrm{d} q^{\nu}=0.$$
Split in temporal and spatial components you get (setting $q^0=t$)
$$g_{00} \mathrm{d} t^2+2g_{0i} \mathrm{d} t \mathrm{d} q^i + g_{ij} \mathrm{d} q^i \mathrm{d} q^j=0.$$
Latin indices run from $1$ to $3$ (i.e., sum over the spatial coordinates).

This has two solutions for $\mathrm{d} t$, and then you define the spatial distance as
$$\mathrm{d} \ell=\frac{\mathrm{d} t_1-\mathrm{d} t_2}{2} = \sqrt{\left (g_{ij}-\frac{g_{i0} g_{j0}}{g_{00}} \right )\mathrm{d} q^i \mathrm{d} q^j}.$$
Then you get the spatial metric as
$$\mathrm{d} \ell^2=\mathrm{d} r^2 + \frac{r^2}{1-\omega^2 r^2} \mathrm{d} \varphi^2+\mathrm{d} r^2.$$
This is the metric of a non-Euclidean 3D Riemannian space (or more strictly speaking for a region of (in this case Minkowskian) spacetime, covered by these coordinates, which are restricted by $\omega r<1$).

Of course, that's a convention, defining the geometry of the observer's 3D spacelike hypersurface, but it's the one which is (for infinitesimal distances) equivalent to the usual Einsteinian description in SR (where he, however, uses the one-way speed of light rather than the round-trip speed of light, but this is problematic for a general (non-static) metric).

 

[Demystifier]

My answer would be that I don't know, because the question refers to a particular formulation of an obsolete axiomatization, and I don't think anyone in the year 2015 should be memorizing that kind of historical trivia (what's postulate #1, what's postulate #2, etc.). I think it's unfortunate if people are still teaching their students SR using Einstein's postulates, because they reinforce various misconceptions, such as the belief that c has something to do with the speed of light, or that light plays some fundamental role in relativity. Since Einstein himself had a view of SR that, looking back from 2015, seems to have been in many ways hazy and incorrect, why would it be of interest to anyone other than historians of science to try to figure out exactly what he had in mind?

In that context I would also recommend reading the book "Einstein's mistakes":
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20
The mistakes include a mistake on a two-way clock synchronization procedure, as well as repeated mistakes in 7 (!) different derivations of $E=mc^2$.

 

[stevendaryl]

My answer would be that I don't know, because the question refers to a particular formulation of an obsolete axiomatization, and I don't think anyone in the year 2015 should be memorizing that kind of historical trivia (what's postulate #1, what's postulate #2, etc.). I think it's unfortunate if people are still teaching their students SR using Einstein's postulates, because they reinforce various misconceptions, such as the belief that c has something to do with the speed of light, or that light plays some fundamental role in relativity. Since Einstein himself had a view of SR that, looking back from 2015, seems to have been in many ways hazy and incorrect, why would it be of interest to anyone other than historians of science to try to figure out exactly what he had in mind?

I think that in learning science, students are simultaneously learning two different things:

1. What is the current, best theory of gravity, relativity, electromagnetism, whatever.
2. How do people go about formulating and testing new theories.

The historical way that a theory developed is not relevant to the first, but it is certainly relevant to the second.

 

[vanhees71]

In that context I would also recommend reading the book "Einstein's mistakes":
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20
The mistakes include a mistake on a two-way clock synchronization procedure, as well as repeated mistakes in 7 (!) different derivations of $E=mc^2$.

Here's a paper (perhaps a short version of the book)?
http://arxiv.org/abs/0805.1400

 

[PeterDonis]

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.

The metric of a single spacelike hypersurface, with a given point chosen as the spatial origin, is positive definite. That's the metric I'm talking about, not the metric of Minkowski spacetime as a whole.

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.

Strictly speaking, yes, but if there are particular physical measurements that happen to match up with the coordinates in a particular way, then sloppy physicists will tend to talk about the coordinates as though they were physical observables instead of labels.

 

[PeterDonis]

The local geometry is rather defined by a metric of a 3D submanifold

But for the case of Langevin observers, there is no single submanifold that is "shared" by all the observers. Each observer has a different one, and the same observer has different ones at different times (and they don't even define a consistent coordinate chart, since the same spacetime event can be contained in multiple such submanifolds, corresponding to different times, for the same observer). The 3D manifold you end up deriving a non-Euclidean metric for is the quotient space I spoke of; it is not a 3D submanifold of 4D Minkowski spacetime.

 

[vanhees71]

Why is this a quotient space? I thought, that's how you generally define the local geometry of any observer in both flat (Minkowski space) and General Relativity. It's analogous to what you do for inertial observers in Minkowski space, but restricted to local (infinitesimal) neighborhoods of any point in the domain of map defined by the coordinates. That's the distance measured by light signals bouncing back and force from a mirror located at the point of interest (sometimes called "radar distance"). I thought, that's a very useful concept when setting up a (1+3)D formalism in general spacetimes. I found the discussion of electrodynamics in curved spacetime in Landau-Lifshitz using this convention very illuminating. I don't see that there is something wrong with this mathematically, it's just a rewriting of the covariant Maxwell equations in a curved (background) spacetime in a not manifestly co-variant way in the sense of a (1+3)D formalism, as one does also in usual Minkowski space in inertial frames to solve more practical problems in E&M. Is there something wrong with it physically (or even mathematically)?

EDIT: Or do you refer to the problem to define distances of finitely separated point in the case of a non-stationary metric, i.e., where the $g_{\mu \nu}$ depend on time, and you cannot define a distance measure for them?

 

[PeterDonis]

Why is this a quotient space? I thought, that's how you generally define the local geometry of any observer in both flat (Minkowski space) and General Relativity.

If we are just using your method to construct local coordinates around the worldline of a single Langevin observer, then yes, each "slice" of constant coordinate time is a 3D submanifold (with the non-Euclidean metric you give) of the 4D manifold covered by the coordinates. But that 4D manifold is not all of Minkowski spacetime; it's just a narrow "world tube" around the chosen worldline.

But if we try to view the non-Euclidean metric you give as the metric of a global "space" that includes all of the Langevin observers--i.e., not limited to a narrow world tube around one observer's worldline--then that "space" is a quotient space; it does not correspond to any 3D submanifold of the 4D manifold that includes the worldlines of all the Langevin observers. (Even that 4D manifold is not, strictly speaking, all of Minkowski spacetime, since it only covers the region of finite radius around the origin in which there are such observers. But that's a much larger region than the narrow world tube of a single observer's worldline.)

 

[loislane]

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.

I was discussing this assumption at the beginning of the thread. So you mean that the footnote that Einstein added to his first sentence in the kinematical part of his seminal 1905 paper was wrong? IOW that in SR inertial coordinates Newtonian mechanics holds good fully, not just to the first approximation?

 

[stevendaryl]

I was discussing this assumption at the beginning of the thread. So you mean that the footnote that Einstein added to his first sentence in the kinematical part of his seminal 1905 paper was wrong? IOW that in SR inertial coordinates Newtonian mechanics holds good fully, not just to the first approximation?

I don't see the connection between Einstein's comment and whether spacetime can be described as time + Euclidean space.

Obviously, Newton's laws don't hold exactly, because SR gives relativistic corrections. I interpret Einstein's words to mean that he's talking about a coordinate system such that Newton's laws hold, in the limit where all velocities are slow compared to the speed of light.