Second postulate of SR quiz question

[loislane]

What vanhees71 described in post #24--first use spacetime symmetries to narrow down the possibilities to either Galilean invariance (no finite invariant speed) or Lorentz invariance (with a finite invariant speed), then adopt a postulate of finite invariant speed (based on experimental evidence) to pick the second of the two--is what I mean by "the modern form". As is evident, a second postulate is needed because spacetime symmetries by themselves (the modern version of the first postulate) leave open two possibilities, not one, so you need a second postulate to choose between them.

It isn't clear to me why the first special relativity postulate doesn't already narrow the possibilities to a finite invariant speed if it is to include the laws of electrodynamics and optics embodied in Maxwell equations that already contained a finite propagation invariant speed.
Contrary to what some say I don't think Einstein's first postulate was equal to Galilei principle of relativity, the latter only referred to mechanics laws with instantaneous influence, the former is extended to the EM laws and it is galilean only in the sense that it uses galilean inertial frames but again valid only to first order while in the classical mechanics case they were considered exactly valid.

 

[PeterDonis]

It doesn't make sense (to me, anyway) to talk about a first or second or third postulate unless you are talking about a specific formulation.

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.

 

[loislane]

Einstein should be referring to the inertial frames of SR. Since Newtonian mechanics holds to first approximation in the inertial frames of SR, that was his way of setting up an operational definition of the inertial frames of SR.

This is fine but my point is that he didn't seemingly apply this operational definition of SR inertial frames to his formulation of the second postulate.

 

[PeterDonis]

It isn't clear to me why the first special relativity postulate doesn't already narrow the possibilities to a finite invariant speed if it is to include the laws of electrodynamics and optics embodied in Maxwell equations that already contained a finite propagation invariant speed.

This would be another example of ambiguity in formulation; in the formulation vanhees71 was describing, the "first postulate" could be viewed as only talking about spacetime symmetries in general, leading to two possibilities, without invoking any specific physical laws or experiments that rule out one of the two possibilities. But a different view could easily be taken, as you say, according to which "all" of the laws of physics being invariant under those symmetries really means "all", so if any of those laws are inconsistent with a given symmetry group (Galilean invariance being inconsistent with Maxwell's Equations in this case), that symmetry group is ruled out. As you and others have pointed out, the same is true of Einstein's original formulation: if the speed of light is part of the laws of physics (since it appears in Maxwell's Equations), then of course it has to be invariant in all inertial frames if the first postulate is true, and you don't need a separate second postulate to say that it is.

 

[PAllen]

I'm not sure quite how Einstein viewed the one-way speed of light. An earlier post in this thread pointed out that the one-way speed of light is dependent on a particular simultaneity convention, and such a convention should not be part of a theoretical postulate. Einstein's original formulation did adopt that simultaneity convention, but I would imagine he viewed the constancy of the one-way speed of light as being deduced from the postulate of the two-way speed being constant, plus the adoption of the Einstein simultaneity convention.

Well, two way speed invariant + isotropy -> one way speed invariant in any coordinates adapted to the isotropy. It is (IMO) the norm in physics to assume isotropy unless it leads to a problem. That is, the philosophical debates about how you prove isotropy [really, really hard, in general] are sidestepped by saying 'physicist isotropy' = does the assumption of isotropy lead to simplification. Then, simultaneity convention is secondary - if you use a convention that assumes isotropy, you find isotropy. If you use a convention that does not, it will work, producing more complex equations. We would question isotropy if the assumption of such led to increased complexity.

 

[bcrowell]

Well, two way speed invariant + isotropy -> one way speed invariant in any coordinates adapted to the isotropy. It is (IMO) the norm in physics to assume isotropy unless it leads to a problem. That is, the philosophical debates about how you prove isotropy [really, really hard, in general] are sidestepped by saying 'physicist isotropy' = does the assumption of isotropy lead to simplification. Then, simultaneity convention is secondary - if you use a convention that assumes isotropy, you find isotropy. If you use a convention that does not, it will work, producing more complex equations. We would question isotropy if the assumption of such led to increased complexity.

I think this may be putting it a little too strongly. We have test theories which include parameters that allow the existence of anisotropy. See, e.g., Clifford Will's thesis, http://thesis.library.caltech.edu/3839/ . Many high-precision tests of relativity, such as clock-comparison tests, can also be interpreted as tests of the isotropy and homogeneity of spacetime.

 

[PAllen]

I think this may be putting it a little too strongly. We have test theories which include parameters that allow the existence of anisotropy. See, e.g., Clifford Will's thesis, http://thesis.library.caltech.edu/3839/ . Many high-precision tests of relativity, such as clock-comparison tests, can also be interpreted as tests of the isotropy and homogeneity of spacetime.

I should have been more precise. Two way light speed measurement isotropy is testable [note, this does not fully prove even two way light speed 'actual' isotropy, because you can posit that different effects cancel so as to always produce an isotropic measurement, e.g. anisotropic ruler behavior + ansisotropic two way light speed can produce isotropic two way measurment]. However, for one way light speed, there are a special class of anisotropic theories that preserve two way isotropy while not having one way isotropy. In these theories, Maxwell's equations become more complex. This latter fact argues for assuming one way isotropy irrespective of simultaneity convention.

Concisely: it is easy to rule out some forms of isotropy experimentally. It is impossible to experimentally to rule out arbitrarily complex conspiratorial anisotropic theories. They are ruled out by assumption due to their excess complexity.

 

[bcrowell]

However, for one way light speed, there are a special class of anisotropic theories that preserve two way isotropy while not having one way isotropy. In these theories, Maxwell's equations become more complex. This latter fact argues for assuming one way isotropy irrespective of simultaneity convention.

Isn't such a theory just Maxwell's equations in an accelerated coordinate system? If so, then Maxwell's equations needn't be any more complex in those coordinates, if you express them tensorially.

Related: https://www.physicsforums.com/threads/differences-between-different-simultaneity-conventions.743972/

 

[atyy]

What vanhees71 described in post #24--first use spacetime symmetries to narrow down the possibilities to either Galilean invariance (no finite invariant speed) or Lorentz invariance (with a finite invariant speed), then adopt a postulate of finite invariant speed (based on experimental evidence) to pick the second of the two--is what I mean by "the modern form". As is evident, a second postulate is needed because spacetime symmetries by themselves (the modern version of the first postulate) leave open two possibilities, not one, so you need a second postulate to choose between them.

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.

I'm citing vanhees71's post as evidence for interpreting the OP to mean "Einstein's second postulate", so as to remove or at least reduce the ambiguity in the question, since the modern form does not have an accepted second postulate (there is no accepted numbering to the modern postulates).

I would still go with PAllen's post #24 and say that it is the one-way speed of light that is postulated. However, as PAllen mentions in post #40, it is also possible to go with the two-way measured speed and consider isotropy as an implicit zeroth assumption.

So do we go with one way or two way? It depends on the interpretation of quantum mechanics

Ignoring the (rest of the) text, Einstein's postulates what the speed of light is, so this supports the idea that the postulate refers to the one way speed of light. This is consistent with his later views that reality is described by laws of physics.

However, Einstein does stress the operational view in his text about procedures to measure the speed of light, so this supports the two way speed of light. This is why Einstein also can be considered a founder of Copenhagen, although he "renounced" this later.

 

[PAllen]

Isn't such a theory just Maxwell's equations in an accelerated coordinate system? If so, then Maxwell's equations needn't be any more complex in those coordinates, if you express them tensorially.

Related: https://www.physicsforums.com/threads/differences-between-different-simultaneity-conventions.743972/

That disguises things, but you still have a choice. There exist coordinates where the connection vanishes and the equations are explicity isotropic (globally, in SR). There are other coordinates with connection coefficients producing anisotropic behavior. Consider the similarity to the older coordinate based definitions of spherical symmetry - if there exists a coordinate transform that takes a general looking metric to one that is manifestly spherically symmetric, we say the spacetime is spherically symmetric. Similarly, if there exists coordinates that display manifest isotropy, you assume (not prove) isotropy.

 

[PAllen]

On the point I am discussing with Bcrowell:

One could imagine in per-relativistic mechanics, someone noting that expressed sufficiently abstractly in tensors (supposing they were invented earlier), that the laws in a rotating frame could take the same form as in an inertial frame. Then, being religious, declaring that the true state of the universe is that it has an axis through Jerusalem (or Mecca, or wherever), and that inertial frames have no special significance, because it is revealed where the center of the universe is, and everything revolves around that center. It is impossible to experimentally disprove this formulation. In my view, Edwards frames amount to the same thing.

The ability to come up with a group of frames manifesting isotropy with laws in simplest form is non-trivial. It is easy to imagine laws for which that is not true. Thus we take the universe to be isotropic as meaning that we can find such a group. It must remain an assumption, but it is strongly motivated by a web of observations. Once accepted, invariant one way speed of light and standard synchronization being preferred follows - because these are the only ways to manifest the assumed isotropy.

 

[atyy]

Question: from the operational viewpoint, is it always the two-way speed of light that is intended? In QFT we postulate Minskowski spacetime, measurement outcomes as events, and no superluminal signalling implemented by spacelike observables commuting. Is that superluminal signalling one way or two way?

The requirement for no superluminal signalling is presumably, since I don't think one runs into any difficuties with killing your grandfather before you are born with one way signalling?

But the implementation of the constraint by requiring commutation of spacelike observables is one way or two way?

 

[stevendaryl]

Question: from the operational viewpoint, is it always the two-way speed of light that is intended? In QFT we postulate Minskowski spacetime, measurement outcomes as events, and no superluminal signalling implemented by spacelike observables commuting. Is that superluminal signalling one way or two way?

Well, the prohibition against FTL signalling just means that your signal can't beat a light signal. So it doesn't matter whether you consider one-way or two-way speed, as long as you use the same criterion for your signal and for light.

 

[atyy]

Well, the prohibition against FTL signalling just means that your signal can't beat a light signal. So it doesn't matter whether you consider one-way or two-way speed, as long as you use the same criterion for your signal and for light.

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?

 

[PeterDonis]

two way speed invariant + isotropy -> one way speed invariant in any coordinates adapted to the isotropy

Yes, and those coordinates are inertial coordinates, with simultaneity defined by the Einstein simultaneity convention. So we're saying the same thing, just in different ways.

simultaneity convention is secondary - if you use a convention that assumes isotropy, you find isotropy

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).

To contrast with this, imagine a family of Rindler observers who want to establish a simultaneity convention. They can do so using a method similar to the Einstein method: they exchange light signals with clock reading information. The only change is that they have to adjust for the different clock rates of different Rindler observers; but making that adjustment, they can establish a common simultaneity convention (this amounts to a physical realization of Rindler coordinates). Once they do this, they will find that the spacelike surfaces of simultaneity thus defined are not isotropic--which is what we expect, physically, since the acceleration of the observers picks out a particular direction in space.

 

[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 ;-).