I Isotropy of the speed of light

[bhobba]

Then hopefully he or she will post said reference by Reichenbach. Personally, what I have read from Reichenbach does not seem to support the post in question. He states that the synchronization choice is a convention but in what I have read he did not claim that the existence of a frame with the standard convention is a matter of convention.

Reichenbach had his own definition of simultaneity, his ε-definition, t2 = t1 + ε(t3 − t1) from which, the standard Einstein criterion falls out as the special case ε = 1/2. He used it in his axiomatisation of SR a few years before he wrote The Philosophy of Space-Time. I have issues with it because of Noether and conservation of angular momentum (except of course for ε = 1/2). But no, as far as I can tell he did not do what you suggest. In researching this to be sure my long ago memory was not playing tricks I came across Malament’s Theorem, which purports to show conventional synchronization (slow clock transport or the Einstein convention which were shown to be equivalent by Eddington) is the only simultaneity relation definable in terms Minkowski spacetime. I have never heard of it before. Does anybody know anything about it? If not I may have to investigate it further myself.

Thanks
Bill

 

[bhobba]

From a modern symmetry-principle point of view it boils down to the idea that Poincare symmetry has to be made a local gauge symmetry.

I just realized, probably because I can be a bit slow, this is exactly the argument used in the U(1) derivation of Maxwell's equations I posted recently. We know the U(1) symmetry is global, but want to see what happens if it is only local. Interesting - GR follows from assuming inertial frames are only local, and you can apply Lovelock's theorem.

Thanks
Bill

 

[vanhees71]

The fact that Einstein did not formulate SR this way in 1905 does not mean it is not possible to formulate SR this way. Basically what I am describing is formulating SR the way we formulate GR, in terms of geometry; "special relativity" is then just the particular solution of the Einstein Field Equation that is Minkowski spacetime. Chapters 2 through 7 of MTW, for example, formulate SR this way.

Nevertheless the paper of 1905 is still ingenious today precisely for the fact that he derives the space-time geometry from physics, i.e., by gedanken experiments how to define (a particularly simple) coordinates, and this delivers the connection between real-world measurements and mathematical abstract space-time models. Also in MTW this connection is very carefully made in both SR and GR!

 

[vanhees71]

I just realized, probably because I can be a bit slow, this is exactly the argument used in the U(1) derivation of Maxwell's equations I posted recently. We know the U(1) symmetry is global, but want to see what happens if it is only local. Interesting - GR follows from assuming inertial frames are only local, and you can apply Lovelock's theorem.

Thanks
Bill

One should say it does not only follow GR but also that it may be necessary to generalize it somewhat to Einstein-Cartan theory, i.e., a manifold with a Lorentzian pseudo-metric and torsion. This is inevitable when you want to consistently describe, e.g., spin-1/2 particles in terms of the corresponding spinors (e.g., the Dirac spinors used in the Standard Model).

For the general formalism, see

R. Utiyama, Invariant theoretical interpretation of
interaction, Phys. Rev. 101, 1597 (1956),
https://doi.org/10.1103/PhysRev.101.1597.

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

or

P. Ramond, Field Theory: A Modern Primer,
Addison-Wesley, Redwood City, Calif., 2 edn. (1989).

 

[Tendex]

I am highly skeptical of this claim. Do you have a reference that makes this claim?

I must say based on your later posts that you seem to misunderstand what I was claiming in 103, but the reference it is based on would have to be Einstein's 1905 article "On the electrodynamics of moving bodies", althoug obviously the same claim is not made verbatim there. In any case if the claim remains to be not understood or considered wrong I have no problem withdrawing it.

 

[Tendex]

Easily: you write all of the laws of physics in tensor form--i.e., equations that are valid in any coordinates you choose, so you don't have to tie your formulation to any choice of coordinates. Your formulation therefore obviously satisfies the first postulate of SR (principle of relativity) without committing you to any choice of coordinates or even to claiming the existence of inertial frames or any other type of frame.

Your formulation will include a constant c in it, but for bonus points, you can choose units in which c=1 , so your formulation now obviously satisfies the second postulate of SR as well, without even having to formulate that postulate in terms of "the speed of light"; instead you formulate it as a postulate about the geometric structure of spacetime, which basically amounts to the postulate that it is possible to choose the units I've just described, in which "space" and "time" have the same units and you can compare lengths along any kinds of curves.

The only part where you seem to be addressing my claim in #103 in these paragraphs is when you say "without... claiming the existence of inertial frames", since I understand and agree with the rest. I am not sure but you seem to be treating inertial frames here just as coordinate systems, and then it is a different use from the one I am using, that has physical content in terms of rigid rulers and ideal clocks, of course this meaning of inertial frame makes already use of the second postulate that you are describing later when you say "your formulation will include a constant c" but it is ambiguous whether you are already including it in you inertial frames of the first paragraph that is referring to the first postulate. I think Dale himself made the distinction I'm referring to earlier in the thread and called 1. the inerfial frame and 2. the coordinate system.
It is important that you specify what you are meaning by "inertial frame" because if it is only a coordinate system it lacks the connection with the coordinated physics of clocks and rulers that comes when the second postulate is added to the first in the absence of contradiction(wich implies a nondegenerate metric tensor, etc). And it is certainly not what I meant by inertial frames and was implicit in my question about connecting the mathematical and physical part.

 

[vanhees71]

Can we agree on the following (formal) definition of a local inertial frame in GR?

An inertial reference frame along a time-like is defined by a non-rotating tetrad with the four-velocity $u^{\mu}$ as the temporal basis vector and parallel-transported space-like basis vectors (along a geodesic parallel transport is equivalent to Fermi-Walker transport thus leading to non-rotating tetrads).

 

[Tendex]

Your (local) here is wrong. SR, considered as a theory in its own right (as opposed to just an approximation) does not just claim these properties locally. It claims them globally. And that global claim is wrong for our actual universe. The spacetime of our actual universe, globally, is not flat Minkowski spacetime.
In a small local patch of spacetime, as I said above, yes, SR is a good enough approximation. But then you cannot make any claims about some ruler in some remote part of spacetime measuring "the same lengths" as your local calipers, or some clock in some remote part of spacetime being synchronized with your local clock.

Yes, for SR the property is global and probably my parenthetical local was unnececesary and a bit confusing there since it is local in the trivial sense that something global affects locally not in the modern sense of local gauge or local inertial frame in GR that is not dependent on the global property, sorry if that nonstandard use of local caused confusion. Also I was not pointing out the difference between SR and our universe here.

 

[Tendex]

Can we agree on the following (formal) definition of a local inertial frame in GR?

An inertial reference frame along a time-like is defined by a non-rotating tetrad with the four-velocity $u^{\mu}$ as the temporal basis vector and parallel-transported space-like basis vectors (along a geodesic parallel transport is equivalent to Fermi-Walker transport thus leading to non-rotating tetrads).

That seems fine to me but I was trying to restrict the discussion to inertial frames of SR in this thread, that's where my questioned claimed is inserted.

 

[Tendex]

For your question, you take a system of moving objects, each with their own clock, radar, and accelerometers (6 degree of freedom type). You measure the object’s proper time, proper acceleration, and relative distance and speed (radar) to each of the other objects. Then you solve the resulting system of equations to determine if there exists an inertial frame that can describe the object’s motion. It may very well turn out that there is no solution to that system

This answer gives me a hint that my point in 103 didn't contradict this, specifically didn't claim that inertial frames are the only existing ones, or that we cannot empirically tell noninertial frames from inertial.

 

[robphy]

Nevertheless the paper of 1905 is still ingenious today precisely for the fact that he derives the space-time geometry from physics, i.e., by gedanken experiments how to define (a particularly simple) coordinates, and this delivers the connection between real-world measurements and mathematical abstract space-time models. Also in MTW this connection is very carefully made in both SR and GR!

Um... I think the often-overlooked Minkowski deserves some credit for the space-time viewpoint.
While the essence of spacetime is there in the 1905 paper, Einstein didn't see it or appreciate it.
(As it often said, hindsight is 20/20.)
As you probably know, Einstein was not very receptive to Minkowski's 1907 spacetime reformulation of relativity. Too bad that Minkowski died in 1909 before General Relativity was fully developed in 1915.
(bolding mine)

https://arxiv.org/abs/1210.6929
https://arxiv.org/ftp/arxiv/papers/1210/1210.6929.pdf#page=2
Max Born, Albert Einstein and Hermann Minkowski's Space-Time Formalism of Special Relativity
Galina Weinstein

Arnold Sommerfeld's recollections of what Einstein said can further indicate his attitude towards
mathematics before 1912: "Strangely enough no personal contacts resulted between his teacher of
mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski built up the special
theory of relativity into his 'world-geometry', Einstein said on one occasion: 'Since the mathematicians
have invaded the theory of relativity, I do not understand it myself any more'. But soon thereafter, at
the time of the conception of the general theory of relativity, he readily acknowledged the
indispensability of the four-dimensional scheme of Minkowski". Sommerfeld, Arnold, "To Albert
Einstein's Seventieth Birthday", in Einstein, 1949, in Schilpp, 1949, pp. 99-105; p. 102.

Abraham Pais also reported that before 1912 Einstein told V. Bergmann that he regarded the
transcription of his theory into tensor form as "überflüssige Gelehrsamkeit" (superfluous learnedness).
Pais, Abraham, Subtle is the Lord. The Science and Life of Albert Einstein, 1982, Oxford: Oxford University Press, p. 152.

 

[vanhees71]

Sure, Minkowski's paper is also highly recommended. I guess there are English translations of it around, but there you learn the mathematicians' point of view rather than how it is constructed via physics gedanken experiments.

An amusing anecdote is that Einstein had the opportunity to listen to Minkowski's math lectures when he was a student at Zürich, but he didn't. Rather, as for most of the lectures he should have attended, he relied on Marcel Grossmann's notes. Then in 1907/08 when Minkowski's famous paper (delivered as a talk at a meeting of the assoziation of natural scientists and physicians), he was famously saying that now he doesn't understand the theory anymore himself after the mathematicians have reformulated it. Later he realized that this reluctance against math was a big mistake, because finally he needed it to get the right formulation of general relativity, where he worked together with his old student friend Marcel Grossmann again ;-)).

 

[PeterDonis]

I am not sure but you seem to be treating inertial frames here just as coordinate systems, and then it is a different use from the one I am using, that has physical content in terms of rigid rulers and ideal clocks

The existence of rigid rulers and ideal clocks, by itself, is not enough to establish the existence of global inertial frames. For the latter, you also need spacetime to be flat. But you do not need spacetime to be flat in order to formulate physics as I described. You can formulate physics as GR does, in general covariant form, and then treat the flatness, or lack thereof, of spacetime, and hence the existence or lack thereof of global inertial frames, as something to be determined by experiment. You do not need to assume it at any point.

 

[PeterDonis]

An inertial reference frame along a time-like

What you are describing is not a local inertial frame since it is not restricted to a small patch of spacetime centered on a particular event.

What you are describing is Fermi normal coordinates centered on a timelike geodesic.

 

[Dale]

I must say based on your later posts that you seem to misunderstand what I was claiming in 103, but the reference it is based on would have to be Einstein's 1905 article "On the electrodynamics of moving bodies", althoug obviously the same claim is not made verbatim there. In any case if the claim remains to be not understood or considered wrong I have no problem withdrawing it.

That is also one of the reasons for asking for references. Sometimes you can have one thing in mind and I understand another, and providing a reference will explain what you mean.

I cannot see any support for your claims in 103 from Einstein's OEMB at all, so I must have definitely misunderstood what you intended. Nothing in that paper claims that the existence of inertial frames is a convention. The closest is his clear introduction of a convention for synchronization in an inertial frame. Is that what you meant?

 

[Dale]

This answer gives me a hint that my point in 103 didn't contradict this, specifically didn't claim that inertial frames are the only existing ones, or that we cannot empirically tell noninertial frames from inertial.

I specifically understood from your 103 that you were explicitly claiming the bolded part. That is the specific claim that I understood from your 103 and objected to.

Again, when references are requested it is best to provide those quickly, it can help resolve misunderstandings.

 

[Tendex]

The existence of rigid rulers and ideal clocks, by itself, is not enough to establish the existence of global inertial frames. For the latter, you also need spacetime to be flat

Certainly, I always assumed explicitly Minkowski spacetime.

But you do not need spacetime to be flat in order to formulate physics as I described. You can formulate physics as GR does, in general covariant form, and then treat the flatness, or lack thereof, of spacetime, and hence the existence or lack thereof of global inertial frames, as something to be determined by experiment. You do not need to assume it at any point.

The discussion at least from my part was restricted to isotropy and inertial frames in SR. General covariance and extending SR to GR are very interesting topics but were not included in my claims.

 

[Tendex]

That is also one of the reasons for asking for references. Sometimes you can have one thing in mind and I understand another, and providing a reference will explain what you mean.

I cannot see any support for your claims in 103 from Einstein's OEMB at all, so I must have definitely misunderstood what you intended. Nothing in that paper claims that the existence of inertial frames is a convention. The closest is his clear introduction of a convention for synchronization in an inertial frame. Is that what you meant?

In the article the inertial frames based on rigid rods of Newton are mentioned and then extended to SR inertial frames using the second postulate through conventional synchronization. Now these extended inertial frames are an infinite family of frames in fact and picking anyone of this family is conventional which gives the relativity of simultaneity. Is this way of wording what I mean by saying that the inertial frames of SR are conventional more undertandable?

 

[Tendex]

I specifically understood from your 103 that you were explicitly claiming the bolded part. That is the specific claim that I understood from your 103 and objected to.

Again, when references are requested it is best to provide those quickly, it can help resolve misunderstandings.

Sorry, I don't have all the free time I wished to respond.

 

[PeterDonis]

I always assumed explicitly Minkowski spacetime.

Yes, but then it's precisely that assumption--that the spacetime geometry is Minkowski spacetime--that leads to the existence of global inertial frames. You don't have to assume them as a separate assumption. They're automatically already there once you've assumed flat Minkowski spacetime.

 

[Tendex]

Yes, but then it's precisely that assumption--that the spacetime geometry is Minkowski spacetime--that leads to the existence of global inertial frames. You don't have to assume them as a separate assumption. They're automatically already there once you've assumed flat Minkowski spacetime.

So in this sense they are conventional in that
it is the only geometry capable of having such global inertial frames coordinating clocks and rulers that incorporates the two postulates of Einstein without contradiction.

 

[PeterDonis]

in this sense they are conventional in that
it is the only geometry capable of having such global inertial frames coordinating clocks and rulers that incorporates the two postulates of Einstein without contradiction.

I'm not sure how that makes them "conventional", since geometric properties are not conventions, but I agree with the geometric property you state here.

 

[Sagittarius A-Star]

In the article the inertial frames based on rigid rods of Newton are mentioned and then extended to SR inertial frames using the second postulate through conventional synchronization. Now these extended inertial frames are an infinite family of frames in fact and picking anyone of this family is conventional which gives the relativity of simultaneity. Is this way of wording what I mean by saying that the inertial frames of SR are conventional more undertandable?

No. The synchronization convention does not determine, if a certain reference frame is inertial (=a frame with no proper acceleration), because of:

Changing coordinates doesn't change the physics, that's what changing simultaneity convention implies there, the transformation is between inertial frames.

 

[Tendex]

No. The synchronization convention does not determine, if a certain reference frame is inertial

That's not what I implied, conventional synchronization helps define the concept of time in SR inertial frames leading to lorentz transformations with relativity of simultaneity.

 

[Sagittarius A-Star]

That's not what I implied, conventional synchronization helps define the concept of time in SR inertial frames leading to lorentz transformations with relativity of simultaneity.

That makes it understandable for me, what you want to imply, and that seems to be a correct statement. But I understand "inertial frames of SR are conventional" as a different and wrong statement, for the before mentioned reason.

 

[Tendex]

That makes it understandable for me, what you want to imply, and that seems to be a correct statement. But I understand "inertial frames of SR are conventional" as a different and wrong statement, for the before mentioned reason.

You are not the only one so I'll have to revise my terminology.

 

[Sagittarius A-Star]

That calculation has nothing whatever to do with your claims that your anisotropic frame is "inertial".

The primed frame under discussion is:
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$
Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

The metric of the inertial, unprimed frame is:
$ds^2 = d(ct)^2 - dx^2- dy^2- dz^2$
=>
$ds^2 = d(ct'-kx')^2 - dx'^2- dy'^2- dz'^2$

Because the spacetime interval is invariant, the metric of the primed frame is:
$ds'^2 = d(ct'-kx')^2 - dx'^2- dy'^2- dz'^2$
=>
The time dilation formula for a clock, moving in x'-direction, in the primed frame is:
$cd \tau = \sqrt{(cdt'(1- \frac{kv}{c}))^2- dx'^2}$
$$d \tau = dt'\sqrt{(1- \frac{kv}{c})^2- \frac{v^2}{c^2}}$$
For a clock at rest ($v=0$) is valid, independent of the x'-coordinate of the clock:
$$d\tau = dt'$$
=> There is no pseudo-gravitational time-dilation => The primed frame is inertial.
========================================================================

P.S.
For comparison the time dilation formula in a Rindler frame. This depends on x and therefore there is a pseudo-gravitational time-dilation:

Yes. Gron derives in his book the following equation (4.50):
$$d\tau = dt\sqrt {(1+ \frac{g\,x} {c^2})^2 - \frac{v^2}{c^2}}$$

 

[PeterDonis]

There is no pseudo-gravitational time-dilation

Yes, agreed.

=> The primed frame is inertial

This depends on the definition of "inertial". I think I am not the only one posting in this thread who believes that "inertial" means more than just "no pseudo-gravitational time dilation". Certainly the primed frame being described is not an "inertial frame" as that term is standardly used in SR textbooks.

 

[Sagittarius A-Star]

Certainly the primed frame being described is not an "inertial frame" as that term is standardly used in SR textbooks.

Yes, but a calculation with non-isotropic one-way speed of light is not SR, because SR, LT and so on relies on the Einstein convention of simultaneity. But you can for example define other theories than SR, that make the same predictions for experiments:

By giving the effects of time dilation and length contraction the exact relativistic value, this test theory is experimentally equivalent to special relativity, independent of the chosen synchronization.

Source:
https://en.wikipedia.org/wiki/Test_theories_of_special_relativity#Theory

 

[PeterDonis]

a calculation with non-isotropic one-way speed of light is not SR

Nonsense. "SR" means "flat spacetime". You can do calculations in flat spacetime with a non-isotropic one-way speed of light. You can do calculations in flat spacetime with any coordinate choice you want.

Also, you appear to have reversed your position. Before you were claiming that the non-isotropic coordinates were an inertial frame, which would indicate that they are "SR" even by your (wrong) definition that "SR" means "inertial frame". But now you are claiming that a non-isotropic one-way speed of light is "not SR".

I think you have not thought this issue through very carefully.

Source

I think you have been here long enough to know that Wikipedia is not a valid reference.