Isotropy of the speed of light

[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.

 

[PeterDonis]

SR, LT and so on relies on the Einstein convention of simultaneity

The Lorentz transformation does, because the Lorentz transformation is defined as transforming between standard inertial frames. But "SR" does not, because, as I have said, you can do calculations in flat spacetime in any coordinates you like; there is no requirement that you have to do them in a standard inertial frame. Of course the transformations between coordinates that aren't standard inertial frames won't be Lorentz transformations, but so what? The physics is still the same.

 

[Sagittarius A-Star]

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".

What else does it mean?

 

[Tendex]

Back to the isotropy of light, but now in GR's generalization, the postulate of light speed constancy is kept only locally, the metric tensor is not Minkowskian but it's still Lorentzian so it preserves locally the relation between what clocks and rulers measure in momentarily at rest local inertial frames compatible with general covariance of all general coordinate transformations as the relevant invariance.

 

[PeterDonis]

What else does it mean?

"Inertial frame" in every SR textbook I have read means a standard inertial frame, in which the one way speed of light is isotropic. Do you have a reference that uses "inertial frame" to mean an anisotropic frame?

 

[Sagittarius A-Star]

"Inertial frame" in every SR textbook I have read means a standard inertial frame, in which the one way speed of light is isotropic.

Do you have a specific example in an SR textbook?

Do you have a reference that uses "inertial frame" to mean an anisotropic frame?

Yes. Rindler differentiates between an inertial frame and a (standard) inertial coordinate system:

The basic principle of clock synchronization is to ensure that the coordinate description of physics is as symmetric as the physics itself. For example, bullets shot off by the same gun at any point and in any direction should always have the same coordinate velocity dr/dt .
...
We should, strictly speaking, differentiate between an inertial frame and an inertial coordinate system, although in sloppy practice one usually calls both IFs. An inertial frame is simply an infinite set of point particles sitting still in space relative to each other. For stability they could be connected by a lattice of rigid rods, but free-floating particles are preferable, since keeping constant distances from each other is also a criterion of the non-rotation of the frame. A standard inertial coordinate system is any set of Cartesian x,y,z axes laid over such an inertial frame, plus synchronized clocks sitting on all the particles, as described above. Standard coordinates always use identical units, say centimeters and seconds.

Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

 

[vanhees71]

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.

Yes, and taking one at a certain point on the geodesic defines a local inertial frame.

 

[PeterDonis]

Do you have a specific example in an SR textbook?

Taylor & Wheeler, Spacetime Physics, Chapter 2 (particularly section 2.6).

Rindler differentiates between an inertial frame and a (standard) inertial coordinate system

Rindler's definition of "inertial frame" is still not your anisotropic one. His definition of "inertial frame" is just the particles, with no clocks at all.

 

[vanhees71]

But as he says, there are also (gedanken) clocks, at rest relative to each other, at each space-time point. Then you synchronize them according to Einstein's description in his paper of 1905. BTW if you have established that there are more than 3 points staying at rest relative to each other I think you already have established an inertial reference frame (at least in SR), cf. Lange's famous work within Newtonian physics, but that should work also within SR (see Laue's famous textbook on relativity).

 

[PeterDonis]

as he says, there are also (gedanken) clocks

As I'm reading what was quoted from Rindler, he treats the clocks as part of a "standard inertial coordinate system", but not as part of an "inertial frame". In other words, he is not using the same definition of the term "inertial frame" that you prefer.

 

[PeterDonis]

if you have established that there are more than 3 points staying at rest relative to each other I think you already have established an inertial reference frame (at least in SR)

I assume that by "points" you mean "freely falling objects that can be idealized as point particles". I agree that more than 3 of these remaining at rest relative to each other can establish a (local) "inertial reference frame" by Rindler's definition, since his definition just includes the particles. But your definition of "inertial reference frame" requires clocks and rulers; where are the clocks and rulers if all we have is more than 3 point particles staying at rest relative to each other? We don't even know how far apart they are and we don't have any clocks.

 

[vanhees71]

If we have more than three particles at rest to each other, and I forgot to say are not all in one plane, we can establish a spatial coordinate system and constructing three spatial tetrads forming a Cartesian basis. Then we can put clocks at a corresponding grid and synchronize them according to the standard procedure a la Einstein 1905. There you have your clocks and rulers. As you say yourself, of course you are free to use any other coordinates and use the usual rules of tensor calculus.

I still don't see, where my point of view is in any way contradicting standard definitions!

 

[Sagittarius A-Star]

Taylor & Wheeler, Spacetime Physics, Chapter 2 (particularly section 2.6).

I can find a definition of "inertial frame" in the T&W summary in section 2.10:

The free-float frame (also called the inertial frame and the Lorentz frame) provides a setting in which to carry out experiments without the presence of so-called "gravitational forces." In such a frame, a particle released from rest remains at rest and a particle in motion continues that motion without change in speed or in direction (Section 2.2), as Newton declared in his First Law of Motion.

In the above primed coordinate system, all the mentioned T&W requirements are fulfilled:

• The "without gravitational forces" I have shown in posting #147.
• The "a particle released from rest remains at rest" follows from $x'=x$, $y'=y$, $z'=z$.
• The "a particle in motion continues that motion without change in speed or in direction" follows from the following:

From $x'=x$ and x'-derivation of $t' = t + \frac{kx}{c}$ follows: $\frac{1}{U'_x} = \frac{1}{U_x} + \frac{k}{c}$

If the x-component of the velocity, $U_x$, is constant in the inertial coordinate system $(x, y, z, t)$, then $U'_x$ must be also constant in the coordinate system $(x', y', z', t')$. For the y- and z-components the same is trivial.

Rindler's definition of "inertial frame" is still not your anisotropic one. His definition of "inertial frame" is just the particles, with no clocks at all.

Yes. The primed coordinate system is laid over an inertial frame. Then the questions follows:

• Shall the primed coordinate system been called "non-standard" (and why)?
• Shall the primed coordinate system been called "non-inertial" (and why)?

 

[PeterDonis]

I can find a definition of "inertial frame" in the T&W summary in section 2.10

Which is not the complete definition. Look at section 2.6.

In the above primed coordinate system, all the mentioned T&W requirements are fulfilled

Not the requirements given in section 2.6. Those requirements explicitly include Einstein clock synchronization.

The primed coordinate system is laid over an inertial frame

The primed coordinate system is not an inertial coordinate system by Rindler's definition. His definition of an inertial coordinate system includes Einstein clock synchronization.

Then the questions follows:

• Shall the primed coordinate system been called "non-standard" (and why)?
• Shall the primed coordinate system been called "non-inertial" (and why)?

These questions are unanswerable because Rindler is no longer around to answer them, and he's the one who wrote the reference. "Unanswerable" does not mean you can just help yourself to the answer "no", which is what would be required to support your claims.

 

[Sagittarius A-Star]

His definition of an inertial coordinate system includes Einstein clock synchronization.

More precise: His definition of a standard inertial coordinate system includes Einstein clock synchronization.

Maybe, the unprimed coordinate system is the standard inertial coordinate system and the primed coordinate system is a non-standard inertial coordinate system (because it is laid over an inertial frame)?

 

[PeterDonis]

More precise: His definition of a standard inertial coordinate system includes Einstein clock synchronization.

That's the only kind of inertial coordinate system he defines.

Maybe, the unprimed coordinate system is the standard inertial coordinate system and the primed coordinate system is a non-standard inertial coordinate system (because it is laid over an inertial frame)?

I have already responded to this: Rindler doesn't say in the article you referenced, and he's not around any more to ask, so the question is unanswerable. So you can't use this reference to support your claim. That would require the reference to give an answer to the question you pose here, and it doesn't.

 

[vanhees71]

Can you again give the reference to Rindler? Just changing coordinates to non-Lorentzian ones don't change necessarily the frame of reference!

 

[PeterDonis]

Can you again give the reference to Rindler?

See post #155.

 

[Dale]

Is this way of wording what I mean by saying that the inertial frames of SR are conventional more undertandable?

Yes, that is helpful. Here is the distinction I am making. What we mean my the term “inertial frame” is a convention, and it includes the synchronization convention described by Einstein. So the definition of “inertial frame” is a convention.

The existence is not. Meaning after having defined, by convention, what we mean by the term it is still a matter of experiment to determine if such frames can accurately describe actual physical measurements of kinematics. That is the part that is not a convention.

 

[vanhees71]

As was worked out by Lange in the 1880ies for Newtonian physics, and this also applies to SR, an inertial reference frame can be operationally defined by having 4 non-coplanar free mass points staying at rest relative to each other. Then, by assumption, for any observer at rest wrt. these mass points space is Euclidean, and you can thus establish Cartesian coordinates or a Cartesian grid. Now you can use Einstein's clock-synchronization convention using a standard clock at (an arbitrarily chosen) origin to synchronize all clocks located at the grid points. In this way you can describe all physics (world lines of mass points in point-particle mechanics, classical fields in continuum mechanics and classical electrodynamics, and quantum fields in relativistic Q(F)T) in terms of the so defined Lorentzian (=pseudo-Cartesian) coordinates.

This is, of course, in a sense a convention, i.e., first assuming that for any inertial observer space is Euclidean and then synchronizing the clocks a la Einstein with light signals. Now you can of course make all kinds of measurements to test the predictions following from this convention, which mathematically of course also leads to the Poincare group as symmetry group of the space-time model, which is a Lorentzian (pseudo-Euclidean) 4D affine space and use it to construct all kinds of theories and you can test the predictions on the dynamics for these theories by measurements, and as is well known, the Minkowski space-time model passes all the tests made at an amazing precision.

It only has to be refined when the gravitational interaction is significant, and there you have to extent SR to GR, which boils down to making Poincare symmetry local (in the sense of a local "gauge symmetry").

 

[Dale]

having 4 non-coplanar free mass points staying at rest relative to each other. Then, by assumption, for any observer at rest wrt. these mass points space is Euclidean

This assumption cannot be simply made but must be experimentally tested. If you have four non-coplanar points and measure the distances and angles between the points then you can determine empirically if the geometry is in fact Euclidean or not.

 

[vanhees71]

That's what I tried to say: You assume a space-time model obeying the special principle of relativity (existence of inertial frames including the Euclidicity of space for any inertial observer). Then you can think about, how to experimentally test this assumption. First you have to establish somehow operationally an inertial frame. As shown in great detail by Lange at the end of the 19th century this can be done by having established that four non-coplanar free particles stay at rest relative to each other (under the assumption about the space-time model as mentioned above!). Then, using the (again assumed!) Euclidicity of space for all points at rest relative to the four "reference particles" you can define a "coordinate grid" (which you can choose as Cartesian for simplicity but also with any coordinates you like).

Now you also need time. For that you assume that there's some standard clock (e.g., nowadays you can simply use an atomic clock as described in the SI definition of the second) for any inertial observer at rest wrt. the reference points.

Now you need some synchronization convention, and here of course the realizability differs between the Newtonian and the SRT case. In Newtonian physics you have rigid rods and you can simply synchronize all clocks with one clock "at the origin", e.g., determined by one of the four reference points, by using a mechanical pulse along the rigid rods connecting all grid points, which instantaneously sets all clocks in motion and so synchronize it with the clock at the origin. It's clear that then it doesn't make a difference whether the clocks move wrt. each other or not, which establishes the "absolute time" of Newtonian mechanics.

In SRT there is instead the postulate of the "constancy of the speed of light" (if you follow Einstein's way) or the existence of an "invariant limiting speed" (if you follow the more modern group-theoretical approach that the special principle of relativity is also consistent with Lorentz rather than Galileo transformations). Then you use light signals (or any signal which propagates at the invariant limiting speed) to synchronize all clocks at rest relative to the clock at the origin as described by Einstein.

Now you have established an inertial frame, a coordinate chart (however you like to call it), and you can build models (like Maxwell's equations for electromagnetism, point-particle or continuum mechanics, etc. etc.) to make predictions for observable facts given the above physically realized inertial frame and check whether they are consistent with the real-world observations. In this sense the above "kinematics" are indeed testable by experiment and of course must be tested to establish that they are (at least in some realm of validity) a space-time model in accordance with Nature.

 

[Dale]

Then you can think about, how to experimentally test this assumption.

Ah, I see. For the purposes of this thread I was reserving the word "assumption" for something that is a convention which can simply be assumed and there is no possible experimental test. You are using it as a synonym for "hypothesis", which is indeed an assumption. Particularly for this thread a hypothesis is a wholly different sort of thing than a convention, even though both can reasonably be called assumptions.

 

[vanhees71]

Well, you can call it a hypothesis too. It's just another word. The difficulty of this age-old problem about the "kinematics" indeed is that on the one hand you need some mathematical model to establish a quantitative description of space (locations of bodies) and time intervals, which you indeed must "assume" or "hypothesize" (however you want to call it) but on the other hand of course to make it a valid physical model/theory you have to find a way to realize the mathematical model with real-world measurement devices. At the end you consider the space-time model valid (maybe within some limited realm of validity as the Newtonian space-time model or SRT in view of the most comprehensive today known space-time model of GR) if it leads to consistent descriptions of the phenomena.

The problem is that you cannot make "assumptions" (in your meaning of the word) without already having a "hypothesis" of a space-time model but on the other hand have to test this very hypothesis doing experiments.

 

[Dale]

This is not nearly as difficult an issue as you are making it out to be. For the purposes of this thread there are two scientifically different types of assumptions that you can make. One is an assumption which has no experimental consequences, and the other is an assumption which has experimental consequences.

Since the former have no experimental consequence you are free to choose any value constrained only by mathematical consistency. No experiment can ever possibly contradict such assumptions. I will call these assumptions “conventions” for clarity.

The second type has experimental consequences, so changing them changes your experimental predictions. Thus experiments can contradict these assumptions. I will call these assumptions “hypotheses” for clarity since making and testing them are central to the scientific method.

To determine if an assumption is a convention or a hypothesis all you need to do is to make different assumptions and see if your experimental measurements would differ.

For example, we can assume that the charge on an electron is negative or positive and we can assume that the charge on a proton is negative or positive. There are four possible permutations of those assumption. Examining the predicted experimental outcomes we find that there is no difference between -e with +p and +e with -p. We also find that there is a predicted difference between -e with +p and +e with +p. So the fact that they have a different sign is a testable hypothesis while the sign itself is an untestable convention.

Similarly with the geometry. That the one way speed of light is c is a convention, it can be changed without changing the predicted experimental results. That the spatial geometry is Euclidean is a hypothesis, changing it changes the experimental results.

It is well understood that many such assumptions (both conventions and hypotheses) will be made in the course of any scientific experiment. Nevertheless, the distinction between conventions and hypotheses is clear and rather straightforward. That you must make assumptions in no way prevents you from distinguishing which assumptions are testable and which are not.

 

[Killtech]

To determine if an assumption is a convention or a hypothesis all you need to do is to make different assumptions and see if your experimental measurements would differ.

That said, isn't the isotropy of light actually a convention and not a hypothesis? I asking this because i find that trying to make a self-consistent hypothetical model which assumes otherwise always ends up in a logical contradiction when attempting to model an experiment measuring the properties of light: it will still show the isotropy contrary to the assumption. The Michelson-Morley experiment null result explanation via aether theory is an example.

The issue seems to be build into a fundamental convention of measurement: in order to be able to measure anything at all one needs to define measures and these ultimately must be defined in terms of real physical entities. Now as it happens both the measure of length and that of time is very directly defined through purely electromagnetic interactions - i.e. light. but if you measure something in terms of multitudes of itself don't expect it to be able to vary.

The SI definitions are conventions which don't just define some units but more importantly the basic physical objects of reference in multitudes of which everything is measured. But you could just as much use a different convention i.e. acoustic lengths (within the domain where it's well defined) that instead uses the speed of sound as a physical reference. Of course both measurement and the theory need to be conducted using the same convention. So if you measure the speed of sound using acoustic lengths it becomes constant while therefore the (acoustic) speed of light won't be anymore - therefore a theory using that convention has different equations of motions while maintaining consistency with measurement i.e. both combined ultimately yield just an equivalent description.

Riemann actually noticed and discussed that issue in his habilitation work. He then abstracted the conventions of measurement into the word of a metric (i.e. the metric tensor; and it that context it is useful to look up the Greek origin of the word). he uses the word magnitude as the physical reference that can be carried forward to compare other magnitudes with.