Relativity on Earth: Understanding Simultaneity & Time Dilation

[analyst5]

Ok, thanks to both of you. But it seems that this 'convention' is avoided in most cases. So regarding simultaneity from Earth's perspective, we can use the coordinate chart that we like with a reasonable definition, but no matter what simultaneity convention we use the relativistic effects and disagreements in simultaneity judgements for different observers under the same convention and for any observer at all will be extremely small for space near Earth since we are moving with very slow velocities compared to the speed of light? It doesn't matter if the system is moving inertially or non-inertially for the fact that at low speeds like on Earth the effects do exist, but aren't really big or noticeable?

 

[Nugatory]

no matter what simultaneity convention we use the relativistic effects and disagreements in simultaneity judgements for different observers under the same convention and for any observer at all will be extremely small for space near Earth since we are moving with very slow velocities compared to the speed of light? It doesn't matter if the system is moving inertially or non-inertially for the fact that at low speeds like on Earth the effects do exist, but aren't really big or noticeable?

Pretty much, yes.

There are a very few exceptions, such as the GPS system, where we do have to make relativistic corrections. The GPS system actually has its own formally defined simultaneity convention which it uses inside itself to coordnate the signals from the various GPS satellites. But whether you're driving 100 km/hr north or 100 km/hr south when you get your GPS position fix, the difference between your time and GPS time (after adding in your timezone) is completely negligible and we don't worry about it.

 

[Dale]

no matter what simultaneity convention we use the relativistic effects

When you are dealing with a non-inertial coordinate system then you need to determine the metric in that coordinate system. It will deviate from the Minkowski metric, but it will nonetheless contain all relativistic effects as well as any fictitious forces or other terms that are not due to relativity per se but simply due to the coordinate system.

 

[analyst5]

When you are dealing with a non-inertial coordinate system then you need to determine the metric in that coordinate system. It will deviate from the Minkowski metric, but it will nonetheless contain all relativistic effects as well as any fictitious forces or other terms that are not due to relativity per se but simply due to the coordinate system.

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

 

[Dale]

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

This gets to be a little tricky with the terminology.

Most of the time we use "reference frame" as a synonym for "coordinate system", but here is a place where the actual distinction makes a difference.

In GR, a reference frame is also known as a tetrad. It is a set of four orthonormal vector fields which represent local rods and clocks. Importantly, these rods and clocks are not synchronized or otherwise coordinated, so you can only make "local" measurements. The combination of orthonormality and local measurements ensures that the speed of light is still c in any reference frame, whether it is inertial or non-inertial.

On the other hand, a non-inertial coordinate system can use arbitrary synchronization conventions as well as arbitrary spatial conventions, so the speed of light can be arbitrary. You are correct that it is similar in principle to choosing the synchronization parameter, but even more general.

 

[analyst5]

This gets to be a little tricky with the terminology.

Most of the time we use "reference frame" as a synonym for "coordinate system", but here is a place where the actual distinction makes a difference.

In GR, a reference frame is also known as a tetrad. It is a set of four orthonormal vector fields which represent local rods and clocks. Importantly, these rods and clocks are not synchronized or otherwise coordinated, so you can only make "local" measurements. The combination of orthonormality and local measurements ensures that the speed of light is still c in any reference frame, whether it is inertial or non-inertial.

On the other hand, a non-inertial coordinate system can use arbitrary synchronization conventions as well as arbitrary spatial conventions, so the speed of light can be arbitrary. You are correct that it is similar in principle to choosing the synchronization parameter, but even more general.

Thanks for the good explanation. So if I'm understanding this well, the question 'what is the speed of light' on Earth (or what is the speed of light that comes to our eyes from everyday objects' does not have a definite and clear answer, and we may choose a different range of possibilites. Therefore different conventions may use a different light speed as some kind of 'basis' for defining the properties of their coordinate systems. Please correct me if I'm wrong.

 

[Dale]

Thanks for the good explanation. So if I'm understanding this well, the question 'what is the speed of light' on Earth (or what is the speed of light that comes to our eyes from everyday objects' does not have a definite and clear answer, and we may choose a different range of possibilites. Therefore different conventions may use a different light speed as some kind of 'basis' for defining the properties of their coordinate systems. Please correct me if I'm wrong.

Yes, except that it isn't a question of being on Earth or not, it is a question of using inertial frames/coordinates or not. I.e. even though it wouldn't be fixed to the surface of the earth, you can certainly use an inertial frame in your analysis, and then the question has a definite answer.

 

[xox]

But the surface of the Earth is not inertial, right? Surely there must be some convention for non-inertial observers that applies to Earth which is spinning and revolving around the Sun. And that also describes lengths and time dilations 'as viewed' from some point, or multiple points on Earth?

Correct, the Earth surface is rotating and revolving meaning that the labs used for experimentation are not inertial. Nevertheless, the effects of Earth's rotation/revolution ARE factored into the experiments used for testing SR. The test theories of SR (both Mansouri-Sexl and Standard Model Extension) take into account the exact motion of the Earth and its effects onto the canonical tests (Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell). The bottom line is that the net effect of rotation is below any experimentally detectable level. This means, that for all intents and purposes, the Earth bound labs can be considered inertial. I believe that this was the gist of your question.

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

So, it seems that I guessed right the gist of your questions. This is a VERY complicated question. In the test theories of SR, one way light speed is anisotropic and depends on a parameters, $$\alpha, \beta, \delta$$. More about it can be found in the famous papers by Mansouri and Sexl. Experimentally, though, the "anisotropy" is constricted to ever reducing limits, all tending towards zero. If you are interested, I can provide you with links to this subject.

 

[analyst5]

Correct, the Earth surface is rotating and revolving meaning that the labs used for experimentation are not inertial. Nevertheless, the effects of Earth's rotation/revolution ARE factored into the experiments used for testing SR. The test theories of SR (both Mansouri-Sexl and Standard Model Extension) take into account the exact motion of the Earth and its effects onto the canonical tests (Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell). The bottom line is that the net effect of rotation is below any experimentally detectable level. This means, that for all intents and purposes, the Earth bound labs can be considered inertial. I believe that this was the gist of your question.
So, it seems that I guessed right the gist of your questions. This is a VERY complicated question. In the test theories of SR, one way light speed is anisotropic and depends on a parameters, $$\alpha, \beta, \delta$$. More about it can be found in the famous papers by Mansouri and Sexl. Experimentally, though, the "anisotropy" is constricted to ever reducing limits, all tending towards zero. If you are interested, I can provide you with links to this subject.

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value. I think it will help me to understand the potential differentiation between the one-way speed of light in a non-inertial frame. For instance, if we use 1/4 the return trip of the light is three times slower than the first leg. So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

And does a different simultaneity convention automatically apply a change in the one-way speed of light?

edit: what really makes this confusing is the fact that the speed of light may be so slow in some circumstances, that we perceive distant past even in the regions of space close to us.

 

[xox]

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value.

The link I provided shows that the most stringent limit is given by the modern re-enactment of the Michelson Morley experiment . It shows a deviation from the two-way light speed, "c" , of the order of $10^{-12}$.

So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

This is WAY wrong. See correct results above.

 

[DrGreg]

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value.

The link I provided shows that the most stringent limit is given by the modern re-enactment of the Michelson Morley experiment . It shows a deviation from the two-way light speed, "c" , of the order of $10^{-12}$.

So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

This is WAY wrong. See correct results above.

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by$$\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}$$

 

[pervect]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by$$\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}$$

I'll go a bit further, and note that using the usual TAI or GPS time synchronization conventions (which are different than the Einstein convention!), on the surface of the Earth the one-way east-west coordinate speed of light is not equal to the one-way west-east coordinate speed.

I'm sure Doctor Greg already knows this, I'm trying to clarify things for readers like analyst5.

 

[xox]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

Not at all, I was quite clear by pointing out to the test theories of SR and to the deviation of OWLS from TWLS , as being constrained by current experiments. I am clearly talking about OWLS. In fact, the test theories of SR (both M-S and SME) employ only two types of clock synchronization: Einstein and slow clock transport. Experimental tests based on either method of synchronization severely constrain the OWLS anisotropy , as I have already pointed out.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞.

It could but, in actual experiments , it doesn't. As pointed out, the measured departure of OWLS from TWLS is of the order of $10^{-12}$.

 

[DrGreg]

Not at all, I was quite clear by pointing out to the test theories of SR and to the deviation of OWLS from TWLS , as being constrained by current experiments. I am clearly talking about OWLS.



It could but, in actual experiments , it doesn't. As pointed out, the measured departure of OWLS from TWLS is of the order of $10^{-12}$.

You've misunderstood what that Wikipedia article is saying. Using the notation of that article, the 1-way speed is determined by $e(v)$. As the article says "The value of $e(v)$ depends only on the choice of clock synchronization and cannot be determined by experiment."

The figure of $10^{-12}$ refers to the variation of the 2-way speed with direction (e.g. North-South v. East-West) which is what the Michelson-Morley experiment measured.

 

[xox]

You've misunderstood what that Wikipedia article is saying. Using the notation of that article, the 1-way speed is determined by $e(v)$. As the article says "The value of $e(v)$ depends only on the choice of clock synchronization and cannot be determined by experiment."

This means that all tests start by CHOOSING a synchronization method (either Einstein or slow clock transport), as I have already explained and PROCEED to constraining OWLS.

The figure of $10^{-12}$ refers to the variation of the 2-way speed with direction (e.g. North-South v. East-West) which is what the Michelson-Morley experiment measured.

I will have to respectfully disagree with you on this issue as well, all tests of light speed anisotropy refer to OWLS, not to TWLS.

Looking at the wiki article I can see what misled you, the sentence:

"Deviations from the two-way (round-trip) speed of light are given by:..."

$$\frac{c}{c'}\sim1+\left(\beta-\delta-\frac{1}{2}\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta+(\alpha-\beta+1)\frac{v^{2}}{c^{2}}$$should actually be corrected to read:

"Deviations of one way light speed, c', from the two-way (round-trip) speed of light ,c, are given by:..."

Now, the sentence: "The value of e(v) depends only on the choice of clock synchronization and cannot be determined by experiment." means exactly what it says, the experiments constraining OWLS do so by constraining $\alpha, \beta , \delta$ as I explained earlier in the thread. The aforementioned experiments are incapable of constraining $e$. In fact, $e$ is FIXED by the CHOICE of the synchronization method upfront, at the setup of the experiment.

 

[analyst5]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by$$\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}$$

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

 

[xox]

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

Actually, $c_1,c_2$ can take any value (if you use unphysical synchronization methods). Experiment tells us that $c_1=c_2$ if we use reasonable synchronization methods (either Einstein or slow clock transport). There is absolutely no reason to think that OWLS is anisotropic.

 

[Jorrie]

Actually, $c_1,c_2$ can take any value (if you use unphysical synchronization methods).

Only if you violate the isotropy of the two-way speed $c$, because as DrGreg said,

$$\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}$$

 

[xox]

Only if you violate the isotropy of the two-way speed $c$, because as DrGreg said,

$$\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}$$

Only if you violate the isotropy of ONE way light speed, not two-way light speed.
There is no experimental evidence supporting this concept, actually all experimental evidence supports the isotropy of OWLS.

 

[Nugatory]

Actually, $c_1,c_2$ can take any value (if you use unphysical synchronization methods).

Only if you violate the isotropy of the two-way speed $c$,

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well), a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right? If so, you two are in violent agreement .

 

[xox]

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well),

I am not aware of any experiments testing TWLS isotropy, all experiments I am aware of test OWLS isotropy, as I pointed out earlier in my response to DrGreg.

a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right?

Clock synchronization is tied to OWLS (actually, to the assumption that OWLS is isotropic, see Einstein synchronization), not to TWLS. It is true that OWLS isotropy results into TWLS isotropy. The reverse is not true, one can have TWLS isotropy without OWLS isotropy.

If so, you two are in violent agreement .

No, we are not in agreement. What Jorrie has posted doesn't even make sense (see above).

 

[xox]

I'll go a bit further, and note that using the usual TAI or GPS time synchronization conventions (which are different than the Einstein convention!), on the surface of the Earth the one-way east-west coordinate speed of light is not equal to the one-way west-east coordinate speed.

I'm sure Doctor Greg already knows this, I'm trying to clarify things for readers like analyst5.

With all respect I have to take exception to this statement. There are no global inertial frames of reference covering the whole circular trajectory of the GPS satellites. As such, there is no point in talking about the isotropy of coordinate light speed. Light speed is certainly isotropic LOCALLY, in a small interval along the trajectory. By contrast, over the WHOLE circle, the Sagnac effect MAKES IT LOOK AS IF the coordinate speeds in the opposing directions of circulation are different. This is a tremendous abuse of language, in reality we know is that what differs is the time taken by the em wavefronts to complete the circle. This effect, listed as the "Sagnac effect" in http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html is well known.

$$t_{\pm}=\frac{2 \pi R}{c \mp \omega R}$$

does not mean that the coordinate speed of the em wave has suddenly become anisotropic ($c \mp \omega R$). The mere notion of coordinate speed of light doesn't make sense in this case because of the absence of an inertial frame of reference covering the whole circle. Besides, one can argue that $c \mp \omega R$ is technically the closing speed between the light front and the receiver, not the coordinate speed.

 

[Jorrie]

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well), a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right? If so, you two are in violent agreement .

No, we are not in agreement. What Jorrie has posted doesn't even make sense (see above).

Then I think we are in non-violent disagreement...

Even ignoring the fact that the thread is about Earth's non-inertial frame and looking at a purely inertial frame, anisotropy of observed light propagation occurs when a non-standard synchrony is used. However, in any "sensible" non-standard synchrony, the observed two-way speed of light remains isotropic (because then synchronization of clocks does not play a role).

 

[DrGreg]

Experiment tells us that $c_1=c_2$ if we use reasonable synchronization methods (either Einstein or slow clock transport)

This seems to be the problem that you are failing to grasp. It's not experiment that tells us that, it's mathematical proof. If you use Einstein or slow clock transport it's mathematically guaranteed that $c_1=c_2$; no experiment required.

Experimental measurement of 1-way speed of light only makes sense if you are using some other sync method, and arguably the test is really whether the other method is equivalent to Einstein's or not.

 

[DrGreg]

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

Provided the 2-way speed is c, and provided your sync method doesn't violate causality (i.e. doesn't allow signals to travel backwards in time), then yes.

It is possible to think up weird coordinate systems where those conditions might not be true, but not (if I understand correctly) in the context you were originally asking.

 

[xox]

This seems to be the problem that you are failing to grasp. It's not experiment that tells us that, it's mathematical proof. If you use Einstein or slow clock transport it's mathematically guaranteed that $c_1=c_2$; no experiment required.

This is not how the experimental tests of SR work, I explained that in my prior answer to you.

Experimental measurement of 1-way speed of light only makes sense if you are using some other sync method,

We do not have "some other sync method" available, we simply ASSUME one method of synchronization (by fixing the value of the parameter $e$) and we RUN the experiment. We use the observed anisotropy in order to constrict the parameters (3 in the case of M-S and 17 for SME). For example, $e=-1$ for Einstein sync. I already explained that to you in my earlier post to you.

and arguably the test is really whether the other method is equivalent to Einstein's or not.

This is definitely not what the tests measure, I will have to refer you again to my corrections to your claims. If you read through the papers dedicated to constraining light speed anisotropy you find that there are no clocks involved, there is just an assumption about the value ascribed to parameter $e$. The reason for that is the fact that the assumption on clock synchronization results into the formula for "anisotropic" OWLS. It is THIS particular formula that is used in developing the theory of the experiment.

 

[Dale]

We do not have "some other sync method" available, we simply ASSUME one method of synchronization

Yes, we do. We can make different assumptions. This is well-known in the literature.

 

[xox]

Yes, we do. We can make different assumptions. This is well-known in the literature.

To my best knowledge, the Mansouri-Sexl test theory provides for the only two sync methods I listed (Einstein and slow clock transport). SME does not employ ANY form of clock synchronization. I would very much like to learn about the other methods that you are referring to, could you list some references?

 

[Dale]

Mansouri-Sexl test theory allows for a wide range of alternative sync methods by varying e.

Yes, I can provide references, but it will have to wait a day or two, apologies.

 

[xox]

Mansouri-Sexl test theory allows for a wide range of alternative sync methods by varying e.

That is true. Nevertheless, the only two methods that I have encountered in both the original (theoretical) papers and in the experimental applications are Einstein and slow clock transport, no experiment uses another $e$ since Clifford Will's proof that the dependency on $e$ cannot be exposed by experiment. Granted, my knowledge is not exhaustive, I haven't read all the papers on the subject, I am always interested in learning new things.

Yes, I can provide references, but it will have to wait a day or two, apologies.

Thank you, I appreciate that, I will await with great interest.

 

[pervect]

With all respect I have to take exception to this statement. There are no global inertial frames of reference covering the whole circular trajectory of the GPS satellites.

I agree, and more to the point there isn't a global inertial frame of reference covering the surface of the Earth (more precisely, the geoid), due to the inability to perform a global Einstein clock synchronization over the geoid,

As such, there is no point in talking about the isotropy of coordinate light speed. Light speed is certainly isotropic LOCALLY, in a small interval along the trajectory. By contrast, over the WHOLE circle, the Sagnac effect MAKES IT LOOK AS IF the coordinate speeds in the opposing directions of circulation are different.

I regard coordinate speeds in general as having little physical significance. Some people get hung up on coordinate speeds by ascribing significance to them they don't have. My view is that in GR, coordinates are perfectly general, as are the coordinate velocities, and that it is an unfortunate error when people expect them to have any particular physical properties that are not derived from the way the coordinates are chosen.

I would disagree in detail with your remarks above, though I don't think our outlooks are actually all that different. I would say that because the TAI coordinate system doesn't use an isotropic clock sychronization scheme (and can't, because any isotropic synchronization scheme won't be global), coordinate speeds are in fact different in different directions, not just "appearing" to be different.

I would agree that using a local clock isotropic synchronization scheme (combined with a suitable local definition of distance), the one way speed of light is constant everywhere and that this is an important principle of relativity. This in no way conflicts with my other point, though it is a good idea to occasioanlly remind readers of it.

I would also agree that the "Sagnac effect" is a good keyword to do more reading on the topic, with the provision that one still needs to use the usual care in evaluating the quality of resources if one is reading on the internet, there are a lot of fringe writings on the topic, some of which use the same language (Sagnac effect, in particular) as the non-fringe writings.

This is a tremendous abuse of language, in reality we know is that what differs is the time taken by the em wavefronts to complete the circle. This effect, listed as the "Sagnac effect" in http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html is well known.

$$t_{\pm}=\frac{2 \pi R}{c \mp \omega R}$$

does not mean that the coordinate speed of the em wave has suddenly become anisotropic ($c \mp \omega R$). The mere notion of coordinate speed of light doesn't make sense in this case because of the absence of an inertial frame of reference covering the whole circle. Besides, one can argue that $c \mp \omega R$ is technically the closing speed between the light front and the receiver, not the coordinate speed.

I don't see why you say this, though I have no argument with your Living Reviews reference (which is a good source, though perhaps to technical for some readers).

Perhaps we disagree on the definition of the coordinate speed of light. I should first specify precisely the coordinates I'm using. These are the time and distance coordinates of the rotating polar form of the ECEF coordinate system, with the time scale set so that clocks on the geoid keep coordinate time, defined by the metric in [22] of http://relativity.livingreviews.org/open?pubNo=lrr-2003-1&page=articlese3.html and accurate to order (1/c^2)

See http://arxiv.org/abs/gr-qc/9508043 for how defining a metric also defines the coordinates (when combined with a suitable set of reference objects).

Equation (1) defines not only the gravitational field that is assumed, but also the coordinate system in which it is presented. There is no other source of information about the coordinates apart from the expression for the metric. It is also not possible to define the coordinate system
unambiguously in any way that does not require a unique expression for the metric. In most cases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearly the choice of coordinates that is being made.

Then, given the coordinate choice, as defined by the metric, it's a simple matter of fact to note that when you solve for the null geodesics, and calculate the value of $d\phi' / dt''$ for east-west and west-east geodesics as the equator, you get different values for this quantity, which is the coordinate velocity.

Mathematically, we can point the finger at the term responsible for the so-called "Sagnac effect":

$2 \omega_E r'^2 sin^2 \theta' d\phi' dt''$

The coordinate velocity IS different in both directions, it's not an "appearance". This can be explained by the fact that in general, pairs of clocks rotating along with the Earth (which at rest in this coordinate system), are not Einstein synchronized, and thus we don't EXPECT coordinate speeds to be isotropic, because the coordinate system itself isn't isotropic.

 

[xox]

I agree, and more to the point there isn't a global inertial frame of reference covering the surface of the Earth (more precisely, the geoid), due to the inability to perform a global Einstein clock synchronization over the geoid,
I regard coordinate speeds in general as having little physical significance. Some people get hung up on coordinate speeds by ascribing significance to them they don't have. My view is that in GR, coordinates are perfectly general, as are the coordinate velocities, and that it is an unfortunate error when people expect them to have any particular physical properties that are not derived from the way the coordinates are chosen.

I would disagree in detail with your remarks above, though I don't think our outlooks are actually all that different. I would say that because the TAI coordinate system doesn't use an isotropic clock sychronization scheme (and can't, because any isotropic synchronization scheme won't be global), coordinate speeds are in fact different in different directions, not just "appearing" to be different.

I would agree that using a local clock isotropic synchronization scheme (combined with a suitable local definition of distance), the one way speed of light is constant everywhere and that this is an important principle of relativity. This in no way conflicts with my other point, though it is a good idea to occasioanlly remind readers of it.

I would also agree that the "Sagnac effect" is a good keyword to do more reading on the topic, with the provision that one still needs to use the usual care in evaluating the quality of resources if one is reading on the internet, there are a lot of fringe writings on the topic, some of which use the same language (Sagnac effect, in particular) as the non-fringe writings.
I don't see why you say this, though I have no argument with your Living Reviews reference (which is a good source, though perhaps to technical for some readers).

Perhaps we disagree on the definition of the coordinate speed of light. I should first specify precisely the coordinates I'm using. These are the time and distance coordinates of the rotating polar form of the ECEF coordinate system, with the time scale set so that clocks on the geoid keep coordinate time, defined by the metric in [22] of http://relativity.livingreviews.org/open?pubNo=lrr-2003-1&page=articlese3.html and accurate to order (1/c^2)

See http://arxiv.org/abs/gr-qc/9508043 for how defining a metric also defines the coordinates (when combined with a suitable set of reference objects).
Then, given the coordinate choice, as defined by the metric, it's a simple matter of fact to note that when you solve for the null geodesics, and calculate the value of $d\phi' / dt''$ for east-west and west-east geodesics as the equator, you get different values for this quantity, which is the coordinate velocity.

Mathematically, we can point the finger at the term responsible for the so-called "Sagnac effect":

$2 \omega_E r'^2 sin^2 \theta' d\phi' dt''$

The coordinate velocity IS different in both directions, it's not an "appearance". This can be explained by the fact that in general, pairs of clocks rotating along with the Earth (which at rest in this coordinate system), are not Einstein synchronized, and thus we don't EXPECT coordinate speeds to be isotropic, because the coordinate system itself isn't isotropic.

I enjoy very much interacting with you, you are not only very knowledgeable but also very pleasant. Looks like we agree on all the main points, I am sorry for being somewhat harsh in my tone referring to the Sagnac effect. I can explain why I consider the speed to be "closing speed" and not "coordinate speed" but I think that it is a very unimportant point , so I'll pass.
As a point of interest, I would only mention that my approach for calculating the "Sagnac effect" in GPS is to start with the Kerr solution and to make $\theta=\frac{\pi}{2}, dr=0$ (circular trajectory at the Equator). This results into the equation degree 2:

$$(r^2+\alpha^2+\frac{r_s r \alpha^2}{\rho^2})(\frac{d \phi}{dt})^2-2\frac{r_s r \alpha^2}{\rho^2}\frac{d \phi}{dt}-(1-\frac{r_s r}{\rho^2})c^2=0$$

with two distinct solutions corresponding to the two distinct "coordinate speeds" of light.

 

[analyst5]

Provided the 2-way speed is c, and provided your sync method doesn't violate causality (i.e. doesn't allow signals to travel backwards in time), then yes.

It is possible to think up weird coordinate systems where those conditions might not be true, but not (if I understand correctly) in the context you were originally asking.

The same applies to non-inertial frames? In one document I red that setting the synchonization parameter to a value different from 1/2 is equivalent to defining simultaneity for a non-inertial frame.

 

[WannabeNewton]

The same applies to non-inertial frames?

That the two-way speed of light is always $c$? No. Not unless one does the radar echo experiment locally. If the radar echo experiment is performed globally then all kinds of weird things can happen with the two-way speed of light. Just take for example the frame of a uniformly accelerating observer in flat space-time.

 

[xox]

In one document I red that setting the synchonization parameter to a value different from 1/2 is equivalent to defining simultaneity for a non-inertial frame.

This cannot be correct since $e=1$ defines Einstein synchronization. Where did you read such a thing?