QM interpretations are not theories. They are a way to think about QM to make it make sense to a person, not QM itself.

We are talking about the same in this case. LET and SR are effectively different interpretations.

In what way does this relate to, and depend on, the definition of the second?

The second provides time as a relative to a specific transition frequency of Caesium atom in the given frame, so in measures time as the number of periods this frequency will do in the same time. So when two time measurements took the same amount of seconds, it does not mean it took the same time but rather the number of periods achieved by the Caesium emissions in those frames over the duration of the measurements was the same.

Think of a simple analogy: the value of a Google equity stock can be measured in USD. The USD here should serve as an analogue of the second. But it can also be measured in EUR. Consider the situation where the stock went down over a week. Its price in EUR might however have risen. We can think of USD and EUR as just two units, but that is not entirely appropriate because the EUR/USD conversion rate changes over time. These are two quite different measures and far more then just units. There are other currencies like the Jordanian Dinar JOD which is fixed to USD with a rate of 0.709 - these numéraires behave like the the Joule to Kilocalorie.

Note that you cannot really measure the price of an asset without a numeraire. You always have to choose one. Doesn't have to be a currency. So when we have a statement like the "price of asset X is constant", it is actually ambiguous and only true for certain interpretations of the price.

Hmm, maybe now you can understand the trick behind rendering the speed of sound an invariant constant?

Edited April 12Apr 12 by Killtech

Killtech

all measurement is relative

Although this is true of most measurements, it is just not true of all measurements.

In particular it is not true of one of the most important SI measurements.

In particular it is not true of one of the most important SI measurements.

Then you gravely misunderstand the definition of the 2019 revision

What the physical real world reference for \(\Delta \nu_{c_s} \) and the second we don't need to discuss. So let's look at the Coulomb, which are defined via e, which again corresponds to the charge of an electron. So you measure charge by how much electrons you would need to get the equivalent amount. For the Ampere you need to additionally to include the second.

Those constant which values SI has fixed, those are not some arbitrary quantities. Each corresponds to some real physical things, the references to which SI measures everything relative to.

Edited April 12Apr 12 by Killtech

Killtech

Then you gravely misunderstand the definition of the 2019 revision

What the physical real world reference for Δνs and the second we don't need to discuss. So let's look at the Coulomb, which are defined via e, which again corresponds to the charge of an electron. So you measure charge by how much electrons you would need to get the equivalent amount. For the Ampere you need to additionally to include the second.

Those constant which values SI has fixed, those are not some arbitrary quantities. Each corresponds to some real physical things, the references to which SI measures everything relative to.

I don't misunderstand anything., but thank you for the kindergarten picture.

Instead of preaching at me I recommend you ask what I mean if you really don't know.

But I suppose as you are not a Chemist you might not.

We are talking about the same in this case. LET and SR are effectively different interpretations.

To the extent that this is true that makes this a philosophical discussion, rather than science.

The second provides time as a relative to a specific transition frequency of Caesium atom in the given frame, so in measures time as the number of periods this frequency will do in the same time. So when two time measurements took the same amount of seconds, it does not mean it took the same time but rather the number of periods achieved by the Caesium emissions in those frames over the duration of the measurements was the same.

This suffers from the same shortcoming as LET; there’s no way to empirically show this, since there’s always a magical fudge factor that equalizes the results.

Think of a simple analogy: the value of a Google equity stock can be measured in USD. The USD here should serve as an analogue of the second. But it can also be measured in EUR. Consider the situation where the stock went down over a week. Its price in EUR might however have risen. We can think of USD and EUR as just two units, but that is not entirely appropriate because the EUR/USD conversion rate changes over time. These are two quite different measures and far more then just units. There are other currencies like the Jordanian Dinar JOD which is fixed to USD with a rate of 0.709 - these numéraires behave like the the Joule to Kilocalorie.

Note that you cannot really measure the price of an asset without a numeraire. You always have to choose one. Doesn't have to be a currency. So when we have a statement like the "price of asset X is constant", it is actually ambiguous and only true for certain interpretations of the price.

“The price of asset X is constant” is a null set, but if we’re idealizing things, you can always convert Euros to dollars and vice-versa, and the transitive property applies here.

Hmm, maybe now you can understand the trick behind rendering the speed of sound an invariant constant?

No, because we can measure the speed of sound and see that it’s not. Unless you invoke a magical medium, but that makes it science fiction.

I can’t help but notice the complete avoidance of discussing/defending the premise of your OP - how unit definitions come into this - despite repeated requests. If you’ve abandoned that you should say so. I don’t care if you want to pursue discussing an invisible pink unicorn interpretation with others, but I focused on your scientific claim and I don’t like bait-and-switch.

But I suppose as you are not a Chemist you might not.

You mean Avogadro constant? Fair enough, this one is not related to anything real or physical to to a convention

I can’t help but notice the complete avoidance of discussing/defending the premise of your OP - how unit definitions come into this - despite repeated requests. If you’ve abandoned that you should say so. I don’t care if you want to pursue discussing an invisible pink unicorn interpretation with others, but I focused on your scientific claim and I don’t like bait-and-switch.

The unit definition is an intermediary step between experimental measurements and what corresponds to the proper time in the model. This is explained in the OP.

In the OP i also mentioned Poincarés critique on these interpretations and the assumptions that c is constant, specifically how that observation was obtained. It was him who also brought LET to its final form which made it equivalent to SR during that time.

As mentioned in the OP any attempt of even thinking of a variation of c causes logical issues. The obvious one where speeds are given in meter per second and the meter is a the distance which light travels through vacuum in a specified time does not allow for any variation of c in any measurement, does it? That problem should be easily to understand. Something similar applies to the second, but less obvious and with more impact.

• To test for a varying c, we’d need a physical framework where such variation makes sense and is not simply reabsorbed into our measurement definitions.

• But how do we define an operational way to measure a changing c, when our time and length units are already tied to its assumed constancy?

The Variation of c

• The fine-structure constant alpha is given by: e² / (4pi epsilon0 hbar c), meaning that if c varied, so would the constant.

• Since the energy levels of Caesium atoms - and thus atomic clock frequencies - depend on alpha, any variation in c would affect the very clocks we use to define the second.

• This creates a self-referential issue: if we use atomic clocks to measure changes in c, but those clocks themselves change due to variations in c, can we even establish whether c is varying in the first place?

Within SI system and SR you cannot resolve this properly to even find something to look for experimentally. The considerations i made really means applying classical QM to the Maxwell-equations of LET to understand what happens to atomic energy levels. And all you get is that atoms have to react to a movement against the aether just so that an atomic clock will get slower same as SR predicts. Additionally we can expand it with a refractive index for the vacuum and more interesting results. But all this leads to is not even a counter-hypothesis but just a different interpretation of the same physics. For those that understand how LET and SR relate to each other, this should be quite simple to understand.

Where it gets more interesting is when consider the speed of light depending additionally on the region. That is something LET did not cover. However it still is good enough to make similar considerations under the assumption that the regional variation appears constant in the local area covered by an atom. And again, this seems to lead to well known predictions, that is the same GR makes around massive bodies.

The pink unicorn we are speaking of is really a GLET, generalized LET which includes gravity, yet other then being a different interpretation of physics is still entirely equivalent to GR.

And if we want to measure GLET explicitly without any costly conversion from GR, we need to use entirely different measurement definitions. If we consider the GR interpretation requiring experiment doing measurements in USD units (to use the analogy from before), then we GLET will require us to measure everything in EUR. Since the conversion is not just a fixed rate, but really depends on the frame and its history, it is far more involved to transform between them.

Edited April 12Apr 12 by Killtech

• But how do we define an operational way to measure a changing c, when our time and length units are already tied to its assumed constancy?

How is asserting that the value of the speed of light in a vacuum is exactly 299792458 m⋅s^–1 any different from asserting that the value of the ground state hyperfine structure transition frequency of the caesium-133 atom is exactly 9192631770 s^–1? The point is that defining any base unit of measurement involves asserting the exact value of some physical quantity. In the case of defining the metre, if one didn't assert that the speed of light in a vacuum is exactly 299792458 m⋅s^–1, then one would have to assert that the length of some other physical notion has some specified exact value.

Edited April 12Apr 12 by KJW

The unit definition is an intermediary step between experimental measurements and what corresponds to the proper time in the model. This is explained in the OP.

Claimed, perhaps. Not explained.

In the OP i also mentioned Poincarés critique on these interpretations and the assumptions that c is constant, specifically how that observation was obtained. It was him who also brought LET to its final form which made it equivalent to SR during that time.

As mentioned in the OP any attempt of even thinking of a variation of c causes logical issues. The obvious one where speeds are given in meter per second and the meter is a the distance which light travels through vacuum in a specified time does not allow for any variation of c in any measurement, does it? That problem should be easily to understand. Something similar applies to the second, but less obvious and with more impact.

The main problem with this claim is that the meter wasn’t defined this way until 1983. Prior to that it was based on the wavelength of light coming from a transition in an isotope of Kr. But relativity worked prior to that. Not surprising, since it has nothing to do with how units are defined. (c wasn’t a defined value until 1983, either)

Atomic clocks measure time without relying on a measurement of length, or the value of c. You’re just comparing the frequency of an oscillator with whatever atomic transition you’re using.

Within SI system and SR you cannot resolve this properly to even find something to look for experimentally. The considerations i made really means applying classical QM to the Maxwell-equations of LET to understand what happens to atomic energy levels. And all you get is that atoms have to react to a movement against the aether just so that an atomic clock will get slower same as SR predicts. Additionally we can expand it with a refractive index for the vacuum and more interesting results. But all this leads to is not even a counter-hypothesis but just a different interpretation of the same physics. For those that understand how LET and SR relate to each other, this should be quite simple to understand.

And you can add in any other effects, just as long as they cancel.

The main problem with this claim is that the meter wasn’t defined this way until 1983. Prior to that it was based on the wavelength of light coming from a transition in an isotope of Kr.

Which still produces the same issue because you still have to claim something else to be constant - and in this case something that behaves still the same in context of relativity, hence you cannot expect it to affect it. But consider this.

How is asserting that the value of the speed of light in a vacuum is exactly 299792458 m⋅s^–1 any different from asserting that the value of the ground state hyperfine structure transition frequency of the caesium-133 atom is exactly 9192631770 s^–1? The point is that defining any base unit of measurement involves asserting the exact value of some physical quantity. In the case of defining the metre, if one didn't assert that the speed of light in a vacuum is exactly 299792458 m⋅s^–1, then one would have to assert that the length of some other physical object has some exact value.

Yes, exactly. And it is worthwhile to explore the implications that we are always forced to define some physical quantities as constant values, and all our previous observations were always based on similar assumptions of the constancy of some other physical quantities. Within such a system we cannot exclude the possibility of those physical quantities actually changing - but on the other hand, this doesn't matter because in our model that physical quantity is described in units of itself, then of course it must remain perfectly constant. So the constancies of these physical quantities is not actually a thing of reality but simply a convention or interpretation that we use.

This is important to understand to how LET may be equivalent to SR. The definition of time the prior uses does translate to assuming some very different physical quantities to be constant in order to achieve time and length not becoming frame dependent.

So let's apply this logic to the the speed of sound (limited to some some limited ideal system where the refractive index does not depend on frequency) and see what happens if we claim it to be constant and hence define measurement and laws of physical based on such an assumption. Sure the resulting theory will look different. But: it will still make the same predictions, hence it will just be a different interpretation of current physics.

Edited April 12Apr 12 by Killtech

Killtech

You mean Avogadro constant? Fair enough, this one is not related to anything real or physical to to a convention

No I did not mean Avogadro's constant, since that is directly related to the definition of mass and its associated units.

The most fundamental quantity I was referring to is exact. I mean of course, count or number.

It takes exactly one oxygen atom and exactly two hydrogen atoms to form one water molecule.

Neither a penny more nor a penny less.

And this is true whether you are measuring in our solar system, or in the Sirius or Alpha Centauri systems or in a spaceship travelling at some relativistic speed between them.

Likewise if you count charges in say an electrolysis reaction.

But note counts are not constants, they are just unaffected by electrodynamical considerations.

Killtech

Within SI system and SR you cannot resolve this properly to even find something to look for experimentally.

Actually most folks would agree that , despite its flaws and it has some, SI is the most self consistent system of units so far devised.

In similar vein you have not understood my comments about the construction of the coordinate system so have answered (when you bothered) inappropriately.

Yes, exactly. And it is worthwhile to explore the implications that we are always forced to define some physical quantities as constant values, and all our previous observations were always based on similar assumptions of the constancy of some other physical quantities. Within such a system we cannot exclude the possibility of those physical quantities actually changing - but on the other hand, this doesn't matter because in our model that physical quantity is described in units of itself, then of course it must remain perfectly constant. So the constancies of these physical quantities is not actually a thing of reality but simply a convention or interpretation that we use.

Or a gauge freedom.

KJW

Or a gauge freedom.

+1

Which still produces the same issue because you still have to claim something else to be constant - and in this case something that behaves still the same in context of relativity, hence you cannot expect it to affect it. But consider this.

Yes, exactly.

But why is that a problem? As you pointed out, all measurements are a comparison. You compare with a standard. The number used is irrelevant. Your arguments still travel back to an alleged variation that’s cancelled by another variation, so that the measurable result is the same.

And it is worthwhile to explore the implications that we are always forced to define some physical quantities as constant values, and all our previous observations were always based on similar assumptions of the constancy of some other physical quantities. Within such a system we cannot exclude the possibility of those physical quantities actually changing - but on the other hand, this doesn't matter because in our model that physical quantity is described in units of itself, then of course it must remain perfectly constant. So the constancies of these physical quantities is not actually a thing of reality but simply a convention or interpretation that we use.

The utility of using atoms is that they are identical, which we know because Fermi-Dirac and Bose-Einstein statistics rely on it, and atoms follow one or the other, depending on their spin.

Which means we can exclude certain differences.

This is important to understand to how LET may be equivalent to SR. The definition of time the prior uses does translate to assuming some very different physical quantities to be constant in order to achieve time and length not becoming frame dependent.

So let's apply this logic to the the speed of sound (limited to some some limited ideal system where the refractive index does not depend on frequency) and see what happens if we claim it to be constant and hence define measurement and laws of physical based on such an assumption. Sure the resulting theory will look different. But: it will still make the same predictions, hence it will just be a different interpretation of current physics.

You keep ignoring the elephant in the room. When we go to measure the speed of sound, we don’t get the same answer when we look at different frames. The speed of sound is empirically not invariant. The beautiful(?) theory slain by an ugly fact.

But why is that a problem? As you pointed out, all measurements are a comparison. You compare with a standard. The number used is irrelevant. Your arguments still travel back to an alleged variation that’s cancelled by another variation, so that the measurable result is the same.

The effect does not cancel out for entirely in the special case when we consider a situation of a regional variation of c. Even so, it cancels out for most things, so it would only be indirectly measurable, namely as a curvature of space time. This would render it indistinguishable from gravity and thus offer an alternative interpretation of it. This might not be too surprising though that one can model gravitational lensing of light in vacuum alternatively by a refractive index.

The utility of using atoms is that they are identical, which we know because Fermi-Dirac and Bose-Einstein statistics rely on it, and atoms follow one or the other, depending on their spin.

Not entirely. Atoms inside a strong electric field are different from atoms outside of it. They change depending on the regional conditions.

You keep ignoring the elephant in the room. When we go to measure the speed of sound, we don’t get the same answer when we look at different frames. The speed of sound is empirically not invariant. The beautiful(?) theory slain by an ugly fact.

You are comparing it to the wrong standard. Consider this: when you look at the acoustic wave equation, you can agree that obviously for every frame there exist coordinates such that the the speed of sound as expressed by a quotient of those coordinates \( \frac{\Delta x}{ \Delta t}) \) yields the same constant value.

Now one could ask if there are physical quantities that correspond to these coordinates, and indeed we can find them. To build a clock, we need an oscillator, so what would happen if we were to used a fixed frequency sinus sound emitter for that? What does happen to that frequency when we are in a frame to which the air is in motion? The doppler effect will change the original frequency so a clock that uses a time standard that bluntly assumes this frequency is a fixed value will therefore slow down. To visualize what that means to a measurement of the periods of that oscillation:
- left the clock is at rest frame of the air, right in a frame where the air is in motion.
This reconstructs the coordinate t in our frame. As for x, we can measure distance using a sonar device but we additionally express distance as the space traversed by a sonic signal within a time span given by our new acoustic clock. What will we expect to measure when we use those devices to measure the speed of sound, in terms of acoustic distance per sonic signal periods?

The effect does not cancel out for entirely in the special case when we consider a situation of a regional variation of c. Even so, it cancels out for most things, so it would only be indirectly measurable, namely as a curvature of space time. This would render it indistinguishable from gravity and thus offer an alternative interpretation of it. This might not be too surprising though that one can model gravitational lensing of light in vacuum alternatively by a refractive index.

Go measure it. I won’t hold my breath.

Not entirely. Atoms inside a strong electric field are different from atoms outside of it. They change depending on the regional conditions.

I said atoms, not atoms in electric fields.

You are comparing it to the wrong standard. Consider this: when you look at the acoustic wave equation, you can agree that obviously for every frame there exist coordinates such that the the speed of sound as expressed by a quotient of those coordinates ΔxΔt) yields the same constant value.

Now one could ask if there are physical quantities that correspond to these coordinates, and indeed we can find them. To build a clock, we need an oscillator, so what would happen if we were to used a fixed frequency sinus sound emitter for that? What does happen to that frequency when we are in a frame to which the air is in motion? The doppler effect will change the original frequency so a clock that uses a time standard that bluntly assumes this frequency is a fixed value will therefore slow down. To visualize what that means to a measurement of the periods of that oscillation:
- left the clock is at rest frame of the air, right in a frame where the air is in motion.
This reconstructs the coordinate t in our frame. As for x, we can measure distance using a sonar device but we additionally express distance as the space traversed by a sonic signal within a time span given by our new acoustic clock. What will we expect to measure when we use those devices to measure the speed of sound, in terms of acoustic distance per sonic signal periods?

Unlike light, it is possible to measure the one-way speed of sound. You would get a different answer if you send a sound source and its medium in motion. In fact, the medium can be moving faster than the speed of sound, and the sound propagates forward.

The effect does not cancel out for entirely in the special case when we consider a situation of a regional variation of c.

Why do you insist that the speed of light in a vacuum can vary? You seem to have a problem with defining its value as exactly 299792458 m⋅s^–1. Would it be less of a problem to you if, instead of defining a specific value of the speed of light in a vacuum, the metre is defined such that the value of the ground state hyperfine structure transition wavelength of the caesium-133 atom is exactly 0.0326122557175 m?

Actually most folks would agree that , despite its flaws and it has some, SI is the most self consistent system of units so far devised.

Why do you insist that the speed of light in a vacuum can vary? You seem to have a problem with defining its value as exactly 299792458 m⋅s^–1. Would it be less of a problem to you if, instead of defining a specific value of the speed of light in a vacuum, the metre is defined such that the value of the ground state hyperfine structure transition wavelength of the caesium-133 atom is exactly 0.0326122557175 m?

Hmm, i think you misunderstand the intension of my thread. The SI system and its definitions are perfectly fine for the purpose of enabling measurements with maximum accuracy. The same goes for GR and its interpretation that are build to work with the SI system.

The question is about how we interpret the physical laws that establish the constancy of the speed of light in vacuum. From these assumptions we derive what physical quantities we set as constant and define their values. This differs from other physical assumptions because this creates a situation where there cannot even exist an experimental way to disprove these assumptions. You brought the term of a gauge freedom into the conversation before, and indeed this is very much appropriate for this.

Whether we define the meter via c or via a atomic transition wavelength, we can deduct that this will produce the same gauge. For a different gauge we have to look further. The invariance of Maxwell is a gauge of SR, but LET uses indeed another one.

Unlike light, it is possible to measure the one-way speed of sound.

That is not entirely true, because there are known cases that contradict this. consider the twin paradox situation in a global geometry of a cylinder without any curvature. In this case the twins travelling in their inertial frames will meet periodically and are able to check who is older. Each twin can also send light signals in both directions around the cylinder and wait which one comes back first to measure the one way speed of light. There are papers exploring this case if you are interested. In fact, if the global geometry has any kind of features, that is it not just the flat unbounded R^4, we can use these features to measure the one-way speed of light. So there is really one special case where the one-way speed is not measurable.

And there is another effect which may offer a new possibility to measure the one-way speed of light. For now we were not able to measure the mass of neutrinos which is needed to explain their oscillation and this is what the interpretation of SR locks us into. But the neutrino is not entirely electro-magnetic in origin. This offers a possibility that the laws it is subject to may be to some degree independent from the laws we assume to define constant for the SI system. Hence there is the hypothetical possibility that it is subject to some physical law which is not Loretz invariant. Any such violation would imply that is has its own idea of proper time and may exhibit oscillation even when traveling at the speed of light.

In such a case the comparison of the neutrino oscillation frequency with the frequency of proper time in a given frame provides a tool to deduct the one way speed of light.

Hmm, i think you misunderstand the intension of my thread. The SI system and its definitions are perfectly fine for the purpose of enabling measurements with maximum accuracy. The same goes for GR and its interpretation that are build to work with the SI system.

The question is about how we interpret the physical laws that establish the constancy of the speed of light in vacuum. From these assumptions we derive what physical quantities we set as constant and define their values. This differs from other physical assumptions because this creates a situation where there cannot even exist an experimental way to disprove these assumptions. You brought the term of a gauge freedom into the conversation before, and indeed this is very much appropriate for this.

Whether we define the meter via c or via a atomic transition wavelength, we can deduct that this will produce the same gauge. For a different gauge we have to look further. The invariance of Maxwell is a gauge of SR, but LET uses indeed another one.

The laws came first, and the definitions have changed over time, so I don’t understand why you continue to bring up units. You have cause and effect reversed. The definition of the meter was chosen because of the invariance of c, not the other way around.

Since the results of SR are correct, I don’t see why any of this matters from a physics perspective. c is functionally invariant if there’s some undetectable effect that cancels out any changes to it.

That is not entirely true, because there are known cases that contradict this. consider the twin paradox situation in a global geometry of a cylinder without any curvature. In this case the twins travelling in their inertial frames will meet periodically and are able to check who is older. Each twin can also send light signals in both directions around the cylinder and wait which one comes back first to measure the one way speed of light. There are papers exploring this case if you are interested. In fact, if the global geometry has any kind of features, that is it not just the flat unbounded R^4, we can use these features to measure the one-way speed of light. So there is really one special case where the one-way speed is not measurable.

1. This has nothing to do with sound

2. Moving around a circular path is not an inertial frame

3. It doesn’t matter if you find exceptions. Invariant means it’s the same in all cases that meet the criteria, not just some of them.

Moving around a circular path is not an inertial frame

who said something about a circle?

here you go, twin paradox in compact spaces: https://arxiv.org/pdf/gr-qc/0101014

who said something about a circle?

What’s the path of “around the cylinder”? (and how do you do that without curvature?)

But the neutrino is not entirely electro-magnetic in origin.

Not at all electromagnetic.

This offers a possibility that the laws it is subject to may be to some degree independent from the laws we assume to define constant for the SI system.

And once again I will point out that you are making a connection between laws and units that simply does not exist. There are no “laws of the SI system” (SI only dates back to 1960)

And saying there are other laws is pure conjecture.

What’s the path of “around the cylinder”? (and how do you do that without curvature?)

here you go: https://en.wikipedia.org/wiki/Flat_manifold

e.g. \(S^1 \times R \) is a flat cylinder even with its the natural metric, i.e. the one you get from embedding it in \(R^3\). For the torus you need to pick a special metric to make it flat.

Edited April 13Apr 13 by Killtech

You do learn so much when asking an AI. Figures, that what i suggest is almost what we do practically do anyway to make any predictions, for example on how light is bend in the solar system.

For such predictions we do use BCRS coordinates (with TCB time), which correct for gravity effects, just like i suggested. When looking at these coordinates in context of LET, we can also identify them as the native spacetime of this interpretation, that where the metric LET chooses becomes trivial ( \( g'^{\mu \nu} = \eta^{\mu \nu} \neq g^{\mu \nu} \) in BCRS) and thus partial and covariant derivatives become the same.

Now, how do we calculate gravitation bending in the solar system? At first the metric in first order approximation (for weak gravity fields) in BCRS coordinates is given as
\[ g_{\mu \nu} = \delta_{\mu \nu} (\pm 1+\frac {2 U(x)} {c^2}) \]
with \( U(x) \) the gravity potential. plugging that into Maxwell and translating the covariant derivative via
\[\nabla_\mu F^{\mu \nu} = \frac {1} {\sqrt {-g}} \partial_\mu (\sqrt {-g} F^{\mu \nu} ) \]
we get the Maxwell equations for the solar system in BCRS, or we can interpret them as Maxwell in LET spacetime. In contrast to the usual Maxwell equation, it has now additional terms.

When looking at how light is bended, we start with a plain wave ansatz and apply it to the new equation. From there we can obtain the Eikonal equation for the plane wave
\[ -(\partial_t \phi(x,t))^2 + |\nabla \phi(x,t)|^2 = 1 + \frac {2 U(x)))} {c^2} = n_{eff}(x) \]
that is an optical equation with an effective refractive index \( n_{eff}(x) \) and hence a local varying coordinate speed of light \( c(x) = \frac c {n_{eff}(x)} \). From there we can calculate the gravitational lensing effect like the bending angle depending on the trajectory. In the LET interpretation the calculation is the same except that we can simply call it the actual speed of light.

The thing is that for any realistic case with a slightly more complicated geometry then Schwarzschild like a many body problem, the calculations are performed in specific coordinates that either are or very closely coincide with the spacetime of LET. So it turns out the pink uniform i was chasing is a simplified interpretation of what we have do in our calculations anyway. Thus the question if c is a constant, is in fact down to the choice of interpretation.

But all that has no consequence for predictions made. Where it starts to matter is when we look at the observed expansion of the universe and the arbitrarily added cosmological constant to fix GR. For LET interpretation, it is very natural that an ether densely packed into a small region of otherwise empty space will start to expand. It turn out reinterpreting GR in terms of LET, it assumes the ether has a constant energy density defined by the cosmological constant and a constant pressure equal to the negative of that density - despite its expansion. in this picture dark energy corresponds to an uniform tension in the ether. Hence, it appears very natural why such a weird attempt at a fix ultimately produces more problems then it solves.

I think the gist of my problem is that we got locked into a singular interpretation, that is becomes almost impossible to talk with people about it. The constancy of the speed of light seems at times more like churches dogma then a practical view. Specifically what bothers me is that most don't seem to understand where it even comes from unaware of the intrinsic circularity in measuring it. Well at least the AIs are smarter...

If one looks at the Schwarzschild metric, it is obvious that the speed of light seen from spatial infinity appears to be [math]c(1-\dfrac{2GM}{c^2r})[/math], but locally it is still c. And considered in terms of proper distance and proper time, regardless of where the observer is, it is still c.

Also, have you checked the deflection of light given by the above? Is it the correct value, or half the correct value? If you only consider gravitational time dilation, you'll only get half the correct value. The full value also includes the curvature of space (the Schwarzschild metric).

Edited May 3May 3 by KJW