Is the premise of the Michelson–Morley argument still valid?

[Killtech]

I've been thinking about the Michelson–Morley experiment lately and how it would play out with acoustic waves clearly showing the presence of a sonic-ether (well, at least in the case it's not enclosed - the air through which the sound travels has to be exposed to the outside wind ofc). And then i compared it to how it works for light. Anyway, argument is pretty straight forward and i remember learning it back in school so many years ago.

But going through the (over a century old) argument again with a little more knowledge i actually find an issue that i didn't see anywhere mentioned: The central premise of the experiment is that the distances of the mirrors in the experiment are fixed and well know. Of course that seems such a trivial thing to assume for an experimenter that it doesn't need to be mentioned... but we are trying to detected a "luminuferious ether" that is supposed to affect the speed of light and therefore the Maxwell equations. And that's actually a problem because the interferometer is build out of many atoms in a solid grid the properties of which are supposed to guarantee to preserve the distances between all mirrors. perhaps not something that was clear when the experiment was performed but since Schrödingers atomic model (discovered 1926 after the ether was dismissed) we know atoms entirely depend on Maxwells electromagnetic interaction to get into shape... so for me it looks like the argument is incomplete if the effect of a hypothetical ether drag on the mirror distances isn't taken into account. And given that the speed of light is a defining property of atoms i just don't understand how it can be neglected.

Unfortunately i don't know of any quantum mechanical model which allows an ether assumption as to properly calculate the impact. But i can try to make a qualitative argument: For example one can look at the electric potential of a point-like charge (i.e. atomic nucleus) under an ether wind assumption. I would think that the planes of equal electric potential should be the same as the planes defined by equal light travel time from the core (one way or rather two way?). That yields an oval (or ellipsoid?) shape with diameters along the different axis being stretched by factors corresponding to the travel times of light along in the interferometer axes. The electron orbitals are the solutions for that potential and define the shape of their atom and should they follow the very same deformation then wouldn't this just exactly cancel out any measurable effect?

One could of course try guarantee the distance of mirrors using gravity instead of solid matter (i.e. a gravity orbit). But unfortunately gravity is assumed to propagate at the very same speed hence leading to the very same dependence on $c$ iff both forces are affected by an ether in identical fashion.

So at a native second glance it seems like the experiment might try to compare the directional properties of the speed of light with itself hence being unable to detect anything. How is that actually resolved in modern physics?

 

[Dale]

The central premise of the experiment is that the distances of the mirrors in the experiment are fixed and well know.

That is exactly the assumption that Lorentz identified. He took the MMX to mean that the length of the parallel arm contracted. This is the basis of the Lorentz transform in the Lorentz aether theory.

As a side note it always bugs me when people say that there is no experimental validation of length contraction. That was actually the original interpretation of the null result of the MMX.

 

[PeterDonis]

it seems like the experiment might try to compare the directional properties of the speed of light with itself hence being unable to detect anything

As @Dale points out, that is exactly how Lorentz (and others, e.g., Fitzgerald) explained the null result: that the forces that determined things like inter-atomic distances in the apparatus were electromagnetic, and that a simple analysis of how those electromagnetic forces would behave in a moving frame led to the result that the apparatus would contract in the direction of motion.

How is that actually resolved in modern physics?

By recognizing that, when you fully unpack Lorentz's argument, you have discovered an argument for why ordinary mechanics--the physics of objects like measuring rods and clocks--should be Lorentz invariant, not Galilean invariant as Newtonian mechanics is. In other words, you realize that you have arrived at Special Relativity, just as Einstein did in 1905.

In other words, the resolution is not that Lorentz's argument for why an object's length should contract in the direction of its motion is "wrong". It's not; as John Bell, for example, emphasized in one of his papers, it is a perfectly valid argument, given that you have chosen a frame in which the object is moving. The point is that you can't stop there; you need to work out all of the implications of looking at things that way, and doing it in every frame, not just one, and when you do, you end up at Special Relativity.

 

[pervect]

There are a couple of possible resolutions.

The first is to say that the lengths do not change in the frame of the instrument, the Earth frame. The underlying philosophy here is that length is frame-dependent. Therefore, when we talk about lengths changing - or not changing - we must specify the frame. So it's possible the lengths do not change in one frame, and do change in another , with this approach. And the important thing is that the lengths do not change in the Earth frame.

The second is to say that the proper length, which is frame independent, does not change.

It's certainly possible to focus on the frame-dependent lengths, but I find it much less confusing to focus on the frame-independent proper lengths.

I've seen bit of a disconnect in how the material is presented. I believe I have seen some sources say that the SI meter defines the unit of proper length, not a frame-dependent length. Unfortunately, I don't have a source for this recollection. It's also common to talk about lengths as being frame dependent quantities. Either approach can work if one is consistent, but I"ve seen both used, and it generates confusion. I personally greatly prefer to deal with frame independent lengths, but I tend to use the term "proper length" to make it clear that that is what I am doing.

 

[anuttarasammyak]

We may say Michelson-Morley experiment in 1887 is revisited with bigger scale and higher accuracy in 2015 by LIGO which detected gravitational wave for the first time. And now Michelson interferometer is an essential apparatus for advanced photonics including photonic quantum computer. In these recent applications I have not heard of challenges you have concern.

 

[Killtech]

Oh, thanks you for all the answers. Turns out its quite common place to skip that important part of the historic argument. So of course first thing i had to do was to look up Lorentz ether theory... and it fills some holes in my knowledge and also rounds up the picture quite well. and it makes the historic development of the theory a lot more intuitive.

Hmm, one problem is still bugging me though: my auto correction say it's "ether" not "aether". I'm not English so i lack the intuition to know which is correct.

That is exactly the assumption that Lorentz identified. He took the MMX to mean that the length of the parallel arm contracted. This is the basis of the Lorentz transform in the Lorentz aether theory.

Reading up on Lorentz theory on Wikipedia i stumbled an interesting remark from Poincaré talking about alternatives of postulating the constancy of speed of light. Were such frameworks ever developed further? For entirely different reasons i have run into an use case for his approach and thought about something similar - mostly because a framework focused around audio signals and i also need those for clock synchronization rather then light.

I think in terms of Riemann geometry the math provides all the tools needed. So technically i think the postulate of a constant $c$ is implicitly selecting/defining a specific Riemann pseudo-metric to work with. But just like in classic Riemann geometry a smooth manifold has a whole Fréchet space of Riemann-metrics to pick from and in my case a different choice seems to be a better one i.e. different postulate. Now the problem with a different metric choice is that it also needs different laws of motions as Poincaré points out... I think calculating these laws of motion is what diff geo calls a pullback from one metric space to the other using the identity function (which should be a diffeomorphism). I tried to get some help with the math of this on the local diff geo forum but it isn't the most active.

The point is that you can't stop there; you need to work out all of the implications of looking at things that way, and doing it in every frame, not just one, and when you do, you end up at Special Relativity.

Do you have to though? I mean, sure you have to work out all implications. But for me it sounds like there are some postulates picked specifically to keep the theory maximally simple, but they aren't required per se and others could be used instead yielding more complicated laws of motion (and messy frame trafos) - just as Poincaré wrote to Rømer which i mentioned above.

've seen bit of a disconnect in how the material is presented. I believe I have seen some sources say that the SI meter defines the unit of proper length, not a frame-dependent length

Well, a definition relying on the travel distances of light depends on your choice of a time measure, i would think.

 

[PeterDonis]

one problem is still bugging me though: my auto correction say it's "ether" not "aether".

Both are technically valid spellings, but "ether" is much more common. (Personally, I would have preferred "aether" to be the more common one, to avoid confusion with ether the anesthetic. )

 

[Ibix]

Ether is more common than aether, but either is fine.

 

[PeterDonis]

for me it sounds like there are some postulates picked specifically to keep the theory maximally simple, but they aren't required per se and others could be used instead yielding more complicated laws of motion (and messy frame trafos) - just as Poincaré wrote to Rømer which i mentioned above.

Poincare's remark was about the one-way speed of light being constant. But that is not the assumption that is made as a postulate of SR. SR only assumes as a postulate the constancy of the round-trip speed of light. The constancy of the one-way speed of light indicates that a particular kind of coordinate choice has been made; it is perfectly possible to do SR in coordinates in which the one-way speed of light is not constant. But, as Poincare correctly remarked, in coordinates like that the laws of motion look more complicated; although Poincare did not include a crucial qualifier, that this apparent additional complication only appears if we do not express the laws of physics in terms of tensors (more precisely, 4-d tensors). Poincare couldn't include that qualifier because nobody had yet investigated how to express the laws of physics in terms of tensors; Einstein's work on General Relativity was the first time that had been attempted (and the full development of it would take decades longer).

 

[Killtech]

Both are technically valid spellings, but "ether" is much more common. (Personally, I would have preferred "aether" to be the more common one, to avoid confusion with ether the anesthetic. )

Oh, indeed. entirely missed the chemical is called the same. English words often tend to have ambiguous meaning. in German its "Äther" and "Ether", so your preference is implemented there.

... anyhow, added aether to auto correction. :)

 

[Killtech]

Poincare's remark was about the one-way speed of light being constant. But that is not the assumption that is made as a postulate of SR. SR only assumes as a postulate the constancy of the round-trip speed of light. The constancy of the one-way speed of light indicates that a particular kind of coordinate choice has been made; it is perfectly possible to do SR in coordinates in which the one-way speed of light is not constant. But, as Poincare correctly remarked, in coordinates like that the laws of motion look more complicated; although Poincare did not include a crucial qualifier, that this apparent additional complication only appears if we do not express the laws of physics in terms of tensors (more precisely, 4-d tensors). Poincare couldn't include that qualifier because nobody had yet investigated how to express the laws of physics in terms of tensors; Einstein's work on General Relativity was the first time that had been attempted (and the full development of it would take decades longer).

Hmm, i have found a translation of the original text and reading it through i don't think it's at all what Poincaré meant. It's clear from the text that he understand this postulate as a working definition, a mean in order for the astronomer to have a measure of length begin with. In modern mathematical terms a Riemann pseudo-metric serves this very purpose. Obviously we cannot do anything without a metric, like the term speed itself cannot be defined without it and much less measured. And he points out that whatever choice is made, it cannot itself create any contradiction i.e. any choice will do. So this is way more fundamental then what you describe. Unfortunately for us, in these old texts the math formalism wasn't that refined yet, and for example the definition of a "metric" was still 8 years into the future.

As for coordinate choice, by itself it doesn't do anything to determine the speed of life. Once you have a set of coordinates picked, the pseudo-metric is still not defined and you have at each point still 15 degrees of freedom left to specify. and only once that is fixed something as crucial as differentials or even lengths begin to exist, so we can define speed and other physical quantities.

Also if the one-way-speed could change you can easily create a case where the two way-speed is not constant. Still thinking with an aether in mid (in 1898) it would be awkward not to take that possibility into account. Anyhow, you can also see how Poincaré speed of light postulate was meant this way: the coordinate maps for a manifold and Riemann pseudo-metrics are independent mathematical objects, so choosing coordinates is insufficient. The postulate however in context of the method of measurement used uniquely specifies 9/15 degrees of freedom of the metric at each point on the manifold (the remaining degrees are left open because they "depend upon the quantitative problem of the measurement of time" (using his words) - i.e. we actually need more postulates to uniquely fix the metric up to isometry, which is what Einstein provided).

Man, it's kind of mind breaking to read all this taking into account at which time this was written. It really speaks to itself how much insight and intuition about physics and more importantly mathematics Poincaré must have had to come up with that understanding at that time.

 

[Dale]

Also if the one-way-speed could change you can easily create a case where the two way-speed is not constant.

While this is true it also misses the point. There do exist multiple ways that the one way speed of light can change without changing the two way speed of light. That is the issue. There is not a unique one way speed of light that is consistent with the data.

 

[PeterDonis]

It's clear from the text that he understand this postulate as a working definition, a mean in order for the astronomer to have a measure of length begin with.

In other words, a choice of coordinates.

In modern mathematical terms a Riemann pseudo-metric serves this very purpose.

No. Length is coordinate-dependent. The spacetime metric determines spacetime intervals, not spatial lengths.

As for coordinate choice, by itself it doesn't do anything to determine the speed of life.

I assume you mean "speed of light".

A coordinate choice affects the one-way speed of light, but not the round-trip speed, as I said.

Once you have a set of coordinates picked, the pseudo-metric is still not defined

You have it backwards. You don't pick a coordinate chart and then go looking for a metric. You find the metric first and then look for different possible coordinate charts on it.

Finding a solution for the metric normally does involve some constraints on the coordinates, but those are driven by assumptions about the spacetime geometry; for example, if you assume that the spacetime is spherically symmetric, it's natural to adopt spherical coordinates and to make some kind of assumption about the radial coordinate (for example, in Schwarzschild coordinates $r$ is the areal radius, which is a geometric invariant). But once you've found a solution (for example, the Schwarzschild solution), you can switch coordinate charts as much as you want; you're not restricted to the chart you originally used to find it.

 

[PeterDonis]

we actually need more postulates to uniquely fix the metric up to isometry, which is what Einstein provided

Einstein's postulates for SR actually don't uniquely fix the metric by themselves, unless you interpret the term "inertial frame" as asserting the existence of global inertial frames, instead of just local ones. But if you only interpret "inertial frame" locally, then general relativity satisfies both postulates, and of course in GR the metric does not have to be the flat Minkowski metric.

 

[Killtech]

While this is true it also misses the point. There do exist multiple ways that the one way speed of light can change without changing the two way speed of light. That is the issue. There is not a unique one way speed of light that is consistent with the data.

I think you missed the point of Poincaré's argument. The data you get from measurement already depends on the speed of light (as the postulates fixing it were implicitly made to be able to derive distances - that is part of his argument). choosing different postulates also means getting different data but also at the same time different laws of motion. because the data changes with the laws of motion in tandem they can remain consistent (which obviously otherwise they could not as you state correctly).

but you are right, in that all data we acquire conforms only to one metric, since we experimentally only have one wave equations that is used to all signal transfer and from which all lengths are derived from. the entirety of physics builds only on that one metric. so it's a bit natural to think of space and metric as a union.

Anyhow around the same time Riemann realized within his study of geometry is that space and the metric are separate and independent entities within the mathematical realm. as a mathematician Poincaré would have been well aware of Riemanns work since he preceded him by only a few decades or so. but since he was also deeply interested in phyisics it would not have escaped him what these insights of the developing geometry meant practically given how all physics merely build on the mathematical framework. Poincaré remarks sound just like explaining that in terms of an astronomers measurements.

I found the full translated text here: https://en.wikisource.org/wiki/The_Measure_of_Time in paragraph XII

 

[Killtech]

Einstein's postulates for SR actually don't uniquely fix the metric by themselves, unless you interpret the term "inertial frame" as asserting the existence of global inertial frames, instead of just local ones. But if you only interpret "inertial frame" locally, then general relativity satisfies both postulates, and of course in GR the metric does not have to be the flat Minkowski metric.

I think Maxwell equations take their simples form only within one unique metric. that's what fixes the metric. it also makes inherently not flat around gravity sources. the constancy of $c$ fixes only the 9 spatial components of the metric while the postulate about the form of Maxwell equations fixes the remaining time-bound components because the equation contains both space and time derivatives (which are defined by the metric). transformation rules between different frames can be derived from there.

 

[PeterDonis]

I think Maxwell equations take their simples form only within one unique metric. that's what fixes the metric.

No. Maxwell's Equations in tensor form have the same form in any spacetime, regardless of its geometry.

it also makes inherently not flat around gravity sources.

I don't know what you mean here. If you mean that flat Minkowski spacetime is not a solution of the coupled Einstein-Maxwell Equations, that's correct; an electromagnetic field has nonzero stress-energy and therefore curves spacetime. However, there are multiple possible curved spacetimes that are solutions of the Einstein-Maxwell Equations; those equations do not fix a single spacetime geometry.

If you are just talking about "gravity sources" in general (i.e., the stress-energy tensor not necessarily having to be that of an EM field), then there are even more possible curved spacetimes that are solutions. All of them are curved, yes.

the constancy of $c$ fixes only the 9 spatial components of the metric

No. The constancy of $c$ does not fix anything about the metric. If you have a locally Lorentzian metric, then you can choose coordinates on a small local patch of spacetime in which $c$ is constant. You can do that in any locally Lorentzian spacetime, regardless of its geometry. But nothing requires you to choose those coordinates. And if you do choose them, they fix all of the metric coefficients: you are choosing a local inertial frame, in which the metric is $\eta_{\alpha \beta}$. There is no freedom of choice left there at all.

 

[pervect]

I think in terms of Riemann geometry the math provides all the tools needed. So technically i think the postulate of a constant $c$ is implicitly selecting/defining a specific Riemann pseudo-metric to work with. But just like in classic Riemann geometry a smooth manifold has a whole Fréchet space of Riemann-metrics to pick from and in my case a different choice seems to be a better one i.e. different postulate. Now the problem with a different metric choice is that it also needs different laws of motions as Poincaré points out... I think calculating these laws of motion is what diff geo calls a pullback from one metric space to the other using the identity function (which should be a diffeomorphism). I tried to get some help with the math of this on the local diff geo forum but it isn't the most active.

The metric can be chosen for convenience, because coordinates can be chosen at convenience. Basically, as far as physics go, choosing a metric specifies the coordinates. See for instance Misner, "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043

GR is "built on top of" SR. And SR defines the Lorentz interval as an invariant for all observers. Most of the questions asked here are really about SR, not GR. However, let's talk about the difference for a bit. GR can be regarded as the geometry of the Lorentz interval, where the Lorentz interval is taken to be constant for all observers. This is the quantity ds, the line element computed via the metric coefficients.

One of the more subtle differences between GR and SR as far as the speed of light - or any velocity - goes, is that in GR, velocities are only defined in the neighborhood of a specific point. Technically, velocities are vectors in the tangent space at a point, and every point has its own tangent space. See for instance Baez, "The Meaning of EInstein's equation". https://arxiv.org/abs/gr-qc/0103044

Before stating Einstein’s equation, we need a little preparation. We assume thereader is somewhat familiar with special relativity — otherwise general relativitywill be too hard. But there are some big differences between special and generalrelativity, which can cause immense confusion if neglected.In special relativity, we cannot talk aboutabsolute velocities, but onlyrel-ativevelocities. For example, we cannot sensibly ask if a particle is at rest,only whether it is at rest relative to another. The reason is that in this theory,velocities are described as vectors in 4-dimensional spacetime. Switching to adifferent inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them pointthe same way.In general relativity, we cannot even talk about relativevelocities, except fortwo particles at the same point of spacetime — that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously thenotion that a vector is a little arrow sitting at a particular point in spacetime.To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning orstretching it is called ‘parallel transport’. When spacetime is curved,the resultof parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vectorunless they are at the same point of spacetime.

I suppose the good news is that parallel transport (at least using the connection GR uses) doesn't change the length of vectors, so I suppose it is possible to talk about the magnitude of the speed of light even in the presence of the ambiguities due to parallel transport.

Back to the original poster:

Do you have to though? I mean, sure you have to work out all implications. But for me it sounds like there are some postulates picked specifically to keep the theory maximally simple, but they aren't required per se and others could be used instead yielding more complicated laws of motion (and messy frame trafos) - just as Poincaré wrote to Rømer which i mentioned above.Well, a definition relying on the travel distances of light depends on your choice of a time measure, i would think.

Indeed, it does. The current standard definition for the second is based on cesium atomic clocks. If someone were to use an older definition of the second, say one based on the mean solar day, they speed of light would be slowly varying, because the number of cesium-clock seconds in a mean solar days varies. We attribute this to the Earth's spin slows down.

One either needs to define both the unit of time and the unit of distance to talk about the speed of light, or avoid the issue by talking about the fine structure constant instead of the speed of light.

There are several equivalent equations for the fine structure constant in terms of other constants, one is
$k_e e^2 / \hbar c$. See for instance wiki, https://en.wikipedia.org/w/index.php?title=Fine-structure_constant&action=history. So if the coulomb constant $k_e$, the charge of the electron e, Planck's reduced constant $\hbar$ are all constant, if c varies, the fine structure constant varies and vica versa. And there is no need to get into the nitty gritty of one's choice of units or standards, because the quantity is dimensionless.

With a dimensonful quantity, I do not believe there is any alternative but to have such a discussion on the experimental realization of the units used, otherwise the dimensionful observation has no meaning. Restricting one's attention to dimensonless but closely related quantites avoids the necessity for this.

 

[PeterDonis]

I found the full translated text here: https://en.wikisource.org/wiki/The_Measure_of_Time in paragraph XII

In modern terms, that paragraph is describing particular choices of coordinates that are made for the convenience of astronomers and other people. (Assumptions about simultaneity and the speed of light are equivalent to choices of coordinates.) It is not saying anything about the geometry of spacetime. It can't be, because what it is saying is equally consistent with both Newtonian mechanics and with relativity; the former does not even have a concept of spacetime geometry, and the latter contains an infinite number of possible spacetime geometries, depending on the distribution of stress-energy.

 

[Dale]

choosing different postulates also means getting different data

This isn’t correct. There are some postulates that you can choose that lead to the same data.

 

[Killtech]

No. Maxwell's Equations in tensor form have the same form in any spacetime, regardless of its geometry.

Okay, i see there is a big misunderstand here that we need to clear up first.

I think i get what you are trying to say. Yes, Maxwell equations remain invariant at any spacetime point in tensor notation, regardless what the curvature at this point is. That is true, but however that's still just one single geometry and metric you are talking about.

See, in mathematics the geometry and metric are global objects defined in every point of the manifold. So the entire pseudo-Riemann manifold has just a single geometry as defined by its metric (which has a tensor representation at every spacetime). And by single i don't mean constant or anything like in Minkowski's case. You however seem to use these terms purely in a localized meaning as in the metric tensor in a single point and the geometry in terms of the tensors defining geometric aspects like curvature at that point. That's not what i was talking about.

So when i use the term metric i always mean the global object. And as for tensors, well they are exactly those affected when pulled back to a different metric with the transformation behavior described here. Therefore it is easy to see that using the identity function $id: x \rightarrow x$ as a diffeomorphic map between two pseudo-Riemann-manifolds induced through a different choices of R.p.-metrics on the same smooth manifold will obviously cause all tensors to change. Well, actually using the identity to do a transformation cannot really change anything but tensors! Iff the metrics aren't isometric the identity map will have non-trivial derivatives which will transform all tensors (hmm, come to think of it, it's a good example showing space-time and the metric are independent entities). Hence it's easy to construct a case where Maxwell doesn't preserve form when mapped to a different geometry (as in the whole p.R.-manifold it was mapped to).

 

[PeterDonis]

the geometry and metric are global objects defined in every point of the manifold. So the entire pseudo-Riemann manifold has just a single geometry as defined by its metric

The metric can vary from point to point, so when you say "global objects" and "a single geometry", you have to be careful to specify what you mean. Even saying "a single solution of the Einstein Field Equations" doesn't quite work, since "solutions" that look mathematically different can still describe the same physical geometry (for example, consider all the different coordinate chart representations of the Schwarzschild geometry).

You however seem to use these terms purely in a localized meaning as in the metric tensor in a single point and the geometry in terms of the tensors defining geometric aspects like curvature at that point

Tensor equations are equations that are valid in the tangent space at every point. So tensor equations are "localized" in that sense. If you want to write down "global" equations, you need to write down integral equations. And you need to be very careful in doing that, because a lot of implicit assumptions that work in flat spacetime break down in curved spacetime. The GR literature is full of examples of people getting bitten by issues of that sort.

when i use the term metric i always mean the global object.

Then you need to be much more specific in defining exactly what you mean by "the global object", for the reasons given above. And I strongly suspect that, when you try to do that, you will run up against a lot of issues that you are not considering.

they are exactly those affected when pulled back to a different metric

"Different metric" is another one of those simple-sounding phrases that hides a lot of things that can bite you. As I noted above, expressions for the metric that look different mathematically can still describe the same physical geometry. So you have to have a concept of "the same physical geometry" that doesn't depend on specific mathematical expressions for the metric. That concept is that you define "the physical geometry" in terms of invariants. Physically speaking, any transformation that preserves invariants preserves the physical geometry; and those transformations will preserve the form of all physical laws. And any transformation that doesn't preserve invariants is physically meaningless, regardless of its mathematical validity.

it's easy to construct a case where Maxwell doesn't preserve form when mapped to a different geometry

Please give a specific, explicit example. I strongly suspect that you won't be able to without getting bitten by the issues I described above.

 

[Killtech]

Then you need to be much more specific in defining exactly what you mean by "the global object", for the reasons given above. And I strongly suspect that, when you try to do that, you will run up against a lot of issues that you are not considering.
[...]
Please give a specific, explicit example. I strongly suspect that you won't be able to without getting bitten by the issues I described above.

You seem to mistake the metric with its metric tensor. The former is defined on the entirety of the manifold an therefore cannot change (unless you consider that in the space of manifolds... which mathematicians ofc have researched, too) while the latter depicts how the metric looks at a specific spacetime point - and of course this object changes depending where on the manifold you are.

i use the terms as i am used to from here: https://www.ime.usp.br/~gorodski/teaching/mat5771/ch1.pdf

but i have found something to read on the trafos i am talking about. it's called conformal transformations / geometry and in context of physics there is an article named conformal gravity. If you think about it, changing the postulates around $c$ as Poincaré discusses effectively means doing an arbitrary Weyl transformation... and being able to foresee the consequences of that 100 years before that stuff was developed is kind of impressive to say the least.

I think in physical terms these transformation could be interpreted that the rescaling function applied to the metric tensor is effectively just slapping a refractive index function onto spacetime and that of course hits all equations and laws of motion pretty hard. In terms of a wave equation like Maxwell it allows for an intuitive interpretation though: mathematically you can bend waves via space geometry/curvature or a refractive index and a conformal trafo translates between both descriptions. but slapping a refractive index to Maxwell in vacuum kind of forces $c$ to change locally in the new geometry - never mind it makes all equations quite a bit more complicated.

This isn’t correct. There are some postulates that you can choose that lead to the same data.

Well, i was talking about postulates that specify the metric only. I though this was clear in the context. i.e. such postulates are more mathematical in nature and cannot be really proved or falsified. Anyhow i found the proper articles for the terminology for this discussion, so this may clear up things a bit.

With a dimensonful quantity, I do not believe there is any alternative but to have such a discussion on the experimental realization of the units used, otherwise the dimensionful observation has no meaning. Restricting one's attention to dimensonless but closely related quantites avoids the necessity for this.

This goes far beyond dimensional quantities. Besides, in natural units $c=1$ doesn't even have a dimension to talk about. The problem is of an entirely different making: in our physical metric an experimenter might measure two distances to have the exact same length (for examples the axis of a MM interferometer) - but once a Weyl-tranfo (as discussed above) is performed and the metric tensor is rescaled those two distances could never have equal length in no frame or whatever (for example if the contraction is postulated to happen, then the metric tensor is scaled linearly along one axis). the length contraction however needs to be postulated because it could never be measured - due to how every measurement is impacted by this very contraction (Poincaré's argument). Anyhow, that rescaling will however cause an aether drag to enter Maxwell equations (as predicted: laws of motion get more complicated) - which then allows to find the atomic electron orbitals under the effect of an aether wind which should result in a matter contraction consistent with the hypothesis. And this is what Poincaré had in mind when he made his argument. So this transformation comes with a whole new level of abstraction and relativity in a sense.

But obviously the Weyl rescaling comes with a huge downside: a complete disconnection to all experimental data once it is performed. All experimental data is simply always bound to a specific metric - i.e. the implicit assumptions made to derive distances from the measurement.

 

[vanhees71]

The metric can be chosen for convenience, because coordinates can be chosen at convenience. Basically, as far as physics go, choosing a metric specifies the coordinates. See for instance Misner, "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043

No, the metric is determined by the physical situation you describe. It's independent of the choice of coordinates. Nothing physical depends on the choice of coordinates.

 

[Dale]

Well, i was talking about postulates that specify the metric only. I though this was clear in the context.

No, actually the context is the problem. The context I responded to was your comment on the one way speed of light.

The one way speed of light is not determined only by the metric. It also requires specification of a coordinate chart. Perhaps specification of the simultaneity convention would be sufficient, but even that is still more than just the metric.

So it definitely is not clear in context that you are talking about the metric only, since what you claimed is about more than the metric.

Whatever class of postulates you are talking about, this statement:

such postulates are more mathematical in nature and cannot be really proved or falsified

Completely contradicts this statement:

choosing different postulates also means getting different data

If changing a postulate leads to different data then that choice of postulate can in fact be falsified.

 

[Killtech]

The one way speed of light is not determined only by the metric. It also requires specification of a coordinate chart. Perhaps specification of the simultaneity convention would be sufficient, but even that is still more than just the metric.

of course it depends on the the coordinates charts just as much as on the metric. but those are independent degrees of freedom. the postulates fixing the metric therefore usually contain assumptions about transformation behavior between frames i.e. breaking away from being coordinate specific.

Whatever class of postulates you are talking about, this statement: Completely contradicts this statement:
If changing a postulate leads to different data then that choice of postulate can in fact be falsified.

Both statements are perfectly consistent but you need to follow Poincarés argumentation and understand that you have basically two separate worlds each with their own theories when undergoing a change of postulates. do never mix these two (unless you have Weyl invariant objects):

I will take two simple examples, the measurement of the velocity of light and the determination of longitude.

When an astronomer tells me that some stellar phenomenon, which his telescope reveals to him at this moment, happened, nevertheless, fifty years ago, I seek his meaning, and to that end I shall ask him first how he knows it, that is, how he has measured the velocity of light.

He has begun by supposing that light has a constant velocity, and in particular that its velocity is the same in all directions. That is a postulate without which no measurement of this velocity could be attempted. This postulate could never be verified directly by experiment; it might be contradicted by it if the results of different measurements were not concordant. We should think ourselves fortunate that this contradiction has not happened and that the slight discordances which may happen can be readily explained.

The postulate, at all events, resembling the principle of sufficient reason, has been accepted by everybody; what I wish to emphasize is that it furnishes us with a new rule for the investigation of simultaneity, entirely different from that which we have enunciated above.

This postulate assumed, let us see how the velocity of light has been measured. You know that Roemer used eclipses of the satellites of Jupiter, and sought how much the event fell behind its prediction. But how is this prediction made? It is by the aid of astronomic laws; for instance Newton's law.

Could not the observed facts be just as well explained if we attributed to the velocity of light a little different value from that adopted, and supposed Newton's law only approximate? Only this would lead to replacing Newton's law by another more complicated. So for the velocity of light a value is adopted, such that the astronomic laws compatible with this value may be as simple as possible. When navigators or geographers determine a longitude, they have to solve just the problem we are discussing; they must, without being at Paris, calculate Paris time. How do they accomplish it? They carry a chronometer set for Paris. The qualitative problem of simultaneity is made to depend upon the quantitative problem of the measurement of time. I need not take up the difficulties relative to this latter problem, since above I have emphasized them at length.

So we have:
1. if the postulates are changed, the measured distances (i.e. the data) have to be recalculated.
2. the recalculation has to be done by applying laws of physics.
3. we have however account for the change is postulates. The way we can describe this in modern terminology is by applying a Wayl-trafo to the metric tensor to get the change of the (one and two-way-) speed of light as desired. To frame this in Poincaré's words it leads "to replacing Newton's law [and all others, too] by another more complicated".
4. the distance data obtained from the new physics can look unrecognizably different.
5. since both the data AND the laws of physics have been transformed by the same conformal change of metric they still agree with each other. i.e the experimental data in the new metric will verify the new laws of physics in the new geometry.
6. However, obviously the data obtained in one metric (and left untransformed) will not agree with laws of physics in a different metric and vice versa. This is perhaps what you have in mind and see a contradiction, but this is not what i described.
7. There is no way to say which of the two postulates (defining the metrics) were correct, because in either case the measured data agrees with the laws of phyiscs in the corresponding geometry. Therefore the choice is left to the user.

 

[Dale]

of course it depends on the the coordinates charts just as much as on the metric. but those are independent degrees of freedom.

So you cannot claim that a comment about the one way speed of light is to be understood in context as about the metric only. It also inherently includes those independent degrees of freedom as well. You cannot ignore my rebuttal above by claiming a limited context because the context was in fact not limited that way.

Your one way speed of light comment simply does not hold, even in its context.

 

[PeterDonis]

I think in physical terms these transformation could be interpreted that the rescaling function applied to the metric tensor is effectively just slapping a refractive index function onto spacetime

Personal theories and personal speculations are off limits here at PF.

A conformal transformation is a mathematical way of defining families of geometries that are related by a conformal factor. It doesn't correspond to any physical transformation at all.

 

[PeterDonis]

if the postulates are changed...

You are very confused. Changing the postulates is something that happens in a model. It's not something that happens out in the physical system being modeled. You can't change a physical system by changing your model of it.

You are also confusing yourself by using the term "data" to refer to model-dependent calculated quantities instead of directly measured quantities. You can't change directly measured quantities by changing your model. Note that Poincare, in what you keep on quoting from him, draws a key distinction between "observed facts" and model-dependent quantities. You do not appear to grasp that distinction.

 

[Killtech]

A conformal transformation is a mathematical way of defining families of geometries that are related by a conformal factor. It doesn't correspond to any physical transformation at all.

You are very confused. Changing the postulates is something that happens in a model. It's not something that happens out in the physical system being modeled.

yes, these transformations are indeed deep math. but it's not like you cannot interpret what they do in practical terms. because the effect they have is to create a new model with different postulates in a new geometry. But you rightly point out that this geometries always remain related through the transformation factor - i.e. everything can always be transformed back. employing this powerful mathematical tool is something that can be made use to help various physical problems for example to removing singularities:
https://academic.oup.com/ptep/article/2020/4/041E01/5825413

But really, if you take your time and visualize what these transformation mean, it's not something that entirely abstract. i actually even have a very particular use case where i need to do such a transformation due to a practical scenario, just because in my case the entire world is perceived through a different class of signals that behave quite differently then light which complicates things a lot (also clock synchronization needs to be done via these signals). Basically i am in the situation of Poincaré's astronomer, except that in my case if i were to use the regular light-metric all calculation become incredibly complicated because i have to do full ray tracing of every signal(!), so taking Poincaré's argument offers a simpler approach: i just can assume my signals to travels at a constant speed in all directions instead so everything fits directly with the "measured" signal data without any need of further transformations - only thing i need however is to find how Newtons laws and a few others look in the new metric.

 

[PeterDonis]

the effect they have is to create a new model with different postulates in a new geometry

Yes, a new model. But there's no actual physical device you can build that will make an actual new geometry according to such a transformation. And the different geometry in the model has nothing whatever to do with the actual geometry of the actual world. So it's irrelevant for the discussion we are having here, which is about analyzing data from the actual world.

employing this powerful mathematical tool is something that can be made use to help various physical problems for example to removing singularities

You can't remove an actual, physical singularity in the actual world (supposing one were to exist) by changing your mathematical model. You are very confused.

i actually even have a very particular use case where i need to do such a transformation due to a practical scenario

Personal research is off limits for PF discussion; that is not what PF is for.

 

[jedishrfu]

Closing thread as the questions posed by the OP have been exhaustively answered and there's nothing more that can be said except thank you for participating here.

Jedi