Constancy of the speed of light

[PeterDonis]

as it is the same set everything formulated on one can be translated to an equivalent on the other via the identity bijection

This is the "identity" for the topological space $\mathbb{R}^2$. But it is not the identity for the manifold "Euclidean plane", because this transformation changes the geometry from the Euclidean plane to something else. So you need to be very clear about exactly what a given transformation does and does not keep the same. At least part of your confusion appears to come from failing to do this.

how is the geometry of space measured?

By the distances between points. In the case of the manifold $\mathbb{R}^2$, specifying the distance according to your chosen metric between all pairs of points fixes the geometry. And your transformation above does not preserve this: if we take two points A and B, the distance between them according to the Euclidean metric is not the same as the distance between them according to your alternate metric. So again, your transformation changes the geometry.

If you are really concerned about keeping things independent of your choice of units, you can rephrase all of the above in terms of ratios of distances between different pairs of points: for example, take two pairs of points, (A, B) and (C, D), and look at the ratio of the distances between them according to the two different metrics. Everything I said still applies: if any of these ratios change, you have changed the geometry.

using the Euclidean metric the distance between $d_2(\begin{pmatrix}0 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 2\end{pmatrix})$ is twice as long as $d_2(\begin{pmatrix}0 \\ 0\end{pmatrix}, \begin{pmatrix}\sqrt 0.5 \\ \sqrt 0.5\end{pmatrix})$ and this ratio has no unit anymore.

Yes.

But using the $d_\infty$ metric the same ratio changes to $2 \sqrt 2$.

Yes. And this is an example of what I said above, that your transformation changes the geometry.

I'm not sure I can make sense of your specific comments about metrology; I think you have confused yourself by failing to pay proper attention to the points I've made above. But I'll try to illustrate what I've said with an example, since you mention SI units: the definition of the SI second in terms of the radiation emitted by a specific hyperfine transition in Cesium is the equivalent of picking a particular pair of points (A, B) in spacetime and calling the spacetime distance between them (which in this case is a time, since the two points are timelike separated) the "standard" distance, and expressing all other spacetime distances as ratios with that standard distance. The SI definition of the meter then just extends this to spacelike distances as well as timelike distances, by fixing the speed of light--which is really the chosen ratio of "space distance" units to "time distance" units--to a particular value. But we could choose different definitions of the second and the meter (and we previously did), without changing any ratios of spacetime distances, i.e., without changing the spacetime geometry. All we would change is which particular spacetime distances we called the "standard" ones.

 

[Killtech]

This is the "identity" for the topological space $\mathbb{R}^2$. But it is not the identity for the manifold "Euclidean plane", because this transformation changes the geometry from the Euclidean plane to something else. So you need to be very clear about exactly what a given transformation does and does not keep the same. At least part of your confusion appears to come from failing to do this.

If you are really concerned about keeping things independent of your choice of units, you can rephrase all of the above in terms of ratios of distances between different pairs of points: for example, take two pairs of points, (A, B) and (C, D), and look at the ratio of the distances between them according to the two different metrics. Everything I said still applies: if any of these ratios change, you have changed the geometry.
[…]
By the distances between points. In the case of the manifold $\mathbb{R}^2$, specifying the distance according to your chosen metric between all pairs of points fixes the geometry. And your transformation above does not preserve this: if we take two points A and B, the distance between them according to the Euclidean metric is not the same as the distance between them according to your alternate metric. So again, your transformation changes the geometry.

Okay, it seems I am missing something here. Perhaps I am not seeing the elephant in the room, but why would a change of geometry be a problem? I see it affecting calculations but not predictions.

I mean sure, a bijection between the two manifold is indeed not entirely trivial as I might have indicated but it exists nevertheless. The identity of the topological space is enough to translate every point $x$ from one manifold to another. Therefore and if the topology is the same it can also translates any scalar function $f(x)$ trivially and all coordinate charts $\phi$ remain conveniently the same, too. Furthermore any vector in the tangent space at $x$ can be decomposed into a linear combination of the chart gradients $d\phi_x$ which yields a bijection for the tangent space since the term is defined on both manifolds. I think in more general context this is called a pushforward in diff geo. Translating equations is a bit trickier since their coordinate free formulation is metric specific – however upon picking coordinates it becomes metric independent which yields a clear translation. The coordinate free formulation can be then derived from there and does not depend on the coordinates chosen for translation (you can do it with the pushforward instead). That leaves us with nothing which we can’t find a bijection for thus I don’t see where any information could get lost or changed irrecoverably in the process.

Therefore purely mathematically speaking I see no problem in changing the geometry. Well, at least as long the metrics are mathematically equivalent (i.e. same topology). So I don’t see any possible way such transformation could have any impact on the physical predictions as there is always an equivalent formulation of everything for both manifolds. It works like a distorting mirror so to speak.

And I would have believed that any kind of transformation that leaves all possible predictions of a model unchanged should be physically acceptable. A change of metric and therefore geometry should have this property or at least I am not able to think of anything that would be predicted wrongly. I mean because anything that can be expressed in coordinates will remain identical, equation must yield identical solutions. And even the distance ratios I used earlier can be predicted correctly in another metric – but since distances/measurement results have a transformation behavior under a metric swap one needs to apply it first: while a distance ratios change, a quotient $\frac {f(\mathbf x, \mathbf y)} { f(\mathbf x, \mathbf z) }$, where $f$ is directly derived from the pushforward and the base metric, does not. Again I don’t see anything I would miscalculate in a different metric.

The above may also explain my prior focus on metrology. The used measurement system always induces a metric those results are valid for – and they need to be transformed first before they can be used in a different geometry. Alternatively one can setup a measurement system that is compliant with the metric chosen. If there is nothing naturally behaving accordingly there is the possibility create one artificially. And even for something as stupid as the maximum norm one can do that in reality if one is stubborn enough: gyroscopes can be used to find the three axis outlined by the metric such that one can construct a device that measures distances optically at first, then decomposes it into the directions provided by the gyros and finally displays only the maximum component. Notably there is no unit transformation between SI meter and such a $d_\infty$ meter since they don’t transform by value (only though the pushforward/pullback instead as a function of the point-pairs).

Hmm, maybe I should try to formulate Newtown’s kinematic equations in the $d_\infty$ metric and see if I get any wrong solutions. Sometimes when just talking about stuff one can be blind for the most obvious error but when one does the calculus it gets hard to miss.

 

[PeterDonis]

why would a change of geometry be a problem? I see it affecting calculations but not predictions.

Then you see it wrong. If you change the geometry, you change the predictions of observable quantities. You can't change the geometry while holding the predictions of all observables constant; the geometry is one of the observables.

purely mathematically speaking I see no problem in changing the geometry.

Both geometries are valid solutions of the mathematical equations, yes. (At least, I'm assuming they are; I haven't checked the particular example of the $d_\infty$ metric you give.) But these two solutions are different physically; they make different predictions about physical observables.

At this point I'm not sure how else to help you. I've said the above before, and so have other people, repeatedly in this thread, but you still don't seem to grasp these fundamental points.

 

[Boing3000]

Therefore purely mathematically speaking I see no problem in changing the geometry.

And we do that all the time

Well, at least as long the metrics are mathematically equivalent (i.e. same topology).

Equivalent ? The metric is the same, or not. It cannot change the geometry without changing the topology. Antarctica is not the biggest continent on earth...

So I don’t see any possible way such transformation could have any impact on the physical predictions as there is always an equivalent formulation of everything for both manifolds. It works like a distorting mirror so to speak.

I think you are mistaking change of coordinate with change of metric. Maybe this link will help you out

The above may also explain my prior focus on metrology. The used measurement system always induces a metric those results are valid for...

Measurements don't induces metric. Metrology is not geometry. It is grounded in physical apparatus... and measurement.

 

[Killtech]

Then you see it wrong. If you change the geometry, you change the predictions of observable quantities. You can't change the geometry while holding the predictions of all observables constant; the geometry is one of the observables.

Both geometries are valid solutions of the mathematical equations, yes. (At least, I'm assuming they are; I haven't checked the particular example of the $d_\infty$ metric you give.) But these two solutions are different physically; they make different predictions about physical observables.

Hmm, that was a great idea to try to express the metric swap in terms of how it affects geometric observations! Basically this means to look at how the geodesic equation of a test particle in the original geometry transforms to the new one. Since each trajectory solution must remain identical (because this transformation is defined via a bijection) while the Christoffel symbols change (even for the identical coordinates) additional new terms appear. These terms exactly compensate the change in geometry. But as these equation has a direct analogy to Newton’s law the (predominant) new term would be identified as a global force vector field on the new manifold. Therefore any real object which movement is described by a geodesic solution in one geometry would still behave identically in the other. However, its description as being in a rest frame falls apart (i.e. not a geodesic solution in the new metric) while all actual observations and predictions that can be made about its time evolution remain untouched.

Besides, I have found that in newer publications there are equivalent formulations of GR in different geometries like in this paper.

As for my remarks made about metrology, the quote I took from the link Boing3000 provided seems to comply with my understanding between measurement and metric. Is it wrong or am I misunderstanding it somehow?

Equivalent ? The metric is the same, or not. It cannot change the geometry without changing the topology. Antarctica is not the biggest continent on earth...

Sorry, the world equivalent was misplaced here. Since the metrics i used in my examples before were based on norms i actually referred to norm equivalence (see line 10). Equivalent norms induce the same topology. Same applies to metrics though the terminology is a bit different. Still, two metrics can yield the same topology and yet a very different geometry. The topology merely governs what a continuous (or smooth) function is but has little impact on the shape of the space apart from global properties like its homotopy group of a manifold (e.g. torus or sphere). The metric transformations i considered would definitively preserve the topology but they would e.g. not preserve the ratios of distances.

I think you are mistaking change of coordinate with change of metric. Maybe this link will help you

Measurements don't induces metric. Metrology is not geometry. It is grounded in physical apparatus... and measurement.

I am sure I do not mistake it with a change of coordinates. But thanks for the link you posted. The best answer to the question asked in your link contains an interesting formulation which very much summarizes exactly what I though the connection between metrology and metric was (it somewhat contradicts your second statement):

Clocks don't measure time, they measure the metric applied the the worldline of their path in 4d spacetime. Rulers don't measure distance, they measure the metric along their path in 4d spacetime. Everything you are used to thinking of as a measurement actually measures the metric.

I tried to generalize this thought by exchanging the arbitrary ruler with what uniquely determines the validity of any distance measuring method including the ruler – which I assumed was in the end the metrology.

On the other hand if rulers measure the metric, then this begs the question if it possible to come up with some device that measures another one. And this seems to be within our technical possibilities to do practically, at least for some simple example cases: See my example (see paragraph 5) for the $d_\infty$ metric. The lack of practical use aside, would it not measure a different metric?

 

[PeterDonis]

Since each trajectory solution must remain identical

You're misusing the word "identical". The only "identity" is the arbitrary designation you make of a point in one manifold being "the same point" as the point in the other manifold that the transformation maps it to. (It seems like you are using "has the same coordinates" as designating the points.) But that is a purely mathematical "identity" that has no physical meaning. The points in the manifolds are abstractions; they aren't real points in a real physical spacetime. The way you identify the real physical points is by real physical observables. Since those observables change when you make your transformation, there is no valid physical sense in which the points mapped to each other by your transformation are "the same".

if rulers measure the metric, then this begs the question if it possible to come up with some device that
measures another one

Now you are misusing the term "metric". The term "metric" means "what rulers and clocks measure". More precisely, the readings on rulers and clocks are the physical observables that the "metric" in the math corresponds to. It makes no sense to say that some other device, that measures different physical observables, measures "a different metric". It measures different physical observables. Period. That's all you can say.

 

[Dale]

Perhaps I am not seeing the elephant in the room, but why would a change of geometry be a problem? I see it affecting calculations but not predictions.

Are you talking about changing coordinate charts on one manifold or changing manifolds? Changing charts does not change predictions, but changing the manifold does.

 

[Boing3000]

Equivalent norms induce the same topology. Same applies to metrics though the terminology is a bit different. Still, two metrics can yield the same topology and yet a very different geometry.

I am kind of nonplussed by your focus on topology. How is it relevant to physics and metrology ? It reminds me the joke of how much coffee can you put in a doughnuts.

I am sure I do not mistake it with a change of coordinates. But thanks for the link you posted. The best answer to the question asked in your link contains an interesting formulation which very much summarizes exactly what I though the connection between metrology and metric was (it somewhat contradicts your second statement):

No, it does not. The article do not event mention metrology. What is said is a physical apparatus don't measure the abstract coordinate system you've made up in your mind to "plot" your event. Those spaces are abstractions. The clocks and rulers aren't.

On the other hand if rulers measure the metric,

But what could it measure otherwise ? The elephant in the room you keep misunderstanding is the "inducing" part. The abstract mathematical metric is not "created by" the apparatus. It is created by human minds that need a "metric" to compute distance anywhere in the manifold. To create such a metric (and measure it with apparatus to see if it actually works) the human mind need some leap of imagination. Pythagoras made some, but his metric that work on a "virtual" 2D sheet of papper, do not work on 2D Earth surface.

then this begs the question if it possible to come up with some device that measures another one.

You mean another type of "distance" metric ? That would be fun. Maybe we'll call it retem and dnoces.
Good luck with persuading the officer that caught you speeding that you actually used another topologically equivalent speedOretem that's better than his

 

[bobob]

Now going back to physics we use a very specific metric to describe the world. I want to understand where this metric originates from and how it is exactly linked to real measurement results - which is needed to make any predictions that are verified experimentally.

Define a "real measurement result." What are you going to assume? If you have a stick lying on the ground and you stand it up, do you expect the stick to be the same length? If so, you've just assumed a principle of invariance and along with it comes mathematics that binds you to that assumption, which can then be tested.

 

[bobob]

and how do you compare? let's assume you have bars that change their length depending on the part of space they are in.

If you want to assume that you can and you can try to create a physical theory around that, but no one knows how to create a physical theory that doesn't assume the laws of physics don't depend on where and when you are.

 

[grandbeauch]

I read in 2014 that the speed of light is not what they once thought. It is beacuse the photons break apart and then come together again so it is slower than posted speed limit. sorry about adding to ur comment it was a mistake.

 

[Matterwave]

I read in 2014 that the speed of light is not what they once thought. It is beacuse the photons break apart and then come together again so it is slower than posted speed limit. sorry about adding to ur comment it was a mistake.

Is there a source for this? Photons are elementary particles and so they can't really "break apart". If they can - and this claim is true - then that's a pretty fundamental change to our understanding of physics.

 

[grandbeauch]

I know I was shocked and I will try to find the article. It was 4 years ago and sorry if I misquoted.

 

[grandbeauch]

I know I was shocked and I will try to find the article. It was 4 years ago and sorry if I misquoted.

found it
Physicist suggests speed of light might be slower than thought
June 26, 2014 by Bob Yirka, Phys.orgRead more at: https://phys.org/news/2014-06-physicist-slower-thought.html#jCp

 

[PeterDonis]

Here is a link to the actual paper by Franson on arxiv.org:

https://arxiv.org/abs/1111.6986

 

[Matterwave]

Thanks for the reference. I can't really comment too much on it since I don't have the time to read it over in detail. It looks like the paper has been cited less than 10 times in the last 7 years though so it seems to have gotten very little traction. That's not a critique of the argument of the paper, but only that it seems it's not well known even within the physics community.