How to rule out that the speed of light was different in the past?

[PeterDonis]

But in the fine structure, there're other "constants" playing other than π and c.

You're looking at it backwards. As has already been pointed out in this thread, the actual physically meaningful constant is the fine structure constant, since it's dimensionless. The "constants" that appear in the formula for the fine structure constant are dependent on your choice of units.

For example, the Universe is expanding and seems the expansion is accelerating.

This has nothing to do with the fine structure constant.

as space is expanding, then for a non-local observer the speed exceeds c if computed globally, but not measured locally.

The "speed" you are talking about is coordinate dependent and has no physical meaning.

What I don't understand (among other billion of things) is that my measuring 1 meter rod is not expanding itself

Correct, because its length is determined by the electromagnetic interactions between its atoms, i.e., by the fine structure constant.

is the space outside the rod that's expanding

This is a common pop science "explanation", but it's coordinate dependent.

or I could not measure the expansion!

If you mean expansion of the 1 meter rod, there is no such expansion to measure.

 

[Dale]

Of course for each of the definition you need also the "mises en pratique" for the units. E.g., to realize the definition of the kg via the defined values Planck action $h$ (and also of the definitions of the s, the, m, and the A) with the Kibble balance what's accurately measured are quantities like the magnetic-flux quantum in superconductors (Josephson constant).

You find the corresponding brochures in English here:

https://www.bipm.org/en/publications/mises-en-pratique

The mises en pratique is, IMO, a rather under appreciated part of the new SI. It is great that they have gotten away from the “artifact in a vault” definitions and are entirely based on physical constants. But without the mises en pratique it would be unclear how to actually measure the units. They provide a very important link between the SI as a theoretical construct and the SI as a practical thing.

 

[vanhees71]

The good thing of the "new SI" is that you can adapt the mises en pratique easily, if better technology becomes available. I guess "soon" (~some years) we'll see a change in how the second is realized since the $\Delta \nu_{\text{Cs}}$ is not so accurately realizable as the intrinsic accuracy of several optical atomic clocks (if I remember right it's an order of magnitude better) or probably soon the optical nuclear Th clock. The problem is of course that you have to get an accurate measurement of $\Delta \nu_{\text{Cs}}$ not to change the definition of the second by more than you can realize the unit right now. For details, see the "mise en pratique" for the second:

https://www.bipm.org/documents/2012...48?version=1.12&t=1643724477633&download=true

and the standard values for various optical atomic-clock frequencies

https://www.bipm.org/en/publication...ies?version=1.4&t=1637238077933&download=true

 

[Nugatory]

What I don't understand (among other billion of things) is that my measuring 1 meter rod is not expanding itself, is the space outside the rod that's expanding - or I could not measure the expansion!

The effect of expansion is that if the two ends of your meter stick were completely isolated, separate from one another and subject to no external forces at all, their worldlines would very slightly diverge as they follow their freefall paths through spacetime. (Exercise: calculate the amount of separation per year under these completely unrealistic hypothetical conditions).

In practice the two ends of the meter stick are not isolated. They are solidly attached to one another, and even if they weren't they would be bound together by the gravity of the earth, by the sun, the galaxy. Even gravity across our entire galactic cluster does more to hold them together than expansion does to separate them.

However, all of this is a digression in this thread. The speed of light in vacuum is 299792458 meters per second for the same reason that is also one light-second per per second and one light-year per year - if we were to measure some other value we would only know that one or both of our clock and our meter stick are out of calibration. However, any change in the physical behavior of propagating electromagnetic radiation would show up as a change in the fine structure constant, which is why you are being told to stop talking about the speed of light and instead focus on the fine structure constant (or other equivalent dimensionless quantities).

 

[hutchphd]

What I don't understand (among other billion of things) is that my measuring 1 meter rod is not expanding itself, is the space outside the rod that's expanding - or I could not measure the expansion!

In order to"measure" your meter rod, you necessarilly must compare it to another meter rod either real or created from a compounded measurement of length
If you find that comparison (ratio) changes in time then something in the chain of inference is not constant.
If the ratio is static then most probably all elements of the measurement are constant:but two or more could be changing in collusion
This argumant holds also for any nondimensional constant in any system

 

[Sagittarius A-Star]

but I can't tell what was the "local" speed then and there where the photon was produced.

GR spacetime is locally flat. That means, locally on earth and locally on the quasar SR is valid. The invariant speed of SR can be defined via a unit system to be $c=1$.

A photon must move with this invariant speed $v=c=1$ in every (local) inertial reference frame because it is massless.
$0=E \sqrt{1 - v^2}$

 

[Lluis Olle]

This has nothing to do with the fine structure constant.

I have no other access to this article, but the "poor's man" first page :), but seems very interesting. As is from 2004, perhaps is outdated nowadays.

In this first page it talks about why you could not detect a change in c, because all you could say is that there's a change in α.

Nevertheless, seems that the expansion of the Universe could have something to do with the fine constant.