Planck's constant, the speed of light and the permeability of free space all have units. So an alien race who measured things differently might not have these constants.
For example humans traditionally measure Energy and Frequency using different units. However we know, from light for example, that Energy and frequency are really the same things and "should" be measured using the same units. A single units of frequency is "really" Joules and this is what Planck's constant records.
Only dimensionless quantities have a real existence outside of our mismatch of units. Hence this whole discussion should only be about the fine-structure constant and not the speed of light, as it is the fundamental quantity.
The basic point is that dimensionful constants can always be eliminated from equations of motion. Basically there is no way to distinguish, really, between a change in 'c' and a change in your units of distance. The only "real" change independent of your unit system, would be a change in the fine structure constant.
Basically any change in units that alters physics can be fundamentally reduced to a change in the fine structure constant or other dimensionless constants. However there are variations of dimensionful constants that are indistinguishable from a change in units and hence have no physical meaning.
I appreciate your patience, but I am still struggling with this idea.
No worries, it's an odd idea and I didn't explain it well.
I see a distinction between c being 3.0 * 10^8 in MKS units and being 6.0 * 10^8 in the same units, because that change would have an affect on matter/energy equivalency.
Let's take two universes with these two different values of c, but with the same value of alpha, the fine structure constant. It would turn out that the laws of physics work out exactly the same, except that in the universe with those laws of physics play out twice as fast. So what takes 2 units of time in the first universe takes 1 unit of time in the second universe.
Hence it's just the same universe viewed using two different measurements of time. The units of one system are worth twice the units of another system.
It's only if you changed alpha that you would get a universe that couldn't be understood as the same universe sped up or slowed down.
A better example might be a world where the North-South distance along the Earth is measured using furlongs and the East-West distance using meters. Then in geographical calculations you'd always have a constant D = 0.005 furlongs/meter, when computing areas. You could ask what would happen if D changed value, but it is meaningless, it would just be a redefinition of the meter or the furlong (or both).
The constants of physics do change (effectively) with Energy, for example high-energy electrons basically have a larger electric charge than low energy ones.
The changes that are being discussed in this thread are changes over time. The problem is that if the constants change in such a way, then they vary from place to place and time to time (They can't just vary in time, according to relativity). Hence they are fields and under quantum mechanics fields produce particles. However we have not seen these particles, so it would seem this idea is ruled out.
However it is possible that the constants are classical fields (they don't obey quantum mechanics) and just vary over space and time in the normal sense. However experiments put very tight bounds on this variations. Current experimental bounds show that if the constants vary, they vary extremely slowly and to such a small degree that they couldn't generate the effects you describe.
Also you are not taking into account what these constants affect. Let's say you want to speed up radioactive decay. To do this you could vary the Weinberg angle or the Electroweak coupling (both fundamental constants which control radioactive decay). However if you adjust these constants to the point where radioactive decay becomes appreciably larger, organic molecules themselves would be highly unstable and the Sun would have stopped functioning. Also the Sun would not have started to work when the constants hit their present values.
Overall, this idea is experimentally ruled out. Interesting in theory, but false.
When you say effective electron charge, what do you mean? Are you referring to the hypothesis of variance of electron fundamental charge?
There are a few ways to define the electric charge of the electron. One of them is the renormalised charge (a better name would be the Coloumb charge). This is basically where you take the electric charge to be the value of the probability that an electron's momentum isn't altered when it interacts with a photon. Although this seems quite abstract it's actually the definition of electric charge that mathematically agrees with one's intuitive idea of charge, i.e. if you define charge this way then charge behaves like you would expect from classical mechanics.
The only subtlety is that this probability changes with energy. For large energies it's larger. So the electric charge changes with energy. We call this a "running of the coupling". The strong force coupling is the opposite, it decreases with energy, a fact known as asymptotic freedom. This because at high energy the strong force charge asymptotically approaches zero. When the charge charge is almost zero, particles don't really interact, hence "freedom".
A citation for my statement would be Weinberg "The Quantum Theory of Fields" Vol. II Chapter 18 or Peskin and Schroeder "An Introduction to Quantum field theory" Chapter 12.
Here is an image on experimental results from different collisions on the strong force coupling:
The vertical axis is the value of the strong force coupling, the horizontal axis is Energy measured in GeV.
I believe alpha radiation also has to do with the strong nuclear force.
Alpha radiation is an odd one, in the sense that it doesn't really have anything to do with any of the forces. A nucleus that is unstable to alpha decays can be considered as "Alpha Particle + the rest". The strong force (or rather the pion force, an artifact of the true strong force) erects an potential energy barrier that prevents the alpha particle from escaping classically. However quantum mechanically the probability field of the alpha particle can just spread to the other side of the barrier and hence when something measures the alpha particle it has a chance to just appear on the other side.
There are still Weak Force contributions to this potential barrier, but they're very small. However in order to speed up the radioactive decay from carbon dating, you need to alter the weak force constants. You have to increase them so much for carbon dating that you'll effect alpha decay by altering this potential. You would also make neutrons very unstable.
When is the Weinberg angle considered a fundamental constant? It may be affected by other fundamental constants but it is not fundamental, as I know it.
It depends on which constants are affected and to what degree. That would be the difference between the decay of uranium for instance verses that of say heavy water.
In the Standard Model, the decay used in Carbon dating involves the electroweak constants much more than the electromagnetic couplings or strong force couplings. The electromagnetic and strong force couplings contribute so little to it, that if you wanted to use them to change carbon decay rates you'd have to alter them enough change: (i) The chemical properties of all elements. (ii) The nuclear stability of several elements.
For instance the square of alpha, the electromagnetic constant, contributes (via virtual electron bubbles) to the energy levels of several atoms. Change alpha enough and glow in the dark stickers stop functioning, the aurora borealis disappear and several other chemical changes (kerosene can no longer burn), that would make the past completely different to the present in an extremely obvious way that we don't see any evidence for.
The only hope for speeding up carbon decay is via the Weak Force, but even though this can effect carbon decay without to much chemical effects, the nuclear effects are far reaching. Stars would shut down.
Actually, the Sun’s formation itself is still not workable physics, regardless of the fundamental constants.
There are small questions over how the area of the nebula which produced the sun collapsed, i.e. how many supernovae instigated it. However we know how the cloud collapsed, numerical simulations of our collapse model match the current orbits of the solar system, including the outer eccentric orbits.
Therefore, I deem the claim it would stop functioning as a mute.
Why? How the gas cloud which formed the Sun collapsed, has nothing to do with the process of nuclear fusion in the Sun itself. Nuclear fusion within giant hydrogen/helium spheres doesn't depend on how the spheres formed. If the Weak Force is altered enough to prevent Nuclear fusion, then Nuclear fusion cannot happen regardless of where the Sun came from.
In other words first describe how it started to work then you can make claims on if it will not work.
(i) You don't need to know where something came from in order to know if it will continue to function if changes are made. If I found a ladder, I've no idea where it came from, but I know it's not going to survive or be functioning after I blow it up. Similarly regardless of how the Sun formed, if you alter the constants such that Nuclear Fusion is impossible inside spheres like the Sun (or much less efficient), then that's it, no Nuclear Fusion.