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RolfAndreassen comments on Local truth - Less Wrong Discussion
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Taboo 'fundamental'; what distinction are you trying to get at? Change the speed of light holding other parameters constant, and the fine structure constant changes; and vice-versa. Whether you want to consider this a "change in the speed of light" or a "change in alpha" seems to me pretty strictly a matter of taste; and the speed of light (especially its constancy) is much more familiar to most people than is alpha. (Which anyway may not be so constant at high energies, in spite of having 'constant' in the name.)
We are pretty far off the original question, but, yeah, there is a distinction. The fine structure constant is indeed more fundamental than the speed of light. How do you change the speed of light and not much else? You have to change at least the fine structure constant (in the IR limit, as you point out) and probably the gravitational coupling strength, and possibly some other free parameters in the standard model, like masses of the electron and quarks (again, in some dimensionless units). Otherwise something will go wrong with the Universe, the common example being carbon not being synthesized by stars. But I agree, these "side effects" of changing the speed of light may not have been important long time ago, or maybe in the galaxy far far away.
But this is just as true if you change alpha! At a minimum, to restore the carbon synthesis rate, you'd have to mess with the proton charge.
The numerical value of a dimensionful constant depends on the system of units you're measuring it in; that of a dimensionless constant doesn't.
Well yes, but who cares about the numerical value? Hold the system of units constant and look at the actual physics!
How would you do that? (The metre is now defined in terms of the second and the speed of light.)
I would define the unit of distance in terms of the size of the King's lower extremities, as God intended. :rolleyes:
How would you distinguish an universe where the, ahem, King's lower extremities grow and everything else stays the same from one where the King's lower extremities stay the same and everything else shrinks? (This is essentially the same point as that of this post but for scaling rather than rototranslations.)
(Well, the last time I had this discussion IIRC I initially had John Baez on my side then the person I was arguing against managed to convert him via private e-mail, but I couldn't understand what he said she said to him.)
Why would I want to?
My point being, you are not under any obligation to define the meter in terms of the speed of light when you are discussing a possible change in the speed of light. You are attempting an argument by definition, and rather a silly one. Define the meter in a way that's convenient to what you're trying to do, and have done.
Like what?
Like using the King's foot, or the curvature of the Earth, or a platinum bar, or indeed anything that isn't defined in terms of the thing whose changes you are trying to talk about.
The fact is that platinum atoms are bound together by the electromagnetic force, so if c = 1/sqrt(mu_0 epsilon_0) changes their distance changes too. (Who the hell is downvoting the entire thread, anyway?)
One of the points is that different frames of reference have different ratios of the length of artifacts, when those artifacts are moving relative to each other.