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army1987 comments on Local truth - Less Wrong Discussion
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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?)
Equally a problem with the fine structure constant, since it directly measures the strength of the electromagnetic force. However, while the length of a platinum bar would, as you say, change under a change in the speed of light, it would not change linearly; so you can measure ratios of lengths and presumably pinpoint the constant that changed. The curvature of the Earth, for example, presumably depends on the strength of gravity in addition to the electromagnetic repulsion of its constituent particles, so it should change more slowly than the platinum bar's length.
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.
Usually when defining lengths in terms of an artefact you use the rest frame of the artefact (i.e. the one in which the artefact is longest).
So what? Stick to one frame of reference; thought it doesn't matter which one you use, that doesn't mean you're not allowed to use one. Every observer will see the same change in the speed of light using their own measure of distance; or at any rate can agree on the change by mathing out the frame-of-reference effects.