E=MC2

[ramparts]

this thread got me thinking about the array of formulas that the speed of light shows up in. It's also neat how it shows up in the macroscopic (classical) world and the quantum realm. Here's just some (all written with c on the left):

(1) In the eq we have been talking about:

c^2 = E/m

(2) As it comes out of the wave equation and relates to the permeability and permittivity of free space (vacuum)

c=1/sqrt(epilson_0*mu_0)

Indeed - that sets the speed of electromagnetic waves as a combination of the constants relevant to electricity and magnetism. I wouldn't take that to mean epsilon_0 and mu_0 define c in any way - rather, for a massless photon in a vacuum (which we know should travel at c), given epsilon_0 or mu_0 the other is set automatically. What this actually is is a very nice way of unifying electricity and magnetism.

(3) In the fine structure constant, which is all over Qunatum Field Theory and shows up in Quantum Mechanics in the Bohr radius of the hydrogen atom and probably other places

c ~= e^2/(137*hbar) ; where e is the fundamental unit of charge, and hbar is Plancks constant

Don't ask this lowly undergrad about the fine structure constant and such fancy things The Bohr radius doesn't need to depend on c, it only shows up in that equation if you couple it with the fine structure constant (to get rid of the e^2). But go ahead and derive the Bohr radius using some fundamental quantum mechanics - no c in there.

(4) As the phase velocity in the wavelength/frequency relation for an electromagnetic wave (and as ramparts pointed out, it's the speed for any particle of zero mass)

c = lambda*f ; where lambda is wavelength, f is frequency

Right, this is what I see as most fundamental about c - if you measure the velocity of any wave with no mass, you'll get c.

(5) Compton wavelength, which represents a fundamental limitation on measuring the position of a particle

c = hbar/(m*lamda)

Nothing really special about this that distinguishes it from the other cases. E=mc^2 (as we've discussed) and also equals h times nu (the frequency) - the de Broglie relation you mention next. Rearrange to get the Compton formula (using c = lambda times nu, which is trivial).

(6) I beleive it shows up in the De Broglie wavelength, which is the wavelength of a matter wave, but I forget the details here.

Well, Planck's constant h is really the important constant in the De Broglie relation. h relates the particle's and its frequency. If it moves at velocity c, then as I said before, it's not very special to note that c relates the frequency and wavelength of the wave.

And, I'm sure there's more I'm just not thinking of off the top of my head or have in memory like Hawkings Entropy Black Hole thingy. Where else?

I'm not sure what this means, other than it's really neat how c is everywhere! Yes, much more than just the speed of a photon.

The black hole entropy involves c in the Planck length (which also uses hbar and G). This shouldn't surprise us - c is fundamentally engrained in spacetime (as the constant that "converts" time to space), so it will appear in practically any result that involves general relativity or quantum field theory. The Schwarzschild radius, the right side of Einstein's field equations, the Friedmann equations, etc. It comes up so often we usually just set it to one so we don't have to deal with it

EDIT: this might belong in the other thread on c, but it looked like the whole f=mv^2 thing killed that thread .... or, was that v^2=f/m?

:lol: don't tell the scientists! The hidebound reactionaries will never listen, with their "dimensional analysis" and resistance to re-arranging equations.

 

[darkmatter4brains]

Good points ramparts. I think your one statement about c being engrained into spacetime is where I was really heading with that. I'd go even further. I think the reason c is so ubiquitous throughout physics is because there is something even more fundamental going on with how it is tied into "reality" (including 4d spacetime AND beyond) than we currently know at this state. I think once we figure it all out, all those equations and their relations will seem "not so special".

I've got my own personal opinions, but it basically assumes that what we now view "light" as is just a shadow of something more fundamental. I think the true phenomena of light and c is something we haven't quite figured out yet. I suspect one of these attempts at unification, like string theory, which involve higher dimensions is probably the way to go and will end up elucidating the matter more. But, we'll see ...

 

[ramparts]

I would take it a step further - I don't think there's anything all that special about light. It was just the first massless particle we were able to detect and measure.

 

[darkmatter4brains]

most definitely, I agree!