Maxwell's equations predict that the speed of light is constant
proofs.wiki
proofs.wiki
[1] homogeneity means that translating your experimental setup to a different location will not change the results of your experiment. Isotropy means that rotating your experimental setup will not change the results of your experiment.
The derivation is correct, but it is not a proof of anything. Maxwell's equations aren't textbook PDEs that exist in a vacuum, devoid of any connection with the surrounding world. They are quantitative approximations of real-life phenomenae, that hold only under very specific circumstances, and break when those conditions (about e.g. distances) aren't met.
The proof that the speed of light in vacuum (which the article "helpfully" fails to mention) is constant is experimental. If the experiment would fail, Maxwell's equations would have to be modified. Any predictions they make are tested against the reality, not vice-versa.
Oh yes: I'm not nitpicking about their failure to mention it's the speed of light in vacuum. The final expression is constant only because \mu and \varepsilon are (supposed to be) constant in vacuum. Precisely the same expression predicts the (correct!) fact that the speed of light is not constant, but depends on the properties of the material it is travelling through.
Edit: this should be very obvious from the fact that the derivation is based entirely on the four laws that constitute Maxwell's equations, not on theorems. Laws are not deduced mathematically, but formulated by experiment. Their predictions cannot constitute a proof. It may be tempting to treat their predictions as such, because in mathematics any correct result obtained by derivation can constitute a proof, but this is not the way to look at a set of physical laws. Maxwell's equations also "prove" that things like squeezed light cannot exist -- despite the fact that it clearly does, since we can generate it. These incorrect predictions are not onthologically different from correct ones.
That aside, is there any specific reason to believe homogeneity holds?
But these were chosen, based precise measurements, to ensure the (old) metre and the Ampere remained relatively unchanged.
As for your question, homogeneity is considered the simplest assumption you could make: you can't get simpler than a no-parameter model. It's also easy to disprove, if you can get data, making it a good scientific hypothesis.
There's some astronomers trying to measure the various fundamental constants in other regions of the galaxy via maser emissions. There was some interesting results a few years ago, but I don't think they're considered definitive. Still, lots of fame if they can do it!
Ehm… one-parameter…
The interesting part of deriving the wave equation from Maxwell's Laws, in my opinion, is how the permittivity and permeability of free space (epsilon_0 and mu_0) can both measured macroscopically in a small laboratory, by measuring forces between electric charges and current-carrying wires. Put these two seemingly unrelated values together and they give you the speed of light!
Another FYI - the author of the wiki proof in step 8 showed the vector calculus proof the 'long' (and boring) way, those vector calculus relations can usually be solved MUCH more easily and concisely using the Levi-Cevita algebra. http://en.m.wikipedia.org/wiki/Levi-Civita_symbol
Probably. Sometimes comments get autokilled and mods will disagree with the algorithm, so they'll unkill them.
http://proofs.wiki/index.php?title=Prerequisite:Copyrights&a... shows the copyright page is from December 6, 2014.
So, I assume this started around two months ago. The sparseness of content is not in conflict with that.
I think I would rather use a site that uses a formal proof checker behind it to verify committed theorems. It may get populated slower, but it can better be trusted.
"[The top-level domain name]... became available to the general public on May 26, 2014"
The "speed of light" isn't an intentional speed limit like the ones we have on highways. It's a mental cruth we use to describe a certain relationship that the elements forming our reality partake in .. Hence you can't reason about the "speed of light" having a different value the same way you could reason about a highway speed limit being altered in one direction or the other.
Is this proof giving a deeper understanding, in the sense that these parameters somehow dissolve?
EDIT: Ok, I've actually read the article and it seems this proof is fairly standard, and was well known long before even the wikipedia page was written. It is nice that this proof is now available in such a clearly written form, though. I wish all the material on wikipedia was so clearly written :)
EDIT 2: It would be nice if this proof could be written down using differential forms, instead of the rather clumsy curl-div-grad notation, that gives rise to the big matrices in this proof. In fact, it bewilders me why the coordinate-dependent curl-div-grad notation is even used anymore in education nowadays, but that is another story.
I take it from your question that you think that there's a way to incorporate the differential forms formalism for electromagnetism into an undergraduate physics major at roughly a 2nd-year level (or maybe 3rd year). I'd be really curious to know what you have in mind: I teach physics professionally, and I have trouble imagining how to do forms justice in a way that matches the intuitive grounding of div/grad/curl (which many students already struggle with, mind you).
FWIIW, this is a terrible proof with bad notation (you don't need to write out the matrix), which doesn't actually prove the interesting piece, which is the fact that the permittivity of free space is the same no matter what your frame of reference is.
"Note that there are two way to interpret this result:
The speed of light was constant with respect to the aether. This theory turned out to be false. Read more about the Luminiferous aether.
The theory of Relativity"
The most they say is just the assertion: "Maxwell's equations are Lorentz invariant"
Before the Michelson-Morley experiment 'disproved' it, it was assumed that the waves propagated through the 'aether', and that c was therefore the speed relative to the aether. So, in addition to Maxwell's equations, extra experimental data was needed.