
Maxwell's equations predict that the speed of light is constant - rhemam
http://proofs.wiki/Maxwell%27s_equations_predict_that_the_speed_of_light_is_constant
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abdullahkhalids
This page is proving that Maxwell's equations _predict_ that the speed of
light is constant. This prediction is dependent on assuming that the vacuum
permeability and permittivity are constant numbers. This is an axiom of
classical electrodynamics, specifically that free space is homogeneous and
isotropic [1]. Therefore, measuring the speed of light is a check on this
axiom.

[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.

~~~
sytelus
Exactly, it doesn't prove "constancy" of c unless you _assume_ constancy of
other constants. Even the isotropic and homogeneous assumption is incomplete
because it doesn't make any assertions on values through time. Is it possible
that u0 or E0 or c had a different value at very start of the big bang? It's a
very tempting thought although professional physicists seem almost universally
consider those ideas a giant crackpot.

~~~
weland
I don't know why you are getting downvoted. I haven't had my morning coffee
yet, so I am cynically going to presume a bunch of folks think you're
nitpicking on the words, but you're not.

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.

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jeffwass
FYI, this is standard fare for any undergraduate-level E&M class.

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](http://en.m.wikipedia.org/wiki/Levi-Civita_symbol)

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thearn4
Aside: I do computational applied math for a living, and haven't yet seen
proofs.wiki before. Is it fairly new? An editable wiki just for formal proofs
does seem like a decent idea.

~~~
nsajko
lawyeard, you seem to be dead.

~~~
wging
I typically see "[dead]" on dead posts. I don't see this on lawyeard's. Am I
missing something, or did the post become alive again?

~~~
saraid216
> or did the post become alive again?

Probably. Sometimes comments get autokilled and mods will disagree with the
algorithm, so they'll unkill them.

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kps
It's just too bad the speed of light is what it is. A few orders of magnitude
higher and we've have a galaxy to explore. A few orders lower and we've have
fun relativistic effects in daily life.

~~~
Peaker
More likely that we just wouldn't be here...

~~~
Padding
Indeed. You'd think of all the people out there, the ones on HN would be a
among the first to recognize an abstraction when they see one..

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.

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loblollyboy
This is awesome. I could never prove anything, but I like to follow along
(especially if the proof pertains to physics) and was thinking of presenting
this in a cool way for people who might think science is lame or just never
have been exposed (I got an undergraduate degree in Mechanical Engineering
without knowing what Maxwell's eqns were). Looks like you have done a good job
at it, I'll share with friends.

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amelius
I always thought that the speed of light was sort-of built into Maxwell's
equations through the permittivity and permeability of the medium (or vacuum),
and comes out as the 1/sqrt of the product of these parameters. See [1].

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.

[1]
[http://en.wikipedia.org/wiki/Maxwell%27s_equations](http://en.wikipedia.org/wiki/Maxwell%27s_equations)

~~~
scottlocklin
Anyone who has taken a JD Jackson based course has derived this result.
Special relativity is a fairly obvious consequence of E&M in hindsight.

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.

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FHorse
Did you prove that the speed of light is constant in different inertial
frames?

~~~
bainsfather
No:

"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.

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troymc
Wikipedia already has this at
[https://en.wikipedia.org/wiki/Electromagnetic_wave_equation](https://en.wikipedia.org/wiki/Electromagnetic_wave_equation)

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nthcolumn
But it is not constant. It was much faster long ago. So there must be shome
mishake.

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ebauch
like

