On Sat, 2 Aug 2008, Salmon Egg wrote:
> I have always heard that the Maxwell equations are relativistically
> invariant but have never seen a proof--certainly not a simple proof. If
> so, do these equations somehow hide the transformation of the
> electromagnetic tensor that combines electric and magnetic fields into
> one unified field?
No. The Maxwell equations are just a bunch of PDEs. They're
relativistically invariant under the Lorentz transformations, but also
under the Galilei transformations. Again, along the lines of your original
post, the Maxwell equations alone are not the whole story.
What makes classical EM "relativistic" is epsilon_0 and mu_0 being the
same in all inertial reference frames. Note that Maxwell and Hertz assumed
that D = e0 E and B = u0 H only held in the rest frame of the ether. See
Maxwell's "G", the velocity of a point (relative to the ether) in his
Treatise, and Hertz's paper on the electrodynamics of moving media. Until
Lorentz, the theory of classical electrodynamics was _not_ relativistic
(in the modern sense of "relativistic"). Of course, until Lorentz, it also
failed to describe electromagnetism in moving media properly ...
Given a field equation, describing how a field A depends on source B,
C(A) = B,
where C is some linear differential operator, and an equation giving the
force (density?) acting on the source,
F = D(B,E)
where D is a function of B and some field E,
and a consitutive relation telling us how E depends on A,
E = a A,
if we assume a to be a Lorentz-invariant scalar (i.e., the same in all
inertial reference frames), what restrictions are there on C and D for the
whole theory to be Lorentz invariant?
I don't know the answer to this, but I do know that the key element in the
"relativistic" part is "a" being Lorentz-invariant. What C and D will tell
you is the relativistic transformation of the fields.
Which is why the Lorentz-relativistic transformation of EM fields differs
from the Galilei-relativistic transformation of EM fields, even though the
EM fields in both cases satisfy the Maxwell equations.
> I already realize that I need F = q E to define the electric field for
> use in Maxwell's equations. What is the equivalent definition for B? Is
> it F = q(E +(v X B)) ?
Yes. In modern terms.
> What is the history of that?
Good question. Perhaps one needs to go back and read Ampere for this. I've
read that Ampere never wrote down what we call "Ampere's law". But perhaps
it's in there. It's there by Maxwell's time.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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