I am well aware that Maxwell’s equations are not invariant under a Galilean transformation; the simplest proof of this is that they predict electromagnetic waves whose speed depends on physical constants that in no way change with observer velocity.
However, some years ago I came across a mathematical derivation of expected effects if Maxwell’s equations were assumed to obey Galilean transformations. I wonder if anyone can point me to this derivation of the fictitious effects.
Here’s a vague recollection: A positive charge moving in the +x direction passes thru a very thin magnetic field lying in the x-z plane which is oriented in the +y diection. Obviously the charge will be deflected in the +z direction, and an observer at rest wrt the magnetic field will deduce this motion was caused by the magnetic field. However, an observer moving at the same speed/direction as the charge will surmise the deflection was caused by an electric field oriented in his/her +z direction, since the charge is initially at-rest wrt him/her.
Now, suppose the magnetic field is reversed for z<0; positive charges which hit the x-z plane at z>0 will be deflected in the +z direction, while those that hit at z<0 will be deflected in the -z direction (if necessary, assume a small “transition” area around z=0). Repeat the experiment, but use two charges, one crossing at z=a, the other at z=-a (Make a large enough to ignore the transition area). Obviously these two charges will deflect away from each other. The observer at rest wrt the field has no trouble explaining this in terms of deflection caused by legal magnetic fields, but the observer moving with the charges must invent an electric field to explain this motion (since the charges are at rest wrt him/her), and that field must have properties which violate Maxwell’s equations (e.g. it must be divergent at z=0 where no charge is present, violating Gauss’ law). Therefore, the original assumption that the equations are invariant under a Galilean transformation is false.
My interpretation is vague and probably indefensible, and of course depends on the erroneous assumption that the Maxwell equations survive a Galilean transform. That’s why I wonder if anyone knows of a mathematical exploration of the assumption, one that derives fields that violate the laws of electromagnetics and hence provides a reductio ad absurdum argument showing Maxwell’s equations are not Galilean invariant.
The simplest constructions I’ve seen involve distributions that are both electrostatic and magnetostatic. For example, take two infinite, parallel sheets of equal, uniform, static areal charge density. Using electrostatics (Gauss’ Law) it is easy to compute the repulsive force (per unit area) between the plates. Now consider a (Galilean) reference frame moving parallel to the plates. The electrostatic force is as before, but now there is also an attractive, magnetostatic force due to the motion of the charges (Ampere’s Law), and the total force between the plates is decreased. You can use this result to argue that Maxwell’s equations predict some combination of length contraction and time dilation.
Are you looking for more examples like this, or a full theoretical treatment?
I appreciate the responses (thx especially to Omphaloskeptic), but I guess what I’m looking for is a mathematical derivation showing that if you apply a Galilean transformation to Maxwell’s eq.'s in free space, you get something other than Maxwell’s eq.'s. I’ve seen it before–I think you get an extra “curl E” term if you do it–but my memory is hazy and I’ve been struggling trying to recreate it without success. Any help would be appreciated…
I think that it goes something like this (hoping the use of Gaussian units and conversion to ASCII isn’t too horribly ugly):
The differential and partial-derivative operators, under a Galilean-relativistic transformation r’=r-vt, transform likedr’ = r - vt
dt’ = dt
del’ = del
@/@t’ = v.del + @/@t(writing del and del’ for the nabla/gradient vector operator in unprimed and primed coordinates). In particular, the time partial derivatives pick up an extra term v.del, so if the Maxwell equations hold in the primed framedel’ . D = (4pi)rho del’ . B = 0 del’ x H = (4pi/c) J’ + (1/c) (@/@t’)D del’ x E = (-1/c) (@/@t’)Bthen in the unprimed frame we havedel . D = del’ . D = (4pi)rho del . B = del’ . B = 0 del x H = del’ x H = (4pi/c) J’ + (1/c) (@/@t’)D = (4pi/c) J’ + (1/c) (@/@t)D + (1/c)(v.del)D del x E = del’ x E = (-1/c) (@/@t’)B = (-1/c) (@/@t)B + (-1/c) (v.del)B
The current density transforms as J=J’+rho v. With use of the vector identity (for constant v)(v.del)A = v(del.A) - del x(v x A)we can cancel the annoying appearance of rho (from J’) in the del x H equation:(v.del)D = v(del . D) - del x(v x D) = (4pi)rho v - del x(v x D)
(v.del)B = v(del . B) - del x(v x B) = - del x(v x B)so the two curl equations becomedel x H = (4pi/c) J + (1/c) (@/@t)D - (1/c)del x(v x D) del x E = (-1/c) (@/@t)B + (1/c) del x(v x B)with an extra del x(v x X) term. Does this look right?
Jackson, the only EM book I have at hand right now, has a brief discussion of Galilean relativity in section 11.1; there he only talks about the transformation of the wave equation (not the four E, B Maxwell equations). I will check a few other books later if I remember.
That looks to be it Omphaloskeptic; I’d forgotten about using the “(v.del)A = v(del.A) - del x(v x A)” vector identity to deal with a current term left over from the first frame-of-reference. Those extra curl (vxD) and curl (vxB) are exactly what I was looking for.
I read once in a book called “Thinking Physics” that magnetism was a relativistic effect. Is that true? It said that when you have current running through two wires, even though the electrons move extremely slowly, the relativistic contraction changes the apparent density of the charges of the electrons, producing a force between the wires. The implication was that all magnetism could be explained as relativistic effects.
I looked through a few other EM and relativity books, but I didn’t find any treatments of this. The only useful reference I found (apart from Jackson) was Chapter 10 of Griffiths, 2nd ed. (it looks like it has become Ch.12 in the third edition), which is mostly about special relativity but has some examples that you might like.