Galilean transform and the maxwell equations

GarageDweller

So I keep hearing that the maxwell equations are variant under Galilean transform. Tired of simply accepting it without seeing the maths, I decided to do the transformation on my own.

To make things easy, I only tried Gauss' law, furthermore I constricted the field to the x axis only. So I have E(x,t).

∇°E(x,t)=ρ(x)/ε

So now I will transform to another inertial frame x' that is moving with speed u with respect to the original frame x.
x'=x-ut
t'=t

What originally was ∂E/∂x=ρ(x)/ε became ∂E/∂x'-(1/u)∂E/∂t=ρ'(x')/ε.
Is this basically what they mean when they say it isn't invariant?

I looked at this again, and noticed that if the electric field is independent of time, then the Galilean transform of this turns out to be invariant, coincidence?

tiny-tim

Hi GarageDweller! Welcome to PF!
Yup!
Not really … if there's no time, then there's no difference between galilean and relativistic, is there?

GarageDweller

oops, missed that

GarageDweller

However, I tried transforming the equation by the Lorentz transform, and yet I'm still getting something different, I think I may have a conceptual error here, my transformation process was:

∂E/∂x=∂E/∂x' * ∂x'/∂x + ∂E/∂t' * ∂t'/∂x

x'=γ(x-ut)
t'=γ(1-ux/c^2)

Which gives me..

∂E/∂x=∂E/∂x' * γ + ∂E/∂t' * (-γu/c^2)

Exactly how does this reduce to ∂E/∂x' ??

tiny-tim

(try using the X² button just above the Reply box )
because (from the ampere-maxwell law) ∂E/∂t = J …

so the RHS ρ * γ + J * (-γu/c²) = ρ'

from the pf library on Maxwell's equations …

(on the RHS, * denotes a pseudovector: a "curl" must be a pseudovector, the dual of a vector)

Changing to units in which [itex]\varepsilon_0[/itex] [itex]\mu_0[/itex] and [itex]c[/itex] are 1, we may combine the two 3-vectors [itex]\mathbf{E}[/itex] and [itex]\mathbf{B}[/itex] into the 6-component Faraday 2-form [itex](\mathbf{E};\mathbf{B})[/itex], or its dual, the Maxwell 2-form [itex](\mathbf{E};\mathbf{B})^*[/itex].

And we may define the current 4-vector J as [itex](Q_f,\mathbf{j}_f)[/itex].

Then the differential versions of Gauss' Law and the Ampère-Maxwell Law can be combined as:

[tex]\nabla \times (\mathbf{E};\mathbf{B})^*\,=\,(\nabla \cdot \mathbf{E}\ ,\ \frac{\partial\mathbf{E}}{\partial t}\,+\,\nabla\times\mathbf{B})^*\,=\,J^*[/tex]

and those of Gauss' Law for Magnetism and Faraday's Law can be combined as:

[tex]\nabla \times (\mathbf{E};\mathbf{B}) = (\nabla \cdot \mathbf{B}\ ,\ \frac{\partial\mathbf{B}}{\partial t}\,+\,\nabla\times\mathbf{E})^*\,=\,0[/tex]

GarageDweller

Ooh right forgot bout the charge density term, thx lol
So basically the lorentz factors all get canceled and the J terms go away on either side?

Oh and one more thing, what's the exact process of changing p into p'?

tiny-tim

sorry, not following you

i always prefer to translate everything into wedge (Λ) products when dealing with maxwell's laws

(E_x,E_y,E_z;B_x,B_y,B_z) becomes E_x(tΛx) + E_y(tΛy) + E_z(tΛz) + B_x(yΛz) + B_y(zΛx) + B_z(xΛy)

(ρ,J_x,J_y,J_z) becomes ρt + J_xx + J_yy + J_zz (and similarly for div)

and you use aΛb = -bΛa, aΛa = 0, xΛyΛz = t*, yΛzΛt = x* etc

ρ is part of the 4-vector (ρ,J_x,J_y,J_z)

Muphrid

To elaborate on what Tim is saying: once you get into relativity and EM, it's helpful to get used to the idea that the EM field isn't a simple vector field. Rather, just as a vector field is a combination of directions, there are bivector fields which are combinations of planes. That's what the EM field is. You can write its six components as

[tex]F = E_x e_t \wedge e_x + E_y e_t \wedge e_y + E_z e_t \wedge e_z + B_x e_y \wedge e_z + B_y e_z \wedge e_x + B_z e_x \wedge e_y[/tex]

Each of the [itex]e_\mu \wedge e_\nu[/itex] represents a plane spanning the [itex]e_\mu, e_\nu[/itex] directions.

Because the EM field is a set of planes, the Lorentz transformation works a little differently. It acts on each basis vector in a wedge, so for example, under a Lorentz transformation [itex]\underline L[/itex], we have the tx-plane [itex]e_t \wedge e_x \mapsto \underline L(e_t) \wedge \underline L(e_x)[/itex]. If the boost itself is in the tx plane, however, it must leave that plane invariant, even though both vectors get rotated. Similarly, since both [itex]e_y, e_z[/itex] are out of the plane, neither get transformed, and they're left invariant.

What's nice about using the Faraday bivector [itex]F[/itex] is that, without matter, Maxwell's equations boil down to a single expression:

[tex]\nabla F = - \mu_0 j[/tex]

(The sign depends on your metric convention and how you assemble [itex]F[/itex], but this is the convention I prefer.)