Galilean transform and the maxwell equations

  1. 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?
     
    Last edited: Jun 23, 2012
  2. jcsd
  3. tiny-tim

    tiny-tim 26,053
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi GarageDweller! Welcome to PF! :smile:
    Yup! :smile:
    Not really … if there's no time, then there's no difference between galilean and relativistic, is there? :wink:
     
  4. oops, missed that
     
  5. 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' ??
     
  6. tiny-tim

    tiny-tim 26,053
    Science Advisor
    Homework Helper

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

    so the RHS ρ * γ + J * (-γu/c2) = ρ' :wink:

    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]​
     
  7. 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'?
     
    Last edited: Jun 23, 2012
  8. tiny-tim

    tiny-tim 26,053
    Science Advisor
    Homework Helper

    sorry, not following you :confused:

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

    (Ex,Ey,Ez;Bx,By,Bz) becomes Ex(tΛx) + Ey(tΛy) + Ez(tΛz) + Bx(yΛz) + By(zΛx) + Bz(xΛy)

    (ρ,Jx,Jy,Jz) becomes ρt + Jxx + Jyy + Jzz (and similarly for div)

    and you use aΛb = -bΛa, aΛa = 0, xΛyΛz = t*, yΛzΛt = x* etc​
    ρ is part of the 4-vector (ρ,Jx,Jy,Jz) :wink:
     
  9. 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.)
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?