How Do You Transform Electromagnetic Dipole Moments When Boosting Orthogonally?

[kuecken]

I just wondered how I transform electromagnetic dipole moments.
For example assuming I have magnetic dipole moment μ in a frame without E dipole moment. Then I boost orthogonal to μ. Now I would like to determine the electric dipole moment.

I could use the magnetic potential and transform it to the new frame. But this quiet tedious to me.
My second guess was to argue that there is a linear dependence of the magnetic fields on μ.
And the E field in the new frame will also have a linear dependence on B and thus on μ. By looking at the boost component I could also find a relation. But this would assume that I have an electric dipole moment in the new frame.

I just wondered whether there is any easier and more elegant method.
I would be very glad about any advise.
Thank you :D

 

[maajdl]

Just a thought: writing the Lagrangian in terms of moments.

 

[vanhees71]

It's just better to start from covariant expressions. As the electromagnetic field itself the electric and magnetic dipole moment densities build an antisymmetric 2nd-rank tensor in Minkowski space-time:

$$M^{\alpha \beta}=\begin{pmatrix} 0 & P_1 & P_2 & P_3 \\ -P_1 & 0 & -M_3 & M_2 \\ -P_2 & M_3 & 0 & -M_1 \\ -P_3 & -M_2 & M_1 & 0 \end{pmatrix},$$
where $\vec{P}$ and $\vec{M}$ are the electric and magnetic dipole density vector fields of the 1+3-dimensional formalism.

They transform under Lorentz transformations as any other 2nd rank tensor components
$$M'^{\gamma \delta} = {\Lambda^{\gamma}}_{\alpha} {\Lambda^{\delta}}_{\beta} M^{\alpha \beta}.$$

 

[kuecken]

oh really? I was looking for sth like this. that's good thank you!

 

[kuecken]

Magnetization-polarization tensor I found it now