# Notes: Magnetostatic multipole expansion using STF tensors

These notes are intended for students (or profs) aware of the multipole expansion for electrostatics in terms of symmetric tracefree (STF) tensors. Standard texts on electrodynamics (like Jackson) hardly mention the STF version, though it is extremely well-known to researchers in GR. After I developed these notes, Julio Parra-Martinez pointed me to a paper by Andreas Ross which implicitly includes these results, though I want to explain a bit more slowly.

# Refresher: Electrostatic STF multipole expansion

Before getting to magnetostatics, we’ll start with electrostatics. This is easier since we only need to solve for the scalar potential, which satisfies

We’re interested in the case where vanishes outside of a compact region. The most efficient way to get to the STF version of the multipole expansion is to start from the Green’s function solution,

We then take the function and perform a multivariate Taylor series expansion about the point , since far away from the source, . This expansion is

In the index tensor , all indices are obviously symmetric; they are also tracefree away from the origin, where we get a delta function, owing to . Since this tensor is symmetric and tracefree (STF), we are free to take only the STF part of the product of direction vectors. We denote this with angle brackets around the relevant indices,

Plugging this in to the Green’s function integral, we get

where we have defined the th STF multipole tensor of the source as

# Magnetostatic multipole expansion

We can apply our results from the electrostatic multipole expansion to magnetostatics, using the potential formulation. In magnetostatics, we are trying to find a magnetic field satisfying

where as usual , and in statics, conservation of charge demands that . Now we go to the potential formulation, , and use our gauge freedom to go to Coulomb gauge, . Plugging in, we are now trying to solve

In Cartesian coordinates, this is just three independent copies of the Poisson equation, one for each Cartesian component . Therefore we can use the Green’s function for the scalar Laplacian’s for each component,

Just like in the electrostatic case, we Taylor expand , pull things out of the integrals, etc. Essentially, we are just making a replacement in the electrostatic case: . This means our multipole moments get an extra index that does not participate in the STF operation. As it stands, our solution is

where the magnetic multipole moments are defined as

which are STF only on the indices after the semicolon.

Notice that the term vanishes – no magnetic monopoles! – by conservation of charge. Integrate and use integration by parts:

The left hand side vanishes since in magnetostatics, . Therefore the magnetic monopole moment vanishes, .

Before handling the arbitrary term, let’s write the dipole in the traditional form seen in e.g. Griffiths. The traditional form for a magnetic dipole is

Here the magnetic dipole pseudo-vector is related to the 2-index magnetic dipole tensor,

This gives the ideal dipole magnetic field

except that we have dropped the singular term.

It seems like we’ve discarded some information — only the antisymmetric part of contributed to . What about the symmetric part? This exactly vanishes, and that generalizes to all higher . The proof follows similarly to why vanished above. Use that , and integrate this divergence against ,

What we found is that the completely symmetric part of vanishes (in fact it vanishes even before removing the traces on the indices after the semicolon).

The next step for understanding these magnetic multipole tensors requires a little knowledge of how Young diagrams classify the index symmetries of tensors (ok, maybe not strictly necessary, but this was how I first realized what to do). We know that the tensor lives in the representation labeled by the diagram of shape ,

Now recall that when we tensor-product a vector with some tensor in a
diagram with shape , we generate tensors in irreps related
by adding one box at the end of any allowed row or as a new row
underneath (the decomposition of tensor products into irreps is
determined by the Littlewood–Richardson
rule;
adding one box where allowed is the simplest case. This is
encapsulated in a *Hasse diagram* called Young’s
lattice, which gives
a partial order on Young diagrams, seen in here:

Now, since lives in the representation, we know that tensoring with can produce content in exactly two representations: the diagram, and the diagram, having shapes

However, above we showed that the completely symmetric part, labeled by , vanishes. Therefore, we have shown that each lives in the diagram, and this means the index is antisymmetric with each index.

Because of the antisymmetry between and any one of the ’s, we are free to insert a projector in the space of 2-forms,

This motivates defining an auxiliary tensor , like in the dipole case,

The two minus signs (which cancel) are here to agree with the traditional notation for the magnetic dipole vector. We can insert the integral expression,

What are the symmetries of ? It is obviously symmetric and tracefree on the ’s. It is also easy to see that tracing with any of the ’s would result in a symmetric pair of indices contracting with the tensor in the definition of , so by symmetry-antisymmetry, is tracefree on all indices.

Now we will show that is symmetric on and thus on with any of the ’s. Suppose we split the tensor into parts that are symmetric and antisymmetric on these two indices, . For the antisymmetric part, we could again insert a projector in the space of 2-forms. While evaluating this projector, we have the dual on . But this is simply a trace of : from Eq. \eqref{eq:mstatic-mpole-and-dual-rels},

which vanishes since is tracefree on all the ’s. Since this antisymmetric part of vanished, we found that is STF on all indices.

We can finally restate and in terms of these magnetic STF moments, after a bit of algebra: