I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and their constituent tensors are invariant under general Lorentz transformations on Minkowski space. Does there exist an expansion of the four-potential where the quadrupole term is truly a tensor in Minkowski space (invariant under arbitrary Lorentz transformations)?
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I don't know about Lorentz invariance, but classically it's generally only the lowest-order nonvanishing multipole moment that is invariant under translations (e.g. the quadrupole moment tensor is in general only invariant if the net charge and dipole moment are both zero), and I would suspect the same would be true in relativity.
