Tensor decomposition of $\partial_\mu A_\nu$ In the decomposition of a rank-2 Minkowski tensor into irreducible representations, I expect the 16 components of the tensor product $M_\mu N_\nu$ to reduce to the sum of a scalar (1), a rank-2 anti-symmetric tensor (6), and a rank-2 traceless symmetric tensor (9).  Applying this to $\partial_\mu A_\nu$, where $A_\nu$ is the electromagnetic 4-potential, I find the usual anti-symmetric field tensor $F_{\mu\nu}$, and the the 4-divergence $\partial\cdot A$ (which vanishes in Lorentz gauge).  But I don't recognize the traceless symmetric tensor.  Have I missed something in the decomposition, messed up my indices, or forgotten something from E & M?
thanks!
 A: I am confused by what is meant by "physical interpretation".  In E&M only gauge invariant quantities have physical interpretations.  And, only the antisymmetric tensor is gauge invariant so no other combination has a physical interpretation, in E&M.  That is a possible answer.  If $A$ is not the gauge field for E&M and/or $\partial$ not the gradient, why the traceless symmetric tensor can have a physical interpretation.  But, that depends on what $A$ and $\partial$ are.  I can not immediately think of good examples of such tensors that have physical interpretations.  In General relativity $\partial x_\mu^\prime/\partial x_\nu$ is an important tensor but not gauge invariant, and so is not really physical. In (Euclidian) elasticity theory the traceless symmetric part of this tensor is the strain, and parts of this tensor also has a similar interpretation in GR if $x$ is a flat-space background and $x^\prime$ is a slight perturbation from that flat space background, I believe.  But, I can not immediately think of traceless symmetric relativistic tensors that are (a) that easy to construct and (b) have physical interpretations.
