I have read the statement that the tracelessness of the energy-momentum tensor is demanded by the condition of photons being massless.
I see how this comes about starting from the canonical energy-momentum tensor, but is there a particular reason why this must be so? After all the energy-momentum tensor is defined only up to an arbitrary divergenceless term.
Thanks in advance.
The Lagrangian for ordinary massless E&M is (in natural units)
\begin{equation}\mathcal{L}=-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}\end{equation}
which gives a canonical Stress-Energy Tensor of
\begin{equation}T^{\mu\nu}=F^{\mu\alpha}F^{\nu}_{\alpha}-\frac{1}{4}\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\end{equation}
Now if we were give the photon a mass, the Lagrangian would be
\begin{equation}\mathcal{L}=-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}+\frac{1}{2}m^{2}A_{\alpha}A^{\alpha}\end{equation}
and the resulting Stress-Energy Tensor would be
\begin{equation}T^{\mu\nu}=F^{\mu\alpha}F^{\nu}_{\alpha}-\frac{1}{4}\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}+\frac{1}{2}\eta^{\mu\nu}m^{2}A_{\alpha}A^{\alpha}\end{equation}
As you can check, this Stress-Energy Tensor is no longer traceless, which is solely due to the inclusion of a massive photon.
