Computing an Energy-Momentum tensor given a Lagrangian

[JD_PM]

Even worse, some of the components are zero.

Ahhh you're right.

This equation

$$\partial_{\mu} \partial^{\mu} A^{\nu} = \eta^{\nu \rho} \partial_{\mu} \partial^{\mu} A_{\rho} = 0$$

It is equivalent to

$$\partial_{\mu} \partial^{\mu} A_{\nu} = \eta_{\nu \rho} \partial_{\mu} \partial^{\mu} A^{\rho} = \partial_{\mu} \partial^{\mu} A_{\nu} = 0$$

Which is what we want.