Understanding EM Stress Energy Tensor: Sources and Current Density

[tim_lou]

I've been trying to obtain the EM stress energy tensor... but I'm not sure if what I got is correct.

In all the websites I googled, I found
$$T^{\mu\nu} =- \frac{1}{4}g^{\mu\nu} F^{\alpha\beta}F_{\alpha\beta} + F^{\mu}\,\!_{\lambda}F^{\nu\lambda}$$

However, is this with the sources or without the sources? Seems to me that this is what I would get for a Lagrangian without current. what happens if I add a current density?

I just need a quick answer... I can't seem to find anything online.

 

[atyy]

Try 't Hooft's notes, Eq. 10.9
http://www.phys.uu.nl/~thooft/lectures/genrel.pdf

 

[tim_lou]

thanks, got my answer, indeed that form above is without any source. A term proportional to the product of AJ are included, as expected.