# Computations with Tensors

I have the following ansatz $$T^{\mu\nu}=AF^{\mu\alpha}F_{\alpha}^\nu+B\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}$$ for some constants $$A,B.$$ Here, $$F_{\mu\nu}$$ is the electromagnetic field tensor, and $$\eta^{\mu\nu}$$ is the (flat?) metric with signature $$(-,+,+,+).$$ I need to show that energy and momentum conservation of the EM field (i.e. $$\partial_\mu T^{\mu\nu}=0$$ for $$\nu=0,1,2,3$$) implies that $$B=A/4$$.

The only idea I have is to expand everything and try to eventually compare coefficients. But this is super tedious and unenlightening. Is there a better, smarter approach to this? I am just hoping for a nudge in the right direction.

• Use $F^{\nu}_{\alpha} = \eta_{\alpha \beta} F^{\nu \beta}$ or the same transformation for covariant components. And then do direct computation of derivatives Oct 9, 2022 at 21:59
• A calculation that takes a few lines doesn’t count as “super tedious”. Oct 10, 2022 at 6:52
• a nudge in the right direction You have to make use of Maxwell’s equations in covariant form. Oct 10, 2022 at 6:55

Your formula does not make sense. The left-hand side has indices $$\mu,\nu$$, and the first term of the right-hand side has indices $$\alpha,\nu$$.