- #1

fluidistic

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**0. Homework Statement**

Hi guys,

I must show that the trace of the stress energy tensor is zero.

The definition of it is ##T^{\mu \nu }=\frac{1}{4\pi} \left ( F^{\mu \sigma } F^{\nu \rho} \eta _{\sigma \rho}-\frac{1}{4} \eta ^{\mu \nu } F^{\sigma \rho} F_{\sigma \rho} \right )##.

**1. The attempt at a solution**

I know it's pure algebraic manipulations but for some reason I get stuck. Trace is ##T^\mu _\mu =\eta_{\mu\nu}T^{\mu\nu}=\frac{1}{4\pi}\left ( \eta_{\mu\nu}F^{\mu\rho}F^{\nu\rho}\eta _{\sigma\rho} - \frac{1}{4} \eta_{\mu\nu}\eta^{\mu\nu}F^{\sigma\rho}F_{\sigma \rho}\right )##

##=\frac{1}{4\pi}\left ( F^\sigma _\nu F^\nu_\sigma-\frac{1}{4} \delta ^\mu _\mu F^{\sigma \rho}F_{\sigma \rho} \right ) = \frac{1}{4\pi} \left ( F^\sigma _\nu F^\nu _\sigma - F^{\sigma\rho}F_{\sigma \rho} \right )##.

This is where I'm stuck. I don't know how to show that the terms in the parenthesis are equal; if they are. I'd appreciate any comment. Thanks.