Divergence of Energy-momentum Tensor

[ClaraOxford]

How do you prove that Maxwell's energy-momentum equation is divergence-free?
I don't know whether or not I have to use Lagrangians or Eistein's tensor, or if there's a simlpler way of expanding out the tensor..

∂_{\mu}T^{\mu\nu}=0

T^{}\mu\nu=F^{}\mu\alphaF^{}\nu_{}\alpha-1/4F^{}\alpha\betaF_{}\alpha\beta\eta^{}\mu\nu

 

[ClaraOxford]

I mean

∂_{\mu}T^{\mu\nu}=0

T^{\mu\nu}=F^{\mu\alpha}F^{\nu}_{\alpha}-1/4F^{\alpha\beta}F_{\alpha\beta}\eta^{\mu\nu}

 

[vela]

Try writing F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu and using the commutativity of the derivatives.