What is the Orthogonality Relation for the Energy-Momentum Tensor in Relativity?

[PineApple2]

Homework Statement

Arrive at the orthogonality relation {T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = K{\delta^{\mu}}_{\nu}
and determine K.

Homework Equations

T_{ij}=T_ji}

The Attempt at a Solution

{T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = {T^{\mu}}_0{T^0}_{\nu}+ {T^{\mu}}_i{T^i}_{\nu}
I am not sure how to continue from here, in which direction I should go...
Thanks!

 

[dextercioby]

Is T appearing in the Einstein equations ? If so, use them.

 

[PineApple2]

No, we haven't even studied yet Einstein's equations (the context is special-relativistic electrodynamics. T is the enrgy-momentum (stress) tensor)

 

[dextercioby]

Aaaa, you should have said that. Ok, then you know how T looks like in terms of F. Then just express the contraction in the LHS in terms of F and regroup it so that you'll get a scalar times unit tensor.