Delta Notation in GR: Replacing vs Raising/Lowering

[binbagsss]

Hi

So we write $g_{ab}g^{ac}=\delta^{(4)c}_b$, but this simply means to replace $b$ with $c$ or vice versa, so, why don't we write $\delta_{bc}$?

Thanks

i.e. the affect is not to replace and raise/lower, it is simply to replace, so I'm a bit confused by the notation...

 

[Ibix]

You could write $g_{ab}g^{ac}\omega_c=\delta^c_b\omega_c$. Not sure exactly why you'd want to do that since it's an identity transform on $\omega_c$, but it's legit. If you wrote $\delta_{bc}$ the right hand side would have to be treated as a special exception to the summation rules. Why have an unnecessary special case?

 

[Orodruin]

If you wrote $\delta_{ab}$ you would end up with inconsistencies in the matching between covariant and contravariant indices.