Tensor equations. Can I change an equation from covariant to contravariant?

Question

Say I have a tensor equation like $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$. Does this also imply that $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?

Answer 1

Yes. If you want to do it explicitly, just multiply both sides with the inverse metric to raise both indices.

Answer 2

I hope this is the correct, full calculation. $$G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R.$$ First, multiply by $g^{ac}$ $$g^{ac}G_{ab}=g^{ac}R_{ab}-g^{ac}\frac{1}{2}g_{ab}R$$ $$G_{b}^{c}=R_{b}^{c}-\frac{1}{2}\delta_{b}^{c}R.$$ Then multiply by $g^{ab}$ $$g^{ab}G_{b}^{c}=g^{ab}R_{b}^{c}-\frac{1}{2}\delta_{b}^{c}g^{ab}R.$$ $$G^{ac}=R^{ac}-\frac{1}{2}g^{ac}R.$$ Relabel $c$ as $b$ $$G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R.$$ Two days later - someone has kindly confirmed this is OK, here: Raising and lowering indices - any chance of checking my work?