Raising index of variation

Question

I know how to prove e.g. $$A^{ik}B_{lk}=A_{k}^iB^{k}_l.\tag{1}$$ (Raising and Lowering Indices Question). Today in a book, I find: $$g^{ik}\delta g_{lk}=-g_{kl}\delta g^{ki}.\tag{2}$$

$g^{ik}$ is the metric tensor. There is a negative sign. If I use the regular method to raise/lower the indices, I cannot get the negative sign. I guess this must be due to the variation $\delta$? Do you know how to prove this?

It then says because of the aforementioned equation, therefore:

$$T^{ik}\delta g_{ik}=-T_{ik}\delta g^{ik}.\tag{3}$$

$T^{ik}$ is the energy momentum tensor. Why they have such relationship?

Answer 1

Remember that the metric tensor $g_{\rho\nu}$ and its inverse $g^{\rho\nu}$ fulfill the relation

$$ g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\nu.$$

In the above equation, if you take the variation on both sides you get

$$ \delta g^{\mu\rho}g_{\rho\nu}+g^{\mu\rho}\delta g_{\rho\nu}=0.$$

From this you get

$$g^{\mu\rho}\delta g_{\rho\nu}=-g_{\rho\nu}\delta g^{\mu\rho}.$$

Which is what you have in Eq. $(2)$. Equation $(3)$ follows from this.