Well, this isn't so much for general raising and lowering of indices. It's a specific step within the formation of the Ricci Tensor in the Linearized Gravity problem. I trying to get from 7.5 to 7.6 in Sean Carrol Spacetime and Geometry, page 275. I'm not matching with the 2nd term in 7.6. I'm pretty sure I'm having a problem when an index is raised on a derivative. 7.6 seems to imply this is somehow equivalent to raising an index on the perturbation. can anybody provide clarification on this or fill in the steps? I've been stuck on this once before, figured out .. now years later, I'm stuck on it again ... frustrating! Any help would be much appreciated!