# Help With Raising and Lowering Indices

1. Aug 15, 2010

### dm4b

This is from Linearized Gravity in General Relativity, where h is the perturbation on the background Minkowski metyric.

Is the following valid?

$$\partial^{\sigma}h_{\sigma}_{\mu}=\eta^{\sigma}^{\epsilon}\partial_{\epsilon}h_{\sigma}_{\mu}=\partial_{\epsilon}h_{\mu}^{\epsilon}=\partial_{\sigma}h_{\mu}^{\sigma}$$

As you can see on the third term, I use the metric (neta) to raise an index on h instead of the partial now. Is that valid?

Since the metric is full of constants in the Minkowski metric, seems like it would be valid to move it inside the partial and operate on h. BUT, seems like this would not be true in general, maybe?

Last edited: Aug 15, 2010
2. Aug 15, 2010

That is valid, because the Minkowski metric is constant (so $\partial_\epsilon \eta^{\sigma \epsilon} = 0$). However, for general metrics g, $\partial_\alpha g^{\alpha \beta}$ is not necessarily zero. The problem disappears if you replace partial derivatives with covariant derivatives, though, because (by definition of the Levi-Civita connection) in this case $\nabla_\alpha g^{\alpha \beta} = 0$.