Help With Raising and Lowering Indices

[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?

 

[adriank]

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 $\nabla_\alpha g^{\alpha \beta} = 0$.

 

[dm4b]

Thanks adriank.

Had this posted in another forum here too and we just came to the same conclusion. It's nice to see it backed up here as well.