Proof of Lorentz Gauge Existence: Help Understanding Schutz 8.3

[epovo]

TL;DR

In the derivation of the proof there is a step that I cannot make sense of

In Schutz 8.3, while proving that a Lorentz gauge exists, it is stated that
$$\bar h^{(new)}_{\mu\nu} = \bar h^{(old)}_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu} + \eta_{\mu\nu}\xi^\alpha_{,\alpha}$$

where $\bar h$ is the trace reverse and $\xi^\alpha$ are the gauge functions. Then it follows with:
"Then the divergence is"
$$\bar h^{(new)\mu\nu}_{\,\,\,\,\,\,\,,\nu} = \bar h^{(old)\mu\nu}_{\,\,\,\,\,\,\,\,,\nu} - \xi^{\mu,\nu}_{\,\,\,,\nu}$$
I can't see why the divergence is that! I've tried and tried but I can't see it. Any help?

 

[George Jones]

I can't see why the divergence is that! I've tried and tried but I can't see it. Any help?

$$-\xi^{\nu,\mu}_{\,\,\,,\nu} + \eta^{\mu\nu} \xi^{\alpha}_{\,\,\,,\alpha \nu} = -\xi^{\nu,\mu}_{\,\,\,,\nu} + \xi^{\alpha,\mu}_{\,\,\,,\alpha} = 0,$$
since $\alpha$ and $\nu$ are both dummy summation indices.

 

[epovo]

True! Thank you