Linearized general relativity problem

1. May 31, 2006

Pietjuh

I've got a problem where I'm not sure my solution is true!
I have to proof that given the vacuum einstein's equation and the lorenz gauge condition imply that the stress energy tensor that generates the gravitational waves must have a vanishing divergence.

The vacuum einstein's equation is given by $$-\partial_{\alpha}\partial^{\alpha}\bar{h}^{\mu\nu} = 16\pi T^{\mu\nu}$$, and the lorentz condition is given by $$\partial^{\nu}\bar{h}_{\mu\nu} = 0$$

Now if I just take the divergence to $\nu[/tex] of this equation I obtain $$-\partial_{\nu}\partial_{\alpha}\partial^{\alpha}\bar{h}^{\mu\nu} = 16\pi \partial_{\nu}T^{\mu\nu}$$ Since we can swap the order of partial differentiation this becomes: $$-\partial_{\alpha}\partial^{\alpha}\left(\partial_{\nu}\bar{h}^{\mu\nu}\right)= 16\pi \partial_{\nu}T^{\mu\nu}$$ Now what I want to proof is that $$\partial_{\nu}\bar{h}^{\mu\nu} = \partial^{\nu}\bar{h}_{\mu\nu}$$ I think that is true because $$\partial_{\gamma}\bar{h}^{\mu\nu} = - \partial^{\gamma}\bar{h}_{\mu\nu}$$. So if I just replace [itex]\gamma$ by $\nu$ this should imply that the divergence of T vanishes.
But I don't know for sure if I could just replace $\gamma$ by $\nu$!