Diffeo-invariant action for a matter covector field

[ergospherical]

I just need a hint to get started, and then I reckon the rest will follow...
We consider a theory where matter is a covector field $\omega_a$ which is described by a diffeomorphism-invariant action $S_m$. Define:$$E^{a} = \frac{1}{\sqrt{-g}} \frac{\delta S_m}{\delta \omega_a}$$Also, $T^{ab} = \tfrac{2}{\sqrt{-g}} \tfrac{\delta S_m}{\delta g_{ab}}$ is defined as per usual. We would like to derive:$$\nabla_a {T^a}_b = E^a \nabla_b \omega_a - \nabla_a(E^a \omega_b)$$The statement that $S_m$ is diffeomorphism invariant: I guess this means, under $x\mapsto x - \xi$, and therefore$$\delta g_{\mu \nu} = (L_{\xi} g)_{\mu \nu} = \nabla_{\mu} \xi_{\nu} + \nabla_{\nu} \xi_{\mu}$$that $S_m$ is invariant... how do I use that? I imagine you can write something down in terms of the Lie derivative, and then manipulate the resulting equation into the result. But how to start??

 

[ergospherical]

Never mind, I figured it out. For sure, you write out$$\delta S_m = \int d^4 x \left[ \frac{\delta S_m}{\delta \omega_a} \delta \omega_a + \frac{\delta S_m}{\delta g_{ab}} \delta g_{ab} \right]$$and then stick in the expressions $\delta \omega = L_{\xi} \omega$ and $\delta g = L_{\xi} g$ in terms of covariant derivatives, as well as replacing $\delta S_m / \delta g_{ab}$ by the expression involving the energy momentum tensor. Then just integration by parts gives what you want, when you make sure it vanishes under arbitrary $\xi$.

Please delete or close the thread if you want... (classic rubber-duck example - write something down and then figure it out)