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I am doing a problem of GR dealing with weak field limit equations. After decoupling the static and dynamical part the usual way ([itex]g^{ab}=\eta^{ab}-h^{ab}[/itex]) I arrived at those field equations:

[itex]

\frac12 \eta^{ab}\partial_a \partial_b h_{\mu\nu} - \frac12 h^{ab}(\partial_a\partial_b h_{\mu\nu} -\partial_a \partial_{\nu} h_{\mu b} - \partial_{\mu}\partial_b h_{a\nu} + \partial_{\nu}\partial_{\mu}h_{ab}) -\frac14 (\partial_a h_{b\nu} +\partial_{\nu}h_{ab} -\partial_b h_{a\nu})(\partial_{\mu}h^{ab} + \partial^a h^{b}_{\mu} - \partial^bh_{\mu}^a) + \mathcal{O}(h^3) = 0

[/itex]

The first term being the operator on the field, the rest should be the source (stress energy tensor of the spin-2 field because we are in vaccum) but I can't derive this stress energy from the naive Lagrangian [itex]\cal{L} = \frac14 (\partial^ah_{\mu\nu})^2[/itex]. Plus one question is: "What are the conditions to interpret that as a stress-energy tensor?"

Some help/tips will be more than welcomed.

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# Energy momentum of linearized gravity.

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