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Hopefully some other people will respond as well.

And since this thread has become long, here is a summary of some of the preceeding.

Using the method described above to try to solve for the divergence of g^{cd}R_{ab}R^{ab}

does it and gets:

[tex]-\frac{1}{2}g^{ab}R_{cd}R^{cd} + 2R^{ca}R_c{}^{b} -2 \nabla^c\nabla^d R_c{}^{(a} \delta_d{}^{b)} + \nabla^c \nabla_c R^{ab} + g^{ab} \nabla^c \nabla^d R_{cd} = \kappa T^{ab}[/tex]

Applying the method, we then get:

[tex]\nabla_b\left(g^{ab}R_{cd}R^{cd}\right) = 2\nabla_b\left(2R^{ca}R_c{}^{b} -2 \nabla^c\nabla^d R_c{}^{(a} \delta_d{}^{b)} + \nabla^c \nabla_c R^{ab} + g^{ab} \nabla^c \nabla^d R_{cd} \right)[/tex]

The problem is, unlike in the f(R) case I worked out explicitly above, I cannot check by hand whether the method gave something which is true

If someone knows how to show mathematically that my method is valid then that would fully answer the question and we'd be done! On the other hand, if someone knows how to show mathematically that my method doesn't hold generally, that would at least put discussion on that to rest and would be immensely helpful as well (especially to see the mathematical details of such a proof).

Basically, if anyone has guidance on how to proceed from here, please do share.

And since this thread has become long, here is a summary of some of the preceeding.

Using the method described above to try to solve for the divergence of g^{cd}R_{ab}R^{ab}

I made mistakes when trying to work out the field equations, but this paper http://arxiv.org/PS_cache/astro-ph/pdf/0410/0410031v2.pdfConsider the action

[tex] S = \int (\frac{1}{2\kappa}R_{ab}R^{ab} + \mathcal{L}_m) \sqrt{-g} d^4x[/tex]

does it and gets:

[tex]-\frac{1}{2}g^{ab}R_{cd}R^{cd} + 2R^{ca}R_c{}^{b} -2 \nabla^c\nabla^d R_c{}^{(a} \delta_d{}^{b)} + \nabla^c \nabla_c R^{ab} + g^{ab} \nabla^c \nabla^d R_{cd} = \kappa T^{ab}[/tex]

Applying the method, we then get:

[tex]\nabla_b\left(g^{ab}R_{cd}R^{cd}\right) = 2\nabla_b\left(2R^{ca}R_c{}^{b} -2 \nabla^c\nabla^d R_c{}^{(a} \delta_d{}^{b)} + \nabla^c \nabla_c R^{ab} + g^{ab} \nabla^c \nabla^d R_{cd} \right)[/tex]

The problem is, unlike in the f(R) case I worked out explicitly above, I cannot check by hand whether the method gave something which is true

*or only true*__in general__*. There is, in my opinion, good heuristic reason to expect that the method gives divergence relations which are true in general (along with evidence in the form of other cases which were checked by hand being true in general). Yet here I cannot check by hand to verify, because that would require me to work out, by hand, what the divergence of [tex]g^{ab}R_{cd}R^{cd}[/tex] is ... which I don't know how to do, as that was the opening question.*__in solutions for this theory's field equations__If someone knows how to show mathematically that my method is valid then that would fully answer the question and we'd be done! On the other hand, if someone knows how to show mathematically that my method doesn't hold generally, that would at least put discussion on that to rest and would be immensely helpful as well (especially to see the mathematical details of such a proof).

Basically, if anyone has guidance on how to proceed from here, please do share.

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