So I am working on the next quadratic Lagrangian:(adsbygoogle = window.adsbygoogle || []).push({});

$$ L = \alpha R_{\mu\nu}R^{\mu\nu} + \beta R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} + \gamma R² $$

I have already derived $$ \delta (R_{\mu\nu}R^{\mu\nu}) = [- \frac{1}{2}g_{\mu\nu}R_{\alpha\beta}R^{\alpha\beta} + 2R^{\alpha\beta}R_{\mu\alpha\nu\beta}]\delta g^{\mu\nu} $$ right?

How can I compute the other variations in order to compute the equations of motion?

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# Gauss - Bonnet Gravity -> Curvature variations

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