GR Lagrangian with higher-order curvature terms

Question

I'm trying to find papers / books / lectures with the derivation of the equations of motion from Lagrangians with higher order in curvature terms, for example with the Kretschmann scalar $R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}$ or the Chern-Pontryagin scalar $^\star R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}$. I want to figure out how the Einstein equations changes if we use the Lagrangian $\int R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma} \sqrt{-g}d^4 x$.

Answer 1

I assume you are familiar with the variation of the Einstein-Hilbert (EH) action. Assuming this, from variation of the EH action, we learn the following rules $$\delta \sqrt { - g} = - \frac{1}{2}\sqrt { - g} {g_{\mu \nu }}\delta {g^{\mu \nu }},$$ and $$\delta \left( {{R^a}_{bcd}} \right) = {\nabla _c}\left( {\delta \Gamma _{db}^a} \right) - {\nabla _d}\left( {\delta \Gamma _{cb}^a} \right),$$ where $$ \delta\Gamma^a_{b c}=\frac{1}{2}g^{a\lambda}(\nabla_c\delta g_{b\lambda}+\nabla_b\delta g_{c \lambda}-\nabla_\lambda\delta g_{b c}), $$ and also $$\delta {g^{\mu \nu }} = - {g^{\mu \alpha }}{g^{\beta \nu }}(\delta {g_{\alpha \beta }}),$$ which are important ingredients in what follows.

Now, let's consider your case, i.e., $\int {{d^4}x\sqrt { - g} {R_{abcd}}{R^{abcd}}}$. To do so, you need to rewrite the Kretschmann scalar as $${R_{abcd}}{R^{abcd}} = {g^{bx}}{g^{cy}}{g^{dz}}{g_{am}}{R^a}_{bcd}{R^m}_{xyz}.$$

Finally, you can simply obtain the following result $$\delta \left( {\sqrt { - g} {R_{abcd}}{R^{abcd}}} \right) = \delta \left( {\sqrt { - g} {g^{bx}}{g^{cy}}{g^{dz}}{g_{am}}{R^a}_{bcd}{R^m}_{xyz}} \right).$$

Straightforwardly, by use of the first four relations in this answer, you can read the resulting field equation of motion. For a full discussion about these kinds of Lagrangians, please see this paper. I highly recommend reading Sect. 4.3 of Spacetime and Geometry by Carroll, which gives you anything you need to deal with Lagrangian formalism and variational principle. Hopefully this helps.