Insights Does Gravity Gravitate? Part 2 - Comments

[vanhees71]

This is the Hilbert action, and it's unique (up to a cosmological constant, which is not included here) in being generally covariant and leading to 2nd-order equations of motion. This is so, because $R$ is the only scalar you can build from the pseudometric tensor that is only linear in the 2nd derivative with coefficients containing no derivatives, which means that via partial integration you can show that the action is a functional of a Lagrangian that only depends on the $g_{\mu \nu}$ and its first derivatives.