Landau & Lifshitz, The Classical Theory of Fields, $\S93$, deals with the action function for the gravitational field. To construct this action, they note:

"To determine this scalar we shall start from the fact that the equations of the gravitational field must contain derivatives of the "potentials (i.e. $g_{\mu\nu}$)" no higher than the second."

Why is this a fact? Why must the gravitational field contain only first, second derivatives of the "potentials"? Of course, without already knowing the Einstein equations.


A believe it is as simple as this: we want to derive any equation and check its low-energy limit later. Of any two cases rendering the same low-energy limit we prefer the simpler.

The requirement for $\mathcal{L}$ to depend on derivatives of $g_{\mu \nu}$ up to 2nd order is just a part of the anzatz for deriving equations. If it fails (it won't), then we shall search for a less simple equation by allowing 3rd order terms.

It is possible that Landau & Lifshitz have a better explanation, but essentially it is handwaving (based on intuition and desperately lacks rigour), because higher-derivative Lagrangians theoretically aren't forbidden: they are considered in modifications of General Relativity.

The final decision is, of course, made a-posteriori by experimental verification.


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