I am trying to understand where the Einstein-Hilbert action comes from. Neglecting a constant multiplier, Wikipedia expresses this action as the integral over all spacetime of - (R + L)* sqrt(-g) where R is the Ricci scalar, g is the determinant of the metric tensor, and L describes the matter fields. (L is, I believe, though Wiki doesn't exactly say this, the stress-energy tensor multiplied by a Lagrangian multiplier. By setting L to 0, you get the vacuum solution for the field equations.) By setting variation of this "action" to 0, the field equations of GR result. My problem is I don't understand how to connect the Einstein-Hilbert action with the underlying physics which I take to be #1 spacetime is a locally Minkowskiian manifold where freely falling particles of matter move along geodesics and #2 for every non-rotating frame moving with a freely falling particle the laws of physics are the same, i.e., they can be expressed using coordinate-free scalars, vectors, tensors, and covariant derivatives. All this is to the best of my knowledge and belief.