general relativity - Can we derive Einstein-Hilbert action through action principle and Levi-Civita connection?

Suppose that we take principle of least action as given. Also assume that any manifold allowed by the action would carry Levi-Civita connection (torsion-free characteristic). Also assume that the local symmetry imposed on the tangent space of each manifold point is that of Poincare group, via general covariance principle.

Would these be sufficient to derive Einstein-Hilbert action, and by corollary Einstein field equations? Or do we need extra conditions to derive the Einstein-Hilbert action?

Edit: If not, then what would be other extra conditions?

Answer

The way to get the Einstein-Hilbert action in 4 dimensions is to take those requirements along with the following ones :

• The stress-energy tensor has a null divergence


• The equation of motion has at most second derivatives in the metric

With those two extra conditions, the Einstein-Hilbert action, plus a cosmological term, is the unique solution in 4 dimensions.