# 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?

• What does it even mean to "derive" an action? What is an action supposed to do that is intrinsic? The field equations are put in by hand, according to what we expect to happen physically. The action comes from that. – Ryan Unger Aug 24 '17 at 3:55

## 2 Answers

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.

• You should clarify whether the "divergence" of the stress-energy tensor is the ordinary or the covariant divergence. – ACuriousMind Aug 24 '17 at 12:07

No, any action that is a scalar would satisfy your requirements. For example, you could have various terms that are functions of the various scalar curvatures $R$, $R_{ab}R^{ab}$, or $R_{abcd}R^{abcd}$, to give a few examples; you could probably invent more. You need some further requirement to fix the Lagrangian to be just $R$.