Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?

Question

There are two ways to do the variation of Einstein-Hilbert action.

First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. Second one is Palatini formalism which takes metric and connection are independent. After variation, we get two equations, first is field equation and second is that connection is Levi-Civita connection.

So my question is why it is so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection and Palatini formalism coincides with Einstein formalism? While for $f(R)$ action they are generally different. Are there some deeper mathematical or physical structures of Einstein-Hilbert action which can account for it.

This post imported from StackExchange Physics at 2014-10-23 07:31 (UTC), posted by SE-user user34669

Answer 1

I) In Palatini $f(R)$ gravity, the Lagrangian density is

$${\cal L}~=~ \sqrt{-g} f(R),$$

with

$$R~:=~ g^{\mu\nu} R_{\mu\nu}(\Gamma),$$

and where $\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}$ is an arbitrary torsionfree$^1$ connection.

II) As OP mentions, the word Palatini refers to that the metric $g_{\mu\nu}$ and the connection $\Gamma^{\lambda}_{\mu\nu}$ are independent variables. We therefore get two types of EL equations:

1. The EL equations for the metric $g_{\mu\nu}$ are the generalization of EFE.

2. The EL equations for the connection $\Gamma^{\lambda}_{\mu\nu}$ turns out to be the metric compatibility condition for a second metric defined as

   $$\hat{g}_{\mu\nu}~:=~f^{\prime}(R) g_{\mu\nu}.$$ In other words, the classical solution for $\Gamma^{\lambda}_{\mu\nu}$ is the Levi-Civita connection for the second metric $\hat{g}_{\mu\nu}$.

III) So Einstein gravity (GR) with a possible cosmological constant

$$f(R)~=~R-2\Lambda,$$

or equivalently

$$f^{\prime}(R)~=~1,$$

corresponds to the special case where the two metrics $g_{\mu\nu}$ and $\hat{g}_{\mu\nu}$ coincide, and hence $\Gamma^{\lambda}_{\mu\nu}$ becomes the Levi-Civita connection for $g_{\mu\nu}$.

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$^1$ One could allow a non-dynamical torsion piece, but we will not pursuit this here for simplicity. For more on torsion, see e.g. also this Phys.SE post.

This post imported from StackExchange Physics at 2014-10-23 07:31 (UTC), posted by SE-user Qmechanic