Einstein-Hilbert Action Derive from Christoffel Symbols and Quadratic Lagrangians?

[Kevin_spencer2]

In Landau's bok "Classical Field Theory" pages: 372-373-374 they manage to get the Einstein-Hilbert action (after integrating by parts and use divergence theorem)

$$\mathcal L = \int dx^{4} \sqrt (-g) g^{ik}(\Gamma^{m}_{il}\Gamma^{l}_{km}-\Gamma^{l}_{ik}\Gamma^{m}_{lm})$$

from this and definition of 'Chrisstoffel symbols' the Lagrangian would be quadratic in the metric and its first derivatives , if we impose the constraint:

$$\mathcal{g}+1 =0$$ (does it has any physical meaning??)

and a Qadratic Lagrangian in the derivatives and fields can be evaluated by means of a Functional integral.

 

[Mentz114]

Please clean up and edit. The last formula makes no sense.