I Einstein-Hilbert Action, Lagrangian Density & Vacuum Fields

[Apashanka]

From the action ∫Ldt =∫ι√|g|d⁴x where |g| is the determinant of the metric .and ι the lagrangian density.
For gravitational field why is this ι is replaced by the Ricci scaler R which yield field equations in vaccum.(R_ij-1/2Rg_ij)=0
Is it that the lagrangian density corresponding to vacuum is the Ricci scaler??

 

[Orodruin]

The action defines the model. Your action must be of the form of an integral over spacetime of some scalar (not "scaler") function multiplied by the spacetime volume element. Having only the metric to play with, you can essentially only end up with invariants constructed from the Riemann tensor. The simplest (apart from just a constant) possibility is then the Ricci scalar. Technically you could input any scalar function constructed from the metric and Riemann tensor, but that will result in a different model and GR works very well.