Lagrangian formulation of the Palatini action

TL;DR

This paper explores the Lagrangian formulation of the Palatini action, emphasizing the necessity of metric compatibility and torsion or tetrad conditions for describing General Relativity, with implications for quantization.

Contribution

It demonstrates that metric compatibility and either torsion-free or tetrad compatibility conditions are essential in the Palatini formulation of General Relativity.

Findings

Metric compatibility is necessary for the Palatini action to describe GR.

Either torsion must be zero or tetrad compatibility must hold.

Results apply to any tetrad and internal connection action describing GR.

Abstract

We work on the Lagrangian formulation of the Palatini action. We find that we must assume the metric compatibility condition for the Palatini action to describe General Relativity, which condition should hold in quantization. We find that we must also assume one of the torsion zero condition or the tetrad compatibility condition. Our results will hold for any action in terms of the tetrad and the internal connection which describes General Relativity.