Palatini Variational Principle for an Extended Einstein-Hilbert Action

TL;DR

This paper explores a generalized Einstein-Hilbert action using the Palatini variation, revealing conditions under which the connection is constrained or unconstrained, thus broadening understanding of the Palatini formulation in gravity theories.

Contribution

It introduces a generalized Einstein-Hilbert action and analyzes the conditions affecting the connection's constraints within the Palatini variation framework.

Findings

Hilbert constraint arises only in special cases

For certain coefficients, the connection remains unconstrained

Relationship between generalized action and standard Palatini formulation

Abstract

We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation.