Extended Palatini action for general relativity and the natural emergence of the cosmological constant

TL;DR

This paper introduces an extended Palatini action for general relativity where the metric and connection are independent variables, revealing that the cosmological constant naturally arises as a condition for the theory's self-consistency.

Contribution

The paper develops a novel extended Palatini formulation treating the metric and connection as independent, showing the cosmological constant's role in theory consistency.

Findings

Reproduces Einstein equations with a single metric tensor.

Demonstrates the cosmological constant's role in ensuring self-consistency.

Provides a new perspective on the origin of the cosmological constant.

Abstract

In the Palatini action of general relativity the connection and the metric are treated as independent dynamical variables. Instead of assuming a relation between these quantities, the desired relation between them is derived through the Euler-Lagrange equations of the Palatini action. In this manuscript we construct an extended Palatini action, where we do not assume any a priori relationship between the connection, the covariant metric tensor, and the contravariant metric tensor. Instead we treat these three quantities as independent dynamical variables. We show that this action reproduces the standard Einstein field equations depending on a single metric tensor. We further show that in this formulation the cosmological constant has an additional theoretical significance. Normally the cosmological constant is added to the Einstein field equations for the purpose of having general relativity be consistent with cosmological observations. In the formulation presented here, the nonvanishing cosmological constant also ensures the self-consistency of the theory.