First-order equivalent to Einstein-Hilbert Lagrangian

TL;DR

This paper introduces a first-order Lagrangian equivalent to the Einstein-Hilbert Lagrangian, dependent on a symmetric connection, with a covariant variational formulation and Hamiltonian analysis.

Contribution

It presents a novel first-order Lagrangian variationally equivalent to Einstein-Hilbert, maintaining covariance and analyzing its Hamiltonian structure.

Findings

The Lagrangian is regular and covariant under diffeomorphisms.

The Hamiltonian formulation associated with the connection is developed.

The approach simplifies the variational problem of general relativity.

Abstract

A first-order Lagrangian $L^\nabla $ variationally equivalent to the second-order Einstein-Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by $L^\nabla $ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to $\nabla $.