Diffeomorphism Invariance in the Hamiltonian formulation of General Relativity

TL;DR

This paper demonstrates that the Hamiltonian formulation of General Relativity, based on the Einstein-Hilbert Lagrangian without modifications, inherently preserves diffeomorphism invariance and general covariance.

Contribution

It shows that the Hamiltonian formulation with metric as a canonical variable naturally leads to gauge transformations consistent with diffeomorphism invariance.

Findings

Hamiltonian formulation preserves general covariance.

First-class constraints generate gauge transformations.

Diffeomorphism invariance is consistent with Hamiltonian approach.

Abstract

It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.