A Novel Hamiltonian Formulation of First Order Einstein-Hilbert Action: Connection with ADM, Diffeomorphism Invariance and Linearized Theory

TL;DR

This paper introduces a new Hamiltonian formulation of the first order Einstein-Hilbert action, explores its connection to ADM formalism and diffeomorphism invariance, and extends it to linearized gravity, revealing novel constraint structures and gauge symmetries.

Contribution

It presents a novel Dirac Hamiltonian formulation of the first order Einstein-Hilbert action with unique constraint algebra and links it to ADM and linearized theories.

Findings

Derived Hamiltonian and momentum constraints from tertiary constraints.

Established the relation between gauge functions and diffeomorphism descriptors.

Extended the formulation to linearized gravity, deriving the Hamiltonian for spin two fields.

Abstract

A novel Dirac Hamiltonian formulation of the first order Einstein-Hilbert (EH) action, in which algebraic constraints are not solved to eliminate fields from the action at the Lagrangian level, has been shown to lead to an action and a constraint structure apparently distinct from the ADM action and the ADM constraint structure in that secondary first class constraints as well as tertiary first class constraints arise with an unusual Poisson Bracket (PB) algebra [24]. By canonical transformations of the fundamental fields we show how from the tertiary constraints one may derive the Hamiltonian and momentum constraints. Special attention is paid to the Hamiltonian formulation of the first order EH action in terms of the variables h, h^i and q^{ij} employed in [19,20]. It is shown that the variables h and h^i are left undetermined in the formalism. This fact is used for a proper gauge fixation of the secondary constraints and reduction to the Faddeev action [19,20]. Considering invariances of the total action, the generator of the gauge transformations of the EH Lagrangian action is derived. Using this generator, the explicit form of the gauge invariance of the field h is obtained, by which the relation between the gauge functions and the descriptors of the diffeomorphism invariance is determined in order for the gauge transformations to correspond to diffeomorphism invariance. By linearizing the novel Hamiltonian formulation of [24], the Hamiltonian formulation of the first order action for the free spin two field [4,24] is derived.