A Hamiltonian functional for the linearized Einstein vacuum field equations

TL;DR

This paper develops a Hamiltonian framework for the linearized Einstein vacuum equations, introducing a conserved functional as the Hamiltonian and establishing a compatible Poisson bracket structure.

Contribution

It presents a novel Hamiltonian formulation for linearized gravitational fields using a conserved functional, distinct from the energy, and derives the associated Poisson structure.

Findings

A conserved functional serving as Hamiltonian for linearized Einstein equations.

A compatible Poisson bracket structure for the field variables.

Generator of spatial translations derived within this framework.

Abstract

By considering the Einstein vacuum field equations linearized about the Minkowski metric, the evolution equations for the gauge-invariant quantities characterizing the gravitational field are written in a Hamiltonian form by using a conserved functional as Hamiltonian; this Hamiltonian is not the analog of the energy of the field. A Poisson bracket between functionals of the field, compatible with the constraints satisfied by the field variables, is obtained. The generator of spatial translations associated with such bracket is also obtained.