Gauge-invariant quadratic approximation of quasi-local mass and its relation with Hamiltonian for gravitational field

TL;DR

This paper develops a gauge-invariant Hamiltonian framework for linearized gravitational fields in a bounded region, linking quasi-local mass definitions with Hamiltonian formulations under specific boundary conditions.

Contribution

It introduces a gauge-invariant quadratic approximation of quasi-local mass and relates it to the Hamiltonian for gravitational fields, extending classical results to a quasi-local context.

Findings

Hamiltonian formulation for linearized gravity with boundary conditions

Reduction of Geroch--Hawking mass to Hamiltonian in weak field limit

Extension of Brill--Deser classical result to quasi-local setting

Abstract

Gauge invariant, Hamiltonian formulation of field dynamics within a compact region $\Sigma$ with boundary $\partial \Sigma$ is given for the gravitational field linearized over a Kottler metric. The boundary conditions which make the system autonomous are discussed and the corresponding Hamiltonian functional $\mathcal{H}_\text{Invariant}$ is calculated. It is shown that, under specific boundary conditions, the quasi-local Geroch--Hawking mass $\mathcal{H}_\text{Hawking}$ reduces to $\mathcal{H}_\text{Invariant}$ in the weak field approximation. This observation is a quasi-local version of the classical Brill--Deser result.