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

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.