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

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. The corresponding Hamiltonian functional $\mathcal{H}\text{\tiny Inv}$ uniquely describes the energy carried by the (linearized) gravitational field. } It is shown that, under specific boundary conditions, the quasi-local Hawking mass $\mathcal{H}\text{\tiny Haw}$ reduces to $\mathcal{H}_\text{\tiny Inv}$ in the weak field approximation. This observation is a quasi-local version of the classical Brill--Deser result [D. R. Brill, S. Deser, Ann.Phys.(N.Y.) \textbf{50}, 3 (1968)] and enables us to declare Hawking mass as the correct expression (at least up to quadratic terms in the Taylor expansion) for the quasi-local mass, which correctly describes energy carried by the gravitational field.