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

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