  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.
