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 Σ with boundary ∂Σ 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 $H_{\text{Inv}}$ uniquely describes the energy carried by the (linearized) gravitational field. It is shown that, under specific boundary conditions, the quasi-local Hawking mass $H_{\text{Haw}}$ reduces to $H_{\text{Inv}}$ in the weak field approximation. This observation is a quasi-local version of the classical Brill–Deser result (Brill and Deser 1968 Ann.Phys.(N.Y.) 50 3) 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. (paper)