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.

10 pages

References in corpus (3)

• Energy in Gravitation and Noether's Theorems
• Localizing Energy in Fierz-Lanczos theory
• Gauge invariant description of weak gravitational field on a spherically symmetric background with cosmological constant