A Simple Derivation of Canonical Structure and Quasi-local Hamiltonians in General Relativity

Abstract

A new method of variation of the gravitational Lagrangian is proposed. This method leads in a simple and straightforward way to the canonical description of the gravitational field dynamics in a finite volume V with boundary. No boundary terms are neglected or subtracted ad hoc. Two different forms of gravitational quasi-local energy are derived. Each of them is equal to the field Hamiltonian, corresponding to a specific way of controlling the field boundary data. They play the role of the “internal energy” and the “free energy” respectively. A relation with the boundary formula governing the thermodynamics of black holes is discussed.