A SIMPLE DERIVATION OF CANONICAL STRUCTURE AND QUASI-LOCAL HAMILTONIANS IN GENERAL-RELATIVITY

A new method of variation of the gravitational Lagrangian is proposed. This method leads in a simple and straightforward way to the canonica l 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. E ach of them is equal to the field Hamiltonian, corresponding to a spec ific way of controlling the field boundary data. They play the role of the ''internal energy'' and the ''free energy'' respectively. A relat ion with the boundary formula governing the thermodynamics of black ho les is discussed.