Summary: ON QUASI-LOCAL HAMILTONIANS IN GENERAL RELATIVITY
MICHAEL T. ANDERSON
Abstract. We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis
of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in
particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is
emphasized here. We argue that a consistent definition of quasi-local energy in GR requires, at a
minimum, a framework based on the (currently unknown) geometric well-posedness of the initial
boundary value problem for the Einstein equations.
The analysis of the gravitational field by Arnowitt-Deser-Misner  has led to a clear and well-
defined construction of the Hamiltonian, and resulting definitions of energy, linear and angular
momentum in the context of asymptotically flat spacetimes. These concepts are obviously of basic
importance in understanding the physics of such (infinite) isolated gravitating systems. Neverthe-
less, infinite systems are idealizations of more realistic physical situations, and it is desirable to
have available a similar analysis in the case of physical systems of finite extent.
However the understanding of this issue for domains of finite extent is much less satisfactory.
Despite numerous proposals, from a number of different viewpoints, a consensus has not yet been
reached on a suitable definition of the Hamiltonian or energy of a finite system, i.e. a quasi-local
Hamiltonian; cf.  for an excellent detailed survey of the current state of the art.
In this paper, we first examine and comment on the approach to the definition of energy of a
finite region of spacetime based on the Hamiltonian formulation of GR. This is essentially based on