## 10 The Hamilton-Jacobi Method

If one is concentrating only on the introduction and study of the properties of the quasi-local quantities,
but not interested in the detailed structure of the quasi-local (Hamiltonian) phase space, then perhaps the
most natural way to derive the general formulae is to follow the Hamilton-Jacobi method. This was done by
Brown and York in deriving their quasi-local energy expression [96, 97]. However, the Hamilton-Jacobi
method in itself does not yield any specific construction. Rather, the resulting general expression is
similar to a superpotential in the Lagrangian approaches, which should be completed by a choice
for the reference configuration and for the generator vector field of the physical quantity (see
Section 3.3.3). In fact, the ‘Brown-York quasi-local energy’ is not a single expression with a single
well-defined prescription for the reference configuration. The same general formula with several other,
mathematically inequivalent definitions for the reference configurations are still called the ‘Brown-York
energy’. A slightly different general expression was used by Kijowski [237], Epp [133], and
Liu and Yau [253]. Although the former follows a different route to derive his expression and
the latter two are not connected directly to the canonical analysis (and, in particular, to the
Hamilton-Jacobi method), the formalism and techniques that are used justify their presentation in this
section.
The present section is based mostly on the original papers [96, 97] by Brown and York. Since, however,
this is the most popular approach to finding quasi-local quantities and is the subject of very active
investigations, especially from the point of view of the applications in black hole physics, this section is
perhaps less complete than the previous ones. The expressions of Kijowski, Epp, and Liu and Yau will be
treated in the formalism of Brown and York.