If one is concentrating only on the introduction and study of the properties of quasi-local quantities, and is 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 [120, 121]. 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 , Epp , Liu and Yau  and Wang and Yau . Although the former follows a different route to derive his expression and the latter three 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 mainly based on the original papers [120, 121] 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, Liu and Yau and Wang and Yau will be treated in the formalism of Brown and York.
Living Rev. Relativity 12, (2009), 4
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