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 [315
], Epp [178
], Liu and Yau [338
] and
Wang and Yau [544
]. 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.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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