10 The Hamilton–Jacobi Method

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 [120Jump To The Next Citation Point, 121Jump To The Next Citation Point]. 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 [315Jump To The Next Citation Point], Epp [178Jump To The Next Citation Point], Liu and Yau [338Jump To The Next Citation Point] and Wang and Yau [544Jump To The Next Citation Point]. 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 [120Jump To The Next Citation Point, 121Jump To The Next Citation Point] 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.

 10.1 The Brown–York expression
  10.1.1 The main idea
  10.1.2 The variation of the action and the surface stress-energy tensor
  10.1.3 The general form of the Brown–York quasi-local energy
  10.1.4 Further properties of the general expressions
  10.1.5 The Hamiltonians
  10.1.6 The flat space and light cone references
  10.1.7 Further properties and the various limits
  10.1.8 Other prescriptions for the reference configuration
 10.2 Kijowski’s approach
  10.2.1 The role of the boundary conditions
  10.2.2 The analysis of the Hilbert action and the quasi-local internal and free energies
 10.3 Epp’s expression
  10.3.1 The general form of Epp’s expression
  10.3.2 The definition of the reference configuration
  10.3.3 The various limits
 10.4 The expression of Liu and Yau
  10.4.1 The Liu–Yau definition
  10.4.2 The main properties of EKLY (𝒮)
  10.4.3 Generalizations of the original construction
 10.5 The expression of Wang and Yau

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