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11.1 The 3 + 1 approaches

There is a lot of literature on the canonical formulation of general relativity both in the traditional ADM and the Møller tetrad (or, recently, the closely related complex Ashtekar) variables. Thus it is quite surprising how little effort has been spent to systematically quasi-localize them. One motivation for the quasi-localization of the ADM-Regge-Teitelboim analysis came from the need for the microscopic understanding of black hole entropy [32Jump To The Next Citation Point31Jump To The Next Citation Point99Jump To The Next Citation Point]: What are the microscopic degrees of freedom behind the phenomenological notion of black hole entropy? Since the aim of the present paper is to review the construction of the quasi-local quantities in classical general relativity, we discuss only the classical 2-surface observables by means of which the ‘quantum edge states’ on the black hole event horizons were intended to be constructed.

11.1.1 The 2-surface observables

If S, the 3-manifold on which the ADM canonical variables h ab, ~pab are defined, has a boundary S := @S, then the usual vacuum constraints

integral { [ ( )] } a --1---3 V~ --- 16pG-- 1-( ab )2 ab ab 3 C [n,n ] := - S n 16pG R |h |+ V~ |h-| 2 ~p hab - p~ ~pab + 2naDb ~p d x (83)
are differentiable with respect to the canonical variables only if the smearing fields n and na and the derivative Dan are vanishing on S 21. However, as Balachandran, Chandar, and Momen [32Jump To The Next Citation Point31Jump To The Next Citation Point] and Carlip [99Jump To The Next Citation Point] realized, the boundary conditions for the smearing fields can be relaxed by adding appropriate boundary terms to the constraints. Namely, for any vector field a V and function T they define
integral gf a ab 3 a ab OM [V ] := ~p ´LVhabd x = C [0,V ]- 2 vaVbp dS, (84) S integral [ ( S )] gf O [T ] := - T --1--3R + 16pG-- 1(~pabh )2 - ~pab~p dS - -1--- Tk dS = H S 16pG |h| 2 ab ab 8pG S 1 gf = C [T,0] - ----- Tk dS, (85) 8pG S
where k is the trace of the extrinsic curvature of S in S and qab is the induced metric on S. Then O [V a] M is functionally differentiable if V a is tangent to S, and O [T ] H is functionally differentiable if DaT is vanishing on S (and hence, in particular, T is a [not necessarily zero] constant on S) and hab is fixed on S. Furthermore, for any two such V a and V 'a coinciding on S, the difference OM[V a]- OM[V 'a] is just a momentum constraint, and, similarly, OH[T ]- OH[T '] is a Hamiltonian constraint. Thus, on the constraint surface, O [V a] M and O [T] H depend only on the value of V a and T on S. A direct calculation shows that their Poisson bracket with the constraints is a constraint, i.e. vanishing on the constraint surface. Therefore, OM[V a] and OH[T ] are well-defined 2-surface observables corresponding to the momentum and the Hamiltonian constraint, respectively. Moreover, the observables OM[V a] form an infinite-dimensional Lie algebra with respect to the Poisson bracket. In this Lie algebra the momentum constraints form an ideal, and the quotient of the algebra of the observables a OM[V ] and the constraint ideal is still infinite-dimensional. (In the asymptotically flat case, cf. [47364].) In fact, OM defines a homomorphism of the Lie algebra of the 2-surface vector fields into the quotient algebra of these observables modulo constraints [32Jump To The Next Citation Point31Jump To The Next Citation Point99Jump To The Next Citation Point]. To understand the meaning of these observables, recall that any vector field a V on S generates a diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum constraint C[0,V a] is just to generate this gauge symmetry in the phase space. However, the boundary S breaks the diffeomorphism invariance of the system, and hence on the boundary the diffeomorphism gauge motions yield the observables O [V a] M and the gauge degrees of freedom give raise to physical degrees of freedom, making it possible to introduce the so-called edge states [32Jump To The Next Citation Point31Jump To The Next Citation Point99Jump To The Next Citation Point]. Evaluating OM[V a] and OH[T ] on the constraint surface we obtain
gf gf a -1--- a -1--- OM [V ] = - 8pG V Aa dS, OH [T ] = - 8pG T kdS, (86) S S
i.e. these are just the integrals of the (unreferenced) Brown-York energy surface density weighted by the constant T and momentum surface density contracted with V a, respectively (see Equation (74View Equation)).

Analogous investigations were done by Husain and Major in [210Jump To The Next Citation Point]. Using Ashtekar’s complex variables [17] they determined all the local boundary conditions for the canonical variables i A a, ~a E i and for the lapse N, the shift N a, and the internal gauge generator N i on S that ensure the functional differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints. Although there are several possibilities, they discussed the two most significant cases. In the first case the generators N, a N, and i N are vanishing on S, whenever there are infinitely many 2-surface observables both from the diffeomorphism and the Gauss constraints, but no observable from the Hamiltonian constraint. The structure of these observables is similar to that of those coming from the ADM diffeomorphism constraint above. The other case considered is when the canonical momentum E~a i (and hence, in particular, the 3-metric) is fixed on the 2-boundary. Then the quasi-local energy could be an observable, as in the ADM analysis above.

All of the papers [323199210] discuss the analogous phenomenon of how the gauge freedoms are getting to be true physical degrees of freedom in the presence of 2-surfaces on the 2-surfaces themselves in the Chern-Simons and BF theories. Weakening the boundary conditions further (allowing certain boundary terms in the variation of the constraints) a more general algebra of ‘observables’ can be obtained [101Jump To The Next Citation Point296Jump To The Next Citation Point]: They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in [101Jump To The Next Citation Point] is based on the so-called covariant Noether charge formalism below.) Since this algebra is well known in conformal field theories, this approach might be a basis of understanding the microscopic origin of the black hole entropy [100Jump To The Next Citation Point101Jump To The Next Citation Point102Jump To The Next Citation Point296Jump To The Next Citation Point103Jump To The Next Citation Point]. However, this quantum issue is beyond the scope of the present review.

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