Another motivation is to try to provide a solid classical basis for the microscopic understanding of black hole entropy [47, 46, 123]: 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 two-surface observables by means of which the ‘quantum edge states’ on the black hole event horizons were intended to be constructed.
If , the three-manifold on which the ADM canonical variables , are defined, has a smooth boundary , then the usual vacuum constraints21 Under these conditions the constraint functions close to a Poisson algebra (the ‘quasi-local constraint algebra’); moreover, the evolution equations preserve these boundary conditions . However, the evolution in the spacetime corresponding to lapses and shifts that are vanishing on the two-boundary yields new Cauchy surfaces in the same Cauchy development of , and during such an evolution the boundary remains pointwise fixed.
A similar analysis  shows that the basic Hamiltonianas a submanifold fixed, but not pointwise. Update
The condition that the area 2-form should be fixed appears to be the part of the ‘ultimate’ boundary condition for the canonical variables. In fact, in a systematic quasi-local Hamiltonian analysis boundary terms appear in the calculation of the Poisson bracket of two Hamiltonians also, which we called Poisson boundary terms in Section 3.3.3. Nevertheless, as we already mentioned there, the quasi-local Hamiltonian analysis of a single real scalar field in Minkowski space shows, these boundary terms represent the infinitesimal flow of energy-momentum and relativistic angular momentum. Thus, they must be gauge invariant . Assuming that in general relativity the Poisson boundary terms should have similar interpretation, their gauge invariance should be expected, and the condition of their gauge invariance can be determined. It is precisely the condition on the lapse and shift that the spacetime vector field built from them on the 2-surface must be divergence free there with respect to the connection of Section 4.1.2, i.e., . However, this is precisely the condition under which the evolution equations preserve the boundary condition . It might also be worth noting that this condition for the lapse and shift is just one of the ten components of the Killing equation: . (For the details, see .)
It should be noted that the area 2-form on the boundary 2-surface appears naturally in connection with the general symplectic structure on the ADM variables on a compact spacelike hypersurface with smooth boundary . In fact, in  an identity is derived for the variation of the ADM canonical variables on and of various geometrical quantities on . Examples are also given to illustrate how the resulting ‘quasi-local energy’ depends on the choice of the boundary conditions.
For the earlier investigations see [47, 46, 123], where stronger boundary conditions, namely fixing the whole three-metric on (but without the requirement ), were used to ensure the functional differentiability.
To understand the meaning of the observables (11.3, recall that any vector field on generates a diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum constraint is just to generate this gauge symmetry in the phase space. However, the boundary breaks the diffeomorphism invariance of the system, and hence, on the boundary the diffeomorphism gauge motions yield the observables and the gauge degrees of freedom give rise to physical degrees of freedom, making it possible to introduce edge states [47, 46, 123].
Analogous investigations were done by Husain and Major in . Using Ashtekar’s complex variables  they determine all the local boundary conditions for the canonical variables , and for the lapse , the shift , and the internal gauge generator on that ensure the functional differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints. Although there are several possibilities, Husain and Major discuss the two most significant cases. In the first case the generators , , and are vanishing on , and thus there are infinitely many two-surface observables, both from the diffeomorphism and the Gauss constraints, but no observables 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 (and hence, in particular, the three-metric) is fixed on the two-boundary. Then the quasi-local energy could be an observable, as in the ADM analysis above.
All of the papers [47, 46, 123, 281] discuss the analogous phenomenon of how the gauge freedoms become true physical degrees of freedom in the presence of two-surfaces on the two-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 [125, 409]. They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in  is based on the covariant Noether-charge formalism below.) Since this algebra is well known in conformal field theories, this approach might be a basis for understanding the microscopic origin of the black hole entropy [124, 125, 126, 409, 127]. However, this quantum issue is beyond the scope of the present review.
Returning to the discussion of above, note first that, though is a gauge potential, by it is boost gauge invariant. Without this condition, Eq. (11.3) would give potentially reasonable physical quantity only if the boost gauge on were geometrically given, e.g., when were a leaf of a physically-distinguished foliation of a physically-distinguished spacelike or timelike hypersurface . In particular, the angular momentum of Brown and York  also takes the form (11.3), and is well defined (because is assumed to be a Killing vector of the intrinsic geometry of ). (In the angular momentum of Liu and Yau  only the gauge invariant part of is present in Eq. (11.3) instead of itself.) Similarly, the expressions in [47, 571] can also be rewritten into the form (11.3), but they should be completed by the condition .
In general Eq. (11.3) is used as a definition of the –component of the angular momentum of quasi-locally defined black holes [40, 97, 227]. This interpretation is supported by the following observations . In axisymmetric spacetimes for axisymmetric surfaces can be rewritten into the Komar integral, the usual definition of angular momentum in axisymmetric spacetimes. Moreover, if extends to spatial infinity, then together with the requirement of the finiteness of the limit of the observable already fix the asymptotic form of , which is precisely the combination of the asymptotic spatial rotation Killing vectors, and reproduces the standard spatial ADM angular momentum. Similarly, at null infinity must be a rotation BMS vector field. However, the null infinity limit of is sensitive to the first two terms (rather than only the leading term) in the asymptotic expansion of , and hence in general radiative spacetime in itself does not yield an unambiguous definition for angular momentum. (But in stationary spacetimes the ambiguities disappear and reproduces the standard formula (4.15).) Thus, additional ideas are needed to restrict the BMS vector field .
Such an idea could be based on the observation that the eigenspinors of the -Dirac operators define -divergence-free vector fields on , and on metric spheres these vector fields built from the eigenspinors with the lowest eigenvalue are just the linear combinations of the three rotation Killing fields . Solving the eigenvalue problem for the -Dirac operators on large spheres near scri in the first two leading orders, a well-defined (ambiguity-free) angular-momentum expression is suggested. The angular momenta associated with different cuts of can be compared, and the angular momentum flux can also be calculated.
It is tempting to interpret as the -component of the quasi-local angular momentum of the gravity + matter system associated with . However, without additional conditions on the integral could be nonzero even in Minkowski spacetime . Hence, must satisfy additional conditions. Cook and Whiting  suggest that one derive from a variational principle on topological two-spheres. Here the action functional is the norm of the Killing operator. (For a viable, general notion of approximate Killing fields see .) Another realization of the approximate Killing fields is given by Beetle in , where the vector field is searched for in the form of the solution of an eigenvalue problem for an equation, derived from the Killing equations. Both prescriptions have versions in which they give -divergence-free . The definition of suggested in  is based on the fact that six of the infinitely many conformal Killing fields on with spherical topology are globally defined, and after an appropriate globally-defined conformal rescaling of the intrinsic metric they become the generators of the standard action on . Then these three are used to define the angular momentum that will be the Killing fields in the rescaled geometry. In general these vector fields are not -divergence-free. Thus, as in the Liu–Yau definition, to keep boost gauge invariance the gauge invariant piece of the connection one-form can be used instead of the itself.
Living Rev. Relativity 12, (2009), 4
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