The 3 + 1 approaches

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 systematically quasi-localizing them. One motivation for the quasi-localization of the ADM–Regge–Teitelboim analysis came from the need for a deeper understanding of the dynamics of subsystems of the universe (and, in particular, the basic results of Friedrich and Nagy [183 ] from a different perspective). Moreover, in this approach we might hope to be able to associate nontrivial observables (and, in particular, conserved quantities) with localized systems in a natural way. Another motivation is to try to provide a solid classical basis for the microscopic understanding of black hole entropy [40

, 39

, 114

]. 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. Update

11.1.1 The quasi-local constraint algebra and the basis Hamiltonian

If $Σ$ , the three-manifold on which the ADM canonical variables $hab$ , $ab p&tidle;$ are defined, has a boundary $𝒮 := ∂ Σ$ , then the usual vacuum constraints

are differentiable with respect to the canonical variables if the fields $N$ and $N a$ are vanishing on $𝒮$ and the area two-form on $𝒮$ , induced from the configuration variable $hab$ , is fixed ¹⁹ . Under these conditions the constraint functions close to a Poisson algebra $𝒞$ (the ‘quasi-local constraint algebra’); moreover, the evolution equations preserve these boundary conditions [460

]. However, the evolution in the spacetime corresponding to lapses and shifts that are vanishing on the two-boundary $𝒮$ yields new Cauchy surfaces in the same Cauchy development $D(Σ )$ of $Σ$ , and during such an evolution the boundary $𝒮$ remains pointwise fixed.

A similar analysis [460 ] shows that the basic Hamiltonian

coming from the Lagrangian $--1-(R + χab χ − χ2 ) 16πG ab$ , is differentiable with respect to the canonical variables if $N$ is vanishing on $𝒮$ , $a N$ is tangent to $𝒮$ on $𝒮$ , and the area two-form on $𝒮$ is fixed. If, in addition, the shift is required to be divergence-free with respect to the connection $δe$ on $𝒮$ , i.e., $δeN e = 0$ , then the evolution equations preserve these boundary conditions, the basic Hamiltonians form a closed Poisson algebra $ℋ0$ in which $𝒞$ is an ideal, and the evaluation of the basic Hamiltonians on the constraint surface,

defines a Lie algebra homomorphism from the Lie algebra of the $δe$ -divergence-free vector fields on $𝒮$ to the quotient Lie algebra $ℋ0 ∕𝒞$ . The evolution with such lapses and shifts in the spacetime is a mapping of the domain of dependence $D(Σ )$ onto itself, keeping the boundary $𝒮$ as a submanifold fixed, but not pointwise.

For the earlier investigations see [40 , 39 , 114 ], where stronger boundary conditions, namely fixing the whole three-metric $hab$ on $𝒮$ (but without the requirement $δeN e = 0$ ), were used to ensure the functional differentiability.

11.1.2 The two-surface observables

To understand the meaning of these observables, recall that any vector field $N a$ on $Σ$ generates a diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum constraint $a C[0,N ]$ 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 $O [N a]$ and the gauge degrees of freedom give rise to physical degrees of freedom, making it possible to introduce edge states [40 , 39 , 114 ].

Analogous investigations were done by Husain and Major in [258 ]. Using Ashtekar’s complex variables [23 ] they determine all the local boundary conditions for the canonical variables $Aia$ , $&tidle;Eai$ 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, Husain and Major discuss the two most significant cases. In the first case the generators $N$ , $N a$ , and $N i$ are vanishing on $𝒮$ , 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 $&tidle;a Ei$ (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 [40 , 39 , 114 , 258 ] 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 [116 , 375 ]. They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in [116 ] 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 [115 , 116 , 117 , 375 , 118 ]. However, this quantum issue is beyond the scope of the present review.

Returning to the discussion of $O [N a]$ above, note first that, though $Ae$ is a gauge potential, by $e δeN = 0$ it is boost gauge invariant. Without this condition, Equation (11.3 ) would give potentially reasonable physical quantity only if the boost gauge on $𝒮$ were geometrically given, e.g., when $𝒮$ were a leaf of a physically-distinguished foliation of a physically-distinguished spacelike or timelike hypersurface [32 ]. In particular, the angular momentum of Brown and York [112 ] also takes the form (11.3 ), and is well defined (because $N a$ is assumed to be a Killing vector of the intrinsic geometry of $𝒮$ ); and in the angular momentum of Liu and Yau [311 ] only the gauge invariant part of $Ae$ is present in Equation (11.3 ) instead of $Ae$ itself. Similarly, the expressions in [40 , 517 ] can be rewritten into the form (11.3 ), but they should be completed by the condition $e δeN = 0$ .

In general Equation (11.3 ) is used as a definition of the $N a$ –component of the angular momentum of quasi-locally defined black holes [33 , 86 , 206 ]. This interpretation is supported by the following observations [460 ]. In axisymmetric spacetimes for axisymmetric surfaces $a O [N ]$ can be rewritten into the Komar expression, the usual definition of angular momentum in axisymmetric spacetimes. Moreover, if $Σ$ extends to spatial infinity, then $δeN e = 0$ together with the requirement of the finiteness of the $r → ∞$ limit of the observable $O [N a]$ already fix the asymptotic form of $N a$ , which is precisely the combination of the asymptotic spatial rotation Killing vectors, and $a O [N ]$ reproduces the standard spatial ADM angular momentum. Similarly, at null infinity $a N$ must be a rotation BMS vector field. However, the null infinity limit of $O [N a]$ is sensitive to the first two terms (rather than only the leading term) in the asymptotic expansion of $N a$ , and hence in general radiative spacetime $O [N a]$ in itself does not yield an unambiguous definition for angular momentum. (But in stationary spacetimes the ambiguities disappear and $a O [N ]$ reproduces the standard formula (4.15 ).) Thus, additional ideas are needed to restrict the BMS vector field $N a$ .

Such an idea could be based on the observation that the eigenspinors of the $δ e$ -Dirac operators define $δe$ -divergence-free vector fields on $𝒮$ , 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 [462 ]. Solving the eigenvalue problem for the $δe$ -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 $+ ℐ$ can be compared, and the angular momentum flux can also be calculated.

It is tempting to interpret $O[N a]$ as the $N a$ -component of the quasi-local angular momentum of the gravity + matter system associated with $𝒮$ . However, without additional conditions on $N a$ the integral $a O [N ]$ could be nonzero even in Minkowski spacetime [462 ]. Hence, $a N$ must satisfy additional conditions. Cook and Whiting [141 ] suggest that one derive $N a$ 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 [330 ].) Another realization of the approximate Killing fields is given by Beetle in [52 ], where the vector field $a N$ 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 $δe$ -divergence-free $N a$ . The definition of $N a$ suggested in [297 ] is based on the fact that six of the infinitely many conformal Killing fields on $𝒮$ 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 $SO (1,3 )$ action on $𝒮$ . 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 $δe$ -divergence-free. Thus, as in the Liu–Yau definition, to keep boost gauge invariance the gauge invariant piece of the connection one-form $Ae$ can be used instead of the $Ae$ itself.