The traditional ADM approach to conserved quantities and the Hamiltonian analysis of general relativity is based on the 3 + 1 decomposition of fields and geometry. Although the results and the content of a theory may be covariant even if their form is not, the manifest spacetime covariance of a formalism may help to find the (spacetime covariant) observables and conserved quantities, boundary conditions, etc. easily. No a posteriori spacetime interpretation of the results is needed. Such a spacetime-covariant Hamiltonian formalism was initiated by Nester [280, 283].

His basic idea is to use (tensor or Dirac spinor valued) differential forms as the basic field variables on the spacetime manifold . Thus his phase space is the collection of fields on the 4-manifold , endowed with the (generalized) symplectic structure of Kijowski and Tulczyjew [239]. He derives the field equations from the Lagrangian 4-form, and for a fixed spacetime vector field finds a Hamiltonian 3-form whose integral on a spacelike hypersurface takes the form

the sum of the familiar ADM constraints and a boundary term. The Hamiltonian is determined from the requirement of the functional differentiability of , i.e. that the variation with respect to the canonical variables should not contain any boundary term on an asymptotically flat (see Sections 2.2.2, 3.2.1, and 3.2.2). For asymptotic translations the boundary term in the Hamiltonian gives the ADM energy-momentum 4-vector. In tetrad variables is essentially Sparling’s 3-form [345], and the 2-component spinor version of is essentially the Nester-Witten 2-form contracted in the name index with the components of (see also Section 3.2.1).The spirit of the first systematic investigations of the covariant phase space of the classical field theories [122, 20, 146, 251] is similar to that of Nester’s. These ideas were recast into the systematic formalism by Wald and Iyer [389, 215, 216], the so-called covariant Noether charge formalism (see also [388, 251]). This formalism generalizes many of the previous approaches: The Lagrangian 4-form may be any diffeomorphism invariant local expression of any finite order derivatives of the field variables. It gives a systematic prescription for the Noether currents, the symplectic structure, the Hamiltonian etc. In particular, the entropy of the stationary black holes turned out to be just a Noether charge derived from Hilbert’s Lagrangian.

These general ideas were applied to general relativity in the tetrad formalism (and also in the Dirac spinor formulation of the theory [109, 105], yielding a Hamiltonian which is slightly different from Equation (87)) as well as in the usual metric formalism [105, 108]. In the latter it is the appropriate projections to of or in some coordinate system that is chosen to be fixed on . Then the dual of the corresponding Dirichlet and Neumann boundary 2-forms, respectively, will be

The first terms are analogous to Freud’s superpotential, while the second ones are analogous to Komar’s superpotential. (Since the boundary 2-form contains only in the form , this is always tensorial. If is chosen to be vanishing, then the first term reduces to Freud’s superpotential.) Because of the Komar-like term, the quasi-local quantities depend not only on the 2-surface data (both in the physical spacetime and the reference configuration), but on the normal directional derivative of as well. The connection between the present expressions and the similar previous results (pseudotensorial, tensorial, and quasi-local) is also discussed in [107, 105]. In particular, the expression based on the Dirichlet-type boundary 2-form (90) gives precisely the Katz-Bicak-Lynden-Bell superpotential [230]. In the spinor formulation of these ideas the vector field would be built from a Dirac spinor (or a pair of Weyl spinors). The main difficulty is, however, to find spinor fields representing both translational and boost-rotational displacements [110]. In the absence of a prescription for the reference configuration (even though that should be defined only on an open neighbourhood of the 2-surface) the construction is still not complete, even if the vector field is chosen to be a Killing vector of the reference spacetime.A nice application of the covariant expression is a derivation of the first law of black hole thermodynamics [107]. The quasi-local energy expressions have been evaluated for several specific 2-surfaces. For round spheres in the Schwarzschild spacetime both the 4-covariant Dirichlet and Neumann boundary terms (with the Minkowski reference spacetime and as the timelike Killing vector ) give at infinity, but at the horizon the former gives and the latter is infinite [107]. The Dirichlet boundary term gives at the spatial infinity in the Kerr-anti-de-Sitter solution the standard and values for the energy and angular momentum, respectively [191]. Also, the center-of-mass is calculated both in the metric and the tetrad formulation of general relativity for the eccentric Schwarzschild solution at the spatial infinity [286, 287], and it was found that the ‘Komar-like term’ is needed to recover the correct, expected value. At the future null infinity of asymptotically flat spacetimes it gives the Bondi-Sachs energy-momentum and the expression of Katz [229, 233] for the angular momentum [192]. The general formulae are evaluated for the Kerr-Vaidya solution too.

Anco continued the investigation of the Dirichlet Hamiltonian in [6], which takes the form

Here the 2-surface is assumed to be mean convex, whenever the boost gauge freedom in the gauge potential can be, and, indeed, is fixed by using the globally defined orthonormal vector basis in the normal bundle obtained by normalizing the mean curvature basis . The vector field is still arbitrary, and is an arbitrary function of the metric on the 2-boundary , i.e. of the boundary data. This is actually assumed to have the structure for as an arbitrary function of . This Hamiltonian is functionally differentiable, gives the correct Einstein equations and, for solutions, its value e.g. with is the general expression of the quasi-local energy of Brown and York. (Compare Equation (92) with Equation (86), or rather with Equations (72, 73, 74).)However, to rule out the dependence of this notion of quasi-local energy on the completely freely specifiable vector field (i.e. on three arbitrary functions on ), Anco makes dynamical by linking it to the vector field . Namely, let , where and are constant, is the area of , and extend this from to in a smooth way. Then Anco proves that, keeping the 2-metric and fixed on ,

is a correct Hamiltonian for the Einstein equations, where is an arbitrary function of . For with the choice the boundary term reduces to the Hawking energy, and for it is the Epp and Kijowski-Liu-Yau energies depending on the choice of (i.e. the definition of the reference term). For general choosing the reference term appropriately Anco gives a 1-parameter generalization of the Hawking and the Epp-Kijowski-Liu-Yau-type quasi-local energies (called the ‘mean curvature masses’). Also, he defines a family of quasi-local angular momenta. Using the positivity of the Kijowski-Liu-Yau energy () it is shown that the higher power () mean curvature masses are bounded from below. Although these masses seem to have the correct large sphere limit at spatial infinity, for general convex 2-surfaces in Minkowski spacetime they do not vanish.

As we discussed briefly in Section 3.3.1, many, apparently different pseudotensors and -gauge dependent energy-momentum density expressions can be recovered from a single differential form defined on the bundle of linear frames over the spacetime manifold: The corresponding superpotentials are the pull-backs to of the various forms of the Nester-Witten 2-from from along the various local sections of the bundle [142, 266, 352, 353]. Thus the different pseudotensors are simply the gauge dependent manifestations of the same geometric object on the bundle in the different gauges. Since, however, is the unique extension of the Nester-Witten 2-form on the principal bundle of normalized spin frames (given in Equation (12)), and the latter has been proven to be connected naturally to the gravitational energy-momentum, the pseudotensors appear to describe the same physics as the spinorial expressions, though in a slightly old fashioned form. That this is indeed the case was demonstrated clearly by Chang, Nester, and Chen [104, 108, 285], by showing an intimate connection between the covariant quasi-local Hamiltonian expressions and the pseudotensors. Writing the Hamiltonian in the form of the sum of the constraints and a boundary term, in a given coordinate system the integrand of this boundary term may be the superpotential of any of the pseudotensors. Then the requirement of the functional differentiability of gives the boundary conditions for the basic variables at . For example, for the Freud superpotential (for Einstein’s pseudotensor) what is fixed on the boundary is a certain piece of .

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