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. more easily. No a posteriori spacetime interpretation of the results is needed. Such a spacetime-covariant Hamiltonian formalism was initiated by Nester [377, 380].

His idea is to use (tensor or Dirac spinor valued) differential forms as the field variables on the spacetime manifold . Thus, his phase space is the collection of fields on the four-manifold , endowed with the (generalized) symplectic structure of Kijowski and Tulczyjew [317]. 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 four-vector. In tetrad variables is essentially Sparling’s 3-form [476], and the two-component spinor version of is essentially the Nester–Witten 2-form contracted in the name index with the components of (see Eq. (3.13), Section 3.2.1 and the introductory paragraphs in Section 8).The spirit of the first systematic investigations of the covariant phase space of the classical field theories [158, 33, 197, 336] is similar to that of Nester’s. These ideas were recast into the systematic formalism by Wald and Iyer [536, 287, 288], the covariant Noether charge formalism (see also [535, 336]). 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 turns out to be just a Noether charge derived from Hilbert’s Lagrangian.

The quasi-local Hamiltonian for a large class of geometric theories, allowing torsion and nonmetricity of the connection, was investigated by Chen, Nester, and Tung [139, 136, 382] in the covariant approach of Nester, above [377, 380]. Starting with a Lagrangian 4-form for a first-order formulation of the theory and an arbitrary vector field , they determine the general form of the Hamiltonian 3-form , including the boundary 2-form . However, in the variation of the corresponding Hamiltonian there will be boundary terms in general. To cancel them, the boundary 2-form has to be modified. Introducing an explicit reference field and canonical momentum (which are solutions of the field equations), Chen, Nester, and Tung suggest (in differential form notation) either of the two four-covariant boundary 2-forms

where the configuration variable is some (tensor-valued) -form and is the interior product of the -form and the vector field , i.e., in the abstract index formalism . Thus, the boundary terms of Chen, Nester and Tung contain not only a general reference term, but the reference values of the canonical variables. Or, in other words, the ‘calibration’ of their quasi-local quantities is made at the level of the basic variables, rather than at the level of the boundary term.The boundary term in the variation of the Hamiltonian with the boundary term (11.5) and (11.6) is the two-surface integral on of and , respectively. Therefore, the Hamiltonian is functionally differentiable with the boundary 2-form if the configuration variable is fixed on , but should be used if is fixed on . Thus, the first boundary 2-form corresponds to a four-covariant Dirichlet-type, while the second corresponds to a four-covariant Neumann-type boundary condition. Obviously, the Hamiltonian evaluated in the reference configuration gives zero. Chen and Nester show [136] that and are the only boundary 2-forms for which the resulting boundary 2-form in the variation of the Hamiltonian 3-form vanishes on , which reflects the type of boundary conditions (i.e., which fields are fixed on the boundary), and is built from the configuration and momentum variables four-covariantly (‘uniqueness’). A further remarkable property of and is that the corresponding Hamiltonian 3-form can be derived directly from appropriate Lagrangians. One possible choice for the vector field is a Killing vector of the reference geometry. This reference geometry is, however, not yet specified, in general.

These general ideas were applied to general relativity in the tetrad formalism (and also in the Dirac spinor formulation of the theory [139, 132], yielding a Hamiltonian, which is slightly different from Eq. (11.4)) as well as in the usual metric formalism [132, 137]. 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 will be, respectively,

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 two-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 [136, 132]. In particular, the expression based on the Dirichlet-type boundary 2-form (11.7) gives precisely the Katz–Bicak–Lynden-Bell superpotential [306]. 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 [140]. In the absence of a prescription for the reference configuration (even though that should be defined only on an open neighborhood of the two-surface) the construction is still not complete, even if the vector field is chosen to be a Killing vector of the reference spacetime. A recent manifestly covariant way of introduction to these ideas is given in [383].A nice application of the covariant expression is a derivation of the first law of black hole thermodynamics [136]. The quasi-local energy expressions have been evaluated for several specific two-surfaces. For round spheres in the Schwarzschild spacetime, both the four-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 [136]. The Dirichlet boundary term gives, at spatial infinity in the Kerr–anti-de Sitter solution, the standard and values for the energy and angular momentum, respectively [257]. The center-of-mass is also calculated, both in the metric and the tetrad formulation of general relativity, for the eccentric Schwarzschild solution at spatial infinity [389, 390], and it was found that the ‘Komar-like term’ is needed to recover the correct, expected value. At future null infinity of asymptotically flat spacetimes it gives the Bondi–Sachs energy-momentum and the expression of Katz [305, 310] for the angular momentum [258]. The general formulae are evaluated for the Kerr–Vaidya solution as well.

The quasi-local energy-momentum is calculated on two-surfaces lying in intrinsically-flat spacelike hypersurfaces in static spherically-symmetric spacetimes [138], and, in particular, for two-surfaces in the slicing of the Schwarzschild solution in the Painlevé–Gullstrand coordinates. Though these hypersurfaces are flat, and hence, the total (ADM type) energy is expected to be vanishing, the quasi-local energy expression based on Eq. (11.7) and a ‘naturally chosen’ frame field gives . (N.B., the Cauchy data on the hypersurfaces do not satisfy the falloff conditions of Section 3.2.1. Though the intrinsic metric is flat, the extrinsic curvature tends to zero only as , while in the expression of the ADM linear momentum a slightly faster than falloff is needed. Thus, the vanishing of the naïvly introduced ADM-type energy does not contradict the rigidity part of the positive energy theorem.)

The null infinity limit of the quasi-local energy and the corresponding outgoing energy flux, based on Eq. (11.5), are calculated in [563]. It is shown that, with Minkowski spacetime as a reference configuration, and even with three different embeddings of the two-surface into the reference spacetime, the null infinity limit of these two quantities are just the standard Bondi energy and Bondi mass-loss, respectively. A more detailed discussion of the general formulae for the quasi-local energy flux, coming from Eqs. (11.5) – (11.6) and the two additional boundary expressions of [137],

is given in [141]. A less technical presentation and further discussions of the energy flux calculations are given in [388].The quasi-local energy flux of spacetime perturbations on a stationary background is calculated by Tung and Yu [531] using the covariant Noether charge formalism and the boundary terms above. As an example they considered the Vaidya spacetime as a time-dependent perturbation of a stationary one with the orthonormal frame field being adapted to the spherical symmetry. At null infinity they recovered the Bondi mass-loss, while for the dynamical horizons they recovered the flux expression of Ashtekar and Krishnan (see Section 13.3.2).

The quasi-local energy-momentum, based on Eq. (11.7) in the tetrad approach to general relativity, is calculated for arbitrary two-surfaces lying in the hypersurfaces of the homogeneity in all the Bianchi cosmological models in [391] (see also [340]). In these calculations the tetrad field was chosen to be the geometrically distinguished triad, being invariant with respect to the global action of the isometry group, and the future-pointing unit timelike normal of the hypersurfaces; while the vector field was chosen to have constant components in this frame. For class A models (i.e., for I, II, VI, VII, VIII and IX Bianchi types) this is zero, and for class B models (III, IV, V, VI and VII Bianchi models) the quasi-local energy is negative, and the energy is proportional to the volume of the domain that is bounded by . (Here a sign error in the previous calculations, reported in [134, 387, 385], is corrected.) The apparent contradiction of the nonpositivity of the energy in the present context and the non-negativity of the energy in general small-sphere calculations indicates that the geometrically distinguished tetrad field in the Bianchi models does not reduce to the ‘natural’ approximate translational Killing fields near a point. Another interpretation of the vanishing and negativity of the quasi-local energy, different from this and those in Section 4.3, is also given.

Instead of the specific boundary terms, So considered a two-parameter family of boundary terms [464], which generalized the special expressions (11.5) – (11.6) and (11.9) – (11.10). The main idea behind this generalization is that one cannot, in general, expect to be able to control only, for example, either the configuration or the momentum variables, rather only a combination of them. Hence, the boundary condition is not purely of a Dirichlet or Neumann type, but rather a more general mixed one. It is shown that, with an appropriate value for these parameters, the resulting energy expression for small spheres is positive definite, even in the holonomic description.

In the general covariant quasi-local Hamiltonians Chen, Nester and Tung left the reference configuration and the boundary conditions unspecified, and hence their construction was not complete. These have been specified in [386]. The key ideas are as follow.

First, because of its correct, advantagous properties (especially its asymptotic behaviour in asymptotically flat spacetimes), Nester, Chen, Liu and Sun choose (11.7) a priori as their Hamiltonian boundary term. Their reference configuration is chosen to be the Minkowski spacetime, and the generator vector field is the general Killing vector (depending on ten parameters).

Next, to match the physical and the reference geometries, they require the two full 4-dimensional metrics to coincide at the points of the two-surface (rather than only the induced two-metrics on ). This condition leaves two unspecified functions in the quasi-local quantities. To find the ‘best matched’ such embedding of into the Minkowski spacetime, Nester, Chen, Liu and Sun propose to choose the one that extremize the quasi-local mass.

This, and some other related strategies have been used to compute quasi-local energy in various spherically symmetric configurations in [135, 341, 561, 562].

Anco and Tung investigated the possible boundary conditions and boundary terms in the quasi-local Hamiltonian using the covariant Noether charge formalism both of general relativity (with the Hilbert Lagrangian and tetrad variables) and of Yang–Mills–Higgs systems [13, 14]. (Some formulae of the journal versions were recently corrected in the latest arXiv versions.) They considered the world tube of a compact spacelike hypersurface with boundary . Thus, the spacetime domain they considered is the same as in the Brown–York approach: . Their evolution vector field is assumed to be tangent to the timelike boundary of the domain . They derived a criterion for the existence of a well-defined quasi-local Hamiltonian. Dirichlet and Neumann-type boundary conditions are imposed. In general relativity, the variations of the tetrad fields are restricted on by requiring in the first case that the induced metric is fixed and the adaptation of the tetrad field to the boundary is preserved, while in the second case that the tetrad components of the extrinsic curvature of is fixed. Then the general allowed boundary condition was shown to be just a mixed Dirichlet–Neumann boundary condition. The corresponding boundary terms of the Hamiltonian, written in the form , were also determined [13]. The properties of the co-vectors and (called the Dirichlet and Neumann symplectic vectors, respectively) were investigated further in [14]. Their part tangential to is not boost gauge invariant, and to evaluate them, the boost gauge determined by the mean extrinsic curvature vector is used (see Section 4.1.2). Both and are calculated for various spheres in several special spacetimes. In particular, for the round spheres of radius in the hypersurface in the Reissner–Nordström solution and , and hence, the Dirichlet and Neumann ‘energies’ with respect to the static observer are and , respectively. Thus, does not reproduce the standard round-sphere expression, while gives the standard round sphere and correct ADM energies only if it is ‘renormalized’ by its own value in Minkowski spacetime [14].

Anco continued the investigation of the Dirichlet Hamiltonian in [11], which takes the form (see also Eqs. (8.1) and (10.20))

Here the two-surface is assumed to be mean convex, in which case 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 assumed to have the structure for as an arbitrary function of . This Hamiltonian 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 Eq. (11.11) with Eq. (11.3), or with Eqs. (10.8), (10.9) and (10.10).)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 dynamic 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 two-metric and fixed on ,

is a correct Hamiltonian for the Einstein equations, where is still 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 one-parameter generalization of Hawking and Epp–Kijowski–Liu–Yau-type quasi-local energies (called the ‘mean curvature masses’). In addition, he defines a family of quasi-local angular momenta. Using the positivity of the Kijowski–Liu–Yau energy () he shows 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 two-surfaces in Minkowski spacetime they do not vanish.The boundary condition on closed untrapped spacelike two-surfaces that make the covariant Hamiltonian functionally differentiable were investigated by Tung [526, 527]. He showed that such a boundary condition might be the following: the area 2-form and the mean curvature vector of are fixed, and the evolution vector field is proportional to the dual mean curvature vector, where the factor of proportionality is a function of the area 2-form. Then, requiring that the value of the Hamiltonian reproduce the ADM energy, he recovers the Hawking energy. If, however, is allowed to have a part tangential to , and is required to be fixed (up to total -divergences), then, though the value of the Hamiltonian is still proportional to the Hawking energy, the factor of proportionality depends on the angular momentum, given by (11.3), as well. With this choice the vector field becomes a generalization of the Kodama vector field [321] (see also Section 4.2.1). The results of [527, 528] are extensions of those in [526].

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 pullbacks to of the various forms of the Nester–Witten 2-from from along the various local sections of the bundle [192, 358, 486, 487]. 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 Eq. (3.10)), 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 [131, 137, 382] 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 .

Living Rev. Relativity 12, (2009), 4
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