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11.3 The covariant approach

11.3.1 The covariant phase space methods

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 [280Jump To The Next Citation Point283Jump To The Next Citation Point].

His basic idea is to use (tensor or Dirac spinor valued) differential forms as the basic field variables on the spacetime manifold M. Thus his phase space is the collection of fields on the 4-manifold M, 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 Ka finds a Hamiltonian 3-form H(K)abc whose integral on a spacelike hypersurface takes the form

integral gf H [K] = -1--- KaGab -1ebcde + B (Ka) , (87) 8pG S 3! @S cd
the sum of the familiar ADM constraints and a boundary term. The Hamiltonian is determined from the requirement of the functional differentiability of H[K], i.e. that the variation dH[K] with respect to the canonical variables should not contain any boundary term on an asymptotically flat S (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 H(K)abc is essentially Sparling’s 3-form [345], and the 2-component spinor version of a B(K )cd is essentially the Nester-Witten 2-form contracted in the name index with the components of Ka (see also Section 3.2.1).

The spirit of the first systematic investigations of the covariant phase space of the classical field theories [12220146251Jump To The Next Citation Point] is similar to that of Nester’s. These ideas were recast into the systematic formalism by Wald and Iyer [389Jump To The Next Citation Point215Jump To The Next Citation Point216Jump To The Next Citation Point], the so-called covariant Noether charge formalism (see also [388251]). 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.

11.3.2 Covariant quasi-local Hamiltonians with explicit reference configurations

UpdateJump To The Next Update Information The quasi-local Hamiltonian for a large class of geometric theories, allowing torsion and non-metricity of the connection, was investigated by Chen, Nester, and Tung [109Jump To The Next Citation Point107Jump To The Next Citation Point285Jump To The Next Citation Point] in the covariant approach of Nester above [280283]. Starting with a Lagrangian 4-form for a first order formulation of the theory and an arbitrary vector field a K, they determine the general form of the Hamiltonian 3-form H(K)abc, including the boundary 2-form B(Ka)cd. 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 f0A and canonical momentum 0 p A (which are solutions of the field equations), Chen, Nester, and Tung suggest (in the differential form notation) either of the two 4-covariant boundary 2-forms
( ) ( ) Bf(Ka) := iKfA /\ pA - p0A - (-)k fA - f0A /\ iKp0A, (88) a 0A ( 0) k ( A 0A) Bp(K ) := iKf /\ pA - p A - (-) f - f / \ iKpA, (89)
where the configuration variable fA is some (tensor valued) k-form and iKfa is the interior product of the k-form A fa1...ak and the vector field a K, i.e. in the abstract index formalism (iKfA)a ...a = kKafA 2 k aa2...ak. Then the boundary term in the variation dH[K] of the Hamiltonian is the 2-surface integral on @S of i (dfA /\ (p - p0)) K A A and i (-(fA - f0A) /\ dp ) K A, respectively. Therefore, the Hamiltonian is functionally differentiable with the boundary 2-form a Bf(K ) if the configuration variable fA is fixed on @S, but Bp(Ka) should be used if pA is fixed on @S. Thus the first boundary 2-form corresponds to a 4-covariant Dirichlet-type, while the second to a 4-covariant Neumann-type boundary condition. Obviously, the Hamiltonian evaluated in the reference configuration 0A 0 (f ,pA) gives zero. Chen and Nester show [107Jump To The Next Citation Point] that a Bf(K ) and a Bp(K ) are the only boundary 2-forms for which the resulting boundary 2-form a C(K )bc in the variation dH(Ka)bcd of the Hamiltonian 3-form vanishes on @S, reflects the type of the boundary conditions (i.e. which fields are fixed on the boundary), and is built from the configuration and momentum variables 4-covariantly (‘uniqueness’). A further remarkable property of B (Ka) f and a Bp(K ) is that the corresponding Hamiltonian 3-form can be derived directly from appropriate Lagrangians. One possible choice for the vector field a K is to be a Killing vector of the reference geometry.

These general ideas were applied to general relativity in the tetrad formalism (and also in the Dirac spinor formulation of the theory [109105Jump To The Next Citation Point], yielding a Hamiltonian which is slightly different from Equation (87View Equation)) as well as in the usual metric formalism [105Jump To The Next Citation Point108Jump To The Next Citation Point]. In the latter it is the appropriate projections to @S of ab 1 V~ ---ab f := 8pG- |g |g or a a pmb := G mb in some coordinate system a {x } that is chosen to be fixed on @S. Then the dual of the corresponding Dirichlet and Neumann boundary 2-forms, respectively, will be

ab e abc( d 0d) ge f ab 0 e( cf 0cf) B f (K ) := ddef(G gc - Ggc) f K + def \~/ cK f( - f ), (90) Babp (Ka) := dadbecf Gdgc - G0gdc f0geKf + daebf \~/ cKe fcf - f0cf . (91)
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 Gamb only in the form Gamb - G0amb, this is always tensorial. If G0a mb 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 Ka as well. The connection between the present expressions and the similar previous results (pseudotensorial, tensorial, and quasi-local) is also discussed in [107Jump To The Next Citation Point105]. In particular, the expression based on the Dirichlet-type boundary 2-form (90View Equation) gives precisely the Katz-Bicak-Lynden-Bell superpotential [230]. In the spinor formulation of these ideas the vector field Ka 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 Ka 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 [107Jump To The Next Citation Point]. 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 Ka as the timelike Killing vector (@/@t)a) give m/G at infinity, but at the horizon the former gives 2m/G and the latter is infinite [107]. The Dirichlet boundary term gives at the spatial infinity in the Kerr-anti-de-Sitter solution the standard m/G and ma/G 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 [286287], 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 [229233] for the angular momentum [192]. The general formulae are evaluated for the Kerr-Vaidya solution too.

11.3.3 Covariant quasi-local Hamiltonians with general reference terms

UpdateJump To The Next Update Information 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 [7Jump To The Next Citation Point8Jump To The Next Citation Point]. (Some formulae of the journal versions were recently corrected in the latest arXiv-versions.) They considered the world tube of a compact spacelike hypersurface S with boundary S := @S. Thus the spacetime domain they considered is the same as in the Brown-York approach: D ~~ S × [t1,t2]. Their evolution vector field Ka is assumed to be tangent to the timelike boundary 3B ~~ @S × [t1,t2] of the domain D. They derived a criterion for the existence of a well-defined quasi-local Hamiltonian. Dirichlet and Neumann-type boundary conditions are imposed (i.e., in general relativity, the variations of the tetrad fields are restricted on 3 B by requiring the induced metric gab to be fixed and the adaptation of the tetrad field to the boundary to be preserved, and the tetrad components QabEba of the extrinsic curvature of 3B to be fixed, respectively). 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 gf a S K Pa dS, were also determined [7]. The properties of the co-vectors PD a and P N a (called the Dirichlet and Neumann symplectic vectors, respectively) were investigated further in [8Jump To The Next Citation Point]. Their part tangential to S is not boost gauge invariant, and to evaluate them the boost gauge determined by the mean extrinsic curvature vector a Q is used (see Section 4.1.2). Both PDa and PaN are calculated for various spheres in several special spacetimes. In particular, for the round spheres of radius r in the t = const. hypersurface in the Reissner-Nordström solution P D = 2(1- 2m/r + e2/r2)d0 a r a and P N= - (m/r2 - e2/r3)d0 a a, and hence the Dirichlet and Neumann ‘energies’ with respect to the static observer a a K = (@/@t) are gf a D 2 Sr K Pa dSr = 8pr - 16p[m - e /(2r)] and gf a N 2 Sr K Pa dSr = - 4p(m - e /r), respectively. Thus N Pa does not reproduce the standard round sphere expression, while PaD gives the standard round sphere and correct ADM energies only if it is ‘renormalized’ by its own value in Minkowski spacetime [8].

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

integral gf ( ) H [K] = --1-- KaGab 1-ebcde - --1-- Ka ~Qa + Aa + Ba dS. (92) 8pG S 3! 8pG @S
Here the 2-surface @S is assumed to be mean convex, whenever the boost gauge freedom in the SO(1, 1) gauge potential Aa can be, and, indeed, is fixed by using the globally defined orthonormal vector basis {ea0,ea1} in the normal bundle obtained by normalizing the mean curvature basis {Q~a, Qa}. The vector field Ka is still arbitrary, and Ba is an arbitrary function of the metric q ab on the 2-boundary @S, i.e. of the boundary data. This Ba is actually assumed to have the structure a a B = e0B for B as an arbitrary function of qab. This Hamiltonian is functionally differentiable, gives the correct Einstein equations and, for solutions, its value e.g. with Ka = ea0 is the general expression of the quasi-local energy of Brown and York. (Compare Equation (92View Equation) with Equation (86View Equation), or rather with Equations (72View Equation, 73View Equation, 74View Equation).)

However, to rule out the dependence of this notion of quasi-local energy on the completely freely specifiable vector field a K (i.e. on three arbitrary functions on S), Anco makes a K dynamical by linking it to the vector field ~a Q. Namely, let | |n-1 a n |~ ~e|-2- ~a K := c0[Area(S)] 2 |QeQ | Q, where c0 and n are constant, Area(S) is the area of S, and extend this Ka from S to S in a smooth way. Then Anco proves that, keeping the 2-metric qab and Ka fixed on S,

integral gf ( | |n+1) H [K] = -1--- KaG 1-eb + -----c0-----[Area(S)]n2 B - ||~Q ~Qe|| 2 dS (93) 8pG S ab3! cde 8pG(n + 1) @S e
is a correct Hamiltonian for the Einstein equations, where B is an arbitrary function of qab. For n = 1 with the choice S B = 2 R the boundary term reduces to the Hawking energy, and for n = 0 it is the Epp and Kijowski-Liu-Yau energies depending on the choice of B (i.e. the definition of the reference term). For general n choosing the reference term B 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 (n = 0) it is shown that the higher power (n > 0) 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.

11.3.4 Pseudotensors and quasi-local quantities

As we discussed briefly in Section 3.3.1, many, apparently different pseudotensors and SO(1, 3)-gauge dependent energy-momentum density expressions can be recovered from a single differential form defined on the bundle L(M ) of linear frames over the spacetime manifold: The corresponding superpotentials are the pull-backs to M of the various forms of the Nester-Witten 2-from k u ab from L(M ) along the various local sections of the bundle [142266352353]. Thus the different pseudotensors are simply the gauge dependent manifestations of the same geometric object on the bundle L(M ) in the different gauges. Since, however, k u ab is the unique extension of the Nester-Witten 2-form K- K-' u(e ,e )ab on the principal bundle of normalized spin frames K- {eA } (given in Equation (12View Equation)), 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 [104108285], by showing an intimate connection between the covariant quasi-local Hamiltonian expressions and the pseudotensors. Writing the Hamiltonian H[K] 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 H[K] gives the boundary conditions for the basic variables at @S. For example, for the Freud superpotential (for Einstein’s pseudotensor) what is fixed on the boundary @S is a certain piece of V~ |g|gab.

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