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3.2 On the global energy-momentum and angular momentum of gravitating systems: The successes

As is well known, in spite of the difficulties with the notion of the gravitational energy-momentum density discussed above, reasonable total energy-momentum and angular momentum can be associated with the whole spacetime provided it is asymptotically flat. In the present section we recall the various forms of them. As we will see, most of the quasi-local constructions are simply ‘quasi-localizations’ of the total quantities. Obviously, the technique used in the ‘quasi-localization’ does depend on the actual form of the total quantities, yielding mathematically inequivalent definitions for the quasi-local quantities. We return to the discussion of the tools needed in the quasi-localization procedures in Sections 4.2 and 4.3. Classical, excellent reviews of global energy-momentum and angular momentum are [151Jump To The Next Citation Point163Jump To The Next Citation Point15Jump To The Next Citation Point289Jump To The Next Citation Point394Jump To The Next Citation Point313Jump To The Next Citation Point], and a recent review of conformal infinity (with special emphasis on its applicability in numerical relativity) is [144]. Reviews of the positive energy proofs from the first third of the eighties are [202Jump To The Next Citation Point314Jump To The Next Citation Point].

3.2.1 Spatial infinity: Energy-momentum

There are several mathematically inequivalent definitions of asymptotic flatness at spatial infinity [151Jump To The Next Citation Point344Jump To The Next Citation Point23Jump To The Next Citation Point48Jump To The Next Citation Point148]. The traditional definition is based on the existence of a certain asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable definition are asymptotically flat in the traditional sense too. A spacelike hypersurface S will be called k-asymptotically flat if for some compact set K < S the complement S - K is diffeomorphic to R3 minus a solid ball, and there exists a (negative definite) metric h 0 ab on S, which is flat on S - K, such that the components of the difference of the physical and the background metrics, hij- 0hij, and of the extrinsic curvature xij in the 0hij-Cartesian coordinate system {xk} fall off as r- k and r-k- 1, respectively, for some k > 0 and r2 := dijxixj [319Jump To The Next Citation Point47Jump To The Next Citation Point]. These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations. S - K together with the metric and extrinsic curvature is called the asymptotic end of S. In a more general definition of asymptotic flatness S is allowed to have finitely many such ends.

As is well known, finite and well-defined ADM energy-momentum [11131214] can be associated with any k-asymptotically flat spacelike hypersurface if 1 k > 2 by taking the value on the constraint surface of the Hamiltonian H[Ka], given for example in [319Jump To The Next Citation Point47Jump To The Next Citation Point], with the asymptotic translations Ka (see [11237291113]). In its standard form this is the r --> oo limit of a 2-surface integral of the first derivatives of the induced 3-metric hab and of the extrinsic curvature xab for spheres of large coordinate radius r. The ADM energy-momentum is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian 4-vector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian coordinates.

The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian analysis of general relativity is based on the 3 + 1 decomposition of the fields and the spacetime. Thus it is not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular momentum and centre-of-mass, discussed below, form an anti-symmetric tensor). One had to check a posteriori that the conserved quantities obtained in the 3 + 1 form are, in fact, Lorentz-covariant. To obtain manifestly Lorentz-covariant quantities one should not do the 3 + 1 decomposition. Such a manifestly Lorentz-covariant Hamiltonian analysis was suggested first by Nester [280Jump To The Next Citation Point], and he was able to recover the ADM energy-momentum in a natural way (see also Section 11.3 below).

Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [163Jump To The Next Citation Point]: Taking the flux integral of the current Ca[K] + T abKb on the spacelike hypersurface S, by Equation (11View Equation) the flux can be rewritten as the r --> oo limit of the 2-surface integral of Møller’s superpotential on spheres of large r with the asymptotic translations Ka. Choosing the tetrad field Ea a to be adapted to the spacelike hypersurface and assuming that the frame a E a tends to a constant Cartesian one as r- k, the integral reproduces the ADM energy-momentum. The same expression can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too: By the standard scenario one can construct the basic Hamiltonian [282]. This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of a ab C [K] + T Kb on S.

A particularly interesting and useful expression for the ADM energy-momentum is possible if the tetrad field is considered to be a frame field built from a normalized spinor dyad {cA-} A, A- = 0,1, on S which is asymptotically constant (see Section 4.2.3 below). (Thus underlined capital Roman indices are concrete name spinor indices.) Then, for the components of the ADM energy-momentum in the constant spinor basis at infinity, Møller’s expression yields the limit of

gf A-B' --1-- i-( B-' A- B' A-) P = 4pG 2 cA'\ ~/ BB'c A - c B' \~/ AA'c B (12) S
as the 2-surface S is blown up to approach infinity. In fact, to recover the ADM energy-momentum in the form (12View Equation), the spinor fields cAA- need not be required to form a normalized spinor dyad, it is enough that they form an asymptotically constant normalized dyad, and we have to use the fact that the generator vector field a K has asymptotically constant components A-A' K in the asymptotically constant frame field A A' cAc A'. Thus a A-A' A A' K = K cA-cA' can be interpreted as an asymptotic translation. The complex-valued 2-form in the integrand of Equation (12View Equation) will be denoted by u(cA-,cB-')ab, and is called the Nester-Witten 2-form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential: For any two spinor fields aA and bA one has
( ) --------- u a, b ab- u (b,a)ab = - i \~/ [aXb], (13) ( ) --------- 1 cd ( C' C) u a, b ab + u (b,a)ab = 2 \~/ cXde ab + i eA'B'a(A\ ~/ B)C'b - eABb(A'\ ~/ B')Ca , (14)
where Xa := aAbA' and overline denotes complex conjugation. Thus, apart from the terms in Equation (14View Equation) involving A \~/ A'Aa and A' \~/ AA'b, the Nester-Witten 2-form u(a,b)ab is just - i2( \~/ [aXb] + i \~/ [cXd] 12ecdab), i.e. the anti-self-dual part of the curl of - i2Xa. (The original expressions by Witten and Nester were given using Dirac rather than two-component Weyl spinors [397Jump To The Next Citation Point279Jump To The Next Citation Point]. The 2-form u(a, b) ab in the present form using the two-component spinors appeared first probably in [205Jump To The Next Citation Point].) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [328Jump To The Next Citation Point329397279202Jump To The Next Citation Point314Jump To The Next Citation Point224], the simplest and most transparent ones are probably those based on the use of 2-component spinors: If the dominant energy condition is satisfied on the k-asymptotically flat spacelike hypersurface S, where k > 1 2, then the ADM energy-momentum is future pointing and non-spacelike (i.e. the Lorentzian length of the energy-momentum vector, the ADM mass, is non-negative), and it is null if and only if the domain of dependence D(S) of S is flat [205Jump To The Next Citation Point320159Jump To The Next Citation Point321Jump To The Next Citation Point70]. Its proof may be based on the Sparling equation [345Jump To The Next Citation Point130Jump To The Next Citation Point313Jump To The Next Citation Point266Jump To The Next Citation Point]: \~/ [au(c, m)bc] = - 12cEmE'Gef 13!efabc + G(c, m)abc. The significance of this equation is that in the exterior derivative of the Nester-Witten 2-form the second derivatives of the metric appear only through the Einstein tensor, thus its structure is similar to that of the superpotential equations in the Lagrangian field theory, and G(c,m)abc, known as the Sparling 3-form, is a quadratic expression of the first derivatives of the spinor fields. If the spinor fields c A and m A solve the Witten equation on a spacelike hypersurface S, then the pull-back of G(c, m)abc to S is positive definite. This theorem has been extended and refined in various ways, in particular by allowing inner boundaries of S that represent future marginally trapped surfaces in black holes [159Jump To The Next Citation Point202Jump To The Next Citation Point314200].

The ADM energy-momentum can also be written as the 2-sphere integral of certain parts of the conformally rescaled spacetime curvature [15Jump To The Next Citation Point1628]. This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [163Jump To The Next Citation Point]): If S is any closed spacelike 2-surface with area element dS, then for any tensor fields wab = w[ab] and mab = m[ab] one can form the integral

gf IS [w,m] := wabRabcdmcddS. (15) S
Since the fall-off of the curvature tensor near spatial infinity is -k-2 r, the integral IS [w, m] at spatial infinity can give finite value precisely when ab cd w m blows up like k r as r --> oo. In particular, for the 1/r fall-off this condition can be satisfied by V~ -------- wabmcd = Area(S) ^wab^mcd, where Area(S) is the area of S and the hatted tensor fields are O(1).

If the spacetime is stationary, then the ADM energy can be recovered as the r --> oo limit of the 2-sphere integral of Komar’s superpotential with the Killing vector a K of stationarity [163Jump To The Next Citation Point], too. On the other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time translation, the Komar expression does not reproduce the ADM energy. However, by Equations (13View Equation, 14View Equation) such an additional restriction might be that Ka should be a constant combination of four future pointing null vector fields of the form aAaA', where the spinor fields aA are required to satisfy the Weyl neutrino equation A \~/ A'Aa = 0. This expression for the ADM energy-momentum was used to give an alternative, ‘4-dimensional’ proof of the positivity of the ADM energy [205Jump To The Next Citation Point].

3.2.2 Spatial infinity: Angular momentum

The value of the Hamiltonian of Beig and Ó Murchadha [47Jump To The Next Citation Point] together with the appropriately defined asymptotic rotation-boost Killing vectors [364Jump To The Next Citation Point] define the spatial angular momentum and centre-of-mass, provided k > 1 and, in addition to the familiar fall-off conditions, certain global integral conditions are also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and Teitelboim [319Jump To The Next Citation Point] on the leading nontrivial parts of the metric hab and extrinsic curvature xab: The components in the Cartesian coordinates {xi} of the former must be even and the components of latter must be odd parity functions of i x /r (see also [47Jump To The Next Citation Point]). Thus in what follows we assume that k = 1. Then the value of the Beig-Ó Murchadha Hamiltonian parameterized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [319Jump To The Next Citation Point], while that parameterized by the asymptotic boost Killing vectors deviate from the centre-of-mass of Beig and Ó Murchadha [47Jump To The Next Citation Point] by a term which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [47Jump To The Next Citation Point], the centre-of-mass of Regge and Teitelboim is not necessarily finite.) The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz 4-tensor, which transforms in the correct way under the 4-translation of the origin of the asymptotically Cartesian coordinate system, and it is conserved by the evolution equations [364Jump To The Next Citation Point].

The centre-of-mass of Beig and Ó Murchadha was reexpressed recently [42Jump To The Next Citation Point] as the r --> oo limit of 2-surface integrals of the curvature in the form (15View Equation) with wabmcd proportional to the lapse N times qacqbd- qadqbc, where qab is the induced 2-metric on S (see Section 4.1.1 below).

A geometric notion of centre-of-mass was introduced by Huisken and Yau [209]. They foliate the asymptotically flat hypersurface S by certain spheres with constant mean curvature. By showing the global uniqueness of this foliation asymptotically, the origin of the leaves of this foliation in some flat ambient Euclidean space R3 defines the centre-of-mass (or rather ‘centre-of-gravity’) of Huisken and Yau. However, no statement on its properties is proven. In particular, it would be interesting to see whether or not this notion of centre-of-mass coincides, for example, with that of Beig and Ó Murchadha.

The Ashtekar-Hansen definition for the angular momentum is introduced in their specific conformal model of the spatial infinity as a certain 2-surface integral near infinity. However, their angular momentum expression is finite and unambiguously defined only if the magnetic part of the spacetime curvature tensor (with respect to the _O_ = const. timelike level hypersurfaces of the conformal factor) falls off faster than would follow from the 1/r fall-off of the metric (but they do not have to impose any global integral, e.g. a parity condition) [23Jump To The Next Citation Point15].

If the spacetime admits a Killing vector of axi-symmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see for example [387Jump To The Next Citation Point]). However, the value of the Komar integral is twice the expected angular momentum. In particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr solution the integral is m/G, for the Killing vector of axi-symmetry it is 2ma/G instead of the expected ma/G (‘factor-of-two anomaly’) [229Jump To The Next Citation Point]. We return to the discussion of the Komar integral in Section 12.1.

3.2.3 Null infinity: Energy-momentum

The study of the gravitational radiation of isolated sources led Bondi to the observation that the 2-sphere integral of a certain expansion coefficient m(u, h, f) of the line element of a radiative spacetime in an asymptotically retarded spherical coordinate system (u, r,h,f) behaves as the energy of the system at the retarded time u: This notion of energy is not constant in time, but decreases with u, showing that gravitational radiation carries away positive energy (‘Bondi’s mass-loss’) [7172]. The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, nowadays known as the BMS group, having a structure very similar to that of the Poincaré group [325]. The only difference is that while the Poincaré group is a semidirect product of the Lorentz group and a 4-dimensional commutative group (of translations), the BMS group is the semidirect product of the Lorentz group and an infinite-dimensional commutative group, called the group of the supertranslations. A 4-parameter subgroup in the latter can be identified in a natural way as the group of the translations. Just at the same time the study of asymptotic solutions of the field equations led Newman and Unti to another concept of energy at null infinity [290Jump To The Next Citation Point]. However, this energy (nowadays known as the Newman-Unti energy) does not seem to have the same significance as the Bondi (or Bondi-Sachs [313Jump To The Next Citation Point] or Trautman-Bondi [115Jump To The Next Citation Point116Jump To The Next Citation Point114Jump To The Next Citation Point]) energy, because its monotonicity can be proven only between special, e.g. stationary, states. The Bondi energy, which is the time component of a Lorentz vector, the so-called Bondi-Sachs energy-momentum, has a remarkable uniqueness property [115116].

Without additional conditions on Ka, Komar’s expression does not reproduce the Bondi-Sachs energy-momentum in non-stationary spacetimes either [395Jump To The Next Citation Point163]: For the ‘obvious’ choice for Ka Komar’s expression yields the Newman-Unti energy. This anomalous behaviour in the radiative regime could be corrected in, at least, two ways. The first is by modifying the Komar integral according to

1 gf ( )1 LS [K] := ----- \~/ [cKd] + a\ ~/ eKe _L ecd-ecdab, (16) 8pG S 2
where _L ecd is the area 2-form on the Lorentzian 2-planes orthogonal to S (see Section 4.1.1) and a is some real constant. For a = 1 the integral LS [K], suggested by Winicour and Tamburino, is called the linkage [395Jump To The Next Citation Point]. In addition, to define physical quantities by linkages associated to a cut of the null infinity one should prescribe how the 2-surface S tends to the cut and how the vector field Ka should be propagated from the spacetime to null infinity into a BMS generator [395Jump To The Next Citation Point394]. The other way is considering the original Komar integral (i.e. a = 0) on the cut of infinity in the conformally rescaled spacetime and by requiring that Ka be divergence-free [153]. For such asymptotic BMS translations both prescriptions give the correct expression for the Bondi-Sachs energy-momentum.

The Bondi-Sachs energy-momentum can also be expressed by the integral of the Nester-Witten 2-form [214Jump To The Next Citation Point255Jump To The Next Citation Point256Jump To The Next Citation Point205Jump To The Next Citation Point]. However, in non-stationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [83Jump To The Next Citation Point]. Thus the spinor fields in the Nester-Witten 2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves be the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [83Jump To The Next Citation Point], was to require the spinor fields to be the solutions of the so-called asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all these, based only on the linear first order differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are determined in [363Jump To The Next Citation Point].

The Bondi-Sachs energy-momentum has a Hamiltonian interpretation as well. Although the fields on a spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable generalization of the standard Hamiltonian analysis could be developed [114Jump To The Next Citation Point] and used to recover the Bondi-Sachs energy-momentum.

Similarly to the ADM case, the simplest proofs of the positivity of the Bondi energy [330] are probably those that are based on the Nester-Witten 2-form [214] and, in particular, the use of two-components spinors [255256205203321]: The Bondi-Sachs mass (i.e. the Lorentzian length of the Bondi-Sachs energy-momentum) of a cut of future null infinity is non-negative if there is a spacelike hypersurface S intersecting null infinity in the given cut such that the dominant energy condition is satisfied on S, and the mass is zero iff the domain of dependence D(S) of S is flat.

3.2.4 Null infinity: Angular momentum

At null infinity there is no generally accepted definition for angular momentum, and there are various, mathematically inequivalent suggestions for it. Here we review only some of those total angular momentum definitions that can be considered as the null infinity limit of some quasi-local expression, and will be discussed in the main part of the review, namely in Section 9.

In their classic paper Bergmann and Thomson [60] raise the idea that while the gravitational energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic O(1, 3) symmetry. Thus, the angular momentum should be analogous with the spin. Based on the tetrad formalism of general relativity and following the prescription of constructing the Noether currents in Yang-Mills theories, Bramson suggested a superpotential for the six conserved currents corresponding to the internal Lorentz-symmetry [84Jump To The Next Citation Point8586Jump To The Next Citation Point]. (For another derivation of this superpotential from Møller’s Lagrangian (9View Equation) see [363Jump To The Next Citation Point].) If {cA-} A, A = 0, 1 --, is a normalized spinor dyad corresponding to the orthonormal frame in Equation (9View Equation), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable 2-surface S is

gf A-B-: --1-- A- B- ' ' JS = 8pG S - ic(Ac B)eA B, (17)
where eA'B' is the symplectic metric on the bundle of primed spinors. We will denote its integrand by w(cA--,cB-)ab, and we call it the Bramson superpotential. To define angular momentum on a given cut of the null infinity by the formula (17View Equation) we should consider its limit when S tends to the cut in question and we should specify the spinor dyad, at least asymptotically. Bramson’s suggestion for the spinor fields was to take the solutions of the asymptotic twistor equation [83Jump To The Next Citation Point]. He showed that this definition yields a well-defined expression, for stationary spacetimes this reduces to the generally accepted formula (34View Equation), and the corresponding Pauli-Lubanski spin, constructed from eA'B'JA-B-+ eA-BJ A'B' and the Bondi-Sachs energy-momentum P AA-' (given for example in the Newman-Penrose formalism by Equation (33View Equation)), is invariant with respect to supertranslations of the cut (‘active supertranslations’). Note that since Bramson’s expression is based on the solutions of a system of partial differential equations on the cut in question, it is independent of the parameterization of the BMS vector fields. Hence, in particular, it is invariant with respect to the supertranslations of the origin cut (‘passive supertranslations’). Therefore, Bramson’s global angular momentum behaves like the spin part of the total angular momentum.

The construction based on the Winicour-Tamburino linkage (16View Equation) can be associated with any BMS vector field [39525230]. In the special case of translations it reproduces the Bondi-Sachs energy-momentum. The quantities that it defines for the proper supertranslations are called the super-momenta. For the boost-rotation vector fields they can be interpreted as angular momentum. However, in addition to the factor-of-two anomaly, this notion of angular momentum contains a huge ambiguity (‘supertranslation ambiguity’): The actual form of both the boost-rotation Killing vector fields of Minkowski spacetime and the boost-rotation BMS vector fields at future null infinity depend on the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively. However, while the set of the origins of Minkowski spacetime is parameterized by four numbers, the set of the origins at null infinity requires a smooth function of the form u : S2 --> R. Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar origin-dependence (containing four parameters), the analogous transformation of the angular momentum defined by using the boost-rotation BMS vector fields depends on an arbitrary smooth real valued function on the 2-sphere. This makes the angular momentum defined at null infinity by the boost-rotation BMS vector fields ambiguous unless a natural selection rule for the origins, making them form a four parameter family of cuts, is found. Such a selection rule could be the suggestion by Dain and Moreschi [125] in the charge integral approach to angular momentum of Moreschi [272273].

Another promising approach might be that of ChruĊ›ciel, Jezierski, and Kijowski [114], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperbolic spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian 4-space of origins, they appear to be the generators with respect to some fixed ‘centre-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.

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