There are several mathematically-inequivalent definitions of asymptotic flatness at spatial infinity [208, 475, 37, 65, 200]. 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 as well. A spacelike hypersurface will be called –asymptotically flat if for some compact set the complement is diffeomorphic to minus a solid ball, and there exists a (negative definite) metric on , which is flat on , such that the components of the difference of the physical and the background metrics, , and of the extrinsic curvature in the -Cartesian coordinate system fall off as and , respectively, for some and [433, 64]. These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations. together with the metric and extrinsic curvature is called the asymptotic end of . In a more general definition of asymptotic flatness is allowed to have finitely many such ends.

As is well known, finite and well-defined ADM energy-momentum [23, 25, 24, 26] can be associated with any –asymptotically flat spacelike hypersurface, if , by taking the value on the constraint surface of the Hamiltonian , given, for example, in [433, 64], with the asymptotic translations (see [144, 52, 399, 145]). In its standard form, this is the limit of a two-surface integral of the first derivatives of the induced three-metric and of the extrinsic curvature for spheres of large coordinate radius . Explicitly:

where is the Levi-Civita derivative oparator determined by , and is the outward pointing unit normal to and tangent to . The ADM energy-momentum, , is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian four-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 center-of-mass, discussed below, form an anti-symmetric tensor). One has 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 [377], and he was able to recover the ADM energy-momentum in a natural way (see Section 11.3).

Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [223]: Taking the flux integral of the current on the spacelike hypersurface , by Eq. (3.7) the flux can be rewritten as the limit of the two-surface integral of Møller’s superpotential on spheres of large with the asymptotic translations . Choosing the tetrad field to be adapted to the spacelike hypersurface and assuming that the frame tends to a constant Cartesian one as , 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 [379]. This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of on .

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 , , on , which is asymptotically constant (see Section 4.2.3). (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

as the two-surface is blown up to approach infinity. In fact, to recover the ADM energy-momentum in the form (3.10), the spinor fields 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 has asymptotically constant components in the asymptotically constant frame field . Thus can be interpreted as an asymptotic translation. The complex-valued 2-form in the integrand of Eq. (3.10) will be denoted by , and is called the Nester–Witten 2-form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential, too. In fact, for any two spinor fields and one has where and the overline denotes complex conjugation. Thus, apart from the terms in Eq. (3.12) involving and , the Nester–Witten 2-form is just , i.e., the anti-self-dual part of the curl of . (The original expressions by Witten and Nester were given using Dirac, rather than two-component Weyl, spinors [559, 376]. The 2-form in the present form using the two-component spinors probably appeared first in [276].) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [444, 445, 559, 376, 273, 427, 300], the simplest and most transparent ones are probably those based on the use of two-component spinors: If the dominant energy condition is satisfied on the –asymptotically flat spacelike hypersurface , where , then the ADM energy-momentum is future pointing and nonspacelike (i.e., the Lorentzian length of the energy-momentum vector, the ADM mass, is non-negative), and is null if and only if the domain of dependence of is flat [276, 434, 217, 436, 88]. Its proof may be based on the Sparling equation [476, 175, 426, 358]: 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 Lagrangian field theory, and , known as the Sparling 3-form, is a homogeneous quadratic expression of the first derivatives of the spinor fields. If the spinor fields and solve the Witten equation on a spacelike hypersurface , then the pullback of to is positive definite. This theorem has been extended and refined in various ways, in particular by allowing inner boundaries of that represent future marginally trapped surfaces in black holes [217, 273, 427, 268].The ADM energy-momentum can also be written as the two-sphere integral of certain parts of the conformally rescaled spacetime curvature [28, 29, 43]. This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [223]): If is any closed spacelike two-surface with area element , then for any tensor fields and one can form the integral

Since the falloff of the curvature tensor near spatial infinity is , the integral at spatial infinity gives finite value when blows up like as . In particular, for the falloff, this condition can be satisfied by , where is the area of and the hatted tensor fields are .If the spacetime is stationary, then the ADM energy can be recovered at the limit of the two-sphere integral of (twice of) Komar’s superpotential with the Killing vector of stationarity [223] (see also [60]), as well. (See also the remark following Eq. (3.15) below.) 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 Eqs. (3.11) and (3.12) such an additional restriction might be that should be a constant combination of four future-pointing null vector fields of the form , where the spinor fields are required to satisfy the Weyl neutrino equation . This expression for the ADM energy-momentum has been used to give an alternative, ‘four-dimensional’ proof of the positivity of the ADM energy [276]. (For a more detailed recent review of the various forms of the ADM energy and linear momentum, see, e.g., [293].)

In stationary spacetime the notion of the mechanical energy with respect to the world lines of stationary observers (i.e., the integral curves of the timelike Killing field) can be introduced in a natural way, and then (by definition) the total (ADM) energy is written as the sum of the mechanical energy and the gravitational energy. Then the latter is shown to be negative for certain classes of systems [308, 348].

The notion of asymptotic flatness at spatial infinity is generalized in [398]; here the background flat metric on is allowed to have a nonzero deficit angle at infinity, i.e., the corresponding line element in spherical polar coordinates takes the form . Then, a canonical analysis of the minimally-coupled Einstein–Higgs field is carried out on such a background, and, following a Regge-Teitelboim–type argumentation, an ADM-type total energy is introduced. It is shown that for appropriately chosen this energy is finite for the global monopole solution, though the standard ADM energy is infinite.

The value of the Hamiltonian of Beig and Ó Murchadha [64], together with the appropriately-defined asymptotic rotation-boost Killing vectors [497], define the spatial angular momentum and center-of-mass, provided and, in addition to the familiar falloff conditions, certain global integral conditions are also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and Teitelboim [433] on the leading nontrivial parts of the metric and extrinsic curvature : The components in the Cartesian coordinates of the former must be even and the components of the latter must be odd parity functions of (see also [64]). Thus, in what follows we assume that . Then the value of the Beig–Ó Murchadha Hamiltonian parametrized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [433], while that parametrized by the asymptotic boost Killing vectors deviates from the center-of-mass of Beig and Ó Murchadha [64] by a term, which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [64], the center-of-mass term of the Hamiltonian of Regge and Teitelboim is not finite on the whole phase space.) The spatial angular momentum and the new center-of-mass form an anti-symmetric Lorentz four-tensor, which transforms in the correct way under the four-translation of the origin of the asymptotically Cartesian coordinate system, and is conserved by the evolution equations [497].

The center-of-mass of Beig and Ó Murchadha was re-expressed recently [57] as the limit of two-surface integrals of the curvature in the form (3.14) with proportional to the lapse times , where is the induced two-metric on (see Section 4.1.1). The geometric notion of center-of-mass introduced by Huisken and Yau [280] is another form of the Beig–Ó Murchadha center-of-mass [156].

The Ashtekar–Hansen definition for the angular momentum is introduced in their specific conformal model of spatial infinity as a certain two-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 timelike level hypersurfaces of the conformal factor) falls off faster than it would fall off in metrics with falloff (but no global integral, e.g., a parity condition had to be imposed) [37, 28].

If the spacetime admits a Killing vector of axisymmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see, e.g., [534]). However, the value of the Komar integral (with the usual normalization) 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 , for the Killing vector of axisymmetry it is instead of the expected (‘factor-of-two anomaly’) [305]. We return to the discussion of the Komar integral in Sections 4.3.1 and 12.1.

The study of the gravitational radiation of isolated sources led Bondi to the observation that the two-sphere integral of a certain expansion coefficient of the line element of a radiative spacetime in an asymptotically-retarded spherical coordinate system behaves as the energy of the system at the retarded time . Indeed, this energy is not constant in time, but decreases with , showing that gravitational radiation carries away positive energy (‘Bondi’s mass-loss’) [91, 92]. The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, currently known as the Bondi–Metzner–Sachs (or BMS) group, having a structure very similar to that of the Poincaré group [440]. The only difference is that while the Poincaré group is a semidirect product of the Lorentz group and a four 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 four-parameter subgroup in the latter can be identified in a natural way as the group of the translations. This makes it possible to compare the Bondi–Sachs four-momenta defined on different cuts of scri, and to calculate the energy-momentum carried away by the gravitational radiation in an unambiguous way. (For further discussion of the flux, see the fourth paragraph of Section 3.2.4.) 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 [394]. However, this energy (currently known as the Newman–Unti energy) does not seem to have the same significance as the Bondi (or Bondi–Sachs [426] or Trautman–Bondi [147, 148, 146]) 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 Bondi–Sachs energy-momentum, has a remarkable uniqueness property [147, 148].

Without additional conditions on , Komar’s expression does not reproduce the Bondi–Sachs energy-momentum in nonstationary spacetimes either [557, 223]: For the ‘obvious’ choice for (twice of) Komar’s expression yields the Newman–Unti energy. This anomalous behavior in the radiative regime could be corrected in at least two ways. The first is by modifying the Komar integral according to

where is the area 2-form on the Lorentzian two-planes orthogonal to (see Section 4.1.1) and is some real constant. For the integral , suggested by Winicour and Tamburino, is called the linkage [557]. (N.B.: The flux integral of the sum of Komar’s gravitational and the matter’s currents on some compact spacelike hypersurface with boundary is , which, for , is half of the linkage.) In addition, to define physical quantities by linkages associated to a cut of the null infinity one should prescribe how the two-surface tends to the cut and how the vector field should be propagated from the spacetime to null infinity into a BMS generator [557, 553]. The other way is to consider the original Komar integral (i.e., ) on the cut of infinity in the conformally-rescaled spacetime and while requiring that be divergence-free [210]. 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 [285, 342, 343, 276]. However, in nonstationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [106]. Thus, the spinor fields in the Nester–Witten 2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves are the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [106], was to require the spinor fields to be the solutions of the asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all of 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 [496].

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 [146] and used to recover the Bondi–Sachs energy-momentum.

Similar to the ADM case, the simplest proofs of the positivity of the Bondi energy [446] are probably those that are based on the Nester–Witten 2-form [285] and, in particular, the use of two-component spinors [342, 343, 276, 274, 436]: 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 intersecting null infinity in the given cut such that the dominant energy condition is satisfied on , and the mass is zero iff the domain of dependence of is flat. Update

Converting the integral of the Nester–Witten 2-form into a (positive definite) 3-dimensional integral on , a strictly positive lower bound can be given both for the ADM and Bondi–Sachs masses. Although total energy-momentum (or mass) in the form of a two-surface integral cannot be a introduced in closed universes (i.e., when is compact with no boundary), a non-negative quantity , based on this positive definite expression, can be associated with . If the matter fields satisfy the dominant energy condition, then if and only if the spacetime is flat and topologically is a 3-torus; moreover its vanishing is equivalent to the existence of non-trivial solutions of Witten’s gauge condition. This turned out to be recoverable as the first eigenvalue of the square of the Sen–Witten operator. It is the usefulness and the applicability of this in practice which tell us if this is a reasonable notion of total mass of closed universes or not [503].

At null infinity we have a generally accepted definition for angular momentum only in stationary or axi-symmetric, but not in general, radiative spacetime, where there are various, mathematically inequivalent suggestions for it (see Section 4.2.4). Here we review only some of those total angular momentum definitions that can be ‘quasi-localized’ or connected somehow to quasi-local expressions, i.e., those that can be considered as the null-infinity limit of some quasi-local expression. We will continue their discussion in the main part of the review, namely in Sections 7.2, 11.1 and 9.

In their classic paper Bergmann and Thomson [78] raise the idea that while the gravitational energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic 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 [107, 108, 109]. (For another derivation of this superpotential from Møller’s Lagrangian (3.5) see [496].) If , , is a normalized spinor dyad corresponding to the orthonormal frame in Eq. (3.5), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable two-surface is

where is the symplectic metric on the bundle of primed spinors. We will denote its integrand by , and we call it the Bramson superpotential. To define angular momentum on a given cut of the null infinity by the formula (3.16), we should consider its limit when 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 [106]. He showed that this definition yields a well-defined expression. For stationary spacetimes this reduces to the generally accepted formula (4.15), and the corresponding Pauli–Lubanski spin, constructed from and the Bondi–Sachs energy-momentum (given, for example, in the Newman–Penrose formalism by Eq. (4.14)), 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 parametrization 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. For a suggestion based on Bramson’s superpotential at the quasi-local level, but using a different prescription for the spinor dyad, see Section 9.The construction based on the Winicour–Tamburino linkage (3.15) can be associated with any BMS vector field [557, 337, 45]. 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 parametrized by four numbers, the set of the origins at null infinity requires a smooth function of the form . 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 two-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.

Motivated by Penrose’s idea that the ‘conserved’ quantities at null infinity should be searched for in the form of a charge integral of the curvature (which will be discussed in detail in Section 7), a general expression , associated with any BMS generator and any cut of scri, was introduced [174]. For real this is real; it is vanishing in Minkowski spacetime; it reproduces the Bondi–Sachs energy-momentum for BMS translations; it yields nontrivial results for proper supertranslations; and for BMS rotations the resulting expressions can be interpreted as angular momentum. It was shown in [453, 173] that the difference for any two cuts and can be written as the integral of some local function on the subset of scri bounded by the cuts and , and this is precisely the flux integral of [44]. Unfortunately, however, the angular momentum introduced in this way still suffers from the same supertranslation ambiguity. A possible resolution of this difficulty could be the suggestion by Dain and Moreschi [169] in the charge integral approach to angular momentum of Moreschi [369, 370]. Their basic idea is that the requirement of the vanishing of the supermomenta (i.e., the quantities corresponding to the proper supertranslations) singles out a four–real-parameter family of cuts, called nice cuts, by means of which the BMS group can be reduced to a Poincaré subgroup that yields a well-defined notion of angular momentum. For further discussion of certain other angular momentum expressions, especially from the points of view of numerical calculations, see also [204].

Another promising approach might be that of Chruลciel, Jezierski, and Kijowski [146], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperboloidal 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 four-space of origins, they appear to be the generators with respect to some fixed ‘center-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.

In addition to the supertranslation ambiguity in the definition of angular momentum, there could be another potential ambiguity, even if the angular momentum is well defined on every cut of future null infinity. In fact, if, for example, the definition of the angular momentum is based on the solutions of some linear partial differential equation on the cut (such as Bramson’s definition, or the ones discussed in Sections 7 and 9), then in general there is no canonical isomorphism between the spaces of the solutions on different cuts, even if the solution spaces, as abstract vector spaces, are isomorphic. Therefore, the angular momenta on two different cuts belong to different vector spaces, and, without any natural correspondence between the solution spaces on the different cuts, it is meaningless to speak about the difference of the angular momenta. Thus, we cannot say anything about, e.g., the angular momentum carried away by gravitational radiation between two retarded time instants represented by two different cuts.

One possible resolution of this difficulty was suggested by Helfer [264]. He followed the twistorial approach presented in Section 7 and used a special bijective map between the two-surface twistor spaces on different cuts. His map is based on the special structures available only at null infinity. Though this map is nonlinear, it is shown that the angular momenta on the different cuts can indeed be compared. Another suggestion for (only) the spatial angular momentum was given in [501]. This is based on the quasi-local Hamiltonian analysis that is discussed in Section 11.1, and the use of the divergence-free vector fields built from the eigenspinors with the smallest eigenvalue of the two-surface Dirac operators. The angular momenta, defined in these ways on different cuts, can also be compared. We give a slightly more detailed discussion of them in Sections 7.2 and 11.1, respectively.

The main idea behind the recent definition of the total angular momentum at future null infinity of Kozameh, Newman and Silva-Ortigoza, suggested in [325, 326], is analogous to finding the center-of-charge (i.e., the time-dependent position vector with respect to which the electric dipole moment is vanishing) in flat-space electromagnetism: By requiring that the dipole part of an appropriate null rotated Weyl tensor component be vanishing, a preferred set of origins, namely a (complex) center-of-mass line can be found in the four–complex-dimensional solution space of the good-cut equation (the -space). Then the asymptotic Bianchi identities take the form of conservation equations, and certain terms in these can (in the given approximation) be identified with angular momentum. The resulting expression is just Eq. (4.15), to which all the other reasonable angular momentum expressions are expected to reduce in stationary spacetimes. A slightly more detailed discussion of the necessary technical background is given in Section 4.2.4.

Living Rev. Relativity 12, (2009), 4
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