1 Introduction
During the last 25 years one of the greatest achievements in classical general relativity is certainly the proof of the positivity of the total gravitational energy, both at spatial and null infinity. It is precisely its positivity that makes this notion not only important (because of its theoretical significance), but a useful tool as well in the everyday practice of working relativists. This success inspired the more ambitious claim to associate energy (or rather energy-momentum and, ultimately, angular momentum too) to extended but finite spacetime domains, i. e. at the quasi-local level. Obviously, the quasi-local quantities could provide a more detailed characterization of the states of the gravitational ‘field’ than the global ones, so they (together with more general quasi-local observables) would be interesting in their own right. Moreover, finding an appropriate notion of energy-momentum and angular momentum would be important from the point of view of applications as well. For example, they may play a central role in the proof of the full Penrose inequality (as they have already played in the proof of the Riemannian version of this inequality). The correct, ultimate formulation of black hole thermodynamics should probably be based on quasi-locally defined internal energy, entropy, angular momentum etc. In numerical calculations conserved quantities (or at least those for which balance equations can be derived) are used to control the errors. However, in such calculations all the domains are finite, i. e. quasi-local. Therefore, a solid theoretical foundation of the quasi-local conserved quantities is needed. However, contrary to the high expectations of the eighties, finding an appropriate quasi-local notion of energy-momentum has proven to be surprisingly difficult. Nowadays, the state of the art is typically postmodern: Although there are several promising and useful suggestions, we have not only no ultimate, generally accepted expression for the energy-momentum and especially for the angular momentum, but there is no consensus in the relativity community even on general questions (for example, what should we mean e. g. by energy-momentum: Only a general expression containing arbitrary functions, or rather a definite one free of any ambiguities, even of additive constants), or on the list of the criteria of reasonableness of such expressions. The various suggestions are based on different philosophies, approaches and give different results in the same situation. Apparently, the ideas and successes of one construction have only very little influence on other constructions. The aim of the present paper is therefore twofold. First, to collect and review the various specific suggestions, and, second, to stimulate the interaction between the different approaches by clarifying the general, potentially common points, issues, questions. Thus we wanted to write not only a ‘who-did-what’ review, but primarily we would like to concentrate on the understanding of the basic questions (such as why should the gravitational energy-momentum and angular momentum, or, more generally, any observable of the gravitational ‘field’, be necessarily quasi-local) and ideas behind the various specific constructions. Consequently, one-third of the present review is devoted to these general questions. We review the specific constructions and their properties only in the second part, and in the third part we discuss very briefly some (potential) applications of the quasi-local quantities. Although this paper is basically a review of known and published results, we believe that it contains several new elements, observations, suggestions etc. Surprisingly enough, most of the ideas and concepts that appear in connection with the gravitational energy-momentum and angular momentum can be introduced in (and hence can be understood from) the theory of matter fields in Minkowski spacetime. Thus, in subsection 2.1 , we review the Belinfante–Rosenfeld procedure that we will apply to gravity in section 3 , introduce the notion of quasi-local energy-momentum and angular momentum of the matter fields and discuss their properties.The philosophy of quasi-locality in general relativity will be demonstrated in Minkowski spacetime where the energy-momentum and angular momentum of the matter fields are treated quasi-locally. Then we turn to the difficulties of gravitational energy-momentum and angular momentum, and we clarify why the gravitational observables should necessarily be quasi-local. The tools needed to construct and analyze the quasi-local quantities are reviewed in the fourth section. This closes the first, the general part of the review. The second part is devoted to the discussion of the specific constructions (sections 5 – 12 ). Since most of the suggestions are constructions, they cannot be given as a short mathematical definition. Moreover, there are important physical ideas behind them, without which the constructions may appear ad hoc. Thus we always try to explain these physical pictures, the motivations and interpretations. Although the present paper is intended to be a non-technical review, the explicit mathematical definitions of the various specific constructions will always be given. Then the properties and the applications are usually summarized only in a nutshell. Sometimes we give a review on technical aspects too, without which it would be difficult to understand even some of the conceptual issues. The list of references connected with this second part is intended to be complete. We apologize to all those whose results were accidentally left out. The list of the (actual and potential) applications of the quasi-local quantities, discussed in section 13 , is far from being complete, and might be a little bit subjective. Here we consider the calculation of gravitational energy transfer, applications in black hole physics and a quasi-local characterization of the pp-wave metrics. We close this paper with a discussion of the successes and deficiencies of the general and (potentially) viable constructions. In contrast to the positivistic style of sections 5 – 12 , section 14 (as well as the choice for the matter of sections 2 – 4 ) reflects our own personal interest and view of the subject. The theory of quasi-local observables in general relativity is far from being complete. The most important open problem is still the trivial one: ‘Find quasi-local energy-momentum and angular momentum expressions satisfying the points of the lists of subsection 4.3 ’. Several specific open questions in connection with the specific definitions are raised both in the corresponding sections and in section 14 , which could be worked out even by graduate students. On the other hand, any of their application to solve physical/geometrical problems (e. g. to some mentioned in section 13 ) would be a real success. In the present paper we adopt the abstract index formalism. The signature of the spacetime metric is ; and the curvature and Ricci tensors and the curvature scalar of the covariant derivative are defined by , and , respectively. Hence Einstein’s equations take the form , where is Newton’s gravitational constant (and the speed of light is ). However, in subsections 13.3 and 13.4 we use the traditional system.2 Energy-Momentum and Angular Momentum of Matter Fields
2.1 Energy-momentum and angular momentum density of matter fields
2.1.1 The symmetric energy-momentum tensor
It is a widely accepted view (appearing e. g. in excellent, standard textbooks on general relativity, too) that the canonical energy-momentum and spin tensors are well defined and have relevance only in flat spacetime, and hence usually are underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus first we introduce these quantities for the matter fields in a general curved spacetime. To specify the state of the matter fields operationally two kinds of devices are needed: The first measures the value of the fields, while the other measures the spatio-temporal location of the first. Correspondingly, the fields on the manifold of events can be grouped into two sharply distinguished classes: The first contains the matter field variables, e. g. finitely many type tensor fields ; whilst the other contains the fields specifying the spacetime geometry, i. e. the metric in Einstein’s theory. Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a Lagrangian : If is the volume integral of on some open domain with compact closure then the equations of motion are , the Euler–Lagrange equations. The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by(1) |
(2) |
2.1.2 The canonical Noether current
Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that for any vector field and the corresponding local 1-parameter family of diffeomorphisms one has for some one parameter family of vector fields . ( is called diffeomorphism invariant if , e. g. when is a scalar.) Let be any smooth vector field on . Then, calculating the divergence to determine the rate of change of the action functional along the integral curves of , by a tedious but straightforward computation one can derive the so-called Noether identity: , where denotes the Lie derivative along and , the so-called Noether current, is given explicitly by(3) |
(4) |
2.2 Quasi-local energy-momentum and angular momentum of the matter fields
In the next section we will see that well defined (i. e. gauge invariant) energy-momentum and angular momentum density cannot be associated with the gravitational ‘field’, and if we want to talk not only about global gravitational energy-momentum and angular momentum, then these quantities must be assigned to extended but finite spacetime domains. In the light of modern quantum field theoretical investigations it has become clear that all physical observables should be associated with extended but finite spacetime domains [164, 163] . Thus observables are always associated with open subsets of spacetime whose closure is compact, i. e. they are quasi-local. Quantities associated with spacetime points or with the whole spacetime are not observable in this sense. In particular, global quantities, such as the total energy or electric charge, should be considered as the limit of quasi-locally defined quantities. Thus the idea of quasi-locality is not new in physics. Although apparently in classical non-gravitational physics this is not obligatory, we adopt this view in talking about energy-momentum and angular momentum even of classical matter fields in Minkowski spacetime. Originally the introduction of these quasi-local quantities was motivated by the analogous gravitational quasi-local quantities [338, 342] . Since, however, many of the basic concepts and ideas behind the various gravitational quasi-local energy-momentum and angular momentum definitions can be understood from the analogous non-gravitational quantities in Minkowski spacetime, we devote the present subsection to the discussion of them and their properties.2.2.1 The definition of the quasi-local quantities
To define the quasi-local conserved quantities in Minkowski spacetime, first observe that for any Killing vector the 3-form is closed, and hence, by the triviality of the third de Rham cohomology class, , it is exact: For some 2-form we have . may be called a superpotential for the conserved current 3-form . (The existence of globally defined superpotentials can be proven even without using the Poincare lemma [370] .) If is (the dual of) another superpotential for the same current , then by and the dual superpotential is unique up to the addition of an exact 2-form. If therefore is any closed orientable spacelike 2-surface in the Minkowski spacetime then the integral of on is free from this ambiguity. Thus if is any smooth compact spacelike hypersurface with smooth 2-boundary , then(5) |
(6) |
(7) |
2.2.2 Hamiltonian introduction of the quasi-local quantities
In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [206, 377] and references therein) the configuration and momentum variables, and , respectively, are fields on a connected 3-manifold , which is interpreted as the typical leaf of a foliation of the spacetime. The foliation can be characterized on by a function , called the lapse. The evolution of the states in the spacetime is described with respect to a vector field (‘evolution vector field’ or ’general time axis’), where is the future directed unit normal to the leaves of the foliation and is some vector field, called the shift, being tangent to the leaves. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Physical states can be only those that are on the constraint surface, specified by the vanishing of certain functions , , of the canonical variables and their derivatives up to some finite order, where is the covariant derivative operator in . Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form(8) |
Sometimes in the literature this requirement is introduced as some new principle in the Hamiltonian formulation of the fields, but its real content is not more than to ensure that the Hamilton equations coincide with the field equations.
2.2.3 Properties of the quasi-local quantities
Suppose that the matter fields satisfy the dominant energy condition. Then is also non-negative for any non-spacelike , and, obviously, is zero precisely when on , and hence, by the conservation laws (see for example page 94 of [170] ), on the whole domain of dependence . Obviously, if and only if is null on . Then by the dominant energy condition it is a future pointing vector field on , and holds. Therefore, on has a null eigenvector with zero eigenvalue, i. e. its algebraic type on is pure radiation. The properties of the quasi-local quantities based on in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [338, 342] that2.2.4 Global energy-momenta and angular momenta
If extends either to spacelike or future null infinity, then, as is well known, the existence of the limit of the quasi-local energy-momentum can be ensured by slightly faster than (for example by ) fall-off of the energy-momentum tensor, where is any spatial radial distance. However, the finiteness of the angular momentum and centre-of-mass is not ensured by the fall-off. Since the typical fall-off of – for example for the electromagnetic field – is , we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the fall-off, six global integral conditions for the leading terms of must be imposed. At the spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’ (as evolution equations for the energy density and momentum density) preserve these fall-off and parity conditions [348] . Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass the fields must be plane waves, furthermore by they must be asymptotically vanishing at the same time. However, a plane wave configuration can be asymptotically vanishing only if it is vanishing.2.2.5 Quasi-local radiative modes of the matter fields
By the results of the previous subsection the vanishing of the quasi-local mass, associated with a closed spacelike 2-surface , implies that the matter must be pure radiation on a four dimensional globally hyperbolic domain . Thus characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present subsection we raise the question if these states on can be characterized completely by data on the 2-surface or not. For the (real or complex) linear massless scalar field and the Maxwell field, represented by the symmetric spinor field , the condition implies that they are completely determined on the whole by the value of these fields on (and is necessarily null: , where is a complex function and is a constant spinor field such that ). Similarly, the null linear zero-rest-mass fields on with any spin, , are completely determined by their value on (assuming, for the sake of simplicity, that is future and past convex in the sense of subsection 4.1.3 below). Technically, these results are based on the unique complex analytic structure of the 2-surfaces foliating , where ; and by the field equations the complex functions turn out to be holomorphic. Thus they are completely determined on (and hence on too) by their value on [342] . Since the independent radiative modes of the linear zero-rest-mass fields are just the -type solutions of their field equations, characterized by the null wavevector and the amplitude in their Fourier expansion, their radiative modes can equivalently be characterized by not only the familiar initial data on three dimensional spacelike hypersurfaces, but by data on closed 2-surfaces as well. Intuitively this result should be clear, because the two radiative (transversal) degrees of freedom of each Fourier mode may be expected to be characterized by two real functions on a two dimensional ‘screen’. This fact may be a specific manifestation in the classical non-gravitational physics of the holographic principle (see subsection 13.4.2 ). A more detailed discussion of these issues will be given elsewhere.3 On the Energy-Momentum and Angular Momentum of Gravitating Systems
3.1 On the gravitational energy-momentum and angular momentum density: the difficulties
3.1.1 The root of the difficulties
The action for the matter fields was a functional of both kinds of fields, thus one could take the variational derivatives both with respect to and . The former gave the field equations, while the latter defined the symmetric energy-momentum tensor. Moreover, provided a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action is, on the other hand, a functional of the metric alone, and its variational derivative with respect to yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of can be traced back to the principle of equivalence [21] , and introduces a huge gauge freedom in the dynamics of because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold endowed with a Lorentzian metric , but the isomorphism class of such pairs, where and are considered to be equivalent for any diffeomorphism of onto itself. Thus we do not have, even in principle, any gravitational analog of the symmetric energy-momentum tensor of the matter fields. In fact, by its very definition, is the source-current for gravity, like the current in Yang–Mills theories (defined by the variational derivative of the action functional of the particles, e. g. of the fermions, interacting with a Yang–Mills field ), rather than energy-momentum. The latter is represented by the Noether currents associated with special spacetime displacements. Thus, in spite of the intimate relation between and the Noether currents, the proper interpretation of is only the source density for gravity, and hence it is not the symmetric energy-momentum tensor whose gravitational counterpart must be searched for. In particular, the Bel–Robinson tensor , given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [318, 317] ), being a quadratic expression of the curvature (and its derivatives), is (are) expected to represent only ‘higher order’ gravitational energy-momentum. In fact, the physical dimension of the Bel–Robinson ‘energy-density’ is , and hence (in the traditional units) there are no powers and such that would have energy-density dimension. Here is the speed of light and is Newton’s gravitational constant. As we will see, the Bel–Robinson ‘energy-momentum density’ appears naturally in connection with the quasi-local energy-momentum and spin-angular momentum expressions for small spheres only in higher order terms. Therefore, if we want to associate energy-momentum and angular momentum with the gravity itself in a Lagrangian framework, then it is the gravitational counterpart of the canonical energy-momentum and spin tensors and the canonical Noether current built from them that should be introduced. Hence it seems natural to apply the Lagrange–Belinfante–Rosenfeld procedure, sketched in the previous section, to gravity too [55, 56, 308, 188, 189, 336] .Since we do not have a third kind of device to specify the spatio-temporal location of the devices measuring the spacetime geometry, we do not have any further operationally defined, maybe non-dynamical background, just in accordance with the principle of equivalence. If there were some non-dynamical background metric on , then by requiring we could reduce the almost arbitrary diffeomorphism (essentially four arbitrary functions of four variables) to a transformation depending on at most ten parameters.
3.1.2 Pseudotensors
The lack of any background geometric structure in the gravitational action yields, first, that any vector field generates a symmetry of the matter plus gravity system. Its second consequence is the need for an auxiliary derivative operator, e. g. the Levi-Civita covariant derivative coming from an auxiliary, non-dynamical background metric (see for example [225, 302] ), or a background (usually torsion free, but not necessarily flat) connection (see for example [209] ), or the partial derivative coming from a local coordinate system (see for example [365] ). Though the natural expectation would be that the final results be independent of these background structures, as is well known, the results do depend on them. In particular [336] , for Hilbert’s second order Lagrangian in a fixed local coordinate system and derivative operator instead of ( 4 ) gives precisely Møller’s energy-momentum pseudotensor , which was defined originally through the superpotential equation , where is the so-called Møller superpotential [260] . (For another simple and natural introduction of Møller’s energy-momentum pseudotensor see [101] .) For the spin pseudotensor ( 2 ) gives which is in fact only pseudotensorial. Similarly, the contravariant form of these pseudotensors and the corresponding canonical Noether current are also pseudotensorial. We saw in subsection 2.1.2 that a specific combination of the canonical energy-momentum and spin tensors gave the symmetric energy-momentum tensor, which is gauge invariant even if the matter fields have gauge freedom, and one might hope that the analogous combination of the energy-momentum and spin pseudotensors gives a reasonable tensorial energy-momentum density for the gravitational field. The analogous expression is, in fact, tensorial, but unfortunately it is just minus the Einstein tensor [336, 337] . Therefore, to use the pseudotensors a ‘natural’ choice for a ‘preferred’ coordinate system would be needed. This could be interpreted as a gauge choice, or reference configuration. A further difficulty is that the different pseudotensors may have different (potential) significance. For example, for any fixed Goldberg’s -th symmetric pseudotensor is defined by (which, for , reduces to the Landau–Lifshitz pseudotensor, the only symmetric pseudotensor which is a quadratic expression of the first derivatives of the metric) [158] . However, by Einstein’s equations this definition implies that . Hence what is (coordinate-)divergence-free (i. e. ‘pseudo-conserved’) cannot be interpreted as the sum of the gravitational and matter energy-momentum densities. Indeed, the latter is , while the second term in the divergence equation has an extra weight . Thus there is only one pseudotensor in this series, , which satisfies the ‘conservation law’ with the correct weight. On the other hand, the pseudotensors coming from some action (the ‘canonical pseudotensors’) appear to be free of this kind of difficulties (see also [336, 337] ). Classical excellent reviews on these (and several other) pseudotensors are [365, 57, 8, 159] , and for some recent ones (using background geometric structures) see for example [133, 134, 77, 150, 151, 222, 302] .) We return to the discussion of pseudotensors in subsections 3.3.1 and 11.3.3 .Since Einstein’s Lagrangian is only weakly diffeomorphism invariant, the situation would even be worse if we used Einstein’s Lagrangian. The corresponding canonical quantities would still be coordinate dependent, though in certain ‘natural’ coordinate system they yield reasonable results (see for example [2] and references therein).
3.1.3 Strategies to avoid pseudotensors I: Background metrics/connections
One way of avoiding the use of the pseudotensorial quantities is to introduce an explicit background connection [209] or background metric [307, 223, 227, 225, 224, 301, 131] . The advantage of this approach would be that we could use the background not only to derive the canonical energy-momentum and spin tensors, but to define the vector fields as the symmetry generators of the background. Then the resulting Noether currents are without doubt tensorial. However, they depend explicitly on the choice of the background connection or metric not only through : The canonical energy-momentum and spin tensors themselves are explicitly background-dependent. Thus, again, the resulting expressions would have to be supplemented by a ‘natural’ choice for the background, and the main question is how to find such a ‘natural’ reference configuration from the infinitely many possibilities.3.1.4 Strategies to avoid pseudotensors II: The tetrad formalism
In the tetrad formulation of general relativity the -orthonormal frame fields , , are chosen to be the gravitational field variables [368, 230] . Re-expressing the Hilbert Lagrangian (i. e. the curvature scalar) in terms of the tetrad field and its partial derivatives in some local coordinate system, one can calculate the canonical energy-momentum and spin by ( 4 ) and ( 2 ), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [261] , the use of the tetrad fields as the field variables instead of the metric makes it possible to introduce a first order, scalar Lagrangian for Einstein’s field equations: If , the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is(9) |
(10) |
(11) |
3.1.5 Strategies to avoid pseudotensors III: Higher derivative currents
Giving up the paradigm that the Noether current should depend only on the vector field and its first derivative – i. e. if we allow a term to be present in the Noether current ( 3 ) even if the Lagrangian is diffeomorphism invariant – one naturally arrives at Komar’s tensorial superpotential and the corresponding Noether current ([235] , see also [57] ). Although its independence of any background structure (viz. its tensorial nature) and uniqueness property (see Komar [235] quoting Sachs) is especially attractive, the vector field is still to be determined.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 subsection 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 subsections 4.2 and 4.3 . Classical, excellent reviews of global energy-momentum and angular momentum are [147, 159, 14, 276, 375, 299] , and a recent review of conformal infinity (with special emphasis on its applicability in numerical relativity) is [140] . Reviews of the positive energy proofs from the first third of the eighties are [196, 300] .3.2.1 Spatial infinity: Energy-momentum
There are several mathematically inequivalent definitions of asymptotic flatness at spatial infinity [147, 328, 22, 47, 144] . 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 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 [304, 46] . 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 [10, 12, 11, 13] 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 [304, 46] , with the asymptotic translations (see [109, 36, 278, 110] ). In its standard form this is the limit of a 2-surface integral of the first derivatives of the induced 3-metric and of the extrinsic curvature for spheres of large coordinate radius . 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 [269] , and he was able to recover the ADM energy-momentum in a natural way (see also subsection 11.3 below). Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [159] : Taking the flux integral of the current on the spacelike hypersurface , by ( 11 ) the flux can be rewritten as the limit of the 2-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 [271] . 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 subsection 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(12) |
(13) |
(14) |
(15) |
3.2.2 Spatial infinity: Angular momentum
The value of the Hamiltonian of Beig and O Murchadha [46] together with the appropriately defined asymptotic rotation–boost Killing vectors [348] define the spatial angular momentum and centre-of-mass, provided 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 [304] 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 latter must be odd parity functions of (see also [46] ). Thus in what follows we assume that . Then the value of the Beig–O Murchadha Hamiltonian parameterized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [304] , while that parameterized by the asymptotic boost Killing vectors deviate from the centre-of-mass of Beig and O Murchadha [46] by a term which is the spatial momentum times the coordinate time. (As Beig and O Murchadha pointed out [46] , 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 [348] . The centre-of-mass of Beig and O Murchadha was reexpressed recently [41] as the limit of 2-surface integrals of the curvature in the form ( 15 ) with proportional to the lapse times , where is the induced 2-metric on (see subsection 4.1.1 below). A geometric notion of centre-of-mass was introduced by Huisken and Yau [203] . They foliate the asymptotically flat hypersurface 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 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 O 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 timelike level hypersurfaces of the conformal factor) falls off faster than would follow from the fall-off of the metric (but they do not have to impose any global integral, e. g. a parity condition) [22, 14] . If the spacetime admits a Killing vector of axis-symmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see for example [369] ). 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 , for the Killing vector of axis-symmetry it is instead of the expected (‘factor-of-two anomaly’) [223] .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 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 : This notion of energy is not constant in time, but decreases with , showing that gravitational radiation carries away positive energy (‘Bondi’s mass-loss’) [69, 70] . 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 Poincare group [310] . The only difference is that while the Poincare 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. 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 [277] . However, this energy (nowadays known as the Newman–Unti energy) does not seem to have the same significance as the Bondi (or Bondi–Sachs [299] or Trautman–Bondi [112, 113, 111] ) 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 [112, 113] . Without additional conditions on , Komar’s expression does not reproduce the Bondi–Sachs energy-momentum in non-stationary spacetimes either [376, 159] : For the ‘obvious’ choice for 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(16) |
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 [58] 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 [82, ?, 83] . (For another derivation of this superpotential from Møller’s Lagrangian ( 9 ) see [347] .) If , , is a normalized spinor dyad corresponding to the orthonormal frame in ( 9 ), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable 2-surface is(17) |
3.3 The necessity of quasi-locality for the observables in general relativity
3.3.1 Non-locality of the gravitational energy-momentum and angular momentum
One reaction to the non-tensorial nature of the gravitational energy-momentum density expressions was to consider the whole problem ill-defined and the gravitational energy-momentum meaningless. However, the successes discussed in the previous subsection show that the global gravitational energy-momenta and angular momenta are useful notions, and hence it could also be useful to introduce them even if the spacetime is not asymptotically flat. Furthermore, the non-tensorial nature of an object does not imply that it is meaningless. For example, the Christoffel symbols are not tensorial, but they do have geometric, and hence physical content, namely the linear connection. Indeed, the connection is a non-local geometric object, connecting the fibres of the vector bundle over different points of the base manifold. Hence any expression of the connection coefficients, in particular the gravitational energy-momentum or angular momentum, must also be non-local. In fact, although the connection coefficients at a given point can be taken zero by an appropriate coordinate/gauge transformation, they cannot be transformed to zero on an open domain unless the connection is flat. Furthermore, the superpotential of many of the classical pseudotensors (e. g. of the Einstein, Bergmann, Møller’s tetrad, Landau–Lifshitz pseudotensors), being linear in the connection coefficients, can be recovered as the pull back to the spacetime manifold of various forms of a single geometric object on the linear frame bundle, namely of the Nester–Witten 2-form, along various local sections [138, 256, 336, 337] , and the expression of the pseudotensors by their superpotentials are the pull backs of the Sparling equation [329, 126, 256] . In addition, Chang, Nester and Chen [101] found a natural quasi-local Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see subsection 11.3.3 ). Therefore, the pseudotensors appear to have been ‘rehabilitated’, and the gravitational energy-momentum and angular momentum are necessarily associated with extended subsets of the spacetime. This fact is a particular consequence of a more general phenomenon [309, 207] : Since the physical spacetime is the isomorphism class of the pairs instead of a single such pair, it is meaningless to speak about the ‘value of a scalar or vector field at a point ’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), 2-surfaces (e. g. the area of a closed 2-surface) etc. Thus, if we want to associate energy-momentum and angular momentum not only to the whole (necessarily asymptotically flat) spacetime, then these quantities must be associated with extended but finite subsets of the spacetime, i. e. must be quasi-local.3.3.2 Domains for quasi-local quantities
The quasi-local quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:(18) |
(19) |
3.3.3 Strategies to construct quasi-local quantities
There are two natural ways of finding the quasi-local energy-momentum and angular momentum. The first is to follow some systematic procedure, while the second is the ‘quasi-localization’ of the global energy-momentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasi-local quantities are integrals of some superpotential derived from the Lagrangian via a Noether-type analysis. The advantage of this approach could be its manifest Lorentz-covariance. On the other hand, since the Noether current is determined only through the Noether identity, which contains only the divergence of the current itself, the Noether current and its superpotential is not uniquely determined. In addition (as in any approach), a gauge reduction (for example in the form of a background metric or reference configuration) and a choice for the ‘translations’ and ‘boost-rotations’ should be made. The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasi-local (covariant or not) Hamiltonian analysis we would have a Hamiltonian, and its value on the constraint surface in the phase space yields the expected quantities. Here the main idea is that of Regge and Teitelboim [304] that the Hamiltonian must reproduce the correct field equations as the flows of the Hamiltonian vector fields, and hence, in particular, the correct Hamiltonian must be functionally differentiable with respect to the canonical variables. This differentiability may restrict the possible ‘translations’ and ‘boost-rotations’ too. However, if we are not interested in the structure of the quasi-local phase space, then, as a short-cut, we can use the Hamilton–Jacobi method to define the quasi-local quantities. The resulting expression is a 2-surface integral. Nevertheless, just as in the Lagrangian approach, this general expression is not uniquely determined, because the action can be modified by adding an (almost freely chosen) boundary term to it. Furthermore, the ‘translations’ and ‘boost-rotations’ are still to be specified. On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasi-local quantities would be some ‘quasi-localization’ of those expressions that gave the global energy-momentum and angular momentum of asymptotically flat spacetimes. Therefore, respecting both strategies, it is also legitimate to consider the Winicour–Tamburino-type (linkage) integrals and the charge integrals of the curvature. Since the global energy-momentum and angular momentum of asymptotically flat spacetimes can be written as 2-surface integrals at infinity (and, as we will see in subsection 7.1.1 , both the energy-momentum and angular momentum of the source in the linear approximation and the gravitational mass in the Newtonian theory of gravity can also be written as 2-surface integrals), the 2-surface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasi-local energy-momentum and angular momentum as 2-surface observables, then three things must be specified:4 Tools to Construct and Analyze the Quasi-Local Quantities
Having accepted that the gravitational energy-momentum and angular momentum should be introduced at the quasi-local level, we next need to discuss the special tools and concepts that are needed in practice to construct (or even to understand the various special) quasi-local expressions. Thus, first, we review the geometry of closed spacelike 2-surfaces, with special emphasis on the so-called 2-surface data. Then, in the remaining two subsections, we discuss the special situations where there is a more or less generally accepted ‘standard’ definition for the energy-momentum (or at least for the mass) and angular momentum. In these situations any reasonable quasi-local quantity should reduce to them.4.1 The geometry of spacelike 2-surfaces
The first systematic study of the geometry of spacelike 2-surfaces from the point of view of quasi-local quantities is probably due to Tod [358, 363] . Essentially, his approach is based on the GHP formalism [148] . Although this is a very effective and flexible formalism [148, 298, 299, 200, 331] , its form is not spacetime covariant. Since in many cases the covariance of a formalism itself already gives some hint how to treat and solve the problem at hand, here we concentrate mainly on a spacetime–covariant description of the geometry of the spacelike 2-surfaces, developed gradually in [339, 341, 342, 343, 143] . The emphasis will be on the geometric structures rather than the technicalities. In the last paragraph, we comment on certain objects appearing in connection with families of spacelike 2-surfaces.4.1.1 The Lorentzian vector bundle
The restriction to the closed, orientable spacelike 2-surface of the tangent bundle of the spacetime has a unique decomposition to the -orthogonal sum of the tangent bundle of and the bundle of the normals, denoted by . Then all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If and are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projection to and is and , respectively. The induced 2-metric and the corresponding area 2-form on will be denoted by and , respectively, while the area 2-form on the normal bundle will be . The bundle together with the fibre metric and the projection will be called the Lorentzian vector bundle over . For the discussion of the global topological properties of the closed orientable 2-manifolds, see for example [5] .4.1.2 Connections
The spacetime covariant derivative operator defines two covariant derivatives on . The first, denoted by , is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: for any section of . Obviously, annihilates both the fibre metric and the projection . However, since for 2-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ , . The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection 1-form on can be characterized, for example, by . Therefore, the connection can be considered as a connection on coming from a connection on the -principal bundle of the -orthonormal frames adapted to . The other connection, , is analogous to the Sen connection [316] , and is defined simply by . This annihilates only the fibre metric, but not the projection. The difference of the connections and turns out to be just the extrinsic curvature tensor: . Here , and and are the standard (symmetric) extrinsic curvatures corresponding to the individual normals and , respectively. The familiar expansion tensors of the future pointing outgoing and ingoing null normals, and , respectively, are and , and the corresponding shear tensors and are defined by their trace-free part. Obviously, and (and hence the expansion and shear tensors , , and ) are boost-gauge dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination is boost-gauge invariant. In particular, it defines a natural normal vector field to by , where , , and are the relevant traces. is called the main extrinsic curvature vector of . If , then their norm is , and they are orthogonal to each other: . It is easy to show that , i. e. is the uniquely pointwise determined direction orthogonal to the 2-surface in which the expansion of the surface is vanishing. If is not null, then defines an orthonormal frame in the normal bundle (see for example [7] ). If is non-zero but (e. g. future pointing) null, then there is a uniquely determined null normal to such that , and hence is a uniquely determined null frame. Therefore, the 2-surface admits a natural gauge choice in the normal bundle unless is vanishing. Geometrically, is a connection coming from a connection on the -principal fibre bundle of the -orthonormal frames. The curvature of the connections and , respectively, are(20) |
(21) |
4.1.3 Convexity conditions
To prove certain statements on quasi-local quantities various forms of the convexity of must be assumed. The convexity of in a 3-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If the embedding space is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of . If is in a Lorentzian spacetime then the weakest convexity conditions are conditions only on the mean null curvatures: will be called weakly future convex if the outgoing null normals are expanding on , i. e. , and weakly past convex if [363] . is called mean convex [177] if on , or, equivalently, if is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions and . Note that although the expansion tensors, and in particular the functions , , and are gauge dependent, their sign is gauge invariant. Then will be called future convex if and , and past convex if and [363, 342] . A different kind of convexity condition, based on global concepts, will be used in subsection 6.1.3 .4.1.4 The spinor bundle
The connections and determine connections on the pull back to of the bundle of unprimed spinors. The natural decomposition defines a chirality on the spinor bundle in the form of the spinor , which is analogous to the matrix in the theory of Dirac spinors. Then the extrinsic curvature tensor above is a simple expression of and (and their complex conjugate), and the two covariant derivatives on are related to each other by . The curvature of can be expressed by the curvature of , the spinor and its -derivative. We can form the scalar invariants of the curvatures according to(22) |
(23) |
4.1.5 Curvature identities
The complete decomposition of into its irreducible parts gives , the Dirac–Witten operator, and , the 2-surface twistor operator. A Sen–Witten type identity for these irreducible parts can be derived. Taking its integral one has(24) |
4.1.6 The GHP formalism
A GHP spin frame on the 2-surface is a normalized spinor basis , , such that the complex null vectors and are tangent to (or, equivalently, the future pointing null vectors and are orthogonal to ). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole . This fact is connected with the non-triviality of the tangent bundle of the 2-surface. For example, on the 2-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors and cannot form a globally defined basis on . Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable 2-surface with globally trivial tangent bundle is the torus. Fixing a GHP spin frame on some , the components of the spinor and tensor fields on will be local representatives of cross sections of appropriate complex line bundles of scalars of type [148, 298] : A scalar is said to be of type if under the rescaling , of the GHP spin frame with some nowhere vanishing complex function the scalar transforms as . For example , , and are of type , , and , respectively. The components of the Weyl and Ricci spinors, , , , . . . , , , . . . , etc., also have definite type. In particular, has type . A global section of is a collection of local cross sections such that forms a covering of and on the non-empty overlappings, e. g. on the local sections are related to each other by , where is the transition function between the GHP spin frames: and . The connection defines a connection on the line bundles [148, 298] . The usual edth operators, and , are just the directional derivatives and on the domain of the GHP spin frame . These locally defined operators yield globally defined differential operators, denoted also by and , on the global sections of . It might be worth emphasizing that the GHP spin coefficients and , which do not have definite -type, play the role of the two components of the connection 1-form, and they are built both from the connection 1-form for the intrinsic Riemannian geometry of and the connection 1-form in the normal bundle. and are elliptic differential operators, thus their global properties, e. g. the dimension of their kernel, are connected with the global topology of the line bundle they act on, and, in particular, with the global topology of . These properties are discussed in [143] for general, and in [128, 42, 340] for spherical topology.4.1.7 Irreducible parts of the derivative operators
Using the projection operators , the irreducible parts and can be decomposed further into their right handed and left handed parts. In the GHP formalism these chiral irreducible parts are(25) |
4.1.8 -connection 1-form versus anholonomicity
Obviously, all the structures we have considered can be introduced on the individual surfaces of oneor two-parameter families of surfaces, too. In particular [176] , let the 2-surface be considered as the intersection of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to , and let the spacetime (or at least a neighbourhood of ) be foliated by two one-parameter families of smooth hypersurfaces and , where , such that and . One can form the two normals, , which are null on and , respectively. Then we can define , for which , where . (If is chosen to be 1 on , then is precisely the connection 1-form above.) Then the so-called anholonomicity is defined by . Since is invariant with respect to the rescalings and of the functions defining the foliations by those functions which preserve , it was claimed in [176] that depends only on . However, this implies only that is invariant with respect to a restricted class of the change of the foliations, and that is invariantly defined only by this class of the foliations rather than the 2-surface. In fact, does depend on the foliation: Starting with a different foliation defined by the functions and for some , the corresponding anholonomicity would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, and , would be different: . Therefore, the anholonomicity is still a gauge dependent quantity.4.2 Standard situations to evaluate the quasi-local quantities
There are exact solutions to the Einstein equations and classes of special (e. g. asymptotically flat) spacetimes in which there is a commonly accepted definition of energy-momentum (or at least mass) and angular momentum. In this subsection we review these situations and recall the definition of these ‘standard’ expressions.4.2.1 Round spheres
If the spacetime is spherically symmetric, then a 2-sphere which is a transitivity surface of the rotation group is called a round sphere. Then in a spherical coordinate system the spacetime metric takes the form , where and are functions of and . (Hence is the so-called area-coordinate). Then with the notations of subsection 4.1 , one obtains . Based on the investigations of Misner, Sharp and Hernandez [258, 193] , Cahill and McVitte [95] found(26) |
(27) |
can be thought of as the 0-component of some quasi-local energy-momentum 4-vector, but, just because of the spherical symmetry, its spatial parts are vanishing. Thus can also be interpreted as the mass, the length of this energy-momentum 4-vector.
4.2.2 Small surfaces
In the literature there are two notions of small surfaces, the first is that of the small spheres (both in the light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [198] , and the other is the concept of the small ellipsoids in some spacelike hypersurface, considered first by Woodhouse in [229] . A small sphere in the light cone is a cut of the future null cone in the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given spacelike hypersurface, whose geodesic distance in the hypersurface from a given point , the centre, is a small given value, and the geometry of this sphere is characterized by data at this centre. Small ellipsoids are 2-surfaces in a spacelike hypersurface with a more general shape. To define the first, let be a point, and a future directed unit timelike vector at . Let , the ‘future null cone of in ’ (i. e. the boundary of the chronological future of ). Let be the future pointing null tangent to the null geodesic generators of such that, at the vertex , . With this condition we fix the scale of the affine parameter on the different generators, and hence by requiring we fix the parameterization completely. Then, in an open neighbourhood of the vertex , is a smooth null hypersurface, and hence for sufficiently small the set is a smooth spacelike 2-surface and homeomorphic to . is called a small sphere of radius with vertex . Note that the condition fixes the boost gauge. Completing to a Newman–Penrose complex null tetrad such that the complex null vectors and are tangent to the 2-surfaces , the components of the metric and the spin coefficients with respect to this basis can be expanded as series in . Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the metric on , the GHP spin coefficients , , , , and and the higher order expansion coefficients of the curvature in terms of the lower order curvature components at . Hence the expression of any quasi-local quantity for the small sphere can be expressed as a series of , where the expansion coefficients are still functions of the coordinates, or , on the unit sphere . If the quasi-local quantity is spacetime–covariant, then the unit sphere integrals of the expansion coefficients must be spacetime covariant expressions of the metric and its derivatives up to some finite order at and the ‘time axis’ . The necessary degree of the accuracy of the solution of the GHP equations depends on the nature of and on whether the spacetime is Ricci-flat in a neighbourhood of or not. These solutions of the GHP equations, with increasing accuracy, are given in [198, 229, 91, 344] . Obviously, we can calculate the small sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime too. In particular [344] , the small sphere expressions for the quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields based on , respectively, are(28) |
(29) |
If, in addition, the spinor constituent of is required to be parallel propagated along , then the tetrad becomes completely fixed, yielding the vanishing of several (combinations of the) spin coefficients.
As we will see soon, the leading term of the small sphere expression of the energy-momenta in non-vacuum is of order , in vacuum it is , while that of the angular momentum is and , respectively.
4.2.3 Large spheres near the spatial infinity
Near spatial infinity we have the a priori and fall-off for the 3-metric and extrinsic curvature , respectively, and both the evolution equations of general relativity and the conservation equation for the matter fields preserve these conditions. The spheres of coordinate radius in are called large spheres if the values of are large enough such that the asymptotic expansions of the metric and extrinsic curvature are legitimate. Introducing some coordinate system, e. g. the complex stereographic coordinates, on one sphere and then extending that to the whole along the normals of the spheres, we obtain a coordinate system on . Let , , be a GHP spinor dyad on adapted to the large spheres in such a way that and are tangent to the spheres and , the future directed unit normal of . These conditions fix the spinor dyad completely, and, in particular, , the outward directed unit normal to the spheres tangent to . The fall-off conditions yield that the spin coefficients tend to their flat spacetime value like and the curvature components to zero like . Expanding the spin coefficients and curvature components as power series of , one can solve the field equations asymptotically (see [47, 43] for a different formalism). However, in most calculations of the large sphere limit of the quasi-local quantities only the leading terms of the spin coefficients and curvature components appear. Thus it is not necessary to solve the field equations for their second or higher order non-trivial expansion coefficients. Using the flat background metric and the corresponding derivative operator we can define a spinor field to be constant if . Obviously, the constant spinors form a two complex dimensional vector space. Then by the fall-off properties . Hence we can define the asymptotically constant spinor fields to be those that satisfy , where is the intrinsic Levi-Civita derivative operator. Note that this implies that, with the notations of ( 25 ), all the chiral irreducible parts, , , and , of the derivative of the asymptotically constant spinor field are .Because of the fall-off, no essential ambiguity in the definition of the large spheres arises from the use of the coordinate radius instead of the physical radial distance.
4.2.4 Large spheres near null infinity
Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [286, 287, 288, 299] (see also [147] ), i. e. the physical spacetime can be conformally compactified by an appropriate boundary . Then future null infinity will be a null hypersurface in the conformally rescaled spacetime. Topologically it is , and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of is the metric of the unit sphere. Fixing such a slice (called ‘the origin cut of ’) the points of can be labelled by a null coordinate, namely the affine parameter along the null geodesic generators of measured from and, for example, the familiar complex stereographic coordinates , defined first on the unit sphere and then extended in a natural way along the null generators to the whole . Then any other cut of can be specified by a function . In particular, the cuts are obtained from by a pure time translation. The coordinates can be extended to an open neighbourhood of in the spacetime in the following way. Let be the family of smooth outgoing null hypersurfaces in a neighbourhood of such that they intersect the null infinity just in the cuts , i. e. . Then let be the affine parameter in the physical metric along the null geodesic generators of . Then forms a coordinate system. The , 2-surfaces (or simply if no confusion can arise) are spacelike topological 2-spheres, which are called large spheres of radius near future null infinity. Obviously, the affine parameter is not unique, its origin can be changed freely: is an equally good affine parameter for any smooth . Imposing certain additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bondi type coordinate system’. In many of the large sphere calculations of the quasi-local quantities the large spheres should be assumed to be large spheres not only in a general null, but in a Bondi type coordinate system. For the detailed discussion of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see for example [277, 276, 82] . In addition to the coordinate system we need a Newman–Penrose null tetrad, or rather a GHP spinor dyad, , , on the hypersurfaces . (Thus boldface indices are referring to the GHP spin frame.) It is natural to choose such that be the tangent of the null geodesic generators of , and itself be constant along . Newman and Unti [277] chose to be parallel propagated along . This choice yields the vanishing of a number of spin coefficients (see for example the review [276] ). The asymptotic solution of the Einstein–Maxwell equations as a series of in this coordinate and tetrad system is given in [277, 130, 298] , where all the non-vanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the -derivative of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces . From the point of view of the large sphere calculations of the quasi-local quantities the choice of Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin frame to the family of the large spheres of constant ‘radius’ , i. e. to require and to be tangents of the spheres. This can be achieved by an appropriate null rotation of the Newman–Unti basis about the spinor . This rotation yields a change of the spin coefficients and the metric and curvature components. As far as the present author is aware of, this rotation with the highest accuracy was done for the solutions of the Einstein–Maxwell system by Shaw [323] . In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [81] . Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i. e. when . In the language of subsection 4.1.7 , this definition would be , , and . However, as Bramson showed [81] , half of these conditions can be imposed. Namely, at future null infinity (and at past null infinity ) can always be imposed asymptotically, and it has two linearly independent solutions , , on (or on , respectively). The space of its solutions turns out to have a natural symplectic metric , and we refer to as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations the future/past asymptotic twistor equations. At asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form for some constant Hermitian matrix . Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are , where are the standard Pauli matrices (divided by ) [347] . Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from , , and too. In the present review we use only the fact that asymptotic spinors can be introduced as anti-holomorphic spinors (see also subsection 8.2.1 ), i. e. the solutions of (and at past null infinity as holomorphic spinors), and as special solutions of the 2-surface twistor equation (see also subsection 7.2.1 ). These operators, together with others reproducing the asymptotic spinors, are discussed in [347] . The Bondi–Sachs energy-momentum given in the Newman–Penrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut of a combination of the leading term of the Weyl spinor component , the asymptotic shear and its -derivative, weighted by the first four spherical harmonics (see for example [276, 299] ):(30) |
(31) |
In the so-called Bondi coordinate system the radial coordinate is the luminosity distance , which tends to the affine parameter asymptotically.
4.2.5 Other special situations
In the weak field approximation of general relativity [365, 21, 369, 299, 221] the gravitational field is described by a symmetric tensor field on Minkowski spacetime , and the dynamics of the field is governed by the linearized Einstein equations, i. e. essentially the wave equation. Therefore, the tools and techniques of the Poincare-invariant field theories, in particular the Noether–Belinfante–Rosenfeld procedure outlined in subsection 2.1 and the ten Killing vectors of the background Minkowski spacetime, can be used to construct the conserved quantities. It turns out that the symmetric energy-momentum tensor of the field is essentially the second order term in the Einstein tensor of the metric . Thus in the linear approximation the field does not contribute to the global energy-momentum and angular momentum of the matter+gravity system, and hence these quantities have the form ( 5 ) with the linearized energy-momentum tensor of the matter fields. However, as we will see in subsection 7.1.1 , this energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized) curvature [333, 200, 299] . pp-waves spacetimes are defined to be those that admit a constant null vector field , and they are interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present then it is necessarily pure radiation with wavevector , i. e. holds [236] . A remarkable feature of the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a two dimensional linear equation for a single function. In contrast to the approach adopted almost exclusively, Aichelburg [3] considered this field equation as an equation for a boundary value problem. As we will see, from the point of view of the quasi-local observables this is a particularly useful and natural standpoint. If a pp-wave spacetime admits an additional spacelike Killing vector with closed orbits, i. e. it is cyclically symmetric too, then and are necessarily commuting and are orthogonal to each other, because otherwise an additional timelike Killing vector would also be admitted [335] . Since the final state of stellar evolution (the neutron star or the black hole state) is expected to be described by an asymptotically flat stationary, axis-symmetric spacetime, the significance of these spacetimes is obvious. It is conjectured that this final state is described by the Kerr–Newman (either outer or black hole) solution with some well defined mass, angular momentum and electric charge parameters [369] . Thus axis-symmetric 2-surfaces in these solutions may provide domains which are general enough but for which the quasi-local quantities are still computable. According to a conjecture by Penrose [291] , the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the Kerr–Newman black hole this area is . Thus, particularly interesting 2-surfaces in these spacetimes are the spacelike cross sections of the event horizon [60] . There is a well defined notion of total energy-momentum not only in the asymptotically flat, but even in the asymptotically anti-de-Sitter spacetimes too. This is the Abbott–Deser energy [1] , whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [157] . The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically anti-de-Sitter spacetimes and to study their general, basic properties in [26] . A comparison and analysis of the various definitions of mass for asymptotically anti-de-Sitter metrics is given in [114] . Thus it is natural to ask whether a specific quasi-local energy-momentum expression is able to reproduce the Abbott–Deser energy-momentum in this limit or not.4.3 On lists of criteria of reasonableness of the quasi-local quantities
In the literature there are various, more or less ad hoc, ‘lists of criteria of reasonableness’ of the quasi-local quantities (see for example [127, 108] ). However, before discussing them, it seems useful to formulate first some general principles that any quasi-local quantity should satisfy.4.3.1 General expectations
In non-gravitational physics the notions of conserved quantities are connected with symmetries of the system, and they are introduced through some systematic procedure in the Lagrangian and/or Hamiltonian formalism. In general relativity the total energy-momentum and angular momentum are 2-surface observables, thus we concentrate on them even at the quasi-local level. These facts motivate our three a priori expectations:(32) |
4.3.2 Pragmatic criteria
Since in certain special situations there are generally accepted definitions for the energy-momentum and angular momentum, it seems reasonable to expect that in these situations the quasi-local quantities reduce to them. One half of the pragmatic criteria is just this expectation, and the other is a list of some a priori requirements on the behaviour of the quasi-local quantities. One such list for the energy-momentum and mass, based mostly on [127, 108] and the properties of the quasi-local energy-momentum of the matter fields of subsection 2.2 , might be the following:4.3.3 Incompatibility of certain ‘natural’ expectations
As Eardley noted in [127] , probably no quasi-local energy definition exists which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasi-local energy definition which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed Friedmann–Robertson–Walker or the spacetimes show explicitly. The points where the monotonicity breaks down are the extremal (maximal or minimal) surfaces, which represent event horizon in the spacetime. Thus one may argue that since the event horizon hides a portion of spacetime, we cannot know the details of the physical state of the matter+gravity system behind the horizon. Hence, in particular, the monotonicity of the quasi-local mass may be expected to break down at the event horizon. However, although for stationary systems (or at the moment of time symmetry of a time symmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general non-stationary systems the concepts of the event and apparent horizons deviate. Thus the causal argument above does not seem possible to be formulated in the hypersurface of subsection 4.3.2 . Actually, the root of the non-monotonicity is the fact that the quasi-local energy is a 2-surface observable in the sense of 1 in subsection 4.3.1 above. This does not mean, of course, that in certain restricted situations the monotonicity (‘local monotonicity’) could not be proven. This local monotonicity may be based, for example, on Lie dragging of the 2-surface along some special spacetime vector field. On the other hand, in the literature sometimes the positivity and the monotonicity requirements are confused, and there is an ‘argument’ that the quasi-local gravitational energy cannot be positive definite, because the total energy of the closed universes must be zero. However, this argument is based on the implicit assumption that the quasi-local energy is associated with a compact three dimensional domain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasi-local energy-momentum is associated with 2-surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression ( 26 ) in the closed Friedmann–Robertson–Walker spacetime, or, more generally, the Dougan–Mason energy-momentum (see subsection 8.2.3 ) are such examples.5 The Bartnik Mass and its Modifications
5.1 The Bartnik mass
5.1.1 The main idea
One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [38, 37] . His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let be a compact, connected 3-manifold with connected boundary , and let be a (negative definite) metric and a symmetric tensor field on such that they, as an initial data set, satisfy the dominant energy condition: If and , then . For the sake of simplicity we denote the triple by . Then let us consider all the possible asymptotically flat initial data sets with a single asymptotic end, denoted simply by , which satisfy the dominant energy condition, have finite ADM energy and are extensions of above through its boundary . The set of these extensions will be denoted by . By the positive energy theorem has non-negative ADM energy , which is zero precisely when is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies, , where the infimum is taken on . Obviously, by the non-negativity of the ADM energies this infimum exists and is non-negative, and it is tempting to define the quasi-local mass of by this infimum. However, it is easy to see that, without further conditions on the extensions of , this infimum is zero. In fact, can be extended to an asymptotically flat initial data set with arbitrarily small ADM energy such that contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and . In particular, in the ‘ -spacetime’, discussed in subsection 4.2.1 on the round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily large round-sphere mass ) has an asymptotically flat extension, the -spacetime itself, with arbitrarily small ADM mass . Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding from the outside. This led Bartnik [38, 37] to formulate his suggestion for the quasi-local mass of . He concentrated on the time-symmetric data sets (i. e. those for which the extrinsic curvature is vanishing), when the horizon appears to be a minimal surface of topology in (see for example [152] ), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature: . Thus, if denotes the set of asymptotically flat Riemannian geometries with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is(33) |
(34) |
Since we take the infimum, we could equally take the ADM masses, which are the minimum values of the zero-th component of the energy-momentum four-vectors in the different Lorentz frames, instead of the energies.
5.1.2 The main properties of
The first immediate consequence of ( 33 ) is the monotonicity of the Bartnik mass: If , then , and hence . Obviously, by definition ( 33 ) one has for any . Thus if is any quasi-local mass functional which is larger than (i. e. which assigns a non-negative real to any such that for any allowed ), furthermore if for any , then by the definition of the infimum in ( 33 ) one has for any . Therefore, is the largest mass functional satisfying for any . Another interesting consequence of the definition of , due to W. Simon, is that if is any asymptotically flat, time symmetric extension of with non-negative scalar curvature satisfying , then there is a black hole in in the form of a minimal surface between and the infinity of (see for example [40] ). As we saw, the Bartnik mass is non-negative, and, obviously, if is flat (and hence is a data set for the flat spacetime), then . The converse of this statement is also true [202] : If , then is locally flat. The Bartnik mass tends to the ADM mass [202] : If is an asymptotically flat Riemannian 3-geometry with non-negative scalar curvature and finite ADM mass , and if , , is a sequence of solid balls of coordinate radius in , then . The proof of these two results is based on the use of the Hawking energy (see subsection 6.1 ), by means of which a positive lower bound for can be given near the non-flat points of . In the proof of the second statement one must use the fact that the Hawking energy tends to the ADM energy, which, in the time symmetric case, is just the ADM mass. The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice application of the Riemannian Penrose inequality [202] : Let be a spherically symmetric Riemannian 3-geometry with spherically symmetric boundary . One can form its ‘standard’ round-sphere energy (see subsection 4.2.1 ), and take its spherically symmetric asymptotically flat vacuum extension (see [38, 40] ). By the Birkhoff theorem the exterior part of is a part of a hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just . Then any asymptotically flat extension of can also be considered as (a part of) an asymptotically flat time symmetric hypersurface with minimal surface, whose area is . Thus by the Riemannian Penrose inequality [202] . Therefore, the Bartnik mass of is just the ‘standard’ round sphere expression .5.1.3 The computability of the Bartnik mass
Since for any given the set of its extensions is a huge set, it is almost hopeless to parameterize it. Thus, by the very definition, it seems very difficult to compute the Bartnik mass for a given, specific . Without some computational method the potentially useful properties of would be lost from the working relativist’s arsenal. Such a computational method might be based on a conjecture of Bartnik [38, 40] : The infimum in definition ( 33 ) of the mass is realized by an extension of such that the exterior region, , is static, the metric is Lipschitz-continuous across the 2-surface , and the mean curvatures of of the two sides are equal. Therefore, to compute for a given , one should find an asymptotically flat, static vacuum metric satisfying the matching conditions on , and the Bartnik mass is the ADM mass of . As Corvino showed [115] , if there is an allowed extension of for which , then the extension is static; furthermore, if , and has an allowed extension for which , then is static. Thus the proof of Bartnik’s conjecture is equivalent to the proof of the existence of such an allowed extension. The existence of such an extension is proven in [257] for geometries close enough to the Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still lacking. Bartnik’s conjecture is that determines this exterior metric uniquely [40] . He conjectures [38, 40] that a similar computation method can be found for the mass , defined in ( 34 ), too, where the exterior metric should be stationary. This second conjecture is also supported by partial results [116] : If is any compact vacuum data set, then it has an asymptotically flat vacuum extension which is a spacelike slice of a Kerr spacetime outside a large sphere near spatial infinity. To estimate one can construct admissible extensions of in the form of the metrics in quasi-spherical form [39] . If the boundary is a metric sphere of radius with non-negative mean curvature , then can be estimated from above in terms of and .5.2 Bray’s modifications
Another, slightly modified definition for the quasi-local mass was suggested by Bray [84, 87] . Here we summarize his ideas. Let be any asymptotically flat initial data set with finitely many asymptotic ends and finite ADM masses, and suppose that the dominant energy condition is satisfied on . Let be any fixed 2-surface in which encloses all the asymptotic ends except one, say the -th (i. e. let be homologous to a large sphere in the -th asymptotic end). The outside region with respect to , denoted by , will be the subset of containing the -th asymptotic end and bounded by , while the inside region, , is the (closure of) . Next Bray defines the ‘extension’ of by replacing by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Similarly, the ‘fill-in’ of is obtained from by replacing by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Finally, the surface will be called outer-minimizing if for any closed 2-surface enclosing one has . Let be outer-minimizing, and let denote the set of extensions of in which is still outer-minimizing, and denote the set of fill-ins of . If and denotes the infimum of the area of the 2-surfaces enclosing all the ends of except the outer one, then Bray defines the outer and inner mass, and , respectively, by6 The Hawking Energy and its Modifications
6.1 The Hawking energy
6.1.1 The definition
Studying the perturbation of the dust-filled Friedmann–Robertson–Walker spacetimes, Hawking found that(35) |
6.1.2 The Hawking energy for spheres
Obviously, for round spheres reduces to the standard expression ( 26 ). This implies, in particular, that the Hawking energy is not monotonic in general. Since for a Killing horizon (e. g. for a stationary event horizon) , the Hawking energy of its spacelike spherical cross sections is . In particular, for the event horizon of a Kerr–Newman black hole it is just the familiar irreducible mass . For a small sphere of radius with centre in non-vacuum spacetimes it is , while in vacuum it is , where is the energy-momentum tensor and is the Bel–Robinson tensor at [198] . The first result shows that in the lowest order the gravitational ‘field’ does not have a contribution to the Hawking energy, that is due exclusively to the matter fields. Thus in vacuum the leading order of must be higher than . Then even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the order term in the power series expansion of is . However, there are no tensorial quantities built from the metric and its derivatives such that the total number of the derivatives involved would be three. Therefore, in vacuum, the leading term is necessarily of order , and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres is positive definite both in non-vacuum (provided the matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in particular, that should be interpreted as energy rather than as mass: For small spheres in a pp-wave spacetime is positive, while, as we saw this for the matter fields in subsection 2.2.3 , a mass expression could be expected to be zero. (We will see in subsections 8.2.3 and 13.5 that, for the Dougan–Mason energy-momentum, the vanishing of the mass characterizes the pp-wave metrics completely.) Using the second expression in ( 35 ) it is easy to see that at future null infinity tends to the Bondi–Sachs energy. A detailed discussion of the asymptotic properties of near null infinity, both for radiative and stationary spacetimes is given in [323, 325] . Similarly, calculating for large spheres near spatial infinity in an asymptotically flat spacelike hypersurface, one can show that it tends to the ADM energy.6.1.3 Positivity and monotonicity properties
In general the Hawking energy may be negative, even in the Minkowski spacetime. Geometrically this should be clear, since for an appropriately general (e. g. concave) 2-surface the integral could be less than . Indeed, in flat spacetime is proportional to by the Gauss equation. For topologically spherical 2-surfaces in the spacelike hyperplane of Minkowski spacetime is real and non-positive, and it is zero precisely for metric spheres; while for 2-surfaces in the timelike cylinder is real and non-negative, and it is zero precisely for metric spheres. If, however, is ‘round enough’ (not to be confused with the round spheres in subsection 4.2.1 , which is some form of a convexity condition, then behaves nicely [108] : will be called round enough if it is a submanifold of a spacelike hypersurface , and if among the two dimensional surfaces in which enclose the same volume as does, has the smallest area. Then it is proven by Christodoulou and Yau [108] that if is round enough in a maximal spacelike slice on which the energy density of the matter fields is non-negative (for example if the dominant energy condition is satisfied), then the Hawking energy is non-negative. Although the Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of 2-surfaces. Hawking considered one-parameter families of spacelike 2-surfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of [166] . These calculations were refined by Eardley [127] . Starting with a weakly future convex 2-surface and using the boost gauge freedom, he introduced a special family of spacelike 2-surfaces in the outgoing null hypersurface , where will be the luminosity distance along the outgoing null generators. He showed that is non-decreasing with , provided the dominant energy condition holds on . Similarly, for weakly past convex and the analogous family of surfaces in the ingoing null hypersurface is non-increasing. Eardley also considered a special spacelike hypersurface, filled by a family of 2-surfaces, for which is non-decreasing. By relaxing the normalization condition for the two null normals to for some , Hayward obtained a flexible enough formalism to introduce a double-null foliation (see subsection 11.2 below) of a whole neighbourhood of a mean convex 2-surface by special mean convex 2-surfaces [177] . (For the more general GHP formalism in which is not fixed, see [298] .) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these 2-surfaces is non-decreasing in the outgoing, and non-increasing in the ingoing direction. In contrast to the special foliations of the null hypersurfaces above, Frauendiener defined a special spacelike vector field, the inverse mean curvature vector in the spacetime [141] . If is a weakly future and past convex 2-surface, then is an outward directed spacelike normal to . Here is the trace of the extrinsic curvature tensor: (see subsection 4.1.2 ). Starting with a single weakly future and past convex 2-surface, Frauendiener gives an argument for the construction of a one-parameter family of 2-surfaces being Lie-dragged along its own inverse mean curvature vector . Hence this family of surfaces would be analogous to the solution of the geodesic equation, where the initial point and direction in that point specify the whole solution, at least locally. Assuming that such a family of surfaces (and hence the vector field on the 3-submanifold swept by ) exists, Frauendiener showed that the Hawking energy is non-decreasing along the vector field if the dominant energy condition is satisfied. However, no investigation has been made to prove the existence of such a family of surfaces. Motivated by this result, Malec, Mars and Simon [251] considered spacelike hypersurfaces with an inverse mean curvature flow of Geroch thereon (see subsection 6.2.2 ). They showed that if the dominant energy condition and certain additional (essentially technical) assumptions hold, then the Hawking energy is monotonic. These two results are the natural adaptations for the Hawking energy of the corresponding results known for some time for the Geroch energy, aiming to prove the Penrose inequality. We return to this latter issue in subsection 13.2 only for a very brief summary.I thank Paul Tod for pointing out this to me.
6.1.4 Two generalizations
Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of the Bondi–Sachs energy-momentum are related to the Bondi energy:(36) |
(37) |
6.2 The Geroch energy
6.2.1 The definition
Suppose that the 2-surface for which is defined is embedded in the spacelike hypersurface . Let be the extrinsic curvature of in and the extrinsic curvature of in . (In subsection 4.1.2 we denoted the latter by .) Then , by means of which(38) |
6.2.2 Monotonicity properties
The Geroch energy has interesting positivity and monotonicity properties along a special flow in [146, 213] . This flow is the so-called inverse mean curvature flow defined as follows. Let be a smooth function such that6.3 The Hayward energy
We saw that can be non-zero even in the Minkowski spacetime. This may motivate considering the following expression7 Penrose’s Quasi-Local Energy-Momentum and Angular Momentum
The construction of Penrose is based on twistor-theoretical ideas, and motivated by the linearized gravity integrals for energy-momentum and angular momentum. Since, however, twistor-theoretical ideas and basic notions are still considered to be some ‘special knowledge’, the review of the basic idea behind the Penrose construction is slightly more detailed than that of the others. The basic references of the field are the volumes [298, 299] by Penrose and Rindler on ‘Spinors and Spacetime’, especially volume 2, the very well readable book by Hugget and Tod [200] and the comprehensive review article [360] by Tod.7.1 Motivations
7.1.1 How do the twistors emerge?
In the Newtonian theory of gravity the mass contained in some finite 3-volume can be expressed as the flux integral of the gravitational field strength on the boundary :(39) |
(40) |
(41) |
(42) |
(43) |
(44) |
The analogous calculations using tensor methods and the real instead of spinors and the anti-self-dual (or, shortly, a.s.d.) part of would be technically more complicated [293, 294, 299, 160] .
7.1.2 Twistor space and the kinematical twistor
Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the pairs of spinor fields, which solve the so-called valence 1 twistor equation . If is a solution of this equation, then is a solution of the corresponding equation in the conformally rescaled spacetime, where and is the conformal factor. In general the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complex parameter family of solutions. Its general solution in the Minkowski spacetime is , where and are constant spinors. Thus the space of valence 1 twistors, the so-called twistor-space, is four complex dimensional, and hence has the structure . admits a natural Hermitian scalar product: If is another twistor, then . Its signature is , it is conformally invariant: , and it is constant on Minkowski spacetime. The metric defines a natural isomorphism between the complex conjugate twistor space, , and the dual twistor space, , by . This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate of the covariant valence 2 twistor can be represented by the so-called conjugate twistor . We should mention two special, higher valence twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by(45) |
(46) |
(47) |
(48) |
(49) |
(50) |
7.2 The original construction for curved spacetimes
7.2.1 2-surface twistors and the kinematical twistor
In general spacetimes the twistor equations have only the trivial solution. Thus to be able to associate a kinematical twistor to a closed orientable spacelike 2-surface in general, the conditions on the spinor field had to be relaxed. Penrose’s suggestion [293, 294] is to consider in ( 40 ) to be the symmetrized product of spinor fields that are solutions of the ‘tangential projection to ’ of the valence 1 twistor equation, the so-called 2-surface twistor equation. (The equation obtained as the ‘tangential projection to ’ of the valence 2 twistor equation ( 43 ) would be under-determined [294] .) Thus the quasi-local quantities are searched for in the form of a charge integral of the curvature:(51) |
7.2.2 The Hamiltonian interpretation of the kinematical twistor
For the solutions and of the 2-surface twistor equation the spinor identity ( 24 ) reduces to Tod’s expression [293, 299, 360] for the kinematical twistor, making it possible to re-express by the integral of the Nester–Witten 2-form [340] . Indeed, if(52) |
7.2.3 The Hermitian scalar product and the infinity twistor
In general the natural pointwise Hermitian scalar product, defined by , is not constant on , thus it does not define a Hermitian scalar product on the 2-surface twistor space. As is shown in [214, 217, 358] , is constant on for any two 2-surface twistors if and only if can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric and extrinsic curvatures. Such 2-surfaces are called non-contorted, while those that cannot be embedded are called contorted. One natural candidate for the Hermitian metric could be the average of on [293] : , which reduces to on non-contorted 2-surfaces. Interestingly enough, can also be re-expressed by the integral ( 52 ) of the Nester–Witten 2-form [340] . Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted 2-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. ( 49 )) is ambiguous unless a natural is found. If is non-contorted, then the scalar product defines the totally anti-symmetric twistor , and for the four independent 2-surface twistors ,. . . , the contraction , and hence by ( 46 ) the determinant , is constant on . Nevertheless, can be constant even for contorted 2-surfaces for which is not. Thus, the totally anti-symmetric twistor can exist even for certain contorted 2-surfaces. Therefore, an alternative definition of the quasi-local mass might be based on ( 50 ) [354] . However, although the two mass definitions are equivalent in the linearized theory, they are different invariants of the kinematical twistor even in de Sitter or anti-de-Sitter spacetimes. Thus, if needed, the former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by ). If we want to have not only the notion of the mass but its reality is also expected, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition ( 48 ), one also needs the infinity twistor. However, is not constant on even if it is non-contorted, thus in general it does not define any twistor on . One might take its average on (which can also be re-expressed by the integral of the Nester–Witten 2-form [340] ), but the resulting twistor would not be simple. In fact, even on 2-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant the natural definition for is [299, 297, 354] , while on round spheres in spherically symmetric spacetimes it is [347] . Thus no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [192] : Even if the spacetime is conformally flat (whenever the Hermitian metric exists) the Hermiticity condition would be fifteen algebraic equations for the (at most) twelve real components of the ‘would be’ infinity twistor. Then, since the possible kinematical twistors form an open set in the space of symmetric twistors, the Hermiticity condition cannot be satisfied even for non-simple s. However, in contrast to the linearized gravity case, the infinity twistor should not be given once and for all on some ‘universal’ twistor space, that may depend on the actual gravitational field. In fact, the 2-surface twistor space itself depends on the geometry of , and hence all the structures thereon also. Since in the Hermiticity condition ( 48 ) only the special combination of the infinity and metric twistors (the so-called ‘bar-hook’ combination) appears, it might still be hoped that an appropriate could be found for a class of 2-surfaces in a natural way [360] . However, as far as the present author is aware of, no real progress has been achieved in this way.7.2.4 The various limits
Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea came from the linearized gravity, the construction gives the correct results in the weak field approximation. The nonrelativistic weak field approximation, i. e. the Newtonian limit, was clarified by Jeffryes [216] . He considers a one parameter family of spacetimes with perfect fluid source such that in the limit of the parameter one gets a Newtonian spacetime, and, in the same limit, the 2-surface lies in a hypersurface of the Newtonian time . In this limit the pointwise Hermitian scalar product is constant, and the norm-mass can be calculated. As could be expected, for the leading order term in the expansion of as a series of he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a order correction. The Penrose definition for the energy-momentum and angular momentum can be applied to the cuts of the future null infinity of an asymptotically flat spacetime [293, 299] . Then every element of the construction is built from conformally rescaled quantities of the non-physical spacetime. Since is shear-free, the 2-surface twistor equations on decouple, and hence the solution space admits a natural infinity twistor . It singles out precisely those solutions whose primary spinor parts span the asymptotic spin space of Bramson (see subsection 4.2.4 ), and they will be the generators of the energy-momentum. Although is contorted, and hence there is no natural Hermitian scalar product, there is a twistor with respect to which is Hermitian. Furthermore, the determinant is constant on , and hence it defines a volume 4-form on the 2-surface twistor space [360] . The energy-momentum coming from is just that of Bondi and Sachs. The angular momentum defined by is, however, new. It has a number of attractive properties. First, in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two anomaly’ between the angular momentum and the energy-momentum. Since its definition is based on the solutions of the 2-surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the 2-surface twistor space on reduces the BMS Lie algebra to one of its Poincare subalgebras. Thus the concept of the ‘translation of the origin’ is moved from null infinity to the twistor space (appearing in the form of a four parameter family of ambiguities in the potential for the shear ), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. As was shown in [125] , Penrose’s angular momentum can be considered as a supertranslation of previous definitions. The corresponding angular momentum flux through a portion of the null infinity between two cuts was calculated in [125, 191] and it was shown that this is precisely that given by Ashtekar and Streubel [28] (see also [321, 322, 124] ). The other way of determining the null infinity limit of the energy-momentum and angular momentum is to calculate them for the large spheres from the physical data, instead of the spheres at null infinity from the conformally rescaled data. These calculations were done by Shaw [323, 325] . At this point it should be noted that the limit of vanishes, and it is that yields the energy-momentum and angular momentum at infinity (see the remarks following eq. ( 15 )). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [354] . The 2-surfaces for which the mass was calculated are the cuts of the geometrically distinguished outgoing null hypersurfaces . Tod found that, for given , the mass is independent of , as could be expected because of the lack of the incoming radiation. The large sphere limit of the 2-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [328] , the Ashtekar–Hansen [22] and the Beig–Schmidt [47] models of spatial infinity in [319, 320, 322] . Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) non-contorted, and both the Hermitian scalar product and the infinity twistor are well defined. Thus the energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar–Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [324] . The Penrose mass in asymptotically anti-de-Sitter spacetimes was studied by Kelly [228] . He calculated the kinematical twistor for spacelike cuts of the infinity , which is now a timelike 3-manifold in the non-physical spacetime. Since admits global 3-surface twistors (see the next subsection), is non-contorted. In addition to the Hermitian scalar product there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding Hermiticity condition. The energy-momentum 4-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [26] . Therefore, the energy-momentum 4-vector is future pointing and timelike if there is a spacelike hypersurface extending to on which the dominant energy condition is satisfied. Consequently, . Kelly showed that is also non-negative and in vacuum it coincides with . In fact [360] , holds.7.2.5 The quasi-local mass of specific 2-surfaces
The Penrose mass has been calculated in a large number of specific situations. Round spheres are always non-contorted [358] , thus the norm-mass can be calculated. (In fact, axis-symmetric 2-surfaces in spacetimes with twist-free rotational Killing vector are non-contorted [217] .) The Penrose mass for round spheres reduces to the standard energy expression discussed in subsection 4.2.1 [354] . Thus every statement given in subsection 4.2.1 for round spheres is valid for the Penrose mass, and we do not repeat them. In particular, for round spheres in a slice of the Kantowski–Sachs spacetime this mass is independent of the 2-surfaces [351] . Interestingly enough, although these spheres cannot be shrunk to a point (thus the mass cannot be interpreted as ‘the 3-volume integral of some mass density’), the time derivative of the Penrose mass looks like the mass conservation equation: It is minus the pressure times the rate of change of the 3-volume of a sphere in flat space with the same area as [359] . In conformally flat spacetimes [354] the 2-surface twistors are just the global twistors restricted to , and the Hermitian scalar product is constant on . Thus the norm-mass is well defined. The construction works nicely even if global twistors exist only on a (say) spacelike hypersurface containing . These twistors are the so-called 3-surface twistors [354, 356] , which are solutions of certain (overdetermined) elliptic partial differential equations, the so-called 3-surface twistor equations, on . These equations are completely integrable (i. e. they admit the maximal number of linearly independent solutions, namely four) if and only if with its intrinsic metric and extrinsic curvature can be embedded, at least locally, into some conformally flat spacetime [356] . Such hypersurfaces are called non-contorted. It might be interesting to note that the non-contorted hypersurfaces can also be characterized as the critical points of the Chern–Simons functional built from the real Sen connection on the Lorentzian vector bundle or from the 3-surface twistor connection on the twistor bundle over [48, 345] . Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a 2-surface in a non-contorted depends only on the homology class of . In particular, if can be shrunk to a point then the corresponding kinematical twistor is vanishing. Since is non-contorted, is also non-contorted, and hence the norm–mass is well defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any non-contorted 2-surface that can be deformed into a round sphere, and it is zero for those that do not link the black hole [358] . Thus, in particular, the Penrose mass can be zero even in curved spacetimes. A particularly interesting class of non-contorted hypersurfaces is that of the conformally flat time-symmetric initial data sets. Tod considered Wheeler’s solution of the time symmetric vacuum constraints describing ‘points at infinity’ (or, in other words, black holes) and 2-surfaces in such a hypersurface [354] . He found that the mass is zero if does not link any black hole, it is the mass of the -th black hole if links precisely the -th hole, it is if links precisely the -th and the -th holes, where is some appropriate measure of the distance of the holes, . . . , etc. Thus, the mass of the -th and -th holes as a single object is less than the sum of the individual masses, in complete agreement with our physical intuition that the potential energy of the composite system should contribute to the total energy with negative sign. Beig studied the general conformally flat time symmetric initial data sets describing ‘points at infinity’ [44] . He found a symmetric trace-free and divergence-free tensor field and, for any conformal Killing vector of the data set, defined the 2-surface flux integral of on . He showed that is conformally invariant, depends only on the homology class of , and, apart from numerical coefficients, for the ten (locally existing) conformal Killing vectors these are just the components of the kinematical twistor derived by Tod in [354] (and discussed in the previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the length of the ’s with respect to the Cartan-Killing metric of the conformal group of the hypersurface. Tod calculated the quasi-local mass for a large class of axis-symmetric 2-surfaces (cylinders) in various LRS Bianchi and Kantowski–Sachs cosmological models [359] and more general cylindrically symmetric spacetimes [361] . In all these cases the 2-surfaces are non-contorted, and the construction works. A technically interesting feature of these calculations is that the 2-surfaces have edges, i. e. they are not smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately, and the resulting spinor fields are required to be continuous at the edges. This matching reduces the number of linearly independent solutions to four. The projection parts of the resulting twistors, the s, are not continuous at the edges. It turns out that the cylinders can be classified invariantly to be hyperbolic, parabolic or elliptic. Then the structure of the quasi-local mass expressions is not simply ‘density’ ‘volume’, but they are proportional to a ‘type factor’ as well, where is the coordinate length of the cylinder. In the hyperbolic, parabolic and elliptic cases this factor is , 1 and , respectively, where is an invariant of the cylinder. The various types are interpreted as the presence of a positive, zero or negative potential energy. In the elliptic case the mass may be zero for finite cylinders. On the other hand, for static perfect fluid spacetimes (hyperbolic case) the quasi-local mass is positive. A particularly interesting spacetime is that describing cylindrical gravitational waves, whose presence is detected by the Penrose mass. In all these cases the determinant-mass has also been calculated and found to coincide with the norm-mass. A numerical investigation of the axis-symmetric Brill waves on the Schwarzschild background was presented in [67] . It was found that the quasi-local mass is positive, and it is very sensitive to the presence of the gravitational waves. Another interesting issue is the Penrose inequality for black holes (see subsection 13.2.1 ). Tod showed [357, 358] that for static black holes the Penrose inequality holds if the mass of the hole is defined to be the Penrose quasi-local mass of the spacelike cross section of the event horizon. The trick here is that is totally geodesic and conformal to the unit sphere, and hence it is non-contorted and the Penrose mass is well defined. Then the Penrose inequality will be a Sobolev type inequality for a non-negative function on the unit sphere. This inequality was tested numerically in [67] . Apart from the cuts of in radiative spacetimes, all the 2-surfaces discussed so far were non-contorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted 2-surface [360] . Thus although the kinematical twistor can be calculated for this, the construction in its original form cannot yield any mass expression. The original construction has to be modified.7.2.6 Small surfaces
The properties of the Penrose construction that we have discussed are very remarkable and promising. However, the small surface calculations showed clearly some unwanted feature of the original construction [355, 229, 379] , and forced its modification. First, although the small spheres are contorted in general, the leading term of the pointwise Hermitian scalar product is constant: for any 2-surface twistors and [355, 229] . Since in non-vacuum spacetimes the kinematical twistor has only the ‘4-momentum part’ in the leading order with , the Penrose mass, calculated with the norm above, is just the expected mass in the leading order. Thus it is positive if the dominant energy condition is satisfied. On the other hand, in vacuum the structure of the kinematical twistor is(53) |
7.3 The modified constructions
Independently of the results of the small sphere calculations, Penrose claimed that in the Schwarzschild spacetime the quasi-local mass expression should yield the same zero value on 2-surfaces, contorted or not, which do not surround the black hole. (For the motivations and the arguments, see [295] .) Thus the original construction should be modified, and the negative results for the small spheres above strengthened this need. A much more detailed review of the various modifications is given by Tod in [360] .7.3.1 The ‘improved’ construction with the determinant
A careful analysis of the roots of the difficulties lead Penrose [295, 299] (see also [355, 229, 360] ) to suggest the modified definition for the kinematical twistor(54) |
7.3.2 Modification through Tod’s expression
These anomalies lead Penrose to modify slightly [296] . This modified form is based on Tod’s form of the kinematical twistor:(55) |
7.3.3 Mason’s suggestions
A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [255] . Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of ( 55 ) is that of the integral of the Nester–Witten 2-form, and the spinor fields and could still be considered as the spinor constituents of the ‘quasi-Killing vectors’ of the 2-surface , their structure is not so simple because the factor itself depends on all of the four independent solutions of the 2-surface twistor equation in a rather complicated way. To have a simple Hamiltonian interpretation Mason suggested further modifications [255, 256] . He considers the four solutions , , of the 2-surface twistor equations, and uses these solutions in the integral ( 52 ) of the Nester–Witten 2-form. Since is a Hermitian bilinear form on the space of the spinor fields (see section 8 below), he obtains 16 real quantities as the components of the Hermitian matrix . However, it is not clear how the four ‘quasi-translations’ of should be found among the 16 vector fields (called ‘quasi-conformal Killing vectors’ of ) for which the corresponding quasi-local quantities could be considered as the quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local quantities.8 Approaches Based on the Nester–Witten 2-Form
We saw in subsection 3.2 that8.1 The Ludvigsen–Vickers construction
8.1.1 The definition
Suppose that the spacetime is asymptotically flat at future null infinity, and the closed spacelike 2-surface can be joined to future null infinity by a smooth null hypersurface . Let , the cut defined by the intersection of with the future null infinity. Then the null geodesic generators of define a smooth bijection between and the cut (and hence, in particular, ). We saw in subsection 4.2.4 that on the cut at the future null infinity we have the asymptotic spin space . The suggestion of Ludvigsen and Vickers [250] for the spin space on is to import the two independent solutions of the asymptotic twistor equations, i. e. the asymptotic spinors, from the future null infinity back to the 2-surface along the null geodesic generators of the null hypersurface . Their propagation equations, given both in terms of spinors and in the GHP formalism, are(56) |
(57) |
8.1.2 Remarks on the validity of the construction
Before discussing the usual questions about the properties of the construction (positivity, monotonicity, the various limits, etc.), we should make some general remarks. First, it is obvious that the Ludvigsen–Vickers energy-momentum in its form above cannot be defined in a spacetime which is not asymptotically flat at null infinity. Thus their construction is not genuinely quasi-local, because it depends not only on the (intrinsic and extrinsic) geometry of , but on the global structure of the spacetime as well. In addition, the requirement of the smoothness of the null hypersurface connecting the 2-surface to the null infinity is a very strong restriction. In fact, for general (even for convex) 2-surfaces in a general asymptotically flat spacetime conjugate points will develop along the (outgoing) null geodesics orthogonal to the 2-surface [290, 170] . Thus either the 2-surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the 2-surface must be nearly spherically symmetric (or the former cannot be ‘very much curved’ and the latter cannot be ‘very much bent’). This limitation yields that in general the original construction above does not have a small sphere limit. However, using the same propagation equations ( 56 )-( 57 ) one could define a quasi-local energy-momentum for small spheres [250, 64] . The basic idea is that there is a spin space at the vertex of the null cone in the spacetime whose spacelike cross section is the actual 2-surface, and the Ludvigsen–Vickers spinors on are defined by propagating these spinors from the vertex to via ( 56 )-( 57 ). This definition works in arbitrary spacetime, but the 2-surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely quasi-local.8.1.3 Monotonicity, mass-positivity and the various limits
Once the Ludvigsen–Vickers spinors are given on a spacelike 2-surface of constant affine parameter in the outgoing null hypersurface , then they are uniquely determined on any other spacelike 2-surface in , too, i. e. the propagation law ( 56 )-( 57 ) defines a natural isomorphism between the space of the Ludvigsen–Vickers spinors on different 2-surfaces of constant affine parameter in the same . ( need not be a Bondi type coordinate.) This makes it possible to compare the components of the Ludvigsen–Vickers energy-momenta on different surfaces. In fact [250] , if the dominant energy condition is satisfied (at least on ), then for any Ludvigsen–Vickers spinor and affine parameter values one has , and the difference can be interpreted as the energy flux of the matter and the gravitational radiation through between and . Thus both and are increasing with (‘mass-gain’). A similar monotonicity property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation law ( 56 )-( 57 ) should be replaced by and . Using these equations the positivity of the Ludvigsen–Vickers mass was proven in various special cases in [250] . Concerning the positivity properties of the Ludvigsen–Vickers mass and energy, first it is obvious by the remarks on the nature of the propagation law ( 56 )-( 57 ) that in Minkowski spacetime the Ludvigsen–Vickers energy-momentum is vanishing. However, in the proof of the non-negativity of the Dougan–Mason energy (discussed in subsection 8.2 ) only the part of the propagation equations is used. Therefore, as realized by Bergqvist [59] , the Ludvigsen–Vickers energy-momenta (both based on the asymptotic and the point spinors) are also future directed and nonspacelike if is the boundary of some compact spacelike hypersurface on which the dominant energy condition is satisfied and is weakly future convex (or at least ). Similarly, the Ludvigsen–Vickers definitions share the rigidity properties proven for the Dougan–Mason energy-momentum [338] : Under the same conditions the vanishing of the energy-momentum implies the flatness of the domain of dependence of . In the weak field approximation [250] the difference is just the integral of on the portion of between the two 2-surfaces, where is the linearized energy-momentum tensor: The increment of on is due only to the flux of the matter energy-momentum. Since the Bondi–Sachs energy-momentum can be written as the integral of the Nester–Witten 2-form on the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first version of the Ludvigsen–Vickers energy-momentum tends to that of Bondi and Sachs. It was shown in [250, 325] that this expectation is, in fact, correct. The Ludvigsen–Vickers mass was calculated for large spheres both for radiative and stationary spacetimes with and accuracy, respectively, in [323, 325] . Finally, on a small sphere of radius in non-vacuum the second definition gives [64] the expected result ( 28 ), while in vacuum [64, 344] it is(58) |
8.2 The Dougan–Mason constructions
8.2.1 Holomorphic/anti-holomorphic spinor fields
The original construction of Dougan and Mason [123] was introduced on the basis of sheaf-theoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [338, 340] . Following Dougan and Mason we define the spinor field to be anti-holomorphic in case , or holomorphic if . Thus, this notion of holomorphicity/anti-holomorphicity is referring to the connection on . While the notion of the holomorphicity/anti-holomorphicity of a function on does not depend on whether the or the operator is used, for tensor or spinor fields it does. Although the vectors and are not uniquely determined (because their phase is not fixed), the notion of the holomorphicity/anti-holomorphicity is well defined, because the defining equations are homogeneous in and . Next suppose that there are at least two independent solutions of . If and are any two such solutions, then , and hence by Liouville’s theorem is constant on . If this constant is not zero, then we call generic, if it is zero then will be called exceptional. Obviously, holomorphic on a generic cannot have any zero, and any two holomorphic spinor fields, e. g. and , span the spin space at each point of (and they can be chosen to form a normalized spinor dyad with respect to on the whole of ). Expanding any holomorphic spinor field in this frame, the expanding coefficients turn out to be holomorphic functions, and hence constant. Therefore, on generic 2-surfaces there are precisely two independent holomorphic spinor fields. In the GHP formalism the condition of the holomorphicity of the spinor field is that its components be in the kernel of . Thus for generic 2-surfaces with the constant would be a natural candidate for the spin space above. For exceptional 2-surfaces the kernel space is either two dimensional but does not inherit a natural spin space structure, or it is higher than two dimensional. Similarly, the symplectic inner product of any two anti-holomorphic spinor fields is also constant, one can define generic and exceptional 2-surfaces as well, and on generic surfaces there are precisely two anti-holomorphic spinor fields. The condition of the anti-holomorphicity of is . Then could also be a natural choice. Note that since the spinor fields whose holomorphicity/anti-holomorphicity is defined are unprimed, and these correspond to the anti-holomorphicity/holomorphicity, respectively, of the primed spinor fields of Dougan and Mason. Thus the main question is whether there exist generic 2-surfaces, and if they do, whether they are ‘really generic’, i. e. whether most of the physically important surfaces are generic or not.8.2.2 The genericity of the generic 2-surfaces
are first order elliptic differential operators on certain vector bundles over the compact 2-surface , and their index can be calculated: , where is the genus of . Therefore, for there are at least two linearly independent holomorphic and at least two linearly independent anti-holomorphic spinor fields. The existence of the holomorphic/anti-holomorphic spinor fields on higher genus 2-surfaces is not guaranteed by the index theorem. Similarly, the index theorem does not guarantee that is generic either: If the geometry of is very special then the two holomorphic/anti-holomorphic spinor fields (which are independent as solutions of ) might be proportional to each other. For example, future marginally trapped surfaces (i. e. for which ) are exceptional from the point of view of holomorphic, and past marginally trapped surfaces ( ) from the point of view of anti-holomorphic spinors. Furthermore, there are surfaces with at least three linearly independent holomorphic/anti-holomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional 2-surface with topology make generic. Finally, we note that several first order differential operators can be constructed from the chiral irreducible parts and of , given explicitly by ( 25 ). However, only four of them, the Dirac–Witten operator , the twistor operator , and the holomorphy and anti-holomorphy operators , are elliptic (which ellipticity, together with the compactness of , would guarantee the finiteness of the dimension of their kernel), and it is only that have two-complex dimensional kernel in the generic case. This purely mathematical result gives some justification for the choices of Dougan and Mason: The spinor fields that should be used in the Nester–Witten 2-form are either holomorphic or anti-holomorphic. The construction does not work for exceptional 2-surfaces.8.2.3 Positivity properties
One of the most important properties of the Dougan–Mason energy-momenta is that they are future pointing nonspacelike vectors, i. e. the corresponding masses and energies are non-negative. Explicitly [123] , if is the boundary of some compact spacelike hypersurface on which the dominant energy condition holds, furthermore if is weakly future convex (in fact, is enough), then the holomorphic Dougan–Mason energy-momentum is a future pointing non-spacelike vector, and, analogously, the anti-holomorphic energy-momentum is future pointing and non-spacelike if . As Bergqvist [59] stressed (and we noted in subsection 8.1.3 ), Dougan and Mason used only the (and in the anti-holomorphic construction the ) half of the ‘propagation law’ in their positivity proof. The other half is needed only to ensure the existence of two spinor fields. Thus that might be ( 56 ) of the Ludvigsen–Vickers construction, or in the holomorphic Dougan–Mason construction, or even for some constant , a ‘deformation’ of the holomorphicity considered by Bergqvist [59] . In fact, the propagation law may even be for any spinor field satisfying . This ensures the positivity of the energy under the same conditions and that