QuasiLocal EnergyMomentum and Angular Momentum in GR: A Review Article
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Abstract
The present status of the quasilocal massenergymomentum and angular momentum constructions in general relativity is reviewed. First the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasilocal quantities are recalled. Then the various specific constructions and their properties (both successes and defects) are discussed. Finally, some of the (actual and potential) applications of the quasilocal concepts and specific constructions are briefly mentioned.
This review is based on the talks given at the Erwin Schrödinger Institute, Vienna, in July 1997, at the Universität Tübingen, in May 1998, and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.
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 energymomentum and, ultimately, angular momentum too) to extended but finite spacetime domains, i.e. at the quasilocal level. Obviously, the quasilocal quantities could provide a more detailed characterization of the states of the gravitational ‘field’ than the global ones, so they (together with more general quasilocal observables) would be interesting in their own right.
Moreover, finding an appropriate notion of energymomentum 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 quasilocally 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. quasilocal. Therefore, a solid theoretical foundation of the quasilocal conserved quantities is needed.
However, contrary to the high expectations of the eighties, finding an appropriate quasilocal notion of energymomentum 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 energymomentum 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 energymomentum: 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 ‘whodidwhat’ review, but primarily we would like to concentrate on the understanding of the basic questions (such as why should the gravitational energymomentum and angular momentum, or, more generally, any observable of the gravitational ‘field’, be necessarily quasilocal) and ideas behind the various specific constructions. Consequently, onethird 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 quasilocal 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 energymomentum and angular momentum can be introduced in (and hence can be understood from) the theory of matter fields in Minkowski spacetime. Thus, in Section 2.1, we review the BelinfanteRosenfeld procedure that we will apply to gravity in Section 3, introduce the notion of quasilocal energymomentum and angular momentum of the matter fields and discuss their properties. The philosophy of quasilocality in general relativity will be demonstrated in Minkowski spacetime where the energymomentum and angular momentum of the matter fields are treated quasilocally. Then we turn to the difficulties of gravitational energymomentum and angular momentum, and we clarify why the gravitational observables should necessarily be quasilocal. The tools needed to construct and analyze the quasilocal 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 nontechnical 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 quasilocal 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 quasilocal characterization of the ppwave 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, 3, and refsec4) reflects our own personal interest and view of the subject.
The theory of quasilocal observables in general relativity is far from being complete. The most important open problem is still the trivial one: ‘Find quasilocal energymomentum and angular momentum expressions satisfying the points of the lists of Section 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 g_{ ab } is −2, and the curvature and Ricci tensors and the curvature scalar of the covariant derivative ∇_{ a } are defined by \(({\nabla _c}{\nabla _d}  {\nabla _d}{\nabla _c}){X^a}: =  {R^a}_{bcd}{X^b},{R_{bd}}: = {R^a}_{bad}\) and \(R: = {R_{bd}}{g^{bd}}\), respectively. Hence Einstein’s equations take the form G_{ ab }+λg_{ ab } := R_{ ab }−½Rg_{ ab }+λg_{ ab } = −8πGT_{ ab }, where G is Newton’s gravitational constant and λ is the cosmological constant (and the speed of light is c = 1). However, apart from special cases stated explicitly, the cosmological constant will be assumed to be vanishing, and in Sections 13.3 and 13.4 we use the traditional cgs system.
2 EnergyMomentum and Angular Momentum of Matter Fields
2.1 Energymomentum and angular momentum density of matter fields
2.1.1 The symmetric energymomentum tensor
It is a widely accepted view (appearing e.g. in excellent, standard textbooks on general relativity, too) that the canonical energymomentum and spin tensors are welldefined 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.
2.1.2 The canonical Noether current
 1.
∇_{ a }T^{ ab } = 0,
 2.
T^{ ab } = θ^{ ab } + ∇_{ c }(σ^{ c }^{[ab]} + σ^{ a }^{[bc]} + σ^{ b }[^{ ac }]),
 3.
C^{ a }[K] = T^{ ab }K_{ b } + ∇_{ c }((σ^{ a }^{[cb]} − σ^{ c }^{[ab]} − σ^{ b }[^{ ac }])K_{ b }), where the second term on the right is an identically conserved (i.e. divergence free) current, and
 4.
C^{ a }[K] is conserved if K^{ a } is a Killing vector.
The interpretation of the conserved currents C^{ a }[K] and T^{ ab }K_{ b } depends on the nature of the Killing vector K^{ a }. In Minkowski spacetime the 10dimensional Lie algebra K of the Killing vectors is well known to split to the semidirect sum of a 4dimensional commutative ideal, T, and the quotient K/T, where the latter is isomorphic to so(1, 3). The ideal T is spanned by the constant Killing vectors, in which a constant orthonormal frame field \(\{{E_{\underline a}^a}\}\) on M, \(\underline a = 0, \ldots, 3\), forms a basis. (Thus the underlined Roman indices \(\underline a, \underline b, \ldots\) are concrete, name indices.) By \({g_{ab}}E_{\underline a}^aE_{\underline b}^b = \eta \underline a \underline b \;: = {\rm{diag}}(1,  1,  1,  1)\) the ideal T inherits a natural Lorentzian vector space structure. Having chosen an origin o ∈ M, the quotient K/T can be identified as the Lie algebra R_{ o } of the boostrotation Killing vectors that vanish at o. Thus K has a ‘4 + 6’ decomposition into translations and boostrotations, where the translations are canonically defined but the boostrotations depend on the choice of the origin o ∈ M. In the coordinate system \(\{{x^{\underline a}}\}\) adapted to \(\{{E_{\underline a}^a}\}\) (i.e. for which the 1form basis dual to \(\{{E_{\underline a}^a}\}\) has the form \(\vartheta _a^{\underline a} = {\nabla _a}{x^{\underline a}}\)) the general form of the Killing vectors (or rather 1forms) is \({K_a} = {T_{\underline a}}\vartheta _a^{\underline a} + {M_{\underline a \underline b}}({x^{\underline a}}\vartheta _a^{\underline b}  {x^{\underline b}}\vartheta _a^{\underline a})\) for some constants \({T_{\underline a}}\) and \({M_{\underline a \underline b}} =  {M_{\underline b \underline a}}\). Then the corresponding canonical Noether current is \({C^e}[K] = E_{\underline e}^e({\theta ^{\underline e \underline a}}{T_{\underline a}}  ({\theta ^{\underline e \underline a}}{x^{\underline b}}  {\theta ^{\underline e \underline b}}{x^{\underline a}}  2{\sigma ^{\underline e [\underline a \underline b ]}}){M_{\underline a \underline b}})\), and the coefficients of the translation and the boostrotation parameters \({T_{\underline a}}\) and \({M_{\underline a \underline b}}\) are interpreted as the density of the energymomentum and the sum of the orbital and spin angular momenta, respectively. Since, however, the difference C^{ a }[K] − T^{ ab }K_{ b } is identically conserved and T^{ ab }K_{ b } has more advantageous properties, it is T^{ ab }K_{ b } that is used to represent the energymomentum and angular momentum density of the matter fields.
Since in the deSitter and antideSitter spacetimes the (ten dimensional) Lie algebra of the Killing vector fields, so(1, 4) and so(2, 3), respectively, are semisimple, there is no such natural notion of translations, and hence no natural ‘4 + 6’ decomposition of the ten conserved currents into energymomentum and (relativistic) angular momentum density.
2.2 Quasilocal energymomentum and angular momentum of the matter fields
In the next Section 3 we will see that welldefined (i.e. gauge invariant) energymomentum and angular momentum density cannot be associated with the gravitational ‘field’, and if we want to talk not only about global gravitational energymomentum 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 [169, 168]. Thus observables are always associated with open subsets of spacetime whose closure is compact, i.e. they are quasilocal. 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 quasilocally defined quantities. Thus the idea of quasilocality is not new in physics. Although apparently in classical nongravitational physics this is not obligatory, we adopt this view in talking about energymomentum and angular momentum even of classical matter fields in Minkowski spacetime. Originally the introduction of these quasilocal quantities was motivated by the analogous gravitational quasilocal quantities [354, 358]. Since, however, many of the basic concepts and ideas behind the various gravitational quasilocal energymomentum and angular momentum definitions can be understood from the analogous nongravitational quantities in Minkowski spacetime, we devote the present section to the discussion of them and their properties.
2.2.1 The definition of the quasilocal quantities
Thus even if there is a gauge invariant and unambiguously defined energymomentum density of the matter fields, it is not a priori clear how the various quasilocal quantities should be introduced. We will see in the second part of the present review that there are specific suggestions for the gravitational quasilocal energy that are analogous to \(P_{\mathcal S}^0\), others to E_{Σ}[t^{ a }] and some to M_{Σ}.
2.2.2 Hamiltonian introduction of the quasilocal quantities
However, if we want to recover the field equations for φ^{ A } (which are partial differential equations on the spacetime with smooth coefficients for the smooth field φ^{ A }) on the phase space as the Hamilton equations and not some of their distributional generalizations, then the functional differentiability of H[K] must be required in the strong sense of [387]^{1}. Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of H[K] requires some boundary conditions on the field variables, and may yield restrictions on the form of Z^{ a }. It may happen that for a given Z^{ a } only too restrictive boundary conditions would be able to ensure the functional differentiability of the Hamiltonian, and hence the ‘quasilocal phase space’ defined with these boundary conditions would contain only very few (or no) solutions of the field equations. In this case Z^{ a } should be modified. In fact, the boundary conditions are connected to the nature of the physical situations considered. For example, in electrodynamics different boundary conditions must be imposed if the boundary is to represent a conducting or an insulating surface. Unfortunately, no universal principle or ‘canonical’ way of finding the ‘correct’ boundary term and the boundary conditions is known.
In the asymptotically flat case the value of the Hamiltonian on the constraint surface defines the total energymomentum and angular momentum, depending on the nature of K^{ a }, in which the total divergence D_{ a }Z^{ a } corresponds to the ambiguity of the superpotential 2form ∪[K]_{ ab }: An identically conserved quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved). The energy density and the momentum density of the matter fields can be recovered as the functional derivative of H[K] with respect to the lapse N and the shift N^{ a }, respectively. In principle, the whole analysis can be repeated quasilocally too. However, apart from the promising achievements of [7, 8, 327] for the KleinGordon, Maxwell, and the YangMillsHiggs fields, as far as we know, such a systematic quasilocal Hamiltonian analysis of the matter fields is still lacking.
2.2.3 Properties of the quasilocal quantities
Suppose that the matter fields satisfy the dominant energy condition. Then E_{Σ}[ξ^{ a }] is also nonnegative for any nonspacelike ξ^{ a }, and, obviously, E_{ Σ }[t^{ a }] is zero precisely when T^{ ab } = 0 on Σ, and hence, by the conservation laws (see for example Page 94 of [175]), on the whole domain of dependence D(Σ). Obviously, M_{Σ} = 0 if and only if L^{ a } := T^{ ab }t_{ b } is null on Σ. Then by the dominant energy condition it is a future pointing vector field on Σ, and L_{ a }T^{ ab } = 0 holds. Therefore, T^{ ab } on Σ has a null eigenvector with zero eigenvalue, i.e. its algebraic type on Σ is pure radiation.
 1.
\(P_{\mathcal S}^{\underline a}\) is a future directed nonspacelike vector, \(m_{\mathcal S}^2 \geq 0\);
 2.
\(P_{\mathcal S}^{\underline a}\) if and only if T_{ ab } = 0 on D(Σ);
 3.
\(m_{\mathcal S}^2 = 0\) if and only if the algebraic type of the matter on D(Σ) is pure radiation, i.e. T_{ ab }L^{ b } = 0 holds for some constant null vector L^{ a }. Then T_{ ab } = rL_{ a }L_{ b } for some nonnegative function τ, whenever \(P_{\mathcal S}^{\underline a} = e{L^{\underline a}}\), where \({L^{\underline a}}: = {L^a}\vartheta _a^{\underline a}\) and \(e: = \int\nolimits_\Sigma \tau {L^a}{1 \over {3!}}{\varepsilon _{abcd}}\);
 4.
For \(m_{\mathcal S}^2 = 0\) the angular momentum has the form \(J_{\mathcal S}^{\underline a \underline b} = {e^{\underline a}}{L^{\underline b}}  {e^{\underline b}}{L^{\underline a}}\), where \({e^{\underline a}}: = \int\nolimits_\Sigma {{x^{\underline a}}} \tau {L^a}{1 \over {3!}}{\varepsilon _{abcd}}\). Thus, in particular, the PauliLubanski spin is zero.
Since \({E_\Sigma}[{t^a}]\) and M_{Σ} are integrals of functions on a hypersurface, they are obviously additive, i.e. for example for any two hypersurfaces Σ_{1} and Σ_{2} (having common points at most on their boundaries \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\)) one has \({E_{{\Sigma _1} \cup {\Sigma _2}}}[{t^a}] = {E_{{\Sigma _1}}}[{t^a}] + {E_{{\Sigma _2}}}[{t^a}]\). On the other hand, the additivity of \(P_{\mathcal S}^{\underline a}\) is a slightly more delicate problem. Namely, \(P_{{{\mathcal S}_1}}^{\underline a}\) and \(P_{{{\mathcal S}_2}}^{\underline a}\) are elements of the dual space of the translations, and hence we can add them and, as in the previous case, we obtain additivity. However, this additivity comes from the absolute parallelism of the Minkowski spacetime: The quasilocal energymomenta of the different 2surfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different 2surfaces, then the energymomenta would belong to different vector spaces, and they could not be added. We will see that the quasilocal quantities discussed in Sections 7, 8, and 9 belong to vector spaces dual to their own ‘quasiKilling vectors’, and there is no natural way of adding the energymomenta of different surfaces.
2.2.4 Global energymomenta and angular momenta
If Σ extends either to spatial or future null infinity, then, as is well known, the existence of the limit of the quasilocal energymomentum can be ensured by slightly faster than \({\mathcal O}({r^{ 3}})\) (for example by \({\mathcal O}({r^{ 4}})\)) falloff of the energymomentum tensor, where r is any spatial radial distance. However, the finiteness of the angular momentum and centreofmass is not ensured by the \({\mathcal O}({r^{ 4}})\) falloff. Since the typical falloff of T_{ ab } — for example for the electromagnetic field — is \({\mathcal O}({r^{ 4}})\), we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the \({\mathcal O}({r^{ 4}})\) falloff, six global integral conditions for the leading terms of T_{ ab } 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’ T^{ ab }_{;b} = 0 (as evolution equations for the energy density and momentum density) preserve these falloff and parity conditions [364].
Although quasilocally 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 \({T_{ab}} = O({r^{ 4}})\) 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 Quasilocal radiative modes and a classical version of the holography for matter fields
By the results of the previous Section 2.2.4 the vanishing of the quasilocal mass, associated with a closed spacelike 2surface \({\mathcal S}\), implies that the matter must be pure radiation on a 4dimensional globally hyperbolic domain D(Σ). Thus \({m_{\mathcal S}} = 0\) characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present section we review how these states on D(Σ) can be characterized completely by data on the 2surface \({\mathcal S}\), and how these states can be used to formulate a classical version of the holographic principle.
For the (real or complex) linear massless scalar field φ and the YangMills fields, represented by the symmetric spinor fields \(\phi _{AB}^\alpha, \alpha = 1, \ldots, N\), α = 1, … N, where N is the dimension of the gauge group, the vanishing of the quasilocal mass is equivalent [365] to plane waves and the ppwave solutions of Coleman [118], respectively. Then the condition T_{ ab }L^{ b } = 0 implies that these fields are completely determined on the whole D(Σ) by their value on \({\mathcal S}\) (whenever the spinor fields \(\phi _{AB}^\alpha\) are necessarily null: \(\phi _{AB}^\alpha = {\phi ^\alpha}{O_A}{O_B}\), where φ^{ α } are complex functions and O_{ a } is a constant spinor field such that L_{ a } = O_{ A }Ō_{a′}). Similarly, the null linear zerorestmass fields φ_{ AB…E } = φO_{ A }O_{ B } … O_{ E } on D(Σ) with any spin and constant spinor O_{ A } are completely determined by their value on \({\mathcal S}\). Technically, these results are based on the unique complex analytic structure of the u = const. 2surfaces foliating Σ, where L_{ a } = ∇_{ a }u, and by the field equations the complex functions φ and φ^{ α } turn out to be antiholomorphic [358]. Assuming, for the sake of simplicity, that \({\mathcal S}\) is future and past convex in the sense of Section 4.1.3 below, the independent boundary data for such a pure radiative solution consist of a constant spinor field on \({\mathcal S}\) and a real function with one and another with two variables. Therefore, the pure radiative modes on D(Σ) can be characterized completely by appropriate data (the socalled holographic data) on the ‘screen’ \({\mathcal S}\).
These ‘quasilocal radiative modes’ can be used to map any continuous spinor field on D(Σ) to a collection of holographic data. Indeed, the special radiative solutions of the form \(\phi {O^A}\) (with fixed constant spinor field O^{ A }) together with their complex conjugate define a dense subspace in the space of all continuous spinor fields on Σ. Thus every such spinor field can be expanded by the special radiative solutions, and hence can also be represented by the corresponding family of holographic data. Therefore, if we fix a foliation of D(Σ) by spacelike Cauchy surfaces Σ_{ t }, then every spinor field on D(Σ) can also be represented on \({\mathcal S}\) by a time dependent family of holographic data, too [365]. This fact may be a specific manifestation in the classical nongravitational physics of the holographic principle (see Section 13.4.2).
3 On the EnergyMomentum and Angular Momentum of Gravitating Systems
3.1 On the gravitational energymomentum and angular momentum density: The difficulties
3.1.1 The root of the difficulties
The action I_{m} for the matter fields was a functional of both kinds of fields, thus one could take the variational derivatives both with respect to \({\Phi _{N_{b \ldots}^{a \ldots}}}\) and g^{ ab }. The former gave the field equations, while the latter defined the symmetric energymomentum tensor. Moreover, g_{ ab } provided a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action I_{g} is, on the other hand, a functional of the metric alone, and its variational derivative with respect to g^{ ab } yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of g^{ ab } can be traced back to the principle of equivalence [22], and introduces a huge gauge freedom in the dynamics of g^{ ab } because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold M endowed with a Lorentzian metric g_{ ab }, but the isomorphism class of such pairs, where (M, g_{ ab }) and (M, φ*g_{ ab }) are considered to be equivalent for any diffeomorphism φ of M onto itself^{2}. Thus we do not have, even in principle, any gravitational analog of the symmetric energymomentum tensor of the matter fields. In fact, by its very definition, T_{ ab } is the sourcecurrent for gravity, like the current \(J_{\bf{A}}^a: = \delta {I_p}/\delta A_a^{\bf{A}}\) in YangMills theories (defined by the variational derivative of the action functional of the particles, e.g. of the fermions, interacting with a YangMills field \(A_a^{\bf{A}}\)), rather than energymomentum. The latter is represented by the Noether currents associated with special spacetime displacements. Thus, in spite of the intimate relation between T_{ ab } and the Noether currents, the proper interpretation of T_{ ab } is only the source density for gravity, and hence it is not the symmetric energymomentum tensor whose gravitational counterpart must be searched for. In particular, the BelRobinson tensor \({T_{abcd}}: = {\psi _{ABCD}}{\bar \psi _{{A}\prime {B}\prime {C}\prime {D}\prime}}\), given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [333, 332]), being a quadratic expression of the curvature (and its derivatives), is (are) expected to represent only ‘higher order’ gravitational energymomentum. (Note that according to the original tensorial definition the BelRobinson tensor is onefourth the expression above. Our convention follows that of Penrose and Rindler [312].) In fact, the physical dimension of the BelRobinson ‘energydensity’ T_{ abcd }t^{ a }t^{ b }t^{ c }t^{ d } is cm^{−4}, and hence (in the traditional units) there are no powers A and B such that c^{ A }G^{ B } T_{ abcd }t^{ a }t^{ b }t^{ c }t^{ d } would have energydensity dimension. Here c is the speed of light and G is Newton’s gravitational constant. As we will see, the BelRobinson ‘energymomentum density’ T_{ abcd }t^{ b }t^{ c }t^{ d } appears naturally in connection with the quasilocal energymomentum and spinangular momentum expressions for small spheres only in higher order terms. Therefore, if we want to associate energymomentum and angular momentum with the gravity itself in a Lagrangian framework, then it is the gravitational counterpart of the canonical energymomentum and spin tensors and the canonical Noether current built from them that should be introduced. Hence it seems natural to apply the LagrangeBelinfanteRosenfeld procedure, sketched in the previous section, to gravity too [56, 57, 323, 193, 194, 352].
3.1.2 Pseudotensors
The lack of any background geometric structure in the gravitational action yields, first, that any vector field K^{ a } generates a symmetry of the matter plus gravity system. Its second consequence is the need for an auxiliary derivative operator, e.g. the LeviCivita covariant derivative coming from an auxiliary, nondynamical background metric (see for example [231, 316]), or a background (usually torsion free, but not necessarily flat) connection (see for example [215]), or the partial derivative coming from a local coordinate system (see for example [382]). 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.
A further difficulty is that the different pseudotensors may have different (potential) significance. For example, for any fixed k ∈ ℝ Goldberg’s 2kth symmetric pseudotensor \(t_{(2k)}^{\alpha \beta}\) is defined by \(2\vert g{\vert ^{k + 1}}(8\pi Gt_{(2k)}^{\alpha \beta}  {G^{\alpha \beta}}): = {\partial _\mu}{\partial _\nu}[\vert g{\vert ^{k + 1}}({g^{\alpha \beta}}{g^{\mu \nu}}  {g^{\mu \nu}}{g^{\beta \mu}})\) (which, for k = 0, reduces to the LandauLifshitz pseudotensor, the only symmetric pseudotensor which is a quadratic expression of the first derivatives of the metric) [162]. However, by Einstein’s equations this definition implies that \({\partial _\alpha}[\vert g{\vert ^{k + 1}}(t_{(2k)}^{\alpha \beta} + {T^{\alpha \beta}})] = 0\). Hence what is (coordinate)divergencefree (i.e. ‘pseudoconserved’) cannot be interpreted as the sum of the gravitational and matter energymomentum densities. Indeed, the latter is \(\vert g{\vert ^{1/2}}{T^{\alpha \beta}}\), while the second term in the divergence equation has an extra weight \(\vert g{\vert ^{k + 1/2}}\). Thus there is only one pseudotensor in this series, \(t_{( 1)}^{\alpha \beta}\), which satisfies the ‘conservation law’ with the correct weight. In particular, the LandauLifshitz pseudotensor \(t_{(0)}^{\alpha \beta}\) also has this defect. On the other hand, the pseudotensors coming from some action (the ‘canonical pseudotensors’) appear to be free of this kind of difficulties (see also [352, 353]). Classical excellent reviews on these (and several other) pseudotensors are [382, 59, 9, 163], and for some recent ones (using background geometric structures) see for example [137, 138, 79, 154, 155, 228, 316]. We return to the discussion of pseudotensors in Sections 3.3.1 and 11.3.4.
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 [215] or background metric [322, 229, 233, 231, 230, 315, 135]. (The superpotential of Katz, Bičák, and LyndelBell [230] has been rediscovered recently by Chen and Nester [108] in a completely different way. We return to the discussion of the latter in Section 11.3.2.) The advantage of this approach would be that we could use the background not only to derive the canonical energymomentum and spin tensors, but to define the vector fields K^{ a } 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 K^{ a }: The canonical energymomentum and spin tensors themselves are explicitly backgrounddependent. 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
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 K^{ a } and its first derivative — i.e. if we allow a term Ḃ^{ a } to be present in the Noether current (3) even if the Lagrangian is diffeomorphism invariant — one naturally arrives at Komar’s tensorial superpotential _{K}∨ [K]^{ ab } := ∇^{[a}K^{ b }] and the corresponding Noether current [242] (see also [59]). Although its independence of any background structure (viz. its tensorial nature) and uniqueness property (see Komar [242] quoting Sachs) is especially attractive, the vector field K^{ a } is still to be determined.
3.2 On the global energymomentum and angular momentum of gravitating systems: The successes
As is well known, in spite of the difficulties with the notion of the gravitational energymomentum density discussed above, reasonable total energymomentum and angular momentum can be associated with the whole spacetime provided it is asymptotically flat. In the present section we recall the various forms of them. As we will see, most of the quasilocal constructions are simply ‘quasilocalizations’ of the total quantities. Obviously, the technique used in the ‘quasilocalization’ does depend on the actual form of the total quantities, yielding mathematically inequivalent definitions for the quasilocal quantities. We return to the discussion of the tools needed in the quasilocalization procedures in Sections 4.2 and 4.3. Classical, excellent reviews of global energymomentum and angular momentum are [151, 163, 15, 289, 394, 313], and a recent review of conformal infinity (with special emphasis on its applicability in numerical relativity) is [144]. Reviews of the positive energy proofs from the first third of the eighties are [202, 314].
3.2.1 Spatial infinity: Energymomentum
There are several mathematically inequivalent definitions of asymptotic flatness at spatial infinity [151, 344, 23, 48, 148]. 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 Σ will be called kasymptotically flat if for some compact set K ⊂ Σ the complement Σ − K is diffeomorphic to ℝ^{3} minus a solid ball, and there exists a (negative definite) metric _{0}h_{ ab } on Σ, which is flat on Σ − K, such that the components of the difference of the physical and the background metrics, h_{ ij } − _{0}h_{ ij }, and of the extrinsic curvature χ_{ ij } in the _{0}h_{ ij }Cartesian coordinate system {x^{ k }} fall off as r^{−k} and r^{−k−1}, respectively, for some k > 0 and r^{2} := δ_{ ij }x^{ i }x^{ j } [319, 47]. These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations. Σ − K together with the metric and extrinsic curvature is called the asymptotic end of Σ. In a more general definition of asymptotic flatness Σ is allowed to have finitely many such ends.
As is well known, finite and welldefined ADM energymomentum [11, 13, 12, 14] can be associated with any kasymptotically flat spacelike hypersurface if k > ½ by taking the value on the constraint surface of the Hamiltonian H[K^{ a }], given for example in [319, 47], with the asymptotic translations K^{ a } (see [112, 37, 291, 113]). In its standard form this is the r → ∞ limit of a 2surface integral of the first derivatives of the induced 3metric h_{ ab } and of the extrinsic curvature χ_{ ab } for spheres of large coordinate radius r. The ADM energymomentum is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian 4vector 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 centreofmass, discussed below, form an antisymmetric tensor). One had to check a posteriori that the conserved quantities obtained in the 3 + 1 form are, in fact, Lorentzcovariant. To obtain manifestly Lorentzcovariant quantities one should not do the 3 + 1 decomposition. Such a manifestly Lorentzcovariant Hamiltonian analysis was suggested first by Nester [280], and he was able to recover the ADM energymomentum in a natural way (see also Section 11.3 below).
Another form of the ADM energymomentum is based on Møller’s tetrad superpotential [163]: Taking the flux integral of the current C^{ a }[K] + T^{ ab }K_{ b } on the spacelike hypersurface Σ, by Equation (11) the flux can be rewritten as the r → ∞ limit of the 2surface integral of Møller’s superpotential on spheres of large r with the asymptotic translations K^{ a }. Choosing the tetrad field \(\{E_{\underline a}^a\}\) to be adapted to the spacelike hypersurface and assuming that the frame \(\{E_{\underline a}^a\}\) tends to a constant Cartesian one as r^{−k}, the integral reproduces the ADM energymomentum. The same expression can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too: By the standard scenario one can construct the basic Hamiltonian [282]. This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of C^{ a }[K] + T^{ ab }K_{ b } on Σ.
If the spacetime is stationary, then the ADM energy can be recovered as the r → ∞ limit of the 2sphere integral of Komar’s superpotential with the Killing vector K^{ a } of stationarity [163], too. On the other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time translation, the Komar expression does not reproduce the ADM energy. However, by Equations (13, 14) such an additional restriction might be that K^{ a } should be a constant combination of four future pointing null vector fields of the form \({\alpha ^A}{\bar \alpha ^{A{\prime}}}\), where the spinor fields α^{ A } are required to satisfy the Weyl neutrino equation ∇_{a′A}α^{ A } = 0. This expression for the ADM energymomentum was used to give an alternative, ‘4dimensional’ proof of the positivity of the ADM energy [205].
3.2.2 Spatial infinity: Angular momentum
The value of the Hamiltonian of Beig and Ó Murchadha [47] together with the appropriately defined asymptotic rotationboost Killing vectors [364] define the spatial angular momentum and centreofmass, provided k ≥ 1 and, in addition to the familiar falloff conditions, certain global integral conditions are also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and Teitelboim [319] on the leading nontrivial parts of the metric h_{ ab } and extrinsic curvature χ_{ ab }: The components in the Cartesian coordinates {x^{ i }} of the former must be even and the components of latter must be odd parity functions of x^{ i }/r (see also [47]). Thus in what follows we assume that k = 1. Then the value of the BeigÓ Murchadha Hamiltonian parameterized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [319], while that parameterized by the asymptotic boost Killing vectors deviate from the centreofmass of Beig and Ó Murchadha [47] by a term which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [47], the centreofmass of Regge and Teitelboim is not necessarily finite.) The spatial angular momentum and the new centreofmass form an antisymmetric Lorentz 4tensor, which transforms in the correct way under the 4translation of the origin of the asymptotically Cartesian coordinate system, and it is conserved by the evolution equations [364].
The centreofmass of Beig and Ó Murchadha was reexpressed recently [42] as the r → ∞ limit of 2surface integrals of the curvature in the form (15) with ω^{ ab }μ^{ cd } proportional to the lapse N times q^{ ac }q^{ bd } − q^{ ad }q^{ bc }, where q_{ ab } is the induced 2metric on S (see Section 4.1.1 below)
A geometric notion of centreofmass was introduced by Huisken and Yau [209]. They foliate the asymptotically flat hypersurface Σ 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 ℝ^{3} defines the centreofmass (or rather ‘centreofgravity’) 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 centreofmass coincides, for example, with that of Beig and Ó Murchadha.
The AshtekarHansen definition for the angular momentum is introduced in their specific conformal model of the spatial infinity as a certain 2surface 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 Ω = const. timelike level hypersurfaces of the conformal factor) falls off faster than would follow from the 1/r falloff of the metric (but they do not have to impose any global integral, e.g. a parity condition) [23, 15].
If the spacetime admits a Killing vector of axisymmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see for example [387]). However, the value of the Komar integral is twice the expected angular momentum. In particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr solution the integral is m/G, for the Killing vector of axisymmetry it is 2ma/G instead of the expected ma/G (‘factoroftwo anomaly’) [229]. We return to the discussion of the Komar integral in Section 12.1.
3.2.3 Null infinity: Energymomentum
The study of the gravitational radiation of isolated sources led Bondi to the observation that the 2sphere integral of a certain expansion coefficient m(u, θ, φ) of the line element of a radiative spacetime in an asymptotically retarded spherical coordinate system (u, r, Θ, φ) behaves as the energy of the system at the retarded time u: This notion of energy is not constant in time, but decreases with u, showing that gravitational radiation carries away positive energy (‘Bondi’s massloss’) [71, 72]. The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, nowadays known as the BMS group, having a structure very similar to that of the Poincaré group [325]. The only difference is that while the Poincaré group is a semidirect product of the Lorentz group and a 4dimensional commutative group (of translations), the BMS group is the semidirect product of the Lorentz group and an infinitedimensional commutative group, called the group of the supertranslations. A 4parameter 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 [290]. However, this energy (nowadays known as the NewmanUnti energy) does not seem to have the same significance as the Bondi (or BondiSachs [313] or TrautmanBondi [115, 116, 114]) 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 socalled BondiSachs energymomentum, has a remarkable uniqueness property [115, 116].
The BondiSachs energymomentum can also be expressed by the integral of the NesterWitten 2form [214, 255, 256, 205]. However, in nonstationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [83]. Thus the spinor fields in the NesterWitten 2form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves be the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [83], was to require the spinor fields to be the solutions of the socalled asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all these, based only on the linear first order differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are determined in [363].
The BondiSachs energymomentum has a Hamiltonian interpretation as well. Although the fields on a spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable generalization of the standard Hamiltonian analysis could be developed [114] and used to recover the BondiSachs energymomentum.
Similarly to the ADM case, the simplest proofs of the positivity of the Bondi energy [330] are probably those that are based on the NesterWitten 2form [214] and, in particular, the use of twocomponents spinors [255, 256, 205, 203, 321]: The BondiSachs mass (i.e. the Lorentzian length of the BondiSachs energymomentum) of a cut of future null infinity is nonnegative if there is a spacelike hypersurface Σ intersecting null infinity in the given cut such that the dominant energy condition is satisfied on Σ, and the mass is zero iff the domain of dependence D(Σ) of Σ is flat.
3.2.4 Null infinity: Angular momentum
At null infinity there is no generally accepted definition for angular momentum, and there are various, mathematically inequivalent suggestions for it. Here we review only some of those total angular momentum definitions that can be considered as the null infinity limit of some quasilocal expression, and will be discussed in the main part of the review, namely in Section 9.
The construction based on the WinicourTamburino linkage (16) can be associated with any BMS vector field [395, 252, 30]. In the special case of translations it reproduces the BondiSachs energymomentum. The quantities that it defines for the proper supertranslations are called the supermomenta. For the boostrotation vector fields they can be interpreted as angular momentum. However, in addition to the factoroftwo anomaly, this notion of angular momentum contains a huge ambiguity (‘supertranslation ambiguity’): The actual form of both the boostrotation Killing vector fields of Minkowski spacetime and the boostrotation BMS vector fields at future null infinity depend on the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively. However, while the set of the origins of Minkowski spacetime is parameterized by four numbers, the set of the origins at null infinity requires a smooth function of the form \(u:{S^2} \rightarrow \mathbb{R}\). Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar origindependence (containing four parameters), the analogous transformation of the angular momentum defined by using the boostrotation BMS vector fields depends on an arbitrary smooth real valued function on the 2sphere. This makes the angular momentum defined at null infinity by the boostrotation BMS vector fields ambiguous unless a natural selection rule for the origins, making them form a four parameter family of cuts, is found. Such a selection rule could be the suggestion by Dain and Moreschi [125] in the charge integral approach to angular momentum of Moreschi [272, 273].
Another promising approach might be that of Chruściel, Jezierski, and Kijowski [114], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperbolic spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian 4space of origins, they appear to be the generators with respect to some fixed ‘centreofthecut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.
3.3 The necessity of quasilocality for the observables in general relativity
3.3.1 Nonlocality of the gravitational energymomentum and angular momentum
One reaction to the nontensorial nature of the gravitational energymomentum density expressions was to consider the whole problem illdefined and the gravitational energymomentum meaningless. However, the successes discussed in the previous Section 3.2.4 show that the global gravitational energymomenta 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 nontensorial 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 nonlocal 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 energymomentum or angular momentum, must also be nonlocal. 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, LandauLifshitz pseudotensors), being linear in the connection coefficients, can be recovered as the pullback to the spacetime manifold of various forms of a single geometric object on the linear frame bundle, namely of the NesterWitten 2form, along various local Sections [142, 266, 352, 353], and the expression of the pseudotensors by their superpotentials are the pullbacks of the Sparling equation [345, 130, 266]. In addition, Chang, Nester, and Chen [104] found a natural quasilocal Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see Section 11.3.4). Therefore, the pseudotensors appear to have been ‘rehabilitated’, and the gravitational energymomentum and angular momentum are necessarily associated with extended subsets of the spacetime.
This fact is a particular consequence of a more general phenomenon [324, 213]: Since the physical spacetime is the isomorphism class of the pairs (M, g_{ ab }) instead of a single such pair, it is meaningless to speak about the ‘value of a scalar or vector field at a point p ∈ M’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), 2surfaces (e.g. the area of a closed 2surface) etc. determined by some body or physical fields. Thus, if we want to associate energymomentum 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 quasilocal.
3.3.2 Domains for quasilocal quantities
 1.
the globally hyperbolic domains D ⊂ M with compact closure,
 2.
the compact spacelike hypersurfaces Σ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and
 3.
the closed, orientable spacelike 2surfaces \({\mathcal S}\) (interpreted as the boundary ∂Σ of Cauchy surfaces for globally hyperbolic domains).
3.3.3 Strategies to construct quasilocal quantities
There are two natural ways of finding the quasilocal energymomentum and angular momentum. The first is to follow some systematic procedure, while the second is the ‘quasilocalization’ of the global energymomentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasilocal quantities are integrals of some superpotential derived from the Lagrangian via a Noethertype analysis. The advantage of this approach could be its manifest Lorentzcovariance. 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 ‘boostrotations’ should be made.
The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasilocal (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 [319] 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 ‘boostrotations’ too. However, if we are not interested in the structure of the quasilocal phase space, then, as a shortcut, we can use the HamiltonJacobi method to define the quasilocal quantities. The resulting expression is a 2surface 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 ‘boostrotations’ are still to be specified.
On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasilocal quantities would be some ‘quasilocalization’ of those expressions that gave the global energymomentum and angular momentum of asymptotically flat spacetimes. Therefore, respecting both strategies, it is also legitimate to consider the WinicourTamburinotype (linkage) integrals and the charge integrals of the curvature.
 1.
an appropriate general 2surface integral (e.g. the integral of a superpotential 2form in the Lagrangian approaches or a boundary term in the Hamiltonian approaches),
 2.
a gauge choice (in the form of a distinguished coordinate system in the pseudotensorial approaches, or a background metric/connection in the background field approaches or a distinguished tetrad field in the tetrad approach), and
 3.
a definition for the ‘quasisymmetries’ of the 2surface (i.e. the ‘generator vector fields’ of the quasilocal quantities in the Lagrangian, and the lapse and the shift in the Hamiltonian approaches, respectively, which, in the case of timelike ‘generator vector fields’, can also be interpreted as a fleet of observers on the 2surface).
In certain approaches the definition of the ‘quasisymmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.
4 Tools to Construct and Analyze the QuasiLocal Quantities
Having accepted that the gravitational energymomentum and angular momentum should be introduced at the quasilocal 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) quasilocal expressions. Thus, first, in Section 4.1 we review the geometry of closed spacelike 2surfaces, with special emphasis on the socalled 2surface data. Then, in the remaining two Sections 4.2 and 4.3, we discuss the special situations where there is a more or less generally accepted ‘standard’ definition for the energymomentum (or at least for the mass) and angular momentum. In these situations any reasonable quasilocal quantity should reduce to them.
4.1 The geometry of spacelike 2surfaces
The first systematic study of the geometry of spacelike 2surfaces from the point of view of quasilocal quantities is probably due to Tod [375, 380]. Essentially, his approach is based on the GHP (GerochHeldPenrose) formalism [152]. Although this is a very effective and flexible formalism [152, 312, 313, 206, 347], 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 spacetimecovariant description of the geometry of the spacelike 2surfaces, developed gradually in [355, 357, 358, 359, 147]. 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 2surfaces.
4.1.1 The Lorentzian vector bundle
The restriction \({{\bf{V}}^a}({\mathcal S})\) to the closed, orientable spacelike 2surface \({\mathcal S}\) of the tangent bundle TM of the spacetime has a unique decomposition to the g_{ ab }orthogonal sum of the tangent bundle \(T{\mathcal S}\) of \({\mathcal S}\) and the bundle of the normals, denoted by \(N{\mathcal S}\). Then all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If t^{ a } and v^{ a } are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projection to \(T{\mathcal S}\) and \(N{\mathcal S}\) is \(\Pi _b^a:\delta _b^a  {t^a}{t_b} + {v^a}{v_b}\) and \(O_b^a: = \delta _b^a  \Pi _b^a\), respectively. The induced 2metric and the corresponding area 2form on \({\mathcal S}\) will be denoted by q_{ ab } = g_{ ab } − t_{ a }t_{ b } + v^{ a }v_{ b } and ε_{ ab } = t^{ c }v^{ d }ε_{ cdab }, respectively, while the area 2form on the normal bundle will be ⊥ε_{ ab } = t_{ a }v_{ b } − t_{ b }v_{ a }. The bundle \({{\bf{V}}^a}({\mathcal S})\) together with the fibre metric g_{ ab } and the projection \(\Pi _b^a\) will be called the Lorentzian vector bundle over \({\mathcal S}\). For the discussion of the global topological properties of the closed orientable 2manifolds, see for example [5].
4.1.2 Connections
The spacetime covariant derivative operator ∇_{ e } defines two covariant derivatives on \({{\bf{V}}^a}({\mathcal S})\). The first, denoted by δ_{ e }, is analogous to the induced (intrinsic) covariant derivative on (onecodimensional) hypersurfaces: \({\delta _e}{X^a}: = \Pi _b^a\Pi _e^f{\nabla _f}(\Pi _c^b{X^c}) + O_b^a\Pi _e^f{\nabla _f}(O_c^b{X^c})\) for any section X^{ a } of \({{\bf{V}}^a}({\mathcal S})\). Obviously, δ_{ e } annihilates both the fibre metric g_{ ab } and the projection \(\Pi _b^a\). However, since for 2surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ t^{ a } ↦ t^{ a } cosh u +v^{ a } sinh u, v^{ a } ↦ t^{ a } sinh u + v^{ a } cosh u. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection 1form on \({\mathcal S}\) can be characterized, for example, by \({A_e}: = \Pi _e^f({\nabla _f}{t_a}){\upsilon ^a}\). Therefore, the connection δ_{ e } can be considered as a connection on \({{\bf{V}}^a}({\mathcal S})\) coming from a connection on the O(2) ⊗ O(1,1)principal bundle of the g_{ ab }orthonormal frames adapted to \({\mathcal S}\).
4.1.3 Convexity conditions
To prove certain statements on quasilocal quantities various forms of the convexity of \({\mathcal S}\) must be assumed. The convexity of \({\mathcal S}\) in a 3geometry 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 \({\mathcal S}\). If \({\mathcal S}\) is in a Lorentzian spacetime then the weakest convexity conditions are conditions only on the mean null curvatures: \({\mathcal S}\) will be called weakly future convex if the outgoing null normals l^{ a } are expanding on \({\mathcal S}\), i.e. \(\theta := {q^{ab}}{\theta _{ab}} > 0\), and weakly past convex if \({\theta {\prime}}: = {q^{ab}}\theta _{ab}{\prime} < 0\) [380]. \({\mathcal S}\) is called mean convex [182] if θθ′ < 0 on \({\mathcal S}\), or, equivalently, if \({{\tilde Q}_a}\) is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions \(D: = \det \left\Vert {{\theta ^a}_b} \right\Vert = {1 \over 2}({\theta _{ab}}{\theta _{cd}}  {\theta _{ac}}{\theta _{bd}}){q^{ab}}{q^{cd}}\) and \({D{\prime}}: = \det \left\Vert {{\theta{{\prime}^a}}_b} \right\Vert = {1 \over 2}(\theta _{ab}{\prime}\theta _{cd}{\prime}  \theta _{ac}{\prime}\theta _{bd}{\prime}){q^{ab}}{q^{cd}}\). Note that although the expansion tensors, and in particular the functions θ, θ′, D, and D′ are gauge dependent, their sign is gauge invariant. Then \({\mathcal S}\) will be called future convex if θ > 0 and D > 0, and past convex if θ′ < 0 and D′ > 0 [380, 358]. These are equivalent to the requirement that the two eigenvalues of \({\theta ^a}_b\) be positive and those of \({\theta {\prime}}{^a_b}\) be negative everywhere on \({\mathcal S}\), respectively. A different kind of convexity condition, based on global concepts, will be used in Section 6.1.3.
4.1.4 The spinor bundle
An interesting decomposition of the SO(1, 1) connection 1form A_{ e }, i.e. the vertical part of the connection δ_{ e }, was given by Liu and Yau [253]: There are real functions α and γ, unique up to additive constants, such that \({A_e} = {\varepsilon _e}^f{\delta _f}\alpha + {\delta _e}\gamma\). α is globally defined on \({\mathcal S}\), but in general γ is defined only on the local trivialization domains of \(N{\mathcal S}\) that are homeomorphic to ℝ^{2}. It is globally defined if \({H^1}({\mathcal S}) = 0\). In this decomposition α is the boostgauge invariant part of A_{ e }, while γ represents its gauge content. Since \({\delta _e}{A^e} = {\delta _e}{\delta ^e}\gamma\), the ‘Coulombgauge condition’ δ_{ e }A_{ e } = 0 uniquely fixes A_{ e } (see also Section 10.4.1).
By the GaussBonnet theorem \(\oint\nolimits_{\mathcal S} f d{\mathcal S} = {\oint\nolimits_{\mathcal S}}^{\mathcal S}Rd{\mathcal S} = 8\pi (1  g)\), where g is the genus of \({\mathcal S}\). Thus geometrically the connection δ_{ e } is rather poor, and can be considered as a part of the ‘universal structure of \({\mathcal S}\)’. On the other hand, the connection Δ_{ e } is much richer, and, in particular, the invariant F carries information on the mass aspect of the gravitational ‘field’. The 2surface data for chargetype quasilocal quantities (i.e. for 2surface observables) are the universal structure (i.e. the intrinsic metric q_{ ab }, the projection \(\Pi _b^a\) and the connection δ_{ e } and the extrinsic curvature tensor \({Q^a}_{eb}\).
4.1.5 Curvature identities
4.1.6 The GHP formalism
A GHP spin frame on the 2surface \({\mathcal S}\) is a normalized spinor basis \(\varepsilon _{\bf{A}}^A: = \{{o^A},{\iota ^A}\}, {\bf{A}} = 0,1\) A = 0, 1, such that the complex null vectors \({m^a}: = {o^A}{{\bar \iota}^A}{\prime}\) and \({{\bar m}^a}: = {\iota ^A}{{\bar o}^A}{\prime}\) are tangent to \({\mathcal S}\) (or, equivalently, the future pointing null vectors \({l^a}: = {o^A}{{\bar o}^{{A{\prime}}}}\) and \({n^a}: = {\iota ^A}{{\bar \iota}^{{A{\prime}}}}\) are orthogonal to \({\mathcal S}\)). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole \({\mathcal S}\). This fact is connected with the nontriviality of the tangent bundle \(T{\mathcal S}\) of the 2surface. For example, on the 2sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors m^{ a } and \({{\bar m}^a}\) cannot form a globally defined basis on \({\mathcal S}\). Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable 2surface with globally trivial tangent bundle is the torus.
Fixing a GHP spin frame \(\{\varepsilon _{\bf{A}}^A\}\) on some open \(U \subset {\mathcal S}\), the components of the spinor and tensor fields on U will be local representatives of cross sections of appropriate complex line bundles E(p, q) of scalars of type (p, q) [152, 312]: A scalar φ is said to be of type (p, q) if under the rescaling \({o^A} \mapsto \lambda {o^A},{\iota ^A} \mapsto {\lambda ^{ 1}}{\iota ^A}\) of the GHP spin frame with some nowhere vanishing complex function λ: U → ℂ the scalar transforms as \(\phi \mapsto {\lambda ^p}{{\bar \lambda}^q}\phi\). For example \(\rho := {\theta _{ab}}{m^a}{{\bar m}^b} =  {1 \over 2}\theta, {\rho {\prime}}: = \theta _{ab}{\prime}{m^a}{{\bar m}^b} =  {1 \over 2}{\theta {\prime}},\sigma := {\theta _{ab}}{m^a}{m^b} = {\sigma _{ab}}{m^a}{m^b}\), and \({\sigma {\prime}}: = \theta _{ab}{\prime}{{\bar m}^a}{{\bar m}^b} = \sigma _{ab}{\prime}{{\bar m}^a}{{\bar m}^b}\) are of type (1, 1), (−1, −1), (3, −1), and (−3, 1), respectively. The components of the Weyl and Ricci spinors, \({\psi _0}: = {\psi _{ABCD}}{o^A}{o^B}{o^C}{o^D},{\psi _1}: = {\psi _{ABCD}}{o^A}{o^B}{o^C}{\iota ^D},{\psi _2}: = {\psi _{ABCD}}{o^A}{o^B}{\iota ^C}{\iota ^D}, \ldots, {\phi _{00}}: = {\phi _{AB{\prime}}}{o^A}{{\bar o}^{{B{\prime}}}},{\phi _{01}}: = {\phi _{AB{\prime}}}{o^A}{{\bar \iota}^{{B{\prime}}}}\), etc., also have definite (p, q)type. In particular, Λ := R/24 has type (0, 0). A global section of E(p, q) is a collection of local cross sections {(U, φ), (U′, φ′), …} such that {U, U′, …} forms a covering of \({\mathcal S}\) and on the nonempty overlappings, e.g. on U ∩ U′ the local sections are related to each other by \(\phi = {\psi ^p}{{\bar \psi}^q}{\phi {\prime}}\). where ψ: U ∩ U′ → ℂ is the transition function between the GHP spin frames: \({o^A} = \psi {o^{{\prime}A}}\) and \({\iota ^A} = {\psi ^{ 1}}{\iota ^{{\prime}A}}\).
The connection δ_{ e } defines a connection ð_{ e } on the line bundles E(p, q) [152, 312]. The usual edth operators, ð and ð′ are just the directional derivatives ð := m^{ a }ð_{ a } and \({\eth\prime}: = {{\bar m}^a}{\eth _a}\) on the domain \(U \subset {\mathcal S}\) of the GHP spin frame \(\{\varepsilon _A^A\}\). These locally defined operators yield globally defined differential operators, denoted also by ð and ð′, on the global sections of E(p, q). It might be worth emphasizing that the GHP spin coefficients β and β′ which do not have definite (p, q)type, play the role of the two components of the connection 1form, and they are built both from the connection 1form for the intrinsic Riemannian geometry of (\({\mathcal S},{q_{ab}}\)) and the connection 1form A_{ e } in the normal bundle. ð and ð′ 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 \({\mathcal S}\). These properties are discussed in [147] for general, and in [132, 43, 356] for spherical topology.
4.1.7 Irreducible parts of the derivative operators
4.1.8 SO(1, 1)connection 1form versus anholonomicity
Obviously, all the structures we have considered can be introduced on the individual surfaces of one or twoparameter families of surfaces, too. In particular [181], let the 2surface \({\mathcal S}\) be considered as the intersection \({{\mathcal N}^ +} \cap {{\mathcal N}^ }\) of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to \({\mathcal S}\), and let the spacetime (or at least a neighbourhood of \({\mathcal S}\)) be foliated by two oneparameter families of smooth hypersurfaces {ν_{+} = const.} and {ν_{−} = const.}, where ν_{±} : M → ℝ, such that \({{\mathcal N}^ +} = \{{\nu _ +} = 0\}\) and \({{\mathcal N}^ } = \{{\nu _ } = 0\}\). One can form the two normals, \({n_{\pm a}}: = {\nabla _a}{\nu _ \pm}\), which are null on \({{\mathcal N}^ +}\) and \({{\mathcal N}^ }\), respectively. Then we can define \({\beta _{\pm e}}: = ({\Delta _e}{n_{\pm a}})n_ \mp ^a\), for which β_{+e} + β_{−e} = Δ_{ e }n^{2}, where \({n^2}: = {g_{ab}}n_ + ^an_  ^b\). (If n^{2} is chosen to be 1 on \({\mathcal S}\), then β_{−e} + β_{+e} is precisely the SO(1, 1) connection 1form A_{ e } above.) Then the socalled anholonomicity is defined by \({w_e}: = {1 \over {2{n^2}}}{[{n_ },{n_ +}]^f}{q_{fe}} = {1 \over {2{n^2}}}({\beta _{+ e}}  {\beta _{ e}})\) Since ωe is invariant with respect to the rescalings ν_{+} ↦ exp(A)ν_{+} and ν_{−} ↦exp(B)ν_{−} of the functions defining the foliations by those functions A, B : M → ℝ which preserve \({\nabla _{[{a^n} \pm b]}} = 0\), it was claimed in [181] that ωe depends only on \({\mathcal S}\). However, this implies only that ωe is invariant with respect to a restricted class of the change of the foliations, and that ωe is invariantly defined only by this class of the foliations rather than the 2surface. In fact, ωe does depend on the foliation: Starting with a different foliation defined by the functions \({{\bar \nu}_ +}: = \exp (\alpha){\nu _ +}\) and \({{\bar \nu}_ }: = \exp (\beta){\nu _ }\) for some α, β : M → ℝ the corresponding anholonomicity \({{\bar \omega}_e}\) would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, ωe and \({{\bar \omega}_e}\), would be different: \({{\bar \omega}_e}  {\omega _e} = {1 \over 2}{\Delta _e}(\alpha  \beta)\). Therefore, the anholonomicity is still a gauge dependent quantity.
4.2 Standard situations to evaluate the quasilocal 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 energymomentum (or at least mass) and angular momentum. In this section we review these situations and recall the definition of these ‘standard’ expressions.
4.2.1 Round spheres
This example suggests a slightly more exotic spherically symmetric spacetime. Its spacelike slice Σ will be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat metrics. The first is a ‘large’ spherically symmetric part of a t = const. hypersurface of the closed FriedmannRobertsonWalker spacetime with the line element \(d{l^2} = \Omega _{{\rm{FRW}}}^2dl_0^2\), where \(dl_0^2\) is the line element for the flat 3space and \(\Omega _{{\rm{FRW}}}^2: = B{(1 + {{{r^2}} \over {4{T^2}}})^{ 2}}\) with some positive constants B and T^{2}, and the range of the Euclidean radial coordinate r is [0, r_{0}], where r_{0} ∈ (2T, ∞). It contains a maximal 2surface at r = 2T with roundsphere mass parameter \(M: = GE(2T) = {1 \over 2}T\sqrt B\) The scalar curvature is R = 6/BT^{2}, and hence, by the constraint parts of the Einstein equations and by the vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the metric of a piece of a t = const. hypersurface in the Schwarzschild solution with mass parameter m (see [156]): \(d{{\bar l}^2} = \Omega _S^2d\bar l_0^2\), where \(\Omega _S^2: = {\left({1 + {m \over {2\bar r}}} \right)^4}\) and the Euclidean radial coordinate \({\bar r}\) runs from \({{\bar r}_0}\) and ∞, where \({{\bar r}_0} \in (0,m/2)\). In this geometry there is a minimal surface at \(\bar r = m/2\), the scalar curvature is zero, and the round sphere energy is \(E(\bar r) = m/G\). These two metrics can be matched to obtain a differentiable metric with Lipschitzcontinuous derivative at the 2surface of the matching (where the scalar curvature has a jump) with arbitrarily large ‘internal mass’ M/G and arbitrarily small ADM mass m/G. (Obviously, the two metrics can be joined smoothly as well by an ‘intermediate’ domain between them.) Since this space looks like a big spherical bubble on a nearly flat 3plane — like the capital Greek letter Ω — for later reference we call it an ‘Ω_{ M },_{ m }spacetime’.
Spherically symmetric spacetimes admit a special vector field, the socalled Kodama vector field K^{ a }, such that K_{ a }G^{ ab } is divergence free [241]. In asymptotically flat spacetimes K^{ a } is timelike in the asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in fact, this is hypersurfaceorthogonal), but in general it is not a Killing vector. However, by ∇_{ a }(G^{ ab }K_{ b }) = 0 the vector field S^{ a } := G^{ ab }K_{ b } has a conserved flux on a spacelike hypersurface Σ. In particular, in the coordinate system (t, r, θ, φ) and line element above K^{ a } = exp[−(α + γ)](∂/∂t)^{ a }. If Σ is the solid ball of radius r, then the flux of S^{ a } is precisely the standard round sphere expression (26) for the 2sphere ∂Σ [278].
An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be given in terms of the energy function E(t, r) above (see for example [408, 262, 185]). In particular, criteria for the existence and the formation of trapped surfaces and the presence and the nature of the central singularity can be given by E(t, r).
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 [204], and the other is the concept of the small ellipsoids in some spacelike hypersurface, considered first by Woodhouse in [235]. 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 p, the centre, is a small given value, and the geometry of this sphere is characterized by data at this centre. Small ellipsoids are 2surfaces in a spacelike hypersurface with a more general shape.
To define the first, let p ∈ M be a point, and t^{ a } a future directed unit timelike vector at p. Let \({{\mathcal N}_p}: = \partial {I^ +}(p)\), the ‘future null cone of p in M’ (i.e. the boundary of the chronological future of p). Let l^{ a } be the future pointing null tangent to the null geodesic generators of \({{\mathcal N}_p}\) such that, at the vertex p, l^{ a }t_{ a } = 1. With this condition we fix the scale of the affine parameter r on the different generators, and hence by requiring r(p) =0 we fix the parameterization completely. Then, in an open neighbourhood of the vertex \(p,{{\mathcal N}_p}  \{p\}\) is a smooth null hypersurface, and hence for sufficiently small r the set \({{\mathcal S}_r}: = \{q \in M\vert r(q) = r\}\) is a smooth spacelike 2surface and homeomorphic to S^{2}. \({{\mathcal S}_r}\) is called a small sphere of radius r with vertex p. Note that the condition l^{ a }t_{ a } = 1 fixes the boost gauge.
Obviously, the same analysis can be repeated for any other quasilocal quantity. For quasilocal angular momentum \({Q_{\mathcal S}}\) has the structure \({\oint\nolimits_{\mathcal S}}({\partial _\mu}{g_{\alpha \beta}})r\,d{\mathcal S}\), while the area of \({\mathcal S}\) is \({\oint\nolimits_{\mathcal S}}\,d{\mathcal S}\). Then the leading term in the expansion of the angular momentum is r^{4} and r^{6} order in nonvacuum and vacuum, respectively, while the first nontrivial correction to the area 4πr^{2} is of order r^{4} and r^{6} in nonvacuum and vacuum, respectively.
On the small geodesic sphere \({{\mathcal S}_r}\) of radius r in the given spacelike hypersurface Σ one can introduce the complex null tangents m^{ a } and \({{\bar m}^a}\) above, and if t^{ a } is the future pointing unit normal of Σ and v^{ a } the outward directed unit normal of \({{\mathcal S}_r}\) in Σ, then we can define l^{ a } := t^{ a } + v^{ a } and 2n^{ a } := t^{ a } − v^{ a }. Then \(\{{l^a},{n^a},{m^a},{{\bar m}^a}\}\) is a NewmanPenrose complex null tetrad, and the relevant GHP equations can be solved for the spin coefficients in terms of the curvature components at p.
The small ellipsoids are defined as follows [235]. If f is any smooth function on Σ with a nondegenerate minimum at p ∈ Σ with minimum value f(p) = 0, then, at least on an open neighbourhood U of p in Σ the level surfaces \({{\mathcal S}_r}: = \{q \in \Sigma \vert 2f(q) = {r^2}\}\) are smooth compact 2surfaces homeomorphic to \({S^2}\). Then, in the r → 0 limit, the surfaces \({{\mathcal S}_r}\) look like small nested ellipsoids centred in p. The function f is usually ‘normalized’ so that h_{ ab }D_{ a }D_{ b }f∣_{ p } = − 3.
4.2.3 Large spheres near the spatial infinity
Near spatial infinity we have the a priori 1/r and 1/r^{2} falloff for the 3metric h_{ ab } and extrinsic curvature χ_{ ab }, respectively, and both the evolution equations of general relativity and the conservation equation T^{ ab }_{;b} = 0 for the matter fields preserve these conditions. The spheres \({{\mathcal S}_r}\) of coordinate radius r in Σ are called large spheres if the values of r are large enough such that the asymptotic expansions of the metric and extrinsic curvature are legitimate^{7}. Introducing some coordinate system, e.g. the complex stereographic coordinates, on one sphere and then extending that to the whole Σ along the normals v^{ a } of the spheres, we obtain a coordinate system \((r,\varsigma, \bar \varsigma)\) on Σ. Let \(\varepsilon _{\bf{A}}^A = \{{o^A},{\iota ^A}\}, {\bf{A}} = 0,1\), A = 0, 1, be a GHP spinor dyad on Σ adapted to the large spheres in such a way that \({m^a}: = {o^A}{{\bar \iota}^{{A{\prime}}}}\) and \({{\bar m}^a}: = {\iota ^A}{{\bar o}^{{A{\prime}}}}\) are tangent to the spheres and \({t^a} = {1 \over 2}{o^A}{{\bar o}^{{A{\prime}}}} + {\iota ^A}{{\bar \iota}^{{A{\prime}}}}\), the future directed unit normal of Σ. These conditions fix the spinor dyad completely, and, in particular, \({\upsilon ^a} = {1 \over 2}{o^A}{{\bar o}^{{A{\prime}}}} + {\iota ^A}{{\bar \iota}^{{A{\prime}}}}\), the outward directed unit normal to the spheres tangent to Σ.
The falloff conditions yield that the spin coefficients tend to their flat spacetime value like 1/r^{2} and the curvature components to zero like 1/r^{3}. Expanding the spin coefficients and curvature components as power series of 1/r, one can solve the field equations asymptotically (see [48, 44] for a different formalism). However, in most calculations of the large sphere limit of the quasilocal 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 nontrivial expansion coefficients.
Using the flat background metric _{0}h_{ ab } and the corresponding derivative operator _{0}D_{ e } we can define a spinor field _{0}λ_{ A } to be constant if _{0}D_{e0}λ_{ A } = 0. Obviously, the constant spinors form a two complex dimensional vector space. Then by the falloff properties \({D_{e0}}{\lambda _A} = {\mathcal O}({r^{ 2}})\). Hence we can define the asymptotically constant spinor fields to be those λ_{ B } that satisfy \({D_e}{\lambda _A} = {\mathcal O}({r^{ 2}})\), where D_{ e } is the intrinsic LeviCivita derivative operator. Note that this implies that, with the notations of Equation (25), all the chiral irreducible parts, \({\Delta ^ +}\lambda, {\Delta ^ }\lambda, {{\mathcal T}^ +}\lambda\) and \({{\mathcal T}^ }\lambda\), of the derivative of the asymptotically constant spinor field λ_{ A } are \({\mathcal O}({r^{ 2}})\).
4.2.4 Large spheres near null infinity
Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [300, 301, 302, 313] (see also [151]), i.e. the physical spacetime can be conformally compactified by an appropriate boundary ℐ^{+}. Then future null infinity ℐ^{+} will be a null hypersurface in the conformally rescaled spacetime. Topologically it is \({\mathbb R} \times {S^2}\), and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of ℐ^{+} is the metric of the unit sphere. Fixing such a slice \({{\mathcal S}_0}\) (called ‘the origin cut of ℐ^{+}’) the points of ℐ^{+} can be labeled by a null coordinate, namely the affine parameter u ∈ ℝ along the null geodesic generators of ℐ^{+} measured from \({{\mathcal S}_0}\) and, for example, the familiar complex stereographic coordinates \((\varsigma, \bar \varsigma) \in {S^2}\), defined first on the unit sphere \({{\mathcal S}_0}\) and then extended in a natural way along the null generators to the whole ℐ^{+}. Then any other cut S of ℐ^{+} can be specified by a function \(u = f(\varsigma, \bar \varsigma)\). In particular, the cuts \({{\mathcal S}_u}: = \{u = {\rm{const}}.\}\) are obtained from \({{\mathcal S}_0}\) by a pure time translation.
The coordinates \((u,\varsigma, \bar \varsigma)\) can be extended to an open neighbourhood of ℐ^{+} in the spacetime in the following way. Let \({{\mathcal N}_u}\) be the family of smooth outgoing null hypersurfaces in a neighbourhood of ℐ^{+} such that they intersect the null infinity just in the cuts \({{\mathcal S}_u}\), i.e. \({{\mathcal N}_u} \cap {\mathscr I}^+ = {{\mathcal S}_u}\). Then let r be the affine parameter in the physical metric along the null geodesic generators of \({{\mathcal N}_u}\). Then (u, r, ζ, \({\bar \varsigma}\)) forms a coordinate system. The u = const., r = const. 2surfaces \({{\mathcal S}_{u,r}}\) (or simply \({{\mathcal S}_r}\) if no confusion can arise) are spacelike topological 2spheres, which are called large spheres of radius r near future null infinity. Obviously, the affine parameter r is not unique, its origin can be changed freely: \(\bar r: = r + g(u,\varsigma, \bar \varsigma)\)) is an equally good affine parameter for any smooth g. Imposing certain additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bonditype coordinate system’^{8}. In many of the large sphere calculations of the quasilocal quantities the large spheres should be assumed to be large spheres not only in a general null, but in a Bonditype 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 [290, 289, 84].
In addition to the coordinate system we need a NewmanPenrose null tetrad, or rather a GHP spinor dyad, \(\varepsilon _{\bf{A}}^A = \{{o^A},{\iota ^A}\}, {\bf{A}} = 0,1\), on the hypersurfaces \({{\mathcal N}_u}\). (Thus boldface indices are referring to the GHP spin frame.) It is natural to choose o^{ A } such that \({l^a}: = {o^A}{{\bar o}^{{A{\prime}}}}\) be the tangent (∂/∂r)^{ a } of the null geodesic generators of \({{\mathcal N}_u}\), and o^{ A } itself be constant along l^{ a }. Newman and Unti [290] chose ι^{ A } to be parallel propagated along l^{ a }. This choice yields the vanishing of a number of spin coefficients (see for example the review [289]). The asymptotic solution of the EinsteinMaxwell equations as a series of 1/r in this coordinate and tetrad system is given in [290, 134, 312], where all the nonvanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the uderivative \({{\dot \sigma}^0}\) of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces \({{\mathcal N}_u}\).
From the point of view of the large sphere calculations of the quasilocal 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’ r, i.e. to require \({m^a}: = {o^A}{{\bar \iota}^{{A{\prime}}}}\) and \({{\bar m}^a}: = {\iota ^A}{{\bar o}^A}{\prime}\) to be tangents of the spheres. This can be achieved by an appropriate null rotation of the NewmanUnti basis about the spinor o^{ A }. 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 EinsteinMaxwell system by Shaw [338].
In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [83]. Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i.e. when \({{\dot \sigma}^0} = 0\). In the language of Section 4.1.7, this definition would be lim_{r→∞} rΔ^{+}λ = 0, lim_{r→∞} rΔ^{−}λ = 0, \({\lim\nolimits_{r \rightarrow \infty}}r{{\mathcal T}^ +}\lambda = 0\) and \({\lim\nolimits_{r \rightarrow \infty}}r{{\mathcal T}^ }\lambda = 0\). However, as Bramson showed [83], half of these conditions can be imposed. Namely, at future null infinity \({{\mathcal C}^ +}\lambda : = ({\Delta ^ +} \oplus {{\mathcal T}^ })\lambda = 0\) (and at past null infinity \({{\mathcal C}^ }\lambda : = ({\Delta ^ } \oplus {{\mathcal T}^ +})\lambda = 0)\) can always be imposed asymptotically, and it has two linearly independent solutions \(\lambda _A^{\underline A},\underline A = 0,1\), on ℐ^{+} (or on ℐ^{}, respectively). The space \({\bf{S}}_\infty ^{\underline A}\) of its solutions turns out to have a natural symplectic metric \({\varepsilon _{\underline A \underline B}}\), and we refer to \({\bf{S}}_\infty ^{\underline A},{\varepsilon _{\underline A \underline B}}\) as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations \({\lim\nolimits_{r \rightarrow \infty}}r{{\mathcal C}^ \pm}\lambda = 0\) the future/past asymptotic twistor equations. At ℐ^{+} asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form \({K^{\underline A {{\underline A}{\prime}}}}\lambda _{\underline A}^A\bar \lambda _{{{\underline A}{\prime}}}^{{A{\prime}}} = {K^{\underline A {{\underline A}{\prime}}}}\lambda _{\underline A}^1\bar \lambda _{{{\underline A}{\prime}}}^{{1{\prime}}}{\iota ^A}{{\bar \iota}^{{A{\prime}}}}\) for some constant Hermitian matrix \({K^{\underline A {{\underline A}{\prime}}}}\). Similarly, (apart from the proper supertranslation content) the components of the antiselfdual part of the boostrotation BMS vector fields are \( \sigma _{\bf{i}}^{\underline A \underline B}\lambda _{\underline A}^1\lambda _{\underline B}^1\), where \({\sigma _{\rm{i}}^{\underline A \underline B}}\) are the standard SU(2) Pauli matrices (divided by \(\sqrt 2\) [363]. Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from Δ^{+}, Δ^{−}, \({{\mathcal T}^ +}\), and \({{\mathcal T}^}\), too. In the present review we use only the fact that asymptotic spinors can be introduced as antiholomorphic spinors (see also Section 8.2.1), i.e. the solutions of \({{\mathcal H}^ }\lambda : = ({\Delta ^ } \oplus {{\mathcal T}^ })\lambda = 0\) (and at past null infinity as holomorphic spinors), and as special solutions of the 2surface twistor equation \({\mathcal T}\lambda : = ({{\mathcal T}^ +} \oplus {{\mathcal T}^ })\lambda = 0\) (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [363].
4.2.5 Other special situations
In the weak field approximation of general relativity [382, 22, 387, 313, 227] the gravitational field is described by a symmetric tensor field h_{ ab } on Minkowski spacetime (ℝ^{4}, \(g_{ab}^0\)), and the dynamics of the field h_{ ab } is governed by the linearized Einstein equations, i.e. essentially the wave equation. Therefore, the tools and techniques of the Poincaréinvariant field theories, in particular the NoetherBelinfanteRosenfeld procedure outlined in Section 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 energymomentum tensor of the field h_{ ab } is essentially the second order term in the Einstein tensor of the metric \({g_{ab}}: = g_{ab}^0 + {h_{ab}}\). Thus in the linear approximation the field h_{ ab } does not contribute to the global energymomentum and angular momentum of the matter + gravity system, and hence these quantities have the form (5) with the linearized energymomentum tensor of the matter fields. However, as we will see in Section 7.1.1, this energymomentum and angular momentum can be reexpressed as a charge integral of the (linearized) curvature [349, 206, 313].
ppwaves spacetimes are defined to be those that admit a constant null vector field L^{ a }, and they are interpreted as describing pure planefronted gravitational waves with parallel rays. If matter is present then it is necessarily pure radiation with wavevector L^{ a }, i.e. T^{ ab }L_{ b } = 0 holds [243]. A remarkable feature of the ppwave 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 quasilocal observables this is a particularly useful and natural standpoint. If a ppwave spacetime admits an additional spacelike Killing vector K^{ a } with closed S^{1} orbits, i.e. it is cyclically symmetric too, then L^{ a } and K^{ a } are necessarily commuting and are orthogonal to each other, because otherwise an additional timelike Killing vector would also be admitted [351].
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, axisymmetric spacetime, the significance of these spacetimes is obvious. It is conjectured that this final state is described by the KerrNewman (either outer or black hole) solution with some welldefined mass, angular momentum and electric charge parameters [387]. Thus axisymmetric 2surfaces in these solutions may provide domains which are general enough but for which the quasilocal quantities are still computable. According to a conjecture by Penrose [305], the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the KerrNewman black hole this area is \(4\pi (2{m^2}  {e^2} + 2m\sqrt {{m^2}  {e^2}  {a^2}})\). Thus, particularly interesting 2surfaces in these spacetimes are the spacelike cross sections of the event horizon [62].
There is a welldefined notion of total energymomentum not only in the asymptotically flat, but even in the asymptotically antideSitter spacetimes too. This is the AbbottDeser energy [1], whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [161]. (In the presence of matter fields, e.g. a selfinteracting scalar field, the falloff properties of the metric can be weakened such that the ‘charges’ defined at infinity and corresponding to the asymptotic symmetry generators remain finite [198].) The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically antideSitter spacetimes and to study their general, basic properties in [27]. A comparison and analysis of the various definitions of mass for asymptotically antideSitter metrics is given in [117]. Thus it is natural to ask whether a specific quasilocal energymomentum expression is able to reproduce the AbbottDeser energymomentum in this limit or not.
4.3 On lists of criteria of reasonableness of the quasilocal quantities
In the literature there are various, more or less ad hoc, ‘lists of criteria of reasonableness’ of the quasilocal quantities (see for example [131, 111]). However, before discussing them, it seems useful to formulate first some general principles that any quasilocal quantity should satisfy.
4.3.1 General expectations
 1.
The quasilocal quantities that are 2surface observables should depend only on the 2surface data, but they cannot depend e.g. on the way that the various geometric structures on \({\mathcal S}\) are extended off the 2surface. There seems to be no a priori reason why the 2surface would have to be restricted to have spherical topology. Thus, in the ideal case, the general construction of the quasilocal energymomentum and angular momentum should work for any closed orientable spacelike 2surface.
 2.
It is desirable to derive the quasilocal energymomentum and angular momentum as the charge integral (Lagrangian interpretation) and/or as the value of the Hamiltonian on the constraint surface in the phase space (Hamiltonian interpretation). If they are introduced in some other way, they should have a Lagrangian and/or Hamiltonian interpretation.
 3.
These quantities should correspond to the ‘quasisymmetries’ of the 2surface. In particular, the quasilocal energymomentum should be expected to be in the dual of the space of the ‘quasitranslations’, and the angular momentum in the dual of the space of the ‘quasirotations’.
To see that these conditions are nontrivial, let us consider the expressions based on the linkage integral (16). \({L_{\mathcal S}}[{\bf{K}}]\) does not satisfy the first part of Requirement 1. In fact, it depends on the derivative of the normal components of K^{ a } in the direction orthogonal to \({\mathcal S}\) for any value of the parameter α. Thus it depends not only on the geometry of \({\mathcal S}\) and the vector field K^{ a } given on the 2surface, but on the way in which K^{ a } is extended off the 2surface. Therefore, \({L_{\mathcal S}}[{\bf{K}}]\) is ‘less quasilocal’ than \({A_{\mathcal S}}[\omega ]\) or \({H_{\mathcal S}}[\lambda, \bar \mu ]\) introduced in Sections 7.2.1 and 7.2.2, respectively.
4.3.2 Pragmatic criteria
Since in certain special situations there are generally accepted definitions for the energymomentum and angular momentum, it seems reasonable to expect that in these situations the quasilocal 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 quasilocal quantities.
 1.1
The quasilocal energymomentum \(P_{\mathcal S}^{\underline a}\) must be a future pointing nonspacelike vector (assuming that the matter fields satisfy the dominant energy condition on some Σ for which \({\mathcal S} = \partial \Sigma\), and maybe some form of the convexity of \({\mathcal S}\) should be required) (‘positivity’).
 1.2
\(P_{\mathcal S}^{\underline a}\) must be zero iff D(Σ) is flat, and null iff D(Σ) has a ppwave geometry with pure radiation (‘rigidity’).
 1.3
\(P_{\mathcal S}^{\underline a}\) must give the correct weak field limit.
 1.4
\(P_{\mathcal S}^{\underline a}\) must reproduce the ADM, BondiSachs and AbbottDeser energymomenta in the appropriate limits (‘correct large sphere behaviour’).
 1.5For small spheres \(P_{\mathcal S}^{\underline a}\) must give the expected results (‘correct small sphere behaviour’):
 1.
\({4 \over 3}\pi {r^3}{T^{ab}}{t_b}\) in nonvacuum and
 2.
kr^{5}T^{ abcd }t^{ b }t^{ c }t^{ d } in vacuum for some positive constant k and the BelRobinson tensor T^{ abcd }.
 1.
 1.6
For round spheres \(P_{\mathcal S}^{\underline a}\) must yield the ‘standard’ round sphere expression.
 1.7
For marginally trapped surfaces the quasilocal mass \({m_{\mathcal S}}\) must be the irreducible mass \(\sqrt {{\rm{Area}}({\mathcal S})/16\pi {G^2}}\).
 2.1
\(J_{\mathcal S}^{\underline a \underline b}\) must give zero for round spheres.
 2.2
For 2surfaces with zero quasilocal mass the PauliLubanski spin should be proportional to the (null) energymomentum 4vector \(P_{\mathcal S}^{\underline a}\).
 2.3
\(J_{\mathcal S}^{\underline a \underline b}\) must give the correct weak field limit.
 2.4
\(J_{\mathcal S}^{\underline a \underline b}\) must reproduce the generally accepted spatial angular momentum at the spatial infinity, and in stationary spacetimes it should reduce to the ‘standard’ expression at the null infinity as well (‘correct large sphere behaviour’).
 2.5
For small spheres the antiselfdual part of \(J_{\mathcal S}^{\underline a \underline b}\), defined with respect to the centre of the small sphere (the ‘vertex’ in Section 4.2.2) is expected to give \({4 \over 3}\pi {r^3}{T_{cd}}{t^c}(r{\varepsilon ^{D({A_t}B){D{\prime}}}})\) in nonvacuum and \(C{r^5}{T_{cdef}}{t^c}{t^d}{t^e}(r{\varepsilon ^{F({A_t}B){F{\prime}}}})\) in vacuum for some constant C (‘correct small sphere behaviour’).
4.3.3 Incompatibility of certain ‘natural’ expectations
As Eardley noted in [131], probably no quasilocal energy definition exists which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasilocal energy definition which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed FriedmannRobertsonWalker or the Ω_{ M,m } 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 quasilocal 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 timesymmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general nonstationary 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 Σ of Section 4.3.2. Actually, the root of the nonmonotonicity is the fact that the quasilocal energy is a 2surface observable in the sense of Expectation 1 in Section 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 2surface 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 quasilocal 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 quasilocal energy is associated with a compact three dimensional dornain, 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 quasilocal energymomentum is associated with 2surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression (26) in the closed FriedmannRobertsonWalker spacetime, or, more generally, the DouganMason energymomentum (see Section 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 quasilocalization of the familiar ADM mass is due to Bartnik [39, 38]. His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let Σ be a compact, connected 3manifold with connected boundary \({\mathcal S}\), and let h_{ ab } be a (negative definite) metric and χ_{ ab } a symmetric tensor field on Σ such that they, as an initial data set, satisfy the dominant energy condition: If 16πG_{ μ } := R + χ^{2} − χ_{ ab }χ^{ ab } and 8πGj^{ a } := D_{ b }(χ^{ ab } − χh^{ ab }), then μ ≥ (− j_{ a }j^{ a })^{1/2}. For the sake of simplicity we denote the triple (Σ, h_{ ab }, χ_{ ab }) by Σ. Then let us consider all the possible asymptotically flat initial data sets \((\hat \Sigma, {{\hat h}_{ab}},{{\hat \chi}_{ab}})\) with a single asymptotic end, denoted simply by \({\hat \Sigma}\), which satisfy the dominant energy condition, have finite ADM energy and are extensions of Σ above through its boundary \({\mathcal S}\). The set of these extensions will be denoted by \(\varepsilon (\Sigma)\). By the positive energy theorem \({\hat \Sigma}\) has nonnegative ADM energy \({E_{{\rm{ADM}}}}(\hat \Sigma)\), which is zero precisely when \({\hat \Sigma}\) is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies, \(\inf \left\{{{E_{{\rm{ADM}}}}(\hat \Sigma)\vert \hat \Sigma \in \varepsilon (\Sigma)} \right\}\), where the infimum is taken on \(\varepsilon (\Sigma)\). Obviously, by the nonnegativity of the ADM energies this infimum exists and is nonnegative, and it is tempting to define the quasilocal mass of Σ by this infimum^{9}. However, it is easy to see that, without further conditions on the extensions of (Σ, h_{ ab }, χ_{ ab }), this infimum is zero. In fact, Σ can be extended to an asymptotically flat initial data set \({\hat \Sigma}\) with arbitrarily small ADM energy such that \({\hat \Sigma}\) contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and Σ. In particular, in the ‘Ω_{ M,m }spacetime’, discussed in Section 4.2.1 on the round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily large roundsphere mass M/G) has an asymptotically flat extension, the Ω_{ M,m }spacetime itself, with arbitrarily small ADM mass m/G.
Of course, to rule out this limitation, one can modify the original definition by considering the set \({{\tilde \varepsilon}_0}({\mathcal S})\) of asymptotically flat Riemannian geometries \(\hat \Sigma = (\hat \Sigma, {{\hat h}_{ab}})\) (with nonnegative scalar curvature, finite ADM energy and with no stable minimal surface) which contain \(({\mathcal S},{q_{ab}})\) as an isometrically embedded Riemannian submanifold, and define \({{\tilde m}_B}({\mathcal S})\) by Equation (36) with \({{\tilde \varepsilon}_0}({\mathcal S})\) instead of ɛ_{0}(Σ). Obviously, this \({{\tilde m}_B}({\mathcal S})\) could be associated with a larger class of 2surfaces than the original m_{B}(Σ) to compact 3manifolds, and \(0 \leq {{\tilde m}_B}(\partial \Sigma) \leq {m_B}(\Sigma)\) holds.
In [208, 41] the set \({{\tilde \varepsilon}_0}({\mathcal S})\) was allowed to include extensions \({\hat \Sigma}\) of Σ having boundaries as compact outermost horizons, whenever the corresponding ADM energies are still nonnegative [159], and hence m_{B}(Σ) is still welldefined and nonnegative. (For another definition for \({{\mathcal E}_0}(\Sigma)\) allowing horizons in the extensions but excluding them between Σ and the asymptotic end, see [87] and Section 5.2 below.)
5.1.2 The main properties of m_{B}(Σ)
The first immediate consequence of Equation (36) is the monotonicity of the Bartnik mass: If Σ_{1} ⊂ Σ_{2}, then \({{\mathcal E}_0}({\Sigma _2}) \subset {{\mathcal E}_0}({\Sigma _1})\), and hence m_{B}(Σ_{1}) ≤ m_{B}(Σ_{2}). Obviously, by definition (36) one has \({m_{\rm{B}}}(\Sigma) \leq {m_{{\rm{ADM}}}}(\hat \Sigma)\) for any \(\hat \Sigma \in {{\mathcal E}_0}(\Sigma)\). Thus if m is any quasilocal mass functional which is larger than m_{B} (i.e. which assigns a nonnegative real to any Σ such that m(Σ) ≥ m_{B}(Σ) for any allowed Σ), furthermore if \(m(\Sigma) \leq {m_{{\rm{ADM}}}}(\hat \Sigma)\) for any \(\hat \Sigma \in {{\mathcal E}_0}(\Sigma)\), then by the definition of the infimum in Equation (36) one has m_{B}(Σ) ≥ m(Σ) − ε ≥ m_{B}(Σ) − ɛ for any ɛ > 0. Therefore, m_{B} is the largest mass functional satisfying \({m_{\rm{B}}}(\Sigma) \leq {m_{{\rm{ADM}}}}(\hat \Sigma)\) for any \(\hat \Sigma \in {{\mathcal E}_0}(\Sigma)\). Another interesting consequence of the definition of m_{B}, due to W. Simon, is that if \({\hat \Sigma}\) is any asymptotically flat, time symmetric extension of Σ with nonnegative scalar curvature satisfying \({m_{{\rm{ADM}}}}(\hat \Sigma) < {m_{\rm{B}}}(\Sigma)\), then there is a black hole in \({\hat \Sigma}\) in the form of a minimal surface between Σ and the infinity of \({\hat \Sigma}\) (see for example [41]).
As we saw, the Bartnik mass is nonnegative, and, obviously, if Σ is flat (and hence is a data set for the flat spacetime), then m_{B}(Σ) = 0. The converse of this statement is also true [208]: If m_{B}(Σ) = 0, then Σ is locally flat. The Bartnik mass tends to the ADM mass [208]: If \((\hat \Sigma, {{\hat h}_{ab}})\) is an asymptotically flat Riemannian 3geometry with nonnegative scalar curvature and finite ADM mass \({m_{{\rm{ADM}}}}(\hat \Sigma)\), and if {Σ_{ n }}, n ∈ ℕ, is a sequence of solid balls of coordinate radius n in \({\hat \Sigma}\), then \({\lim\nolimits _{n \rightarrow \infty}}{m_{\rm{B}}}({\Sigma _n}) = {m_{{\rm{ADM}}}}(\hat \Sigma)\). The proof of these two results is based on the use of the Hawking energy (see Section 6.1), by means of which a positive lower bound for m_{B}(Σ) can be given near the nonflat points of Σ. In the proof of the second statement one must use the fact that the Hawking energy tends to the ADM energy, which, in the timesymmetric 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 [208]: Let Σ be a spherically symmetric Riemannian 3geometry with spherically symmetric boundary \({\mathcal S}: = \partial \Sigma\). One can form its ‘standard’ roundsphere energy \(E({\mathcal S})\) (see Section 4.2.1), and take its spherically symmetric asymptotically flat vacuum extension \({{\hat \Sigma}_{{\rm{SS}}}}\) (see [39, 41]). By the Birkhoff theorem the exterior part of \({{\hat \Sigma}_{{\rm{SS}}}}\) is a part of a t = const. hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just \(E({\mathcal S})\). Then any asymptotically flat extension \({\hat \Sigma}\) of Σ can also be considered as (a part of) an asymptotically flat timesymmetric hypersurface with minimal surface, whose area is \(16\pi {G^2}{E_{{\rm{ADM}}}}({{\hat \Sigma}_{{\rm{SS}}}})\). Thus by the Riemannian Penrose inequality [208] \({E_{{\rm{ADM}}}}(\hat \Sigma) \geq {E_{{\rm{ADM}}}}({{\hat \Sigma}_{{\rm{SS}}}}) = E({\mathcal S})\). Therefore, the Bartnik mass of Σ is just the ‘standard’ round sphere expression \(E({\mathcal S})\).
5.1.3 The computability of the Bartnik mass
Since for any given Σ the set \({{\mathcal E}_0}(\Sigma)\) 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 (Σ, h_{ ab }). Without some computational method the potentially useful properties of m_{B}(Σ) would be lost from the working relativist’s arsenal.
Such a computational method might be based on a conjecture of Bartnik [39, 41]: The infimum in definition (36) of the mass m_{B}(Σ) is realized by an extension \((\hat \Sigma, {{\hat h}_{ab}})\) of (Σ, h_{ ab }) such that the exterior region, \((\hat \Sigma  \Sigma, {{\hat h}_{ab}}{\vert _{\hat \Sigma  \Sigma}})\), is static, the metric is Lipschitzcontinuous across the 2surface \(\partial \Sigma \subset \hat \Sigma\), and the mean curvatures of ∂Σ of the two sides are equal. Therefore, to compute m_{B} for a given (Σ, h_{ ab }), one should find an asymptotically flat, static vacuum metric ĥ_{ ab } satisfying the matching conditions on ∂Σ, and the Bartnik mass is the ADM mass of ĥ_{ ab }. As Corvino showed [119], if there is an allowed extension \({\hat \Sigma}\) of Σ for which \({m_{{\rm{ADM}}}}(\hat \Sigma) = {m_{\rm{B}}}(\Sigma)\), then the extension \(\hat \Sigma  \overline \Sigma\) is static; furthermore, if Σ_{1} ⊂ Σ_{2}, m_{B}(Σ_{1}) = m_{B}(Σ_{2}) and Σ_{2} has an allowed extension \({\hat \Sigma}\) for which \({m_{\rm{B}}}({\Sigma _2}) = {m_{{\rm{ADM}}}}(\hat \Sigma)\), then \({\Sigma _2}  \overline {{\Sigma _1}}\) 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 [267] for geometries (Σ, h_{ ab }) 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 (Σ, h_{ ab }) determines this exterior metric uniquely [41]. He conjectures [39, 41] that a similar computation method can be found for the mass \({m_{\rm{B}}}({\mathcal S})\), defined in Equation (37), too, where the exterior metric should be stationary. This second conjecture is also supported by partial results [120]: If (Σ, h_{ ab }, χ_{ ab }) 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 m_{B}(Σ) one can construct admissible extensions of (Σ, h_{ ab }) in the form of the metrics in quasispherical form [40]. If the boundary ∂Σ is a metric sphere of radius r with nonnegative mean curvature k, then m_{B}(Σ) can be estimated from above in terms of r and k.
5.2 Bray’s modifications
Another, slightly modified definition for the quasilocal mass was suggested by Bray [87, 90]. Here we summarize his ideas.
Let Σ = (Σ, h_{ ab }, χ_{ ab }) 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 Σ. Let \({\mathcal S}\) be any fixed 2surface in Σ which encloses all the asymptotic ends except one, say the ith (i.e. let \({\mathcal S}\) be homologous to a large sphere in the ith asymptotic end). The outside region with respect to \({\mathcal S}\), denoted by \(O({\mathcal S})\), will be the subset of Σ containing the ith asymptotic end and bounded by \(O({\mathcal S})\), while the inside region, \(I({\mathcal S})\), is the (closure of) \(\Sigma  O({\mathcal S})\). Next Bray defines the ‘extension’ \({{\hat \Sigma}_e}\) of \({\mathcal S}\) by replacing \(O({\mathcal S})\) by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Similarly, the ‘fillin’ \({{\hat \Sigma}_{\rm{f}}}\) of \({\mathcal S}\) is obtained from Σ by replacing \(I({\mathcal S})\) by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Finally, the surface \({\mathcal S}\) will be called outerminimizing if for any closed 2surface \({\tilde {\mathcal S}}\) enclosing \({\mathcal S}\) one has \({\rm{Area}}({\mathcal S}) \leq {\rm{Area}}(\tilde {\mathcal S})\).
A simple consequence of the definitions is the monotonicity of these masses: If \({{\mathcal S}_2}\) and \({{\mathcal S}_1}\) are outerminimizing 2surfaces such that \({{\mathcal S}_2}\) encloses \({{\mathcal S}_2}\), then \({m_{{\rm{in}}}}({{\mathcal S}_2}) \geq {m_{{\rm{in}}}}({{\mathcal S}_1})\) and \({m_{{\rm{out}}}}({{\mathcal S}_2}) \geq {m_{{\rm{out}}}}({{\mathcal S}_1})\). Furthermore, if the Penrose inequality holds (for example in a timesymmetric data set, for which the inequality has been proved), then for outerminimizing surfaces \({m_{{\rm{out}}}}({\mathcal S}) \geq {m_{{\rm{in}}}}({\mathcal S})\) [87, 90]. Furthermore, if Σ_{ i } is a sequence such that the boundaries ∂Σ_{ i } shrink to a minimal surface \({\mathcal S}\), then the sequence m_{out}(∂Σ_{ i }) tends to the irreducible mass \(\sqrt {{\rm{Area}}({\mathcal S})/(16\pi {G^2})}\) [41]. Bray defines the quasilocal mass of a surface not simply to be a number, but the whole closed interval \([{m_{{\rm{in}}}}({\mathcal S}),{m_{{\rm{out}}}}({\mathcal S})]\). If \({\mathcal S}\) encloses the horizon in the Schwarzschild data set, then the inner and outer masses coincide, and Bray expects that the converse is also true: If \({m_{{\rm{in}}}}({\mathcal S}) = {m_{{\rm{out}}}}({\mathcal S})\) then \({\mathcal S}\) can be embedded into the Schwarzschild spacetime with the given 2surface data on \({\mathcal S}\) [90].
6 The Hawking Energy and its Modifications
6.1 The Hawking energy
6.1.1 The definition
The Hawking energy has the following clear physical interpretation even in a general spacetime, and, in fact, E_{H} can be introduced in this way. Starting with the rough idea that the massenergy surrounded by a spacelike 2sphere \({\mathcal S}\) should be the measure of bending of the ingoing and outgoing light rays orthogonal to \({\mathcal S}\), and recalling that under a boost gauge transformation l^{ a } ↦ αl^{ a }, n^{ a } ↦ α^{−1}n^{ a } the convergences ρ and ρ′ transform as ρ ↦ αρ and ρ′ ↦ α^{−1} ρ′, respectively, the energy must have the form \(C + D{\oint\nolimits_{\mathcal S}}\rho {\rho {\prime}}\,d{\mathcal S}\), where the unspecified parameters C and D can be determined in some special situations. For metric 2spheres of radius r in the Minkowski spacetime, for which ρ = −1/r and ρ′ = 1/2r, we expect zero energy, thus D = C/2(π). For the event horizon of a Schwarzschild black hole with mass parameter m, for which ρ = 0 = ρ′, we expect m/G, which can be expressed by the area of \({\mathcal S}\). Thus \({C^2} = {\rm{Area}}({\mathcal S})/(16\pi {G^2})\), and hence we arrive at Equation (38).
6.1.2 The Hawking energy for spheres
Obviously, for round spheres E_{H} 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) ρ = 0, the Hawking energy of its spacelike spherical cross sections \({\mathcal S}\) is \(\sqrt {{\rm{Area}}({\mathcal S})/(16\pi {G^2})}\). In particular, for the event horizon of a KerrNewman black hole it is just the familiar irreducible mass \(\sqrt {2{m^2}  {e^2} + 2m\sqrt {{m^2}  {e^2}  {a^2}}}/(2G)\).
For a small sphere of radius r with centre p ∈ M in nonvacuum spacetimes it is \({{4\pi} \over 3}{r^3}{T_{ab}}{t^a}{t^b}\) while in vacuum it is \({2 \over {45G}}{r^5}{T_{abcd}}{t^a}{t^b}{t^c}{t^d}\), where T_{ ab } is the energymomentum tensor and T_{ abcd } is the BelRobinson tensor at p [204]. 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 E_{H} must be higher than r^{3}. Then even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the r^{ k } order term in the power series expansion of E_{H} is (k−1). 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 r^{5}, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres E_{H} is positive definite both in nonvacuum (provided the matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in particular, that E_{H} should be interpreted as energy rather than as mass: For small spheres in a ppwave spacetime E_{H} is positive, while, as we saw this for the matter fields in Section 2.2.3, a mass expression could be expected to be zero. (We will see in Sections 8.2.3 and 13.5 that, for the DouganMason energymomentum, the vanishing of the mass characterizes the ppwave metrics completely.)
Using the second expression in Equation (38) it is easy to see that at future null infinity E_{H} tends to the BondiSachs energy. A detailed discussion of the asymptotic properties of E_{H} near null infinity, both for radiative and stationary spacetimes is given in [338, 340]. Similarly, calculating E_{H} 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) 2surface \({\mathcal S}\) the integral \({\oint\nolimits_{\mathcal S}}\rho {\rho {\prime}}d{\mathcal S}\) could be less than −2π. Indeed, in flat spacetime E_{H} is proportional to \({\oint\nolimits_{\mathcal S}}(\sigma {\sigma {\prime}} + \bar \sigma {{\bar \sigma}{\prime}})d{\mathcal S}\) by the Gauss equation. For topologically spherical 2surfaces in the t = const. spacelike hyperplane of Minkowski spacetime σσ′ is real and nonpositive, and it is zero precisely for metric spheres, while for 2surfaces in the r = const. timelike cylinder σσ′ is real and nonnegative, and it is zero precisely for metric spheres^{10}. If, however, \({\mathcal S}\) is ‘round enough’ (not to be confused with the round spheres in Section 4.2.1), which is some form of a convexity condition, then E_{H} behaves nicely [111]: \({\mathcal S}\) will be called round enough if it is a submanifold of a spacelike hypersurface Σ, and if among the 2dimensional surfaces in Σ which enclose the same volume as \({\mathcal S}\) does, \({\mathcal S}\) has the smallest area. Then it is proven by Christodoulou and Yau [111] that if \({\mathcal S}\) is round enough in a maximal spacelike slice Σ on which the energy density of the matter fields is nonnegative (for example if the dominant energy condition is satisfied), then the Hawking energy is nonnegative.
Although the Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of 2surfaces. Hawking considered oneparameter families of spacelike 2surfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of E_{H} [171]. These calculations were refined by Eardley [131]. Starting with a weakly future convex 2surface \({\mathcal S}\) and using the boost gauge freedom, he introduced a special family \({{\mathcal S}_r}\) of spacelike 2surfaces in the outgoing null hypersurface \({\mathcal N}\), where r will be the luminosity distance along the outgoing null generators. He showed that \({E_{\rm{H}}}({{\mathcal S}_r})\) is nondecreasing with r, provided the dominant energy condition holds on \({\mathcal N}\). Similarly, for weakly past convex \({\mathcal S}\) and the analogous family of surfaces in the ingoing null hypersurface \({E_{\rm{H}}}({{\mathcal S}_r})\) is nonincreasing. Eardley also considered a special spacelike hypersurface, filled by a family of 2surfaces, for which \({E_{\rm{H}}}({{\mathcal S}_r})\) is nondecreasing. By relaxing the normalization condition l_{ a }n^{ a } = 1 for the two null normals to l_{ a }n^{ a } = exp(f) for some \(f:{\mathcal S} \rightarrow \mathbb R\), Hayward obtained a flexible enough formalism to introduce a doublenull foliation (see Section 11.2 below) of a whole neighbourhood of a mean convex 2surface by special mean convex 2surfaces [182]. (For the more general GHP formalism in which l_{ a }n^{ a } is not fixed, see [312].) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these 2surfaces is nondecreasing in the outgoing, and nonincreasing 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 [145]. If \({\mathcal S}\) is a weakly future and past convex 2surface, then q^{ a } := 2Q^{ a }/(Q_{ b }Q^{ b }) = − [1/(2ρ)]l^{ a } − [1/(2ρ′)]n^{ a } is an outward directed spacelike normal to \({\mathcal S}\). Here Q_{ b } is the trace of the extrinsic curvature tensor: \({Q_b}: = {Q^a}_{ab}\) (see Section 4.1.2). Starting with a single weakly future and past convex 2surface, Frauendiener gives an argument for the construction of a oneparameter family \({{\mathcal S}_t}\) of 2surfaces being Liedragged along its own inverse mean curvature vector q^{ a }. 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 q^{ a } on the 3submanifold swept by \({{\mathcal S}_t}\)) exists, Frauendiener showed that the Hawking energy is nondecreasing along the vector field q^{ a } 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 [261] considered spacelike hypersurfaces with an inverse mean curvature flow of Geroch thereon (see Section 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 Section 13.2 only for a very brief summary.
6.1.4 Two generalizations
Hawking considered the extension of the definition of \({E_{\rm{H}}}({\mathcal S})\) to higher genus 2surfaces also by the second expression in Equation (38). Then in the expression analogous to the first one in Equation (38) the genus of \({\mathcal S}\) appears.
6.2 The Geroch energy
6.2.1 The definition
The calculation of the small sphere limit of the Geroch energy was saved by observing [204] that, by Equation (41), the difference of the Hawking and the Geroch energies is proportional to \(\sqrt {{\rm{Area}}({\mathcal S})} \times \oint {_{\mathcal S}{{({\chi _{ab}}{q^{ab}})}^2}d{\mathcal S}}\). Since, however, χ_{ ab }q^{ ab } — for the family of small spheres \({{\mathcal S}_r}\) — does not tend to zero in the r → 0 limit, in general this difference is \({\mathcal O}({r^3})\). It is zero if Σ is spanned by spacelike geodesics orthogonal to t^{ a } at p. Thus, for general Σ, the Geroch energy does not give the expected \({{4\pi} \over 3}{r^3}{T_{ab}}{t^a}{t^b}\) result. Similarly, in vacuum the Geroch energy deviates from the BelRobinson energy in r^{5} order even if Σ is geodesic at p.
Since \({E_{\rm{H}}}({\mathcal S}) \geq {E_{\rm{G}}}({\mathcal S})\) and since the Hawking energy tends to the ADM energy, the large sphere limit of \({E_{\rm{G}}}({\mathcal S})\) in an asymptotically flat Σ cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [150].
6.2.2 Monotonicity properties
 1.
its level surfaces, \({{\mathcal S}_t}: = \{q \in \Sigma \vert t(q) = t\}\), are homeomorphic to S^{2},
 2.
there is a point p ∈ Σ such that the surfaces \({{\mathcal S}_t}\) are shrinking to p in the limit t → −∞, and
 3.
they form a foliation of Σ − {p}.
The existence and the properties of the original inverse mean curvature foliation of (Σ, h_{ ab }) above were proven and clarified by Huisken and Ilmanen [207, 208], giving the first complete proof of the Riemannian Penrose inequality, and, as proved by Schoen and Yau [328], Jang’s quasilinear elliptic equation admits a global solution.
6.3 The Hayward energy
In the literature there is another modification of the Hawking energy, due to Hayward [183]. His suggestion is essentially \(I({\mathcal S})\) with the only difference that the integrands above contain an additional term, namely the square of the anholonomicity −ω_{ a }ω^{ a } (see Sections 4.1.8 and 11.2.1). However, we saw that ω_{ a } is a boost gauge dependent quantity, thus the physical significance of this suggestion is questionable unless a natural boost gauge choice, e.g. in the form of a preferred foliation, is made. (Such a boost gauge might be that given by the main extrinsic curvature vector Q_{ a } and \({{\tilde Q}_a}\) discussed in Section 4.1.2.) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [63, 65] seems to be gauge invariant, this is due only to an implicit gauge choice. The correct, general GHP form of the extra term is \( {\omega _a}{\omega ^a} = 2(\beta  {{\bar \beta}\prime})(\bar \beta  {\beta \prime})\). If, however, the GHP spinor dyad is fixed as in the large sphere or in the small sphere calculations, then \(\beta  {{\bar \beta}\prime} = \tau =  {{\bar \tau}\prime}\), and hence the extra term is, in fact, \(2\tau \bar \tau\).
Taking into account that \(\tau = {\mathcal O}({r^{ 2}})\) near the future null infinity (see for example [338]), it is immediate from the remark on the asymptotic behaviour of \(I({\mathcal S})\) above that the Hayward energy tends to the NewmanUnti instead of the BondiSachs energy at the future null infinity. The Hayward energy has been calculated for small spheres both in nonvacuum and vacuum [63]. In nonvacuum it gives the expected value \({{4\pi} \over 3}{r^3}{T_{ab}}{t^a}{t^b}\). However, in vacuum it is \( {8 \over {45G}}{r^5}{T_{abcd}}{t^a}{t^b}{t^c}{t^d}\), which is negative.
7 Penrose’s QuasiLocal EnergyMomentum and Angular Momentum
The construction of Penrose is based on twistortheoretical ideas, and motivated by the linearized gravity integrals for energymomentum and angular momentum. Since, however, twistortheoretical 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 [312, 313] by Penrose and Rindler on ‘Spinors and Spacetime’, especially volume 2, the very well readable book by Hugget and Tod [206] and the comprehensive review article [377] by Tod.
7.1 Motivations
7.1.1 How do the twistors emerge?
7.1.2 Twistor space and the kinematical twistor
7.2 The original construction for curved spacetimes
7.2.1 2surface twistors and the kinematical twistor
The 2surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation \({{\mathcal T}_{{E\prime}EA}}^B{\lambda _B} = 0\). By Equation (25) its GHP form is \({\mathcal T}\lambda : = ({{\mathcal T}^ +} \oplus {{\mathcal T}^ })\lambda = 0\), which is a first order elliptic system, and its index is 4(1 − g), where g is the genus of \({\mathcal S}\) [43]. Thus there are at least four (and in the generic case precisely four) linearly independent solutions to \({\mathcal T}\lambda = 0\) on topological 2spheres. However, there are ‘exceptional’ 2spheres for which there exist at least five linearly independent solutions [221]. For such ‘exceptional’ 2spheres (and for higher genus 2surfaces for which the twistor equation has only the trivial solution in general) the subsequent construction does not work. (The concept of quasilocal charges in YangMills theory can also be introduced in an analogous way [370]). The space \(T_{\mathcal S}^\alpha\) of the solutions to \({{\mathcal T}_{{E\prime}EA}}^B{\lambda _B} = 0\) is called the 2surface twistor space. In fact, in the generic case this space is 4complexdimensional, and under conformal rescaling the pair Z^{ α } = (λ^{ A }, iΔ_{ A′A }λ^{ A }) transforms like a valence 1 contravariant twistor. Z^{ α } is called a 2surface twistor determined by λ^{ A }. If \({\lambda ^A}\) is another generic 2surface with the corresponding 2surface twistor space \(T_{\mathcal S}^\alpha\), then although \(T_{\mathcal S}^\alpha\) and \(T_{\mathcal S}^\alpha\) are isomorphic as vector spaces, there is no canonical isomorphism between them. The kinematical twistor A_{ αβ } is defined to be the symmetric twistor determined by \({A_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {A_{\mathcal S}}[\lambda, \, \omega ]\) for any Z^{ α } = (λ^{ A }, iΔ_{ A′A }λ^{ A }) and W^{ α } = (ω^{ A }, iΔ_{ A′A }ω^{ A }) from \(T_{\mathcal S}^\alpha\). Note that \({A_{\mathcal S}}[\lambda, \, \omega ]\) is constructed only from the 2surface data on \({\mathcal S}\).
7.2.2 The Hamiltonian interpretation of the kinematical twistor
7.2.3 The Hermitian scalar product and the infinity twistor
In general the natural pointwise Hermitian scalar product, defined by \(\left\langle {Z,\bar W} \right\rangle : =  {\rm{i(}}{\lambda ^A}{\Delta _{A{A\prime}}}{{\bar \omega}^{{A\prime}}}  {{\bar \omega}^{{A\prime}}}{\Delta _{A{A\prime}}}{\lambda ^A}{\rm{)}}\), is not constant on \({\mathcal S}\), thus it does not define a Hermitian scalar product on the 2surface twistor space. As is shown in [220, 223, 375], \(\left\langle {Z,\bar W} \right\rangle\) is constant on \({\mathcal S}\) for any two 2surface twistors if and only if \({\mathcal S}\) can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric and extrinsic curvatures. Such 2surfaces are called noncontorted, while those that cannot be embedded are called contorted. One natural candidate for the Hermitian metric could be the average of \(\left\langle {Z,\bar W} \right\rangle\) on \({\mathcal S}\) [307]: \({H_{\alpha {\beta \prime}}}{Z^\alpha}{{\bar W}^{{\beta \prime}}}: = {[{\rm{Area}}({\mathcal S})]^{ {1 \over 2}}}\oint\nolimits_{\mathcal S} {{\mathcal S}\left\langle {Z,\bar W} \right\rangle} d{\mathcal S}\), which reduces to \(\left\langle {Z,\bar W} \right\rangle\) on noncontorted 2surfaces. Interestingly enough, \(\oint\nolimits_{\mathcal S} {\left\langle {Z,\bar W} \right\rangle} d{\mathcal S}\) can also be reexpressed by the integral (55) of the NesterWitten 2form [356]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted 2surfaces, the definition of the quasilocal mass as the norm of the kinematical twistor (cf. Equation (52)) is ambiguous unless a natural H_{ αβ′ } is found.
If \({\mathcal S}\) is noncontorted, then the scalar product \(\left\langle {Z,\bar W} \right\rangle\) defines the totally antisymmetric twistor ε_{ αβγδ }, and for the four independent 2surface twistors \(Z_1^\alpha, \ldots, Z_4^\alpha\) the contraction \({\varepsilon _{\alpha \beta \gamma \delta}}Z_1^\alpha Z_2^\beta Z_3^\gamma Z_4^\delta\), and hence by Equation (49) the determinant ν, is constant on \({\mathcal S}\). Nevertheless, ν can be constant even for contorted 2surfaces for which \(\left\langle {Z,\bar W} \right\rangle\) is not. Thus, the totally antisymmetric twistor ε_{ αβγδ } can exist even for certain contorted 2surfaces. Therefore, an alternative definition of the quasilocal mass might be based on Equation (53) [371]. 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 antideSitter spacetimes. Thus, if needed, the former notion of mass will be called the normmass, the latter the determinantmass (denoted by m_{D}).
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 (51), one also needs the infinity twistor. However, −ɛ^{ A′B } Δ_{ A′A }λ^{ A }Δ_{ B′B }ω^{ B } is not constant on \({\mathcal S}\) even if it is noncontorted, thus in general it does not define any twistor on \({\rm{T}}_{\mathcal S}^\alpha\). One might take its average on \({\mathcal S}\) (which can also be reexpressed by the integral of the NesterWitten 2form [356]), but the resulting twistor would not be simple. In fact, even on 2surfaces in de Sitter and antide Sitter spacetimes with cosmological constant λ the natural definition for I_{ αβ } is I_{ αβ } := diag(λε_{ AB }, ε^{ A′B′ }) [313, 311, 371], while on round spheres in spherically symmetric spacetimes it is \({I_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {1 \over {2{r^2}}}(1 + 2{r^2}\rho {\rho \prime}){\varepsilon _{AB}}{\lambda ^A}{\omega ^B}  {\varepsilon ^{{A\prime}{B\prime}}}{\Delta _{{A\prime}A}}{\lambda ^A}{\Delta _{{B\prime}B}}{\omega ^B}\) [363]. Thus no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [197]: 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 nonsimple I^{ αβ }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 2surface twistor space itself depends on the geometry of \({\mathcal S}\), and hence all the structures thereon also.
Since in the Hermiticity condition (51) only the special combination \({H^\alpha}_{{\beta \prime}}: = {I^{\alpha \beta}}{H_{\beta {\beta \prime}}}\) of the infinity and metric twistors (the socalled ‘barhook’ combination) appears, it might still be hoped that an appropriate \({H^\alpha}_{{\beta \prime}}\) could be found for a class of 2surfaces in a natural way [377]. 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 [222]. He considers a 1parameter family of spacetimes with perfect fluid source such that in the λ → 0 limit of the parameter λ one gets a Newtonian spacetime, and, in the same limit, the 2surface \({\mathcal S}\) lies in a t = const. hypersurface of the Newtonian time t. In this limit the pointwise Hermitian scalar product is constant, and the normmass can be calculated. As could be expected, for the leading λ^{2} order term in the expansion of m as a series of λ he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a λ^{4} order correction.
The Penrose definition for the energymomentum and angular momentum can be applied to the cuts \({\mathcal S}\) of the future null infinity ℐ^{+} of an asymptotically flat spacetime [307, 313]. Then every element of the construction is built from conformally rescaled quantities of the nonphysical spacetime. Since \({\mathscr I}^+\) is shearfree, the 2surface twistor equations on \({\mathcal S}\) decouple, and hence the solution space admits a natural infinity twistor I_{ αβ }. It singles out precisely those solutions whose primary spinor parts span the asymptotic spin space of Bramson (see Section 4.2.4), and they will be the generators of the energymomentum. Although \({\mathcal S}\) is contorted, and hence there is no natural Hermitian scalar product, there is a twistor \({H^\alpha}_{{\beta \prime}}\) with respect to which A_{ αβ } is Hermitian. Furthermore, the determinant ν is constant on \({\mathcal S}\), and hence it defines a volume 4form on the 2surface twistor space [377]. The energymomentum coming from A_{ αβ } is just that of Bondi and Sachs. The angular momentum defined by A_{αβ} 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 ‘factoroftwo anomaly’ between the angular momentum and the energymomentum. Since its definition is based on the solutions of the 2surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boostrotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the 2surface twistor space on \({\mathcal S}\) reduces the BMS Lie algebra to one of its Poincaré 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 4parameter family of ambiguities in the potential for the shear σ), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. As was shown in [129], 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 [129, 196] and it was shown that this is precisely that given by Ashtekar and Streubel [29] (see also [336, 337, 128]).
The other way of determining the null infinity limit of the energymomentum 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 [338, 340]. At this point it should be noted that the r → ∞ limit of A_{ αβ } vanishes, and it is \(\sqrt {{\rm{Area}}({{\mathcal S}_r})} {A_{\alpha \beta}}\) that yields the energymomentum and angular momentum at infinity (see the remarks following Equation (15)). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [371]. The 2surfaces for which the mass was calculated are the r = const. cuts of the geometrically distinguished outgoing null hypersurfaces u = const. Tod found that, for given u, the mass m is independent of r, as could be expected because of the lack of the incoming radiation.
The large sphere limit of the 2surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [344], the AshtekarHansen [23], and the BeigSchmidt [48] models of spatial infinity in [334, 335, 337]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) noncontorted, and both the Hermitian scalar product and the infinity twistor are welldefined. Thus the energymomentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the AshtekarHansen expression for the energymomentum 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 [339].
The Penrose mass in asymptotically antideSitter spacetimes was studied by Kelly [234]. He calculated the kinematical twistor for spacelike cuts \({\mathcal S}\) of the infinity \({\mathscr I}\), which is now a timelike 3manifold in the nonphysical spacetime. Since \({\mathscr I}\) admits global 3surface twistors (see the next Section 7.2.5), \({\mathcal S}\) is noncontorted. In addition to the Hermitian scalar product there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding Hermiticity condition. The energymomentum 4vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [27]. Therefore, the energymomentum 4vector is future pointing and timelike if there is a spacelike hypersurface extending to \({\mathscr I}\) on which the dominant energy condition is satisfied. Consequently, m^{2} ≥ 0. Kelly showed that \(m_D^2\) is also nonnegative and in vacuum it coincides with m^{2}. In fact [377], m ≥ m_{D} ≥ 0 holds.
7.2.5 The quasilocal mass of specific 2surfaces
The Penrose mass has been calculated in a large number of specific situations. Round spheres are always noncontorted [375], thus the normmass can be calculated. (In fact, axisymmetric 2surfaces in spacetimes with twistfree rotational Killing vector are noncontorted [223].) The Penrose mass for round spheres reduces to the standard energy expression discussed in Section 4.2.1 [371]. Thus every statement given in Section 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 t = const. slice of the KantowskiSachs spacetime this mass is independent of the 2surfaces [368]. Interestingly enough, although these spheres cannot be shrunk to a point (thus the mass cannot be interpreted as ‘the 3volume 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 3volume of a sphere in flat space with the same area as \({\mathcal S}\) [376]. In conformally flat spacetimes [371] the 2surface twistors are just the global twistors restricted to \({\mathcal S}\), and the Hermitian scalar product is constant on \({\mathcal S}\). Thus the normmass is welldefined.
The construction works nicely even if global twistors exist only on a (say) spacelike hypersurface Σ containing \({\mathcal S}\). These twistors are the socalled 3surface twistors [371, 373], which are solutions of certain (overdetermined) elliptic partial differential equations, the socalled 3surface twistor equations, on Σ. These equations are completely integrable (i.e. they admit the maximal number of linearly independent solutions, namely four) if and only if Σ with its intrinsic metric and extrinsic curvature can be embedded, at least locally, into some conformally flat spacetime [373]. Such hypersurfaces are called noncontorted. It might be interesting to note that the noncontorted hypersurfaces can also be characterized as the critical points of the ChernSimons functional built from the real Sen connection on the Lorentzian vector bundle or from the 3surface twistor connection on the twistor bundle over Σ [49, 361]. Returning to the quasilocal mass calculations, Tod showed that in vacuum the kinematical twistor for a 2surface \({\mathcal S}\) in a noncontorted Σ depends only on the homology class of \({\mathcal S}\). In particular, if \({\mathcal S}\) can be shrunk to a point then the corresponding kinematical twistor is vanishing. Since Σ is noncontorted, \({\mathcal S}\) is also noncontorted, and hence the normmass is welldefined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any noncontorted 2surface that can be deformed into a round sphere, and it is zero for those that do not link the black hole [375]. Thus, in particular, the Penrose mass can be zero even in curved spacetimes.
A particularly interesting class of noncontorted hypersurfaces is that of the conformally flat timesymmetric initial data sets. Tod considered Wheeler’s solution of the timesymmetric vacuum constraints describing n ‘points at infinity’ (or, in other words, n − 1 black holes) and 2surfaces in such a hypersurface [371]. He found that the mass is zero if \({\mathcal S}\) does not link any black hole, it is the mass M_{ i } of the ith black hole if \({\mathcal S}\) links precisely the ith hole, it is \({M_i} + {M_j}  {M_i}{M_j}/{d_{ij}} + {\mathcal O}(1/d_{ij}^2)\) if \({\mathcal S}\) links precisely the ith and the jth holes, where d_{ ij } is some appropriate measure of the distance of the holes, …, etc. Thus, the mass of the ith and jth 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 timesymmetric initial data sets describing n ‘points at infinity’ [45]. He found a symmetric tracefree and divergencefree tensor field T^{ ab } and, for any conformal Killing vector ξ^{ a } of the data set, defined the 2surface flux integral P(ξ) of T^{ ab }ξ_{ b } on \({\mathcal S}\). He showed that P(ξ) is conformally invariant, depends only on the homology class of \({\mathcal S}\), 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 [371] (and discussed in the previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the length of the P’s with respect to the CartanKilling metric of the conformal group of the hypersurface.
Tod calculated the quasilocal mass for a large class of axisymmetric 2surfaces (cylinders) in various LRS Bianchi and KantowskiSachs cosmological models [376] and more general cylindrically symmetric spacetimes [378]. In all these cases the 2surfaces are noncontorted, and the construction works. A technically interesting feature of these calculations is that the 2surfaces 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 iΔ_{ A′A }λ^{ A }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 quasilocal mass expressions is not simply ‘density’ × ‘volume’, but they are proportional to a ‘type factor’ f(L) as well, where L is the coordinate length of the cylinder. In the hyperbolic, parabolic, and elliptic cases this factor is sinhωL/(ωL), 1, and sinωL/(ωL), respectively, where ω 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 quasilocal 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 determinantmass has also been calculated and found to coincide with the normmass. A numerical investigation of the axisymmetric Brill waves on the Schwarzschild background was presented in [69]. It was found that the quasilocal 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 Section 13.2.1). Tod showed [374, 375] that for static black holes the Penrose inequality holds if the mass of the hole is defined to be the Penrose quasilocal mass of the spacelike cross section \({\mathcal S}\) of the event horizon. The trick here is that \({\mathcal S}\) is totally geodesic and conformal to the unit sphere, and hence it is noncontorted and the Penrose mass is welldefined. Then the Penrose inequality will be a Sobolevtype inequality for a nonnegative function on the unit sphere. This inequality was tested numerically in [69].
Apart from the cuts of \({\mathscr I}^+\) in radiative spacetimes, all the 2surfaces discussed so far were noncontorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted 2surface [377]. 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 [372, 235, 398], and forced its modification.
7.3 The modified constructions
Independently of the results of the small sphere calculations, Penrose claimed that in the Schwarzschild spacetime the quasilocal mass expression should yield the same zero value on 2surfaces, contorted or not, which do not surround the black hole. (For the motivations and the arguments, see [309].) 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 [377].
7.3.1 The ‘improved’ construction with the determinant
7.3.2 Modification through Tod’s expression
7.3.3 Mason’s suggestions
A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [265]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Equation (58) is that of the integral of the NesterWitten 2form, and the spinor fields \(\sqrt \eta {\lambda ^A}\) and \({\rm{i}}{\Delta _{{A\prime}A}}(\sqrt \eta {\lambda ^A})\) could still be considered as the spinor constituents of the ‘quasiKilling vectors’ of the 2surface \({\mathcal S}\), their structure is not so simple because the factor η itself depends on all of the four independent solutions of the 2surface twistor equation in a rather complicated way.
To have a simple Hamiltonian interpretation Mason suggested further modifications [265, 266]. He considers the four solutions \(\lambda _i^A,\,i = 1, \ldots, 4\), of the 2surface twistor equations, and uses these solutions in the integral (55) of the NesterWitten 2form. Since \({H_{\mathcal S}}\) 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 4 × 4 Hermitian matrix \({E_{ij}}: = {H_{\mathcal S}}[{\lambda _i},{{\bar \lambda}_j}]\). However, it is not clear how the four ‘quasitranslations’ of \({H_{\mathcal S}}\) should be found among the 16 vector fields \(\lambda _i^A\bar \lambda _i^{{A\prime}}\) (called ‘quasiconformal Killing vectors’ of \({H_{\mathcal S}}\)) for which the corresponding quasilocal quantities could be considered as the quasilocal energymomentum. Nevertheless, this suggestion leads us to the next class of quasilocal quantities.
8 Approaches Based on the NesterWitten 2Form

both the ADM and BondiSachs energymomenta can be reexpressed by the integral of the NesterWitten 2form \(u{(\lambda, \, \bar \mu)_{ab}}\),

the proof of the positivity of the ADM and BondiSachs masses is relatively simple in terms of the 2component spinors, and

the integral of Møller’s tetrad superpotential for the energymomentum, coming from his tetrad Lagrangian (9), is just the integral of \(u{({\lambda ^{\underline A}},{{\bar \lambda}^{\underline B}}\prime)_{ab}}\), where \(\{\lambda _A^{\underline A}\}\) is a normalized spinor dyad.
If \({\mathcal S}\) is any closed, orientable spacelike 2surface and λ_{ A }, μ_{ A } are arbitrary spinor fields, then in the integral \({H_{\mathcal S}}[\lambda, \, \bar \mu ]\), defined by Equation (55), only the tangential derivative of λ_{ A } appears. (μ_{ A } is involved in \({H_{\mathcal S}}[\lambda, \, \bar \mu ]\) algebraically.) Thus, by Equation (13), \({H_{\mathcal S}}:{C^\infty}({\mathcal S},{{\bf{S}}_A}) \times {C^\infty}({\mathcal S},{{\bf{S}}_A}) \rightarrow {\mathbb C}\) is a Hermitian scalar product on the (infinitedimensional complex) vector space of smooth spinor fields on \({\mathcal S}\). Thus, in particular, the spinor fields in \({H_{\mathcal S}}[\lambda, \, \bar \mu ]\) need be defined only on \({\mathcal S}\), and \({H_{\mathcal S}}[\lambda, \, \bar \mu ] = {H_{\mathcal S}}[\mu, \, \overline \lambda ]\) holds. A remarkable property of \({H_{\mathcal S}}\) is that if λ_{ A } is a constant spinor field on \({\mathcal S}\) with respect to the covariant derivative Δ_{ e }, then \({H_{\mathcal S}}[\lambda, \, \bar \mu ] = 0\) for any smooth spinor field μ_{ A } on \({\mathcal S}\). Furthermore, if \(\lambda _A^{\underline A} = (\lambda _A^0,\,\lambda _A^1)\) is any pair of smooth spinor fields on \({\mathcal S}\), then for any constant SL(2, ℂ) matrix \({\Lambda _{\underline A}}^{\underline B}\) one has \({H_{\mathcal S}}[{\lambda ^{\underline C}}{\Lambda _{\underline C}}^{\underline A},{{\bar \lambda}^{{{\underline D}\prime}}}{\Lambda _{{{\underline D}\prime}}}^{{{\underline B}\prime}}] = {H_{\mathcal S}}[{\lambda ^{\underline C}},{{\bar \lambda}^{{{\underline D}\prime}}}]{\Lambda _{\underline C}}^{\underline A}{{\bar \Lambda}_{{{\underline D}\prime}}}^{{{\underline B}\prime}}\), i.e. the integrals \({H_{\mathcal S}}[{\lambda ^{\underline A}},{\lambda ^{{{\underline B}\prime}}}]\) transform as the spinor components of a real Lorentz vector over the twocomplex dimensional space spanned by \(\lambda _A^0\,and\,\lambda _A^1\). Therefore, to have a welldefined quasilocal energymomentum vector we have to specify some 2dimensional subspace \({{\bf{S}}^{\underline A}}\) of the infinitedimensional space \({C^\infty}({\mathcal S},\,{{\bf{S}}_A})\) and a symplectic metric \({\varepsilon _{\underline A \underline B}}\) thereon. Thus underlined capital Roman indices will be referring to this space. The elements of this subspace would be interpreted as the spinor constituents of the ‘quasitranslations’ of the surface \({\mathcal S}\). Since in Møller’s tetrad approach it is natural to choose the orthonormal vector basis to be a basis in which the translations have constant components (just as the constant orthonormal bases in Minkowski spacetime which are bases in the space of translations), the spinor fields \({\lambda _A}^{\underline A}\) could also be interpreted as the spinor basis that should be used to construct the orthonormal vector basis in Møller’s superpotential (10). In this sense the choice of the subspace \({{\bf{S}}^{\underline A}}\) and the metric \({\varepsilon _{\underline A \underline B}}\) is just a gauge reduction, or a choice for the ‘reference configuration’ of Section 3.3.3.
Once the spin space \(({{\bf{S}}^{\underline A}},\,{\varepsilon _{\underline A \underline B}})\) is chosen, the quasilocal energymomentum is defined to be \({P_{\mathcal S}}^{\underline A {{\underline B}\prime}}:{H_{\mathcal S}}[{\lambda ^{\underline A}},{{\bar \lambda}^{{{\underline B}\prime}}}]\) and the corresponding quasilocal mass \({m_{\mathcal S}}\) is \(m_{\mathcal S}^2: = {\varepsilon _{\underline A \underline B}}{\varepsilon _{{{\underline A}\prime}{{\underline B}\prime}}}P_{\mathcal S}^{\underline A {{\underline A}\prime}}P_{\mathcal S}^{\underline B {{\underline B}\prime}}\). In particular, if one of the spinor fields \(\lambda _A^{\underline A}\), e.g. \(\lambda _A^0\), is constant on \({\mathcal S}\) (which means that the geometry of \({\mathcal S}\) is considerably restricted), then \(P_{\mathcal S}^{{{00}\prime}} = P_{\mathcal S}^{{{01}\prime}} = P_{\mathcal S}^{{{10}\prime}} = 0\), and hence the corresponding mass \({m_{\mathcal S}}\) is zero. If both \(\lambda _A^0\) and \(\lambda _A^1\) are constant (in particular, when they are the restrictions to \({\mathcal S}\) of the two constant spinor fields in the Minkowski spacetime), then \(P_{\mathcal S}^{\underline A {{\underline B}\prime}}\) itself is vanishing.
Therefore, to summarize, the only thing that needs to be specified is the spin space \(({{\bf{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\), and the various suggestions for the quasilocal energymomentum based on the integral of the NesterWitten 2form correspond to the various choices for this spin space.
8.1 The LudvigsenVickers construction
8.1.1 The definition
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 LudvigsenVickers energymomentum in its form above cannot be defined in a spacetime which is not asymptotically flat at null infinity. Thus their construction is not genuinely quasilocal, because it depends not only on the (intrinsic and extrinsic) geometry of \({\mathcal S}\), but on the global structure of the spacetime as well. In addition, the requirement of the smoothness of the null hypersurface \({\mathcal N}\) connecting the 2surface to the null infinity is a very strong restriction. In fact, for general (even for convex) 2surfaces in a general asymptotically flat spacetime conjugate points will develop along the (outgoing) null geodesics orthogonal to the 2surface [304, 175]. Thus either the 2surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the 2surface 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 (59, 60) one could define a quasilocal energymomentum for small spheres [259, 66]. The basic idea is that there is a spin space at the vertex p of the null cone in the spacetime whose spacelike cross section is the actual 2surface, and the LudvigsenVickers spinors on \({\mathcal S}\) are defined by propagating these spinors from the vertex p to \({\mathcal S}\) via Equations (59, 60). This definition works in arbitrary spacetime, but the 2surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely quasilocal.
8.1.3 Monotonicity, masspositivity and the various limits
Once the LudvigsenVickers spinors are given on a spacelike 2surface \({{\mathcal S}_r}\) of constant affine parameter r in the outgoing null hypersurface \({\mathcal N}\), then they are uniquely determined on any other spacelike 2surface \({{\mathcal S}_{{r\prime}}}\) in \({\mathcal N}\), too, i.e. the propagation law (59, 60) defines a natural isomorphism between the space of the LudvigsenVickers spinors on different 2surfaces of constant affine parameter in the same \({\mathcal N}\). (r need not be a Bonditype coordinate.) This makes it possible to compare the components of the LudvigsenVickers energymomenta on different surfaces. In fact [259], if the dominant energy condition is satisfied (at least on \({\mathcal N}\)), then for any LudvigsenVickers spinor λ^{ A } and affine parameter values r_{1} ≤ r_{2} one has \({H_{{{\mathcal S}_{{r_1}}}}}[\lambda, \bar \lambda ] \leq {H_{{{\mathcal S}_{{r_2}}}}}[\lambda, \bar \lambda ]\), and the difference \({H_{{{\mathcal S}_{{r_2}}}}}[\lambda, \bar \lambda ]  {H_{{{\mathcal S}_{{r_1}}}}}[\lambda, \bar \lambda ] \geq 0\) can be interpreted as the energy flux of the matter and the gravitational radiation through \({\mathcal N}\) between \({{\mathcal S}_{{r_1}}}\) and \({{\mathcal S}_{{r_2}}}\). Thus both \(P_{{{\mathcal S}_r}}^{{{00}\prime}}\) and \(P_{{{\mathcal S}_r}}^{{{11}\prime}}\) are increasing with r (‘massgain’). A similar monotonicity property (‘massloss’) can be proven on ingoing null hypersurfaces, but then the propagation law (59, 60) should be replaced by Ϸ′λ_{1} = 0 and \( {\Delta ^ }\lambda : = \eth{\lambda _1} + {\rho \prime}{\lambda _0} = 0\). Using these equations the positivity of the LudvigsenVickers mass was proven in various special cases in [259].
Concerning the positivity properties of the LudvigsenVickers mass and energy, first it is obvious by the remarks on the nature of the propagation law (59, 60) that in Minkowski spacetime the LudvigsenVickers energymomentum is vanishing. However, in the proof of the nonnegativity of the DouganMason energy (discussed in Section 8.2) only the λ_{ A } ∈ ker Δ^{+} part of the propagation equations is used. Therefore, as realized by Bergqvist [61], the LudvigsenVickers energymomenta (both based on the asymptotic and the point spinors) are also future directed and nonspacelike if \({\mathcal S}\) is the boundary of some compact spacelike hypersurface Σ on which the dominant energy condition is satisfied and \({\mathcal S}\) is weakly future convex (or at least ρ ≤ 0). Similarly, the LudvigsenVickers definitions share the rigidity properties proven for the DouganMason energymomentum [354]: Under the same conditions the vanishing of the energymomentum implies the flatness of the domain of dependence D(Σ) of Σ.
In the weak field approximation [259] the difference \({H_{{{\mathcal S}_{{r_2}}}}}[\lambda, \bar \lambda ]  {H_{{{\mathcal S}_{{r_1}}}}}[\lambda, \bar \lambda ]\) is just the integral of \(4\pi G{T_{ab}}\,{l^a}{\lambda ^B}{{\bar \lambda}^{{B\prime}}}\) on the portion of \({\mathcal N}\) between the two 2surfaces, where T_{ ab } is the linearized energymomentum tensor: The increment of \({H_{{{\mathcal S}_r}}}[\lambda, \bar \lambda ]\) on \({\mathcal N}\) is due only to the flux of the matter energymomentum.
Since the BondiSachs energymomentum can be written as the integral of the NesterWitten 2form on the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first version of the LudvigsenVickers energymomentum tends to that of Bondi and Sachs. It was shown in [259, 340] that this expectation is, in fact, correct. The LudvigsenVickers mass was calculated for large spheres both for radiative and stationary spacetimes with r^{−2} and r^{−3} accuracy, respectively, in [338, 340].
8.2 The DouganMason constructions
8.2.1 Holomorphic/antiholomorphic spinor fields
The original construction of Dougan and Mason [127] was introduced on the basis of sheaftheoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [354, 356].
Following Dougan and Mason we define the spinor field λ_{ A } to be antiholomorphic in case m^{ e }∇_{ e }λ_{ A } = m^{ e }Δ_{ e }λ_{ A } = 0, or holomorphic if \({{\bar m}^e}{\nabla _e}{\lambda _A} = {{\bar m}^e}{\nabla _e}{\lambda _A} = 0\). Thus, this notion of holomorphicity/antiholomorphicity is referring to the connection Δ_{ e } on \({\mathcal S}\). While the notion of the holomorphicity/antiholomorphicity of a function on \({\mathcal S}\) does not depend on whether the Δ_{ e } or the δ_{ e } operator is used, for tensor or spinor fields it does. Although the vectors m^{ a } and \({{\bar m}^a}\) are not uniquely determined (because their phase is not fixed), the notion of the holomorphicity/antiholomorphicity is welldefined, because the defining equations are homogeneous in m^{ a } and \({{\bar m}^a}\). Next suppose that there are at least two independent solutions of \({{\bar m}^e}{\nabla _e}{\lambda _A} = 0\). If λ_{ A } and μ_{ A } are any two such solutions, then \({{\bar m}^e}{\nabla _e}({\lambda _A}{\mu _B}{\varepsilon ^{AB}}) = 0\), and hence by Liouville’s theorem \({\lambda _A}{\mu _B}{\varepsilon ^{AB}}\) is constant on \({\mathcal S}\). If this constant is not zero, then we call \({\mathcal S}\) generic, if it is zero then \({\mathcal S}\) will be called exceptional. Obviously, holomorphic λ_{ A } on a generic \({\mathcal S}\) cannot have any zero, and any two holomorphic spinor fields, e.g. \(\lambda _A^0\) and \(\lambda _A^1\), span the spin space at each point of \({\mathcal S}\) (and they can be chosen to form a normalized spinor dyad with respect to ɛ_{ AB } on the whole of \({\mathcal S}\)). Expanding any holomorphic spinor field in this frame, the expanding coefficients turn out to be holomorphic functions, and hence constant. Therefore, on generic 2surfaces there are precisely two independent holomorphic spinor fields. In the GHP formalism the condition of the holomorphicity of the spinor field λ_{ A } is that its components (λ_{0}, λ_{1}) be in the kernel of \({{\mathcal H}^ +}: = {\Delta ^ +} \oplus {{\mathcal T}^ +}\). Thus for generic 2surfaces \({{\mathcal H}^ +}\) with the constant \({\varepsilon _{\underline A \underline B}}\) would be a natural candidate for the spin space \(({{\bf{S}}^{\underline A}},\,{\varepsilon _{\underline A \underline B}})\) above. For exceptional 2surfaces the kernel space \({{\mathcal H}^ +}\) is either 2dimensional but does not inherit a natural spin space structure, or it is higher than two dimensional. Similarly, the symplectic inner product of any two antiholomorphic spinor fields is also constant, one can define generic and exceptional 2surfaces as well, and on generic surfaces there are precisely two antiholomorphic spinor fields. The condition of the antiholomorphicity of λ_{ A } is \(\lambda \in \ker {{\mathcal H}^ }: = \ker ({\Delta ^ } \oplus {{\mathcal T}^ })\). Then \({{\bf{S}}^{\underline A}} = \ker {{\mathcal H}^ }\) could also be a natural choice. Note that since the spinor fields whose holomorphicity/antiholomorphicity is defined are unprimed, and these correspond to the antiholomorphicity/holomorphicity, respectively, of the primed spinor fields of Dougan and Mason. Thus the main question is whether there exist generic 2surfaces, 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 2surfaces
\({{\mathcal H}^ \pm}\) are first order elliptic differential operators on certain vector bundles over the compact 2surface \({\mathcal S}\), and their index can be calculated: index \(({{\mathcal H}^ \pm}) = 2(1  g)\), where g is the genus of \({\mathcal S}\). Therefore, for \({\mathcal S} \approx {S^2}\) there are at least two linearly independent holomorphic and at least two linearly independent antiholomorphic spinor fields. The existence of the holomorphic/antiholomorphic spinor fields on higher genus 2surfaces is not guaranteed by the index theorem. Similarly, the index theorem does not guarantee that \({\mathcal S} \approx {S^2}\) is generic either: If the geometry of \({\mathcal S}\) is very special then the two holomorphic/antiholomorphic spinor fields (which are independent as solutions of \({{\mathcal H}^ \pm}\lambda = 0\)) might be proportional to each other. For example, future marginally trapped surfaces (i.e. for which ρ = 0) are exceptional from the point of view of holomorphic, and past marginally trapped surfaces (ρ′ = 0) from the point of view of antiholomorphic spinors. Furthermore, there are surfaces with at least three linearly independent holomorphic/antiholomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional 2surface \({\mathcal S}\) with S^{2} topology make \({\mathcal S}\) generic.
Finally, we note that several first order differential operators can be constructed from the chiral irreducible parts Δ^{±} and \({{\mathcal T}^ \pm}\) of Δ_{ e }, given explicitly by Equation (25). However, only four of them, the DiracWitten operator Δ := Δ^{+} ⊕ Δ^{−}, the twistor operator \({\mathcal T}: = {{\mathcal T}^ \pm} \oplus {{\mathcal T}^ }\), and the holomorphy and antiholomorphy operators \({{\mathcal H}^ \pm}\), are elliptic (which ellipticity, together with the compactness of \({\mathcal S}\), would guarantee the finiteness of the dimension of their kernel), and it is only \({{\mathcal H}^ \pm}\) that have 2complexdimensional kernel in the generic case. This purely mathematical result gives some justification for the choices of Dougan and Mason: The spinor fields \(\lambda _A^{\underline A}\) that should be used in the NesterWitten 2form are either holomorphic or antiholomorphic. The construction does not work for exceptional 2surfaces.
8.2.3 Positivity properties
One of the most important properties of the DouganMason energymomenta is that they are future pointing nonspacelike vectors, i.e. the corresponding masses and energies are nonnegative. Explicitly [127], if \({\mathcal S}\) is the boundary of some compact spacelike hypersurface Σ on which the dominant energy condition holds, furthermore if \({\mathcal S}\) is weakly future convex (in fact, ρ ≤ 0 is enough), then the holomorphic DouganMason energymomentum is a future pointing nonspacelike vector, and, analogously, the antiholomorphic energymomentum is future pointing and nonspacelike if ρ′ ≥ 0. As Bergqvist [61] stressed (and we noted in Section 8.1.3), Dougan and Mason used only the Δ^{+}λ = 0 (and in the antiholomorphic construction the Δ^{−} λ = 0) 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 Equation (59) of the LudvigsenVickers construction, or \({{\mathcal T}^ +}\lambda = 0\) in the holomorphic DouganMason construction, or even \({{\mathcal T}^ +}\lambda = k{\sigma \prime}\psi _2\prime{\lambda _0}\) for some constant k, a ‘deformation’ of the holomorphicity considered by Bergqvist [61]. In fact, the propagation law may even be \({{\bar m}^a}{\Delta _a}{\lambda _B} = {{\tilde f}_B}^C{\lambda _C}\) for any spinor field \({{\tilde f}_B}^C\) satisfying \({\pi ^{ B}}_A{{\tilde f}_B}^C = {{\tilde f}_A}^B{\pi ^{+ C}}_B = 0\). This ensures the positivity of the energy under the same conditions and that \({\varepsilon ^{AB}}{\lambda _A}{\mu _B}\) is still constant on \({\mathcal S}\) for any two solutions λ_{ A } and μ_{ A }, making it possible to define the norm of the resulting energymomentum, i.e. the mass.
 1.
\(P_{\mathcal S}^{\underline A {{\underline B}\prime}}\) is zero iff D(Σ) is flat, which is also equivalent to the vanishing of the quasilocal energy, \({E_{\mathcal S}}: = {1 \over {\sqrt 2}}(P_{\mathcal S}^{{{00}\prime}} + P_{\mathcal S}^{{{11}\prime}}) = 0\), and
 2.
\(P_{\mathcal S}^{\underline A {{\underline B}\prime}}\) is null (i.e. the quasilocal mass is zero) iff D(Σ) is a ppwave geometry and the matter is pure radiation.
These results show some sort of rigidity of the matter + gravity system (where the latter satisfies the dominant energy condition) even at the quasilocal level, which is much more manifest from the following equivalent form of Results 1 and 2: Under the same conditions D(Σ) is flat if and only if there exist two linearly independent spinor fields on \({\mathcal S}\) which are constant with respect to Δ_{ e }, and D(Σ) is a ppwave geometry and the matter is pure radiation if and only if there exists a Δ_{ e }constant spinor field on \({\mathcal S}\) [356]. Thus the full information that D(Σ) is flat/ppwave is completely encoded not only in the usual initial data on \(\Sigma\), but in the geometry of the boundary of Σ, too. In Section 13.5 we return to the discussion of this phenomenon, where we will see that, assuming that \({\mathcal S}\) is future and past convex, the whole line element of D(Σ) (and not only the information that it is some ppwave geometry) is determined by the 2surface data on \({\mathcal S}\).
Comparing Results 1 and 2 above with the properties of the quasilocal energymomentum (and angular momentum) listed in Section 2.2.3, the similarity is obvious: \(P_{\mathcal S}^{\underline A {{\underline B}\prime}} = 0\) characterizes the ‘quasilocal vacuum state’ of general relativity, while \({m_{\mathcal S}} = 0\) is equivalent to ‘pure radiative quasilocal states’. The equivalence of \({E_{\mathcal S}} = 0\) and the flatness of D(Σ) shows that curvature always yields positive energy, or, in other words, with this notion of energy no classical symmetry breaking can occur in general relativity: The ‘quasilocal ground states’ (defined by \({E_{\mathcal S}} = 0\)) are just the ‘quasilocal vacuum states’ (defined by the trivial value of the field variables on D(Σ)) [354], in contrast, for example, to the well known ϕ^{4} theories.
8.2.4 The various limits
Both definitions give the same standard expression for round spheres [126]. Although the limit of the DouganMason masses for round spheres in ReissnerNordström spacetime gives the correct irreducible mass of the ReissnerNordström black hole on the horizon, the constructions do not work on the surface of bifurcation itself, because that is an exceptional 2surface. Unfortunately, without additional restrictions (e.g. the spherical symmetry of the 2surfaces in a spherically symmetric spacetime) the mass of the exceptional 2surfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic 2surfaces approaching the exceptional one [356].
Both definitions give the same, expected results in the weak field approximation and for large spheres at spatial infinity: Both tend to the ADM energymomentum [127]. In nonvacuum both definitions give the same, expected expression (28) for small spheres, in vacuum they coincide in the r^{5} order with that of Ludvigsen and Vickers, but in the r^{6} order they differ from each other: The holomorphic definition gives Equation (61), but in the analogous expression for the antiholomorphic energymomentum the numerical coefficient 4/(45G) is replaced by 1/(9G) [126]. The DouganMason energymomenta have also been calculated for large spheres of constant Bonditype radial coordinate value r near future null infinity [126]. While the antiholomorphic construction tends to the BondiSachs energymomentum, the holomorphic one diverges in general. In stationary spacetimes they coincide and both give the BondiSachs energymomentum. At the past null infinity it is the holomorphic construction which reproduces the BondiSachs energymomentum and the antiholomorphic diverges.
We close this section with some caution and general comments on a potential gauge ambiguity in the calculation of the various limits. By the definition of the holomorphic and antiholomorphic spinor fields they are associated with the 2surface \({\mathcal S}\) only. Thus if \({{\mathcal S}\prime}\) is another 2surface, then there is no natural isomorphism between the space — for example of the antiholomorphic spinor fields ker \({{\mathcal H}^ }({\mathcal S})\) on \({{\mathcal S} }\) — and ker \({{\mathcal H}^ }({\mathcal S\prime})\) on \({{\mathcal S}\prime}\), even if both surfaces are generic and hence there are isomorphisms between them^{12}. This (apparently ‘only theoretical’) fact has serious pragmatic consequences. In particular, in the small or large sphere calculations we compare the energymomenta, and hence the holomorphic or antiholomorphic spinor fields also, on different surfaces. For example [360], in the small sphere approximation every spin coefficient and spinor component in the GHP dyad and metric component in some fixed coordinate system \((\zeta, \overline \zeta)\) is expanded as a series of r, like \({\lambda _{\bf{A}}}(r,\zeta, \bar \zeta)\vert = {\lambda _{\bf{A}}}^{(0)}(\zeta, \bar \zeta) + r{\lambda _{\bf{A}}}^{(1)}(\zeta, \bar \zeta) + \cdots + {r^k}{\lambda _{\bf{A}}}^{(k)}(\zeta, \bar \zeta) + {\mathcal O}({r^{k + 1}})\). Substituting all such expansions and the asymptotic solutions of the Bianchi identities for the spin coefficients and metric functions into the differential equations defining the holomorphic/antiholomorphic spinors, we obtain a hierarchical system of differential equations for the expansion coefficients λ_{ A }^{(0)}, λ_{ A }^{(1)}, …, etc. It turns out that the solutions of this system of equations with accuracy r^{ k } form a 2k rather than the expected two complex dimensional space. 2(k−1) of these 2k solutions are ‘gauge’ solutions, and they correspond in the approximation with given accuracy to the unspecified isomorphism between the space of the holomorphic/antiholomorphic spinor fields on surfaces of different radii. Obviously, similar ‘gauge’ solutions appear in the large sphere expansions, too. Therefore, without additional gauge fixing, in the expansion of a quasilocal quantity only the leading nontrivial term will be gaugeindependent. In particular, the r^{6} order correction in Equation (61) for the DouganMason energymomenta is welldefined only as a consequence of a natural gauge choice^{13}. Similarly, the higher order corrections in the large sphere limit of the antiholomorphic DouganMason energymomentum are also ambiguous unless a ‘natural’ gauge choice is made. Such a choice is possible in stationary spacetimes.
8.3 A specific construction for the Kerr spacetime
Logically, this specific construction perhaps would have to be presented only in Section 12, but the technique that it is based on may justify its placing here.
By investigating the propagation law (59, 60) of Ludvigsen and Vickers, for the Kerr spacetimes Bergqvist and Ludvigsen constructed a natural flat, (but nonsymmetric) metric connection [67]. Writing the new covariant derivative in the form \({{\tilde \nabla}_{A{A\prime}}}{\lambda _B} = {\nabla _{A{A\prime}}}{\lambda _B} + {\Gamma _{A{A\prime}B}}^C{\lambda _C}\), the ‘correction’ term \({\Gamma _{A{A\prime}B}}^C\) could be given explicitly in terms of the GHP spinor dyad (adapted to the two principal null directions), the spin coefficients ρ, τ and τ′, and the curvature component \({\psi _2}\). \({\Gamma _{A{A\prime}B}}^C\) admits a potential [68]: \({\Gamma _{A{A\prime}BC}} =  {\nabla _{({C^{{B\prime}}}}}{H_{B)A{A\prime}{B\prime}}}\), where \({H_{AB{A\prime}{B\prime}}}:{1 \over 2}{\rho ^{ 3}}(\rho + \bar \rho){\psi _2}{o_A}{o_B}{{\bar o}_{{A\prime}}}{{\bar o}_{{B\prime}}}\). However, this potential has the structure H_{ ab } = fl_{ a }l_{ b } appearing in the form of the metric \({g_{ab}} = g_{ab}^0 + f{l_a}{l_b}\) for the KerrSchild spacetimes, where \(g_{ab}^0\) is the flat metric. In fact, the flat connection \({{\tilde \nabla}_e}\) above could be introduced for general KerrSchild metrics [170], and the corresponding ‘correction term’ Γ_{ AA′BC } could be used to find easily the Lánczos potential for the Weyl curvature [10].
Since the connection \({{\tilde \nabla}_{A{A\prime}}}\) is flat and annihilates the spinor metric ɛ_{ AB }, there are precisely two linearly independent spinor fields, say \(\lambda _A^0\,and\,\lambda _A^1\), that are constant with respect to \({{\tilde \nabla}_{A{A\prime}}}\) and form a normalized spinor dyad. These spinor fields are asymptotically constant. Thus it is natural to choose the spin space \(({{\bf{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\) to be the space of the \({{\tilde \nabla}_a}\)constant spinor fields, independently of the 2surface \({\mathcal S}\).
A remarkable property of these spinor fields is that the NesterWitten 2form built from them is closed: \(du({\lambda ^{\underline A}},{{\bar \lambda}^{\underline {{B\prime}}}}) = 0\). This implies that the quasilocal energymomentum depends only on the homology class of \({\mathcal S}\), i.e. if \({{\mathcal S}_1}\,and\,{{\mathcal S}_2}\) are 2surfaces such that they form the boundary of some hypersurface in M, then \(P_{{{\mathcal S}_1}}^{\underline A {{\underline B}\prime}} = P_{{{\mathcal S}_2}}^{\underline A {{\underline B}\prime}}\), and if \({\mathcal S}\) is the boundary of some hypersurface, then \(P_{\mathcal S}^{\underline A {{\underline B}\prime}} = 0\). In particular, for twospheres that can be shrunk to a point the energymomentum is zero, but for those that can be deformed to a cut of the future null infinity the energymomentum is that of Bondi and Sachs.
9 QuasiLocal SpinAngular Momentum
In this section we review three specific quasilocal spinangular momentum constructions that are (more or less) ‘quasilocalizations’ of Bramson’s expression at null infinity. Thus the quasilocal spinangular momentum for the closed, orientable spacelike 2surface \({\mathcal S}\) will be sought in the form (17). Before considering the specific constructions themselves we summarize the most important properties of the general expression of Equation (17). Since the most detailed discussion of Equation (17) is given probably in [360, 363], the subsequent discussions will be based on them.
First, observe that the integral depends on the spinor dyad algebraically, thus it is enough to specify the dyad only at the points of \({\mathcal S}\). Obviously, \(J_{\mathcal S}^{\underline A \underline B}\) transforms like a symmetric second rank spinor under constant SL(2, C) transformations of the dyad \(\{\lambda _A^{\underline A}\}\). Second, suppose that the spacetime is flat, and let \(\{\lambda _A^{\underline A}\}\) be constant. Then the corresponding 1form basis \(\{\vartheta _a^{\underline a}\}\) is the constant Cartesian one, which consists of exact 1forms. Then since the Bramson superpotential \(w {({\lambda ^{\underline A}},{\lambda ^{\underline B}})_{ab}}\) is the antiselfdual part (in the name indices) of \(\vartheta _a^{\underline a}\vartheta _b^{\underline b}  \vartheta _b^{\underline a}\vartheta _a^{\underline b}\), which is also exact, for such spinor bases Equation (17) gives zero. Therefore, the integral of Bramson’s superpotential (17) measures the nonintegrability of the 1form basis \(\vartheta _a^{\underline A {{\underline A}\prime}} = \lambda _A^{\underline A}\bar \lambda _{{A\prime}}^{\underline {{A\prime}}}\), i.e. \(J_{\mathcal S}^{\underline A \underline B}\) is a measure of how much the actual 1form basis is ‘distorted’ by the curvature relative to the constant basis of Minkowski spacetime.
Thus the only question is how to specify a spin frame on \({\mathcal S}\) to be able to interpret \(J_{\mathcal S}^{\underline A \underline B}\) as angular momentum. It seems natural to choose those spinor fields that were used in the definition of the quasilocal energymomenta in the previous Section 8. At first sight this may appear to be only an ad hoc idea, but, recalling that in Section 8 we interpreted the elements of the spin spaces \(({{\bf{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\) as the ‘spinor constituents of the quasitranslations of \({\mathcal S}\prime\), we can justify such a choice. Based on our experience with the superpotentials for the various conserved quantities, the quasilocal angular momentum can be expected to be the integral of something like ‘superpotential’ × ‘quasirotation generator’, and the ‘superpotential’ is some expression in the first derivative of the basic variables, actually the tetrad or spinor basis. Since, however, Bramson’s superpotential is an algebraic expression of the basic variables, and the number of the derivatives in the expression for the angular momentum should be one, the angular momentum expressions based on Bramson’s superpotential must contain the derivative of the ‘quasirotations’, i.e. (possibly a combination of) the ‘quasitranslations’. Since, however, such an expression cannot be sensitive to the ‘change of the origin’, they can be expected to yield only the spin part of the angular momentum.
The following two specific constructions differ from each other only in the choice for the spin space \(({{\bf{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\), and correspond to the energymomentum constructions of the previous Section 8. The third construction (valid only in the Kerr spacetimes) is based on the sum of two terms, where one is Bramson’s expression, and uses the spinor fields of Section 8.3. Thus the present section is not independent of Section 8, and for the discussion of the choice of the spin spaces \(({{\bf{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\) we refer to that.
Another suggestion for the quasilocal spatial angular momentum, proposed by Liu and Yau [253], will be introduced in Section 10.4.1.
9.1 The LudvigsenVickers angular momentum
Under the conditions that ensured the LudvigsenVickers construction for the energymomentum would work in Section 8.1, the definition of their angular momentum is straightforward [259]. Since in Minkowski spacetime the LudvigsenVickers spinors are just the restriction to \({\mathcal S}\) of the constant spinor fields, by the general remark above the LudvigsenVickers spinangular momentum is zero in Minkowski spacetime.
9.2 Holomorphic/antiholomorphic spinangular momenta
Both the holomorphic and antiholomorphic spinangular momenta were calculated for small spheres [360]. In nonvacuum the holomorphic spinangular momentum reproduces the expected result (29), and, apart from a minus sign, the antiholomorphic construction does also. In vacuum both definitions give exactly Equation (62).
In general the antiholomorphic and the holomorphic spinangular momenta are diverging near the future null infinity of EinsteinMaxwell spacetimes as r and r^{2}, respectively. However, the coefficient of the diverging term in the antiholomorphic expression is just the spatial part of the BondiSachs energymomentum. Thus the antiholomorphic spinangular momentum is finite in the centreofmass frame, and hence it seems to describe only the spin part of the gravitational field. In fact, the PauliLubanski spin (63) built from this spinangular momentum and the antiholomorphic DouganMason energymomentum is always finite, free of ‘gauge’ ambiguities discussed in Section 8.2.4, and is built only from the gravitational data even in the presence of electromagnetic fields. In stationary spacetimes both constructions are finite and coincide with the ‘standard’ expression (34). Thus the antiholomorphic spinangular momentum defines an intrinsic angular momentum at the future null infinity. Note that this angular momentum is free of supertranslation ambiguities, because it is defined on the given cut in terms of the solutions of elliptic differential equations. These solutions can be interpreted as the spinor constituents of certain boostrotation BMS vector fields, but the definition of this angular momentum is not based on them [363].
9.3 A specific construction for the Kerr spacetime
The angular momentum of Bergqvist and Ludvigsen [68] for the Kerr spacetime is based on their special flat, nonsymmetric but metric connection explained briefly in Section 8.3, but their idea is not simply the use of the two \({{\tilde \nabla}_e}\)constant spinor fields in Bramson’s superpotential. Rather, in the background of their approach there are twistortheoretical ideas. (The twistortheoretic aspects of the analogous flat connection for the general KerrSchild class are discussed in [170].)
10 The HamiltonJacobi Method
If one is concentrating only on the introduction and study of the properties of the quasilocal quantities, but not interested in the detailed structure of the quasilocal (Hamiltonian) phase space, then perhaps the most natural way to derive the general formulae is to follow the HamiltonJacobi method. This was done by Brown and York in deriving their quasilocal energy expression [96, 97]. However, the HamiltonJacobi method in itself does not yield any specific construction. Rather, the resulting general expression is similar to a superpotential in the Lagrangian approaches, which should be completed by a choice for the reference configuration and for the generator vector field of the physical quantity (see Section 3.3.3). In fact, the ‘BrownYork quasilocal energy’ is not a single expression with a single welldefined prescription for the reference configuration. The same general formula with several other, mathematically inequivalent definitions for the reference configurations are still called the ‘BrownYork energy’. A slightly different general expression was used by Kijowski [237], Epp [133], and Liu and Yau [253]. Although the former follows a different route to derive his expression and the latter two are not connected directly to the canonical analysis (and, in particular, to the HamiltonJacobi method), the formalism and techniques that are used justify their presentation in this section.
The present section is based mostly on the original papers [96, 97] by Brown and York. Since, however, this is the most popular approach to finding quasilocal quantities and is the subject of very active investigations, especially from the point of view of the applications in black hole physics, this section is perhaps less complete than the previous ones. The expressions of Kijowski, Epp, and Liu and Yau will be treated in the formalism of Brown and York.
10.1 The BrownYork expression
10.1.1 The main idea
10.1.2 The variation of the action and the surface stressenergy tensor
The main idea of Brown and York [96, 97] is to calculate the analogous variation of an appropriate first order action of general relativity (or of the coupled matter + gravity system) and isolate the boundary term that could be analogous to the energy E above. To formulate this idea mathematically, they considered a compact spacetime domain D with topology Σ × [t_{1}, t_{2}] such that Σ × {t} correspond to compact spacelike hypersurfaces Σ_{ t }; these form a smooth foliation of D and the 2surfaces \({{\mathcal S}_t}: = \partial {\Sigma _t}\) (corresponding to ∂Σ × {t}) form a foliation of the timelike 3boundary ^{3}B of D. Note that this D is not a globally hyperbolic domain^{14}. To ensure the compatibility of the dynamics with this boundary, the shift vector is usually chosen to be tangent to \({{\mathcal S}_t}\) on ^{3}B. The orientation of ^{3}B is chosen to be outward pointing, while the normals both of \({\Sigma _1}:{\Sigma _{{t_1}}}\) and \({\Sigma _2}:{\Sigma _{{t_2}}}\) to be future pointing. The metric and extrinsic curvature on Σ_{ t } will be denoted, respectively, by h_{ ab } and χ_{ ab }, those on ^{3}B by γ_{ ab } and Θ_{ ab }.
Clearly, the traceχ action cannot be recovered as the volume integral of some scalar Lagrangian, because it is the Hilbert action plus a boundary integral of the trace χ, and the latter depends on the location of the boundary itself. Such a Lagrangian was found by Pons [317]. This depends on the coordinate system adapted to the boundary of the domain D of integration. An interesting feature of this Lagrangian is that it is second order in the derivatives of the metric, but it depends only on the first time derivative. A detailed analysis of the variational principle, the boundary conditions and the conserved charges is given. In particular, the asymptotic properties of this Lagrangian is similar to that of the ΓΓ Lagrangian of Einstein, rather than to that of Hilbert’s.
10.1.3 The general form of the BrownYork quasilocal energy
10.1.4 Further properties of the general expressions
Lau repeated the general analysis above using the tetrad (in fact, triad) variables and the Ashtekar connection on the timelike boundary instead of the traditional ADMtype variables [245]. Here the energy and momentum surface densities are reexpressed by the superpotential \({\vee _b}^{ae}\), given by Equation (10), in a frame adapted to the 2surface. (Lau called the corresponding superpotential 2form the ‘Sparling 2form’.) However, in contrast to the usual Ashtekar variables on a spacelike hypersurface [17], the time gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every orientable 3manifold Σ is parallelizable [297], and hence a globally defined orthonormal triad can be given on Σ, the only parallelizable closed orientable 2surface is the torus. Thus, on ^{3}B, we cannot impose the global time gauge condition with respect to any spacelike 2surface \({\mathcal S}\) in ^{3}B unless \({\mathcal S}\) is a torus. Similarly, the global radial gauge condition in the spacelike hypersurfaces Σ_{ t } (even on a small open neighbourhood of the whole 2surfaces \({{\mathcal S}_t}\) in Σ_{ t }) can be imposed on a triad field only if the 2boundaries \({{\mathcal S}_t} = \partial {\Sigma _t}\) are all tori. Obviously, these gauge conditions can be imposed on every local trivialization domain of the tangent bundle \(T{{\mathcal S}_t}\) of \({{\mathcal S}_t}\). However, since in Lau’s local expressions only geometrical objects (like the extrinsic curvature of the 2surface) appear, they are valid even globally (see also [246]). On the other hand, further investigations are needed to clarify whether or not the quasilocal Hamiltonian, using the Ashtekar variables in the radialtime gauge [247], is globally welldefined.
In general the BrownYork quasilocal energy does not have any positivity property even if the matter fields satisfy the dominant energy conditions. However, as G. Hayward pointed out [179], for the variations of the metric around the vacuum solutions that extremalize the Hamiltonian, called the ‘ground states’, the quasilocal energy cannot decrease. On the other hand, the interpretation of this result as a ‘quasilocal dominant energy condition’ depends on the choice of the time gauge above, which does not exist globally on the whole 2surface \({\mathcal S}\).
Booth and Mann [77] shifted the emphasis from the foliation of the domain D to the foliation of the boundary ^{3}B. (These investigations were extended to include charged black holes in [78], where the gauge dependence of the quasilocal quantities is also examined.) In fact, from the point of view of the quasilocal quantities defined with respect to the observers with world lines in ^{3}B and orthogonal to \({\mathcal S}\) it is irrelevant how the spacetime domain D is foliated. In particular, the quasilocal quantities cannot depend on whether or not the leaves Σ_{ t } of the foliation of D are orthogonal to ^{3}B. As a result, they recovered the quasilocal charge and energy expressions of Brown and York derived in the ‘orthogonal boundary’ case. However, they suggested a new prescription for the definition of the reference configuration (see Section 10.1.8). Also, they calculated the quasilocal energy for round spheres in the spherically symmetric spacetimes with respect to several moving observers, i.e., in contrast to Equation (73), they did not link the generator vector field ξ^{ a } to the normal t^{ a } of \({{\mathcal S}_t}\). In particular, the world lines of the observers are not integral curves of (∂/∂t) in the coordinate basis given in Section 4.2.1 on the round spheres.
Using an explicit, nondynamical background metric \(g_{ab}^0\), one can construct a covariant, first order Lagrangian \(L({g_{ab}},g_{ab}^0)\) for general relativity [230], and one can use the action \({I_D}[{g_{ab}},g_{ab}^0]\) based on this Lagrangian instead of the trace χ action. Fatibene, Ferraris, Francaviglia, and Raiteri [135] clarified the relationship between the two actions, I_{ D }[g_{ ab }] and \({I_D}[{g_{ab}},g_{ab}^0]\), and the corresponding quasilocal quantities: Considering the reference term S^{0} in the BrownYork expression as the action of the background metric \(g_{ab}^0\) (which is assumed to be a solution of the field equations), they found that the two first order actions coincide if the spacetime metrics g_{ ab } and \(g_{ab}^0\) coincide on the boundary ∂D. Using \(L({g_{ab}},g_{ab}^0)\), they construct the conserved Noether current for any vector field ξ^{ a } and, by taking its flux integral, define charge integrals \({Q_{\mathcal S}}[{\xi ^a},{g_{ab}},g_{ab}^0]\) on 2surfaces \({\mathcal S}\)^{15}. Again, the BrownYork quasilocal quantity E_{ t }[ξ^{ a },t^{ a }] and \({Q_{{{\mathcal S}_t}}}[{\xi ^a},{g_{ab}},g_{ab}^0]\) coincide if the spacetime metrics coincide on the boundary ∂D and ξ^{ a } has some special form. Therefore, although the two approaches are basically equivalent under the boundary condition above, this boundary condition is too strong both from the points of view of the variational principle and the quasilocal quantities. We will see in Section 10.1.8 that even the weaker boundary condition that only the induced 3metrics on ^{3}B from g_{ ab } and from \(g_{ab}^0\) be the same is still too strong.
10.1.5 The Hamiltonians
If we can write the action I[q(t)] of our mechanical system into the canonical form \(\int\nolimits_{{t_1}}^{{t_2}} {\left[ {{p_a}{{\dot q}^a}  H({q^a},{p_a},t)} \right]dt}\), then it is straightforward to read off the Hamiltonian of the system. Thus, having accepted the trace χ action as the action for general relativity, it is natural to derive the corresponding Hamiltonian in the analogous way. Following this route Brown and York derived the Hamiltonian, corresponding to the ‘basic’ (or nonreferenced) action I^{1} too [97]. They obtained the familiar integral of the sum of the Hamiltonian and the momentum constraints, weighted by the lapse N and the shift N^{ a }, respectively, plus E_{ t }[Nt^{ a } + N^{ a }, t^{ a }], given by Equation (72), as a boundary term. This result is in complete agreement with the expectations, as their general quasilocal quantities can also be recovered as the value of the Hamiltonian on the constraint surface (see also [77]). This Hamiltonian was investigated further in [95]. Here all the boundary terms that appear in the variation of their Hamiltonian are determined and decomposed with respect to the 2surface ∂Σ. It is shown that the change of the Hamiltonian under a boost of yields precisely the boosts of the energy and momentum surface density discussed above.
Booth and Fairhurst [73] reexamined the general form of the BrownYork energy and angular momentum from a Hamiltonian point of view^{16}. Their starting point is the observation that the domain D is not isolated from its environment, thus the quasilocal Hamiltonian cannot be time independent. Therefore, instead of the standard Hamiltonian formalism for the autonomous systems, a more general formalism, based on the extended phase space, must be used. This phase space consists of the usual bulk configuration and momentum variables \(({h_{ab}},{\tilde p^{ab}})\) on the typical 3manifold Σ and the time coordinate t, the space coordinates x^{ A } on the 2boundary \({\mathcal S} = \partial \sum\), and their conjugate momenta π and π_{ A }, respectively.
Their second important observation is that the BrownYork boundary conditions are too restrictive: The 2metric, the lapse, and the shift need not to be fixed but their variations corresponding to diffeomorphisms on the boundary must be allowed. Otherwise diffeomorphisms that are not isometries of the 3metric γ_{ ab } on ^{3}B cannot be generated by any Hamiltonian. Relaxing the boundary conditions appropriately, they show that there is a Hamiltonian on the extended phase space which generates the correct equations of motions, and the quasilocal energy and angular momentum expression of Brown and York are just (minus) the momentum π conjugate to the time coordinate t. The only difference between the present and the original BrownYork expressions is the freedom in the functional form of the unspecified reference term: Because of the more restrictive boundary conditions of Brown and York their reference term is less restricted. Choosing the same boundary conditions in both approaches the resulting expressions coincide completely.
10.1.6 The flat space and light cone references
The quasilocal quantities introduced above become welldefined only if the subtraction term S^{0} in the principal function is specified. The usual interpretation of a choice for S^{0} is the calibration of the quasilocal quantities, i.e. fixing where to take their zero value.
The only restriction on S^{0} that we had is that it must be a functional of the metric γ_{ ab } on the timelike boundary ^{3}B. To specify S^{0}, it seems natural to expect that the principal function S be zero in Minkowski spacetime [158, 96]. Then S^{0} would be the integral of the trace Θ^{0} of the extrinsic curvature of ^{3}B if it were embedded in Minkowski spacetime with the given intrinsic metric γ_{ ab }. However, a general Lorentzian 3manifold (^{3}B, γ_{ ab }) cannot be isometrically embedded, even locally, into the Minkowski spacetime. (For a detailed discussion of this embeddability, see [96] and Section 10.1.8.)
Another assumption on S^{0} might be the requirement of the vanishing of the quasilocal quantities, or of the energy and momentum surface densities, or only of the energy surface density ε, in some reference spacetime, e.g. in Minkowski or in antideSitter spacetime. Assuming that S^{0} depends on the lapse N and shift N^{ a } linearly, the functional derivatives (∂S^{0}/∂N) and ((∂S^{0}/∂N^{ a }) depend only on the 2metric q_{ ab } and on the boostgauge that ^{3}B defined on \({{\mathcal S}_t}\). Therefore, ε and j_{ a } take the form (74), and by the requirement of the vanishing of ε in the reference spacetime it follows that k^{0} should be the trace of the extrinsic curvature of \({{\mathcal S}_t}\) in the reference spacetime. Thus it would be natural to fix k^{0} as the trace of the extrinsic curvature of \({{\mathcal S}_t}\) when \(({{\mathcal S}_t},{q_{ab}})\) is embedded isometrically into the reference spacetime. However, this embedding is far from being unique (since, in particular, there are two independent normals of \({\mathcal S_t}\) in the spacetime and it would not be fixed which normal should be used to calculate k^{0}), and hence the construction would be ambiguous. On the other hand, one could require \(({{\mathcal S}_t},{q_{ab}})\) to be embedded into flat Euclidean 3space, i.e. into a spacelike hyperplane of Minkowski spacetime^{17}. This is the choice of Brown and York [96, 97]. In fact, at least for a large class of 2surfaces \(({{\mathcal S}_t},{q_{ab}})\), such an embedding exists and is unique: If \({\mathcal S_t} \approx {S_t}\) and the metric is C^{2} and has everywhere positive scalar curvature, then there is an isometric embedding of \(({\mathcal S_t},{q_{ab}})\) into the flat Euclidean 3space [195], and apart from rigid motions this embedding is unique [346]. The requirement that the scalar curvature of the 2surface must be positive can be interpreted as some form of the convexity, as in the theory of surfaces in the Euclidean space. However, there are counterexamples even to local isometric embeddability when this convexity condition is violated [276]. A particularly interesting 2surface that cannot be isometrically embedded into the flat 3space is the event horizon of the Kerr black hole if the angular momentum parameter a exceeds the irreducible mass (but is still not greater than the mass parameter m), i.e. if \(\sqrt 3 m<2\vert a \vert <2m\) [343]. Thus, the construction works for a large class of 2surfaces, but certainly not for every potentially interesting 2surface. The convexity condition is essential.
It is known that the (local) isometric embeddability of \(({\mathcal S_t},{q_{ab}})\) into flat 3space with extrinsic curvature \(k_{ab}^0\) is equivalent to the GaussCodazziMainardi equations \({\delta _a}({k^{0a}}_b  \delta _b^a{k^0}) = 0\) and \(^{\mathcal S}R  {({k^0})^2} + k_{ab}^0{k^{0ab}} = 0\). Here δ_{ a } is the intrinsic LeviCivita covariant derivative and \(^{\mathcal S}R\) is the corresponding curvature scalar on \({\mathcal S}\) determined by q_{ ab }. Thus, for given q_{ ab } and (actually the flat) embedding geometry, these are three equations for the three components of \(k_{ab}^0\), and hence, if the embedding exists, q_{ ab } determines k^{0}. Therefore, the subtraction term k^{0} can also be interpreted as a solution of an underdetermined elliptic system which is constrained by a nonlinear algebraic equation. In this form the definition of the reference term is technically analogous to the definition of those in Sections 7, 8, and 9, but, by the nonlinearity of the equations, in practice it is much more difficult to find the reference term k^{0} than the spinor fields in the constructions of Sections 7, 8, and 9.
Accepting this choice for the reference configuration, the reference SO(1, 1) gauge potential \(A_a^0\) will be zero in the boostgauge in which the timelike normal of \({{\mathcal S}_t}\) in the reference Minkowski spacetime is orthogonal to the spacelike 3plane, because this normal is constant. Thus, to summarize, for convex 2surfaces the flat space reference of Brown and York is uniquely determined, k^{0} is determined by this embedding, and \(A_a^0 = 0\). Then \(8\pi G{S^0} =  \int\nolimits_{{{\mathcal S}_t}} {N{k^0}d{{\mathcal S}_t}}\), from which s^{ ab } can be calculated (if needed). The procedure is similar if, instead of a spacelike hyperplane of Minkowski spacetime, a spacelike hypersurface of constant curvature (for example in the deSitter or antideSitter spacetime) is used. The only difference is that extra (known) terms appear in the GaussCodazziMainardi equations.
Brown, Lau, and York considered another prescription for the reference configuration as well [94, 248, 249]. In this approach the 2surface \(({{\mathcal S}_t},{q_{ab}})\) is embedded into the light cone of a point of the Minkowski or antide Sitter spacetime instead of a spacelike hypersurface of constant curvature. The essential difference between the new (‘light cone reference’) and the previous (‘flat space reference’) prescriptions is that the embedding into the light cone is not unique, but the reference term k^{0} may be given explicitly, in a closed form. The positivity of the Gauss curvature of the intrinsic geometry of \(({\mathcal S},{q_{ab}})\) is not needed. In fact, by a result of Brinkmann [91], every locally conformally flat Riemannian ngeometry is locally isometric to an appropriate cut of a light cone of the n + 2 dimensional Minkowski spacetime (see also [133]). To achieve uniqueness some extra condition must be imposed. This may be the requirement of the vanishing of the ‘normal momentum density’ \(j_\vdash^0 = 0\) in the reference spacetime [248, 249], yielding \({k^0} = \sqrt {{2^\mathcal S}R + 4/{{\rm{\lambda}}^2}}\), where \(^{\mathcal S}R\) is the Ricci scalar of \(({\mathcal S},{q_{ab}})\) and λ is the cosmological constant of the reference spacetime. The condition \(j_\vdash^0 = 0\) defines something like a ‘rest frame’ in the reference spacetime. Another, considerably more complicated choice for the light cone reference term is used in [94].
10.1.7 Further properties and the various limits
Although the general, nonreferenced expressions are additive, the prescription for the reference term k^{0} destroys the additivity in general. In fact, if \({\mathcal S}\prime\) and \({\mathcal S}\prime \prime\) are 2surfaces such that \({\mathcal S}\prime \cap {\mathcal S}\prime \prime\) is connected and 2dimensional (more precisely, it has a nonempty open interior for example in \({\mathcal S}\prime\)), then in general \(\overline {{\mathcal S}\prime \cup {\mathcal S}\prime \prime  {\mathcal S}\prime \cap {\mathcal S}\prime \prime}\) (overline means topological closure) is not guaranteed to be embeddable into the flat 3space, and even if it is embeddable then the resulting reference term k^{0} differs from the reference terms k′^{0} and k″^{0} determined from the individual embeddings.
As it is noted in [77], the BrownYork energy with the flat space reference configuration is not zero in Minkowski spacetime in general. In fact, in the standard spherical polar coordinates let Σ_{1} be the spacelike hyperboloid \(t =  \sqrt {{\rho ^2} + {r^2}} ,{\Sigma _0}\) the hyperplane t = −T = const. < − ρ < 0 and \({\mathcal S}: = {\Sigma _0} \cap {\Sigma _1}\), the sphere of radius \(\sqrt {{T^2}  {\rho ^2}}\) in the t = −T hyperplane. Then the trace of the extrinsic curvature of \({\mathcal S}\) in Σ_{0} and in Σ_{1} is \(2/\sqrt {{T^2}  {\rho ^2}}\) and \(2T/\rho \sqrt {{T^2}  {\rho ^2}}\), respectively. Therefore, the BrownYork quasilocal energy (with the flat 3space reference) associated with \({\mathcal S}\) and the normals of Σ_{1} on \({\mathcal S}\) is \( \sqrt {(T + \rho){{(T  \rho)}^3}}/(\rho G)\). Similarly, the BrownYork quasilocal energy with the light cone references in [248] and in [94] is also negative for such surfaces with the boosted observers.
Recently, Shi, and Tam [341] proved interesting theorems in Riemannian 3geometries, which can be used to prove positivity of the BrownYork energy if the 2surface \({\mathcal S}\) is a boundary of some timesymmetric spacelike hypersurface on which the dominant energy condition holds. In the timesymmetric case this energy condition is just the condition that the scalar curvature be nonnegative. The key theorem of Shi and Tam is the following: Let Σ be a compact, smooth Riemannian 3manifold with nonnegative scalar curvature and smooth 2boundary \({\mathcal S}\) such that each connected component \({{\mathcal S}_i}\) of \({\mathcal S}\) is homeomorphic to S^{2} and the scalar curvature of the induced 2metric on \({{\mathcal S}_i}\) is strictly positive. Then for each component \({\oint\nolimits_{{{\mathcal S}_i}}}kd{{\mathcal S}_i} \leq {\oint\nolimits_{{{\mathcal S}_i}}}{k^0}d{{\mathcal S}_i}\) holds, where k is the trace of the extrinsic curvature of \({\mathcal S}\) in Σ with respect to the outward directed normal, and k^{0} is the trace of the extrinsic curvature of \({{\mathcal S}_i}\) in the flat Euclidean 3space when \({{\mathcal S}_i}\) is isometrically embedded. Furthermore, if in these inequalities the equality holds for at least one \({{\mathcal S}_i}\), then \({\mathcal S}\) itself is connected and Σ is flat. This result is generalized in [342] by weakening the energy condition, whenever lower estimates of the BrownYork energy can still be given.
The energy expression for round spheres in spherically symmetric spacetimes was calculated in [97, 77]. In the spherically symmetric metric discussed in Section 4.2.1, on the round spheres the BrownYork energy with the flat space reference and fleet of observers ∂/∂t on \({\mathcal S}\) is \(G{E_{{\rm{BY}}}}[{{\mathcal S}_r},{(\partial/\partial t)^a}] = r(1  \exp ( {\rm{\alpha))}}\). In particular, it is \(r[1  \sqrt {1  (2m/r)]}\) for the Schwarzschild solution. This deviates from the standard round sphere expression, and, for the horizon of the Schwarzschild black hole it is 2m (instead of the expected m). (The energy has also been calculated explicitly for boosted foliations of the Schwarzschild solution and for round spheres in isotropic cosmological models [95].) The Newtonian limit can be derived from this by assuming that m is the mass of a fluid ball of radius r and m/r is small: It is \(G{E_{{\rm{BY}}}} = m + ({m^2}/2r) + {\mathcal O}({r^{ 2}})\). The first term is simply the mass defined at infinity, and the second term is minus the Newtonian potential energy associated with building a spherical shell of mass m and radius r from individual particles, bringing them together from infinity. However, taking into account that on the Schwarzschild horizon GE_{BY} = 2m while at the spatial infinity it is just m, the BrownYork energy is monotonically decreasing with r. Also, the first law of black hole mechanics for spherically symmetric black holes can be recovered by identifying E_{BY} with the internal energy [96, 97]. The thermodynamics of the SchwarzschildantideSitter black holes was investigated in terms of the quasilocal quantities in [92]. Still considering E_{BY} to be the internal energy, the temperature, surface pressure, heat capacity, etc. are calculated (see Section 13.3.1). The energy has also been calculated for the EinsteinRosen cylindrical waves [95].
The energy is explicitly calculated for three different kinds of 2spheres in the t = const. slices (in the BoyerLindquist coordinates) of the slow rotation limit of the Kerr black hole spacetime with the flat space reference [264]. These surfaces are the r = const. surfaces (such as the outer horizon), spheres whose intrinsic metric (in the given slow rotation approximation) is of a metric sphere of radius R with surface area 4πR^{2}, and the ergosurface (i.e. the outer boundary of the ergosphere). The slow rotation approximation is defined such that ∣a∣/R ≪ 1, where R is the typical spatial measure of the 2surface. In the first two cases the angular momentum parameter a enters the energy expression only in the m^{2}a^{2}/R^{3} order. In particular, the energy for the outer horizon \({r_ +}: = m + \sqrt {{m^2}  {a^2}}\) is \(2m[1  {a^2}/(8{m^2}) + {\mathcal O}({a^4}/r_ + ^4)]\), which is twice the irreducible mass of the black hole. An interesting feature of this calculation is that the energy cannot be calculated for the horizon directly, because, as we noted in the previous point, the horizon itself cannot be isometrically embedded into a flat 3space if the angular momentum parameter exceeds the irreducible mass [343]. The energy for the ergosurface is positive, as for the other two kinds of surfaces.
The spacelike infinity limit of the charges interpreted as the energy, spatial momentum, and spatial angular momentum are calculated in [95] (see also [176]). Here the flat space reference configuration and the asymptotic Killing vectors of the spacetime are used, and the limits coincide with the standard ADM energy, momentum, and spatial angular momentum. The analogous calculation for the centreofmass is given in [42]. It is shown that the corresponding large sphere limit is just the centreofmass expression of Beig and Ó Murchadha [47]. Here the centreofmass integral in terms of a charge integral of the curvature is also given.
Although the prescription for the reference configuration by Hawking and Horowitz cannot be imposed for a general timelike 3boundary ^{3}B (see Section 10.1.8), asymptotically, when ^{3}B is pushed out to infinity, this prescription can be used, and coincides with the prescription of Brown and York. Choosing the background metric \(g_{ab}^0\) to be the antideSitter one, Hawking and Horowitz [176] calculated the limit of the quasilocal energy, and they found it to tend to the AbbottDeser energy. (For the spherically symmetric, SchwarzschildantideSitter case see also [92].) In [93] the null infinity limit of the integral of N(k^{0}−k)/(8πG) was calculated both for the lapses N generating asymptotic time translations and supertranslations at the null infinity, and the fleet of observers was chosen to tend to the BMS translation. In the former case the BondiSachs energy, in the latter case Geroch’s supermomenta are recovered. These calculations are based directly on the Bondi form of the spacetime metric, and do not use the asymptotic solution of the field equations. In a slightly different formulation Booth and Creighton calculated the energy flux of outgoing gravitational radiation [76] (see also Section 13.1) and they recovered the BondiSachs massloss.
However, the calculation of the small sphere limit based on the flat space reference configuration gave strange results [249]. While in nonvacuum the quasilocal energy is the expected (4π/3)r^{3}T_{ ab }t^{ a }t^{ b }, in vacuum it is proportional to 4E_{ ab }E^{ ab } + H_{ ab }H^{ ab } instead of the BelRobinson ‘energy’ T_{ abcd }t^{ a }t^{ b }t^{ c }t^{ d }. (Here E_{ ab } and H_{ ab } are, respectively, the conformal electric and conformal magnetic curvatures, and t^{ a } plays a double role: It defines the 2sphere of radius r [as is usual in the small sphere calculations], and defines the fleet of observers on the 2sphere.) On the other hand, the special light cone reference used in [94, 249] reproduces the expected result in nonvacuum, and yields [1/(90G)]r^{5}T_{ abcd }t^{ a }t^{ b }t^{ c }t^{ d } in vacuum.
The light cone reference \({k^0} = \sqrt {{2^{\mathcal S}}R + 4/{\lambda ^2}}\) was shown to work in the large sphere limit near the null and spatial infinities of asymptotically flat, and near the infinity of asymptotically antideSitter spacetimes [248]. Namely, the BrownYork quasilocal energy expression with this null cone reference term tends to the BondiSachs, the ADM, and AbbottDeser energies, respectively. The supermomenta of Geroch at null infinity can also be recovered in this way. The proof is simply a demonstration of the fact that this light cone and the flat space prescriptions for the subtraction term have the same asymptotic structure up to order \({\mathcal O}({r^{ 3}})\). This choice seems to work properly only in the asymptotics, because for small ellipsoids in the Minkowski spacetime this definition yields nonzero energy and for small spheres in vacuum it does not yield the BelRobinson ‘energy’ [250].
10.1.8 Other prescriptions for the reference configuration
As we noted above, Hawking, Horowitz, and Hunter [176, 177] defined their reference configuration by embedding the Lorentzian 3manifold (^{3}B, γ_{ ab }) isometrically into some given Lorentzian spacetime, e.g. into the Minkowski spacetime (see also [158]). However, for the given intrinsic 3metric γ_{ ab } and the embedding 4geometry the corresponding Gauss and CodazziMainardi equations form a system of 6 + 8 = 14 equations for the six components of the extrinsic curvature Θ_{ ab } [96]. Thus, in general, this is a highly overdetermined system, and hence it may be expected to have a solution only in exceptional cases. However, even if such an embedding existed, then even the small perturbations of the intrinsic metric h_{ ab } would break the conditions of embeddability. Therefore, in general this prescription for the reference configuration can work only if the 3surface ^{3}B is ‘pushed out to infinity’ but does not work for finite 3surfaces [96].
To rule out the possibility that the BrownYork energy can be nonzero even in Minkowski spacetime (on 2surfaces in the boosted flat data set), Booth and Mann [77] suggested to embed \(({\mathcal S},{q_{ab}})\) isometrically into a reference spacetime \(({M^0},g_{ab}^0)\) (mostly into the Minkowski spacetime) instead of a spacelike slice of it, and to map the evolution vector field ξ^{ a } = Nt^{ a } + N^{ a } of the dynamics, tangent to ^{3}B, to a vector field ξ^{0a} in M^{0} such that \({\rlap{\!}L}_{\xi}{q_{ab}} = {\phi ^\ast}({\rlap{\!}L}_{\xi 0}q_{ab}^0)\) and \({\xi ^a}{\xi _{^a}} = {\phi ^\ast}({\xi ^{0a}}\xi _a^0)\). Here ϕ is a diffeomorphism mapping an open neighbourhood U of \({\mathcal S}\) in M into M^{0} such that \(\phi {\vert_{\mathcal S}}\), the restriction of ϕ to \({\mathcal S}\), is an isometry, and Ł_{ ξ }q_{ ab } denotes the Lie derivative of q_{ ab } along This condition might be interpreted as some local version of that of Hawking, Horowitz, and Hunter. However, Booth and Mann did not investigate the existence or the uniqueness of this choice.
10.2 Kijowski’s approach
10.2.1 The role of the boundary conditions
In the BrownYork approach the leading principle was the claim to have a welldefined variational principle. This led them to modify the Hilbert action to the traceχaction and the boundary condition that the induced 3metric on the boundary of the domain D of the action is fixed.
However, as stressed by Kijowski [237, 239], the boundary conditions have much deeper content. For example in thermodynamics the different definitions of the energy (internal energy, enthalpy, free energy, etc.) are connected with different boundary conditions. Fixing the pressure corresponds to enthalpy, but fixing the temperature to free energy. Thus the different boundary conditions correspond to different physical situations, and, mathematically, to different phase spaces^{18}. Therefore, to relax the a priori boundary conditions, Kijowski abandoned the variational principle and concentrated on the equations of motions. However, to treat all possible boundary conditions on an equal footing he used the enlarged phase space of Tulczyjew (see for example [239])^{19}. The boundary condition of Brown and York is only one of the possible boundary conditions.
10.2.2 The analysis of the Hilbert action and the quasilocal internal and free energies
Starting with the variation of Hilbert’s Lagrangian (in fact, the corresponding HamiltonJacobi principal function on a domain D above), and defining the Hamiltonian by the standard Legendre transformation on the typical compact spacelike 3manifold Φ and its boundary \({\mathcal S} = \partial \Sigma\) too, Kijowski arrived at a variation formula involving the value on \({\mathcal S}\) of the variation of the canonical momentum, \({{\tilde \pi}^{ab}}: =  {1 \over {16\pi G}}\sqrt {\vert\gamma \vert} ({\Theta ^{ab}}  \Theta {\gamma ^{ab}})\), conjugate to γ_{ ab }. (Apart from a numerical coefficient and the subtraction term, this is essentially the surface stressenergy tensor τ^{ ab } given by Equation (67).) Since, however, it is not clear whether or not the initial + boundary value problem for the Einstein equations with fixed canonical momenta (i.e. extrinsic curvature) is well posed, he did not consider the resulting Hamiltonian as the appropriate one, and made further Legendre transformations on the boundary \({\mathcal S}\).
10.3 The expression of Epp
10.3.1 The general form of Epp’s expression
10.3.2 The definition of the reference configuration
The subtraction term in Equation (80) is defined through an isometric embedding of \(({\mathcal S},{q_{ab}})\) into some reference spacetime instead of a 3space. This spacetime is usually Minkowski or antideSitter spacetime. Since the 2surface data consist of the metric, the two extrinsic curvatures and the SO(1, 1)gauge potential, for given \(({\mathcal S},{q_{ab}})\) and ambient spacetime \(({M^0},g_{ab}^0)\) the conditions of the isometric embedding form a system of six equations for eight quantities, namely for the two extrinsic curvatures and the gauge potential A_{ e } (see Section 4.1.2, and especially Equations (20, 21)). Therefore, even a naive function counting argument suggests that the embedding exists, but is not unique. To have uniqueness, additional conditions must be imposed. However, since A_{ e } is a gauge field, one condition might be a gauge fixing in the normal bundle, and Epp’s suggestion is to require that the curvature of the connection 1form A_{ e } in the reference spacetime and in the physical spacetime be the same [133]. Or, in other words, not only the intrinsic metric q_{ ab } of \({\mathcal S}\) is required to be preserved in the embedding, but the whole curvature \({f^a}_{bcd}\) of the connection δ_{ e } as well. In fact, in the connection δ_{ e } on the spinor bundle \({{\rm{S}}^A}({\mathcal S})\) both the LeviCivita and the SO(1, 1) connection coefficients appear on an equal footing. (Recall that we interpreted the connection δ_{ e } to be a part of the universal structure of \({\mathcal S}\).) With this choice of the reference configuration \({E_{\rm{E}}}({\mathcal S})\) depends not only on the intrinsic 2metric q_{ ab } of \({\mathcal S}\), but on the connection δ_{ e } on the normal bundle as well.
Suppose that \({\mathcal S}\) is a 2surface in M such that k^{2} > l^{2} with k > 0, and, in addition, \(({\mathcal S},{q_{ab}})\) can be embedded into the flat 3space with k^{0} ≥ 0. Then there is a boost gauge (the ‘quasilocal rest frame’) in which \({E_{\rm{E}}}({\mathcal S})\) coincides with the BrownYork energy \({E_{{\rm{BY}}}}({\mathcal S},{t^a})\) in the particular boostgauge t^{ a } for which t^{ a }Q_{ a } = 0. Consequently, every statement stated for the latter is valid for \({E_{\rm{E}}}({\mathcal S})\), and every example calculated for \({E_{{\rm{BY}}}}({\mathcal S},{t^a})\) is an example for \({E_{\rm{E}}}({\mathcal S})\) as well [133]. A clear and careful discussion of the potential alternative choices for the reference term, especially their potential connection with the angular momentum, is also given there.
10.3.3 The various limits
First, it should be noted that Epp’s quasilocal energy is vanishing in Minkowski spacetime for any 2surface, independently of any fleet of observers. In fact, if \({\mathcal S}\) is a 2surface in Minkowski spacetime, then the same physical Minkowski spacetime defines the reference spacetime as well, and hence \({E_{\rm{E}}}({\mathcal S}) = 0\). For round spheres in the Schwarzschild spacetime it yields the result that E_{BY} gave. In particular, for the horizon it is 2m/G (instead of m/G), and at infinity it is m/G [133]. Thus, in particular, E_{E} is also monotonically decreasing with r in Schwarzschild spacetime.
Epp calculated the various limits of his expression too [133]. In the large sphere limit near spatial infinity he recovered the AshtekarHansen form of the ADM energy, at future null infinity the BondiSachs energy. The technique that is used in the latter calculations is similar to that of [93]. In nonvacuum in the small sphere limit \({E_{\rm{E}}}({\mathcal S})\) reproduces the standard \({{4\pi} \over 3}{r^3}{T_{ab}}{t^a}{t^b}\) result, but the calculations for the vacuum case are not completed. The leading term is still probably of order r^{5}, but its coefficient has not been calculated. Although in these calculations t^{ a } plays the role only of fixing the 2surfaces, as a result we got energy seen by the observer t^{ a } instead of mass. It is this reason why \({E_{\rm{E}}}({\mathcal S})\) is considered to be energy rather than mass. In the asymptotically antideSitter spacetime (with the antideSitter spacetime as the reference spacetime) E_{E} gives zero. This motivated Epp to modify his expression to recover the mass parameter of the SchwarzschildantideSitter spacetime at the infinity. The modified expression is, however, not boostgauge invariant. Here the potential connection with the AdS/CFT correspondence is also discussed (see also [33]).
10.4 The expression of Liu and Yau
10.4.1 The LiuYau definition
10.4.2 The main properties of \({E_{{\rm{KLY}}}}({\mathcal S})\)
The most important property of the quasilocal energy definition (81) is its positivity. Namely [253], let \(\sum\) be a compact spacelike hypersurface with smooth boundary ∂Σ, consisting of finitely many connected components \({{\mathcal S}_1}, \ldots, {{\mathcal S}_k}\) such that each of them has positive intrinsic curvature. Suppose that the matter fields satisfy the dominant energy condition on Σ. Then \(\sum\) is strictly positive unless the spacetime is flat along Σ. In this case ∂Σ is necessarily connected. The proof is based on the use of Jang’s equation [217], by means of which the general case can be reduced to the results of Shi and Tam in the timesymmetric case [341], stated in Section 10.1.7 (see also [399]). This positivity result is generalized in [254]: It is shown that \({E_{{\rm{KLY}}}}({{\mathcal S}_i})\) is nonnegative for all i = 1,…, k, and if \({E_{{\rm{KLY}}}}({{\mathcal S}_i}) = 0\) for some i then the spacetime is flat along Σ and ∂Σ is connected. (In fact, since E_{KLY} (∂Σ) depends only on ∂Σ but is independent of the actual Σ, if the energy condition is satisfied on the domain of dependence D(Σ) then E_{KLY}(∂Σ) = 0 implies the flatness of the spacetime along every Cauchy surface for D(Σ), i.e. the flatness of the whole domain of dependence too.)
However, \({E_{{\rm{KLY}}}}({\mathcal S})\) can be positive even if \({\mathcal S}\) is in the Minkowski spacetime. In fact, for given intrinsic metric q_{ ab } on \({\mathcal S}\) (with positive scalar curvature) \({\mathcal S}\) can be embedded into the flat ℝ^{3}; this embedding is unique, and the trace of the extrinsic curvature k^{0} is determined by q_{ ab }. On the other hand, the isometric embedding of \({\mathcal S}\) in the Minkowski spacetime is not unique: The equations of the embedding (i.e. the Gauss, the CodazziMainardi, and the Ricci equations) form a system of six equations for the six components of the two extrinsic curvatures k_{ ab } and l_{ ab } and the two components of the SO(1, 1) gauge potential A_{ e }. Thus, even if we impose a gauge condition for the connection 1form A_{ e }, we have only six equations for the seven unknown quantities, leaving enough freedom to deform \({\mathcal S}\) (with given, fixed intrinsic metric) in the Minkowski spacetime to get positive KijowskiLiuYau energy. Indeed, specific 2surfaces in the Minkowski spacetime are given in [292] for which \({E_{{\rm{KLY}}}}({\mathcal S}) > 0\).
11 Towards a Full Hamiltonian Approach
The HamiltonJacobi method is only one possible strategy to define the quasilocal quantities in a large class of approaches, called the Hamiltonian or canonical approaches. Thus there is a considerable overlap between the various canonical methods, and hence the cutting of the material into two parts (Section 10 and Section 11) is, in some sense, artifical. In the previous Section 10 we reviewed those approaches that are based on the analysis of the action, while in the present we discuss those that are based primarily of the analysis of the Hamiltonian in the spirit of Regge and Teitelboim [319]^{20}.
By a full Hamiltonian analysis we mean a detailed study of the structure of the quasilocal phase space, including the constraints, the smearing fields, the symplectic structure and the Hamiltonian itself, according to the standard or some generalized Hamiltonian scenarios, in the traditional 3+1 or in the fully Lorentzcovariant form, or even in the 2+2 form, using the metric or triad/tetrad variables (or even the Weyl or Dirac spinors). In the literature of canonical general relativity (at least in the asymptotically flat context) there are examples for all these possibilities, and we report on the quasilocal investigations on the basis of the decomposition they use. Since the 2+2 decomposition of the spacetime is less known, we also summarize its basic idea.
11.1 The 3+1 approaches
There is a lot of literature on the canonical formulation of general relativity both in the traditional ADM and the Møller tetrad (or, recently, the closely related complex Ashtekar) variables. Thus it is quite surprising how little effort has been spent to systematically quasilocalize them. One motivation for the quasilocalization of the ADMReggeTeitelboim analysis came from the need for the microscopic understanding of black hole entropy [32, 31, 99]: What are the microscopic degrees of freedom behind the phenomenological notion of black hole entropy? Since the aim of the present paper is to review the construction of the quasilocal quantities in classical general relativity, we discuss only the classical 2surface observables by means of which the ‘quantum edge states’ on the black hole event horizons were intended to be constructed.
11.1.1 The 2surface observables
Analogous investigations were done by Husain and Major in [210]. Using Ashtekar’s complex variables [17] they determined all the local boundary conditions for the canonical variables \(A_a^{\rm{i}},\tilde E_{\rm{i}}^a\) and for the lapse N, the shift N^{ a }, and the internal gauge generator N^{ i } on \({\mathcal S}\) that ensure the functional differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints. Although there are several possibilities, they discussed the two most significant cases. In the first case the generators N, N^{ a }, and N^{ i } are vanishing on \({\mathcal S}\), whenever there are infinitely many 2surface observables both from the diffeomorphism and the Gauss constraints, but no observable from the Hamiltonian constraint. The structure of these observables is similar to that of those coming from the ADM diffeomorphism constraint above. The other case considered is when the canonical momentum \(\tilde E_{\rm{i}}^a\) (and hence, in particular, the 3metric) is fixed on the 2boundary. Then the quasilocal energy could be an observable, as in the ADM analysis above.
All of the papers [32, 31, 99, 210] discuss the analogous phenomenon of how the gauge freedoms are getting to be true physical degrees of freedom in the presence of 2surfaces on the 2surfaces themselves in the ChernSimons and BF theories. Weakening the boundary conditions further (allowing certain boundary terms in the variation of the constraints) a more general algebra of ‘observables’ can be obtained [101, 296]: They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in [101] is based on the socalled covariant Noether charge formalism below.) Since this algebra is well known in conformal field theories, this approach might be a basis of understanding the microscopic origin of the black hole entropy [100, 101, 102, 296, 103]. However, this quantum issue is beyond the scope of the present review.
11.2 Approaches based on the doublenull foliations
11.2.1 The 2+2 decomposition
The decomposition of the spacetime in a 2+2 way with respect to two families of null hypersurfaces is as old as the study of gravitational radiation and the concept of the characteristic initial value problem (see for example [326, 306]). The basic idea is that we foliate an open subset U of the spacetime by a 2parameter family of (e.g. closed) spacelike 2surfaces: If \({\mathcal S}\) is the typical 2surface, then this foliation is defined by a smooth embedding \(\phi: {\mathcal S} \times ( \epsilon, \epsilon) \times ( \epsilon, \epsilon) \rightarrow U:(p,{\nu _ +}{\nu _ }) \mapsto \phi (p,{\nu _ +},{\nu _ })\). Then, keeping ν_{+} fixed and varying ν_{−}, or keeping ν_{−} fixed and varying ν_{+}, respectively, \({{\mathcal S}_{{\nu _ +},{\nu _ }}}: = \phi ({\mathcal S},{\nu _ +},{\nu _ })\) defines two 1parameter families of hypersurfaces Σ_{ν+} and Σ_{ν−}. Requiring one (or both) of the hypersurfaces Σ_{ν±} to be null, we get a socalled null (or doublenull, respectively) foliation of U. (In Section 4.1.8 we required the hypersurfaces Σ_{ν±} to be null only for the special value ν_{±} = 0 of the parameters.) As is well known, because of the conjugate points, in the null or double null cases the foliation can be welldefined only locally. For fixed ν_{+} and \(p \in {\mathcal S}\) the prescription ν_{−} ↦ ϕ(p, ν_{+}, ν_{−}) defines a curve through \(\phi (p,{\nu _ +},0) \in {{\mathcal S}_{{\nu _ +},0}}\) in Σ_{ν+}, and hence a vector field \(\xi _ + ^a: = {(\partial/\partial {\nu _ })^a}\) tangent everywhere to Σ_{+} on U. The Lie bracket of \(\xi _ + ^a\) and the analogously defined \(\xi _  ^a\) is zero. There are several inequivalent ways of introducing coordinates or rigid frame fields on U, which are fit naturally to the null or double null foliation \(\{{{\mathcal S}_{{\nu _ +},{\nu _ }}}\}\), in which the (vacuum) Einstein equations and Bianchi identities take a relatively simple form [326, 152, 123, 348, 381, 180, 165, 82, 189].
Defining the ‘time derivative’ to be the Lie derivative, for example, along the vector field \(\xi _ + ^a\), the Hilbert action can be rewritten according to the 2+2 decomposition. Then the 2+2 form of the Einstein equations can be derived from the corresponding action as the EulerLagrange equations provided the fact that the foliation is null is imposed only after the variation has made. (Otherwise, the variation of the action with respect to the less than ten nontrivial components of the metric would not yield all the 10 Einstein equations.) One can form the corresponding Hamiltonian, in which the null character of the foliation should appear as a constraint. Then the formal Hamilton equations are just the Einstein equations in their 2+2 form [123, 381, 180, 189]. However, neither the boundary terms in this Hamiltonian nor the boundary conditions that could ensure its functional differentiability were considered. Therefore, this Hamiltonian can be ‘correct’ only up to boundary terms. Such a Hamiltonian was used by Hayward [180, 183] as the basis of his quasilocal energy expression discussed already in Section 6.3. (A similar energy expression was derived by Ikumi and Shiromizi [211], starting with the idea of the ‘freely falling 2surfaces’.)
11.2.2 The 2+2 quasilocalization of the BondiSachs massloss
As we mentioned in Section 6.1.3, this doublenull foliation was used by Hayward [182] to quasilocalize the BondiSachs massloss (and massgain) by using the Hawking energy. Thus we do not repeat the review of his results here.
Yoon investigated the vacuum field equations in a coordinate system based on a null 2+2 foliation. Thus one family of hypersurfaces was (outgoing) null, e.g. N_{ u }, but the other was timelike, say B_{ v }. The former defined a foliation of the latter in terms of the spacelike 2surfaces \({{\mathcal S}_{u,\upsilon}}: = {{\mathcal N}_u} \cap {B_\upsilon}\). Yoon found [400, 401] a certain 2surface integral on S_{ u,v }, denoted by Ẽ(u, v), for which the difference Ẽ(u_{2}, v) − Ẽ(u_{1}, v), u_{1} < u_{2}, could be expressed as a flux integral on the portion of the timelike hypersurface B_{ v } between \({{\mathcal S}_{{u_1},\upsilon}}\) and \({{\mathcal S}_{{u_2},\upsilon}}\). In general this flux does not have a definite sign, but Yoon showed that asymptotically, when B_{ v } is ‘pushed out to null infinity’ (i.e. in the v → ∞ limit in an asymptotically flat spacetime), it becomes negative definite. In fact, ‘renormalizing’ Ẽ(u, v) by a subtraction term, \(E(u,\upsilon): = \tilde E(u,\upsilon)  \sqrt {{\rm{Area(}}{{\mathcal S}_{0,\upsilon}})/(16\pi {G^2})}\) tends to the Bondi energy, and the flux integral tends to the Bondi massloss between the cuts u = u_{1} and u = u_{2} [400, 401]. These investigations were extended for other integrals in [402, 403, 404], which are analogous to spatial momentum and angular momentum. However, all these integrals, including Ẽ(u, v) above, depend not only on the geometry of the spacelike 2surface \({{\mathcal S}_{u,\upsilon}}\) but on the 2+2 foliation on an open neighbourhood of \({{\mathcal S}_{u,\upsilon}}\) too.
11.3 The covariant approach
11.3.1 The covariant phase space methods
The traditional ADM approach to conserved quantities and the Hamiltonian analysis of general relativity is based on the 3+1 decomposition of fields and geometry. Although the results and the content of a theory may be covariant even if their form is not, the manifest spacetime covariance of a formalism may help to find the (spacetime covariant) observables and conserved quantities, boundary conditions, etc. easily. No a posteriori spacetime interpretation of the results is needed. Such a spacetimecovariant Hamiltonian formalism was initiated by Nester [280, 283].
The spirit of the first systematic investigations of the covariant phase space of the classical field theories [122, 20, 146, 251] is similar to that of Nester’s. These ideas were recast into the systematic formalism by Wald and Iyer [389, 215, 216], the socalled covariant Noether charge formalism (see also [388, 251]). This formalism generalizes many of the previous approaches: The Lagrangian 4form may be any diffeomorphism invariant local expression of any finite order derivatives of the field variables. It gives a systematic prescription for the Noether currents, the symplectic structure, the Hamiltonian etc. In particular, the entropy of the stationary black holes turned out to be just a Noether charge derived from Hilbert’s Lagrangian.
11.3.2 Covariant quasilocal Hamiltonians with explicit reference configurations
A nice application of the covariant expression is a derivation of the first law of black hole thermodynamics [107]. The quasilocal energy expressions have been evaluated for several specific 2surfaces. For round spheres in the Schwarzschild spacetime both the 4covariant Dirichlet and Neumann boundary terms (with the Minkowski reference spacetime and K^{ a } as the timelike Killing vector (∂/∂t)^{ a }) give m/G at infinity, but at the horizon the former gives 2m/G and the latter is infinite [107]. The Dirichlet boundary term gives at the spatial infinity in the KerrantideSitter solution the standard m/G and ma/G values for the energy and angular momentum, respectively [191]. Also, the centerofmass is calculated both in the metric and the tetrad formulation of general relativity for the eccentric Schwarzschild solution at the spatial infinity [286, 287], and it was found that the ‘Komarlike term’ is needed to recover the correct, expected value. At the future null infinity of asymptotically flat spacetimes it gives the BondiSachs energymomentum and the expression of Katz [229, 233] for the angular momentum [192]. The general formulae are evaluated for the KerrVaidya solution too.
11.3.3 Covariant quasilocal Hamiltonians with general reference terms
Anco and Tung investigated the possible boundary conditions and boundary terms in the quasilocal Hamiltonian using the covariant Noether charge formalism both of general relativity (with the Hilbert Lagrangian and tetrad variables) and of YangMillsHiggs systems [7, 8]. (Some formulae of the journal versions were recently corrected in the latest arXivversions.) They considered the world tube of a compact spacelike hypersurface Σ with boundary \({\mathcal S}: = \partial \Sigma\). Thus the spacetime domain they considered is the same as in the BrownYork approach: D ≈ Σ × [t_{1}, t_{2}]. Their evolution vector field K^{ a } is assumed to be tangent to the timelike boundary ^{3}B ≈ ∂Σ × [t_{1}, t_{2}] of the domain D. They derived a criterion for the existence of a welldefined quasilocal Hamiltonian. Dirichlet and Neumanntype boundary conditions are imposed (i.e., in general relativity, the variations of the tetrad fields are restricted on ^{3}B by requiring the induced metric γ_{ ab } to be fixed and the adaptation of the tetrad field to the boundary to be preserved, and the tetrad components \({\Theta _{ab}}E_{\underline a}^b\) of the extrinsic curvature of ^{3}B to be fixed, respectively). Then the general allowed boundary condition was shown to be just a mixed DirichletNeumann boundary condition. The corresponding boundary terms of the Hamiltonian, written in the form \({\oint\nolimits_{\mathcal S}}{K^a}{p_a}d{\mathcal S}\) were also determined [7]. The properties of the covectors \(P_a^D\) and \(P_a^N\) (called the Dirichlet and Neumann symplectic vectors, respectively) were investigated further in [8]. Their part tangential to \({\mathcal S}\) is not boost gauge invariant, and to evaluate them the boost gauge determined by the mean extrinsic curvature vector Q^{ a } is used (see Section 4.1.2). Both \(P_a^D\) and \(P_a^N\) are calculated for various spheres in several special spacetimes. In particular, for the round spheres of radius r in the t = const. hypersurface in the ReissnerNordström solution \(P_a^{\rm{D}} = {2 \over r}(1  2m/r + {e^2}/{r^2})\delta _a^0\) and \(P_a^{\rm{N}} =  (m/{r^2}  {e^2}/{r^3})\delta _a^0\), and hence the Dirichlet and Neumann ‘energies’ with respect to the static observer K^{ a } = (∂/∂t)^{ a } are \({\oint\nolimits_{{{\mathcal S}_r}}}{K^a}P_a^{\rm{D}}d{{\mathcal S}_r} = 8\pi r  16\pi [m  {e^2}/(2r)]\) and \({\oint\nolimits_{{{\mathcal S}_r}}}{K^a}P_a^{\rm{N}}d{{\mathcal S}_r} =  4\pi (m  {e^2}/r)\), respectively. Thus \(P_a^{\rm{N}}\) does not reproduce the standard round sphere expression, while \(P_a^{\rm{D}}\) gives the standard round sphere and correct ADM energies only if it is ‘renormalized’ by its own value in Minkowski spacetime [8].
11.3.4 Pseudotensors and quasilocal quantities
As we discussed briefly in Section 3.3.1, many, apparently different pseudotensors and SO(1, 3)gauge dependent energymomentum density expressions can be recovered from a single differential form defined on the bundle L(M) of linear frames over the spacetime manifold: The corresponding superpotentials are the pullbacks to M of the various forms of the NesterWitten 2from \(u_{ab}^{\underline k}\) from L(M) along the various local sections of the bundle [142, 266, 352, 353]. Thus the different pseudotensors are simply the gauge dependent manifestations of the same geometric object on the bundle L(M) in the different gauges. Since, however, \(u_{ab}^{\underline k}\) is the unique extension of the NesterWitten 2form \(u{({\varepsilon ^{\underline K}},{{\bar \varepsilon}^{\underline K \prime}})_{ab}}\) on the principal bundle of normalized spin frames \(\left\{{\varepsilon {K \over A}} \right\}\) (given in Equation (12)), and the latter has been proven to be connected naturally to the gravitational energymomentum, the pseudotensors appear to describe the same physics as the spinorial expressions, though in a slightly old fashioned form. That this is indeed the case was demonstrated clearly by Chang, Nester, and Chen [104, 108, 285], by showing an intimate connection between the covariant quasilocal Hamiltonian expressions and the pseudotensors. Writing the Hamiltonian H[K] in the form of the sum of the constraints and a boundary term, in a given coordinate system the integrand of this boundary term may be the superpotential of any of the pseudotensors. Then the requirement of the functional differentiability of H[K] gives the boundary conditions for the basic variables at ∂Σ. For example, for the Freud superpotential (for Einstein’s pseudotensor) what is fixed on the boundary ∂Σ is a certain piece of \(\sqrt {\vert g\vert} {g^{\alpha \beta}}\).
12 Constructions for Special Spacetimes
12.1 The Komar integral for spacetimes with Killing vectors
Although the Komar integral (and, in general, the linkage (16) for some α) does not satisfy our general requirements discussed in Section 4.3.1, and it does not always give the standard values in specific situations (see for example the ‘factoroftwo anomaly’ or the examples below), in the presence of a Killing vector the Komar integral, built from the Killing field, could be a very useful tool in practice. (For Killing fields the linkage \({L_{\mathcal S}}[{\bf{K}}]\) reduces to the Komar integral for any α.) One of its most important properties is that in vacuum \({L_{\mathcal S}}[{\bf{K}}]\) depends only on the homology class of the 2surface (see for example [387]): If \({\mathcal S}\) and \({\mathcal S}\prime\) are any two 2surfaces such that \({\mathcal S}  {\mathcal S}\prime = \partial \Sigma\) for some compact 3dimensional hypersurface Σ on which the energymomentum tensor of the matter fields is vanishing, then \({L_{\mathcal S}}[{\bf{K}}] = {L_{\mathcal S}}_\prime[{\bf{K}}]\). In particular, the Komar integral for the static Killing field in the Schwarzschild spacetime is the mass parameter m of the solution for any 2surface \({\mathcal S}\) surrounding the black hole, but it is zero if \({\mathcal S}\) does not.
On the other hand [371], the analogous integral in the ReissnerNordström spacetime on a metric 2sphere of radius r is m − e^{2}/r, which deviates from the generally accepted roundsphere value m − e^{2}/(2r). Similarly, in Einstein’s static universe for the spheres of radius r in a t = const. hypersurface \({L_{\mathcal S}}[{\bf{K}}]\) is zero instead of the round sphere result \({{4\pi} \over 3}{r^3}[\mu + \lambda/8\pi G]\), where μ is the energy density of the matter and λ is the cosmological constant.
12.2 The effective mass of Kulkarni, Chellathurai, and Dadhich for the Kerr spacetime
The KulkarniChellathuraiDadhich [244] effective mass for the Kerr spacetime is obtained from the Komar integral (i.e. the linkage with α = 0) using a hypersurface orthogonal vector field X^{ a } instead of the Killing vector T^{ a } of stationarity. The vector field X^{ a } is defined to be T^{ a }+ωΦ^{ a }, where Φ^{ a } is the Killing vector of axisymmetry and the function ω is −g(T, Φ)/g(Φ, Φ). This is timelike outside the horizon, it is the asymptotic time translation at infinity, and coincides with the null tangent on the event horizon. On the event horizon r = r_{+} it yields \({M_{{\rm{KCD}}}} = \sqrt {{m^2}  {a^2}}\), while in the limit r → ∞ it is the mass parameter m of the solution. The effective mass is computed for the KerrNewman spacetime in [106].
12.3 The KatzLyndenBellIsrael energy for static spacetimes
In asymptotically flat spacetimes \({E_{{\rm{KLI}}}}({{\mathcal S}_{\rm{K}}})\) tends to the ADM energy [232]. However, it does not reduce to the roundsphere energy in spherically symmetric spacetimes [277], and, in particular, gives zero for the event horizon of a Schwarzschild black hole.
13 Applications in General Relativity
In this part we give a very short review of some of the potential applications of the paradigm of quasilocality in general relativity. This part of the review is far from being complete, and our claim here is not to discuss the problems considered in detail, but rather to give a collection of problems that are (effectively or potentially) related to quasilocal ideas, tools, notions, etc. In some of these problems the various quasilocal expressions and techniques have been used successfully, but others may provide new and promising areas of their application.
13.1 Calculation of tidal heating
According to astronomical observations, there is an intensive volcanism on the moon Io of Jupiter. One possible explanation of this phenomenon is that Jupiter is heating Io via the gravitational tidal forces (like the Moon, whose gravitational tidal forces raise the ocean’s tides on the Earth). To check whether this could be really the case, one must be able to calculate how much energy is pumped into Io. However, gravitational energy (both in Newtonian theory and in general relativity) is only ambiguously defined (and hence cannot be localized), while the phenomena mentioned above cannot depend on the mathematics that we use to describe them. The first investigations intended to calculate the tidal work (or heating) of a compact massive body were based on the use of the various gravitational pseudotensors [318, 136]. It has been shown that although in the given (slow motion and isolated body) approximation the interaction energy between the body and its companion is ambiguous, the tidal work that the companion does on the body via the tidal forces is not. This is independent both of the gauge conditions [318] and the actual pseudotensor (Einstein, M0ller, Bergmann, or LandauLifshitz) [136].
Recently, these calculations were repeated using quasilocal concepts by Booth and Creighton [76]. They calculated the time derivative of the BrownYork energy, given by Equations (72, 73). Assuming the form of the metric used in the pseudotensorial calculations, for the tidal work they recovered the gauge invariant expressions obtained in [318, 136]. In these approximate calculations the precise form of the boundary conditions (or reference configurations) is not essential, because the results obtained by using different boundary conditions deviate from each other only in higher order.
13.2 Geometric inequalities for black holes
13.2.1 On the Penrose inequality
To rule out a certain class of potential counterexamples to the (weak) cosmic censorship hypothesis [303], Penrose derived an inequality that any asymptotically flat initial data set with (outermost) apparent horizon \({\mathcal S}\) must satisfy [305]: The ADM mass m_{ADM} of the data set cannot be less than the socalled irreducible mass of the horizon, \(M: = \sqrt {{\rm{Area(}}{\mathcal S})/(16\pi {G^2})}\) (see also [156], and for a recent review of the problem and the relevant literature see [90]). However, as stressed by BenDov [58], the more careful formulation of the inequality, due to Horowitz [202], is needed: Assuming that the dominant energy condition is satisfied, the ADM mass of the data set cannot be less than the irreducible mass of the 2surface \({{\mathcal S}_{\min}}\), where \({{\mathcal S}_{\min}}\) has the minimum area among the 2surfaces enclosing the apparent horizon \({\mathcal S}\). In [58] a spherically symmetric asymptotically flat data set with future apparent horizon is given which violates the first, but not the second version of the Penrose inequality.
The inequality has been proven for the outermost future apparent horizons outside the outermost past apparent horizon in maximal data sets in spherically symmetric spacetimes [262] (see also [408, 185, 186]), for static black holes (using the Penrose mass, as we mentioned in Section 7.2.5) [374, 375], and for the perturbed ReissnerNordström spacetimes [225] (see also [226]). Although the original specific potential counterexample has been shown not to violate the Penrose inequality [157], the inequality has not been proven for a general data set. (For the limitations of the proof of the Penrose inequality for the area of trapped surface and the Bondi mass at past null infinity [258], see [64].) If the inequality were true, then this would be a strengthened version of the positive mass theorem, providing a positive lower bound for the ADM mass.
On the other hand, for timesymmetric data sets the Penrose inequality has been proven, even in the presence of more than one black hole. The proof is based on the use of some quasilocal energy expression, mostly of Geroch or of Hawking. First it is shown that these expressions are monotonic along the normal vector field of a special foliation of the timesymmetric initial hypersurface (see Sections 6.1.3 and 6.2, and also [143]), and then the global existence of such a foliation between the apparent horizon and the 2sphere at infinity is proven. The first complete proof of the latter was given by Huisken and Ilmanen [207, 208]. (Recently Bray used a conformal technique to give an alternative proof [87, 88, 89].)
A more general form of the conjecture, containing the electric charge e of the black hole, was formulated by Gibbons [156]: The ADM mass is claimed not to be exceeded by M + e^{2}/(4G^{2}M). Although the weaker form of the inequality, the socalled Bogomolny inequality m_{ADM} ≥ ∣e∣/G, has been proven (under assumptions on the matter content, see for example [160, 369, 257, 159, 274, 156]), Gibbons’ inequality for the electric charge has been proven for special cases (for spherically symmetric spacetimes see for example [186]), and for timesymmetric initial data sets using Geroch’s inverse mean curvature flow [218]. As a consequence of the results of [207, 208] the latter has become a complete proof. However, this inequality does not seem to work in the presence of more than one black hole: For a timesymmetric data set describing n > 1 nearly extremal ReissnerNodström black holes, M+e^{2}/(4G^{2}M) can be greater than the ADM mass, where 16πGM^{2} is either the area of the outermost marginally trapped surface [393], or the sum of the areas of the individual black hole horizons [124]. On the other hand, the weaker inequality derived from the cosmic censorship assumption, does not seem to be violated even in the presence of more than one black hole^{22}.
13.2.2 On the hoop conjecture
In connection with the formation of black holes and the weak cosmic censorship hypothesis another geometric inequality has also been formulated. This is the hoop conjecture of Thorne [367, 269], saying that ‘black holes with horizons form when and only when a mass m gets compacted into a region whose circumference C in every direction is C ≤ 4πGm’ (see also [139, 391]). Mathematically, this conjecture is not precisely formulated: Neither the mass nor the notion of the circumference is welldefined. In certain situations the mass might be the ADM or the Bondi mass, but might be the integral of some locally defined ‘mass density’ as well [139, 35, 260, 240]. The most natural formulation of the hoop conjecture would be based on some spacelike 2surface \({\mathcal S}\) and some reasonable notion of the quasilocal mass, and the trapped nature of the surface would be characterized by the mass and the ‘circumference’ of \({\mathcal S}\). In fact, for round spheres outside the outermost trapped surface and the standard round sphere definition of the quasilocal energy (26) one has 4πGE = 2πr[1 − exp(−2α)] < 2πr = C, where we used the fact that r is an areal radius (see Section 4.2.1). If, however, \({\mathcal S}\) is not axisymmetric then there is no natural definition (or, there are several inequivalent ‘natural’ definitions) of the circumference of \({\mathcal S}\). Interesting necessary and also sufficient conditions for the existence of averaged trapped surfaces in nonspherically symmetric cases, both in special asymptotically flat and cosmological spacetimes, are found in [260, 240]. For the investigations of the hoop conjecture in the GibbonsPenrose spacetime of the collapsing thin matter shell see [36, 35, 379, 298], and for colliding black holes see [407].
13.3 Quasilocal laws of black hole dynamics
13.3.1 Quasilocal thermodynamics of black holes
Black holes are usually introduced in asymptotically flat spacetimes [172, 173, 175, 387], and hence it was natural to derive the formal laws of black hole mechanics/thermodynamics in the asymptotically flat context (see for example [34, 50, 51], and for a recent review see [392]). The discovery of the Hawking radiation [174] showed that the laws of black hole thermodynamics are not only analogous to the laws of thermodynamics, but black holes are genuine thermodynamical objects: The black hole temperature is a physical temperature, that is ℏc/(2πk) times the surface gravity, and the entropy is a physical entropy, kc^{3}/(4Għ) times the area of the horizon (in the traditional units with the Boltzmann constant k, speed of light c, Newton’s gravitational constant G, and Planck’s constant ħ) (see also [390]). Apparently, the detailed microscopic (quantum) theory of gravity is not needed to derive the black hole entropy, and it can be derived even from the general principles of a conformal field theory on the horizon of the black holes [100, 101, 102, 296, 103].
However, black holes are localized objects, thus one must be able to describe their properties and dynamics even at the quasilocal level. Nevertheless, beyond this rather theoretic claim, there are pragmatic reasons that force us to quasilocalize the laws of black hole dynamics. In particular, it is well known that the Schwarzschild black hole, fixing its temperature at infinity, has negative heat capacity. Similarly, in an asymptotically antideSitter spacetime fixing the black hole temperature via the normalization of the timelike Killing vector at infinity is not justified because there is no such physically distinguished Killing field (see [92]). These difficulties lead to the need of a quasilocal formulation of black hole thermodynamics. In [92] Brown, Creighton, and Mann investigated the thermal properties of the SchwarzschildantideSitter black hole. They used the quasilocal approach of Brown and York to define the energy of the black hole on a spherical 2surface \({\mathcal S}\) outside the horizon. Identifying the BrownYork energy with the internal (thermodynamical) energy and (in the k = ħ = c =1 units) 1/(4G) times the area of the event horizon with the entropy, they calculated the temperature, surface pressure, and heat capacity. They found that these quantities do depend on the location of the surface \({\mathcal S}\). In particular, there is a critical value T_{0} such that for temperatures T greater than T_{0} there are two black hole solutions, one with positive and one with negative heat capacity, but there are no SchwarzschildantideSitter black holes with temperature T less than T_{0}. In [121] the BrownYork analysis is extended to include dilaton and YangMills fields, and the results are applied to stationary black holes to derive the first law of black hole thermodynamics. The socalled Noether charge formalism of Wald [389], and Iyer and Wald [215] can be interpreted as a generalization of the BrownYork approach from general relativity to any diffeomorphism invariant theory to derive quasilocal quantities [216]. However, this formalism gave a general expression for the black hole entropy as well: That is the Noether charge derived from the Hilbert Lagrangian corresponding to the null normal of the horizon, and explicitly this is still 1/(4G) times the area of the horizon. (For some recent related works see for example [149, 188]).
There is an extensive literature of the quasilocal formulation of the black hole dynamics and relativistic thermodynamics in the spherically symmetric context (see for example [185, 187, 186, 190] and for nonspherically symmetric cases [275, 189, 74]). However, one should see clearly that while the laws of black hole thermodynamics above refer to the event horizon, which is a global concept in the spacetime, the subject of the recent quasilocal formulations is to describe the properties and the evolution of the socalled trapping horizon, which is a quasilocally defined notion. (On the other hand, the investigations of [183, 181, 184] are based on energy and angular momentum definitions that are gauge dependent; see also Sections 4.1.8 and 6.3.)
13.3.2 On the isolated horizons
The idea of the isolated horizons (more precisely, the gradually more restrictive notions of the nonexpanding, the weakly isolated and isolated horizons, and the special weakly isolated horizon called the rigidly rotating horizons) is to generalize the notion of Killing horizons by keeping their basic properties without the existence of any Killing vector in general. (For a recent review see [19] and references therein, especially [21, 18].) The phase space for asymptotically flat spacetimes containing an isolated horizon is based on a 3manifold with an asymptotic end (or finitely many such ends) and an inner boundary. The boundary conditions on the inner boundary are determined by the precise definition of the isolated horizon. Then, obviously, the Hamiltonian will be the sum of the constraints and boundary terms, corresponding both to the ends and the horizon. Thus, by the appearance of the boundary term on the inner boundary makes the Hamiltonian partly quasilocal. It is shown that the condition of the Hamiltonian evolution of the states on the inner boundary along the evolution vector field is precisely the first law of black hole mechanics [21, 18].
Booth [75] applied the general idea of Brown and York to a domain D whose boundary consists not only of two spacelike submanifolds Σ_{1} and Σ_{2} and a timelike one ^{3}B, but a further, internal boundary Δ as well, which is null. Thus he made the investigations of the isolated horizons fully quasilocal. Therefore, the topology of Σ_{1} and Σ_{2} is \({S^2} \times [a,b]\), and the inner (null) boundary is interpreted as (a part of) a nonexpanding horizon. Then to have a welldefined variational principle on D, the Hilbert action had to be modified by appropriate boundary terms. However, requiring Δ to be a socalled rigidly rotating horizon, the boundary term corresponding to Δ and the allowed variations are considerably restricted. This made it possible to derive the ‘first law of rigidly rotating horizon mechanics’ quasilocally, an analog of the first law of black hole mechanics. The first law for rigidly rotating horizons was also derived by Allemandi, Francaviglia, and Raiteri in the EinsteinMaxwell theory [4] using their ReggeTeitelboimlike approach [141].
Another concept is the notion of a dynamical horizon [25, 26]. This is a smooth spacelike hypersurface that can be foliated by a preferred family of marginally trapped 2spheres. By an appropriate definition of the energy and angular momentum balance equations for these quantities, carried by gravitational waves, are derived. Isolated horizons are the asymptotic state of dynamical horizons.
13.4 Entropy bounds
13.4.1 On Bekenstein’s bounds for the entropy
13.4.2 On the holographic hypothesis
In the literature there is another kind of upper bound for the entropy for a localized system, the socalled holographic bound. The holographic principle [366, 350, 81] says that, at the fundamental (quantum) level, one should be able to characterize the state of any physical system located in a compact spatial domain by degrees of freedom on the surface of the domain too, analogously to the holography by means of which a three dimensional image is encoded into a 2dimensional surface. Consequently, the number of physical degrees of freedom in the domain is bounded from above by the area of the boundary of the domain instead of its volume, and the number of physical degrees of freedom on the 2surface is not greater onefourth of the area of the surface measured in Planckarea units \(L_{\rm{P}}^2: = G\hbar/{c^3}\). This expectation is formulated in the (spacelike) holographic entropy bound [81]: Let Σ be a compact spacelike hypersurface with boundary \({\mathcal S}\). Then the entropy S(Σ) of the system in Σ should satisfy \(S(\Sigma) \leq k\,{\rm{Area(}}{\mathcal S})/(4L_{\rm{P}}^2)\). Formally, this bound can be obtained from the Bekenstein bound with the assumption that 2E ≤ Rc^{4}/G, i.e. that R is not less than the Schwarzschild radius of E. Also, as with the Bekenstein bounds, this inequality can be violated in specific situations (see also [392, 81]).
On the other hand, there is another formulation of the holographic entropy bound, due to Bousso [80, 81]. Bousso’s socalled covariant entropy bound is much more quasilocal than the previous formulations, and is based on spacelike 2surfaces and the null hypersurfaces determined by the 2surfaces in the spacetime. Its classical version has been proved by Flanagan, Marolf, and Wald [140]: If \({\mathcal N}\) is an everywhere noncontracting (or nonexpanding) null hypersurface with spacelike cuts \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\), then, assuming that the local entropy density of the matter is bounded by its energy density, the entropy flux \({{\mathcal S}_{\mathcal N}}\) through \({\mathcal N}\) between the cuts \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) is bounded: \({{\mathcal S}_{\mathcal N}} \leq k\vert{\rm{Area(}}{{\mathcal S}_2})  {\rm{Area(}}{{\mathcal S}_1})\vert/(4L_{\rm{P}}^2)\). For a detailed discussion see [392, 81]. For still another, quasilocal formulation of the holographic principle see Section 2.2.5 and [365].
13.5 Quasilocal radiative modes of GR
In Section 8.2.3 we discussed the properties of the DouganMason energymomenta, and we saw that, under the conditions explained there, the energymomentum is vanishing iff D(Σ) is flat, and it is null iff D(Σ) is a ppwave geometry with pure radiative matter, and that these properties of the domain of dependence D(Σ) are completely encoded into the geometry of the 2surface \({\mathcal S}\). However, there is an important difference between these two statements: While in the former case we know the metric of D(Σ), that is flat, in the second we know only that the geometry admits a constant null vector field, but we do not know the line element itself. Thus the question arises as whether the metric of D(Σ) is also determined by the geometry of \({\mathcal S}\) even in the zero quasilocal mass case.
In [358] it was shown that under the condition above there is a complex valued function Φ on \({\mathcal S}\), describing the deviation of the antiholomorphic and the holomorphic spinor dyads from each other, which plays the role of a potential for the curvature \({F^A}_{Bcd}\) on \({\mathcal S}\). Then, assuming that \({\mathcal S}\) is future and past convex and the matter is an Ntype zerorestmass field, Φ and the value ϕ of the matter field on \({\mathcal S}\) determine the curvature of D(Σ). Since the field equations for the metric of D(Σ) reduce to Poissonlike equations with the curvature as the source, the metric of D(Σ) is also determined by Φ and ϕ on \({\mathcal S}\). Therefore, the (purely radiative) ppwave geometry and matter field on D(Σ) are completely encoded in the geometry of \({\mathcal S}\) and complex functions defined on \({\mathcal S}\), respectively, in complete agreement with the holographic principle of the previous Section 13.4.
As we saw in Section 2.2.5, the radiative modes of the zerorestmassfields in Minkowski spacetime, defined by their Fourier expansion, can be characterized quasilocally on the globally hyperbolic subset D(Σ) of the spacetime by the value of the Fourier modes on the appropriately convex spacelike 2surface \({\mathcal S} = \partial \Sigma\). Thus the two transversal radiative modes of them are encoded in certain fields on \({\mathcal S} = \partial \Sigma\). On the other hand, because of the nonlinearity of the Einstein equations it is difficult to define the radiative modes of general relativity. It could be done when the field equations become linear, i.e. near the null infinity, in the linear approximation and for ppwaves. In the first case the gravitational radiation is characterized on a cut \({{\mathcal S}_\infty}\) of the null infinity ℐ^{+} by the uderivative \({\dot \sigma ^0}\) of the asymptotic shear of the outgoing null hypersurface \({\mathcal N}\) for which \({S_\infty} = N \cap{\mathscr I}^+\), i.e. by a complex function on \({{\mathcal S}_\infty}\). It is remarkable that it is precisely this complex function which yields the deviation of the holomorphic and antiholomorphic spin frames at the null infinity (see for example [363]). The linear approximation of Einstein’s theory is covered by the analysis of Section 2.2.5, thus those radiative modes can be characterized quasilocally, while for the ppwaves the result of [358], reported above, gives such a quasilocal characterization in terms of a complex function measuring the deviation of the holomorphic and antiholomorphic spin frames. However, the deviation of the holomorphic and antiholomorphic structures on \({\mathcal S}\) can be defined even for generic 2surfaces in generic spacetimes too, which might yield the possibility of introducing the radiative modes quasilocally in general.
14 Summary: Achievements, Difficulties, and Open Issues
In the previous sections we tried to give an objective review of the present state of the art. This section is, however, less positivistic: We close the present review by a critical discussion, evaluating those strategies, approaches etc. that are explicitly and unambiguously given and (at least in principle) applicable in any generic spacetime.
14.1 On the Bartnik mass and the Hawking energy
Although in the literature the notions mass and energy are used almost synonymously, in the present review we have made a distinction between them. By energy we meant the time component of the energymomentum fourvector, i.e. a reference frame dependent quantity, while by mass the length of the energymomentum, i.e. an invariant. In fact, these two have different properties: The quasilocal energy (both for the matter fields and for gravity according to the DouganMason definition) is vanishing precisely for the ‘ground state’ of the theory (i.e. for vanishing energymomentum tensor in the domain of dependence D(Σ) and flatness of D(Σ) respectively, see Sections 2.2.5 and 8.2.3). In particular, for configurations describing pure radiation (purely radiative matter fields and ppwaves, respectively) the energy is positive. On the other hand, the vanishing of the quasilocal mass does not characterize the ‘ground state’, rather that is equivalent just to these purely radiative configurations.
The Bartnik mass is a natural quasilocalization of the ADM mass, and its monotonicity and positivity makes it a potentially very useful tool in proving various statements on the spacetime, because it fully characterizes the nontriviality of the finite Cauchy data by a single scalar. However, our personal opinion is that, just by its strict positivity for nonflat 3dimensional domains, it overestimates the ‘physical’ quasilocal mass. In fact, if (Σ, h_{ ab }, χ_{ ab }) is a finite data set for a ppwave geometry (i.e. a compact subset of the data set for a ppwave metric), then it probably has an asymptotically flat extension \((\hat \Sigma, {{\hat h}_{ab}},{{\hat \chi}_{ab}})\) satisfying the dominant energy condition with bounded ADM energy and no apparent horizon between ∂Σ and infinity. Thus while the DouganMason mass of ∂Σ is zero, the Bartnik mass m_{B}(Σ) is strictly positive unless (Σ, h_{ ab }, χ_{ ab }) is trivial. Thus, this example shows that it is the procedure of taking the asymptotically flat extension that gives strictly positive mass. Indeed, one possible proof of the rigidity part of the positive energy theorem [24] (see also [354]) is to prove first that the vanishing of the ADM mass implies, through the Witten equation, that the spacetime admits a constant spinor field, i.e. it is a ppwave spacetime, and then that the only asymptotically flat spacetime that admits a constant null vector field is the Minkowski spacetime. Therefore, it is just the global condition of the asymptotic flatness that rules out the possibility of nontrivial spacetimes with zero ADM mass. Hence it would be instructive to calculate the Bartnik mass for a compact part of a ppwave data set. It might also be interesting to calculate its small surface limit to see its connection with the local fields (energymomentum tensor and probably the BelRobinson tensor).
The other very useful definition is the Hawking energy (and its slightly modified version, the Geroch energy). Its advantage is its simplicity, calculability, and monotonicity for special families of 2surfaces, and it has turned out to be a very effective tool in practice in proving for example the Penrose inequality. The small sphere limit calculation shows that it is energy rather than mass, so in principle one should be able to complete this to an energymomentum 4vector. One possibility is Equation (39, 40), but, as far as we are aware, its properties have not been investigated. Unfortunately, although the energy can be defined for 2surfaces with nonzero genus, it is not clear how the 4momentum could be extended for such surfaces. Although the Hawking energy is a welldefined 2surface observable, it has not been linked to any systematic (Lagrangian or Hamiltonian) scenario. Perhaps it does not have any such interpretation, and it is simply a natural (but in general spacetimes for quite general 2surfaces not quite viable) generalization of the standard round sphere expression (27). This view appears to be supported by the fact that the Hawking energy has strange properties for nonspherical surfaces, e.g. for 2surfaces in Minkowski spacetime which are not metric spheres.
14.2 On the Penrose mass
Penrose’s suggestion for the quasilocal mass (or, more generally, energymomentum and angular momentum) was based on a promising and farreaching strategy to use twistors at the fundamental level. The basic object of the construction, the socalled kinematical twistor, is intended to comprise both the energymomentum and angular momentum, and is a welldefined quasilocal quantity on generic spacelike surfaces homeomorphic to S^{2}. It can be interpreted as the value of a quasilocal Hamiltonian, and the four independent 2surface twistors play the role of the quasitranslations and quasirotations. The kinematical twistor was calculated for a large class of special 2surfaces and gave acceptable results.
However, the construction is not complete. First, the construction does not work for 2surfaces whose topology is different from S^{2}, and does not work even for certain topological 2spheres for which the 2surface twistor equation admits more than four independent solutions (‘exceptional 2surfaces’). Second, two additional objects, the socalled infinity twistor and a Hermitian inner product on the space of 2surface twistors, are needed to get the energymomentum and angular momentum from the kinematical twistor and to ensure their reality. The latter is needed if we want to define the quasilocal mass as a norm of the kinematical twistor. However, no natural infinity twistor has been found, and no natural Hermitian scalar product can exist if the 2surface cannot be embedded into a conformally flat spacetime. In addition, in the small surface calculations the quasilocal mass may be complex. If, however, we do not want to form invariants of the kinematical twistor (e.g. the mass), but we want to extract the energymomentum and angular momentum from the kinematical twistor and we want them to be real, then only a special combination of the infinity twistor and the Hermitian scalar product, the socalled ‘barhook combination’ (see Equation (51)), would be needed.
To save the main body of the construction, the definition of the kinematical twistor was modified. Nevertheless, the mass in the modified constructions encountered an inherent ambiguity in the small surface approximation. One can still hope to find an appropriate ‘barhook’, and hence real energymomentum and angular momentum, but invariants, such as norms, could not be formed.
14.3 On the DouganMason energymomenta and the holomorphic/antiholomorphic spin angular momenta
From pragmatic points of view the DouganMason energymomenta (see Section 8.2) are certainly among the most successful definitions: The energypositivity and the rigidity (zero energy implies flatness), and the intimate connection between the ppwaves and the vanishing of the masses make these definitions potentially useful quasilocal tools as the ADM and BondiSachs energymomenta in the asymptotically flat context. Similar properties are proven for the quasilocal energymomentum of the matter fields, in particular for the nonAbelian YangMills fields, too. They depend only on the 2surface data on \({\mathcal S}\), they have a clear Lagrangian interpretation, and the spinor fields that they are based on can be considered as the spinor constituents of the quasitranslations of the 2surface. In fact, in the Minkowski spacetime the corresponding spacetime vectors are precisely the restriction to \({\mathcal S}\) of the constant Killing vectors. These notions of energymomentum are linked completely to the geometry of \({\mathcal S}\), and are independent of any ad hoc choice for the ‘fleet of observers’ on it. On the other hand, the holomorphic/antiholomorphic spinor fields determine a six real parameter family of orthonormal frame fields on \({\mathcal S}\), which can be interpreted as some distinguished class of observers. In addition, they reproduce the expected, correct limits in a number of special situations. In particular, these energymomenta appear to have been completed by spinangular momenta (see Section 9.2) in a natural way.
However, in spite of their successes, the DouganMason energymomenta and the spinangular momenta based on Bramson’s superpotential and the holomorphic/antiholomorphic spinor fields have some unsatisfactory properties, too (see the lists of our expectations in Section 4.3). First, they are defined only for topological 2spheres (but not for other topologies, e.g. for the torus S^{1} × S^{1}), and they are not welldefined even for certain topological 2spheres either. Such surfaces are, for example, past marginally trapped surfaces in the antiholomorphic (and future marginally trapped surfaces in the holomorphic) case. Although the quasilocal mass associated with a marginally trapped surface \({\mathcal S}\) is expected to be its irreducible mass \(\sqrt {{\rm{Area(}}{\mathcal S})/16\pi {G^2})}\), neither of the DouganMason masses is welldefined for the bifurcation surfaces of the KerrNewman (or even Schwarzschild) black hole. Second, the role and the physical content of the holomorphicity/antiholomorphicity of the spinor fields is not clear. The use of the complex structure is justified a posteriori by the nice physical properties of the constructions and the pure mathematical fact that it is only the holomorphy and antiholomorphy operators in a large class of potentially acceptable first order linear differential operators acting on spinor fields that have a 2dimensional kernel. Furthermore, since the holomorphic and antiholomorphic constructions are not equivalent, we have two constructions instead of one, and it is not clear why we should prefer for example holomorphicity instead of antiholomorphicity even at the quasilocal level.
The angular momentum based on Bramson’s superpotential and the antiholomorphic spinors together with the antiholomorphic DouganMason energymomentum give acceptable PauliLubanski spin for axisymmetric zeromass Cauchy developments, for small spheres, and at future null infinity, but the global angular momentum at the future null infinity is finite and welldefined only if the spatial 3momentum part of the BondiSachs 4momentum is vanishing, i.e. only in the centreofmass frame. (The spatial infinity limit of the spinangular momenta has not been calculated.)
Thus the NesterWitten 2form appears to serve as an appropriate framework for defining the energymomentum, and it is the two spinor fields which should probably be changed and a new choice would be needed. The holomorphic/antiholomorphic spinor fields appears to be ‘too rigid’. In fact, it is the topology of \({\mathcal S}\), namely the zero genus of \({\mathcal S}\), that restricts the solution space to two complex dimensions, instead of the local properties of the differential equations. (Thus, the situation is the same as in the twistorial construction of Penrose.) On the other hand, Bramson’s superpotential is based on the idea of Bergmann and Thomson that the angular momentum of gravity is analogous to the spin. Thus the question arises as to whether this picture is correct, or the gravitational angular momentum also has an orbital part, whenever Bramson’s superpotential describes only (the general form of) its spin part. The fact that our antiholomorphic construction gives the correct, expected results for small spheres but unacceptable ones for large spheres near future null infinity in frames that are not centreofmass frames may indicate the lack of such an orbital term. This term could be neglected for small spheres, but certainly not for large spheres. For example, in the special quasilocal angular momentum of Bergqvist and Ludvigsen for the Kerr spacetime (see Section 9.3) it is the sum of Bramson’s expression and a term that can be interpreted as the orbital angular momentum.
14.4 On the BrownYorktype expressions
The idea of Brown and York that the quasilocal conserved quantities should be introduced via the canonical formulation of the theory is quite natural. In fact, as we saw, one could arrive at their general formulae from different points of departure (functional differentiability of the Hamiltonian, 2surface observables). If the a priori requirement that we should have a welldefined action principle for the traceχaction yielded undoubtedly well behaving quasilocal expressions, then the results would a posteriori justify this basic requirement (like the holomorphicity or antiholomorphicity of the spinor fields in the DouganMason definitions). However, if not, then that might be considered as an unnecessarily restrictive assumption, and the question arises whether the present framework is wide enough to construct reasonable quasilocal energymomentum and angular momentum.
Indeed, the basic requirement automatically yields the boundary condition that the 3metric γ_{ ab } should be fixed on the boundary \({\mathcal S}\), and that the boundary term in the Hamiltonian should be built only from the surface stress tensor τ_{ ab }. Since the boundary conditions are given, no Legendre transformation of the canonical variables on the 2surface is allowed (see the derivation of Kijowski’s expression in Section 10.2). The use of τ_{ ab } has important consequences. First, the quasilocal quantities depend not only on the geometry of the 2surface \({\mathcal S}\), but on an arbitrarily chosen boost gauge, interpreted as a ‘fleet of observers t^{ a } being at rest with respect to \({\mathcal S}\)’, too. This leaves a huge ambiguity in the BrownYork energy (three arbitrary functions of two variables, corresponding to the three boost parameters at each point of \({\mathcal S}\)) unless a natural gauge choice is prescribed^{23}. Second, since τ_{ ab } does not contain the extrinsic curvature of \({\mathcal S}\) in the direction t^{ a }, which is a part of the 2surface data, this extrinsic curvature is ‘lost’ from the point of view of the quasilocal quantities. Moreover, since τ_{ ab } is a tensor only on the 3manifold ^{3}B, the integral of K^{ a }τ_{ ab }t^{ b } on \({\mathcal S}\) is not sensitive to the component of K^{ a } normal to ^{3}B. The normal piece v^{ a }v_{ b }K^{ b } of the generator K^{ a } is ‘lost’ from the point of view of the quasilocal quantities.
The other important ingredient of the BrownYork construction is the prescription of the subtraction term. Considering the GaussCodazziMainardi equations of the isometric embedding of the 2surface into the flat 3space (or rather into a spacelike hyperplane of Minkowski spacetime) only as a system of differential equations for the reference extrinsic curvature, this prescription — contrary to frequently appearing opinions — is as explicit as the condition of the holomorphicity/antiholomorphicity of the spinor fields in the DouganMason definition. (One essential, and from pragmatic points of view important, difference is that the GaussCodazziMainardi equations form an underdetermined elliptic system constrained by a nonlinear algebraic equation.) Similarly to the DouganMason definitions, the general BrownYork formulae are valid for arbitrary spacelike 2surfaces, but solutions to the equations defining the reference configuration exist certainly only for topological 2spheres with strictly positive intrinsic scalar curvature. Thus there are exceptional 2surfaces here, too. On the other hand, the BrownYork expressions (both for the flat 3space and the light cone references) work properly for large spheres.
At first sight, this choice for the definition of the subtraction term seems quite natural. However, we do not share this view. If the physical spacetime is the Minkowski one, then we expect that the geometry of the 2surface in the reference Minkowski spacetime be the same as in the physical Minkowski spacetime. In particular, if \({\mathcal S}\) — in the physical Minkowski spacetime — does not lie in any spacelike hyperplane, then we think that it would be unnatural to require the embedding of \({\mathcal S}\) into a hyperplane of the reference Minkowski spacetime. Since in the two Minkowski spacetimes the extrinsic curvatures can be quite different, the quasilocal energy expressions based on this prescription of the reference term can be expected to yield a nonzero value even in flat spacetime. Indeed, there are explicit examples showing this defect. (Epp’s definition is free of this difficulty, because he embeds the 2surface into the Minkowski spacetime by preserving its ‘universal structure’; see Section 4.1.4.)
Another objection against the embedding into flat 3space is that it is not Lorentz covariant. As we discussed in Section 4.2.2, Lorentz covariance (together with the positivity requirement) was used to show that the quasilocal energy expression for small spheres in vacuum is of order r^{5} with the BelRobinson ‘energy’ as the factor of proportionality. The BrownYork expression (even with the light cone reference \({k^0} = \sqrt {{2^{\mathcal S}}R}\)) fails to give the BelRobinson ‘energy’.^{24}
Finally, in contrast to the DouganMason definitions, the BrownYork type expressions are welldefined on marginally trapped surfaces. However, they yield just twice the expected irreducible mass, and they do not reproduce the standard round sphere expression, which, for nontrapped surfaces, comes out from all the other expressions discussed in the present section (including Kijowski’s definition). It is remarkable that the derivation of the first law of black hole thermodynamics, based on the identification of the thermodynamical internal energy with the BrownYork energy, is independent of the definition of the subtraction term.
Footnotes
 1.
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.
Since we do not have a third kind of device to specify the spatiotemporal location of the devices measuring the spacetime geometry, we do not have any further operationally defined, maybe nondynamical background, just in accordance with the principle of equivalence. If there were some nondynamical background metric \(g_{ab}^0\) on M, then by requiring \(g_{ab}^0 = {\phi ^\ast}g_{ab}^0\), we could reduce the almost arbitrary diffeomorphism φ (essentially four arbitrary functions of four variables) to a transformation depending on at most ten parameters.
 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).
 4.
\(E({\mathcal S})\) can be thought of as the 0component of some quasilocal energymomentum 4vector, but, just because of the spherical symmetry, its spatial parts are vanishing. Thus \(E({\mathcal S})\) can also be interpreted as the mass, the length of this energymomentum 4vector.
 5.
If, in addition, the spinor constituent o^{ A } of \({l^a} = {o^A}\bar oA\prime\) is required to be parallel propagated along l^{ a }, then the tetrad becomes completely fixed, yielding the vanishing of several (combinations of the) spin coefficients.
 6.
As we will see soon, the leading term of the small sphere expression of the energymomenta in nonvacuum is of order r^{3}, in vacuum it is r^{5}, while that of the angular momentum is r^{4} and r^{6}, respectively.
 7.
Because of the falloff, no essential ambiguity in the definition of the large spheres arises from the use of the coordinate radius instead of the physical radial distance.
 8.
In the socalled Bondi coordinate system the radial coordinate is the luminosity distance r_{D} := −1/ρ, which tends to the affine parameter r asymptotically.
 9.
Since we take the infimum, we could equally take the ADM masses, which are the minimum values of the zeroth component of the energymomentum fourvectors in the different Lorentz frames, instead of the energies.
 10.
I thank Paul Tod for pointing this out to me.
 11.
 12.
Recall that, similarly, we did not have any natural isomorphism between the 2surface twistor spaces, discussed in Section 7.2.1, on different 2surfaces.
 13.
Clearly, for the LudvigsenVickers energymomentum no such ambiguity is present, because the part (59) of their propagation law defines a natural isomorphism between the space of the LudvigsenVickers spinors on the different 2surfaces.
 14.
In the original papers Brown and York assumed that the leaves Σ_{ t } of the foliation of D were orthogonal to ^{3}B (‘orthogonal boundaries assumption’).
 15.
The paper gives a clear, well readable summary of these earlier results.
 16.
Thus, in principle, we would have to report on their investigations in the next Section 11. Nevertheless, since essentially they rederive and justify the results of Brown and York following only a different route, we discuss their results here.
 17.
The problem to characterize this embeddability is known as the Weyl problem of differential geometry.
 18.
According to this view the quasilocal energy is similar to E_{Φ} of Equation (6), rather than to the charges which are connected somehow to some ‘absolute’ element of the spacetime structure.
 19.
This phase space is essentially T*TQ, the cotangent bundle of the tangent bundle of the configuration manifold Q, endowed with the natural symplectic structure, and can be interpreted as the collection of quadruples \(({q^a},{{\dot q}^a},{p_a},{{\dot p}_a})\). The usual Lagrangian (or velocity) phase space TQ and the Hamiltonian (or momentum) phase space T*Q are special submanifolds of T*TQ.
 20.
In fact, Kijowski’s results could have been presented here, but the technique that he uses may justify their inclusion in the previous Section 10.
 21.
Here we concentrate only on the genuine, finite boundary of Σ. The analysis is straightforward even in the presence of ‘boundaries at infinity’ at the asymptotic ‘ends’ of asymptotically flat Σ.
 22.
I am grateful to Sergio Dain for pointing out this to me.
 23.
It could be interesting to clarify the consequences of the boost gauge choice that is based on the main extrinsic curvature vector Q_{ a }, discussed in Section 4.1.2. This would rule out the arbitrary element of the construction.
 24.
It might be interesting to see the small sphere expansion of the Kijowski and KijowskiLiuYau expressions in vacuum.
Notes
Acknowledgements
I am grateful to Peter Aichelburg, Herbert Balasin, Robert Bartnik, Robert Beig, ChiangMei Chen, Piotr Chruściel, Sergio Dain, Jörg Frauendiener, Sean Hayward, Jacek Jezierski, Jerzy Kijowski, Stephen Lau, Lionel Mason, Niall Ó Murchadha, James Nester, Ezra Newman, Alexander Petrov, Walter Simon, George Sparling, Paul Tod, Roh Tung, and Helmuth Urbantke for their valuable comments, remarks, and stimulating questions. Special thanks to Jorg Frauendiener for continuous and fruitful discussions in the last eight years, to James Nester for the critical reading of an earlier version of the present manuscript, whose notes and remarks considerably improved its clarity, and to the two referees whose constructive criticism helped to make the present review more accurate and complete. Thanks are due to the Erwin Schrödinger Institute, Vienna, the Stefan Banach Center, Warsaw, the Universität Tübingen, the MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, the National Center for Theoretical Sciences, Hsinchu, and the National Central University, Chungli, for hospitality, where parts of the present work were done and/or could be presented. This work was partially supported by the Hungarian Scientific Research Fund grant OTKA T042531.
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