QuasiLocal EnergyMomentum and Angular Momentum in General Relativity
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Abstract
The present status of the quasilocal mass, energymomentum and angularmomentum 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 deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasilocal concepts and specific constructions are briefly mentioned.
1 Introduction
Over the last 35 years, one of the greatest achievements in classical general relativity has certainly been 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 also a useful tool in the everyday practice of working relativists. This success inspired the more ambitious claim to associate energy (or rather energymomentum and, ultimately, angular momentum as well) 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 1980s, 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 not only have no ultimate, generally accepted expression for the energymomentum and especially for the angular momentum, but there is not even a consensus in the relativity community on general questions (for example, what do we mean by energymomentum? just 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 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, potentiallycommon points, issues and questions. Thus, we wanted not only to write a ‘whodidwhat’ review, but 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, one third of the present review is devoted to these general questions. We review the specific constructions and their properties only in the second part, and in the third part we discuss very briefly some (potential) applications of the quasilocal quantities. Although this paper is at heart 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 (general) part of the review (Sections 2–4).
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, while the properties and applications are usually summarized only. Sometimes we give a review of technical aspects as well, 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 bit subjective. Here we consider the calculation of gravitational energy transfer, applications to black hole physics and cosmology, 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 of subject matter of Sections 2–4) 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; these are simple enough to be worked out by graduate students. On the other hand, applying them to solve physical/geometrical problems (e.g., to some mentioned in Section 13) would be a real achievement.
In the present paper we adopt the abstract index formalism. The signature of the spacetime metric g_{ ab } is −2, and the curvature Ricci tensors and 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 \({G_{ab}} + \lambda {g_{ab}}: = {R_{ab}}  {1 \over 2}R{g_{ab}} + \lambda {g_{ab}} =  8\pi G{T_{ab}}\), 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 3.1.1, 13.3 and 13.4 we use the traditional cgs system.
2 EnergyMomentum and Angular Momentum of Matter Fields
2.1 Energymomentum and angularmomentum density of matter fields
2.1.1 The symmetric energymomentum tensor
It is a widely accepted view that the canonical energymomentum and spin tensors are well defined and have relevance only in flat spacetime, and, hence, are usually underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus, we first 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^{c[ab]} + σ^{c[ab]}),
 3.
C^{ a }[K] = T^{ ab }K_{ b } + ∇_{ c }((σ^{c[ab]} − σ^{c[ab]} − σ^{c[ab]}K_{ b }), where the second term on the right is an identicallyconserved (i.e., divergencefree) 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 tendimensional Lie algebra K of the Killing vectors is well known to split into the semidirect sum of a fourdimensional 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 \(\{E_{\underline a}^a\} {\rm{on}}\,M{\rm{,}}\,\underline a = 0, \ldots, 3\), forms a basis. (Thus, the underlined Roman indices \(\underline a, \underline b\), … 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 boost rotations, 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 oneform 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 oneforms) 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 \({C^e}[{\bf{K}}] = E_{\underline e}^e({\theta ^{\underline e \underline a}}{T_{\underline a}}  ({\theta ^{\underline e \underline a}}{x^{\underline b}}  {\theta ^{\underline e \underline b}}{x^{\underline a}}  2{\sigma ^{\underline e [\underline a \underline {b]}}}){M_{\underline a \underline b}})\), 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 of 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 angularmomentum density of the matter fields.
Since in de Sitter and antide Sitter spacetimes the (tendimensional) 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 Section 3 we will see that welldefined (i.e., gaugeinvariant) energymomentum and angularmomentum density cannot be associated with the gravitational ‘field’, and if we do not want to talk 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 quantumfieldtheory investigations, it has become clear that all physical observables should be associated with extended but finite spacetime domains [232, 231]. 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 quasilocallydefined quantities. Thus, the idea of quasilocality is not new in physics. Although 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 [488, 492]. 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 Section 2.2 to the discussion of them and their properties.
2.2.1 The definition of quasilocal quantities
Thus, even if there is a gaugeinvariant and unambiguouslydefined 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 this 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 [534].^{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 identicallyconserved 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 [13, 14, 442] 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, e.g., page 94 of [240]), 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 futurepointing 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 } = τL_{ a }L_{ b } for some nonnegative function τ. In this case \(P_{\mathcal S}^{\underline a} = e{L^{\underline a}}\), where \({L^{\underline a}}: = {L^a}\vartheta _a^{\underline a}\)
 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_{Σ}[t^{ a }] and M_{Σ} are integrals of functions on a hypersurface, they are obviously additive, e.g., 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 twosurfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different twosurfaces, 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 centerofmass is not ensured by the \({\mathcal O}({r^{ 4}})\) falloff. Since the typical falloff of T_{ ab } — for the electromagnetic field, for example — 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 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 [497].
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 \({T_{ab}} = {\mathcal O}({r^{ 4}})\), they must be asymptotically vanishing at the same time. However, a planewave 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 Section 2.2.4, the vanishing of the quasilocal mass, associated with a closed spacelike twosurface \({\mathcal S}\), implies that the matter must be pure radiation on a fourdimensional 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 twosurface \({\mathcal S}\), 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 \(\phi _{AB}^\alpha, \alpha = 1, \ldots, N\), where N is the dimension of the gauge group, the vanishing of the quasilocal mass is equivalent [498] to plane waves and the ppwave solutions of Coleman [152], 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 \({\mathcal S}\) (in which case the spinor fields \(\phi _{AB}^\alpha\) are necessarily null: \(\phi _{AB}^\alpha = {\phi ^\alpha}{O_A}{O_B}\), whereϕ^{ α } are complex functions and O_{ A } is a constant spinor field such that L_{ a } = O_{ A }O_{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. twosurfaces foliating Σ, where L_{ a } = ∇_{ a }u, and, by the field equations, the complex functions ϕ and ϕ^{ α } turn out to be antiholomorphic [492]. 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 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 ϕO^{ A } (with fixed constantspinor 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 timedependent family of holographic data, as well [498]. This fact may be a specific manifestation in 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: Gravitational energy in Newton’s theory
By the negative definiteness of U, outside the source the quasilocal energy E_{ D } is a decreasing set function, i.e., if D_{1} ⊂ D_{2} and D_{2} − D_{1} is source free, then \({E_{{D_2}}} \leq {E_{{D_1}}}\). In particular, for a 2sphere of radius r surrounding a localized spherically symmetric homogeneous source with negligible internal energy, the quasilocal energy is \({E_{{D_r}}} = {{{c^4}} \over G}{\rm{m}}(1 + {1 \over 2}{{\rm{m}} \over r}) + O({r^{ 2}})\), where the mass parameter is \({\rm{m: =}}{{GM} \over {{c^2}}}(1  {3 \over 5}{{GM} \over {{c^2}R}}) + O({c^{ 6}})\) and M is the rest mass and R is the radius of the source. For a more detailed discussion of the energy in the (relativistically corrected) Newtonian theory, see [199].
3.1.2 The root of the difficulties: Gravitational energymomentum in Einstein’s theory
The action I_{m} for the matter fields is a functional of both kinds of fields, thus one can take the variational derivatives both with respect to \({\Phi _N}_{b \ldots}^{a \ldots}\) and \({g^{ab}}\). The former give the field equations, while the latter define the symmetric energymomentum tensor. Moreover, g_{ ab } provides 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 [36] (i.e., the GalileoEötvös experiment), 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 source density for gravity, like the current \(J_A^a: = \delta {I_p}/\delta A_a^A\) 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 \(A_a^A\)), 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 [449, 448]), being a quadratic expression of the curvature (and its derivatives), is (are) expected to represent only ‘higherorder’ gravitational energymomentum. (Note that according to the original tensorial definition the BelRobinson tensor is onefourth of the expression above. Our convention follows that of Penrose and Rindler [425].) 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. 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 spin angular momentum expressions for small spheres only in higherorder 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 2.1, to gravity too [73, 74, 438, 259, 260, 486].
3.1.3 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 matterplusgravity system. Its second consequence is the need for an auxiliary derivative operator, e.g., the LeviCivita covariant derivative coming from an auxiliary, nondynamic background metric (see, e.g., [307, 430]), or a background (usually torsion free, but not necessarily flat) connection (see, e.g., [287]), or the partial derivative coming from a local coordinate system (see, e.g., [525]). 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 2k^{ th } symmetric pseudotensor \(t_{(2k)}^{\alpha \beta}\) is defined by \(2\vert g{\vert ^{k + 1}}(8\pi Gt_{(2k)}^{\alpha \beta}  {G^{\alpha \beta}}): = {\partial _\mu}{\partial _\nu}[\vert g{\vert ^{k + 1}}({g^{\alpha \beta}}{g^{\mu \nu}}  {g^{\alpha \nu}}{g^{\beta \mu}})]\) (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) [222]. 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 g^{1/2} T^{ αβ }, while the second term in the divergence equation has an extra weight g^{k+1/2}. Thus, there is only one pseudotensor in this series, which satisfies the ‘conservation law’ with the correct weight. In particular, the LandauLifshitz pseudotensor 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 difficulty (see also [486, 487]). Excellent classical reviews on these (and several other) pseudotensors are [525, 77, 15, 223], and for some recent ones (using background geometric structures) see, e.g., [186, 187, 102, 211, 212, 304, 430].
A particularly useful and comprehensive recent review with many applications and an extended bibliography is that of Petrov [428]. We return to the discussion of pseudotensors in Sections 3.3.1, 4.2.2 and 11.3.5.
3.1.4 Strategies to avoid pseudotensors I: Background metrics/connections
One way of avoiding the use of pseudotensorial quantities is to introduce an explicit background connection [287] or background metric [437, 305, 310, 307, 306, 429, 184]. (The superpotential of Katz, Bičák, and LyndenBell [306] has been rediscovered recently by Chen and Nester [137] in a completely different way. We return to a discussion of the approach of Chen and Nester 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. A particularly interesting special bimetric approach was suggested in [407] (see also [408]), in which the background (flat) metric is also fixed by using Synge’s world function.
3.1.5 Strategies to avoid pseudotensors II: The tetrad formalism
In general, the frame field \(\{E_{\underline a}^a\}\) is defined only on an open subset U ⊂ M. If the domain of the frame field can be extended to the whole M, then M is called parallelizable. For time and spaceorientable spacetimes this is equivalent to the existence of a spinor structure [206], which is known to be equivalent to the vanishing of the second StiefelWhitney class of M [364], a global topological condition on M.
The discussion of how Møller’s superpotential \({\vee _e}^{ab}\) is related to the NesterWitten 2form, by means of which an alternative form of the ADM energymomentum is given and and by means of which several quasilocal energymomentum expressions are defined, is given in Section 3.2.1 and in the first paragraphs of Section 8.
3.1.6 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 (2.3), 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 \({C^a}[{\bf{K}}]: = {G^a}_b{K^b} + {\nabla _b}{\nabla ^{[a}}{K^{b]}}\) [322] (see also [77]). Although its independence of any background structure (viz. its tensorial nature) and its uniqueness property (see Komar [322] quoting Sachs) is especially attractive, the vector field K^{ a } is still to be determined. A new suggestion for the approximate spacetime symmetries that can, in principle, be used in Komar’s expression, both near a point and a world line, is given in [235]. This is a generalization of the affine collineations (including the homotheties and the Killing symmetries). We continue the discussion of the Komar expression in Sections 3.2.2, 3.2.3, 4.3.1 and 12.1, and of the approximate spacetime symmetries in Section 11.1.
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 mathematicallyinequivalent 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 [208, 223, 28, 393, 553, 426], and a recent review of conformal infinity (with special emphasis on its applicability in numerical relativity) is [195]. Reviews of the positive energy proofs from the early 1980s are [273, 427].
3.2.1 Spatial infinity: Energymomentum
There are several mathematicallyinequivalent definitions of asymptotic flatness at spatial infinity [208, 475, 37, 65, 200]. The traditional definition is based on the existence of a certain asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable definition are asymptotically flat in the traditional sense as well. A spacelike hypersurface 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 } [433, 64]. These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations. 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.
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 centerofmass, discussed below, form an antisymmetric tensor). One has 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 [377], and he was able to recover the ADM energymomentum in a natural way (see Section 11.3).
Another form of the ADM energymomentum is based on Møller’s tetrad superpotential [223]: Taking the flux integral of the current C^{ a } [K] + T^{ ab }K_{ b } on the spacelike hypersurface Σ, by Eq. (3.7) the flux can be rewritten as the r → ∞ limit of the twosurface 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 [379]. 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 at the r → ∞ limit of the twosphere integral of (twice of) Komar’s superpotential with the Killing vector K^{ a } of stationarity [223] (see also [60]), as well. (See also the remark following Eq. (3.15) below.) On the other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time translation, the Komar expression does not reproduce the ADM energy. However, by Eqs. (3.11) and (3.12) such an additional restriction might be that K^{ a } should be a constant combination of four futurepointing null vector fields of the form \({\alpha ^A}{{\bar \alpha}^{{A{\prime}}}}\), where the spinor fields a^{ A } are required to satisfy the Weyl neutrino equation ∇_{ A′A }α^{ A } = 0. This expression for the ADM energymomentum has been used to give an alternative, ‘fourdimensional’ proof of the positivity of the ADM energy [276]. (For a more detailed recent review of the various forms of the ADM energy and linear momentum, see, e.g., [293].)
In stationary spacetime the notion of the mechanical energy with respect to the world lines of stationary observers (i.e., the integral curves of the timelike Killing field) can be introduced in a natural way, and then (by definition) the total (ADM) energy is written as the sum of the mechanical energy and the gravitational energy. Then the latter is shown to be negative for certain classes of systems [308, 348].
The notion of asymptotic flatness at spatial infinity is generalized in [398]; here the background flat metric _{0}h_{ ab } on Σ − K is allowed to have a nonzero deficit angle α at infinity, i.e., the corresponding line element in spherical polar coordinates takes the form −dr^{2} − r^{2}(1 − α)(dθ^{2} + sin^{2} (θ) dϕ^{2}). Then, a canonical analysis of the minimallycoupled EinsteinHiggs field is carried out on such a background, and, following a ReggeTeitelboimtype argumentation, an ADMtype total energy is introduced. It is shown that for appropriately chosen α this energy is finite for the global monopole solution, though the standard ADM energy is infinite.
3.2.2 Spatial infinity: Angular momentum
The value of the Hamiltonian of Beig and Ó Murchadha [64], together with the appropriatelydefined asymptotic rotationboost Killing vectors [497], define the spatial angular momentum and centerofmass, 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 [433] 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 the latter must be odd parity functions of x^{ i }/r (see also [64]). Thus, in what follows we assume that k = 1. Then the value of the BeigÓ Murchadha Hamiltonian parametrized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [433], while that parametrized by the asymptotic boost Killing vectors deviates from the centerofmass of Beig and Ó Murchadha [64] by a term, which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [64], the centerofmass term of the Hamiltonian of Regge and Teitelboim is not finite on the whole phase space.) The spatial angular momentum and the new centerofmass form an antisymmetric Lorentz fourtensor, which transforms in the correct way under the fourtranslation of the origin of the asymptotically Cartesian coordinate system, and is conserved by the evolution equations [497].
The centerofmass of Beig and Ó Murchadha was reexpressed recently [57] as the r → ∞ limit of twosurface integrals of the curvature in the form (3.14) 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 twometric on \({\mathcal S}\) (see Section 4.1.1). The geometric notion of centerofmass introduced by Huisken and Yau [280] is another form of the BeigÓ Murchadha centerofmass [156].
The AshtekarHansen definition for the angular momentum is introduced in their specific conformal model of spatial infinity as a certain twosurface 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 it would fall off in metrics with 1/r falloff (but no global integral, e.g., a parity condition had to be imposed) [37, 28].
If the spacetime admits a Killing vector of axisymmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see, e.g., [534]). However, the value of the Komar integral (with the usual normalization) is twice the expected angular momentum. In particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr solution the integral is m/G, for the Killing vector of axisymmetry it is 2ma/G instead of the expected ma/G (‘factoroftwo anomaly’) [305]. We return to the discussion of the Komar integral in Sections 4.3.1 and 12.1.
3.2.3 Null infinity: Energymomentum
The study of the gravitational radiation of isolated sources led Bondi to the observation that the twosphere integral of a certain expansion coefficient m(u, θ, ϕ) of the line element of a radiative spacetime in an asymptoticallyretarded spherical coordinate system (u, r, θ, ϕ) behaves as the energy of the system at the retarded time u. Indeed, this energy is not constant in time, but decreases with u, showing that gravitational radiation carries away positive energy (‘Bondi’s massloss’) [91, 92]. The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, currently known as the BondiMetznerSachs (or BMS) group, having a structure very similar to that of the Poincaré group [440]. The only difference is that while the Poincaré group is a semidirect product of the Lorentz group and a four dimensional commutative group (of translations), the BMS group is the semidirect product of the Lorentz group and an infinitedimensional commutative group, called the group of the supertranslations. A fourparameter subgroup in the latter can be identified in a natural way as the group of the translations. This makes it possible to compare the BondiSachs fourmomenta defined on different cuts of scri, and to calculate the energymomentum carried away by the gravitational radiation in an unambiguous way. (For further discussion of the flux, see the fourth paragraph of Section 3.2.4.) At the same time the study of asymptotic solutions of the field equations led Newman and Unti to another concept of energy at null infinity [394]. However, this energy (currently known as the NewmanUnti energy) does not seem to have the same significance as the Bondi (or BondiSachs [426] or TrautmanBondi [147, 148, 146]) energy, because its monotonicity can be proven only between special, e.g., stationary, states. The Bondi energy, which is the time component of a Lorentz vector, the BondiSachs energymomentum, has a remarkable uniqueness property [147, 148].
The BondiSachs energymomentum can also be expressed by the integral of the NesterWitten 2form [285, 342, 343, 276]. However, in nonstationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [106]. Thus, the spinor fields in the NesterWitten 2form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves are the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [106], was to require the spinor fields to be the solutions of the asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all of these, based only on the linear firstorder differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are determined in [496].
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 [146] and used to recover the BondiSachs energymomentum.
Similar to the ADM case, the simplest proofs of the positivity of the Bondi energy [446] are probably those that are based on the NesterWitten 2form [285] and, in particular, the use of twocomponent spinors [342, 343, 276, 274, 436]: 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.
Converting the integral of the NesterWitten 2form into a (positive definite) 3dimensional integral on Σ, a strictly positive lower bound can be given both for the ADM and BondiSachs masses. Although total energymomentum (or mass) in the form of a twosurface integral cannot be a introduced in closed universes (i.e., when Σ is compact with no boundary), a nonnegative quantity m, based on this positive definite expression, can be associated with Σ. If the matter fields satisfy the dominant energy condition, then \({\rm{m}}\,{\rm{=}}\,{\rm{0}}\) if and only if the spacetime is flat and topologically Σ is a 3torus; moreover its vanishing is equivalent to the existence of nontrivial solutions of Witten’s gauge condition. This m turned out to be recoverable as the first eigenvalue of the square of the SenWitten operator. It is the usefulness and the applicability of this m in practice which tell us if this is a reasonable notion of total mass of closed universes or not [503].
3.2.4 Null infinity: Angular momentum
At null infinity we have a generally accepted definition for angular momentum only in stationary or axisymmetric, but not in general, radiative spacetime, where there are various, mathematically inequivalent suggestions for it (see Section 4.2.4). Here we review only some of those total angular momentum definitions that can be ‘quasilocalized’ or connected somehow to quasilocal expressions, i.e., those that can be considered as the nullinfinity limit of some quasilocal expression. We will continue their discussion in the main part of the review, namely in Sections 7.2, 11.1 and 9.
The construction based on the WinicourTamburino linkage (3.15) can be associated with any BMS vector field [557, 337, 45]. 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 parametrized by four numbers, the set of the origins at null infinity requires a smooth function of the form \(u:{S^2} \rightarrow {\rm{\mathbb R}}\). 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 twosphere. 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.
Motivated by Penrose’s idea that the ‘conserved’ quantities at null infinity should be searched for in the form of a charge integral of the curvature (which will be discussed in detail in Section 7), a general expression \({Q_{\mathcal S}}[{K^a}]\), associated with any BMS generator K^{ a } and any cut \({\mathcal S}\) of scri, was introduced [174]. For real K^{ a } this is real; it is vanishing in Minkowski spacetime; it reproduces the BondiSachs energymomentum for BMS translations; it yields nontrivial results for proper supertranslations; and for BMS rotations the resulting expressions can be interpreted as angular momentum. It was shown in [453, 173] that the difference \({Q_{{{\mathcal S}{\prime}}}}[{K^a}]  {Q_{{{\mathcal S}{{\prime\prime}}}}}[{K^a}]\) for any two cuts \({{\mathcal S}{\prime}}\) and \({{\mathcal S}{{\prime\prime}}}\) can be written as the integral of some local function on the subset of scri bounded by the cuts \({{\mathcal S}{\prime}}\) and \({{\mathcal S}{{\prime\prime}}}\), and this is precisely the flux integral of [44]. Unfortunately, however, the angular momentum introduced in this way still suffers from the same supertranslation ambiguity. A possible resolution of this difficulty could be the suggestion by Dain and Moreschi [169] in the charge integral approach to angular momentum of Moreschi [369, 370]. Their basic idea is that the requirement of the vanishing of the supermomenta (i.e., the quantities corresponding to the proper supertranslations) singles out a fourrealparameter family of cuts, called nice cuts, by means of which the BMS group can be reduced to a Poincaré subgroup that yields a welldefined notion of angular momentum. For further discussion of certain other angular momentum expressions, especially from the points of view of numerical calculations, see also [204].
Another promising approach might be that of Chruściel, Jezierski, and Kijowski [146], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperboloidal spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian fourspace of origins, they appear to be the generators with respect to some fixed ‘centerofthecut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.
In addition to the supertranslation ambiguity in the definition of angular momentum, there could be another potential ambiguity, even if the angular momentum is well defined on every cut of future null infinity. In fact, if, for example, the definition of the angular momentum is based on the solutions of some linear partial differential equation on the cut (such as Bramson’s definition, or the ones discussed in Sections 7 and 9), then in general there is no canonical isomorphism between the spaces of the solutions on different cuts, even if the solution spaces, as abstract vector spaces, are isomorphic. Therefore, the angular momenta on two different cuts belong to different vector spaces, and, without any natural correspondence between the solution spaces on the different cuts, it is meaningless to speak about the difference of the angular momenta. Thus, we cannot say anything about, e.g., the angular momentum carried away by gravitational radiation between two retarded time instants represented by two different cuts.
One possible resolution of this difficulty was suggested by Helfer [264]. He followed the twistorial approach presented in Section 7 and used a special bijective map between the twosurface twistor spaces on different cuts. His map is based on the special structures available only at null infinity. Though this map is nonlinear, it is shown that the angular momenta on the different cuts can indeed be compared. Another suggestion for (only) the spatial angular momentum was given in [501]. This is based on the quasilocal Hamiltonian analysis that is discussed in Section 11.1, and the use of the divergencefree vector fields built from the eigenspinors with the smallest eigenvalue of the twosurface Dirac operators. The angular momenta, defined in these ways on different cuts, can also be compared. We give a slightly more detailed discussion of them in Sections 7.2 and 11.1, respectively.
The main idea behind the recent definition of the total angular momentum at future null infinity of Kozameh, Newman and SilvaOrtigoza, suggested in [325, 326], is analogous to finding the centerofcharge (i.e., the timedependent position vector with respect to which the electric dipole moment is vanishing) in flatspace electromagnetism: By requiring that the dipole part of an appropriate null rotated Weyl tensor component \(\psi _1^0\) be vanishing, a preferred set of origins, namely a (complex) centerofmass line can be found in the fourcomplexdimensional solution space of the goodcut equation (the Hspace). Then the asymptotic Bianchi identities take the form of conservation equations, and certain terms in these can (in the given approximation) be identified with angular momentum. The resulting expression is just Eq. (4.15), to which all the other reasonable angular momentum expressions are expected to reduce in stationary spacetimes. A slightly more detailed discussion of the necessary technical background is given in Section 4.2.4.
3.3 The necessity of quasilocality for 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 ill defined and the gravitational energymomentum meaningless. However, the successes discussed in Section 3.2 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 fibers 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 to 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 cross sections [192, 358, 486, 487], and the expression of the pseudotensors by their superpotentials are the pullbacks of the Sparling equation [476, 175, 358]. In addition, Chang, Nester, and Chen [131] found a natural quasilocal Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see Section 11.3.5). 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 [76, 439, 284]: Since (in the absence of any nondynamical geometric background) 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), twosurfaces (e.g., the area of a closed twosurface) etc. determined by some body or physical fields. In addition, as Torre showed [523] (see also [524]), in spatiallyclosed vacuum spacetimes there can be no nontrivial observable, built as spatial integrals of local functions of the canonical variables and their finitely many derivatives. 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.
The results of Friedrich and Nagy [202] show that under appropriate boundary conditions the initial boundary value problem for the vacuum Einstein equations, written into a firstorder symmetric hyperbolic form, has a unique solution. Thus, there is a solid mathematical basis for the investigations of the evolution of subsystems of the universe, and hence, it is natural to ask about the observables, and in particular the conserved quantities, of their dynamics.
3.3.2 Domains for quasilocal quantities
 1.
the globally hyperbolic domains D ⊂ M with compact closure,
 2.
the compact spacelike (in fact, acausal) hypersurfaces Σ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and
 3.
the closed, orientable spacelike twosurfaces \({\mathcal S}\) (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 one of the main ideas is that of Regge and Teitelboim [433], 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. Another idea is the expectation, based on the study of the quasilocal Hamiltonian dynamics of a single scalar field, that the boundary terms appearing in the calculation of the Poisson brackets of two Hamiltonians (the ‘Poisson boundary terms’), represent the infinitesimal flow of energymomentum and angular momentum between the physical system and the rest of the universe [502]. Therefore, these boundary terms must be gauge invariant in every sense. This requirement restricts the potential boundary terms in the Hamiltonian as well as the boundary conditions for the canonical variables and the lapse and shift. However, if we are not interested in the structure of the quasilocal phase space, then, as a short cut, we can use the HamiltonJacobi method to define the quasilocal quantities. The resulting expression is a twosurface 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 twosurface integral (e.g., in the Lagrangian approaches the integral of a superpotential 2form, or in the Hamiltonian approaches a boundary term together with the boundary conditions for the canonical variables),
 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 twosurface (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 twosurface).
4 Tools to Construct and Analyze 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 twosurfaces, with special emphasis on twosurface data. Then, in Sections 4.2 and 4.3, we discuss the special situations where there is a moreorless 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 twosurfaces
The first systematic study of the geometry of spacelike twosurfaces from the point of view of quasilocal quantities is probably due to Tod [514, 519]. Essentially, his approach is based on the GerochHeldPenrose (GHP) formalism [209]. Although this is a very effective and flexible formalism [209, 425, 426, 277, 479], its form is not spacetime covariant. Since in many cases the covariance of a formalism itself already gives some hint as to how to treat and solve the problem at hand, we concentrate here mainly on a spacetimecovariant description of the geometry of the spacelike twosurfaces, developed gradually in [489, 491, 492, 493, 198, 500]. 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 twosurfaces. Our standard differential geometric reference is [318, 319].
4.1.1 The Lorentzian vector bundle
The restriction \({{\rm{V}}^a}({\mathcal S})\) to the closed, orientable spacelike twosurface \({\mathcal S}\) of the tangent bundle TM of the spacetime has a unique decomposition to the g_{ ab }orthogonal sum of the tangent bundle TS of \({\mathcal S}\) and the bundle of the normals, denoted by NS. 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 projections to \(T{\mathcal S}\) and \(N{\mathcal S}\) are \(\Pi _b^a: = \delta _b^a  {t^a}{t_b} + {\upsilon ^a}{\upsilon _b}\) and \(O_b^a: = \delta _b^a  \Pi _b^a\), respectively. The induced twometric 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 \({{\rm{V}}^a}({\mathcal S})\) together with the fiber 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 twomanifolds, see, e.g., [10, 500].
4.1.2 Connections
The spacetime covariant derivative operator ∇_{ e } defines two connections on \({{\rm{V}}^a}({\mathcal S})\). The first covariant derivative, 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 \({{\rm{V}}^a}({\mathcal S})\). Obviously, δ_{ e } annihilates both the fiber metric g_{ ab } and the projection \(\Pi _b^a\). However, since for twosurfaces 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 oneform 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 \({{\rm{V}}^a}({\mathcal S})\) 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 Embeddings and convexity conditions
To prove certain statements about quasilocal quantities, various forms of the convexity of \({\mathcal S}\) must be assumed. The convexity of \({\mathcal S}\) in a threegeometry is defined by the positive definiteness of its extrinsic curvature tensor. If, in addition, the threegeometry is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of \({\mathcal S}\). It is this convexity condition that appears in the solution of the Weyl problem of differential geometry [397]: if \(({S^2},{q_{ab}})\) is a C^{4} Riemannian twomanifold with positive scalar curvature, then this can be isometrically embedded (i.e., realized as a closed convex twosurface) in the Euclidean threespace ℝ^{3}, and this embedding is unique up to rigid motions [477]. However, there are counterexamples even to local isometric embedability, when the convexity condition, i.e., the positivity of the scalar curvature, is violated [373]. We continue the discussion of this embedding problem in Section 10.1.6.
In the context of general relativity the isometric embedding of a closed orientable twosurface into the Minkowski spacetime ℝ^{1,3} is perhaps more interesting. However, even a naïve function counting shows that if such an embedding exists then it is not unique. An existence theorem for such an embedding, \(i:{\mathcal S} \rightarrow {{\rm{{\mathbb R}}}^{1,3}}\), (with S^{2} topology) was given by Wang and Yau [543], and they controlled these isometric embeddings in terms of a single function τ on the twosurface. This function is just \({x^{\underline a}}{T_{\underline a}}\), the ‘time function’ of the surface in the Cartesian coordinates of the Minkowski space in the direction of a constant unit timelike vector field \({T_{\underline a}}\). Interestingly enough, \(({\mathcal S},{q_{ab}})\) is not needed to have positive scalar curvature, only the sum of the scalar curvature and a positive definite expression of the derivative δ_{ e }τ is required to be positive. This condition is just the requirement that the surface must have a convex ‘shadow’ in the direction \({T^{\underline a}}\), i.e., the scalar curvature of the projection of the twosurface \(i({\mathcal S}) \subset {{\rm{{\mathbb R}}}^{1,3}}\) to the spacelike hyperplane orthogonal to \({T^{\underline a}}\) is positive. The Laplacian δ_{ e }δ^{ e }τ of the ‘time function’ gives the mean curvature vector of \(i({\mathcal S})\) in ℝ^{1,3} in the direction \({T^{\underline a}}\).
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., θ:= q^{ ab }θ_{ ab } > 0, and weakly past convex if θ′:= q^{ ab }θ′_{ ab } < 0 [519]. \({\mathcal S}\) is called mean convex [247] 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 \(D: = \det \Vert {\theta ^a}_b\Vert \, = \,{1 \over 2}({\theta _{ab}}{\theta _{cd}}  {\theta _{ac}}{\theta _{bd}}){q^{ab}}{q^{cd}}\) and \({D{\prime}}: = \det \Vert{\theta{\prime}^{a}}_b\Vert \, = \,{1 \over 2}(\theta _{ab}{\prime}\theta _{cd}{\prime}  \theta _{ac}{\prime}\theta _{cd}{\prime}){q^{ab}}{q^{cd}}\). Note that, although the expansion tensors, and in particular the functions θ, θ′, D, and D′ are boostgaugedependent, 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 [519, 492]. 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 oneform A_{ e }, i.e., the vertical part of the connection δ_{ e }, was given by Liu and Yau [338]: There are real functions α and γ, unique up to additive constants, such that A_{ e } = ε_{ e }fδ_{ f }α + δ_{ e }γ. 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 boostgaugeinvariant part of A_{ e }, while γ represents its gauge content. Since δ_{ e }A^{ e } = δ_{ e }δ^{ e }γ, the ‘Coulombgauge condition’ δ_{ e }A^{ e } = 0 uniquely fixes A_{ e } (see also Section 10.4.1).
By the GaussBonnet theorem one has \(\oint\nolimits_{\mathcal S} {f\,d{\mathcal S} =} \oint\nolimits_{\mathcal S} {^{\mathcal S}Rd{\mathcal S} = 8\pi (1  g)}\), 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 twosurface data for chargetype quasilocal quantities (i.e., for twosurface 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 twosurface \({\mathcal S}\) is a normalized spinor basis \(\varepsilon _{\rm{A}}^A: = \{{o^A},\,{\iota ^A}\}, \, {\bf{A}} = 0,1\), 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 futurepointing null vectors l^{ a } := o^{ A }ō^{A′} 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 twosurface. For example, on the twosphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors m^{ a } and \({{\bar m}^a}\) cannot form a globallydefined basis on \({\mathcal S}\). Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable twosurface with a globallytrivial tangent bundle is the torus.
Fixing a GHP spin frame \(\{\varepsilon _{\rm{A}}^A\}\) 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) [209, 425]: A scalar ϕ is said to be of type (p, q) if, under the rescaling o^{ A } − λo^{ A }, ι^{ A } ↦ λ^{−1} ι^{ A } of the GHP spin frame with some nowherevanishing complex function λ: U → ℂ, the scalar transforms as \(\phi \mapsto {\lambda ^p}{{\bar \lambda}^q}\phi\). For example, \(\rho: = {\theta _{ab}}{m^a}{{\bar m}^b} =  {1 \over 2}\theta, \,{\rho \prime}: = \theta _{ab}\prime{m^a}{{\bar m}^b} = \theta  {1 \over 2}{\theta \prime},\,\sigma := {\theta _{ab}}{m^a}{m^b} = {\sigma _{ab}}{m^a}{m^b}\) and \(\sigma := \theta _{ab}\prime{{\bar m}^a}{{\bar m}^b}\) are of type (1,1), (−1, −1), (3, −1), and (−3, 1), respectively. The components of the Weyl and Ricci spinors, \({\psi _0}: = {\psi _{ABCD}}{o^A}{o^B}{o^C}{o^D},{\psi _1}: = {\psi _{ABCD}}{o^A}{o^B}{o^C}{\iota ^D},\,{\psi _2}: = {\psi _{ABCD}}{o^A}{o^B}{\iota ^C}{\iota ^D},\, \ldots, \,{\phi _{00}}: = {\phi _{A{B\prime}}}{o^A}{{\bar o}^{{B\prime}}},\,{\phi _{01}}: = {\phi _{A{B\prime}}}{o^A}{{\bar \iota}^{{B\prime}}},\, \ldots\), 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 } = ψo^{′A} and ι^{ A } = ψ^{−1}ι^{′A}.
The connection δ_{ e } defines a connection ð_{ e } on the line bundles E(p,q) [209, 425]. 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 \(\{\varepsilon _{\bf{A}}^A\}\). These locallydefined operators yield globallydefined 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 oneform, and are built both from the connection oneform for the intrinsic Riemannian geometry of \(({\mathcal S},\,{q_{ab}})\) and the connection oneform 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 [198] in general, and in [177, 58, 490] for spherical topology.
4.1.7 Irreducible parts of the derivative operators
4.1.8 SO(1, 1)connection oneform versus anholonomicity
Obviously, all the structures we have considered can be introduced on the individual surfaces of one or twoparameter families of surfaces, as well. In particular [246], let the twosurface \({\mathcal S}\) 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 neighborhood of \({\mathcal S}\)) be foliated by two oneparameter families of smooth hypersurfaces {ν_{+} = const.} and {ν_{−} = const.}, where ν_{ ± }: M → ℝ, such that \({{\mathcal N}^ +} = \{{v_ +} = 0\}\) and \({{\mathcal N}^ } = \{{v_ } = 0\}\). One can form the two normals, n_{ ±a }:= ∇_{ a }ν_{ ± }, 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 oneform A_{ e } above.) Then the anholonomicity is defined by \({\omega _e}: = {1 \over {2{n^2}}}{[{n_ },\,{n_ +}]^f}{q_{fe}} = {1 \over {2{n^2}}}({\beta _{+ e}}  {\beta _{ e}})\). Since ω_{ e } 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 [246] that ω_{ e } depends only on \({\mathcal S}\). However, this implies only that ω_{ e } is invariant with respect to a restricted class of the change of the foliations, and that ω_{ e } is invariantly defined only by this class of the foliations rather than the twosurface. In fact, ω_{ e } does depend on the foliation: Starting with a different foliation defined by the functions \({{\bar v}_ +}: = \exp (\alpha){v_ +}\) and \({{\bar v}_ }: = \exp (\beta){v_ }\) 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, ω_{ e } and \({{\bar \omega}_e}\), would be different: \({{\bar \omega}_e}  {\omega _e} = {1 \over 2}{\Delta _e}(\alpha  \beta)\). Therefore, the anholonomicity is a gaugedependent 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
\(E({\mathcal S})\) contains a contribution from the gravitational ‘field’ too. For example, for fluids it is not simply the volume integral of the energy density μ of the fluid, because that would be \(4\pi \int\nolimits_0^r {{r\prime^{2}}\exp (\alpha)\mu \,d{r\prime}}\). This deviation can be interpreted as the contribution of the gravitational potential energy to the total energy. Consequently, \(E({\mathcal S})\) is not a globally monotonic function of r, even if μ ≥ 0. For example, in the closed FriedmannRobertsonWalker spacetime (where, to cover the whole threespace, r cannot be chosen to be the arearadius and \(r \in [0,\pi ])\,E({\mathcal S})\) is increasing for r ∈ [0, π/2), taking its maximal value at r = π/2, and decreasing for r ∈ [π/2, π].
This example suggests a slightly more exotic sphericallysymmetric 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’ sphericallysymmetric 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 threespace and \(d{l^2} = \Omega _{{\rm{FRW}}}^2: = B{(1 + {{{r^2}} \over {4{T^2}}})^{ 2}}\) with 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 twosurface 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 [213]): \(d{{\bar l}^2} = \Omega _S^2d\bar l_0^2\), where \(\Omega _S^2: = {(1 + {m \over {2\bar r}})^4}\) and the Euclidean radial coordinate \({\bar r}\) runs from \({{\bar r}_0}\) to ∞, 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 roundsphere energy is \(E(\bar r) = m/G\). These two metrics can be matched to obtain a differentiable metric with a Lipschitzcontinuous derivative at the twosurface 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 threeplane — like the capital Greek letter Ω — for later reference we will call it an ‘Ω_{ M,m }spacetime’.
Sphericallysymmetric spacetimes admit a special vector field, called the Kodama vector field K^{ a }, such that K_{ a }G^{ ab } is divergence free [321]. 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 in the line element given in the first paragraph above K^{ a } = exp[−(α + γ)](∂/∂t)^{ a }. If Σ is a solid ball of radius r, then the flux of S_{ a } is precisely the standard roundsphere expression (4.7) for the twosphere ∂Σ [375].
An interesting characterization of the dynamics of the sphericallysymmetric gravitational fields can be given in terms of the energy function E(t, r) given by (4.7) (or by (4.8)) (see, e.g., [578, 352, 250]). In particular, criteria for the existence and formation of trapped surfaces and for the presence and nature of the central singularity can be given by E(t, r). Other interesting quasilocallydefined quantities are introduced and used to study nonlinear perturbations and backreaction in a wide class of sphericallysymmetric spacetimes in [483]. For other applications of E(t, r) in cosmology see, e.g., [484, 130].
4.2.2 Small surfaces
In the literature there are two kinds 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 [275], and the other is the concept of small ellipsoids in a spacelike hypersurface, considered first by Woodhouse in [313]. 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 center, is a small given value, and the geometry of this sphere is characterized by data at this center. Small ellipsoids are twosurfaces in a spacelike hypersurface with a more general shape.
To define the first, let p ∈ M be a point, and t^{ a } a futuredirected 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 parametrization completely. Then, in an open neighborhood 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 twosurface and is homeomorphic to \({{\mathcal S}^2}\). \({{\mathcal S}_r}\) is called a small sphere of radius r with vertex p. Note that the condition l^{ a }t_{ a } = 1 fixes the boost gauge, too.
Obviously, the same analysis can be repeated for any other quasilocal quantity. For the energymomentum, \({Q_{\mathcal S}}\) has the structure \(\oint\nolimits_{\mathcal S} {\mathcal Q} ({\partial _\mu}{g_{\alpha \beta}})\,d{\mathcal S}\), for angular momentum it is \(\oint\nolimits_{\mathcal S} {\mathcal Q} ({\partial _\mu}{g_{\alpha \beta}})r\, d{\mathcal S}\), while the area of \({\mathcal S}\) is \(\oint\nolimits_{\mathcal S} {d{\mathcal S}}\). Therefore, the leading term in the expansion of the angular momentum is r^{4} and r^{6} order in nonvacuum and vacuum with the energymomentum and the BelRobinson tensors, respectively, while the first nontrivial correction to the area 4πr^{2} is of order r^{A} 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 futurepointing 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 [313]. 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 neighborhood U of p in Σ, the level surfaces \({{\mathcal S}_r}: = \{q \in \Sigma 2f(q) = {r^2}\}\) are smooth compact twosurfaces homeomorphic to S^{2}. Then, in the r → 0 limit, the surfaces \({{\mathcal S}_r}\) look like small nested ellipsoids centered at p. The function f is usually ‘normalized’ so that h^{ ab }D_{ a }D_{ b }f_{ p } = −3.
A slightly different framework for calculations in small regions was used in [327, 170, 235]. Instead of the NewmanPenrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or Fermi type normal) coordinates on an open neighborhood U of a point p ∈ M or a timelike curve γ are used, in which the metric, as well as the Christoffel symbols on U, are expressed by the coordinates on U and the components of the Riemann tensor at p or on γ. In these coordinates and the corresponding frames, the various pseudotensorial and tetrad expressions for the energymomentum have been investigated. It has been shown that a quadratic expression of these coordinates with the BelRobinson tensor as their coefficient appears naturally in the local conservation law for the matter energymomentum tensor [327]; the BelRobinson tensor can be recovered as some ‘double gradient’ of a special combination of the Einstein and the LandauLifshitz pseudotensors [170]; Møller’s tetrad expression, as well as certain combinations of several other classical pseudotensors, yield the BelRobinson tensor [473, 470, 471]. In the presence of some nondynamical (background) metric a 11parameter family of combinations of the classical pseudotensors exists, which, in vacuum, yields the BelRobinson tensor [472, 474]. (For this kind of investigation see also [465, 468, 466, 467, 469]).
In [235] a new kind of approximate symmetries, namely approximate affine collineations, are introduced both near a point and a world line, and used to introduce Komartype ‘conserved’ currents. (For a readable text on the nonKilling type symmetries see, e.g., [233].) These symmetries turn out to yield a nontrivial gravitational contribution to the matter energymomentum, even in the leading r^{3} order.
4.2.3 Large spheres near spatial infinity
Near spatial infinity we have the a priori 1/r and 1/r^{2} falloff for the threemetric 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.^{6} 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 \((r,\zeta, \,\bar \zeta)\) on Σ. Let \(\varepsilon _{\bf{A}}^A = \{{o^A},{\iota ^A}\}, \, {\bf{A}} = 0,\, 1\), be a GHP spinor dyad on Σ adapted to the large spheres in such a way that m^{ a } := o^{ A }ῑ^{A′} and \({{\bar m}^a} = {\iota ^A}{{\bar o}^{{A\prime}}}\) are tangent to the spheres and are tangent to the spheres and, the future directed unit normal of Σ. These conditions fix the spinor dyad completely, and, in particular, \({v^a} = _2^1{o^A}{{\bar o}^{{A\prime}}}  {\iota ^A}{{\bar \iota}^{{A\prime}}}\), 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 as 1/r^{2} and the curvature components to zero like 1/r^{3}. Expanding the spin coefficients and curvature components as a power series of 1/r, one can solve the field equations asymptotically (see [65, 61] 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 higherorder 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 twocomplexdimensional vector space. Then, by the falloff properties \({D_{e0}}{\lambda _A} = {\mathcal O}({r^{ 2}})\). Thus, we can define the asymptotically constant spinor fields to be those λ_{ A } that satisfy \({D_e}{\lambda _A} = {\mathcal O}({r^{ 2}})\), where D_{ e } is the intrinsic LeviCivita derivative operator on Σ. Note that this implies that, with the notation of Eq. (4.6), 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 [413, 414, 415, 426] (see also [208]), 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 \({\rm{\mathbb R}} \times {S^2}\), 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 \((\zeta, \bar \zeta) \in {S^2}\), defined first on the origin cut \({{\mathcal S}_0}\) and then extended in a natural way along the null generators to the whole ℐ^{+}. Then any other cut \({\mathcal S}\) of ℐ^{+} can be specified by a function \(u = f(\zeta, \bar \zeta)\). In particular, the cuts \({{\mathcal S}_u}: = \{u = {\rm{const}}.\}\) are obtained from \({{\mathcal S}_0}\) by a pure time translation.
The coordinates \((u,\zeta, \bar \zeta)\) can be extended to an open neighborhood of ℐ^{+} in the spacetime in the following way. Let \({{\mathcal N}_u}\) be the family of smooth outgoing null hypersurfaces in a neighborhood 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 \((u,r,\zeta, \bar \zeta)\) forms a coordinate system. The u = const., r = const. twosurfaces \({{\mathcal S}_{u,r}}\) (or simply \({{\mathcal S}_r}\) if no confusion can arise) are spacelike topological twospheres, 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: \(\bar r: = r + g(u,\zeta, \bar \zeta)\) 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’.^{7} In many of the largesphere 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 a detailed discussion of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see, for example, [394, 393, 107].
In addition to the coordinate system, we need a NewmanPenrose null tetrad, or rather a GHP spinor dyad, \(\varepsilon _{\rm{A}}^A = \{{o^A},{\iota ^A}\}, \,{\rm{A = 0,1}}\), 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 }ō^{A′} 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 [394] chose ι^{ A } to be parallelly propagated along l^{ a }. This choice yields the vanishing of a number of spin coefficients (see, for example, the review [393]). The asymptotic solution of the EinsteinMaxwell equations as a series of 1/r in this coordinate and tetrad system is given in [394, 179, 425], 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 }ῑ^{A′} 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, the rotation with the highest accuracy was done for the solutions of the EinsteinMaxwell system by Shaw [455].
In contrast to the spatialinfinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [106]. 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 [106], 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 has two linearlyindependent 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 \({K^{\underline A {{\underline A}{\prime}}}}\lambda _{\underline A}^A\bar \lambda _{{{\underline A}{\prime}}}^{{A{\prime}}} = {K^{\underline A {{\underline A}{\prime}}}}\lambda _A^1\bar \lambda _{\underline {{A{\prime}}}}^{{1{\prime}}}{\iota ^A}{{\bar \iota}^{{A{\prime}}}}\) 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 \( \sigma _{\rm{i}}^{\underline A \underline B}\lambda _{\underline A}^1\lambda _{\underline B}^1\), where \(\sigma _{\rm{i}}^{\underline A \underline B}\) are the standard SU(2) Pauli matrices (divided by \(\sqrt 2)\) [496]. 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 twosurface twistor equation \({\mathcal N}\lambda : = ({{\mathcal T}^ +} \oplus {{\mathcal T}^ })\lambda = 0\) (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [496].
Instead of the Bondi type coordinates above, one can introduce other ‘natural’ coordinates in a neighborhood of ℐ^{+}. Such is the one based on the outgoing asymptoticallyshearfree null geodesics [27]. While the Bonditype coordinate system is based on the null geodesic generators of the outgoing null hypersurfaces \({{\mathcal N}_u}\), and hence, in the rescaled metric these generators are orthogonal to the cuts \({{\mathcal S}_u}\), the new coordinate system is based on the use of outgoing null geodesic congruences that extend to ℐ^{+} but are not orthogonal to the cuts of ℐ^{+} (and hence, in general, they have twist). The definition of the new coordinates \((u,r,\zeta, \bar \zeta)\) is analogous to that of the Bonditype coordinates: \((u, \zeta, \bar \zeta)\) labels the intersection point of the actual geodesic and ℐ^{+}, while r is the affine parameter along the geodesic. The tangent \({{\tilde l}^a}\) of this null congruence is asymptotically null rotated about n^{ a }: In the NP basis \(\{{l^a},{n^a},{m^a},{{\bar m}^a}\}\) above \({{\tilde l}^a} = {l^a} + b{{\bar m}^a} + \bar b{m^a} + b\bar b{m^a}\), where \(b =  L(u,\zeta, \bar \zeta)/r + {\mathcal O}({r^{ 2}})\) and \(L = L(u,\zeta, \bar \zeta)\) is a complex valued function (with spin weight one) on ℐ^{+}. Then Aronson and Newman show in [27] that if L is chosen to satisfy \(\eth L + L\dot L = {\sigma ^0}\), then the asymptotic shear of the congruence is, in fact, of order r^{−3}, and by an appropriate choice for the other vectors of the NP basis many spin coefficients can be made zero. In this framework it is the function L that plays a role analogous to that of σ^{0}, and, indeed, the asymptotic solution of the field equations is given in terms of L in [27]. This L can be derived from the solution Z of the goodcut equation, which, however, is not uniquely determined, but depends on four complex parameters: \(Z = Z({Z^{\underline a}},\zeta, \bar \zeta)\). It is this freedom that is used in [325, 326] to introduce the angular momentum at future null infinity (see Section 3.2.4). Further discussion of these structures, in particular their connection with the solutions of the goodcut equation and the Hspace, as well as their applications, is given in [324, 325, 326, 5].
4.2.5 Other special situations
In the weak field approximation of general relativity [525, 36, 534, 426, 303] the gravitational field is described by a symmetric tensor field h_{ ab } on Minkowski spacetime (\(({{\rm{R}}^4},g_{ab}^0)\)), 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 secondorder 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 (2.5) 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 [481, 277, 426].
ppwaves spacetimes are defined to be those that admit a constant null vector field L^{ a }, and they 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 [478]. A remarkable feature of the ppwave metrics is that, in the usual coordinate system, the Einstein equations become a twodimensional linear equation for a single function. In contrast to the approach adopted almost exclusively, Aichelburg [8] 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 [485].
Since the final state of stellar evolution (the neutron star or 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 [534]. Thus, axisymmetric twosurfaces 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 [418], 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 twosurfaces in these spacetimes are the spacelike cross sections of the event horizon [80].
There is a welldefined notion of total energymomentum not only in the asymptotically flat, but even in the asymptotically antide Sitter spacetimes as well. 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 [220]. (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 [265].) The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically antide Sitter spacetimes and to study their general, basic properties in [42]. A comparison and analysis of the various definitions of mass for asymptotically antide Sitter metrics is given in [150].
Extending the spinorial proof [349] of the positivity of the total energy in asymptotically antide Sitter spacetime, Chruściel, Maerten and Tod [149] give an upper bound for the angular momentum and centerofmass in terms of the total mass and the cosmological constant. (Analogous investigations show that there is a similar bound at the future null infinity of asymptotically flat spacetimes with no outgoing energy flux, provided the spacetime contains a constantmeancurvature, hyperboloidal, initialdata set on which the dominant energy condition is satisfied. In this bound the role of the cosmological constant is played by the (constant) mean curvature of the hyperboloidal spacelike hypersurface [151].) Thus, it is natural to ask whether or not a specific quasilocal energymomentum or angular momentum expression has the correct limit for large spheres in asymptotically antide Sitter spacetimes.
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, [176, 143]). However, before discussing them, it seems useful to first formulate some general principles that any quasilocal quantity should satisfy.
4.3.1 General expectations
 1.
The quasilocal quantities that are twosurface observables should depend only on the twosurface data, but they cannot depend, e.g., on the way that the various geometric structures on \({\mathcal S}\) are extended off the twosurface. There seems to be no a priori reason why the twosurface would have to be restricted to spherical topology. Thus, in the ideal case, the general construction of the quasilocal energymomentum and angular momentum should work for any closed orientable spacelike twosurface.
 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 twosurface, which quasisymmetries are special spacetime vector fields on the twosurface. 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’.
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 behavior of the quasilocal quantities.
 1.1
The quasilocal energymomentum \(P_{\mathcal S}^{\underline a}\) must be a futurepointing 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 largesphere 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’ MisnerSharp roundsphere expression.
 1.7
For marginally trapped surfaces the quasilocal mass \({m_{\mathcal S}}\) must be the irreducible mass \(\sqrt {{\rm{Area(}}{\mathcal S}{\rm{)/16}}\pi {G^2}}\).
Usually, \({m_{\mathcal S}}\) is expected to be monotonicgally increasing in some appropriate sense [143]. For example, if \({{\mathcal S}_1} = \partial \Sigma\) for some achronal (and hence spacelike or null) hypersurface Σ in which \({{\mathcal S}_2}\) is a spacelike closed twosurface and the dominant energy condition is satisfied on Σ, then \({m_{{{\mathcal S}_1}}} \geq {m_{{{\mathcal S}_2}}}\) seems to be a reasonable expectation [176]. (However, see also Section 4.3.3.) A further, and, in fact, a related issue is the (post) Newtonian limit of the quasilocal mass expressions. In item 1.4 we expected, in particular, that the quasilocal mass tends to the ADM mass at spatial infinity. However, near spatial infinity the radiation and the dynamics of the fields and the geometry die off rapidly. Hence, in vacuum asymptotically flat spacetimes in the asymptotic regime the gravitational ‘field’ approaches the Newtonian one, and hence its contribution to the total energy of the system is similar to that of the negative definite binding energy [400, 199]. Therefore, it seems natural to expect that the quasilocal mass tends to the ADM mass as a monotonically decreasing function (see also sections 3.1.1 and 12.3.3).
In contrast to the energymomentum and angular momentum of the matter fields on the Minkowski spacetime, the additivity of the energymomentum (and angular momentum) is not expected. In fact, if \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) are two connected twosurfaces, then, for example, the corresponding quasilocal energymomenta would belong to different vector spaces, namely to the dual of the space of the quasitranslations of the first and second twosurface, respectively. Thus, even if we consider the disjoint union \({{\mathcal S}_1} \cup {{\mathcal S}_2}\) to surround a single physical system, we can add the energymomentum of the first to that of the second only if there is some physically/geometrically distinguished rule defining an isomorphism between the different vector spaces of the quasitranslations. Such an isomorphism would be provided for example by some naturallychosen globallydefined flat background. However, as we discussed in Section 3.1.2, general relativity itself does not provide any background. The use of such a background would contradict the complete diffeomorphism invariance of the theory. Nevertheless, the quasilocal mass and the length of the quasilocal PauliLubanski spin of different surfaces can be compared, because they are scalar quantities.
 2.1
\(J_{\mathcal S}^{\underline a \underline b}\) must give zero for round spheres.
 2.2
For twosurfaces with zero quasilocal mass, the PauliLubanski spin should be proportional to the (null) energymomentum fourvector \(P_{\mathcal S}^{\underline a}\).
 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 generallyaccepted spatial angular momentum at spatial infinity, and in stationary and in axisymmetric spacetimes it should reduce to the ‘standard’ expressions at the null infinity as well (‘correct largesphere behaviour’).
 2.5
For small spheres the antiselfdual part of \(J_{\mathcal S}^{\underline a \underline b}\), defined with respect to the center 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 Cr^{5}T_{ cdef }t^{ c }t^{ d }t^{ e }(rε^{F(A}t^{B)F′}) in vacuum for some constant C (‘correct small sphere behaviour’).
4.3.3 Incompatibility of certain ‘natural’ expectations
As Eardley noted in [176], 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 an 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, it does not seem possible to formulate the causal argument of Section 4.3.2 in the hypersurface Σ. Actually, the root of the nonmonotonicity is the fact that the quasilocal energy is a twosurface observable in the sense of requirement 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 twosurface along some special spacetime vector field.
If the quasilocal mass should, in fact, tend to the ADM mass as a monotonically deceasing function in the asymptotic region of asymptotically flat spacetimes, then neither item 1.6 nor 1.7 can be expected to hold. In fact, if the dominant energy condition is satisfied, then the standard roundsphere MisnerSharp energy is a monotonically increasing or constant (rather than strictly decreasing) function of the area radius r. For example, the MisnerSharp energy in the Schwarzschild spacetime is the constant function <monospace>m</monospace>/G. The Schwarzschild solution provides a conterexample to item 1.7, too: Since both its ADM mass and the irreducible mass of the black hole are <monospace>m</monospace>/G, any quasilocal mass function of the radius r which is strictly decreasing for large r and coincides with them at infinity and on the horizon, respectively, would have to take its maximal value on some twosurface outside the horizon. However, it does not seem why such a gemetrically, and hence physically distinguished twosurface should exist.
In the literature the positivity and monotonicity requirements are sometimes 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 threedimensional domain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasilocal energymomentum is associated with twosurfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression (4.7) 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 [54, 53]. His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let Σ be a compact, connected threemanifold 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 16πGμ ≔ 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 \({\mathcal E}(\Sigma)\). 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 \(\{{E_{{\rm{ADM}}}}(\hat \Sigma)\vert \hat \Sigma \; \in \;{\mathcal E}(\Sigma)\}\), where the infimum is taken on \({\mathcal E}(\Sigma)\). 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.^{8} 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 round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarilylarge roundsphere mass M/G) has an asymptotically flat extension, the complete spacelike hypersurface of the data set for the ΩM,mspacetime 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 \({{\tilde {\mathcal E}}_0}(\mathcal S)\) 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 isometricallyembedded Riemannian submanifold, and define \({{\tilde m}_{\rm{B}}}({\mathcal S})\) by Eq. (5.1) with \({{\mathcal E}_0}({\mathcal S})\) instead of \({{\mathcal E}_0}(\Sigma)\). Obviously, this \({{\tilde m}_{\rm{B}}}({\mathcal S})\) could be associated with a larger class of twosurfaces than the original m_{B}(Σ) can be to compact threemanifolds, and \(0 \leq {{\tilde m}_{\rm{B}}}(\partial \Sigma) \leq {m_{\rm{B}}}(\Sigma)\) holds.
In [279, 56] the set \({{\mathcal E}_0}(\Sigma)\) was allowed to include extensions \({\hat \Sigma}\) of Σ having boundaries as compact outermost horizons, when the corresponding ADM energies are still nonnegative [217], and hence m_{B}(Σ) is still well defined and nonnegative. (For another description of \({{\mathcal E}_0}(\Sigma)\) allowing horizons in the extensions but excluding them between Σ and the asymptotic end, see [110] and Section 5.2 of this paper.)
5.1.2 The main properties of m_{B}(Σ)
The first immediate consequence of Eq. (5.1) 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 (5.1) 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 that is larger than m_{B} (i.e., that 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 Eq. (5.1) one has m_{B}(Σ) ≥ m(Σ) −ε ≥ m_{B}(Σ) − ε for any ε < 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 Simon (see [56]), is that if \({\hat \Sigma}\) is any asymptotically flat, timesymmetric 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 \({\hat \Sigma}\). For further discussion of m_{B}(Σ) from the point of view of black holes, as well as the relationship between the Bartnik mass and other expressions (e.g., the Hawking energy), see [460].
As we saw, the Bartnik mass is nonnegative, and, obviously, if Σ is flat (and hence is a data set for flat spacetime), then m_{B}(Σ) = 0. The converse of this statement is also true [279]: If m_{B}(Σ) = 0, then Σ is locally flat. The Bartnik mass tends to the ADM mass [279]: If \((\hat \Sigma, {\hat h_{ab}})\) is an asymptotically flat Riemannian threegeometry 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 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 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 [279]. Let Σ be a sphericallysymmetric Riemannian threegeometry with sphericallysymmetric boundary \({\mathcal S}: = \partial \Sigma\). One can form its ‘standard’ roundsphere energy \(E({\mathcal S})\) (see Section 4.2.1), and take its sphericallysymmetric asymptotically flat vacuum extension \({{\hat \Sigma}_{{\rm{SS}}}}\) (see [54, 56]). 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 [279] \({E_{{\rm{ADM}}}}(\hat \Sigma) \geq {E_{{\rm{ADM}}}}({{\hat \Sigma}_{{\rm{SS}}}}) = E({\mathcal S})\). Therefore, the Bartnik mass of Σ is just the ‘standard’ roundsphere 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 parametrize it. Thus, by its 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 [54, 56]: The infimum in definition (5.1) 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 twosurface \(\partial \Sigma \subset \hat \Sigma\), 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 where the Bartnik mass is the ADM mass of ĥ_{ ab }. As Corvino shows [154], if there is an allowed extension \({\hat \Sigma}\) of Σ for which \({m_{{\rm{ADM}}}}(\hat \Sigma) = {m_{\rm{B}}}(\Sigma)\), then the extension \(\hat \Sigma  \bar \Sigma\) 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 [360] 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. (For further partial existence results see [17].) Bartnik’s conjecture is that (Σ, h_{ ab }) determines this exterior metric uniquely [56]. He conjectures [54, 56] that a similar computation method can be found for the mass \({m_{\rm{B}}}({\mathcal S})\), defined in Eq. (5.2), as well, where the exterior metric should be stationary. This second conjecture is also supported by partial results [155]: 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 [55]. 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 is suggested by Bray [110, 113]. Here we summarize his ideas.
Let Σ = (Σ, h_{ ab }, χ_{ ab }) be any asymptotically flat initial data set with finitelymany asymptotic ends and finite ADM masses, and suppose that the dominant energy condition is satisfied on Σ. Let \({\mathcal S}\) be any fixed twosurface 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 \({\mathcal S}\), while the inside region, \(I({\mathcal S})\), is the (closure of) \(\Sigma  O({\mathcal S})\). Next, Bray defines the ‘extension’ \({{\hat \Sigma}_{\rm{e}}}\) 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 twosurface \({\tilde {\mathcal S}}\) 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 twosurfaces such that \({{\mathcal S}_2}\) encloses \({{\mathcal S}_1}\), 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 proven), then for outerminimizing surfaces \({m_{{\rm{out}}}}({\mathcal S}) \geq {m_{{\rm{in}}}}({\mathcal S})\) [110, 113]. 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 \(\sqrt {{\rm{Area}}({\mathcal S})/(16\pi {G^2})}\) [56]. 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 twosurface data on \({\mathcal S}\) [113].
For further modification of Bartnik’s original ideas, see [311].
6 The Hawking Energy and its Modifications
6.1 The Hawking energy
6.1.1 The definition
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 twosphere \({\mathcal S}\) 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 twospheres of radius r in the Minkowski spacetime, for which ρ = −1/r and ρ′ = 1/2r, we expect zero energy, thus, D = C/(2π). 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 Eq. (6.1).
6.1.2 Hawking energy for spheres
Obviously, for round spheres, E_{H} reduces to the standard expression (4.7). 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 more general surfaces Hawking energy is calculated numerically in [272].
For a small sphere of radius r with center 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 [275]. The first result shows that in the lowest order the gravitational ‘field’ does not have a contribution to 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 for 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 Eq. (6.1) 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 [455, 457]. 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, Hawking energy may be negative, even in Minkowski spacetime. Geometrically this should be clear, since for an appropriately general (e.g., concave) twosurface \({\mathcal S}\), the integral \(\oint\nolimits_{\mathcal S} {\rho {\rho \prime}s} {\mathcal S}\) 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 topologicallyspherical twosurfaces in the t = const. spacelike hyperplane of Minkowski spacetime σσ′ is real and nonpositive, and it is zero precisely for metric spheres, while for twosurfaces in the r = const. timelike cylinder σσ′ is real and nonnegative, and it is zero precisely for metric spheres.^{9} 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 [143]: \({\mathcal S}\) will be called round enough if it is a submanifold of a spacelike hypersurface Σ, and if among the twodimensional surfaces in Σ, which enclose the same volume as \({\mathcal S}\) does, \({\mathcal S}\) has the smallest area. It is proven by Christodoulou and Yau [143] 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 Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of twosurfaces. Hawking considered oneparameter families of spacelike twosurfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of E_{H} [236]. These calculations were refined by Eardley [176]. Starting with a weakly future convex twosurface \({\mathcal S}\) and using the boost gauge freedom, he introduced a special family \({{\mathcal S}_r}\) of spacelike twosurfaces in the outgoing null hypersurface \({\mathcal N}\), where r will be the luminosity distance along the outgoing null generators. He showed that \({E_H}({{\mathcal S}_r})\) 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 \({E_H}({{\mathcal S}_r})\) is nonincreasing. Eardley also considered a special spacelike hypersurface, filled by a family of twosurfaces, for which \({E_H}({{\mathcal S}_r})\) 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 neighborhood of a mean convex twosurface by special mean convex twosurfaces [247]. (For the more general GHP formalism in which l_{ a }n^{ a } is not fixed, see [425].) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these twosurfaces 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 [194]. If \({\mathcal S}\) is a weakly future and past convex twosurface, then q^{ a } ≔ 2Q^{ a }/(Q_{ b }Q^{ b }) = −[1/(2ρ)]l_{ a } − [1/(2ρ′)]n^{ a } is an outwarddirected spacelike normal to \({\mathcal S}\). Here Q_{ b } is the trace of the extrinsic curvature tensor: \({Q_b}: = {Q^b}_{ab}\) (see Section 4.1.2). Starting with a single weakly future and past convex twosurface, Frauendiener gives an argument for the construction of a oneparameter family \({{\mathcal S}_t}\) of twosurfaces being Liedragged along its own inverse mean curvature vector q^{ a }. Assuming that such a family of surfaces (and hence, the vector field q^{ a } on the threesubmanifold 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. This family of surfaces would be analogous to the solution of the geodesic equation, where the initial point and direction at that point specify the whole solution, at least locally. However, it is known (Frauendiener, private communication) that the corresponding flow is based on a system of parabolic equations such that it does not admit a wellposed initial value formulation.^{10} Motivated by this result, Malec, Mars, and Simon [351] considered the inverse mean curvature flow of Geroch on spacelike hypersurfaces (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.) The necessary conditions on flows of twosurfaces on null, as well as spacelike, hypersurfaces ensuring the monotonicity of the Hawking energy are investigated in [114]. The monotonicity property of the Hawking energy under another geometric flows is discussed in [89].
For a discussion of the relationship between Hawking energy and other expressions (e.g., the Bartnik mass and the BrownYork energy), see [460]. For the first attempts to introduce quasilocal energy oparators, in particular the Hawking energy oparator, in loop quantum gravity, see [565].
6.1.4 Two generalizations
Hawking considered the extension of the definition of \({E_H}({\mathcal S})\) to higher genus twosurfaces as well by the second expression in Eq. (6.1). Then, in the expression analogous to the first one in Eq. (6.1), the genus of \({\mathcal S}\) appears. For recent generalizations of the Hawking energy for twosurfaces foliating the stationary and dynamical untrapped hypersurfaces, see [527, 528] and Section 11.3.4.
6.2 The Geroch energy
6.2.1 The definition
In the last step we use the GaussBonnet theorem for \({\mathcal S} \approx {S^2}\). \({E_G}({\mathcal S})\) is known as the Geroch energy [207]. Thus, it is not greater than the Hawking energy, and, in contrast to E_{H}, it depends not only on the twosurface \({\mathcal S}\), but on the hypersurface Σ as well.
The calculation of the small sphere limit of the Geroch energy was saved by observing [275] that, by Eq. (6.4), the difference of the Hawking and the Geroch energies is proportional to \(\sqrt {{\rm{Area}}({\mathcal S})} \times \oint\nolimits_{\mathcal S} {{{({\chi _{ab}}{q^{ab}})}^2}d{\mathcal S}}\). 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 \({E_H}({\mathcal S}) \geq {E_G}({\mathcal S})\) and since the Hawking energy tends to the ADM energy, the large sphere limit of \({E_G}({\mathcal S})\) in an asymptotically flat Σ cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [207].
For a definition of Geroch’s energy as a quasilocal energy oparator in loop quantum gravity, see [565].
6.2.2 Monotonicity properties
 1.
its level surfaces, \({{\mathcal S}_t}: = \{q \in \Sigma \left\vert {t(q) = t} \right.\}\), 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 inversemeancurvature foliation of (Σ, h_{ ab }) above were proven and clarified by Huisken and Ilmanen [278, 279], giving the first complete proof of the Riemannian Penrose inequality, and, as proven by Schoen and Yau [444], Jang’s quasilinear elliptic equation admits a global solution.
6.3 The Hayward energy
In the literature there is another modification of Hawking energy, due to Hayward [248]. His suggestion is essentially \(I({\mathcal S})\) with the only difference being that the integrands of Eq. (6.5) 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 boostgaugedependent 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 mean extrinsic curvature vector Q_{ a } and \({{\bar Q}_a}\) discussed in Section 4.1.2.) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [81, 83] 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 or small sphere calculations, then \(\beta  {{\bar \beta}\prime} = \tau =  {{\bar \tau}\prime}\), and hence, the extra term is, in fact, the gauge invariant \(2\tau \bar \tau\).
Taking into account that \(\tau = {\mathcal O}({r^{ 2}})\) near the future null infinity (see, e.g., [455]), it is obvious from the remark on the asymptotic behavior 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 [81]. 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 ‘special knowledge’, the review here of the basic idea behind the Penrose construction is slightly more detailed than that of the others. The main introductory references of the field are the volumes [425, 426] by Penrose and Rindler on ‘Spinors and Spacetime’, especially volume 2, the very readable book by Hugget and Tod [277] and the comprehensive review article [516] by Tod.
7.1 Motivations
7.1.1 How do the twistors emerge?
7.1.2 Twistor space and the kinematical twistor
Thus, to summarize, the various spinor parts of the kinematical twistor A_{ αβ } are the energymomentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, are needed to be able to ‘isolate’ its energymomentum and angular momentum parts, and, in particular, to define the mass and express the PauliLubanski spin. Furthermore, the Hermiticity condition ensuring that A_{ αβ } has the correct number of components (ten reals) is also formulated in terms of these additional structures.
7.2 The original construction for curved spacetimes
7.2.1 Twosurface twistors and the kinematical twistor
The twosurface twistor equation that the spinor fields should satisfy is just the covariant spinor equation \({\mathcal{T}_{E'EA}}{{\mkern 1mu} ^B}{\lambda _B} = 0\). By Eq. (4.6) its GHP form is \({\mathcal T}\lambda : = ({{\mathcal T}^ +} \oplus {{\mathcal T}^ })\lambda = 0\), which is a firstorder elliptic system, and its index is 4(1 − g), where g is the genus of \({\mathcal S}\) [58]. Thus, there are at least four (and in the generic case precisely four) linearlyindependent solutions to \({\mathcal T}\lambda = 0\) on topological twospheres. However, there are ‘exceptional’ twospheres for which there exist at least five linearly independent solutions [297]. For such ‘exceptional’ twospheres (and for highergenus twosurfaces 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 [509, 183]). The space of the solutions to \({\rm{T}}_{\mathcal S}^\alpha\) is called the twosurface twistor space. In fact, in the generic case this space is fourcomplexdimensional, and under conformal rescaling the pair Z^{ α } = (λ^{ A }, iΔ_{ A′A }λ^{ A }) transforms like a valence one contravariant twistor. Z^{ α } is called a twosurface twistor determined by λ^{ A }. If \({{\mathcal S}\prime}\) is another generic twosurface with the corresponding twosurface twistor space \({\rm{T}}_{{{\mathcal S}\prime}}^\alpha\), then although \({\rm{T}}_{\mathcal S}^\alpha\) and \({\rm{T}}_{{{\mathcal S}\prime}}^\alpha\) 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 \({\rm{T}}_{\mathcal S}^\alpha\). Note that \({A_{\mathcal S}}[\lambda, \,\omega ]\) is constructed only from the twosurface 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 \(\left\langle {Z,\,\bar W} \right\rangle : =  {\rm{i(}}{\lambda ^A}{\Delta _{A{A\prime}}}{{\bar \omega}^{{A\prime}}}  {{\bar \omega}^{{A\prime}}}{\Delta _{A{A\prime}}}{\lambda ^A})\), is not constant on \({\mathcal S}\), thus, it does not define a Hermitian scalar product on the twosurface twistor space. As is shown in [296, 299, 514], \(\left\langle {Z,\,\bar W} \right\rangle\) is constant on \({\mathcal S}\) for any two twosurface 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 twosurfaces 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 \({\mathcal S}\) [420]: \({H_{\alpha {\beta \prime}}}{Z^\alpha}{{\bar W}^{{\beta \prime}}}: = [{\rm{Area(}}{\mathcal S}{{\rm{)}}^{ {1 \over 2}}}\oint\nolimits_{\mathcal S} {\left\langle {Z,\,\bar W} \right\rangle \,d{\mathcal S}}\), which reduces to \(\left\langle {Z,\,\bar W} \right\rangle\) on noncontorted twosurfaces. Interestingly enough, \(\oint\nolimits_{\mathcal S} {\left\langle {Z,\,\bar W} \right\rangle \,d{\mathcal S}}\) can also be reexpressed by the integral (7.14) of the NesterWitten 2form [490]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted twosurfaces, the definition of the quasilocal mass as the norm of the kinematical twistor (cf. Eq. (7.10)) 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_{ εαβγ }, and for the four independent twosurface 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 Eq. (7.7), the determinant ν, is constant on \({\mathcal S}\). Nevertheless, ν can be constant even for contorted twosurfaces for which \(\left\langle {Z,\,\bar W} \right\rangle\) is not. Thus, the totally antisymmetric twistor ε_{ εαβγ } can exist even for certain contorted twosurfaces. Therefore, an alternative definition of the quasilocal mass might be based on Eq. (7.11) [510]. 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 antide Sitter 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 as well, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (7.9), one also needs the infinity twistor. However, −ε^{ 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 [490]), but the resulting twistor would not be simple. In fact, even on twosurfaces 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′ }) [426, 424, 510], while on round spheres in sphericallysymmetric spacetimes it is \({I_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {1 \over {2{r^2}}}(1 + 2{r^2}\rho {\rho {\prime}}){\varepsilon _{AB}}{\lambda ^A}{\omega ^B}  {\varepsilon ^{{A{\prime}}{B{\prime}}}}{\Delta _{{A{\prime}}A}}{\lambda ^A}{\Delta _{{B{\prime}}B}}{\omega ^B}\) [496]. Thus, no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [263]: even if the spacetime is conformally flat (in which case 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 twosurface twistor space itself depends on the geometry of \({\mathcal S}\), and hence all its structures also.
Since in the Hermiticity condition (7.9) only the special combination \({H^\alpha}_{{\beta {\prime}}}: = {I^{\alpha \beta}}{H_{\beta {\beta {\prime}}}}\) of the infinity and metric twistors (the ‘barhook’ combination) appears, it might still be hoped that an appropriate \({H^\alpha}_{{\beta {\prime}}}\) could be found for a class of twosurfaces in a natural way [516]. However, as far as the present author is aware, 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 comes from 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 [298]. He considers a oneparameter 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 twosurface \({\mathcal S}\) 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 [420, 426]. Then every element of the construction is built from conformallyrescaled quantities of the nonphysical spacetime. Since ℐ^{+} is shearfree, the twosurface 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 twosurface twistor space [516]. 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 twosurface 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 twosurface 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 fourparameter 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’. It is shown in [174] that Penrose’s angular momentum can be considered as a supertranslation of previous definitions.
The other way of determining the null infinity limit of the energymomentum and angular momentum is to calculate them for large spheres from the physical data, instead of for the spheres at null infinity from the conformallyrescaled data. These calculations were done by Shaw [455, 457]. 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 Eq. (3.14)). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [510]. The twosurfaces for which the mass was calculated are the r = const. cuts of the geometricallydistinguished 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 incoming radiation.
In [264] Helfer suggested a bijective nonlinear map between the twosurface twistor spaces on the different cuts of ℐ^{+}, by means of which he got something like a ‘universal twistor space’. Then he extends the kinematical twistor to this space, and in this extension the shear potential (i.e., the complex function for which the asymptotic shear can be written as σ = ð^{2} S) appears explicitly. Using Eq. (7.12) as the definition of the intrinsicspin angular momentum at scri, Helfer derives an explicit formula for the spin. In addition to the expected PauliLubanski type term, there is an extra term, which is proportional to the imaginary part of the shear potential. Since the twistor spaces on the different cuts of scri have been identified, the angular momentum flux can be, and has in fact been, calculated. (For an earlier attempt to calculate this flux, see [262].)
The large sphere limit of the twosurface twistor space and the Penrose construction were investigated by Shaw in the Sommers [475], AshtekarHansen [37], and BeigSchmidt [65] models of spatial infinity in [451, 452, 454]. 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 well defined. 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 [456].
The Penrose mass in asymptotically antide Sitter spacetimes was studied by Kelly [312]. He calculated the kinematical twistor for spacelike cuts \({\mathcal S}\) of the infinity ℐ^{+}, which is now a timelike threemanifold in the nonphysical spacetime. Since ℐ admits global threesurface 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 fourvector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [42]. Therefore, the energymomentum fourvector is future pointing and timelike if there is a spacelike hypersurface extending to ℐ on which the dominant energy condition is satisfied. Consequently, m^{2} ≥ 0. Kelly shows that \(m_{\rm{D}}^2\) is also nonnegative and in vacuum it coincides with m^{2}. In fact [516], m ≥ m_{D} ≥ 0 holds.
7.2.5 The quasilocal mass of specific twosurfaces
The Penrose mass has been calculated in a large number of specific situations. Round spheres are always noncontorted [514], thus, the normmass can be calculated. (In fact, axisymmetric twosurfaces in spacetimes with twistfree rotational Killing vectors are noncontorted [299].) The Penrose mass for round spheres reduces to the standard energy expression discussed in Section 4.2.1 [510]. 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 twosurfaces [507]. Interestingly enough, although these spheres cannot be shrunk to a point (thus, the mass cannot be interpreted as ‘the threevolume 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 threevolume of a sphere in flat space with the same area as \({\mathcal S}\) [515]. In conformallyflat spacetimes [510] the twosurface twistors are just the global twistors restricted to \({\mathcal S}\), and the Hermitian scalar product is constant on \({\mathcal S}\). Thus, the normmass is well defined.
The construction works nicely, even if global twistors exist only on a, e.g., spacelike hypersurface Σ containing \({\mathcal S}\). These are the threesurface twistors [510, 512], which are solutions of certain (overdetermined) elliptic partialdifferential equations, called the threesurface twistor equations, on Σ. These equations are completely integrable (i.e., they admit the maximal number of linearlyindependent solutions, namely four) if and only if Σ, with its intrinsic metric and extrinsic curvature, can be embedded, at least locally, into some conformallyflat spacetime [512]. 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 threesurface twistor connection on the twistor bundle over Σ [66, 495]. Returning to the quasilocal mass calculations, Tod showed that in vacuum the kinematical twistor for a twosurface \({\mathcal S}\) 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 well defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any noncontorted twosurface that can be deformed into a round sphere, and it is zero for those that do not go round the black hole [514]. Thus, in particular, the Penrose mass can be zero even in curved spacetimes.
A particularly interesting class of noncontorted hypersurfaces is that of the conformallyflat 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 twosurfaces in such a hypersurface [510]. He found that the mass is zero if \({\mathcal S}\) does not go around any black hole, it is the mass M_{ i } of the ith black hole if \({\mathcal S}\) links precisely the ith black 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 black holes, where d_{ ij } is some appropriate measure of the distance between the black 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 conformallyflat timesymmetric initial data sets describing n ‘points at infinity’ [62]. 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 twosurface 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 (locallyexisting) conformal Killing vectors, these are just the components of the kinematical twistor derived by Tod in [510] (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 twosurfaces (cylinders) in various LRS Bianchi and KantowskiSachs cosmological models [515] and more general cylindricallysymmetric spacetimes [517]. In all these cases the twosurfaces are noncontorted, and the construction works. A technically interesting feature of these calculations is that the twosurfaces 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 linearlyindependent solutions to four. The projection parts of the resulting twistors, the \({\rm{i}}{\Delta _{{A\prime}A}}{\lambda ^A}{\rm{s}}\), 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 is proportional to a ‘type factor’ f(L) as well, where is the coordinate length of the cylinder. In the hyperbolic, parabolic, and elliptic cases this factor is sinh ωL/(ωL), 1, and sin ωL/(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 is presented in [87]. 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 shows [513, 514] that for static black holes the Penrose inequality holds if the mass of the black 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 well defined. Then, the Penrose inequality will be a Sobolevtype inequality for a nonnegative function on the unit sphere. This inequality is tested numerically in [87].
Apart from the cuts of ℐ^{+} in radiative spacetimes, all the twosurfaces discussed so far were noncontorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted twosurface [516]. 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 clearly show some unwanted features of the original construction [511, 313, 560], and force its modification.
7.3 The modified constructions
Independently of the results of the smallsphere calculations, Penrose claims that in the Schwarzschild spacetime the quasilocal mass expression should yield the same zero value on twosurfaces, contorted or not, which do not surround the black hole. (For the motivations and the arguments, see [422].) 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 [516].
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 [357]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Eq. (7.17) 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 twosurface \({\mathcal S}\), their structure is not so simple, because the factor η itself depends on all four of the independent solutions of the twosurface twistor equation in a rather complicated way.
To have a simple Hamiltonian interpretation, Mason suggested further modifications [357, 358]. He considers the four solutions \(\lambda _i^A,i = 1, \ldots, 4\), of the twosurface twistor equations, and uses these solutions in the integral (7.14) of the NesterWitten 2form. Since \({H_{\mathcal S}}\) is a Hermitian bilinear form on the space of the spinor fields (see Section 8), 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 \({\mathcal S}\) should be found among the 16 vector fields \(\lambda _i^A\bar \lambda _j^{{A\prime}}\) (called ‘quasiconformal Killing vectors’ of \({\mathcal S}\)) for which the corresponding quasilocal quantities could be considered as the components of 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 Bondi—Sachs masses is relatively simple in terms of the twocomponent spinors.

the integral of Møller’s tetrad superpotential for the energymomentum, coming from his tetrad Lagrangian (3.5), 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.
Next we will discuss some general properties of the integral of \(u{(\lambda, \bar \mu)_{ab}}\), where λ_{ A } and μ_{ A } are arbitrary spinor fields on \({\mathcal S}\). Then, in the integral \({H_{\mathcal S}}[\lambda, \bar \mu ]\), defined by Eq. (7.14), only the tangential derivative of λ_{ A } appears. (μ_{ A } is involved in \({H_{\mathcal S}}[\lambda, \bar \mu ]\) algebraically.) Thus, by Eq. (3.11), \({H_{\mathcal S}}:{C^\infty}({\mathcal S},{{\rm{S}}_A}) \times {C^\infty}({\mathcal S},{{\rm{S}}_A}) \rightarrow {\rm{\mathbb C}}\) 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 \(\overline {{H_{\mathcal S}}[\lambda, \bar \mu ]}\) 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 \({H_{\mathcal S}}[{\lambda ^{\underline C}}{\Lambda _{\underline C}}^{\underline A},{{\bar \lambda}^{\underline {{D{\prime}}}}}{{\bar \Lambda}_{\underline {{D{\prime}}}}}^{{{\underline B}{\prime}}}] = {H_{\mathcal S}}[{\lambda ^{\underline C}},{{\bar \lambda}^{{{\underline D}{\prime}}}}]{\Lambda _{\underline C}}^{\underline A}{{\bar \Lambda}_{{{\underline D}{\prime}}}}^{{{\underline B}{\prime}}}\), i.e., the integrals \({H_{\mathcal S}}[{\lambda ^{\underline A}},{{\bar \lambda}^{{{\underline B}{\prime}}}}]\) transform as the spinor components of a real Lorentz vector over the twocomplexdimensional space spanned by \(\lambda _A^0\) and \(\lambda _A^1\). Therefore, to have a welldefined quasilocal energymomentum vector we have to specify some twodimensional subspace \({{\bf{S}}^{\underline A}}\) of the infinitedimensional space \({C^\infty}({\mathcal S},{{\rm{S}}_A})\) 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 \({\mathcal S}\). Note, however, that in general the symplectic metric \({\varepsilon _{\underline A \underline B}}\) need not be related to the pointwise symplectic metric ε_{ AB } on the spinor spaces, i.e., the spinor fields \(\lambda _A^0\) and \(\lambda _A^1\) that span \({{\bf{S}}^{\underline A}}\) are not expected to form a normalized spin frame on \({\mathcal S}\). 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 like 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 (3.6). 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 (see Section 3.3.3).
Once the spin space \({\rm{(}}{{\rm{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\) 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 \(({{\rm{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\), 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 above form 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 twosurface to the null infinity is a very strong restriction. In fact, for general (even for convex) twosurfaces in a general asymptotically flat spacetime, conjugate points will develop along the (outgoing) null geodesics orthogonal to the twosurface [417, 240]. Thus, either the twosurface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the twosurface 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 (8.3) and (8.4) one could define a quasilocal energymomentum for small spheres [346, 84]. 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 twosurface, and the LudvigsenVickers spinors on \({\mathcal S}\) are defined by propagating these spinors from the vertex p to \({\mathcal S}\) via Eqs. (8.3) and (8.4). This definition works in arbitrary spacetimes, but the twosurface 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 twosurface \({{\mathcal S}_r}\) of constant affine parameter r in the outgoing null hypersurface \({\mathcal N}\), then they are uniquely determined on any other spacelike twosurface \({{\mathcal S}_{{r{\prime}}}}\) in \({\mathcal N}\), as well, i.e., the propagation law, Eqs. (8.3) and (8.4), defines a natural isomorphism between the space of the LudvigsenVickers spinors on different twosurfaces 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 [346], 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 \({H_{{{\mathcal S}_{{r_2}}}}}[\lambda, \bar \lambda ] \leq {H_{{{\mathcal S}_{{r_1}}}}}[\lambda, \bar \lambda ] \geq 0\) 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 equations (8.3) and (8.4) should be replaced by ϸ′λ_{1} = 0 and − Δ^{−}λ ≔ ðλ_{1} + ρ′λ_{0} = 0. Using these equations the positivity of the LudvigsenVickers mass was proven in various special cases in [346].
Concerning the positivity properties of the LudvigsenVickers mass and energy, first it is obvious by the remarks on the nature of the propagation equations (8.3) and (8.4) 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 [79], 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 Γ 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 [488]. 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 [346] 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 twosurfaces, 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 [346, 457] 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 [455, 457].
8.2 The DouganMason constructions
8.2.1 Holomorphic/antiholomorphic spinor fields
The original construction of Dougan and Mason [172] was introduced on the basis of sheaftheoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [488, 490].
Following Dougan and Mason we define the spinor field λ_{ A } to be antiholomorphic when m^{ e }∇_{ e }λ_{ A } = m^{ e }Δ_{ e }λ_{ A } = 0, or holomorphic if \({\bar m^e}{\nabla _e}{\lambda _A} = {\bar m^e}{\Delta _e}{\lambda _A} = 0\). 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 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 holomorphicity/antiholomorphicity is well defined, because the defining equations are homogeneous in m^{ a } and \({{\bar m}^a}\). Next, suppose that there are at least two independent solutions of \({\bar m^e}{\Delta _e}{\lambda _A} = 0\). If λ_{ A } and μ_{ A } are any two such solutions, then \({\bar m^e}{\Delta _e}({\lambda _A}{\mu _B}{\varepsilon ^{AB}}) = 0\), and hence by Liouville’s theorem λ_{ A }μ_{ B }ε^{ 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., λ_{ A } and λ_{ A }, span the spin space at each point of \({\mathcal S}\) (and they can be chosen to form a normalized spinor dyad with respect to ε_{ 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 twosurfaces 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 twosurfaces ker \({{\mathcal H}^ +}\) with the constant \({\varepsilon _{\underline A \underline B}}\) would be a natural candidate for the spin space \(\left({{{\bf{S}}^{\underline A}},\,{\varepsilon _{\underline A \underline B}}} \right)\) above. For exceptional twosurfaces, the kernel space ker \({{\mathcal H}^ +}\) is either twodimensional 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 twosurfaces 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 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 twosurfaces, 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 twosurfaces
\({{\mathcal H}^ \pm}\) are firstorder elliptic differential operators on certain vector bundles over the compact twosurface \({\mathcal S}\), and their index can be calculated: \({\rm{index}}({{\mathcal H}^ \pm}) = 2(1  g)\), where g is the genus of \({\mathcal S}\). Therefore, for \({\mathcal S} \approx {S^2}\) there are at least two linearlyindependent holomorphic and at least two linearlyindependent antiholomorphic spinor fields. The existence of the holomorphic/antiholomorphic spinor fields on highergenus twosurfaces 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 marginallytrapped surfaces (i.e., for which ρ = 0) are exceptional from the point of view of holomorphic spinors, and past marginallytrapped surfaces (ρ′ = 0) from the point of view of antiholomorphic spinors. Furthermore, there are surfaces with at least three linearlyindependent holomorphic/antiholomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional twosurface \({\mathcal S}\) with S^{2} topology make \({\mathcal S}\) generic.
Finally, we note that several firstorder differential operators can be constructed from the chiral irreducible parts Δ^{±} and \({{\mathcal T}^ \pm}\) of Δ_{ e }, given explicitly by Eq. (4.6). However, only four of them, the DiracWitten operator Δ ≔ Δ^{+} ⊕ Δ^{−}, the twistor operator \({\mathcal T}: = {{\mathcal T}^ +} \oplus {{\mathcal T}^ }\), 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 a twocomplexdimensional 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. This construction does not work for exceptional twosurfaces.
8.2.3 Positivity properties
One of the most important properties of the DouganMason energymomenta is that they are futurepointing nonspacelike vectors, i.e., the corresponding masses and energies are nonnegative. Explicitly [172], 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 futurepointing nonspacelike vector, and, analogously, the antiholomorphic energymomentum is future pointing and nonspacelike if ρ′ ≥ 0. (For the functional analytic techniques and tools to give a complete positivity proof, see, e.g., [182].) As Bergqvist [79] 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 Eq. (8.3) of the LudvigsenVickers construction, or \({{\mathcal T}^ +}\lambda = 0\) in the holomorphic DouganMason construction, or even \({{\mathcal T}^ +}\lambda = k\sigma {\prime}{\psi{\prime}_2}{\lambda _0}\) for some constant k, a ‘deformation’ of the holomorphicity considered by Bergqvist [79]. 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 ε^{ AB } λ_{ AμB } 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 the results 1 and 2. Under the same conditions D(Σ) is flat if and only if there exist two linearlyindependent spinor fields on \({\mathcal S}\), which are constant with respect to Δ_{ e }, and D(Σ) is a ppwave geometry; the matter is pure radiation if and only if there exists a Δ_{ e }constant spinor field on \({\mathcal S}\) [490]. Thus, the full information that D(Σ) is flat/ppwave is completely encoded, not only in the usual initial data on, but in the geometry of the boundary of Σ, as well. In Section 13.5 we return to the discussion of this phenomenon, where we will see that, assuming \({\mathcal S}\) 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 twosurface 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) show 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)) [488], 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 [171]. 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 twosurface. Unfortunately, without additional restrictions (e.g., the spherical symmetry of the twosurfaces in a sphericallysymmetric spacetime) the mass of the exceptional twosurfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic twosurfaces approaching the exceptional one [490].
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 [172]. (The Newtonian limit in the covariant Newtonian spacetime was studied in [564].) In nonvacuum both definitions give the same, expected expression (4.9) 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 Eq. (8.5), but in the analogous expression for the antiholomorphic energymomentum, the numerical coefficient 4/(45G) is replaced by 1/(9G) [171]. The DouganMason energymomenta have also been calculated for large spheres of constant Bonditype radial coordinate value r near future null infinity [171]. 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 construction 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 twosurface \({\mathcal S}\) only. Thus, if \({\mathcal S}{\prime}\) is another twosurface, 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 as well, on different surfaces. For example [494], in the smallsphere approximation every spin coefficient and spinor component in the GHP dyad and metric component in some fixed coordinate system \((\zeta, \,\bar \zeta)\) is expanded as a series of r, as \({\lambda _{\mathbf{A}}}(r,\,\zeta, \,\bar \zeta) = {\lambda _{\mathbf{A}}}^{(0)}(\zeta, \,\bar \zeta) + r{\lambda _{\mathbf{A}}}^{(1)}(\zeta, \,\bar \zeta) + \cdots + {r^k}{\lambda _{\bf{A}}}^{(k)}(\zeta, \,\bar \zeta) + {\mathcal O}({r^{k + 1}})\). 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 form a 2k, rather than the expected twocomplexdimensional, 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 Eq. (8.5) for the DouganMason energymomenta is well defined only as a consequence of a natural gauge choice.^{13} Similarly, the higherorder 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 should be presented in Section 12, but the technique that it is based on justifies its placement here.
By investigating the propagation law, Eqs. (8.3) and (8.4) of Ludvigsen and Vickers for the Kerr spacetimes, Bergqvist and Ludvigsen constructed a natural flat, (but nonsymmetric) metric connection [85]. Writing the new covariant derivative in the form \({\tilde \nabla _{AA{\prime}}}{\lambda _B} = {\nabla _{AA{\prime}}}{\lambda _B} + {\Gamma _{AA{\prime}B}}^C{\lambda _C}\), the ‘correction’ term \({\Gamma _{AA\prime B}}^C\) could be given explicitly in terms of the GHP spinor dyad (adapted to the two principal null directions), the spin coefficients ρ, τ and ρ′, and the curvature component ψ_{2}. \({\Gamma _{AA\prime B}}^C\) admits a potential [86]: \({\Gamma _{AA\prime BC}} =  {\nabla _{(C}}^{B{\prime}}{H_{B)}}_{AA{\prime}B{\prime}}\), where \({H_{ABA{\prime}B{\prime}}}: = {1 \over 2}{\rho ^{ 3}}(\rho + \bar \rho){\psi _2}{o_A}{o_B}{\bar o_{A{\prime}}}{\bar o_{B{\prime}}}\). 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 [234], and the corresponding ‘correction term’ Γ_{ AA′BC } could be used to easily find the Lánczos potential for the Weyl curvature [18].
Since the connection \({\tilde \nabla _{AA{\prime}}}\) is flat and annihilates the spinor metric ε_{ AB }, there are precisely two linearlyindependent 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 \(({{\mathbf{S}}^{\underline A}},\,{\varepsilon _{\underline A \underline B}})\) to be the space of the \({\tilde \nabla _a}\)constant spinor fields, irrespectively of the twosurface \({\mathcal S}\).
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 twosurfaces, 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 Spin Angular Momentum
In this section we review three specific quasilocal spinangularmomentum constructions that are (more or less) ‘quasilocalizations’ of Bramson’s expression at null infinity. Thus, the quasilocal spin angular momentum for the closed, orientable spacelike twosurface \({\mathcal S}\) will be sought in the form (3.16). Before considering the specific constructions themselves, we summarize the most important properties of the general expression of Eq. (3.16). Since the most detailed discussion of Eq. (3.16) is probably given in [494, 496], 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 secondrank 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 oneform basis \(\{\vartheta _a^{\underline a}\}\) is the constant Cartesian one, which consists of exact oneforms. Then, since the Bramson superpotential \(w({\lambda ^{\underline A}},{\lambda ^{\underline B}})\) 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, Eq. (3.16) gives zero. Therefore, the integral of Bramson’s superpotential (3.16) measures the nonintegrability of the oneform basis \(\vartheta _a^{{\underline A}{\underline A'}} = \lambda _A^{\underline A}\bar \lambda _{A'}^{{\underline A'}}\), i.e., \(J_{\mathcal S}^{\underline A\underline B}\) is a measure of how much the actual oneform 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 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 \({\mathcal S}\)’, 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 [338], 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 [346]. 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 spin angular momentum is zero in Minkowski spacetime.
9.2 Holomorphic/antiholomorphic spin angular momenta
Both the holomorphic and antiholomorphic spin angular momenta were calculated for small spheres [494]. In nonvacuum the holomorphic spin angular momentum reproduces the expected result (4.10), and, apart from a minus sign, the antiholomorphic construction does also. In vacuum, both definitions give exactly Eq. (9.1).
In general the antiholomorphic and the holomorphic spin angular 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 spin angular momentum is finite in the centerofmass frame, and hence it seems to describe only the spin part of the gravitational field. In fact, the PauliLubanski spin (9.2) built from this spin angular momentum and the antiholomorphic DouganMason energymomentum is always finite, free of the ‘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 (4.15). Thus, the antiholomorphic spin angular 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 [496].
9.3 A specific construction for the Kerr spacetime
The angular momentum of Bergqvist and Ludvigsen [86] 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 [234].)
10 The HamiltonJacobi Method
If one is concentrating only on the introduction and study of the properties of quasilocal quantities, and is 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 [120, 121]. 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, mathematicallyinequivalent definitions for the reference configurations are still called the ‘BrownYork energy’. A slightly different general expression was used by Kijowski [315], Epp [178], Liu and Yau [338] and Wang and Yau [544]. Although the former follows a different route to derive his expression and the latter three 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 mainly based on the original papers [120, 121] 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, Liu and Yau and Wang 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 [120, 121] is to calculate the analogous variation of an appropriate firstorder action of general relativity (or of the coupled matter + gravity system) and isolate the boundary term that could be analogous to the energy above. To formulate this idea mathematically, Brown and York 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 twosurfaces \({{\mathcal S}_t}: = \partial {\Sigma _t}\) (corresponding to ∂Σ × {t}) form a foliation of the timelike threeboundary ^{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 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 of \({\Sigma _2}: = {\Sigma _{{t_2}}}\), are chosen to be future pointing. The metric and extrinsic curvature on Σ_{ t } will be denoted, respectively, by h_{ ab } and χ_{ ab }, and 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 [431]. 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.
10.1.3 The general form of the BrownYork quasilocal energy
For a definition of the BrownYork energy as a quasilocal energy oparator in loop quantum gravity, see [565].
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 [331]. Here the energy and momentum surface densities are reexpressed by the superpotential \({\vee _b}^{ae}\), given by Eq. (3.6), in a frame adapted to the twosurface. (Lau called the corresponding superpotential 2form the ‘Sparling 2form’.) However, in contrast to the usual Ashtekar variables on a spacelike hypersurface [30], the time gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every orientable threemanifold Σ is parallelizable [410], and hence, a globallydefined orthonormal triad can be given on Σ, the only parallelizable, closed, orientable twosurface is the torus. Thus, on ^{3}B, we cannot impose the global time gauge condition with respect to any spacelike twosurface \({\mathcal S}\) in ^{3}B unless \({\mathcal S}\) is a torus. Similarly, the global radial gauge condition in the spacelike hypersurfaces Σ_{ t } (even in a small open neighborhood of the whole twosurfaces \({{\mathcal S}_t}\) in Σ_{ t }) can be imposed on a triad field only if the twoboundaries \({{\mathcal S}_t} = \partial {\Sigma _t}\) 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 twosurface) appear, they are valid even globally (see also [332]). On the other hand, further investigations are needed to clarify whether or not the quasilocal Hamiltonian, using the Ashtekar variables in the radialtime gauge [333], is globally well defined.
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 [244], 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 twosurface \({\mathcal S}\).
Booth and Mann [100] 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 [101], 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, Booth and Mann 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 sphericallysymmetric spacetimes with respect to several moving observers, i.e., in contrast to Eq. (10.9), 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, nondynamic background metric \(g_{ab}^0\), one can construct a covariant firstorder Lagrangian \(L({g_{ab}},g_{ab}^0)\) for general relativity [306], 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 [184] 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 firstorder 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 twosurfaces \({\mathcal S}\).^{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 if 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 from both the point of view of the variational principle and that of the quasilocal quantities. We will see in Section 10.1.8 that even the weaker boundary condition, that requires only the induced threemetrics on ^{3}B fromg_{ ab } and from \(g_{ab}^0\) to 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 \(\int\nolimits_{{t_1}}^{{t_2}} {[{p_a}{{\dot q}^a}  H({q^a},{p_a},t)]}\), 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} as well [121]. 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 Eq. (10.8), 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 [100]). This Hamiltonian was investigated further in [119]. Here all the boundary terms that appear in the variation of their Hamiltonian are determined and decomposed with respect to the twosurface ∂Σ. It is shown that the change of the Hamiltonian under a boost of Σ yields precisely the boosts of the energy and momentum surface density discussed above.
Booth and Fairhurst [95] 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 threemanifold Σ and the time coordinate t, the space coordinates x^{ A } on the twoboundary \({\mathcal S} = \partial \Sigma\), and their conjugate momenta π and π_{ a }.
The second important observation of Booth and Fairhurst is that the BrownYork boundary conditions are too restrictive. The twometric, lapse, and shift need not be fixed, but their variations corresponding to diffeomorphisms on the boundary must be allowed. Otherwise diffeomorphisms that are not isometries of the threemetric 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 well defined 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 [216, 120]. 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 threemanifold (^{3}B, γ_{ ab }) cannot be isometrically embedded, even locally, into the Minkowski spacetime. (For a detailed discussion of this embedability, see [120] 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 antide Sitter 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 twometric q_{ ab } and on the boostgauge that ^{3}B defined on \({{\mathcal S}_t}\). Therefore, ε and j_{ a } take the form (10.10), 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 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 threespace, i.e., into a spacelike hyperplane of Minkowski spacetime. This is the choice of Brown and York [120, 121]. In fact, as we already noted in Section 4.1.3, for twosurfaces with everywhere positive scalar curvature, such an embedding exists and is unique. (The order of the differentiability of the metric is reduced in [261] to C^{2}.) A particularly interesting twosurface that cannot be isometrically embedded into the flat threespace 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 \(\sqrt 3 m < 2\vert a\vert \; < 2m\) [463]. (On the other hand, for its global isometric embedding into ℝ^{4}, see [203].) Thus, the construction works for a large class of twosurfaces, but certainly not for every potentially interesting twosurface. The convexity condition is essential.
It is known that the (local) isometric embedability of (\({\mathcal S}\), q_{ ab }) into flat threespace 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 threeplane, because this normal is constant. Thus, to summarize, for convex twosurfaces, 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 de Sitter or antide Sitter 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 [118, 334, 335]. In this approach the twosurface (\({{\mathcal S}_t}\), q_{ ab }) is embedded into the light cone of a point of the Minkowski or antide Sitter spacetime instead of into 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 [115], every locallyconformallyflat Riemannian ngeometry is locally isometric to an appropriate cut of a light cone of the n + 2 dimensional Minkowski spacetime (see, also, [178]). To achieve uniqueness some extra condition must be imposed. This may be the requirement of the vanishing of the ‘normal momentum density’ \(j_\vdash^0\) in the reference spacetime [334, 335], yielding \({k^0} = \sqrt {{2^{\mathcal S}}R + 4/{\lambda ^2}}\), 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 [118].
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 twosurfaces such that \({{\mathcal S}{\prime}} \cap {{\mathcal S}^{^{\prime\prime}}}\) is connected and twodimensional (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, the flat threespace, 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 noted in [100], 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 threespace 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 [334] and in [118] is also negative for such surfaces with the boosted observers.
Recently, Shi and Tam [458] have proven interesting theorems in Riemannian threegeometries, which can be used to prove positivity of the BrownYork energy if the twosurface \({\mathcal S}\) 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 threemanifold with nonnegative scalar curvature and smooth twoboundary \({\mathcal S}\) such that each connected component \({{\mathcal S}_i}\) of \({\mathcal S}\) is homeomorphic to S^{2} and the scalar curvature of the induced twometric 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 outwarddirected normal, and k^{0} is the trace of the extrinsic curvature of \({{\mathcal S}_i}\) in the flat Euclidean threespace 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 [459] by weakening the energy condition, in which case lower estimates of the BrownYork energy can still be given. For some rigidity theorems connected with this positivity result, see [461]; and for their generalization for higher dimensional spin manifolds, see [329].
The energy expression for round spheres was calculated in [121, 100]. In the sphericallysymmetric metric discussed in Section 4.2.1, on round spheres the BrownYork energy with the flat space reference and fleet of observers ∂/∂t on \({\mathcal S}\) is \(G{E_{{\rm{BY}}}}[{{\mathcal S}_r}{(\partial/\partial t)^a}] = r(1  \exp ( \alpha))\). 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 [119].) Still in the sphericallysymmetric context the definition of the BrownYork energy is extended to spherical twosurfaces beyond the event horizon in [347] (see also [443]). A remarkable result is that while the total energy of the electrostatic field of a point charge in any finite threevolume surrounding the point charge in Minkowski spacetime is always infinite, the negative gravitational binding energy compensates the electrostatic energy so that the quasilocal energy is negative within a certain radius under the event horizon in the ReissnerNordström spacetime and tends to −e as r → 0. The BrownYork energy is discussed from the point of view of observers in sphericallysymmetric spacetimes (e.g., the connection between this energy and the effective energy in the geodesic equation for radial geodesics) in [90, 576]. The explicit calculation of the BrownYork energy with the (implicitly assumed) flatspace reference in FriedmannRobertsonWalker spacetimes (as particular examples for the general round sphere case) is given in [6].
The Newtonian limit can be derived from the round sphere expression 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 from individual particles, bringing them together from infinity. (For the calculation of the Newtonian limit in the covariant Newtonian spacetime, see [564].) However, taking into account that on the Schwarzschild horizon \(G{E_{{\rm{BY}}}} = 2m\), while at spatial infinity it is just m, the BrownYork energy is monotonically decreasing with r. Also, the first law of black hole mechanics for sphericallysymmetric black holes can be recovered by identifying E_{BY} with the internal energy [120, 121]. The thermodynamics of the Schwarzschildantide Sitter black holes was investigated in terms of the quasilocal quantities in [116]. 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 [119].
The energy is explicitly calculated for three different kinds of twospheres 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 [356]. 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 twosurface. In the first two cases the angular momentum parameter enters the energy expression only in the m^{2}a^{2}/R^{3} order. In particular, the energy for the outer horizon \({r_ +}: = m + \sqrt {{m^2}  {a^2}}\), 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 previously noted, the horizon itself cannot be isometrically embedded into a flat threespace if the angular momentum parameter exceeds the irreducible mass [463]. 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 [119] (see also [241]). Here the flatspace 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 centerofmass is given in [57]. It is shown that the corresponding large sphere limit is just the centerofmass expression of Beig and Ó Murchadha [64]. Here the centerofmass integral is also given in terms of a charge integral of the curvature. The large sphere limit of the energy for metrics with the weakest possible falloff conditions is calculated in [181, 462]. A further demonstration that the spatial infinity limit of the BrownYork energy in an asymptotically Schwarzschild spacetime is the ADM energy is given in [180].
Although the prescription for the reference configuration by Hawking and Horowitz cannot be imposed for a general timelike threeboundary ^{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 antide Sitter one, Hawking and Horowitz [241] calculated the limit of the quasilocal energy, and they found it to tend to the AbbottDeser energy. (For the sphericallysymmetric Schwarzschildantide Sitter case see also [116].) In [117] 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. (The limit of the BrownYork energy on general asymptotically hyperboloidal hypersurfaces is calculated in [330].) In a slightly different formulation Booth and Creighton calculated the energy flux of outgoing gravitational radiation [94] (see also Section 13.1) and they recovered the BondiSachs massloss.
However, the calculation of the small sphere limit based on the flatspace reference configuration gave strange results [335]. 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_{ abc }dt^{ 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 twosphere of radius r [as is usual in the small sphere calculations], and defines the fleet of observers on the twosphere.) On the other hand, the special light cone reference used in [118, 335] 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 small sphere limit was also calculated in [181] for small geodesic spheres in a time symmetric spacelike hypersurface.
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 spacetimes and near the infinity of asymptotically antide Sitter spacetimes [334]. Namely, the BrownYork quasilocal energy expression with this nullcone reference term tends to the BondiSachs, the ADM, and AbbottDeser energies. 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’.^{17}
A formulation and a proof of a version of Thorne’s hoop conjecture for spherically symmetric configuarations in terms of E_{BY} are given in [402], and will be discussed in Section 13.2.2.
10.1.8 Other prescriptions for the reference configuration
As previously noted, Hawking, Horowitz, and Hunter [241, 242] defined their reference configuration by embedding the Lorentzian threemanifold (^{3}B, γ_{ ab }) isometrically into some given Lorentzian spacetime, e.g., into the Minkowski spacetime (see also [216]). However, for the given intrinsic threemetric γ_{ ab } and the embedding fourgeometry the corresponding Gauss and CodazziMainardi equations form a system of 6 + 8 = 14 equations for the six components of the extrinsic curvature Θ_{ ab } [120]. 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, even the small perturbations of the intrinsic metric h_{ ab } would break the conditions of embedability. Therefore, in general, this prescription for the reference configuration can work only if the threesurface ^{3}B is ‘pushed out to infinity’, but does not work for finite threesurfaces [120].
To rule out the possibility that the BrownYork energy can be nonzero even in Minkowski spacetime (on twosurfaces in the boosted flat data set), Booth and Mann [100] suggested that one 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 \({\!\!\!\!\! L_\xi}{q_{ab}} = {\phi ^{\ast}}({\!\!\!\!\! L_{\xi 0}}q_{ab}^0)\) and \({\xi ^a}{\xi _a} = {\phi ^{\ast}}({\xi ^{0a}}\xi _a^0)\). Here ϕ is a diffeomorphism mapping an open neighborhood 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 \({\!\!\!\!\! L_\xi}{q_{ab}}\) denotes the Lie derivative of q_{ ab } along ξ^{ a }. 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 (i) to modify the Hilbert action to the traceχaction and (ii) to the boundary condition that the induced threemetric on the boundary of the domain D of the action is fixed.
However, as stressed by Kijowski [315, 317, 229], 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 corresponds 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, [317]).^{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 threemanifold Σ and its boundary \({\mathcal S} = \partial \Sigma\) as well, 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 Eq. (10.3).) 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 Epp’s expression
10.3.1 The general form of Epp’s expression
10.3.2 The definition of the reference configuration
The subtraction term in Eq. (10.16) is defined through an isometric embedding of \(({\mathcal S},{q_{ab}})\) into some reference spacetime instead of a threespace. This spacetime is usually Minkowski or antide Sitter spacetime. Since the twosurface 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 Eqs. (4.1) and (4.2)). Therefore, even a naïve 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 oneform A_{ e } in the reference spacetime and in the physical spacetime be the same [178]. 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 \({{\bf{S}}^A}({\mathcal S})\) 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 reference configuration \({E_{\rm{E}}}({\mathcal S})\) depends not only on the intrinsic twometric q_{ ab } of \({\mathcal S}\), but on the connection δ_{ e } on the normal bundle as well.
Suppose that \({\mathcal S}\) is a twosurface in M such that k^{2} > l^{2} with k > 0, and, in addition, \(({\mathcal S},{q_{ab}})\) can be embedded into the flat threespace 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 [178]. 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 twosurface, independent of any fleet of observers. In fact, if \({\mathcal S}\) is a twosurface 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 [178]. Thus, in particular, E_{E} is also monotonically decreasing with r in Schwarzschild spacetime. The explicit calculation of Epp’s energy in FriedmannRobertsonWalker spacetimes is given in [6].
Epp calculated the various limits of his expression as well [178]. In the large sphere limit, near spatial infinity, he recovered the AshtekarHansen form of the ADM energy, and at future null infinity, the BondiSachs energy. The technique that is used in the latter calculation is similar to that of [117]. 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 only the role of fixing the twosurfaces, as a result we get the energy seen by the observer t^{ a } instead of mass. This is why \({E_{\rm{E}}}({\mathcal S})\) is considered to be energy rather than mass. In the asymptotically antide Sitter spacetime (with the antide Sitter spacetime as the reference spacetime) E_{E} gives zero. This motivated Epp to modify his expression to recover the mass parameter of the Schwarzschildantide Sitter spacetime at infinity. The modified expression is, however, not boostgauge invariant. Here the potential connection with the AdS/CFT correspondence is also discussed (see also [48]).
10.4 The expression of Liu and Yau
10.4.1 The LiuYau definition
For a definition of the KijowskiLiuYau energy as a quasilocal energy oparator in loop quantum gravity, see [565].
10.4.2 The main properties of. \({E_{{\rm{KLY}}}}({\mathcal S})\)
The most important property of the quasilocal energy (10.17) is its positivity. Namely [338], let Σ 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 \({E_{{\rm{KLY}}}}(\partial \Sigma): = \sum\nolimits_{i = 1}^k {{E_{{\rm{KLY}}}}({{\mathcal S}_i})}\) 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 [289], by means of which the general case can be reduced to the results of Shi and Tam in the timesymmetric case [458], stated in Section 10.1.7 (see also [566]). This positivity result is generalized in [339], Namely, \({{E_{{\rm{KLY}}}}({{\mathcal S}_i})}\) is shown to be 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 as well.) A potential spinorial proof of the positivity of \({{E_{{\rm{KLY}}}}({{\mathcal S}_i})}\) is suggested in [12]. This is based on the use of the NesterWitten 2form and a Witten type argumentation. However, the spinor field solving the Witten equation on the spacelike hypersurface Σ would have to satisfy a nonlinear boundary condition.
However, \({{E_{{\rm{KLY}}}}({\mathcal S})}\) can be positive even if \({\mathcal S}\) is in the Minkowski spacetime. In fact, for a 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, CodazziMainardi, and 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 oneform 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 twosurfaces in the Minkowski spacetime are given in [401], for which \({{E_{{\rm{KLY}}}}({\mathcal S}) > 0}\). Moreover, it is shown in [361] that the KijowskiLiuYau energy for a closed twosurface \({\mathcal S}\) in Minkowski spacetime strictly positive unless \({\mathcal S}\) lies in a spacelike hyperplane. On the applicability of \({{E_{{\rm{KLY}}}}({\mathcal S})}\) in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2.
10.4.3 Generalizations of the original construction
In the definition of \({E_{{\rm{KLY}}}}({\mathcal S})\) one of the assumptions is the positivity of the scalar curvature of the intrinsic metric on the twosurface \({\mathcal S}\). Thus, it is natural to ask if this condition can be relaxed and whether or not the quasilocal mass can be associated with a wider class of surfaces. Moreover, though in certain circumstances \({E_{{\rm{KLY}}}}({\mathcal S})\) behaves as energy (see [400, 575]), it is the (renormalized) integral of the length of the mean curvature vector, i.e., it is analogous to mass (compare with Eq. (2.7)). Hence, it is natural to ask if a energymomentum fourvector can be introduced in this way. In addition, in the calculation of the large sphere limit of \({E_{{\rm{KLY}}}}({\mathcal S})\) in asymptotically antide Sitter spacetimes it seems natural to choose the reference configuration by embedding \({\mathcal S}\) into a hyperbolic rather than Euclidean threespace. These issues motivate the following generalization [542] of the KijowskiLiuYau expression.
The proof of the nonspacelike nature of (10.19) is based on a Witten type argumentation, in which ‘the mass with respect to a Dirac spinor ϕ_{0} on \({\mathcal S}\)’ takes the form of an integral of \(({k^0}  \sqrt {{k^2}  {l^2}})\) weighted by the norm of ϕ_{0}. Thus, the norm of ϕ_{0} appears to be a nontrivial lapse function. The suggestion of [580] for a quasilocal masslike quantity is based on an analogous expression. Let \({\mathcal S}\) be the boundary of some spacelike hypersurface Σ on which the intrinsic scalar curvature is positive, let us isometrically embed \({\mathcal S}\) into the Euclidean threespace, and let ϕ_{0} be the pull back to \({\mathcal S}\) of a constant spinor field. Suppose that the dominant energy condition is satisfied on Σ, and consider the solution ϕ of the Witten equation on Σ with one of the chiral boundary conditions Π_{±}(ϕ − ϕ_{0}) = 0, where Π_{±} are the projections to the space of the right/left handed Dirac spinors, built from the projections κ^{±A}_{ B } of Section 4.1.7. Then, by the SenWitten identity, a positive definite boundary expression is introduced, and interpreted as the ‘quasilocal mass’ associated with \({\mathcal S}\). In contrast to BrownYork type expressions, this mass, associated with the twospheres of radius in the t = const. hypersurfaces in Schwarzschild spacetime, is an increasing function of the radial coordinate, and tends to the ADM mass. In general, however, this limit is E_{ADM} − P_{adm}, rather than the expected ADM mass. This construction is generalized in [581] by embedding \({\mathcal S}\) into some \(\mathbb H_{ {\kappa ^2}}^3\) instead of ℝ^{3}. A modified version of these constructions is given in [582], which tends to the ADM energy and mass at spatial infinity.
Suggestion (11.12), due to Anco [11], can also be considered as a generalization of the KijowskiLiuYau mass.
10.5 The expression of Wang and Yau
Since A_{ e } is boostgauge dependent and Eq. (10.20) in itself does not yield, e.g., the correct ADM energy in asymptotically flat spacetime, a boost gauge and a restriction on the vector field K^{ a } and/or a ‘renormalization’ of Eq. (10.20) (in the form of an appropriate reference term) must be given. Wang and Yau suggest that one determine these by embedding the spacelike twosurface \({\mathcal S}\) isometrically into the Minkowski spacetime in an appropriate way.
To prove, e.g., the positivity of this energy, or to ensure that in flat spacetime the energy be zero, further conditions must be satisfied. Wang and Yau formulate these conditions in the notion of admissible pairs \((i,{T^{\underline a}})\) should have a convex shadow in the direction \({T^{\underline a}},i({\mathcal S})\) must be the boundary of some spacelike hypersurface in ℝ^{1,3} on which the Dirichlet boundary value problem for the Jang equation can be solved with the time function τ discussed in Section 4.1.3, and the connection 1form and the mean curvature in a certain gauge must satisfy an inequality. (For the precise definition of the admissible pairs see [544]; for the geometrical background see [543] and Section 4.1.3.) Then it is shown that if the dominant energy condition holds and \({\mathcal S}\) has a spacelike mean curvature vector, then for the admissible pairs the quasilocal energy (10.21) is nonnegative. Therefore, if the set of the admissible pairs is not empty (e.g., when the scalar curvature of (\({\mathcal S}\), q_{ ab }) is positive), then the infimum \({m_{{\rm{WY}}}}({\mathcal S})\) of \({E_{{\rm{WY}}}}({\mathcal S};i,{T^{\underline a}})\) among all admissible pairs is nonnegative, and is called the quasilocal mass. If this infimum is achieved by the pair \((i,{T^{\underline a}})\), i.e., by an embedding i and a timelike \({T^{\underline a}}\), then \({P^{\underline a}}: = {m_{{\rm{WY}}}}({\mathcal S}){T^{\underline a}}\) is called the quasilocal energymomentum, which is then future pointing and timelike. It is still an open question that if the quasilocal mass \({m_{{\rm{WY}}}}({\mathcal S})\) is vanishing, then the domain of dependence D(Σ) of the spacelike hypersurface Σ with boundary \({\mathcal S}\) can be curved (e.g., a ppwave geometry with pure radiation) or not. If not, then the quasilocal energymomentum would be expected to be null.
The quasilocal energymomentum associated with any twosurface in Minkowski spacetime with a convex shadow in some direction is clearly zero. The mass has been calculated for round spheres in the Schwarzschild spacetime. It is \(r(1  \sqrt {1  (2m/r)})/G\), and hence, for the event horizon it gives 2m/G. m_{WY} has been calculated for large spheres and it has the expected limits at the spatial and null infinities [545, 142]. Also, it has the correct small sphere limit both in nonvacuum and vacuum [544]. Upper and lower estimates of the WangYau energy are derived, and its critical points are investigated in [362] and [363], respectively. On the applicability of \({E_{{\rm{WY}}}}({\mathcal S})\) in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2. A recent review of the results in connection with the WangYau energy see [541].
11 Towards a Full Hamiltonian Approach
The HamiltonJacobi method is only one possible strategy for defining 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, artificial. In Section 10 we reviewed those approaches that are based on the analysis of the action, while in this section we discuss those that are based primarily on the analysis of the Hamiltonian in the spirit of Regge and Teitelboim [433].^{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 systematically quasilocalizing them. One motivation for the quasilocalization of the ADMReggeTeitelboim analysis came from the need for a deeper understanding of the dynamics of subsystems of the universe. In particular, such a systematic Hamiltonian formalism would shed new light on the basic results on the initial boundary value problem in general relativity, initiated by Friedrich and Nagy [202] (see also [201, 554, 555] and, for some recent reviews, see [435, 556] and references therein), and would yield the interpretation of their boundary conditions from a different perspective. Conversely, quasilocal Hamiltonian techniques could potentially be used to identify a large class of boundary conditions that are compatible with the evolution equation. (For a discussion of such a potential link between the two appraches, see e.g., [502, 16]). Moreover, in the quasilocal Hamiltonian approach we might hope to be able to associate nontrivial observables (and, in particular, conserved quantities) with localized systems in a natural way.
Another motivation is to try to provide a solid classical basis for the microscopic understanding of black hole entropy [47, 46, 123]: 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 twosurface observables by means of which the ‘quantum edge states’ on the black hole event horizons were intended to be constructed.
11.1.1 The quasilocal constraint algebra and the basic Hamiltonian
The condition that the area 2form ε_{ ab } should be fixed appears to be the part of the ‘ultimate’ boundary condition for the canonical variables. In fact, in a systematic quasilocal Hamiltonian analysis boundary terms appear in the calculation of the Poisson bracket of two Hamiltonians also, which we called Poisson boundary terms in Section 3.3.3. Nevertheless, as we already mentioned there, the quasilocal Hamiltonian analysis of a single real scalar field in Minkowski space shows, these boundary terms represent the infinitesimal flow of energymomentum and relativistic angular momentum. Thus, they must be gauge invariant [502]. Assuming that in general relativity the Poisson boundary terms should have similar interpretation, their gauge invariance should be expected, and the condition of their gauge invariance can be determined. It is precisely the condition on the lapse and shift that the spacetime vector field K^{ a } = Nt^{ a } + N^{ a } built from them on the 2surface must be divergence free there with respect to the connection Δ_{ a } of Section 4.1.2, i.e., Δ_{ a }K^{ a } = 0. However, this is precisely the condition under which the evolution equations preserve the boundary condition δε_{ ab } = 0. It might also be worth noting that this condition for the lapse and shift is just one of the ten components of the Killing equation: 0 = 2Δ_{ a }K^{ a } = q^{ ab }(∇_{ a }K_{ b } + ∇_{ b }K_{ a }). (For the details, see [502].)
It should be noted that the area 2form on the boundary 2surface \({\mathcal S}\) appears naturally in connection with the general symplectic structure on the ADM variables on a compact spacelike hypersurface Σ with smooth boundary \({\mathcal S}\). In fact, in [229] an identity is derived for the variation of the ADM canonical variables on Σ and of various geometrical quantities on \({\mathcal S}\). Examples are also given to illustrate how the resulting ‘quasilocal energy’ depends on the choice of the boundary conditions.
For the earlier investigations see [47, 46, 123], where stronger boundary conditions, namely fixing the whole threemetric h_{ ab } on \({\mathcal S}\) (but without the requirement δ_{ e }N^{ e } = 0), were used to ensure the functional differentiability.
11.1.2 The twosurface observables
To understand the meaning of the observables (11.3, recall that any vector field N^{ a } on Σ generates a diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum constraint C[0, N^{ a }] is just to generate this gauge symmetry in the phase space. However, the boundary \({\mathcal S}\) breaks the diffeomorphism invariance of the system, and hence, on the boundary the diffeomorphism gauge motions yield the observables O[N^{ a }] and the gauge degrees of freedom give rise to physical degrees of freedom, making it possible to introduce edge states [47, 46, 123].
Analogous investigations were done by Husain and Major in [281]. Using Ashtekar’s complex variables [30] they determine 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, Husain and Major discuss the two most significant cases. In the first case the generators N, N^{ a }, and N^{ i } are vanishing on \({\mathcal S}\), and thus there are infinitely many twosurface observables, both from the diffeomorphism and the Gauss constraints, but no observables 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 \({\mathcal S}\) (and hence, in particular, the threemetric) is fixed on the twoboundary. Then the quasilocal energy could be an observable, as in the ADM analysis above.
All of the papers [47, 46, 123, 281] discuss the analogous phenomenon of how the gauge freedoms become true physical degrees of freedom in the presence of twosurfaces on the twosurfaces 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 [125, 409]. They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in [125] is based on the covariant Noethercharge formalism below.) Since this algebra is well known in conformal field theories, this approach might be a basis for understanding the microscopic origin of the black hole entropy [124, 125, 126, 409, 127]. However, this quantum issue is beyond the scope of the present review.
Returning to the discussion of O[N^{ a }] above, note first that, though A_{ e } is a gauge potential, by δ_{ e }N^{ e } = 0 it is boost gauge invariant. Without this condition, Eq. (11.3) would give potentially reasonable physical quantity only if the boost gauge on \({\mathcal S}\) were geometrically given, e.g., when \({\mathcal S}\) were a leaf of a physicallydistinguished foliation of a physicallydistinguished spacelike or timelike hypersurface [39]. In particular, the angular momentum of Brown and York [121] also takes the form (11.3), and is well defined (because N^{ a } is assumed to be a Killing vector of the intrinsic geometry of \({\mathcal S}\)). (In the angular momentum of Liu and Yau [338] only the gauge invariant part of A_{ e } is present in Eq. (11.3) instead of A_{ e } itself.) Similarly, the expressions in [47, 571] can also be rewritten into the form (11.3), but they should be completed by the condition δ_{ e }N^{ e } = 0.
In general Eq. (11.3) is used as a definition of the N^{ a }component of the angular momentum of quasilocally defined black holes [40, 97, 227]. This interpretation is supported by the following observations [499]. In axisymmetric spacetimes for axisymmetric surfaces O[N^{ a }] can be rewritten into the Komar integral, the usual definition of angular momentum in axisymmetric spacetimes. Moreover, if Σ extends to spatial infinity, then δ_{ e }N^{ e } = 0 together with the requirement of the finiteness of the r → ∞ limit of the observable O[N^{ a }] already fix the asymptotic form of N^{ a }, which is precisely the combination of the asymptotic spatial rotation Killing vectors, and O[N^{ a }] reproduces the standard spatial ADM angular momentum. Similarly, at null infinity N^{ a } must be a rotation BMS vector field. However, the null infinity limit of O[N^{ a }] is sensitive to the first two terms (rather than only the leading term) in the asymptotic expansion of N^{ a }, and hence in general radiative spacetime O[N^{ a }] in itself does not yield an unambiguous definition for angular momentum. (But in stationary spacetimes the ambiguities disappear and O[N^{ a }] reproduces the standard formula (4.15).) Thus, additional ideas are needed to restrict the BMS vector field N^{ a }.
Such an idea could be based on the observation that the eigenspinors of the δ_{ e }Dirac operators define δ_{ e }divergencefree vector fields on \({\mathcal S}\), and on metric spheres these vector fields built from the eigenspinors with the lowest eigenvalue are just the linear combinations of the three rotation Killing fields [501]. Solving the eigenvalue problem for the δ_{ e }Dirac operators on large spheres near scri in the first two leading orders, a welldefined (ambiguityfree) angularmomentum expression is suggested. The angular momenta associated with different cuts of \({\mathcal S}\) can be compared, and the angular momentum flux can also be calculated.
It is tempting to interpret O[N^{ a }] as the N^{ a }component of the quasilocal angular momentum of the gravity + matter system associated with \({\mathcal S}\). However, without additional conditions on N^{ a } the integral O[N^{ a }] could be nonzero even in Minkowski spacetime [501]. Hence, N^{ a } must satisfy additional conditions. Cook and Whiting [153] suggest that one derive N^{ a } from a variational principle on topological twospheres. Here the action functional is the norm of the Killing operator. (For a viable, general notion of approximate Killing fields see [359].) Another realization of the approximate Killing fields is given by Beetle in [59], where the vector field N^{ a } is searched for in the form of the solution of an eigenvalue problem for an equation, derived from the Killing equations. Both prescriptions have versions in which they give δ_{ e }divergencefree N^{ a }. The definition of N^{ a } suggested in [323] is based on the fact that six of the infinitely many conformal Killing fields on \({\mathcal S}\) with spherical topology are globally defined, and after an appropriate globallydefined conformal rescaling of the intrinsic metric they become the generators of the standard SO(1, 3) action on \({\mathcal S}\). Then these three are used to define the angular momentum that will be the Killing fields in the rescaled geometry. In general these vector fields are not δ_{ e }divergencefree. Thus, as in the LiuYau definition, to keep boost gauge invariance the gauge invariant piece of the connection oneform A_{ e } can be used instead of the A_{ e } itself.
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, e.g., [441, 419]). The basic idea is that we foliate an open subset U of the spacetime by a twoparameter family of (e.g., closed) spacelike twosurfaces. If \({\mathcal S}\) is the typical twosurface, 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 \({\nu _ +},{{\mathcal S}_{{\nu _ +},{\nu _ }}}: = \phi ({\mathcal S},{\nu _ +},{\nu _ })\) defines two oneparameter families of hypersurfaces Σ_{ν+} and Σ_{ν−} respectively. Requiring one (or both) of the hypersurfaces Σ_{ν+} to be null, we get a null (or doublenull, respectively) foliation of U. (In Section 4.1.8 we require 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 well defined 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 analogouslydefined \(\xi _  ^a\) are 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[441, 209, 160, 480, 522, 245, 225, 105, 254].
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 Euler—Lagrange equations, provided the fact that the foliation is null is imposed only after the variation has been made. (Otherwise, the variation of the action with respect to the lessthanten nontrivial components of the metric would not yield all ten 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 [160, 522, 245, 254]. 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 [245, 248] as the basis of his quasilocal energy expression discussed already in Section 6.3. (A similar energy expression was derived by Ikumi and Shiromizi [282], starting with the idea of the ‘freely falling twosurfaces’.)
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 [247] 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., \({{\mathcal N}_u}\), but the other was timelike, e.g., B_{ v }. The former defined a foliation of the latter in terms of the spacelike twosurfaces \({{\mathcal S}_{u,\,\upsilon}}: = {{\mathcal N}_u} \cap {B_\upsilon}\). Yoon found [567, 568] a certain twosurface integral on \({{\mathcal S}_{u,\,\upsilon}}\), 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, \(\tilde E(u,\,\upsilon)\) tends to the Bondi energy, and the flux integral tends to the Bondi massloss between the cuts u = u_{1} and u = u_{2} [567, 568]. These investigations were extended for other integrals in [569, 570, 571], 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 twosurface \({{\mathcal S}_{u,\,\upsilon}}\) but on the 2 + 2 foliation on an open neighborhood of \({{\mathcal S}_{u,\,\upsilon}}\) as well.
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. more easily. No a posteriori spacetime interpretation of the results is needed. Such a spacetimecovariant Hamiltonian formalism was initiated by Nester [377, 380].
The spirit of the first systematic investigations of the covariant phase space of the classical field theories [158, 33, 197, 336] is similar to that of Nester’s. These ideas were recast into the systematic formalism by Wald and Iyer [536, 287, 288], the covariant Noether charge formalism (see also [535, 336]). This formalism generalizes many of the previous approaches. The Lagrangian 4form may be any diffeomorphisminvariant local expression of any finiteorder 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 turns out to be just a Noether charge derived from Hilbert’s Lagrangian.
11.3.2 The general expressions of Chen, Nester and Tung: Covariant quasilocal Hamiltonians with explicit reference configurations
The boundary term in the variation δH[K] of the Hamiltonian with the boundary term (11.5) and (11.6) is the twosurface integral on ∂Σ of \({\iota _{\rm{K}}}(\delta {\phi ^A} \wedge ({\pi _A}  \pi _A^0))\) and \({\iota _{\rm{K}}}( ({\phi ^A}  {\phi ^{0A}}) \wedge \delta {\pi _A}\), respectively. Therefore, the Hamiltonian is functionally differentiable with the boundary 2form B_{ ϕ }(K^{ a }) if the configuration variable ϕ^{ A } is fixed on ∂Σ, but B_{ π }(K^{ a }) should be used if π_{ A } is fixed on ∂Σ. Thus, the first boundary 2form corresponds to a fourcovariant Dirichlettype, while the second corresponds to a fourcovariant Neumanntype boundary condition. Obviously, the Hamiltonian evaluated in the reference configuration \(({\phi ^{0A}},\,\pi _A^0)\) gives zero. Chen and Nester show [136] that B_{ ϕ }(K^{ a }) and B_{ π }(K^{ a }) are the only boundary 2forms for which the resulting boundary 2form C(K^{ a })_{ bc } in the variation δH(K^{ a })_{ bcd } of the Hamiltonian 3form vanishes on ∂Σ, which reflects the type of boundary conditions (i.e., which fields are fixed on the boundary), and is built from the configuration and momentum variables fourcovariantly (‘uniqueness’). A further remarkable property of B_{ ϕ }(K^{ a }) and B_{ π }(K^{ a }) is that the corresponding Hamiltonian 3form can be derived directly from appropriate Lagrangians. One possible choice for the vector field K^{ a } is a Killing vector of the reference geometry. This reference geometry is, however, not yet specified, in general.
A nice application of the covariant expression is a derivation of the first law of black hole thermodynamics [136]. The quasilocal energy expressions have been evaluated for several specific twosurfaces. For round spheres in the Schwarzschild spacetime, both the fourcovariant 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 [136]. The Dirichlet boundary term gives, at spatial infinity in the Kerrantide Sitter solution, the standard m/G and ma/G values for the energy and angular momentum, respectively [257]. The centerofmass is also calculated, both in the metric and the tetrad formulation of general relativity, for the eccentric Schwarzschild solution at spatial infinity [389, 390], and it was found that the ‘Komarlike term’ is needed to recover the correct, expected value. At future null infinity of asymptotically flat spacetimes it gives the BondiSachs energymomentum and the expression of Katz [305, 310] for the angular momentum [258]. The general formulae are evaluated for the KerrVaidya solution as well.
The quasilocal energymomentum is calculated on twosurfaces lying in intrinsicallyflat spacelike hypersurfaces in static sphericallysymmetric spacetimes [138], and, in particular, for twosurfaces in the τ = const. slicing of the Schwarzschild solution in the PainlevéGullstrand coordinates. Though these hypersurfaces are flat, and hence, the total (ADM type) energy is expected to be vanishing, the quasilocal energy expression based on Eq. (11.7) and a ‘naturally chosen’ frame field gives 2m/G. (N.B., the Cauchy data on the τ = const hypersurfaces do not satisfy the falloff conditions of Section 3.2.1. Though the intrinsic metric is flat, the extrinsic curvature tends to zero only as \({r^{ {3 \over 2}}}\), while in the expression of the ADM linear momentum a slightly faster than \({r^{ {3 \over 2}}}\) falloff is needed. Thus, the vanishing of the naïvly introduced ADMtype energy does not contradict the rigidity part of the positive energy theorem.)
The quasilocal energy flux of spacetime perturbations on a stationary background is calculated by Tung and Yu [531] using the covariant Noether charge formalism and the boundary terms above. As an example they considered the Vaidya spacetime as a timedependent perturbation of a stationary one with the orthonormal frame field being adapted to the spherical symmetry. At null infinity they recovered the Bondi massloss, while for the dynamical horizons they recovered the flux expression of Ashtekar and Krishnan (see Section 13.3.2).
The quasilocal energymomentum, based on Eq. (11.7) in the tetrad approach to general relativity, is calculated for arbitrary twosurfaces \({\mathcal S}\) lying in the hypersurfaces of the homogeneity in all the Bianchi cosmological models in [391] (see also [340]). In these calculations the tetrad field was chosen to be the geometrically distinguished triad, being invariant with respect to the global action of the isometry group, and the futurepointing unit timelike normal of the hypersurfaces; while the vector field K^{ a } was chosen to have constant components in this frame. For class A models (i.e., for I, II, VI_{0}, VII_{0}, VIII and IX Bianchi types) this is zero, and for class B models (III, IV, V, VI_{ h } and VII_{ h } Bianchi models) the quasilocal energy is negative, and the energy is proportional to the volume of the domain that is bounded by \({\mathcal S}\). (Here a sign error in the previous calculations, reported in [134, 387, 385], is corrected.) The apparent contradiction of the nonpositivity of the energy in the present context and the nonnegativity of the energy in general smallsphere calculations indicates that the geometrically distinguished tetrad field in the Bianchi models does not reduce to the ‘natural’ approximate translational Killing fields near a point. Another interpretation of the vanishing and negativity of the quasilocal energy, different from this and those in Section 4.3, is also given.
Instead of the specific boundary terms, So considered a twoparameter family of boundary terms [464], which generalized the special expressions (11.5)–(11.6) and (11.9)–(11.10). The main idea behind this generalization is that one cannot, in general, expect to be able to control only, for example, either the configuration or the momentum variables, rather only a combination of them. Hence, the boundary condition is not purely of a Dirichlet or Neumann type, but rather a more general mixed one. It is shown that, with an appropriate value for these parameters, the resulting energy expression for small spheres is positive definite, even in the holonomic description.
11.3.3 The reference configuration of Nester, Chen, Liu and Sun
In the general covariant quasilocal Hamiltonians Chen, Nester and Tung left the reference configuration and the boundary conditions unspecified, and hence their construction was not complete. These have been specified in [386]. The key ideas are as follow.
First, because of its correct, advantagous properties (especially its asymptotic behaviour in asymptotically flat spacetimes), Nester, Chen, Liu and Sun choose (11.7) a priori as their Hamiltonian boundary term. Their reference configuration is chosen to be the Minkowski spacetime, and the generator vector field is the general Killing vector (depending on ten parameters).
Next, to match the physical and the reference geometries, they require the two full 4dimensional metrics to coincide at the points of the twosurface \({\mathcal S}\) (rather than only the induced twometrics on \({\mathcal S}\)). This condition leaves two unspecified functions in the quasilocal quantities. To find the ‘best matched’ such embedding of \({\mathcal S}\) into the Minkowski spacetime, Nester, Chen, Liu and Sun propose to choose the one that extremize the quasilocal mass.
This, and some other related strategies have been used to compute quasilocal energy in various spherically symmetric configurations in [135, 341, 561, 562].
11.3.4 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 [13, 14]. (Some formulae of the journal versions were recently corrected in the latest arXiv versions.) 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. In general relativity, the variations of the tetrad fields are restricted on ^{3}B by requiring in the first case that the induced metric γ_{ ab } is fixed and the adaptation of the tetrad field to the boundary is preserved, while in the second case that the tetrad components \({\Theta _{ab}}E_{\underline a}^b\) of the extrinsic curvature of ^{3}B is fixed. 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 [13]. The properties of the covectors \(P_a^{\rm{D}}\) and \(P_a^{\rm{N}}\) (called the Dirichlet and Neumann symplectic vectors, respectively) were investigated further in [14]. 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 \(P_a^{\rm{D}}\) and \(P_a^{\rm{N}}\) 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 \(P_a^{\rm{N}} =  (m/{r^2}  {e^2}/{r^2})\delta _a^0\), 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 \(\oint\nolimits_{{{\mathcal S}_r}} {{K^a}P_a^{\rm{D}}d{{\mathcal S}_r} = 4\pi (m  {e^2}/2r)}\), respectively. Thus, \(P_a^{\rm{N}}\) does not reproduce the standard roundsphere 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 [14].
The boundary condition on closed untrapped spacelike twosurfaces that make the covariant Hamiltonian functionally differentiable were investigated by Tung [526, 527]. He showed that such a boundary condition might be the following: the area 2form and the mean curvature vector of \({\mathcal S}\) are fixed, and the evolution vector field K^{ a } is proportional to the dual mean curvature vector, where the factor of proportionality is a function of the area 2form. Then, requiring that the value of the Hamiltonian reproduce the ADM energy, he recovers the Hawking energy. If, however, K^{ a } is allowed to have a part tangential to \({\mathcal S}\), and K^{ a }A_{ a } is required to be fixed (up to total δ_{ e }divergences), then, though the value of the Hamiltonian is still proportional to the Hawking energy, the factor of proportionality depends on the angular momentum, given by (11.3), as well. With this choice the vector field K^{ a } becomes a generalization of the Kodama vector field [321] (see also Section 4.2.1). The results of [527, 528] are extensions of those in [526].
11.3.5 Pseudotensors and quasilocal quantities
As we discussed briefly in Section 3.3.1, many, apparently different, pseudotensors and SO(1, 3)gaugedependent 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 \(u{k \over {ab}}\) from L(M) along the various local sections of the bundle [192, 358, 486, 487]. Thus, the different pseudotensors are simply the gaugedependent manifestations of the same geometric object on the bundle L(M) in the different gauges. Since, however, \(u{k \over {ab}}\) is the unique extension of the NesterWitten 2form \(u{({\varepsilon ^{\underline K}},\,{{\bar \varepsilon}^{\underline K}})_{ab}}\), on the principal bundle of normalized spin frames \(\{\varepsilon {K \over A}\}\) (given in Eq. (3.10)), 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 [131, 137, 382] 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 \(\sqrt {\left\vert g \right\vert} {g^{\alpha \beta}}\).
12 Constructions for Special Spacetimes
12.1 The Komar integral for spacetimes with Killing vectors
Although the Komar integral (and, in general, the linkage (3.15) for some α) does not satisfy our general requirements discussed in Section 4.3.1, and 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 twosurface (see, e.g., [534]). This follows directly from the explicit form of Komar’s canonical Noether current: 8 \(8\pi G{C^a}[{\bf{K}}] = {G^a}_b{K^b} + {\nabla _b}{\nabla ^{[a}}{K^{b]}} =  {1 \over 2}R{K^a}  {\nabla _b}({\nabla ^{(a}}{K^{b)}}  {g^{ab}}{\nabla _c}{K^c})\). In fact, if \({\mathcal S}\) and \({{\mathcal S}{\prime}}\) are any two twosurfaces such that \({\mathcal S}  {{\mathcal S}{\prime}} = \partial \Sigma\) for some compact threedimensional hypersurface Σ on which the energymomentum tensor of the matter fields is vanishing and K^{ a } is a Killing vector, then \({L_{\mathcal S}}[{\bf{K}}] = {L_{{{\mathcal S}{\prime}}}}[{\bf{K}}]\). (Note that, as we already stressed, the structure of the Noether current above dictates that the numerical coefficient in the definition (3.15) of the linkage would have to be \({1 \over {16\pi G}}\) rather than \({1 \over {8\pi G}}\), i.e., the one that gives the correct value of angular momentum (rather than the mass) in Kerr spacetime.) In particular, the Komar integral for the static Killing field in the Schwarzschild spacetime is the mass parameter m of the solution for any twosurface \({\mathcal S}\) surrounding the black hole, but it is zero if \({\mathcal S}\) does not surround it. The explicit form of the current shows that, for timelike Killing field K^{ a }, the small sphere expression of Komar’s quasilocal energy in the first nontrivial order is \( {{2\pi} \over 3}{r^3}{T_{ab}}{g^{ab}}{t_c}{K^c}\), i.e., it does not reproduce the expected result (4.9); moreover, in vacuum it always gives zero rather than, e.g., the BelRobinson ‘energy’ (see Section 4.2.2).
Furthermore [510], the analogous integral in the ReissnerNordström spacetime on a metric twosphere 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 spheres of radius r on a t = const. hypersurface, \({L_{\mathcal S}}[{\mathbf{K}}]\) 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.
Accurate numerical calculations show that in stationary, axisymmetric asymptotically flat spacetimes describing a black hole or a rigidlyrotating dust disc surrounded by a perfect fluid ring the Komar energy of the black hole or the dust disc could be negative, even though the conditions of the positive energy theorem hold [21]. Moreover, the central black hole’s event horizon can be distorted by the ring so that the black hole’s Komar angular momentum is greater than the square of its Komar energy [20].
12.2 The effective mass of Kulkarni, Chellathurai, and Dadhich for the Kerr spacetime
The KulkarniChellathuraiDadhich [328] 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 [133].
12.3 Expressions in static spacetimes
12.3.1 Tolman’s energy for static spacetimes
Let K^{ a } be a hypersurfaceorthogonal timelike Killing vector field, Σ a spacelike hypersurface to which K^{ a } is orthogonal, and f^{2} ≔ K_{ a }K^{ a }. Then \( {D_a}{D^a}f = 4\pi G(\pi + 3p  {\lambda \over {4\mu G}})f\), a field equation for f, follows from Einstein equations (see, e.g., pp. 71–74 of [240] or [199]). Here μ ≔ T_{ ab }t^{ a }t^{ b } and \(p: =  {1 \over 3}{T_{ab}}{h^{ab}}\), the energy density and the average spatial pressure of the matter fields, respectively, seen by the observer at rest with respect to Σ (or K^{ a }).
The Tolman energy appeared to be a useful tool in practice: By means of E_{T} Abreu and Visser gave remarkable entropy bounds for localized, but uncollapsed bodies [2, 3]. (We discuss this bound in Section 13.4.3.)
12.3.2 The KatzLyndenBellIsrael energy for static spacetimes
In asymptotically flat spacetimes \({E_{{\rm{KLI}}}}({{\mathcal S}_{\rm{K}}})\) tends to the ADM energy [309]. However, it does not reduce to the roundsphere energy in sphericallysymmetric spacetimes [374], and, in particular, gives zero for the event horizon of a Schwarzschild black hole.
12.3.3 Static spacetimes and postNewtonian approximation
Though E_{ D } is negative in the vacuum regime, for spherically symmetric configurations, when the material source of the gravitational ‘field’ is contained in D, it is positive if an energy condition is satisfied; and it is zero if and only if the domain of dependence of D in the spacetime is flat. (For the details see [199].)
13 Applications in General Relativity
In this section 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 complete, and our aim 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 for their application. For a recent review of the applications of these ideas, especially in black hole physics, with an extended bibliography, see [294, 293].
13.1 Calculation of tidal heating
According to astronomical observations, there is intense volcanic activity on the moon Io of Jupiter. One possible explanation of this phenomenon is that Jupiter is heating Io via gravitational tidal forces (like the Moon, whose gravitational tidal forces raise the ocean’s tides on the Earth). To check if this is 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 various gravitational pseudotensors [432, 185]. 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 of both the gauge conditions [432] and the actual pseudotensor (Einstein, Møller, Bergmann, or LandauLifshitz) [185].
Recently, these calculations were repeated using quasilocal concepts by Booth and Creighton [94]. They calculated the time derivative of the BrownYork energy, given by Eqs. (10.8) and (10.9). Assuming the form of the metric used in the pseudotensorial calculations, for the tidal work they recovered the gauge invariant expressions obtained in [432, 185]. 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 [416], Penrose derived an inequality that any asymptotically flat initial data set with (outermost) apparent horizon \({\mathcal S}\) must satisfy [418]: The ADM mass m_{ADM} of the data set cannot be less than the irreducible mass of the horizon, \(M: = \sqrt {{\rm{Area(}}{\mathcal S}{\rm{)/(16}}\pi {G^2})}\) (see, also, [213, 113, 354]). However, as stressed by BenDov [75], the more careful formulation of the inequality, due to Horowitz [273], 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 twosurface \({{\mathcal S}_{\min}}\), where \({{\mathcal S}_{\min}}\) has the minimum area among the twosurfaces enclosing the apparent horizon \({\mathcal S}\). In [75] a sphericallysymmetric 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 sphericallysymmetric spacetimes [352] (see, also, [578, 250, 251]), for static black holes (using the Penrose mass, as mentioned in Section 7.2.5) [513, 514] and for the perturbed ReissnerNordström spacetimes [301] (see, also, [302]). Although the original specific potential counterexample has been shown not to violate the Penrose inequality [214], 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 a trapped surface and the Bondi mass at past null infinity [345], see [82].) 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 [193]), and then the global existence of such a foliation between the apparent horizon and the twosphere at infinity is proven. The first complete proof of the latter was given by Huisken and Ilmanen [278, 279]. (An alternative proof, using a conformal technique, was given by Bray [110, 111, 112].) A simple (but complete) proof of the Riemannian Penrose inequality is given in the special case of axisymmetric timesymmetric data sets by using Brill’s energy positivity proof [218].
A more general form of the conjecture, containing the electric charge parameter e of the black hole, was formulated by Gibbons [213]: The ADM mass is claimed not to be exceeded by M + e^{2}/ (4G^{2}M). Although the weaker form of the inequality, the Bogomolny inequality m_{ADM} ≥ e /G, has been proven (under assumptions on the matter content, see, e.g., [219, 508, 344, 217, 371, 213]), Gibbons’ inequality for the electric charge has been proven for special cases (for sphericallysymmetric spacetimes see, e.g., [251]), and for timesymmetric initial data sets using Geroch’s inverse mean curvature flow [290]. As a consequence of the results of [278, 279] 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 k > 1 nearlyextremal ReissnerNordström 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 marginallytrapped surface [546], or the sum of the areas of the individual black hole horizons. On the other hand, the weaker inequality (13.1) below, derived from the cosmic censorship assumption, does not seem to be violated, even in the presence of more than one black hole.^{22}
The structure of Eqs. (13.2) and (13.3) suggests another interpretation, too. In fact, since M is a quasilocally defined property of the black hole itself, it is natural to ask if the lower bound for the ADM mass can be given only in terms of quasilocally defined quantities. In the absence of charges outside the horizon, q is just the charge measured at \({{\mathcal S}_{\min}}\), and if, in addition, the spacetime is axisymmetric and vacuum, then J_{ADM} coincides with the Komar angular momentum also at \({{\mathcal S}_{\min}}\). However, in general it is not clear what J^{2} would have to be: The magnitude of some quasilocally defined relativistic angular momentum, or only of the spatial part of the angular momentum, or even the PauliLubanski spin?
Penroselike inequalities are studied numerically in [295], while counterexamples to a new version, and to a generalized form (including charge) of the Penrose inequality are given in [129] and [166], respectively. Reviews of the Penrose inequality with an extended bibliography are [354, 355].
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 [506, 366], 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, [188, 538]). Mathematically, this conjecture is not precisely formulated. Neither the mass nor the notion of the circumference is well defined. In certain situations the mass might be the ADM or the Bondi mass, but might be the integral of some locallydefined ‘mass density’, as well [188, 50, 350, 320]. The most natural formulation of the hoop conjecture would be based on some spacelike twosurface \({\mathcal S}\) 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 roundsphere definition of the quasilocal energy (4.7) one has 4πGE = 2πr[1 − exp(− 2α)] < 2πr = C, where we use the fact that r is an areal radius (see Section 4.2.1).
Another formulation of the hoop conjecture, also for the spherically symmetric configurations, was given by Ó Murchadha, Tung, Xie and Malec in [402] using the BrownYork energy. They showed that a spherical 2surface, which is embedded in a spherically symmetric asymptotically flat 3slice with a regular center and which satisfies C < 2πGE_{BY}, is trapped. Moreover, if C > 2πGE_{BY} holds for all embeddings, then the surface is not trapped. The root of the deviation of the numerical coefficient in front of the quasilocal energy E_{BY} here (viz. 2π) from the one in Thorne’s original formulation (i.e., 4π) is the fact that E_{BY} on the event horizon of a Schwarzschild black hole is 2m, rather than the expected m. It is also shown in [402] that no analogous statement can be proven in terms of the KijowskiLiuYau or the WangYau energies.
If, however, \({\mathcal S}\) is not axisymmetric, then there is no natural definition (or, there are several inequivalent ‘natural’ definitions) for the circumference of \({\mathcal S}\). Interesting, necessary and also sufficient conditions for the existence of averaged trapped surfaces in nonsphericallysymmetric cases, both in special asymptotically flat and cosmological spacetimes, are found in [350, 320]. For the investigations of the hoop conjecture in the GibbonsPenrose spacetime of the collapsing thin matter shell see [51, 50, 518, 411], and for colliding black holes see [574]. One reformulation of the hoop conjecture, using the new concept of the ‘trapped circle’ instead of the illdefined circumference, is suggested by Senovilla [450]. Another version of the hoop conjecture was suggested by Gibbons in terms of the ADM mass and the Birkhoff invariant of horizon of spherical topology, and this form of the conjecture was proved in a number a special cases [215, 159].
13.2.3 On the Dain inequality
The KerrNewman solution describes a black hole precisely when the mass parameter dominates the angular momentum and the charge parameters: m^{2} ≥ a^{2} + e^{2}. Thus, it is natural to ask whether or not an analogous inequality holds for more general, dynamic black holes. As Dain has proven, in the axisymmetric, vacuum case there is an analogous inequality, a consequence of an extremality property of Brill’s form of the ADM mass. Namely, it is shown in [165], that the unique absolute minimum of the ADM mass functional on the set of the vacuum Brill data sets with fixed ADM angular momentum is the extreme Kerr data set. Here a Brill data set is an axisymmetric, asymptotically flat, maximal, vacuum data set, which, in addition, satisfies certain global conditions (viz. the form of the metric is given globally, and nontrivial boundary conditions are imposed) [218, 165]. The key tool is a manifestly positive definite expression of the ADM energy in the form of a threedimensional integral, given in globally defined coordinates. If the angular momentum is nonzero, then by the assumption of axisymmetry and vacuum, the data set contains a black hole (or black holes), and hence, the extremality property of the ADM energy implies that the ADM mass of this (in general, nonstationary) black hole cannot be less than its ADM angular momentum. For further discussion of this inequality, in particular its role analogous to that of the Penrose inequality, see [164]; and for earlier versions of the extremality result above, see [163, 162, 161].
Since in the above result the spacetime is axisymmetric and vacuum, the ADM angular momentum could be written as the Komar integral built from the Killing vector of axisymmetry on any closed spacelike spherical twosurface homologous to the large sphere near the actual infinity. Thus, the angular momentum in Dain’s inequality can be considered as a quasilocal expression. Hence, it is natural to ask if the whole inequality is a condition on quasilocally defined quantities or not. However, as already noted in Section 12.1, in the stationary axisymmetric but nonvacuum case it is possible to arrange the matter outside the horizon in such a way that the Komar angular momentum on the horizon is greater than the Komar energy there, or the latter can even be negative [20, 21]. Therefore, if a massangular momentum inequality is expected to hold quasilocally at the horizon, then it is not obvious which definitions for the quasilocal mass and angular momentum should be used. In the stationary axisymmetric case, the angular momentum could still be the Komar expression, but the mass is the area of the event horizon [266]: Area(\({\rm{Area(}}{\mathcal S}{\rm{)}} \geq {\rm{8}}\pi G{J_{\rm{K}}}\)) ≥ 8πGJ_{K}. For the extremal case (even in the presence of Maxwell fields), see [22]. (For the extremality of black holes formulated in terms of isolated and dynamic horizons, see [99] and Section 13.3.2.)
For a recent, very wellreadable and comprehensive review of the Dain inequality with the extended bibliography, where both the old and the recent results are summarized, see the topical review of Dain himself in [167].
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 [237, 238, 240, 534], and hence, it is natural to derive the formal laws of black hole mechanics/thermodynamics in the asymptotically flat context (see, e.g., [49, 67, 68], and for a comprehensive review, [539]). The discovery of Hawking radiation [239] showed that the laws of black hole thermodynamics are not only analogous to the laws of thermodynamics, but black holes are genuine thermodynamic objects: black hole temperature is a physical temperature, that is ħc/(2πk) times the surface gravity, and its 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, [537]). Apparently, the detailed microscopic (quantum) theory of gravity is not needed to derive black hole entropy, and it can be derived even from the general principles of a conformal field theory on the horizon of black holes [124, 125, 126, 409, 127, 128].
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 theoretical 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 antide Sitter spacetime, fixing black hole temperature via the normalization of the timelike Killing vector at infinity is not justified because there is no such physicallydistinguished Killing field (see [116]). These difficulties lead to the need of a quasilocal formulation of black hole thermodynamics. In [116], Brown, Creighton, and Mann investigated the thermal properties of the Schwarzschildantide Sitter black hole. They used the quasilocal approach of Brown and York to define the energy of the black hole on a spherical twosurface \({\mathcal S}\) outside the horizon. Identifying the BrownYork energy with the internal (thermodynamic) 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 Schwarzschildantide Sitter black holes with temperature T less than T_{0}. In [157] 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 Noether charge formalism of Wald [536], and Iyer and Wald [287] can be interpreted as a generalization of the BrownYork approach from general relativity to any diffeomorphism invariant theory to derive quasilocal quantities [288]. 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 related work see, e.g., [205, 253]). A comparison of the various proposals for the surface gravity of dynamic black holes in sphericallysymmetric black hole spacetimes is given by Nielsen and Yoon [396].
There is extensive literature on the quasilocal formulation of the black hole dynamics and relativistic thermodynamics in the sphericallysymmetric context (see, e.g., [250, 252, 251, 256] and for nonsphericallysymmetric cases [372, 254, 96]). These investigations are based on the quasilocally defined notion of trapping horizons [246]. A trapping horizon is a smooth hypersurface that can be foliated by (e.g., future) marginallytrapped surfaces such that the expansion of the outgoing null normals is decreasing along the incoming null normals. (On the other hand, the investigations of [248, 246, 249] are based on gaugedependent energy and angular momentum definitions; see also Sections 4.1.8 and 6.3.) For reviews of the quasilocal formulations and the various aspects of black hole dynamics based on the notion of trapping horizons, see [41, 294, 395, 255], and, for a recent one with an extended bibliography, see [292].
13.3.2 On isolated and dynamic horizons
The idea of isolated horizons (more precisely, the gradually more restrictive notion of nonexpanding, weakly isolated and isolated horizons, and the special weakly isolated horizon called rigidly rotating) generalizes the notion of Killing horizons by keeping their basic properties without the existence of any Killing vector in general. Thus, while the black hole is thought to be settled down to its final state, the spacetime outside the black hole may still be dynamic. (For a review see [32, 41] and references therein, especially [34, 31].) The phase space for asymptotically flat spacetimes containing an isolated horizon is based on a threemanifold 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 the Hamiltonian is the sum of the constraints and boundary terms, corresponding both to the ends and the horizon. Thus, 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 [34, 31].
Booth [93] 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} × [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, by requiring Δ to be a rigidlyrotating 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 rigidlyrotating horizons was also derived by Allemandi, Francaviglia, and Raiteri in the EinsteinMaxwell theory [9] using their ReggeTeitelboimlike approach [191]. The first law for ‘slowly evolving horizons’ was derived in [96].
Another concept is the notion of a dynamic horizon [39, 40]. This is a smooth spacelike hypersurface that can be foliated by a geometrically distinguished family of (e.g., future) marginallytrapped surfaces, i.e., it is a generalization of the trapping horizon above. The isolated horizons are thought to be the asymptotic state of dynamic horizons. The local existence of such horizons was proven by Andersson, Mars and Simon [19]: If \({\mathcal S}\) is a (strictly stably outermost) marginally trapped surface lying in a leaf, e.g., Σ_{0}, of a foliation Σ_{ t } of the spacetime, then there exists a hypersurface ℋ (the ‘horizon’) such that \({\mathcal H}\) lies in ℋ, and which is foliated by marginally outertrapped surfaces. (For the related uniqueness properties of the structure of the dynamic horizons see [35]). This structure of the dynamic horizons makes it possible to derive balance equations for the areal radius of the surfaces \({\mathcal S}\) and the angular momentum given by Eq. (11.3) [32, 40] (see also [41]). In particular, the difference of the areal radius of two marginallytrapped surfaces of the foliation, e.g., \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\), is just the flux integral on the portion of \({\mathcal H}\) between \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) of a positive definite expression: This is the flux of the energy current of the matter fields and terms that can be interpreted as the energy flux carried by the gravitational waves. Interestingly enough, the generator vector field in this flux expression is proportional to the geometrically distinguished outward null normal of the surfaces \({\mathcal S}\), just as in the derivation of black hole entropy as a Noether charge by Wald [536] and Iyer and Wald [287] above. Thus, the second law of black hole mechanics is proven for dynamic horizons. Moreover, this supports the view that the energy that we should associate with marginallytrapped surfaces is the irreducible mass. For further discussion (and generalizations) of the basic flux expressions see [227, 228]. For a different calculation of the energy flux in the Vaidya spacetime, see [531].
In [97, 98] Booth and Fairhurst extended their previous investigations [93, 95] (see above and Section 10.1.5). In [97] a canonical analysis, based on the extended phase space, is given such that the underlying threemanifold has an inner boundary, which can be any of the horizon types above. Though the formalism does not give any explicit expression for the energy on the horizons, an argument is given that supports the expectation that this must be the irreducible mass of the horizon. The variations of marginally trapped surfaces, generated by vector fields orthogonal to the surfaces, are investigated and the corresponding variations of various geometric objects (intrinsic metric, expansions, connection oneform on the normal bundle, etc.) on the surfaces are calculated in [98]. In terms of these, several basic properties of marginally trapped or future outer trapped surfaces (and hence, of the horizons themselves) are derived in a straightforward way.
13.4 Entropy bounds
13.4.1 On Bekenstein’s bounds for the entropy
One should stress, however, that in general curved spacetimes the notion of energy, angular momentum, and radial distance appearing in Eq. (13.4) are not yet well defined. Perhaps it is just the quasilocal ideas that should be used to make them well defined, and there is a deep connection between the GibbonsPenrose inequality and the Bekenstein bound. The former is the geometric manifestation of the latter for black holes.
13.4.2 On the holographic hypothesis
In the literature there is another kind of upper bound for the entropy of a localized system, the holographic bound. The holographic principle [504, 482, 104] 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 as well, analogous to the holography by means of which a threedimensional image is encoded into a twodimensional 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 twosurface is not greater than 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 [104]. 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 [539, 104]).
On the other hand, there is another formulation of the holographic entropy bound, due to Bousso [103, 104]. Bousso’s covariant entropy bound is much more quasilocal than the previous formulations, and is based on spacelike twosurfaces and the null hypersurfaces determined by the twosurfaces in the spacetime. Its classical version has been proven by Flanagan, Marolf, and Wald [189]. 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: \({{\mathcal S}_{\mathcal N}} \leq k\vert \mathrm {Area}({{\mathcal S}_2})  \mathrm {Area}({{\mathcal S}_1})\vert/(4L_{\mathrm {P}}^2)\). For a detailed discussion see [539, 104]. For another, quasilocal formulation of the holographic principle see Section 2.2.5 and [498].
13.4.3 Entropy bounds of Abreu and Visser for uncollapsed bodies
13.5 Quasilocal radiative modes of general relativity
In Section 8.2.3 we discuss the properties of the DouganMason energymomenta, and we see 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 twosurface \({\mathcal S}\). However, there is an important difference between these two statements. While in the former case we know the metric of D(Σ) 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 to whether the metric of D(Σ) is also determined by the geometry of \({\mathcal S}\) even in the zero quasilocalmass case.
In [492] it is shown that under the condition above there is a complex valued function Φ on \({\mathcal S}\), describing the deviation of the antiholomorphic and 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 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 twosurface \({\mathcal S} = \partial \Sigma\). Thus, the two transversal radiative modes of these fields are encoded in certain fields on \({\mathcal S}\). 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 \({{\mathcal S}_\infty} = {\mathcal N} \cap {{\mathscr I}^ +}\), 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, [496]). 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 [492], reported above, gives just 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 twosurfaces in generic spacetimes as well, which might yield the possibility of introducing the radiative modes quasilocally in general.
13.6 Potential applications in cosmology
The systematic deviation of the observed luminosityredshift values for type Ia supernovae for large red shift from the expected ones in the standard FriedmannRobertsonWalker model is usually interpreted as evidence that the expansion of the universe is accelerating. To generate this acceleration, a hypothetical matter field, the dark matter violating the strong energy condition, is postulated. Here the homogeneity and isotropy of the space, i.e., the use of the FriedmannRobertsonWalker line element, seems to be justified by the isotropy and the thermal nature of the cosmic microwave background radiation. Nevertheless, as is well known, the observed matter distribution is far from being homogeneous. There are huge voids and the matter is distributed as walls between the voids, as in as foam; and hence, the homogeneity of the universe is expected only after an averaging at a larger scale.
However, motivated by quasilocal energymomentum ideas, Wiltshire [547, 548, 551] suggested a new averaging procedure (see also [550, 549]). Since by general relativistic redshift clocks in the voids run significantly faster than in the presence of matter (i.e., in the walls), the average should be taken in the voids and in the walls separately, and the model of the universe is built from these two like Swiss cheese. Then cosmic acceleration is explained only as an apparent phenomenon, due to the naïve averaging above, in which the general relativistic clock effect was not taken into account, and hence, no dark energy is needed. A wellreadable review of the key ideas is [552].
14 Summary: Achievements, Difficulties, and Open Issues
In the previous sections we have tried to give an objective review of the present state of the art. This section is, however, more subjective: We close the present review with a critical discussion, evaluating 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 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 mean the time component of the energymomentum fourvector, i.e., a referenceframedependent quantity, while by mass we mean the length of the energymomentum, i.e., an invariant. In fact, these two have different properties. The quasilocal energy (both for matter fields and for gravity according to the DouganMason definition) is vanishing precisely for the ‘ground state’ of the theory (i.e., for the vanishing energymomentum tensor in the domain of dependence D(Σ) and the flatness of D(Σ), see Sections 2.2.5 and 8.2.3, respectively). 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 only 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, by its strict positivity requirement for nonflat threedimensional 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 [38] (see also [488]) 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 only 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 twosurfaces, 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 the Hawking energy is, in fact, energy rather than mass, so, in principle, one should be able to complete this by a linear momentum to an energymomentum fourvector. One possibility is Eq. (6.2), but, as far as we are aware, its properties have not been investigated. Unfortunately, although the energy can be defined for twosurfaces with nonzero genus, it is not clear how the fourmomentum could be extended for such surfaces. Although Hawking energy is a welldefined twosurface 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 twosurfaces, not quite viable) generalization of the standard round sphere expression (4.8). This view appears to be supported by the fact that Hawking energy has strange properties for nonspherical surfaces, e.g., for twosurfaces 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 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 twosurface twistors play the role of the quasitranslations and quasirotations. The kinematical twistor was calculated for a large class of special twosurfaces and gave acceptable results.
However, the construction is not complete. First, the construction does not work for twosurfaces, whose topology is different from S^{2}, and does not work even for certain topological twospheres for which the twosurface twistor equation admits more than four independent solutions (‘exceptional twosurfaces’). Second, two additional objects, the infinity twistor and a Hermitian inner product on the space of twosurface 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 twosurface cannot be embedded into a conformally flat spacetime. In addition, in 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 do 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 ‘barhook combination’ (see Eq. (7.9)), 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, cannot 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 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 such 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. The properties depend only on the twosurface 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 twosurface. 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 sixrealparameter 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 spin angular momenta (see Section 9.2) in a natural way.
However, in spite of their successes, the DouganMason energymomenta and the spin angular momenta based on Bramson’s superpotential and the holomorphic/antiholomorphic spinor fields have some unsatisfactory properties, as well (see the lists of our expectations in Section 4.3). First, they are defined only for topological twospheres (but not for other topologies, e.g., for the torus S^{1} × S^{1}), and, even for certain topological twospheres, they are not well defined. 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 well defined 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 firstorder linear differential operators acting on spinor fields that have a twodimensional 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 well defined only if the spatial threemomentum part of the BondiSachs fourmomentum is vanishing, i.e., only in the centerofmass frame. (The spatial infinity limit of the spin angular 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 if the gravitational angular momentum also has an orbital part, in which case 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 centerofmass 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 twosurface 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 as to whether the present framework is wide enough to construct reasonable quasilocal energymomenta and angular momenta.
Indeed, the basic requirement automatically yields the boundary condition that the threemetric 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 twosurface 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 twosurface \({\mathcal S}\), but on an arbitrarily chosen boost gauge, interpreted as a ‘fleet of observers t^{a} being at rest with respect to \({\mathcal S}\prime\), as well. 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 twosurface 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 threemanifold ^{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 υ^{ a }υ_{ 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 twosurface into the flat threespace (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.) Similar to the DouganMason definitions, the general BrownYork formulae are valid for arbitrary spacelike twosurfaces, but solutions to the equations defining the reference configuration exist certainly only for topological twospheres with strictly positive intrinsic scalar curvature. Thus, there are exceptional twosurfaces here, too. On the other hand, the BrownYork expressions (both for the flat threespace 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 twosurface in the reference Minkowski spacetime would 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 twosurface into the Minkowski spacetime by preserving its ‘universal structure’; see Section 4.1.4.)
Another objection against the embedding into flat threespace 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’.
Finally, in contrast to the DouganMason definitions, the BrownYork type expressions are well defined 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, arises 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 thermodynamic 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 nondynamic background, just in accordance with the principle of equivalence. If there were some nondynamic 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 be even worse if we used Einstein’s Lagrangian. The corresponding canonical quantities would still be coordinate dependent, though in certain ‘natural’ coordinate systems they yield reasonable results (see, e.g., [7] and references therein).
 4.
\(E({\mathcal S})\) can be thought of as the 0component of some quasilocal energymomentum fourvector, but, 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 fourvector.
 5.
As we will soon see, the leading term of the smallsphere expression of the energymomenta in nonvacuum is of order r^{3}, in vacuum it is of order r^{5}, while those of the relativistic angular momentum are r^{4} and r^{6}, respectively.
 6.
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.
 7.
In the Bondi coordinate system the radial coordinate is the luminosity distance r_{ D }: = −1/p, which tends to the affine parameter r asymptotically.
 8.
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.
 9.
I thank Paul Tod for pointing this out to me.
 10.
I am grateful to Jörg Frauendiener and one of the referees for clarifying this point.
 11.
 12.
Recall that, similarly, we did not have any natural isomorphism between the twosurface twistor spaces, discussed in Section 7.2.1, on different twosurfaces.
 13.
Clearly, for the LudvigsenVickers energymomentum no such ambiguity is present, because the part (8.3) of their propagation law defines a natural isomorphism between the space of the LudvigsenVickers spinors on the different twosurfaces.
 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 [184] gives a clear, 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.
Lau, S.R., personal communication (July 2003)
 18.
According to this view the quasilocal energy is similar to E_{ Σ } of Eq. (2.6), 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 justifies their inclusion in 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 this out 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.
Notes
Acknowledgments
I am grateful to Peter Aichelburg, Thomas Bäckdahl, Herbert Balasin, Robert Bartnik, Robert Beig, Florian Beyer, ChiangMei Chen, Piotr Chruściel, Sergio Dain, Jörg Frauendiener, Helmut Friedrich, Sean Hayward, Jacek Jezierski, Jerzy Kijowski, Stephen Lau, Marc Mars, Lionel Mason, Niall Ó Murchadha, James Nester, Ezra Newman, Alexander Petrov, Walter Simon, George Sparling, Jacek Tafel, Paul Tod, RohSuan Tung, Helmuth Urbantke, Juan Antonio ValienteKroon, James Vickers, MuTao Wang, Jong Yoon and HoiLai Yu for their valuable comments and stimulating questions. Special thanks to Jörg Frauendiener for continuous and fruitful discussions in the last twenty years, and to James Nester for the critical reading of an earlier version of the present manuscript, whose notes and remarks considerably improved its clarity. 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 presented.
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