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.
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.
EnergyMomentum and Angular Momentum of Matter Fields
Energymomentum and angularmomentum density of matter fields
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.
To specify the state of the matter fields operationally, two kinds of devices are needed: the first measures the value of the fields, while the other measures the spatiotemporal location of the first. Correspondingly, the fields on the manifold M of events can be grouped into two sharplydistinguished classes. The first contains the matter field variables, e.g., finitely many (r, s)type tensor fields \({\Phi _N}_{{b_1} \ldots {b_{\mathcal S}}}^{{a_1} \ldots {a_r}}\), whilst the second contains the fields specifying the spacetime geometry, i.e., the metric g_{ab} in Einstein’s theory. Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a Lagrangian \({L_{\rm{m}}} = {L_{\rm{m}}}({g^{ab}},{\Phi _N},{\nabla _e}{\Phi _N}, \ldots, {\nabla _{{e_1} \ldots}}{\nabla _{{e_k}}}{\Phi _N})\). If I_{m}[g^{ab}, Φ_{N}] is the action functional, i.e., the volume integral of L_{m} on some open domain D with compact closure, then the equations of motion are
the EulerLagrange equations. (Here, of course, \(\delta {I_{\rm{m}}}/\delta {\Phi _N}_{b \ldots}^{a \ldots}\) denotes the formal variational derivative of I_{m} with respect to the field variable \({\Phi _N}_{b \ldots}^{a \ldots}\).) The symmetric (or dynamical) energymomentum tensor is defined (and is given explicitly) by
where we introduced the canonical spin tensor
(The terminology will be justified in Section 2.2.) Here \(\Delta _{b{d_1} \ldots {d_q}{h_1} \ldots {h_p}}^{a{c_1} \ldots {c_p}{g_1} \ldots {g_q}}\) is the (p + q + 1, p + q + 1)type invariant tensor, built from the Kronecker deltas, appearing naturally in the expression of the Lie derivative of the (p, q)type tensor fields in terms of the torsion free covariant derivatives: \({{\!\!\!\! L}}_{\rm{K}}\Phi _{d \ldots}^{c \ldots} = {\nabla _{\rm{K}}}\Phi _{d \ldots}^{c \ldots}  {\nabla _a}{K^b}\nabla _{bd \ldots h \ldots}^{ac \ldots g \ldots}\Phi _{g \ldots}^{h \ldots}\). (For the general idea behind the derivation of T_{ab} and Eq. (2.2), see, e.g., Section 3 of [240].)
The canonical Noether current
Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that, for any vector field K^{a} and the corresponding local oneparameter family of diffeomorphisms ϕ_{t}, one has
for some oneparameter family of vector fields \(B_t^e = B_t^e({g^{ab}},{\Phi _N}, \ldots)\). (L_{m} is called diffeomorphism invariant if \({\nabla _e}B_t^e = 0\), e.g., when L_{m} is a scalar.) Let K^{a} be any smooth vector field on M. Then, calculating the divergence ∇_{a}(L_{m}K^{a}) to determine the rate of change of the action functional I_{m} along the integral curves of K^{a}, by a tedious but straightforward computation, one can derive the Noether identity: \({E^N}_{a \ldots}^{b \ldots}{{\!\!\!\! L}_{\rm{K}}}{\Phi _N}_{b \ldots}^{a \ldots} + {1 \over 2}{T_{ab}}{{\!\!\!\! L}_{\rm{K}}}{g^{ab}} + {\nabla _e}{C^e}[{\rm{K]}}\,{\rm{=}}\,{\rm{0}}\), where Ł_{K} denotes the Lie derivative along K^{a}, and C^{e}[K], the Noether current, is given explicitly by
Here Ḃ^{e} is the derivative of \(B_t^e\) with respect to t at t = 0, which may depend on K_{a} and its derivatives, and \({\theta ^a}_b\), the canonical energymomentum tensor, is defined by
Note that, apart from the term Ḃ^{e}, the current C^{e}[K] does not depend on higher than the first derivative of K^{a}, and the canonical energymomentum and spin tensors could be introduced as the coefficients of K_{a} and its first derivative, respectively, in C^{e}[K]. (For the original introduction of these concepts, see [73, 74, 438]. If the torsion \({\Theta ^c}_{ab}\) is not vanishing, then in the Noether identity there is a further term, \({1 \over 2}{S^{ab}}_c{{\!\!\!\! L}_{\rm{K}}}{\Theta ^c}_{ab}\), where the dynamic spin tensor \({S^{ab}}_c\) is defined by \(\sqrt {\vert g\vert} {S^{ab}}_c: = 2\delta {I_{\rm{m}}}/\delta {\Theta ^c}_{ab}\), and the Noether current has a slightly different structure [259, 260].) Obviously, C^{e}[K] is not uniquely determined by the Noether identity, because that contains only its divergence, and any identicallyconserved current may be added to it. In fact, \(B_t^e\) may be chosen to be an arbitrary nonzero (but divergence free) vector field, even for diffeomorphisminvariant Lagrangians. Thus, to be more precise, if Ḃ^{e} = 0, then we call the specific combination (2.3) the canonical Noether current. Other choices for the Noether current may contain higher derivatives of K^{a}, as well (see, e.g., [304]), but there is a specific one containing K^{a} algebraically (see points 3 and 4 below).
However, C^{a}[K] is sensitive to total divergences added to the Lagrangian, and, if the matter fields have gauge freedom (e.g., if the matter is a Maxwell or YangMills field), then in general it is not gauge invariant, even if the Lagrangian is. On the other hand, T^{ab} is gauge invariant and is independent of total divergences added to L_{m} because it is the variational derivative of the gauge invariant action with respect to the metric. Provided the field equations are satisfied, the Noether identity implies [73, 74, 438, 259, 260] that

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.
Hence, T^{ab}K_{b} is also conserved and can equally be considered as a Noether current. (For a formally different, but essentially equivalent, introduction of the Noether current and identity, see [536, 287, 191].)
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.
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.
The definition of quasilocal quantities
To define the quasilocal conserved quantities in Minkowski spacetime, first observe that, for any Killing vector K^{a} ∈ K, the 3form ω_{abc}:= K_{e}T^{ef} ε_{fabc} is closed, and hence, by the triviality of the third de Rham cohomology group, H^{3}(ℝ^{4}) = 0, it is exact: For some 2form ⋃[K]_{ab} we have \({K_e}{T^{ef}}{\varepsilon _{fabc}} = 3{\nabla _{[a}} \cup {[{\bf{K}}]_{bc] \cdot}}\,{\vee ^{cd}}: =  {1 \over 2} \cup {[{\bf{K}}]_{ab}}{\varepsilon ^{abcd}}\) may be called a ‘superpotential’ for the conserved current 3form ω_{abc}. (However, note that while the superpotential for the gravitational energymomentum expressions of Section 3 is a local function of the general field variables, the existence of this ‘superpotential’ is a consequence of the field equations and the Killing nature of the vector field K^{a}. The existence of globallydefined superpotentials that are local functions of the field variables can be proven even without using the Poincaré lemma [535].) If \(\tilde \cup {[{\bf{K}}]_{ab}}\) is (the dual of) another superpotential for the same current ω_{abc}, then by \({\nabla _{[a}}(\cup {[{\bf{K}}]_{bc]}}  \tilde \cup {[{\bf{K}}]_{bc]}}) = 0\) and H^{2}(ℝ^{4}) = 0 the dual superpotential is unique up to the addition of an exact 2form. If, therefore, \({\mathcal S}\) is any closed orientable spacelike twosurface in the Minkowski spacetime then the integral of ⋃[K]_{ab} on \({\mathcal S}\) is free from this ambiguity. Thus, if Σ is any smooth compact spacelike hypersurface with smooth twoboundary \({\mathcal S}\), then
depends only on \({\mathcal S}\). Hence, it is independent of the actual Cauchy surface Σ of the domain of dependence D(Σ) because all the spacelike Cauchy surfaces for D(Σ) have the same common boundary \({\mathcal S}\). Thus, \({Q_{\mathcal S}}[{\bf{K}}]\) can equivalently be interpreted as being associated with the whole domain of dependence D(Σ), and, hence, it is quasilocal in the sense of [232, 231] above. It defines the linear maps \({P_{\mathcal S}}:{\rm{T}} \rightarrow {\rm{{\mathbb R}}}\), and \({J_{\mathcal S}}:{{\rm{R}}_o} \rightarrow {\rm{\mathbb R}}\,{\rm{by}}\,{{\rm{Q}}_{\mathcal S}}[{\bf{K}}] =: {T_{\underline a}}P_{\mathcal S}^{\underline a} + {M_{\underline a \underline b}}J_{\mathcal S}^{\underline a \underline b}\) i.e., they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates \(P_{\mathcal S}^{\underline a}\) and \(J_{\mathcal S}^{\underline a \underline b}\) transform as a Lorentz vector and antisymmetric tensor, respectively. Under the translation \({x^{\underline a}} \mapsto {a^{\underline a}} + {\eta ^{\underline a}}\) of the origin \(P_{\mathcal S}^{\underline a}\) is unchanged, but \(J_{\mathcal S}^{\underline a \underline b}\) transforms as \(J_{\mathcal S}^{\underline a \underline b} \mapsto J_{\mathcal S}^{\underline a \underline b} + 2{\eta ^{[\underline a}}P_{\mathcal S}^{\underline b ]}\). Thus, \(P_{\mathcal S}^{\underline a}\) and \(J_{\mathcal S}^{\underline a \underline b}\) may be interpreted as the quasilocal energymomentum and angular momentum of the matter fields associated with the spacelike twosurface \({\mathcal S}\), or, equivalently, to D(Σ). Then the quasilocal mass and PauliLubanski spin are defined, respectively, by the usual formulae \(m_{\mathcal S}^2: = {\eta _{\underline a \underline b}}P_{\mathcal S}^{\underline a}P_{\mathcal S}^{\underline b}\) and \(S_{\mathcal S}^{\underline a}: = {1 \over 2}{\varepsilon ^{\underline a}}_{\underline b \underline c \underline d}P_{\mathcal S}^{\underline b}J_{\mathcal S}^{\underline c \underline d}\). (If m^{2} ≠ 0, then the dimensionallycorrect definition of the PauliLubanski spin is \({1 \over m}S_{\mathcal S}^{\underline a}\).) As a consequence of the definitions, \({\eta _{\underline a \underline b}}P_{\mathcal S}^{\underline a}S_{\mathcal S}^b = 0\) holds, i.e., if \(P_{\mathcal S}^{\underline a}\) is timelike then \(S_{\mathcal S}^{\underline a}\) is spacelike or zero, but if \(P_{\mathcal S}^{\underline a}\) is null (i.e., \(m_{\mathcal S}^2 = 0\)) then \(S_{\mathcal S}^{\underline a}\) is spacelike or proportional to \(P_{\mathcal S}^{\underline a}\).
Obviously we can form the flux integral of the current T^{ab}ξ_{b} on the hypersurface even if ξ^{a} is not a Killing vector, even in general curved spacetime:
then, however, the integral E_{Σ}[ξ^{a}] does depend on the hypersurface, because it is not connected with the spacetime symmetries. In particular, the vector field ξ^{a} can be chosen to be the unit timelike normal t^{a} of Σ. Since the component μ:= T_{ab}t^{a}t^{b} of the energymomentum tensor is interpreted as the energydensity of the matter fields seen by the local observer t^{a}, it would be legitimate to interpret the corresponding integral E_{Σ}[t^{a}] as ‘the quasilocal energy of the matter fields seen by the fleet of observers being at rest with respect to Σ’. Thus, EΣ[t^{a}] defines a different concept of the quasilocal energy: While that based on \({Q_{\mathcal S}}[{\bf{K}}]\) is linked to some absolute element, namely to the translational Killing symmetries of the spacetime, and the constant timelike vector fields can be interpreted as the observers ‘measuring’ this energy, E_{Σ}[t^{a}] is completely independent of any absolute element of the spacetime and is based exclusively on the arbitrarily chosen fleet of observers. Thus, while \(P_{\mathcal S}^{\underline a}\) is independent of the actual normal t^{a} of \({\mathcal S}\), E_{Σ}[ξ^{a}] (for nonKilling ξ^{a}) depends on t^{a} intrinsically and is a genuine threehypersurface rather than a twosurface integral.
If \(P_b^{\underline a}: = \delta _b^a  {t^a}{t_b}\), the orthogonal projection to Σ, then the part \({j^a}: = P_b^a{T^{bc}}{t_c}\) of the energymomentum tensor is interpreted as the momentum density seen by the observer t^{a}. Hence,
is the square of the mass density of the matter fields, where h_{ab} is the spatial metric in the plane orthogonal to t^{a}. If T^{ab} satisfies the dominant energy condition (i.e., T^{ab}V_{b} is a future directed nonspacelike vector for any future directed nonspacelike vector V^{a}, see, e.g., [240]), then this is nonnegative, and hence,
can also be interpreted as the quasilocal mass of the matter fields seen by the fleet of observers being at rest with respect to Σ, even in general curved spacetime. However, although in Minkowski spacetime E_{Σ}[K] for the four translational Killing vectors gives the four components of the energymomentum \(P_{\mathcal S}^{\underline a}\), the mass M_{Σ} is different from \({m_{\mathcal S}}\). In fact, while \({m_{\mathcal S}}\) is defined as the Lorentzian norm of \(P_{\mathcal S}^{\underline a}\) with respect to the metric on the space of the translations, in the definition of M_{Σ} the norm of the current T^{ab}t_{b} is first taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of more advantageous properties (see Section 2.2.3), we prefer to represent the quasilocal energy(momentum and angular momentum) of the matter fields in the form \({Q_{\mathcal S}}[{\bf{K}}]\) instead of E_{Σ}[ξ^{a}].
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}.
Hamiltonian introduction of the quasilocal quantities
In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see, e.g., [283, 558] and references therein) the configuration and momentum variables, ϕ^{A} and π_{A}, respectively, are fields on a connected threemanifold Σ, which is interpreted as the typical leaf of a foliation Σ_{t} of the spacetime. The foliation can be characterized on Σ by a function N, called the lapse. The evolution of the states in the spacetime is described with respect to a vector field K^{a} = Nt^{a} + N^{a} (‘evolution vector field’ or ‘general time axis’), where t^{a} is the futuredirected unit normal to the leaves of the foliation and N^{a} is some vector field, called the shift, being tangent to the leaves. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Physical states can be only those that are on the constraint surface, specified by the vanishing of certain functions C_{i} = C_{i}(ϕ^{A}, D_{e}ϕ^{A},…, π_{A}, D_{e}π_{A},…), i = 1,…, n, of the canonical variables and their derivatives up to some finite order, where D_{e} is the covariant derivative operator in Σ. Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form
Here dΣ is the induced volume element, the coefficients μ and j_{a} are local functions of the canonical variables and their derivatives up to some finite order, the N^{i}s are functions on Σ, and Z^{a} is a local function of the canonical variables and is a linear function of the lapse, the shift, the functions N^{i}, and their derivatives up to some finite order. The part C_{i}N^{i} of the Hamiltonian generates gauge motions in the phase space, and the functions N^{i} are interpreted as the freely specifiable ‘gauge generators’.
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].^{Footnote 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.
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.
The properties of the quasilocal quantities based on \({Q_{\mathcal S}}[{\bf{K}}]\) in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [488, 492] that

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.
Therefore, the vanishing of the quasilocal energymomentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasilocal mass is equivalent to special configurations representing pure radiation.
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.
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.
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).
On the EnergyMomentum and Angular Momentum of Gravitating Systems
On the gravitational energymomentum and angular momentum density: The difficulties
The root of the difficulties: Gravitational energy in Newton’s theory
In Newton’s theory the gravitational field is represented by a singe scalar field ϕ on the flat 3space Σ ≈ ℝ^{3} satisfying the Poisson equation −h^{ab}D_{a}D_{b}ϕ = 4πGρ. (Here h_{ab} is the flat (negative definite) metric, D_{a} is the corresponding LeviCivita covariant derivative operator and ρ is the (nonnegative) mass density of the matter source.) Hence, the mass of the source contained in some finite threevolume D ⊂ Σ can be expressed as the flux integral of the gravitational field strength on the boundary \({\mathcal S}: = \partial D\)
where v^{a} is the outwarddirected unit normal to \({\mathcal S}\). If \({\mathcal S}\) is deformed in Σ through a sourcefree region, then the mass does not change. Thus, the rest mass of the source is analogous to charge in electrostatics. Following the analogy with electrostatics, we can introduce the energy density and the spatial stress of the gravitational field, respectively, by
Note that since gravitation is always attractive, U is a binding energy, and hence it is negative definite. However, by the GalileoEötvös experiment, i.e., the principle of equivalence, there is an ambiguity in the gravitational force: It is determined only up to an additive constant covector field a_{e}, and hence by an appropriate transformation D_{e}ϕ ↦ D_{e}ϕ + a_{e} the gravitational force D_{e}ϕ at a given point p ∈ Σ can be made zero. Thus, at this point both the gravitational energy density and the spatial stress have been made vanishing. On the other hand, they can be made vanishing on an open subset U ⊂ Σ only if the tidal force, D_{a}D_{b}ϕ, is vanishing on U. Therefore, the gravitational energy and the spatial stress cannot be localized to a point, i.e., they suffer from the ambiguity in the gravitational force above.
In a relativistically corrected Newtonian theory both the internal energy density u of the (matter) source and the energy density U of the gravitational field itself contribute to the source of gravity. Thus (in the traditional units, when c is the speed of light) the corrected field equation could be expected to be the genuinely nonlinear equation
(Note that, together with additional corrections, this equation with the correct sign of U can be recovered from Einstein’s equations applied to static configurations [199] in the first postNewtonian approximation. Note, however, that the theory defined by (3.3) and the usual formula for the force density, is internally inconsistent [221]. A thorough analysis of this theory, and in particular its inconsistency, is given by Giulini [221].) Therefore, by (3.3)
i.e., now it is the energy of the source plus gravity system in the domain D that can be rewritten into the form of a twosurface integral on the boundary of the domain D. Note that the gravitational energy reduces the source term in (3.3) (and hence the energy E_{D} also), and, more importantly, the quasilocal energy E_{D} of the source + gravity system is free of the ambiguity that is present in the gravitational energy density. This in itself already justifies the introduction and use of the quasilocal concept of energy in the study of gravitating systems.
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].
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.^{Footnote 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].
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.
In particular [486], for Hilbert’s secondorder Lagrangian L_{H}:= R/16πG in a fixed local coordinate system {x^{α}} and derivative operator ∂_{μ} instead of ∇_{e}, Eq. (2.4) gives precisely Møller’s energymomentum pseudotensor \({{\rm{M}}^{{\theta ^\alpha}}}\beta\), which was defined originally through the superpotential equation \(\sqrt {\vert g\vert} (8\pi {G_{\rm{M}}}{\theta ^\alpha}_\beta  {G^\alpha}_\beta): = {\partial _{\mu {\rm{M}}}}{\cup _\beta}^{\alpha \mu}\), where \(_{\rm{M}}{\cup _\beta}^{\alpha \mu}: = \sqrt {\vert g\vert} {g^{\alpha \rho}}{g^{\mu \omega}}({\partial _{[\omega}}{g_{\rho ]\beta}})\) is the Møller superpotential [367]. (For another simple and natural introduction of Møller’s energymomentum pseudotensor, see [131].) For the spin pseudotensor, Eq. (2.2) gives
which is, in fact, only pseudotensorial. Similarly, the contravariant form of these pseudotensors and the corresponding canonical Noether current are also pseudotensorial. We saw in Section 2.1.2 that a specific combination of the canonical energymomentum and spin tensors gave the symmetric energymomentum tensor, which is gauge invariant even if the matter fields have gauge freedom, and one might hope that the analogous combination of the energymomentum and spin pseudotensors gives a reasonable tensorial energymomentum density for the gravitational field. The analogous expression is, in fact, tensorial, but unfortunately it is just the negative of the Einstein tensor [486, 487].^{Footnote 3} Therefore, to use the pseudotensors, a ‘natural’ choice for a ‘preferred’ coordinate system would be needed. This could be interpreted as a gauge choice, or a choice for the reference configuration.
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.
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.
Strategies to avoid pseudotensors II: The tetrad formalism
In the tetrad formulation of general relativity, the g_{ab}orthonormal frame fields \(\{E_{\underline a}^a\}, \underline a = 0, \ldots, 3\), are chosen to be the gravitational field variables [533, 314]. Reexpressing the Hilbert Lagrangian (i.e., the curvature scalar) in terms of the tetrad field and its partial derivatives in some local coordinate system, one can calculate the canonical energymomentum and spin by Eqs. (2.4) and (2.2), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [368], the use of the tetrad fields as the field variables instead of the metric makes it possible to introduce a firstorder, scalar Lagrangian for Einstein’s field equations: If \(\gamma _{\underline e \underline b}^{\underline a}: = E_{\underline e}^e\gamma _{e\underline b}^{\underline a}: = E_{\underline e}^e\vartheta _a^{\underline a}{\nabla _e}E_{\underline b}^a\), the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is
(Here \(\left\{{\vartheta _a^{\underline a}} \right\}\) is the oneform basis dual to \(\left\{{E_{\underline a}^a} \right\}\).) Although L depends on the actual tetrad field \(\left\{{E_{\underline a}^a} \right\}\), it is weakly O(1, 3)invariant. Møller’s Lagrangian has a nice uniqueness property [412]: Any firstorder scalar Lagrangian built from the tetrad fields, whose EulerLagrange equations are the Einstein equations, is Møller’s Lagrangian. (Using Dirac spinor variables Nester and Tung found a firstorder spinor Lagrangian [392], which turned out to be equivalent to Møller’s Lagrangian [530]. Another firstorder spinor Lagrangian, based on the use of the twocomponent spinors and the antiselfdual connection, was suggested by Tung and Jacobson [529]. Both Lagrangians yield a welldefined Hamiltonian, reproducing the standard ADM energymomentum in asymptotically flat spacetimes.) The canonical energymomentum θ^{aβ} derived from Eq. (3.5) using the components of the tetrad fields in some coordinate system as the field variables is still pseudotensorial, but, as Møller realized, it has a tensorial superpotential:
The canonical spin turns out to be essentially \({\vee _b}^{ae}\), i.e., a tensor. The tensorial nature of the superpotential makes it possible to introduce a canonical energymomentum tensor for the gravitational ‘field’. Then, the corresponding canonical Noether current C^{a}[K] will also be tensorial and satisfies
Therefore, the canonical Noether current derived from Møller’s tetrad Lagrangian is independent of the background structure (i.e., the coordinate system) that we used to do the calculations (see also [486]). However, C^{a}[K] depends on the actual tetrad field, and hence, a preferred class of frame fields, i.e., an O(1, 3)gauge reduction, is needed. Thus, the explicit background dependence of the final result of other approaches has been transformed into an internal O(1, 3)gauge dependence. It is important to realize that this difficulty always appears in connection with the gravitational energymomentum and angular momentum, at least in disguise. In particular, the Hamiltonian approach in itself does not yield a well defined energymomentum density for the gravitational ‘field’ (see, e.g., [379, 353]). Thus in the tetrad approach the canonical Noether current should be supplemented by a gauge condition for the tetrad field. Such a gauge condition could be some spacetime version of Nester’s gauge conditions (in the form of certain partial differential equations) for the orthonormal frames of Riemannian manifolds [378, 381]. (For the existence and the potential obstruction to the existence of the solutions to this gauge condition on spacelike hypersurfaces, see [384, 196].) Furthermore, since C^{a}[K] + T^{ab}K_{b} is conserved for any vector field K^{a}, in the absence of the familiar Killing symmetries of the Minkowski spacetime it is not trivial to define the ‘translations’ and ‘rotations’, and hence the energymomentum and angular momentum. To make them well defined, additional ideas would be needed. For recent reviews of the tetrad formalism of general relativity, including an extended bibliography, see, e.g., [486, 487, 403, 286].
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.
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.
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].
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.
As is well known, finite and welldefined ADM energymomentum [23, 25, 24, 26] can be associated with any kasymptotically flat spacelike hypersurface, if \(k > {1 \over 2}\), by taking the value on the constraint surface of the Hamiltonian H[K^{a}], given, for example, in [433, 64], with the asymptotic translations K^{a} (see [144, 52, 399, 145]). In its standard form, this is the r → ∞ limit of a twosurface integral of the first derivatives of the induced threemetric h_{ab} and of the extrinsic curvature χ_{ab} for spheres \({\mathcal S_r}\) of large coordinate radius r. Explicitly:
where _{0}D_{e} is the LeviCivita derivative oparator determined by _{0}h_{ab}, and v^{a} is the outward pointing unit normal to \({{\mathcal S}_r}\) and tangent to Σ. The ADM energymomentum, \({P^{\underline a}} = (E,{P^{\rm{i}}}\), is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian fourvector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian coordinates.
The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian analysis of general relativity is based on the 3+1 decomposition of the fields and the spacetime. Thus, it is not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular momentum and 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.
A particularly interesting and useful expression for the ADM energymomentum is possible if the tetrad field is considered to be a frame field built from a normalized spinor dyad \(\{\lambda _A^{\underline A}\}, \underline A = 0,1\), on Σ, which is asymptotically constant (see Section 4.2.3). (Thus, underlined capital Roman indices are concrete name spinor indices.) Then, for the components of the ADM energymomentum in the constant spinor basis at infinity, Møller’s expression yields the limit of
as the twosurface \({\mathcal S}\) is blown up to approach infinity. In fact, to recover the ADM energymomentum in the form (3.10), the spinor fields \(\lambda _A^{\underline A}\) need not be required to form a normalized spinor dyad, it is enough that they form an asymptotically constant normalized dyad, and we have to use the fact that the generator vector field K^{a} has asymptotically constant components \({K^{\underline A {{\underline A}{\prime}}}}\) in the asymptotically constant frame field \(\lambda _{\underline A}^A\bar \lambda _{\underline {{A{\prime}}}}^{{A{\prime}}}\). Thus \({K^a} = {K^{\underline A {{\underline A}{\prime}}}}\lambda _{\underline A}^A\bar \lambda _{\underline A}^{{A{\prime}}}\) can be interpreted as an asymptotic translation. The complexvalued 2form in the integrand of Eq. (3.10) will be denoted by \(u{({\lambda ^{\underline A}},{{\bar \lambda}^{\underline {{B{\prime}}}}})_{ab}}\), and is called the NesterWitten 2form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential, too. In fact, for any two spinor fields α^{A} and β^{A} one has
where \({X_a}: = {\alpha _A}{{\bar \beta}_{{A{\prime}}}}\) and the overline denotes complex conjugation. Thus, apart from the terms in Eq. (3.12) involving ∇_{A′A}α^{A} and \({\nabla _{A{A{\prime}}}}{{\bar \beta}^{{A{\prime}}}}\), the NesterWitten 2form \(u{(\alpha, \bar \beta)_{ab}}\) is just \( {{\rm{i}} \over 2}({\nabla _{[a}}{X_{b]}} + {\rm{i}}{\nabla _{[c}}{X_{d]}}{1 \over 2}{\varepsilon ^{cd}}_{ab})\), i.e., the antiselfdual part of the curl of \( {{\rm{i}} \over 2}{X_a}\) (The original expressions by Witten and Nester were given using Dirac, rather than twocomponent Weyl, spinors [559, 376]. The 2form \(u{(\alpha, \bar \beta)_{ab}}\) in the present form using the twocomponent spinors probably appeared first in [276].) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [444, 445, 559, 376, 273, 427, 300], the simplest and most transparent ones are probably those based on the use of twocomponent spinors: If the dominant energy condition is satisfied on the kasymptotically flat spacelike hypersurface Σ, where \(k > {1 \over 2}\), then the ADM energymomentum is future pointing and nonspacelike (i.e., the Lorentzian length of the energymomentum vector, the ADM mass, is nonnegative), and is null if and only if the domain of dependence D(Σ) of Σ is flat [276, 434, 217, 436, 88]. Its proof may be based on the Sparling equation [476, 175, 426, 358]:
The significance of this equation is that, in the exterior derivative of the NesterWitten 2form, the second derivatives of the metric appear only through the Einstein tensor, thus its structure is similar to that of the superpotential equations in Lagrangian field theory, and \(\Gamma {(\lambda, \mu)_{abc}}\), known as the Sparling 3form, is a homogeneous quadratic expression of the first derivatives of the spinor fields. If the spinor fields λ_{A} and μ_{A} solve the Witten equation on a spacelike hypersurface Σ, then the pullback of \(\Gamma {(\lambda, \bar \mu)_{abc}}\) to Σ is positive definite. This theorem has been extended and refined in various ways, in particular by allowing inner boundaries of Σ that represent future marginally trapped surfaces in black holes [217, 273, 427, 268].
The ADM energymomentum can also be written as the twosphere integral of certain parts of the conformally rescaled spacetime curvature [28, 29, 43]. This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [223]): If \({\mathcal S}\) is any closed spacelike twosurface with area element \(d{\mathcal S}\), then for any tensor fields ω_{ab} = ω_{[ab]} and μ_{ab} = μ_{[ab]} one can form the integral
Since the falloff of the curvature tensor near spatial infinity is r^{−k−2}, the integral \({I_{\mathcal S}}[\omega, \mu ]\) at spatial infinity gives finite value when ω^{ab}μ^{cd} blows up like r^{k} as r → ∞. In particular, for the 1/r falloff, this condition can be satisfied by \({\omega ^{ab}}{\mu ^{cd}} = \sqrt {{\rm{Area(}}{\mathcal S}{\rm{)}}} {{\hat \omega}^{ab}}{{\hat \mu}^{cd}}\), where Area(\(({\mathcal S})\)) is the area of \({\mathcal S}\) and the hatted tensor fields are \({\mathcal O}(1)\).
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.
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.
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].
Without additional conditions on K^{a}, Komar’s expression does not reproduce the BondiSachs energymomentum in nonstationary spacetimes either [557, 223]: For the ‘obvious’ choice for K^{a}(twice of) Komar’s expression yields the NewmanUnti energy. This anomalous behavior in the radiative regime could be corrected in at least two ways. The first is by modifying the Komar integral according to
where ^{⊥}ε_{cd} is the area 2form on the Lorentzian twoplanes orthogonal to \({\mathcal S}\) (see Section 4.1.1) and α is some real constant. For α =1 the integral \({L_{\mathcal S}}[{\bf{K}}]\), suggested by Winicour and Tamburino, is called the linkage [557]. (N.B.: The flux integral of the sum \({C^a}[{\bf{K}}] + {T^a}_b{K^b}\) of Komar’s gravitational and the matter’s currents on some compact spacelike hypersurface Σ with boundary \({\mathcal S}\) is \({1 \over {16\pi G}}\oint {_{\mathcal S}} {\nabla ^{[a}}{K^{b]}}{1 \over 2}{\varepsilon _{abcd}}\), which, for α = 0, is half of the linkage.) In addition, to define physical quantities by linkages associated to a cut of the null infinity one should prescribe how the twosurface \({\mathcal S}\) tends to the cut and how the vector field K^{a} should be propagated from the spacetime to null infinity into a BMS generator [557, 553]. The other way is to consider the original Komar integral (i.e., α = 0) on the cut of infinity in the conformallyrescaled spacetime and while requiring that K^{a} be divergencefree [210]. For such asymptotic BMS translations both prescriptions give the correct expression for the BondiSachs energymomentum.
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].
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.
In their classic paper Bergmann and Thomson [78] raise the idea that while the gravitational energymomentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic O(1, 3) symmetry. Thus, the angular momentum should be analogous with the spin. Based on the tetrad formalism of general relativity and following the prescription of constructing the Noether currents in YangMills theories, Bramson suggested a superpotential for the six conserved currents corresponding to the internal Lorentzsymmetry [107, 108, 109]. (For another derivation of this superpotential from Møller’s Lagrangian (3.5) see [496].) If \(\{\lambda _A^{\underline A}\}, \underline A = 0,1\), is a normalized spinor dyad corresponding to the orthonormal frame in Eq. (3.5), then the integral of the spinor form of the antiselfdual part of this superpotential on a closed orientable twosurface \({\mathcal S}\) is
where ε_{A′B′} is the symplectic metric on the bundle of primed spinors. We will denote its integrand by \(w{({\lambda ^{\underline A}},{\lambda ^{\underline B}})_{ab}}\), and we call it the Bramson superpotential. To define angular momentum on a given cut of the null infinity by the formula (3.16), we should consider its limit when \({\mathcal S}\) tends to the cut in question and we should specify the spinor dyad, at least asymptotically. Bramson’s suggestion for the spinor fields was to take the solutions of the asymptotic twistor equation [106]. He showed that this definition yields a welldefined expression. For stationary spacetimes this reduces to the generally accepted formula (4.15), and the corresponding PauliLubanski spin, constructed from \({\varepsilon ^{\underline {{A{\prime}}} \underline {{B{\prime}}}}}{J^{\underline A \underline B}} + {\varepsilon ^{\underline A \underline B}} + {{\bar J}^{\underline A \underline {{\prime}{B{\prime}}}}}\) and the BondiSachs energymomentum \({P^{\underline A \underline {{A{\prime}}}}}\) (given, for example, in the NewmanPenrose formalism by Eq. (4.14)), is invariant with respect to supertranslations of the cut (‘active supertranslations’). Note that since Bramson’s expression is based on the solutions of a system of partial differential equations on the cut in question, it is independent of the parametrization of the BMS vector fields. Hence, in particular, it is invariant with respect to the supertranslations of the origin cut (‘passive supertranslations’). Therefore, Bramson’s global angular momentum behaves like the spin part of the total angular momentum. For a suggestion based on Bramson’s superpotential at the quasilocal level, but using a different prescription for the spinor dyad, see Section 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.
The necessity of quasilocality for observables in general relativity
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.
Domains for quasilocal quantities
The quasilocal quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:

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).
A typical example of type 3 is any charge integral expression: The quasilocal quantity is the integral of some superpotential 2form built from the data given on the twosurface, as in Eq. (3.10), or the expression \({Q_{\mathcal S}}[{\bf{K}}]\) for the matter fields given by (2.5). An example of type 2 might be the integral of the BelRobinson ‘momentum’ on the hypersurface Σ:
This quantity is analogous to the integral E_{Σ}[ξ^{a}] for the matter fields given by Eq. (2.6) (though, by the remarks on the BelRobinson ‘energy’ in Section 3.1.2, its physical dimension cannot be of energy). If ξ^{a} is a futurepointing nonspacelike vector then E_{Σ}[ξ^{a}] ≥ 0. Obviously, if such a quantity were independent of the actual hypersurface Σ, then it could also be rewritten as a charge integral on the boundary ∂Σ. The gravitational Hamiltonian provides an interesting example for the mixture of type 2 and 3 expressions, because the form of the Hamiltonian is the threesurface integral of the constraints on Σ and a charge integral on its boundary ∂Σ, and thus, if the constraints are satisfied then the Hamiltonian reduces to a charge integral. Finally, an example of type 1 might be
the infimum of the ‘quasilocal BelRobinson energies’, where the infimum is taken on the set of all the Cauchy surfaces Σ for D with given boundary ∂Σ. (The infimum always exists because the BelRobinson ‘energy density’ T_{abcd}t^{a}t^{b}t^{c}t^{d} is nonnegative.) Quasilocality in any of these three senses is compatible with the quasilocality of Haag and Kastler [231, 232]. The specific quasilocal energymomentum constructions provide further examples both for chargeintegraltype expressions and for those based on spacelike hypersurfaces.
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.
Since the global energymomentum and angular momentum of asymptotically flat spacetimes can be written as twosurface integrals at infinity (and, as we saw in Section 3.1.1 that the mass of the source in Newtonian theory, and as we will see in Section 7.1.1 that both the energymomentum and angular momentum of the source in the linearized Einstein theory can also be written as twosurface integrals), the twosurface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasilocal energymomentum and angular momentum as twosurface observables, then three things must be specified:

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).
In certain approaches the definition of the ‘quasisymmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.
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.
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].
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].
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 \({\mathcal S}\).
The other connection, Δ_{e}, is analogous to the Sen connection [447], and is defined simply by \({\Delta _e}{X^a}: = \Pi _e^f{\Delta _f}{X^a}\). This annihilates only the fiber metric, but not the projection. The difference of the connections Δ_{e} and δ_{e} turns out to be just the extrinsic curvature tensor: \({\Delta _e}{X^a} = {\delta _e}{X^a} + {Q^a}_{eb}{X^b}  {X^b}{Q_{be}}^a\). Here \({Q^a}_{eb}: =  \Pi _c^a{\Delta _e}\Pi _b^c = {\tau ^a}_e{t_b}  {v^a}_e{\upsilon _b}\), and \({\tau _{ab}}: = \Pi _a^c\Pi _b^d{\nabla _c}{t_d}\) and \({v_{ab}}: = \Pi _a^c\Pi _b^d{\nabla _c}{\upsilon _d}\) are the standard (symmetric) extrinsic curvatures corresponding to the individual normals t_{a} and v_{a}, respectively. The familiar expansion tensors of the futurepointing outgoing and ingoing null normals, l^{a} := t^{a} + υ^{a} and \({n^a}: = {1 \over 2}({t^a}  {\upsilon ^a})\), respectively, are θ_{ab} = Q_{abc}l^{c} and θ′_{ab} = Q_{abc}n^{c}, and the corresponding shear tensors σ_{ab} and σ′_{ab} are defined by their tracefree part. Obviously, τ_{ab} and ν_{ab} (and hence the expansion and shear tensors θ_{ab}, θ′_{ab}, σ_{ab}, and σ′_{ab}) are boostgaugedependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination \({Q^a}_{eb}\) is boostgauge invariant. In particular, it defines a natural normal vector field to \({\mathcal S}\) as \({Q_b}: = {Q^a}_{ab} = \tau {t_b}  v{\upsilon _b} = {\theta {\prime}}{l_b} + \theta {n_b}\) and θ′ are the relevant traces. Q_{a} is called the mean extrinsic curvature vector of \({\mathcal S}\). If \({{\tilde Q}_b}:{= ^ \bot}{\varepsilon ^a}_b{Q^b} = v{t_b}  \tau {\upsilon _b} =  {\theta {\prime}}{l_a} + \theta {n_a}\), called the dual mean curvature vector, then the norm of Q_{a} and Q_{a} is \({Q_a}{Q_b}{g^{ab}} =  {{\tilde Q}_a}{{\tilde Q}_b}{g^{ab}} = {\tau ^2}  {v^2} = 2\theta {\theta {\prime}}\), and they are orthogonal to each other: \({Q_a}{Q_b}{g^{ab}} = 0\). It is easy to show that \({\Delta _a}{{\tilde Q}^a} = 0,\,{\rm{i}}{\rm{.e}}{\rm{.,}}\,{{\tilde Q}^a}\) is the uniquely pointwisedetermined direction orthogonal to the twosurface in which the expansion of the surface is vanishing. If Q_{a} is not null, then \(\{{Q_a},{{\tilde Q}_a}\}\) defines an orthonormal frame in the normal bundle (see, e.g., [14]). If Q_{a} is nonzero, but (e.g., futurepointing) null, then there is a uniquely determined null normal S_{a} to \({\mathcal S}\), such that Q_{a}S^{a} = 1, and hence, {Q_{a}, S_{a}} is a uniquely determined null frame. Therefore, the twosurface admits a natural gauge choice in the normal bundle, unless Q_{a} is vanishing. Geometrically, Δ_{e} is a connection coming from a connection on the O(1, 3)principal fiber bundle of the g_{ab}orthonormal frames. The curvature of the connections δ_{e} and Δ_{e}, respectively, are
where \(^{\mathcal S}R\) is the curvature scalar of the familiar intrinsic LeviCivita connection of \(^{\mathcal S}R\). The curvature of Δ_{e} is just the pullback to \({\mathcal S}\) of the spacetime curvature 2form: \({F^a}_{bcd} = {R^a}_{bef}\Pi _c^e\Pi _d^f\). Therefore, the wellknown Gauss, CodazziMainardi, and Ricci equations for the embedding of \({\mathcal S}\) in M are just the various projections of Eq. (4.2).
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.
The spinor bundle
The connections δ_{e} and Δ_{e} determine connections on the pullback \({{\rm{S}}^A}({\mathcal S})\) to \({\mathcal S}\) of the bundle of unprimed spinors. The natural decomposition \({{\rm{V}}^a}({\mathcal S}) = T{\mathcal S} \oplus N{\mathcal S}\) defines a chirality on the spinor bundle \({{\rm{S}}^A}({\mathcal S})\) in the form of the spinor \({\gamma ^A}_B: = 2{t^{A{A{\prime}}}}{\upsilon _{B{A{\prime}}}}\), which is analogous to the γ_{5} matrix in the theory of Dirac spinors. Then, the extrinsic curvature tensor above is a simple expression of \({Q^A}_{eB}: = {1 \over 2}({\Delta _e}{\gamma ^A}_C){\gamma ^C}_B\) and \({\gamma ^A}_B\) (and their complex conjugate), and the two covariant derivatives on \({{\rm{S}}^A}({\mathcal S})\) are related to each other by \({\Delta _e}{\lambda ^A} = {\delta _e}{\lambda ^A} + {Q^A}_{eB}{\lambda ^B}\). The curvature \({F^A}_{Bcd}\) of Δ_{e} can be expressed by the curvature \({f^A}_{Bcd}\) of δ_{e}, the spinor \({Q^A}_{eB}\), and its δ_{e}derivative. We can form the scalar invariants of the curvatures according to
f is four times the complex Gauss curvature [425] of \({\mathcal S}\), by means of which the whole curvature \({f^A}_{Bcd}\) can be characterized: \({f^A}_{Bcd} =  {i \over 4}f{\gamma ^A}_B{\varepsilon _{cd}}\) If the spacetime is space and time orientable, at least on an open neighborhood of \({\mathcal S}\), then the normals t_{a} and v_{a} can be chosen to be globally well defined, and hence, \(N{\mathcal S}\) is globally trivializable and the imaginary part of f is a total divergence of a globally welldefined vector field.
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 \({Q^a}_{eb}\).
Curvature identities
The complete decomposition of Δ_{AA′}λ_{B} into its irreducible parts gives Δ_{AA′}λ^{A}, the DiracWitten operator, and \({{\mathcal T}_{{E\prime}EA}}^B{\lambda _B}: = {\Delta _{{E\prime}(E}}{\lambda _{A)}} + {1 \over 2}\gamma EA{\gamma ^{CD}}{\Delta _{{E\prime}C}}{\lambda _D}\), the twosurface twistor operator. The former is essentially the antisymmetric part Δ_{A′[A}λ_{B]}, the latter is the symmetric and (with respect to the complex metric γ_{AB} tracefree part of the derivative. (The trace \({\gamma ^{AB}}{\Delta _{{A\prime}A}}{\lambda _B}\) can be shown to be the DiracWitten operator, too.) A SenWittentype identity for these irreducible parts can be derived. Taking its integral one has
where λ_{A} and μ_{A} are two arbitrary spinor fields on \({\mathcal S}\), and the righthand side is just the charge integral of the curvature \({F^A}_{Bcd}\) on \({\mathcal S}\).
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.
Irreducible parts of the derivative operators
Using the projection operators \({\pi ^{\pm A}}_B: = {1 \over 2}(\delta _B^A \pm {\gamma ^A}_B)\), the irreducible parts Δ_{a′a}λ^{A} and \({{\mathcal T}_{E \prime EA}}^B{\lambda _B}\) can be decomposed further into their righthanded and lefthanded parts. In the GHP formalism these chiral irreducible parts are
where λ:= (λ_{0},λ_{1}) and the spinor components are defined by λ_{A} =: λ_{1}o_{A} − λ_{0}ι_{A}. The various firstorder linear differential operators acting on spinor fields, e.g., the twosurface twistor operator, the holomorphy/antiholomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [106], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [58, 490, 496].
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.
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.
Round spheres
If the spacetime is spherically symmetric, then a twosphere, which is a transitivity surface of the rotation group, is called a round sphere. Then in a spherical coordinate system (t, r, θ, ϕ) the spacetime metric takes the form g_{ab} = diag(exp(2γ), − exp(2α), −r^{2}, −r^{2} sin^{2} θ), where γ and α are functions of t and r. (Hence, r is called the areacoordinate.) Then, with the notation of Section 4.1, one obtains \({R_{abcd}}{\varepsilon ^{ab}}{\varepsilon ^{cd}} = {4 \over {{r^2}}}(1  \exp ( 2\alpha))\). Based on the investigations of Misner, Sharp, and Hernandez [365, 267], Cahill and McVitte [122] found
to be an appropriate (and hence, suggested to be the general) notion of energy, the MisnerSharp energy, contained in the twosphere \({\mathcal S}: = \{t = const.,\,r = const.\}\). (For another expression of E(t, r) in terms of the norm of the Killing fields and the metric, see [577].) In particular, for the ReissnerNordström solution GE(t, r) = m − e^{2}/2r, while for the isentropic fluid solutions \(E(t,\,r) = 4\pi \int\nolimits_0^r {{r\prime^{2}}\mu (t,\,{r\prime})d{r\prime}}\), where and are the usual parameters of the ReissnerNordstroïm solutions and μ is the energy density of the fluid [365, 267] (for the static solution, see, e.g., Appendix B of [240]). Using Einstein’s equations, simple equations can be derived for the derivatives ∂_{t}E(t, r) and ∂_{t}E(t, r), and if the energymomentum tensor satisfies the dominant energy condition, then ∂_{r}E(t, r) > 0. Thus, E(t, r) is a monotonic function of r, provided r is the areacoordinate. Since, by spherical symmetry all the quantities with nonzero spin weight, in particular the shears σ and σ′, are vanishing and ψ_{2} is real, by the GHP form of Eqs. (4.3), (4.4) the energy function E(t, r) can also be written as
Any of these expressions is considered to be the ‘standard’ definition of the energy for round spheres.^{Footnote 4} The last of these expressions does not depend on whether r is an areacoordinate or not.
\(E({\mathcal S})\) 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].
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.
Completing l^{a} to get a NewmanPenrose complex null tetrad \(\{{l^a},{n^a},{m^a},{{\bar m}^a}\}\) such that the complex null vectors m^{a} and \({{\bar m}^a}\) are tangent to the twosurfaces \({{\mathcal S}_r}\), the components of the metric and the spin coefficients with respect to this basis can be expanded as a series in r. If, in addition, the spinor constituent o^{A} of l^{a} = o^{A}ō^{A′} is required to be parallelly propagated along l^{a}, then the tetrad becomes completely fixed, yielding the vanishing of several (combinations of the) spin coefficients. Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the metric q_{ab} on \({{\mathcal S}_r}\), the GHP spin coefficients ρ, σ, τ, p′, σ′ and β, and the higherorder expansion coefficients of the curvature in terms of the lowerorder curvature components at p. Hence, the expression of any quasilocal quantity \({Q_{{{\mathcal S}_r}}}\) for the small sphere \(_{{{\mathcal S}_r}}\) can be expressed as a series of r,
where the expansion coefficients Q^{(k)} are still functions of the coordinates, \((\zeta, \,\bar \zeta)\) or (θ,ϕ), on the unit sphere \({\mathcal S}\). If the quasilocal quantity Q is spacetimecovariant, then the unit sphere integrals of the expansion coefficients Q^{(k)} must be spacetime covariant expressions of the metric and its derivatives up to some finite order at p and the ‘time axis’ t^{a}. The necessary degree of the accuracy of the solution of the GHP equations depends on the nature of \({Q_{{{\mathcal S}_r}}}\) and on whether the spacetime is Ricciflat in the neighborhood of p or not.^{Footnote 5} These solutions of the GHP equations, with increasing accuracy, are given in [275, 313, 118, 494].
Obviously, we can calculate the smallsphere limit of various quasilocal quantities built from the matter fields in the Minkowski spacetime, as well. In particular [494], the smallsphere expressions for the quasilocal energymomentum and the (antiselfdual part of the) quasilocal angular momentum of the matter fields based on \({Q_{\mathcal S}}[{\bf{K}}]\), are, respectively,
where \(\{{\mathcal E}{A \over A}\}, \,\underline A = 0,\,1\), is the ‘Cartesian spin frame’ at p and the origin of the Cartesian coordinate system is chosen to be the vertex p. Here \(K_a^{\underline A \,{{\underline B}\prime}} = {\mathcal E}_A^{\underline A}\bar {\mathcal E}_{{A\prime}}^{{{\underline B}\prime}}\) can be interpreted as the translation oneforms, while \(K_a^{\underline A \,\underline B} = r{t_{{A\prime}}}^E{\mathcal E}_{(E}^{\underline A}{\mathcal E}_{A)}^{\underline B}\) is an average on the unit sphere of the boostrotation Killing oneforms that vanish at the vertex p. Thus, \(P_{{{\mathcal S}_r}}^{\underline A \,{{\underline B}\prime}}\) and \(J_{{{\mathcal S}_r}}^{\underline A \,\underline B}\) are the threevolume times the energymomentum and angular momentum density with respect to p, respectively, that the observer with fourvelocity t^{a} sees at p.
Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in a large class of quasilocal spacetime covariant energymomentum and angular momentum expressions. In fact, if \({Q_{\mathcal S}}\) is any coordinateindependent quasilocal quantity built from the first derivatives ∂_{μ}g_{a}β of the spacetime metric, then in its expansion the difference of the power of r and the number of the derivatives in every term must be one, i.e., it must have the form
where Q_{i}[A, B, …], i = 2, 3, …, are scalars. They are polynomial expressions of t^{a}, g_{ab} and ε_{abcd} at the vertex p, and they depend linearly on the tensors that are constructed at p from \({g_{\alpha \beta}},\,{g^{\alpha \beta}}\) and linearly from the coordinatedependent quantities A, B, …. Since there is no nontrivial tensor built from the first derivative ∂_{μ}g_{αβ} and g_{αβ}, the leading term is of order r^{3}. Its coefficient Q_{3}[∂^{2}g, (dg)^{2}] must be a linear expression of R_{ab} and C_{abcd}, and polynomial in t^{a}, g_{ab} and ε_{abcd}. In particular, if \({Q_{\mathcal S}}\) is to represent energymomentum with generator K^{c} at p, then the leading term must be
for some unspecified constants a, b, and c, where \(P_b^a: = \delta _b^a  {t^a}{t_b}\), the projection to the subspace orthogonal to t^{a}. If, in addition to the coordinateindependence of \({Q_{\mathcal S}}\), it is Lorentzcovariant, i.e., it does not, for example, depend on the choice for a normal to \({\mathcal S}\) (e.g., in the smallsphere approximation on t^{a}) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer t^{a} in the same way. Therefore, a = c, in which case the first and the third terms can in fact be written as r^{3} at^{a}G_{ab}K^{b}. Then, comparing Eq. (4.11) with Eq. (4.9), we see that a = −1/(6G), and hence the term r^{3} bRt_{a}K^{a} would have to be interpreted as the contribution of the gravitational ‘field’ to the quasilocal energymomentum of the matter + gravity system. However, this contributes only to energy, but not to linear momentum in any frame defined by the observer t^{a}, even in a general spacetime. This seems to be quite unacceptable. Furthermore, even if the matter fields satisfy the dominant energy condition, \({Q_{{{\mathcal S}_r}}}\) given by Eq. (4.11) can be negative, even for c = a, unless b = 0. Thus, in the leading r^{3} order in nonvacuum, any coordinate and Lorentzcovariant quasilocal energymomentum expression which is nonspacelike and future pointing, should be proportional to the energymomentum density of the matter fields seen by the observer t^{a} times the Euclidean volume of the threeball of radius r. No contribution from the gravitational ‘field’ is expected at this order. In fact, this result is compatible the with the principle of equivalence, and the particular results obtained in the relativistically corrected Newtonian theory (considered in Section 3.1.1) and in the weak field approximation (see Sections 4.2.5 and 7.1.1 below). Interestingly enough, even for a timelike Killing field K^{e}, the well known expression of Komar does not satisfy this criterion. (For further discussion of Komar’s expression see also Section 12.1.)
If the neighborhood of p is vacuum, then the r^{3}order term is vanishing, and the fourthorder term must be built from ∇_{e}C_{abcd}. However, the only scalar polynomial expression of t^{a}, g_{ab}, ε_{abcd}, ∇_{e}C_{abcd} and the generator vector K^{a}, depending linearly on the latter two, is the zero tensor field. Thus, the r^{4}order term in vacuum is also vanishing. At the fifth order the only nonzero terms are quadratic in the various parts of the Weyl tensor, yielding
for constants a, b, c, and d, where E_{ab}: = C_{aebf}t^{e}t^{f} is the electric part and \({H_{ab}}: = {\ast} {C_{aebf}}{t^e}{t^f}: = {1 \over 2}{\varepsilon _{ae}}^{cd}{C_{cdbf}}{t^e}{t^f}\) is the magnetic part of the Weyl curvature, and ε_{abc}:=ε_{abcd}t^{d} is the induced volume 3form. However, using the identities C_{abcd}C^{abcd} = 8(E_{ab}E^{ab} − H_{ab}H^{ab}), C_{abcd} * C^{abcd} = 16E_{ab}H^{ab}, 4T_{abcd}t^{a}t^{b}tH^{d} = E_{ab}E^{ab} + H_{ab}H^{ab} and \(2{T_{abcd}}{t^a}{t^b}{t^c}P_e^d = {E_{ab}}{H^a}_c{\varepsilon ^{bc}}_e\), we can rewrite the above formula to be
Again, if \({Q_{\mathcal S}}\) does not depend on t^{a} intrinsically, then d = (a + b), in which case the first and the fourth terms together can be written into the Lorentz covariant form 2r^{5} dT_{abcd}t^{a}t^{b}t^{c}K^{d}. In a general expression the curvature invariants C_{abcd}C^{abcd} and C_{abcd} * C^{abcd} may be present. Since, however, E_{ab} and H_{ab} at a given point are independent, these invariants can be arbitrarily large positive or negative, and hence, for a ≠ b or c ≠ 0 the quasilocal energymomentum could not be future pointing and nonspacelike. Therefore, in vacuum in the leading r^{5} order any coordinate and Lorentzcovariant quasilocal energymomentum expression, which is nonspacelike and future pointing must be proportional to the BelRobinson ‘momentum’ T_{abcd}t^{a}t^{b}t^{c}.
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.
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.^{Footnote 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 \({\mathcal O}({r^{ 2}})\).
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’.^{Footnote 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].
The BondiSachs energymomentum given in the NewmanPenrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut \({\mathcal S}\) of a combination of the leading term \(\psi _2^0\) of the Weyl spinor component \({\psi _2}\), the asymptotic shear σ^{0} and its uderivative, weighted by the first four spherical harmonics (see, for example, [393, 426]):
where \(\lambda _0^{\underline A}: = \lambda _A^{\underline A}{o^A},\underline A = 0,1\), are the o^{A}component of the vectors of a spin frame in the space of the asymptotic spinors. (For the various realizations of these spinors see, e.g., [496].) The minimal assumptions on the physical Ricci tensor that already ensure that the BondiSachs energymomentum and Bondi’s massloss are well defined are determined by Tafel [505]. The expression of the BondiSachs energymomentum in terms of the conformal factor is also given there.
Similarly, the various definitions for angular momentum at null infinity could be rewritten in this formalism. Although there is no generally accepted definition for angular momentum at null infinity in general spacetimes, in stationary and in axisymmetric spacetimes there is. The former is the unit sphere integral on the cut \({\mathcal S}\) of the leading term of the Weyl spinor component \({{\bar \psi}_{{1{\prime}}}}\), weighted by appropriate (spinweighted) spherical harmonics:
In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [109]. If the spacetime is axisymmetric, then the generally accepted definition of angular momentum is that of Komar with the numerical coefficient \({1 \over {16\pi G}}\) (rather than \({1 \over {8\pi G}}\)) and α = 0 in (3.15). This view is supported by the partial results of a quasilocal canonical analysis of general relativity given in [499], too.
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].
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.
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.
General expectations
In nongravitational physics the notions of conserved quantities are connected with symmetries of the system, and they are introduced through some systematic procedure in the Lagrangian and/or Hamiltonian formalism. In general relativity the total energymomentum and angular momentum are twosurface observables, thus, we concentrate on them even at the quasilocal level. These facts motivate our three a priori 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’.
To see that these conditions are nontrivial, let us consider the expressions based on the linkage integral (3.15). \({L_{\mathcal S}}[{\bf{K}}]\) does not satisfy the first part of our first requirement. In fact, it depends on the derivative of the normal components of K^{a} in the direction orthogonal to \({\mathcal S}\) for any value of the parameter α. Thus, it depends not only on the geometry of \({\mathcal S}\) and the vector field K^{a} given on the twosurface, but on the way in which K^{a} is extended off the twosurface. Therefore, \({L_{\mathcal S}}[{\bf{K}}]\) is ‘less quasilocal’ than \({A_{\mathcal S}}[\omega ]\) or \({H_{\mathcal S}}[\lambda, \bar \mu ]\) that will be introduced in Sections 7.2.1 and 7.2.2, respectively.
We will see that the Hawking energy satisfies our first requirement, but not the second and the third ones. The Komar integral (i.e., half of the linkage for α = 0) has the form of the charge integral of a superpotential, \({1 \over {16\pi G}}\oint\nolimits_{\mathcal S} {{\nabla ^{[a}}{K^{b]}}{1 \over 2}{\varepsilon _{abcd}}}\), i.e., it has a Lagrangian interpretation. The corresponding conserved Komarcurrent was defined by 8 \(8\pi G{C^a}[{\bf{K}}]: = {G^a}_b{K^b} + {\nabla _b}{\nabla ^{[a}}{K^{b]}}\). However, its flux integral on some compact spacelike hypersurface with boundary \({\mathcal S}: = \partial \Sigma\) cannot be a Hamiltonian on the ADM phase space in general. In fact, it is
Here c and c_{a} are, respectively, the Hamiltonian and momentum constraints of the vacuum theory, t^{a} is the futuredirected unit normal to Σ, v^{a} is the outwarddirected unit normal to \({\mathcal S}\) in Σ, and N and N^{a} are the lapse and shift part of K^{a}, respectively, defined by K^{a} =: Nt^{a} + N^{a}. Thus, _{K}H[K] is a welldefined function of the configuration and velocity variables (N, N^{a}, h_{ab}) and (Ṅ, Ṅ^{a}, ḣ_{ab}), respectively. However, since the velocity Ṅ^{a} cannot be expressed by the canonical variables (see e.g. [558, 63]), _{K}H[K] can be written as a function on the ADM phase space only if the boundary conditions at ∂Σ ensure the vanishing of the integral of v_{a}Ṅ^{a}/N.
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.
One such list for the energymomentum and mass, based mostly on [176, 143] and the properties of the quasilocal energymomentum of the matter fields of Section 2.2, might be the following:

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’).

1.5
For 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 \(\sqrt {{\rm{Area(}}{\mathcal S}{\rm{)/16}}\pi {G^2}}\).
For a different view on the positivity of the quasilocal energy see [391]. Item 1.7 is motivated by the expectation that the quasilocal mass associated with the apparent horizon of a black hole (i.e., the outermost marginallytrapped surface in a spacelike slice) be just the irreducible mass [176, 143].
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.
Similarly, any reasonable quasilocal angular momentum expression \(J_{\mathcal S}^{\underline a \underline b}\) may be expected to satisfy the following:

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’).
Since there is no generally accepted definition for the angular momentum at null infinity, we cannot expect anything definite there in nonstationary, nonaxisymmetric spacetimes. Similarly, there are inequivalent suggestions for the centerofmass at spatial infinity (see Sections 3.2.2 and 3.2.4).
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.
The Bartnik Mass and its Modifications
The Bartnik mass
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.^{Footnote 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.
Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding Σ from the outside. This led Bartnik [54, 53] to formulate his suggestion for the quasilocal mass of Σ. He concentrated on timesymmetric data sets (i.e., those for which the extrinsic curvature η_{ab} is vanishing), when the horizon appears to be a minimal surface of topology S^{2} in \({\hat \Sigma}\) (see, e.g., [213]), and the dominant energy condition is just the requirement of the nonnegativity of the scalar curvature of the spatial metric: R ≥ 0. Thus, if \({{\mathcal E}_0}(\Sigma)\) denotes the set of asymptotically flat Riemannian geometries \(\hat \Sigma = (\hat \Sigma, {{\hat h}_{ab}})\) with nonnegative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is
The ‘nohorizon’ condition on \({\hat \Sigma}\) implies that topologically Σ is a threeball. Furthermore, the definition of \({{\mathcal E}_0}(\Sigma)\) in its present form does not allow one to associate the Bartnik mass to those threegeometries (Σ, h_{ab}) that contain minimal surfaces inside Σ. Although formally the maximal twosurfaces inside Σ are not excluded, any asymptotically flat extension of such a Σ would contain a minimal surface. In particular, the sphericallysymmetric threegeometry, with line element dl^{2} = − dr^{2} − sin^{2} r(dθ^{2} + sin^{2} θ dϕ^{2}) with (θ, ϕ) ∈ S^{2} and r ∈ [0, r_{0}], π/2 < r_{0} < π, has a maximal twosurface at r = π/2, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than 4π sin^{2} r_{0}. Thus, the Bartnik mass (according to the original definition given in [54, 53]) cannot be associated with every compact timesymmetric data set (Σ, h_{ab}), even if Σ is topologically trivial. Since for 0 < r_{0} < π/2 this data set can be extended without any difficulty, this example shows that m_{B} is associated with the threedimensional data set Σ, and not only to the twodimensional boundary ∂Σ.
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.)
Bartnik suggests a definition for the quasilocal mass of a spacelike twosurface \({\mathcal S}\) (together with its induced metric and the two extrinsic curvatures), as well [54]. He considers those globallyhyperbolic spacetimes \(\hat M: = (\hat M,{{\hat g}_{ab}})\) that satisfy the dominant energy condition, admit an asymptotically flat (metricallycomplete) Cauchy surface \({\hat \Sigma}\) with finite ADM energy, have no event horizon and in which \({\mathcal S}\) can be embedded with its first and second fundamental forms. Let \({{\mathcal E}_0}({\mathcal S})\) denote the set of these spacetimes. Since the ADM energy \({E_{{\rm{ADM}}}}(\hat M)\) is nonnegative for any \(\hat M \in \;{{\mathcal E}_0}({\mathcal S})\) (and is zero precisely for flat \({\hat M}\)), the infimum
exists and is nonnegative. Although it seems plausible that m_{B}(∂Σ) is only the ‘spacetime version’ of m_{B}(Σ), without the precise form of the nohorizon conditions in \({{\mathcal E}_0}(\Sigma)\) and that in \({{\mathcal E}_0}({\mathcal S})\) they cannot be compared, even if the extrinsic curvature were allowed in the extensions \({\hat \Sigma}\) of Σ.
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 \(E({\mathcal S})\).
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.
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 \({\rm{Area}}({\mathcal S}) \leq {\rm{Area}}(\tilde {\mathcal S})\).
Let \({\mathcal S}\) be outerminimizing, and let \({\mathcal E}({\mathcal S})\) denote the set of extensions of \({\mathcal S}\) in which \({\mathcal S}\) is still outerminimizing, and \({\mathcal F}({\mathcal S})\) denote the set of fillins of \({\mathcal S}\). If \({{\hat \Sigma}_{\rm{f}}} \in {\mathcal F}({\mathcal S})\) and \({A_{{{\hat \Sigma}_{\rm{f}}}}}\) denotes the infimum of the area of the twosurfaces enclosing all the ends of \({{\hat \Sigma}_{\rm{f}}}\) except the outer one, then Bray defines the outer and inner mass, \({m_{{\rm{out}}}}({\mathcal S})\) and \({m_{{\rm{in}}}}({\mathcal S})\), respectively, by
\({m_{{\rm{out}}}}({\mathcal S})\) deviates slightly from Bartnik’s mass (5.1) even if the latter would be defined for nontimesymmetric data sets, because Bartnik’s ‘nohorizon condition’ excludes apparent horizons from the extensions, while Bray’s condition is that \({\mathcal S}\) be outerminimizing.
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].
The Hawking Energy and its Modifications
The Hawking energy
The definition
Studying the perturbation of the dustfilled k = −1 FriedmannRobertsonWalker spacetimes, Hawking found that
behaves as an appropriate notion of energy surrounded by the spacelike topological twosphere \({\mathcal S}\) [236]. Here we used the GaussBonnet theorem and the GHP form of Eqs. (4.3) and (4.4) for F to express ρρ′ by the curvature components and the shears. Thus, Hawking energy is genuinely quasilocal.
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).
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.
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.^{Footnote 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.^{Footnote 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].
Two generalizations
Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of BondiSachs energymomentum are related to Bondi energy:
where \({W^{\underline a}},\,a = 0,\, \ldots, \,3\), are essentially the first four spherical harmonics:
Here ζ and \({\bar \zeta}\) are the standard complex stereographic coordinates on \({\mathcal S} \approx {S^2}\).
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.
The Geroch energy
The definition
Suppose that the twosurface \({\mathcal S}\) for which E_{H} is defined is embedded in the spacelike hypersurface Σ. Let χ_{ab} be the extrinsic curvature of Σ in M and k_{ab} the extrinsic curvature of \(\Sigma\) in Σ. (In Section 4.1.2 we denote the latter by ν_{ab}.) Then 8ρρ′ = (χ_{ab}q^{ab})^{2} − (k_{ab}q^{ab})^{2}, by means of which
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].
Monotonicity properties
The Geroch energy has interesting positivity and monotonicity properties along a special flow in Σ [207, 291]. This flow is the inverse mean curvature flow defined as follows. Let t: Σ → ℝ be a smooth function such that

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}.
Let n be the lapse function of this foliation, i.e., if v^{a} is the outward directed unit normal to \({{\mathcal S}_t}\) in Σ, then nv^{a}D_{a}t = 1. Denoting the integral on the righthand side in Eq. (6.4) by W_{t}, we can calculate its derivative with respect to t. In general this derivative does not seem to have any remarkable properties. If, however, the foliation is chosen in a special way, namely if the lapse is just the inverse mean curvature of the foliation, n = 1/k where k ≔ k_{ab}q^{ab}, and furthermore Σ is maximal (i.e., χ = 0) and the energy density of the matter is nonnegative, then, as shown by Geroch [207], W_{t} ≥ 0 holds. Jang and Wald [291] modified the foliation slightly, such that t ∈ [0, ∞), and the surface \({{\mathcal S}_0}\) was assumed to be future marginally trapped (i.e., ρ = 0 and ρ′ ≥ 0). Then they showed that, under the conditions above, \(\sqrt {{\rm{Area}}({{\mathcal S}_0})} {W_0} \leq \sqrt {{\rm{Area}}({{\mathcal S}_t})} {W_t}\). Since \({E_G}({{\mathcal S}_t})\) tends to the ADM energy as t → ∞, these considerations were intended to argue that the ADM energy should be nonnegative (at least for maximal Σ) and not less than \(\sqrt {{\rm{Area}}({{\mathcal S}_0})/(16\pi {G^2})}\) (at least for timesymmetric Σ), respectively. Later Jang [289] showed that, if a certain quasilinear elliptic differential equation for a function w on a hypersurface Σ admits a solution (with given asymptotic behavior), then w defines a mapping between the data set (Σ, h_{ab}, χ_{ab}) on Σ and a maximal data set \((\Sigma, \,{{\bar h}_{ab}},\,{{\bar \chi}_{ab}})\) (i.e., for which \({{\bar \chi}_{ab}}{{\bar h}^{ab}} = 0\)) such that the corresponding ADM energies coincide. Then Jang shows that a slightly modified version of the Geroch energy is monotonic (and tends to the ADM energy) with respect to a new, modified version of the inverse mean curvature foliation of \((\Sigma, \,{{\bar h}_{ab}})\).
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.
The Hayward energy
We saw that E_{H} can be nonzero, even in the Minkowski spacetime. This may motivate us to consider the following expression
(Thus, the integrand is \({1 \over 4}(F + \bar F)\), where F is given by Eq. (4.4).) By the Gauss equation, this is zero in flat spacetime, furthermore, it is not difficult to see that its limit at spatial infinity is still the ADM energy. However, using the second expression of \(I({\mathcal S})\), one can see that its limit at the future null infinity is the NewmanUnti, rather than the BondiSachs 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.
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.
Motivations
How do the twistors emerge?
We saw in Section 3.1.1 that in the Newtonian theory of gravity the mass of the source in D can be expressed as the flux integral of the gravitational field strength on the boundary \({\mathcal S}: = \partial D\). Similarly, in the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the gravitational field (i.e., the linearized energymomentum tensor) is still analogous to charge. In fact, the total energymomentum and angular momentum of the source can be expressed as appropriate twosurface integrals of the curvature at infinity [481]. Thus, it is natural to expect that the energymomentum and angular momentum of the source in a finite threevolume Σ, given by Eq. (2.5), can also be expressed as the charge integral of the curvature on the twosurface \({\mathcal S}\). However, the curvature tensor can be integrated on \({\mathcal S}\) only if at least one pair of its indices is annihilated by some tensor via contraction, i.e., according to Eq. (3.14) if some ω^{ab} = ω^{[ab]} is chosen and μ^{ab} = ε^{ab}. To simplify the subsequent analysis, ω^{ab} will be chosen to be antiselfdual: ω^{ab} = ε^{A′B′} ω^{AB} with ωAB = ω^{(AB)}.^{Footnote 11} Thus, our goal is to find an appropriate spinor field ω^{AB} on \({\mathcal S}\) such that
Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual of the 8πG times the integrand on the left, respectively, is
expressions (7.2) and (7.3) are equal if ω^{AB} satisfies
This equation in its symmetrized form, \({\nabla ^{{A\prime}(A}}{\omega ^{BC)}} = 0\), is the valence 2 twistor equation, a specific example for the general twistor equation \({\nabla ^{{A\prime}(A}}{\omega ^{BC \ldots E)}} = 0\) for ω^{BC.…E} = ω^{(BC.…E)}. Thus, as could be expected, ω^{AB} depends on the Killing vector K^{a}, and, in fact, K^{a} can be recovered from ω^{AB} as \({K^{{A\prime}A}} = {2 \over 3}{\rm{i}}\nabla _B^{{A\prime}}{\omega ^{AB}}\). Thus, ω^{AB} plays the role of a potential for the Killing vector K^{A′A}. However, as a consequence of Eq. (7.4), K_{a} is a selfdual Killing 1form in the sense that its derivative is a selfdual (s.d.) 2form: In fact, the general solution of Eq. (7.4) and the corresponding Killing vector are
where \({{\bar M}_{{A\prime}{B\prime}}},\,{T^{A{A\prime}}}\), and Ω^{AB} are constant spinors, and using the notation \({x^{A{A\prime}}}: = {x^{\underline a}}\sigma _{\underline a}^{\underline A \,{{\underline A}\prime}}{\mathcal E}_{\underline A}^A\bar {\mathcal E}_{{{\underline A}\prime}}^{{A\prime}}\), where \(\{{\mathcal E}_{\underline {\rm{A}}}^{\rm{A}}\}\) is a constant spin frame (the ‘Cartesian spin frame’) and \(\sigma _{\underline a}^{\underline A \,{{\underline A}\prime}}\) are the standard SL(2, ℂ) Pauli matrices (divided by \(\sqrt 2\)). These yield that K_{a} is, in fact, selfdual, \({\nabla _{A{A\prime}}}{K_{B{B\prime}}} = {\varepsilon _{AB}}{{\bar M}_{{A\prime}{B\prime}}},\,{T^{A{A\prime}}}\) is a translation and \({{\bar M}_{{A\prime}{B\prime}}}\) generates selfdual rotations. Then \({Q_{\mathcal S}}[{\bf{K}}] = {T_{A{A\prime}}}{P^{A{A\prime}}} + 2{{\bar M}_{{A\prime}{B\prime}}}{J^{{A\prime}{B\prime}}}\), implying that the charges corresponding to Ω^{AB} are vanishing, the four components of the quasilocal energymomentum correspond to the real T^{AA′} s, and the spatial angular momentum and centerofmass are combined into the three complex components of the selfdual angular momentum \({{\bar J}^{{A\prime}{B\prime}}}\), generated by \({{\bar M}_{{A\prime}{B\prime}}}\).
Twistor space and the kinematical twistor
Recall that the space of the contravariant valenceone twistors of Minkowski spacetime is the set of the pairs Z^{α} ≔ (λ^{A}, π_{A′}) of spinor fields, which solve the valenceonetwistor equation ∇^{A′A}λ^{B} = −iε^{AB}π_{A′}. If Z^{α} is a solution of this equation, then Ẑ^{α} ≔ (α^{A}, π_{A′} + iϒ^{A′a}λ^{A}) is a solution of the corresponding equation in the conformallyrescaled spacetime, where ϒ_{a} ≔ Ω^{−1}∇_{a}Ω and Ω is the conformal factor. In general, the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complexparameter family of solutions. Its general solution in the Minkowski spacetime is λ^{A} = Λ^{A} − ix^{AA′} π_{A′}, where Λ^{A} and π_{A′} are constant spinors. Thus, the space T^{α} of valenceone twistors, called the twistor space, is fourcomplexdimensional, and hence, has the structure \({{\rm{T}}^\alpha} = {{\rm{S}}^A} \oplus {{{\rm{\bar S}}}_{{A\prime}}}\). T^{α} admits a natural Hermitian scalar product: if W^{β} = (ω^{B}, σ_{B′}) is another twistor, then \({H_{\alpha {\beta \prime}}}{Z^\alpha}{{\bar W}^{{\beta \prime}}}: = {\lambda ^A}{{\bar \sigma}_A} + {\pi _{{A\prime}}}{{\bar \omega}^{{A\prime}}}\). Its signature is (+, +, −, −), it is conformally invariant, \({H_{\alpha {\beta \prime}}}{{\hat Z}^\alpha}{{\bar \hat W}^{{\beta \prime}}}: = {H_{\alpha {\beta \prime}}}{Z^\alpha}{{\bar W}^{{\beta \prime}}}\), and it is constant on Minkowski spacetime. The metric H_{αβ′} defines a natural isomorphism between the complex conjugate twistor space, \({{{\rm{\bar T}}}^\alpha}\prime\), and the dual twistor space, \({{\rm{T}}_\beta}: = {{\rm{S}}_B} \oplus {{\rm{\bar S}}^{{B\prime}}}\), by \(({{\bar \lambda}^{{A\prime}}},\,{{\bar \pi}_A}) \mapsto ({{\bar \pi}_A},\,{{\bar \lambda}^{{A\prime}}})\). This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate Ā_{α′β′} of the covariant valencetwo twistor A_{αβ} can be represented by the conjugate twistor A^{αβ} ≔ A_{α′β′}H^{α′β}H^{β′β}. We should mention two special, highervalence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by
The other is the completely antisymmetric twistor ε_{εαβγ}, whose component ε_{0123} in an H_{αβ′}orthonormal basis is required to be one. The only nonvanishing spinor parts of ε_{εαβγ} are those with two primed and two unprimed spinor indices: \({\varepsilon ^{A{\prime}B{\prime}}}_{CD} = {\varepsilon ^{A{\prime}B{\prime}}}{\varepsilon _{CD}},{\varepsilon ^{A{\prime}}}_B{\,^{C{\prime}}}_D =  {\varepsilon ^{A{\prime}C{\prime}}}{\varepsilon _{BD}},{\varepsilon _{AB}}^{C{\prime}D{\prime}} = {\varepsilon _{AB}}{\varepsilon ^{C{\prime}D{\prime}}}\). Thus, for any four twistors \(Z_i^\alpha = (\lambda _i^A,\,\pi _{{A\prime}}^i),\,i = 1,\, \ldots, \,4\), the determinant of the 4×4 matrix, whose ith column is \((\lambda _i^0,\,\lambda _i^1,\,\pi _0^i,\,\pi _1^i)\), where the \(\lambda _i^0,\, \ldots, \,\pi _1^i\), are the components of the spinors \(\lambda _i^A\) and \(\pi _A^i\), in some spin frame, is
where \({\epsilon ^{ij}}_{kl}\) is the totally antisymmetric LeviCivita symbol. Then I^{αβ} and I_{αβ} are dual to each other in the sense that \({I^{\alpha \beta}} = {1 \over 2}{\varepsilon ^{\alpha \beta \gamma \delta}}{I_{\gamma \delta}}\), and by the simplicity of I^{αβ} one has ε_{αβγδ}I^{αβ}I^{γδ} = 0.
The solution ω^{AB} of the valencetwo twistor equation, given by Eq. (7.5), can always be written as a linear combination of the symmetrized product \({\lambda ^{(A}}{\omega ^B})\) of the solutions λ^{A} and ω^{A} of the valenceone twistor equation. ω^{AB} uniquely defines a symmetric twistor ω^{αβ} (see, e.g., [426]). Its spinor parts are
However, Eq. (7.1) can be interpreted as a ℂlinear mapping of ω^{αβ} into ℂ, i.e., Eq. () defines a dual twistor, the (symmetric) kinematical twistor A_{αβ}, which therefore has the structure
Thus, the quasilocal energymomentum and selfdual angular momentum of the source are certain spinor parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor, it has only ten real components as a consequence of its structure (its spinor part A_{AB} is identically zero) and the reality of P^{AA′}. These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on A_{αβ}: the vanishing of the spinor part Aab is equivalent to A_{αβ}IαγI^{βδ} = 0 and the energy momentum is the \({A_{\alpha \beta}}{Z^\alpha}{I^{\beta \gamma}}{H_{\gamma {\gamma \prime}}}{{\bar Z}^{{\gamma \prime}}}\) part of the kinematical twistor, while the whole reality condition (ensuring both A_{AB} = 0 and the reality of the energymomentum) is equivalent to
Using the conjugate twistors, this can be rewritten (and, in fact, usually is written) as \({A_{\alpha \beta}}{I^{\beta \gamma}} = ({H^{\gamma {\alpha \prime}}}\,{{\bar A}_{{\alpha \prime}{\beta \prime}}}{H^{{\beta \prime}\delta}})\,({H_{\delta {\delta \prime}}}{{\bar I}^{{\delta \prime}{\gamma \prime}}}{H_{{\gamma \prime}\alpha}}) = {{\bar A}^{\gamma \delta}}{I_{\delta \alpha}}\). The quasilocal mass can also be expressed by the kinematical twistor as its Hermitian norm [420] or as its determinant [510]:
Similarly, as Helfer shows [264], the various components of the PauliLubanski spin vector \({S_a}: = {1 \over 2}{\varepsilon _{abcd}}{P^b}{J^{cd}}\) can also be expressed by the kinematic and infinity twistors and by certain special null twistors: if Z^{α} = (−ix^{AB′} π_{B′}, π_{A′}) and W^{α} = (−ix^{AB′} σ_{B′}, σ_{A′}) are two different (null) twistors such that A_{αβ}Z^{α}Z^{β} = 0 and A^{αβ}W^{α}W^{β} = 0, then
(ℜ on the right means ‘real part’.)
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.
The original construction for curved spacetimes
Twosurface twistors and the kinematical twistor
In general spacetimes, the twistor equations have only the trivial solution. Thus, to be able to associate a kinematical twistor with a closed orientable spacelike twosurface \({\mathcal S}\) in general, the conditions on the spinor field ω^{AB} have to be relaxed. Penrose’s suggestion [420, 421] is to consider ω^{AB} in Eq. (7.1) to be the symmetrized product \({\lambda ^{(A}}{\omega ^\beta})\) of spinor fields that are solutions of the ‘tangential projection to \({\mathcal S}\)’ of the valenceone twistor equation, the twosurface twistor equation. (The equation obtained as the ‘tangential projection to \({\mathcal S}\)’ of the valencetwo twistor equation (7.4) would be underdetermined [421].) Thus, the quasilocal quantities are searched for in the form of a charge integral of the curvature:
where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the twosurface \({\mathcal S}\). Since the indices c and d on the right of the first expression are tangential to \({\mathcal S}\), this is just the charge integral of F_{ABcd} in the spinor identity (4.5) of Section 4.1.5.
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 \({\mathcal S}\).
The Hamiltonian interpretation of the kinematical twistor
For the solutions λ^{A} and ω^{A} of the twosurface twistor equation, the spinor identity (4.5) reduces to Tod’s expression [420, 426, 516] for the kinematical twistor, making it possible to reexpress \({\mathcal S}\) by the integral of the NesterWitten 2form [490]. Indeed, if
then, with the choice \({{\bar \mu}_{{A\prime}}}: = {\rm{i}}{\Delta _{{A\prime}}}^A{\omega _A}\), this gives Penrose’s charge integral by Eq. (4.5): \({A_{\mathcal S}}[\lambda, \,\omega ] = {H_{\mathcal S}}[\lambda, \,\bar \mu ]\). Then, extending the spinor fields λ^{A} and ω^{A} from \({\mathcal S}\) to a spacelike hypersurface \(\Sigma\) with boundary \({\mathcal S}\) in an arbitrary way, by the Sparling equation it is straightforward to rewrite \({A_{\mathcal S}}[\lambda, \,\omega ]\) in the form of the integral of the energymomentum tensor of the matter fields and the Sparling form on Σ. Since such an integral of the Sparling form can be interpreted as the Hamiltonian of general relativity, this is a quick rederivation of Mason’s [357, 358] Hamiltonian interpretation of Penrose’s kinematical twistor: \({A_{\mathcal S}}[\lambda, \,\omega ]\) is just the boundary term in the total Hamiltonian of the matter + gravity system, and the spinor fields λ^{A} and ω^{A} (together with their ‘projection parts’ iΔ_{A′A}λ^{A} and iΔ_{A′A}ω^{A}) on \({\mathcal S}\) are interpreted as the spinor constituents of the special lapse and shift, called the ‘quasitranslations’ and ‘quasirotations’ of the twosurface, on the twosurface itself.
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.
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.
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.
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.
First, although the small spheres are contorted in general, the leading term of the pointwise Hermitian scalar product is constant: \({\lambda ^A}{\Delta _{A{A\prime}}}{{\bar \omega}^{{A\prime}}}  {{\bar \omega}^{{A\prime}}}{\Delta _{{A\prime}A}}{\lambda ^A}\) for any twosurface twistors Z^{α} = (λ^{A},iΔ_{A′A}λ^{A}) and W^{α} = (ω^{A},iΔ_{A′A}ω^{A}) [511, 313]. Since in nonvacuum spacetimes the kinematical twistor has only the ‘fourmomentum part’ in the leading \({\mathcal O}({r^3})\)order with \({P_a} = {{4\pi} \over 3}{r^3}{T_{ab}}{t^b}\), the Penrose mass, calculated with the norm above, is just the expected mass in the leading \({\mathcal O}({r^3})\) order. Thus, it is positive if the dominant energy condition is satisfied. On the other hand, in vacuum the structure of the kinematical twistor is
where \({\lambda _{AB}} = {\mathcal O}({r^5})$${P_{A{A\prime}}} = {2 \over {45G}}{r^5}{\psi _{ABCD}}{\chi _{{A\prime}{B\prime}{C\prime}{D\prime}}}{t^{B{B\prime}}}{t^{CC}}{t^{D{D\prime}}}\) with \({\chi _{ABCD}}: = {\psi _{ABCD}}  4{{\bar \psi}_{{A\prime}{B\prime}{C\prime}{D\prime}}}{t^{{A\prime}}}{\,_A}{t^{{B\prime}}}_B{t^{{C\prime}}}_C{t^{{D\prime}}}_D\). In particular, in terms of the familiar conformal electric and magnetic parts of the curvature the leading term in the time component of the fourmomentum is \({P_{A{A\prime}}}{t^{A{A\prime}}} = {1 \over {45G}}{H_{ab}}({H^{ab}}  {\rm{i}}{E^{ab}})\). Then, the corresponding normmass, in the leading order, can even be complex! For an \({{\mathcal S}_r}\) in the t = const. hypersurface of the Schwarzschild spacetime, this is zero (as it must be inlight of the results of Section 7.2.5, because this is a noncontorted spacelike hypersurface), but for a general small twosphere not lying in such a hypersurface, P_{AA′} is real and spacelike, and hence, m^{2} < 0. In the Kerr spacetime, P_{AA′} itself is complex [511, 313].
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].
The ‘improved’ construction with the determinant
A careful analysis of the roots of the difficulties lead Penrose [422, 426] (see also [511, 313, 516]) to suggest the modified definition for the kinematical twistor
where η is a constant multiple of the determinant in Eq. (7.7). Since on noncontorted twosurfaces the determinant ν is constant, for such surfaces A′_{αβ} reduces to A_{αβ}, and hence, all the nice properties proven for the original construction on noncontorted twosurfaces are shared by A′_{αβ}. The quasilocal mass calculated from Eq. (7.16) for small spheres (in fact, for small ellipsoids [313]) in vacuum is vanishing in the fifth order. Thus, apparently, the difficulties have been resolved. However, as Woodhouse pointed out, there is an essential ambiguity in the (nonvanishing, sixthorder) quasilocal mass [560]. In fact, the structure of the modified kinematical twistor has the form (7.15) with vanishing \({P^{{A\prime}}}_B\) and \({P_A}^{{B\prime}}\) but with nonvanishing λ_{AB} in the fifth order. Then, in the quasilocal mass (in the leading sixth order) there will be a term coming from the (presumably nonvanishing) sixthorder part of \({P^{{A\prime}}}_B\) and \({P_A}^{{B\prime}}\) and the constant part of the Hermitian scalar product, and the fifthorder λ_{AB} and the still ambiguous \({\mathcal O}(r)\)order part of the Hermitian metric.
Modification through Tod’s expression
These anomalies lead Penrose to modify A′_{αβ} slightly [423]. This modified form is based on Tod’s form of the kinematical twistor:
The quasilocal mass on small spheres coming from A″_{αβ} is positive [516].
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.
Approaches Based on the NesterWitten 2Form
We saw in Section 3.2 that

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.
Thus, from a pragmatic point of view, it seems natural to search for the quasilocal energymomentum in the form of the integral of the NesterWitten 2form. Now we will show that

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.
Hence, all the quasilocal energymomenta based on the integral of the NesterWitten 2form have a natural Lagrangian interpretation in the sense that they are charge integrals of the canonical Noether current derived from Møller’s firstorder tetrad Lagrangian.
If \({\mathcal S}\) is any closed, orientable spacelike twosurface and an open neighborhood of \({\mathcal S}\) is time and space orientable, then an open neighborhood of \({\mathcal S}\) is always a trivialization domain of both the orthonormal and the spin frame bundles [500]. Therefore, the orthonormal frame \(\{E_{\underline a}^a\}\) can be chosen to be globally defined on \({\mathcal S}\), and the integral of the dual of Møller’s superpotential, \({1 \over 2}{K^e}{\vee_e}^{ab}{1 \over 2}{\varepsilon _{abcd}}\), appearing on the righthand side of the superpotential Eq. (3.7), is well defined. If (t^{a}, v^{a}) is a pair of globallydefined normals of \({\mathcal S}\) in the spacetime, then in terms of the geometric objects introduced in Section 4.1, this integral takes the form
The first term on the right is just the dual mean curvature vector of \({\mathcal S}\), the second is the connection oneform on the normal bundle, while the remaining terms are explicitly SO(1, 3) gauge dependent. On the other hand, this is boost gauge invariant (the boost gauge dependence of the second term is compensated by the last one), and depends on the tetrad field and the vector field K^{a} given only on \({\mathcal S}\), but is independent in the way in which they are extended off the surface. As we will see, the general form of other quasilocal energymomentum expressions show some resemblance to Eq. (8.1).
Then, suppose that the orthonormal basis is built from a normalized spinor dyad, i.e., \(E_{\underline a}^a = \sigma _{\underline a}^{\underline A {{\underline B}{\prime}}}\varepsilon _{\underline A}^A\bar \varepsilon _{{{\underline B}{\prime}}}^{{A{\prime}}}\), where \(\sigma _{\underline a}^{\underline A {{\underline B}{\prime}}}\) are the SL(2, ℂ) Pauli matrices (divided by \(\sqrt 2)\)) and \(\{\varepsilon _{\underline A}^A\}, \underline A = 0,1\), is a normalized spinor basis. A straightforward calculation yields the following remarkable expression for the dual of Møller’s superpotential:
where the overline denotes complex conjugation. Thus, the real part of the NesterWitten 2form, and hence, by Eq. (3.11), apart from an exact 2form, the NesterWitten 2form itself, built from the spinors of a normalized spinor basis, is just the superpotential 2form derived from Møller’s firstorder tetrad Lagrangian [500].
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.
The LudvigsenVickers construction
The definition
Suppose that spacetime is asymptotically flat at future null infinity, and the closed spacelike twosurface \({\mathcal S}\) can be joined to future null infinity by a smooth null hypersurface \({\mathcal N}\). Let \({{\mathcal S}_\infty}: = {\mathcal N} \cap {{\mathscr I}^ +}\), the cut defined by the intersection of \({\mathcal N}\) with future null infinity. Then, the null geodesic generators of \({\mathcal N}\) define a smooth bijection between \({\mathcal S}\) and the cut \({{\mathcal S}_\infty}\) (and hence, in particular, \({\mathcal S} \approx {S^2}\)). We saw in Section 4.2.4 that on the cut \({{\mathcal S}_\infty}\) at the future null infinity we have the asymptotic spin space \((S_\infty ^{\underline A},{\varepsilon _{\underline A \underline B}})\). The suggestion of Ludvigsen and Vickers [346] for the spin space \(({{\rm{S}}^{\underline A}},{\varepsilon _{\underline A \underline B}})\) on \({\mathcal S}\) is to import the two independent solutions of the asymptotic twistor equations, i.e., the asymptotic spinors, from the future null infinity back to the twosurface along the null geodesic generators of the null hypersurface \({\mathcal N}\). Their propagation equations, given both in terms of spinors and in the GHP formalism, are
Here \(\varepsilon _{\rm{A}}^A = \{{o^A},{\iota ^A}\}\) is the GHP spin frame introduced in Section 4.2.4, and by Eq. (4.6) the second half of these equations is just Δ^{+}λ = 0. It should be noted that the choice of Eqs. (8.3) and (8.4) for the propagation law of the spinors is ‘natural’ in the sense that in flat spacetime they reduce to the condition of parallel propagation, and Eq. (8.4) is just the appropriate part of the asymptotic twistor equation of Bramson. We call the spinor fields obtained by using Eqs. (8.3) and (8.4) the LudvigsenVickers spinors on \({\mathcal S}\). Thus, given an asymptotic spinor at infinity, we propagate its zeroth components (with respect to the basis \(\varepsilon _{\rm{A}}^A\)) to \({\mathcal S}\) by Eq. (8.3). This will be the zeroth component of the LudvigsenVickers spinor. Then, its first component will be determined by Eq. (8.4), provided ρ is not vanishing on any open subset of \({\mathcal S}\). If \(\lambda _A^0\) and \(\lambda _A^1\) are LudvigsenVickers spinors on \({\mathcal S}\) obtained by Eqs. (8.3) and (8.4) from two asymptotic spinors that formed a normalized spin frame, then, by considering \(\lambda _A^0\) and \(\lambda _A^1\) to be normalized in \({{\bf{S}}^{\underline A}}\), we define the symplectic metric \({\varepsilon _{\underline A \underline B}}\) on \({{\rm{S}}^{\underline A}}\) to be that with respect to which \(\lambda _A^0\) and \(\lambda _A^1\) form a normalized spin frame. Note, however, that this symplectic metric is not connected with the symplectic fiber metric ε_{AB} of the spinor bundle \({{\bf{S}}^A}({\mathcal S})\) over \({\mathcal S}\). Indeed, in general, \(\lambda _A^{\underline A}\lambda _B^{\underline B}{\varepsilon ^{AB}}\) is not constant on \({\mathcal S}\), and hence, ε_{AB} does not determine any symplectic metric on the space \({{\bf{S}}^{\underline A}}\) of the LudvigsenVickers spinors. In Minkowski spacetime the two LudvigsenVickers spinors are just the restriction to \({\mathcal S}\) of the two constant spinors.
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.
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].
Finally, on a small sphere of radius r in nonvacuum the second definition gives [84] the expected result (4.9), while in vacuum [84, 494] it is
Thus, its leading term is the energymomentum of the matter fields and the BelRobinson momentum, respectively, seen by the observer t^{a} at the vertex p. Thus, assuming that the matter fields satisfy the dominant energy condition, for small spheres this is an explicit proof that the LudvigsenVickers quasilocal energymomentum is future pointing and nonspacelike.
The DouganMason constructions
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.
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.
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.
In the asymptotically flat spacetimes the positive energy theorems have a rigidity part as well, namely the vanishing of the energymomentum (and, in fact, even the vanishing of the mass) implies flatness. There are analogous theorems for the DouganMason energymomenta as well [488, 490]. Namely, under the conditions of the positivity proof

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.
In particular [498], for a coupled EinsteinYangMills system (with compact, semisimple gauge groups) the zero quasilocal mass configurations are precisely the ppwave solutions found by Güven [230]. Therefore, in contrast to the asymptotically flat cases, the vanishing of the mass does not imply the flatness of D(Σ). Since, as we will see below, the DouganMason masses tend to the ADM mass at spatial infinity, there is a seeming contradiction between the rigidity part of the positive mass theorems and the result 2 above. However, this is only an apparent contradiction. In fact, according to one of the possible positive mass proofs [38], the vanishing of the ADM mass implies the existence of a constant null vector field on D(Σ), and then the flatness follows from the incompatibility of the conditions of the asymptotic flatness and the existence of a constant null vector field: The only asymptotically flat spacetime admitting a constant null vector field is flat spacetime.
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.
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.^{Footnote 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.^{Footnote 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.
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.
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.
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.
Using the asymptotic solution of the EinsteinMaxwell equations in a Bonditype coordinate system it has been shown in [346] that the LudvigsenVickers spin angular momentum tends to that of Bramson at future null infinity. For small spheres [494] in nonvacuum it reproduces precisely the expected result (4.10), and in vacuum it is
We stress that in both the vacuum and nonvacuum cases, the factor \(r{t^{D'E}}{\varepsilon ^{DF}}\;{\mathcal E}_{(E}^{\underline A}{\mathcal E}_{F)}^{\underline B}\), interpreted in Section 4.2.2 as an average of the boostrotation Killing fields that vanish at p, emerges naturally. No (approximate) boostrotation Killing field was put into the general formulae by hand.
Holomorphic/antiholomorphic spin angular momenta
Obviously, the spinangularmomentum expressions based on the holomorphic and antiholomorphic spinor fields [492] on generic twosurfaces are genuinely quasilocal. Since, in Minkowski spacetime the restriction of the two constant spinor fields to any twosurface is constant, and hence holomorphic and antiholomorphic at the same time, both the holomorphic and antiholomorphic spin angular momenta are vanishing. Similarly, for round spheres both definitions give zero [496], as would be expected in a sphericallysymmetric system. The antiholomorphic spin angular momentum has already been calculated for axisymmetric twosurfaces \({\mathcal S}\), for which the antiholomorphic DouganMason energymomentum is null, i.e., for which the corresponding quasilocal mass is zero. (As we saw in Section 8.2.3, this corresponds to a ppwave geometry and pure radiative matter fields on D(Σ) [488, 490].) This null energymomentum vector turned out to be an eigenvector of the antisymmetric spinangularmomentum tensor \(J_{\mathcal S}^{\underline A\underline B}\), which, together with the vanishing of the quasilocal mass, is equivalent to the proportionality of the (null) energymomentum vector and the PauliLubanski spin [492], where the latter is defined by
This is a known property of the zerorestmass fields in Poincaré invariant quantum field theories [231].
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].
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].)
The main idea is that, while the energymomentum is a single fourvector in the dual of the Hermitian subspace of \({{\bf{S}}^{\underline A}} \otimes {{{\bf{\bar S}}}^{\underline B{\prime}}}\), the angular momentum is not only an antisymmetric tensor over the same space, but should depend on the ‘origin’, a point in a fourdimensional affine space M_{0} as well, and should transform in a specific way under the translation of the ‘origin’. Bergqvist and Ludvigsen defined the affine space M_{0} to be the space of the solutions X_{a} of \({{\tilde \nabla}_a}{X_b} = {g_{ab}}  {H_{ab}}\), and showed that M_{0} is, in fact, a real, fourdimensional affine space. Then, for a given X_{aa′}, to each \({{\tilde \nabla}_a}\)constant spinor field λ^{A} they associate a primed spinor field by μ_{A′} ≔ X_{a′a}λ^{A}. This μ_{A′} turns out to satisfy the modified valenceone twistor equation \({{\tilde \nabla}_{A(A{\prime}}}{\mu _{B{\prime})}} =  {H_{AA{\prime}BB{\prime}}}{\lambda ^B}\). Finally, they form the 2form
and define the angular momentum \(J_{\mathcal S}^{\underline A\underline B}(X)\) with respect to the origin X_{a} as 1/(8πG) times the integral of \(W{(X,{\lambda ^{\underline A}},{\lambda ^{\underline B}})_{ab}}\) on some closed, orientable spacelike twosurface \({\mathcal S}\). Since this W_{ab} is closed, Δ_{[a}W_{bc]} = 0 (similar to the NesterWitten 2form in Section 8.3), the integral \(J_{\mathcal S}^{\underline A\underline B}(X)\) depends only on the homology class of \({\mathcal S}\). Under the ‘translation’ X_{e} ↦ X_{e} + a_{e} of the ‘origin’ by a \({{\tilde \nabla}_a}\)constant oneform a_{e}, it transforms as \(J_{\mathcal S}^{\underline A\underline B}(\tilde X) = J_{\mathcal S}^{\underline A\underline B}(X) + {a^{(\underline A}}_{\underline B{\prime}}P_{\mathcal S}^{\underline B)\underline B{\prime}}\), where the components \({a_{\underline A\underline B{\prime}}}\) are taken with respect to the basis \(\{\lambda _A^{\underline A}\}\) in the solution space. Unfortunately, no explicit expression for the angular momentum in terms of the Kerr parameters m and a is given.
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.
The BrownYork expression
The main idea
To motivate the main idea behind the BrownYork definition [120, 121], let us first consider a classical mechanical system of n degrees of freedom with configuration manifold Q and Lagrangian L: TQ × ℝ → ℝ (i.e., the Lagrangian is assumed to be first order and may depend on time explicitly). For given initial and final configurations, \((q_1^a,{t_1})\) and \((q_2^a,{t_2})\), respectively, the corresponding action functional is \({I^1}[q(t)]\;: = \int\nolimits_{{t_1}}^{{t_2}} {L({q^a}(t),{{\dot q}^a}(t),t)\;dt}\), where q^{a}(t) is a smooth curve in Q from \({q^a}({t_1}) = q_1^a\) to \({q^a}({t_2}) = q_2^a\) with tangent \({{\dot q}^a}(t)\) at t. (The pair (q^{a}(t), t) may be called a history or world line in the ‘spacetime’ Q × ℝ.) Let (q^{a}(u, t(u)), t(u)) be a smooth oneparameter deformation of this history, for which (q^{a}(0, t(0)), t(0)) = (q^{a}(t), t), and u ∈ (−ϵ, ϵ) for some ϵ > 0. Then, denoting the derivative with respect to the deformation parameter u at u = 0 by δ, one has the well known expression
Therefore, introducing the HamiltonJacobi principal function \({S^1}(q_1^a,{t_1};q_2^a,{t_2})\) as the value of the action on the solution q^{a}(t) of the equations of motion from \((q_1^a,{t_1})\) to \((q_2^a,{t_2})\), the derivative of S^{1} with respect to \(q_2^a\) gives the canonical momenta \(p_a^1: = (\partial L/\partial {{\dot q}^a})\), while its derivative with respect to t_{2} gives minus the energy, \( {E^1} =  (p_a^1{{\dot q}^a}  L)\), at t_{2}. Obviously, neither the action I^{1} nor the principal function S^{1} are unique: I[q(t)] ≔ I^{1}[q(t)] − I^{0}[q(t)] for any I^{0}[q(t)] of the form \( {E^1} =  (p_a^1{{\dot q}^a}  L)\) (dh/dt) dt with arbitrary smooth function h = h(q^{a}(t), t) is an equally good action for the same dynamics. Clearly, the subtraction term I^{0}[q(t)] alters both the canonical momenta and the energy according to \(p_a^1 \mapsto {p_a} = p_a^1  (\partial h/\partial {q^a})\) and E^{1} ↦ E = E^{1} + (∂h/∂t), respectively.
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.^{Footnote 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}.
The primary requirement of Brown and York on the action is to provide a welldefined variational principle for the Einstein theory. This claim leads them to choose for I^{1} the ‘trace K action’ (or, in the present notation, the ‘trace χ action’) for general relativity [572, 573, 534], and the action for the matter fields may be included. (For minimal, nonderivative couplings, the presence of the matter fields does not alter the subsequent expressions.) However, as Geoff Hayward pointed out [243], to have a welldefined variational principle, the ‘trace χ action’ should in fact be completed by two twosurface integrals, one on \({{\mathcal S}_1}\) and the other on \({{\mathcal S}_2}\). Otherwise, as a consequence of the edges \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\), called the ‘joints’ (i.e., the nonsmooth parts of the boundary ∂D), the variation of the metric at the points of the edges \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) could not be arbitrary. (See also [242, 315, 100, 119], where the ‘orthogonal boundaries assumption’ is also relaxed.) Let η_{1} and η_{2} be the scalar product of the outwardpointing normal of ^{3}B and the futurepointing normal of Σ_{1} and of Σ_{2}, respectively. Then, varying the spacetime metric (for the variation of the corresponding principal function S^{1}) they obtained the following:
The first two terms together correspond to the term \(p_a^1\delta {q^a}\vert _{{t_1}}^{{t_2}}\) of Eq. (10.1), and, in fact, the familiar ADM expression for the canonical momentum \({{\tilde p}^{ab}}\) is just \({1 \over {16\pi G}}\sqrt {\vert h\vert} ({\chi ^{ab}}  \chi {h^{ab}})\). The last two terms give the effect of the presence of the nondifferentiable ‘joints’. Therefore, it is the third term that should be analogous to the third term of Eq. (10.1). In fact, roughly, this is proportional to the proper time separation of the ‘instants’ Σ_{1} and Σ_{2}, and it is reasonable to identify its coefficient as some (quasilocal) analog of the energy. However, just as in the case of the mechanical system, the action (and the corresponding principal function) is not unique, and the principal function should be written as S ≔ S^{1} − S^{0}, where S^{0} is assumed to be an arbitrary function of the threemetric on the boundary ∂D = Σ_{2} ∪^{3}B ∪ Σ_{1}. Then
defines a symmetric tensor field on the timelike boundary ^{3}B, and is called the surface stressenergy tensor. (Since our signature for γ_{ab} on ^{3}B is (+, −, −) rather than (−, +, +), we should define τ^{ab} with the extra minus sign, according to Eq. (2.1).) Its divergence with respect to the connection ^{3} D_{e} on ^{3}B determined by γ_{ab} is proportional to the part γ^{ab}T_{bc}υ^{c} of the energymomentum tensor, and hence, in particular, τ^{ab} is divergencefree in vacuum. Therefore, if (^{3}B, γ_{ab}) admits a Killing vector, say K^{a}, then, in vacuum
the flux integral of τ^{ab}K_{b} on any spacelike cross section \({\mathcal S}\) of ^{3}B, is independent of the cross section itself, and hence, defines a conserved charge. If K^{a} is timelike, then the corresponding charge is called a conserved mass, while for spacelike K^{a} with closed orbits in \({\mathcal S}\) the charge is called angular momentum. (Here \({\mathcal S}\) is not necessarily an element of the foliation \({{\mathcal S}_t}\)_{t} of ^{3}B, and \({{\bar t}^a}\) is the unit normal to \({\mathcal S}\) tangent to ^{3}B.)
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.
The general form of the BrownYork quasilocal energy
The 3 + 1 decomposition of the spacetime metric yields a 2 + 1 decomposition of the metric γ_{ab}, as well. Let N and N^{a} be the lapse and the shift of this decomposition on ^{3}B. Then the corresponding decomposition of τ^{ab} defines the energy, momentum, and spatialstress surface densities according to
where q_{ab} is the spacelike twometric, A_{e} is the SO(1,1) vector potential on \({{\mathcal S}_t}\), \(\Pi _b^a\) is the projection to \({{\mathcal S}_t}\) introduced in Section 4.1.2, k_{ab} is the extrinsic curvature of \({{\mathcal S}_t}\) corresponding to the normal v^{a} orthogonal to ^{3}B, and k is its trace. The timelike boundary ^{3}B defines a boostgauge on the twosurfaces \({{\mathcal S}_t}\) (which coincides with that determined by the foliation Σ_{t} in the ‘orthogonal boundaries’ case). The gauge potential A_{e} is taken in this gauge. Thus, although ε and j_{a} on \({{\mathcal S}_t}\) are built from the twosurface data (in a particular boostgauge), the spatial surface stress depends on the part t^{a}(∇_{a}t_{b})v^{b} of the acceleration of the foliation Σ_{t} as well. Let ξ^{a} be any vector field on ^{3}B tangent to ^{3}B, and ξ^{a} = nt^{a} + n^{a} its 2 + 1 decomposition. Then we can form the charge integral (10.4) for the leaves \({{\mathcal S}_t}\) of the foliation of ^{3}B
Obviously, in general E_{t}[ξ^{a}, t^{a}] is not conserved, and depends not only on the vector field ξ^{a} and the twosurface data on the particular \({{\mathcal S}_t}\), but on the boostgauge that ^{3}B defines on \({t^a}\), i.e., the timelike normal t^{a} as well. Brown and York define the general form of their quasilocal energy on \({\mathcal S}: = {{\mathcal S}_t}\) by
i.e., they link the ‘quasitimetranslation’ (i.e., the ‘generator of the energy’) to the preferred unit normal t^{a} of \({{\mathcal S}_t}\). Since the preferred unit normals t^{a} are usually interpreted as a fleet of observers who are at rest with respect to \({{\mathcal S}_t}\), in their spirit the BrownYorktype quasilocal energy expressions are similar to E_{Σ}[t^{a}] given by Eq. (2.6) for the matter fields or Eq. (3.17) for the gravitational ‘field’ rather than to the charges \({Q_{\mathcal S}}[{\bf{K}}]\). For vector fields ξ^{a} = n^{a} with closed integral curved in \({{\mathcal S}_t}\) the quantity E_{t}[ξ^{a}, t^{a}] might be interpreted as angular momentum corresponding to ξ^{a}.
The quasilocal energy is still not completely determined, because the ‘subtraction term’ S^{0} in the principal function has not been specified. This term is usually interpreted as our freedom to shift the zero point of the energy. Thus, the basic idea of fixing the subtraction term is to choose a ‘reference configuration’, i.e., a spacetime in which we want to obtain zero quasilocal quantities E_{t}[ξ^{a}, t^{a}] (in particular zero quasilocal energy), and identify S^{0} with the S^{1} of the reference spacetime. Thus, by Eq. (10.5) and (10.6) we obtain that
where k^{0} and \(A_a^0\) are the reference values of the trace of the extrinsic curvature and SO(1, 1)gauge potential, respectively. Note that to ensure that k^{0} and \(A_a^0\) really be the trace of the extrinsic curvature and SO(1, 1)gauge potential, respectively, in the reference spacetime, they cannot depend on the lapse N and the shift N^{a}. This can be ensured by requiring that S^{0} be a linear functional of them. We return to the discussion of the reference term in the various specific constructions below.
For a definition of the BrownYork energy as a quasilocal energy oparator in loop quantum gravity, see [565].
Further properties of the general expressions
As we noted, ε, j_{a}, and s_{ab} depend on the boostgauge that the timelike boundary defines on \({{\mathcal S}_t}\). Lau clarified how these quantities change under a boost gauge transformation, where the new boostgauge is defined by the timelike boundary ^{3}B′ of another domain D′such that the particular twosurface S_{t} is a leaf of the foliation of ^{3}B′ as well [333]. If \(\{{{\bar \Sigma}_t}\}\) is another foliation of D such that \(\partial {{\bar \Sigma}_t} = {{\mathcal S}_t}\) and \({{\bar \Sigma}_t}\) is orthogonal to ^{3}B, then the new ε′, j′_{a}, and \(s_{ab}{\prime}\) are built from the old ε, j_{a}, and s_{ab} and the 2 + 1 pieces on \({{\mathcal S}_t}\) of the canonical momentum \({{\bar \tilde p}^{ab}}\), defined on \({{\bar \Sigma}_t}\). Apart from the contribution of S^{0}, these latter quantities are
where l_{ab} is the extrinsic curvature of \({{\mathcal S}_t}\) corresponding to its normal t^{a} (we denote this by τ_{ab} in Section 4.1.2), and l is its trace. (By Eq. (10.12) \({{\hat j}_a}\) is not an independent quantity, that is just j_{a}. These quantities were originally introduced as the variational derivatives of the principal function with respect to the lapse, the shift and the twometric of the radial foliation of Σ_{t} [333, 119], which are, in fact, essentially the components of the canonical momentum.) Thus, the required transformation formulae for ε, j_{a}, and s_{ab} follow from the definitions and those for the extrinsic curvature and the SO(1, 1) gauge potential of Section 4.1.2. The various boostgauge invariant quantities that can be built from ε, j_{a}, s_{ab}, j_{⊢}, and t_{ab} are also discussed in [333, 119].
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 \({\mathcal S}\).^{Footnote 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.
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.
Hawking, Horowitz, and Hunter also derived the Hamiltonian from the trace χ action \(I_D^1[{g_{ab}}]\) both with the orthogonal [241] and nonorthogonal boundary assumptions [242]. They allowed matter fields Φ_{N}, whose dynamics is governed by a firstorder action \(I_{{\rm{m}}D}^1[{g_{ab}},{\Phi _N}]\), to be present. However, they treated the reference configuration in a different way. In the traditional canonical analysis of the fields and the geometry based on a noncompact Σ (for example in the asymptotically flat case) one has to impose certain falloff conditions that ensure the finiteness of the action, the Hamiltonian, etc. This finiteness requirement excludes several potentially interesting field + gravity configurations from our investigations. In fact, in the asymptotically flat case we compare the actual matter + gravity configurations with the flat spacetime + vanishing matter fields configuration. Hawking and Horowitz generalized this picture by choosing a static, but otherwise arbitrary, solution \(g_{ab}^0\), \(\Phi _N^0\) of the field equations, considered the timelike boundary ^{3}B of D to be a timelike cylinder ‘near the infinity’, and considered the action
and those matter + gravity configurations that induce the same value on ^{3}B as and \(\Phi _N^0\) and \(g_{ab}^0\). Its limit as ^{3}B is ‘pushed out to infinity’ can be finite, even if the limit of the original (i.e., nonreferenced) action is infinite. Although in the nonorthogonal boundaries case the Hamiltonian derived from the nonreferenced action contains terms coming from the ‘joints’, by the boundary conditions at ^{3}B they are canceled from the referenced Hamiltonian. This latter Hamiltonian coincides with that obtained in the orthogonal boundaries case. Both the ADM and the AbbottDeser energy can be recovered from this Hamiltonian [241], and the quasilocal energy for spheres in domains with nonorthogonal boundaries in the Schwarzschild solution is also calculated [242]. A similar Hamiltonian, including the ‘joints’ or ‘corner’ terms, was obtained by Francaviglia and Raiteri [191] for the vacuum Einstein theory (and for EinsteinMaxwell systems in [9]), using a Noether charge approach. Their formalism, using the language of jet bundles, is, however, slightly more sophisticated than that common in general relativity.
Booth and Fairhurst [95] reexamined the general form of the BrownYork energy and angular momentum from a Hamiltonian point of view.^{Footnote 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.
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