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Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

The Original Version of this article was published on 01 December 2004

Abstract

The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local 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 quasi-local 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 energy-momentum and, ultimately, angular momentum as well) to extended, but finite, spacetime domains, i.e., at the quasi-local level. Obviously, the quasi-local quantities could provide a more detailed characterization of the states of the gravitational ‘field’ than the global ones, so they (together with more general quasi-local observables) would be interesting in their own right.

Moreover, finding an appropriate notion of energy-momentum and angular momentum would be important from the point of view of applications as well. For example, they may play a central role in the proof of the full Penrose inequality (as they have already played in the proof of the Riemannian version of this inequality). The correct, ultimate formulation of black hole thermodynamics should probably be based on quasi-locally defined internal energy, entropy, angular momentum, etc. In numerical calculations, conserved quantities (or at least those for which balance equations can be derived) are used to control the errors. However, in such calculations all the domains are finite, i.e., quasi-local. Therefore, a solid theoretical foundation of the quasi-local conserved quantities is needed.

However, contrary to the high expectations of the 1980s, finding an appropriate quasi-local notion of energy-momentum has proven to be surprisingly difficult. Nowadays, the state of the art is typically postmodern: although there are several promising and useful suggestions, we not only have no ultimate, generally accepted expression for the energy-momentum 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 energy-momentum? 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, potentially-common points, issues and questions. Thus, we wanted not only to write a ‘who-did-what’ review, but to concentrate on the understanding of the basic questions (such as why should the gravitational energy-momentum and angular momentum, or, more generally, any observable of the gravitational ‘field’, be necessarily quasi-local) and ideas behind the various specific constructions. Consequently, one third of the present review is devoted to these general questions. We review the specific constructions and their properties only in the second part, and in the third part we discuss very briefly some (potential) applications of the quasi-local quantities. Although this paper is 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 energy-momentum and angular momentum can be introduced in (and hence can be understood from) the theory of matter fields in Minkowski spacetime. Thus, in Section 2.1, we review the Belinfante-Rosenfeld procedure that we will apply to gravity in Section 3, introduce the notion of quasi-local energy-momentum and angular momentum of the matter fields and discuss their properties. The philosophy of quasi-locality in general relativity will be demonstrated in Minkowski spacetime where the energy-momentum and angular momentum of the matter fields are treated quasi-locally. Then we turn to the difficulties of gravitational energy-momentum and angular momentum, and we clarify why the gravitational observables should necessarily be quasi-local. The tools needed to construct and analyze the quasi-local quantities are reviewed in the fourth section. This closes the first (general) part of the review (Sections 24).

The second part is devoted to the discussion of the specific constructions (Sections 512). 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 quasi-local 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 quasi-local characterization of the pp-wave metrics. We close this paper with a discussion of the successes and deficiencies of the general and (potentially) viable constructions. In contrast to the positivistic style of Sections 512, Section 14 (as well as the choice of subject matter of Sections 24) reflects our own personal interest and view of the subject.

The theory of quasi-local observables in general relativity is far from being complete. The most important open problem is still the trivial one: ‘Find quasi-local energy-momentum and angular momentum expressions satisfying the points of the lists of 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 gab 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.

Energy-Momentum and Angular Momentum of Matter Fields

Energy-momentum and angular-momentum density of matter fields

The symmetric energy-momentum tensor

It is a widely accepted view that the canonical energy-momentum 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 spatio-temporal location of the first. Correspondingly, the fields on the manifold M of events can be grouped into two sharply-distinguished 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 gab 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 Im[gab, ΦN] is the action functional, i.e., the volume integral of Lm on some open domain D with compact closure, then the equations of motion are

$$E_{\;\;\;a \ldots}^{Nb \ldots}: = {1 \over {\sqrt {\vert g \vert}}}{{\delta {I_{\rm{m}}}} \over {\delta {\Phi _{N_{b \ldots}^{a \ldots}}}}} = \sum\limits_{n = 0}^k {{{(-)}^n}{\nabla _{{e_n}}} \ldots {\nabla _{{e_1}}}\left({{{\partial {L_{\rm{m}}}} \over {\partial \left({{\nabla _{{e_1}}} \ldots {\nabla _{{e_n}}}{\Phi _{N_{b \ldots}^{a \ldots}}}} \right)}}} \right) =} 0,$$

the Euler-Lagrange equations. (Here, of course, \(\delta {I_{\rm{m}}}/\delta {\Phi _N}_{b \ldots}^{a \ldots}\) denotes the formal variational derivative of Im with respect to the field variable \({\Phi _N}_{b \ldots}^{a \ldots}\).) The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by

$${T_{ab}}: = {1 \over {\sqrt {\vert g\vert}}}{{\delta {I_{\rm{m}}}} \over {\delta {g^{ab}}}} = 2{{\partial {L_{\rm{m}}}} \over {\partial {g^{ab}}}} - {L_{\rm{m}}}{g_{ab}} + {1 \over 2}{\nabla ^e}({\sigma _{abe}} + {\sigma _{bae}} - {\sigma _{aeb}} - {\sigma _{bea}} - {\sigma _{eab}} - {\sigma _{eba}}),$$
(2.1)

where we introduced the canonical spin tensor

$${\sigma ^{ea}}_b: = \sum\limits_{n = 1}^k {\sum\limits_{i = 1}^n {{{( - )}^i}\delta _{{e_i}}^e{\nabla _{{e_{i - 1}}}}...{\nabla _{{e_1}}}\left( {\frac{{\partial {L_m}}}{{\partial ({\nabla _{{e_1}}}...{\nabla _{{e_n}}}{\Phi _N}_{d...}^{c...})}}} \right)}} \;\Delta_{be_{i+1...}e_{n}d...}^{ac...}{_{h...}^{f_{i+1...}f_{n}g}}\Delta_{f_{i+1}}...\Delta_{f_{n}}\;\Phi_{N_{g...}^{h...}}.$$
(2.2)

(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 Tab 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 Ka and the corresponding local one-parameter family of diffeomorphisms ϕt, one has

$$(\phi _t^{\ast} {L_{\rm{m}}})({g^{ab}},{\Phi _N},{\nabla _e}{\Phi _N}, \ldots) - {L_{\rm{m}}}(\phi _t^{\ast} {g^{ab}},\phi _t^{\ast} {\Phi _N},\phi _t^{\ast} {\nabla _e}{\Phi _N}, \ldots) = {\nabla _e}B_t^e,$$

for some one-parameter family of vector fields \(B_t^e = B_t^e({g^{ab}},{\Phi _N}, \ldots)\). (Lm is called diffeomorphism invariant if \({\nabla _e}B_t^e = 0\), e.g., when Lm is a scalar.) Let Ka be any smooth vector field on M. Then, calculating the divergence a(LmKa) to determine the rate of change of the action functional Im along the integral curves of Ka, 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 ŁK denotes the Lie derivative along Ka, and Ce[K], the Noether current, is given explicitly by

$${C^e}[{\bf{K}}] = {\dot B^e} + {\theta ^{ea}} + {K_a} + \left({{\sigma ^{e[ab]}} + {\sigma ^{a[be]}} + {\sigma ^{b[ae]}}} \right){\nabla _a}{K_b}.$$
(2.3)

Here e is the derivative of \(B_t^e\) with respect to t at t = 0, which may depend on Ka and its derivatives, and \({\theta ^a}_b\), the canonical energy-momentum tensor, is defined by

$${\theta ^a}_b: = - {L_{\rm{m}}}\delta _b^a - \sum\limits_{n = 1}^k {\sum\limits_{i = 1}^n {{{(-)}^i}\delta _{{e_i}}^a{\nabla _{{e_{i - 1}}}} \ldots {\nabla _{{e_1}}}\left({{{\partial {L_{\rm{m}}}} \over {\partial ({\nabla _{{e_1}}} \ldots {\nabla _{{e_n}}}{\Phi _{N_{d \ldots}^{c \ldots}}})}}} \right)}} {\nabla _b}{\nabla _{{e_{i + 1}}}} \ldots {\nabla _{{e_n}}}{\Phi _N}_{d \ldots}^{c \ldots}.$$
(2.4)

Note that, apart from the term e, the current Ce[K] does not depend on higher than the first derivative of Ka, and the canonical energy-momentum and spin tensors could be introduced as the coefficients of Ka and its first derivative, respectively, in Ce[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, Ce[K] is not uniquely determined by the Noether identity, because that contains only its divergence, and any identically-conserved 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 diffeomorphism-invariant Lagrangians. Thus, to be more precise, if 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 Ka, as well (see, e.g., [304]), but there is a specific one containing Ka algebraically (see points 3 and 4 below).

However, Ca[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 Yang-Mills field), then in general it is not gauge invariant, even if the Lagrangian is. On the other hand, Tab is gauge invariant and is independent of total divergences added to Lm 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. 1.

    aTab = 0,

  2. 2.

    Tab = θab + ∇c(σc[ab] + σc[ab] + σc[ab]),

  3. 3.

    Ca[K] = TabKb + ∇c((σc[ab]σc[ab]σc[ab]Kb), where the second term on the right is an identically-conserved (i.e., divergence-free) current, and

  4. 4.

    Ca[K] is conserved if Ka is a Killing vector.

Hence, TabKb 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, Ca[K] and TabKb, depends on the nature of the Killing vector, Ka. In Minkowski spacetime the ten-dimensional Lie algebra K of the Killing vectors is well known to split into the semidirect sum of a four-dimensional 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 Ro of the boost-rotation 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 boost-rotations 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 one-form 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 one-forms) 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 boost-rotation parameters \({T_{\underline a}}\) and \({M_{\underline a \underline b}}\) are interpreted as the density of the energy-momentum and of the sum of the orbital and spin angular momenta, respectively. Since, however, the difference Ca[K] − TabKb is identically conserved and TabKb has more advantageous properties, it is TabKb, that is used to represent the energy-momentum and angular-momentum density of the matter fields.

Since in de Sitter and anti-de Sitter spacetimes the (ten-dimensional) Lie algebra of the Killing vector fields, so(1, 4) and so(2, 3), respectively, are semisimple, there is no such natural notion of translations, and hence no natural ‘4 + 6’ decomposition of the ten conserved currents into energy-momentum and (relativistic) angular momentum density.

Quasi-local energy-momentum and angular momentum of the matter fields

In Section 3 we will see that well-defined (i.e., gauge-invariant) energy-momentum and angular-momentum density cannot be associated with the gravitational ‘field’, and if we do not want to talk only about global gravitational energy-momentum and angular momentum, then these quantities must be assigned to extended, but finite, spacetime domains.

In the light of modern quantum-field-theory 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 quasi-local. Quantities associated with spacetime points or with the whole spacetime are not observable in this sense. In particular, global quantities, such as the total energy or electric charge, should be considered as the limit of quasi-locally-defined quantities. Thus, the idea of quasi-locality is not new in physics. Although in classical nongravitational physics this is not obligatory, we adopt this view in talking about energy-momentum and angular momentum even of classical matter fields in Minkowski spacetime. Originally, the introduction of these quasi-local quantities was motivated by the analogous gravitational quasi-local quantities [488, 492]. Since, however, many of the basic concepts and ideas behind the various gravitational quasi-local energy-momentum and angular momentum definitions can be understood from the analogous nongravitational quantities in Minkowski spacetime, we devote Section 2.2 to the discussion of them and their properties.

The definition of quasi-local quantities

To define the quasi-local conserved quantities in Minkowski spacetime, first observe that, for any Killing vector Ka ∈ K, the 3-form ωabc:= KeTef εfabc is closed, and hence, by the triviality of the third de Rham cohomology group, H3(ℝ4) = 0, it is exact: For some 2-form ⋃[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 3-form ωabc. (However, note that while the superpotential for the gravitational energy-momentum 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 Ka. The existence of globally-defined 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 ωabc, then by \({\nabla _{[a}}(\cup {[{\bf{K}}]_{bc]}} - \tilde \cup {[{\bf{K}}]_{bc]}}) = 0\) and H2(ℝ4) = 0 the dual superpotential is unique up to the addition of an exact 2-form. If, therefore, \({\mathcal S}\) is any closed orientable spacelike two-surface 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 two-boundary \({\mathcal S}\), then

$${Q_{\mathcal S}}[{\bf{K}}]: = {\textstyle{1 \over 2}}\oint\nolimits_{\mathcal S} {\cup {{[{\bf{K}}]}_{ab}}} = \int\nolimits_\Sigma {{K_e}{T^{ef}}{\textstyle{1 \over {3!}}}{\varepsilon _{f\;abc}}}$$
(2.5)

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 quasi-local 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 anti-symmetric 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 quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike two-surface \({\mathcal S}\), or, equivalently, to D(Σ). Then the quasi-local mass and Pauli-Lubanski 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 m2 ≠ 0, then the dimensionally-correct definition of the Pauli-Lubanski 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 Tabξb on the hypersurface even if ξa is not a Killing vector, even in general curved spacetime:

$${E_\Sigma}[{\xi ^a}]: = \int\nolimits_\Sigma {{\xi _e}{T^{e\;f}}{\textstyle{1 \over {3!}}}{\varepsilon _{f\;abc}}}.$$
(2.6)

then, however, the integral EΣ[ξa] does depend on the hypersurface, because it is not connected with the spacetime symmetries. In particular, the vector field ξa can be chosen to be the unit timelike normal ta of Σ. Since the component μ:= Tabtatb of the energy-momentum tensor is interpreted as the energy-density of the matter fields seen by the local observer ta, it would be legitimate to interpret the corresponding integral EΣ[ta] as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to Σ’. Thus, EΣ[ta] defines a different concept of the quasi-local 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, EΣ[ta] 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 ta of \({\mathcal S}\), EΣ[ξa] (for non-Killing ξa) depends on ta intrinsically and is a genuine three-hypersurface rather than a two-surface 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 energy-momentum tensor is interpreted as the momentum density seen by the observer ta. Hence,

$$({t_a}{T^{ab}})({t_c}{T^{cd}}){g_{bd}} = {\mu ^2} + {h_{ab}}{j^a}{j^b} = {\mu ^2} - \vert {j^a}{\vert ^2}$$

is the square of the mass density of the matter fields, where hab is the spatial metric in the plane orthogonal to ta. If Tab satisfies the dominant energy condition (i.e., TabVb is a future directed nonspacelike vector for any future directed nonspacelike vector Va, see, e.g., [240]), then this is non-negative, and hence,

$${M_\Sigma}: = \int\nolimits_\Sigma {\sqrt {{\mu ^2} - \vert {j^e}{\vert ^2}} {\textstyle{1 \over {3!}}}{t^f}{\varepsilon _{f\;abc}}}$$
(2.7)

can also be interpreted as the quasi-local 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 EΣ[K] for the four translational Killing vectors gives the four components of the energy-momentum \(P_{\mathcal S}^{\underline a}\), the mass MΣ 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 MΣ the norm of the current Tabtb is first taken with respect to the pointwise physical metric of the space-time, and then its integral is taken. Nevertheless, because of more advantageous properties (see Section 2.2.3), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form \({Q_{\mathcal S}}[{\bf{K}}]\) instead of EΣ[ξa].

Thus, even if there is a gauge-invariant and unambiguously-defined energy-momentum density of the matter fields, it is not a priori clear how the various quasi-local quantities should be introduced. We will see in the second part of this review that there are specific suggestions for the gravitational quasi-local energy that are analogous to \(P_{\mathcal S}^0\), others to EΣ[ta], and some to MΣ.

Hamiltonian introduction of the quasi-local quantities

In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see, e.g., [283, 558] and references therein) the configuration and momentum variables, ϕA and πA, respectively, are fields on a connected three-manifold Σ, which is interpreted as the typical leaf of a foliation Σt 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 Ka = Nta + Na (‘evolution vector field’ or ‘general time axis’), where ta is the future-directed unit normal to the leaves of the foliation and Na 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 Ci = Ci(ϕA, DeϕA,…, πA, DeπA,…), i = 1,…, n, of the canonical variables and their derivatives up to some finite order, where De 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

$$H\;[{\bf{K}}] = \int\nolimits_\Sigma {(\mu N + {j_a}{N^a} + {C_i}{N^i} + {D_a}{Z^a})} \;d\Sigma.$$
(2.8)

Here dΣ is the induced volume element, the coefficients μ and ja are local functions of the canonical variables and their derivatives up to some finite order, the Nis are functions on Σ, and Za is a local function of the canonical variables and is a linear function of the lapse, the shift, the functions Ni, and their derivatives up to some finite order. The part CiNi of the Hamiltonian generates gauge motions in the phase space, and the functions Ni are interpreted as the freely specifiable ‘gauge generators’.

However, if we want to recover the field equations for ϕA (which are partial differential equations on the spacetime with smooth coefficients for the smooth field ϕA) 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 Za. It may happen that, for a given Za, only too restrictive boundary conditions would be able to ensure the functional differentiability of the Hamiltonian, and, hence, the ‘quasi-local phase space’ defined with these boundary conditions would contain only very few (or no) solutions of the field equations. In this case, Za 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 energy-momentum and angular momentum, depending on the nature of Ka, in which the total divergence DaZa corresponds to the ambiguity of the superpotential 2-form ⋃[K]ab: An identically-conserved quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved). The energy density and the momentum density of the matter fields can be recovered as the functional derivative of H[K] with respect to the lapse N and the shift Na, respectively. In principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising achievements of [13, 14, 442] for the Klein-Gordon, Maxwell, and the Yang-Mills-Higgs fields, as far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still lacking.

Properties of the quasi-local quantities

Suppose that the matter fields satisfy the dominant energy condition. Then EΣ[ξa] is also non-negative for any nonspacelike ξa, and, obviously, EΣ[ta] is zero precisely when Tab = 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, MΣ = 0 if and only if \({L^a}: = {T^{ab}}{t_b}\) is null on Σ. Then, by the dominant energy condition it is a future-pointing vector field on Σ, and LaTab = 0 holds. Therefore, Tab on Σ has a null eigenvector with zero eigenvalue, i.e., its algebraic type on Σ is pure radiation.

The properties of the quasi-local 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. 1.

    \(P_{\mathcal S}^{\underline a}\) is a future directed nonspacelike vector, \(m_{\mathcal S}^2 \geq 0\)

  2. 2.

    \(P_{\mathcal S}^{\underline a}\) if and only if Tab = 0 on D(Σ);

  3. 3.

    \(m_{\mathcal S}^2 = 0\) if and only if the algebraic type of the matter on D(Σ) is pure radiation, i.e., TabLb = 0 holds for some constant null vector La. Then Tab = τLaLb for some non-negative 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. 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 Pauli-Lubanski spin is zero.

Therefore, the vanishing of the quasi-local energy-momentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasi-local mass is equivalent to special configurations representing pure radiation.

Since EΣ[ta] and MΣ are integrals of functions on a hypersurface, they are obviously additive, e.g., for any two hypersurfaces Σ1 and Σ2 (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 quasi-local energy-momenta of the different two-surfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different two-surfaces, then the energy-momenta would belong to different vector spaces, and they could not be added. We will see that the quasi-local quantities discussed in Sections 7, 8, and 9 belong to vector spaces dual to their own ‘quasi-Killing vectors’, and there is no natural way of adding the energy-momenta of different surfaces.

Global energy-momenta 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 quasi-local energy-momentum can be ensured by slightly faster than \({\mathcal O}({r^{- 3}})\) (for example by \({\mathcal O}({r^{- 4}})\) falloff of the energy-momentum tensor, where r is any spatial radial distance. However, the finiteness of the angular momentum and center-of-mass is not ensured by the \({\mathcal O}({r^{- 4}})\) falloff. Since the typical falloff of Tab — 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 Tab must be imposed. At spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’ Tab;b = 0 (as evolution equations for the energy density and momentum density) preserve these falloff and parity conditions [497].

Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass, the fields must be plane waves, furthermore, by \({T_{ab}} = {\mathcal O}({r^{- 4}})\), they must be asymptotically vanishing at the same time. However, a plane-wave configuration can be asymptotically vanishing only if it is vanishing.

Quasi-local radiative modes and a classical version of the holography for matter fields

By the results of Section 2.2.4, the vanishing of the quasi-local mass, associated with a closed spacelike two-surface \({\mathcal S}\), implies that the matter must be pure radiation on a four-dimensional 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 two-surface \({\mathcal S}\), 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 Yang-Mills 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 quasi-local mass is equivalent [498] to plane waves and the pp-wave solutions of Coleman [152], respectively. Then, the condition TabLb = 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: \(\phi _{AB}^\alpha = {\phi ^\alpha}{O_A}{O_B}\), whereϕα are complex functions and OA is a constant spinor field such that La = OAOA). Similarly, the null linear zero-rest-mass fields ϕAB…E = ϕOAOBOE on D(Σ) with any spin and constant spinor OA are completely determined by their value on \({\mathcal S}\). Technically, these results are based on the unique complex analytic structure of the u = const. two-surfaces foliating Σ, where La = ∇au, and, by the field equations, the complex functions ϕ and ϕα 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 ‘quasi-local 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 ϕOA (with fixed constant-spinor field OA), 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 Σt, then every spinor field on D(Σ) can also be represented on \({\mathcal S}\) by a time-dependent 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 Energy-Momentum and Angular Momentum of Gravitating Systems

On the gravitational energy-momentum 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 3-space Σ ≈ ℝ3 satisfying the Poisson equation −habDaDbϕ = 4πGρ. (Here hab is the flat (negative definite) metric, Da is the corresponding Levi-Civita covariant derivative operator and ρ is the (non-negative) mass density of the matter source.) Hence, the mass of the source contained in some finite three-volume D ⊂ Σ can be expressed as the flux integral of the gravitational field strength on the boundary \({\mathcal S}: = \partial D\)

$${m_D} = {1 \over {4\pi G}}\oint\nolimits_{\mathcal S} {{\upsilon ^a}({D_a}\phi)\;d{\mathcal S}},$$
(3.1)

where va is the outward-directed unit normal to \({\mathcal S}\). If \({\mathcal S}\) is deformed in Σ through a source-free 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

$$\begin{array}{*{20}c}{U: = {1 \over {8\pi G}}{h^{cd}}({D_c}\phi)\,({D_d}\phi),} & {{\Sigma _{ab}}: = {1 \over {4\pi G}}\left({({D_a}\phi)\,({D_b}\phi) - {1 \over 2}{h_{ab}}{h^{cd}}({D_c}\phi)\,({D_d}\phi)} \right).} \\ \end{array}$$
(3.2)

Note that since gravitation is always attractive, U is a binding energy, and hence it is negative definite. However, by the Galileo-Eötvös 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 ae, and hence by an appropriate transformation DeϕDeϕ + ae the gravitational force Deϕ 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, DaDbϕ, 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 non-linear equation

$$- {h^{ab}}{D_a}{D_b}\phi = 4\pi G\left({\rho + {1 \over {{c^2}}}\left({u + U} \right)} \right).$$
(3.3)

(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 post-Newtonian 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)

$${E_D}: = \int\nolimits_D {({c^2}\rho + u + U){\rm{d}}\Sigma} = {{{c^2}} \over {4\pi G}}\oint\nolimits_{\mathcal S} {{\upsilon ^a}({D_a}\phi)\;\,d{\mathcal S}},$$
(3.4)

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 two-surface integral on the boundary of the domain D. Note that the gravitational energy reduces the source term in (3.3) (and hence the energy ED also), and, more importantly, the quasi-local energy ED 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 quasi-local concept of energy in the study of gravitating systems.

By the negative definiteness of U, outside the source the quasi-local energy ED is a decreasing set function, i.e., if D1D2 and D2D1 is source free, then \({E_{{D_2}}} \leq {E_{{D_1}}}\). In particular, for a 2-sphere of radius r surrounding a localized spherically symmetric homogeneous source with negligible internal energy, the quasi-local 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 energy-momentum in Einstein’s theory

The action Im 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 energy-momentum tensor. Moreover, gab provides a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action Ig is, on the other hand, a functional of the metric alone, and its variational derivative with respect to gab yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of gab can be traced back to the principle of equivalence [36] (i.e., the Galileo-Eötvös experiment), and introduces a huge gauge freedom in the dynamics of gab because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold M endowed with a Lorentzian metric gab, but the isomorphism class of such pairs, where (M, gab) and (M, ϕ*gab) 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 energy-momentum tensor of the matter fields. In fact, by its very definition, Tab is the source density for gravity, like the current \(J_A^a: = \delta {I_p}/\delta A_a^A\) in Yang-Mills theories (defined by the variational derivative of the action functional of the particles, e.g., of the fermions, interacting with a Yang-Mills field \(A_a^A\)), rather than energy-momentum. The latter is represented by the Noether currents associated with special spacetime displacements. Thus, in spite of the intimate relation between Tab and the Noether currents, the proper interpretation of Tab is only the source density for gravity, and hence it is not the symmetric energy-momentum tensor whose gravitational counterpart must be searched for. In particular, the Bel-Robinson tensor \({T_{abcd}}: = {\psi _{ABCD}}{{\bar \psi}_{{A{\prime}}{B{\prime}}{C{\prime}}{D{\prime}}}}\), 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 ‘higher-order’ gravitational energy-momentum. (Note that according to the original tensorial definition the Bel-Robinson tensor is one-fourth of the expression above. Our convention follows that of Penrose and Rindler [425].) In fact, the physical dimension of the Bel-Robinson ‘energy-density’ Tabcdtatbtctd is cm−4, and hence (in the traditional units) there are no powers A and B such that cAGB Tabcdtatbtctd would have energy-density dimension. As we will see, the Bel-Robinson ‘energy-momentum density’ Tabcdtbtctd appears naturally in connection with the quasi-local energy-momentum and spin angular momentum expressions for small spheres only in higher-order terms. Therefore, if we want to associate energy-momentum and angular momentum with the gravity itself in a Lagrangian framework, then it is the gravitational counterpart of the canonical energy-momentum and spin tensors and the canonical Noether current built from them that should be introduced. Hence it seems natural to apply the Lagrange-Belinfante-Rosenfeld procedure, sketched in the previous Section 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 Ka generates a symmetry of the matter-plus-gravity system. Its second consequence is the need for an auxiliary derivative operator, e.g., the Levi-Civita covariant derivative coming from an auxiliary, 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 second-order Lagrangian LH:= R/16πG in a fixed local coordinate system {xα} and derivative operator μ instead of e, Eq. (2.4) gives precisely Møller’s energy-momentum 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 energy-momentum pseudotensor, see [131].) For the spin pseudotensor, Eq. (2.2) gives

$$8\pi G{\sqrt {\vert g\vert} _{\rm{M}}}{\sigma ^{\mu \alpha}}_\beta = {- _{\rm{M}}}{\cup _\beta}^{\alpha \mu} + \;{\partial _\nu}\left({\sqrt {\vert g\vert} \delta _\beta ^{\left[ \mu \right.}{g^{\left. \nu \right]\alpha}}} \right),$$

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 energy-momentum and spin tensors gave the symmetric energy-momentum tensor, which is gauge invariant even if the matter fields have gauge freedom, and one might hope that the analogous combination of the energy-momentum and spin pseudotensors gives a reasonable tensorial energy-momentum density for the gravitational field. The analogous expression is, in fact, tensorial, but unfortunately it is just 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 kR Goldberg’s 2kth symmetric pseudotensor \(t_{(2k)}^{\alpha \beta}\) is defined by \(2\vert g{\vert ^{k + 1}}(8\pi Gt_{(2k)}^{\alpha \beta} - {G^{\alpha \beta}}): = {\partial _\mu}{\partial _\nu}[\vert g{\vert ^{k + 1}}({g^{\alpha \beta}}{g^{\mu \nu}} - {g^{\alpha \nu}}{g^{\beta \mu}})]\) (which, for k = 0, reduces to the Landau-Lifshitz 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-)divergence-free (i.e., ‘pseudo-conserved’) cannot be interpreted as the sum of the gravitational and matter energy-momentum densities. Indeed, the latter is |g|1/2 Tαβ, 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 Landau-Lifshitz 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 Lynden-Bell [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 energy-momentum and spin tensors, but to define the vector fields Ka 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 Ka: The canonical energy-momentum and spin tensors themselves are explicitly background-dependent. Thus, again, the resulting expressions would have to be supplemented by a ‘natural’ choice for the background, and the main question is how to find such a ‘natural’ reference configuration from the infinitely many possibilities. 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 gab-orthonormal frame fields \(\{E_{\underline a}^a\}, \underline a = 0, \ldots, 3\), are chosen to be the gravitational field variables [533, 314]. Re-expressing the Hilbert Lagrangian (i.e., the curvature scalar) in terms of the tetrad field and its partial derivatives in some local coordinate system, one can calculate the canonical energy-momentum and spin by 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 first-order, 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

$$L: = {1 \over {16\pi G}}\left[ {R - 2{\nabla _a}(E_{\underline a}^a{\eta ^{\underline a \underline b}}\gamma _{\underline c \underline b}^{\underline c})} \right] = {1 \over {16\pi G}}\left({E_{\underline a}^aR_{\underline b}^b - E_{\underline a}^bE_{\underline b}^a} \right)\gamma _{a\underline c}^{\underline a}\gamma _b^{\underline c \underline b}.$$
(3.5)

(Here \(\left\{{\vartheta _a^{\underline a}} \right\}\) is the one-form 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 first-order scalar Lagrangian built from the tetrad fields, whose Euler-Lagrange equations are the Einstein equations, is Møller’s Lagrangian. (Using Dirac spinor variables Nester and Tung found a first-order spinor Lagrangian [392], which turned out to be equivalent to Møller’s Lagrangian [530]. Another first-order spinor Lagrangian, based on the use of the two-component spinors and the anti-self-dual connection, was suggested by Tung and Jacobson [529]. Both Lagrangians yield a well-defined Hamiltonian, reproducing the standard ADM energy-momentum in asymptotically flat spacetimes.) The canonical energy-momentum θ 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:

$${\vee _b}^{ae}: = 2\left({- \gamma _{\underline b \underline c}^{\underline a}{\eta ^{\underline c \underline e}} + \gamma _{\underline d \underline c}^{\underline d}{\eta ^{\underline c \underline s}}\left({\delta _{\underline b}^{\underline a}\delta _{\underline s}^{\underline e} - \delta _{\underline s}^{\underline a}\delta _{\underline b}^{\underline e}} \right)} \right)\;\vartheta _b^{\underline b}E_{\underline a}^aE_{\underline e}^e = {\vee _b}^{[ae]}.$$
(3.6)

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 energy-momentum tensor for the gravitational ‘field’. Then, the corresponding canonical Noether current Ca[K] will also be tensorial and satisfies

$$8\pi G{C^a}[{\bf{K}}] = {G^{ab}}{K_b} + {\textstyle{1 \over 2}}{\nabla _c}({K^b}{\vee _b}^{ac}).$$
(3.7)

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, Ca[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 energy-momentum and angular momentum, at least in disguise. In particular, the Hamiltonian approach in itself does not yield a well defined energy-momentum 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 Ca[K] + TabKb is conserved for any vector field Ka, 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 energy-momentum 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 UM. If the domain of the frame field can be extended to the whole M, then M is called parallelizable. For time and space-orientable spacetimes this is equivalent to the existence of a spinor structure [206], which is known to be equivalent to the vanishing of the second Stiefel-Whitney class of M [364], a global topological condition on M.

The discussion of how Møller’s superpotential \({\vee _e}^{ab}\) is related to the Nester-Witten 2-form, by means of which an alternative form of the ADM energy-momentum is given and and by means of which several quasi-local energy-momentum 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 Ka and its first derivative — i.e., if we allow a term 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:= ∇[aKb] 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 Ka 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 energy-momentum and angular momentum of gravitating systems: The successes

As is well known, in spite of the difficulties with the notion of the gravitational energy-momentum density discussed above, reasonable total energy-momentum and angular momentum can be associated with the whole spacetime, provided it is asymptotically flat. In the present section we recall the various forms of them. As we will see, most of the quasi-local constructions are simply ‘quasi-localizations’ of the total quantities. Obviously, the technique used in the ‘quasi-localization’ does depend on the actual form of the total quantities, yielding mathematically-inequivalent definitions for the quasi-local quantities. We return to the discussion of the tools needed in the quasi-localization procedures in Sections 4.2 and 4.3. Classical, excellent reviews of global energy-momentum and angular momentum are [208, 223, 28, 393, 553, 426], and a recent review of con-formal 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: Energy-momentum

There are several mathematically-inequivalent definitions of asymptotic flatness at spatial infinity [208, 475, 37, 65, 200]. The traditional definition is based on the existence of a certain asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable definition are asymptotically flat in the traditional sense as well. A spacelike hypersurface Σ will be called k-asymptotically flat if for some compact set K ⊂ Σ the complement Σ − K is diffeomorphic to ℝ3 minus a solid ball, and there exists a (negative definite) metric 0hab on Σ, which is flat on Σ − K, such that the components of the difference of the physical and the background metrics, hij0hij, and of the extrinsic curvature χij in the 0hij-Cartesian coordinate system {xk} fall off as rk and rk−1, respectively, for some k > 0 and r2:= δijxixj [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 well-defined ADM energy-momentum [23, 25, 24, 26] can be associated with any k-asymptotically flat spacelike hypersurface, if \(k > {1 \over 2}\), by taking the value on the constraint surface of the Hamiltonian H[Ka], given, for example, in [433, 64], with the asymptotic translations Ka (see [144, 52, 399, 145]). In its standard form, this is the r → ∞ limit of a two-surface integral of the first derivatives of the induced three-metric hab and of the extrinsic curvature χab for spheres \({\mathcal S_r}\) of large coordinate radius r. Explicitly:

$$E = {1 \over {16\pi G}}\;\underset {r \rightarrow \infty}{\lim} \oint\nolimits_{{{\mathcal S}_r}} {{\upsilon ^a}{{{(_0}{D_c}{h_{da}}{- _0}{D_a}{h_{cd}})}_0}{h^{cd}}{\rm{d}}{{\mathcal S}_r}},$$
(3.8)
$${P^{\bf{i}}} = - {1 \over {8\pi G}} \underset {r \rightarrow \infty}{\lim} \oint\nolimits_{{{\mathcal S}_r}} {{\upsilon ^a}({\chi _a}^b - \chi \delta _a^b){\;_0}D{x^{\bf{i}}}{\rm{d}}{{\mathcal S}_r}},$$
(3.9)

where 0De is the Levi-Civita derivative oparator determined by 0hab, and va is the outward pointing unit normal to \({{\mathcal S}_r}\) and tangent to Σ. The ADM energy-momentum, \({P^{\underline a}} = (E,{P^{\rm{i}}}\), is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian four-vector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian coordinates.

The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian analysis of general relativity is based on the 3+1 decomposition of the fields and the spacetime. Thus, it is not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular momentum and center-of-mass, discussed below, form an antisymmetric tensor). One has to check a posteriori that the conserved quantities obtained in the 3 + 1 form are, in fact, Lorentz-covariant. To obtain manifestly Lorentz-covariant quantities one should not do the 3 + 1 decomposition. Such a manifestly Lorentz-covariant Hamiltonian analysis was suggested first by Nester [377], and he was able to recover the ADM energy-momentum in a natural way (see Section 11.3).

Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [223]: Taking the flux integral of the current Ca [K] + TabKb on the spacelike hypersurface Σ, by Eq. (3.7) the flux can be rewritten as the r → ∞ limit of the two-surface integral of Møller’s superpotential on spheres of large r with the asymptotic translations Ka. 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 rk, the integral reproduces the ADM energy-momentum. The same expression can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too: By the standard scenario one can construct the basic Hamiltonian [379]. This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of Ca[K] + TabKb, on Σ.

A particularly interesting and useful expression for the ADM energy-momentum is possible if the tetrad field is considered to be a frame field built from a normalized spinor dyad \(\{\lambda _A^{\underline A}\}, \underline A = 0,1\), 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 energy-momentum in the constant spinor basis at infinity, Møller’s expression yields the limit of

$${P^{\underline A \underline {B{\prime}}}} = {1 \over {4\pi G}}\oint\nolimits_{\mathcal S} {{{\rm{i}} \over 2}} \left({\overline \lambda _{A{\prime}}^{\underline {B{\prime}}}{\nabla _{BB{\prime}}}\lambda _A^{\underline A} - \overline \lambda _{B{\prime}}^{\underline {B{\prime}}}{\nabla _{AA{\prime}}}\lambda _B^{\underline A}} \right),$$
(3.10)

as the two-surface \({\mathcal S}\) is blown up to approach infinity. In fact, to recover the ADM energy-momentum 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 Ka 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 complex-valued 2-form 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 Nester-Witten 2-form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential, too. In fact, for any two spinor fields αA and βA one has

$$u{\left({\alpha, \overline \beta} \right)_{ab}} - \overline {u{{(\beta, \overline \alpha)}_{ab}}} = - {\rm{i}}{\nabla _{\left[ a \right.}}{X_{\left. b \right]}},$$
(3.11)
$$u{\left({\alpha, \bar \beta} \right)_{ab}} - \overline {u{{(\beta, \bar \alpha)}_{ab}}} = {\textstyle{1 \over 2}}{\nabla _c}{X_d}{\varepsilon ^{cd}}_{ab} + {\rm{i}}\left({{\varepsilon _{A{\prime}B{\prime}}}{\alpha _{\left(A \right.}}{\nabla _{\left. B \right)C{\prime}}}{{\overline \beta}^{C{\prime}}} - {\varepsilon _{A\,B}}{{\overline \beta}_{{{\left(A{\prime}\right.}}}}{\nabla _{\left. {B{\prime}} \right)C}}{\alpha ^C}} \right),$$
(3.12)

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′AαA and \({\nabla _{A{A{\prime}}}}{{\bar \beta}^{{A{\prime}}}}\), the Nester-Witten 2-form \(u{(\alpha, \bar \beta)_{ab}}\) 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 anti-self-dual part of the curl of \(- {{\rm{i}} \over 2}{X_a}\) (The original expressions by Witten and Nester were given using Dirac, rather than two-component Weyl, spinors [559, 376]. The 2-form \(u{(\alpha, \bar \beta)_{ab}}\) in the present form using the two-component spinors probably appeared first in [276].) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [444, 445, 559, 376, 273, 427, 300], the simplest and most transparent ones are probably those based on the use of two-component spinors: If the dominant energy condition is satisfied on the k-asymptotically flat spacelike hypersurface Σ, where \(k > {1 \over 2}\), then the ADM energy-momentum is future pointing and nonspacelike (i.e., the Lorentzian length of the energy-momentum vector, the ADM mass, is non-negative), and is null if and only if the domain of dependence D(Σ) of Σ is flat [276, 434, 217, 436, 88]. Its proof may be based on the Sparling equation [476, 175, 426, 358]:

$${\nabla _{\left[ a \right.}}u{(\lambda, \overline \mu)_{\left. {bc} \right]}} = - {1 \over 2}{\lambda _E}{\overline \mu _{E{\prime}}}{G^{e\;f}}{1 \over {3!}}{\varepsilon _{f\;abc}} + \Gamma {(\lambda, \overline \mu)_{abc}}.$$
(3.13)

The significance of this equation is that, in the exterior derivative of the Nester-Witten 2-form, the second derivatives of the metric appear only through the Einstein tensor, thus its structure is similar to that of the superpotential equations in Lagrangian field theory, and \(\Gamma {(\lambda, \mu)_{abc}}\), known as the Sparling 3-form, is a homogeneous quadratic expression of the first derivatives of the spinor fields. If the spinor fields λA and μA 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 energy-momentum can also be written as the two-sphere integral of certain parts of the conformally rescaled spacetime curvature [28, 29, 43]. This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [223]): If \({\mathcal S}\) is any closed spacelike two-surface with area element \(d{\mathcal S}\), then for any tensor fields ωab = ω[ab] and μab = μ[ab] one can form the integral

$${I_{\mathcal S}}[\omega, \mu ]: = \oint\nolimits_{\mathcal S} {{\omega ^{ab}}{R_{abcd}}{\mu ^{cd}}d{\mathcal S}}.$$
(3.14)

Since the falloff of the curvature tensor near spatial infinity is rk−2, the integral \({I_{\mathcal S}}[\omega, \mu ]\) at spatial infinity gives finite value when ωabμcd blows up like rk 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 two-sphere integral of (twice of) Komar’s superpotential with the Killing vector Ka 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 Ka should be a constant combination of four future-pointing null vector fields of the form \({\alpha ^A}{{\bar \alpha}^{{A{\prime}}}}\), where the spinor fields aA are required to satisfy the Weyl neutrino equation ∇A′AαA = 0. This expression for the ADM energy-momentum has been used to give an alternative, ‘four-dimensional’ proof of the positivity of the ADM energy [276]. (For a more detailed recent review of the various forms of the ADM energy and linear momentum, see, e.g., [293].)

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

The notion of asymptotic flatness at spatial infinity is generalized in [398]; here the background flat metric 0hab 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 −dr2r2(1 − α)(2 + sin2 (θ) 2). Then, a canonical analysis of the minimally-coupled Einstein-Higgs field is carried out on such a background, and, following a Regge-Teitelboim-type argumentation, an ADM-type total energy is introduced. It is shown that for appropriately chosen 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 appropriately-defined asymptotic rotation-boost Killing vectors [497], define the spatial angular momentum and center-of-mass, 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 hab and extrinsic curvature χab: The components in the Cartesian coordinates {xi} of the former must be even and the components of the latter must be odd parity functions of xi/r (see also [64]). Thus, in what follows we assume that k = 1. Then the value of the Beig-Ó Murchadha Hamiltonian parametrized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [433], while that parametrized by the asymptotic boost Killing vectors deviates from the center-of-mass of Beig and Ó Murchadha [64] by a term, which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [64], the center-of-mass term of the Hamiltonian of Regge and Teitelboim is not finite on the whole phase space.) The spatial angular momentum and the new center-of-mass form an anti-symmetric Lorentz four-tensor, which transforms in the correct way under the four-translation of the origin of the asymptotically Cartesian coordinate system, and is conserved by the evolution equations [497].

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

The Ashtekar-Hansen definition for the angular momentum is introduced in their specific conformal model of spatial infinity as a certain two-surface integral near infinity. However, their angular momentum expression is finite and unambiguously defined only if the magnetic part of the spacetime curvature tensor (with respect to the 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 (‘factor-of-two anomaly’) [305]. We return to the discussion of the Komar integral in Sections 4.3.1 and 12.1.

Null infinity: Energy-momentum

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

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

$${L_{\mathcal S}}[{\bf{K}}]: = {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {\left({{\nabla ^{\left[ c \right.}}{K^{\left. d \right]}} + \alpha {\nabla _e}{K^{e\; \bot}}{\varepsilon ^{cd}}} \right){1 \over 2}{\varepsilon _{cdab}}},$$
(3.15)

where εcd is the area 2-form on the Lorentzian two-planes 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 two-surface \({\mathcal S}\) tends to the cut and how the vector field Ka 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 conformally-rescaled spacetime and while requiring that Ka be divergence-free [210]. For such asymptotic BMS translations both prescriptions give the correct expression for the Bondi-Sachs energy-momentum.

The Bondi-Sachs energy-momentum can also be expressed by the integral of the Nester-Witten 2-form [285, 342, 343, 276]. However, in nonstationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [106]. Thus, the spinor fields in the Nester-Witten 2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves are the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [106], was to require the spinor fields to be the solutions of the asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all of these, based only on the linear first-order differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are determined in [496].

The Bondi-Sachs energy-momentum has a Hamiltonian interpretation as well. Although the fields on a spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable generalization of the standard Hamiltonian analysis could be developed [146] and used to recover the Bondi-Sachs energy-momentum.

Similar to the ADM case, the simplest proofs of the positivity of the Bondi energy [446] are probably those that are based on the Nester-Witten 2-form [285] and, in particular, the use of two-component spinors [342, 343, 276, 274, 436]: The Bondi-Sachs mass (i.e., the Lorentzian length of the Bondi-Sachs energy-momentum) of a cut of future null infinity is non-negative if there is a spacelike hypersurface 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 Nester-Witten 2-form into a (positive definite) 3-dimensional integral on Σ, a strictly positive lower bound can be given both for the ADM and Bondi-Sachs masses. Although total energy-momentum (or mass) in the form of a two-surface integral cannot be a introduced in closed universes (i.e., when Σ is compact with no boundary), a non-negative 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 3-torus; moreover its vanishing is equivalent to the existence of non-trivial solutions of Witten’s gauge condition. This m turned out to be recoverable as the first eigenvalue of the square of the Sen-Witten operator. It is the usefulness and the applicability of this 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 axi-symmetric, but not in general, radiative spacetime, where there are various, mathematically inequivalent suggestions for it (see Section 4.2.4). Here we review only some of those total angular momentum definitions that can be ‘quasi-localized’ or connected somehow to quasi-local expressions, i.e., those that can be considered as the null-infinity limit of some quasi-local expression. We will continue their discussion in the main part of the review, namely in Sections 7.2, 11.1 and 9.

In their classic paper Bergmann and Thomson [78] raise the idea that while the gravitational energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic 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 Yang-Mills theories, Bramson suggested a superpotential for the six conserved currents corresponding to the internal Lorentz-symmetry [107, 108, 109]. (For another derivation of this superpotential from Møller’s Lagrangian (3.5) see [496].) If \(\{\lambda _A^{\underline A}\}, \underline A = 0,1\), is a normalized spinor dyad corresponding to the orthonormal frame in Eq. (3.5), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable two-surface \({\mathcal S}\) is

$$J_{\mathcal S}^{\underline A \underline B}: = {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {- {\rm{i}}\lambda _{\left(A \right.}^{\underline A}\lambda _{\left. B \right)}^{\underline B}{\varepsilon _{A{\prime}B{\prime}}}},$$
(3.16)

where εA′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 well-defined expression. For stationary spacetimes this reduces to the generally accepted formula (4.15), and the corresponding Pauli-Lubanski 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 Bondi-Sachs energy-momentum \({P^{\underline A \underline {{A{\prime}}}}}\) (given, for example, in the Newman-Penrose formalism by Eq. (4.14)), is invariant with respect to supertranslations of the cut (‘active supertranslations’). Note that since Bramson’s expression is based on the solutions of a system of partial differential equations on the cut in question, it is independent of the parametrization of the BMS vector fields. Hence, in particular, it is invariant with respect to the supertranslations of the origin cut (‘passive supertranslations’). Therefore, Bramson’s global angular momentum behaves like the spin part of the total angular momentum. For a suggestion based on Bramson’s superpotential at the quasi-local level, but using a different prescription for the spinor dyad, see Section 9.

The construction based on the Winicour-Tamburino linkage (3.15) can be associated with any BMS vector field [557, 337, 45]. In the special case of translations it reproduces the Bondi-Sachs energy-momentum. The quantities that it defines for the proper supertranslations are called the super-momenta. For the boost-rotation vector fields they can be interpreted as angular momentum. However, in addition to the factor-of-two anomaly, this notion of angular momentum contains a huge ambiguity (‘supertranslation ambiguity’): The actual form of both the boost-rotation Killing vector fields of Minkowski spacetime and the boost-rotation BMS vector fields at future null infinity depend on the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively. However, while the set of the origins of Minkowski spacetime is parametrized by four numbers, the set of the origins at null infinity requires a smooth function of the form \(u:{S^2} \rightarrow {\rm{\mathbb R}}\). Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar origin-dependence (containing four parameters), the analogous transformation of the angular momentum defined by using the boost-rotation BMS vector fields depends on an arbitrary smooth real valued function on the two-sphere. This makes the angular momentum defined at null infinity by the boost-rotation BMS vector fields ambiguous unless a natural selection rule for the origins, making them form a four parameter family of cuts, is found.

Motivated by Penrose’s idea that the ‘conserved’ quantities at null infinity should be searched for in the form of a charge integral of the curvature (which will be discussed in detail in Section 7), a general expression \({Q_{\mathcal S}}[{K^a}]\), associated with any BMS generator Ka and any cut \({\mathcal S}\) of scri, was introduced [174]. For real Ka this is real; it is vanishing in Minkowski spacetime; it reproduces the Bondi-Sachs energy-momentum for BMS translations; it yields nontrivial results for proper supertranslations; and for BMS rotations the resulting expressions can be interpreted as angular momentum. It was shown in [453, 173] that the difference \({Q_{{{\mathcal S}{\prime}}}}[{K^a}] - {Q_{{{\mathcal S}{{\prime\prime}}}}}[{K^a}]\) 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 four-real-parameter family of cuts, called nice cuts, by means of which the BMS group can be reduced to a Poincaré subgroup that yields a well-defined notion of angular momentum. For further discussion of certain other angular momentum expressions, especially from the points of view of numerical calculations, see also [204].

Another promising approach might be that of 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 four-space of origins, they appear to be the generators with respect to some fixed ‘center-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.

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

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

The main idea behind the recent definition of the total angular momentum at future null infinity of Kozameh, Newman and Silva-Ortigoza, suggested in [325, 326], is analogous to finding the center-of-charge (i.e., the time-dependent position vector with respect to which the electric dipole moment is vanishing) in flat-space electromagnetism: By requiring that the dipole part of an appropriate null rotated Weyl tensor component \(\psi _1^0\) be vanishing, a preferred set of origins, namely a (complex) center-of-mass line can be found in the four-complex-dimensional solution space of the good-cut equation (the H-space). Then the asymptotic Bianchi identities take the form of conservation equations, and certain terms in these can (in the given approximation) be identified with angular momentum. The resulting expression is just Eq. (4.15), to which all the other reasonable angular momentum expressions are expected to reduce in stationary spacetimes. A slightly more detailed discussion of the necessary technical background is given in Section 4.2.4.

The necessity of quasi-locality for observables in general relativity

Nonlocality of the gravitational energy-momentum and angular momentum

One reaction to the nontensorial nature of the gravitational energy-momentum density expressions was to consider the whole problem ill defined and the gravitational energy-momentum meaningless. However, the successes discussed in Section 3.2 show that the global gravitational energy-momenta and angular momenta are useful notions, and hence, it could also be useful to introduce them even if the spacetime is not asymptotically flat. Furthermore, the 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 energy-momentum 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, Landau-Lifshitz 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 Nester-Witten 2-form, along various local cross sections [192, 358, 486, 487], and the expression of the pseudotensors by their super-potentials are the pullbacks of the Sparling equation [476, 175, 358]. In addition, Chang, Nester, and Chen [131] found a natural quasi-local 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 energy-momentum and angular momentum are necessarily associated with extended subsets of the spacetime.

This fact is a particular consequence of a more general phenomenon [76, 439, 284]: Since (in the absence of any non-dynamical geometric background) the physical spacetime is the isomorphism class of the pairs (M, gab) (instead of a single such pair), it is meaningless to speak about the ‘value of a scalar or vector field at a point pM’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), two-surfaces (e.g., the area of a closed two-surface) etc. determined by some body or physical fields. In addition, as Torre showed [523] (see also [524]), in spatially-closed 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 energy-momentum and angular momentum not only to the whole (necessarily asymptotically flat) spacetime, then these quantities must be associated with extended but finite subsets of the spacetime, i.e., must be quasi-local.

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 first-order 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 quasi-local quantities

The quasi-local quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:

  1. 1.

    the globally hyperbolic domains DM with compact closure,

  2. 2.

    the compact spacelike (in fact, acausal) hypersurfaces Σ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and

  3. 3.

    the closed, orientable spacelike two-surfaces \({\mathcal S}\) (interpreted as the boundary Σ of Cauchy surfaces for globally hyperbolic domains).

A typical example of type 3 is any charge integral expression: The quasi-local quantity is the integral of some superpotential 2-form built from the data given on the two-surface, 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 Bel-Robinson ‘momentum’ on the hypersurface Σ:

$${E_\Sigma}[{\xi ^a}]: = \int\nolimits_\Sigma {{\xi ^d}{T_{de\,f\,g}}{t^e}{t^f}{\textstyle{1 \over {3!}}}{\varepsilon ^g}_{abc}}.$$
(3.17)

This quantity is analogous to the integral EΣ[ξa] for the matter fields given by Eq. (2.6) (though, by the remarks on the Bel-Robinson ‘energy’ in Section 3.1.2, its physical dimension cannot be of energy). If ξa is a future-pointing nonspacelike vector then EΣ[ξa] ≥ 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 three-surface 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

$${E_D}: = \inf \;\{{E_\Sigma}[{\bf{t}}]\vert \Sigma \;{\rm{is}}\;{\rm{a}}\;{\rm{Cauchy}}\;{\rm{surface}}\;{\rm{for}}\;D\},$$
(3.18)

the infimum of the ‘quasi-local Bel-Robinson 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 Bel-Robinson ‘energy density’ Tabcdtatbtctd is non-negative.) Quasi-locality in any of these three senses is compatible with the quasi-locality of Haag and Kastler [231, 232]. The specific quasi-local energy-momentum constructions provide further examples both for charge-integraltype expressions and for those based on spacelike hypersurfaces.

Strategies to construct quasi-local quantities

There are two natural ways of finding the quasi-local energy-momentum and angular momentum. The first is to follow some systematic procedure, while the second is the ‘quasi-localization’ of the global energy-momentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasi-local quantities are integrals of some superpotential derived from the Lagrangian via a Noether-type analysis. The advantage of this approach could be its manifest Lorentz-covariance. On the other hand, since the Noether current is determined only through the Noether identity, which contains only the divergence of the current itself, the Noether current and its superpotential is not uniquely determined. In addition (as in any approach), a gauge reduction (for example in the form of a background metric or reference configuration) and a choice for the ‘translations’ and ‘boost-rotations’ should be made.

The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasi-local (covariant or not) Hamiltonian analysis we would have a Hamiltonian, and its value on the constraint surface in the phase space yields the expected quantities. Here 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 ‘boost-rotations’ too. Another idea is the expectation, based on the study of the quasi-local 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 energy-momentum 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 quasi-local phase space, then, as a short cut, we can use the Hamilton-Jacobi method to define the quasi-local quantities. The resulting expression is a two-surface integral. Nevertheless, just as in the Lagrangian approach, this general expression is not uniquely determined, because the action can be modified by adding an (almost freely chosen) boundary term to it. Furthermore, the ‘translations’ and ‘boost-rotations’ are still to be specified.

On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasi-local quantities would be some ‘quasi-localization’ of those expressions that gave the global energy-momentum and angular momentum of asymptotically flat spacetimes. Therefore, respecting both strategies, it is also legitimate to consider the Winicour-Tamburino-type (linkage) integrals and the charge integrals of the curvature.

Since the global energy-momentum and angular momentum of asymptotically flat spacetimes can be written as two-surface 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 energy-momentum and angular momentum of the source in the linearized Einstein theory can also be written as two-surface integrals), the two-surface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasi-local energy-momentum and angular momentum as two-surface observables, then three things must be specified:

  1. 1.

    an appropriate general two-surface integral (e.g., in the Lagrangian approaches the integral of a superpotential 2-form, or in the Hamiltonian approaches a boundary term together with the boundary conditions for the canonical variables),

  2. 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. 3.

    a definition for the ‘quasi-symmetries’ of the two-surface (i.e., the ‘generator vector fields’ of the quasi-local 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 two-surface).

In certain approaches the definition of the ‘quasi-symmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.

Tools to Construct and Analyze Quasi-Local Quantities

Having accepted that the gravitational energy-momentum and angular momentum should be introduced at the quasi-local level, we next need to discuss the special tools and concepts that are needed in practice to construct (or even to understand) the various special quasi-local expressions. Thus, first, in Section 4.1 we review the geometry of closed spacelike two-surfaces, with special emphasis on two-surface data. Then, in Sections 4.2 and 4.3, we discuss the special situations where there is a more-or-less generally accepted ‘standard’ definition for the energy-momentum (or at least for the mass) and angular momentum. In these situations any reasonable quasi-local quantity should reduce to them.

The geometry of spacelike two-surfaces

The first systematic study of the geometry of spacelike two-surfaces from the point of view of quasi-local quantities is probably due to Tod [514, 519]. Essentially, his approach is based on the Geroch-Held-Penrose (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 spacetime-covariant description of the geometry of the spacelike two-surfaces, 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 two-surfaces. Our standard differential geometric reference is [318, 319].

The Lorentzian vector bundle

The restriction \({{\rm{V}}^a}({\mathcal S})\) to the closed, orientable spacelike two-surface \({\mathcal S}\) of the tangent bundle TM of the spacetime has a unique decomposition to the gab-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 ta and va 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 two-metric and the corresponding area 2-form on \({\mathcal S}\) will be denoted by qab = gabtatb + vavb and εab = tcvdεcdab, respectively, while the area 2-form on the normal bundle will be ⊥εab = tavbtbva. The bundle \({{\rm{V}}^a}({\mathcal S})\) together with the fiber metric gab 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 two-manifolds, 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 δe, is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) 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 Xa of \({{\rm{V}}^a}({\mathcal S})\). Obviously, δe annihilates both the fiber metric gab and the projection \(\Pi _b^a\). However, since for two-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ tata cosh u + va sinh u, vata sinh u + va cosh u. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection one-form on \({\mathcal S}\) can be characterized, for example, by \({A_e}: = \Pi _e^f({\nabla _f}{t_a}){\upsilon ^a}\). Therefore, the connection δe 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 gab-orthonormal frames adapted to \({\mathcal S}\).

The other connection, Δe, 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 Δe and δe 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 ta and va, respectively. The familiar expansion tensors of the future-pointing outgoing and ingoing null normals, la := ta + υa and \({n^a}: = {1 \over 2}({t^a} - {\upsilon ^a})\), respectively, are θab = Qabclc and θab = Qabcnc, and the corresponding shear tensors σab and σab are defined by their trace-free part. Obviously, τab and νab (and hence the expansion and shear tensors θab, θab, σab, and σab) are boost-gauge-dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination \({Q^a}_{eb}\) is boost-gauge 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. Qa 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 Qa and Qa 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 pointwise-determined direction orthogonal to the two-surface in which the expansion of the surface is vanishing. If Qa is not null, then \(\{{Q_a},{{\tilde Q}_a}\}\) defines an orthonormal frame in the normal bundle (see, e.g., [14]). If Qa is nonzero, but (e.g., future-pointing) null, then there is a uniquely determined null normal Sa to \({\mathcal S}\), such that QaSa = 1, and hence, {Qa, Sa} is a uniquely determined null frame. Therefore, the two-surface admits a natural gauge choice in the normal bundle, unless Qa is vanishing. Geometrically, Δe is a connection coming from a connection on the O(1, 3)-principal fiber bundle of the gab-orthonormal frames. The curvature of the connections δe and Δe, respectively, are

$${f^a}_{bcd} = {- ^ \bot}{\varepsilon ^a}_b({\delta _c}{A_d} - {\delta _d}{A_c}) + {\textstyle{1 \over 2}}{}^{\mathcal S}R(\Pi _c^a{q_{bd}} - \Pi _d^a{q_{bc}}),$$
(4.1)
$$\begin{array}{*{20}c} {{F^a}_{bcd} = {f^a}_{bcd} - {\delta _c}({Q^a}_{db} - {Q_{bd}}^a) + {\delta _d}({Q^a}_{cb} - {Q_{bc}}^a) +} \\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {Q^a}_{ce}{Q_{bd}}^e + {Q_{ec}}^a{Q^e}_{db} - {Q^a}_{de}{Q_{bc}}^e - {Q_{ed}}^a{Q^e}_{cb},} \\ \end{array}$$
(4.2)

where \(^{\mathcal S}R\) is the curvature scalar of the familiar intrinsic Levi-Civita connection of \(^{\mathcal S}R\). The curvature of Δe is just the pullback to \({\mathcal S}\) of the spacetime curvature 2-form: \({F^a}_{bcd} = {R^a}_{bef}\Pi _c^e\Pi _d^f\). Therefore, the well-known Gauss, Codazzi-Mainardi, 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 quasi-local quantities, various forms of the convexity of \({\mathcal S}\) must be assumed. The convexity of \({\mathcal S}\) in a three-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If, in addition, the three-geometry 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 C4 Riemannian two-manifold with positive scalar curvature, then this can be isometrically embedded (i.e., realized as a closed convex two-surface) in the Euclidean three-space ℝ3, 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 two-surface into the Minkowski spacetime ℝ1,3 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 S2 topology) was given by Wang and Yau [543], and they controlled these isometric embeddings in terms of a single function τ on the two-surface. 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 δeτ 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 two-surface \(i({\mathcal S}) \subset {{\rm{{\mathbb R}}}^{1,3}}\) to the spacelike hyperplane orthogonal to \({T^{\underline a}}\) is positive. The Laplacian δeδeτ of the ‘time function’ gives the mean curvature vector of \(i({\mathcal S})\) in ℝ1,3 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 la are expanding on \({\mathcal S}\), i.e., θ:= qabθab > 0, and weakly past convex if θ′:= qabθ′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 boost-gauge-dependent, their sign is gauge invariant. Then \({\mathcal S}\) will be called future convex if θ > 0 and D > 0, and past convex if θ′ < 0 and D′ > 0 [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 δe and Δe 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 γ5 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 Δe can be expressed by the curvature \({f^A}_{Bcd}\) of δe, the spinor \({Q^A}_{eB}\), and its δe-derivative. We can form the scalar invariants of the curvatures according to

$$f: = {f_{abcd}}{1 \over 2}({\varepsilon ^{ab}} - {{\rm{i}}^ \bot}{\varepsilon ^{ab}})\;{\varepsilon ^{cd}} = {\rm{i}}{\gamma ^A}_B{f^B}_{Acd}{\varepsilon ^{cd}}{= ^{\mathcal S}}R - 2{\rm{i}}{\delta _c}({\varepsilon ^{cd}}{A_d}),$$
(4.3)
$$F: = {F_{abcd}}{1 \over 2}({\varepsilon ^{ab}} - {{\rm{i}}^ \bot}{\varepsilon ^{ab}}){\varepsilon ^{cd}} = {\rm{i}}{\gamma ^A}_B{F^B}_{Acd}{\varepsilon ^{cd}} = f + \theta \theta {\prime}- 2{\sigma {\prime}_{ea}}{\sigma ^e}_b({q^{ab}} + {\rm{i}}{\varepsilon ^{ab}}).$$
(4.4)

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 ta and va 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 well-defined vector field.

An interesting decomposition of the SO(1, 1) connection one-form Ae, i.e., the vertical part of the connection δe, was given by Liu and Yau [338]: There are real functions α and γ, unique up to additive constants, such that Ae = εefα + δeγ. 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 ℝ2. It is globally defined if \({H^1}({\mathcal S}) = 0\). In this decomposition α is the boost-gauge-invariant part of Ae, while γ represents its gauge content. Since δeAe = δeδeγ, the ‘Coulomb-gauge condition’ δeAe = 0 uniquely fixes Ae (see also Section 10.4.1).

By the Gauss-Bonnet 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 δe is rather poor, and can be considered as a part of the ‘universal structure of \({\mathcal S}\)’. On the other hand, the connection Δe is much richer, and, in particular, the invariant F carries information on the mass aspect of the gravitational ‘field’. The two-surface data for charge-type quasi-local quantities (i.e., for two-surface observables) are the universal structure (i.e., the intrinsic metric qab, the projection \(\Pi _b^a\) and the connection δe) and the extrinsic curvature tensor \({Q^a}_{eb}\).

Curvature identities

The complete decomposition of ΔAAλB into its irreducible parts gives ΔAAλA, the Dirac-Witten 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 two-surface twistor operator. The former is essentially the anti-symmetric part ΔA′[AλB], the latter is the symmetric and (with respect to the complex metric γAB trace-free part of the derivative. (The trace \({\gamma ^{AB}}{\Delta _{{A\prime}A}}{\lambda _B}\) can be shown to be the Dirac-Witten operator, too.) A Sen-Witten-type identity for these irreducible parts can be derived. Taking its integral one has

$$\oint\nolimits_{\mathcal S} {{{\overline \gamma}^{A{\prime}B{\prime}}}[({\Delta _{A{\prime}A}}{\gamma ^A})({\Delta _{B{\prime}B}}{\mu ^B}) + ({\tau _{A{\prime}CD}}^E{\lambda _E})({\tau _{B{\prime}}}^{CDF}{\mu _F})]\;\;d{\mathcal S}} = - {\textstyle{{\rm{i}} \over 2}}\oint\nolimits_{\mathcal S} {{\lambda ^A}{\mu ^B}{F_{A\,Bcd}}},$$
(4.5)

where λA and μA are two arbitrary spinor fields on \({\mathcal S}\), and the right-hand side is just the charge integral of the curvature \({F^A}_{Bcd}\) on \({\mathcal S}\).

The GHP formalism

A GHP spin frame on the two-surface \({\mathcal S}\) 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 future-pointing null vectors la := oAōA 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 two-surface. For example, on the two-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors ma and \({{\bar m}^a}\) cannot form a globally-defined basis on \({\mathcal S}\). Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable two-surface with a globally-trivial 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 oAλoA, ιAλ−1 ιA of the GHP spin frame with some nowhere-vanishing complex function λ: U → ℂ, the scalar transforms as \(\phi \mapsto {\lambda ^p}{{\bar \lambda}^q}\phi\). For example, \(\rho: = {\theta _{ab}}{m^a}{{\bar m}^b} = - {1 \over 2}\theta, \,{\rho \prime}: = \theta _{ab}\prime{m^a}{{\bar m}^b} = \theta - {1 \over 2}{\theta \prime},\,\sigma := {\theta _{ab}}{m^a}{m^b} = {\sigma _{ab}}{m^a}{m^b}\) 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 UU′, the local sections are related to each other by \(\phi = {\psi ^p}{{\bar \psi}^q}{\phi \prime}\), where ψ: UU′ → ℂ is the transition function between the GHP spin frames: oA = ψoA and ιA = ψ−1ιA.

The connection δe defines a connection ðe on the line bundles E(p,q) [209, 425]. The usual edth operators, ð and ð′, are just the directional derivatives ð:= maða 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 locally-defined operators yield globally-defined differential operators, denoted also by ð and ð′, on the global sections of E(p, q). It might be worth emphasizing that the GHP spin coefficients β and β′, which do not have definite (p, q)-type, play the role of the two components of the connection one-form, and are built both from the connection one-form for the intrinsic Riemannian geometry of \(({\mathcal S},\,{q_{ab}})\) and the connection one-form Ae 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 Δa′aλA and \({{\mathcal T}_{E \prime EA}}^B{\lambda _B}\) can be decomposed further into their right-handed and left-handed parts. In the GHP formalism these chiral irreducible parts are

$$\begin{array}{*{20}c} {- {\Delta ^ -}\lambda : = \;\eth{\lambda _1} + \rho {\prime}{\lambda _0},} & {{\Delta ^ +}\lambda : = \;\eth{\prime}{\lambda _0} + \rho {\lambda _1},} \\ {{\tau ^ -}\lambda : = \;\eth{\lambda _0} + \sigma {\lambda _1},} & {- {\tau ^ +}\lambda : = \;\eth{\prime}{\lambda _1} + \sigma {\prime}{\lambda _0},} \\ \end{array}$$
(4.6)

where λ:= (λ0,λ1) and the spinor components are defined by λA =: λ1oAλ0ιA. The various first-order linear differential operators acting on spinor fields, e.g., the two-surface 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 one-form versus anholonomicity

Obviously, all the structures we have considered can be introduced on the individual surfaces of one or two-parameter families of surfaces, as well. In particular [246], let the two-surface \({\mathcal S}\) 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 one-parameter 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:= ∇aν±, 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 + βe = Δen2, where \({n^2}: = {g_{ab}}n_ + ^an_ - ^b\). (If n2 is chosen to be 1 on \({\mathcal S}\), then βe = −β+e is precisely the SO(1, 1)-connection one-form Ae 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 ωe is invariant with respect to the rescalings ν+ ↦ exp(A)ν+ and νexp(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 ωe depends only on \({\mathcal S}\). However, this implies only that ωe is invariant with respect to a restricted class of the change of the foliations, and that ωe is invariantly defined only by this class of the foliations rather than the two-surface. In fact, ωe 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, ωe and \({{\bar \omega}_e}\), would be different: \({{\bar \omega}_e} - {\omega _e} = {1 \over 2}{\Delta _e}(\alpha - \beta)\). Therefore, the anholonomicity is a gauge-dependent quantity.

Standard situations to evaluate the quasi-local quantities

There are exact solutions to the Einstein equations and classes of special (e.g., asymptotically flat) spacetimes in which there is a commonly accepted definition of energy-momentum (or at least mass) and angular momentum. In this section we review these situations and recall the definition of these ‘standard’ expressions.

Round spheres

If the spacetime is spherically symmetric, then a two-sphere, 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 gab = diag(exp(2γ), − exp(2α), −r2, −r2 sin2 θ), where γ and α are functions of t and r. (Hence, r is called the area-coordinate.) 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

$$E(t,r): = {1 \over {8G}}{r^3}{R_{abcd}}{\varepsilon ^{ab}}{\varepsilon ^{cd}} = {r \over {2G}}(1 - {e^{- 2\alpha}})$$
(4.7)

to be an appropriate (and hence, suggested to be the general) notion of energy, the Misner-Sharp energy, contained in the two-sphere \({\mathcal S}: = \{t = const.,\,r = const.\}\). (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 Reissner-Nordström solution GE(t, r) = me2/2r, 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 Reissner-Nordstroïm 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 tE(t, r) and tE(t, r), and if the energy-momentum tensor satisfies the dominant energy condition, then rE(t, r) > 0. Thus, E(t, r) is a monotonic function of r, provided r is the area-coordinate. Since, by spherical symmetry all the quantities with nonzero spin weight, in particular the shears σ and σ′, are vanishing and ψ2 is real, by the GHP form of Eqs. (4.3), (4.4) the energy function E(t, r) can also be written as

$$E({\mathcal S}) = {1 \over G}{r^3}\left({{1 \over 4}{}^{\mathcal S}R + \rho \rho {\prime}} \right) = {1 \over G}{r^3}(- {\psi _2} + {\phi _{11}} + \Lambda) = \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \left({1 + {1 \over {2\pi}}\oint\nolimits_{\mathcal S} {\rho \rho {\prime}\;d{\mathcal S}}} \right).$$
(4.8)

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 area-coordinate 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 Friedmann-Robertson-Walker spacetime (where, to cover the whole three-space, r cannot be chosen to be the area-radius 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 spherically-symmetric spacetime. Its spacelike slice Σ will be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat metrics. The first is a ‘large’ spherically-symmetric part of a t = const. hypersurface of the closed Friedmann-Robertson-Walker 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 three-space and \(d{l^2} = \Omega _{{\rm{FRW}}}^2: = B{(1 + {{{r^2}} \over {4{T^2}}})^{- 2}}\) with positive constants B and T2, and the range of the Euclidean radial coordinate r is [0, r0], where r0 ∈ (2T, ∞). It contains a maximal two-surface at r = 2T with round-sphere mass parameter \(M: = GE(2T) = {1 \over 2}T\sqrt B\). The scalar curvature is R = 6/BT2, 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 round-sphere energy is \(E(\bar r) = m/G\). These two metrics can be matched to obtain a differentiable metric with a Lipschitz-continuous derivative at the two-surface 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 three-plane — like the capital Greek letter Ω — for later reference we will call it an ‘ΩM,m-spacetime’.

Spherically-symmetric spacetimes admit a special vector field, called the Kodama vector field Ka, such that KaGab is divergence free [321]. In asymptotically flat spacetimes Ka is timelike in the asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in fact, this is hypersurface-orthogonal), but, in general, it is not a Killing vector. However, by ∇a(GabKb) = 0, the vector field Sa := GabKb 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 Ka = exp[−(α + γ)](∂/∂t)a. If Σ is a solid ball of radius r, then the flux of Sa is precisely the standard round-sphere expression (4.7) for the two-sphere Σ [375].

An interesting characterization of the dynamics of the spherically-symmetric 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 quasi-locally-defined quantities are introduced and used to study nonlinear perturbations and backreaction in a wide class of spherically-symmetric 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 two-surfaces in a spacelike hypersurface with a more general shape.

To define the first, let pM be a point, and ta a future-directed unit timelike vector at p. Let \({{\mathcal N}_p}: = \partial {I^ +}(p)\), the ‘future null cone of p in M’ (i.e., the boundary of the chronological future of p). Let la be the future pointing null tangent to the null geodesic generators of \({{\mathcal N}_p}\), such that, at the vertex p, lata = 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 two-surface 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 lata = 1 fixes the boost gauge, too.

Completing la to get a Newman-Penrose complex null tetrad \(\{{l^a},{n^a},{m^a},{{\bar m}^a}\}\) such that the complex null vectors ma and \({{\bar m}^a}\) are tangent to the two-surfaces \({{\mathcal S}_r}\), 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 oA of la = oAōA is required to be parallelly propagated along la, 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 qab on \({{\mathcal S}_r}\), the GHP spin coefficients ρ, σ, τ, p′, σ′ and β, and the higher-order expansion coefficients of the curvature in terms of the lower-order curvature components at p. Hence, the expression of any quasi-local quantity \({Q_{{{\mathcal S}_r}}}\) for the small sphere \(_{{{\mathcal S}_r}}\) can be expressed as a series of r,

$${Q_{{{\mathcal S}_r}}} = \oint\nolimits_{\mathcal S} {\left({{Q^{\left(0 \right)}} + r{Q^{\left(1 \right)}} + {\textstyle{1 \over 2}}{r^2}{Q^{\left(2 \right)}} + \cdots} \right)\;\;d{\mathcal S}},$$

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 quasi-local quantity Q is spacetime-covariant, 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’ ta. 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 Ricci-flat 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 small-sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime, as well. In particular [494], the small-sphere expressions for the quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields based on \({Q_{\mathcal S}}[{\bf{K}}]\), are, respectively,

$$P_{{{\mathcal S}_r}}^{\underline A \underline {B{\prime}}} = {{4\pi} \over 3}{r^3}{T^{AA{\prime}\,BB{\prime}}}{t_{AA{\prime}}}\varepsilon _B^{\underline A}\bar \varepsilon _{B{\prime}}^{\underline {B{\prime}}} + {\mathcal O}\left({{r^4}} \right),$$
(4.9)
$$J_{{{\mathcal S}_r}}^{\underline A \underline B} = {{4\pi} \over 3}{r^3}{T_{AA{\prime}BB{\prime}}}{t^{AA{\prime}}}\left({r{t^{B{\prime}E}}{{\textstyle\varepsilon} ^{BF}}\varepsilon _{\left(E \right.}^{\underline A}\varepsilon _{\left. F \right)}^{\underline B}} \right) + {\mathcal O}\,({r^5}),$$
(4.10)

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 one-forms, 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 boost-rotation Killing one-forms 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 three-volume times the energy-momentum and angular momentum density with respect to p, respectively, that the observer with four-velocity ta sees at p.

Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in a large class of quasi-local spacetime covariant energy-momentum and angular momentum expressions. In fact, if \({Q_{\mathcal S}}\) is any coordinate-independent quasi-local quantity built from the first derivatives μgaβ 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

$$\begin{array}{*{20}c} {{Q_{{{\mathcal S}_r}}} = {Q_2}[\partial g]\;{r^2}+{Q_3}\left[ {{\partial ^2}g,{{(\partial g)}^2}} \right]\;{r^3} + {Q_4}\left[ {{\partial ^3}g,({\partial ^2}g)\;(\partial g),{{(\partial g)}^3}} \right]\;{r^4} +} \\ {+ {Q_5}\left[ {{\partial ^4}g,({\partial ^3}g)\;(\partial g),{{({\partial ^2}g)}^2},({\partial ^2}g)\;{{(\partial g)}^2},{{(\partial g)}^4}} \right]\;{r^5} + \ldots,} \\ \end{array}$$

where Qi[A, B, …], i = 2, 3, …, are scalars. They are polynomial expressions of ta, gab and εabcd 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 coordinate-dependent quantities A, B, …. Since there is no nontrivial tensor built from the first derivative μgαβ and gαβ, the leading term is of order r3. Its coefficient Q3[2g, (dg)2] must be a linear expression of Rab and Cabcd, and polynomial in ta, gab and εabcd. In particular, if \({Q_{\mathcal S}}\) is to represent energy-momentum with generator Kc at p, then the leading term must be

$${Q_{{{\mathcal S}_r}}}[{\bf{K}}] = {r^3}\left[ {a\left({{G_{ab}}{t^a}{t^b}} \right){t_c} + bR{t_c} + c\left({{G_{ab}}{t^a}P_c^b} \right)} \right]{K^c} + {\mathcal O}\left({{r^4}} \right)$$
(4.11)

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 ta. If, in addition to the coordinate-independence of \({Q_{\mathcal S}}\), it is Lorentz-covariant, i.e., it does not, for example, depend on the choice for a normal to \({\mathcal S}\) (e.g., in the small-sphere approximation on ta) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer ta in the same way. Therefore, a = c, in which case the first and the third terms can in fact be written as r3 ataGabKb. Then, comparing Eq. (4.11) with Eq. (4.9), we see that a = −1/(6G), and hence the term r3 bRtaKa would have to be interpreted as the contribution of the gravitational ‘field’ to the quasi-local energy-momentum of the matter + gravity system. However, this contributes only to energy, but not to linear momentum in any frame defined by the observer ta, 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 r3 order in nonvacuum, any coordinate and Lorentz-covariant quasi-local energy-momentum expression which is nonspacelike and future pointing, should be proportional to the energy-momentum density of the matter fields seen by the observer ta times the Euclidean volume of the three-ball 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 Ke, 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 r3-order term is vanishing, and the fourth-order term must be built from ∇eCabcd. However, the only scalar polynomial expression of ta, gab, εabcd, ∇eCabcd and the generator vector Ka, depending linearly on the latter two, is the zero tensor field. Thus, the r4-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

$${Q_{{{\mathcal S}_r}}}[{\bf{K}}] = {r^5}\;[(a{E_{ab}}{E^{ab}} + b{H_{ab}}{H^{ab}} + c{E_{ab}}{H^{ab}}){t_c} + d{E_{ae}}{H^e}_b{\varepsilon ^{ab}}_c]\;{K^c} + {\mathcal O}\;({r^6})$$
(4.12)

for constants a, b, c, and d, where Eab: = Caebftetf 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 εabc:=εabcdtd is the induced volume 3-form. However, using the identities CabcdCabcd = 8(EabEabHabHab), Cabcd * Cabcd = 16EabHab, 4TabcdtatbtHd = EabEab + HabHab 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

$$\begin{array}{*{20}c} {{Q_{{{\mathcal S}_r}}}[{\bf{K}}] = {r^5}\;\left[ {\left({2(a + b){T_{abcd}}{t^a}{t^b}{t^c}{t^d} + {\textstyle{1 \over {16}}}(a - b){C_{abcd}}{C^{abcd}} +} \right.} \right.\quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {\left. {+ {\textstyle{1 \over {16}}}c{C_{abcd}} {\ast} {C^{abcd}}} \right){t_e} + 2d{T_{abcd}}{t^a}{t^b}{t^c}P_e^d} \right]\;{K^e} + {\mathcal O}\;({r^6}).} \\ \end{array}$$
(4.13)

Again, if \({Q_{\mathcal S}}\) does not depend on ta intrinsically, then d = (a + b), in which case the first and the fourth terms together can be written into the Lorentz covariant form 2r5 dTabcdtatbtcKd. In a general expression the curvature invariants CabcdCabcd and Cabcd * Cabcd may be present. Since, however, Eab and Hab at a given point are independent, these invariants can be arbitrarily large positive or negative, and hence, for ab or c ≠ 0 the quasi-local energy-momentum could not be future pointing and nonspacelike. Therefore, in vacuum in the leading r5 order any coordinate and Lorentz-covariant quasi-local energy-momentum expression, which is nonspacelike and future pointing must be proportional to the Bel-Robinson ‘momentum’ Tabcdtatbtc.

Obviously, the same analysis can be repeated for any other quasi-local quantity. For the energy-momentum, \({Q_{\mathcal S}}\) 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 r4 and r6 order in nonvacuum and vacuum with the energy-momentum and the Bel-Robinson tensors, respectively, while the first nontrivial correction to the area 4πr2 is of order rA and r6 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 ma and \({{\bar m}^a}\) above, and if ta is the future-pointing unit normal of Σ and va the outward directed unit normal of \({{\mathcal S}_r}\) in Σ, then we can define la := ta + va and 2na:= tava. Then \(\{{l^a},{n^a},{m^a},{{\bar m}^a}\}\) is a Newman-Penrose 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 two-surfaces homeomorphic to S2. 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 habDaDbf|p = −3.

A slightly different framework for calculations in small regions was used in [327, 170, 235]. Instead of the Newman-Penrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or Fermi type normal) coordinates on an open neighborhood U of a point pM 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 energy-momentum have been investigated. It has been shown that a quadratic expression of these coordinates with the Bel-Robinson tensor as their coefficient appears naturally in the local conservation law for the matter energy-momentum tensor [327]; the Bel-Robinson tensor can be recovered as some ‘double gradient’ of a special combination of the Einstein and the Landau-Lifshitz pseudotensors [170]; Møller’s tetrad expression, as well as certain combinations of several other classical pseudotensors, yield the Bel-Robinson tensor [473, 470, 471]. In the presence of some non-dynamical (background) metric a 11-parameter family of combinations of the classical pseudotensors exists, which, in vacuum, yields the Bel-Robinson 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 Komar-type ‘conserved’ currents. (For a readable text on the non-Killing type symmetries see, e.g., [233].) These symmetries turn out to yield a nontrivial gravitational contribution to the matter energy-momentum, even in the leading r3 order.

Large spheres near spatial infinity

Near spatial infinity we have the a priori 1/r and 1/r2 falloff for the three-metric hab and extrinsic curvature χab, 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 va 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 ma := oAA 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/r2 and the curvature components to zero like 1/r3. 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 quasi-local quantities, only the leading terms of the spin coefficients and curvature components appear. Thus, it is not necessary to solve the field equations for their second or higher-order nontrivial expansion coefficients.

Using the flat background metric 0hab and the corresponding derivative operator 0De we can define a spinor field 0λA to be constant if 0De0λA = 0. Obviously, the constant spinors form a two-complex-dimensional vector space. Then, by the falloff properties \({D_{e0}}{\lambda _A} = {\mathcal O}({r^{- 2}})\). Thus, we can define the asymptotically constant spinor fields to be those λA that satisfy \({D_e}{\lambda _A} = {\mathcal O}({r^{- 2}})\), where De is the intrinsic Levi-Civita 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 λA 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. two-surfaces \({{\mathcal S}_{u,r}}\) (or simply \({{\mathcal S}_r}\) if no confusion can arise) are spacelike topological two-spheres, 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 ‘Bondi-type coordinate system’.Footnote 7 In many of the large-sphere calculations of the quasi-local quantities the large spheres should be assumed to be large spheres not only in a general null, but in a Bondi-type coordinate system. For 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 Newman-Penrose 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 oA such that la := oAōA be the tangent (∂/∂r)a of the null geodesic generators of \({{\mathcal N}_u}\), and oA itself be constant along la. Newman and Unti [394] chose ιA to be parallelly propagated along la. This choice yields the vanishing of a number of spin coefficients (see, for example, the review [393]). The asymptotic solution of the Einstein-Maxwell 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 u-derivative \({{\dot \sigma}^0}\) 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 quasi-local quantities, the choice of Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin frame to the family of the large spheres of constant ‘radius’ r, i.e., to require ma := oAA 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 Newman-Unti basis about the spinor oA. 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 Einstein-Maxwell system by Shaw [455].

In contrast to the spatial-infinity 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 limr→∞rΔ+λ = 0, limr→∞ rΔλ = 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 linearly-independent solutions \(\lambda _A^{\underline A},\underline A = 0,1\), on ℐ+ (or on ℐ, respectively). The space \({\bf{S}}_\infty ^{\underline A}\) of its solutions turns out to have a natural symplectic metric \({\varepsilon _{\underline A \underline B}}\), and we refer to \(({\bf{S}}_\infty ^{\underline A},{\varepsilon _{\underline A \underline B}})\) as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations \({\lim\nolimits_{r \rightarrow \infty}}r{{\mathcal C}^ \pm}\lambda = 0\), the future/past asymptotic twistor equations. At ℐ+ asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form \({K^{\underline A {{\underline A}{\prime}}}}\lambda _{\underline A}^A\bar \lambda _{{{\underline A}{\prime}}}^{{A{\prime}}} = {K^{\underline A {{\underline A}{\prime}}}}\lambda _A^1\bar \lambda _{\underline {{A{\prime}}}}^{{1{\prime}}}{\iota ^A}{{\bar \iota}^{{A{\prime}}}}\) for some constant Hermitian matrix \({K^{\underline A {{\underline A}{\prime}}}}\). Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation 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 two-surface 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 Bondi-Sachs energy-momentum given in the Newman-Penrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut \({\mathcal S}\) of a combination of the leading term \(\psi _2^0\) of the Weyl spinor component \({\psi _2}\), the asymptotic shear σ0 and its u-derivative, weighted by the first four spherical harmonics (see, for example, [393, 426]):

$$P_{B\,S}^{\underline A \underline {B{\prime}}} = - {1 \over {4\pi G}}\oint {\left({\psi _2^0 + {\sigma ^0}{{\dot \bar \sigma}^0}} \right)\lambda _0^{\underline A}\bar \lambda _{0{\prime}}^{\underline {B{\prime}}}\;d{\mathcal S}},$$
(4.14)

where \(\lambda _0^{\underline A}: = \lambda _A^{\underline A}{o^A},\underline A = 0,1\), are the oA-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 Bondi-Sachs energy-momentum and Bondi’s mass-loss are well defined are determined by Tafel [505]. The expression of the Bondi-Sachs energy-momentum 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 axi-symmetric 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 (spin-weighted) spherical harmonics:

$${J^{\underline A \underline B}} = {1 \over {8\pi G}}\oint {\bar \psi _1^0,\lambda _0^{\underline A}\lambda _0^{\underline B}\,d{\mathcal S}}.$$
(4.15)

In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [109]. If the spacetime is axi-symmetric, 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 quasi-local 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 asymptotically-shear-free null geodesics [27]. While the Bondi-type 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 na: 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 σ0, 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 good-cut 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 good-cut equation and the H-space, 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 hab on Minkowski spacetime (\(({{\rm{R}}^4},g_{ab}^0)\)), and the dynamics of the field hab is governed by the linearized Einstein equations, i.e., essentially the wave equation. Therefore, the tools and techniques of the Poincaré-invariant field theories, in particular the Noether-Belinfante-Rosenfeld 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 energy-momentum tensor of the field hab is essentially the second-order term in the Einstein tensor of the metric \({g_{ab}}: = g_{ab}^0 + {h_{ab}}\). Thus, in the linear approximation the field hab does not contribute to the global energy-momentum and angular momentum of the matter + gravity system, and hence these quantities have the form (2.5) with the linearized energy-momentum tensor of the matter fields. However, as we will see in Section 7.1.1, this energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized) curvature [481, 277, 426].

pp-waves spacetimes are defined to be those that admit a constant null vector field La, and they interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present, then it is necessarily pure radiation with wave-vector La, i.e., TabLb = 0 holds [478]. A remarkable feature of the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a two-dimensional linear equation for a single function. In contrast to the approach adopted almost exclusively, Aichelburg [8] considered this field equation as an equation for a boundary value problem. As we will see, from the point of view of the quasi-local observables this is a particularly useful and natural standpoint. If a pp-wave spacetime admits an additional spacelike Killing vector Ka with closed S1 orbits, i.e., it is cyclically symmetric too, then La and Ka 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 Kerr-Newman (either outer or black hole) solution with some well-defined mass, angular momentum and electric charge parameters [534]. Thus, axisymmetric two-surfaces in these solutions may provide domains, which are general enough but for which the quasi-local quantities are still computable. According to a conjecture by Penrose [418], the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the Kerr-Newman black hole this area is \(4\pi (2{m^2} - {e^2} + 2m\sqrt {{m^2} - {e^2} - {a^2}})\). Thus, particularly interesting two-surfaces in these spacetimes are the spacelike cross sections of the event horizon [80].

There is a well-defined notion of total energy-momentum not only in the asymptotically flat, but even in the asymptotically anti-de Sitter spacetimes as well. This is the Abbott-Deser energy [1], whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [220]. (In the presence of matter fields, e.g., a self-interacting 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 anti-de Sitter spacetimes and to study their general, basic properties in [42]. A comparison and analysis of the various definitions of mass for asymptotically anti-de Sitter metrics is given in [150].

Extending the spinorial proof [349] of the positivity of the total energy in asymptotically anti-de Sitter spacetime, Chruściel, Maerten and Tod [149] give an upper bound for the angular momentum and center-of-mass 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 constant-mean-curvature, hyperboloidal, initial-data 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 quasi-local energy-momentum or angular momentum expression has the correct limit for large spheres in asymptotically anti-de Sitter spacetimes.

On lists of criteria of reasonableness of the quasi-local quantities

In the literature there are various, more or less ad hoc, ‘lists of criteria of reasonableness’ of the quasi-local quantities (see, for example, [176, 143]). However, before discussing them, it seems useful to first formulate some general principles that any quasi-local 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 energy-momentum and angular momentum are two-surface observables, thus, we concentrate on them even at the quasi-local level. These facts motivate our three a priori expectations:

  1. 1.

    The quasi-local quantities that are two-surface observables should depend only on the two-surface data, but they cannot depend, e.g., on the way that the various geometric structures on \({\mathcal S}\) are extended off the two-surface. There seems to be no a priori reason why the two-surface would have to be restricted to spherical topology. Thus, in the ideal case, the general construction of the quasi-local energy-momentum and angular momentum should work for any closed orientable spacelike two-surface.

  2. 2.

    It is desirable to derive the quasi-local energy-momentum 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. 3.

    These quantities should correspond to the ‘quasi-symmetries’ of the two-surface, which quasisymmetries are special spacetime vector fields on the two-surface. In particular, the quasilocal energy-momentum 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 ‘quasi-rotations’.

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 Ka 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 Ka given on the two-surface, but on the way in which Ka is extended off the two-surface. Therefore, \({L_{\mathcal S}}[{\bf{K}}]\) is ‘less quasi-local’ 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 Komar-current 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

$$\begin{array}{*{20}c} {{}_KH\;[{\bf{K}}]: = \int\nolimits_\Sigma {{C^a}[{\bf{K}}]\,{t_a}\;d\Sigma} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\;} \\ {= \int\nolimits_\Sigma {(cN + {c_a}{N^a})\;d\Sigma + {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {{\upsilon _a}\left({{\chi ^a}_b{N^b} - {D^a}N + {1 \over {2N}}{{\dot N}^a}} \right)\;d{\mathcal S}}.}} \\ \end{array}$$
(4.16)

Here c and ca are, respectively, the Hamiltonian and momentum constraints of the vacuum theory, ta is the future-directed unit normal to Σ, va is the outward-directed unit normal to \({\mathcal S}\) in Σ, and N and Na are the lapse and shift part of Ka, respectively, defined by Ka =: Nta + Na. Thus, KH[K] is a well-defined function of the configuration and velocity variables (N, Na, hab) and (, a, ab), respectively. However, since the velocity a cannot be expressed by the canonical variables (see e.g. [558, 63]), KH[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 vaa/N.

Pragmatic criteria

Since in certain special situations there are generally accepted definitions for the energy-momentum and angular momentum, it seems reasonable to expect that in these situations the quasi-local quantities reduce to them. One half of the pragmatic criteria is just this expectation, and the other is a list of some a priori requirements on the behavior of the quasi-local quantities.

One such list for the energy-momentum and mass, based mostly on [176, 143] and the properties of the quasi-local energy-momentum of the matter fields of Section 2.2, might be the following:

  1. 1.1

    The quasi-local energy-momentum \(P_{\mathcal S}^{\underline a}\) must be a future-pointing nonspacelike vector (assuming that the matter fields satisfy the dominant energy condition on some Σ for which \({\mathcal S} = \partial \Sigma\), and maybe some form of the convexity of \({\mathcal S}\) should be required) (‘positivity’).

  2. 1.2

    \(P_{\mathcal S}^{\underline a}\) must be zero iff D(Σ) is flat, and null iff D(Σ) has a pp-wave geometry with pure radiation (‘rigidity’).

  3. 1.3

    \(P_{\mathcal S}^{\underline a}\) must give the correct weak field limit.

  4. 1.4

    \(P_{\mathcal S}^{\underline a}\) must reproduce the ADM, Bondi-Sachs and Abbott-Deser energy-momenta in the appropriate limits (‘correct large-sphere behaviour’).

  5. 1.5

    For small spheres \(P_{\mathcal S}^{\underline a}\) must give the expected results (‘correct small sphere behaviour’):

    1. 1.

      \({4 \over 3}\pi {r^3}{T^{ab}}{t_b}\) in nonvacuum and

    2. 2.

      kr5Tabcdtbtctd in vacuum for some positive constant k and the Bel-Robinson tensor Tabcd.

  6. 1.6

    For round spheres \(P_{\mathcal S}^{\underline a}\) must yield the ‘standard’ Misner-Sharp round-sphere expression.

  7. 1.7

    For marginally trapped surfaces the quasi-local 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 quasi-local energy see [391]. Item 1.7 is motivated by the expectation that the quasi-local mass associated with the apparent horizon of a black hole (i.e., the outermost marginally-trapped 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 two-surface 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 quasi-local mass expressions. In item 1.4 we expected, in particular, that the quasi-local 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 quasi-local 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 energy-momentum and angular momentum of the matter fields on the Minkowski spacetime, the additivity of the energy-momentum (and angular momentum) is not expected. In fact, if \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) are two connected two-surfaces, then, for example, the corresponding quasi-local energy-momenta would belong to different vector spaces, namely to the dual of the space of the quasi-translations of the first and second two-surface, 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 energy-momentum 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 quasi-translations. Such an isomorphism would be provided for example by some naturally-chosen globally-defined 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 quasi-local mass and the length of the quasi-local Pauli-Lubanski spin of different surfaces can be compared, because they are scalar quantities.

Similarly, any reasonable quasi-local angular momentum expression \(J_{\mathcal S}^{\underline a \underline b}\) may be expected to satisfy the following:

  1. 2.1

    \(J_{\mathcal S}^{\underline a \underline b}\) must give zero for round spheres.

  2. 2.2

    For two-surfaces with zero quasi-local mass, the Pauli-Lubanski spin should be proportional to the (null) energy-momentum four-vector \(P_{\mathcal S}^{\underline a}\).

  3. 2.3

    \(J_{\mathcal S}^{\underline a \underline b}\) must give the correct weak field limit.

  4. 2.4

    \(J_{\mathcal S}^{\underline a \underline b}\) must reproduce the generally-accepted spatial angular momentum at spatial infinity, and in stationary and in axi-symmetric spacetimes it should reduce to the ‘standard’ expressions at the null infinity as well (‘correct large-sphere behaviour’).

  5. 2.5

    For small spheres the anti-self-dual 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 Cr5Tcdeftctdte(F(AtB)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, non-axi-symmetric spacetimes. Similarly, there are inequivalent suggestions for the center-of-mass at spatial infinity (see Sections 3.2.2 and 3.2.4).

Incompatibility of certain ‘natural’ expectations

As Eardley noted in [176], probably no quasi-local energy definition exists, which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasi-local energy definition, which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed Friedmann-Robertson-Walker or the ΩM,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 quasi-local mass may be expected to break down at the event horizon. However, although for stationary systems (or at the moment of time symmetry of a time-symmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general 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 quasi-local energy is a two-surface 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 two-surface along some special spacetime vector field.

If the quasi-local 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 round-sphere Misner-Sharp energy is a monotonically increasing or constant (rather than strictly decreasing) function of the area radius r. For example, the Misner-Sharp 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 quasi-local 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 two-surface outside the horizon. However, it does not seem why such a gemetrically, and hence physically distinguished two-surface should exist.

In the literature the positivity and monotonicity requirements are sometimes confused, and there is an ‘argument’ that the quasi-local gravitational energy cannot be positive definite, because the total energy of the closed universes must be zero. However, this argument is based on the implicit assumption that the quasi-local energy is associated with a compact three-dimensional domain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasi-local energy-momentum is associated with two-surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression (4.7) in the closed FriedmannRobertson-Walker spacetime, or, more generally, the Dougan-Mason energy-momentum (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 quasi-localization 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 three-manifold with connected boundary \({\mathcal S}\), and let hab be a (negative definite) metric and χab a symmetric tensor field on Σ, such that they, as an initial data set, satisfy the dominant energy condition: if 16πGμR + χ2χabχab and 8πGjaDb(χabχhab), then μ ≥ (−jaja)1/2. For the sake of simplicity we denote the triple (Σ, hab, χab) 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 non-negative 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 non-negativity of the ADM energies, this infimum exists and is non-negative, and it is tempting to define the quasi-local mass of Σ by this infimum.Footnote 8 However, it is easy to see that, without further conditions on the extensions of (Σ, hab, χab), 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 ‘ΩM,m-spacetime’ discussed in Section 4.2.1 on round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily-large round-sphere mass M/G) has an asymptotically flat extension, the complete spacelike hypersurface of the data set for the ΩM,m-spacetime 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 quasi-local mass of Σ. He concentrated on time-symmetric data sets (i.e., those for which the extrinsic curvature ηab is vanishing), when the horizon appears to be a minimal surface of topology S2 in \({\hat \Sigma}\) (see, e.g., [213]), and the dominant energy condition is just the requirement of the non-negativity 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 non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

$${m_{\rm{B}}}(\Sigma): = \inf \left\{{{E_{{\rm{ADM}}}}(\hat \Sigma)\vert \hat \Sigma \in {\varepsilon _0}(\Sigma)} \right\}.$$
(5.1)

The ‘no-horizon’ condition on \({\hat \Sigma}\) implies that topologically Σ is a three-ball. Furthermore, the definition of \({{\mathcal E}_0}(\Sigma)\) in its present form does not allow one to associate the Bartnik mass to those three-geometries (Σ, hab) that contain minimal surfaces inside Σ. Although formally the maximal two-surfaces inside Σ are not excluded, any asymptotically flat extension of such a Σ would contain a minimal surface. In particular, the spherically-symmetric three-geometry, with line element dl2 = − dr2 − sin2 r(2 + sin2 θ dϕ2) with (θ, ϕ) ∈ S2 and r ∈ [0, r0], π/2 < r0 < π, has a maximal two-surface at r = π/2, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than 4π sin2 r0. Thus, the Bartnik mass (according to the original definition given in [54, 53]) cannot be associated with every compact time-symmetric data set (Σ, hab), even if Σ is topologically trivial. Since for 0 < r0 < π/2 this data set can be extended without any difficulty, this example shows that mB is associated with the three-dimensional data set Σ, and not only to the two-dimensional 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 non-negative scalar curvature, finite ADM energy and with no stable minimal surface), which contain \(({\mathcal S},{q_{ab}})\) as an isometrically-embedded Riemannian submanifold, and define \({{\tilde m}_{\rm{B}}}({\mathcal S})\) 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 two-surfaces than the original mB(Σ) can be to compact three-manifolds, 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 non-negative [217], and hence mB(Σ) is still well defined and non-negative. (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 quasi-local mass of a spacelike two-surface \({\mathcal S}\) (together with its induced metric and the two extrinsic curvatures), as well [54]. He considers those globally-hyperbolic spacetimes \(\hat M: = (\hat M,{{\hat g}_{ab}})\) that satisfy the dominant energy condition, admit an asymptotically flat (metrically-complete) 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 non-negative for any \(\hat M \in \;{{\mathcal E}_0}({\mathcal S})\) (and is zero precisely for flat \({\hat M}\)), the infimum

$${m_{\rm{B}}}({\mathcal S}): = \inf \left\{{{E_{{\rm{ADM}}}}(\hat M)\vert \hat M \in {\varepsilon _0}({\mathcal S})} \right\}$$
(5.2)

exists and is non-negative. Although it seems plausible that mB(Σ) is only the ‘spacetime version’ of mB(Σ), without the precise form of the no-horizon 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 mB(Σ)

The first immediate consequence of Eq. (5.1) is the monotonicity of the Bartnik mass. If Σ1 ⊂ Σ2, then \({{\mathcal E}_0}({\Sigma _2}) \subset {{\mathcal E}_0}({\Sigma _1})\), and hence, mB1) ≤ mB2). 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 quasi-local mass functional that is larger than mB (i.e., that assigns a non-negative real to any Σ such that m(Σ) ≥ mB(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 mB(Σ) ≥ m(Σ) −εmB(Σ) − ε for any ε < 0. Therefore, mB 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 mB, due to Simon (see [56]), is that if \({\hat \Sigma}\) is any asymptotically flat, time-symmetric extension of Σ with non-negative 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 mB(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 non-negative, and, obviously, if Σ is flat (and hence is a data set for flat spacetime), then mB(Σ) = 0. The converse of this statement is also true [279]: If mB(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 three-geometry with non-negative scalar curvature and finite ADM mass \({m_{{\rm{ADM}}}}(\hat \Sigma)\), and if {Σn}, 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 mB(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 time-symmetric case, is just the ADM mass.

The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice application of the Riemannian Penrose inequality [279]. Let Σ be a spherically-symmetric Riemannian three-geometry with spherically-symmetric boundary \({\mathcal S}: = \partial \Sigma\). One can form its ‘standard’ round-sphere energy \(E({\mathcal S})\) (see Section 4.2.1), and take its spherically-symmetric 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 time-symmetric 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’ round-sphere 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, hab). Without some computational method the potentially useful properties of mB(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 mB(Σ) is realized by an extension \((\hat \Sigma, {{\hat h}_{ab}})\) of (Σ, hab) such that the exterior region, \((\hat \Sigma - \Sigma, {{\hat h}_{ab}}{\vert _{\hat \Sigma - \Sigma}})\), is static, the metric is Lipschitz-continuous across the two-surface \(\partial \Sigma \subset \hat \Sigma\), and the mean curvatures of Σ of the two sides are equal. Therefore, to compute mB for a given (Σ, hab), one should find an asymptotically flat, static vacuum metric ĥab satisfying the matching conditions on Σ, and where the Bartnik mass is the ADM mass of ĥab. 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 Σ1 ⊂ Σ2, mB1) = mB2) and Σ2 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, hab) 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, hab) 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, hab, χab) 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 mB(Σ) one can construct admissible extensions of (Σ, hab) in the form of the metrics in quasi-spherical form [55]. If the boundary Σ is a metric sphere of radius r with non-negative mean curvature k, then mB(Σ) can be estimated from above in terms of r and k.

Bray’s modifications

Another, slightly modified definition for the quasi-local mass is suggested by Bray [110, 113]. Here we summarize his ideas.

Let Σ = (Σ, hab, χab) be any asymptotically flat initial data set with finitely-many asymptotic ends and finite ADM masses, and suppose that the dominant energy condition is satisfied on Σ. Let \({\mathcal S}\) be any fixed two-surface in Σ, which encloses all the asymptotic ends except one, say the i-th (i.e., let \({\mathcal S}\) be homologous to a large sphere in the i-th asymptotic end). The outside region with respect to \({\mathcal S}\), denoted by \(O({\mathcal S})\), will be the subset of Σ containing the i-th 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 ‘fill-in’ \({{\hat \Sigma}_{\rm{f}}}\) 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 outer-minimizing if, for any closed two-surface \({\tilde {\mathcal S}}\) enclosing \({\mathcal S}\), one has \({\rm{Area}}({\mathcal S}) \leq {\rm{Area}}(\tilde {\mathcal S})\).

Let \({\mathcal S}\) be outer-minimizing, and let \({\mathcal E}({\mathcal S})\) denote the set of extensions of \({\mathcal S}\) in which \({\mathcal S}\) is still outer-minimizing, and \({\mathcal F}({\mathcal S})\) denote the set of fill-ins 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 two-surfaces 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

$$\begin{array}{*{20}c} {{m_{{\rm{out}}}}({\mathcal S}): = \inf \left\{{{m_{{\rm{ADM}}}}({{\hat \Sigma}_e})\vert {{\hat \Sigma}_e} \in {\mathcal E} \,({\mathcal S})} \right\},} \\ {{m_{{\rm{in}}}}({\mathcal S}): = \sup \left\{{\sqrt {{{{A_{{{\hat \Sigma}_{\rm{f}}}}}} \over {16\pi G}}} \vert {{\hat \Sigma}_{\rm{f}}} \in {\mathcal F}\,({\mathcal S})} \right\}.} \\ \end{array}$$

\({m_{{\rm{out}}}}({\mathcal S})\) deviates slightly from Bartnik’s mass (5.1) even if the latter would be defined for non-time-symmetric data sets, because Bartnik’s ‘no-horizon condition’ excludes apparent horizons from the extensions, while Bray’s condition is that \({\mathcal S}\) be outer-minimizing.

A simple consequence of the definitions is the monotonicity of these masses: If \({{\mathcal S}_2}\) and \({{\mathcal S}_1}\) are outer-minimizing two-surfaces 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 time-symmetric data set, for which the inequality has been proven), then for outer-minimizing surfaces \({m_{{\rm{out}}}}({\mathcal S}) \geq {m_{{\rm{in}}}}({\mathcal S})\) [110, 113]. Furthermore, if Σi is a sequence such that the boundaries Σi shrink to a minimal surface \({\mathcal S}\), then the sequence mout(Σi) tends to the irreducible mass \(\sqrt {{\rm{Area}}({\mathcal S})/(16\pi {G^2})}\) [56]. Bray defines the quasi-local 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 two-surface 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 dust-filled k = −1 Friedmann-Robertson-Walker spacetimes, Hawking found that

$$\begin{array}{*{20}c} {{E_{\rm{H}}}({\mathcal S}): = \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \left({1 + {1 \over {2\pi}}\oint\nolimits_{\mathcal S} {\rho \rho {\prime}\;d{\mathcal S}}} \right) = \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {= \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} {1 \over {4\pi}}\oint\nolimits_{\mathcal S} {(\sigma \sigma {\prime}+ \bar \sigma \bar \sigma {\prime}- {\psi _2} - {{\bar \psi}_{2{\prime}}} + 2{\phi _{11}} + 2\Lambda)\;d{\mathcal S}}} \\ \end{array}$$
(6.1)

behaves as an appropriate notion of energy surrounded by the spacelike topological two-sphere \({\mathcal S}\) [236]. Here we used the Gauss-Bonnet 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 quasi-local.

Hawking energy has the following clear physical interpretation even in a general spacetime, and, in fact, EH can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a spacelike two-sphere \({\mathcal S}\) 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 laαla, naα−1na the convergences ρ and ρ′ transform as ραρ and ρ′ ↦ α−1ρ′, 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 two-spheres 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, EH 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 Kerr-Newman 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 pM 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 Tab is the energy-momentum tensor and Tabcd is the Bel-Robinson 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 EH must be higher than r3. Then, even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the rk-order term in the power series expansion of EH 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 r5, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres EH is positive definite both in nonvacuum (provided the matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in particular, that EH should be interpreted as energy rather than as mass: For small spheres in a pp-wave spacetime EH 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 Dougan-Mason energy-momentum, the vanishing of the mass characterizes the pp-wave metrics completely.)

Using the second expression in Eq. (6.1) it is easy to see that at future null infinity EH tends to the Bondi-Sachs energy. A detailed discussion of the asymptotic properties of EH near null infinity both for radiative and stationary spacetimes is given in [455, 457]. Similarly, calculating EH 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) two-surface \({\mathcal S}\), the integral \(\oint\nolimits_{\mathcal S} {\rho {\rho \prime}s} {\mathcal S}\) could be less than −2π. Indeed, in flat spacetime EH is proportional to \(\oint\nolimits_{\mathcal S} {(\sigma {\sigma \prime} + \bar \sigma {{\bar \sigma}\prime})d} {\mathcal S}\) by the Gauss equation. For topologically-spherical two-surfaces in the t = const. spacelike hyperplane of Minkowski spacetime σσ′ is real and nonpositive, and it is zero precisely for metric spheres, while for two-surfaces in the r = const. timelike cylinder σσ′ is real and non-negative, 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 EH behaves nicely [143]: \({\mathcal S}\) will be called round enough if it is a submanifold of a spacelike hypersurface Σ, and if among the two-dimensional 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 non-negative (for example, if the dominant energy condition is satisfied), then the Hawking energy is non-negative.

Although Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of two-surfaces. Hawking considered one-parameter families of spacelike two-surfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of EH [236]. These calculations were refined by Eardley [176]. Starting with a weakly future convex two-surface \({\mathcal S}\) and using the boost gauge freedom, he introduced a special family \({{\mathcal S}_r}\) of spacelike two-surfaces 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 two-surfaces, for which \({E_H}({{\mathcal S}_r})\) is nondecreasing. By relaxing the normalization condition lana = 1 for the two null normals to lana = exp(f) for some \(f:{\mathcal S} \rightarrow {\mathbb R}\), Hayward obtained a flexible enough formalism to introduce a double-null foliation (see Section 11.2 below) of a whole neighborhood of a mean convex two-surface by special mean convex two-surfaces [247]. (For the more general GHP formalism in which lana is not fixed, see [425].) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these two-surfaces 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 two-surface, then qa ≔ 2Qa/(QbQb) = −[1/(2ρ)]la − [1/(2ρ′)]na is an outward-directed spacelike normal to \({\mathcal S}\). Here Qb 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 two-surface, Frauendiener gives an argument for the construction of a one-parameter family \({{\mathcal S}_t}\) of two-surfaces being Lie-dragged along its own inverse mean curvature vector qa. Assuming that such a family of surfaces (and hence, the vector field qa on the three-submanifold swept by \({{\mathcal S}_t}\)) exists, Frauendiener showed that the Hawking energy is nondecreasing along the vector field qa 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 well-posed 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 two-surfaces 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 Brown-York energy), see [460]. For the first attempts to introduce quasi-local 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 Bondi-Sachs energy-momentum are related to Bondi energy:

$$P_{\rm{H}}^{\underline a}({\mathcal S}) = \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} {1 \over {4\pi}}\oint\nolimits_{\mathcal S} {(\sigma \sigma {\prime}+ \bar \sigma \bar \sigma {\prime}- {\psi _2} - {{\bar \psi}_{2{\prime}}} + 2{\phi _{11}} + 2\Lambda)\,{W^{\underline a}}\,d{\mathcal S}},$$
(6.2)

where \({W^{\underline a}},\,a = 0,\, \ldots, \,3\), are essentially the first four spherical harmonics:

$$\begin{array}{*{20}c} {{W^0} = 1,} & {{W^1} = {{\zeta + \bar \zeta} \over {1 + \zeta \bar \zeta}},} & {{W^2} = {1 \over {\rm{i}}}{{\zeta - \bar \zeta} \over {1 + \zeta \bar \zeta}},} & {{W^3} = {{1 - \zeta \bar \zeta} \over {1 + \zeta \bar \zeta}}.} \\ \end{array}$$
(6.3)

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 two-surfaces 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 two-surfaces foliating the stationary and dynamical untrapped hypersurfaces, see [527, 528] and Section 11.3.4.

The Geroch energy

The definition

Suppose that the two-surface \({\mathcal S}\) for which EH is defined is embedded in the spacelike hypersurface Σ. Let χab be the extrinsic curvature of Σ in M and kab the extrinsic curvature of \(\Sigma\) in Σ. (In Section 4.1.2 we denote the latter by νab.) Then 8ρρ′ = (χabqab)2 − (kabqab)2, by means of which

$$\begin{array}{*{20}c} {{E_{\rm{H}}}({\mathcal S}) = \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \left({1 - {1 \over {16\pi}}\oint\nolimits_{\mathcal S} {{{({k_{ab}}{q^{ab}})}^2}\;d{\mathcal S}} + {1 \over {16\pi}}\oint\nolimits_{\mathcal S} {{{({\chi _{ab}}{q^{ab}})}^2}\;d{\mathcal S}}} \right) \geq} \\ {\; \geq \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \left({1 - {1 \over {16\pi}}\oint\nolimits_{\mathcal S} {{{({k_{ab}}{q^{ab}})}^2}\;d{\mathcal S}}} \right) = \quad \quad \quad \quad \quad \quad} \\ {\;\; = {1 \over {16\pi}}\sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \oint\nolimits_{\mathcal S} {\left({{2^{\mathcal S}}R - {{({k_{ab}}{q^{ab}})}^2}} \right)\;d{\mathcal S}} = :{E_{\rm{G}}}({\mathcal S}).\quad \quad} \\ \end{array}$$
(6.4)

In the last step we use the Gauss-Bonnet 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 EH, it depends not only on the two-surface \({\mathcal S}\), 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, χabqab — 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 ta 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 Bel-Robinson energy in r5 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 quasi-local 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. 1.

    its level surfaces, \({{\mathcal S}_t}: = \{q \in \Sigma \left\vert {t(q) = t} \right.\}\), are homeomorphic to S2,

  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. 3.

    they form a foliation of Σ − {p}.

Let n be the lapse function of this foliation, i.e., if va is the outward directed unit normal to \({{\mathcal S}_t}\) in Σ, then nvaDat = 1. Denoting the integral on the right-hand side in Eq. (6.4) by Wt, 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 kkabqab, and furthermore Σ is maximal (i.e., χ = 0) and the energy density of the matter is non-negative, then, as shown by Geroch [207], Wt ≥ 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 non-negative (at least for maximal Σ) and not less than \(\sqrt {{\rm{Area}}({{\mathcal S}_0})/(16\pi {G^2})}\) (at least for time-symmetric Σ), respectively. Later Jang [289] showed that, if a certain quasi-linear 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, hab, χab) 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 inverse-mean-curvature foliation of (Σ, hab) 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 quasi-linear elliptic equation admits a global solution.

The Hayward energy

We saw that EH can be nonzero, even in the Minkowski spacetime. This may motivate us to consider the following expression

$$\begin{array}{*{20}c} {I({\mathcal S}): = \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} \left({1 + {1 \over {4\pi}}\oint\nolimits_{\mathcal S} {(2\rho \rho {\prime}- \sigma \sigma {\prime}- \bar \sigma \bar \sigma {\prime})\;d{\mathcal S}}} \right)} \\ {\quad \;\;\;= \sqrt {{{{\rm{Area}}({\mathcal S})} \over {16\pi {G^2}}}} {1 \over {4\pi}}\oint\nolimits_{\mathcal S} {(- {\psi _2} - {{\bar \psi}_{2{\prime}}} + 2{\phi _{11}} + 2\Lambda)\;d{\mathcal S}}.} \\ \end{array}$$
(6.5)

(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 Newman-Unti, rather than the Bondi-Sachs 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 −ωaωa (see Sections 4.1.8 and 11.2.1). However, we saw that ωa is a boost-gauge-dependent quantity, thus, the physical significance of this suggestion is questionable unless a natural boost gauge choice, e.g., in the form of a preferred foliation, is made. (Such a boost gauge might be that given by the mean extrinsic curvature vector Qa 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 Newman-Unti, instead of the Bondi-Sachs, 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 Quasi-Local Energy-Momentum and Angular Momentum

The construction of Penrose is based on twistor-theoretical ideas, and motivated by the linearized gravity integrals for energy-momentum and angular momentum. Since, however, twistor-theoretical ideas and basic notions are still considered ‘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 energy-momentum tensor) is still analogous to charge. In fact, the total energy-momentum and angular momentum of the source can be expressed as appropriate two-surface integrals of the curvature at infinity [481]. Thus, it is natural to expect that the energy-momentum and angular momentum of the source in a finite three-volume Σ, given by Eq. (2.5), can also be expressed as the charge integral of the curvature on the two-surface \({\mathcal S}\). 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 ωab = ω[ab] is chosen and μab = εab. To simplify the subsequent analysis, ωab will be chosen to be anti-self-dual: ωab = εA′B′ ωAB with ωAB = ω(AB).Footnote 11 Thus, our goal is to find an appropriate spinor field ωAB on \({\mathcal S}\) such that

$${Q_{\mathcal S}} = [{\bf{K}}]: = \int\nolimits_\Sigma {{K_a}{T^{ab}}{1 \over {3!}}{\varepsilon _{bcde}}} = {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {{\omega ^{A\,B}}{R_{A\,Bcd}}} = :{A_{\mathcal S}}[\omega ].$$
(7.1)

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

$${\varepsilon ^{ecdf}}{\nabla _e}({\omega ^{A\,B}}{R_{A\,Bcd}}) = - 2{\rm{i}}{\psi ^F}_{A\,BC}{\nabla ^{F{\prime}\left(A \right.}}{\omega ^{\left. {BC} \right)}} + 2{\phi _{A\,B\,E{\prime}}}^{F{\prime}}{\rm{i}}{\nabla ^{E{\prime}F}}{\omega ^{A\,B}} + 4\Lambda {\rm{i}}\nabla _A^{F{\prime}}{\omega ^{F\,A}},$$
(7.2)
$$- 8\pi G{K_a}{T^{a\,f}} = 2{\phi ^{FAF{\prime}A{\prime}}}{K_{AA{\prime}}} + 6\Lambda {K^{FF{\prime}}}.$$
(7.3)

expressions (7.2) and (7.3) are equal if ωAB satisfies

$${\nabla ^{A{\prime}A}}{\omega ^{BC}} = - {\rm{i}}{\varepsilon ^{A\left(B \right.}}{K^{\left. C \right)A{\prime}}}.$$
(7.4)

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 ωBC.…E = ω(BC.…E). Thus, as could be expected, ωAB depends on the Killing vector Ka, and, in fact, Ka can be recovered from ωAB as \({K^{{A\prime}A}} = {2 \over 3}{\rm{i}}\nabla _B^{{A\prime}}{\omega ^{AB}}\). Thus, ωAB plays the role of a potential for the Killing vector KA′A. However, as a consequence of Eq. (7.4), Ka is a self-dual Killing 1-form in the sense that its derivative is a self-dual (s.d.) 2-form: In fact, the general solution of Eq. (7.4) and the corresponding Killing vector are

$$\begin{array}{*{20}c} {{\omega ^{A\,B}} = - {\rm{i}}{x^{AA{\prime}}}{x^{BB{\prime}}}{{\bar M}_{A{\prime}B{\prime}}} + {\rm{i}}{x^{\left(A \right.}}_{A{\prime}}{T^{\left. B \right)A{\prime}}} + {\Omega ^{A\,B}},} \\ {{K^{AA{\prime}}} = {T^{AA{\prime}}} + 2{x^{A\,B{\prime}}}\bar M_{B{\prime}}^{A{\prime}},\quad \quad \quad \quad \quad \quad \;\;\;} \\ \end{array}$$
(7.5)

where \({{\bar M}_{{A\prime}{B\prime}}},\,{T^{A{A\prime}}}\), and ΩAB 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 Ka is, in fact, self-dual, \({\nabla _{A{A\prime}}}{K_{B{B\prime}}} = {\varepsilon _{AB}}{{\bar M}_{{A\prime}{B\prime}}},\,{T^{A{A\prime}}}\) is a translation and \({{\bar M}_{{A\prime}{B\prime}}}\) generates self-dual 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 ΩAB are vanishing, the four components of the quasi-local energy-momentum correspond to the real TAA′ s, and the spatial angular momentum and center-of-mass are combined into the three complex components of the self-dual 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 valence-one twistors of Minkowski spacetime is the set of the pairs Zα ≔ (λA, πA′) of spinor fields, which solve the valence-one-twistor equation ∇A′AλB = −iεABπA′. If Zα is a solution of this equation, then α ≔ (αA, πA′ + iϒA′aλA) is a solution of the corresponding equation in the conformally-rescaled spacetime, where ϒaΩ−1aΩ 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 complex-parameter family of solutions. Its general solution in the Minkowski spacetime is λA = ΛA − ixAA′ πA′, where ΛA and πA′ are constant spinors. Thus, the space Tα of valence-one twistors, called the twistor space, is four-complex-dimensional, and hence, has the structure \({{\rm{T}}^\alpha} = {{\rm{S}}^A} \oplus {{{\rm{\bar S}}}_{{A\prime}}}\). Tα admits a natural Hermitian scalar product: if Wβ = (ωB, σB′) 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 Hαβ′ 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 Āα′β′ of the covariant valence-two twistor Aαβ can be represented by the conjugate twistor AαβAα′β′Hα′βHβ′β. We should mention two special, higher-valence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by

$$\begin{array}{*{20}c} {{I^{\alpha \beta}}: = \left({\begin{array}{*{20}c} {{\varepsilon ^{A\,B}}} & 0 \\ 0 & 0 \\ \end{array}} \right),} & {{I_{\alpha \beta}}: = {{\bar I}^{\alpha {\prime}\beta {\prime}}}{H_{\alpha {\prime}\alpha}}{H_{\beta {\prime}\beta}} = \left({\begin{array}{*{20}c} 0 & 0 \\ 0 & {{\varepsilon ^{A{\prime}\,B{\prime}}}} \\ \end{array}} \right)} \\ \end{array}.$$
(7.6)

The other is the completely anti-symmetric twistor εεαβγ, whose component ε0123 in an Hαβ′-orthonormal basis is required to be one. The only nonvanishing spinor parts of εεαβγ 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 i-th 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

$$\nu : = \det \left({\begin{array}{*{20}c} {\lambda _1^{\bf{0}}} & {\lambda _2^{\bf{0}}} & {\lambda _3^{\bf{0}}} & {\lambda _4^{\bf{0}}} \\ {\lambda _1^{\bf{1}}} & {\lambda _2^{\bf{1}}} & {\lambda _3^{\bf{1}}} & {\lambda _4^{\bf{1}}} \\ {\pi _{\bf{0\prime}}^1} & {\pi _{\bf{0\prime}}^2} & {\pi _{\bf{0\prime}}^3} & {\pi _{\bf{0\prime}}^4} \\ {\pi _{\bf{1\prime}}^1} & {\pi _{\bf{1\prime}}^2} & {\pi _{\bf{1\prime}}^3} & {\pi _{\bf{1\prime}}^4} \\ \end{array}} \right) = {\textstyle{1 \over 4}}{{\epsilon}^{ij}}_{kl}\lambda _i^A\lambda _j^B\pi _{A{\prime}}^k\pi _{B{\prime}}^l{\varepsilon _{A\,B}}{\varepsilon ^{A{\prime}B{\prime}}} = {\textstyle{1 \over 4}}{\varepsilon _{\alpha \beta \gamma \delta}}Z_1^\alpha Z_2^\beta Z_3^\gamma Z_4^\delta,$$
(7.7)

where \({\epsilon ^{ij}}_{kl}\) is the totally antisymmetric Levi-Civita symbol. Then Iαβ and Iαβ 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 Iαβ one has εαβγδIαβIγδ = 0.

The solution ωAB of the valence-two 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 λA and ωA of the valence-one twistor equation. ωAB uniquely defines a symmetric twistor ωαβ (see, e.g., [426]). Its spinor parts are

$${\omega ^{\alpha \beta}} = \left({\begin{array}{*{20}c} {{\omega ^{A\,B}}} & {- {\textstyle{1 \over 2}}{K^A}_{B{\prime}}} \\ {- {\textstyle{1 \over 2}}{K_{A{\prime}}}^B} & {- {\rm{i}}{{\bar M}_{A{\prime}B{\prime}}}} \\ \end{array}} \right).$$

However, Eq. (7.1) can be interpreted as a ℂ-linear mapping of ωαβ into ℂ, i.e., Eq. () defines a dual twistor, the (symmetric) kinematical twistor Aαβ, which therefore has the structure

$${A_{\alpha \beta}} = \left({\begin{array}{*{20}c} 0 & {{P_A}^{B{\prime}}} \\ {{P^{A{\prime}}}_B} & {2{\rm{i}}{{\bar J}^{A{\prime}B{\prime}}}} \\ \end{array}} \right).$$
(7.8)

Thus, the quasi-local energy-momentum and self-dual 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 AAB is identically zero) and the reality of PAA′. These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on Aαβ: the vanishing of the spinor part Aab is equivalent to AαβIαγIβδ = 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 AAB = 0 and the reality of the energy-momentum) is equivalent to

$${A_{\alpha \beta}}{I^{\beta \gamma}}{H_{\gamma \delta {\prime}}} = {\bar A_{\delta {\prime}\beta {\prime}}}{\bar I^{\beta {\prime}\gamma {\prime}}}{H_{\gamma {\prime}\alpha}}.$$
(7.9)

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 quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [420] or as its determinant [510]:

$${m^2} = - {P_A}^{A{\prime}}{P^A}_{A{\prime}} = - {1 \over 2}{A_{\alpha \beta}}{\bar A_{\alpha {\prime}\beta {\prime}}}{H^{\alpha \alpha {\prime}}}{H^{\beta \beta {\prime}}} = - {1 \over 2}{A_{\alpha \beta}}{\bar A^{\alpha \beta}},$$
(7.10)
$${m^4} = 4\det {A_{\alpha \beta}} = {\textstyle{1 \over {3!}}}{\varepsilon ^{\alpha \beta \gamma \delta}}{\varepsilon ^{\mu \nu \,\rho \sigma}}{A_{\alpha \mu}}{A_{\beta \nu}}{A_{\gamma \rho}}{A_{\delta \sigma}}.$$
(7.11)

Similarly, as Helfer shows [264], the various components of the Pauli-Lubanski 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 Zα = (−ixAB′ πB′, πA′) and Wα = (−ixAB′ σB′, σA′) are two different (null) twistors such that AαβZαZβ = 0 and AαβWαWβ = 0, then

$${(2{P^e}{\pi _{E{\prime}}}{\bar \pi _E}{P^f}{\sigma _{F{\prime}}}{\bar \sigma _F})^{- 1}}{\bar \pi _A}{\pi _{A{\prime}}}{\bar \sigma _B}{\bar \sigma _{B{\prime}}}({S^a}{P^b} - {S^b}{P^a}) = - \Re \left({{{{A_{\alpha \beta}}{Z^\alpha}{W^\beta}} \over {{I_{\gamma \delta}}{Z^\gamma}{W^\delta}}}} \right).$$
(7.12)

(ℜ on the right means ‘real part’.)

Thus, to summarize, the various spinor parts of the kinematical twistor Aαβ are the energy-momentum 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 energy-momentum and angular momentum parts, and, in particular, to define the mass and express the Pauli-Lubanski spin. Furthermore, the Hermiticity condition ensuring that Aαβ has the correct number of components (ten reals) is also formulated in terms of these additional structures.

The original construction for curved spacetimes

Two-surface twistors and the kinematical twistor

In general spacetimes, the twistor equations have only the trivial solution. Thus, to be able to associate a kinematical twistor with a closed orientable spacelike two-surface \({\mathcal S}\) in general, the conditions on the spinor field ωAB have to be relaxed. Penrose’s suggestion [420, 421] is to consider ωAB 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 valence-one twistor equation, the two-surface twistor equation. (The equation obtained as the ‘tangential projection to \({\mathcal S}\)’ of the valence-two twistor equation (7.4) would be under-determined [421].) Thus, the quasi-local quantities are searched for in the form of a charge integral of the curvature:

$$\begin{array}{*{20}c} {{A_{\mathcal S}}[\lambda, \omega ]: = {{- 1} \over {8\pi G}}\oint\nolimits_{\mathcal S} {{\lambda ^A}{\omega ^B}{R_{A\,Bcd}}} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {= {{\rm{i}} \over {4\pi G}}\oint\nolimits_{\mathcal S} {[{\lambda ^0}{\omega ^0}({\phi _{01}} - {\psi _1}) + ({\lambda ^0}{\omega ^1} + {\lambda ^1}{\omega ^0})\,({\phi _{11}} + \Lambda - {\psi _2}) + {\lambda ^1}{\omega ^1}({\phi _{21}} - {\psi _3})]\;d{\mathcal S},}} \\ \end{array}$$
(7.13)

where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the two-surface \({\mathcal S}\). 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 FABcd in the spinor identity (4.5) of Section 4.1.5.

The two-surface 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 first-order 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) linearly-independent solutions to \({\mathcal T}\lambda = 0\) on topological two-spheres. However, there are ‘exceptional’ two-spheres for which there exist at least five linearly independent solutions [297]. For such ‘exceptional’ two-spheres (and for higher-genus two-surfaces for which the twistor equation has only the trivial solution in general) the subsequent construction does not work. (The concept of quasi-local charges in Yang-Mills 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 two-surface twistor space. In fact, in the generic case this space is four-complex-dimensional, and under conformal rescaling the pair Zα = (λA, iΔA′AλA) transforms like a valence one contravariant twistor. Zα is called a two-surface twistor determined by λA. If \({{\mathcal S}\prime}\) is another generic two-surface with the corresponding two-surface 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 Aαβ is defined to be the symmetric twistor determined by \({A_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {A_{\mathcal S}}[\lambda, \,\omega ]\) for any Zα = (λA, iΔA′AλA) and Wα = (ωA, iΔA′AωA from \({\rm{T}}_{\mathcal S}^\alpha\). Note that \({A_{\mathcal S}}[\lambda, \,\omega ]\) is constructed only from the two-surface data on \({\mathcal S}\).

The Hamiltonian interpretation of the kinematical twistor

For the solutions λA and ωA of the two-surface twistor equation, the spinor identity (4.5) reduces to Tod’s expression [420, 426, 516] for the kinematical twistor, making it possible to re-express \({\mathcal S}\) by the integral of the Nester-Witten 2-form [490]. Indeed, if

$${H_{\mathcal S}}[\lambda, \bar \mu ]: = {1 \over {4\pi G}}\oint\nolimits_{\mathcal S} {u{{(\lambda, \bar \mu)}_{ab}}} = - {1 \over {4\pi G}}\oint\nolimits_{\mathcal S} {{{\bar \gamma}^{A{\prime}B{\prime}}}{{\bar \mu}_{A{\prime}}}{\Delta _{B{\prime}B}}{\lambda ^B}\,d{\mathcal S}},$$
(7.14)

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 λA and ωA 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 energy-momentum 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 re-derivation 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 λA and ωA (together with their ‘projection parts’ iΔA′AλA and iΔA′AωA) on \({\mathcal S}\) are interpreted as the spinor constituents of the special lapse and shift, called the ‘quasi-translations’ and ‘quasi-rotations’ of the two-surface, on the two-surface 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 two-surface twistor space. As is shown in [296, 299, 514], \(\left\langle {Z,\,\bar W} \right\rangle\) is constant on \({\mathcal S}\) for any two two-surface 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 two-surfaces 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 two-surfaces. 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 Nester-Witten 2-form [490]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted two-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. Eq. (7.10)) is ambiguous unless a natural Hαβ′ is found.

If \({\mathcal S}\) is noncontorted, then the scalar product \(\left\langle {Z,\,\bar W} \right\rangle\) defines the totally anti-symmetric twistor εεαβγ, and for the four independent two-surface 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 two-surfaces for which \(\left\langle {Z,\,\bar W} \right\rangle\) is not. Thus, the totally anti-symmetric twistor εεαβγ can exist even for certain contorted two-surfaces. Therefore, an alternative definition of the quasi-local 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 anti-de Sitter spacetimes. Thus, if needed, the former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by mD).

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, −εA′B Δ A′AλAΔB′BωB 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 re-expressed by the integral of the Nester-Witten 2-form [490]), but the resulting twistor would not be simple. In fact, even on two-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant λ the natural definition for Iαβ is Iαβ ≔ diag(λεAB, εA′B′) [426, 424, 510], while on round spheres in spherically-symmetric spacetimes it is \({I_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {1 \over {2{r^2}}}(1 + 2{r^2}\rho {\rho {\prime}}){\varepsilon _{AB}}{\lambda ^A}{\omega ^B} - {\varepsilon ^{{A{\prime}}{B{\prime}}}}{\Delta _{{A{\prime}}A}}{\lambda ^A}{\Delta _{{B{\prime}}B}}{\omega ^B}\) [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 IαβS. 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 two-surface 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 ‘bar-hook’ combination) appears, it might still be hoped that an appropriate \({H^\alpha}_{{\beta {\prime}}}\) could be found for a class of two-surfaces 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 one-parameter 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 two-surface \({\mathcal S}\) lies in a t = const. hypersurface of the Newtonian time t. In this limit the pointwise Hermitian scalar product is constant, and the norm-mass can be calculated. As could be expected, for the leading λ2-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 λ4-order correction.

The Penrose definition for the energy-momentum 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 conformally-rescaled quantities of the nonphysical spacetime. Since ℐ+ is shear-free, the two-surface twistor equations on \({\mathcal S}\) decouple, and hence, the solution space admits a natural infinity twistor Iαβ. 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 energy-momentum. 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 Aαβ is Hermitian. Furthermore, the determinant ν is constant on \({\mathcal S}\), and hence it defines a volume 4-form on the two-surface twistor space [516]. The energy-momentum coming from Aαβ is just that of Bondi and Sachs. The angular momentum defined by Aαβ is, however, new. It has a number of attractive properties. First, in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two anomaly’ between the angular momentum and the energy-momentum. Since its definition is based on the solutions of the two-surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the two-surface 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 four-parameter 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 energy-momentum and angular momentum is to calculate them for large spheres from the physical data, instead of for the spheres at null infinity from the conformally-rescaled data. These calculations were done by Shaw [455, 457]. At this point it should be noted that the r → ℞ limit of Aαβ vanishes, and it is \(\sqrt {{\rm{Area(}}{{\mathcal S}_r})} {A_{\alpha \beta}}\) that yields the energy-momentum 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 two-surfaces for which the mass was calculated are the r = const. cuts of the geometrically-distinguished outgoing null hypersurfaces u = const. Tod found that, for given u, the mass m is independent of r, as could be expected because of the lack of incoming radiation.

In [264] Helfer suggested a bijective nonlinear map between the two-surface 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 = ð2 S) appears explicitly. Using Eq. (7.12) as the definition of the intrinsic-spin angular momentum at scri, Helfer derives an explicit formula for the spin. In addition to the expected Pauli-Lubanski 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 two-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [475], Ashtekar-Hansen [37], and Beig-Schmidt [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 energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar-Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [456].

The Penrose mass in asymptotically anti-de 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 three-manifold in the nonphysical spacetime. Since ℐ admits global three-surface 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 energy-momentum four-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [42]. Therefore, the energy-momentum four-vector is future pointing and timelike if there is a spacelike hypersurface extending to ℐ on which the dominant energy condition is satisfied. Consequently, m2 ≥ 0. Kelly shows that \(m_{\rm{D}}^2\) is also non-negative and in vacuum it coincides with m2. In fact [516], mmD ≥ 0 holds.

The quasi-local mass of specific two-surfaces

The Penrose mass has been calculated in a large number of specific situations. Round spheres are always noncontorted [514], thus, the norm-mass can be calculated. (In fact, axisymmetric two-surfaces in spacetimes with twist-free 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 Kantowski-Sachs spacetime, this mass is independent of the two-surfaces [507]. Interestingly enough, although these spheres cannot be shrunk to a point (thus, the mass cannot be interpreted as ‘the three-volume integral of some mass density’), the time derivative of the Penrose mass looks like the mass conservation equation. It is, minus the pressure times the rate of change of the three-volume of a sphere in flat space with the same area as \({\mathcal S}\) [515]. In conformally-flat spacetimes [510] the two-surface twistors are just the global twistors restricted to \({\mathcal S}\), and the Hermitian scalar product is constant on \({\mathcal S}\). Thus, the norm-mass 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 three-surface twistors [510, 512], which are solutions of certain (overdetermined) elliptic partial-differential equations, called the three-surface twistor equations, on Σ. These equations are completely integrable (i.e., they admit the maximal number of linearly-independent solutions, namely four) if and only if Σ, with its intrinsic metric and extrinsic curvature, can be embedded, at least locally, into some conformally-flat spacetime [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 Chern-Simons functional, built from the real Sen connection on the Lorentzian vector bundle or from the three-surface twistor connection on the twistor bundle over Σ [66, 495]. Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a two-surface \({\mathcal S}\) 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 norm-mass is well defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any noncontorted two-surface 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 conformally-flat time-symmetric initial data sets. Tod considered Wheeler’s solution of the time-symmetric vacuum constraints describing n ‘points at infinity’ (or, in other words, n − 1 black holes) and two-surfaces 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 Mi of the i-th black hole if \({\mathcal S}\) links precisely the i-th 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 i-th and the j-th black holes, where dij is some appropriate measure of the distance between the black holes, …, etc. Thus, the mass of the i-th and j-th holes as a single object is less than the sum of the individual masses, in complete agreement with our physical intuition that the potential energy of the composite system should contribute to the total energy with negative sign.

Beig studied the general conformally-flat time-symmetric initial data sets describing n ‘points at infinity’ [62]. He found a symmetric trace-free and divergence-free tensor field Tab and, for any conformal Killing vector ξa of the data set, defined the two-surface flux integral P(ξ) of Tabξb on \({\mathcal S}\). He showed that P(ξ) is conformally invariant, depends only on the homology class of \({\mathcal S}\), and, apart from numerical coefficients, for the ten (locally-existing) conformal Killing vectors, these are just the components of the kinematical twistor derived by Tod in [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 Cartan-Killing metric of the conformal group of the hypersurface.

Tod calculated the quasi-local mass for a large class of axisymmetric two-surfaces (cylinders) in various LRS Bianchi and Kantowski-Sachs cosmological models [515] and more general cylindrically-symmetric spacetimes [517]. In all these cases the two-surfaces are noncontorted, and the construction works. A technically interesting feature of these calculations is that the two-surfaces have edges, i.e., they are not smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately, and the resulting spinor fields are required to be continuous at the edges. This matching reduces the number of linearly-independent solutions to four. The projection parts of the resulting twistors, the \({\rm{i}}{\Delta _{{A\prime}A}}{\lambda ^A}{\rm{s}}\), 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 quasi-local 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 quasi-local mass is positive. A particularly interesting spacetime is that describing cylindrical gravitational waves, whose presence is detected by the Penrose mass. In all these cases the determinant-mass has also been calculated and found to coincide with the norm-mass. A numerical investigation of the axisymmetric Brill waves on the Schwarzschild background is presented in [87]. It was found that the quasi-local mass is positive, and it is very sensitive to the presence of the gravitational waves.

Another interesting issue is the Penrose inequality for black holes (see 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 quasi-local 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 Sobolev-type inequality for a non-negative function on the unit sphere. This inequality is tested numerically in [87].

Apart from the cuts of ℐ+ in radiative spacetimes, all the two-surfaces discussed so far were noncontorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted two-surface [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 two-surface twistors Zα = (λA,iΔA′AλA) and Wα = (ωA,iΔA′AωA) [511, 313]. Since in nonvacuum spacetimes the kinematical twistor has only the ‘four-momentum 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

$${A_{\alpha \beta}} = \left({\begin{array}{*{20}c} {2{\rm{i}}{\lambda _{AB}}} & {{P_A}^{B{\prime}}} \\ {{P^{A{\prime}}}_B} & 0 \\ \end{array}} \right) + {\mathcal O}\,({r^6}),$$
(7.15)

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 four-momentum is \({P_{A{A\prime}}}{t^{A{A\prime}}} = {1 \over {45G}}{H_{ab}}({H^{ab}} - {\rm{i}}{E^{ab}})\). Then, the corresponding norm-mass, 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 two-sphere not lying in such a hypersurface, PAA′ is real and spacelike, and hence, m2 < 0. In the Kerr spacetime, PAA′ itself is complex [511, 313].

The modified constructions

Independently of the results of the small-sphere calculations, Penrose claims that in the Schwarzschild spacetime the quasi-local mass expression should yield the same zero value on two-surfaces, 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

$${A{\prime}_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {{\rm{i}} \over {8\pi G}}\oint\nolimits_{\mathcal S} {\eta \,{\lambda ^A}{\omega ^B}{R_{A\,Bcd}}},$$
(7.16)

where η is a constant multiple of the determinant in Eq. (7.7). Since on noncontorted two-surfaces the determinant ν is constant, for such surfaces A′αβ reduces to Aαβ, and hence, all the nice properties proven for the original construction on noncontorted two-surfaces are shared by A′αβ. The quasi-local 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, sixth-order) quasi-local 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 λAB in the fifth order. Then, in the quasi-local mass (in the leading sixth order) there will be a term coming from the (presumably nonvanishing) sixth-order part of \({P^{{A\prime}}}_B\) and \({P_A}^{{B\prime}}\) and the constant part of the Hermitian scalar product, and the fifth-order λAB 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:

$${A^{{\prime}{\prime}}_{\alpha \beta}}{Z^\alpha}{W^\beta}: = {1 \over {4\pi G}}\oint\nolimits_{\mathcal S} {{{\bar \gamma}^{A{\prime}B{\prime}}}[{\rm{i}}{\Delta _{A{\prime}A}}(\sqrt \eta {\lambda ^A})]\;[{\rm{i}}{\Delta _{B{\prime}B}}(\sqrt \eta {\omega ^B})]\;d{\mathcal S}}.$$
(7.17)

The quasi-local 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 Nester-Witten 2-form, 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 ‘quasi-Killing vectors’ of the two-surface \({\mathcal S}\), their structure is not so simple, because the factor η itself depends on all four of the independent solutions of the two-surface 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 two-surface twistor equations, and uses these solutions in the integral (7.14) of the Nester-Witten 2-form. 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 ‘quasi-translations’ of \({\mathcal S}\) should be found among the 16 vector fields \(\lambda _i^A\bar \lambda _j^{{A\prime}}\) (called ‘quasi-conformal Killing vectors’ of \({\mathcal S}\)) for which the corresponding quasi-local quantities could be considered as the components of the quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local quantities.

Approaches Based on the Nester-Witten 2-Form

We saw in Section 3.2 that

  • both the ADM and Bondi-Sachs energy-momenta can be re-expressed by the integral of the Nester-Witten 2-form \(u{(\lambda, \bar \mu)_{ab}}\),

  • the proof of the positivity of the ADM and Bondi—Sachs masses is relatively simple in terms of the two-component spinors.

Thus, from a pragmatic point of view, it seems natural to search for the quasi-local energy-momentum in the form of the integral of the Nester-Witten 2-form. Now we will show that

  • the integral of Møller’s tetrad superpotential for the energy-momentum, 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 quasi-local energy-momenta based on the integral of the Nester-Witten 2-form have a natural Lagrangian interpretation in the sense that they are charge integrals of the canonical Noether current derived from Møller’s first-order tetrad Lagrangian.

If \({\mathcal S}\) is any closed, orientable spacelike two-surface 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 right-hand side of the superpotential Eq. (3.7), is well defined. If (ta, va) is a pair of globally-defined normals of \({\mathcal S}\) in the spacetime, then in terms of the geometric objects introduced in Section 4.1, this integral takes the form

$$\begin{array}{*{20}c} {Q\,[{\bf{K}}]: = {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {{1 \over 2}{K^e}{\vee _e}^{ab}{1 \over 2}{\varepsilon _{abcd}}} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {= {1 \over {8\pi G}}\oint\nolimits_{\mathcal S} {{K^e}\left({- {}^ \bot {\varepsilon _{ea}}{Q_b}^{ba} - {A_e} - {}^ \bot {\varepsilon _{ea}}({\delta _b}E_{\underline b}^b){\eta ^{\underline b \underline a}}E_{\underline a}^a + {\delta _e}({t_a}E_{\underline a}^a){\eta ^{\underline a \underline b}}E_{\underline b}^b{\upsilon _b}} \right)d{\mathcal S}.}} \\ \end{array}$$
(8.1)

The first term on the right is just the dual mean curvature vector of \({\mathcal S}\), the second is the connection one-form 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 Ka 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 quasi-local energy-momentum 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:

$${1 \over 4}\sigma _{\underline A \,\underline {B{\prime}}}^{\underline a}E_{\underline a}^e{\vee _\varepsilon}^{ab}{1 \over 2}{\varepsilon _{abcd}} = u\,{({\varepsilon _{\underline A}},{\bar \varepsilon _{\underline {B{\prime}}}})_{cd}} + \overline {u{{({\varepsilon _{\underline B}},{{\bar \varepsilon}_{\underline {A{\prime}}}})}_{cd}}},$$
(8.2)

where the overline denotes complex conjugation. Thus, the real part of the Nester-Witten 2-form, and hence, by Eq. (3.11), apart from an exact 2-form, the Nester-Witten 2-form itself, built from the spinors of a normalized spinor basis, is just the superpotential 2-form derived from Møller’s first-order tetrad Lagrangian [500].

Next we will discuss some general properties of the integral of \(u{(\lambda, \bar \mu)_{ab}}\), where λA and μA 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 λA appears. (μA 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 (infinite-dimensional 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 λA is a constant spinor field on \({\mathcal S}\) with respect to the covariant derivative Δe, then \({H_{\mathcal S}}[\lambda, \bar \mu ] = 0\) for any smooth spinor field μA 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 two-complex-dimensional space spanned by \(\lambda _A^0\) and \(\lambda _A^1\). Therefore, to have a well-defined quasi-local energy-momentum vector we have to specify some two-dimensional subspace \({{\bf{S}}^{\underline A}}\) of the infinite-dimensional 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 ‘quasi-translations’ 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 εAB 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 quasi-local energy-momentum 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 quasi-local 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 quasi-local energy-momentum based on the integral of the Nester-Witten 2-form correspond to the various choices for this spin space.

The Ludvigsen-Vickers construction

The definition

Suppose that spacetime is asymptotically flat at future null infinity, and the closed spacelike two-surface \({\mathcal S}\) 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 two-surface 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

(8.3)
$${\iota ^A}{\bar o^{A{\prime}}}({\nabla _{AA{\prime}}}{\lambda _B})\,{o^B} = \;{\prime}{\lambda _0} + \rho {\lambda _1} = 0.$$
(8.4)

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 Ludvigsen-Vickers spinors on \({\mathcal S}\). Thus, given an asymptotic spinor at infinity, we propagate its zero-th components (with respect to the basis \(\varepsilon _{\rm{A}}^A\)) to \({\mathcal S}\) by Eq. (8.3). This will be the zero-th component of the Ludvigsen-Vickers 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 Ludvigsen-Vickers 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 εAB 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, εAB does not determine any symplectic metric on the space \({{\bf{S}}^{\underline A}}\) of the Ludvigsen-Vickers spinors. In Minkowski spacetime the two Ludvigsen-Vickers 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 Ludvigsen-Vickers energy-momentum in its above form cannot be defined in a spacetime, which is not asymptotically flat at null infinity. Thus, their construction is not genuinely quasi-local, because it depends not only on the (intrinsic and extrinsic) geometry of \({\mathcal S}\), 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 two-surface to the null infinity is a very strong restriction. In fact, for general (even for convex) two-surfaces in a general asymptotically flat spacetime, conjugate points will develop along the (outgoing) null geodesics orthogonal to the two-surface [417, 240]. Thus, either the two-surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the two-surface must be nearly spherically symmetric (or the former cannot be ‘very much curved’ and the latter cannot be ‘very much bent’).

This limitation yields that, in general, the original construction above does not have a small sphere limit. However, using the same propagation equations (8.3) and (8.4) one could define a quasi-local energy-momentum 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 two-surface, and the Ludvigsen-Vickers 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 two-surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely quasi-local.

Monotonicity, mass-positivity and the various limits

Once the Ludvigsen-Vickers spinors are given on a spacelike two-surface \({{\mathcal S}_r}\) of constant affine parameter r in the outgoing null hypersurface \({\mathcal N}\), then they are uniquely determined on any other spacelike two-surface \({{\mathcal S}_{{r{\prime}}}}\) 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 Ludvigsen-Vickers spinors on different two-surfaces of constant affine parameter in the same \({\mathcal N}\). (r need not be a Bondi-type coordinate.) This makes it possible to compare the components of the Ludvigsen-Vickers energy-momenta on different surfaces. In fact [346], if the dominant energy condition is satisfied (at least on \({\mathcal N}\)), then for any Ludvigsen-Vickers spinor λA and affine parameter values r1r2, 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 (‘mass-gain’). A similar monotonicity property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation equations (8.3) and (8.4) should be replaced by ϸ′λ1 = 0 and − Δλ ≔ ðλ1 + ρ′λ0 = 0. Using these equations the positivity of the Ludvigsen-Vickers mass was proven in various special cases in [346].

Concerning the positivity properties of the Ludvigsen-Vickers 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 Ludvigsen-Vickers energy-momentum is vanishing. However, in the proof of the non-negativity of the Dougan-Mason energy (discussed in Section 8.2) only the λA ∈ ker Δ+ part of the propagation equations is used. Therefore, as realized by Bergqvist [79], the Ludvigsen-Vickers energy-momenta (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 Ludvigsen-Vickers definitions share the rigidity properties proven for the Dougan-Mason energy-momentum [488]. Under the same conditions the vanishing of the energy-momentum 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 two-surfaces, where Tab is the linearized energy-momentum tensor. The increment of \({H_{{{\mathcal S}_r}}}[\lambda, \bar \lambda ]\) on \({\mathcal N}\) is due only to the flux of the matter energy-momentum.

Since the Bondi-Sachs energy-momentum can be written as the integral of the Nester-Witten 2-form on the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first version of the Ludvigsen-Vickers energy-momentum tends to that of Bondi and Sachs. It was shown in [346, 457] that this expectation is, in fact, correct. The Ludvigsen-Vickers 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

$$P_{{{\mathcal S}_r}}^{\underline A \underline {B{\prime}}} = {1 \over {10G}}{r^5}{T^a}_{bcd}{t^b}{t^c}{t^d}\varepsilon _A^{\underline A}\bar \varepsilon _{A{\prime}}^{\underline {B{\prime}}} + {4 \over {45G}}{r^6}{t^e}({\nabla _e}{T^a}_{bcd}){t^b}{t^c}{t^d}\varepsilon _A^{\underline A}\bar \varepsilon _{A{\prime}}^{\underline {B{\prime}}} + {\mathcal O}({r^7}).$$
(8.5)

Thus, its leading term is the energy-momentum of the matter fields and the Bel-Robinson momentum, respectively, seen by the observer ta 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 Ludvigsen-Vickers quasi-local energy-momentum is future pointing and nonspacelike.

The Dougan-Mason constructions

Holomorphic/antiholomorphic spinor fields

The original construction of Dougan and Mason [172] was introduced on the basis of sheaf-theoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [488, 490].

Following Dougan and Mason we define the spinor field λA to be antiholomorphic when meeλA = meΔeλA = 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 Δe on \({\mathcal S}\). While the notion of the holomorphicity/antiholomorphicity of a function on \({\mathcal S}\) does not depend on whether the Δe or δe operator is used, for tensor or spinor fields it does. Although the vectors ma 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 ma and \({{\bar m}^a}\). Next, suppose that there are at least two independent solutions of \({\bar m^e}{\Delta _e}{\lambda _A} = 0\). If λA and μA are any two such solutions, then \({\bar m^e}{\Delta _e}({\lambda _A}{\mu _B}{\varepsilon ^{AB}}) = 0\), and hence by Liouville’s theorem λAμBεAB 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 λA on a generic \({\mathcal S}\) cannot have any zero, and any two holomorphic spinor fields, e.g., λA and λA, span the spin space at each point of \({\mathcal S}\) (and they can be chosen to form a normalized spinor dyad with respect to εAB 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 two-surfaces there are precisely two independent holomorphic spinor fields. In the GHP formalism, the condition of the holomorphicity of the spinor field λA is that its components (λ0, λ1) be in the kernel of \({{\mathcal H}^ +}: = {\Delta ^ +} \oplus {{\mathcal T}^ +}\). Thus, for generic two-surfaces 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 two-surfaces, the kernel space ker \({{\mathcal H}^ +}\) is either two-dimensional but does not inherit a natural spin space structure, or it is higher than two dimensional.

Similarly, the symplectic inner product of any two antiholomorphic spinor fields is also constant, one can define generic and exceptional two-surfaces as well, and on generic surfaces there are precisely two antiholomorphic spinor fields. The condition of the antiholomorphicity of λA 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 two-surfaces, 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 two-surfaces

\({{\mathcal H}^ \pm}\) are first-order elliptic differential operators on certain vector bundles over the compact two-surface \({\mathcal S}\), 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 linearly-independent holomorphic and at least two linearly-independent antiholomorphic spinor fields. The existence of the holomorphic/antiholomorphic spinor fields on higher-genus two-surfaces is not guaranteed by the index theorem. Similarly, the index theorem does not guarantee that \({\mathcal S} \approx {S^2}\) is generic either. If the geometry of \({\mathcal S}\) is very special, then the two holomorphic/antiholomorphic spinor fields (which are independent as solutions of \({{\mathcal H}^ \pm}\lambda = 0\)) might be proportional to each other. For example, future marginally-trapped surfaces (i.e., for which ρ = 0) are exceptional from the point of view of holomorphic spinors, and past marginally-trapped surfaces (ρ′ = 0) from the point of view of antiholomorphic spinors. Furthermore, there are surfaces with at least three linearly-independent holomorphic/antiholomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional two-surface \({\mathcal S}\) with S2 topology make \({\mathcal S}\) generic.

Finally, we note that several first-order differential operators can be constructed from the chiral irreducible parts Δ± and \({{\mathcal T}^ \pm}\) of Δe, given explicitly by Eq. (4.6). However, only four of them, the Dirac-Witten 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 two-complex-dimensional kernel in the generic case. This purely mathematical result gives some justification for the choices of Dougan and Mason. The spinor fields \(\lambda _A^{\underline A}\) that should be used in the Nester-Witten 2-form are either holomorphic or antiholomorphic. This construction does not work for exceptional two-surfaces.

Positivity properties

One of the most important properties of the Dougan-Mason energy-momenta is that they are future-pointing nonspacelike vectors, i.e., the corresponding masses and energies are non-negative. Explicitly [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 Dougan-Mason energy-momentum is a future-pointing nonspacelike vector, and, analogously, the antiholomorphic energy-momentum 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 Δλ = 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 Ludvigsen-Vickers construction, or \({{\mathcal T}^ +}\lambda = 0\) in the holomorphic Dougan-Mason 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 εAB λAμB is still constant on \({\mathcal S}\) for any two solutions λA and μA, making it possible to define the norm of the resulting energy-momentum, i.e., the mass.

In the asymptotically flat spacetimes the positive energy theorems have a rigidity part as well, namely the vanishing of the energy-momentum (and, in fact, even the vanishing of the mass) implies flatness. There are analogous theorems for the Dougan-Mason energy-momenta as well [488, 490]. Namely, under the conditions of the positivity proof

  1. 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 quasi-local energy, \({E_{\mathcal S}}: = {1 \over {\sqrt 2}}(P_{\mathcal S}^{00{\prime}} + P_{\mathcal S}^{11{\prime}}) = 0\), and

  2. 2.

    \(P_{\mathcal S}^{{\underline A}{{\underline B}\prime}}\) is null (i.e., the quasi-local mass is zero) iff D(Σ) is a pp-wave geometry and the matter is pure radiation.

In particular [498], for a coupled Einstein-Yang-Mills system (with compact, semisimple gauge groups) the zero quasi-local mass configurations are precisely the pp-wave 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 Dougan-Mason 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 quasi-local 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 linearly-independent spinor fields on \({\mathcal S}\), which are constant with respect to Δe, and D(Σ) is a pp-wave geometry; the matter is pure radiation if and only if there exists a Δe-constant spinor field on \({\mathcal S}\) [490]. Thus, the full information that D(Σ) is flat/pp-wave 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 pp-wave geometry) is determined by the two-surface data on \({\mathcal S}\).

Comparing results 1 and 2 above with the properties of the quasi-local energy-momentum (and angular momentum) listed in Section 2.2.3, the similarity is obvious: \(P_{\mathcal S}^{{\underline A}{{\underline B}\prime}} = 0\) characterizes the ‘quasi-local vacuum state’ of general relativity, while \({m_{\mathcal S}} = 0\) is equivalent to ‘pure radiative quasi-local 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 ‘quasi-local ground states’ (defined by \({E_{\mathcal S}} = 0\)) are just the ‘quasi-local vacuum states’ (defined by the trivial value of the field variables on D(Σ)) [488], in contrast, for example, to the well known ϕ4 theories.

The various limits

Both definitions give the same standard expression for round spheres [171]. Although the limit of the Dougan-Mason masses for round spheres in Reissner-Nordström spacetime gives the correct irreducible mass of the Reissner-Nordström black hole on the horizon, the constructions do not work on the surface of bifurcation itself, because that is an exceptional two-surface. Unfortunately, without additional restrictions (e.g., the spherical symmetry of the two-surfaces in a spherically-symmetric spacetime) the mass of the exceptional two-surfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic two-surfaces 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 energy-momentum [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 r5 order with that of Ludvigsen and Vickers, but in the r6 order they differ from each other. The holomorphic definition gives Eq. (8.5), but in the analogous expression for the antiholomorphic energy-momentum, the numerical coefficient 4/(45G) is replaced by 1/(9G) [171]. The Dougan-Mason energy-momenta have also been calculated for large spheres of constant Bondi-type radial coordinate value r near future null infinity [171]. While the antiholomorphic construction tends to the Bondi-Sachs energy-momentum, the holomorphic one diverges in general. In stationary spacetimes they coincide and both give the Bondi-Sachs energy-momentum. At the past null infinity it is the holomorphic construction, which reproduces the Bondi-Sachs energy-momentum, 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 two-surface \({\mathcal S}\) only. Thus, if \({\mathcal S}{\prime}\) is another two-surface, 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 energy-momenta, and hence, the holomorphic or antiholomorphic spinor fields as well, on different surfaces. For example [494], in the small-sphere 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 λA(0), λA(1), …, etc. It turns out that the solutions of this system of equations with accuracy form a 2k, rather than the expected two-complex-dimensional, space. 2(k − 1) of these 2k solutions are ‘gauge’ solutions, and they correspond in the approximation with given accuracy to the unspecified isomorphism between the space of the holomorphic/antiholomorphic spinor fields on surfaces of different radii. Obviously, similar ‘gauge’ solutions appear in the large sphere expansions, too. Therefore, without additional gauge fixing, in the expansion of a quasi-local quantity only the leading nontrivial term will be gauge-independent. In particular, the r6-order correction in Eq. (8.5) for the Dougan-Mason energy-momenta is well defined only as a consequence of a natural gauge choice.Footnote 13 Similarly, the higher-order corrections in the large sphere limit of the antiholomorphic Dougan-Mason energy-momentum 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 ψ2. \({\Gamma _{AA\prime B}}^C\) 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 Hab = flalb appearing in the form of the metric \({g_{ab}} = g_{ab}^0 + f{l_a}{l_b}\) for the Kerr-Schild spacetimes, where \(g_{ab}^0\) is the flat metric. In fact, the flat connection \({\tilde \nabla _e}\) above could be introduced for general Kerr-Schild metrics [234], and the corresponding ‘correction term’ ΓAA′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 εAB, there are precisely two linearly-independent spinor fields, say \(\lambda _A^0\) and \(\lambda _A^1\), that are constant with respect to \({\tilde \nabla _{A{A\prime}}}\) and form a normalized spinor dyad. These spinor fields are asymptotically constant. Thus, it is natural to choose the spin space \(({{\mathbf{S}}^{\underline A}},\,{\varepsilon _{\underline A \underline B}})\) to be the space of the \({\tilde \nabla _a}\)-constant spinor fields, irrespectively of the two-surface \({\mathcal S}\).

A remarkable property of these spinor fields is that the Nester-Witten 2-form built from them is closed: \(du({\lambda ^{\underline A}},\,{\bar \lambda ^{{{\underline B}\prime}}}) = 0\). This implies that the quasi-local energy-momentum depends only on the homology class of \({\mathcal S}\), i.e., if \({{\mathcal S}_1}\) and \({{\mathcal S}_2}\) are two-surfaces, 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 two-spheres that can be shrunk to a point, the energy-momentum is zero, but for those that can be deformed to a cut of the future null infinity, the energy-momentum is that of Bondi and Sachs.

Quasi-Local Spin Angular Momentum

In this section we review three specific quasi-local spin-angular-momentum constructions that are (more or less) ‘quasi-localizations’ of Bramson’s expression at null infinity. Thus, the quasi-local spin angular momentum for the closed, orientable spacelike two-surface \({\mathcal S}\) 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 second-rank spinor under constant SL(2, ℂ) transformations of the dyad \(\{\lambda _A^{\underline A}\}\). Second, suppose that the spacetime is flat, and let \(\{\lambda _A^{\underline A}\}\) be constant. Then the corresponding one-form basis \(\{\vartheta _a^{\underline a}\}\) is the constant Cartesian one, which consists of exact one-forms. Then, since the Bramson superpotential \(w({\lambda ^{\underline A}},{\lambda ^{\underline B}})\) is the anti-self-dual 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 one-form 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 one-form 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 quasi-local energy-momenta 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 quasi-translations of \({\mathcal S}\)’, we can justify such a choice. Based on our experience with the superpotentials for the various conserved quantities, the quasi-local angular momentum can be expected to be the integral of something like ‘superpotential’ × ‘quasi-rotation 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 ‘quasi-rotations’, i.e., (possibly a combination of) the ‘quasi-translations’. 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 energy-momentum 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 quasi-local spatial angular momentum, proposed by Liu and Yau [338], will be introduced in Section 10.4.1.

The Ludvigsen-Vickers angular momentum

Under the conditions that ensured the Ludvigsen-Vickers construction for the energy-momentum would work in Section 8.1, the definition of their angular momentum is straightforward [346]. Since in Minkowski spacetime the Ludvigsen-Vickers spinors are just the restriction to \({\mathcal S}\) of the constant spinor fields, by the general remark above the Ludvigsen-Vickers spin angular momentum is zero in Minkowski spacetime.

Using the asymptotic solution of the Einstein-Maxwell equations in a Bondi-type coordinate system it has been shown in [346] that the Ludvigsen-Vickers 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

$$J_{{{\mathcal S}_r}}^{\underline A \underline B} = {4 \over {45G}}{r^5}{T_{AA{\prime}BB{\prime}CC{\prime}DD{\prime}}}{t^{AA{\prime}}}{t^{BB{\prime}}}{t^{CC{\prime}}}\left({r{t^{D{\prime}E}}{\varepsilon ^{DF}}\varepsilon _{\left(E \right.}^{\underline A}\varepsilon _{\left. F \right)}^{\underline B}} \right) + {\mathcal O}({r^7}).$$
(9.1)

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 boost-rotation Killing fields that vanish at p, emerges naturally. No (approximate) boost-rotation Killing field was put into the general formulae by hand.

Holomorphic/antiholomorphic spin angular momenta

Obviously, the spin-angular-momentum expressions based on the holomorphic and antiholomorphic spinor fields [492] on generic two-surfaces are genuinely quasi-local. Since, in Minkowski spacetime the restriction of the two constant spinor fields to any two-surface 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 spherically-symmetric system. The antiholomorphic spin angular momentum has already been calculated for axisymmetric two-surfaces \({\mathcal S}\), for which the antiholomorphic Dougan-Mason energy-momentum is null, i.e., for which the corresponding quasi-local mass is zero. (As we saw in Section 8.2.3, this corresponds to a pp-wave geometry and pure radiative matter fields on D(Σ) [488, 490].) This null energy-momentum vector turned out to be an eigenvector of the anti-symmetric spin-angular-momentum tensor \(J_{\mathcal S}^{\underline A\underline B}\), which, together with the vanishing of the quasi-local mass, is equivalent to the proportionality of the (null) energy-momentum vector and the Pauli-Lubanski spin [492], where the latter is defined by

$$S_{\mathcal S}^{\underline a}: = {\textstyle{1 \over 2}}{\varepsilon ^{\underline a}}_{\underline b \underline c \underline d}P_{\mathcal S}^{\underline b}J_{\mathcal S}^{\underline c \underline d}.$$
(9.2)

This is a known property of the zero-rest-mass 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 Einstein-Maxwell spacetimes as r and r2, respectively. However, the coefficient of the diverging term in the antiholomorphic expression is just the spatial part of the Bondi-Sachs energy-momentum. Thus, the antiholomorphic spin angular momentum is finite in the center-of-mass frame, and hence it seems to describe only the spin part of the gravitational field. In fact, the Pauli-Lubanski spin (9.2) built from this spin angular momentum and the antiholomorphic Dougan-Mason energy-momentum 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 boost-rotation 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 twistor-theoretical ideas. (The twistor-theoretic aspects of the analogous flat connection for the general Kerr-Schild class are discussed in [234].)

The main idea is that, while the energy-momentum is a single four-vector 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 anti-symmetric tensor over the same space, but should depend on the ‘origin’, a point in a four-dimensional affine space M0 as well, and should transform in a specific way under the translation of the ‘origin’. Bergqvist and Ludvigsen defined the affine space M0 to be the space of the solutions Xa of \({{\tilde \nabla}_a}{X_b} = {g_{ab}} - {H_{ab}}\), and showed that M0 is, in fact, a real, four-dimensional affine space. Then, for a given Xaa′, to each \({{\tilde \nabla}_a}\)-constant spinor field λA they associate a primed spinor field by μA′Xa′aλA. This μA′ turns out to satisfy the modified valence-one twistor equation \({{\tilde \nabla}_{A(A{\prime}}}{\mu _{B{\prime})}} = - {H_{AA{\prime}BB{\prime}}}{\lambda ^B}\). Finally, they form the 2-form

$$W\,{(X,{\lambda ^{\underline A}},{\lambda ^{\underline B}})_{ab}}: = {\rm{i}}\left[ {\lambda _A^{\underline A}{\nabla _{B\,B{\prime}}}\left({{X_{A{\prime}C}}{\varepsilon ^{CD}}\lambda _D^{\underline B}} \right) - \lambda _B^{\underline A}{\nabla _{A\,A{\prime}}}\left({{X_{B{\prime}C}}{\varepsilon ^{CD}}\lambda _D^{\underline B}} \right) + {\varepsilon _{A{\prime}B{\prime}}}\lambda _{\left(A \right.}^{\underline A}\lambda _{\left. B \right)}^{\underline B}} \right],$$
(9.3)

and define the angular momentum \(J_{\mathcal S}^{\underline A\underline B}(X)\) with respect to the origin Xa as 1/(8πG) times the integral of \(W{(X,{\lambda ^{\underline A}},{\lambda ^{\underline B}})_{ab}}\) on some closed, orientable spacelike two-surface \({\mathcal S}\). Since this Wab is closed, Δ[aWbc] = 0 (similar to the Nester-Witten 2-form 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’ XeXe + ae of the ‘origin’ by a \({{\tilde \nabla}_a}\)-constant one-form ae, 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 Hamilton-Jacobi Method

If one is concentrating only on the introduction and study of the properties of quasi-local quantities, and is not interested in the detailed structure of the quasi-local (Hamiltonian) phase space, then perhaps the most natural way to derive the general formulae is to follow the Hamilton-Jacobi method. This was done by Brown and York in deriving their quasi-local energy expression [120, 121]. However, the Hamilton-Jacobi 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 ‘Brown-York quasi-local energy’ is not a single expression with a single well-defined prescription for the reference configuration. The same general formula with several other, mathematically-inequivalent definitions for the reference configurations are still called the ‘Brown-York 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 Hamilton-Jacobi 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 quasi-local 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 Brown-York expression

The main idea

To motivate the main idea behind the Brown-York 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 qa(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 (qa(t), t) may be called a history or world line in the ‘spacetime’ Q × ℝ.) Let (qa(u, t(u)), t(u)) be a smooth one-parameter deformation of this history, for which (qa(0, t(0)), t(0)) = (qa(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

$$\delta {I^1}[q(t)] = \int\nolimits_{{t_1}}^{{t_2}} {\left({{{\partial L} \over {\partial {q^a}}} - {d \over {dt}}{{\partial L} \over {\partial {{\dot q}^a}}}} \right)} \;(\delta {q^a} - {\dot q^a}\delta t)\;dt + {{\partial L} \over {\partial {{\dot q}^a}}}\delta {q^a}\vert _{{t_1}}^{{t_2}} - \left({{{\partial L} \over {\partial {{\dot q}^a}}}{{\dot q}^a} - L} \right)\;\delta t\vert _{{t_1}}^{{t_2}}.$$
(10.1)

Therefore, introducing the Hamilton-Jacobi principal function \({S^1}(q_1^a,{t_1};q_2^a,{t_2})\) as the value of the action on the solution qa(t) of the equations of motion from \((q_1^a,{t_1})\) to \((q_2^a,{t_2})\), the derivative of S1 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 t2 gives minus the energy, \(- {E^1} = - (p_a^1{{\dot q}^a} - L)\), at t2. Obviously, neither the action I1 nor the principal function S1 are unique: I[q(t)] ≔ I1[q(t)] − I0[q(t)] for any I0[q(t)] of the form \(- {E^1} = - (p_a^1{{\dot q}^a} - L)\) (dh/dt) dt with arbitrary smooth function h = h(qa(t), t) is an equally good action for the same dynamics. Clearly, the subtraction term I0[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 E1E = E1 + (∂h/∂t), respectively.

The variation of the action and the surface stress-energy tensor

The main idea of Brown and York [120, 121] is to calculate the analogous variation of an appropriate first-order action of general relativity (or of the coupled matter + gravity system) and isolate the boundary term that could be analogous to the energy above. To formulate this idea mathematically, Brown and York considered a compact spacetime domain D with topology Σ × [t1,t2] such that Σ × {t} correspond to compact spacelike hypersurfaces Σt; these form a smooth foliation of D and the two-surfaces \({{\mathcal S}_t}: = \partial {\Sigma _t}\) (corresponding to Σ × {t}) form a foliation of the timelike three-boundary 3B 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 St on 3B. The orientation of 3B 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 Σt will be denoted, respectively, by hab and χab, and those on 3B by γab and Θab.

The primary requirement of Brown and York on the action is to provide a well-defined variational principle for the Einstein theory. This claim leads them to choose for I1 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 well-defined variational principle, the ‘trace χ action’ should in fact be completed by two two-surface 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 η1 and η2 be the scalar product of the outward-pointing normal of 3B and the future-pointing normal of Σ1 and of Σ2, respectively. Then, varying the spacetime metric (for the variation of the corresponding principal function S1) they obtained the following:

$$\begin{array}{*{20}c} {\delta {S^1} = \int\nolimits_{{\Sigma _2}} {{1 \over {16\pi G}}\sqrt {\vert h\vert} \,({\chi ^{ab}} - \chi {h^{ab}})\;\delta {h_{ab}}{d^3}x -} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- \int\nolimits_{{\Sigma _1}} {{1 \over {16\pi G}}\sqrt {\vert h\vert} \,({\chi ^{ab}} - \chi {h^{ab}})\;\delta {h_{ab}}{d^3}x -} \quad \quad \quad \quad \quad \quad \quad} \\ {- \int\nolimits_{{}^3B} {{1 \over {16\pi G}}\sqrt {\vert \gamma \vert} \,({\Theta ^{ab}} - \Theta {\gamma ^{ab}})\;\delta {\gamma _{ab}}\,{d^3}x} - \quad \quad \quad \quad \quad \quad \;\;\;} \\ {\quad - {1 \over {8\pi G}}\oint\nolimits_{{{\mathcal S}_2}} {{{\tanh}^{- 1}}{\eta _2}\delta \sqrt {\vert q\vert} {d^2}x} + {1 \over {8\pi G}}\oint\nolimits_{{{\mathcal S}_1}} {{{\tanh}^{- 1}}{\eta _1}\delta \sqrt {\vert q\vert} {d^2}x}.} \\ \end{array}$$
(10.2)

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’ Σ1 and Σ2, and it is reasonable to identify its coefficient as some (quasi-local) 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 SS1S0, where S0 is assumed to be an arbitrary function of the three-metric on the boundary ∂D = Σ23B ∪ Σ1. Then

$${\tau ^{ab}}: = - {2 \over {\sqrt {\vert \gamma \vert}}}{{\delta S} \over {\delta {\gamma _{ab}}}} = {1 \over {8\pi G}}({\Theta ^{ab}} - \Theta {\gamma ^{ab}}) + {2 \over {\sqrt {\vert \gamma \vert}}}{{\delta {S^0}} \over {\delta {\gamma _{ab}}}}$$
(10.3)

defines a symmetric tensor field on the timelike boundary 3B, and is called the surface stress-energy tensor. (Since our signature for γab on 3B is (+, −, −) rather than (−, +, +), we should define τab with the extra minus sign, according to Eq. (2.1).) Its divergence with respect to the connection 3 De on 3B determined by γab is proportional to the part γabTbcυc of the energy-momentum tensor, and hence, in particular, τab is divergence-free in vacuum. Therefore, if (3B, γab) admits a Killing vector, say Ka, then, in vacuum

$${Q_{\mathcal S}}\,[{\bf{K}}]: = \oint\nolimits_{\mathcal S} {{K_a}{\tau ^{ab}}{{\bar t}_b}\,d{\mathcal S}},$$
(10.4)

the flux integral of τabKb on any spacelike cross section \({\mathcal S}\) of 3B, is independent of the cross section itself, and hence, defines a conserved charge. If Ka is timelike, then the corresponding charge is called a conserved mass, while for spacelike Ka 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 3B, and \({{\bar t}^a}\) is the unit normal to \({\mathcal S}\) tangent to 3B.)

Clearly, the trace-χ 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 Brown-York quasi-local energy

The 3 + 1 decomposition of the spacetime metric yields a 2 + 1 decomposition of the metric γab, as well. Let N and Na be the lapse and the shift of this decomposition on 3B. Then the corresponding decomposition of τab defines the energy, momentum, and spatial-stress surface densities according to

$$\varepsilon : = {t_a}{t_b}{\tau ^{ab}} = - {1 \over {8\pi G}}k + {1 \over {\sqrt {\vert q\vert}}}{{\delta {S^0}} \over {\delta N}},$$
(10.5)
$${j_a}: = - {q_{ab}}{t_c}{\tau ^{bc}} = {1 \over {8\pi G}}{A_a} + {1 \over {\sqrt {\vert q\vert}}}{{\delta {S^0}} \over {\delta {N^a}}},$$
(10.6)
$${s}^{ab}: = \Pi _c^a\Pi _d^b{\tau ^{cd}} = {1 \over {8\pi G}}\left[ {{k^{ab}} - k{q^{ab}} + {q^{ab}}{t^e}({\nabla _e}{t_f})\;{\upsilon ^f}} \right] + {2 \over {\sqrt {\vert q\vert}}}{{\delta {S^0}} \over {\delta {q_{ab}}}},$$
(10.7)

where qab is the spacelike two-metric, Ae 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, kab is the extrinsic curvature of \({{\mathcal S}_t}\) corresponding to the normal va orthogonal to 3B, and k is its trace. The timelike boundary 3B defines a boost-gauge on the two-surfaces \({{\mathcal S}_t}\) (which coincides with that determined by the foliation Σt in the ‘orthogonal boundaries’ case). The gauge potential Ae is taken in this gauge. Thus, although ε and ja on \({{\mathcal S}_t}\) are built from the two-surface data (in a particular boost-gauge), the spatial surface stress depends on the part ta(∇atb)vb of the acceleration of the foliation Σt as well. Let ξa be any vector field on 3B tangent to 3B, and ξa = nta + na its 2 + 1 decomposition. Then we can form the charge integral (10.4) for the leaves \({{\mathcal S}_t}\) of the foliation of 3B

$${E_t}[{\xi ^a},{t^a}]: = \oint\nolimits_{{{\mathcal S}_t}} {{\xi _a}{\tau ^{ab}}{t_b}\,d{{\mathcal S}_t}} = \oint\nolimits_{{{\mathcal S}_t}} {(n\varepsilon - {n^a}{j_a})\;d{{\mathcal S}_t}}.$$
(10.8)

Obviously, in general Et[ξa, ta] is not conserved, and depends not only on the vector field ξa and the two-surface data on the particular \({{\mathcal S}_t}\), but on the boost-gauge that 3B defines on \({t^a}\), i.e., the timelike normal ta as well. Brown and York define the general form of their quasi-local energy on \({\mathcal S}: = {{\mathcal S}_t}\) by

$${E_{{\rm{BY}}}}({\mathcal S},{t^a}): = {E_t}\;[{t^a},{t^a}],$$
(10.9)

i.e., they link the ‘quasi-time-translation’ (i.e., the ‘generator of the energy’) to the preferred unit normal ta of \({{\mathcal S}_t}\). Since the preferred unit normals ta are usually interpreted as a fleet of observers who are at rest with respect to \({{\mathcal S}_t}\), in their spirit the Brown-York-type quasi-local energy expressions are similar to EΣ[ta] 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 ξa = na with closed integral curved in \({{\mathcal S}_t}\) the quantity Et[ξa, ta] might be interpreted as angular momentum corresponding to ξa.

The quasi-local energy is still not completely determined, because the ‘subtraction term’ S0 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 quasi-local quantities Et[ξa, ta] (in particular zero quasi-local energy), and identify S0 with the S1 of the reference spacetime. Thus, by Eq. (10.5) and (10.6) we obtain that

$$\begin{array}{*{20}c} {\varepsilon = - {1 \over {8\pi G}}(k - {k^0}),} & {{j_a} = {1 \over {8\pi G}}({A_a} - A_a^0),} \\ \end{array}$$
(10.10)

where k0 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 k0 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 Na. This can be ensured by requiring that S0 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 Brown-York energy as a quasi-local energy oparator in loop quantum gravity, see [565].

Further properties of the general expressions

As we noted, ε, ja, and sab depend on the boost-gauge that the timelike boundary defines on \({{\mathcal S}_t}\). Lau clarified how these quantities change under a boost gauge transformation, where the new boost-gauge is defined by the timelike boundary 3B′ of another domain D′such that the particular two-surface St is a leaf of the foliation of 3B′ 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 3B, then the new ε′, ja, and \(s_{ab}{\prime}\) are built from the old ε, ja, and sab 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 S0, these latter quantities are

$${j_ \vdash}: = {2 \over {\sqrt {\vert h\vert}}}{\upsilon _a}{\upsilon _b}{\bar \tilde p^{ab}} = {1 \over {8\pi G}}l,$$
(10.11)
$${\hat j_a}: = {2 \over {\sqrt {\vert h\vert}}}{q_{ab}}{\upsilon _c}{\bar \tilde p^{bc}} = {1 \over {8\pi G}}{A_a},$$
(10.12)
$${t_{ab}}: = {2 \over {\sqrt {\vert h\vert}}}{q_{ac}}{q_{bd}}{\bar \tilde p^{cd}} = {1 \over {8\pi G}}\;[{l_{ab}} - {q_{ab}}\;(l + {\upsilon ^e}({\nabla _e}{\upsilon _f}){t^e})],$$
(10.13)

where lab is the extrinsic curvature of \({{\mathcal S}_t}\) corresponding to its normal ta (we denote this by τab 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 ja. These quantities were originally introduced as the variational derivatives of the principal function with respect to the lapse, the shift and the two-metric of the radial foliation of Σt [333, 119], which are, in fact, essentially the components of the canonical momentum.) Thus, the required transformation formulae for ε, ja, and sab follow from the definitions and those for the extrinsic curvature and the SO(1, 1) gauge potential of Section 4.1.2. The various boost-gauge invariant quantities that can be built from ε, ja, sab, j, and tab 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 ADM-type variables [331]. Here the energy and momentum surface densities are re-expressed by the superpotential \({\vee _b}^{ae}\), given by Eq. (3.6), in a frame adapted to the two-surface. (Lau called the corresponding superpotential 2-form the ‘Sparling 2-form’.) 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 three-manifold Σ is parallelizable [410], and hence, a globally-defined orthonormal triad can be given on Σ, the only parallelizable, closed, orientable two-surface is the torus. Thus, on 3B, we cannot impose the global time gauge condition with respect to any spacelike two-surface \({\mathcal S}\) in 3B unless \({\mathcal S}\) is a torus. Similarly, the global radial gauge condition in the spacelike hypersurfaces Σt (even in a small open neighborhood of the whole two-surfaces \({{\mathcal S}_t}\) in Σt) can be imposed on a triad field only if the two-boundaries \({{\mathcal S}_t} = \partial {\Sigma _t}\) 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 two-surface) appear, they are valid even globally (see also [332]). On the other hand, further investigations are needed to clarify whether or not the quasi-local Hamiltonian, using the Ashtekar variables in the radial-time gauge [333], is globally well defined.

In general, the Brown-York quasi-local 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 quasi-local energy cannot decrease. On the other hand, the interpretation of this result as a ‘quasi-local dominant energy condition’ depends on the choice of the time gauge above, which does not exist globally on the whole two-surface \({\mathcal S}\).

Booth and Mann [100] shifted the emphasis from the foliation of the domain D to the foliation of the boundary 3B. (These investigations were extended to include charged black holes in [101], where the gauge dependence of the quasi-local quantities is also examined.) In fact, from the point of view of the quasi-local quantities defined with respect to the observers with world lines in 3B and orthogonal to \({\mathcal S}\), it is irrelevant how the spacetime domain D is foliated. In particular, the quasi-local quantities cannot depend on whether or not the leaves Σt of the foliation of D are orthogonal to 3B. As a result, Booth and Mann recovered the quasi-local 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 quasi-local energy for round spheres in the spherically-symmetric spacetimes with respect to several moving observers, i.e., in contrast to Eq. (10.9), they did not link the generator vector field ξa to the normal ta 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 first-order 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 quasi-local quantities. Considering the reference term S0 in the Brown-York expression as the action of the background metric \(g_{ab}^0\) (which is assumed to be a solution of the field equations), they found that the two first-order actions coincide if the spacetime metrics gab 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 ξa and, by taking its flux integral, define charge integrals \({Q_{\mathcal S}}[{\xi ^a},{g_{ab}},g_{ab}^0]\) on two-surfaces \({\mathcal S}\).Footnote 15 Again, the Brown-York quasi-local quantity Et[ξa, ta] and \({Q_{{{\mathcal S}_t}}}[{\xi ^a},{g_{ab}},g_{ab}^0]\) coincide if the spacetime metrics coincide on the boundary ∂D and if ξa 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 quasi-local quantities. We will see in Section 10.1.8 that even the weaker boundary condition, that requires only the induced three-metrics on 3B fromgab 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 I1 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 Na, respectively, plus Et[Nta + Na, ta], given by Eq. (10.8), as a boundary term. This result is in complete agreement with the expectations, as their general quasi-local 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 two-surface 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 ΦN, whose dynamics is governed by a first-order 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 3B of D to be a timelike cylinder ‘near the infinity’, and considered the action

$${I_D}\,[{g_{ab}},{\Phi _N}]: = I_D^1\,[{g_{ab}}] + I_{{\rm{m}}D}^1\,[{g_{ab}},{\Phi _N}] - I_D^1\left[ {g_{ab}^0} \right] - I_{{\rm{m}}D}^1\,[g_{ab}^0,\Phi _N^0]$$

and those matter + gravity configurations that induce the same value on 3B as and \(\Phi _N^0\) and \(g_{ab}^0\). Its limit as 3B 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 3B they are canceled from the referenced Hamiltonian. This latter Hamiltonian coincides with that obtained in the orthogonal boundaries case. Both the ADM and the Abbott-Deser energy can be recovered from this Hamiltonian [241], and the quasi-local 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 Einstein-Maxwell 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 Brown-York 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 quasi-local 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 three-manifold Σ and the time coordinate t, the space coordinates xA on the two-boundary \({\mathcal S} = \partial \Sigma\), and their conjugate momenta π and πa.

The second important observation of Booth and Fairhurst is that the Brown-York boundary conditions are too restrictive. The two-metric, 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 three-metric γab on 3B 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 quasi-local 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 Brown-York 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 quasi-local quantities introduced above become well defined only if the subtraction term S0 in the principal function is specified. The usual interpretation of a choice for S0 is the calibration of the quasi-local quantities, i.e., fixing where to take their zero value.

The only restriction on S0 that we had is that it must be a functional of the metric γab on the timelike boundary 3B. To specify S0, it seems natural to expect that the principal function S be zero in Minkowski spacetime [216, 120]. Then S0 would be the integral of the trace Θ0 of the extrinsic curvature of 3B, if it were embedded in Minkowski spacetime with the given intrinsic metric γab. However, a general Lorentzian three-manifold (3B, γab) 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 S0 might be the requirement of the vanishing of the quasi-local 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 anti-de Sitter spacetime. Assuming that S0 depends on the lapse N and shift Na linearly, the functional derivatives (∂S0/∂N) and (∂S0/∂Na) depend only on the two-metric qab and on the boost-gauge that 3B defined on \({{\mathcal S}_t}\). Therefore, ε and ja take the form (10.10), and, by the requirement of the vanishing of ε in the reference spacetime it follows that k0 should be the trace of the extrinsic curvature of \({{\mathcal S}_t}\) in the reference spacetime. Thus, it would be natural to fix k0 as the trace of the extrinsic curvature of \({{\mathcal S}_t}\), when (\({{\mathcal S}_t}\), qab) 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 k0), and hence the construction would be ambiguous. On the other hand, one could require (\({{\mathcal S}_t}\), qab) to be embedded into flat Euclidean three-space, 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 two-surfaces 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 C2.) A particularly interesting two-surface that cannot be isometrically embedded into the flat three-space 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 ℝ4, see [203].) Thus, the construction works for a large class of two-surfaces, but certainly not for every potentially interesting two-surface. The convexity condition is essential.

It is known that the (local) isometric embedability of (\({\mathcal S}\), qab) into flat three-space with extrinsic curvature \(k_{ab}^0\) is equivalent to the Gauss-Codazzi-Mainardi 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 δa is the intrinsic Levi-Civita covariant derivative and \(^{\mathcal S}R\) is the corresponding curvature scalar on \({\mathcal S}\) determined by qab. Thus, for given qab and (actually the flat) embedding geometry, these are three equations for the three components of \(k_{ab}^0\), and hence, if the embedding exists, qab determines k0. Therefore, the subtraction term k0 can also be interpreted as a solution of an under-determined 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 k0 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 boost-gauge in which the timelike normal of \({{\mathcal S}_t}\) in the reference Minkowski spacetime is orthogonal to the spacelike three-plane, because this normal is constant. Thus, to summarize, for convex two-surfaces, the flat space reference of Brown and York is uniquely determined, k0 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 sab can be calculated (if needed). The procedure is similar if, instead of a spacelike hyperplane of Minkowski spacetime, a spacelike hypersurface of