Living Rev. Relativity,12,(2009),4 LIVING REVIEWS http://www.livingreviews.org/lrr-2009-4 (Updateoflrr-2004-4) in relativity Quasi-Local Energy-Momentum and Angular Momentum in General Relativity La´szl´o B. Szabados ResearchInstituteforParticleandNuclearPhysics oftheHungarianAcademyofSciences P.O.Box49 H-1525Budapest114 Hungary email: [email protected] http://www.rmki.kfki.hu/~lbszab/ Living Reviews in Relativity ISSN1433-8351 Acceptedon24March2009 Publishedon19June2009 Abstract The present status of the quasi-local mass, energy-momentum and angular-momentum constructionsingeneralrelativityisreviewed. First,thegeneralideas,concepts,andstrategies, aswellasthenecessarytoolstoconstructandanalyzethequasi-localquantities,arerecalled. 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. This review is based on talks given at the Erwin Schro¨dinger Institute, Vienna in July 1997, at the Universita¨t Tu¨bingen in May 1998, and at the National Center for Theoretical SciencesinHsinchu,TaiwanandattheNationalCentralUniversity,Chungli,Taiwan,inJuly 2000. ThisreviewislicensedunderaCreativeCommons Attribution-Non-Commercial-NoDerivs3.0GermanyLicense. http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck InstituteforGravitationalPhysics,AmMu¨hlenberg1,14476Potsdam,Germany. ISSN1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License: http://creativecommons.org/licenses/by-nc-nd/3.0/de/ BecauseaLiving Reviewsarticlecanevolveovertime, werecommendtocitethearticleasfollows: L´aszl´o B. Szabados, “Quasi-Local Energy-Momentum and Angular Momentum in General Relativity”, Living Rev. Relativity, 12, (2009), 4. 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For detailed documentation of an article’s evolution, please refer always to the history document of the article’s online version at http://www.livingreviews.org/lrr-2009-4. 17 June 2009: Recentdevelopmentsofthefieldaswellassomeoldclassicalresultsareincluded. Almost 120 new references are added, the text is improved and corrected at several points, and the bibliography is updated. The major changes are as follows: Page 28: Fournewparagraphs,devotedtotherecentideashowtheangularmomentumatnull infinity should be defined, and several new references are added. A more detailed discussion of the flux integrals is given. Page 34: Two paragraphs on the embedding problem, both into the Euclidean three-space and the Minkowski spacetime, and two references are added. Page 42: TwonewparagraphsoncalculationsofthesmallspherelimitinRiemannandFermi normal coordinates and on approximate symmetries with eight new references are added. Page 44: A new paragraph on the Aronson–Newman coordinates and four references are added. Page 65: A new paragraph (with a new reference) is added about a suggestion of the identi- fication of the two-surface twistor spaces on different cuts of scri and of the Pauli–Lubanski spin at null infinity. Page 69: Two new paragraphs devoted to a more detailed discussion of the Nester–Witten form and the Møller superpotential are added, and one reference is changed. Page 95: This new subsection on the generalizations of the Kijowski–Liu–Yau energy with three new references is added. Page 96: This new subsection on a new quasi-local energy-momentum expression is added. Page 98: Several new (and a few classical) references are added; Section 11.1 of the previous versionhasbeensplitintotwosubsections. Amoredetaileddiscussionofthequasi-localization of the canonical formulation of general relativity, and of a class of two-surface observables are given. Page 108: A new paragraph with two new references on results of numerical calculations of the Komar expression is added. Page 110: AmoredetaileddiscussionofthePenroseinequalityinvolvingtheangularmomen- tum is given. Four new references have been added. Page 113: This new subsection, mostly on Dain’s inequality, and several new references are added. Page 117: Thisnewsubsectiononpotentialapplicationsofthequasi-localideasincosmology, and several new references are. Contents 1 Introduction 9 2 Energy-Momentum and Angular Momentum of Matter Fields 11 2.1 Energy-momentum and angular-momentum density of matter fields . . . . . . . . . 11 2.1.1 The symmetric energy-momentum tensor . . . . . . . . . . . . . . . . . . . 11 2.1.2 The canonical Noether current . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Quasi-local energy-momentum and angular momentum of the matter fields . . . . 13 2.2.1 The definition of quasi-local quantities . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Hamiltonian introduction of the quasi-local quantities . . . . . . . . . . . . 15 2.2.3 Properties of the quasi-local quantities . . . . . . . . . . . . . . . . . . . . . 16 2.2.4 Global energy-momenta and angular momenta . . . . . . . . . . . . . . . . 17 2.2.5 Quasi-local radiative modes and a classical version of the holography for matter fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 On the Energy-Momentum and Angular Momentum of Gravitating Systems 19 3.1 On the gravitational energy-momentum and angular momentum density: The diffi- culties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 The root of the difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 Pseudotensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.3 Strategies to avoid pseudotensors I: Background metrics/connections . . . . 21 3.1.4 Strategies to avoid pseudotensors II: The tetrad formalism. . . . . . . . . . 21 3.1.5 Strategies to avoid pseudotensors III: Higher derivative currents . . . . . . 22 3.2 On the global energy-momentum and angular momentum of gravitating systems: The successes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 Spatial infinity: Energy-momentum . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Spatial infinity: Angular momentum . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3 Null infinity: Energy-momentum . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.4 Null infinity: Angular momentum . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 The necessity of quasi-locality for observables in general relativity . . . . . . . . . 29 3.3.1 Nonlocality of the gravitational energy-momentum and angular momentum 29 3.3.2 Domains for quasi-local quantities . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.3 Strategies to construct quasi-local quantities . . . . . . . . . . . . . . . . . 31 4 Tools to Construct and Analyze Quasi-Local Quantities 33 4.1 The geometry of spacelike two-surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 The Lorentzian vector bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.3 Embeddings and convexity conditions . . . . . . . . . . . . . . . . . . . . . 34 4.1.4 The spinor bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1.5 Curvature identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.6 The GHP formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.7 Irreducible parts of the derivative operators . . . . . . . . . . . . . . . . . . 37 4.1.8 𝑆𝑂(1,1)-connection one-form versus anholonomicity . . . . . . . . . . . . . 37 4.2 Standard situations to evaluate the quasi-local quantities . . . . . . . . . . . . . . 37 4.2.1 Round spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.2 Small surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.3 Large spheres near spatial infinity . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.4 Large spheres near null infinity . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.5 Other special situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 On lists of criteria of reasonableness of the quasi-local quantities . . . . . . . . . . 46 4.3.1 General expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 Pragmatic criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.3 Incompatibility of certain ‘natural’ expectations . . . . . . . . . . . . . . . 49 5 The Bartnik Mass and its Modifications 50 5.1 The Bartnik mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1.1 The main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1.2 The main properties of 𝑚 (Σ) . . . . . . . . . . . . . . . . . . . . . . . . . 51 B 5.1.3 The computability of the Bartnik mass . . . . . . . . . . . . . . . . . . . . 52 5.2 Bray’s modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6 The Hawking Energy and its Modifications 54 6.1 The Hawking energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.1.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.1.2 Hawking energy for spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.1.3 Positivity and monotonicity properties . . . . . . . . . . . . . . . . . . . . . 55 6.1.4 Two generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2 The Geroch energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2.2 Monotonicity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.3 The Hayward energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7 Penrose’s Quasi-Local Energy-Momentum and Angular Momentum 59 7.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.1.1 How do the twistors emerge? . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.1.2 Twistor space and the kinematical twistor . . . . . . . . . . . . . . . . . . . 60 7.2 The original construction for curved spacetimes . . . . . . . . . . . . . . . . . . . . 62 7.2.1 Two-surface twistors and the kinematical twistor . . . . . . . . . . . . . . . 62 7.2.2 The Hamiltonian interpretation of the kinematical twistor . . . . . . . . . . 62 7.2.3 The Hermitian scalar product and the infinity twistor . . . . . . . . . . . . 63 7.2.4 The various limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.2.5 The quasi-local mass of specific two-surfaces. . . . . . . . . . . . . . . . . . 65 7.2.6 Small surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3 The modified constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3.1 The ‘improved’ construction with the determinant . . . . . . . . . . . . . . 68 7.3.2 Modification through Tod’s expression . . . . . . . . . . . . . . . . . . . . . 68 7.3.3 Mason’s suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8 Approaches Based on the Nester–Witten Two-Form 69 8.1 The Ludvigsen–Vickers construction . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.1.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.1.2 Remarks on the validity of the construction . . . . . . . . . . . . . . . . . . 71 8.1.3 Monotonicity, mass-positivity and the various limits . . . . . . . . . . . . . 71 8.2 The Dougan–Mason constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.2.1 Holomorphic/antiholomorphic spinor fields . . . . . . . . . . . . . . . . . . 72 8.2.2 The genericity of the generic two-surfaces . . . . . . . . . . . . . . . . . . . 73 8.2.3 Positivity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.2.4 The various limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.3 A specific construction for the Kerr spacetime . . . . . . . . . . . . . . . . . . . . . 76 9 Quasi-Local Spin Angular Momentum 77 9.1 The Ludvigsen–Vickers angular momentum . . . . . . . . . . . . . . . . . . . . . . 77 9.2 Holomorphic/antiholomorphic spin angular momenta . . . . . . . . . . . . . . . . . 78 9.3 A specific construction for the Kerr spacetime . . . . . . . . . . . . . . . . . . . . . 79 10 The Hamilton–Jacobi Method 80 10.1 The Brown–York expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.1.1 The main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.1.2 The variation of the action and the surface stress-energy tensor . . . . . . . 81 10.1.3 The general form of the Brown–York quasi-local energy . . . . . . . . . . . 82 10.1.4 Further properties of the general expressions . . . . . . . . . . . . . . . . . 83 10.1.5 The Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 10.1.6 The flat space and light cone references . . . . . . . . . . . . . . . . . . . . 86 10.1.7 Further properties and the various limits. . . . . . . . . . . . . . . . . . . . 87 10.1.8 Other prescriptions for the reference configuration . . . . . . . . . . . . . . 90 10.2 Kijowski’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10.2.1 The role of the boundary conditions . . . . . . . . . . . . . . . . . . . . . . 90 10.2.2 TheanalysisoftheHilbertactionandthequasi-localinternalandfreeenergies 91 10.3 Epp’s expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10.3.1 The general form of Epp’s expression. . . . . . . . . . . . . . . . . . . . . . 92 10.3.2 The definition of the reference configuration . . . . . . . . . . . . . . . . . . 92 10.3.3 The various limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.4 The expression of Liu and Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.4.1 The Liu–Yau definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.4.2 The main properties of 𝐸 (𝒮) . . . . . . . . . . . . . . . . . . . . . . . . 94 KLY 10.4.3 Generalizations of the original construction . . . . . . . . . . . . . . . . . . 95 10.5 The expression of Wang and Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 11 Towards a Full Hamiltonian Approach 98 11.1 The 3+1 approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 11.1.1 The quasi-local constraint algebra and the basis Hamiltonian . . . . . . . . 98 11.1.2 The two-surface observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.2 Approaches based on the double-null foliations . . . . . . . . . . . . . . . . . . . . 101 11.2.1 The 2+2 decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.2.2 The 2+2 quasi-localization of the Bondi–Sachs mass-loss . . . . . . . . . . 101 11.3 The covariant approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 11.3.1 The covariant phase space methods. . . . . . . . . . . . . . . . . . . . . . . 102 11.3.2 The general expressions of Chen, Nester and Tung: Covariant quasi-local Hamiltonians with explicit reference configurations . . . . . . . . . . . . . . 102 11.3.3 Covariant quasi-local Hamiltonians with general reference terms . . . . . . 105 11.3.4 Pseudotensors and quasi-local quantities . . . . . . . . . . . . . . . . . . . . 106 12 Constructions for Special Spacetimes 108 12.1 The Komar integral for spacetimes with Killing vectors. . . . . . . . . . . . . . . . 108 12.2 The effective mass of Kulkarni, Chellathurai, and Dadhich for the Kerr spacetime . 108 12.3 The Katz–Lynden-Bell–Israel energy for static spacetimes . . . . . . . . . . . . . . 108 13 Applications in General Relativity 110 13.1 Calculation of tidal heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 13.2 Geometric inequalities for black holes. . . . . . . . . . . . . . . . . . . . . . . . . . 110 13.2.1 On the Penrose inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 13.2.2 On the hoop conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 13.2.3 Other inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 13.3 Quasi-local laws of black hole dynamics . . . . . . . . . . . . . . . . . . . . . . . . 113 13.3.1 Quasi-local thermodynamics of black holes . . . . . . . . . . . . . . . . . . 113 13.3.2 On isolated and dynamic horizons . . . . . . . . . . . . . . . . . . . . . . . 114 13.4 Entropy bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 13.4.1 On Bekenstein’s bounds for the entropy . . . . . . . . . . . . . . . . . . . . 115 13.4.2 On the holographic hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 116 13.5 Quasi-local radiative modes of general relativity . . . . . . . . . . . . . . . . . . . . 117 13.6 Potential applications in cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 14 Summary: Achievements, Difficulties, and Open Issues 119 14.1 On the Bartnik mass and Hawking energy . . . . . . . . . . . . . . . . . . . . . . . 119 14.2 On the Penrose mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 14.3 OntheDougan–Masonenergy-momentaandtheholomorphic/antiholomorphicspin angular momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 14.4 On the Brown–York–type expressions . . . . . . . . . . . . . . . . . . . . . . . . . 121 15 Acknowledgments 124 References 125 Quasi-Local Energy-Momentum and Angular Momentum in General Relativity 9 1 Introduction Over the last 30 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 inspiredthemoreambitiousclaimtoassociateenergy(orratherenergy-momentumand,ultimately, angularmomentumaswell)toextended,butfinite,spacetimedomains,i.e.,atthequasi-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 cen- tral 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 ther- modynamics 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 bal- ance 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 notionofenergy-momentumhasproventobesurprisinglydifficult. Nowadays,thestateoftheartis typicallypostmodern;althoughthereareseveralpromisingandusefulsuggestions,wenotonlyhave noultimate,generallyacceptedexpressionfortheenergy-momentumandespeciallyfortheangular momentum,butthereisnotevenaconsensusintherelativitycommunityongeneralquestions(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 grav- itational energy-momentum and angular momentum can be introduced in (and hence can be un- derstoodfrom)thetheoryofmatterfieldsinMinkowskispacetime. Thus,inSection2.1,wereview the Belinfante–Rosenfeld procedure that we will apply to gravity in Section 3, introduce the no- tionofquasi-localenergy-momentumandangularmomentumofthematterfieldsanddiscusstheir properties.Thephilosophyofquasi-localityingeneralrelativitywillbedemonstratedinMinkowski 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 mo- mentum, and we clarify why the gravitational observables should necessarily be quasi-local. The Living Reviews in Relativity http://www.livingreviews.org/lrr-2009-4 10 L´aszl´o B. Szabados toolsneededtoconstructandanalyzethequasi-localquantitiesarereviewedinthefourthsection. This closes the first (general) part of the review (Sections 2–4). Thesecondpartisdevotedtothediscussionofthespecificconstructions(Sections5–12). Since most of the suggestions are constructions, they cannot be given as a short mathematical defini- tion. 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 prop- erties and applications are usually summarized. 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 5–12, Section 14 (as well as the choice of subject matter of Sections 2- 4) reflects our own personal interest and view of the subject. The theory of 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 𝑔 is −2, and the curvature Ricci tensors and curvature scalar of the covariant derivative 𝑎𝑏 ∇ are defined by (∇ ∇ −∇ ∇ )𝑋𝑎 :=−𝑅𝑎 𝑋𝑏, 𝑅 :=𝑅𝑎 and 𝑅:=𝑅 𝑔𝑏𝑑, respectively. 𝑎 𝑐 𝑑 𝑑 𝑐 𝑏𝑐𝑑 𝑏𝑑 𝑏𝑎𝑑 𝑏𝑑 Hence, Einstein’s equations take the form 𝐺 +𝜆𝑔 := 𝑅 − 1𝑅𝑔 +𝜆𝑔 = −8𝜋𝐺𝑇 , where 𝑎𝑏 𝑎𝑏 𝑎𝑏 2 𝑎𝑏 𝑎𝑏 𝑎𝑏 𝐺 is Newton’s gravitational constant and 𝜆 is the cosmological constant (and the speed of light is 𝑐 = 1). However, apart from special cases stated explicitly, the cosmological constant will be assumed to be vanishing, and in Sections 13.3 and 13.4 we use the traditional cgs system. Living Reviews in Relativity http://www.livingreviews.org/lrr-2009-4