A review of total energy-momenta in GR with a positive cosmological constant
aa r X i v : . [ g r- q c ] S e p A review of total energy-momenta in GR with apositive cosmological constant ∗ László B. SzabadosWigner Research Centre for Physics,H-1525 Budapest 114, P. O. Box 49, European UnionPaul TodMathematical Institute, Oxford University,Oxford OX2 6GGSeptember 13, 2018
Abstract
A review is given of the various approaches to and expressions for total energy-momentum and mass in the presence of a positive cosmological constant in Einstein’sfield equations, together with a discussion of the key conceptual questions, mainideas and techniques behind them.
As a consequence of the equivalence principle, following the lesson of the Galileo–Eötvösexperiments, gravity is known to have a universal nature and the spacetime metric plays adouble role: both as the field variable for gravity and as the geometric background for thedynamics of the matter fields at the same time. A direct consequence of the lack of anynon-dynamical geometric background is that any local expression for the gravitationalenergy-momentum is necessarily pseudotensorial or, in the tetrad formulation of thetheory, Lorentz gauge dependent (see e.g. [1, 2, 3]). Although all these local quantitiescan be recovered from (various forms of) a single geometric object on the linear framebundle (or of its subbundles) (see e.g. [4, 5, 6, 7]), the gravitational energy-momentumdensity in the spacetime is not well defined – it cannot be localized to points. Thegravitational energy-momentum is non-local in nature, and can be well defined only ifit is associated to extended domains in spacetime. Thus, it must be a measure of total energy-momentum, i.e. when the domain is an infinitely large part of the spacetime (for ∗ Dedicated to our friend, Jörg Frauendiener, on the occasion of his 60th birthday.
Ach, wie schön muss sich’s ergehen,Dort in Newmans Himmelsraum.Und Lichtkugeln auf den Höhen –O wie lab’n sie meinen Traum ! – from Ein theoretisch–physikalischer Schiller–Traum by N. V. Mitskievitch quasi-local , when the domain is only a compactsubset of the spacetime.From pragmatic points of view the significance of these quantities is given by the pos-itivity of the corresponding total (or quasi-local) energy/mass, or at least their bound-edness from below. In fact, some of these quantities have already been proven to beuseful tools in geometric analysis (e.g. of asymptotic structure of spacetime or black-hole uniqueness), in stability investigations of general relativistic gravitating systems, innumerical calculations (e.g. to control errors), and so on.The usual form of the total energy-momentum of a localized gravitating system isa 2-surface integral of some local ‘superpotential’ U , which also depends (linearly) onsome vector field K a or (quadratically) on some spinor field λ A representing ‘asymptotictranslations’ in the spacetime: P a K a := I S U ( K )d S , or P a σ aA B ′ λ A ¯ λ B ′ := I S U ( λ, ¯ λ )d S . (1.1)Here K a and λ A are the components of the ‘asymptotic translation’ K a and its spinorconstituent λ A , respectively, in some basis of the space of (candidate) asymptotic trans-lations , and σ aA B ′ , a = 0 , ..., , A , B = 0 , , are the standard SL (2 , C ) Pauli matrices.Thus, the energy-momentum 4-vector is an element of the dual space of the space ofasymptotic translations. Therefore, to have a well defined total energy-momentum ex-pression, we should specify: (1) the domain of integration S (i.e. the choice of what toconsider as the physical system); (2) the ‘superpotential’ U ; and (3) the ‘generator’ ( K a or λ A ) of the quantity in question (i.e. the definition of the asymptotic translations).Different choices for these yield different expressions with different properties. Total mass (rather than energy-momentum) can be associated not only with localizedsources, but with closed universes with non-negative Λ as well. In this case the domainof integration is a Cauchy surface (rather than a closed 2-surface).The aim of the present paper is to review the total energy-momentum/mass construc-tions in the presence of a strictly positive cosmological constant Λ . However, to put themin perspective and to see the roots of certain key ideas behind the actual constructions,and also to motivate how to choose the domain, the superpotential and the generatorfield in items (1)-(3) above, we briefly recall the analogous constructions in the Λ = 0 and Λ < cases where these ideas appeared first. This part of the review is far frombeing complete. For a review of the quasi-local constructions, see e.g. [8]. Λ = 0
If the cosmological constant is zero, then the spacetime describing the gravitational ‘field’of a localized source is expected to be ‘asymptotically flat’ in some well-defined sense,and hence its global structure at large distances is expected to be similar to that ofthe Minkowski spacetime. (For the global properties of the latter, see e.g. [9].) Infact, formally the conserved
Arnowitt–Deser–Misner (ADM) energy-momentum [10] isbased on a 2-surface integral on the boundary at infinity of an (appropriately defined)asymptotically flat spacelike hypersurface that extends to spatial infinity . In Minkowskispacetime the t = const hyperplanes, denoted by Σ t , provide a foliation of the spacetime As we shall see, this can be a problematic notion with Λ = 0 .
2y such global Cauchy surfaces, while their (common) boundary at infinity is the r → ∞ limit of the 2-spheres S t,r := { t = const , r = const } (see Fig. 1.i.). Clearly, this boundaryis spatially separated from the world tube of the source. i i − i + S t t ` − ` + i i − i + S t t u ` + ` − i i − i + S T T ( t ,0,0,0) ` − ` + Figure 1: Foliations of the Minkowski spacetime: i. The hypersurfaces Σ t form a globalfoliation. Each of the leaves is a global Cauchy surface with R topology, and all theseextend to spatial infinity i . These hypersurfaces are both intrinsically and extrinsicallyasymptotically flat. The t = const hyperplanes in the Cartesian coordinates are likethis. ii. The hypersurfaces Σ τ , which are only partial Cauchy surfaces, extend to thefuture null infinity I + and foliate both the spacetime and I + . These hypersurfaces areintrinsically asymptotically hyperboloidal and the extrinsic curvature is asymptoticallyproportional to the intrinsic metric. For fixed T > , in the Cartesian coordinates, thehypersurfaces τ := t − √ T + r = const have this character. iii. However, for fixed τ and variable T , the hypersurfaces Σ T , given by T := ( t − τ ) − r = const , foliate onlythe chronological future of the point ( τ, , , but do not provide a foliation of (even anopen subset of) I + .On the other hand, if the localized system emits radiation and the total energy-momentum is expected to reflect the dynamical aspects of the system, e.g. to give ac-count of the energy carried away by the radiation, then that should be associated witha hypersurface extending to the null infinity of the spacetime. In fact, the Trautman–Bondi–Sachs (TBS) energy-momentum [11, 12, 13] is an integral on the boundary atinfinity of an outgoing null hypersurface that defines a retarded time instant. One mayalso think of this 2-surface as the boundary at infinity of an asymptotically hyperboloidal spacelike hypersurface whose asymptote is just the null hypersurface. For example, inMinkowski spacetime, the family of spacelike hypersurfaces τ := t − √ T + r = const for some constant T > , denoted by Σ τ , provides a foliation of the whole spacetime (seeFig. 1.ii.). Σ τ intersects future null infinity at the retarded time instant u = τ . The in-duced metric h ab on them is of constant curvature with the scalar curvature R = − /T ,and the extrinsic curvature is χ ab = h ab /T . Hence, these are hyperboloidal , but only par-tial Cauchy surfaces . The r → ∞ limit of the 2-spheres S τ,r := { τ = const , r = const } is null separated from the source in the centre r = 0 at t = τ . This energy-momentum It might be interesting to note that the line element of the flat metric with the global foliations Σ t and Σ τ takes the Friedman–Robertson–Walker (FRW) form with k = 0 and k = − , respectively. On theother hand, for fixed τ and variable T the hypersurfaces Σ T := { T = const } foliate only the chronological u that the hypersurface defines, and, in particular,the mass, defined to be the Lorentzian length of the TBS energy-momentum 4-vector, isa monotonically decreasing function of u (‘Bondi’s mass-loss’). To derive this, a foliationof the null infinity and the asymptotic solutions of the field equations are needed.In the conformal approach of Penrose [14, 15] the future null infinity, as a smoothboundary I + , is attached to the conformal spacetime. In this framework the asymptoticflatness is defined by the compactifiability of the spacetime by such a boundary. Since Λ = 0 , I + is a null hypersurface in the conformal spacetime, its null geodesic generatorsare shear-free (and, in a certain conformal gauge, divergence-free, too); and it is assumedto have S × R topology . The advantage of this approach to asymptotic flatness isthat the techniques of local differential geometry can be used to study the asymptoticproperties of the fields and spacetime (see also [17, 18, 15, 19]). In particular, the r → ∞ limit of the large spheres S τ,r is the regular 2-surface Σ τ ∩ I + , a so-called cut, of the nullinfinity I + . The difference of the energy-momenta on two different cuts can be writtenas a flux integral on the piece I ⊂ I + between the two cuts [20], and its vanishing, i.e.the absence of outgoing gravitational radiation, is equivalent to the vanishing of the newsfunction [12] or equivalently of the magnetic part of the rescaled Weyl curvature, definedby the null normal of I + , on I [21].Nevertheless, the existence of null infinity as a smooth boundary of the conformalspacetime restricts the fall-off rate of the physical metric: it should tend to the flat metricas /r . For a treatment of asymptotically flat spacetimes with slower (e.g. logarithmic)fall-off metrics, see. e.g. [22]. In Minkowski spacetime the four independent translational Killing fields are defined in ageometric/algebraic way: they are constant vector fields ; and, also, they form a commuta-tive ideal in the Poincaré algebra of the Killing fields. At spatial infinity their restrictionto the boundary 2-surface at infinity yields constant vector fields there. Since the isome-tries of the spacetime must take I + to itself, the vector fields that they determine on I + must be tangent to I + . In particular, the vector fields that the translations yield on I + are all proportional to the tangent of the null geodesic generators of I + . However,in contrast to the spatial infinity case, the factor of proportionality is not constant, butrather is a linear combination of the first four ordinary spherical harmonics.In fact, these vector fields, the so-called Bondi–Metzner–Sachs (BMS) translationalvector fields, can be characterized completely in terms of the conformal structure of I + .This yields the notion of asymptotic translations even in general spacetimes admitting I + [13, 17, 18, 15]; and these coincide with the (equivalence classes of the) asymptoticallyconstant solutions of the asymptotic Killing equation [20]. The BMS translational vectorfields can be characterized by their Weyl or , too: theyare proportional to the constituent spinor of the tangent of the null geodesic generators of I + , and the factor of proportionality is a linear combination of spin-weight / sphericalharmonics. These can be recovered as the solutions of various linear partial differentialequations on the cut. (For a list of these, see the appendix of [23].) future of the point with Cartesian coordinates ( τ, , , . All the leaves of this latter foliation intersectthe future null infinity at one and the same cut u = τ , rather than providing a foliation of the nullinfinity (see Fig. 1.iii.). This can be proved, given certain causality assumptions; see [16]. .1.3 The superpotential In the literature several different forms of the integrand in (1.1) are known both forthe ADM [10] and TBS [11, 12, 13] energy-momenta: they can be given (1) by certainexpansion coefficients in the asymptotic expansion of the components of the (spatial orspacetime) metric as a series of /r in an asymptotic Cartesian or retarded null coordi-nate system; or (2) by the traditional superpotential of some classical (e.g. Einstein’s)energy-momentum pseudotensor (or, in the tetrad formalism of the theory, by an SO (1 , gauge dependent energy-momentum complex) in some appropriate coordinate system (orLorentz frame). Also, (3) one can write the spacetime metric as the sum of the flatspacetime metric ¯ g ab and some ‘correction field’ γ ab , and then rewrite the exact Einsteinequations by the linearized Einstein tensor, built from γ ab on the background ¯ g ab , andan effective energy-momentum tensor. The U in (1.1) is the contraction of the superpo-tential for this effective energy-momentum tensor and a translational Killing field of thebackground metric. (4) The 2-surface integral (1.1) can also be sought in the form of theboundary term in the Hamiltonian of the theory, in which the lapse and the shift are thetimelike and spacelike part of K a , respectively. (5) These total energy-momenta can alsobe written as a Komar or linkage integral, based on the vector field K a (see [24]); or,(6) using the field equations, they can be re-expressed by certain parts of the curvaturetensor.However, (7) there is a ‘universal superpotential’, built from a Dirac spinor or apair of 2-component spinors, by means of which the classical (e.g. the pseudotensorial)superpotentials could be recovered (see the Introduction); and both the ADM and TBSenergy-momenta can be re-expressed. This is the Nester–Witten 2-form [25, 26], given interms of any two 2-component spinors [28] by u ( λ, ¯ µ ) ab := i2 (cid:0) ¯ µ A ′ ∇ BB ′ λ A − ¯ µ B ′ ∇ AA ′ λ B (cid:1) . (1.2)Although this is a complex-valued 2-form, its integral on any closed, orientable 2-surface S defines a Hermitian bilinear form on the space of the spinor fields on S .To ensure the existence of the integrals in (1.1), and, in the traditional formu-lation, also the independence of the ADM energy-momentum of the coordinate sys-tem/background Minkowski metric, non-trivial fall-off conditions should be imposed bothon the matter fields and the geometric data ( h ab , χ ab ) on the hypersurface. The lattershould tend asymptotically to the trivial flat or hyperboloidal data on the hypersurfaces Σ t and Σ τ , respectively.The key properties of the ADM and TBS energy-momenta are that they are futurepointing and timelike vectors (a property we will call ‘positivity’), provided the energy-momentum tensor of the matter fields satisfies the dominant energy condition on thespacelike hypersurface Σ whose boundary at infinity is S ; and the vanishing of theseenergy-momenta is equivalent to the flatness of the domain of dependence D (Σ) of Σ andthe vanishing of the matter fields (‘rigidity’) [27, 26, 25, 28, 29]. These results together,known as the positive energy or positive mass theorem, hold true even in the presence ofblack holes [30]. Probably the simplest proof of this theorem is based on the use of spinors,the superpotential (1.2) and the Witten-type gauge condition for the spinor fields on thehypersurface Σ [29]. The rigorous mathematical proof of the existence and uniqueness ofthe solution of the Witten equation on asymptotically flat spacelike hypersurfaces couldbe based on the techniques of [31]. The analogous results on asymptotically hyperboloidal hypersurfaces are given in [32]. 5he spacetimes describing the history of closed universes are defined to be thoseglobally hyperbolic spacetimes that admit compact Cauchy surfaces. Clearly, these do not represent the gravitational field of localized sources, and total energy-momentum ofthe form (1.1) cannot be associated with them. Nevertheless, their total mass can beintroduced using the (common) of the ADM and TBS energy-momenta. Although its non-negativity is trivial by construction (given the assumptionthat the dominant energy condition holds), still it has a non-trivial rigidity property:this total mass is zero precisely when the spacetime is locally flat, the matter fields arevanishing and the Cauchy surface is a 3-torus [33, 34]. Λ < If Λ < , then the gravitational field of a localized source is expected to be representedby a spacetime of asymptotically constant negative curvature, i.e. by some ‘asymptot-ically anti-de Sitter’ spacetime. The anti-de Sitter spacetime itself [9] is the universalcovering space of the vacuum spacetime with constant negative curvature R = 4Λ < .Its conformal boundary I is a timelike hypersurface in its conformal completion withtopology S × R . In the standard global coordinates ( t, r, θ, φ ) [9] the t = const spacelikehypersurfaces, denoted by Σ t , are partial Cauchy surfaces and intersect I in 2-spheres.They provide a foliation of I as well (see Fig. 2.i.). The induced metric on them isof constant negative curvature with scalar curvature R = 2Λ and vanishing extrinsiccurvature, i.e. these hypersurfaces are intrinsically hyperboloidal and extrinsically flat .The traditional definition of asymptotically anti-de Sitter spacetimes is based on thedecomposition of the spacetime metric into the sum of the anti-de Sitter metric ¯ g ab andsome ‘correction term’ γ ab [35]; but the latter tensor field should be required to satisfyappropriate fall-off conditions. However, adapting the key ideas of the conformal approachto infinity to the present case, the asymptotically anti-de Sitter spacetimes could bedefined in a fully geometric way as in [36], by the conformal compactifiability of thespacetime. Since Λ < the conformal boundary I is timelike , the trace-free part ofits extrinsic curvature is vanishing and the trace can be made to be vanishing in someconformal gauge. However, its intrinsic metric at this point is not restricted further,and could still be completely general . Thus, to restrict the asymptotic properties ofthe spacetime to be similar to that of the anti-de Sitter, the conformal boundary I is usually required to be topologically S × R and intrinsically conformally flat [36].The latter condition is called the ‘reflective boundary condition’ [37], and is equivalentto the vanishing of the magnetic part of the conformally rescaled Weyl curvature [38],determined by the spacelike normal of I , on I . Nevertheless, this boundary conditionexcludes any gravitational energy-flux through the conformal boundary, and, as Friedrichstresses (see [39], and especially [40]), this boundary condition is one choice among many.Other ‘natural’ boundary conditions, when incoming and/or outgoing energy flux through I is allowed, are also possible [40].The domain S of integration in (1.1) is a closed spacelike 2-surface, which may alsobe called a ‘cut’, in I . In the exact anti-de Sitter spacetime such a cut is, for example, The anti-de Sitter line element can be rewritten in FRW form with k = − , but the leaves of thislatter foliation would foliate only globally hyperbolic proper subsets of the spacetime, say D (Σ t ) forgiven t (see Fig. 2.ii.). S t t `S T T Figure 2: Foliations of the anti-de Sitter spacetime: i. The hypersurfaces Σ t form aglobal foliation. The leaves Σ t are partial Cauchy surfaces with R topology, and thesefoliate the (timelike) conformal boundary I , too. The t = const hypersurfaces in theglobal coordinates of the anti-de Sitter spacetime are like this. These hypersurfaces areintrinsically hyperboloidal and extrinsically flat. ii. The hypersurfaces Σ T foliate neitherthe whole spacetime nor its conformal boundary I . They foliate only a globally hy-perbolic open subset. These hypersurfaces are intrinsically asymptotically hyperboloidal,and their extrinsic curvature is asymptotically proportional to the intrinsic metric. Sucha T , whose level sets are the hypersurfaces Σ T , is the time coordinate in the FRW formof the anti-de Sitter metric. (See also Figure 20 of [9].)the intersection Σ t ∩ I . Since, however, I is timelike , there is no way to distinguishthe ADM and TBS type energy-momenta: the specific properties of the total energy-momentum associated with the cuts of I depend crucially on the boundary conditionson I . The Lie algebra of the Killing fields is the anti-de Sitter Lie algebra so (2 , . Since,however, so (2 , is semi-simple , there is no way to single out ‘translations’ in a canonical algebraic way; nor, since this spacetime does not admit any non-trivial constant vectorfield, is there a way to single out ‘translations’ in a canonical geometric manner either.Nevertheless, the coordinate t is timelike and the components of the metric in the globalcoordinates ( t, r, θ, φ ) are independent of t , so the Killing field ( ∂/∂t ) a is often interpretedas the time translational Killing field (though, for example, it is not a geodesic vectorfield). The Killing fields of the spacetime extend to conformal Killing fields on infinity,and, in particular, ( ∂/∂t ) a extends to a timelike one on I .If, however, the ‘asymptotically anti-de Sitter spacetimes’ are defined simply by theexistence of a (timelike) conformal boundary I (possibly with S × R topology) but without any further condition on its intrinsic conformal geometry, then in general I doesnot admit any conformal isometry. Hence, the spacetime cannot admit any ‘asymptoticKilling vector’ either. On the other hand, the requirement of the intrinsic conformal7atness of I , i.e. the reflective boundary condition, guarantees the maximal number ofconformal Killing fields, and their Lie algebra is isomorphic to the anti-de Sitter algebra so (2 , [36]. As in the
Λ = 0 case (in subsection 1.1.3), several different superpotentials can (and, infact, have been) used in (1.1) to associate energy-momentum with cuts of the timelikeconformal boundary I . For example, that could be based on the use of the effectiveenergy-momentum tensor built from γ ab mentioned in (3) of subsection 1.1.3 [35]; oron another explicitly given superpotential [41], obtained from a Hamiltonian. Also, theprecise fall-off conditions for γ ab are determined which make the resulting expressionsunambiguously defined. (Without this, the expression of [35] would suffer from coordi-nate ambiguities.) The integrand used in [36] is the electric part of the rescaled Weylcurvature, contracted with a conformal Killing vector of I ; and this was shown in [42]to coincide with the ‘renormalized’ quasi-local mass of Penrose [15] (after subtracting thecosmological constant term), calculated on a cut of I . The latter two are manifestlycoordinate-free, and all these expressions, when they are well-defined, give the same re-sult. The Nester–Witten 2-form (1.2) in its ‘renormalized’ form is used in [30], where aWitten-type proof of the positivity of energy, and also of rigidity, are also given. In allthese investigations the intrinsic conformal flatness of I was assumed. Λ > The simplest explanation of the observed accelerating expansion of the Universe [43, 44]is the presence of a strictly positive cosmological constant in Einstein’s field equations.Thus, the future history of our Universe is asymptotically de Sitter. Also, the positivityof Λ is the basis of the conformal cyclic cosmological (CCC) model of Penrose [45]. Inthe analysis of the asymptotic structure of these spacetimes, the total energy-momentacould still provide useful tools. In fact, the need (and three potential expressions) for thetotal TBS type energy was already raised by Penrose in [46]. Since then several papersdevoted to the question of total energy-momentum in asymptotically de Sitter spacetimeshave appeared. The aim of the present paper is to give a review of these results.In section 2 we review the properties of general asymptotically de Sitter spacetimes;and discuss the important special spacetimes that are asymptotic to de Sitter, withspecial emphasis on their foliations, symmetries and the fields on the background de Sitterspacetime. Then, in section 3, we discuss ADM type energy-momentum expressions,while sections 4 and 5 are devoted to the TBS type expressions and a suggestion forthe total mass in closed universes, respectively. The signature of the spacetime metric ischosen to be (+ , − , − , − ) , and Einstein’s equations are written in the form R ab − Rg ab = − κT ab − Λ g ab with κ = 8 πG . 8 The structure of asymptotically de Sitter spacetimes
The de Sitter spacetime is the constant positive curvature solution of the vacuum Einsteinequations with scalar curvature R = 4Λ > . (For a summary of its key geometricproperties, see [47], and for a detailed discussion of its global structure, especially itsPenrose diagram, see [9].) Its line element in the global coordinates ( t, r, θ, φ ) is ds = dt − α cosh ( t/α ) (cid:16) dr + sin r (cid:0) dθ + sin θ dφ (cid:1)(cid:17) , where α := 3 / Λ , and the range of the coordinates are t ∈ R , r ∈ [0 , π ] and ( θ, φ ) ∈ S .This metric has the FRW form with k = 1 , scale function S ( t ) = α cosh( t/α ) and foliationby global Cauchy surfaces Σ t := { t = const } with S topology (see Fig. 3.i.). Then, inthe coordinates ( τ, r, θ, φ ) with τ := 2 arctan(exp( t/α )) − π/ , the future/past conformalboundary, I ± , is given by τ = ± π/ and ds = S (cid:16) dτ − dr − sin r (cid:0) dθ + sin θ dφ (cid:1)(cid:17) . ` − ` + S t t ` − ` + i S t ^ t ^ Figure 3: Foliations of the de Sitter spacetime: i. The hypersurfaces Σ t form a globalfoliation. The leaves Σ t are global Cauchy surfaces with S topology. The t = const hypersurfaces in the global coordinates of the de Sitter spacetime are like this, which aremetric spheres. ii. The hypersurfaces Σ ˆ t foliate the ‘steady state’ part of (or ‘Poincarépatch’ in) the (half) de Sitter spacetime. Their topology is R , they are intrinsicallyasymptotically flat and their extrinsic curvature is asymptotic to a nonzero value. Theirone-point compactification yields a ‘spatial infinity’ i , which is a point of the futureconformal boundary I + . These hypersurfaces do not foliate I + . (See also Figure 17 of[9].)The (half) de Sitter spacetime, often called the ‘steady state universe’ or ‘Poincarépatch’, has another foliation with intrinsically flat spacelike hypersurfaces as well [9] (seeFig. 3.ii.). These are the Σ ˆ t := { ˆ t = const } hypersurfaces, where9 t := α ln (cid:16) sinh( tα ) + cosh( tα ) cos r (cid:17) , ˆ x := α tanh( tα ) + cos r sin r sin θ cos φ, ˆ y := α tanh( tα ) + cos r sin r sin θ sin φ, ˆ z := α tanh( tα ) + cos r sin r cos θ. (2.1)This coordinate system covers only the ‘steady state’ part of the de Sitter spacetime, forwhich sinh( t/α ) + cosh( t/α ) cos r > , and the line element takes the form ds = d ˆ t − exp(2ˆ t/α ) (cid:16) d ˆ x + d ˆ y + d ˆ z (cid:17) . Thus, this line element has the FRW form with k = 0 and scale function S (ˆ t ) = exp(ˆ t/α ) .Hence, the extrinsic curvature of the hypersurfaces Σ ˆ t is a pure trace, and the meancurvature is the positive constant χ = 3 /α . All these hypersurfaces reach I + at the point r = π , anti-podal to the origin, where they are, in fact, tangential to I + . Thus, the anti-podal point provides a one-point-compactification of these hypersurfaces, and this can beinterpreted as the ‘spatial infinity’ of the flat 3-spaces. Clearly, this foliation is analogousto the foliation of the Minkowski spacetime by the t = const hyperplanes. Nevertheless,the hypersurfaces Σ ˆ t are extrinsically hyperboloidal ; and none of these hypersurfaces is aglobal Cauchy surface, because no future inextendible non-spacelike curve terminating atthe anti-podal point intersects any of these Σ ˆ t . These are Cauchy surfaces only for thesteady state part of the de Sitter spacetime. ` − ` + i S t ^ t ^ ` − ` + S T T Figure 4: Foliations of the de Sitter spacetime: iii. The hypersurfaces Σ ˆ τ foliate both the ‘steady state’ part of the de Sitter spacetime and the (spacelike) future conformalboundary I + (minus i ). The leaves Σ ˆ τ are intrinsically asymptotically hyperboloidal,and their extrinsic curvature is asymptotically proportional to the intrinsic metric. iv.The hypersurfaces Σ T foliate the disjoint union of the past domain of dependence of the r ∈ [0 , π/ and the r ∈ ( π/ , π ] hemispheres of the conformal boundary I + ≈ S . Theydo not provide a foliation of the conformal boundary itself.In contrast to this foliation, a neighbourhood of I + − { ( π, , } can be foliatedby spacelike hypersurfaces which foliate I + − { ( π, , } , too (see e.g. [32]). Such afoliation can be based on the coordinate system ( u, w, θ, φ ) , where, in terms of the globalcoordinates above, u := α ( τ − r ) and w := ( π/ − τ ) /α . Then I + is just the w = 0 hypersurface, and the u = 0 ‘origin cut’ of I + is the r = π/ maximal 2-sphere. Then, for10 fixed positive W and ˆ τ := u − W w , let us define Σ ˆ τ := { ˆ τ = const } . These are spacelikehypersurfaces, − απ/ ≤ ˆ τ ≤ απ/ , and Σ ˆ τ ∩ I + is the cut r = π/ − ˆ τ /α . (Althoughthese hypersurfaces are not smooth at r = 0 [in the original coordinates], these conicalsingularities can be smoothed out.) Hence, { Σ ˆ τ } provides, in fact, a foliation of theconformal boundary (see Fig. 4.iii.). One can show [32] that in the de Sitter spacetimetheir intrinsic geometry is asymptotically hyperboloidal and their extrinsic curvature isasymptotically proportional to their intrinsic metric.The analog of the three special foliations shown by Figures 3.i., 3.ii. and 4.iii. are usedin asymptotically de Sitter spacetimes in a total mass construction in closed universes(section 5), and in the definition of ADM type (section 3) and of TBS type energy-momenta (section 4), respectively. The foliation shown by Fig. 4.iv. (like the ones shownby Fig. 1.ii. in the Minkowski and by Fig. 2.ii. in the anti-de Sitter cases) does notseem to provide an appropriate framework in which the mass-loss property of a TBS typeenergy-momentum could potentially be proven. The de Sitter spacetime is of maximal symmetry: the Lie algebra of its Killing andconformal Killing vectors is so (1 , , the so-called de Sitter Lie algebra, and so (2 , , re-spectively. Since both are semi-simple , generators of ‘translations’ cannot be defined in anatural geometric or algebraic way. On the other hand, there is an analytical character-ization of a basis of proper conformal Killing vectors: these are gradients (see e.g. [47])and any conformal Killing vector is a sum of a gradient conformal Killing vector and aKilling vector.Clearly, the six independent Killing vectors that are tangent to the hypersurfaces Σ t are spacelike, and the remaining four are timelike on some open subset, and spacelike onthe interior of its complement. However, since I + is spacelike and the isometries map I + to itself, every spacetime Killing field must be spacelike or zero on I + . For example,the Killing field K e = − cos r ( ∇ e t ) + α sinh (cid:0) tα (cid:1) cosh (cid:0) tα (cid:1) sin r ( ∇ e r ) , given in the global coordinates, is timelike precisely on the domain where cos r > tanh ( t/α ) holds. In the t → ∞ limit the closure of this domain reduces to the twopoints (0 , , and ( π, , of the future conformal boundary, where K e vanishes. Theremaining three Killing vectors behave in a similar way. Hence, no Killing field is timelikeon any open neighbourhood of some asymptotic end of the hypersurfaces Σ ˆ t of the in-trinsically flat foliation of the steady state part of the de Sitter spacetime. In particular,the Killing field K a := ( ∂∂ ˆ t ) a + ˆ x ( ∂∂ ˆ x ) a + ˆ y ( ∂∂ ˆ y ) a + ˆ z ( ∂∂ ˆ z ) a , (2.2)given in the coordinates (2.1) on the steady state part of the de Sitter spacetime, istimelike precisely for exp(2ˆ t/α )(ˆ x + ˆ y + ˆ z ) < . A simple (although less explicit) The hypersurfaces Σ T := { T = const } , given in the coordinates ( T, R, θ, φ ) defined by cosh( T /α ) :=cosh( t/α ) cos r , sinh( T /α ) cosh R := sinh( t/α ) (see e.g. [48]), foliate only the disjoint union of the pastdomain of dependence of the r ∈ [0 , π/ and the r ∈ ( π/ , π ] pieces of I + . Moreover, all of thesehypersurfaces intersect I + in the 2-sphere r = π/ , rather than giving a foliation of I + (see Fig. 4.iv.). globally timelike Killing field in a neighbourhoodof I + is given in [47].Clearly, one of the five conformal Killing fields, namely K e = cosh( t/α )( ∇ e t ) , iseverywhere timelike, but the remaining four are timelike only on some open subset. Forexample, K e = sinh (cid:0) tα (cid:1) cos r ( ∇ e t ) − α cosh (cid:0) tα (cid:1) sin r ( ∇ e r ) is timelike precisely when cosh ( t/α ) cos r > . In the t → ∞ limit this reduces to thedisjoint union of the r ∈ [0 , π/ and r ∈ ( π/ , π ] hemispheres of I + . This vector fieldis future pointing on the r ∈ [0 , π/ , but past pointing on the r ∈ ( π/ , π ] hemisphere.The remaining three conformal Killing fields behave in a similar way. Rewriting the con-travariant form of them in the ( τ, r, θ, φ ) coordinates we find that all these are orthogonalto I + in the conformal spacetime: K a = 1 α ( ∂∂τ ) a , K a ≈ cos r K a , (2.3) K a ≈ sin r sin θ cos φ K a , K a ≈ sin r sin θ sin φ K a , K a ≈ sin r cos θ K a ; where ≈ means ‘equal at the points of I + ’. Therefore, the structure of the conformalKilling vectors at the conformal boundary is similar to that of the BMS translationsat future null infinity in asymptotically flat spacetimes, though now the number of theconformal Killing vectors is five. Another (twistor theoretical) demonstration of thisstatement is given in [47], where all the 15 conformal Killing vectors are constructedfrom the independent solutions of the twistor equation, too. Using Dirac spinors, someof these results are also discussed in [49]. Some of the ADM type energy-momenta arebased on the use of the asymptotic Killing vectors, while others depend on the use of theasymptotic conformal Killing vectors. In Yang–Mills theory on the de Sitter background the meaning of the vanishing of themagnetic field strength on I + , i.e. the meaning of the analog of the condition of theintrinsic conformal flatness of I + , is investigated in [21]. It is shown that this conditionremoves half the degrees of freedom of the Yang–Mills field; and all the ten de Sitterfluxes, built from the energy-momentum tensor and the ten conformal Killing fields of I + ,vanish. A systematic investigation of the Maxwell fields on and of the linear gravitationalperturbation of the (steady state part of the) de Sitter spacetime are given in [50]. Thecovariant phase space method is used to obtain a Hamiltonian for them, in which the‘time translation’ is a Killing field which is future pointing timelike only in the intersectionof the chronological past of the point (0 , , of I + and the steady state universe, butspacelike on the interior of the rest of the steady state universe. The resulting energyflux through I + can be negative and arbitrarily large . The requirement of the vanishingof the magnetic field strength, and of the magnetic part of the rescaled Weyl curvatureon I + , removes half the degrees of freedom of the Maxwell and linear gravitationalperturbations, respectively, and yields vanishing energy flux through I + [50]. Herethe ‘energy’ is defined by using the Killing field (2.2). For a single, isolated source12ocated at the origin, and in the absence of incoming gravitational waves through thenull boundary of the steady state part of the de Sitter spacetime, which is the physicalsituation of principal interest in the body of work here referred to, this energy is provento be positive. Under the same assumptions, and following a tour de force of calculation,in [51] a generalisation of the Einstein quadrupole formula is derived and discussed. Thesummary of these results is given in [52, 53]. For the calculation of the Penrose mass ata cut of I + in the linearized theory, see [47] and subsection 4.1 below. The geometric definition of asymptotically de Sitter spacetimes is based on the idea ofconformal compactifiability of the physical spacetime by attaching a boundary hyper-surface to it [15]. Explicitly, it is assumed that there is a manifold ˜ M with non-emptyboundary ∂M , a Lorentzian metric ˜ g ab on ˜ M and a smooth function Ω : ˜ M → [0 , ∞ ) such that (1) ˜ M − ∂M is diffeomorphic to (and hence identified with) M ; (2) ˜ g ab = Ω g ab on M ; (3) the boundary is just ∂M = { Ω = 0 } and ˜ ∇ a Ω is nowhere vanishing on ∂M ;(4) g ab solves Einstein’s equations, R ab − Rg ab = − κT ab − Λ g ab , on M , where Λ > ; (5) ˜ T ab := Ω − T ab can be extended to ˜ M as a smooth field.A systematic investigation of the asymptotic structure of spacetimes satisfying theseconditions is given in [32, 21]. In particular, the asymptotic form of spacetime metric, theNewman–Penrose spin coefficients, the various pieces of the curvature and the energy-momentum tensor, the geometry of asymptotically hyperboloidal hypersurfaces, etc; andalso the asymptotic field equations for them are determined. The conformal boundary ∂M is necessarily spacelike , so it consists of the future and past boundary, I + and I − ,respectively. (In what follows, we concentrate only on I + . In a cosmological setting I − will usually not be present, being instead replaced by an initial singularity.) Thetrace-free part of its extrinsic curvature is vanishing and, in an appropriate conformalgauge, its trace can be made zero; but its intrinsic conformal geometry is still completelyunrestricted . Therefore, such a general asymptotically de Sitter spacetime cannot admitany of the usual ten de Sitter Killing fields even asymptotically, otherwise they would ex-tend to conformal Killing fields on I + . Thus, as the term is usually used, asymptoticallyde Sitter spacetimes are not the spacetimes that are asymptotic to de Sitter spacetime .The latter form a very special subset of the former.In fact, I + admits ten linearly independent conformal Killing vectors precisely whenits intrinsic geometry is conformally flat ; which is equivalent to the vanishing of the mag-netic part of the rescaled Weyl tensor on I + . This condition appears to be unreasonablystrong: investigations of linear gravitational perturbations of the de Sitter backgroundreveal that this condition would remove half the gravitational degrees of freedom [21, 50](see subsection 2.2.1 above) and in the full (nonlinear) theory this is confirmed. Theexistence of a large number of spacetimes with positive Λ admitting a general future con-formal boundary has been demonstrated by Friedrich [54]. He shows that the free datafor the Einstein equations consist of a pair of symmetric tensors ( h ab , E ab ) with h ab theRiemannian metric of I + and E ab the electric part of the rescaled Weyl tensor at I + ,subject to h ab E ab = 0 , D a E ab = 0 and with the freedom ( h ab , E ab ) (Θ h ab , Θ − E ab ) for positive functions Θ on I + . (Here D a is the covariant derivative on I + defined by h ab .) These results, together with those13n [39, 40], show the essential difference between the role of the boundary conditions inthe Λ < and Λ > cases: while in the Λ < case some boundary condition must bespecified on I to have a well posed initial-boundary value problem for the field equations,in the Λ > case similar boundary conditions on I + restrict the freely specifiable partof the Cauchy data.A detailed discussion of the various global and asymptotic properties of some spe-cial asymptotically de Sitter solutions (viz. the Schwarzschild–de Sitter, Kerr–de Sitter,Vaidya–de Sitter and Friedman–Robertson–Walker) is also given in [21]. In particular,the existence and properties of the horizons, the global topology of the conformal bound-ary, the group of the globally defined isometries and so on are clarified. Motivated bythese special cases, one may wish to impose further conditions, for example the geodesiccompleteness of the conformal boundary. The definition of the ADM type energy-momenta are based on spacelike hypersurfacesthat extend to ‘spatial infinity’; i.e. which are analogous to the foliation of the steadystate part of the de Sitter spacetime by intrinsically flat hypersurfaces, given by (2.1).Thus, strictly speaking, such an ADM type energy-momentum is associated with a point p of the conformal boundary I + (still conveniently thought of as ‘spatial infinity’), andthe construction itself is based on a spacelike hypersurface Σ that is a Cauchy surfacefor the past domain of dependence of I + − { p } in the spacetime. These hypersurfacesare tangent to I + at p , and hence Σ is asymptotically intrinsically flat and extrinsicallyasymptotically hyperboloidal. If the intrinsic conformal geometry of I + is homogeneous,e.g. when it is conformally flat, then the energy-momentum is independent of p , but ingeneral it may depend on it. It is obvious how to generalize these ideas to allow finitelymany ‘spatial infinities’. The Abbott–Deser energy-momentum [35] is based on the decomposition g ab = ¯ g ab + γ ab of the spacetime metric into the sum of the de Sitter metric ¯ g ab and some tensor field γ ab . Then Einstein’s equations can be written in the form L G ab = Λ γ ab − κt ab , where L G ab denotes the linearized Einstein tensor on the de Sitter background, built from γ ab ,and t ab is the sum of the matter energy-momentum tensor and the correction to thelinearized Einstein tensor, being quadratic and higher order in γ ab . By the contractedBianchi identity ¯ ∇ a t ab = 0 holds, where ¯ ∇ a is the covariant derivative in the de Sittergeometry and index raising and lowering are defined by the background metric. Thus, t ab plays the role of an effective energy-momentum tensor . Therefore, for any Killing vector ¯ K a of the de Sitter background the contraction ¯ K a t ab is ¯ ∇ b -divergence-free, and hencethe integral Q [ ¯ K ] := Z Σ ¯ K a t ab ¯ t b d ¯Σ (3.1)is independent of the hypersurface provided it is finite and free of the ambiguity in thedecomposition g ab = ¯ g ab + γ ab , which seems unlikely in general. Here ¯ t a is the ¯ g ab -unitnormal to, and d ¯Σ is the induced volume element on Σ . In addition to the implicitassumption that Q [ ¯ K ] is well defined, it is also assumed that the conformal boundary14 + is intrinsically conformally flat; otherwise the Killing fields ¯ K a could not extendto conformal Killing vectors of I + and they, as solutions of the asymptotic Killingequations in a neighbourhood of I + , would be ambiguously defined. By the conservation ¯ ∇ b ( ¯ K a t ab ) = 0 the integral Q [ ¯ K ] can be rewritten as a 2-surface integral of an appropriatesuperpotential on the ‘spatial infinity’ of Σ (see also [55, 56, 57]).According to the traditional prescription [35], the so-called Abbott–Deser (AD) en-ergy is Q [ ¯ K ] in which ¯ K a is the ‘time translational Killing field’, given explicitly e.g. by(2.2). However, this Killing field is not timelike on any neighbourhood of some asymp-totic end of Σ . (In fact, as we saw in subsection 2.1.2, this holds for any Killing field in aneighbourhood of I + .) Nevertheless, this can still be called ‘energy’, but its interpreta-tion is not so obvious. For alternative forms of this energy expression see [21] and (underadditional conditions) [57, 48].Although in certain special cases the positivity of the AD energy can be proven [55,57, 48], in general it can have any sign [56, 58]. Indeed, in the light of the results of [50]on the energy flux through I + of the linear gravitational perturbations preserving theintrinsic conformal flatness of I + (see subsection 2.2.1), this notion of energy does notseem to have the rigidity property. To resolve these and above conceptual difficulties, ithas already been suggested to use the conformal Killing vector K a , given in (2.3), as thegenerator of the total de Sitter energy [57]. This conformal Killing field is used in thenext subsection, too. Using the original Dirac spinor form of the Nester–Witten form [26, 25], the Witten-typeenergy positivity proof of [30], originally given for Λ < , has been successfully adaptedto the Λ > case and yielded a positivity argument by Kastor and Traschen in [49](see also [57]). Here, the energy-momentum tensor of the matter fields is assumed tosatisfy the dominant energy condition, and, as the boundary condition for the (modified)Witten equation, the Dirac spinor constituents Ψ of the conformal Killing vectors of thebackground de Sitter spacetime are used.A generalization C a of the conserved current ¯ K b t ba of subsection 3.1, built from t ab ,the ‘perturbation’ γ ab and an arbitrary vector field ξ a , is also given in [49]: this C a is ¯ ∇ a -divergence free, and if ξ a is chosen to be a Killing vector of the de Sitter background,then it reduces to ¯ K a t ab . Then, it is shown that the flux integral of C b on Σ with the conformal Killing vector ξ a determined by the Dirac spinor Ψ coincides with the spinorialconstruction above. Therefore, the present spinorial expression of these authors is a non-trivial generalization of the original construction in [35]. Chruściel with a range of collaborators has been investigating definitions of mass in avariety of space-times with Λ positive, negative or zero for many years. The investiga-tions are typically motivated by adherence to a Hamiltonian approach to space-times,following the monograph [59], analysing Cauchy data to arrive at definitions of mass andmomentum. In this subsection we review their work on asymptotically-Schwarzschild–deSitter data and asymptotically-Kerr–de Sitter data. These are discussed in [60] and [61]15espectively and these in turn rely on families of initial data for the Einstein equationsshown to exist in [62] and [63] respectively.The Schwarzschild–de Sitter metric, sometimes called the Kottler solution, can bedefined in space-time dimension n + 1 as ds = − V dt + dr V + r ˜ h, (3.2)where ˜ h is the metric of the standard (round) unit ( n − -sphere and V = 1 − mr n − − r ℓ , (3.3)with m, ℓ real positive constants. It is straightforward to check that the metric (3.2)satisfies the Einstein equations with cosmological constant Λ = n ( n − / (2 ℓ ) .Typically one wants ∂/∂t to be a time-like Killing vector at least somewhere, so that V must be positive for some r , and it will be, with two positive simple zeroes, provided m Λ n − < c n := 1 n (cid:18) ( n − n − (cid:19) n − . (3.4)On a surface of constant t the metric (3.2) defines the spatial metric g = dr V + r ˜ h, (3.5)which is conformally-flat (by spherical symmetry) with constant scalar curvature R = n ( n − /ℓ and necessarily satisfies the Einstein constraint equations with vanishingsecond fundamental form (so this is time-symmetric data). Given (3.4), the function V has two real distinct positive zeroes in r as noted, but the metric can be extended throughboth of these (in the style of the Kruskal extension) to obtain a complete, spherically-symmetric, conformally-flat metric with constant scalar curvature. Such metrics arecalled Delaunay metrics in [62] by analogy with Delaunay surfaces which are complete,rotationally-symmetric, constant mean curvature surfaces in R . The Delaunay metricsalso arise from singular solutions of the Yamabe problem on S n (see the discussion in[62]).A similar but more general class of solutions of the Einstein equations than those in(3.2) was published by Birmingham [64]. The metric looks the same but ˜ h is taken to bea Riemannian Einstein metric on a compact ( n − -manifold with scalar curvature ˜ R ,and V is changed to V = ˜ R ( n − n − − mr − r ℓ . With positive m , we need positive ˜ R to have V anywhere positive and then constantrescaling and redefinition of r can be used to set ˜ R = ( n − n − to recover theprevious expression for V . Now one can introduce a notion of generalised Delaunaymetrics with this ˜ h in place of the previous ˜ h .In [62] the authors use gluing techniques to produce new time-symmetric data forthe Einstein equations which have many exactly Delaunay or generalised Delaunay ends,which is to say that the ends are exactly Delaunay or generalised Delaunay for infinite Here the Einstein equations are taken to be G ab + Λ g ab = 0 m in the appropriate metric (3.5) (and a fortiori it ispositive).In [63], Cortier generalised the work in [62] to produce initial data which are deforma-tions of that for Kerr–de Sitter (but only in space-time dimension 4, so the data surfacehas dimension 3). Again one has an infinite periodic metric but now with a particularnonzero periodic second fundamental form as well. Then in [61], the authors use theHamiltonian method to assign both mass and angular momentum to these data.This work, in associating a mass with each Delaunay or generalised Delaunay end,is reminiscent of the calculation of Penrose’s quasi-local mass for various 2-spheres inconformally-flat, time-symmetric initial data for the vacuum Einstein equations (so Λ =0 ) in [65]. In particular, the mass defined can be thought of as a ‘mass at spatial infinity’,given a Delaunay or generalised Delaunay end that extends to spatial infinity, but it isnot a mass at null infinity.
In contrast to the ADM type energy-momenta, the TBS type expressions are associatedwith closed orientable 2-surfaces S , that is ‘cuts’, rather than points of I + . Then,to be potentially able to prove a formula analogous to Bondi’s mass-loss for the TBStype expressions, we should have, in fact, a foliation of I + by such cuts. Thus, thespacelike hypersurfaces Σ that intersect I + in the given cuts should be analogous to thehypersurfaces Σ ˆ τ discussed at the end of subsection 2.1.1. In [47] some general twistor theory of de Sitter spacetime is recorded, sufficient to describePenrose’s quasi-local mass construction at any space-like 2-sphere for linear gravitationalperturbations of de Sitter. This includes 2-spheres at I + , and it is pointed out that theabsence of timelike Killing fields near I + entails that this quasi-local mass does not havea positivity property, in contrast to the Λ = 0 and Λ < cases. It can be the basis of adefinition in asymptotically-de Sitter space-times but it won’t have positivity or rigidityproperties. In [46] Penrose considered the problem of defining a cosmological total mass in the pres-ence of a positive cosmological constant. He was motivated, at least in part, by his Con-formal Cyclic Cosmology (or CCC) rather than by a consideration of isolated systems,which was the motivation of Ashtekar and colleagues considered above. Thus Penrosesought a definition at I + of one aeon which was suitable for carrying through to the nextaeon. Interestingly, Penrose considered but rejected the idea of working in a Poincarépatch and using the Killing vector of the steady-state universe to define energy (section 3of [46]) because galaxies beyond the cosmological event horizon, where this Killing vectoris space-like, would be represented as having super-luminal velocities. This could be a17roblem in a cosmological setting but one that Ashtekar et. al. avoid by considering onlyisolated bodies.Penrose makes three concrete suggestions:(1) To seek an expression motivated by the conserved currents that one constructsfrom a trace-free energy-momentum tensor and a conformal Killing vector of de Sitterspace. In CCC one expects matter to become massless near I + , so that T ab will becometrace-free, and in de Sitter space there do exist time-like conformal Killing vectors. Thissuggestion is not worked out in detail but the insight also underlay the earlier work of[49], [55] and [57].(2) To use his quasi-local energy-momentum construction, as discussed above (seesubsection 4.1), at I + . This construction is defined for a space-like 2-surface S , usuallya topological sphere. One first solves an elliptic system for the 2-surface twistors on S ,which are particular 2-component spinor fields, then forms an integrand linear in thespacetime curvature but with the cosmological constant term removed and integratesover S . This is a well-defined procedure but the result does not have any positivity ortherefore rigidity property. In particular it can be zero even in the presence of non-trivialcurvature [32]: applied to 2-spheres on the I + of the Schwarzschild–de Sitter space-timeit gives zero if the 2-sphere S is homologous to a point on I + and a non-zero constantbut of either sign if S surrounds the source.(3) To use the original TBS energy expression, given in the Newman–Penrose form as E = 2 κ I S (cid:0) σσ ′ − ψ (cid:1) d S , (4.1)where ψ is the o A o B ι C ι D component of the Weyl curvature spinor and σ and σ ′ arethe asymptotic shear of the two null geodesic congruences hitting the cut S of I + or-thogonally. This is the basis of the suggestion taken up by Saw [68, 69, 73], followingFrauendiener [66, 67]. The key property of the TBS mass in the asymptotically flat context is mass-loss: themass, as a function of the retarded time coordinate, is a monotonically decreasing func-tion. Thus, accepting (4.1) as the definition of the TBS energy in the Λ > case, thenatural question is whether or not the analogue of the mass-loss formula can be derived.Such a formula could be based directly on the analysis of the Bianchi identities, or onthe general integral formula of Frauendiener [66, 67]. However, to derive this, one shouldsolve the Newman–Penrose spin coefficient equations to an appropriately high order. Thissolution, both for the vacuum and electro-vacuum in the physical spacetime, is given bySaw in [68] and [69], respectively (see also [70]). The analogue of the mass-loss formulais derived from the Bianchi identity in [68, 69], using an integral identity in [73]. In [73],with E as in (4.1), he finds ˙ E = − κ I S (cid:16) | ˙ σ | + Λ2 | ð ′ σ | − Λ6 | ð σ | + 2Λ9 | σ | + Λ ℜ (¯ σψ ) (cid:17) d S , (4.2)where the dot denotes derivative with respect to the parameter u which labels the actualcut S of the foliation of I + by 2-surfaces S u , ℜ denotes ‘real part’, ð and ð ′ are thestandard GHP edth and edth-prime operators, respectively, and ψ is the o A o B o C o D component of the Weyl curvature spinor. Saw shows that, in the absence of incoming18adiation, so that ψ = 0 , and for purely quadrupole gravitational radiation (which hereis taken to mean that ð σ = 0 , which would imply, for a sphere with constant Gausscurvature, that σ is proportional to a combination of the Y m spin weighted sphericalharmonics) ˙ E is nonpositive and vanishes only for zero σ . He also notes, in [68], that thevanishing of σ on I + is equivalent to the intrinsic conformal flatness of I + , in agreementwith [21].Saw also makes use of another expression for ˙ E , equivalent to (4.2) by an identity forfunctions on a sphere, namely ˙ E = − κ I S K | σ | d S − κ I S (cid:16) | ˙ σ | + Λ3 | ð ′ σ | + 2Λ9 | σ | + Λ ℜ (¯ σψ ) (cid:17) d S , (4.3)where K is the Gauss curvature of S (which need not have fixed sign). This is the basisof a suggestion he makes for a different analogue of the TBS mass in the Λ > case,namely E Λ ( u ) := E ( u ) + 2Λ3 κ Z uu (cid:16)I S ¯ u K | σ | d S ¯ u (cid:17) d¯ u with E ( u ) as in (4.1), and for some u . Now the u -derivative of E Λ consists of just thesecond integral in (4.3) and in the absence of incoming radiation, but this time withoutthe restriction to quadrupole radiation, it is therefore negative (strictly speaking it isnonnegative and vanishes only for zero σ ). In this mass-loss formula there is a termproportional to | ˙ σ | that is familiar from the case of Λ = 0 , but there are also termsinvolving the shear and its angular derivatives but undotted. However, this proposednew expression depends not only on the instantaneous state of the physical system at u , but on the whole history of the system until u . A summary of the results of theinvestigations of Saw prior to [73] is given in [71, 72]. As noted above, a key property of the TBS energy in the asymptotically flat case is itspositivity and rigidity. Since in the
Λ = 0 and Λ < cases the total energy-momentacould be recovered from a unified form based on the integral of the Nester–Witten form(1.2), and moreover the simplest energy positivity proof is arguably the one based on theuse of 2-component spinors and Witten-type arguments [28, 30], it seems natural to try toformulate the TBS energy-momentum in the presence of a positive Λ in this framework,too. This was done in [32]. However, since it is not a priori clear what the ‘asymptotictime translations’ near I + should be, in the investigations of [32] the spinor fields at thecut are initially left unspecified. Surprisingly enough, the requirement of the finitenessof the resulting integral together with the desire to have a Witten-type energy positivityproof determine the spinor fields in (1.2) at the cut: They should solve Penrose’s 2-surfacetwistor equations.
Thus, the boundary conditions for the Witten spinors, i.e. the spinorconstituents of what one could considered to be the ‘asymptotic translations’, come outof the formalism .The integral of the Nester–Witten form with these boundary conditions defines a × complex Hermitian matrix with the structure H = (cid:18) P Q − ¯ Q ¯ P (cid:19) , P is a × Hermitian, and Q is a complex anti-symmetric matrix. If the matterfields satisfy the dominant energy condition, then it has been shown that H is positivedefinite (‘positivity’), and it is vanishing precisely when the past domain of dependence ofthe (asymptotically hyperboloidal) spacelike hypersurface Σ for which Σ ∩ I + is just thegiven cut S is locally isometric to the de Sitter spacetime (‘rigidity’). The symmetry groupof the 2-surface twistor space T on S is (0 , ∞ ) times the spin group of SO (1 , . If thereexists a volume 4-form on T , then the TBS-type mass can be defined as the determinantof H . For example, if there is a scalar product on T (which with (+ , + , − , − ) signatureis guaranteed e.g. when I + is intrinsically conformally flat), then such a volume formexists. In these special cases the symmetry group reduces to the spin group of SO (1 , and to the spin group of the de Sitter group SO (1 , , respectively.The same general analysis can be repeated in the Λ < and Λ = 0 cases, too [74]. If Λ < , then the result is similar but the symmetry group of T is (0 , ∞ ) times the spingroup of SO (3 , , which, in the presence of the reflective boundary condition, reduces tothe spin group of the anti-de Sitter group SO (2 , . If Λ = 0 , then T splits to the directsum T ⊕ T , where T is the space of the spinor constituents of the BMS translations, Q in H is zero, P reduces to the TBS 4-momentum, and the symmetry group of T is SL (2 , C ) × SL (2 , C ) . Therefore, the above construction is a natural generalization of theTBS 4-momentum of asymptotically flat spacetimes to the Λ > case. It has positivityand rigidity, at the cost of being a matrix with six independent real components ratherthan a Lorentz scalar or vector. In [75] Chruściel and Ifsits define a mass from data for the Einstein equations given on anoutgoing null hypersurface N , which could be the null cone of a point, in a conformally-compactifiable ( n +1) -dimensional space-time. The main interest in [75] is with Λ > butall values of Λ are allowed, and the method is to construct a Bondi coordinate systemat N and solve for the space-time metric and connection at N in terms of free data,making assumptions about asymptotic decay rates, for example of any matter content,as required. The paper proposes a definition of mass from a consideration of the TBSmass defined for Λ = 0 . The generalised definition is checked against the
Λ = 0 case,for which a corresponding calculation was presented in [76], and, by consideration of avariety of examples, with the Λ < case, which is fairly well understood.The calculation leading to the definition of mass naturally leads to the definition ofa renormalised volume for N . This is defined from the integral V ( r ) of the area A ( r ) ofsections of N of constant r where r is an affine parameter along the generators of N .Given an origin for r , which would be the vertex of N if N were a light-cone but otherchoices are possible, the authors obtain an expansion: V ( r ) = Z r A ( r ′ ) dr ′ = a r + a r + a r + a ℓ log r + a + a − r − + o ( r − ) , for coefficients a k given in terms of the data and quantities obtained from the data. Thenthe coefficient a = V ren is the renormalised volume.In discussion of the result in the last section of the paper it is observed that themass defined is geometric and gauge-invariant and coincides in cases of Λ ≤ with other In space-time dimension n + 1 , A is the ( n − -dimensional volume of these sections. rigid in thesense that vanishing mass implies that the space-time is exactly de Sitter inside the cone N . The construction of the total mass for closed universes, introduced first for
Λ = 0 (andmentioned at the end of subsection 1.1.3), can be generalized in a straightforward wayto closed universes with positive Λ [77]. The basis of this construction is the observationthat, in the Witten type gauge, the λ A ¯ λ A ′ -component of the hypersurface integral formof the spinorial expression of all the energy-momentum expressions above (independentlyof the sign of Λ ) takes the manifestly positive definite expression P ( λ ) := √ κ kD ( AB λ C ) k L + Z Σ t a T ab λ B ¯ λ B ′ dΣ . Here Σ is the asymptotically flat/hyperboloidal hypersurface whose boundary ‘at infinity’is just the 2-surface S , the L -norm is defined on this Σ with √ t AA ′ as the pointwisepositive definite Hermitian scalar product on the spinor spaces, and D AB is the unitaryspinor form of the Sen derivative operator on Σ (see [38]).However, if Σ is a compact Cauchy surface in a closed universe, e.g. when Σ isanalogous to the leaves Σ t of the global foliation of the de Sitter spacetime given insubsection 2.1.1, then the above quantity can be formed even when T ab is replaced by T ab + g ab Λ /κ with positive Λ and even for any spinor field λ A with the normalization k λ k L = √ . The total mass M (in fact, mass density) associated with Σ is defined to bejust the infimum of P ( λ ) on the set of spinor fields satisfying the normalization above.Then, the spinor fields λ A for which P ( λ ) = M holds are precisely the eigenspinors in theeigenvalue equation D AA ′ D A ′ B λ B = (3 κ/ M λ A . Thus, the mass M could have been definedas the first eigenvalue in this eigenvalue problem. Clearly, κ M ≥ Λ by construction, but ithas the non-trivial rigidity property : κ M = Λ if and only if the whole spacetime is locallyisometric to the de Sitter spacetime and the Cauchy surface Σ is homeomorphic to S /G ,where G is a discrete subgroup of SU (2) ≈ S .The authors are grateful to György Wolf for drawing the figures. References [1] A. Trautman, Conservation laws in general relativity, in
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