4.2 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.

4.2.1 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,๐œƒ,ฯ•) the spacetime metric takes the form 2 2 2 gab = diag(exp (2γ),− exp(2α ),− r ,− r sin ๐œƒ), 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 Rabcd๐œ€ab๐œ€cd = 42(1 − exp(− 2α)) r. Based on the investigations of Misner, Sharp, and Hernandez [365Jump To The Next Citation Point, 267Jump To The Next Citation Point], Cahill and McVitte [122] found

E(t,r) := -1-r3Rabcd๐œ€ab๐œ€cd = -r- (1 − e−2α) (4.7 ) 8G 2G
to be an appropriate (and hence, suggested to be the general) notion of energy, the Misner–Sharp energy, contained in the two-sphere : ๐’ฎ = {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) = m − e2โˆ•2r, while for the isentropic fluid solutions ∫ E(t,r) = 4π r0 r′2μ(t,r′)dr′, where m and e are the usual parameters of the Reissner–Nordström solutions and μ is the energy density of the fluid [365, 267] (for the static solution, see, e.g., Appendix B of [240Jump To The Next Citation Point]). Using Einstein’s equations, simple equations can be derived for the derivatives ∂tE(t,r) and ∂rE(t,r), and if the energy-momentum tensor satisfies the dominant energy condition, then ∂ E (t,r) > 0 r. 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. (4.3View Equation), (4.4View Equation) the energy function E(t,r) can also be written as
( ) โˆ˜ ---------( โˆฎ ) E (๐’ฎ ) = 1-r3 1๐’ฎR + ρρ′ = 1-r3(− ψ2 + ฯ•11 + Λ) = Area(๐’ฎ-) 1 + -1- ρρ′d๐’ฎ . (4.8 ) G 4 G 16 πG2 2π ๐’ฎ
Any of these expressions is considered to be the ‘standard’ definition of the energy for round spheres.4 The last of these expressions does not depend on whether r is an area-coordinate or not.

E(๐’ฎ ) 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 ∫ 4π rr′2 exp(α)μdr ′ 0. This deviation can be interpreted as the contribution of the gravitational potential energy to the total energy. Consequently, E (๐’ฎ ) 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 ∈ [0,π]) E (๐’ฎ) is increasing for r ∈ [0,πโˆ•2 ), taking its maximal value at r = πโˆ•2, and decreasing for r ∈ (πโˆ•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 dl2 = Ω2FRWdl20, where dl20 is the line element for the flat three-space and Ω2 := B(1 + -r22)− 2 FRW 4T 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 1 √ -- M := GE (2T ) = 2T B. The scalar curvature is R = 6โˆ•BT 2, and hence, by the constraint parts of the Einstein equations and by the vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the metric of a piece of a t = const. hypersurface in the Schwarzschild solution with mass parameter m (see [213Jump To The Next Citation Point]): ¯2 2 ¯2 dl = ΩSd l0, where 2 m-4 Ω S := (1 + 2¯r) and the Euclidean radial coordinate ¯r runs from ¯r0 to ∞, where ¯r0 ∈ (0,m โˆ•2). In this geometry there is a minimal surface at ¯r = m โˆ•2, the scalar curvature is zero, and the round-sphere energy is E (r¯) = m โˆ•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 โˆ•G and arbitrarily small ADM mass m โˆ•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 [321Jump To The Next Citation Point]. 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 ab ∇a (G Kb ) = 0, the vector field a ab S := G Kb has a conserved flux on a spacelike hypersurface Σ. In particular, in the coordinate system (t,r,๐œƒ,ฯ•) and in the line element given in the first paragraph above Ka = exp[− (α + γ)](∂โˆ•∂t)a. If Σ is a solid ball of radius r, then the flux of Sa is precisely the standard round-sphere expression (4.7View Equation) 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 (4.7View Equation) (or by (4.8View Equation)) (see, e.g., [578Jump To The Next Citation Point, 352Jump To The Next Citation Point, 250Jump To The Next Citation Point]). 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].

4.2.2 Small surfaces

In the literature there are two kinds of small surfaces. The first is that of the small spheres (both in the light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [275Jump To The Next Citation Point], and the other is the concept of small ellipsoids in a spacelike hypersurface, considered first by Woodhouse in [313Jump To The Next Citation Point]. 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 p ∈ M be a point, and ta a future-directed unit timelike vector at p. Let + ๐’ฉp := ∂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 ๐’ฉp, such that, at the vertex p, lat = 1 a. 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, ๐’ฉp − {p} is a smooth null hypersurface, and hence, for sufficiently small r, the set ๐’ฎr := {q ∈ M | r (q) = r} is a smooth spacelike two-surface and is homeomorphic to S2. ๐’ฎr is called a small sphere of radius r with vertex p. Note that the condition lata = 1 fixes the boost gauge, too.

Completing a l to get a Newman–Penrose complex null tetrad a a a a {l,n ,m , ¯m } such that the complex null vectors a m and a m¯ are tangent to the two-surfaces ๐’ฎr, 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 = oA ¯oA′ 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 ๐’ฎr, the GHP spin coefficients ρ, σ, τ, ρ ′, 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 ๐’ฎr for the small sphere ๐’ฎ r can be expressed as a series of r,

โˆฎ ( (0) (1) 1 2 (2) ) Q ๐’ฎr = Q + rQ + 2r Q + ⋅⋅⋅ d ๐’ฎ, ๐’ฎ

where the expansion coefficients (k) Q are still functions of the coordinates, ¯ (ζ,ζ) or (๐œƒ,ฯ•), on the unit sphere ๐’ฎ. 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 ๐’ฎr and on whether the spacetime is Ricci-flat in the neighborhood of p or not.5 These solutions of the GHP equations, with increasing accuracy, are given in [275Jump To The Next Citation Point, 313Jump To The Next Citation Point, 118Jump To The Next Citation Point, 494Jump To The Next Citation Point].

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 [494Jump To The Next Citation Point], 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 Q๐’ฎ [K ], are, respectively,

′ 4π ′ ( ) P A๐’ฎB- = ---r3T AA′BB′tAA′โ„ฐABโ„ฐ¯BB′ + ๐’ช r4 , (4.9 ) r 3 ( ) J AB-= 4π-r3T ′ ′tAA ′ rtB′E๐œ€BF โ„ฐA-โ„ฐB- + ๐’ช (r5), (4.10 ) ๐’ฎr 3 AA BB (E F)
where {โ„ฐA-} A, 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 AB′ A- B′ Ka---= โ„ฐA ¯โ„ฐA′ can be interpreted as the translation one-forms, while KABa--= rtA′E โ„ฐA(Eโ„ฐBA) is an average on the unit sphere of the boost-rotation Killing one-forms that vanish at the vertex p. Thus, AB′ P๐’ฎr and AB-- J๐’ฎr 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 ๐’ฎ is any coordinate-independent quasi-local quantity built from the first derivatives ∂ μgαβ 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

[ ] [ ( ) ] Q ๐’ฎr = Q2 [∂g ]r2 + Q3 ∂2g,(∂g)2 r3 + Q4 ∂3g, ∂2g (∂g ),(∂g)3 r4 + [ ( ) ( )2 ( ) 2 4] +Q5 ∂4g, ∂3g (∂g), ∂2g , ∂2g (∂g) ,(∂g ) r5 + ...,
where Qi[A,B, ...], i = 2,3,..., are scalars. They are polynomial expressions of ta, gab and ๐œ€abcd at the vertex p, and they depend linearly on the tensors that are constructed at p from gαβ, gαβ 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 3 r. Its coefficient 2 2 Q3 [∂ g,(∂g) ] must be a linear expression of Rab and Cabcd, and polynomial in ta, gab and ๐œ€abcd. In particular, if Q ๐’ฎ is to represent energy-momentum with generator Kc at p, then the leading term must be
Q [K ] = r3[a (G tatb)t + bRt + c(G taP b)]Kc + ๐’ช (r4) (4.11 ) ๐’ฎr ab c c ab c
for some unspecified constants a, b, and c, where Pba:= δab − tatb, the projection to the subspace orthogonal to ta. If, in addition to the coordinate-independence of Q๐’ฎ, it is Lorentz-covariant, i.e., it does not, for example, depend on the choice for a normal to ๐’ฎ (e.g., in the small-sphere approximation on a t) 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 r3ataGabKb. Then, comparing Eq. (4.11View Equation) with Eq. (4.9View Equation), we see that a = − 1โˆ• (6G ), and hence the term 3 a r bRtaK 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 ๐’ฎr given by Eq. (4.11View Equation) can be negative, even for c = a, unless b = 0. Thus, in the leading 3 r 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 3 r-order term is vanishing, and the fourth-order term must be built from ∇eCabcd. However, the only scalar polynomial expression of ta, gab, ๐œ€abcd, ∇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 [K ] = r5[(aE Eab + bH Hab + cE Hab )t + dE He ๐œ€ab ]Kc + ๐’ช (r6) (4.12 ) ๐’ฎr ab ab ab c ae b c
for constants a, b, c, and d, where Eab := Caebftetf is the electric part and Hab := ∗Caebftetf := 1๐œ€aecdCcdbftetf 2 is the magnetic part of the Weyl curvature, and ๐œ€abc := ๐œ€abcdtd is the induced volume 3-form. However, using the identities C Cabcd = 8(E Eab − H Hab ) abcd ab ab, abcd ab Cabcd ∗ C = 16EabH, a b cd ab ab 4Tabcdtt t t = EabE + HabH and ab c d a bc 2Tabcdt tt Pe = EabH c๐œ€ e, we can rewrite the above formula to be
[( Q ๐’ฎr [K ] = r5 2(a + b)Tabcdtatbtctd + 116(a − b)CabcdCabcd+ 1- abcd) a bc d] e ( 6) + 16 cCabcd ∗ C te + 2dTabcdt tt Pe K + ๐’ช r . (4.13 )
Again, if Q ๐’ฎ 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 2r5dT tatbtcKd abcd. In a general expression the curvature invariants abcd CabcdC and abcd Cabcd ∗ C 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 a โ„= b or c โ„= 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’ a b c Tabcdtt t.

Obviously, the same analysis can be repeated for any other quasi-local quantity. For the energy-momentum, Q ๐’ฎ has the structure โˆฎ ๐’ฎ ๐’ฌ (∂μgαβ)d๐’ฎ, for angular momentum it is โˆฎ ๐’ฎ ๐’ฌ (∂ μgαβ)rd๐’ฎ, while the area of ๐’ฎ is โˆฎ d๐’ฎ ๐’ฎ. 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 r4 and r6 in nonvacuum and vacuum, respectively.

On the small geodesic sphere ๐’ฎr of radius r in the given spacelike hypersurface Σ one can introduce the complex null tangents ma and m¯a above, and if ta is the future-pointing unit normal of Σ and va the outward directed unit normal of ๐’ฎr in Σ, then we can define la := ta + va and 2na := ta − va. Then {la,na, ma,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 [313Jump To The Next Citation Point]. 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 ๐’ฎ := {q ∈ Σ | 2f(q) = r2} r are smooth compact two-surfaces homeomorphic to 2 S. Then, in the r → 0 limit, the surfaces ๐’ฎr 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 [327Jump To The Next Citation Point, 170Jump To The Next Citation Point, 235Jump To The Next Citation Point]. 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 p ∈ M or a timelike curve γ are used, in which the metric, as well as the Christoffel symbols on U, are expressed by the coordinates on U and the components of the Riemann tensor at p or on γ. In these coordinates and the corresponding frames, the various pseudotensorial and tetrad expressions for the 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.

4.2.3 Large spheres near spatial infinity

Near spatial infinity we have the a priori 1โˆ•r and 1โˆ•r2 falloff for the three-metric hab and extrinsic curvature χab, respectively, and both the evolution equations of general relativity and the conservation equation Tab;b = 0 for the matter fields preserve these conditions. The spheres ๐’ฎ r of coordinate radius r in Σ are called large spheres if the values of r are large enough, such that the asymptotic expansions of the metric and extrinsic curvature are legitimate.6 Introducing some coordinate system, e.g., the complex stereographic coordinates, on one sphere and then extending that to the whole Σ along the normals va of the spheres, we obtain a coordinate system (r,ζ, ¯ζ) on Σ. Let ๐œ€AA = {oA, ιA }, A = 0,1, be a GHP spinor dyad on Σ adapted to the large spheres in such a way that ma := oA¯ιA′ and m¯a = ιA¯oA ′ are tangent to the spheres and a 1 A A′ A A′ t = 2o ¯o + ι ¯ι, the future directed unit normal of Σ. These conditions fix the spinor dyad completely, and, in particular, ′ ′ va = 12oA¯oA − ιA¯ιA, 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 1โˆ•r2 and the curvature components to zero like 3 1โˆ•r. Expanding the spin coefficients and curvature components as a power series of 1โˆ•r, one can solve the field equations asymptotically (see [65Jump To The Next Citation Point, 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 De0λA = ๐’ช (r−2). Thus, we can define the asymptotically constant spinor fields to be those λ A that satisfy − 2 De λA = ๐’ช(r ), where De is the intrinsic Levi-Civita derivative operator on Σ. Note that this implies that, with the notation of Eq. (4.6View Equation), all the chiral irreducible parts, + Δ λ, Δ − λ, ๐’ฏ + λ, and ๐’ฏ − λ of the derivative of the asymptotically constant spinor field λA are ๐’ช (r−2).

4.2.4 Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [413, 414, 415, 426Jump To The Next Citation Point] (see also [208]), i.e., the physical spacetime can be conformally compactified by an appropriate boundary โ„ +. Then future null infinity โ„ + will be a null hypersurface in the conformally rescaled spacetime. Topologically it is โ„ × S2, and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of + โ„ is the metric of the unit sphere. Fixing such a slice ๐’ฎ0 (called ‘the origin cut of + โ„’) the points of + โ„ can be labeled by a null coordinate, namely the affine parameter u ∈ โ„ along the null geodesic generators of โ„ + measured from ๐’ฎ0 and, for example, the familiar complex stereographic coordinates (ζ, ¯ζ) ∈ S2, defined first on the origin cut ๐’ฎ0 and then extended in a natural way along the null generators to the whole + โ„. Then any other cut ๐’ฎ of + โ„ can be specified by a function u = f(ζ, ¯ζ). In particular, the cuts ๐’ฎu := {u = const.} are obtained from ๐’ฎ0 by a pure time translation.

The coordinates ¯ (u,ζ,ζ) can be extended to an open neighborhood of + โ„ in the spacetime in the following way. Let ๐’ฉu be the family of smooth outgoing null hypersurfaces in a neighborhood of โ„ +, such that they intersect the null infinity just in the cuts ๐’ฎu, i.e., ๐’ฉu ∩ โ„ + = ๐’ฎu. Then let r be the affine parameter in the physical metric along the null geodesic generators of ๐’ฉ u. Then (u,r,ζ,ζ¯) forms a coordinate system. The u = const., r = const. two-surfaces ๐’ฎu,r (or simply ๐’ฎr 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: ¯r := r + g(u,ζ, ¯ζ) 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’.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, [394Jump To The Next Citation Point, 393Jump To The Next Citation Point, 107].

In addition to the coordinate system, we need a Newman–Penrose null tetrad, or rather a GHP spinor dyad, A A A ๐œ€A = {o ,ι }, A = 0,1, on the hypersurfaces ๐’ฉu. (Thus, boldface indices are referring to the GHP spin frame.) It is natural to choose A o such that a A A′ l := o ¯o be the tangent a (∂โˆ•∂r ) of the null geodesic generators of ๐’ฉu, and oA itself be constant along la. Newman and Unti [394Jump To The Next Citation Point] chose ιA to be parallelly propagated along la. This choice yields the vanishing of a number of spin coefficients (see, for example, the review [393Jump To The Next Citation Point]). The asymptotic solution of the Einstein–Maxwell equations as a series of 1โˆ•r in this coordinate and tetrad system is given in [394, 179, 425Jump To The Next Citation Point], 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 σห™0 of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces ๐’ฉu.

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 := oA¯ιA′ and m¯a = ιA¯oA ′ to be tangents of the spheres. This can be achieved by an appropriate null rotation of the Newman–Unti basis about the spinor A o. 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 [455Jump To The Next Citation Point].

In contrast to the spatial-infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [106Jump To The Next Citation Point]. Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i.e., when ห™σ0 = 0. In the language of Section 4.1.7, this definition would be limr→ ∞ rΔ+ λ = 0, limr → ∞ rΔ − λ = 0, limr → ∞ r๐’ฏ +λ = 0 and limr →∞ r๐’ฏ − λ = 0. However, as Bramson showed [106], half of these conditions can be imposed. Namely, at future null infinity + + − ๐’ž λ := (Δ ⊕ ๐’ฏ )λ = 0 (and at past null infinity ๐’ž − λ := (Δ − ⊕ ๐’ฏ +)λ = 0) can always be imposed asymptotically, and has two linearly-independent solutions λA- A, A = 0, 1 --, on โ„ + (or on โ„ −, respectively). The space SA- ∞ of its solutions turns out to have a natural symplectic metric ๐œ€AB-, and we refer to A- (S ∞,๐œ€AB-) as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations limr →∞ r๐’ž ±λ = 0, the future/past asymptotic twistor equations. At โ„ + asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form KAA--′λA ¯λA ′′ = KAA-′λ1 ¯λ1′′ιA¯ιA′ A-A- A-A- for some constant Hermitian matrix AA′ K. Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are − σAiB-λ1A λ1B -- --, where σABi-- are the standard SU (2) Pauli matrices (divided by √ -- 2[496Jump To The Next Citation Point]. Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from + Δ, − Δ, + ๐’ฏ, and − ๐’ฏ, 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 โ„‹ − λ := (Δ − ⊕ ๐’ฏ − )λ = 0 (and at past null infinity as holomorphic spinors), and as special solutions of the two-surface twistor equation ๐’ฏ λ := (๐’ฏ + ⊕ ๐’ฏ − )λ = 0 (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [496Jump To The Next Citation Point].

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 ๐’ฎ of a combination of the leading term ψ0 2 of the Weyl spinor component ψ2, the asymptotic shear σ0 and its u-derivative, weighted by the first four spherical harmonics (see, for example, [393, 426Jump To The Next Citation Point]):

AB-′ 1 โˆฎ ( 0 0 0) A- B′ P BS = − ----- ψ2 + σ ¯σห™ λ0 ¯λ0′ d๐’ฎ, (4.14 ) 4πG
where λA0-:= λAAoA, 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., [496Jump To The Next Citation Point].) 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 ๐’ฎ of the leading term of the Weyl spinor component ψ¯′ 1, weighted by appropriate (spin-weighted) spherical harmonics:

1 โˆฎ A B JAB--= ----- ¯ψ01′λ0λ0-d๐’ฎ. (4.15 ) 8πG
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-- 16πG (rather than -1-- 8πG) and α = 0 in (3.15View Equation). This view is supported by the partial results of a quasi-local canonical analysis of general relativity given in [499Jump To The Next Citation Point], too.

Instead of the Bondi type coordinates above, one can introduce other ‘natural’ coordinates in a neighborhood of โ„ +. Such is the one based on the outgoing asymptotically–shear-free null geodesics [27Jump To The Next Citation Point]. While the Bondi-type coordinate system is based on the null geodesic generators of the outgoing null hypersurfaces ๐’ฉu, and hence, in the rescaled metric these generators are orthogonal to the cuts ๐’ฎu, the new coordinate system is based on the use of outgoing null geodesic congruences that extend to โ„ + but are not orthogonal to the cuts of โ„ + (and hence, in general, they have twist). The definition of the new coordinates (u,r,ζ,ζ¯) is analogous to that of the Bondi-type coordinates: ¯ (u,ζ, ζ) labels the intersection point of the actual geodesic and + โ„, while r is the affine parameter along the geodesic. The tangent a &tidle;l of this null congruence is asymptotically null rotated about na: In the NP basis {la,na,ma, ¯ma } above &tidle;la = la + b¯ma + ¯bma + b¯bna, where b = − L (u, ζ, ¯ζ)โˆ•r + ๐’ช (r−2) and L = L(u,ζ,ζ¯) is a complex valued function (with spin weight one) on โ„ +. Then Aronson and Newman show in [27Jump To The Next Citation Point] that if L is chosen to satisfy ห™ 0 ???L + LL = σ, then the asymptotic shear of the congruence is, in fact, of order −3 r, 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: a ¯ Z = Z (z ,ζ,ζ). It is this freedom that is used in [325Jump To The Next Citation Point, 326Jump To The Next Citation Point] 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].

4.2.5 Other special situations

In the weak field approximation of general relativity [525, 36, 534Jump To The Next Citation Point, 426Jump To The Next Citation Point, 303] the gravitational field is described by a symmetric tensor field hab on Minkowski spacetime (โ„4,g0 ) ab, and the dynamics of the field h ab is governed by the linearized Einstein equations, i.e., essentially the wave equation. Therefore, the tools and techniques of the 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 : 0 gab = gab + hab. 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 (2.5View Equation) 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 [481Jump To The Next Citation Point, 277Jump To The Next Citation Point, 426Jump To The Next Citation Point].

pp-waves spacetimes are defined to be those that admit a constant null vector field a L, 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 [534Jump To The Next Citation Point]. 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 [418Jump To The Next Citation Point], 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 2 2 √ --2---2----2- 4π(2m − e + 2m m − e − a ). 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 [42Jump To The Next Citation Point]. 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, Chruล›ciel, 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.


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