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 the spacetime metric takes the form , where and are functions of and . (Hence, is called the area-coordinate.) Then, with the notation of Section 4.1, one obtains . Based on the investigations of Misner, Sharp, and Hernandez [365, 267], Cahill and McVitte [122] found

to be an appropriate (and hence, suggested to be the general) notion of energy, the Misner–Sharp energy, contained in the two-sphere . (For another expression of in terms of the norm of the Killing fields and the metric, see [577].) In particular, for the Reissner–Nordström solution , while for the isentropic fluid solutions , where and are the usual parameters of the Reissner–Nordström solutions and is the energy density of the fluid [365, 267] (for the static solution, see, e.g., Appendix B of [240]). Using Einstein’s equations, simple equations can be derived for the derivatives and , and if the energy-momentum tensor satisfies the dominant energy condition, then . Thus, is a monotonic function of , provided is the area-coordinate. Since, by spherical symmetry all the quantities with nonzero spin weight, in particular the shears and , are vanishing and is real, by the GHP form of Eqs. (4.3), (4.4) the energy function can also be written as Any of these expressions is considered to be the ‘standard’ definition of the energy for round spheres.contains a contribution from the gravitational ‘field’ too. For example, for fluids it is not simply the volume integral of the energy density of the fluid, because that would be . This deviation can be interpreted as the contribution of the gravitational potential energy to the total energy. Consequently, is not a globally monotonic function of , even if . For example, in the closed Friedmann–Robertson–Walker spacetime (where, to cover the whole three-space, cannot be chosen to be the area–radius and ) is increasing for , taking its maximal value at , and decreasing for .

This example suggests a slightly more exotic spherically-symmetric spacetime. Its spacelike slice 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 hypersurface of the closed Friedmann–Robertson–Walker spacetime with the line element , where is the line element for the flat three-space and with positive constants and , and the range of the Euclidean radial coordinate is , where . It contains a maximal two-surface at with round-sphere mass parameter . The scalar curvature is , 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 hypersurface in the Schwarzschild solution with mass parameter (see [213]): , where and the Euclidean radial coordinate runs from to , where . In this geometry there is a minimal surface at , the scalar curvature is zero, and the round-sphere energy is . 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’ and arbitrarily small ADM mass . (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 – for later reference we will call it an‘-spacetime’.

Spherically-symmetric spacetimes admit a special vector field, called the Kodama vector field , such that is divergence free [321]. In asymptotically flat spacetimes 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 , the vector field has a conserved flux on a spacelike hypersurface . In particular, in the coordinate system and in the line element given in the first paragraph above . If is a solid ball of radius , then the flux of is precisely the standard round-sphere expression (4.7) for the two-sphere [375].

An interesting characterization of the dynamics of the spherically-symmetric gravitational fields can be given in terms of the energy function given by (4.7) (or by (4.8)) (see, e.g., [578, 352, 250]). In particular, criteria for the existence and formation of trapped surfaces and for the presence and nature of the central singularity can be given by . 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 in cosmology see, e.g., [484, 130].

In the literature there are two kinds of small surfaces. The first is that of the small spheres (both in the light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [275], and the other is the concept of small ellipsoids in a spacelike hypersurface, considered first by Woodhouse in [313]. A small sphere in the light cone is a cut of the future null cone in the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given spacelike hypersurface, whose geodesic distance in the hypersurface from a given point , 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 be a point, and a future-directed unit timelike vector at . Let , the ‘future null cone of in ’ (i.e., the boundary of the chronological future of ). Let be the future pointing null tangent to the null geodesic generators of , such that, at the vertex , . With this condition we fix the scale of the affine parameter on the different generators, and hence, by requiring , we fix the parametrization completely. Then, in an open neighborhood of the vertex , is a smooth null hypersurface, and hence, for sufficiently small , the set is a smooth spacelike two-surface and is homeomorphic to . is called a small sphere of radius with vertex . Note that the condition fixes the boost gauge, too.

Completing to get a Newman–Penrose complex null tetrad such that the complex null vectors and are tangent to the two-surfaces , the components of the metric and the spin coefficients with respect to this basis can be expanded as a series in . If, in addition, the spinor constituent of is required to be parallelly propagated along , 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 on , the GHP spin coefficients , , , , and , and the higher-order expansion coefficients of the curvature in terms of the lower-order curvature components at . Hence, the expression of any quasi-local quantity for the small sphere can be expressed as a series of ,

where the expansion coefficients are still functions of the coordinates, or ,
on the unit sphere . If the quasi-local quantity is spacetime-covariant, then the unit
sphere integrals of the expansion coefficients must be spacetime covariant expressions
of the metric and its derivatives up to some finite order at and the ‘time axis’ . The
necessary degree of the accuracy of the solution of the GHP equations depends on the
nature of and on whether the spacetime is Ricci-flat in the neighborhood of or
not.^{5}
These solutions of the GHP equations, with increasing accuracy, are given in [275, 313, 118, 494].

Obviously, we can calculate the small-sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime, as well. In particular [494], the small-sphere expressions for the quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields based on , are, respectively,

where , , is the ‘Cartesian spin frame’ at and the origin of the Cartesian coordinate system is chosen to be the vertex . Here can be interpreted as the translation one-forms, while is an average on the unit sphere of the boost-rotation Killing one-forms that vanish at the vertex . Thus, and are the three-volume times the energy-momentum and angular momentum density with respect to , respectively, that the observer with four-velocity sees at .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 is any coordinate-independent quasi-local quantity built from the first derivatives of the spacetime metric, then in its expansion the difference of the power of and the number of the derivatives in every term must be one, i.e., it must have the form

If the neighborhood of is vacuum, then the -order term is vanishing, and the fourth-order term must be built from . However, the only scalar polynomial expression of , , , and the generator vector , depending linearly on the latter two, is the zero tensor field. Thus, the -order term in vacuum is also vanishing. At the fifth order the only nonzero terms are quadratic in the various parts of the Weyl tensor, yielding

for constants , , , and , where is the electric part and is the magnetic part of the Weyl curvature, and is the induced volume 3-form. However, using the identities , , and , we can rewrite the above formula to be Again, if does not depend on intrinsically, then , in which case the first and the fourth terms together can be written into the Lorentz covariant form . In a general expression the curvature invariants and may be present. Since, however, and at a given point are independent, these invariants can be arbitrarily large positive or negative, and hence, for or the quasi-local energy-momentum could not be future pointing and nonspacelike. Therefore, in vacuum in the leading 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’ .Obviously, the same analysis can be repeated for any other quasi-local quantity. For the energy-momentum, has the structure , for angular momentum it is , while the area of is . Therefore, the leading term in the expansion of the angular momentum is and order in nonvacuum and vacuum with the energy-momentum and the Bel–Robinson tensors, respectively, while the first nontrivial correction to the area is of order and in nonvacuum and vacuum, respectively.

On the small geodesic sphere of radius in the given spacelike hypersurface one can introduce the complex null tangents and above, and if is the future-pointing unit normal of and the outward directed unit normal of in , then we can define and . Then 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 .

The small ellipsoids are defined as follows [313]. If is any smooth function on with a nondegenerate minimum at with minimum value , then, at least on an open neighborhood of in , the level surfaces are smooth compact two-surfaces homeomorphic to . Then, in the limit, the surfaces look like small nested ellipsoids centered at . The function is usually ‘normalized’ so that .

A slightly different framework for calculations in small regions was used in [327, 170, 235]. Instead of the Newman–Penrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or Fermi type normal) coordinates on an open neighborhood of a point or a timelike curve are used, in which the metric, as well as the Christoffel symbols on , are expressed by the coordinates on and the components of the Riemann tensor at or on . 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 order.

Near spatial infinity we have the a priori and falloff for the three-metric and
extrinsic curvature , respectively, and both the evolution equations of general relativity
and the conservation equation for the matter fields preserve these conditions. The
spheres of coordinate radius in are called large spheres if the values of 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 along the normals of the spheres, we obtain a coordinate system
on . Let , , be a GHP spinor dyad on adapted to the large
spheres in such a way that and are tangent to the spheres and
, the future directed unit normal of . These conditions fix the spinor dyad
completely, and, in particular, , the outward directed unit normal to the spheres
tangent to .

The falloff conditions yield that the spin coefficients tend to their flat spacetime value as and the curvature components to zero like . Expanding the spin coefficients and curvature components as a power series of , one can solve the field equations asymptotically (see [65, 61] for a different formalism). However, in most calculations of the large sphere limit of the quasi-local quantities, only the leading terms of the spin coefficients and curvature components appear. Thus, it is not necessary to solve the field equations for their second or higher-order nontrivial expansion coefficients.

Using the flat background metric and the corresponding derivative operator we can define a spinor field to be constant if . Obviously, the constant spinors form a two–complex-dimensional vector space. Then, by the falloff properties . Thus, we can define the asymptotically constant spinor fields to be those that satisfy , where is the intrinsic Levi-Civita derivative operator on . Note that this implies that, with the notation of Eq. (4.6), all the chiral irreducible parts, , , , and of the derivative of the asymptotically constant spinor field are .

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [413, 414, 415, 426] (see also [208]), i.e., the physical spacetime can be conformally compactified by an appropriate boundary . Then future null infinity will be a null hypersurface in the conformally rescaled spacetime. Topologically it is , and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of is the metric of the unit sphere. Fixing such a slice (called ‘the origin cut of ’) the points of can be labeled by a null coordinate, namely the affine parameter along the null geodesic generators of measured from and, for example, the familiar complex stereographic coordinates , defined first on the origin cut and then extended in a natural way along the null generators to the whole . Then any other cut of can be specified by a function . In particular, the cuts are obtained from by a pure time translation.

The coordinates can be extended to an open neighborhood of in the spacetime in the
following way. Let be the family of smooth outgoing null hypersurfaces in a neighborhood of ,
such that they intersect the null infinity just in the cuts , i.e., . Then let be the
affine parameter in the physical metric along the null geodesic generators of . Then forms
a coordinate system. The , two-surfaces (or simply if no confusion can
arise) are spacelike topological two-spheres, which are called large spheres of radius near future null
infinity. Obviously, the affine parameter is not unique, its origin can be changed freely:
is an equally good affine parameter for any smooth . 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, [394, 393, 107].

In addition to the coordinate system, we need a Newman–Penrose null tetrad, or rather a GHP spinor dyad, , , on the hypersurfaces . (Thus, boldface indices are referring to the GHP spin frame.) It is natural to choose such that be the tangent of the null geodesic generators of , and itself be constant along . Newman and Unti [394] chose to be parallelly propagated along . This choice yields the vanishing of a number of spin coefficients (see, for example, the review [393]). The asymptotic solution of the Einstein–Maxwell equations as a series of in this coordinate and tetrad system is given in [394, 179, 425], where all the nonvanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the -derivative of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces .

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’ , i.e., to require and to be tangents of the spheres. This can be achieved by an appropriate null rotation of the Newman–Unti basis about the spinor . This rotation yields a change of the spin coefficients and the metric and curvature components. As far as the present author is aware, the rotation with the highest accuracy was done for the solutions of the Einstein–Maxwell system by Shaw [455].

In contrast to the spatial-infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [106]. Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i.e., when . In the language of Section 4.1.7, this definition would be , , and . However, as Bramson showed [106], half of these conditions can be imposed. Namely, at future null infinity (and at past null infinity ) can always be imposed asymptotically, and has two linearly-independent solutions , , on (or on , respectively). The space of its solutions turns out to have a natural symplectic metric , and we refer to as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations , the future/past asymptotic twistor equations. At asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form for some constant Hermitian matrix . Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are , where are the standard Pauli matrices (divided by ) [496]. Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from , , , and , 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 (and at past null infinity as holomorphic spinors), and as special solutions of the two-surface twistor equation (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [496].

The Bondi–Sachs energy-momentum given in the Newman–Penrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut of a combination of the leading term of the Weyl spinor component , the asymptotic shear and its -derivative, weighted by the first four spherical harmonics (see, for example, [393, 426]):

where , , are the -component of the vectors of a spin frame in the space of the asymptotic spinors. (For the various realizations of these spinors see, e.g., [496].) The minimal assumptions on the physical Ricci tensor that already ensure that the Bondi–Sachs energy-momentum and Bondi’s mass-loss are well defined are determined by Tafel [505]. The expression of the Bondi–Sachs energy-momentum in terms of the conformal factor is also given there.Similarly, the various definitions for angular momentum at null infinity could be rewritten in this formalism. Although there is no generally accepted definition for angular momentum at null infinity in general spacetimes, in stationary and in axi-symmetric spacetimes there is. The former is the unit sphere integral on the cut of the leading term of the Weyl spinor component , weighted by appropriate (spin-weighted) spherical harmonics:

In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [109]. If the spacetime is axi-symmetric, then the generally accepted definition of angular momentum is that of Komar with the numerical coefficient (rather than ) and in (3.15). This view is supported by the partial results of a quasi-local canonical analysis of general relativity given in [499], too.Instead of the Bondi type coordinates above, one can introduce other ‘natural’ coordinates in a neighborhood of . Such is the one based on the outgoing asymptotically–shear-free null geodesics [27]. While the Bondi-type coordinate system is based on the null geodesic generators of the outgoing null hypersurfaces , and hence, in the rescaled metric these generators are orthogonal to the cuts , the new coordinate system is based on the use of outgoing null geodesic congruences that extend to but are not orthogonal to the cuts of (and hence, in general, they have twist). The definition of the new coordinates is analogous to that of the Bondi-type coordinates: labels the intersection point of the actual geodesic and , while is the affine parameter along the geodesic. The tangent of this null congruence is asymptotically null rotated about : In the NP basis above , where and is a complex valued function (with spin weight one) on . Then Aronson and Newman show in [27] that if is chosen to satisfy , then the asymptotic shear of the congruence is, in fact, of order , 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 that plays a role analogous to that of , and, indeed, the asymptotic solution of the field equations is given in terms of in [27]. This can be derived from the solution of the good-cut equation, which, however, is not uniquely determined, but depends on four complex parameters: . It is this freedom that is used in [325, 326] to introduce the angular momentum at future null infinity (see Section 3.2.4). Further discussion of these structures, in particular their connection with the solutions of the good-cut equation and the -space, as well as their applications, is given in [324, 325, 326, 5].

In the weak field approximation of general relativity [525, 36, 534, 426, 303] the gravitational field is described by a symmetric tensor field on Minkowski spacetime , and the dynamics of the field 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 is essentially the second-order term in the Einstein tensor of the metric . Thus, in the linear approximation the field does not contribute to the global energy-momentum and angular momentum of the matter + gravity system, and hence these quantities have the form (2.5) with the linearized energy-momentum tensor of the matter fields. However, as we will see in Section 7.1.1, this energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized) curvature [481, 277, 426].

pp-waves spacetimes are defined to be those that admit a constant null vector field , 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 , i.e., 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 with closed orbits, i.e., it is cyclically symmetric too, then and are necessarily commuting and are orthogonal to each other, because otherwise an additional timelike Killing vector would also be admitted [485].

Since the final state of stellar evolution (the neutron star or black hole state) is expected to be described by an asymptotically flat, stationary, axisymmetric spacetime, the significance of these spacetimes is obvious. It is conjectured that this final state is described by the Kerr–Newman (either outer or black hole) solution with some well-defined mass, angular momentum and electric charge parameters [534]. Thus, axisymmetric two-surfaces in these solutions may provide domains, which are general enough but for which the quasi-local quantities are still computable. According to a conjecture by Penrose [418], the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the Kerr–Newman black hole this area is . Thus, particularly interesting two-surfaces in these spacetimes are the spacelike cross sections of the event horizon [80].

There is a well-defined notion of total energy-momentum not only in the asymptotically flat, but even in the asymptotically anti-de Sitter spacetimes as well. This is the Abbott–Deser energy [1], whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [220]. (In the presence of matter fields, e.g., a self-interacting scalar field, the falloff properties of the metric can be weakened such that the ‘charges’ defined at infinity and corresponding to the asymptotic symmetry generators remain finite [265].) The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically anti-de Sitter spacetimes and to study their general, basic properties in [42]. A comparison and analysis of the various definitions of mass for asymptotically anti-de Sitter metrics is given in [150].

Extending the spinorial proof [349] of the positivity of the total energy in asymptotically anti-de Sitter spacetime, Chruลciel, 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.

Living Rev. Relativity 12, (2009), 4
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