### 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 2-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 the so-called area-coordinate). Then with the notations of Section 4.1, one obtains . Based on the investigations of Misner, Sharp, and Hernandez [268199], Cahill and McVitte [98] found

to be an appropriate (and hence suggested to be the general) notion of energy contained in the 2-sphere . 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 [268199] (for the static solution, see e.g. Appendix B of [175]). Using Einstein’s equations nice and 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 the spherical symmetry all the quantities with non-zero spin weight, in particular the shears and , are vanishing and is real, by the GHP form of Equations (22, 23) the energy function can also be written as
This expression is considered to be the ‘standard’ definition of the energy for round spheres. Its last expression does not depend on whether is an area-coordinate or not. contains a contribution from the gravitational ‘field’ too. In fact, 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 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 3-space and with some positive constants and , and the range of the Euclidean radial coordinate is , where . It contains a maximal 2-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 [156]): , where and the Euclidean radial coordinate runs from and , 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 Lipschitz-continuous derivative at the 2-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 3-plane - like the capital Greek letter - for later reference we call it an ‘-spacetime’.

Spherically symmetric spacetimes admit a special vector field, the so-called Kodama vector field , such that is divergence free [241]. 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 line element above . If is the solid ball of radius , then the flux of is precisely the standard round sphere expression (26) for the 2-sphere  [278].

An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be given in terms of the energy function above (see for example [408262185]). In particular, criteria for the existence and the formation of trapped surfaces and the presence and the nature of the central singularity can be given by .

#### 4.2.2 Small surfaces

In the literature there are two notions 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 [204], and the other is the concept of the small ellipsoids in some spacelike hypersurface, considered first by Woodhouse in [235]. 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 centre, is a small given value, and the geometry of this sphere is characterized by data at this centre. Small ellipsoids are 2-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 parameterization completely. Then, in an open neighbourhood of the vertex , is a smooth null hypersurface, and hence for sufficiently small the set is a smooth spacelike 2-surface and homeomorphic to . is called a small sphere of radius with vertex . Note that the condition fixes the boost gauge.

Completing to a Newman-Penrose complex null tetrad such that the complex null vectors and are tangent to the 2-surfaces , the components of the metric and the spin coefficients with respect to this basis can be expanded as series in  . 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 a neighbourhood of or not. These solutions of the GHP equations, with increasing accuracy, are given in [20423594360]. Obviously, we can calculate the small sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime, too. In particular [360], 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 , respectively, are

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 1-forms, while is an average on the unit sphere of the boost-rotation Killing 1-forms that vanish at the vertex . Thus and are the 3-volume times the energy-momentum and angular momentum density with respect to , respectively, that the observer with 4-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 metric, i.e. , then its expansion is

where is a scalar. It depends linearly on the tensors constructed from and linearly from the coordinate dependent quantities , , …, and it is a polynomial expression of , and at the vertex . Since there is no non-trivial tensor built from the first derivative and , the leading term is of order . Its coefficient must be a linear expression of and , and polynomial in , and . In particular, if is to represent energy-momentum with generator at , then the leading term must be
for some unspecified constants , , and , where , the projection to the subspace orthogonal to . If, in addition to the coordinate-independence of , it is Lorentz-covariant, i.e. for example it does not depend on the choice for a normal to (e.g. in the small sphere approximation on ) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer in the same way. Therefore, , whenever the first and the third terms can in fact be written as . Then, comparing Equation (30) with Equation (28), we see that , and then the term 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 , even in a general spacetime. This seems to be quite unreasonable. Furthermore, even if the matter fields satisfy the dominant energy condition, given by Equation (30) can be negative even for unless . Thus, in the leading order in non-vacuum any coordinate and Lorentz-covariant quasi-local energy-momentum expression which is non-spacelike and future pointing should be proportional to the energy-momentum density of the matter fields seen by the observer times the Euclidean volume of the 3-ball of radius .

If a neighbourhood 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 on the latter two linearly, is the zero. Thus the order term in vacuum is also vanishing. In the fifth order the only non-zero terms are quadratic in the various parts of the Weyl tensor, yielding

for some constants , , , and , where is the electric 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
Again, if does not depend on intrinsically, then , whenever 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 non-spacelike. Therefore, in vacuum in the leading order any coordinate and Lorentz-covariant quasi-local energy-momentum expression which is non-spacelike 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 quasi-local angular momentum has the structure , while the area of is . Then the leading term in the expansion of the angular momentum is and order in non-vacuum and vacuum, respectively, while the first non-trivial correction to the area is of order and in non-vacuum 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 [235]. If is any smooth function on with a non-degenerate minimum at with minimum value , then, at least on an open neighbourhood of in the level surfaces are smooth compact 2-surfaces homeomorphic to . Then, in the limit, the surfaces look like small nested ellipsoids centred in . The function is usually ‘normalized’ so that .

#### 4.2.3 Large spheres near the spatial infinity

Near spatial infinity we have the a priori and fall-off for the 3-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. 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 fall-off conditions yield that the spin coefficients tend to their flat spacetime value like and the curvature components to zero like . Expanding the spin coefficients and curvature components as power series of , one can solve the field equations asymptotically (see [4844] 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 non-trivial 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 fall-off properties . Hence we can define the asymptotically constant spinor fields to be those that satisfy , where is the intrinsic Levi-Civita derivative operator. Note that this implies that, with the notations of Equation (25), all the chiral irreducible parts, , , , and , of the derivative of the asymptotically constant spinor field are .

#### 4.2.4 Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [300301302313] (see also [151]), 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 unit sphere 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 neighbourhood of in the spacetime in the following way. Let be the family of smooth outgoing null hypersurfaces in a neighbourhood 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 , 2-surfaces (or simply if no confusion can arise) are spacelike topological 2-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’. 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 the detailed discussion of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see for example [29028984]. 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 [290] chose to be parallel propagated along . This choice yields the vanishing of a number of spin coefficients (see for example the review [289]). The asymptotic solution of the Einstein-Maxwell equations as a series of in this coordinate and tetrad system is given in [290134312], where all the non-vanishing 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 of, this rotation with the highest accuracy was done for the solutions of the Einstein-Maxwell system by Shaw [338].

In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [83]. 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 [83], half of these conditions can be imposed. Namely, at future null infinity (and at past null infinity ) can always be imposed asymptotically, and it 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 [363]. 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 anti-holomorphic 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 2-surface twistor equation (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [363].

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 [289313]):

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 for example [363].)

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 spacetimes there is. It 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 [86].

#### 4.2.5 Other special situations

In the weak field approximation of general relativity [38222387313227] 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 (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 [349206313].

pp-waves spacetimes are defined to be those that admit a constant null vector field , and they are interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present then it is necessarily pure radiation with wavevector , i.e.  holds [243]. 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 [3] 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 [351].

Since the final state of stellar evolution (the neutron star or the black hole state) is expected to be described by an asymptotically flat stationary, axi-symmetric 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 [387]. Thus axi-symmetric 2-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 [305], 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 2-surfaces in these spacetimes are the spacelike cross sections of the event horizon [62]. 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 too. 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 [161]. (In the presence of matter fields, e.g. a self-interacting scalar field, the fall-off properties of the metric can be weakened such that the ‘charges’ defined at infinity and corresponding to the asymptotic symmetry generators remain finite [198].) 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 [27]. A comparison and analysis of the various definitions of mass for asymptotically anti-de-Sitter metrics is given in [117]. Thus it is natural to ask whether a specific quasi-local energy-momentum expression is able to reproduce the Abbott-Deser energy-momentum in this limit or not.