Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

The symmetric energy-momentum tensor

It is a widely accepted view (appearing e.g. in excellent, standard textbooks on general relativity, too) that the canonical energy-momentum and spin tensors are well-defined and have relevance only in flat spacetime, and hence usually are underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus first we introduce these quantities for the matter fields in a general curved spacetime.

To specify the state of the matter fields operationally two kinds of devices are needed: The first measures the value of the fields, while the other measures the spatio-temporal location of the first. Correspondingly, the fields on the manifold M of events can be grouped into two sharply distinguished classes: The first contains the matter field variables, e.g. finitely many (r, s)-type tensor fields , whilst the other contains the fields specifying the spacetime geometry, i.e. the metric g_ab in Einstein’s theory. Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a Lagrangian : If I_m[g^ab, Φ_n] is the volume integral of L_m on some open domain D with compact closure then the equations of motion are , the Euler-Lagrange equations. The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by

1

where we introduced the so-called canonical spin tensor

2

(The terminology will be justified in the next Section 2.2.) Here is the (p + q + 1, p + q + 1)-type invariant tensor, built from the Kronecker deltas, appearing naturally in the expression of the Lie derivative of the (p, q)-type tensor fields in terms of the torsion free covariant derivatives: . (For the general idea of deriving T_ab and Equation (2), see e.g. Section 3 of [175].)

The canonical Noether current

Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that for any vector field K^a and the corresponding local 1-parameter family of diffeomorphisms φ_t one has

for some 1-parameter family of vector fields . (L_m is called diffeomorphism invariant if , e.g. when L_m is a scalar.) Let K^a be any smooth vector field on M. Then, calculating the divergence ∇_a(L_mK^a) to determine the rate of change of the action functional I_m along the integral curves of K^a, by a tedious but straightforward computation one can derive the so-called Noether identity: , where Ł_K denotes the Lie derivative along K^a, and C^a[K], the so-called Noether current, is given explicitly by

3

Here Ḃ^e is the derivative of with respect to t at t = 0, which may depend on K_a and its derivatives, and , the so-called canonical energy-momentum tensor, is defined by

4

Note that, apart from the term Ḃ^e, the current C^e[K] does not depend on higher than the first derivative of K^a, and the canonical energy-momentum and spin tensors could be introduced as the coefficients of K_a and its first derivative, respectively, in C^e[K]. (For the original introduction of these concepts, see [56, 57, 323]. If the torsion is not vanishing, then in the Noether identity there is a further term, , where the so-called dynamical spin tensor is defined by , and the Noether current has a slightly different structure [193, 194].) Obviously, C^e[K] is not uniquely determined by the Noether identity, because that contains only its divergence, and any identically conserved current may be added to it. In fact, may be chosen to be an arbitrary non-zero (but divergence free) vector field even for diffeomorphism invariant Lagrangians. Thus, to be more precise, if Ḃ^e = 0, then we call the specific combination (3) the canonical Noether current. Other choices for the Noether current may contain higher derivatives of K^a, too (see e.g. [228]), but there is a specific one containing K^a algebraically (see the Points 3 and 4 below). However, C^a[K] is sensitive to total divergences added to the Lagrangian, and, if the matter fields have gauge freedom, then in general it is not gauge invariant even if the Lagrangian is. On the other hand, T^ab is gauge invariant and is independent of total divergences added to L_m because it is the variational derivative of the gauge invariant action with respect to the metric. Provided the field equations are satisfied, the Noether identity implies [56, 57, 323, 193, 194] that

1. ∇_aT^ab = 0,

2. T^ab = θ^ab + ∇_c(σ^c[ab] + σ^a[bc] + σ^b[^ac]),

3. C^a[K] = T^abK_b + ∇_c((σ^a[cb] − σ^c[ab] − σ^b[^ac])K_b), where the second term on the right is an identically conserved (i.e. divergence free) current, and

4. C^a[K] is conserved if K^a is a Killing vector.

Hence T^abK_b is also conserved and can equally be considered as a Noether current. (For a formally different, but essentially equivalent introduction of the Noether current and identity, see [389, 215, 141].)

The interpretation of the conserved currents C^a[K] and T^abK_b depends on the nature of the Killing vector K^a. In Minkowski spacetime the 10-dimensional Lie algebra K of the Killing vectors is well known to split to the semidirect sum of a 4-dimensional commutative ideal, T, and the quotient K/T, where the latter is isomorphic to so(1, 3). The ideal T is spanned by the constant Killing vectors, in which a constant orthonormal frame field on M, , forms a basis. (Thus the underlined Roman indices are concrete, name indices.) By the ideal T inherits a natural Lorentzian vector space structure. Having chosen an origin o ∈ M, the quotient K/T can be identified as the Lie algebra R_o of the boost-rotation Killing vectors that vanish at o. Thus K has a ‘4 + 6’ decomposition into translations and boost-rotations, where the translations are canonically defined but the boost-rotations depend on the choice of the origin o ∈ M. In the coordinate system adapted to (i.e. for which the 1-form basis dual to has the form ) the general form of the Killing vectors (or rather 1-forms) is for some constants and . Then the corresponding canonical Noether current is , and the coefficients of the translation and the boost-rotation parameters and are interpreted as the density of the energy-momentum and the sum of the orbital and spin angular momenta, respectively. Since, however, the difference C^a[K] − T^abK_b is identically conserved and T^abK_b has more advantageous properties, it is T^abK_b that is used to represent the energy-momentum and angular momentum density of the matter fields.

Since in the de-Sitter and anti-de-Sitter spacetimes the (ten dimensional) Lie algebra of the Killing vector fields, so(1, 4) and so(2, 3), respectively, are semisimple, there is no such natural notion of translations, and hence no natural ‘4 + 6’ decomposition of the ten conserved currents into energy-momentum and (relativistic) angular momentum density.

The definition of the quasi-local quantities

To define the quasi-local conserved quantities in Minkowski spacetime, first observe that for any Killing vector K^a ∈ K the 3-form ω_abc := K_eT^efε_fabc is closed, and hence, by the triviality of the third de Rham cohomology class, H³(ℝ⁴) = 0, it is exact: For some 2-form ∪[K]_ab we have may be called a ‘superpotential’ for the conserved current 3-form ω_abc. (However, note that while the superpotential for the gravitational energy-momentum expressions of the next Section 3 is a local function of the general field variables, the existence of this ‘superpotential’ is a consequence of the field equations and the Killing nature of the vector field K^a. The existence of globally defined superpotentials that are local functions of the field variables can be proven even without using the Poincaré lemma [388].) If Ũ[K]_ab is (the dual of) another superpotential for the same current ω_abc, then by Ũ_[a(∪[K]_bc] − Ũ[K]_bc]) = 0 and H²(ℝ⁴) = 0 the dual superpotential is unique up to the addition of an exact 2-form. If therefore is any closed orientable spacelike 2-surface in the Minkowski spacetime then the integral of ∪[K]_ab on is free from this ambiguity. Thus if Σ is any smooth compact spacelike hypersurface with smooth 2-boundary , then

5

depends only on . Hence it is independent of the actual Cauchy surface Σ of the domain of dependence D(Σ) because all the spacelike Cauchy surfaces for D(Σ) have the same common boundary . Thus can equivalently be interpreted as being associated with the whole domain of dependence D(Σ), and hence quasi-local in the sense of [169, 168] above. It defines the linear maps and by , i.e. they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates, and transform as a Lorentz vector and anti-symmetric tensor, respectively, whilst under the translation of the origin, is unchanged while . Thus and may be interpreted as the quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike 2-surface , or, equivalently, to D(Σ). Then the quasi-local mass and Pauli-Lubanski spin are defined, respectively, by the usual formulae and . (If m² ≠ 0, then the dimensionally correct definition of the Pauli-Lubanski spin is .) As a consequence of the definitions holds, i.e. if is timelike then is spacelike or zero, but if is null (i.e. ) then is spacelike or proportional to .

Obviously, we can form the flux integral of the current T^abξ_b on the hypersurface even if ξ^a is not a Killing vector, even in general curved spacetime:

6

Then, however, the integral E_Σ[ξ^a] does depend on the hypersurface, because this is not connected with the spacetime symmetries. In particular, the vector field ξ^a can be chosen to be the unit timelike normal t^a of Σ. Since the component μ := T_abt^at^b of the energy-momentum tensor is interpreted as the energy-density of the matter fields seen by the local observer t^a, it would be legitimate to interpret the corresponding integral E_Σ[t^a] as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to Σ’. Thus E_Σ[t^a] defines a different concept of the quasi-local energy: While that based on is linked to some absolute element, namely to the translational Killing symmetries of the spacetime and the constant timelike vector fields can be interpreted as the observers ‘measuring’ this energy, E_Σ[t^a] is completely independent of any absolute element of the spacetime and is based exclusively on the arbitrarily chosen fleet of observers. Thus, while is independent of the actual normal t^a of , E_Σ[ξ^a] (for non-Killing ξ^a) depends on t^a intrinsically and is a genuine 3-hypersurface rather than a 2-surface integral.

If , the orthogonal projection to Σ, then the part of the energy-momentum tensor is interpreted as the momentum density seen by the observer t^a. Hence

is the square of the mass density of the matter fields, where h_ab is the spatial metric in the plane orthogonal to t^a. If T^ab satisfies the dominant energy condition (i.e. T^abV_b is a future directed non-spacelike vector for any future directed non-spacelike vector V^a, see for example [175]), then this is non-negative, and hence

7

can also be interpreted as the quasi-local mass of the matter fields seen by the fleet of observers being at rest with respect to Σ, even in general curved spacetime. However, although in Minkowski spacetime E_Σ[K] for the four translational Killing vectors gives the four components of the energy-momentum , the mass M_Σ is different from . In fact, while is defined as the Lorentzian norm of with respect to the metric on the space of the translations, in the definition of M_Σ first the norm of the current T^abt_b is taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of the more advantageous properties (see Section 2.2.3 below), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form instead of E_Σ[ξ^a].

Thus even if there is a gauge invariant and unambiguously defined energy-momentum density of the matter fields, it is not a priori clear how the various quasi-local quantities should be introduced. We will see in the second part of the present review that there are specific suggestions for the gravitational quasi-local energy that are analogous to , others to E_Σ[t^a] and some to M_Σ.

Hamiltonian introduction of the quasi-local quantities

In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [212, 396] and references therein) the configuration and momentum variables, φ^A and π_A, respectively, are fields on a connected 3-manifold Σ, which is interpreted as the typical leaf of a foliation Σ_t of the spacetime. The foliation can be characterized on Σ by a function N, called the lapse. The evolution of the states in the spacetime is described with respect to a vector field K^a = Nt^a + N^a (‘evolution vector field’ or ‘general time axis’), where t^a is the future directed unit normal to the leaves of the foliation and N^a is some vector field, called the shift, being tangent to the leaves. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Physical states can be only those that are on the constraint surface, specified by the vanishing of certain functions C_i = C_i(φ^A, D_eφ^A,…, π_A, D_eπ_A,…), i = 1,…, n, of the canonical variables and their derivatives up to some finite order, where D_e is the covariant derivative operator in Σ. Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form

8

Here dΣ is the induced volume element, the coefficients μ and j_a are local functions of the canonical variables and their derivatives up to some finite order, the Nⁱ’s are functions on Σ, and Z^a is a local function of the canonical variables, the lapse, the shift, the functions Nⁱ, and their derivatives up to some finite order. The part C_iNⁱ of the Hamiltonian generates gauge motions in the phase space, and the functions Nⁱ are interpreted as the freely specifiable ‘gauge generators’.

However, if we want to recover the field equations for φ^A (which are partial differential equations on the spacetime with smooth coefficients for the smooth field φ^A) on the phase space as the Hamilton equations and not some of their distributional generalizations, then the functional differentiability of H[K] must be required in the strong sense of [387]1. Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of H[K] requires some boundary conditions on the field variables, and may yield restrictions on the form of Z^a. It may happen that for a given Z^a only too restrictive boundary conditions would be able to ensure the functional differentiability of the Hamiltonian, and hence the ‘quasi-local phase space’ defined with these boundary conditions would contain only very few (or no) solutions of the field equations. In this case Z^a should be modified. In fact, the boundary conditions are connected to the nature of the physical situations considered. For example, in electrodynamics different boundary conditions must be imposed if the boundary is to represent a conducting or an insulating surface. Unfortunately, no universal principle or ‘canonical’ way of finding the ‘correct’ boundary term and the boundary conditions is known.

In the asymptotically flat case the value of the Hamiltonian on the constraint surface defines the total energy-momentum and angular momentum, depending on the nature of K^a, in which the total divergence D_aZ^a corresponds to the ambiguity of the superpotential 2-form ∪[K]_ab: An identically conserved quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved). The energy density and the momentum density of the matter fields can be recovered as the functional derivative of H[K] with respect to the lapse N and the shift N^a, respectively. In principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising achievements of [7, 8, 327] for the Klein-Gordon, Maxwell, and the Yang-Mills-Higgs fields, as far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still lacking.

Properties of the quasi-local quantities

Suppose that the matter fields satisfy the dominant energy condition. Then E_Σ[ξ^a] is also nonnegative for any non-spacelike ξ^a, and, obviously, E_Σ[t^a] is zero precisely when T^ab = 0 on Σ, and hence, by the conservation laws (see for example Page 94 of [175]), on the whole domain of dependence D(Σ). Obviously, M_Σ = 0 if and only if L^a := T^abt_b is null on Σ. Then by the dominant energy condition it is a future pointing vector field on Σ, and L_aT^ab = 0 holds. Therefore, T^ab on Σ has a null eigenvector with zero eigenvalue, i.e. its algebraic type on Σ is pure radiation.

The properties of the quasi-local quantities based on in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [354, 358] that

1. is a future directed nonspacelike vector, ;

2. if and only if T_ab = 0 on D(Σ);

3. if and only if the algebraic type of the matter on D(Σ) is pure radiation, i.e. T_abL^b = 0 holds for some constant null vector L^a. Then T_ab = rL_aL_b for some non-negative function τ, whenever , where and ;

4. For the angular momentum has the form , where . Thus, in particular, the Pauli-Lubanski spin is zero.

Therefore, the vanishing of the quasi-local energy-momentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasi-local mass is equivalent to special configurations representing pure radiation.

Since and M_Σ are integrals of functions on a hypersurface, they are obviously additive, i.e. for example for any two hypersurfaces Σ₁ and Σ₂ (having common points at most on their boundaries and ) one has . On the other hand, the additivity of is a slightly more delicate problem. Namely, and are elements of the dual space of the translations, and hence we can add them and, as in the previous case, we obtain additivity. However, this additivity comes from the absolute parallelism of the Minkowski spacetime: The quasi-local energy-momenta of the different 2-surfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different 2-surfaces, then the energy-momenta would belong to different vector spaces, and they could not be added. We will see that the quasi-local quantities discussed in Sections 7, 8, and 9 belong to vector spaces dual to their own ‘quasi-Killing vectors’, and there is no natural way of adding the energy-momenta of different surfaces.

Global energy-momenta and angular momenta

If Σ extends either to spatial or future null infinity, then, as is well known, the existence of the limit of the quasi-local energy-momentum can be ensured by slightly faster than (for example by ) fall-off of the energy-momentum tensor, where r is any spatial radial distance. However, the finiteness of the angular momentum and centre-of-mass is not ensured by the fall-off. Since the typical fall-off of T_ab — for example for the electromagnetic field — is , we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the fall-off, six global integral conditions for the leading terms of T_ab must be imposed. At the spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’ T^ab_;b = 0 (as evolution equations for the energy density and momentum density) preserve these fall-off and parity conditions [364].

Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass the fields must be plane waves, furthermore by they must be asymptotically vanishing at the same time. However, a plane wave configuration can be asymptotically vanishing only if it is vanishing.

Quasi-local radiative modes and a classical version of the holography for matter fields

By the results of the previous Section 2.2.4 the vanishing of the quasi-local mass, associated with a closed spacelike 2-surface , implies that the matter must be pure radiation on a 4-dimensional globally hyperbolic domain D(Σ). Thus characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present section we review how these states on D(Σ) can be characterized completely by data on the 2-surface , and how these states can be used to formulate a classical version of the holographic principle.

For the (real or complex) linear massless scalar field φ and the Yang-Mills fields, represented by the symmetric spinor fields , α = 1, … N, where N is the dimension of the gauge group, the vanishing of the quasi-local mass is equivalent [365] to plane waves and the pp-wave solutions of Coleman [118], respectively. Then the condition T_abL^b = 0 implies that these fields are completely determined on the whole D(Σ) by their value on (whenever the spinor fields are necessarily null: , where φ^α are complex functions and O_a is a constant spinor field such that L_a = O_AŌ_a′). Similarly, the null linear zero-rest-mass fields φ_AB…E = φO_AO_B … O_E on D(Σ) with any spin and constant spinor O_A are completely determined by their value on . Technically, these results are based on the unique complex analytic structure of the u = const. 2-surfaces foliating Σ, where L_a = ∇_au, and by the field equations the complex functions φ and φ^α turn out to be anti-holomorphic [358]. Assuming, for the sake of simplicity, that is future and past convex in the sense of Section 4.1.3 below, the independent boundary data for such a pure radiative solution consist of a constant spinor field on and a real function with one and another with two variables. Therefore, the pure radiative modes on D(Σ) can be characterized completely by appropriate data (the so-called holographic data) on the ‘screen’ .

These ‘quasi-local radiative modes’ can be used to map any continuous spinor field on D(Σ) to a collection of holographic data. Indeed, the special radiative solutions of the form (with fixed constant spinor field O^A) together with their complex conjugate define a dense subspace in the space of all continuous spinor fields on Σ. Thus every such spinor field can be expanded by the special radiative solutions, and hence can also be represented by the corresponding family of holographic data. Therefore, if we fix a foliation of D(Σ) by spacelike Cauchy surfaces Σ_t, then every spinor field on D(Σ) can also be represented on by a time dependent family of holographic data, too [365]. This fact may be a specific manifestation in the classical non-gravitational physics of the holographic principle (see Section 13.4.2).

The root of the difficulties

The action I_m for the matter fields was a functional of both kinds of fields, thus one could take the variational derivatives both with respect to and g^ab. The former gave the field equations, while the latter defined the symmetric energy-momentum tensor. Moreover, g_ab provided a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action I_g is, on the other hand, a functional of the metric alone, and its variational derivative with respect to g^ab yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of g^ab can be traced back to the principle of equivalence [22], and introduces a huge gauge freedom in the dynamics of g^ab because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold M endowed with a Lorentzian metric g_ab, but the isomorphism class of such pairs, where (M, g_ab) and (M, φ*g_ab) are considered to be equivalent for any diffeomorphism φ of M onto itself2. Thus we do not have, even in principle, any gravitational analog of the symmetric energy-momentum tensor of the matter fields. In fact, by its very definition, T_ab is the source-current for gravity, like the current in Yang-Mills theories (defined by the variational derivative of the action functional of the particles, e.g. of the fermions, interacting with a Yang-Mills field ), rather than energy-momentum. The latter is represented by the Noether currents associated with special spacetime displacements. Thus, in spite of the intimate relation between T_ab and the Noether currents, the proper interpretation of T_ab is only the source density for gravity, and hence it is not the symmetric energy-momentum tensor whose gravitational counterpart must be searched for. In particular, the Bel-Robinson tensor , given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [333, 332]), being a quadratic expression of the curvature (and its derivatives), is (are) expected to represent only ‘higher order’ gravitational energy-momentum. (Note that according to the original tensorial definition the Bel-Robinson tensor is one-fourth the expression above. Our convention follows that of Penrose and Rindler [312].) In fact, the physical dimension of the Bel-Robinson ‘energy-density’ T_abcdt^at^bt^ct^d is cm⁻⁴, and hence (in the traditional units) there are no powers A and B such that c^AG^B T_abcdt^at^bt^ct^d would have energy-density dimension. Here c is the speed of light and G is Newton’s gravitational constant. As we will see, the Bel-Robinson ‘energy-momentum density’ T_abcdt^bt^ct^d appears naturally in connection with the quasi-local energy-momentum and spin-angular momentum expressions for small spheres only in higher order terms. Therefore, if we want to associate energy-momentum and angular momentum with the gravity itself in a Lagrangian framework, then it is the gravitational counterpart of the canonical energy-momentum and spin tensors and the canonical Noether current built from them that should be introduced. Hence it seems natural to apply the Lagrange-Belinfante-Rosenfeld procedure, sketched in the previous section, to gravity too [56, 57, 323, 193, 194, 352].

Pseudotensors

The lack of any background geometric structure in the gravitational action yields, first, that any vector field K^a generates a symmetry of the matter plus gravity system. Its second consequence is the need for an auxiliary derivative operator, e.g. the Levi-Civita covariant derivative coming from an auxiliary, non-dynamical background metric (see for example [231, 316]), or a background (usually torsion free, but not necessarily flat) connection (see for example [215]), or the partial derivative coming from a local coordinate system (see for example [382]). Though the natural expectation would be that the final results be independent of these background structures, as is well known, the results do depend on them.

In particular [352], for Hilbert’s second order Lagrangian L_H := R/16_πG in a fixed local coordinate system {x^α} and derivative operator ∂_μ instead of ∇_e, Equation (4) gives precisely Møller’s energy-momentum pseudotensor , which was defined originally through the superpotential equation , where is the so-called Møller superpotential [270]. (For another simple and natural introduction of Møller’s energy-momentum pseudotensor see [104].) For the spin pseudotensor Equation (2) gives

which is in fact only pseudotensorial. Similarly, the contravariant form of these pseudotensors and the corresponding canonical Noether current are also pseudotensorial. We saw in Section 2.1.2 that a specific combination of the canonical energy-momentum and spin tensors gave the symmetric energy-momentum tensor, which is gauge invariant even if the matter fields have gauge freedom, and one might hope that the analogous combination of the energy-momentum and spin pseudotensors gives a reasonable tensorial energy-momentum density for the gravitational field. The analogous expression is, in fact, tensorial, but unfortunately it is just minus the Einstein tensor [352, 353]3. Therefore, to use the pseudotensors a ‘natural’ choice for a ‘preferred’ coordinate system would be needed. This could be interpreted as a gauge choice, or reference configuration.

A further difficulty is that the different pseudotensors may have different (potential) significance. For example, for any fixed k ∈ ℝ Goldberg’s 2k-th symmetric pseudotensor is defined by (which, for k = 0, reduces to the Landau-Lifshitz pseudotensor, the only symmetric pseudotensor which is a quadratic expression of the first derivatives of the metric) [162]. However, by Einstein’s equations this definition implies that . Hence what is (coordinate-)divergence-free (i.e. ‘pseudo-conserved’) cannot be interpreted as the sum of the gravitational and matter energy-momentum densities. Indeed, the latter is , while the second term in the divergence equation has an extra weight . Thus there is only one pseudotensor in this series, , which satisfies the ‘conservation law’ with the correct weight. In particular, the Landau-Lifshitz pseudotensor also has this defect. On the other hand, the pseudotensors coming from some action (the ‘canonical pseudotensors’) appear to be free of this kind of difficulties (see also [352, 353]). Classical excellent reviews on these (and several other) pseudotensors are [382, 59, 9, 163], and for some recent ones (using background geometric structures) see for example [137, 138, 79, 154, 155, 228, 316]. We return to the discussion of pseudotensors in Sections 3.3.1 and 11.3.4.

Strategies to avoid pseudotensors I: Background metrics/connections

One way of avoiding the use of the pseudotensorial quantities is to introduce an explicit background connection [215] or background metric [322, 229, 233, 231, 230, 315, 135]. (The superpotential of Katz, Bičák, and Lyndel-Bell [230] has been rediscovered recently by Chen and Nester [108] in a completely different way. We return to the discussion of the latter in Section 11.3.2.) The advantage of this approach would be that we could use the background not only to derive the canonical energymomentum and spin tensors, but to define the vector fields K^a as the symmetry generators of the background. Then the resulting Noether currents are without doubt tensorial. However, they depend explicitly on the choice of the background connection or metric not only through K^a: The canonical energy-momentum and spin tensors themselves are explicitly background-dependent. Thus, again, the resulting expressions would have to be supplemented by a ‘natural’ choice for the background, and the main question is how to find such a ‘natural’ reference configuration from the infinitely many possibilities.

Strategies to avoid pseudotensors II: The tetrad formalism

In the tetrad formulation of general relativity the g_ab-orthonormal frame fields , are chosen to be the gravitational field variables [386, 236]. Re-expressing the Hilbert Lagrangian (i.e. the curvature scalar) in terms of the tetrad field and its partial derivatives in some local coordinate system, one can calculate the canonical energy-momentum and spin by Equations (4) and (2), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [271], the use of the tetrad fields as the field variables instead of the metric makes it possible to introduce a first order, scalar Lagrangian for Einstein’s field equations: If , the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is

9

(Here is the 1-form basis dual to .) Although L depends on the actual tetrad field , it is weakly O(1, 3)-invariant. Møller’s Lagrangian has a nice uniqueness property [299]: Any first order scalar Lagrangian built from the tetrad fields, whose Euler-Lagrange equations are the Einstein equations, is Møller’s Lagrangian. Using Dirac spinor variables Nester and Tung found a first order spinor Lagrangian [288], which turned out to be equivalent to Møller’s Lagrangian [383]. Another first order spinor Lagrangian, based on the use of the two-component spinors and the anti-self-dual connection, was suggested by Tung and Jacobson [384]. Both Lagrangians yield a well-defined Hamiltonian, reproducing the standard ADM energy-momentum in asymptotically flat spacetimes. The canonical energy-momentum derived from Equation (9) using the components of the tetrad fields in some coordinate system as the field variables is still pseudotensorial, but, as Møller realized, it has a tensorial superpotential:

10

The canonical spin turns out to be essentially , i.e. a tensor. The tensorial nature of the superpotential makes it possible to introduce a canonical energy-momentum tensor for the gravitational ‘field’. Then the corresponding canonical Noether current C^a[K] will also be tensorial and satisfies

11

Therefore, the canonical Noether current derived from Møller’s tetrad Lagrangian is independent of the background structure (i.e. the coordinate system) that we used to do the calculations (see also [352]). However, C^a[K] depends on the actual tetrad field, and hence a preferred class of frame fields, i.e. an O(1, 3)-gauge reduction, is needed. Thus the explicit background-dependence of the final result of other approaches has been transformed into an internal O(1, 3)-gauge dependence. It is important to realize that this difficulty always appears in connection with the gravitational energy-momentum and angular momentum, at least in disguise. In particular, the Hamiltonian approach in itself does not yield well-defined energy-momentum density for the gravitational ‘field’ (see for example [282, 263]). Thus in the tetrad approach the canonical Noether current should be supplemented by a gauge condition for the tetrad field. Such a gauge condition could be some spacetime version of Nester’s gauge conditions (in the form of certain partial differential equations) for the orthonormal frames of Riemannian manifolds [281, 284]. Furthermore, since C^a[K] + T^abK_b is conserved for any vector field K^a, in the absence of the familiar Killing symmetries of the Minkowski spacetime it is not trivial to define the ‘translations’ and ‘rotations’, and hence the energy-momentum and angular momentum. To make them well-defined additional ideas would be needed.

Strategies to avoid pseudotensors III: Higher derivative currents

Giving up the paradigm that the Noether current should depend only on the vector field K^a and its first derivative — i.e. if we allow a term Ḃ^a to be present in the Noether current (3) even if the Lagrangian is diffeomorphism invariant — one naturally arrives at Komar’s tensorial superpotential _K∨ [K]^ab := ∇^[aK^b] and the corresponding Noether current [242] (see also [59]). Although its independence of any background structure (viz. its tensorial nature) and uniqueness property (see Komar [242] quoting Sachs) is especially attractive, the vector field K^a is still to be determined.

Spatial infinity: Energy-momentum

There are several mathematically inequivalent definitions of asymptotic flatness at spatial infinity [151, 344, 23, 48, 148]. The traditional definition is based on the existence of a certain asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable definition are asymptotically flat in the traditional sense too. A spacelike hypersurface Σ will be called k-asymptotically flat if for some compact set K ⊂ Σ the complement Σ − K is diffeomorphic to ℝ³ minus a solid ball, and there exists a (negative definite) metric ₀h_ab on Σ, which is flat on Σ − K, such that the components of the difference of the physical and the background metrics, h_ij − ₀h_ij, and of the extrinsic curvature χ_ij in the ₀h_ij-Cartesian coordinate system {x^k} fall off as r^−k and r^−k−1, respectively, for some k > 0 and r² := δ_ijxⁱx^j [319, 47]. These conditions make it possible to introduce the notion of asymptotic spacetime Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations. Σ − K together with the metric and extrinsic curvature is called the asymptotic end of Σ. In a more general definition of asymptotic flatness Σ is allowed to have finitely many such ends.

As is well known, finite and well-defined ADM energy-momentum [11, 13, 12, 14] can be associated with any k-asymptotically flat spacelike hypersurface if k > ½ by taking the value on the constraint surface of the Hamiltonian H[K^a], given for example in [319, 47], with the asymptotic translations K^a (see [112, 37, 291, 113]). In its standard form this is the r → ∞ limit of a 2-surface integral of the first derivatives of the induced 3-metric h_ab and of the extrinsic curvature χ_ab for spheres of large coordinate radius r. The ADM energy-momentum is an element of the space dual to the space of the asymptotic translations, and transforms as a Lorentzian 4-vector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian coordinates.

The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian analysis of general relativity is based on the 3+1 decomposition of the fields and the spacetime. Thus it is not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular momentum and centre-of-mass, discussed below, form an antisymmetric tensor). One had to check a posteriori that the conserved quantities obtained in the 3 + 1 form are, in fact, Lorentz-covariant. To obtain manifestly Lorentz-covariant quantities one should not do the 3 + 1 decomposition. Such a manifestly Lorentz-covariant Hamiltonian analysis was suggested first by Nester [280], and he was able to recover the ADM energy-momentum in a natural way (see also Section 11.3 below).

Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [163]: Taking the flux integral of the current C^a[K] + T^abK_b on the spacelike hypersurface Σ, by Equation (11) the flux can be rewritten as the r → ∞ limit of the 2-surface integral of Møller’s superpotential on spheres of large r with the asymptotic translations K^a. Choosing the tetrad field to be adapted to the spacelike hypersurface and assuming that the frame tends to a constant Cartesian one as r^−k, the integral reproduces the ADM energy-momentum. The same expression can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too: By the standard scenario one can construct the basic Hamiltonian [282]. This Hamiltonian, evaluated on the constraints, turns out to be precisely the flux integral of C^a[K] + T^abK_b on Σ.

A particularly interesting and useful expression for the ADM energy-momentum is possible if the tetrad field is considered to be a frame field built from a normalized spinor dyad , , on Σ which is asymptotically constant (see Section 4.2.3 below). (Thus underlined capital Roman indices are concrete name spinor indices.) Then, for the components of the ADM energy-momentum in the constant spinor basis at infinity, Møller’s expression yields the limit of

12

as the 2-surface is blown up to approach infinity. In fact, to recover the ADM energy-momentum in the form (12), the spinor fields need not be required to form a normalized spinor dyad, it is enough that they form an asymptotically constant normalized dyad, and we have to use the fact that the generator vector field K^a has asymptotically constant components in the asymptotically constant frame field . Thus can be interpreted as an asymptotic translation. The complex-valued 2-form in the integrand of Equation (12) will be denoted by , and is called the Nester-Witten 2-form. This is ‘essentially Hermitian’ and connected with Komar’s superpotential: For any two spinor fields α^A and β^A one has

13

14

where and overline denotes complex conjugation. Thus, apart from the terms in Equation (14) involving and , the Nester-Witten 2-form u(α, β)_ab is just , i.e. the anti-self-dual part of the curl of . (The original expressions by Witten and Nester were given using Dirac rather than two-component Weyl spinors [397, 279]. The 2-form in the present form using the two-component spinors appeared first probably in [205].) Although many interesting and original proofs of the positivity of the ADM energy are known even in the presence of black holes [328, 329, 397, 279, 202, 314, 224], the simplest and most transparent ones are probably those based on the use of 2-component spinors: If the dominant energy condition is satisfied on the k-asymptotically flat spacelike hypersurface Σ, where k > ½, then the ADM energy-momentum is future pointing and non-spacelike (i.e. the Lorentzian length of the energy-momentum vector, the ADM mass, is non-negative), and it is null if and only if the domain of dependence D(Σ) of Σ is flat [205, 320, 159, 321, 70]. Its proof may be based on the Sparling equation [345, 130, 313, 266]: . The significance of this equation is that in the exterior derivative of the Nester-Witten 2-form the second derivatives of the metric appear only through the Einstein tensor, thus its structure is similar to that of the superpotential equations in the Lagrangian field theory, and , known as the Sparling 3-form, is a quadratic expression of the first derivatives of the spinor fields. If the spinor fields λ_a and μ_a solve the Witten equation on a spacelike hypersurface Σ, then the pull-back of to Σ is positive definite. This theorem has been extended and refined in various ways, in particular by allowing inner boundaries of Σ that represent future marginally trapped surfaces in black holes [159, 202, 314, 200].

The ADM energy-momentum can also be written as the 2-sphere integral of certain parts of the conformally rescaled spacetime curvature [15, 16, 28]. This expression is a special case of the more general ‘Riemann tensor conserved quantities’ (see [163]): If is any closed spacelike 2-surface with area element , then for any tensor fields ω_ab = ω_[ab] and μ_ab = μ_[ab] one can form the integral

15

Since the fall-off of the curvature tensor near spatial infinity is r^−k−2, the integral at spatial infinity can give finite value precisely when ω^abμ^cd blows up like r^k as r ← ∞. In particular, for the 1/r fall-off this condition can be satisfied by , where Area is the area of and the hatted tensor fields are O(1)

If the spacetime is stationary, then the ADM energy can be recovered as the r → ∞ limit of the 2-sphere integral of Komar’s superpotential with the Killing vector K^a of stationarity [163], too. On the other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time translation, the Komar expression does not reproduce the ADM energy. However, by Equations (13, 14) such an additional restriction might be that K^a should be a constant combination of four future pointing null vector fields of the form , where the spinor fields α^A are required to satisfy the Weyl neutrino equation ∇_a′Aα^A = 0. This expression for the ADM energy-momentum was used to give an alternative, ‘4-dimensional’ proof of the positivity of the ADM energy [205].

Spatial infinity: Angular momentum

The value of the Hamiltonian of Beig and Ó Murchadha [47] together with the appropriately defined asymptotic rotation-boost Killing vectors [364] define the spatial angular momentum and centre-of-mass, provided k ≥ 1 and, in addition to the familiar fall-off conditions, certain global integral conditions are also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and Teitelboim [319] on the leading nontrivial parts of the metric h_ab and extrinsic curvature χ_ab: The components in the Cartesian coordinates {xⁱ} of the former must be even and the components of latter must be odd parity functions of xⁱ/r (see also [47]). Thus in what follows we assume that k = 1. Then the value of the Beig-Ó Murchadha Hamiltonian parameterized by the asymptotic rotation Killing vectors is the spatial angular momentum of Regge and Teitelboim [319], while that parameterized by the asymptotic boost Killing vectors deviate from the centre-of-mass of Beig and Ó Murchadha [47] by a term which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed out [47], the centre-of-mass of Regge and Teitelboim is not necessarily finite.) The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz 4-tensor, which transforms in the correct way under the 4-translation of the origin of the asymptotically Cartesian coordinate system, and it is conserved by the evolution equations [364].

The centre-of-mass of Beig and Ó Murchadha was reexpressed recently [42] as the r → ∞ limit of 2-surface integrals of the curvature in the form (15) with ω^abμ^cd proportional to the lapse N times q^acq^bd − q^adq^bc, where q_ab is the induced 2-metric on S (see Section 4.1.1 below)

A geometric notion of centre-of-mass was introduced by Huisken and Yau [209]. They foliate the asymptotically flat hypersurface Σ by certain spheres with constant mean curvature. By showing the global uniqueness of this foliation asymptotically, the origin of the leaves of this foliation in some flat ambient Euclidean space ℝ³ defines the centre-of-mass (or rather ‘centre-of-gravity’) of Huisken and Yau. However, no statement on its properties is proven. In particular, it would be interesting to see whether or not this notion of centre-of-mass coincides, for example, with that of Beig and Ó Murchadha.

The Ashtekar-Hansen definition for the angular momentum is introduced in their specific conformal model of the spatial infinity as a certain 2-surface integral near infinity. However, their angular momentum expression is finite and unambiguously defined only if the magnetic part of the spacetime curvature tensor (with respect to the Ω = const. timelike level hypersurfaces of the conformal factor) falls off faster than would follow from the 1/r fall-off of the metric (but they do not have to impose any global integral, e.g. a parity condition) [23, 15].

If the spacetime admits a Killing vector of axi-symmetry, then the usual interpretation of the corresponding Komar integral is the appropriate component of the angular momentum (see for example [387]). However, the value of the Komar integral is twice the expected angular momentum. In particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr solution the integral is m/G, for the Killing vector of axi-symmetry it is 2ma/G instead of the expected ma/G (‘factor-of-two anomaly’) [229]. We return to the discussion of the Komar integral in Section 12.1.

Null infinity: Energy-momentum

The study of the gravitational radiation of isolated sources led Bondi to the observation that the 2-sphere integral of a certain expansion coefficient m(u, θ, φ) of the line element of a radiative spacetime in an asymptotically retarded spherical coordinate system (u, r, Θ, φ) behaves as the energy of the system at the retarded time u: This notion of energy is not constant in time, but decreases with u, showing that gravitational radiation carries away positive energy (‘Bondi’s massloss’) [71, 72]. The set of transformations leaving the asymptotic form of the metric invariant was identified as a group, nowadays known as the BMS group, having a structure very similar to that of the Poincaré group [325]. The only difference is that while the Poincaré group is a semidirect product of the Lorentz group and a 4-dimensional commutative group (of translations), the BMS group is the semidirect product of the Lorentz group and an infinite-dimensional commutative group, called the group of the supertranslations. A 4-parameter subgroup in the latter can be identified in a natural way as the group of the translations. Just at the same time the study of asymptotic solutions of the field equations led Newman and Unti to another concept of energy at null infinity [290]. However, this energy (nowadays known as the Newman-Unti energy) does not seem to have the same significance as the Bondi (or Bondi-Sachs [313] or Trautman-Bondi [115, 116, 114]) energy, because its monotonicity can be proven only between special, e.g. stationary, states. The Bondi energy, which is the time component of a Lorentz vector, the so-called BondiSachs energy-momentum, has a remarkable uniqueness property [115, 116].

Without additional conditions on K^a, Komar’s expression does not reproduce the Bondi-Sachs energy-momentum in non-stationary spacetimes either [395, 163]: For the ‘obvious’ choice for K^a Komar’s expression yields the Newman-Unti energy. This anomalous behaviour in the radiative regime could be corrected in, at least, two ways. The first is by modifying the Komar integral according to

16

where ^⊥ε_cd is the area 2-form on the Lorentzian 2-planes orthogonal to (see Section 4.1.1) and α is some real constant. For α = 1 the integral , suggested by Winicour and Tamburino, is called the linkage [395]. In addition, to define physical quantities by linkages associated to a cut of the null infinity one should prescribe how the 2-surface tends to the cut and how the vector field K^a should be propagated from the spacetime to null infinity into a BMS generator [395, 394]. The other way is considering the original Komar integral (i.e. α = 0) on the cut of infinity in the conformally rescaled spacetime and by requiring that K^a be divergence-free [153]. For such asymptotic BMS translations both prescriptions give the correct expression for the Bondi-Sachs energy-momentum.

The Bondi-Sachs energy-momentum can also be expressed by the integral of the Nester-Witten 2-form [214, 255, 256, 205]. However, in non-stationary spacetimes the spinor fields that are asymptotically constant at null infinity are vanishing [83]. Thus the spinor fields in the Nester-Witten 2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves be the spinor constituents of the BMS translations. The first such condition, suggested by Bramson [83], was to require the spinor fields to be the solutions of the so-called asymptotic twistor equation (see Section 4.2.4). One can impose several such inequivalent conditions, and all these, based only on the linear first order differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are determined in [363].

The Bondi-Sachs energy-momentum has a Hamiltonian interpretation as well. Although the fields on a spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable generalization of the standard Hamiltonian analysis could be developed [114] and used to recover the Bondi-Sachs energy-momentum.

Similarly to the ADM case, the simplest proofs of the positivity of the Bondi energy [330] are probably those that are based on the Nester-Witten 2-form [214] and, in particular, the use of two-components spinors [255, 256, 205, 203, 321]: The Bondi-Sachs mass (i.e. the Lorentzian length of the Bondi-Sachs energy-momentum) of a cut of future null infinity is non-negative if there is a spacelike hypersurface Σ intersecting null infinity in the given cut such that the dominant energy condition is satisfied on Σ, and the mass is zero iff the domain of dependence D(Σ) of Σ is flat.

Null infinity: Angular momentum

At null infinity there is no generally accepted definition for angular momentum, and there are various, mathematically inequivalent suggestions for it. Here we review only some of those total angular momentum definitions that can be considered as the null infinity limit of some quasi-local expression, and will be discussed in the main part of the review, namely in Section 9.

In their classic paper Bergmann and Thomson [60] raise the idea that while the gravitational energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be connected with its intrinsic O(1, 3) symmetry. Thus, the angular momentum should be analogous with the spin. Based on the tetrad formalism of general relativity and following the prescription of constructing the Noether currents in Yang-Mills theories, Bramson suggested a superpotential for the six conserved currents corresponding to the internal Lorentz-symmetry [84, 85, 86]. (For another derivation of this superpotential from Møller’s Lagrangian (9) see [363].) , , is a normalized spinor dyad corresponding to the orthonormal frame in Equation (9), then the integral of the spinor form of the anti-self-dual part of this superpotential on a closed orientable 2-surface is

17

where ε_a′b′ is the symplectic metric on the bundle of primed spinors. We will denote its integrand by , and we call it the Bramson superpotential. To define angular momentum on a given cut of the null infinity by the formula (17) we should consider its limit when tends to the cut in question and we should specify the spinor dyad, at least asymptotically. Bramson’s suggestion for the spinor fields was to take the solutions of the asymptotic twistor equation [83]. He showed that this definition yields a well-defined expression, for stationary spacetimes this reduces to the generally accepted formula (34), and the corresponding Pauli-Lubanski spin, constructed from and the Bondi-Sachs energy-momentum (given for example in the Newman-Penrose formalism by Equation (33)), is invariant with respect to supertranslations of the cut (‘active supertranslations’). Note that since Bramson’s expression is based on the solutions of a system of partial differential equations on the cut in question, it is independent of the parameterization of the BMS vector fields. Hence, in particular, it is invariant with respect to the supertranslations of the origin cut (‘passive supertranslations’). Therefore, Bramson’s global angular momentum behaves like the spin part of the total angular momentum.

The construction based on the Winicour-Tamburino linkage (16) can be associated with any BMS vector field [395, 252, 30]. In the special case of translations it reproduces the Bondi-Sachs energy-momentum. The quantities that it defines for the proper supertranslations are called the super-momenta. For the boost-rotation vector fields they can be interpreted as angular momentum. However, in addition to the factor-of-two anomaly, this notion of angular momentum contains a huge ambiguity (‘supertranslation ambiguity’): The actual form of both the boost-rotation Killing vector fields of Minkowski spacetime and the boost-rotation BMS vector fields at future null infinity depend on the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively. However, while the set of the origins of Minkowski spacetime is parameterized by four numbers, the set of the origins at null infinity requires a smooth function of the form . Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar origin-dependence (containing four parameters), the analogous transformation of the angular momentum defined by using the boost-rotation BMS vector fields depends on an arbitrary smooth real valued function on the 2-sphere. This makes the angular momentum defined at null infinity by the boost-rotation BMS vector fields ambiguous unless a natural selection rule for the origins, making them form a four parameter family of cuts, is found. Such a selection rule could be the suggestion by Dain and Moreschi [125] in the charge integral approach to angular momentum of Moreschi [272, 273].

Another promising approach might be that of Chruściel, Jezierski, and Kijowski [114], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperbolic spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian 4-space of origins, they appear to be the generators with respect to some fixed ‘centre-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.

Non-locality of the gravitational energy-momentum and angular momentum

One reaction to the non-tensorial nature of the gravitational energy-momentum density expressions was to consider the whole problem ill-defined and the gravitational energy-momentum meaningless. However, the successes discussed in the previous Section 3.2.4 show that the global gravitational energy-momenta and angular momenta are useful notions, and hence it could also be useful to introduce them even if the spacetime is not asymptotically flat. Furthermore, the non-tensorial nature of an object does not imply that it is meaningless. For example, the Christoffel symbols are not tensorial, but they do have geometric, and hence physical content, namely the linear connection. Indeed, the connection is a non-local geometric object, connecting the fibres of the vector bundle over different points of the base manifold. Hence any expression of the connection coefficients, in particular the gravitational energy-momentum or angular momentum, must also be non-local. In fact, although the connection coefficients at a given point can be taken zero by an appropriate coordinate/gauge transformation, they cannot be transformed to zero on an open domain unless the connection is flat.

Furthermore, the superpotential of many of the classical pseudotensors (e.g. of the Einstein, Bergmann, Møller’s tetrad, Landau-Lifshitz pseudotensors), being linear in the connection coefficients, can be recovered as the pull-back to the spacetime manifold of various forms of a single geometric object on the linear frame bundle, namely of the Nester-Witten 2-form, along various local Sections [142, 266, 352, 353], and the expression of the pseudotensors by their superpotentials are the pull-backs of the Sparling equation [345, 130, 266]. In addition, Chang, Nester, and Chen [104] found a natural quasi-local Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see Section 11.3.4). Therefore, the pseudotensors appear to have been ‘rehabilitated’, and the gravitational energy-momentum and angular momentum are necessarily associated with extended subsets of the spacetime.

This fact is a particular consequence of a more general phenomenon [324, 213]: Since the physical spacetime is the isomorphism class of the pairs (M, g_ab) instead of a single such pair, it is meaningless to speak about the ‘value of a scalar or vector field at a point p ∈ M’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), 2-surfaces (e.g. the area of a closed 2-surface) etc. determined by some body or physical fields. Thus, if we want to associate energy-momentum and angular momentum not only to the whole (necessarily asymptotically flat) spacetime, then these quantities must be associated with extended but finite subsets of the spacetime, i.e. must be quasi-local.

Domains for quasi-local quantities

The quasi-local quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:

1. the globally hyperbolic domains D ⊂ M with compact closure,

2. the compact spacelike hypersurfaces Σ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and

3. the closed, orientable spacelike 2-surfaces (interpreted as the boundary ∂Σ of Cauchy surfaces for globally hyperbolic domains).

A typical example fo Type 3 is any charge integral expression: The quasi-local quantity is the integral of some superpotential 2-form built from the data given on the 2-surface, as in Equation (12), or the expression for the matter fields given by Equation (5). An example for Type 2 might be the integral of the Bel-Robinson ‘momentum’ on the hypersurface Σ:

18

This quantity is analogous to the integral E_Σ[ξ^a] for the matter fields given by Equation (6) (though, by the remarks on the Bel-Robinson ‘energy’ in Section 3.1.1, its physical dimension cannot be of energy). If ξ^a is a future pointing nonspacelike vector then E_Σ[ξ^a] ≥ 0. Obviously, if such a quantity were independent of the actual hypersurface Σ, then it could also be rewritten as a charge integral on the boundary ∂Σ. The gravitational Hamiltonian provides an interesting example for the mixture of Type 2 and 3 expressions, because the form of the Hamiltonian is the 3-surface integral of the constraints on Σ and a charge integral on its boundary ∂Σ, thus if the constraints are satisfied then the Hamiltonian reduces to a charge integral. Finally, an example for Type 1 might be

19

the infimum of the ‘quasi-local Bel-Robinson energies’, where the infimum is taken on the set of all the Cauchy surfaces Σ for D with given boundary ∂Σ. (The infimum always exists because the Bel-Robinson ‘energy density’ T_abcdt^at^bt^ct^d is non-negative.) Quasi-locality in any of these three senses agrees with the quasi-locality of Haag and Kastler [168, 169]. The specific quasi-local energy-momentum constructions provide further examples both for charge-integral-type expressions and those based on spacelike hypersurfaces.

Strategies to construct quasi-local quantities

There are two natural ways of finding the quasi-local energy-momentum and angular momentum. The first is to follow some systematic procedure, while the second is the ‘quasi-localization’ of the global energy-momentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasi-local quantities are integrals of some superpotential derived from the Lagrangian via a Noether-type analysis. The advantage of this approach could be its manifest Lorentz-covariance. On the other hand, since the Noether current is determined only through the Noether identity, which contains only the divergence of the current itself, the Noether current and its superpotential is not uniquely determined. In addition (as in any approach), a gauge reduction (for example in the form of a background metric or reference configuration) and a choice for the ‘translations’ and ‘boost-rotations’ should be made.

The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasi-local (covariant or not) Hamiltonian analysis we would have a Hamiltonian, and its value on the constraint surface in the phase space yields the expected quantities. Here the main idea is that of Regge and Teitelboim [319] that the Hamiltonian must reproduce the correct field equations as the flows of the Hamiltonian vector fields, and hence, in particular, the correct Hamiltonian must be functionally differentiable with respect to the canonical variables. This differentiability may restrict the possible ‘translations’ and ‘boost-rotations’ too. However, if we are not interested in the structure of the quasi-local phase space, then, as a short-cut, we can use the Hamilton-Jacobi method to define the quasi-local quantities. The resulting expression is a 2-surface integral. Nevertheless, just as in the Lagrangian approach, this general expression is not uniquely determined, because the action can be modified by adding an (almost freely chosen) boundary term to it. Furthermore, the ‘translations’ and ‘boost-rotations’ are still to be specified.

On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasi-local quantities would be some ‘quasi-localization’ of those expressions that gave the global energy-momentum and angular momentum of asymptotically flat spacetimes. Therefore, respecting both strategies, it is also legitimate to consider the Winicour-Tamburino-type (linkage) integrals and the charge integrals of the curvature.

Since the global energy-momentum and angular momentum of asymptotically flat spacetimes can be written as 2-surface integrals at infinity (and, as we will see in Section 7.1.1, both the energy-momentum and angular momentum of the source in the linear approximation and the gravitational mass in the Newtonian theory of gravity can also be written as 2-surface integrals), the 2-surface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasi-local energy-momentum and angular momentum as 2-surface observables, then three things must be specified:

1. an appropriate general 2-surface integral (e.g. the integral of a superpotential 2-form in the Lagrangian approaches or a boundary term in the Hamiltonian approaches),

2. a gauge choice (in the form of a distinguished coordinate system in the pseudotensorial approaches, or a background metric/connection in the background field approaches or a distinguished tetrad field in the tetrad approach), and

3. a definition for the ‘quasi-symmetries’ of the 2-surface (i.e. the ‘generator vector fields’ of the quasi-local quantities in the Lagrangian, and the lapse and the shift in the Hamiltonian approaches, respectively, which, in the case of timelike ‘generator vector fields’, can also be interpreted as a fleet of observers on the 2-surface).

In certain approaches the definition of the ‘quasi-symmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.

The Lorentzian vector bundle

The restriction to the closed, orientable spacelike 2-surface of the tangent bundle TM of the spacetime has a unique decomposition to the g_ab-orthogonal sum of the tangent bundle of and the bundle of the normals, denoted by . Then all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If t^a and v^a are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projection to and is and , respectively. The induced 2-metric and the corresponding area 2-form on will be denoted by q_ab = g_ab − t_at_b + v^av_b and ε_ab = t^cv^dε_cdab, respectively, while the area 2-form on the normal bundle will be ⊥ε_ab = t_av_b − t_bv_a. The bundle together with the fibre metric g_ab and the projection will be called the Lorentzian vector bundle over . For the discussion of the global topological properties of the closed orientable 2-manifolds, see for example [5].

Connections

The spacetime covariant derivative operator ∇_e defines two covariant derivatives on . The first, denoted by δ_e, is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: for any section X^a of . Obviously, δ_e annihilates both the fibre metric g_ab and the projection . However, since for 2-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ t^a ↦ t^a cosh u +v^a sinh u, v^a ↦ t^a sinh u + v^a cosh u. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection 1-form on can be characterized, for example, by . Therefore, the connection δ_e can be considered as a connection on coming from a connection on the O(2) ⊗ O(1,1)-principal bundle of the g_ab-orthonormal frames adapted to .

The other connection, Δ_e, is analogous to the Sen connection [331], and is defined simply by . This annihilates only the fibre metric, but not the projection. The difference of the connections Δ_e and δ_e turns out to be just the extrinsic curvature tensor: . Here , and and are the standard (symmetric) extrinsic curvatures corresponding to the individual normals t_a and v_a, respectively. The familiar expansion tensors of the future pointing outgoing and ingoing null normals, l^a := t^a + v^a and n^a := ½(t^a − v^a), respectively, are θ_ab = Q_abcl^c and and the corresponding shear tensors σ_ab and are defined by their trace-free part. Obviously, τ_ab and ν_ab (and hence the expansion and shear tensors θ_ab, , σ_ab, and ) are boost-gauge dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination is boost-gauge invariant. In particular, it defines a natural normal vector field to by where τ, ν, θ, θ′ are the relevant traces. Q_a is called the main extrinsic curvature vector of . If then the norm of Q_a and is , and they are orthogonal to each other: . It is easy to show that , i.e. is the uniquely pointwise determined direction orthogonal to the 2-surface in which the expansion of the surface is vanishing. If Q_a is not null, then defines an orthonormal frame in the normal bundle (see for example [8]). If Q_a is non-zero but (e.g. future pointing) null, then there is a uniquely determined null normal S_a to such that Q_aS^a = 1, and hence {Q_a, S_a} is a uniquely determined null frame. Therefore, the 2-surface admits a natural gauge choice in the normal bundle unless Q_a is vanishing. Geometrically, Δ_e is a connection coming from a connection on the O(1, 3)-principal fibre bundle of the g_ab-orthonormal frames. The curvature of the connections δ_e and Δ_e, respectively, are

20

21

where is the curvature scalar of the familiar intrinsic Levi-Civita connection of . The curvature of Δ_e is just the pull-back to of the spacetime curvature 2-form: . Therefore, the well known Gauss, Codazzi-Mainardi, and Ricci equations for the embedding of in M are just the various projections of Equation (21).

Convexity conditions

To prove certain statements on quasi-local quantities various forms of the convexity of must be assumed. The convexity of in a 3-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If the embedding space is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of . If is in a Lorentzian spacetime then the weakest convexity conditions are conditions only on the mean null curvatures: will be called weakly future convex if the outgoing null normals l^a are expanding on , i.e. , and weakly past convex if [380]. is called mean convex [182] if θθ′ < 0 on , or, equivalently, if is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions and . Note that although the expansion tensors, and in particular the functions θ, θ′, D, and D′ are gauge dependent, their sign is gauge invariant. Then will be called future convex if θ > 0 and D > 0, and past convex if θ′ < 0 and D′ > 0 [380, 358]. These are equivalent to the requirement that the two eigenvalues of be positive and those of be negative everywhere on , respectively. A different kind of convexity condition, based on global concepts, will be used in Section 6.1.3.

The spinor bundle

The connections δ_e and Δ_e determine connections on the pull-back to of the bundle of unprimed spinors. The natural decomposition defines a chirality on the spinor in the form of the spinor , which is analogous to the γ₅ matrix in the theory of Dirac spinors. Then the extrinsic curvature tensor above is a simple expression of and (and their complex conjugate), and the two covariant derivatives on are related to each other by The curvature of Δ_e can be expressed by the curvature of δ_e, the spinor , and its δ_e-derivative. We can form the scalar invariants of the curvatures according to

22

23

f is four times the so-called complex Gauss curvature [312] of , by means of which the whole curvature can be characterized: . If the spacetime is space and time orientable, at least on an open neighbourhood of , then the normals ta and v_a can be chosen to be globally well-defined, and hence is globally trivializable and the imaginary part of f is a total divergence of a globally well-defined vector field.

An interesting decomposition of the SO(1, 1) connection 1-form A_e, i.e. the vertical part of the connection δ_e, was given by Liu and Yau [253]: There are real functions α and γ, unique up to additive constants, such that . α is globally defined on , but in general γ is defined only on the local trivialization domains of that are homeomorphic to ℝ². It is globally defined if . In this decomposition α is the boost-gauge invariant part of A_e, while γ represents its gauge content. Since , the ‘Coulomb-gauge condition’ δ_eA_e = 0 uniquely fixes A_e (see also Section 10.4.1).

By the Gauss-Bonnet theorem , where g is the genus of . Thus geometrically the connection δ_e is rather poor, and can be considered as a part of the ‘universal structure of ’. On the other hand, the connection Δ_e is much richer, and, in particular, the invariant F carries information on the mass aspect of the gravitational ‘field’. The 2-surface data for charge-type quasi-local quantities (i.e. for 2-surface observables) are the universal structure (i.e. the intrinsic metric q_ab, the projection and the connection δ_e and the extrinsic curvature tensor .

Curvature identities

The complete decomposition of into its irreducible parts gives , the Dirac-Witten operator, and , the 2-surface twistor operator. A Sen-Witten-type identity for these irreducible parts can be derived. Taking its integral one has

24

where λ_A and μ_A are two arbitrary spinor fields on , and the right hand side is just the charge integral of the curvature on .

The GHP formalism

A GHP spin frame on the 2-surface is a normalized spinor basis A = 0, 1, such that the complex null vectors and are tangent to (or, equivalently, the future pointing null vectors and are orthogonal to ). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole . This fact is connected with the non-triviality of the tangent bundle of the 2-surface. For example, on the 2-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors m^a and cannot form a globally defined basis on . Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable 2-surface with globally trivial tangent bundle is the torus.

Fixing a GHP spin frame on some open , the components of the spinor and tensor fields on U will be local representatives of cross sections of appropriate complex line bundles E(p, q) of scalars of type (p, q) [152, 312]: A scalar φ is said to be of type (p, q) if under the rescaling of the GHP spin frame with some nowhere vanishing complex function λ: U → ℂ the scalar transforms as . For example , and are of type (1, 1), (−1, −1), (3, −1), and (−3, 1), respectively. The components of the Weyl and Ricci spinors, , etc., also have definite (p, q)-type. In particular, Λ := R/24 has type (0, 0). A global section of E(p, q) is a collection of local cross sections {(U, φ), (U′, φ′), …} such that {U, U′, …} forms a covering of and on the non-empty overlappings, e.g. on U ∩ U′ the local sections are related to each other by . where ψ: U ∩ U′ → ℂ is the transition function between the GHP spin frames: and .

The connection δ_e defines a connection ð_e on the line bundles E(p, q) [152, 312]. The usual edth operators, ð and ð′ are just the directional derivatives ð := m^að_a and on the domain of the GHP spin frame . These locally defined operators yield globally defined differential operators, denoted also by ð and ð′, on the global sections of E(p, q). It might be worth emphasizing that the GHP spin coefficients β and β′ which do not have definite (p, q)-type, play the role of the two components of the connection 1-form, and they are built both from the connection 1-form for the intrinsic Riemannian geometry of () and the connection 1-form A_e in the normal bundle. ð and ð′ are elliptic differential operators, thus their global properties, e.g. the dimension of their kernel, are connected with the global topology of the line bundle they act on, and, in particular, with the global topology of . These properties are discussed in [147] for general, and in [132, 43, 356] for spherical topology.

Irreducible parts of the derivative operators

Using the projection operators , the irreducible parts and can be decomposed further into their right handed and left handed parts. In the GHP formalism these chiral irreducible parts are

25

where λ := (λ₀, λ₁) and the spinor components are defined by . The various first order linear differential operators acting on spinor fields, e.g. the 2-surface twistor operator, the holomorphy/anti-holomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [83], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [43, 356, 363].

SO(1, 1)-connection 1-form versus anholonomicity

Obviously, all the structures we have considered can be introduced on the individual surfaces of one- or two-parameter families of surfaces, too. In particular [181], let the 2-surface be considered as the intersection of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to , and let the spacetime (or at least a neighbourhood of ) be foliated by two one-parameter families of smooth hypersurfaces {ν₊ = const.} and {ν₋ = const.}, where ν_± : M → ℝ, such that and . One can form the two normals, , which are null on and , respectively. Then we can define , for which β_+e + β_−e = Δ_en², where . (If n² is chosen to be 1 on , then β_−e + β_+e is precisely the SO(1, 1) connection 1-form A_e above.) Then the so-called anholonomicity is defined by Since ωe is invariant with respect to the rescalings ν₊ ↦ exp(A)ν₊ and ν₋ ↦exp(B)ν₋ of the functions defining the foliations by those functions A, B : M → ℝ which preserve , it was claimed in [181] that ωe depends only on . However, this implies only that ωe is invariant with respect to a restricted class of the change of the foliations, and that ωe is invariantly defined only by this class of the foliations rather than the 2-surface. In fact, ωe does depend on the foliation: Starting with a different foliation defined by the functions and for some α, β : M → ℝ the corresponding anholonomicity would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, ωe and , would be different: . Therefore, the anholonomicity is still a gauge dependent quantity.

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 (t, r, θ, φ) the spacetime metric takes the form g_ab = diag(exp(2γ), − exp(2α), −r², −r² sin² θ, where γ and α are functions of t and r. (Hence r 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 [268, 199], Cahill and McVitte [98] found

26

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 GE(t, r) = m − e²/2r, while for the isentropic fluid solutions , where m and e are the usual parameters of the Reissner-Nordström solutions and μ is the energy density of the fluid [268, 199] (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 ∂_tE(t, r) and ∂_rE(t, r), and if the energy-momentum tensor satisfies the dominant energy condition then ∂_rE(t, r) > 0. Thus E(t, r) is a monotonic function of r provided r is the area-coordinate. Since by the spherical symmetry all the quantities with non-zero spin weight, in particular the shears σ and σ′, are vanishing and ψ₂ is real, by the GHP form of Equations (22, 23) the energy function E(t, r) can also be written as

27

This expression is considered to be the ‘standard’ definition of the energy for round spheres4. Its last expression does not depend on whether r 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 μ 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 r, even if μ ≥ 0. For example, in the closed Friedmann-Robertson-Walker spacetime (where, to cover the whole space, r cannot be chosen to be the area-radius and is increasing for r ∈ [0, π/2), taking its maximal value at r = π/2, and decreasing for r ∈ (π/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 , where is the line element for the flat 3-space and with some positive constants B and T², and the range of the Euclidean radial coordinate r is [0, r₀], where r₀ ∈ (2T, ∞). It contains a maximal 2-surface at r = 2T with round-sphere mass parameter The scalar curvature is R = 6/BT², 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 [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’ M/G and arbitrarily small ADM mass m/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 3-plane — like the capital Greek letter Ω — for later reference we call it an ‘Ω_M,_m-spacetime’.

Spherically symmetric spacetimes admit a special vector field, the so-called Kodama vector field K^a, such that K_aG^ab is divergence free [241]. In asymptotically flat spacetimes K^a 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 ∇_a(G^abK_b) = 0 the vector field S^a := G^abK_b has a conserved flux on a spacelike hypersurface Σ. In particular, in the coordinate system (t, r, θ, φ) and line element above K^a = exp[−(α + γ)](∂/∂t)^a. If Σ is the solid ball of radius r, then the flux of S^a is precisely the standard round sphere expression (26) for the 2-sphere ∂Σ [278].

An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be given in terms of the energy function E(t, r) above (see for example [408, 262, 185]). 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 E(t, r).

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 p, 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 p ∈ M be a point, and t^a a future directed unit timelike vector at p. Let , the ‘future null cone of p in M’ (i.e. the boundary of the chronological future of p). Let l^a be the future pointing null tangent to the null geodesic generators of such that, at the vertex p, l^at_a = 1. 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 parameterization completely. Then, in an open neighbourhood of the vertex is a smooth null hypersurface, and hence for sufficiently small r the set is a smooth spacelike 2-surface and homeomorphic to S². is called a small sphere of radius r with vertex p. Note that the condition l^at_a = 1 fixes the boost gauge.

Completing l^a to a Newman-Penrose complex null tetrad such that the complex null vectors m^a 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 r5. Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the metric q_ab on S_r, 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 for the small sphere can be expressed as a series of r,

where the expansion coefficients Q^(k) are still functions of the coordinates, or (θ, φ), on the unit sphere . 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’ t^a. 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 p or not6. These solutions of the GHP equations, with increasing accuracy, are given in [204, 235, 94, 360].

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

28

29

where , is the ‘Cartesian spin frame’ at p and the origin of the Cartesian coordinate system is chosen to be the vertex p. 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 p. Thus and are the 3-volume times the energy-momentum and angular momentum density with respect to p, respectively, that the observer with 4-velocity t^a 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 is any coordinate-independent quasi-local quantity, built from the first derivatives of the metric, i.e. , then its expansion is

where [A, B, …] is a scalar. It depends linearly on the tensors constructed from g_αβ and linearly from the coordinate dependent quantities A, B, …, and it is a polynomial expression of t^a, g_ab and ε_abcd at the vertex p. Since there is no non-trivial tensor built from the first derivative ∂_μg_αβ and g_αβ, the leading term is of order r³. Its coefficient [∂²g, (∂g)²] must be a linear expression of R_ab and C_abcd, and polynomial in t^a, g_ab and ε_abcd. In particular, if is to represent energy-momentum with generator K^c at p, then the leading term must be

30

for some unspecified constants a, b, and c, where , the projection to the subspace orthogonal to t^a. 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 t^a) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer t^a in the same way. Therefore, a = c, whenever the first and the third terms can in fact be written as r³at^aG_abK^b. Then, comparing Equation (30) with Equation (28), we see that a = −1/(6G), and then the term r³b Rt_aK^a 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 t^a, 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 c = a unless b = 0. Thus, in the leading r³ 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 t^a times the Euclidean volume of the 3-ball of radius r.

If a neighbourhood of p is vacuum, then the r³ order term is vanishing, and the fourth order term must be built from ∇_eC_abcd. However, the only scalar polynomial expression of t^a, g_ab, ε_abcd, ∇_eC_abcd and the generator vector K^a, depending on the latter two linearly, is the zero. Thus the r⁴ 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

31

for some constants a, b, c, and d, where E_ab := C_aebft^et^f is the electric and is the magnetic part of the Weyl curvature, and ε_abc := ε_abcdt^d is the induced volume 3-form. However, using the identities C_abcdC^abcd = 8(E_abE^ab − H_abH^ab), C_abcd * C^abcd = 16E_abH^ab4T_abcdt^at^bt^ct^d and , we can rewrite the above formula to

32

Again, if does not depend on t^a intrinsically, then d = (a + b), whenever the first and the fourth terms together can be written into the Lorentz covariant form 2r⁵ dT_abcdt^at^bt^cK^d. In a general expression the curvature invariants C_abcdC^abcd and C_abcd * C^abcd may be present. Since, however, E_ab and H_ab at a given point are independent, these invariants can be arbitrarily large positive or negative, and hence for a ≠ b or c ≠ 0 the quasi-local energy-momentum could not be future pointing and non-spacelike. Therefore, in vacuum in the leading r⁵ 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’ T_abcdt^at^bt^c

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 r⁴ and r⁶ order in non-vacuum and vacuum, respectively, while the first non-trivial correction to the area 4πr² is of order r⁴ and r⁶ in non-vacuum and vacuum, respectively.

On the small geodesic sphere of radius r in the given spacelike hypersurface Σ one can introduce the complex null tangents m^a and above, and if t^a is the future pointing unit normal of Σ and v^a the outward directed unit normal of in Σ, then we can define l^a := t^a + v^a and 2n^a := t^a − v^a. 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 p.

The small ellipsoids are defined as follows [235]. If f is any smooth function on Σ with a non-degenerate minimum at p ∈ Σ with minimum value f(p) = 0, then, at least on an open neighbourhood U of p in Σ the level surfaces are smooth compact 2-surfaces homeomorphic to . Then, in the r → 0 limit, the surfaces look like small nested ellipsoids centred in p. The function f is usually ‘normalized’ so that h_abD_aD_bf∣_p = − 3.

Large spheres near the spatial infinity

Near spatial infinity we have the a priori 1/r and 1/r² fall-off for the 3-metric h_ab and extrinsic curvature χ_ab, respectively, and both the evolution equations of general relativity and the conservation equation T^ab_;b = 0 for the matter fields preserve these conditions. The spheres 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 legitimate7. Introducing some coordinate system, e.g. the complex stereographic coordinates, on one sphere and then extending that to the whole Σ along the normals v^a of the spheres, we obtain a coordinate system on Σ. Let , A = 0, 1, be a GHP spinor dyad on Σ adapted to the large spheres in such a way that and are tangent to the spheres and , the future directed unit normal of Σ. These conditions fix the spinor dyad completely, and, in particular, , the outward directed unit normal to the spheres tangent to Σ.

The fall-off conditions yield that the spin coefficients tend to their flat spacetime value like 1/r² and the curvature components to zero like 1/r³. Expanding the spin coefficients and curvature components as power series of 1/r, one can solve the field equations asymptotically (see [48, 44] 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 ₀h_ab and the corresponding derivative operator ₀D_e we can define a spinor field ₀λ_A to be constant if ₀D_e0λ_A = 0. 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 λ_B that satisfy , where D_e 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 λ_A are .

Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [300, 301, 302, 313] (see also [151]), i.e. the physical spacetime can be conformally compactified by an appropriate boundary ℐ⁺. Then future null infinity ℐ⁺ 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 ℐ⁺ is the metric of the unit sphere. Fixing such a slice (called ‘the origin cut of ℐ⁺’) the points of ℐ⁺ can be labeled by a null coordinate, namely the affine parameter u ∈ ℝ along the null geodesic generators of ℐ⁺ 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 ℐ⁺. Then any other cut S of ℐ⁺ 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 ℐ⁺ in the spacetime in the following way. Let be the family of smooth outgoing null hypersurfaces in a neighbourhood of ℐ⁺ such that they intersect the null infinity just in the cuts , i.e. . Then let r be the affine parameter in the physical metric along the null geodesic generators of . Then (u, r, ζ, ) forms a coordinate system. The u = const., r = const. 2-surfaces (or simply if no confusion can arise) are spacelike topological 2-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: ) 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’8. 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 [290, 289, 84].

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 o^A such that be the tangent (∂/∂r)^a of the null geodesic generators of , and o^A itself be constant along l^a. Newman and Unti [290] chose ι^A to be parallel propagated along l^a. 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 1/r in this coordinate and tetrad system is given in [290, 134, 312], where all the non-vanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the u-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’ r, 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 o^A. 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 lim_r→∞ rΔ⁺λ = 0, lim_r→∞ rΔ⁻λ = 0, 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 ℐ⁺ (or on ℐ^-, 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 ℐ⁺ 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 SU(2) Pauli matrices (divided by [363]. Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from Δ⁺, Δ⁻, , 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 ψ₂, the asymptotic shear σ⁰ and its u-derivative, weighted by the first four spherical harmonics (see for example [289, 313]):

33

where , are the o^A-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:

34

In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [86].

Other special situations

In the weak field approximation of general relativity [382, 22, 387, 313, 227] the gravitational field is described by a symmetric tensor field h_ab on Minkowski spacetime (ℝ⁴, ), 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 h_ab is essentially the second order term in the Einstein tensor of the metric . Thus in the linear approximation the field h_ab 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 [349, 206, 313].

pp-waves spacetimes are defined to be those that admit a constant null vector field L^a, 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 L^a, i.e. T^abL_b = 0 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 K^a with closed S¹ orbits, i.e. it is cyclically symmetric too, then L^a and K^a 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.

General expectations

In non-gravitational physics the notions of conserved quantities are connected with symmetries of the system, and they are introduced through some systematic procedure in the Lagrangian and/or Hamiltonian formalism. In general relativity the total energy-momentum and angular momentum are 2-surface observables, thus we concentrate on them even at the quasi-local level. These facts motivate our three a priori expectations:

1. The quasi-local quantities that are 2-surface observables should depend only on the 2-surface data, but they cannot depend e.g. on the way that the various geometric structures on are extended off the 2-surface. There seems to be no a priori reason why the 2-surface would have to be restricted to have spherical topology. Thus, in the ideal case, the general construction of the quasi-local energy-momentum and angular momentum should work for any closed orientable spacelike 2-surface.

2. It is desirable to derive the quasi-local energy-momentum and angular momentum as the charge integral (Lagrangian interpretation) and/or as the value of the Hamiltonian on the constraint surface in the phase space (Hamiltonian interpretation). If they are introduced in some other way, they should have a Lagrangian and/or Hamiltonian interpretation.

3. These quantities should correspond to the ‘quasi-symmetries’ of the 2-surface. In particular, the quasi-local energy-momentum should be expected to be in the dual of the space of the ‘quasi-translations’, and the angular momentum in the dual of the space of the ‘quasi-rotations’.

To see that these conditions are non-trivial, let us consider the expressions based on the linkage integral (16). does not satisfy the first part of Requirement 1. In fact, it depends on the derivative of the normal components of K^a in the direction orthogonal to for any value of the parameter α. Thus it depends not only on the geometry of and the vector field K^a given on the 2-surface, but on the way in which K^a is extended off the 2-surface. Therefore, is ‘less quasi-local’ than or introduced in Sections 7.2.1 and 7.2.2, respectively.

We will see that the Hawking energy satisfies Requirement 1, but not Requirements 2 and 3. The Komar integral (i.e. the linkage for α = 0) has the form of the charge integral of a superpotential: i.e. it has a Lagrangian interpretation. The corresponding conserved Komar-current is defined by . However, its flux integral on some compact spacelike hypersurface with boundary cannot be a Hamiltonian on the ADM phase space in general. In fact, it is

35

Here c and c_a are, respectively, the Hamiltonian and momentum constraints of the vacuum theory, t^a is the future directed unit normal to Σ, v^a is the outward directed unit normal to in Σ, and N and N^a are the lapse and shift part of K^a, respectively, defined by K^a =: Nt^a + N^a. Thus _KH[K] is a well-defined function of the configuration and velocity variables (N, N^a, h_ab) and (Ṅ, Ṅ^a, ḣ_ab), respectively. However, since the velocity Ṅ^a cannot be expressed by the canonical variables [396, 46], _KH[K] can be written as a function on the ADM phase space only if the boundary conditions at ∂Σ ensure the vanishing of the integral of v_aṄ^a/N.

Pragmatic criteria

Since in certain special situations there are generally accepted definitions for the energy-momentum and angular momentum, it seems reasonable to expect that in these situations the quasi-local quantities reduce to them. One half of the pragmatic criteria is just this expectation, and the other is a list of some a priori requirements on the behaviour of the quasi-local quantities.

One such list for the energy-momentum and mass, based mostly on [131, 111] and the properties of the quasi-local energy-momentum of the matter fields of Section 2.2, might be the following:

• 1.1

  The quasi-local energy-momentum must be a future pointing nonspacelike vector (assuming that the matter fields satisfy the dominant energy condition on some Σ for which , and maybe some form of the convexity of should be required) (‘positivity’).

• 1.2

  must be zero iff D(Σ) is flat, and null iff D(Σ) has a pp-wave geometry with pure radiation (‘rigidity’).

• 1.3

  must give the correct weak field limit.

• 1.4

  must reproduce the ADM, Bondi-Sachs and Abbott-Deser energy-momenta in the appropriate limits (‘correct large sphere behaviour’).

• 1.5

  For small spheres must give the expected results (‘correct small sphere behaviour’):

  1. in non-vacuum and

  2. kr⁵T^abcdt^bt^ct^d in vacuum for some positive constant k and the Bel-Robinson tensor T^abcd.

• 1.6

  For round spheres must yield the ‘standard’ round sphere expression.

• 1.7

  For marginally trapped surfaces the quasi-local mass must be the irreducible mass .

Item 1.7 is motivated by the expectation that the quasi-local mass associated with the apparent horizon of a black hole (i.e. the outermost marginally trapped surface in a spacelike slice) be just the irreducible mass [131, 111]. Usually, is expected to be monotonic in some appropriate sense [111]. For example, if for some achronal (and hence spacelike or null) hypersurface Σ in which is a spacelike closed 2-surface and the dominant energy condition is satisfied on Σ, then seems to be a reasonable expectation [131]. (But see also the next Section 4.3.3.) On the other hand, in contrast to the energy-momentum and angular momentum of the matter fields on the Minkowski spacetime, the additivity of the energy-momentum (and angular momentum) is not expected. In fact, if and are two connected 2-surfaces, then, for example, the corresponding quasi-local energy-momenta would belong to different vector spaces, namely to the dual of the space of the quasi-translations of the first and of the second 2-surface, respectively. Thus, even if we consider the disjoint union to surround a single physical system, then we can add the energy-momentum of the first to that of the second only if there is some physically/geometrically distinguished rule defining an isomorphism between the different vector spaces of the quasi-translations. Such an isomorphism would be provided for example by some naturally chosen globally defined flat background. However, as we discussed in Section 3.1.1, general relativity itself does not provide any background: The use of such a background contradicts the complete diffeomorphism invariance of the theory. Nevertheless, the quasi-local mass and the length of the quasi-local Pauli-Lubanski spin of different surfaces can be compared, because they are scalar quantities.

Similarly, any reasonable quasi-local angular momentum expression may be expected to satisfy the following:

• 2.1

  must give zero for round spheres.

• 2.2

  For 2-surfaces with zero quasi-local mass the Pauli-Lubanski spin should be proportional to the (null) energy-momentum 4-vector .

• 2.3

  must give the correct weak field limit.

• 2.4

  must reproduce the generally accepted spatial angular momentum at the spatial infinity, and in stationary spacetimes it should reduce to the ‘standard’ expression at the null infinity as well (‘correct large sphere behaviour’).

• 2.5

  For small spheres the anti-self-dual part of , defined with respect to the centre of the small sphere (the ‘vertex’ in Section 4.2.2) is expected to give in non-vacuum and in vacuum for some constant C (‘correct small sphere behaviour’).

Since there is no generally accepted definition for the angular momentum at null infinity, we cannot expect anything definite there in non-stationary spacetimes. Similarly, there are inequivalent suggestions for the centre-of-mass at the spatial infinity (see Sections 3.2.2 and 3.2.4).

Incompatibility of certain ‘natural’ expectations

As Eardley noted in [131], probably no quasi-local energy definition exists which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasi-local energy definition which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed Friedmann-Robertson-Walker or the Ω_M,m spacetimes show explicitly. The points where the monotonicity breaks down are the extremal (maximal or minimal) surfaces, which represent event horizon in the spacetime. Thus one may argue that since the event horizon hides a portion of spacetime, we cannot know the details of the physical state of the matter + gravity system behind the horizon. Hence, in particular, the monotonicity of the quasi-local mass may be expected to break down at the event horizon. However, although for stationary systems (or at the moment of time symmetry of a time-symmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general non-stationary systems the concepts of the event and apparent horizons deviate. Thus the causal argument above does not seem possible to be formulated in the hypersurface Σ of Section 4.3.2. Actually, the root of the non-monotonicity is the fact that the quasi-local energy is a 2-surface observable in the sense of Expectation 1 in Section 4.3.1 above. This does not mean, of course, that in certain restricted situations the monotonicity (‘local monotonicity’) could not be proven. This local monotonicity may be based, for example, on Lie dragging of the 2-surface along some special spacetime vector field.

On the other hand, in the literature sometimes the positivity and the monotonicity requirements are confused, and there is an ‘argument’ that the quasi-local gravitational energy cannot be positive definite, because the total energy of the closed universes must be zero. However, this argument is based on the implicit assumption that the quasi-local energy is associated with a compact three dimensional dornain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasi-local energy-momentum is associated with 2-surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression (26) in the closed Friedmann-Robertson-Walker spacetime, or, more generally, the Dougan-Mason energy-momentum (see Section 8.2.3) are such examples.

The main idea

One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [39, 38]. His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let Σ be a compact, connected 3-manifold with connected boundary , and let h_ab be a (negative definite) metric and χ_ab a symmetric tensor field on Σ such that they, as an initial data set, satisfy the dominant energy condition: If 16πG_μ := R + χ² − χ_abχ^ab and 8πGj^a := D_b(χ^ab − χh^ab), then μ ≥ (− j_aj^a)^1/2. For the sake of simplicity we denote the triple (Σ, h_ab, χ_ab) by Σ. Then let us consider all the possible asymptotically flat initial data sets with a single asymptotic end, denoted simply by , which satisfy the dominant energy condition, have finite ADM energy and are extensions of Σ above through its boundary . The set of these extensions will be denoted by . By the positive energy theorem has non-negative ADM energy , which is zero precisely when is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies, , where the infimum is taken on . Obviously, by the non-negativity of the ADM energies this infimum exists and is non-negative, and it is tempting to define the quasi-local mass of Σ by this infimum9. However, it is easy to see that, without further conditions on the extensions of (Σ, h_ab, χ_ab), this infimum is zero. In fact, Σ can be extended to an asymptotically flat initial data set with arbitrarily small ADM energy such that contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and Σ. In particular, in the ‘Ω_M,m-spacetime’, discussed in Section 4.2.1 on the round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily large round-sphere mass M/G) has an asymptotically flat extension, the Ω_M,m-spacetime itself, with arbitrarily small ADM mass m/G.

Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding Σ from the outside. This led Bartnik [39, 38] to formulate his suggestion for the quasi-local mass of Σ. He concentrated on the time-symmetric data sets (i.e. those for which the extrinsic curvature χ_ab is vanishing), when the horizon appears to be a minimal surface of topology S² in (see for example [156]), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature: R ≥ 0. Thus, if denotes the set of asymptotically flat Riemannian geometries with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

36

The ‘no-horizon’ condition on implies that topologically Σ is a 3-ball. Furthermore, the definition of in its present form does not allow one to associate the Bartnik mass to those 3-geometries (Σ, h_ab) that contain minimal surfaces inside Σ. Although formally the maximal 2-surfaces inside Σ are not excluded, any asymptotically flat extension of such a Σ would contain a minimal surface. In particular, the spherically symmetric 3-geometry with line element dl² = − dr² − sin² r(dθ² + sin² θdϕ²) with (θ, ϕ) ∈ S² and r ∈ [0, r₀], π/2 < r₀ < π, has a maximal 2-surface at r = π/2, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than 4π sin² r₀. Thus the Bartnik mass (according to the original definition given in [39, 38]) cannot be associated with every compact time-symmetric data set (Σ, h_ab) even if Σ is topologically trivial. Since for 0 < r₀ < π/2 this data set can be extended without any difficulty, this example shows that m_B is associated with the 3-dimensional data set Σ and not only to the 2-dimensional boundary ∂Σ.

Of course, to rule out this limitation, one can modify the original definition by considering the set of asymptotically flat Riemannian geometries (with non-negative scalar curvature, finite ADM energy and with no stable minimal surface) which contain as an isometrically embedded Riemannian submanifold, and define by Equation (36) with instead of ɛ₀(Σ). Obviously, this could be associated with a larger class of 2-surfaces than the original m_B(Σ) to compact 3-manifolds, and holds.

In [208, 41] the set was allowed to include extensions of Σ having boundaries as compact outermost horizons, whenever the corresponding ADM energies are still non-negative [159], and hence m_B(Σ) is still well-defined and non-negative. (For another definition for allowing horizons in the extensions but excluding them between Σ and the asymptotic end, see [87] and Section 5.2 below.)

Bartnik suggested a definition for the quasi-local mass of a spacelike 2-surface (together with its induced metric and the two extrinsic curvatures), too [39]. He considered those globally hyperbolic spacetimes that satisfy the dominant energy condition, admit an asymptotically flat (metrically complete) Cauchy surface with finite ADM energy, have no event horizon and in which can be embedded with its first and second fundamental forms. Let denote the set of these spacetimes. Since the ADM energy is non-negative for any (and is zero precisely for flat ), the infimum

37

exists and is non-negative. Although it seems plausible that m_B(∂Σ) is only the ‘spacetime version’ of m_B(Σ), without the precise form of the no-horizon conditions in and that in they cannot be compared even if the extrinsic curvature were allowed in the extensions of Σ.

The main properties of m_B(Σ)

The first immediate consequence of Equation (36) is the monotonicity of the Bartnik mass: If Σ₁ ⊂ Σ₂, then , and hence m_B(Σ₁) ≤ m_B(Σ₂). Obviously, by definition (36) one has for any . Thus if m is any quasi-local mass functional which is larger than m_B (i.e. which assigns a non-negative real to any Σ such that m(Σ) ≥ m_B(Σ) for any allowed Σ), furthermore if for any , then by the definition of the infimum in Equation (36) one has m_B(Σ) ≥ m(Σ) − ε ≥ m_B(Σ) − ɛ for any ɛ > 0. Therefore, m_B is the largest mass functional satisfying for any . Another interesting consequence of the definition of m_B, due to W. Simon, is that if is any asymptotically flat, time symmetric extension of Σ with non-negative scalar curvature satisfying , then there is a black hole in in the form of a minimal surface between Σ and the infinity of (see for example [41]).

As we saw, the Bartnik mass is non-negative, and, obviously, if Σ is flat (and hence is a data set for the flat spacetime), then m_B(Σ) = 0. The converse of this statement is also true [208]: If m_B(Σ) = 0, then Σ is locally flat. The Bartnik mass tends to the ADM mass [208]: If is an asymptotically flat Riemannian 3-geometry with non-negative scalar curvature and finite ADM mass , and if {Σ_n}, n ∈ ℕ, is a sequence of solid balls of coordinate radius n in , then . The proof of these two results is based on the use of the Hawking energy (see Section 6.1), by means of which a positive lower bound for m_B(Σ) can be given near the non-flat points of Σ. In the proof of the second statement one must use the fact that the Hawking energy tends to the ADM energy, which, in the time-symmetric case, is just the ADM mass.

The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice application of the Riemannian Penrose inequality [208]: Let Σ be a spherically symmetric Riemannian 3-geometry with spherically symmetric boundary . One can form its ‘standard’ round-sphere energy (see Section 4.2.1), and take its spherically symmetric asymptotically flat vacuum extension (see [39, 41]). By the Birkhoff theorem the exterior part of is a part of a t = const. hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just . Then any asymptotically flat extension of Σ can also be considered as (a part of) an asymptotically flat time-symmetric hypersurface with minimal surface, whose area is . Thus by the Riemannian Penrose inequality [208] . Therefore, the Bartnik mass of Σ is just the ‘standard’ round sphere expression .

The computability of the Bartnik mass

Since for any given Σ the set of its extensions is a huge set, it is almost hopeless to parameterize it. Thus, by the very definition, it seems very difficult to compute the Bartnik mass for a given, specific (Σ, h_ab). Without some computational method the potentially useful properties of m_B(Σ) would be lost from the working relativist’s arsenal.

Such a computational method might be based on a conjecture of Bartnik [39, 41]: The infimum in definition (36) of the mass m_B(Σ) is realized by an extension of (Σ, h_ab) such that the exterior region, , is static, the metric is Lipschitz-continuous across the 2-surface , and the mean curvatures of ∂Σ of the two sides are equal. Therefore, to compute m_B for a given (Σ, h_ab), one should find an asymptotically flat, static vacuum metric ĥ_ab satisfying the matching conditions on ∂Σ, and the Bartnik mass is the ADM mass of ĥ_ab. As Corvino showed [119], if there is an allowed extension of Σ for which , then the extension is static; furthermore, if Σ₁ ⊂ Σ₂, m_B(Σ₁) = m_B(Σ₂) and Σ₂ has an allowed extension for which , then is static. Thus the proof of Bartnik’s conjecture is equivalent to the proof of the existence of such an allowed extension. The existence of such an extension is proven in [267] for geometries (Σ, h_ab) close enough to the Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still lacking. Bartnik’s conjecture is that (Σ, h_ab) determines this exterior metric uniquely [41]. He conjectures [39, 41] that a similar computation method can be found for the mass , defined in Equation (37), too, where the exterior metric should be stationary. This second conjecture is also supported by partial results [120]: If (Σ, h_ab, χ_ab) is any compact vacuum data set, then it has an asymptotically flat vacuum extension which is a spacelike slice of a Kerr spacetime outside a large sphere near spatial infinity.

To estimate m_B(Σ) one can construct admissible extensions of (Σ, h_ab) in the form of the metrics in quasi-spherical form [40]. If the boundary ∂Σ is a metric sphere of radius r with non-negative mean curvature k, then m_B(Σ) can be estimated from above in terms of r and k.

The definition

Studying the perturbation of the dust-filled k = −1 Friedmann-Robertson-Walker spacetimes, Hawking found that

38

behaves as an appropriate notion of energy surrounded by the spacelike topological 2-sphere [171]. Here we used the Gauss-Bonnet theorem and the GHP form of Equations (22, 23) for F to express ρρ′ by the curvature components and the shears. Thus the Hawking energy is genuinely quasi-local.

The Hawking energy has the following clear physical interpretation even in a general spacetime, and, in fact, E_H can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a spacelike 2-sphere should be the measure of bending of the ingoing and outgoing light rays orthogonal to , and recalling that under a boost gauge transformation l^a ↦ αl^a, n^a ↦ α⁻¹n^a the convergences ρ and ρ′ transform as ρ ↦ αρ and ρ′ ↦ α⁻¹ ρ′, respectively, the energy must have the form , where the unspecified parameters C and D can be determined in some special situations. For metric 2-spheres of radius r in the Minkowski spacetime, for which ρ = −1/r and ρ′ = 1/2r, we expect zero energy, thus D = C/2(π). For the event horizon of a Schwarzschild black hole with mass parameter m, for which ρ = 0 = ρ′, we expect m/G, which can be expressed by the area of . Thus , and hence we arrive at Equation (38).

The Hawking energy for spheres

Obviously, for round spheres E_H reduces to the standard expression (26). This implies, in particular, that the Hawking energy is not monotonic in general. Since for a Killing horizon (e.g. for a stationary event horizon) ρ = 0, the Hawking energy of its spacelike spherical cross sections is . In particular, for the event horizon of a Kerr-Newman black hole it is just the familiar irreducible mass .

For a small sphere of radius r with centre p ∈ M in non-vacuum spacetimes it is while in vacuum it is , where T_ab is the energy-momentum tensor and T_abcd is the Bel-Robinson tensor at p [204]. The first result shows that in the lowest order the gravitational ‘field’ does not have a contribution to the Hawking energy, that is due exclusively to the matter fields. Thus in vacuum the leading order of E_H must be higher than r³. Then even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the r^k order term in the power series expansion of E_H is (k−1). However, there are no tensorial quantities built from the metric and its derivatives such that the total number of the derivatives involved would be three. Therefore, in vacuum, the leading term is necessarily of order r⁵, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres E_H is positive definite both in non-vacuum (provided the matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in particular, that E_H should be interpreted as energy rather than as mass: For small spheres in a pp-wave spacetime E_H is positive, while, as we saw this for the matter fields in Section 2.2.3, a mass expression could be expected to be zero. (We will see in Sections 8.2.3 and 13.5 that, for the Dougan-Mason energy-momentum, the vanishing of the mass characterizes the pp-wave metrics completely.)

Using the second expression in Equation (38) it is easy to see that at future null infinity E_H tends to the Bondi-Sachs energy. A detailed discussion of the asymptotic properties of E_H near null infinity, both for radiative and stationary spacetimes is given in [338, 340]. Similarly, calculating E_H for large spheres near spatial infinity in an asymptotically flat spacelike hypersurface, one can show that it tends to the ADM energy.

Positivity and monotonicity properties

In general the Hawking energy may be negative, even in the Minkowski spacetime. Geometrically this should be clear, since for an appropriately general (e.g. concave) 2-surface the integral could be less than −2π. Indeed, in flat spacetime E_H is proportional to by the Gauss equation. For topologically spherical 2-surfaces in the t = const. spacelike hyperplane of Minkowski spacetime σσ′ is real and non-positive, and it is zero precisely for metric spheres, while for 2-surfaces in the r = const. timelike cylinder σσ′ is real and non-negative, and it is zero precisely for metric spheres10. If, however, is ‘round enough’ (not to be confused with the round spheres in Section 4.2.1), which is some form of a convexity condition, then E_H behaves nicely [111]: will be called round enough if it is a submanifold of a spacelike hypersurface Σ, and if among the 2-dimensional surfaces in Σ which enclose the same volume as does, has the smallest area. Then it is proven by Christodoulou and Yau [111] that if is round enough in a maximal spacelike slice Σ on which the energy density of the matter fields is non-negative (for example if the dominant energy condition is satisfied), then the Hawking energy is non-negative.

Although the Hawking energy is not monotonic in general, it has interesting monotonicity properties for special families of 2-surfaces. Hawking considered one-parameter families of spacelike 2-surfaces foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of E_H [171]. These calculations were refined by Eardley [131]. Starting with a weakly future convex 2-surface and using the boost gauge freedom, he introduced a special family of spacelike 2-surfaces in the outgoing null hypersurface , where r will be the luminosity distance along the outgoing null generators. He showed that is non-decreasing with r, provided the dominant energy condition holds on . Similarly, for weakly past convex and the analogous family of surfaces in the ingoing null hypersurface is non-increasing. Eardley also considered a special spacelike hypersurface, filled by a family of 2-surfaces, for which is non-decreasing. By relaxing the normalization condition l_an^a = 1 for the two null normals to l_an^a = exp(f) for some , Hayward obtained a flexible enough formalism to introduce a double-null foliation (see Section 11.2 below) of a whole neighbourhood of a mean convex 2-surface by special mean convex 2-surfaces [182]. (For the more general GHP formalism in which l_an^a is not fixed, see [312].) Assuming that the dominant energy condition holds, he showed that the Hawking energy of these 2-surfaces is non-decreasing in the outgoing, and non-increasing in the ingoing direction.

In contrast to the special foliations of the null hypersurfaces above, Frauendiener defined a special spacelike vector field, the inverse mean curvature vector in the spacetime [145]. If is a weakly future and past convex 2-surface, then q^a := 2Q^a/(Q_bQ^b) = − [1/(2ρ)]l^a − [1/(2ρ′)]n^a is an outward directed spacelike normal to . Here Q_b is the trace of the extrinsic curvature tensor: (see Section 4.1.2). Starting with a single weakly future and past convex 2-surface, Frauendiener gives an argument for the construction of a one-parameter family of 2-surfaces being Lie-dragged along its own inverse mean curvature vector q^a. Hence this family of surfaces would be analogous to the solution of the geodesic equation, where the initial point and direction in that point specify the whole solution, at least locally. Assuming that such a family of surfaces (and hence the vector field q^a on the 3-submanifold swept by ) exists, Frauendiener showed that the Hawking energy is non-decreasing along the vector field q^a if the dominant energy condition is satisfied. However, no investigation has been made to prove the existence of such a family of surfaces. Motivated by this result, Malec, Mars, and Simon [261] considered spacelike hypersurfaces with an inverse mean curvature flow of Geroch thereon (see Section 6.2.2). They showed that if the dominant energy condition and certain additional (essentially technical) assumptions hold, then the Hawking energy is monotonic. These two results are the natural adaptations for the Hawking energy of the corresponding results known for some time for the Geroch energy, aiming to prove the Penrose inequality. We return to this latter issue in Section 13.2 only for a very brief summary.

Two generalizations

Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of the Bondi-Sachs energy-momentum are related to the Bondi energy:

39

where , are essentially the first four spherical harmonics:

40

Here ζ and are the standard complex stereographic coordinates on .

Hawking considered the extension of the definition of to higher genus 2-surfaces also by the second expression in Equation (38). Then in the expression analogous to the first one in Equation (38) the genus of appears.

The definition

Suppose that the 2-surface for which E_H is defined is embedded in the spacelike hypersurface Σ. Let χ_ab be the extrinsic curvature of Σ in M and k_ab the extrinsic curvature of in Σ. (In Section 4.1.2 we denoted the latter by ν_ab.) Then 8ρρ′ = (χ_abq^ab)² − (k_abq^ab)², by means of which

41

In the last step we used the Gauss-Bonnet theorem for . is known as the Geroch energy [150]. Thus it is not greater than the Hawking energy, and, in contrast to E_H, it depends not only on the 2-surface , but the hypersurface Σ as well.

The calculation of the small sphere limit of the Geroch energy was saved by observing [204] that, by Equation (41), the difference of the Hawking and the Geroch energies is proportional to . Since, however, χ_abq^ab — for the family of small spheres — does not tend to zero in the r → 0 limit, in general this difference is . It is zero if Σ is spanned by spacelike geodesics orthogonal to t^a at p. Thus, for general Σ, the Geroch energy does not give the expected result. Similarly, in vacuum the Geroch energy deviates from the Bel-Robinson energy in r⁵ order even if Σ is geodesic at p.

Since and since the Hawking energy tends to the ADM energy, the large sphere limit of in an asymptotically flat Σ cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [150].

Monotonicity properties

The Geroch energy has interesting positivity and monotonicity properties along a special flow in Σ [150, 219]. This flow is the so-called inverse mean curvature flow defined as follows. Let t: Σ → ℝ be a smooth function such that

1. its level surfaces, , are homeomorphic to S²,

2. there is a point p ∈ Σ such that the surfaces are shrinking to p in the limit t → −∞, and

3. they form a foliation of Σ − {p}.

Let n be the lapse function of this foliation, i.e. if v^a is the outward directed unit normal to in Σ, then nv^aD_at = 1. Denoting the integral on the right hand side in Equation (41) by W_t, we can calculate its derivative with respect to t. In general this derivative does not seem to have any remarkable property. If, however, the foliation is chosen in a special way, namely if the lapse is just the inverse mean curvature of the foliation, n = 1/k where k := k_abq^ab, furthermore Σ is maximal (i.e. χ = 0) and the energy density of the matter is non-negative, then, as shown by Geroch [150], W_t ≥ 0 holds. Jang and Wald [219] modified the foliation slightly such that t ∈ [0, ∞), and the surface was assumed to be future marginally trapped (i.e. ρ = 0 and ρ′ ≥ 0). Then they showed that, under the conditions above, . Since tends to the ADM energy as t → ∞, these considerations were intended to argue that the ADM energy should be non-negative (at least for maximal Σ) and not less than (at least for time-symmetric Σ), respectively. Later Jang [217] showed that if a certain quasi-linear elliptic differential equation for a function w on a hypersurface Σ admits a solution (with given asymptotic behaviour), then w defines a mapping between the data set (Σ, h_ab, χ_ab) on Σ and a maximal data set (i.e. for which ) such that the corresponding ADM energies coincide. Then Jang shows that a slightly modified version of the Geroch energy is monotonic (and tends to the ADM energy) with respect to a new, modified version of the inverse mean curvature foliation of .

The existence and the properties of the original inverse mean curvature foliation of (Σ, h_ab) above were proven and clarified by Huisken and Ilmanen [207, 208], giving the first complete proof of the Riemannian Penrose inequality, and, as proved by Schoen and Yau [328], Jang’s quasi-linear elliptic equation admits a global solution.

How do the twistors emerge?

In the Newtonian theory of gravity the mass contained in some finite 3-volume Σ can be expressed as the flux integral of the gravitational field strength on the boundary :

42

where ϕ is the gravitational potential and v^a is the outward directed unit normal to . If is deformed in through a source-free region, then the mass does not change. Thus the mass m_Σ is analogous to charge in electrostatics.

In the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the gravitational field, i.e. the linearized energy-momentum tensor, is still analogous to charge. In fact, the total energy-momentum and angular momentum of the source can be expressed as appropriate 2-surface integrals of the curvature at infinity [349]. Thus it is natural to expect that the energy-momentum and angular momentum of the source in a finite 3-volume Σ, given by Equation (5), can also be expressed as the charge integral of the curvature on the 2-surface . However, the curvature tensor can be integrated on only if at least one pair of its indices is annihilated by some tensor via contraction, i.e. according to Equation (15) if some ω^ab = ω^[ab] is chosen and μ^ab = ɛ^ab. To simplify the subsequent analysis ω^ab will be chosen to be anti-self-dual: ω^ab = ɛ^A′B′ ω^ab with ω^AB = ω^(AB)11. Thus our claim is to find an appropriate spinor field ω^AB on such that

43

Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual of the 8πG times the integrand on the left, respectively, is

44

45

Equations (44) and (45) are equal if ω^AB satisfies

46

This equation in its symmetrized form, , is the valence 2 twistor equation, a specific example for the general twistor equation for ω^BC…E = ω^(BC…E). Thus, as could be expected, ω^AB depends on the Killing vector K^a, and, in fact, K^a can be recovered from ω^AB as . Thus ω^AB plays the role of a potential for the Killing vector K^A′A. However, as a consequence of Equation (46), K_a is a self-dual Killing 1-form in the sense that its derivative is a self-dual (or s.d.) 2-form: In fact, the general solution of equation (46) and the corresponding Killing vector are

47

where , and Ω^AB are constant spinors, using the notation , where is a constant spin frame (the ‘Cartesian spin frame’) and are the standard Pauli matrices (divided by ). These yield that K_a is, in fact, self-dual, , and T^AA′ is a translation and generates self-dual rotations. Then , implying that the charges corresponding to Ω^AB are vanishing; the four components of the quasi-local energy-momentum correspond to the real T^AA′ s, and the spatial angular momentum and centre-of-mass are combined into the three complex components of the self-dual angular momentum , generated by .

Twistor space and the kinematical twistor

Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the pairs Z^α := (λ^A,π_A′) of spinor fields, which solve the so-called valence 1 twistor equation ∇^A′Aλ^B = −iε^ABπ^A′. If Z^α is a solution of this equation, then is a solution of the corresponding equation in the conformally rescaled spacetime, where ϒ_a := Ω⁻¹∇_aΩ and Ω is the conformal factor. In general the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complex parameter family of solutions. Its general solution in the Minkowski spacetime is λ^A = Λ^A − ix^AA′ π_a′, where Λ^A and π_A′ are constant spinors. Thus the space T^α of valence 1 twistors, the so-called twistor-space, is 4-complex-dimensional, and hence has the structure . T^α admits a natural Hermitian scalar product: If W^β = (ω^B, σ_B′) is another twistor, then . Its signature is (+, +, −, −), it is conformally invariant, , and it is constant on Minkowski spacetime. The metric H_αβ′ defines a natural isomorphism between the complex conjugate twistor space, , and the dual twistor space, , by . This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate Ā_Σ′β′ of the covariant valence 2 twistor A_αβ can be represented by the so-called conjugate twistor . We should mention two special, higher valence twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by

48

The other is the completely anti-symmetric twistor ε_αβγδ, whose component ε₀₁₂₃ in an H_αβ′-orthonormal basis is required to be one. The only non-vanishing spinor parts of ε_αβγδ are those with two primed and two unprimed spinor indices: . Thus for any four twistors , i = 1,…, 4, the determinant of the 4 × 4 matrix whose i-th column is , where the , are the components of the spinors and in some spin frame, is

49

where is the totally antisymmetric Levi-Civita symbol. Then I^αβ and I_αβ are dual to each other in the sense that , and by the simplicity of I^αβ one has ɛ_αβγδI^αβI^γδ = 0.

The solution ω^AB of the valence 2 twistor equation, given by Equation (47), can always be written as a linear combination of the symmetrized product λ^(Aω^B) of the solutions λ^A and ω^A of the valence 1 twistor equation. ω^AB defines uniquely a symmetric twistor ω^αβ (see for example [313]). Its spinor parts are

However, Equation (43) can be interpreted as a ℂ-linear mapping of ω^αβ into , i.e. Equation (43) defines a dual twistor, the (symmetric) kinematical twistor A_αβ, which therefore has the structure

50

Thus the quasi-local energy-momentum and self-dual angular momentum of the source are certain spinor parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor, it has only ten real components as a consequence of its structure (its spinor part A_AB is identically zero) and the reality of P^AA′. These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on A_αβ: The vanishing of the spinor part A_AB is equivalent to A_αβI^αγI^βδ = 0 and the energy momentum is the part of the kinematical twistor, while the whole reality condition (ensuring both A_AB = 0 and the reality of the energy-momentum) is equivalent to

51

Using the conjugate twistors this can be rewritten (and, in fact, usually it is written) as . Finally, the quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [307] or as its determinant [371]:

52

53

Thus, to summarize, the various spinor parts of the kinematical twistor A_αβ are the energy-momentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, were needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and, in particular, to define the mass. Furthermore, the Hermiticity condition ensuring A_αβ to have the correct number of components (ten reals) were also formulated in terms of these additional structures.

2-surface twistors and the kinematical twistor

In general spacetimes the twistor equations have only the trivial solution. Thus to be able to associate a kinematical twistor to a closed orientable spacelike 2-surface in general, the conditions on the spinor field ω^AB had to be relaxed. Penrose’s suggestion [307, 308] is to consider ω^AB in Equation (43) to be the symmetrized product λ^(Aω^B) of spinor fields that are solutions of the ‘tangential projection to ’ of the valence 1 twistor equation, the so-called 2-surface twistor equation. (The equation obtained as the ‘tangential projection to ’ of the valence 2 twistor equation (46) would be under-determined [308].) Thus the quasi-local quantities are searched for in the form of a charge integral of the curvature:

54

where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the 2-surface . Since the indices c and d on the right of the first expression are tangential to , this is just the charge integral of F_ABcd in the spinor identity (24) of Section 4.1.5.

The 2-surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation . By Equation (25) its GHP form is , which is a first order elliptic system, and its index is 4(1 − g), where g is the genus of [43]. Thus there are at least four (and in the generic case precisely four) linearly independent solutions to on topological 2-spheres. However, there are ‘exceptional’ 2-spheres for which there exist at least five linearly independent solutions [221]. For such ‘exceptional’ 2-spheres (and for higher genus 2-surfaces for which the twistor equation has only the trivial solution in general) the subsequent construction does not work. (The concept of quasi-local charges in Yang-Mills theory can also be introduced in an analogous way [370]). The space of the solutions to is called the 2-surface twistor space. In fact, in the generic case this space is 4-complex-dimensional, and under conformal rescaling the pair Z^α = (λ^A, iΔ_A′Aλ^A) transforms like a valence 1 contravariant twistor. Z^α is called a 2-surface twistor determined by λ^A. If is another generic 2-surface with the corresponding 2-surface twistor space , then although and are isomorphic as vector spaces, there is no canonical isomorphism between them. The kinematical twistor A_αβ is defined to be the symmetric twistor determined by for any Z^α = (λ^A, iΔ_A′Aλ^A) and W^α = (ω^A, iΔ_A′Aω^A) from . Note that is constructed only from the 2-surface data on .

The Hamiltonian interpretation of the kinematical twistor

For the solutions λ^A and ω^A of the 2-surface twistor equation, the spinor identity (24) reduces to Tod’s expression [307, 313, 377] for the kinematical twistor, making it possible to re-express by the integral of the Nester-Witten 2-form [356]. Indeed, if

55

then with the choice this gives Penrose’s charge integral by Equation (24): . Then, extending the spinor fields λ^A and ω^A from to a spacelike hypersurface Σ with boundary in an arbitrary way, by the Sparling equation it is straightforward to rewrite in the form of the integral of the energy-momentum tensor of the matter fields and the Sparling form on Σ. Since such an integral of the Sparling form can be interpreted as the Hamiltonian of general relativity, this is a quick re-derivation of Mason’s [265, 266] Hamiltonian interpretation of Penrose’s kinematical twistor: is just the boundary term in the total Hamiltonian of the matter + gravity system, and the spinor fields λ^A and ω^A (together with their ‘projection parts’ and ) on are interpreted as the spinor constituents of the special lapse and shift, the so-called ‘quasi-translations’ and ‘quasi-rotations’ of the 2-surface, on the 2-surface itself.

The Hermitian scalar product and the infinity twistor

In general the natural pointwise Hermitian scalar product, defined by , is not constant on , thus it does not define a Hermitian scalar product on the 2-surface twistor space. As is shown in [220, 223, 375], is constant on for any two 2-surface twistors if and only if can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric and extrinsic curvatures. Such 2-surfaces are called non-contorted, while those that cannot be embedded are called contorted. One natural candidate for the Hermitian metric could be the average of on [307]: , which reduces to on non-contorted 2-surfaces. Interestingly enough, can also be re-expressed by the integral (55) of the Nester-Witten 2-form [356]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted 2-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. Equation (52)) is ambiguous unless a natural H_αβ′ is found.

If is non-contorted, then the scalar product defines the totally anti-symmetric twistor ε_αβγδ, and for the four independent 2-surface twistors the contraction , and hence by Equation (49) the determinant ν, is constant on . Nevertheless, ν can be constant even for contorted 2-surfaces for which is not. Thus, the totally anti-symmetric twistor ε_αβγδ can exist even for certain contorted 2-surfaces. Therefore, an alternative definition of the quasi-local mass might be based on Equation (53) [371]. However, although the two mass definitions are equivalent in the linearized theory, they are different invariants of the kinematical twistor even in de Sitter or anti-de-Sitter spacetimes. Thus, if needed, the former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by m_D).

If we want to have not only the notion of the mass but its reality is also expected, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (51), one also needs the infinity twistor. However, −ɛ^A′B Δ_A′Aλ^AΔ_B′Bω^B is not constant on even if it is non-contorted, thus in general it does not define any twistor on . One might take its average on (which can also be re-expressed by the integral of the Nester-Witten 2-form [356]), but the resulting twistor would not be simple. In fact, even on 2-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant λ the natural definition for I_αβ is I_αβ := diag(λε_AB, ε^A′B′) [313, 311, 371], while on round spheres in spherically symmetric spacetimes it is [363]. Thus no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [197]: Even if the spacetime is conformally flat (whenever the Hermitian metric exists) the Hermiticity condition would be fifteen algebraic equations for the (at most) twelve real components of the ‘would be’ infinity twistor. Then, since the possible kinematical twistors form an open set in the space of symmetric twistors, the Hermiticity condition cannot be satisfied even for non-simple I^αβs. However, in contrast to the linearized gravity case, the infinity twistor should not be given once and for all on some ‘universal’ twistor space, that may depend on the actual gravitational field. In fact, the 2-surface twistor space itself depends on the geometry of , and hence all the structures thereon also.

Since in the Hermiticity condition (51) only the special combination of the infinity and metric twistors (the so-called ‘bar-hook’ combination) appears, it might still be hoped that an appropriate could be found for a class of 2-surfaces in a natural way [377]. However, as far as the present author is aware of, no real progress has been achieved in this way.

The various limits

Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea came from the linearized gravity, the construction gives the correct results in the weak field approximation. The nonrelativistic weak field approximation, i.e. the Newtonian limit, was clarified by Jeffryes [222]. He considers a 1-parameter family of spacetimes with perfect fluid source such that in the λ → 0 limit of the parameter λ one gets a Newtonian spacetime, and, in the same limit, the 2-surface lies in a t = const. hypersurface of the Newtonian time t. In this limit the pointwise Hermitian scalar product is constant, and the norm-mass can be calculated. As could be expected, for the leading λ² order term in the expansion of m as a series of λ he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a λ⁴ order correction.

The Penrose definition for the energy-momentum and angular momentum can be applied to the cuts of the future null infinity ℐ⁺ of an asymptotically flat spacetime [307, 313]. Then every element of the construction is built from conformally rescaled quantities of the non-physical spacetime. Since is shear-free, the 2-surface twistor equations on decouple, and hence the solution space admits a natural infinity twistor I_αβ. It singles out precisely those solutions whose primary spinor parts span the asymptotic spin space of Bramson (see Section 4.2.4), and they will be the generators of the energy-momentum. Although is contorted, and hence there is no natural Hermitian scalar product, there is a twistor with respect to which A_αβ is Hermitian. Furthermore, the determinant ν is constant on , and hence it defines a volume 4-form on the 2-surface twistor space [377]. The energy-momentum coming from A_αβ is just that of Bondi and Sachs. The angular momentum defined by A_αβ is, however, new. It has a number of attractive properties. First, in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two anomaly’ between the angular momentum and the energy-momentum. Since its definition is based on the solutions of the 2-surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the 2-surface twistor space on reduces the BMS Lie algebra to one of its Poincaré subalgebras. Thus the concept of the ‘translation of the origin’ is moved from null infinity to the twistor space (appearing in the form of a 4-parameter family of ambiguities in the potential for the shear σ), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. As was shown in [129], Penrose’s angular momentum can be considered as a supertranslation of previous definitions. The corresponding angular momentum flux through a portion of the null infinity between two cuts was calculated in [129, 196] and it was shown that this is precisely that given by Ashtekar and Streubel [29] (see also [336, 337, 128]).

The other way of determining the null infinity limit of the energy-momentum and angular momentum is to calculate them for the large spheres from the physical data, instead of the spheres at null infinity from the conformally rescaled data. These calculations were done by Shaw [338, 340]. At this point it should be noted that the r → ∞ limit of A_αβ vanishes, and it is that yields the energy-momentum and angular momentum at infinity (see the remarks following Equation (15)). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [371]. The 2-surfaces for which the mass was calculated are the r = const. cuts of the geometrically distinguished outgoing null hypersurfaces u = const. Tod found that, for given u, the mass m is independent of r, as could be expected because of the lack of the incoming radiation.

The large sphere limit of the 2-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [344], the Ashtekar-Hansen [23], and the Beig-Schmidt [48] models of spatial infinity in [334, 335, 337]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) non-contorted, and both the Hermitian scalar product and the infinity twistor are well-defined. Thus the energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar-Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [339].

The Penrose mass in asymptotically anti-de-Sitter spacetimes was studied by Kelly [234]. He calculated the kinematical twistor for spacelike cuts of the infinity , which is now a timelike 3-manifold in the non-physical spacetime. Since admits global 3-surface twistors (see the next Section 7.2.5), is non-contorted. In addition to the Hermitian scalar product there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding Hermiticity condition. The energy-momentum 4-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [27]. Therefore, the energy-momentum 4-vector is future pointing and timelike if there is a spacelike hypersurface extending to on which the dominant energy condition is satisfied. Consequently, m² ≥ 0. Kelly showed that is also non-negative and in vacuum it coincides with m². In fact [377], m ≥ m_D ≥ 0 holds.

The quasi-local mass of specific 2-surfaces

The Penrose mass has been calculated in a large number of specific situations. Round spheres are always non-contorted [375], thus the norm-mass can be calculated. (In fact, axi-symmetric 2-surfaces in spacetimes with twist-free rotational Killing vector are non-contorted [223].) The Penrose mass for round spheres reduces to the standard energy expression discussed in Section 4.2.1 [371]. Thus every statement given in Section 4.2.1 for round spheres is valid for the Penrose mass, and we do not repeat them. In particular, for round spheres in a t = const. slice of the Kantowski-Sachs spacetime this mass is independent of the 2-surfaces [368]. Interestingly enough, although these spheres cannot be shrunk to a point (thus the mass cannot be interpreted as ‘the 3-volume integral of some mass density’), the time derivative of the Penrose mass looks like the mass conservation equation: It is minus the pressure times the rate of change of the 3-volume of a sphere in flat space with the same area as [376]. In conformally flat spacetimes [371] the 2-surface twistors are just the global twistors restricted to , and the Hermitian scalar product is constant on . Thus the norm-mass is well-defined.

The construction works nicely even if global twistors exist only on a (say) spacelike hyper-surface Σ containing . These twistors are the so-called 3-surface twistors [371, 373], which are solutions of certain (overdetermined) elliptic partial differential equations, the so-called 3-surface twistor equations, on Σ. These equations are completely integrable (i.e. they admit the maximal number of linearly independent solutions, namely four) if and only if Σ with its intrinsic metric and extrinsic curvature can be embedded, at least locally, into some conformally flat spacetime [373]. Such hypersurfaces are called non-contorted. It might be interesting to note that the non-contorted hypersurfaces can also be characterized as the critical points of the Chern-Simons functional built from the real Sen connection on the Lorentzian vector bundle or from the 3-surface twistor connection on the twistor bundle over Σ [49, 361]. Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a 2-surface in a non-contorted Σ depends only on the homology class of . In particular, if can be shrunk to a point then the corresponding kinematical twistor is vanishing. Since Σ is non-contorted, is also non-contorted, and hence the norm-mass is well-defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any non-contorted 2-surface that can be deformed into a round sphere, and it is zero for those that do not link the black hole [375]. Thus, in particular, the Penrose mass can be zero even in curved spacetimes.

A particularly interesting class of non-contorted hypersurfaces is that of the conformally flat time-symmetric initial data sets. Tod considered Wheeler’s solution of the time-symmetric vacuum constraints describing n ‘points at infinity’ (or, in other words, n − 1 black holes) and 2-surfaces in such a hypersurface [371]. He found that the mass is zero if does not link any black hole, it is the mass M_i of the i-th black hole if links precisely the i-th hole, it is if links precisely the i-th and the j-th holes, where d_ij is some appropriate measure of the distance of the holes, …, etc. Thus, the mass of the i-th and j-th holes as a single object is less than the sum of the individual masses, in complete agreement with our physical intuition that the potential energy of the composite system should contribute to the total energy with negative sign.

Beig studied the general conformally flat time-symmetric initial data sets describing n ‘points at infinity’ [45]. He found a symmetric trace-free and divergence-free tensor field T^ab and, for any conformal Killing vector ξ^a of the data set, defined the 2-surface flux integral P(ξ) of T^abξ_b on . He showed that P(ξ) is conformally invariant, depends only on the homology class of , and, apart from numerical coefficients, for the ten (locally existing) conformal Killing vectors these are just the components of the kinematical twistor derived by Tod in [371] (and discussed in the previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the length of the P’s with respect to the Cartan-Killing metric of the conformal group of the hypersurface.

Tod calculated the quasi-local mass for a large class of axi-symmetric 2-surfaces (cylinders) in various LRS Bianchi and Kantowski-Sachs cosmological models [376] and more general cylindrically symmetric spacetimes [378]. In all these cases the 2-surfaces are non-contorted, and the construction works. A technically interesting feature of these calculations is that the 2-surfaces have edges, i.e. they are not smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately, and the resulting spinor fields are required to be continuous at the edges. This matching reduces the number of linearly independent solutions to four. The projection parts of the resulting twistors, the iΔ_A′Aλ^As, are not continuous at the edges. It turns out that the cylinders can be classified invariantly to be hyperbolic, parabolic, or elliptic. Then the structure of the quasi-local mass expressions is not simply ‘density’ × ‘volume’, but they are proportional to a ‘type factor’ f(L) as well, where L is the coordinate length of the cylinder. In the hyperbolic, parabolic, and elliptic cases this factor is sinhωL/(ωL), 1, and sinωL/(ωL), respectively, where ω is an invariant of the cylinder. The various types are interpreted as the presence of a positive, zero, or negative potential energy. In the elliptic case the mass may be zero for finite cylinders. On the other hand, for static perfect fluid spacetimes (hyperbolic case) the quasi-local mass is positive. A particularly interesting spacetime is that describing cylindrical gravitational waves, whose presence is detected by the Penrose mass. In all these cases the determinant-mass has also been calculated and found to coincide with the norm-mass. A numerical investigation of the axi-symmetric Brill waves on the Schwarzschild background was presented in [69]. It was found that the quasi-local mass is positive, and it is very sensitive to the presence of the gravitational waves.

Another interesting issue is the Penrose inequality for black holes (see Section 13.2.1). Tod showed [374, 375] that for static black holes the Penrose inequality holds if the mass of the hole is defined to be the Penrose quasi-local mass of the spacelike cross section of the event horizon. The trick here is that is totally geodesic and conformal to the unit sphere, and hence it is non-contorted and the Penrose mass is well-defined. Then the Penrose inequality will be a Sobolev-type inequality for a non-negative function on the unit sphere. This inequality was tested numerically in [69].

Apart from the cuts of in radiative spacetimes, all the 2-surfaces discussed so far were non-contorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted 2-surface [377]. Thus although the kinematical twistor can be calculated for this, the construction in its original form cannot yield any mass expression. The original construction has to be modified.

Small surfaces

The properties of the Penrose construction that we have discussed are very remarkable and promising. However, the small surface calculations showed clearly some unwanted feature of the original construction [372, 235, 398], and forced its modification.

First, although the small spheres are contorted in general, the leading term of the pointwise Hermitian scalar product is constant: for any 2-surface twistors Z^α = (λ^A, iΔ_a′aλ^A) and W^α = (ω^A, iΔ_a′aω^A) [372, 235]. Since in non-vacuum space-times the kinematical twistor has only the ‘4-momentum part’ in the leading order with , the Penrose mass, calculated with the norm above, is just the expected mass in the leading order. Thus it is positive if the dominant energy condition is satisfied. On the other hand, in vacuum the structure of the kinematical twistor is

56

where and with . In particular, in terms of the familiar conformal electric and magnetic parts of the curvature the leading term in the time component of the 4-momentum is . Then the corresponding norm-mass, in the leading order, can even be complex! For an in the t = const. hypersurface of the Schwarzschild spacetime this is zero (as it must be in the light of the results of the previous Section 7.2.5, because this is a non-contorted spacelike hypersurface), but for a general small 2-sphere not lying in such a hypersurface P_aa′ is real and spacelike, and hence m² < 0. In the Kerr spacetime P_aa′ itself is complex [372, 235].

The ‘improved’ construction with the determinant

A careful analysis of the roots of the difficulties lead Penrose [309, 313] (see also [372, 235, 377]) to suggest the modified definition for the kinematical twistor

57

where η is a constant multiple of the determinant ν in Equation (49). Since on non-contorted 2-surfaces the determinant ν is constant, for such surfaces reduces to A_αβ, and hence all the nice properties proven for the original construction on non-contorted 2-surfaces are shared by too. The quasi-local mass calculated from Equation (57) for small spheres (in fact, for small ellipsoids [235]) in vacuum is vanishing in the fifth order. Thus, apparently, the difficulties have been resolved. However, as Woodhouse pointed out, there is an essential ambiguity in the (non-vanishing, sixth order) quasi-local mass [398]. In fact, the structure of the modified kinematical twistor has the form (56) with vanishing and but with non-vanishing λ_AB in the fifth order. Then in the quasi-local mass (in the leading sixth order) there will be a term coming from the (presumably non-vanishing) sixth order part of and and the constant part of the Hermitian scalar product, and the fifth order λ_AB and the still ambiguous order part of the Hermitian metric.

Modification through Tod’s expression

These anomalies lead Penrose to modify slightly [310]. This modified form is based on Tod’s form of the kinematical twistor:

58

The quasi-local mass on small spheres coming from is positive [377].

Mason’s suggestions

A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [265]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Equation (58) is that of the integral of the Nester-Witten 2-form, and the spinor fields and could still be considered as the spinor constituents of the ‘quasi-Killing vectors’ of the 2-surface , their structure is not so simple because the factor η itself depends on all of the four independent solutions of the 2-surface twistor equation in a rather complicated way.

To have a simple Hamiltonian interpretation Mason suggested further modifications [265, 266]. He considers the four solutions , of the 2-surface twistor equations, and uses these solutions in the integral (55) of the Nester-Witten 2-form. Since is a Hermitian bilinear form on the space of the spinor fields (see Section 8 below), he obtains 16 real quantities as the components of the 4 × 4 Hermitian matrix . However, it is not clear how the four ‘quasi-translations’ of should be found among the 16 vector fields (called ‘quasi-conformal Killing vectors’ of ) for which the corresponding quasi-local quantities could be considered as the quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local quantities.

The definition

Suppose that the spacetime is asymptotically flat at future null infinity, and the closed spacelike 2-surface can be joined to future null infinity by a smooth null hypersurface . Let , the cut defined by the intersection of with the future null infinity. Then the null geodesic generators of define a smooth bijection between and the cut (and hence, in particular, ). We saw in Section 4.2.4 that on the cut at the future null infinity we have the asymptotic spin space . The suggestion of Ludvigsen and Vickers [259] for the spin space on is to import the two independent solutions of the asymptotic twistor equations, i.e. the asymptotic spinors, from the future null infinity back to the 2-surface along the null geodesic generators of the null hypersurface . Their propagation equations, given both in terms of spinors and in the GHP formalism, are

59

60

Here is the GHP spin frame introduced in Section 4.2.4, and by Equation (25) the second half of these equations is just Δ⁺λ = 0. It should be noted that the choice (59, 60) for the propagation law of the spinors is ‘natural’ in the sense that in flat spacetime (59, 60) reduce to the condition of parallel propagation, and Equation (60) is just the appropriate part of the asymptotic twistor equation of Bramson. We call the spinor fields obtained by using Equations (59, 60) the Ludvigsen-Vickers spinors on . Thus, given an asymptotic spinor at infinity, we propagate its zero-th components (with respect to the basis ) to by Equation (59). This will be the zero-th component of the Ludvigsen-Vickers spinor. Then its first component will be determined by Equation (60), provided ρ is not vanishing on any open subset of . If and are Ludvigsen-Vickers spinors on obtained by Equations (59, 60) from two asymptotic spinors that formed a normalized spin frame, then, by considering and to be normalized in , we define the symplectic metric on to be that with respect to which and form a normalized spin frame. Note, however, that this symplectic metric is not connected with the symplectic fibre metric ɛ_AB of the spinor bundle over . Indeed, in general, is not constant on , and hence ɛ_AB does not determine any symplectic metric on the space of the Ludvigsen-Vickers spinors. In Minkowski spacetime the two Ludvigsen-Vickers spinors are just the restriction to of the two constant spinors.

Remarks on the validity of the construction

Before discussing the usual questions about the properties of the construction (positivity, monotonicity, the various limits, etc.), we should make some general remarks. First, it is obvious that the Ludvigsen-Vickers energy-momentum in its form above cannot be defined in a spacetime which is not asymptotically flat at null infinity. Thus their construction is not genuinely quasi-local, because it depends not only on the (intrinsic and extrinsic) geometry of , but on the global structure of the spacetime as well. In addition, the requirement of the smoothness of the null hypersurface connecting the 2-surface to the null infinity is a very strong restriction. In fact, for general (even for convex) 2-surfaces in a general asymptotically flat spacetime conjugate points will develop along the (outgoing) null geodesics orthogonal to the 2-surface [304, 175]. Thus either the 2-surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the 2-surface must be nearly spherically symmetric (or the former cannot be ‘very much curved’ and the latter cannot be ‘very much bent’).

This limitation yields that in general the original construction above does not have a small sphere limit. However, using the same propagation equations (59, 60) one could define a quasilocal energy-momentum for small spheres [259, 66]. The basic idea is that there is a spin space at the vertex p of the null cone in the spacetime whose spacelike cross section is the actual 2-surface, and the Ludvigsen-Vickers spinors on are defined by propagating these spinors from the vertex p to via Equations (59, 60). This definition works in arbitrary spacetime, but the 2-surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely quasi-local.

Monotonicity, mass-positivity and the various limits

Once the Ludvigsen-Vickers spinors are given on a spacelike 2-surface of constant affine parameter r in the outgoing null hypersurface , then they are uniquely determined on any other spacelike 2-surface in , too, i.e. the propagation law (59, 60) defines a natural isomorphism between the space of the Ludvigsen-Vickers spinors on different 2-surfaces of constant affine parameter in the same . (r need not be a Bondi-type coordinate.) This makes it possible to compare the components of the Ludvigsen-Vickers energy-momenta on different surfaces. In fact [259], if the dominant energy condition is satisfied (at least on ), then for any Ludvigsen-Vickers spinor λ^A and affine parameter values r₁ ≤ r₂ one has , and the difference can be interpreted as the energy flux of the matter and the gravitational radiation through between and . Thus both and are increasing with r (‘mass-gain’). A similar monotonicity property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation law (59, 60) should be replaced by Ϸ′λ₁ = 0 and . Using these equations the positivity of the Ludvigsen-Vickers mass was proven in various special cases in [259].

Concerning the positivity properties of the Ludvigsen-Vickers mass and energy, first it is obvious by the remarks on the nature of the propagation law (59, 60) that in Minkowski spacetime the Ludvigsen-Vickers energy-momentum is vanishing. However, in the proof of the non-negativity of the Dougan-Mason energy (discussed in Section 8.2) only the λ_A ∈ ker Δ⁺ part of the propagation equations is used. Therefore, as realized by Bergqvist [61], the Ludvigsen-Vickers energy-momenta (both based on the asymptotic and the point spinors) are also future directed and nonspacelike if is the boundary of some compact spacelike hypersurface Σ on which the dominant energy condition is satisfied and is weakly future convex (or at least ρ ≤ 0). Similarly, the Ludvigsen-Vickers definitions share the rigidity properties proven for the Dougan-Mason energy-momentum [354]: Under the same conditions the vanishing of the energy-momentum implies the flatness of the domain of dependence D(Σ) of Σ.

In the weak field approximation [259] the difference is just the integral of on the portion of between the two 2-surfaces, where T_ab is the linearized energy-momentum tensor: The increment of on is due only to the flux of the matter energy-momentum.

Since the Bondi-Sachs energy-momentum can be written as the integral of the Nester-Witten 2-form on the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first version of the Ludvigsen-Vickers energy-momentum tends to that of Bondi and Sachs. It was shown in [259, 340] that this expectation is, in fact, correct. The Ludvigsen-Vickers mass was calculated for large spheres both for radiative and stationary spacetimes with r⁻² and r⁻³ accuracy, respectively, in [338, 340].

Finally, on a small sphere of radius r in non-vacuum the second definition gives [66] the expected result (28), while in vacuum [66, 360] it is

61

Thus its leading term is the energy-momentum of the matter fields and the Bel-Robinson momentum, respectively, seen by the observer t^a at the vertex p. Thus, assuming that the matter fields satisfy the dominant energy condition, for small spheres this is an explicit proof that the Ludvigsen-Vickers quasi-local energy-momentum is future pointing and nonspacelike.

Holomorphic/anti-holomorphic spinor fields

The original construction of Dougan and Mason [127] was introduced on the basis of sheaf-theoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [354, 356].

Following Dougan and Mason we define the spinor field λ_A to be anti-holomorphic in case m^e∇_eλ_A = m^eΔ_eλ_A = 0, or holomorphic if . Thus, this notion of holomorphicity/anti-holomorphicity is referring to the connection Δ_e on . While the notion of the holomorphicity/anti-holomorphicity of a function on does not depend on whether the Δ_e or the δ_e operator is used, for tensor or spinor fields it does. Although the vectors m^a and are not uniquely determined (because their phase is not fixed), the notion of the holomorphicity/anti-holomorphicity is well-defined, because the defining equations are homogeneous in m^a and . Next suppose that there are at least two independent solutions of . If λ_A and μ_A are any two such solutions, then , and hence by Liouville’s theorem is constant on . If this constant is not zero, then we call generic, if it is zero then will be called exceptional. Obviously, holomorphic λ_A on a generic cannot have any zero, and any two holomorphic spinor fields, e.g. and , span the spin space at each point of (and they can be chosen to form a normalized spinor dyad with respect to ɛ_AB on the whole of ). Expanding any holomorphic spinor field in this frame, the expanding coefficients turn out to be holomorphic functions, and hence constant. Therefore, on generic 2-surfaces there are precisely two independent holomorphic spinor fields. In the GHP formalism the condition of the holomorphicity of the spinor field λ_A is that its components (λ₀, λ₁) be in the kernel of . Thus for generic 2-surfaces with the constant would be a natural candidate for the spin space above. For exceptional 2-surfaces the kernel space is either 2-dimensional but does not inherit a natural spin space structure, or it is higher than two dimensional. Similarly, the symplectic inner product of any two anti-holomorphic spinor fields is also constant, one can define generic and exceptional 2-surfaces as well, and on generic surfaces there are precisely two anti-holomorphic spinor fields. The condition of the anti-holomorphicity of λ_A is . Then could also be a natural choice. Note that since the spinor fields whose holomorphicity/anti-holomorphicity is defined are unprimed, and these correspond to the anti-holomorphicity/holomorphicity, respectively, of the primed spinor fields of Dougan and Mason. Thus the main question is whether there exist generic 2-surfaces, and if they do, whether they are ‘really generic’, i.e. whether most of the physically important surfaces are generic or not.

The genericity of the generic 2-surfaces

are first order elliptic differential operators on certain vector bundles over the compact 2-surface , and their index can be calculated: index , where g is the genus of . Therefore, for there are at least two linearly independent holomorphic and at least two linearly independent anti-holomorphic spinor fields. The existence of the holomorphic/anti-holomorphic spinor fields on higher genus 2-surfaces is not guaranteed by the index theorem. Similarly, the index theorem does not guarantee that is generic either: If the geometry of is very special then the two holomorphic/anti-holomorphic spinor fields (which are independent as solutions of ) might be proportional to each other. For example, future marginally trapped surfaces (i.e. for which ρ = 0) are exceptional from the point of view of holomorphic, and past marginally trapped surfaces (ρ′ = 0) from the point of view of anti-holomorphic spinors. Furthermore, there are surfaces with at least three linearly independent holomorphic/anti-holomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional 2-surface with S² topology make generic.

Finally, we note that several first order differential operators can be constructed from the chiral irreducible parts Δ^± and of Δ_e, given explicitly by Equation (25). However, only four of them, the Dirac-Witten operator Δ := Δ⁺ ⊕ Δ⁻, the twistor operator , and the holomorphy and anti-holomorphy operators , are elliptic (which ellipticity, together with the compactness of , would guarantee the finiteness of the dimension of their kernel), and it is only that have 2-complex-dimensional kernel in the generic case. This purely mathematical result gives some justification for the choices of Dougan and Mason: The spinor fields that should be used in the Nester-Witten 2-form are either holomorphic or anti-holomorphic. The construction does not work for exceptional 2-surfaces.

Positivity properties

One of the most important properties of the Dougan-Mason energy-momenta is that they are future pointing nonspacelike vectors, i.e. the corresponding masses and energies are non-negative. Explicitly [127], if is the boundary of some compact spacelike hypersurface Σ on which the dominant energy condition holds, furthermore if is weakly future convex (in fact, ρ ≤ 0 is enough), then the holomorphic Dougan-Mason energy-momentum is a future pointing non-spacelike vector, and, analogously, the anti-holomorphic energy-momentum is future pointing and non-spacelike if ρ′ ≥ 0. As Bergqvist [61] stressed (and we noted in Section 8.1.3), Dougan and Mason used only the Δ⁺λ = 0 (and in the anti-holomorphic construction the Δ⁻ λ = 0) half of the ‘propagation law’ in their positivity proof. The other half is needed only to ensure the existence of two spinor fields. Thus that might be Equation (59) of the Ludvigsen-Vickers construction, or in the holomorphic Dougan-Mason construction, or even for some constant k, a ‘deformation’ of the holomorphicity considered by Bergqvist [61]. In fact, the propagation law may even be for any spinor field satisfying . This ensures the positivity of the energy under the same conditions and that is still constant on for any two solutions λ_A and μ_A, making it possible to define the norm of the resulting energy-momentum, i.e. the mass.

In the asymptotically flat spacetimes the positive energy theorems have a rigidity part too, namely the vanishing of the energy-momentum (and, in fact, even the vanishing of the mass) implies flatness. There are analogous theorems for the Dougan-Mason energy-momenta too [354, 356]. Namely, under the conditions of the positivity proof

1. is zero iff D(Σ) is flat, which is also equivalent to the vanishing of the quasi-local energy, , and

2. is null (i.e. the quasi-local mass is zero) iff D(Σ) is a pp-wave geometry and the matter is pure radiation.

In particular [365], for a coupled Einstein-Yang-Mills system (with compact, semisimple gauge groups) the zero quasi-local mass configurations are precisely the pp-wave solutions found by Güven [167]. Therefore, in contrast to the asymptotically flat cases, the vanishing of the mass does not imply the flatness of D(Σ). Since, as we will see below, the Dougan-Mason masses tend to the ADM mass at spatial infinity, seemingly there is a contradiction between the rigidity part of the positive mass theorems and the Result 2 above. However, this contradiction is only apparent. In fact, according to one of the possible positive mass proofs [24], the vanishing of the ADM mass implies the existence of a constant null vector field on D(Σ), and then the flatness follows from the incompatibility of the conditions of the asymptotic flatness and the existence of a constant null vector field: The only asymptotically flat spacetime admitting a constant null vector field is the flat spacetime.

These results show some sort of rigidity of the matter + gravity system (where the latter satisfies the dominant energy condition) even at the quasi-local level, which is much more manifest from the following equivalent form of Results 1 and 2: Under the same conditions D(Σ) is flat if and only if there exist two linearly independent spinor fields on which are constant with respect to Δ_e, and D(Σ) is a pp-wave geometry and the matter is pure radiation if and only if there exists a Δ_e-constant spinor field on [356]. Thus the full information that D(Σ) is flat/pp-wave is completely encoded not only in the usual initial data on , but in the geometry of the boundary of Σ, too. In Section 13.5 we return to the discussion of this phenomenon, where we will see that, assuming that is future and past convex, the whole line element of D(Σ) (and not only the information that it is some pp-wave geometry) is determined by the 2-surface data on .

Comparing Results 1 and 2 above with the properties of the quasi-local energy-momentum (and angular momentum) listed in Section 2.2.3, the similarity is obvious: characterizes the ‘quasi-local vacuum state’ of general relativity, while is equivalent to ‘pure radiative quasi-local states’. The equivalence of and the flatness of D(Σ) shows that curvature always yields positive energy, or, in other words, with this notion of energy no classical symmetry breaking can occur in general relativity: The ‘quasi-local ground states’ (defined by ) are just the ‘quasi-local vacuum states’ (defined by the trivial value of the field variables on D(Σ)) [354], in contrast, for example, to the well known ϕ⁴ theories.

The various limits

Both definitions give the same standard expression for round spheres [126]. Although the limit of the Dougan-Mason masses for round spheres in Reissner-Nordström spacetime gives the correct irreducible mass of the Reissner-Nordström black hole on the horizon, the constructions do not work on the surface of bifurcation itself, because that is an exceptional 2-surface. Unfortunately, without additional restrictions (e.g. the spherical symmetry of the 2-surfaces in a spherically symmetric spacetime) the mass of the exceptional 2-surfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic 2-surfaces approaching the exceptional one [356].

Both definitions give the same, expected results in the weak field approximation and for large spheres at spatial infinity: Both tend to the ADM energy-momentum [127]. In non-vacuum both definitions give the same, expected expression (28) for small spheres, in vacuum they coincide in the r⁵ order with that of Ludvigsen and Vickers, but in the r⁶ order they differ from each other: The holomorphic definition gives Equation (61), but in the analogous expression for the anti-holomorphic energy-momentum the numerical coefficient 4/(45G) is replaced by 1/(9G) [126]. The Dougan-Mason energy-momenta have also been calculated for large spheres of constant Bondi-type radial coordinate value r near future null infinity [126]. While the anti-holomorphic construction tends to the Bondi-Sachs energy-momentum, the holomorphic one diverges in general. In stationary spacetimes they coincide and both give the Bondi-Sachs energy-momentum. At the past null infinity it is the holomorphic construction which reproduces the Bondi-Sachs energy-momentum and the anti-holomorphic diverges.

We close this section with some caution and general comments on a potential gauge ambiguity in the calculation of the various limits. By the definition of the holomorphic and anti-holomorphic spinor fields they are associated with the 2-surface only. Thus if is another 2-surface, then there is no natural isomorphism between the space — for example of the anti-holomorphic spinor fields ker on — and ker on , even if both surfaces are generic and hence there are isomorphisms between them12. This (apparently ‘only theoretical’) fact has serious pragmatic consequences. In particular, in the small or large sphere calculations we compare the energy-momenta, and hence the holomorphic or anti-holomorphic spinor fields also, on different surfaces. For example [360], in the small sphere approximation every spin coefficient and spinor component in the GHP dyad and metric component in some fixed coordinate system is expanded as a series of r, like . Substituting all such expansions and the asymptotic solutions of the Bianchi identities for the spin coefficients and metric functions into the differential equations defining the holomorphic/anti-holomorphic spinors, we obtain a hierarchical system of differential equations for the expansion coefficients λ_A⁽⁰⁾, λ_A⁽¹⁾, …, etc. It turns out that the solutions of this system of equations with accuracy r^k form a 2k rather than the expected two complex dimensional space. 2(k−1) of these 2k solutions are ‘gauge’ solutions, and they correspond in the approximation with given accuracy to the unspecified isomorphism between the space of the holomorphic/anti-holomorphic spinor fields on surfaces of different radii. Obviously, similar ‘gauge’ solutions appear in the large sphere expansions, too. Therefore, without additional gauge fixing, in the expansion of a quasi-local quantity only the leading non-trivial term will be gauge-independent. In particular, the r⁶ order correction in Equation (61) for the Dougan-Mason energy-momenta is well-defined only as a consequence of a natural gauge choice13. Similarly, the higher order corrections in the large sphere limit of the anti-holomorphic Dougan-Mason energy-momentum are also ambiguous unless a ‘natural’ gauge choice is made. Such a choice is possible in stationary spacetimes.

The main idea

To motivate the main idea behind the Brown-York definition [96, 97], let us consider first a classical mechanical system of n degrees of freedom with configuration manifold Q and Lagrangian L : TQ×ℝ → ℝ (i.e. the Lagrangian is assumed to be first order and may depend on time explicitly). For given initial and final configurations, and , respectively, the corresponding action functional is , where q^a(t) is a smooth curve in Q from to with tangent at t. (The pair (q^a(t), t) may be called a history or world line in the ‘spacetime’ .) Let (q^a(u, t(u)), t(u)) be a smooth 1-parameter deformation of this history, i.e. for which (q^a(0, t(0)), t(0)) = (q^a(t), t), and u ∈ (−ϵ, ϵ) for some ϵ > 0. Then, denoting the derivative with respect to the deformation parameter u at u = 0 by δ, one has the well known expression

65

Therefore, introducing the Hamilton-Jacobi principal function as the value of the action on the solution q^a(t) of the equations of motion from to , the derivative of S¹ with respect to gives the canonical momenta , while its derivative with respect to t₂ gives minus the energy, , at t₂. Obviously, neither the action I¹ nor the principal function S¹ are unique: I[q(t)] := I¹[q(t)] − I⁰[q(t)] for any I⁰[q(t)] of the form with arbitrary smooth function h = h(q^a(t), t) is an equally good action for the same dynamics. Clearly, the subtraction term I⁰[q(t)] alters both the canonical momenta and the energy according to and E¹ ↦ E = E¹ + (∂h/∂t), respectively.

The variation of the action and the surface stress-energy tensor

The main idea of Brown and York [96, 97] is to calculate the analogous variation of an appropriate first order action of general relativity (or of the coupled matter + gravity system) and isolate the boundary term that could be analogous to the energy E above. To formulate this idea mathematically, they considered a compact spacetime domain D with topology Σ × [t₁, t₂] such that Σ × {t} correspond to compact spacelike hypersurfaces Σ_t; these form a smooth foliation of D and the 2-surfaces (corresponding to ∂Σ × {t}) form a foliation of the timelike 3-boundary ³B of D. Note that this D is not a globally hyperbolic domain14. To ensure the compatibility of the dynamics with this boundary, the shift vector is usually chosen to be tangent to on ³B. The orientation of ³B is chosen to be outward pointing, while the normals both of and to be future pointing. The metric and extrinsic curvature on Σ_t will be denoted, respectively, by h_ab and χ_ab, those on ³B by γ_ab and Θ_ab.

The primary requirement of Brown and York on the action is to provide a well-defined variational principle for the Einstein theory. This claim leads them to choose for I¹ the ‘trace K action’ (or, in the present notation, rather the ‘trace χ action’) for general relativity [405, 406, 387], and the action for the matter fields may be included. (For the minimal, non-derivative couplings the presence of the matter fields does not alter the subsequent expressions.) However, as Geoff Hayward pointed out [178], to have a well-defined variational principle, the ‘trace χ action’ should in fact be completed by two 2-surface integrals, one on and the other on . Otherwise, as a consequence of the edges and , called the ‘joints’ (i.e. the non-smooth parts of the boundary ∂D), the variation of the metric at the points of the edges and could not be arbitrary. (See also [177, 237, 77, 95], where the ‘orthogonal boundaries assumption’ is also relaxed.) Let η₁ and η₂ be the scalar product of the outward pointing normal of ³B and the future pointing normal of Σ₁ and of Σ₂, respectively. Then, varying the spacetime metric, for the variation of the corresponding principal function S¹ they obtained

66

The first two terms together correspond to the term of Equation (65), and, in fact, the familiar ADM expression for the canonical momentum is just . The last two terms give the effect of the presence of the non-differentiable ‘joints’. Therefore, it is the third term that should be analogous to the third term of Equation (65). In fact, roughly, this is proportional to the proper time separation of the ‘instants’ Σ₁ and Σ₂, and it is reasonable to identify its coefficient as some (quasi-local) analog of the energy. However, just as in the case of the mechanical system, the action (and the corresponding principal function) is not unique, and the principal function should be written as S := S¹ − S⁰, where S⁰ is assumed to be an arbitrary function of the 3-metric on the boundary ∂D = Σ₂ ∪ ³B ∪ Σ₁. Then

67

defines a symmetric tensor field on the timelike boundary ³B, and is called the surface stress-energy tensor. (Since our signature for γ_ab on ³B is (+, −, −) rather than (−, +, +), we should define τ^ab with the extra minus sign, just according to Equation (1).) Its divergence with respect to the connection ³D_e on ³B determined by γ_ab is proportional to the part γ^abT_bcv^c of the energy-momentum tensor, and hence, in particular, τ^ab is divergence-free in vacuum. Therefore, if (³B, γ_ab) admits a Killing vector, say K^a, then in vacuum

68

the flux integral of τ^abK_b on any spacelike cross section of ³B, is independent of the cross section itself, and hence defines a conserved charge. If K^a is timelike, then the corresponding charge is called a conserved mass, while for spacelike K^a with closed orbits in the charge is called angular momentum. (Here is not necessarily an element of the foliation of ³B, and is the unit normal to tangent to ³B.)

Clearly, the trace-χ action cannot be recovered as the volume integral of some scalar Lagrangian, because it is the Hilbert action plus a boundary integral of the trace χ, and the latter depends on the location of the boundary itself. Such a Lagrangian was found by Pons [317]. This depends on the coordinate system adapted to the boundary of the domain D of integration. An interesting feature of this Lagrangian is that it is second order in the derivatives of the metric, but it depends only on the first time derivative. A detailed analysis of the variational principle, the boundary conditions and the conserved charges is given. In particular, the asymptotic properties of this Lagrangian is similar to that of the ΓΓ Lagrangian of Einstein, rather than to that of Hilbert’s.

The general form of the Brown-York quasi-local energy

The 3+1 decomposition of the spacetime metric yields a 2+1 decomposition of the metric γ_ab, too. Let N and N^a be the lapse and the shift of this decomposition on ³B. Then the corresponding decomposition of τ^ab defines the energy, momentum, and spatial stress surface densities according to

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70

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where q_ab is the spacelike 2-metric, A_e is the SO(1, 1) vector potential on and is the projection to introduced in Section 4.1.2, and k_ab is the extrinsic curvature of corresponding to the normal v^a orthogonal to ³B, and k is its trace. The timelike boundary ³B defines a boost-gauge on the 2-surfaces (which coincides with that determined by the foliation Σ_t in the ‘orthogonal boundaries’ case). The gauge potential A_e is taken in this gauge. Thus, although ε and j_a on are built from the 2-surface data (in a particular boost-gauge), the spatial surface stress depends on the part t^a(∇_at_b)v^b of the acceleration of the foliation Σ_t too. Let ξ^a be any vector field on ³B tangent to ³B, and ξ^a = nt^a + n^a its 2+1 decomposition. Then we can form the charge integral (68) for the leaves of the foliation of ³B