3.1 On the gravitational energy-momentum and angular momentum density: The difficulties

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3.1.1 The root of the difficulties: Gravitational energy in Newton’s theory

In Newton’s theory the gravitational field is represented by a singe scalar field ϕ on the flat 3-space Σ ≈ ℝ3 satisfying the Poisson equation − habDaDb ϕ = 4πG ρ. (Here hab is the flat (negative definite) metric, Da is the corresponding Levi-Civita covariant derivative operator and ρ is the (non-negative) mass density of the matter source.) Hence, the mass of the source contained in some finite three-volume D ⊂ Σ can be expressed as the flux integral of the gravitational field strength on the boundary 𝒮 := ∂D:

∮ m = --1-- va(D ϕ) d𝒮, (3.1 ) D 4πG 𝒮 a
where a v is the outward-directed unit normal to 𝒮. If 𝒮 is deformed in Σ through a source-free region, then the mass does not change. Thus, the rest mass mD of the source is analogous to charge in electrostatics. Following the analogy with electrostatics, we can introduce the energy density and the spatial stress of the gravitational field, respectively, by
-1---cd( )( ) --1--(( )( ) 1- cd( )( )) U := 8πG h Dc ϕ Dd ϕ , Σab := 4πG Da ϕ Db ϕ − 2 habh Dcϕ Dd ϕ . (3.2 )
Note that since gravitation is always attractive, U is a binding energy, and hence it is negative definite. However, by the Galileo–Eötvös experiment, i.e., the principle of equivalence, there is an ambiguity in the gravitational force: It is determined only up to an additive constant covector field ae, and hence by an appropriate transformation De ϕ ↦→ De ϕ + ae the gravitational force De ϕ at a given point p ∈ Σ can be made zero. Thus, at this point both the gravitational energy density and the spatial stress have been made vanishing. On the other hand, they can be made vanishing on an open subset U ⊂ Σ only if the tidal force, D D ϕ a b, is vanishing on U. Therefore, the gravitational energy and the spatial stress cannot be localized to a point, i.e., they suffer from the ambiguity in the gravitational force above.

In a relativistically corrected Newtonian theory both the internal energy density u of the (matter) source and the energy density U of the gravitational field itself contribute to the source of gravity. Thus (in the traditional units, when c is the speed of light) the corrected field equation could be expected to be the genuinely non-linear equation

( 1( )) − habDaDb ϕ = 4 πG ρ + -- u + U . (3.3 ) c2
(Note that, together with additional corrections, this equation with the correct sign of U can be recovered from Einstein’s equations applied to static configurations [199Jump To The Next Citation Point] in the first post-Newtonian approximation. Note, however, that the theory defined by (3.3View Equation) and the usual formula for the force density, is internally inconsistent [221Jump To The Next Citation Point]. A thorough analysis of this theory, and in particular its inconsistency, is given by Giulini [221].) Therefore, by (3.3View Equation)
∫ 2 ∮ : ( 2 ) -c--- a ED = D c ρ + u + U d Σ = 4πG 𝒮 v (Daϕ )d𝒮, (3.4 )
i.e., now it is the energy of the source plus gravity system in the domain D that can be rewritten into the form of a two-surface integral on the boundary of the domain D. Note that the gravitational energy reduces the source term in (3.3View Equation) (and hence the energy ED also), and, more importantly, the quasi-local energy ED of the source + gravity system is free of the ambiguity that is present in the gravitational energy density. This in itself already justifies the introduction and use of the quasi-local concept of energy in the study of gravitating systems.

By the negative definiteness of U, outside the source the quasi-local energy ED is a decreasing set function, i.e., if D1 ⊂ D2 and D2 − D1 is source free, then ED2 ≤ ED1. In particular, for a 2-sphere of radius r surrounding a localized spherically symmetric homogeneous source with negligible internal energy, the quasi-local energy is EDr = c4m(1 + 1 m) + O (r−2) G 2 r, where the mass parameter is m := GM-(1 − 3GM-) + O(c−6) c2 5c2R and M is the rest mass and R is the radius of the source. For a more detailed discussion of the energy in the (relativistically corrected) Newtonian theory, see [199Jump To The Next Citation Point].

3.1.2 The root of the difficulties: Gravitational energy-momentum in Einstein’s theory

The action Im for the matter fields is a functional of both kinds of fields, thus one can take the variational derivatives both with respect to ΦN a... b... and gab. The former give the field equations, while the latter define the symmetric energy-momentum tensor. Moreover, g ab provides a metrical geometric background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The gravitational action Ig is, on the other hand, a functional of the metric alone, and its variational derivative with respect to gab yields the gravitational field equations. The lack of any further geometric background for describing the dynamics of gab can be traced back to the principle of equivalence [36Jump To The Next Citation Point] (i.e., the Galileo–Eötvös experiment), and introduces a huge gauge freedom in the dynamics of gab because that should be formulated on a bare manifold: The physical spacetime is not simply a manifold M endowed with a Lorentzian metric gab, but the isomorphism class of such pairs, where (M, gab) and (M, ϕ ∗gab) are considered to be equivalent for any diffeomorphism ϕ of M onto itself.2 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 density for gravity, like the current a A JA := δIp∕δA a 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 AAa), 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 Tab and the Noether currents, the proper interpretation of Tab 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 Tabcd := ψABCD ¯ψA ′B′C′D′, given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [449, 448]), 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 of the expression above. Our convention follows that of Penrose and Rindler [425Jump To The Next Citation Point].) In fact, the physical dimension of the Bel–Robinson ‘energy-density’ T tatbtctd abcd is cm −4, and hence (in the traditional units) there are no powers A and B such that A B a b cd c G Tabcdt t tt would have energy-density dimension. As we will see, the Bel–Robinson ‘energy-momentum density’ Tabcdtbtctd 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 2.1, to gravity too [73, 74, 438, 259, 260, 486Jump To The Next Citation Point].

3.1.3 Pseudotensors

The lack of any background geometric structure in the gravitational action yields, first, that any vector field Ka 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, nondynamic background metric (see, e.g., [307Jump To The Next Citation Point, 430Jump To The Next Citation Point]), or a background (usually torsion free, but not necessarily flat) connection (see, e.g., [287Jump To The Next Citation Point]), or the partial derivative coming from a local coordinate system (see, e.g., [525Jump To The Next Citation Point]). 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 [486Jump To The Next Citation Point], for Hilbert’s second-order Lagrangian LH := R ∕16πG in a fixed local coordinate system {xα} and derivative operator ∂μ instead of ∇e, Eq. (2.4View Equation) gives precisely Møller’s energy-momentum pseudotensor 𝜃α M β, which was defined originally through the superpotential equation ∘ --- α α αμ |g|(8πGM 𝜃 β − G β) := ∂μM ∪β, where αμ ∘ --- αρ μω M ∪ β := |g|g g (∂[ωgρ]β) is the Møller superpotential [367]. (For another simple and natural introduction of Møller’s energy-momentum pseudotensor, see [131Jump To The Next Citation Point].) For the spin pseudotensor, Eq. (2.2View Equation) gives

∘ --- ( ∘ --- ) 8πG |g|Mσ μαβ = − M ∪β αμ + ∂ν |g|δβ[μgν]α ,

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 the negative of the Einstein tensor [486Jump To The Next Citation Point, 487Jump To The Next Citation Point].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 a choice for the reference configuration.

A further difficulty is that the different pseudotensors may have different (potential) significance. For example, for any fixed k ∈ ℝ Goldberg’s 2kth symmetric pseudotensor tαβ (2k) is defined by k+1 αβ k+1 2 |g| (8πGt (2k) − Gαβ) := ∂μ∂ν[|g| (gαβgμν − gανgβμ)] (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) [222]. However, by Einstein’s equations, this definition implies that k+1 αβ αβ ∂α[|g| (t(2k) + T )] = 0. 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 |g|1∕2 Tαβ, while the second term in the divergence equation has an extra weight k+1∕2 |g|. Thus, there is only one pseudotensor in this series, αβ t(− 1), which satisfies the ‘conservation law’ with the correct weight. In particular, the Landau–Lifshitz pseudotensor tαβ (0) 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 difficulty (see also [486Jump To The Next Citation Point, 487Jump To The Next Citation Point]). Excellent classical reviews on these (and several other) pseudotensors are [525Jump To The Next Citation Point, 77Jump To The Next Citation Point, 15, 223Jump To The Next Citation Point], and for some recent ones (using background geometric structures) see, e.g., [186, 187, 102, 211, 212, 304, 430].

A particularly useful and comprehensive recent review with many applications and an extended bibliography is that of Petrov [428]. We return to the discussion of pseudotensors in Sections 3.3.1, 4.2.2 and 11.3.5.

3.1.4 Strategies to avoid pseudotensors I: Background metrics/connections

One way of avoiding the use of pseudotensorial quantities is to introduce an explicit background connection [287Jump To The Next Citation Point] or background metric [437, 305Jump To The Next Citation Point, 310Jump To The Next Citation Point, 307, 306Jump To The Next Citation Point, 429, 184Jump To The Next Citation Point]. (The superpotential of Katz, Bičák, and Lynden-Bell [306Jump To The Next Citation Point] has been rediscovered recently by Chen and Nester [137Jump To The Next Citation Point] in a completely different way. We return to a discussion of the approach of Chen and Nester in Section 11.3.2.) The advantage of this approach would be that we could use the background not only to derive the canonical energy-momentum and spin tensors, but to define the vector fields a K 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 Ka: 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. A particularly interesting special bimetric approach was suggested in [407] (see also [408]), in which the background (flat) metric is also fixed by using Synge’s world function.

3.1.5 Strategies to avoid pseudotensors II: The tetrad formalism

In the tetrad formulation of general relativity, the gab-orthonormal frame fields a {E a}, a-= 0,...,3, are chosen to be the gravitational field variables [533, 314]. 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 Eqs. (2.4View Equation) and (2.2View Equation), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [368], 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 γaeb := Eeeγaeb := Eee𝜗aa∇eEab -- - - - -, the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is

[ ( )] L := --1--- R − 2∇ Ea ηabγc = --1---(Ea Eb − Eb Ea )γa γcb. (3.5 ) 16πG a a cb 16πG a b a b ac b
(Here { 𝜗a} a is the one-form basis dual to {Ea} a.) Although L depends on the actual tetrad field a {E a}, it is weakly O (1,3)-invariant. Møller’s Lagrangian has a nice uniqueness property [412]: 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 [392], which turned out to be equivalent to Møller’s Lagrangian [530]. 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 [529]. Both Lagrangians yield a well-defined Hamiltonian, reproducing the standard ADM energy-momentum in asymptotically flat spacetimes.) The canonical energy-momentum 𝜃αβ derived from Eq. (3.5View Equation) 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:
ae ( a ce d cs ( a e a e)) b a e [ae] ∨b := 2 − γbcη + γdcη δbδs − δsδb 𝜗bE aEe = ∨b . (3.6 )
The canonical spin turns out to be essentially ae ∨b, 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 Ca[K ] will also be tensorial and satisfies
a ab 1 ( b ac) 8πGC [K ] = G Kb + 2∇c K ∨b . (3.7 )
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 [486Jump To The Next Citation Point]). However, a C [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 a well defined energy-momentum density for the gravitational ‘field’ (see, e.g., [379Jump To The Next Citation Point, 353]). 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 [378, 381]. (For the existence and the potential obstruction to the existence of the solutions to this gauge condition on spacelike hypersurfaces, see [384, 196].) Furthermore, since Ca [K ] + TabKb is conserved for any vector field Ka, 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. For recent reviews of the tetrad formalism of general relativity, including an extended bibliography, see, e.g., [486Jump To The Next Citation Point, 487Jump To The Next Citation Point, 403, 286].

In general, the frame field a {Ea } is defined only on an open subset U ⊂ M. If the domain of the frame field can be extended to the whole M, then M is called parallelizable. For time and space-orientable spacetimes this is equivalent to the existence of a spinor structure [206], which is known to be equivalent to the vanishing of the second Stiefel–Whitney class of M [364], a global topological condition on M.

The discussion of how Møller’s superpotential ab ∨e is related to the Nester–Witten 2-form, by means of which an alternative form of the ADM energy-momentum is given and and by means of which several quasi-local energy-momentum expressions are defined, is given in Section 3.2.1 and in the first paragraphs of Section 8.

3.1.6 Strategies to avoid pseudotensors III: Higher derivative currents

Giving up the paradigm that the Noether current should depend only on the vector field a K and its first derivative – i.e., if we allow a term B˙a to be present in the Noether current (2.3View Equation), even if the Lagrangian is diffeomorphism invariant – one naturally arrives at Komar’s tensorial superpotential ∨ [K ]ab := ∇ [aKb ] K and the corresponding Noether current Ca [K ] := Ga Kb + ∇ ∇[aKb ] b b [322Jump To The Next Citation Point] (see also [77]). Although its independence of any background structure (viz. its tensorial nature) and its uniqueness property (see Komar [322] quoting Sachs) is especially attractive, the vector field a K is still to be determined. A new suggestion for the approximate spacetime symmetries that can, in principle, be used in Komar’s expression, both near a point and a world line, is given in [235Jump To The Next Citation Point]. This is a generalization of the affine collineations (including the homotheties and the Killing symmetries). We continue the discussion of the Komar expression in Sections 3.2.2, 3.2.3, 4.3.1 and 12.1, and of the approximate spacetime symmetries in Section 11.1.

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