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

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

3.1.1 The root of the difficulties

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 $Φ a... Nb...$ and $gab$ . The former give the field equations, while the latter define the symmetric energy-momentum tensor. Moreover, $gab$ 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 $ab g$ can be traced back to the principle of equivalence [29 ], and introduces a huge gauge freedom in the dynamics of $ab g$ 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 $(M, g ) ab$ 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, $Tab$ is the source-current for gravity, like the current $JaA := δIp∕δAAa$ 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 [414 , 413 ]), 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 [391 ].) In fact, the physical dimension of the Bel–Robinson ‘energy-density’ $Tabcdtatbtctd$ is $cm −4$ , and hence (in the traditional units) there are no powers $A$ and $B$ such that $cAGBT tatbtctd abcd$ 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’ $bc d Tabcdtt t$ 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 [65 , 66 , 403 , 237 , 238 , 447 ].

3.1.2 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, for example, [282 , 396 ]), or a background (usually torsion free, but not necessarily flat) connection (see, for example, [264 ]), or the partial derivative coming from a local coordinate system (see, for example, [482 ]). 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 [447 ], for Hilbert’s second-order Lagrangian $LH := R ∕16πG$ in a fixed local coordinate system ${xα}$ and derivative operator $∂μ$ instead of $∇e$ , Equation (2.4 ) gives precisely Møller’s energy-momentum pseudotensor $M𝜃αβ$ , which was defined originally through the superpotential equation $∘ |g|(8πG 𝜃α − G α ) := ∂ ∪ αμ M β β μM β$ , where $∪ αμ := ∘ |g|gαρgμω(∂ g ) M β [ω ρ]β$ is the Møller superpotential [335 ]. (For another simple and natural introduction of Møller’s energy-momentum pseudotensor, see [121 ].) For the spin pseudotensor, Equation (2.2 ) gives

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

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 [447 , 448 ]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 $th 2k$ symmetric pseudotensor $αβ t(2k)$ is defined by $2 |g|k+1 (8πGt α(2βk) − Gαβ) := ∂μ∂ν[|g|k+1 (gαβgμν − gανgβμ)]$ (which, for $k = 0$ , reduces to the Landau–Lifshitz pseudotensor, the only symmetric pseudotensor that is a quadratic expression of the first derivatives of the metric) [201 ]. However, by Einstein’s equations, this definition implies that $∂α[|g|k+1 (tαβ + T αβ)] = 0 (2k)$ . 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 $1∕2 |g| Tαβ$ , while the second term in the divergence equation has an extra weight $|g|k+1∕2$ . Thus, there is only one pseudotensor in this series, $tαβ (− 1)$ , which satisfies the ‘conservation law’ with the correct weight. In particular, the Landau–Lifshitz pseudotensor $αβ t(0)$ 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 [447 , 448 ]). Excellent classical reviews on these (and several other) pseudotensors are [482 , 69 , 10 , 202 ], and for some recent ones (using background geometric structures) see, for example, [170 , 171 , 93 , 192 , 193 , 279 , 396 ].

A particularly useful and comprehensive recent review with many applications and an extended bibliography is that of Petrov [394 ]. We return to the discussion of pseudotensors in Sections 3.3.1 and 11.3.4.

3.1.3 Strategies to avoid pseudotensors I: Background metrics/connections

One way of avoiding the use of pseudotensorial quantities is to introduce an explicit background connection [264 ] or background metric [402 , 280 , 285 , 282 , 281 , 395 , 168 ]. (The superpotential of Katz, Bičák, and Lyndel-Bell [281 ] has been rediscovered recently by Chen and Nester [126 ] 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 [373 ] (see also [374 ]), in which the background (flat) metric is also fixed by using Synge’s world function.

3.1.4 Strategies to avoid pseudotensors II: The tetrad formalism

In the tetrad formulation of general relativity, the $gab$ -orthonormal frame fields ${Eaa}$ , $a-= 0,...,3$ , are chosen to be the gravitational field variables [489 , 288 ]. 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 (2.4 ) and (2.2 ), respectively. Not surprisingly at all, we recover the pseudotensorial quantities that we obtained in the metric formulation above. However, as realized by Møller [336 ], 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

(Here ${ 𝜗aa}$ is the one-form basis dual to ${Eaa}$ .) Although $L$ depends on the actual tetrad field ${Eaa} -$ , it is weakly $O (1,3)$ -invariant. Møller’s Lagrangian has a nice uniqueness property [378 ]: 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 [359 ], which turned out to be equivalent to Møller’s Lagrangian [486 ]. 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 [485 ]. Both Lagrangians yield a well-defined Hamiltonian, reproducing the standard ADM energy-momentum in asymptotically flat spacetimes.) The canonical energy-momentum $α 𝜃 β$ derived from Equation (3.1

) 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:

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

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 [447

]). However, $Ca [K ]$ 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, for example, [347

, 325 ]). 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 [346 , 349 ]. (For the existence and the potential obstruction to the existence of this gauge condition, see [352 ].) Furthermore, since $Ca [K ] + T abKb$ 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., [447

, 448

, 369 , 263 ].

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 [187 ], which is known to be equivalent to the vanishing of the second Stiefel–Whitney class of $M$ [332 ], a global topological condition on $M$ .

3.1.5 Strategies to avoid pseudotensors III: Higher derivative currents

Giving up the paradigm that the Noether current should depend only on the vector field $Ka$ and its first derivative – i.e., if we allow a term $˙a B$ to be present in the Noether current (2.3 ), even if the Lagrangian is diffeomorphism invariant – one naturally arrives at Komar’s tensorial superpotential $K ∨ [K ]ab := ∇ [aKb ]$ and the corresponding Noether current [296 ] (see also [69 ]). Although its independence of any background structure (viz. its tensorial nature) and its uniqueness property (see Komar [296 ] 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 [213 ]. 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 11.1.