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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 was a functional of both kinds of fields, thus one could take the variational derivatives both with respect to a... PN b... and ab g. The former gave the field equations, while the latter defined the symmetric energy-momentum tensor. Moreover, gab provided 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 [22Jump To The Next Citation Point], 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, f*gab) are considered to be equivalent for any diffeomorphism f 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 a A JA := dIp/dAa 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 A A a), 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 := yABCDyA'B'C'D', given in terms of the Weyl spinor, (and its generalizations introduced by Senovilla [333332]), 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 [312Jump To The Next Citation Point].) 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 cAGB Tabcdtatbtctd 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 tbtctd abcd 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 [5657323193194352Jump To The Next Citation Point].

3.1.2 Pseudotensors

The lack of any background geometric structure in the gravitational action yields, first, that any vector field a K 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 [231Jump To The Next Citation Point316Jump To The Next Citation Point]), or a background (usually torsion free, but not necessarily flat) connection (see for example [215Jump To The Next Citation Point]), or the partial derivative coming from a local coordinate system (see for example [382Jump 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 [352Jump To The Next Citation Point], for Hilbert’s second order Lagrangian LH := R/16pG in a fixed local coordinate system {xa} and derivative operator @ m instead of \~/ e, Equation (4View Equation) gives precisely Møller’s energy-momentum pseudotensor a Mh b, which was defined originally through the superpotential equation V~ --- |g|(8pG Mhab - Gab) = @mM U b am, where V~ --- M U b am := |g|gargmw(@[wgr]b) is the so-called Møller superpotential [270]. (For another simple and natural introduction of Møller’s energy-momentum pseudotensor see [104Jump To The Next Citation Point].) For the spin pseudotensor Equation (2View Equation) gives

V ~ --- ma am ( V~ --- [m n]a) 8pG |g|Ms b = - M U b + @n |g|db 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 minus the Einstein tensor [352Jump To The Next Citation Point353Jump 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 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 tab (2k) is defined by k+1 ab k+1 2 |g| (8pGt (2k)- Gab) := @m@n[|g| (gabgmn - gangbm)] (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 k+1 ab ab @a[| 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 1/2 ab |g| T, 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, ta(-b1), which satisfies the ‘conservation law’ with the correct weight. In particular, the Landau-Lifshitz pseudotensor ab 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 difficulties (see also [352Jump To The Next Citation Point353Jump To The Next Citation Point]). Classical excellent reviews on these (and several other) pseudotensors are [382Jump To The Next Citation Point59Jump To The Next Citation Point9163Jump To The Next Citation Point], and for some recent ones (using background geometric structures) see for example [13713879154155228316]. 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 the pseudotensorial quantities is to introduce an explicit background connection [215Jump To The Next Citation Point] or background metric [322229Jump To The Next Citation Point233Jump To The Next Citation Point231230Jump To The Next Citation Point315135Jump To The Next Citation Point]. (The superpotential of Katz, Bičák, and Lyndel-Bell [230Jump To The Next Citation Point] has been rediscovered recently by Chen and Nester [108Jump To The Next Citation Point] 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 energy-momentum and spin tensors, but to define the vector fields Ka 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 a K: 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.

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 [386236]. 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 (4View Equation) and (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 [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 ga- := Ee ga := Ee ha- \~/ eEa eb e eb e a b, the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is

[ ( )] L := --1--- R - 2 \~/ Ea jabgc- = --1---(Ea Eb - EbEa )ga- gcb. (9) 16pG a a cb 16pG a b a b ac b
(Here {ha-} a is the 1-form basis dual to {Ea } a.) Although L depends on the actual tetrad field {Ea } a, 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]. UpdateJump To The Next Update Information 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 hab derived from Equation (9View 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 - gbcj + gdcj db ds- ds db hbE aE e = \/ b . (10)
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 ( b ac) 8pGC [K] = G Kb + 12 \~/ c K \/ b . (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 [352Jump To The Next Citation Point]). 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 well-defined energy-momentum density for the gravitational ‘field’ (see for example [282Jump To The Next Citation Point263]). 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 [281284]. Furthermore, since a ab C [K] + T Kb 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.

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 Ba to be present in the Noether current (3View Equation) even if the Lagrangian is diffeomorphism invariant - one naturally arrives at Komar’s tensorial superpotential ab [a b] K \/ [K] := \~/ K and the corresponding Noether current [242Jump To The Next Citation Point] (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 Ka is still to be determined.


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