The action for the matter fields was a functional of both kinds of fields, thus one could take the
variational derivatives both with respect to and . The former gave the field equations, while
the latter defined the symmetric energy-momentum tensor. Moreover, provided a metrical geometric
background, in particular a covariant derivative, for carrying out the analysis of the matter fields. The
gravitational action is, on the other hand, a functional of the metric alone, and its variational
derivative with respect to yields the gravitational field equations. The lack of any further
geometric background for describing the dynamics of can be traced back to the principle of
equivalence [22], and introduces a huge gauge freedom in the dynamics of because that
should be formulated on a bare manifold: The physical spacetime is not simply a manifold
endowed with a Lorentzian metric , but the isomorphism class of such pairs, where
and are considered to be equivalent for any diffeomorphism of 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, 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 and the Noether currents, the proper interpretation of
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’ is , and hence (in the
traditional units) there are no powers and such that would have
energy-density dimension. Here is the speed of light and is Newton’s gravitational
constant. As we will see, the Bel-Robinson ‘energy-momentum density’ 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].

The lack of any background geometric structure in the gravitational action yields, first, that any vector field 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 in a fixed local coordinate system and derivative operator instead of , 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 Goldberg’s -th symmetric pseudotensor is defined
by (which, for , 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.

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 energy-momentum and spin tensors, but to define the vector fields 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 : 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.

In the tetrad formulation of general relativity the -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

(Here is the 1-form basis dual to .) Although depends on the actual tetrad field , it is weakly -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]. Update 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: 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 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 [352]). However, depends on the actual tetrad field, and hence a preferred class of frame fields, i.e. an -gauge reduction, is needed. Thus the explicit background-dependence of the final result of other approaches has been transformed into an internal -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 is conserved for any vector field , 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.

Giving up the paradigm that the Noether current should depend only on the vector field and its first derivative - i.e. if we allow a term to be present in the Noether current (3) even if the Lagrangian is diffeomorphism invariant - one naturally arrives at Komar’s tensorial superpotential 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 is still to be determined.

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