In Newton’s theory the gravitational field is represented by a singe scalar field on the flat 3-space
satisfying the Poisson equation
. (Here
is the flat (negative definite)
metric,
is the corresponding Levi-Civita covariant derivative operator and
is the (non-negative)
mass density of the matter source.) Hence, the mass of the source contained in some finite three-volume
can be expressed as the flux integral of the gravitational field strength on the boundary
:
In a relativistically corrected Newtonian theory both the internal energy density of the (matter)
source and the energy density
of the gravitational field itself contribute to the source of gravity. Thus
(in the traditional units, when
is the speed of light) the corrected field equation could be expected to be
the genuinely non-linear equation
By the negative definiteness of , outside the source the quasi-local energy
is a decreasing set
function, i.e., if
and
is source free, then
. In particular, for a 2-sphere of
radius
surrounding a localized spherically symmetric homogeneous source with negligible internal
energy, the quasi-local energy is
, where the mass parameter is
and
is the rest mass and
is the radius of the source. For a
more detailed discussion of the energy in the (relativistically corrected) Newtonian theory,
see [199
].
The action for the matter fields is a functional of both kinds of fields, thus one can take the variational
derivatives both with respect to
and
. The former give the field equations, while the latter
define the symmetric energy-momentum tensor. Moreover,
provides 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 [36
] (i.e., the
Galileo–Eötvös experiment), 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 density 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 [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 [425
].) 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. 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 2.1, to gravity
too [73, 74, 438, 259, 260, 486
].
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,
nondynamic background metric (see, e.g., [307
, 430
]), or a background (usually torsion free, but
not necessarily flat) connection (see, e.g., [287
]), or the partial derivative coming from a local
coordinate system (see, e.g., [525
]). 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 [486], for Hilbert’s second-order Lagrangian
in a fixed local coordinate
system
and derivative operator
instead of
, Eq. (2.4
) gives precisely Møller’s
energy-momentum pseudotensor
, which was defined originally through the superpotential equation
, where
is the Møller
superpotential [367]. (For another simple and natural introduction of Møller’s energy-momentum
pseudotensor, see [131
].) For the spin pseudotensor, Eq. (2.2
) 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 the negative of the Einstein
tensor [486, 487
].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 Goldberg’s
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) [222]. 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 difficulty
(see also [486
, 487
]). Excellent classical reviews on these (and several other) pseudotensors
are [525
, 77
, 15, 223
], 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.
One way of avoiding the use of pseudotensorial quantities is to introduce an explicit background
connection [287] or background metric [437, 305
, 310
, 307, 306
, 429, 184
]. (The superpotential of Katz,
Bičák, and Lynden-Bell [306
] has been rediscovered recently by Chen and Nester [137
] 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
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. 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.
In the tetrad formulation of general relativity, the -orthonormal frame fields
,
,
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. (2.4
) 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 [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
, the Ricci rotation coefficients, then Møller’s tetrad Lagrangian is
In general, the frame field is defined only on an open subset
. If the domain of the
frame field can be extended to the whole
, then
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
[364], a global topological
condition on
.
The discussion of how Møller’s superpotential 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.
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 (2.3
), even if the
Lagrangian is diffeomorphism invariant – one naturally arrives at Komar’s tensorial superpotential
and the corresponding Noether current
[322
] (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
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 [235
]. 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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