It is a widely accepted view that the canonical energy-momentum and spin tensors are well defined and have relevance only in flat spacetime, and, hence, are usually underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus, we first introduce these quantities for the matter fields in a general curved spacetime.
To specify the state of the matter fields operationally, two kinds of devices are needed: the first measures
the value of the fields, while the other measures the spatio-temporal location of the first. Correspondingly,
the fields on the manifold of events can be grouped into two sharply-distinguished classes. The first
contains the matter field variables, e.g., finitely many
-type tensor fields
, whilst the
second contains the fields specifying the spacetime geometry, i.e., the metric
in Einstein’s theory.
Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a
Lagrangian
. If
is the action functional, i.e., the
volume integral of
on some open domain
with compact closure, then the equations of motion
are
Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that, for any vector
field and the corresponding local one-parameter family of diffeomorphisms
, one
has
for some one-parameter family of vector fields . (
is called diffeomorphism
invariant if
, e.g., when
is a scalar.) Let
be any smooth vector field on
. Then,
calculating the divergence
to determine the rate of change of the action functional
along the integral curves of
, by a tedious but straightforward computation, one
can derive the Noether identity:
, where
denotes the Lie derivative along
, and
, the Noether current, is given explicitly by
However, is sensitive to total divergences added to the Lagrangian, and, if the matter fields
have gauge freedom (e.g., if the matter is a Maxwell or Yang–Mills field), then in general it is not gauge
invariant, even if the Lagrangian is. On the other hand,
is gauge invariant and is independent of
total divergences added to
because it is the variational derivative of the gauge invariant
action with respect to the metric. Provided the field equations are satisfied, the Noether identity
implies [73
, 74
, 438
, 259
, 260
] that
Hence, is also conserved and can equally be considered as a Noether current. (For a formally different,
but essentially equivalent, introduction of the Noether current and identity, see [536
, 287
, 191
].)
The interpretation of the conserved currents, and
, depends on the nature of the
Killing vector,
. In Minkowski spacetime the ten-dimensional Lie algebra
of the Killing vectors is
well known to split into the semidirect sum of a four-dimensional commutative ideal,
, and the quotient
, where the latter is isomorphic to
. The ideal
is spanned by the constant
Killing vectors, in which a constant orthonormal frame field
on
,
,
forms a basis. (Thus, the underlined Roman indices
,
, … are concrete, name indices.) By
the ideal
inherits a natural Lorentzian vector space structure.
Having chosen an origin
, the quotient
can be identified as the Lie algebra
of
the boost-rotation Killing vectors that vanish at
. Thus,
has a ‘4 + 6’ decomposition
into translations and boost rotations, where the translations are canonically defined but the
boost-rotations depend on the choice of the origin
. In the coordinate system
adapted to
(i.e., for which the one-form basis dual to
has the form
), the
general form of the Killing vectors (or rather one-forms) is
for
some constants
and
. Then, the corresponding canonical Noether current is
, and the coefficients of the translation and the
boost-rotation parameters
and
are interpreted as the density of the energy-momentum and of
the sum of the orbital and spin angular momenta, respectively. Since, however, the difference
is identically conserved and
has more advantageous properties, it is
that is used to represent the energy-momentum and angular-momentum density of the matter
fields.
Since in de Sitter and anti-de Sitter spacetimes the (ten-dimensional) Lie algebra of the Killing vector
fields, and
, respectively, are semisimple, there is no such natural notion of translations,
and hence no natural ‘4 + 6’ decomposition of the ten conserved currents into energy-momentum and
(relativistic) angular momentum density.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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