It is a widely accepted view (appearing e.g. in excellent, standard textbooks on general relativity, too) that the canonical energy-momentum and spin tensors are well-defined and have relevance only in flat spacetime, and hence usually are underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus first we 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 other 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 volume integral
of
on some open domain
with compact closure then the equations of motion are
, the Euler-Lagrange
equations. The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by
Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that for any vector field
and the corresponding local 1-parameter family of diffeomorphisms
one has
for some 1-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
so-called Noether identity:
, where
denotes the
Lie derivative along
, and
, the so-called Noether current, is given explicitly by
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 [389
, 215
, 141
].)
The interpretation of the conserved currents and
depends on the nature of the Killing
vector
. In Minkowski spacetime the 10-dimensional Lie algebra
of the Killing vectors is well
known to split to the semidirect sum of a 4-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 1-form basis dual to
has the form
) the general
form of the Killing vectors (or rather 1-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
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 the 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.
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