We saw in Section 3.2 that
Thus, from a pragmatic point of view, it seems natural to search for the quasi-local energy-momentum in the form of the integral of the Nester–Witten 2-form. Now we will show that
Hence, all the quasi-local energy-momenta based on the integral of the Nester–Witten 2-form have a natural Lagrangian interpretation in the sense that they are charge integrals of the canonical Noether current derived from Møller’s first-order tetrad Lagrangian.
If is any closed, orientable spacelike two-surface and an open neighborhood of
is time and space
orientable, then an open neighborhood of
is always a trivialization domain of both the orthonormal and
the spin frame bundles [500
]. Therefore, the orthonormal frame
can be chosen to be globally
defined on
, and the integral of the dual of Møller’s superpotential,
, appearing on the
right-hand side of the superpotential Eq. (3.7
), is well defined. If
is a pair of globally-defined
normals of
in the spacetime, then in terms of the geometric objects introduced in Section 4.1, this
integral takes the form
Then, suppose that the orthonormal basis is built from a normalized spinor dyad, i.e., ,
where
are the
Pauli matrices (divided by
) and
,
, is a normalized
spinor basis. A straightforward calculation yields the following remarkable expression for the dual of
Møller’s superpotential:
Next we will discuss some general properties of the integral of , where
and
are arbitrary spinor fields on
. Then, in the integral
, defined by Eq. (7.14
),
only the tangential derivative of
appears. (
is involved in
algebraically.)
Thus, by Eq. (3.11
),
is a Hermitian scalar product on the
(infinite-dimensional complex) vector space of smooth spinor fields on
. Thus, in particular, the
spinor fields in
need be defined only on
, and
holds. A
remarkable property of
is that if
is a constant spinor field on
with respect to
the covariant derivative
, then
for any smooth spinor field
on
.
Furthermore, if
is any pair of smooth spinor fields on
, then for any constant
matrix
one has
, i.e., the
integrals
transform as the spinor components of a real Lorentz vector over the
two–complex-dimensional space spanned by
and
. Therefore, to have a well-defined
quasi-local energy-momentum vector we have to specify some two-dimensional subspace
of the
infinite-dimensional space
and a symplectic metric
thereon. Thus, underlined
capital Roman indices will be referring to this space. The elements of this subspace would be
interpreted as the spinor constituents of the ‘quasi-translations’ of the surface
. Note, however,
that in general the symplectic metric
need not be related to the pointwise symplectic
metric
on the spinor spaces, i.e., the spinor fields
and
that span
are not
expected to form a normalized spin frame on
. Since, in Møller’s tetrad approach it is natural
to choose the orthonormal vector basis to be a basis in which the translations have constant
components (just like the constant orthonormal bases in Minkowski spacetime, which are bases in the
space of translations), the spinor fields
could also be interpreted as the spinor basis that
should be used to construct the orthonormal vector basis in Møller’s superpotential (3.6
). In
this sense the choice of the subspace
and the metric
is just a gauge reduction (see
Section 3.3.3).
Once the spin space is chosen, the quasi-local energy-momentum is defined to be
and the corresponding quasi-local mass
is
. In
particular, if one of the spinor fields
, e.g.,
, is constant on
(which means that the geometry of
is considerably restricted), then
, and hence, the corresponding mass
is zero. If both
and
are constant (in particular, when they are the restrictions
to
of the two constant spinor fields in the Minkowski spacetime), then
itself is
vanishing.
Therefore, to summarize, the only thing that needs to be specified is the spin space , and the
various suggestions for the quasi-local energy-momentum based on the integral of the Nester–Witten 2-form
correspond to the various choices for this spin space.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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