Thus, both from conceptual and pragmatic points of views, it seems natural to search for the quasi-local energy-momentum in the form of the integral of the Nester-Witten 2-form. All the quasi-local energy-momenta based on the integral of the Nester-Witten 2-form have a natural Lagrangian interpretation [362]. Thus first let us discuss briefly the most important properties of such integrals.
If is any closed, orientable spacelike 2-surface and
,
are arbitrary spinor fields, then in the
integral
, defined by Equation (55
), only the tangential derivative of
appears. (
is
involved in
algebraically.) Thus, by Equation (13
),
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 2-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
. 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 as 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 (10
). In this sense the choice of the subspace
and the metric
is just a gauge reduction, or a choice for the ‘reference configuration’ of
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.
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