## 8 Approaches Based on the Nester-Witten 2-Form

We saw in Section 3.2 that
• both the ADM and Bondi-Sachs energy-momenta can be re-expressed by the integral of the Nester-Witten 2-form ,
• the proof of the positivity of the ADM and Bondi-Sachs masses is relatively simple in terms of the 2-component spinors, and
• the integral of Møller’s tetrad superpotential for the energy-momentum, coming from his tetrad Lagrangian (9), is just the integral of , where is a normalized spinor dyad.

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.