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 . 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.
Living Rev. Relativity 12, (2009), 4
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