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8 Approaches Based on the Nester-Witten 2-Form

We saw in Section 3.2 that

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 S is any closed, orientable spacelike 2-surface and cA, mA are arbitrary spinor fields, then in the integral H [c,m] S, defined by Equation (55View Equation), only the tangential derivative of c A appears. (m A is involved in HS [c, m] algebraically.) Thus, by Equation (13View Equation), o o o o HS : C (S,SA) × C (S, SA) --> C is a Hermitian scalar product on the (infinite-dimensional complex) vector space of smooth spinor fields on S. Thus, in particular, the spinor fields in HS [c, m] need be defined only on S, and -------- HS [c,m] = HS [m,c] holds. A remarkable property of HS is that if cA is a constant spinor field on S with respect to the covariant derivative D e, then H [c,m] = 0 S for any smooth spinor field m A on S. Furthermore, if A- 0 1 c A = (cA, cA) is any pair of smooth spinor fields on S, then for any constant SL(2, C) matrix /\A-B- one has ' ' ' ' HS [cC-/\C-A,cD--/\D-'B-] = HS [cC-,cD--]/\C--A/\D-'B-, i.e. the integrals ' HS [cA-,cB--] transform as the spinor components of a real Lorentz vector over the two-complex dimensional space spanned by c0 A and c1 A. Therefore, to have a well-defined quasi-local energy-momentum vector we have to specify some 2-dimensional subspace A- S of the infinite-dimensional space o o C (S, SA) and a symplectic metric eA-B- 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 S. 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 cA- A could also be interpreted as the spinor basis that should be used to construct the orthonormal vector basis in Møller’s superpotential (10View Equation). In this sense the choice of the subspace SA- and the metric eAB- is just a gauge reduction, or a choice for the ‘reference configuration’ of Section 3.3.3.

Once the spin space (SA-,eAB-) is chosen, the quasi-local energy-momentum is defined to be AB-' A- B-' P S := HS [c ,c ] and the corresponding quasi-local mass mS is 2 ' ' A-A' B-B-' m S := eA-BeA-B-P S PS. In particular, if one of the spinor fields A cA-, e.g. c0A, is constant on S (which means that the geometry of S is considerably restricted), then P0S0'= P0S1'= P1S0'= 0, and hence the corresponding mass mS is zero. If both c0 A and c1 A are constant (in particular, when they are the restrictions to S of the two constant spinor fields in the Minkowski spacetime), then AB-' PS itself is vanishing.

Therefore, to summarize, the only thing that needs to be specified is the spin space (SA-,eAB-), 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.

 8.1 The Ludvigsen-Vickers construction
  8.1.1 The definition
  8.1.2 Remarks on the validity of the construction
  8.1.3 Monotonicity, mass-positivity and the various limits
 8.2 The Dougan-Mason constructions
  8.2.1 Holomorphic/anti-holomorphic spinor fields
  8.2.2 The genericity of the generic 2-surfaces
  8.2.3 Positivity properties
  8.2.4 The various limits
 8.3 A specific construction for the Kerr spacetime

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