However, in spite of their successes, the Dougan–Mason energy-momenta and the spin angular momenta based on Bramson’s superpotential and the holomorphic/antiholomorphic spinor fields have some unsatisfactory properties, as well (see the lists of our expectations in Section 4.3). First, they are defined only for topological two-spheres (but not for other topologies, e.g., for the torus ), and, even for certain topological two-spheres, they are not well defined. Such surfaces are, for example, past marginally trapped surfaces in the antiholomorphic (and future marginally trapped surfaces in the holomorphic) case. Although the quasi-local mass associated with a marginally trapped surface is expected to be its irreducible mass , neither of the Dougan–Mason masses is well defined for the bifurcation surfaces of the Kerr–Newman (or even Schwarzschild) black hole. Second, the role and the physical content of the holomorphicity/antiholomorphicity of the spinor fields is not clear. The use of the complex structure is justified a posteriori by the nice physical properties of the constructions and the pure mathematical fact that it is only the holomorphy and antiholomorphy operators in a large class of potentially acceptable first-order linear differential operators acting on spinor fields that have a two-dimensional kernel. Furthermore, since the holomorphic and antiholomorphic constructions are not equivalent, we have two constructions instead of one, and it is not clear why we should prefer, for example, holomorphicity instead of antiholomorphicity, even at the quasi-local level.
The angular momentum based on Bramson’s superpotential and the antiholomorphic spinors together with the antiholomorphic Dougan–Mason energy-momentum give acceptable Pauli–Lubanski spin for axisymmetric zero-mass Cauchy developments, for small spheres, and at future null infinity, but the global angular momentum at the future null infinity is finite and well defined only if the spatial three-momentum part of the Bondi–Sachs four-momentum is vanishing, i.e., only in the center-of-mass frame. (The spatial infinity limit of the spin angular momenta has not been calculated.)
Thus, the Nester–Witten 2-form appears to serve as an appropriate framework for defining the energy-momentum, and it is the two spinor fields, which should probably be changed, and a new choice would be needed. The holomorphic/antiholomorphic spinor fields appears to be ‘too rigid’. In fact, it is the topology of , namely the zero genus of , that restricts the solution space to two complex dimensions, instead of the local properties of the differential equations. (Thus, the situation is the same as in the twistorial construction of Penrose.) On the other hand, Bramson’s superpotential is based on the idea of Bergmann and Thomson, that the angular momentum of gravity is analogous to the spin. Thus, the question arises as to whether this picture is correct, or if the gravitational angular momentum also has an orbital part, in which case Bramson’s superpotential describes only (the general form of) its spin part. The fact that our antiholomorphic construction gives the correct, expected results for small spheres, but unacceptable ones for large spheres near future null infinity in frames that are not center-of-mass frames, may indicate the lack of such an orbital term. This term could be neglected for small spheres, but certainly not for large spheres. For example, in the special quasi-local angular momentum of Bergqvist and Ludvigsen for the Kerr spacetime (see Section 9.3), it is the sum of Bramson’s expression and a term that can be interpreted as the orbital angular momentum.
Living Rev. Relativity 12, (2009), 4
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