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14.3 On the Dougan-Mason energy-momenta and the holomorphic/anti-holomorphic spin angular momenta

From pragmatic points of view the Dougan-Mason energy-momenta (see Section 8.2) are certainly among the most successful definitions: The energy-positivity and the rigidity (zero energy implies flatness), and the intimate connection between the pp-waves and the vanishing of the masses make these definitions potentially useful quasi-local tools as the ADM and Bondi-Sachs energy-momenta in the asymptotically flat context. Similar properties are proven for the quasi-local energy-momentum of the matter fields, in particular for the non-Abelian Yang-Mills fields, too. They depend only on the 2-surface data on S, they have a clear Lagrangian interpretation, and the spinor fields that they are based on can be considered as the spinor constituents of the quasi-translations of the 2-surface. In fact, in the Minkowski spacetime the corresponding spacetime vectors are precisely the restriction to S of the constant Killing vectors. These notions of energy-momentum are linked completely to the geometry of S, and are independent of any ad hoc choice for the ‘fleet of observers’ on it. On the other hand, the holomorphic/anti-holomorphic spinor fields determine a six real parameter family of orthonormal frame fields on S, which can be interpreted as some distinguished class of observers. In addition, they reproduce the expected, correct limits in a number of special situations. In particular, these energy-momenta appear to have been completed by spin-angular momenta (see Section 9.2) in a natural way.

However, in spite of their successes, the Dougan-Mason energy-momenta and the spin-angular momenta based on Bramson’s superpotential and the holomorphic/anti-holomorphic spinor fields have some unsatisfactory properties, too (see the lists of our expectations in Section 4.3). First, they are defined only for topological 2-spheres (but not for other topologies, e.g. for the torus 1 1 S × S), and they are not well-defined even for certain topological 2-spheres either. Such surfaces are, for example, past marginally trapped surfaces in the anti-holomorphic (and future marginally trapped surfaces in the holomorphic) case. Although the quasi-local mass associated with a marginally trapped surface S is expected to be its irreducible mass V~ ---------------2-- Area(S)/(16pG ), 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/anti-holomorphicity 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 anti-holomorphy operators in a large class of potentially acceptable first order linear differential operators acting on spinor fields that have a 2-dimensional kernel. Furthermore, since the holomorphic and anti-holomorphic 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 anti-holomorphicity even at the quasi-local level.

The angular momentum based on Bramson’s superpotential and the anti-holomorphic spinors together with the anti-holomorphic Dougan-Mason energy-momentum give acceptable Pauli-Lubanski spin for axi-symmetric 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 3-momentum part of the Bondi-Sachs 4-momentum is vanishing, i.e. only in the centre-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/anti-holomorphic spinor fields appears to be ‘too rigid’. In fact, it is the topology of S, namely the zero genus of S, 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 the gravitational angular momentum also has an orbital part, whenever Bramson’s superpotential describes only (the general form of) its spin part. The fact that our anti-holomorphic construction gives the correct, expected results for small spheres but unacceptable ones for large spheres near future null infinity in frames that are not centre-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.

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