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9.2 Holomorphic/anti-holomorphic spin-angular momenta

Obviously, the spin-angular momentum expressions based on the holomorphic and anti-holomorphic spinor fields [358Jump To The Next Citation Point] on generic 2-surfaces are genuinely quasi-local. Since in Minkowski spacetime the restriction of the two constant spinor fields to any 2-surface are constant, and hence holomorphic and anti-holomorphic at the same time, both the holomorphic and anti-holomorphic spin-angular momenta are vanishing. Similarly, for round spheres both definitions give zero [363Jump To The Next Citation Point], as it could be expected in a spherically symmetric system. The anti-holomorphic spin-angular momentum has already been calculated for axi-symmetric 2-surfaces S for which the anti-holomorphic Dougan-Mason energy-momentum is null, i.e. for which the corresponding quasi-local mass is zero. (As we saw in Section 8.2.3, this corresponds to a pp-wave geometry and pure radiative matter fields on D(S) [354Jump To The Next Citation Point356].) This null energy-momentum vector turned out to be an eigenvector of the anti-symmetric spin-angular momentum tensor ab J S, which, together with the vanishing of the quasi-local mass, is equivalent to the proportionality of the (null) energy-momentum vector and the Pauli-Lubanski spin [358Jump To The Next Citation Point], where the latter is defined by
a 1 a b cd SS := 2e bcdPS JS . (63)
This is a known property of the zero-rest-mass fields in Poincaré invariant quantum field theories [168].

Both the holomorphic and anti-holomorphic spin-angular momenta were calculated for small spheres [360]. In non-vacuum the holomorphic spin-angular momentum reproduces the expected result (29View Equation), and, apart from a minus sign, the anti-holomorphic construction does also. In vacuum both definitions give exactly Equation (62View Equation).

In general the anti-holomorphic and the holomorphic spin-angular momenta are diverging near the future null infinity of Einstein-Maxwell spacetimes as r and 2 r, respectively. However, the coefficient of the diverging term in the anti-holomorphic expression is just the spatial part of the Bondi-Sachs energy-momentum. Thus the anti-holomorphic spin-angular momentum is finite in the centre-of-mass frame, and hence it seems to describe only the spin part of the gravitational field. In fact, the Pauli-Lubanski spin (63View Equation) built from this spin-angular momentum and the anti-holomorphic Dougan-Mason energy-momentum is always finite, free of ‘gauge’ ambiguities discussed in Section 8.2.4, and is built only from the gravitational data even in the presence of electromagnetic fields. In stationary spacetimes both constructions are finite and coincide with the ‘standard’ expression (34View Equation). Thus the anti-holomorphic spin-angular momentum defines an intrinsic angular momentum at the future null infinity. Note that this angular momentum is free of supertranslation ambiguities, because it is defined on the given cut in terms of the solutions of elliptic differential equations. These solutions can be interpreted as the spinor constituents of certain boost-rotation BMS vector fields, but the definition of this angular momentum is not based on them [363Jump To The Next Citation Point].


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