Both the holomorphic and anti-holomorphic spin-angular momenta were calculated for small spheres . In non-vacuum the holomorphic spin-angular momentum reproduces the expected result (29), and, apart from a minus sign, the anti-holomorphic construction does also. In vacuum both definitions give exactly Equation (62).
In general the anti-holomorphic and the holomorphic spin-angular momenta are diverging near the future null infinity of Einstein-Maxwell spacetimes as and , 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 (63) 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 (34). 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 .
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