Both the holomorphic and antiholomorphic spin angular momenta were calculated for small spheres . In nonvacuum the holomorphic spin angular momentum reproduces the expected result (4.10), and, apart from a minus sign, the antiholomorphic construction does also. In vacuum, both definitions give exactly Eq. (9.1).
In general the antiholomorphic 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 antiholomorphic expression is just the spatial part of the Bondi–Sachs energy-momentum. Thus, the antiholomorphic spin angular momentum is finite in the center-of-mass frame, and hence it seems to describe only the spin part of the gravitational field. In fact, the Pauli–Lubanski spin (9.2) built from this spin angular momentum and the antiholomorphic Dougan–Mason energy-momentum is always finite, free of the ‘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 (4.15). Thus, the antiholomorphic 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 .
Living Rev. Relativity 12, (2009), 4
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