9.2 Holomorphic/antiholomorphic spin angular momenta

Obviously, the spin–angular-momentum expressions based on the holomorphic and antiholomorphic spinor fields [492Jump To The Next Citation Point] on generic two-surfaces are genuinely quasi-local. Since, in Minkowski spacetime the restriction of the two constant spinor fields to any two-surface is constant, and hence holomorphic and antiholomorphic at the same time, both the holomorphic and antiholomorphic spin angular momenta are vanishing. Similarly, for round spheres both definitions give zero [496Jump To The Next Citation Point], as would be expected in a spherically-symmetric system. The antiholomorphic spin angular momentum has already been calculated for axisymmetric two-surfaces 𝒮, for which the antiholomorphic 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 (Σ ) [488Jump To The Next Citation Point, 490].) This null energy-momentum vector turned out to be an eigenvector of the anti-symmetric spin–angular-momentum tensor ab J𝒮, 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 [492Jump To The Next Citation Point], where the latter is defined by
a 1 a b cd S𝒮 := 2𝜀 bcdP𝒮J 𝒮 . (9.2 )
This is a known property of the zero-rest-mass fields in Poincaré invariant quantum field theories [231].

Both the holomorphic and antiholomorphic spin angular momenta were calculated for small spheres [494]. In nonvacuum the holomorphic spin angular momentum reproduces the expected result (4.10View Equation), and, apart from a minus sign, the antiholomorphic construction does also. In vacuum, both definitions give exactly Eq. (9.1View Equation).

In general the antiholomorphic 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 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 (9.2View Equation) 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 (4.15View Equation). 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 [496Jump To The Next Citation Point].

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