### 9.2 Holomorphic/anti-holomorphic spin-angular momenta

Obviously, the spin-angular momentum expressions based on the holomorphic and anti-holomorphic
spinor fields [358] 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 [363], 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 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 [354, 356].) This null energy-momentum
vector turned out to be an eigenvector of the anti-symmetric spin-angular momentum tensor , 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 [358], where the latter is defined by
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 (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 [363].