The original construction of Dougan and Mason  was introduced on the basis of sheaf-theoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [354, 356].
Following Dougan and Mason we define the spinor field to be anti-holomorphic in case , or holomorphic if . Thus, this notion of holomorphicity/anti-holomorphicity is referring to the connection on . While the notion of the holomorphicity/anti-holomorphicity of a function on does not depend on whether the or the operator is used, for tensor or spinor fields it does. Although the vectors and are not uniquely determined (because their phase is not fixed), the notion of the holomorphicity/anti-holomorphicity is well-defined, because the defining equations are homogeneous in and . Next suppose that there are at least two independent solutions of . If and are any two such solutions, then , and hence by Liouville’s theorem is constant on . If this constant is not zero, then we call generic, if it is zero then will be called exceptional. Obviously, holomorphic on a generic cannot have any zero, and any two holomorphic spinor fields, e.g. and , span the spin space at each point of (and they can be chosen to form a normalized spinor dyad with respect to on the whole of ). Expanding any holomorphic spinor field in this frame, the expanding coefficients turn out to be holomorphic functions, and hence constant. Therefore, on generic 2-surfaces there are precisely two independent holomorphic spinor fields. In the GHP formalism the condition of the holomorphicity of the spinor field is that its components be in the kernel of . Thus for generic 2-surfaces with the constant would be a natural candidate for the spin space above. For exceptional 2-surfaces the kernel space is either 2-dimensional but does not inherit a natural spin space structure, or it is higher than two dimensional. Similarly, the symplectic inner product of any two anti-holomorphic spinor fields is also constant, one can define generic and exceptional 2-surfaces as well, and on generic surfaces there are precisely two anti-holomorphic spinor fields. The condition of the anti-holomorphicity of is . Then could also be a natural choice. Note that since the spinor fields whose holomorphicity/anti-holomorphicity is defined are unprimed, and these correspond to the anti-holomorphicity/holomorphicity, respectively, of the primed spinor fields of Dougan and Mason. Thus the main question is whether there exist generic 2-surfaces, and if they do, whether they are ‘really generic’, i.e. whether most of the physically important surfaces are generic or not.
are first order elliptic differential operators on certain vector bundles over the compact 2-surface , and their index can be calculated: , where is the genus of . Therefore, for there are at least two linearly independent holomorphic and at least two linearly independent anti-holomorphic spinor fields. The existence of the holomorphic/anti-holomorphic spinor fields on higher genus 2-surfaces is not guaranteed by the index theorem. Similarly, the index theorem does not guarantee that is generic either: If the geometry of is very special then the two holomorphic/anti-holomorphic spinor fields (which are independent as solutions of ) might be proportional to each other. For example, future marginally trapped surfaces (i.e. for which ) are exceptional from the point of view of holomorphic, and past marginally trapped surfaces () from the point of view of anti-holomorphic spinors. Furthermore, there are surfaces with at least three linearly independent holomorphic/anti-holomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional 2-surface with topology make generic.
Finally, we note that several first order differential operators can be constructed from the chiral irreducible parts and of , given explicitly by Equation (25). However, only four of them, the Dirac-Witten operator , the twistor operator , and the holomorphy and anti-holomorphy operators , are elliptic (which ellipticity, together with the compactness of , would guarantee the finiteness of the dimension of their kernel), and it is only that have 2-complex-dimensional kernel in the generic case. This purely mathematical result gives some justification for the choices of Dougan and Mason: The spinor fields that should be used in the Nester-Witten 2-form are either holomorphic or anti-holomorphic. The construction does not work for exceptional 2-surfaces.
One of the most important properties of the Dougan-Mason energy-momenta is that they are future pointing nonspacelike vectors, i.e. the corresponding masses and energies are non-negative. Explicitly , if is the boundary of some compact spacelike hypersurface on which the dominant energy condition holds, furthermore if is weakly future convex (in fact, is enough), then the holomorphic Dougan-Mason energy-momentum is a future pointing non-spacelike vector, and, analogously, the anti-holomorphic energy-momentum is future pointing and non-spacelike if . As Bergqvist  stressed (and we noted in Section 8.1.3), Dougan and Mason used only the (and in the anti-holomorphic construction the ) half of the ‘propagation law’ in their positivity proof. The other half is needed only to ensure the existence of two spinor fields. Thus that might be Equation (59) of the Ludvigsen-Vickers construction, or in the holomorphic Dougan-Mason construction, or even for some constant , a ‘deformation’ of the holomorphicity considered by Bergqvist . In fact, the propagation law may even be for any spinor field satisfying . This ensures the positivity of the energy under the same conditions and that is still constant on for any two solutions and , making it possible to define the norm of the resulting energy-momentum, i.e. the mass.
In the asymptotically flat spacetimes the positive energy theorems have a rigidity part too, namely the vanishing of the energy-momentum (and, in fact, even the vanishing of the mass) implies flatness. There are analogous theorems for the Dougan-Mason energy-momenta too [354, 356]. Namely, under the conditions of the positivity proof
Comparing Results 1 and 2 above with the properties of the quasi-local energy-momentum (and angular momentum) listed in Section 2.2.3, the similarity is obvious: characterizes the ‘quasi-local vacuum state’ of general relativity, while is equivalent to ‘pure radiative quasi-local states’. The equivalence of and the flatness of shows that curvature always yields positive energy, or, in other words, with this notion of energy no classical symmetry breaking can occur in general relativity: The ‘quasi-local ground states’ (defined by ) are just the ‘quasi-local vacuum states’ (defined by the trivial value of the field variables on ) , in contrast, for example, to the well known theories.
Both definitions give the same standard expression for round spheres . Although the limit of the Dougan-Mason masses for round spheres in Reissner-Nordström spacetime gives the correct irreducible mass of the Reissner-Nordström black hole on the horizon, the constructions do not work on the surface of bifurcation itself, because that is an exceptional 2-surface. Unfortunately, without additional restrictions (e.g. the spherical symmetry of the 2-surfaces in a spherically symmetric spacetime) the mass of the exceptional 2-surfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic 2-surfaces approaching the exceptional one .
Both definitions give the same, expected results in the weak field approximation and for large spheres at spatial infinity: Both tend to the ADM energy-momentum . In non-vacuum both definitions give the same, expected expression (28) for small spheres, in vacuum they coincide in the order with that of Ludvigsen and Vickers, but in the order they differ from each other: The holomorphic definition gives Equation (61), but in the analogous expression for the anti-holomorphic energy-momentum the numerical coefficient is replaced by . The Dougan-Mason energy-momenta have also been calculated for large spheres of constant Bondi-type radial coordinate value near future null infinity . While the anti-holomorphic construction tends to the Bondi-Sachs energy-momentum, the holomorphic one diverges in general. In stationary spacetimes they coincide and both give the Bondi-Sachs energy-momentum. At the past null infinity it is the holomorphic construction which reproduces the Bondi-Sachs energy-momentum and the anti-holomorphic diverges.
We close this section with some caution and general comments on a potential gauge ambiguity in the calculation of the various limits. By the definition of the holomorphic and anti-holomorphic spinor fields they are associated with the 2-surface only. Thus if is another 2-surface, then there is no natural isomorphism between the space - for example of the anti-holomorphic spinor fields on - and on , even if both surfaces are generic and hence there are isomorphisms between them12. This (apparently ‘only theoretical’) fact has serious pragmatic consequences. In particular, in the small or large sphere calculations we compare the energy-momenta, and hence the holomorphic or anti-holomorphic spinor fields also, on different surfaces. For example , in the small sphere approximation every spin coefficient and spinor component in the GHP dyad and metric component in some fixed coordinate system is expanded as a series of , like . Substituting all such expansions and the asymptotic solutions of the Bianchi identities for the spin coefficients and metric functions into the differential equations defining the holomorphic/anti-holomorphic spinors, we obtain a hierarchical system of differential equations for the expansion coefficients , , …, etc. It turns out that the solutions of this system of equations with accuracy form a rather than the expected two complex dimensional space. of these solutions are ‘gauge’ solutions, and they correspond in the approximation with given accuracy to the unspecified isomorphism between the space of the holomorphic/anti-holomorphic spinor fields on surfaces of different radii. Obviously, similar ‘gauge’ solutions appear in the large sphere expansions, too. Therefore, without additional gauge fixing, in the expansion of a quasi-local quantity only the leading non-trivial term will be gauge-independent. In particular, the order correction in Equation (61) for the Dougan-Mason energy-momenta is well-defined only as a consequence of a natural gauge choice13. Similarly, the higher order corrections in the large sphere limit of the anti-holomorphic Dougan-Mason energy-momentum are also ambiguous unless a ‘natural’ gauge choice is made. Such a choice is possible in stationary spacetimes.
© Max Planck Society and the author(s)