The original construction of Dougan and Mason [172] was introduced on the basis of sheaf-theoretical arguments. Here we follow a slightly different, more ‘pedestrian’ approach, based mostly on [488, 490].

Following Dougan and Mason we define the spinor field to be antiholomorphic when , or holomorphic if . Thus, this notion of holomorphicity/antiholomorphicity is referring to the connection on . While the notion of the holomorphicity/antiholomorphicity of a function on does not depend on whether the or 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 holomorphicity/antiholomorphicity 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 two-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 two-surfaces with the constant would be a natural candidate for the spin space above. For exceptional two-surfaces, the kernel space is either two-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 antiholomorphic spinor fields is also constant, one can define generic and exceptional two-surfaces as well, and on generic surfaces there are precisely two antiholomorphic spinor fields. The condition of the antiholomorphicity of is . Then could also be a natural choice. Note that the spinor fields, whose holomorphicity/antiholomorphicity is defined, are unprimed, and these correspond to the antiholomorphicity/holomorphicity, respectively, of the primed spinor fields of Dougan and Mason. Thus, the main question is whether there exist generic two-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 two-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 antiholomorphic spinor fields. The existence of the holomorphic/antiholomorphic spinor fields on higher-genus two-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/antiholomorphic 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 spinors, and past marginally-trapped surfaces () from the point of view of antiholomorphic spinors. Furthermore, there are surfaces with at least three linearly-independent holomorphic/antiholomorphic spinor fields. However, small generic perturbations of the geometry of an exceptional two-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 Eq. (4.6). However, only four of them, the Dirac–Witten operator , the twistor operator , and the holomorphy and antiholomorphy 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 a two-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 antiholomorphic. This construction does not work for exceptional two-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 [172], 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 nonspacelike vector, and, analogously, the antiholomorphic energy-momentum is future pointing and nonspacelike if . (For the functional analytic techniques and tools to give a complete positivity proof, see, e.g., [182].) As Bergqvist [79] stressed (and we noted in Section 8.1.3), Dougan and Mason used only the (and, in the antiholomorphic 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 Eq. (8.3) 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 [79]. 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 as well, 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 as well [488, 490]. Namely, under the conditions of the positivity proof

- is zero iff is flat, which is also equivalent to the vanishing of the quasi-local energy, , and
- is null (i.e., the quasi-local mass is zero) iff is a pp-wave geometry and the matter is pure radiation.

In particular [498], for a coupled Einstein–Yang–Mills system (with compact, semisimple gauge groups) the zero quasi-local mass configurations are precisely the pp-wave solutions found by Güven [230]. Therefore, in contrast to the asymptotically flat cases, the vanishing of the mass does not imply the flatness of . Since, as we will see below, the Dougan–Mason masses tend to the ADM mass at spatial infinity, there is a seeming contradiction between the rigidity part of the positive mass theorems and the result 2 above. However, this is only an apparent contradiction. In fact, according to one of the possible positive mass proofs [38], the vanishing of the ADM mass implies the existence of a constant null vector field on , and then the flatness follows from the incompatibility of the conditions of the asymptotic flatness and the existence of a constant null vector field: The only asymptotically flat spacetime admitting a constant null vector field is flat spacetime.

These results show some sort of rigidity of the matter + gravity system (where the latter satisfies the dominant energy condition), even at the quasi-local level, which is much more manifest from the following equivalent form of the results 1 and 2. Under the same conditions is flat if and only if there exist two linearly-independent spinor fields on , which are constant with respect to , and is a pp-wave geometry; the matter is pure radiation if and only if there exists a -constant spinor field on [490]. Thus, the full information that is flat/pp-wave is completely encoded, not only in the usual initial data on , but in the geometry of the boundary of , as well. In Section 13.5 we return to the discussion of this phenomenon, where we will see that, assuming is future and past convex, the whole line element of (and not only the information that it is some pp-wave geometry) is determined by the two-surface data on .

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 show 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 ) [488], in contrast, for example, to the well known theories.

Both definitions give the same standard expression for round spheres [171]. 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 two-surface. Unfortunately, without additional restrictions (e.g., the spherical symmetry of the two-surfaces in a spherically-symmetric spacetime) the mass of the exceptional two-surfaces cannot be defined in a limiting process, because, in general, the limit depends on the family of generic two-surfaces approaching the exceptional one [490].

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 [172]. (The Newtonian limit in the covariant Newtonian spacetime was studied in [564].) In nonvacuum both definitions give the same, expected expression (4.9) 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 Eq. (8.5), but in the analogous expression for the antiholomorphic energy-momentum, the numerical coefficient is replaced by [171]. The Dougan–Mason energy-momenta have also been calculated for large spheres of constant Bondi-type radial coordinate value near future null infinity [171]. While the antiholomorphic 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 antiholomorphic construction 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 antiholomorphic spinor fields they
are associated with the two-surface only. Thus, if is another two-surface, then there is no natural
isomorphism between the space – for example of the antiholomorphic spinor fields on –
and on , even if both surfaces are generic and hence, there are isomorphisms between
them.^{12}
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 antiholomorphic spinor fields as well, on different surfaces. For example [494], 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 , as
. 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/antiholomorphic 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/antiholomorphic 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
nontrivial term will be gauge-independent. In particular, the -order correction in Eq. (8.5) for
the Dougan–Mason energy-momenta is well defined only as a consequence of a natural gauge
choice.^{13}
Similarly, the higher-order corrections in the large sphere limit of the antiholomorphic Dougan–Mason
energy-momentum are also ambiguous unless a ‘natural’ gauge choice is made. Such a choice is possible in
stationary spacetimes.

Living Rev. Relativity 12, (2009), 4
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