The original construction of Dougan and Mason [127] 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 [127],
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 [61
]
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 [61]. 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
) [354
], in contrast, for example, to the well
known
theories.
Both definitions give the same standard expression for round spheres [126]. 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 [356
].
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 [127]. 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
[126
]. The Dougan-Mason energy-momenta have
also been calculated for large spheres of constant Bondi-type radial coordinate value
near
future null infinity [126]. 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 [360
], 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.
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