In [358] it was shown that under the condition above there is a complex valued function
on
,
describing the deviation of the anti-holomorphic and the holomorphic spinor dyads from each other, which
plays the role of a potential for the curvature
on
. Then, assuming that
is future and past
convex and the matter is an N-type zero-rest-mass field,
and the value
of the matter field on
determine the curvature of
. Since the field equations for the metric of
reduce to Poisson-like equations with the curvature as the source, the metric of
is also
determined by
and
on
. Therefore, the (purely radiative) pp-wave geometry and matter
field on
are completely encoded in the geometry of
and complex functions defined
on
, respectively, in complete agreement with the holographic principle of the previous
Section 13.4.
As we saw in Section 2.2.5, the radiative modes of the zero-rest-mass-fields in Minkowski
spacetime, defined by their Fourier expansion, can be characterized quasi-locally on the globally
hyperbolic subset of the spacetime by the value of the Fourier modes on the appropriately
convex spacelike 2-surface
. Thus the two transversal radiative modes of them are
encoded in certain fields on
. On the other hand, because of the non-linearity of the Einstein
equations it is difficult to define the radiative modes of general relativity. It could be done when the
field equations become linear, i.e. near the null infinity, in the linear approximation and for
pp-waves. In the first case the gravitational radiation is characterized on a cut
of the
null infinity
by the
-derivative
of the asymptotic shear of the outgoing null
hypersurface
for which
, i.e. by a complex function on
. It is remarkable
that it is precisely this complex function which yields the deviation of the holomorphic and
anti-holomorphic spin frames at the null infinity (see for example [363]). The linear approximation of
Einstein’s theory is covered by the analysis of Section 2.2.5, thus those radiative modes can be
characterized quasi-locally, while for the pp-waves the result of [358], reported above, gives such
a quasi-local characterization in terms of a complex function measuring the deviation of the
holomorphic and anti-holomorphic spin frames. However, the deviation of the holomorphic
and anti-holomorphic structures on
can be defined even for generic 2-surfaces in generic
spacetimes too, which might yield the possibility of introducing the radiative modes quasi-locally in
general.
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