In [492] it is shown that under the condition above there is a complex valued function
on
,
describing the deviation of the antiholomorphic and 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
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 two-surface
. Thus, the two transversal radiative modes of these fields are
encoded in certain fields on
. On the other hand, because of the nonlinearity 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
antiholomorphic spin frames at the null infinity (see, for example, [496]). 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 [492], reported above, gives just such
a quasi-local characterization in terms of a complex function measuring the deviation of the
holomorphic and antiholomorphic spin frames. However, the deviation of the holomorphic and
antiholomorphic structures on
can be defined even for generic two-surfaces in generic spacetimes
as well, which might yield the possibility of introducing the radiative modes quasi-locally in
general.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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