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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