13.5 Quasi-local radiative modes of general relativity

In Section 8.2.3 we discuss the properties of the Dougan–Mason energy-momenta, and we see that, under the conditions explained there, the energy-momentum is vanishing iff D (Σ) is flat, and it is null iff D (Σ ) is a pp-wave geometry with pure radiative matter, and that these properties of the domain of dependence D (Σ ) are completely encoded into the geometry of the two-surface 𝒮. However, there is an important difference between these two statements. While in the former case we know the metric of D (Σ ) is flat, in the second we know only that the geometry admits a constant null vector field, but we do not know the line element itself. Thus, the question arises as to whether the metric of D (Σ) is also determined by the geometry of 𝒮 even in the zero quasi-local–mass case.

In [492Jump To The Next Citation Point] it is shown that under the condition above there is a complex valued function Φ on 𝒮, describing the deviation of the antiholomorphic and holomorphic spinor dyads from each other, which plays the role of a potential for the curvature A F Bcd on 𝒮. Then, assuming that 𝒮 is future and past convex and the matter is an N-type zero-rest-mass field, Φ and the value ϕ of the matter field on 𝒮 determine the curvature of D (Σ ). Since the field equations for the metric of D (Σ) reduce to Poisson-like equations with the curvature as the source, the metric of D (Σ ) is also determined by Φ and ϕ on 𝒮. Therefore, the (purely radiative) pp-wave geometry and matter field on D (Σ ) are completely encoded in the geometry of 𝒮 and complex functions defined on 𝒮, 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 D (Σ ) of the spacetime by the value of the Fourier modes on the appropriately convex spacelike two-surface 𝒮 = ∂Σ. Thus, the two transversal radiative modes of these fields are encoded in certain fields on 𝒮. 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 𝒮 ∞ of the null infinity ℐ + by the u-derivative ˙σ0 of the asymptotic shear of the outgoing null hypersurface 𝒩 for which 𝒮 ∞ = 𝒩 ∩ ℐ +, i.e., by a complex function on 𝒮 ∞. 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 𝒮 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.

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