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9.4 The characteristic initial value problem

In the standard Cauchy problem, which has been the basic set-up for all the previous sections, initial data are given on a spacelike hypersurface. However, there is also another possibility, where data are given on one or more null hypersurfaces. This is the characteristic initial value problem. It has the advantage over the Cauchy problem that the constraints reduce to ordinary differential equations. One variant is to give initial data on two smooth null hypersurfaces that intersect transversely in a spacelike surface. A local existence theorem for the Einstein equations with an initial configuration of this type was proved in [284]. Another variant is to give data on a light cone. In that case local existence for the Einstein equations has not been proved, although it has been proved for a class of quasilinear hyperbolic equations that includes the reduced Einstein equations in harmonic coordinates [133]. For some new work on the global characteristic initial value problem see [73].

Another existence theorem that does not use the standard Cauchy problem, and which is closely connected to the use of null hypersurfaces, concerns the Robinson-Trautman solutions of the vacuum Einstein equations. In that case the Einstein equations reduce to a parabolic equation. Global existence for this equation has been proved by ChruĊ›ciel [115].


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