### 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 ChruĊciel [115].