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 which intersect
transversely in a spacelike surface. A local existence theorem
for the Einstein equations with an initial configuration of this
type was proved in . 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 which includes the reduced Einstein equations in
harmonic coordinates .
Another existence theorem which 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 Chrusciel .
||Local and Global Existence Theorems for the Einstein
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351