(i) There are no elliptic constraint equations. This eliminates the need for time consuming iterative methods to enforce constraints or to otherwise test the challenge of constraint free evolution.
(ii) No second time derivatives appear so that the number of basic variables is half the number for the corresponding version of the Cauchy problem.
(iii) The main Einstein equations form a system of coupled
ordinary differential equations with respect to the parameter
varying along the characteristics. This allows construction of
an evolution algorithm in terms of a simple march along the
characteristics.
(iv) In problems with isolated sources, the radiation zone can be compactified into a finite grid boundary using Penrose's conformal technique. Because the Penrose boundary is a null hypersurface, no extraneous outgoing radiation condition or other artificial boundary condition is required.
(v) The grid domain is exactly the region in which waves propagate, which is ideally efficient for radiation studies. Since each null hypersurface of the foliation extends to infinity, the radiation is calculated immediately (in retarded time) with no need to propagate it across the grid.
(vi) In black hole space-times, a large redshift at null infinity relative to internal sources is an indication of the formation of an event horizon and can be used to limit the evolution to the exterior region of space-time.
Characteristic schemes also share as a common disadvantage the
necessity either to deal with caustics or to avoid them
altogether. The scheme to tackle the caustics head on by
including their development as part of the evolution is perhaps a
great idea still ahead of its time, one that should not be
forgotten. There are only a handful of structurally stable
caustics, and they have well known algebraic properties. This
makes it possible to model their singular structure in terms of
Pade approximants. The structural stability of the singularities
should in principle make this possible, and algorithms to evolve
the elementary caustics have been proposed [19,
20]. In the axisymmetric case, cusps and folds are the only stable
caustics, and they have already been identified in the horizon
formation occurring in simulations of head-on collisions of black
holes and in the temporarily toroidal horizons occurring in
collapse of rotating matter [21,
22].
![]() |
Characteristic Evolution and Matching
Jeffrey Winicour http://www.livingreviews.org/lrr-1998-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |