6.4 The constraints in the initial-boundary value problem
A completely different situation arises if one is considering an initial-boundary value problem for
Einstein’s equations. Although this problem has not been solved in its full generality for Einstein’s
equations, it is clear that in order for the constraints to be satisfied during evolution, some of the boundary
values have to be chosen in a special way. It is here that the type of equations the constraints satisfy is most
important. In particular, if they also form a hyperbolic system, then a study of its principal part at the
boundary would tell which conditions are needed to guarantee uniqueness of the solutions, in particular the
trivial solution, and so which are the boundary conditions we must force upon the evolution system
for the dynamical fields. Since most numerical simulations are in fact initial-boundary value
problems, the problem of well posedness and the problem of the propagation of the constraints are
central.
