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.

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