6.2 The constraints in the new systems: Theoretical considerations
In the new hyperbolic systems, where covariance is lost, one solves only for the evolution equations.
Thus, the question of whether the constraint equations hold during evolution if they hold at the initial
surface arises again. If the problem is about the evolution of the whole space-time, or about evolution on
the domain of dependence of some space-like surface, then there is a good argument showing that the
constraint equations would be satisfied as a consequence of the uniqueness of the system under
consideration:
Assume initial data is given at some space-like hypersurface which satisfies the constraints there. We use
the new evolution system and get a solution in the domain of dependence of the system, (which, if gauges
propagate at speeds greater than light, might be smaller than the domain given by the metric). But using
the harmonic gauge I know that there is a solution to the Einstein equation on a maximally extended
domain of dependence. If one can diffeomorphically transform metric corresponding to that solution into
one satisfying the gauge used for the evolution with the new system, then, since it satisfies all the equations,
including the constraints, it follows that it will also satisfy the equations of the new system. Uniqueness of
solutions of the new system implies it must be the one found initially and so it also satisfies
the constraints. Thus we see that no particular consideration for the constraint equations is
needed.
