Evolution of the constraint equations in general relativity

We consider the evolution of initial data in general relativity. The Bianchi identity guarantees that data which initially satisfy the constraint equations will always satisfy the constraints. But we often expect only the approximate satisfaction of the constraint equations, for example, in numerical analysis. Here we study the evolution of only approximately good initial data. Under rather general circumstances the evolution drives the data away from good data; and we give a simple example where the traditional methods employed in numerical relativity would be likely to give erroneous results. We also present a minor modification of these traditional methods which would be likely to remedy this difficulty.