In order to formulate the initial value problem for the Einstein equations as nonlinear wave equations, we express the Einstein equations in terms of a partial dierential equation along with a gauge condition.
We suppose that is a Lorentzian manifold of the dimension
. The dummy indices will be from
up to
. In a coordinate system that will be fixed from now on, we have
as Christoffel symbols for the metric .
By lower order terms we mean terms consisting of either no derivative or first order derivative of the metric . As such, terms consisting of derivatives of order higher than two will be called high order terms.
Let us now introduce the following notation
It is obvious to see that
Since
and