Mapping onto Solutions of the Gravitational Initial Value Problem

Two approaches to the Einstein initial value problem for vacuum gravitational fields are considered. In the first, the metric of a spacelike slice is prescribed arbitrarily and it is shown that momenta satisfying the constraints can be constructed by exploiting the well‐known relation of three of the four constraints to the three‐dimensional coordinate transformation group. Specifically, it is shown that there exists a coordinate mapping of a certain specified set of functions onto momenta satisfying the constraints for a specified 3‐metric. A further interpretation of this procedure is discussed. In the second approach the 3‐geometry of a spacelike slice is specified up to a conformal factor. It is shown that, using a coordinate transformation method similar to the above, transverse traceless momenta can be constructed and that this construction depends essentially only on the conformal geometry of the spacelike hypersurface. The remaining constraint is satisfied by a choice of the conformal factor. As a result, it follows that the initial value equations can be satisfied by mapping certain specified sets of functions onto solutions by using coordinate transformations and a group of scale transformations which include conformal transformation of the metric. This is significant because the unconstrained initial data (gravitational degrees of freedom) are represented by a pair of scale‐invariant transverse, traceless tensors of weight$\frac{5}{3}$ . These objects, in turn, give irreducible representations of the coordinate and scaling groups which are used to effect solutions of the initial value equations.