Left-flat spaces and their associated space-times

Abstract

It is already known that for an asymptotically flat space-time the metric coefficients and the other Newman-Penrose variables (in a suitable frame) can be constructed, in principle, by specifying certain initial data at conformal null infinity (and one further function on another null hypersurface), integrating the Newman-Penrose equations in the conformally rescaled “unphysical” space, and then transforming the results back to the physical space-time. If this is done approximately near ℐ+, for vacuum, the well-known Newman-Unti expansion is obtained. In this paper, after complexifying null infinity ℐ+ we generate, in a similar fashion, a left-flat spaceH using as much of the initial data of a given asymptotically flat space-timeM as possible, and show that the left-flat spaceH thus constructed is, in fact, the H-space corresponding toM. The advantage of our method is that it allows a reversal of procedure. Under suitable conditions we can generate from a given left-flat spaceH a class of physical space-times whose H-space is precisely the given left-flat spaceH. We shall see that the formal procedure requires only the local but not the global properties of ℊ+.