Conformal treatment of infinity

In the late 1950’s F. A. E. Pirani, A. Trautman, R. K. Sachs, H. Bondi, E. T. Newman, R. Penrose and others began exploring possibilities to establish in Einstein’s nonlinear theory a notion of gravitational radiation which is covariant and not based on approximations. The proposal they finally came up with relies, however, on an idealization. The solutions to Einstein’s field equations were assumed to be asymptotically flat at null infinity in the sense that they admit distinguished coordinate systems which include a null coordinate whose level hypersurfaces open up in the future so that the generating null geodesics are future complete and the curvature decays to zero in the infinite future. Apart from subtle questions about the precise decay conditions, there was a general agreement that one had arrived at the right concept. A survey of this development and of the results supporting this view is given by R. K. Sachs in the same volume which contains the article reprinted here [1]. Shortly after this development Roger Penrose put forward a remarkable idea. Seeking a formulation that emphasizes the role of the null cone or, equivalently, the conformal structure, which are the basic geometric elements underlying the earlier considerations, he proposes to characterize the required fall-off behaviour of the fields at null infinity by the condition that the conformal structure of the space–time admits an extension across null infinity of a certain smoothness. This provides a geometric notion