2.1 Asymptotic flatness and 𝕴+

Ever since the work of Bondi [16Jump To The Next Citation Point] illustrated the importance of null hypersurfaces in the study of outgoing gravitational radiation, the study of asymptotically-flat spacetimes has been one of the more important research topics in GR. Qualitatively speaking, a spacetime can be thought of as (future) asymptotically flat if the curvature tensor vanishes at an appropriate rate as infinity is approached along the future-directed null geodesics of the null hypersurfaces. The type of physical situation we have in mind is an arbitrary compact gravitating source (perhaps with an electric charge and current distribution), with the associated gravitational (and electromagnetic) field. The task is to gain information about the interior of the spacetime from the study of far-field features, multipole moments, gravitational and electromagnetic radiation, etc. [60Jump To The Next Citation Point]. The arena for this study is on what is referred to as future null infinity, ℑ+, the future boundary of the spacetime. The intuitive picture of this boundary is the set of all endpoints of future-directed null geodesics.

A precise definition of null asymptotic flatness and the boundary was given by Penrose [62Jump To The Next Citation Point, 63Jump To The Next Citation Point], whose basic idea was to rescale the spacetime metric by a conformal factor, which approaches zero asymptotically: the zero value defining future null infinity. This process leads to the boundary being a null hypersurface for the conformally-rescaled metric. When this boundary can be attached to the interior of the rescaled manifold in a regular way, then the spacetime is said to be asymptotically flat.

As the details of this formal structure are not used here, we will rely largely on the intuitive picture. A thorough review of this subject can be found in [23]. However, there are a number of important properties of ℑ+ arising from Penrose’s construction that we rely on [60Jump To The Next Citation Point, 62, 63]:

(A): For both the asymptotically-flat vacuum Einstein equations and the Einstein–Maxwell equations, + ℑ is a null hypersurface of the conformally rescaled metric.

(B): ℑ+ is topologically S2 × ℝ.

(C): The Weyl tensor Cabcd vanishes at ℑ+, with the peeling theorem describing the speed of its falloff (see below).

Property (B) allows an easy visualization of the boundary, + ℑ, as the past light cone of the point + I, future timelike infinity. As mentioned earlier, + ℑ will be the stage for our study of asymptotically shear-free NGCs.


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