### 2.1 Asymptotic flatness and

Ever since the work of Bondi [16] 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. [60]. The arena for
this study is on what is referred to as future null infinity, , 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 [62, 63], 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 arising from Penrose’s construction that we rely on [60, 62, 63]:

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

(B): is topologically .

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

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