A precise definition of null asymptotic flatness and the boundary was given by Penrose [46, 47], 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 . However, there are a number of important properties of arising from Penrose’s construction that we rely on [44, 46, 47]:
(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.
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