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.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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