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Again assuming that the Bondi shear is sufficiently small and the -space complex world line is not
too far from the “real”, the solution to the good-cut equation (4.15
), i.e., Eq. (4.38
), with
, is
decomposed into real and imaginary parts,
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As in the flat case, for fixed ,
has values on a line segment bounded between some
and
. The allowed values of
are again on a ribbon in the
-plane (i.e., region which is
topologically
for an interval
); all values of
and allowed values on the
-line
segments.
Each level curve of the function constant on the
-sphere (closed
curves or isolated points) determines a specific subset of the null directions and associated null
geodesics on the light-cone of the complex point
that intersect the real
. These geodesics will be referred to as ‘real’ geodesics. As
moves over all allowed
values of its segment, we obtain the set of
-space points,
and their
collection of ‘real’ geodesics. From Eq. (4.39
), these ‘real’ geodesics intersect
on the real cut
As varies we obtain a one-parameter family of cuts. If these cuts do not intersect with each other we
say that the complex world line
is by definition ‘timelike.’ This occurs when the time
component of the real part of the complex velocity vector,
, is sufficiently
large.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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