4.4 Summary of Real Structures
To put the ideas of this section into perspective we collect the claims.
- In Minkowski space, the future directed light-cones emanating from a real timelike world
line, , intersect future null infinity, , on a one-parameter family of spherical
non-intersecting cuts.
- The complex light-cones emanating from a timelike complex analytic curve in complex
Minkowski space, parametrized by the complex parameter , has for
each fixed value of and a limited set of null geodesics that reach real . However,
for a ribbon in the complex -plane (i.e., a region topologically , with and
), there will be many null geodesics intersecting . Such null geodesics
were referred to as ‘real’ geodesics . More specifically, for a fixed , there is a specific range
of such that all the real null geodesics intersect in a full cut, leading to a
one-parameter family of real (distorted sphere) slicings of . The ribbon is the generalization
of the real world line and the slicings are the analogues of the spherical slicings. When the ribbon
shrinks to a line it degenerates to the real case. We can consider the ribbon as a generalized
world line and the ‘real’ null geodesics from a constant portion of the ribbon as a generalized
light-cone.
- For the case of asymptotically flat spacetimes, the real light-cones from interior points are
replaced by the virtual light-cones generated by the asymptotically shear-free NGCs. These
cones emanate from a complex virtual world line in the associated -space. As
in the case of complex Minkowski space, there is a ribbon in the -plane where the ‘real’ null
geodesics originate from. The ‘real’ null geodesics coming from a cross-section of the strip at
fixed (as in the complex Minkowski case), intersect in a cut; the collection of cuts
yielding a one-parameter family. The situation is exactly the same as in the complex Minkowski
space case except that the spherical harmonic decomposition of these cuts is in general more
complicated.
Example: the (charged) Kerr metric
Considering the Kerr or the charged Kerr metrics (or even more
generally any asymptotically flat stationary metric), we have immediately that the Bondi shear
vanishes and hence the associated -space is complex Minkowski space (cf. [3, 6] and Appendix F).
From the stationarity and a real origin shift and rotation, the complex world line can be put into the form
with being the Kerr parameter. The complex cut function is then
so that the angle fields of 3.33-3.34 are
Using in Eq. (4.42), the reality condition on the cut function is that
so that on the -ribbon, ranges between and the real slices from the ribbon becomes
simply .
Though we are certainly not making the claim that one can in reality ‘observe’ these complex world lines
that arise from (asymptotically) shear-free congruences, we nevertheless claim that they can be observed in
a different sense. In the next two sections our goal will be to show that, just as a complex center of
charge world line in can be selected, so too can a complex center of mass world line be
singled out in -space. As we will see, some surprising physical identifications arise from
this program, and it is in this sense which the footprints of these complex world lines can be
observed.