Throughout this work, the Good-Cut Equation (GCEq) has played a major role in allowing us to study
shear-free and asymptotically shear-free NGCs in asymptotically flat spacetimes. In this context, the GCEq
is a partial differential equation on a topologically cut of
; due to the freedom in the choice of
conformal factor on the two-sphere in the conformal compactification of asymptotically flat spacetimes, we
can always take the space of null generators of
to be a metric two-spheres. However, one can imagine
solving the GCEq on a surface which is only conformal to a metric two-sphere, we refer to such a PDE as
the ‘Generalized’ GCEq, or G2CEq for short. In this appendix, we briefly motivate why one could be
interested in the G2CEq, and then prove that it can be reduced to the GCEq on the metric two-sphere
by a coordinate transformation (this is essentially a proof of the conformal invariance of the
GCEq) [6].
The study of horizons in the interior of spacetime is an important topic in a variety of areas, particularly
quantum gravity. One interesting class of null horizons are the so-called ‘vacuum non-expanding
horizons’, which are null 3-surfaces in a spacetime that have vanishing divergence and shear,
and are topologically [13, 14]. In analogy with the setting on
discussed in the
body of this review, one can look for null geodesic congruences in the interior of a spacetime
which have vanishing shear at their intersection with a vacuum non-expanding horizon. It has
been shown that such ‘horizon-shear-free’ NGCs are described, where they ‘cut’ the horizon, by
a good-cut equation on the topologically
cut. Since we cannot freely rescale objects in
the interior of the spacetime, this means that horizon-shear-free NGCs are described by the
G2CEq [5].
Consider an arbitrary vacuum non-expanding horizon with associated G2CEq. As in the asymptotic
case, we consider the complexification
of the horizon when looking for solutions to the G2CEq, and
make use of local Bondi-like coordinates
. The (
), which label the null generators of
are
the stereographic coordinates on the
portion of
(
need not be a metric sphere); while the
coordinate
parametrizes the cross-sections of
For
, the
is allowed to take complex values
close to the real, while
goes over to an independent variable close to the complex conjugate of
. The context should make it clear when
is actually the complex conjugate of
. The
distinction between the GCEq and the G2CEq is that the former lives on a 3-surface
whose
cross-sections are metric spheres, while for the latter equation the 2-surface metric is
arbitrary.
As mentioned earlier, the 3-surface is described by an
worth of null geodesics with
the cross sections given by
= constant. The metric of the two-surface cross-sections are
expressed in stereographic coordinates (
) so that the metric takes the conformally flat form:
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while in general we write
The G2CEq contains the general For the most general situation, the G2CEq can be written as a differential equation for the function
:
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When the arbitrary spin-weight-2 function, vanishes, we have the homogeneous G
CEq:
It is now shown how, by a coordinate transformation of the (independent) complex stereographic
coordinates (), G2CEq can be transformed into the GCEq. It must be remembered from our
notation that
(or
is close to, but is not necessarily, the complex conjugate of
(or
).
First rewrite the GCEq with stereographic coordinates () as
We now apply the coordinate transformation
with Hence, we see that the G2CEq is equivalent to the GCEq via the coordinate transformation (F.9). This
means that the study of the G2CEq on a general 3-surface
can be reduced to the study of the
properties of the GCEq on a 3-surface whose cross-sections are metric spheres.
Remark 14. As in the main text, solutions to the GCEq or G2CEq, , known as
‘good-cut functions’, describe cross-sections of
that are referred to as ‘good cuts.’ From the
tangents to these good cuts,
, one can construct null directions (pointing out of
) into
the spacetime itself that determine a NGC whose shear vanishes at
.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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