Throughout this work, the GoodCut Equation (GCEq) has played a major role in allowing us to study shearfree and asymptotically shearfree 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 twosphere in the conformal compactification of asymptotically flat spacetimes, we can always take the space of null generators of to be a metric twospheres. However, one can imagine solving the GCEq on a surface which is only conformal to a metric twosphere, we refer to such a PDE as the ‘Generalized’ GCEq, or G^{2}CEq for short. In this appendix, we briefly motivate why one could be interested in the G^{2}CEq, and then prove that it can be reduced to the GCEq on the metric twosphere 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 socalled ‘vacuum nonexpanding horizons’, which are null 3surfaces 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 nonexpanding horizon. It has been shown that such ‘horizonshearfree’ NGCs are described, where they ‘cut’ the horizon, by a goodcut equation on the topologically cut. Since we cannot freely rescale objects in the interior of the spacetime, this means that horizonshearfree NGCs are described by the G^{2}CEq [5].
Consider an arbitrary vacuum nonexpanding horizon with associated G^{2}CEq. As in the asymptotic case, we consider the complexification of the horizon when looking for solutions to the G^{2}CEq, and make use of local Bondilike 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 crosssections 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 G^{2}CEq is that the former lives on a 3surface whose crosssections are metric spheres, while for the latter equation the 2surface metric is arbitrary.
As mentioned earlier, the 3surface is described by an worth of null geodesics with the cross sections given by = constant. The metric of the twosurface crosssections are expressed in stereographic coordinates () so that the metric takes the conformally flat form:
with an arbitrary smooth nonvanishing function on the sphere, the extended complex plane (Riemann sphere). In the special case of a metric sphere we take

while in general we write
The G^{2}CEq contains the general , while the special case using yields the GCEq.For the most general situation, the G^{2}CEq can be written as a differential equation for the function :
or When we have the GCEq:

When the arbitrary spinweight2 function, vanishes, we have the homogeneous GCEq:
It is now shown how, by a coordinate transformation of the (independent) complex stereographic coordinates (), G^{2}CEq 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
and the G^{2}CEq asWe now apply the coordinate transformation
with (a spinweight 1 function) defined fromHence, we see that the G^{2}CEq is equivalent to the GCEq via the coordinate transformation (F.9). This means that the study of the G^{2}CEq on a general 3surface can be reduced to the study of the properties of the GCEq on a 3surface whose crosssections are metric spheres.
Remark 14. As in the main text, solutions to the GCEq or G^{2}CEq, , known as ‘goodcut functions’, describe crosssections 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 .
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Living Rev. Relativity 15, (2012), 1
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