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) .
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 .
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:metric sphere we take
while in general we write2CEq contains the general , while the special case using yields the GCEq.
For the most general situation, the G2CEq can be written as a differential equation for the function :
When the arbitrary spin-weight-2 function, vanishes, we have the homogeneous GCEq:
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 () as2CEq as
We now apply the coordinate transformation
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 .
Living Rev. Relativity 15, (2012), 1
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