F The Generalized Good-Cut Equation

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 2 S cut of + ℑ; 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 ℑ+ 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 ℝ × S2 [13, 14]. In analogy with the setting on ℐ+ 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 S2 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 ℌ with associated G2CEq. As in the asymptotic case, we consider the complexification ℌ ℂ of the horizon when looking for solutions to the G2CEq, and make use of local Bondi-like coordinates -- (u,ζ,ζ ). The (-- ζ, ζ), which label the null generators of ℌ, are the stereographic coordinates on the S2 portion of ℌ (S2 need not be a metric sphere); while the coordinate u parametrizes the cross-sections of ℌ. For ℌ ℂ, the u is allowed to take complex values close to the real, while -- ζ goes over to an independent variable close to the complex conjugate of ζ. The context should make it clear when -- ζ is actually the complex conjugate of ζ. The distinction between the GCEq and the G2CEq is that the former lives on a 3-surface ℌ whose u = constant cross-sections are metric spheres, while for the latter equation the 2-surface metric is arbitrary.

As mentioned earlier, the 3-surface ℌ is described by an S2 worth of null geodesics with the cross sections given by u = constant. The metric of the two-surface cross-sections are expressed in stereographic coordinates (-- ζ, ζ) so that the metric takes the conformally flat form:

-- 2 -4dζdζ-- ds = 2 -- , (F.1 ) P (ζ,ζ )
with -- P (u,ζ,ζ ) an arbitrary smooth nonvanishing function on the -- (ζ,ζ)-sphere, the extended complex plane (Riemann sphere). In the special case of a metric sphere we take
-- P = P0 ≡ 1 + ζζ,

while in general we write

-- P = V (u,ζ,ζ )P0. (F.2 )
The G2CEq contains the general P, while the special case using P0 yields the GCEq.

For the most general situation, the G2CEq can be written as a differential equation for the function -- u = G(ζ,ζ):

--2 2 2 -- -- ∂ G ≡ ∂ ζ(V P 0∂ζG ) = σ (G,ζ, ζ), (F.3 )
or
-- P 20∂2ζG + 2[P20V −1∂-V + P0ζ]∂ζG = V −2σ(G, ζ,ζ). (F.4 ) ζ
When V = 1 we have the GCEq:
-2 2 - -- -- ∂0G ≡ ∂ζ(P0∂ζG ) = σ(G, ζ,ζ).

When the arbitrary spin-weight-2 function, σ-(G, ζ, ¯ζ) vanishes, we have the homogeneous G2CEq:

2 - ∂ ζ(P ∂ζG ) = 0. (F.5 )

It is now shown how, by a coordinate transformation of the (independent) complex stereographic coordinates (-- ζ,ζ), G2CEq can be transformed into the GCEq. It must be remembered from our notation that -- ζ∗ (or -- ζ) is close to, but is not necessarily, the complex conjugate of ζ∗ (or ζ).

First rewrite the GCEq with stereographic coordinates (∗ -∗ ζ ,ζ) as

--2 -- -∗ ∂ 0∗G = ∂ζ∗(P0∗2∂ ζ∗G ) = σ∗(G, ζ∗,ζ ), (F.6 ) ∗ ∗-∗ P0 = 1 + ζ ζ , (F.7 )
and the G2CEq as
-2 2 2 - -- -- ∂ G = ∂ζ(V P 0∂ζG ) = σ(G, ζ,ζ). (F.8 )

We now apply the coordinate transformation

-- -- -- ζ∗ = ζ-+-W---≡ N (ζ,ζ), (F.9 ) 1 − W ζ ζ∗ = ζ, (F.10 )
with W (a spin-weight 1 function) defined from
-- V −2 = 1 + ∂0W = 1 + P0∂-W − W ζ, -- ζ P0 = 1 + ζζ,
to Eq. (F.8View Equation). Substituting the derived relations,
-- -∗ 1 + ζζ P0 P∗0 = 1 + ζζ = --------= --------, 1 − W ζ 1 − W ζ ∂ ζG = ∂ζ∗G ⋅ ∂ζN, 2 2 - 2 - 2 ∂ζG = ∂ζy∗G ⋅ (∂ζN ) + ∂ζ∗G ⋅ ∂ζN, V − 2 − W ζ ∂ζN = ----------2, (1 − W ζ) 2 2 ζ[V −2 − 1]∂ζW ζ∂ζW 2V− 3∂ζV ∂ζN = -------------3-- + ---------2 − ---------2, (1 − W ζ) (1 − W ζ) (1 − W ζ)
into Eq. (F.8View Equation), we have, after a bit of algebra,
-2 ∗2 ∂0∗G = ∂ ζ∗(P 0 ∂ζ∗G) = F (ζ∗,ζ∗)σ(G, ζ(ζ∗,ζ∗),ζ∗(ζ∗,ζ∗)) -- -∗ ≡ σ ∗(G, ζ∗,ζ ),
namely Eq. (F.6View Equation), the GCEq.

Hence, we see that the G2CEq is equivalent to the GCEq via the coordinate transformation (F.9View Equation). This means that the study of the G2CEq on a general 3-surface ℌ 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, u = G (ζ, ¯ζ), known as ‘good-cut functions’, describe cross-sections of ℌ that are referred to as ‘good cuts.’ From the tangents to these good cuts, -- L = ∂G, one can construct null directions (pointing out of ℌ) into the spacetime itself that determine a NGC whose shear vanishes at ℌ.


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