Generality of QLHS features beyond head-on collisions
Ascertain whether the qualitative features of quasi-local horizon segments observed in head-on collisions of non-spinning black holes—including the behavior and spectrum of the stability operator—persist generically when spins are allowed and beyond head-on configurations.
References
Open Issue 7 (OI-7) Do qualitative features of the QLHSs of progenitors and the remnant, including the spectrum of the stability operator, continue to hold generically beyond the head-on collision, especially even when one allows for spin?
Several questions remain open. Ref. reaches the first MOTS but does not describe its subsequent evolution. In the present setting, $\operatorname{tr}\underline{\chi}<0$ at the first MOTS, and a generic first MOTS is strictly stable. If $\lambda_1>0$ and $\varphi>0$ are the principal eigenvalue and eigenfunction of the stability operator $\mathcal L_{\mathrm{MOTS}$, then an inward deformation with velocity $-\varphi\nu$ gives $\left.\frac{d}{d\varepsilon}\operatorname{tr}\chi\right|_{\varepsilon=0}=-\lambda_1\varphi<0$. Hence, for sufficiently small $\varepsilon>0$, the deformed surfaces satisfy $\operatorname{tr}\chi<0$ and $\operatorname{tr}\underline{\chi}<0$, and are therefore trapped. Under the standard hypotheses of Penrose's theorem, this implies future null geodesic incompleteness. The degenerate case $\lambda_1=0$ requires a separate rigidity analysis. Thus, the main remaining issue is the evolution of the resulting marginally trapped tube and trapped surface.