Interior evolution through the angular-characteristic degeneracy surface

Determine whether and how scalar and metric perturbations can be evolved through the interior surface where the angular characteristic coefficient vanishes, namely where \((q/(rX))^2=3B_\infty^2/B^2\), for the three-dimensional scalar–tensor Einstein–Gauss–Bonnet black-hole solutions on the \(D>0\) branch.

Background

The paper establishes the exterior perturbation theory and a short extension across the future metric horizon for the $D>0$ black-hole branch. In this region, the propagating scalar degree of freedom has a positive kinetic coefficient, a hyperbolic equation, and positive bulk spatial energy for compactly supported perturbations.

The authors explicitly identify an interior surface at which the angular characteristic coefficient vanishes. Their analysis does not determine whether the perturbation equations remain well posed there or whether solutions can be continued through it, making evolution across this surface an unresolved problem.