Establish persistence of the unstable branches with gravitational backreaction

Determine whether the static zero modes and unstable branches of the Lorentz-violating Kalb–Ramond field persist when gravitational backreaction and metric perturbations are included, and whether the additional Kalb–Ramond modes mix with gravitational quasinormal modes and produce signatures in black-hole ringdown.

Background

The analysis treats the Schwarzschild geometry as fixed and neglects the Kalb–Ramond field’s gravitational backreaction. Consequently, the reported zero modes, unstable branches, and oscillatory instabilities belong to a probe-field problem rather than to the fully coupled gravity–Kalb–Ramond theory.

The authors identify coupling metric perturbations to Kalb–Ramond fluctuations as an important unresolved extension. Such an analysis would test the robustness of the static and unstable modes and determine whether their interactions modify observable black-hole ringdown.