Hod’s conjecture

Hod’s conjecture proposes a universal upper bound on the damping rate of a black hole’s least-damped quasinormal mode in terms of its Hawking temperature. Public work has tested the bound in several specialized black-hole models, with mixed numerical evidence but no general proof or definitive counterexample.

Known results

• Test-scalar perturbations in five-dimensional Gauss–Bonnet gravity showed formal violations, but the authors said gravitational perturbations were needed before treating this as a counterexample (2019).
• A specified regime of charged black holes satisfies Hod’s proposal and a stronger bound, without addressing the general conjecture (2022).
• A five-dimensional Einstein–Gauss–Bonnet Bumblebee model reported violations for small Lorentz-violation parameter $\lambda$ and larger Gauss–Bonnet coupling $\alpha$ (2024).

August 2026 numerical validation

Qin and Long cross-validated quasinormal-mode and transmission-probability calculations for charged black holes in Kalb–Ramond gravity with perfect-fluid dark matter. Their tested parameter scan found no violation of $\lvert\operatorname{Im}(\omega)\rvert\le\pi T_H$, providing model-specific evidence rather than a proof.

Current status (as of August 2026): The general conjecture remains open; numerical tests provide both supporting evidence and model-dependent apparent violations, none decisive.