Dynamical viability across the Legendre-degeneracy radius

Analyze the characteristics of the first-order Hamiltonian action for the nonlinear-electrodynamics theory across the radius at which the Legendre map between the $P$ and $F$ frames degenerates, and determine whether the corresponding regular black-hole backgrounds are acceptable solutions with well-posed dynamics.

Background

For the electrically charged regular black holes considered in the paper, the Legendre map P↦F=PHP2P\mapsto F=PH_P^2 becomes degenerate at a radius rbr_{\rm b}. In dimensions D≥5D\geq5, this radius lies outside the event horizon for a range of near-extremal black holes. At the degeneracy, the effective metric governing electromagnetic perturbations becomes singular and the FF-frame description develops a branch point.

The thermodynamic identities established in the paper concern only static background geometries and do not resolve whether the underlying matter theory has well-posed dynamics across this degeneracy. The authors explicitly leave the characteristic analysis of the Hamiltonian action for future work.

References

The question whether these backgrounds are acceptable solutions of the field theory with the well-posed dynamics requires the analysis of the characteristics of (\ref{action}) across $r_{\rm b}$, which is beyond the scope of the present paper.

— Regular black holes do not violate the first law of thermodynamics  (2610.08099 - Toshmatov et al., 6 Oct 2026) in Section 6, “Remark on the matter theory”