On the Gravitational Energy of Axial Perturbations in Regular Black Holes

The article deals with the gravitational energy associated with axial perturbations of regular black holes. We review the stability of the geometry under odd-parity perturbations and the corresponding quasinormal modes, previously obtained for this class of spacetimes. The perturbative functions describing the metric fluctuations are reconstructed from the master equation. To evaluate the energy content of these perturbations, we employ the Teleparallel Equivalent of General Relativity (TEGR), which provides a well-defined expression for gravitational energy. The gravitational energy is computed up to second order in the perturbation parameter and expressed in terms of the quasinormal mode functions. Our results establish a direct connection between the dynamical response of regular black holes and the energy carried by their gravitational perturbations.

💡 Research Summary

The paper investigates the gravitational energy carried by axial (odd‑parity) perturbations of regular (non‑singular) black holes, using the Teleparallel Equivalent of General Relativity (TEGR) as a framework that provides a well‑defined, tensorial expression for gravitational energy. The authors begin by reviewing the motivation for regular black holes: classical solutions of Einstein’s equations typically contain curvature singularities (e.g., the Schwarzschild singularity), which raise conceptual and physical problems. Regular black holes, such as the Bardeen family, avoid a central singularity by introducing a parameter α that modifies the metric function
(f(r)=1-\frac{2Mr^{2}}{(r^{2}+α^{2})^{3/2}}).
For α≠0 the spacetime is regular at r=0 while still possessing an event horizon defined by f(r)=0.

Next, the authors consider axial perturbations of this background. They write the perturbed metric as (g_{\mu\nu}= \bar g_{\mu\nu}+ε h_{\mu\nu}) and keep only the off‑diagonal components (\delta g_{0φ}=ε h_{0}(r,t)h(θ)) and (\delta g_{rφ}=ε h_{1}(r,t)h(θ)). Assuming a harmonic time dependence (e^{-iωt}) and Legendre angular dependence, the Einstein equations reduce to a single master equation for the variable (\phi(r)=f(r)\tilde h_{1}(r) r):
(\frac{d^{2}\phi}{dx^{2}}+