On the Gravitational Energy of Axial Perturbations in Regular Black Holes

TL;DR

This work addresses the gravitational energy carried by axial perturbations of a regular black hole within the Teleparallel Equivalent of General Relativity (TEGR). The authors derive the axial master equation for a Bardeen-type regular black hole, compute the quasinormal modes via a third-order WKB method, and reconstruct the perturbation functions $h_0$ and $h_1$ from the master variable $\phi$, enabling a second-order expression for the energy variation $\delta \mathcal{E}(r)$. Using the TEGR energy–momentum framework, they obtain a concrete formula for $\delta \mathcal{E}(r)$ that depends on the perturbations and background metric, and they show numerically that the regularization parameter $\alpha$ predominantly scales the energy density while the quasinormal frequencies determine the oscillatory phase and damping (with the Schwarzschild limit recovered at $\alpha=0$). The results link the dynamical response of regular black holes to the gravitational energy carried by axial perturbations and motivate extending the analysis to polar perturbations for a complete energy accounting. $\text{(Key equations involve } f(r), V(r), \phi, \omega, \alpha, \ell, n, \delta\mathcal{E}(r)\text{ and their interrelations.)}$

Abstract

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.

Figures (4)

• Figure 1: Radial profile of the real part of the gravitational energy variation $\mathrm{Re}\,\delta\mathcal{E}(r)$ for the fundamental axial quasinormal mode with $\ell=2$ and $n=0$, evaluated at fixed time $t=5$. The mass is set to $M=1$ and different curves correspond to distinct values of the regularization parameter $\alpha$.
• Figure 2: Radial profile of the real part of the gravitational energy variation $\mathrm{Re}\,\delta\mathcal{E}(r)$ for several axial quasinormal modes, evaluated at fixed time $t=5$ with $M=1$ and $\alpha=0.3$. The different curves correspond to modes with $(\ell,n)=(2,0)$, $(2,1)$, $(3,0)$, and $(4,0)$, whose frequencies determine the spatial oscillation pattern.
• Figure 3: Temporal evolution of the real part of the gravitational energy variation $\mathrm{Re}\,\delta\mathcal{E}(t)$ evaluated at fixed radius $r=5$ for the fundamental axial quasinormal mode with $\ell=2$ and $n=0$. The mass is set to $M=1$, and the curves correspond to different values of the regularization parameter $\alpha$.
• Figure 4: Temporal evolution of the real part of the gravitational energy variation $\mathrm{Re}\,\delta\mathcal{E}(t)$ evaluated at fixed radius $r=5$ with $M=1$ and $\alpha=0.3$. The curves correspond to different axial quasinormal modes with $(\ell,n)=(2,0)$, $(2,1)$, $(3,0)$, and $(4,0)$.