On the Gravitational Energy of Axial Perturbations in Regular Black Holes

This paper utilizes the Teleparallel Equivalent of General Relativity to compute the second-order gravitational energy of axial perturbations in regular black holes, establishing a direct link between their dynamical response via quasinormal modes and the energy carried by these fluctuations.

The Gist

Imagine the universe as a giant, stretchy trampoline. In the center of this trampoline sits a heavy ball, representing a black hole. Usually, when we talk about black holes in physics, we imagine the trampoline stretching so infinitely deep that it tears right through the fabric of reality at the very center. This "tear" is called a singularity, and it's a place where our current laws of physics break down and stop making sense.

But what if the trampoline didn't tear? What if, instead of a sharp, infinite point, the center was just a very smooth, rounded bump? This is the idea behind a "Regular Black Hole." It looks like a black hole from the outside (it has an event horizon, a point of no return), but the dangerous, physics-breaking tear at the center is gone.

The Big Question: How Do These Objects "Sing"?

The authors of this paper wanted to know: If you poke a regular black hole, how does it react?

Think of a bell. If you hit a bell, it doesn't just stay still; it vibrates. It rings with a specific tone that slowly fades away. In physics, these vibrations are called Quasinormal Modes. They are the "notes" a black hole plays when it gets disturbed.

The paper asks two main things:

1. Do these regular black holes have a stable "ring"? (Yes, they do. They vibrate and settle down, just like a normal black hole.)
2. How much "energy" is carried by these vibrations?

The Problem with Measuring Gravity's Energy

Here is where it gets tricky. In Einstein's theory of gravity, measuring the energy of the gravitational field itself is notoriously difficult. It's like trying to weigh the wind. For a long time, physicists couldn't agree on a single, clear way to measure this energy without getting confused numbers.

To solve this, the authors used a special tool called TEGR (Teleparallel Equivalent of General Relativity). You can think of TEGR as a different pair of glasses. When you look at gravity through standard glasses, the energy is blurry and hard to define. When you look through TEGR glasses, the energy becomes sharp, clear, and easy to calculate. It's like switching from a fuzzy map to a high-definition GPS.

What They Did

The team took the mathematical description of a regular black hole (the smooth bump on the trampoline) and imagined a small ripple moving across it. They used their "high-definition GPS" (TEGR) to calculate exactly how much energy is contained in that ripple.

They didn't just look at the energy at one spot; they looked at how the energy moves:

• Across space (Radial): They checked how the energy is distributed from the center of the black hole out to the edge.
• Over time (Temporal): They watched how the energy pulses and fades away as the black hole settles down.

What They Found

1. The "Core" Matters: The smooth center (the "regular" part) changes the loudness of the vibration. If you make the center smoother (changing a parameter they call $\alpha$), the energy of the ripple gets weaker or stronger, but the basic pattern of the wave stays the same. It's like changing the material of the bell; the tone might get softer, but the song is still the same song.
2. The "Note" Matters: The specific frequency of the vibration (the "note" the black hole is singing) changes how the energy ripples through space. Higher notes create tighter, faster ripples.
3. It Fades Away: Just like a real bell, the energy doesn't last forever. The vibrations are "damped," meaning they lose energy over time and the black hole returns to a calm state. This proves that these regular black holes are stable; they don't fall apart when poked.

The Bottom Line

This paper connects the "sound" a black hole makes when disturbed to the actual energy it carries. By using a special mathematical method (TEGR), the authors showed that regular black holes behave very much like normal black holes in terms of stability, but the smoothness of their center subtly changes how much energy is involved in their vibrations.

In short: Regular black holes are stable, they have a "voice," and we now have a clearer way to measure the energy of that voice.

Technical Summary

Technical Summary: On the Gravitational Energy of Axial Perturbations in Regular Black Holes

Problem Statement
While General Relativity admits solutions containing physical singularities, "regular" black hole models (such as the Bardeen solution) offer singularity-free alternatives that possess event horizons. Although the stability of these spacetimes under axial (odd-parity) gravitational perturbations has been established via the existence of quasinormal modes (QNMs), a complete characterization of their energetic content remains lacking. Specifically, there is no established analysis determining the gravitational energy associated with these perturbations. This is a non-trivial problem because defining gravitational energy in General Relativity (GR) is historically fraught with difficulties, often resulting in non-covariant pseudotensors. Furthermore, while the stability of regular black holes is known, the energy carried by their dynamical response has not been quantified within a framework that provides a well-defined local energy density.

Methodology
The authors employ a two-pronged approach combining perturbation theory in curved spacetime with the Teleparallel Equivalent of General Relativity (TEGR).

1. Perturbation Framework: The study utilizes the Bardeen class of regular black holes, described by a metric with a regularization parameter $\alpha$ that controls the deviation from the Schwarzschild geometry. Axial perturbations are introduced as odd-parity metric fluctuations. The authors reconstruct the perturbation functions ($h_0$ and $h_1$) from a master equation of the Schrödinger type, derived previously in Ref. [14]. The quasinormal frequencies ($\omega$) are determined using a third-order WKB approximation, and the radial wave functions are expressed in terms of these frequencies and the effective potential.
2. Energy Definition (TEGR): To evaluate the energy content, the authors adopt the Teleparallel Equivalent of General Relativity (TEGR). Unlike standard GR, TEGR utilizes a curvature-free Weitzenb"ock connection with non-zero torsion. This framework allows for a well-defined, covariant expression for the gravitational energy-momentum vector, avoiding the ambiguities of pseudotensors. The energy is computed by expanding the tetrad field and the associated superpotential in powers of a perturbation parameter $\epsilon$.
3. Calculation Strategy: The gravitational energy is calculated up to second order in the perturbation parameter ($\epsilon^2$). The first-order contribution vanishes identically. The authors define the energy variation $\delta E$ as the difference between the second-order perturbed energy and the background (unperturbed) energy. The calculation involves substituting the reconstructed perturbation functions into the TEGR energy-momentum integral over a closed two-surface.

Key Contributions and Results

• Explicit Energy Expression: The paper derives an explicit algebraic expression for the gravitational energy variation $\delta E(r)$ associated with axial perturbations, expressed in terms of the master variable $\phi(r)$ and the quasinormal mode functions.
• Numerical Analysis of Energy Profiles: The authors perform a numerical evaluation of the real part of the energy variation to analyze its spatial and temporal distribution:
  • Radial Dependence: The analysis reveals that the regularization parameter $\alpha$ primarily modulates the amplitude of the energy density without significantly altering the spatial oscillatory pattern. In contrast, changes in the quasinormal frequency (specifically the real part of $\omega$) primarily affect the phase and wavelength of the radial oscillations.
  • Temporal Evolution: The energy exhibits the characteristic damped oscillatory behavior expected from quasinormal modes. The temporal analysis confirms that the energy signal's oscillation frequency and damping rate are directly encoded by the complex quasinormal frequencies of the background geometry.
  • Consistency Check: In the limit where the regularization parameter $\alpha \to 0$, the results reduce to the known Schwarzschild case (reproducing the qualitative behavior found in Ref. [17]), serving as a validation of the calculation.
• Complex Nature of Energy: The study highlights that because quasinormal modes possess complex frequencies, the gravitational energy density is also complex. The plots focus on the real part to visualize the oscillatory spatial structure.

Significance and Claims
The paper establishes a direct connection between the dynamical response of regular black holes (their quasinormal modes) and the gravitational energy carried by these perturbations. By utilizing TEGR, the authors provide a consistent, non-pseudotensorial definition of this energy, demonstrating that regular black holes possess a well-defined energetic signature during their relaxation phase.

The authors modestly frame the work as a necessary step in the broader understanding of regular black hole physics. They note that while the axial sector is now characterized regarding energy, a corresponding analysis for polar (even-parity) perturbations remains an open issue, as a Zerilli-type master equation for regular black holes has not yet been established. Consequently, a complete understanding of the gravitational energy for all perturbative sectors of regular black hole spacetimes awaits the resolution of the polar perturbation problem. The current work serves as a foundational link between the stability properties and the energy content of these singularity-free geometries.