On the Gravitational Angular Momentum of Axial Perturbations of a Regular Black Hole

This paper derives a closed expression for the gravitational angular momentum of axial perturbations in a Bardeen regular black hole using the teleparallel equivalent of general relativity, revealing a multipolar selection rule where the angular momentum vanishes for odd multipole indices but is nonzero for even ones.

The Gist

Imagine a black hole not as a terrifying, infinitely dense point that tears space apart, but as a perfectly smooth, ultra-dense sphere. This is the "Bardeen regular black hole" the authors are studying. It's a theoretical object that behaves like a black hole (it has an event horizon) but avoids the mathematical "crash" or singularity at its center.

The paper asks a specific question: If you poke this smooth black hole, how much "twisting" energy (angular momentum) does that poke create?

Here is the breakdown of their findings using simple analogies:

1. The Setup: Poking the Black Hole

Think of the black hole as a giant, still pond. The authors are studying what happens when you drop a stone in it, but instead of water ripples, they are looking at "axial perturbations."

• The Analogy: Imagine spinning a top. If you nudge it slightly, it wobbles. The authors are calculating the "wobble" of the black hole's gravity.
• The Tool: They used a specific mathematical toolkit called TEGR (Teleparallel Equivalent of General Relativity). You can think of this as a different pair of glasses for looking at gravity. While standard Einstein gravity looks at how space curves, TEGR looks at how space "twists" (torsion). This toolkit allows them to measure the "twisting energy" very precisely.

2. The Big Discovery: The Odd/Even Rule

The most surprising result of the paper is a strict "selection rule" regarding the shape of the wobble.

• The Analogy: Imagine the black hole is a drum. You can hit it in different patterns. Some patterns are "odd" (like a wobble that flips upside down), and some are "even" (like a symmetric bulge).
• The Result:
  • Odd Patterns (Odd numbers): If the wobble has an "odd" shape (mathematically, an odd number called $\ell$), the black hole creates zero twisting energy. It's like trying to spin a perfectly balanced wheel by pushing it from the exact center; nothing happens.
  • Even Patterns (Even numbers): If the wobble has an "even" shape, the black hole does generate twisting energy.

The authors found that only the even-numbered wobbles carry angular momentum. The odd ones are "silent" in terms of rotation.

3. How They Measured It

The authors didn't just guess; they did the math using the "Hamiltonian definition" from their toolkit.

• The Surface Term: They found that the total twisting energy is determined entirely by what happens at the "surface" or edge of the region they are measuring, rather than deep inside the volume.
• The Calculation: They plugged in known "ringing" patterns (called quasinormal modes) of the Bardeen black hole. These are the specific frequencies at which the black hole vibrates after being disturbed, similar to how a bell rings at specific notes after being struck.

4. What the Graphs Show

The paper includes several graphs showing how this twisting energy behaves over time and distance:

• Distance: As you move further away from the black hole, the twisting energy builds up and oscillates (wiggles up and down) before settling.
• Time: Over time, the twisting energy vibrates and slowly fades away, just like the sound of a bell dying out.
• The "Smoothness" Factor: The Bardeen black hole has a "smoothness parameter" (called $\alpha$). The authors found that if this smoothness parameter is small, the black hole behaves almost exactly like a standard, "rough" (singular) black hole. The twisting energy looks nearly identical in both cases.

5. Why This Matters (According to the Paper)

The authors conclude that this "Odd/Even Rule" is a new way to test black holes.

• The Limitation: Currently, we can't easily tell the difference between a "smooth" black hole (Bardeen) and a "rough" one (standard General Relativity) just by listening to their ringdown frequencies (the notes they play). They sound too similar.
• The New Clue: However, the amount of twisting energy they carry depends on the shape of the wobble in a very specific way (the even/odd rule). This provides a new, concrete target for future experiments. If we can measure the angular momentum of a real black hole's wobble, we might finally be able to tell if it has a smooth center or a singular one.

In summary: The paper shows that for a smooth, regular black hole, gravity only "spins up" when the disturbance has a specific symmetric shape (even numbers). If the disturbance is asymmetric (odd numbers), no spin is generated. This rule offers a new, precise way to distinguish between different types of black holes in the future.

Technical Summary

Technical Summary: On the Gravitational Angular Momentum of Axial Perturbations of a Regular Black Hole

Problem Statement
Current observational data regarding compact astrophysical objects, while detailed, cannot directly probe the internal structure of strong-field regions or definitively discriminate between singular solutions of General Relativity (GR) and nonsingular alternatives like regular black holes. While the stability of regular black holes (specifically the Bardeen solution) under axial gravitational perturbations has been established via quasinormal mode (QNM) analysis, the characterization of the conserved quantities associated with these perturbations remains incomplete. Specifically, there is a need to determine the gravitational angular momentum carried by axial (odd-parity) perturbations within a framework that provides unambiguous definitions for conserved charges.

Methodology
The authors investigate the gravitational angular momentum of axial perturbations of the Bardeen regular black hole using the Teleparallel Equivalent of General Relativity (TEGR). The methodology proceeds as follows:

1. Framework: The study employs the Hamiltonian formulation of TEGR, where gravitational energy-momentum and angular momentum arise as generators of spacetime translations and global Lorentz transformations, respectively. These quantities are defined as well-defined surface integrals associated with spacetime symmetries.
2. Background and Perturbations: The background spacetime is the spherically symmetric Bardeen metric, characterized by mass $M$ and a regularization parameter $\alpha$. Axial (odd-parity) perturbations are introduced via metric components $\delta g_{0\phi}$ and $\delta g_{r\phi}$, governed by linearized Regge–Wheeler-type equations derived in previous work.
3. Tetrad Construction: To calculate the angular momentum, a tetrad field adapted to static observers in the Bardeen spacetime is constructed. This tetrad is chosen such that the observer four-velocity corresponds to the timelike component of the inverse tetrad.
4. Derivation: The gravitational angular momentum tensor $L_{ab}$ is expressed as a volume integral of the angular momentum density. By restricting to spatial indices and utilizing the specific tetrad, the authors derive an explicit expression for the axial component of the angular momentum, $J^{(3)}$.
5. Perturbative Expansion: The perturbative angular momentum $\delta J$ is defined as the first-order term in the expansion parameter $\epsilon$. The calculation involves integrating the angular momentum density over a spatial hypersurface, utilizing the properties of Legendre polynomials $P_\ell(\cos \theta)$ to perform the angular integration analytically.
6. Numerical Illustration: The resulting closed-form expression is evaluated using known axial quasinormal mode solutions for the Bardeen spacetime (obtained via third-order WKB approximation) to illustrate the radial and temporal behavior of $\delta J$.

Key Contributions and Results
The primary contribution of this work is the derivation of a closed, simple expression for the perturbative gravitational angular momentum $\delta J$ in terms of the axial perturbation function $h_0(r, t)$. The analysis yields a distinct multipolar selection rule:

• Odd $\ell$ Modes: The gravitational angular momentum vanishes identically for all odd values of the multipole index $\ell$.
• Even $\ell$ Modes: Nonzero contributions to the angular momentum arise only for even values of $\ell$.

The derived expression for $\delta J$ depends explicitly on the background metric function $f(r)$ and the parity of $\ell$. For $\ell \geq 1$, the angular momentum is given by an integral over the radial coordinate $r$ involving the perturbation function $h_0$ and the metric function.

Numerical evaluations using the fundamental axial mode ($\ell=2, n=0$) and higher overtones demonstrate:

• Radial Behavior: The angular momentum exhibits an oscillatory radial structure reflecting the spatial profiles of the axial modes. The accumulation of $\delta J$ is weakly sensitive to the regularization parameter $\alpha$ for small values, remaining close to the Schwarzschild ($\alpha=0$) case.
• Temporal Evolution: The time dependence shows the expected oscillatory decay characteristic of quasinormal frequencies.
• Degeneracy: The time evolution of $\delta J$ for small $\alpha$ closely mimics the behavior of the singular Schwarzschild case, suggesting that regular and singular geometries may produce nearly degenerate signatures in perturbative gravitational angular momentum.

Significance
The paper claims that this result provides a new characterization of gravitational perturbations in regular black holes, complementing previous analyses based solely on quasinormal frequencies. The multipolar selection rule (vanishing for odd $\ell$) is presented as a consequence of the axial angular dependence within the TEGR definition of angular momentum, rather than an independent dynamical property of the Bardeen background.

The authors emphasize that while ringdown observations are limited by degeneracies among different effective black-hole descriptions, the multipolar selection rule for gravitational angular momentum offers a concrete, complementary target for experimental tests. It extends beyond standard spectroscopic checks based on frequencies alone, potentially aiding in distinguishing between different perturbative sectors or geometries, although the paper notes that current observational uncertainties may still render regular and singular geometries indistinguishable in the perturbative angular momentum sector. The work does not propose new experimental setups but highlights the theoretical utility of TEGR-conserved charges in analyzing black hole spectroscopy.