Stability Analysis of Canonical Acoustic Black Hole

Technical Summary: Stability Analysis of Canonical Acoustic Black Hole

Problem Statement
This paper investigates the stability of the "canonical acoustic black hole" (a convergent spherical acoustic black hole) under perturbations from three distinct fields: a non-minimally coupled scalar field, an electromagnetic field, and a Dirac field. While the stability of general relativistic black holes and their analogues (such as the draining bathtub model) has been studied, this work specifically focuses on the canonical metric derived from a spherically symmetric, incompressible, convergent flow. The primary objective is to determine if this acoustic system remains stable when subjected to these perturbations and to characterize the resulting quasi-normal modes (QNMs).

Methodology
The authors employ a theoretical framework based on analogue gravity, where the acoustic metric is derived from fluid dynamics equations (continuity and Euler's equations). The study proceeds in the following steps:

1. Metric Formulation: The acoustic line element for a convergent flow is established, yielding a metric similar to the Schwarzschild solution but with a specific function $f(r) = 1 - (r_0/r)^4$, where $r_0$ represents the acoustic horizon.
2. Field Equations:
   • Scalar Field: The dynamics are governed by a modified Klein-Gordon equation incorporating a non-minimal coupling parameter $\zeta$ to the Ricci scalar $R$.
   • Electromagnetic Field: The evolution is governed by Maxwell's equations in the curved acoustic background.
   • Dirac Field: The dynamics are described by the massless Dirac equation in curved spacetime, utilizing a tetrad formalism to separate variables.
3. Reduction to Schrödinger-like Equations: Through standard ansatzes (separation of variables using spherical harmonics or spin-weighted harmonics) and the introduction of the tortoise coordinate ($r_*$), the authors reduce the field equations for all three cases into a Schrödinger-like wave equation of the form:
   $$\frac{d^2\phi}{dr_*^2} + (\omega^2 - V(r))\phi = 0$$
   where $V(r)$ is the effective potential and $\omega$ is the complex frequency.
4. Stability Analysis:
   • Potential Analysis: The authors analyze the sign and behavior of the effective potential $V(r)$ outside the horizon.
   • QNMs Calculation: The quasi-normal mode frequencies are computed using the third-order WKB (Wentzel-Kramers-Brillouin) approximation method. This involves calculating derivatives of the potential at its maximum point to determine the complex frequency $\omega = \omega_R + i\omega_I$.

Key Contributions and Results

• Scalar Field Perturbation:

  • The effective potential $V(r)$ is found to be strictly positive outside the horizon. It depends on the angular quantum number $l$, the overtone number $n$, and the coupling parameter $\zeta$.
  • As $\zeta$ and $l$ increase, the peak value of the potential increases, and the peak location shifts inward (decreases).
  • QNMs: The imaginary parts of the QNMs ($\omega_I$) are negative for all tested parameters, indicating stability. The real part ($\omega_R$) and the magnitude of the imaginary part increase with the coupling parameter $\zeta$. This implies that stronger coupling leads to faster oscillation frequencies and more rapid damping of perturbations.

• Electromagnetic Field Perturbation:

  • The effective potential is positive everywhere outside the horizon and depends on $l$. Both the potential value and the peak position increase with $l$.
  • QNMs: The imaginary parts of the frequencies are negative, confirming stability. The results show that for a fixed $l$, higher overtone numbers ($n$) correspond to lower real frequencies and larger absolute imaginary parts (faster damping).
  • Comparison: When compared to the Schwarzschild black hole, the canonical acoustic black hole exhibits significantly larger absolute values for the imaginary part of the QNMs, suggesting that acoustic perturbations decay much faster than gravitational perturbations in the Schwarzschild case.

• Dirac Field Perturbation:

  • The effective potential is positive and depends on the total angular momentum quantum number $|k|$. As $|k|$ increases, both the potential value and the peak position increase (the peak moves to the right).
  • QNMs: The imaginary parts of the frequencies are negative, confirming stability. Both the real and imaginary parts of the frequency increase with the mode order $n$, meaning higher-order modes oscillate faster and decay more rapidly.

Significance and Claims
The paper claims that the canonical acoustic black hole is stable under perturbations from scalar, electromagnetic, and Dirac fields. This conclusion is supported by two consistent lines of evidence:

1. The effective potentials for all three cases are positive outside the event horizon, which, according to Chandrasekhar's arguments, precludes the existence of unstable modes.
2. The calculated QNMs possess negative imaginary parts, which physically corresponds to the damping of perturbations rather than exponential growth.

The authors emphasize that the stability of this analogue system mirrors the behavior of general relativistic black holes (such as the Kerr and Schwarzschild solutions) regarding linear perturbations. The study provides specific quantitative data on how the non-minimal coupling parameter $\zeta$ influences the stability and damping rates of scalar fields, a detail specific to this analogue model. The work reinforces the utility of acoustic black holes as laboratory-scale tools for probing the stability and characteristic frequencies of black hole physics in curved spacetime.