Absorption and quasinormal modes by rotating acoustic black holes in Lorentz-violating background

This paper investigates a rotating acoustic black hole in a (2+1)-dimensional Lorentz-violating background, demonstrating that symmetry violation enhances the absorption cross section across all energy scales while accelerating the damping of quasinormal modes by reducing their real frequencies and increasing the magnitude of their imaginary parts.

The Gist

Imagine you are standing by a whirlpool in a bathtub. If you throw a rubber duck into the water, it gets sucked in. But what if the water itself had a secret "glitch" in the laws of physics that made it behave differently than we expect? That is essentially what this paper explores, but instead of a bathtub, the scientists are looking at a sound-based black hole.

Here is a simple breakdown of their discovery:

1. The Setup: A "Sound" Black Hole

Real black holes are made of gravity and light. But it's hard to study them in a lab. So, physicists use analog models. They create a fluid (like water or a gas) that flows so fast it creates a "sonic horizon." Once sound waves enter this fast-flowing zone, they can't swim back out, just like light can't escape a real black hole. This is a Rotating Acoustic Black Hole.

2. The Twist: Breaking the Rules (Lorentz Violation)

In our universe, there is a rule called Lorentz symmetry. Think of it like a universal speed limit and a rule that says physics looks the same no matter how you are moving or which way you are facing.

This paper asks: "What if that rule is slightly broken?"
The authors introduce a "glitch" (a parameter they call $\alpha$) into the equations. It's like adding a tiny bit of friction or a strange magnetic field to the fluid that shouldn't be there according to standard physics. They wanted to see how this glitch changes the behavior of the sound-black hole.

3. The Findings: What Happens When the Rules Break?

A. The "Vacuum Cleaner" Gets Stronger (Absorption)

Imagine the black hole is a vacuum cleaner sucking in sound waves.

• The Discovery: When they turned on the "glitch" (Lorentz violation), the vacuum cleaner got stronger.
• The Analogy: Normally, a vacuum cleaner might struggle to suck up a heavy rug. But with this glitch, it suddenly sucks up everything, even the heavy stuff, much more efficiently.
• The Result: The "absorption cross-section" (a fancy way of saying "how big the target is for the black hole to catch waves") got bigger at all energy levels. Even the rotation of the black hole (spinning like a top) helped the glitch make the black hole eat more sound.

B. The "Echoes" Die Faster (Quasinormal Modes)

When you hit a bell, it rings and then slowly fades away. Black holes do something similar when they are disturbed; they "ring" with specific frequencies before fading out. These are called Quasinormal Modes.

• The Discovery: The glitch changed the sound of the bell.
  • The Pitch (Real Part): The pitch of the ring got slightly lower.
  • The Fade (Imaginary Part): The sound died out much faster.
• The Analogy: Imagine a bell that usually rings for 10 seconds. With this glitch, it rings for only 2 seconds and sounds a bit deeper. The black hole becomes "damped" or "muted" more quickly. The energy of the disturbance is drained away faster.

C. The Shadow Gets Bigger

Black holes cast a "shadow" (a dark spot where light/sound can't escape).

• The Discovery: Because the glitch makes the black hole a better absorber, its "shadow" effectively gets larger.
• The Analogy: If you hold a net to catch fish, and you make the net sticky, you catch more fish. The "sticky net" (the glitch) makes the black hole's capture zone wider.

4. Why Does This Matter?

You might ask, "Why study a sound hole in a lab?"

• Testing the Universe: We don't know if the laws of physics are perfect. Maybe at incredibly high energies (like inside a star or right after the Big Bang), these rules break down. This paper simulates that scenario.
• Quark-Gluon Plasma: The authors mention that this could help us understand Quark-Gluon Plasma (a super-hot soup of particles that existed right after the Big Bang). If that plasma has these "glitches," it might behave like the sound black holes in their experiment.
• Safety Net: It helps us understand how black holes would behave if the universe wasn't exactly how we think it is.

Summary

In short, the authors built a mathematical model of a spinning black hole made of sound. They added a "rule-breaker" to the mix. They found that breaking the rules makes the black hole a voracious eater (it absorbs more sound) and a quieter singer (its vibrations die out faster).

It's a reminder that even tiny changes in the fundamental laws of physics can have huge, dramatic effects on how the universe (or a bathtub) behaves.

Technical Summary

1. Problem Statement

The paper investigates the physical consequences of Lorentz symmetry violation (LV) on the dynamics of a rotating acoustic black hole in $(2+1)$ dimensions. Specifically, the authors aim to determine how LV affects:

• The absorption cross-section ($\sigma_{abs}$) of scalar waves across different energy regimes (low and high frequency).
• The quasinormal modes (QNMs) (complex frequencies characterizing the damping and oscillation of perturbations) of the system.

The study is motivated by the potential manifestation of LV effects in high-energy regimes, such as the formation of quark-gluon plasma (QGP), and the utility of analog gravity models (acoustic black holes) to simulate these phenomena in controlled laboratory environments.

2. Methodology

A. Theoretical Framework

• Model Origin: The acoustic metric is derived from the Abelian Higgs model extended with a Lorentz-violating term in the Lagrangian:
  $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + |D_\mu\phi|^2 + m^2|\phi|^2 - b|\phi|^4 + k_{\mu\nu}D_\mu\phi^* D^\nu\phi$$
  where $k_{\mu\nu}$ is a constant symmetric tensor breaking Lorentz symmetry.
• Metric Construction: The authors consider a specific case where the tensor components are defined by parameters $\alpha$ and $\beta$. They focus on the scenario where $\beta=0$ and $\alpha \neq 0$, representing a specific type of LV.
• Acoustic Metric: In the non-relativistic limit ($v^2 \ll c_s^2$) for an incompressible, axially symmetric fluid with radial drainage ($v_r \propto 1/r$) and vortex ($v_\phi \propto 1/r$), the line element for the rotating acoustic black hole is derived as:
  $$ds^2 = -F(r)(1+\alpha)d\tau^2 - 2B d\phi d\tau + G^{-1}(r)dr^2 + \gamma^2(r)d\phi^2$$
  Here, $A$ represents the drainage parameter, $B$ the rotation (vortex) parameter, and $\alpha$ the LV parameter. The presence of $\alpha$ modifies the horizon radius ($r_h$) and the ergoregion.

B. Analytical Approaches

1. Low-Frequency Regime:
   • The massless scalar field is governed by the Klein-Gordon equation on the derived metric.
   • Using partial wave analysis, the radial equation is transformed into a Schrödinger-type equation using a "tortoise" coordinate.
   • The authors perform an asymptotic expansion for $\omega \to 0$ to derive an analytical expression for the transmission coefficient and the absorption cross-section.
2. High-Frequency Regime:
   • The geodesic method is employed. The authors analyze null geodesics (sound rays) to determine the critical impact parameters ($b_c$) that separate scattering from capture.
   • The classical absorption cross-section is calculated as the sum of the critical impact parameters for co-rotating and counter-rotating trajectories.

C. Numerical and Approximation Methods

• Numerical Integration: The radial wave equation is integrated numerically from the event horizon (imposing purely ingoing boundary conditions) to spatial infinity to extract reflection ($R$) and transmission ($T$) coefficients for the full frequency spectrum.
• Quasinormal Modes (QNMs): The authors employ the sixth-order WKB approximation (Konoplya's method) to calculate the complex QNM frequencies ($\omega = \omega_R + i\omega_I$). This method is applied to the effective potential barrier of the system.

3. Key Contributions and Results

A. Absorption Cross-Section ($\sigma_{abs}$)

• Low-Frequency Limit: The analytical derivation reveals that the absorption cross-section acquires an explicit dependence on the rotation parameter $B$ due to the presence of $\alpha$.
  $$\sigma_{abs}^{(0)} \propto \frac{A}{\sqrt{1+\alpha}} \sqrt{1 + \frac{2\alpha\sqrt{1+\alpha}(A+B)}{A}}$$
  Result: Lorentz violation ($\alpha > 0$) increases the absorption cross-section compared to the standard case ($\alpha=0$).
• High-Frequency Limit: The geodesic analysis shows that the critical impact parameters are modified by $\alpha$.
  Result: The critical capture radius increases, leading to a larger classical absorption cross-section.
• Numerical Verification: Numerical integration confirms the analytical limits. The total absorption cross-section increases monotonically with $\alpha$ across the entire frequency spectrum.
  • Superradiance: The study observes that increasing $\alpha$ reduces the superradiance regime (where $|R|^2 > 1$), effectively suppressing the amplification of waves caused by rotation.

B. Quasinormal Modes (QNMs)

• Frequency Shifts: The presence of the LV parameter $\alpha$ systematically alters the complex frequencies:
  • Real Part ($\text{Re}(\omega)$): Decreases as $\alpha$ increases. This implies a reduction in the oscillation frequency of the perturbations.
  • Imaginary Part ($\text{Im}(\omega)$): The magnitude of the imaginary part increases (becomes more negative).
• Physical Interpretation: An increased magnitude of the imaginary part indicates faster damping of the oscillations. Lorentz violation makes the acoustic black hole more efficient at dissipating perturbations.
• Rotation Coupling:
  • For co-rotating modes ($m > 0$), the effect of $\alpha$ on the damping rate intensifies as the rotation parameter $B$ increases.
  • For counter-rotating modes ($m < 0$), the influence of $\alpha$ on the damping is attenuated at higher rotation values, suggesting that rotation dominates the dynamics in this regime.

4. Significance and Conclusion

• Analog Gravity Validation: The work demonstrates that Lorentz symmetry violation, a feature of various quantum gravity theories, leaves distinct imprints on analog black hole systems. Specifically, it enhances the "capture" capability of the black hole (larger shadow/absorption) and accelerates the decay of perturbations.
• Experimental Relevance: Since acoustic black holes can be realized in laboratory fluids (e.g., Bose-Einstein condensates or water tanks), these results suggest that measuring absorption cross-sections or QNM damping rates could provide a pathway to constrain or detect Lorentz-violating parameters in a controlled setting.
• Astrophysical Analogy: The findings offer insights into how fundamental symmetry violations might affect the gravitational wave signatures (ringdown) and shadow sizes of astrophysical black holes, potentially aiding in the interpretation of data from detectors like LIGO/Virgo and telescopes like the Event Horizon Telescope (EHT).

In summary, the paper establishes that Lorentz violation acts as a mechanism to increase the effective size of the acoustic black hole's capture region and to dampen its oscillatory modes more rapidly, with a notable interplay between the LV parameter and the system's rotation.