Acoustic Black Hole in Hayward Spacetime: Shadow, Quasinormal Modes and Analogue Hawking Radiation

Here is an explanation of the paper "Acoustic Black Hole in Hayward Spacetime," translated into simple, everyday language with creative analogies.

The Big Idea: Building a Black Hole in a Bathtub

Imagine you want to study a black hole. The problem is, they are far away, incredibly hot, and involve gravity so strong it breaks our current laws of physics. You can't bring one into a lab.

So, physicists have a clever trick: Analogue Gravity. Instead of using actual gravity, they use sound.

Think of a river flowing toward a waterfall. If the water flows faster than the speed of sound, a sound wave trying to swim upstream will never make it past the edge. To the sound wave, that edge acts exactly like a black hole's "event horizon"—a point of no return. This is called an Acoustic Black Hole.

In this paper, the authors didn't just build a standard acoustic black hole. They built one inside a very specific, "smooth" theoretical universe called Hayward Spacetime.

1. The Setting: A "Smooth" Black Hole (Hayward Spacetime)

Usually, when we talk about black holes, we imagine a "singularity" at the center—a point where matter is crushed into infinite density and the laws of physics break down. It's like a hole in the fabric of reality.

The Hayward Black Hole is a special kind of theoretical black hole that avoids this crash.

The Analogy: Imagine a standard black hole is a sharp, jagged pit in the ground. The Hayward black hole is like a smooth, rounded bowl. It still has a deep center, but the bottom isn't a sharp point; it's a gentle curve.
Why it matters: This makes the math "regular" (no infinities), which is easier to work with and might be closer to what real black holes actually look like if we account for quantum mechanics.

2. The Experiment: Tuning the Flow

The authors created a model where a fluid (like a super-cooled gas) flows around this smooth Hayward black hole. They introduced a "Tuning Parameter" ( $\xi$ ).

The Analogy: Think of the fluid flow like a faucet.
- Low $\xi$ : The water flows gently. The "acoustic horizon" (where sound gets trapped) is close to the actual black hole.
- High $\xi$ : You turn the faucet up to maximum. The water rushes so fast that the "trap" for sound moves far away from the center. The whole universe becomes a place where sound cannot escape.

3. What They Found: The Three Main Discoveries

The paper looked at three things: the Shadow, the Vibrations (Quasinormal Modes), and the Radiation (Hawking Radiation).

A. The Shadow (The Silhouette)

When light (or sound) passes a black hole, it gets bent. If you look from far away, you see a dark circle (a shadow) against the background.

The Finding: As they turned up the "Tuning Parameter" (made the fluid flow faster), the shadow got bigger.
The Metaphor: Imagine holding a spoon in a stream. If the water is calm, the spoon casts a small shadow. If the water is raging, the turbulence pushes the "shadow" further out. The faster the flow, the larger the dark circle appears to an observer.

B. The Vibrations (Quasinormal Modes)

When you knock on a bell, it rings with a specific pitch and then fades away. Black holes do the same. If you poke a black hole, it "rings" in gravitational waves. These are called Quasinormal Modes (QNMs).

The Finding: The "Acoustic Hayward Black Hole" was more stable than a regular one. As they increased the tuning parameter, the "ringing" became slower and faded away more gently.
The Metaphor: A regular black hole is like a glass bell; if you hit it, it rings loudly and sharply. The acoustic version in this paper is like a muffled drum. It still vibrates, but the sound is softer, lower-pitched, and dies out more smoothly. This suggests the system is very stable and won't explode or collapse easily.

C. The Radiation (Hawking Radiation)

Stephen Hawking predicted that black holes aren't truly black; they leak tiny amounts of energy (radiation) because of quantum effects.

The Finding: As the tuning parameter increased, the black hole became a better emitter. It let more energy escape.
The Metaphor: Think of the black hole as a leaky bucket.
- With a low tuning parameter, the bucket has a tiny pinhole leak.
- As they increased the parameter, the hole in the bucket got bigger, and the water (energy) poured out faster.
- Interestingly, the "smoothness" of the Hayward black hole (the parameter $L$ ) didn't change the radiation much. The flow speed (tuning parameter) was the real driver.

4. Why This Matters

This paper is a bridge between two worlds:

Theoretical Physics: It proves that we can study complex, "smooth" black holes (Hayward) using sound waves in a lab.
Future Observations: Astronomers are currently taking pictures of real black holes (like M87* and Sagittarius A*). By understanding how "acoustic" black holes behave, scientists can create better models to interpret what they see in the sky.

In Summary:
The authors built a virtual "sound trap" inside a smooth, non-crashing black hole. They found that by speeding up the "wind" (the tuning parameter), the shadow gets bigger, the vibrations get calmer and more stable, and the energy leak gets stronger. It's like discovering that if you blow harder on a specific type of flute, the note changes in a predictable, stable way, giving us new clues about how the universe's most mysterious objects might actually work.

Here is a detailed technical summary of the paper "Acoustic Black Hole in Hayward Spacetime: Shadow, Quasinormal Modes and Analogue Hawking Radiation" by Hui, Cheng, and Sun.

1. Problem Statement

The paper addresses the intersection of analogue gravity and regular black hole physics. While analogue black holes (acoustic black holes) have been extensively studied in flat spacetime and singular backgrounds (like Schwarzschild), their behavior in regular black hole spacetimes (which lack central curvature singularities) remains underexplored. Specifically, the authors aim to:

Construct an acoustic black hole within the Hayward spacetime, a well-known model of a regular black hole free of intrinsic singularities.
Investigate observable quantities associated with this system, specifically the acoustic shadow, quasinormal modes (QNMs), and analogue Hawking radiation (grey-body factors and energy emission rates).
Determine how the tuning parameter of the fluid flow and the Hayward regularization parameter influence these physical properties.

2. Methodology

A. Theoretical Framework

Background Spacetime: The study utilizes the Hayward metric, a regular black hole solution derived from nonlinear electrodynamics. The metric is defined by a lapse function $f(r)$ containing a mass parameter $M$ (via Schwarzschild radius $r_s$ ) and a regularization length scale $L$ .
Analogue Gravity Model: The authors employ the relativistic Gross-Pitaevskii (GP) theory to describe a fluid in this curved background. By considering small fluctuations of a complex scalar field $\phi$ (representing the fluid), they derive an effective metric $G_{\mu\nu}$ for phonon propagation.
Acoustic Metric Construction: A fluid falling radially from rest at infinity is considered. The effective acoustic line element is derived by embedding the GP effective metric into the Hayward background. This introduces a tuning parameter $\xi$ related to the fluid's velocity profile.

B. Analytical and Numerical Techniques

Horizon Analysis: The locations of the Hayward event horizon ( $r_H$ ) and the acoustic horizons ( $r'_\pm$ ) are determined by solving cubic equations derived from the vanishing of the metric functions.
Acoustic Shadow: The shadow radius is calculated by analyzing critical null geodesics of the acoustic metric. The authors derive the radius of the acoustic photon sphere ( $r_A$ ) and the resulting shadow radius ( $r_S$ ) for a distant observer.
Quasinormal Modes (QNMs): The covariant scalar field equation for phonons is reduced to a Schrödinger-like wave equation with an effective potential $V(r)$ . The QNM frequencies ( $\omega$ ) are computed numerically using the WKB (Wentzel-Kramers-Brillouin) approximation up to the 9th order. Results are validated using the Asymptotic Iteration Method (AIM).
Hawking Radiation: The grey-body factors ( $|A_l(\omega)|^2$ ) and energy emission rates are calculated using the WKB method applied to scattering boundary conditions. The analogue Hawking temperature ( $T_H$ ) is derived from the surface gravity at the acoustic horizon.

3. Key Contributions

First Construction in Regular Spacetime: This work presents the first extension of acoustic black hole theory to a regular black hole background (Hayward), bridging the gap between analogue gravity and singularity-free gravity models.
Comprehensive Observable Analysis: Unlike previous studies focusing on single aspects, this paper provides a unified analysis of three distinct observational signatures: shadow geometry, QNM stability, and Hawking radiation spectra.
Parameter Dependence Mapping: The study systematically maps the behavior of these observables against two key parameters:
- $\xi$ (Tuning Parameter): Controls the fluid flow strength and the existence of the acoustic horizon.
- $L$ (Hayward Parameter): Controls the degree of regularization (deviation from Schwarzschild geometry).

4. Key Results

A. Horizon Structure

The acoustic Hayward black hole possesses an outer acoustic horizon ( $r'_+$ ) and an inner acoustic horizon ( $r'_-$ ).
Condition for Existence: An acoustic horizon exists only if the tuning parameter $\xi \geq 4$ .
Hierarchy: The horizons satisfy the inequality $r'_+ > r'_- > r_H$ . As $\xi \to \infty$ , the outer acoustic horizon moves to infinity, implying sound cannot escape the entire spacetime.

B. Acoustic Shadow

Shadow Radius ( $r_S$ ): The shadow radius increases monotonically as the tuning parameter $\xi$ increases.
Dependence on $L$ : The shadow size shows a slight decrease as the Hayward parameter $L$ increases, but this effect is minimal compared to the influence of $\xi$ .
Geometry: The shadow images are circular (due to spherical symmetry), and their size is directly correlated with the acoustic horizon radius.

C. Quasinormal Modes (QNMs)

Stability: All computed QNM frequencies have a positive real part ( $\text{Re}(\omega) > 0$ ) and a negative imaginary part ( $\text{Im}(\omega) < 0$ ), confirming the stability of the acoustic Hayward black hole against scalar perturbations.
Effect of $\xi$ : As $\xi$ increases, both the oscillation frequency ( $\text{Re}(\omega)$ ) and the decay rate ( $|\text{Im}(\omega)|$ ) decrease. This indicates that the acoustic black hole becomes "more stable" (slower decay) as the fluid flow intensifies.
Comparison: The QNM amplitudes for the acoustic black hole ( $\xi \geq 4$ ) are significantly smaller than those of the standard Hayward black hole ( $\xi = 0$ ), suggesting weaker detectability of acoustic signals compared to gravitational ones.
Correlation: The variations in QNM frequencies correlate directly with the behavior of the effective potential barrier, which becomes lower and wider as $\xi$ increases.

D. Analogue Hawking Radiation

Temperature ( $T_H$ ): The analogue Hawking temperature initially increases with $\xi$ (for $\xi \geq 4$ ) and then decreases. It decreases monotonically as $L$ increases.
Grey-body Factors & Emission Rate:
- Both the grey-body factor and the energy emission rate increase as the tuning parameter $\xi$ increases. This is attributed to the lowering of the effective potential barrier, facilitating easier tunneling of phonons.
- The dominant contribution to radiation comes from the $l=0$ (s-wave) mode.
- Variations in the Hayward parameter $L$ have a negligible effect on the radiation spectrum due to the constrained range of $L$ required for horizon existence.

5. Significance and Implications

Theoretical Insight: The study demonstrates that regular black hole spacetimes can support stable acoustic horizons with distinct dynamical properties. The "smoothing" of the potential barrier in the acoustic scenario leads to more stable QNMs compared to the singular background.
Observational Potential: The acoustic shadow provides a directly observable quantity. The strong dependence of the shadow size and radiation rates on the tuning parameter $\xi$ suggests that laboratory experiments with Bose-Einstein Condensates (BECs) or optical analogues could potentially distinguish between different fluid flow configurations.
Astrophysical Relevance: While the acoustic model is a laboratory analogue, the results offer a framework for understanding how acoustic horizons (if they exist in astrophysical fluid environments surrounding real black holes) might alter the dynamics near the gravitational horizon.
Future Directions: The authors propose extending this work to rotating acoustic regular black holes, investigating holographic connections, and developing observational diagnostics to distinguish between regular and singular acoustic black holes based on their QNM and shadow signatures.

In summary, this paper successfully constructs a robust theoretical model of an acoustic black hole in a regular spacetime, providing quantitative predictions for its shadow, stability, and radiation, thereby enriching the field of analogue gravity and offering new avenues for testing gravity theories in laboratory settings.