⚛️ general relativity

Acoustic Schwarzschild-like black hole and analogous Hawking radiation in Einstein-bumblebee gravity

This paper derives an acoustic Schwarzschild-like black hole within Einstein-bumblebee gravity using Gross-Pitaevskii theory, revealing that while both gravitational and acoustic Hawking temperatures depend on the Lorentz violation coupling constant, the acoustic temperature and grey-body factors exhibit a distinct, experimentally valuable dependence on the tuning parameter $\xi$ that increases with $\xi$ but decreases with the coupling constant.

Original authors: Chikun Ding, Yuebing Zhou, Yu Shi, Yuehua Xiao, Jiliang Jing

Published 2026-10-27

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CC BY 4.0

Original authors: Chikun Ding, Yuebing Zhou, Yu Shi, Yuehua Xiao, Jiliang Jing

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Black holes are the most extreme laboratories nature offers, places where the rules of gravity become so intense that even light cannot escape. For decades, physicists have wondered if these cosmic traps also emit a faint, ghostly glow known as Hawking radiation, a prediction that would link the physics of the very large with the physics of the very small. However, this glow is so incredibly faint for real black holes that no telescope has ever detected it. To solve this problem, scientists have turned to "analogue gravity," a field where they recreate the mathematical behavior of black holes using fluids in a laboratory. By speeding up a fluid until it flows faster than sound, they create a "dumb hole," a place where sound waves, like light in a real black hole, get trapped and cannot escape. This setup allows researchers to test the core ideas of black hole physics in a controlled environment, offering a glimpse into how gravity and quantum mechanics might interact.

Building on this concept, a team of researchers has taken the study of these laboratory black holes a step further by placing them inside a theoretical framework that challenges one of physics' most fundamental assumptions: that the laws of nature look the same to everyone, everywhere. This assumption is called Lorentz invariance. The researchers explored a theory called Einstein-bumblebee gravity, which suggests that at very high energies, this symmetry might be broken, subtly altering the shape of space and time. They used a mathematical model of a super-cold fluid, known as a Bose-Einstein condensate, to simulate a black hole existing within this modified spacetime. Their goal was to see how this potential break in symmetry would change the behavior of the acoustic black hole and the radiation it emits.

The team constructed a model of an acoustic black hole that possesses two distinct types of boundaries, or horizons. The first is the standard gravitational horizon, the point of no return for light, located at a specific distance determined by the mass of the black hole. The second is the acoustic horizon, the point where the fluid flows so fast that sound waves can no longer swim upstream. In their model, the researchers found that the position of this acoustic horizon is not fixed; it shifts depending on two specific factors. One factor is a tuning parameter that controls how fast the fluid flows, and the other is a constant that measures the strength of the symmetry breaking in the fabric of spacetime. They discovered that by adjusting these two knobs, they could move the acoustic horizon closer to or further away from the gravitational horizon, creating a complex region where the two interact.

When the researchers calculated the temperature of the radiation coming from these horizons, they found a striking difference between the two. The temperature of the gravitational horizon depends only on the symmetry-breaking constant, while the temperature of the acoustic horizon depends on both the flow speed and that same constant. Most notably, they found that the acoustic temperature reaches a specific maximum value when the flow speed is tuned to a precise setting. At this peak, the acoustic temperature is exactly one-eighth of the gravitational temperature. This result suggests that the "dumb hole" is cooler than the "black hole" it mimics, but the relationship between the two is governed by the specific conditions of the fluid and the underlying spacetime structure.

The study then looked at how particles, specifically sound waves in this fluid, escape from the black hole. In the real world, black holes are not perfectly black; they have a "grey-body" factor, meaning they are slightly better at letting some frequencies of radiation escape than others. The researchers calculated this factor for their acoustic black hole and found that it changes in opposite ways depending on which parameter is adjusted. When they increased the speed of the fluid flow, the grey-body factor went up, meaning more sound waves could escape. However, when they increased the strength of the symmetry breaking in the spacetime, the grey-body factor went down, making it harder for the waves to escape. This opposite behavior is a crucial discovery because it provides a way to tell the difference between a change caused by the fluid's motion and a change caused by a fundamental shift in the laws of physics.

Finally, the team examined the rate at which energy is emitted from the acoustic black hole. They found that this emission rate follows a pattern similar to the thermal radiation emitted by a hot object, with a specific peak frequency where the energy output is highest. Just like the grey-body factor, the total energy emitted increases with faster fluid flow but decreases as the symmetry-breaking effect gets stronger. The peak frequency of this emission also shifts depending on these parameters. The researchers conclude that this opposing relationship between the fluid flow and the spacetime modification is the most valuable feature of their model. In a laboratory setting, scientists could independently adjust the fluid flow and the background conditions to see which one is responsible for any observed changes in radiation. This offers a potential method to distinguish between hydrodynamic effects and new physics, providing a clear path to test whether the symmetry of spacetime is truly broken, a question that remains one of the biggest mysteries in modern physics.

Technical Summary: Acoustic Schwarzschild-like Black Hole and Analogous Hawking Radiation in Einstein-Bumblebee Gravity

Problem Statement
While analogue gravity systems have successfully simulated black hole horizons in flat spacetime (e.g., Bose-Einstein condensates), they typically assume exact Lorentz invariance. Modern quantum gravity frameworks suggest Lorentz symmetry may be broken at high energies, yet constructing analogue models that incorporate both curved spacetime backgrounds and Lorentz violation (LV) remains a challenge. Furthermore, real astrophysical black holes exist in curved spacetime, potentially surrounded by accretion flows or quantum fluids that could mimic acoustic horizons. The paper addresses the need to construct an acoustic black hole within a curved spacetime background that includes Lorentz symmetry breaking, specifically to study how such violations affect Hawking radiation and grey-body factors.

Methodology
The authors employ the Gross-Pitaevskii theory to model a fluid condensate in the background of a Schwarzschild-like spacetime derived from Einstein-bumblebee gravity.

Background Spacetime: They utilize an exact Schwarzschild-like solution in Einstein-bumblebee theory, characterized by a mass $M$ and a Lorentz violation coupling constant $\ell$ . The metric includes a deformation term $(1+\ell)$ in the radial component.
Acoustic Construction: Using the Madelung representation ( $\phi = \sqrt{\rho}e^{i\theta}$ ), they perturb the fluid density and phase around a stationary background. In the long-wavelength limit, the phase fluctuations obey a wave equation governed by an effective acoustic metric $G_{\mu\nu}$ .
Tuning Parameters: A dimensionless tuning parameter $\xi$ is introduced to define the radial flow velocity ( $v_r = \sqrt{2M\xi/r}$ ), ensuring the acoustic horizon exists outside the gravitational event horizon.
Radiation Analysis: The authors analyze the propagation of massless scalar particles through this acoustic metric. They derive the effective potential $V(r)$ , calculate the grey-body factors using the sixth-order WKB approximation, and determine the energy emission rates for the Hawking radiation.

Key Contributions

Derivation of a Dual-Horizon System: The paper derives an acoustic Schwarzschild-like black hole possessing two distinct types of horizons:
- The gravitational event horizon ( $r_h = 2M$ ), where neither light nor sound can escape.
- Acoustic horizons (internal $r_{ac-}$ and external $r_{ac}$ ), where only sound (phonons) cannot escape. These depend on both the LV coupling $\ell$ and the tuning parameter $\xi$ .
Temperature Analysis: The study distinguishes between the Hawking temperature at the gravitational horizon ( $T_h$ $T_{h}$ ) and the "dumb hole" temperature at the acoustic horizon ( $T_{ac}$ $T_{a c}$ ).
- $T_h$ depends solely on the LV parameter $\ell$ .
- $T_{ac}$ depends on both $\ell$ and $\xi$ . The authors find a maximum acoustic temperature $T_{acm} = T_h/8$ occurring at a specific tuning value $\xi = 16(1+\ell)/3$ .
Grey-Body and Emission Characteristics: The paper calculates the grey-body factors and energy emission rates, revealing distinct dependencies on the system parameters.

Results

Horizon Structure: The acoustic horizons exist only when $\xi \geq 4(1+\ell)$ . As $\xi$ increases, the acoustic horizon moves outward ( $r_{ac} \geq 4M$ ). When $\xi = 9(1+\ell)/2$ , the acoustic horizon coincides with the innermost stable circular orbit ( $6M$ ) of the Schwarzschild spacetime.
Grey-Body Factors ( $|A|^2$ ):
- In the low-frequency regime, the grey-body factor increases with the tuning parameter $\xi$ but decreases with the Lorentz violation parameter $\ell$ .
- In the high-frequency limit, the grey-body factor approaches unity for all parameters.
- Physically, increasing $\xi$ lowers the effective potential barrier, allowing more phonons to escape, while increasing $\ell$ raises the barrier due to enhanced gravitational drag effects from the spacetime deformation.
Energy Emission Rate: The emission rate follows a Planck-like blackbody spectrum with a peak frequency $\omega_m$ $ω_{m}$ .
- The emission rate increases with $\xi$ .
- The emission rate decreases with $\ell$ .
- The peak frequency $\omega_m$ decreases as the acoustic temperature $T_{ac}$ decreases.

Significance and Claims
The paper claims that the opposite dependence of the grey-body factor and energy emission rate on the hydrodynamic parameter ( $\xi$ ) versus the Lorentz-violating parameter ( $\ell$ ) is the system's most experimentally valuable feature.

Experimental Discrimination: In laboratory analogue experiments, $\xi$ (fluid flow) and $\ell$ (effective spacetime background) can be adjusted independently. This allows researchers to disentangle hydrodynamic effects from Lorentz-violating gravitational corrections, a distinction that is impossible in traditional flat-spacetime analogue models where these effects are coupled.
Astrophysical Benchmarking: The authors suggest these findings can be benchmarked against observations of real astrophysical black holes (e.g., via Event Horizon Telescope or LIGO). If observed radiation rates deviate from General Relativity predictions, the direction of the deviation could indicate whether the cause is a variation in accretion flow velocity profiles ( $\xi$ ) or a positive correction from Lorentz symmetry breaking ( $\ell$ ).

The authors maintain a modest stance, noting that while the acoustic metric shares the kinematics of General Relativity (Hawking radiation, temperature), it is not a solution to Einstein's equations itself. The work serves as a theoretical framework to explore the interplay between analogue gravity and Lorentz symmetry breaking.

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