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Emergence of an acoustic black hole in a lattice model

This paper demonstrates the emergence of an acoustic black hole with a horizon and ringdown behavior in a lattice-Boltzmann model without imposing an acoustic metric, while analyzing how cell-scale lattice effects influence horizon absorption, superradiance, and surface gravity.

Original authors: Antonino Flachi

Published 2026-10-30

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CC BY 4.0

Original authors: Antonino Flachi

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

In the vast landscape of modern physics, there is a persistent fascination with the idea that the universe might be built from something far smaller and more discrete than the smooth, continuous fabric we perceive. This line of thinking, known as the analogue-gravity program, explores a remarkable coincidence: the way sound waves ripple through a moving fluid follows the exact same mathematical rules as light waves traveling through the warped space around a black hole. In a fluid, if the flow moves faster than the speed of sound, sound waves cannot escape; they are swept downstream, trapped behind a point of no return. This creates an "acoustic black hole," a laboratory-scale mimic of the cosmic event horizon where nothing, not even sound, can get out. Scientists have built these mimics in water tanks, in clouds of super-cooled atoms, and in optical fibers to study how black holes behave, particularly how they might emit radiation or amplify waves. The ultimate goal is to see if the strange predictions of black hole physics hold up when tested against the messy, real-world details of a physical system, rather than just on the clean pages of a theoretical equation.

A researcher at Keio University has taken this concept a step further by constructing an acoustic black hole not in a fluid tank, but inside a computer simulation of a grid. They used a method called the lattice-Boltzmann model, which treats a fluid not as a continuous substance, but as a collection of tiny cells, each holding a specific amount of fluid that moves and collides with its neighbors according to simple, local rules. By programming these cells to drain toward a central point while swirling around it, the researcher created a flow that naturally evolved into a state where the fluid moved faster than the speed of sound in the center. Crucially, they did not program a black hole into the system; they simply set the rules for the fluid's motion and let the black hole emerge on its own. The geometry of the trap, the location of the point of no return, and the behavior of the waves were all measured directly from the simulation's output, treating the computer grid as if it were a physical laboratory experiment.

The results confirmed that a stable horizon did indeed form. The researcher mapped the flow and found a clear boundary where the fluid speed crossed the sound speed, creating a ring of no return. To verify this, they traced the paths of sound waves, or "phonons," through the measured flow. They found that waves launched from inside this ring were trapped, while those from outside could escape, exactly as predicted for a black hole. When they disturbed the fluid with a small pulse, the system settled back down by ringing like a bell, emitting a specific set of fading tones known as quasinormal frequencies. These tones matched the theoretical predictions for a draining bathtub vortex with surprising accuracy, differing by only a few percent. This small difference was not a failure of the model but a signature of the grid itself; the square shape of the cells introduced a tiny, four-fold asymmetry in the flow that would not exist in a smooth, continuous fluid.

However, the study also revealed a subtle but important distinction between this digital black hole and the ideal ones described in textbooks. In a perfect theoretical black hole, the event horizon is a one-way membrane that absorbs everything. In this grid-based model, the horizon is a thin layer of cells where the fluid is being drained away. The researcher found that the way this layer absorbs energy depends on the frame of reference. While the emergent geometry of the flow was capable of amplifying waves in a specific way known as superradiance—a process where a rotating black hole can steal energy from passing waves—the actual simulation did not show this amplification happening dynamically. The reason lies in the nature of the drain: the grid cells that absorb the fluid are fixed in place, not rotating with the flow. Because the absorber is stationary relative to the grid, it does not meet the specific conditions required to trigger the amplification, even though the swirling flow around it suggests it should.

The researcher tested this by modeling the horizon in two different ways. One model treated the horizon as a perfect reflector, while the other treated it as a point that absorbs energy in its own moving frame. The second model, which accounted for the fact that the absorption happens in the frame of the stationary grid, described the behavior of the system best. This suggests that while the large-scale shape of the black hole is correct, the microscopic details of how the horizon interacts with waves are dictated by the underlying structure of the grid. The study concludes that the lattice model successfully reproduces the kinematics of an analogue black hole, including its ringing tones and the formation of a horizon, but that the specific microphysics of the grid—its discrete, cell-based nature—leaves a distinct imprint on how the system behaves. Whether a more sophisticated version of this model could eventually demonstrate the dynamic amplification of waves remains an open question, but the work provides a concrete example of how relativistic phenomena can emerge from simple, local rules on a discrete substrate.

Technical Summary: Emergence of an Acoustic Black Hole in a Lattice Model

Problem and Context
The analogue-gravity program posits that sound waves on a moving fluid background obey the same wave equation as a massless scalar field on a curved Lorentzian spacetime, characterized by an emergent "acoustic metric." While experimental realizations exist in water tanks, Bose–Einstein condensates, and optical fibers, theoretical analogues typically rely on a prescribed metric as an input. Conversely, lattice treatments of horizons (e.g., lattice field theories or quantum walks) usually impose a black-hole background geometry to study phenomena like Hawking radiation.

This paper addresses the gap between theoretical analogues and experimental realizations by constructing an acoustic black hole entirely from the local, microscopic rules of a lattice model. The goal is to demonstrate that an analogue black hole, complete with an emergent horizon, quasinormal modes, and superradiance potential, can arise as an output of a discrete, deterministic cellular automaton without any pre-imposed acoustic metric.

Methodology
The author employs the standard D2Q9 (two-dimensional, nine-velocity) lattice-Boltzmann (LB) method. The system evolves via a "collide-and-stream" rule:

Collision: Populations relax toward a local equilibrium distribution defined by density ( $\rho$ ) and momentum ( $\rho \mathbf{u}$ ).
Streaming: Populations move to adjacent lattice cells.

To generate a draining-bathtub flow ( $v_{db} = (-A\hat{r} + B\hat{\theta})/r$ ), the author imposes specific boundary conditions and a static radial body force ( $g_r = -(A^2 + B^2)/r^3$ ) in the bulk. This force ensures that a transonic steady state with a well-resolved horizon forms in the bulk, rather than being pinned to the central sink. The simulation is treated as "experimental data": no metric is prescribed. Instead, the acoustic metric, horizon location, surface gravity, and wave dynamics are measured directly from the steady-state flow fields ( $\rho, \mathbf{v}$ ) generated by the simulation.

Key Results

Emergent Horizon and Geometry:
A stable sonic horizon emerges where the radial Mach number crosses unity. The location is confirmed by two independent methods: measuring the Mach profile and tracing phonon rays on the measured metric. The rays behave as expected for a black hole: those launched inside the horizon are trapped, while those outside escape. The horizon is anchored to the dissipation edge of the central absorber.
- Lattice Artifacts: At coarse resolution, the square lattice breaks rotational invariance, manifesting as four small "supersonic islands" in the transonic layer aligned with the lattice axes. These are causally inert and vanish with lattice refinement.
Surface Gravity and Hawking Temperature:
The author identifies a discrepancy in defining surface gravity ( $\kappa$ ) for the lattice hole.
1. Exterior Flow: Fitting the exterior flow yields a $\kappa$ consistent with the continuum prediction ( $c_s^2/r_h$ ).
2. Local Gradient: Measuring the gradient across the transonic layer yields a $\kappa$ significantly larger (by a factor of 2 to 6) due to the finite thickness of the layer (2–3 cells).
  Consequently, the Hawking temperature is not well-defined for this specific model, as the value depends on the dispersion relation at the layer scale, which is beyond the scope of this classical study.
Quasinormal Modes (QNM):
Perturbing the steady density with a localized Gaussian and recording the ringdown reveals a discrete set of damped quasinormal frequencies.
- The measured frequencies match the theoretical predictions for an ideal draining bathtub (computed via Leaver's continued-fraction method) and the continuum wave equation on the realized flow to within a few percent (non-rotating) and about 10% (rotating).
- The residual offsets are attributed to the cell-scale structure of the horizon and the finite lattice spacing acting as an ultraviolet cutoff.
- The lattice hole is shown to be classically stable, with all modes damped.
Superradiance:
The study examines superradiance (wave amplification by a rotating black hole) in two contexts:
1. Loss-free Scattering: When solving the scattering problem on the measured metric with the horizon absorbing in its own frame (the sonic point), the emergent geometry amplifies waves in the band $0 < \omega < m\Omega_H$ . The gain matches the ideal bathtub prediction to within 4%.
2. Dynamic Realization: Whether the lattice hole dynamically amplifies waves remains an open question. The horizon in the lattice model is a dissipative sink fixed to the lattice ( $\Omega_{abs} = 0$ ), whereas an ideal horizon absorbs in its rotating frame ( $\Omega_{abs} = \Omega_H$ ). In the lattice frame, the wave frequency is always positive, suggesting no amplification occurs dynamically.
  The author models the horizon microphysics using a "frame parameter" $\alpha$ . While the ringdown frequencies are largely insensitive to $\alpha$ , the width of the superradiant band is not. The current lattice setup corresponds to $\alpha = 1$ (absorbing in the lattice frame), which theoretically suppresses amplification.

Significance and Claims
The paper claims to provide a concrete instance of the emergence of relativistic phenomenology (a black hole with a horizon and ringdown) from local, deterministic, discrete rules. Its significance lies in:

Output vs. Input: Unlike previous lattice treatments where the metric is an input, here the geometry is an emergent output of the simulation.
Substrate Physics: It explicitly demonstrates how the "substrate" (the lattice) leaves imprints on the emergent physics, such as the fourfold anisotropy of the sonic surface and the dual definition of surface gravity.
Decoupling of Ingredients: The study separates the emergent geometry (which supports superradiance in a loss-free calculation) from the horizon microphysics (which determines whether that amplification occurs dynamically).
Toy Model for Discrete Spacetime: The model serves as a toy example for investigating which continuum features survive on a discrete substrate and which become substrate-specific, relevant to programs exploring deterministic cellular automata as a basis for quantum mechanics.

The author remains modest regarding the dynamical observation of superradiance, noting that it has not been observed in the current toy model and that whether it requires a more sophisticated model or higher-resolution simulations remains to be seen.

Technical Summary: Emergence of an Acoustic Black Hole in a Lattice Model

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