Analog regular black holes and black hole mimickers for surface-gravity waves in fluids

This paper proposes a fluid-based analogue gravity setup using surface-gravity waves to experimentally emulate the instabilities of regular black holes and horizonless mimickers, concluding that while theoretically feasible, alternative media like Bose-Einstein condensates may be more practical for capturing these specific physical features.

The Gist

Imagine you are a scientist trying to understand the most mysterious objects in the universe: Black Holes.

For a long time, we thought black holes were simple: a point of infinite density (a singularity) surrounded by a point of no return (the event horizon). But recent theories suggest reality might be more complex. Maybe black holes aren't "singed" at the center, but have a smooth, regular core. Or maybe they aren't black holes at all, but ultra-dense "mimickers" that look like black holes but have no horizon.

The problem? We can't fly inside a real black hole to check. The gravity is too strong, and the physics is too extreme.

So, what do scientists do? They build a miniature, safe version in a bathtub.

The Big Idea: The "Water Black Hole"

This paper proposes building a laboratory experiment using shallow water to simulate the physics of these exotic black holes.

Think of it like this:

• Real Black Hole: A massive object in space that bends light and time.
• Lab Version: A pool of water with a drain in the middle.
• The Analogy: When water flows toward a drain, it speeds up. If it flows fast enough, surface waves (ripples) can't swim upstream against the current. To the waves, the drain looks like a "horizon" they can't escape. This is the Acoustic Black Hole.

The Two Exotic Scenarios

The authors want to test two specific, weird theories about what's inside these objects:

1. The "Regular" Black Hole (The Safe Core)

• The Theory: Instead of a terrifying singularity, the center is a smooth, bouncing ball of space (like a trampoline). It has an inner horizon and an outer horizon.
• The Danger: The inner horizon is unstable. Imagine a hallway where sound echoes perfectly. If you shout, the echo comes back, hits the wall, bounces again, and gets louder and louder until the walls explode. This is called Mass Inflation. The authors want to see if their water model shows this "explosion" of energy.

2. The "Mimicker" (The Trap)

• The Theory: This object has no horizon at all. It's just a super-dense ball. But it has a "light ring" (a place where light or water waves get stuck in a perfect circle).
• The Danger: Imagine a hamster wheel. If you put a hamster in it, it runs forever. If you put many hamsters in, they pile up and the wheel breaks. Similarly, waves get trapped in this ring, pile up, and could destabilize the whole object.

How They Plan to Build It

To make water behave like these complex black holes, the scientists need to control the water flow very precisely.

• The Setup: A large, circular tank of water.
• The Drain: In the center, they need a drain that sucks water in at a specific speed.
• The Trick: The water depth and the speed of the drain must change in a very specific mathematical way as you move from the center to the edge.
  • Near the center: The water needs to flow in a way that mimics the "smooth core" (no singularity).
  • Near the edge: The water needs to slow down to mimic the empty space far away from a black hole.

The Hurdle: It's Harder Than It Looks

The paper is a mix of "Yes, we can do this" and "Actually, it's really tricky."

• The Math Problem: To make the water flow exactly right, the scientists have to solve a puzzle where they have too many rules and not enough knobs to turn. It's like trying to bake a cake where you must control the temperature, the flour, the sugar, and the oven humidity, but you only have one dial to adjust.
• The Solution: They found that if they keep the water depth almost perfectly flat (which is hard to do near a drain), the math works out.
• The Alternative: They suggest that while water is great for some things, Bose-Einstein Condensates (a weird state of matter made of super-cold atoms) might be even better. It's like switching from a wooden toy boat to a high-tech submarine; the atoms are easier to control precisely to mimic the complex physics.

Why Does This Matter?

If they succeed, they can watch these "instabilities" happen in real-time in a lab.

• Will the inner horizon explode? (Mass Inflation)
• Will the trapped waves break the object? (Light Ring Instability)

This won't just tell us about black holes; it might tell us how the universe behaves when gravity gets crazy. It's a way to test the laws of physics without needing a spaceship to the edge of a black hole.

The Bottom Line

This paper is a blueprint for building a black hole simulator in a swimming pool. It admits that the engineering is tough and might require switching to "quantum water" (cold atoms) to get it right, but if they pull it off, we could finally see how the universe handles its most extreme objects without getting crushed.

Technical Summary

1. Problem Statement

Recent observational advances in black hole physics (gravitational waves and Event Horizon Telescope imaging) have renewed interest in testing General Relativity (GR) against alternatives, specifically Regular Black Holes (RBHs) and Horizonless Black Hole Mimickers (BHMs).

• RBHs possess a regular core (often de Sitter-like) surrounded by an inner and outer horizon. They are theoretically plagued by mass inflation and semiclassical instabilities at the inner (Cauchy) horizon.
• BHMs (or Ultracompact Objects, UCOs) lack horizons but possess a stable inner light ring. This stability can lead to the accumulation of energy and nonlinear dynamical instabilities via long-lived quasinormal modes (QNMs) and "echoes" in gravitational wave signals.

The core challenge is that these instabilities are difficult to probe directly in astrophysical settings due to timescales and the inability to manipulate the background spacetime. While analogue gravity (using fluids to simulate curved spacetime) has successfully modeled horizons and ergoregions, it has not yet successfully simulated the entire background spacetime of a static, spherically symmetric UCO/RBH, including the specific core structure required to study these inner-region instabilities.

2. Methodology

The authors propose an analogue gravity platform using surface-gravity waves in a shallow-water basin filled with an incompressible, irrotational fluid (e.g., water or superfluid helium).

A. Theoretical Framework

1. Metric Mapping: They establish a correspondence between the target gravitational metric (static, spherically symmetric UCO/RBH in Painlevé–Gullstrand coordinates) and the effective acoustic metric of the fluid.
   • The key identification condition is: $\frac{v_r(r)}{c_{sw}(r)} = -\sqrt{\frac{2m(r)}{r}}$, where $v_r$ is the radial fluid velocity, $c_{sw}$ is the surface wave speed, and $m(r)$ is the mass function characterizing the regularization.
2. Fluid Dynamics: The system is governed by the continuity equation, the Bernoulli equation, and kinematic boundary conditions. The authors derive a system of Ordinary Differential Equations (ODEs) for the radial velocity $v_r(r)$ and fluid height $h(r)$.
3. The Overdetermination Problem: A critical theoretical hurdle identified is that the fluid equations (continuity + Bernoulli) plus the identification equation (to match the target metric) create an overdetermined system. There are more equations than independent variables ($v_r$ and $h$), meaning a specific target spacetime $m(r)$ cannot be realized exactly without approximations or additional control parameters.

B. Experimental Strategy

To resolve the overdetermination and realize the spacetime, the authors analyze specific experimental configurations:

1. Central Drainage: A standard setup where fluid drains at the center ($r=0$). They analyze the behavior near the origin to simulate the regular core.
2. Global Gradient Drainage: To simulate the asymptotic flat region ($r \to \infty$), they propose a spatially varying drainage profile $v_{z=0}(r)$ across the container floor.
3. Approximations: They demonstrate that to make the system tractable, the fluid height profile $h(r)$ must be approximately constant ($h' \approx 0, h'' \approx 0$) in the region of interest. This simplifies the acoustic metric and reduces the ODEs to algebraic relations.

3. Key Contributions

• Full Spacetime Simulation: Unlike previous works focusing only on horizons, this paper provides a roadmap to simulate the entire static UCO/RBH geometry, specifically the transition from the regular core to the asymptotic exterior.
• Resolution of Overdetermination: The authors rigorously analyze the mathematical constraints of the fluid equations. They show that realizing a specific $m(r)$ requires either:
  • Approximating the fluid height as constant (valid for specific regimes).
  • Introducing a non-flat container bottom (varying curvature $f(r)$) as a third tunable variable to balance the equations (proposed as a future direction).
  • Switching to Bose-Einstein Condensates (BECs), where the compressibility and external potentials provide enough degrees of freedom to avoid overdetermination entirely.
• Experimental Feasibility Analysis: They provide numerical estimates for water and superfluid helium experiments, calculating the required parameters (drainage velocity $U$, container size, fluid depth) to simulate specific regularization scales ($\ell$) and ADM masses ($M$).

4. Results

• Core Simulation: A non-rotating central drainage setup with a constant fluid height approximation can successfully simulate the inner core of a UCO/RBH. The regularization parameter $\ell$ is directly related to the central drainage velocity $U$ and fluid depth $h_0$ via $\ell \propto h_0/U$.
• Asymptotic Region: A "global gradient drainage" (where the drainage velocity follows a power law $v_{z=0} \propto r^{-3/2}$) can theoretically reproduce the asymptotic behavior required for the ADM mass $M$.
• Instability Probing:
  • Inner Light Rings (BHMs): The setup allows for the simulation of stable inner light rings. The authors argue that the accumulation of energy near these rings (leading to echoes or nonlinear instability) could be observed as a "sloshing" in the analogue signal.
  • Inner Horizons (RBHs): While the inner horizon is behind an outer horizon in the RBH branch, the kinematic features (blueshift) can be studied. The authors note that while the full backreaction cannot be simulated, the linear dynamics leading to instability can be tested.
• Limitations:
  • Scale: For water experiments, the required container sizes to simulate astrophysically relevant compactness (e.g., $M \sim 0.25$ m) are large (meters), though feasible. For superfluid helium, the required scales are currently out of reach due to small container sizes.
  • Refilling: Implementing a global gradient drainage requires a complex refilling mechanism that does not disturb the flow, which is a significant experimental engineering challenge.
  • Rotation: The current analysis is for non-rotating spacetimes. Including rotation (simulating a slowly rotating UCO) introduces significant difficulties in matching the azimuthal velocity profiles.

5. Significance

• Bridging Theory and Experiment: This work moves analogue gravity from testing isolated features (like Hawking radiation) to testing global spacetime structures and their associated instabilities.
• Testing Quantum Gravity Effects: By simulating the regular core and the resulting instabilities (mass inflation or light-ring accumulation), these experiments could provide empirical data on how UV modifications (regularization) affect black hole dynamics, potentially distinguishing between singular and regular black hole scenarios.
• Platform Comparison: The paper highlights that while surface-gravity waves are a viable proof-of-concept, Bose-Einstein Condensates (BECs) offer a superior theoretical framework for this specific problem due to their compressibility and tunable external potentials, which naturally resolve the overdetermination issue found in incompressible fluids.
• Future Directions: The authors propose that even if a perfect analytic model ($m(r)$) cannot be realized, a "bottom-up" approach is valid: if the experiment achieves the necessary regularity conditions ($m(r) \sim r^3$) and compactness, the resulting QNM and echo spectra can be measured to infer the nature of the simulated object, regardless of whether it matches a specific theoretical model.

In conclusion, the paper establishes that while challenging, simulating the full spacetime of regular black holes and mimickers in fluid basins is theoretically possible and provides a concrete pathway to experimentally probe the stability of inner horizons and light rings, offering a unique window into the physics of compact objects beyond standard General Relativity.