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Effective Lorentz Factor and Analogue Horizon in a Giant Quantum Vortex of Superfluid Helium-4: Design of a Laboratory-Scale Pre- Experiment with an Optional Cryogenic Module for a Dual-Capsule Apparatus

This paper proposes a laboratory-scale pre-experiment using a giant quantum vortex in superfluid helium-4 to generate an effective Lorentz factor for surface waves and an analogue horizon, while clarifying that this effect does not produce measurable time dilation for clocks or organisms and noting a mathematical correspondence between the author's Börekci-2 metric and the Painlevé–Gullstrand flow.

Original authors: Hasan Börekci

Published 2026-09-29

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CC BY 4.0

Original authors: Hasan Börekci

Original paper licensed under CC BY 4.0 (https://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

Imagine a universe where the rules of space and time are not fixed, but can be bent and stretched by the movement of a fluid. This is the realm of analogue gravity, a field of physics that studies how sound waves moving through a liquid can mimic the behavior of light near a black hole. In the real cosmos, a black hole is so massive that nothing, not even light, can escape once it crosses a certain boundary called an event horizon. In a laboratory, scientists can create a similar effect using a swirling vortex of liquid. If the liquid spins and drains fast enough, the surface waves on top of it get caught in the flow, unable to swim upstream against the current. To these waves, the liquid acts like a curved space-time, creating a miniature version of a black hole's edge. This allows researchers to study the strange physics of the universe's most extreme objects using a tank of liquid in a room-sized lab, without needing to travel to the edge of a star.

A researcher named Hasan Börekci has designed a detailed plan for a new experiment to create one of these miniature black holes using superfluid helium. This is not just any liquid; it is helium cooled to a temperature so low that it becomes a superfluid, a state where it flows with absolutely no friction. The experiment involves a large, cylindrical tank filled with this super-cold liquid. Inside, a propeller stirs the helium to create a giant whirlpool, while a drain in the center pulls the liquid downward. The goal is to spin the liquid fast enough that the surface waves, which normally ripple across the surface, are dragged along by the flow. When the flow speed exceeds the speed of the waves, a region forms where the waves cannot escape, creating what the paper calls an "analogue horizon."

The design is precise and grounded in the known properties of superfluid helium. The proposed tank is 200 millimeters wide and holds liquid to a depth of 40 millimeters. To achieve the necessary conditions, the experiment requires a vortex carrying 40,000 units of circulation, a specific measure of how much the fluid is spinning. At this speed, the point where the flow becomes too fast for the waves to escape—the analogue horizon—would sit at a radius of 1.92 millimeters from the center. Just outside this, at a radius of 6.70 millimeters, lies the "ergoregion," a zone where the flow is so strong that it drags everything along with it. The paper calculates that at a distance just slightly larger than the horizon, the effect on the waves is equivalent to a Lorentz factor of 3.28, a measure of how much the flow distorts the wave's experience of time and space.

Crucially, the paper makes a sharp distinction between what happens to the waves and what happens to the atoms themselves. The intense flow creates a dramatic effect for the surface waves, making them behave as if they are in a warped universe. However, this is not a real time dilation for the liquid or for any living thing. If a clock or a person were placed in the tank, they would not age slower or faster; the effect is strictly limited to the ripples on the surface. The helium atoms are moving far too slowly to affect the speed of light, which governs the passage of time for matter. The experiment is purely about how waves travel through a moving medium, offering a way to test theories of gravity using sound instead of light.

The design includes a complete system for keeping the helium at the required temperature of 1.6 Kelvin, just a fraction of a degree above absolute zero. This involves a sophisticated cryostat, or deep-freeze chamber, with multiple layers of insulation and a system to pump away the gas that naturally evaporates from the liquid. The apparatus is built to be stable, with vibration isolation to ensure that the delicate measurements of the surface waves are not disturbed by the hum of pumps or the movement of the building. The plan also details how to measure the waves using high-speed cameras and projected light patterns, allowing researchers to map exactly how the flow speed changes from the center of the vortex to the edges.

The paper also considers a more ambitious, hypothetical scenario where this helium tank is placed inside a much larger, complex machine designed to test other theories of physics. This larger machine, described as a dual-capsule apparatus, would generate intense energy fields. The author analyzes whether placing the helium experiment inside this machine would reveal any new effects. The conclusion is clear: standard physics predicts no change. The intense fields of the larger machine would not alter the behavior of the helium waves in a way that could be measured. While the experiment could be physically installed inside the larger machine, the paper argues that it would not add any new sensitivity to the tests being performed there. The most valuable path forward is to build the helium experiment on its own, in a standard laboratory, to prove that the analogue horizon can be created and measured.

Ultimately, this work is a blueprint for a tangible, buildable experiment. It does not claim to have discovered a new law of the universe, but rather offers a concrete method to observe a known mathematical relationship in a new setting. By creating a giant vortex in superfluid helium, the experiment aims to show that the geometry of space-time, usually reserved for the study of black holes, can be mimicked in a tank of liquid. If successful, it would provide a powerful tool for physicists to study the behavior of waves in curved space, bridging the gap between the abstract mathematics of the cosmos and the concrete reality of a laboratory bench.

Technical Summary: Effective Lorentz Factor and Analogue Horizon in a Giant Quantum Vortex of Superfluid Helium-4

Problem Statement
The paper addresses the feasibility of generating and measuring a high effective Lorentz factor ( $\gamma_{\text{eff}}$ ) and an analogue event horizon within a laboratory-scale system. While analogue gravity has been demonstrated in Bose–Einstein condensates and water vortices, superfluid helium-4 (He II) offers a unique medium due to its natural vanishing viscosity and "vacuum-like" behavior. The central challenge is to determine if a stable "giant quantum vortex" in He II can produce flow speeds exceeding the phase speed of surface waves (creating an ergoregion and horizon) without violating the Landau critical velocity limit for bulk sound. Furthermore, the paper investigates whether this effective relativistic factor, which applies to waves in the fluid, can be integrated into a larger, speculative "dual-capsule" apparatus (Stage 1) to test a specific "black-hole-matter" hypothesis regarding time dilation, or if such integration is physically redundant.

Methodology
The study is a comprehensive design proposal for a "Stage 0" pre-experiment, detailed down to the engineering specifications of a cryostat, flow loop, and metrology system.

Theoretical Framework: The design relies on the acoustic metric analogy, where sound waves in a moving fluid obey equations similar to fields in curved spacetime. The effective Lorentz factor is defined as $\gamma_{\text{eff}} = (1 - v^2/c_w^2)^{-1/2}$ $γ_{eff} = (1 - v^{2} / c_{w}^{2})^{- 1/2}$ , where $v$ $v$ is the fluid flow speed and $c_w$ $c_{w}$ is the wave speed.
- Bulk Sound: Limited by the Landau critical velocity ( $v_L \approx 58.8$ m/s), yielding a maximum $\gamma_{\text{eff}} \approx 1.032$ , which is insufficient for horizon formation.
- Surface Waves: Capillary-gravity waves in He II have a minimum phase speed ( $c_{\text{min}} \approx 9.89$ cm/s). This low threshold allows the flow to exceed the wave speed at much lower velocities, enabling the formation of an ergoregion and an analogue horizon.
Apparatus Design (Stage 0):
- Cryostat: A helium-bath cryostat operating at $\approx 1.6$ K, utilizing a closed He II flow loop.
- Vortex Cell: A 200 mm diameter copper cell with a 40 mm liquid depth. A "draining bathtub" flow is generated by a propeller-driven circulation loop with tangential inlet vanes and a central 10 mm drain tube.
- Target Parameters: The design targets a vortex carrying $N = 4 \times 10^4$ circulation quanta. This is expected to create an ergoregion radius ( $r_e$ ) of 6.70 mm and an analogue horizon radius ( $r_h$ ) of 1.92 mm.
- Metrology: Surface waves are excited by a piezoelectric paddle near the minimum phase speed frequency ( $\approx 31.6$ Hz). Surface height is measured via Fourier-transform profilometry using a high-speed camera and fringe projection, allowing for the reconstruction of the velocity field and $\gamma_{\text{eff}}$ map.
Integration Analysis (Stage 1): The paper analyzes the optional installation of this module into the inner capsule of the author's "BEF" (Börekci Energy Field) dual-capsule apparatus. This involves assessing thermal loads, magnetic field interference (up to 2 T), and vibration isolation. Two operational modes are proposed: a closed-loop recondensing circuit (Option A) or a sealed autonomous module (Option B).

Key Contributions

Detailed Engineering Design: The paper provides a complete, traceable design for a laboratory-scale experiment capable of producing an analogue horizon in superfluid helium, including specific calculations for heat loads, pumping speeds, and cryogenic recovery.
Clarification of Effective vs. Real Time Dilation: The author explicitly distinguishes between the effective Lorentz factor for fluid waves and real time dilation for clocks or biological matter. The paper asserts that while $\gamma_{\text{eff}}$ can reach values like 1.67 or 3.28 for surface waves, the real metric for atoms and clocks remains unaffected by the fluid flow, as the helium's mass and flow velocity are negligible relative to the speed of light.
Theoretical Connection: The work re-expresses the "Börekci-2 metric" node-spacing function as the inverse of the effective Lorentz factor of a Painlevé–Gullstrand river flow, linking the author's specific hypothesis to established analogue-gravity mathematics.
Feasibility Assessment of Stage 1: The paper provides a rigorous assessment of integrating the helium module into the larger dual-capsule apparatus, concluding that while technically possible, it offers no meaningful improvement in testing the phenomenological parameter $\xi$ compared to existing clock-based bounds.

Calculated Results
Based on the design parameters ( $N = 4 \times 10^4$ , $D/C = 0.3$ ):

Ergoregion Radius ( $r_e$ ): 6.70 mm.
Analogue Horizon Radius ( $r_h$ ): 1.92 mm.
Hollow Core Radius: 0.75 mm.
Effective Lorentz Factor:
- At $r = 1.25 r_e$ : $\gamma_{\text{eff}} \approx 1.67$ .
- At $r = 1.05 r_e$ : $\gamma_{\text{eff}} \approx 3.28$ .
- At the ergoregion boundary ( $r_e$ ): $\gamma_{\text{eff}} \to \infty$ .
Thermal Performance: The system is designed to operate for over 160 hours on a single helium fill with a heat load budget of 0.15 W.

Significance and Claims
The paper positions itself as a "design of a laboratory-scale pre-experiment" rather than a report of completed experimental data. Its primary significance lies in:

Demonstrating Reachability: Showing that the conditions for an analogue horizon in He II are achievable with current low-temperature technology and that the required circulation quanta ( $N \approx 4 \times 10^4$ ) are within the range of recent experimental reports (e.g., Švančara et al.).
Modest Scope on Hypothesis Testing: The author explicitly states that the integration of this module into the dual-capsule apparatus (Stage 1) is unlikely to yield new constraints on the "black-hole-matter" hypothesis. The paper concludes that Stage 0 should be conducted independently first, and Stage 1 should only be considered as a secondary demonstration channel if Stage 0 succeeds.
Analogue vs. Fundamental Physics: The paper maintains a strict boundary between analogue gravity (kinematics of waves in a medium) and fundamental dynamics (how geometry arises from matter). It claims the experiment supports the idea that spacetime can be modeled as a medium for waves, but does not claim to prove that spacetime is a medium made of black-hole matter.

The work emphasizes that the "effective Lorentz factor" is a property of the wave propagation in the fluid, not a real relativistic time dilation for the helium atoms or any potential living subject, thereby avoiding misconceptions regarding the physical effects on matter within the proposed apparatus.

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