Euler-Korteweg vortices: A fluid-mechanical analogue to the Schrödinger and Klein-Gordon equations

This paper demonstrates that by applying specific assumptions regarding angular momentum and Korteweg capillary stress to an Euler-Korteweg vortex model, the resulting fluid-mechanical equations become mathematically equivalent to the Schrödinger and Klein-Gordon equations, thereby establishing a formal analogy that reproduces fundamental quantum mechanical relations such as the de Broglie wavelength and the uncertainty principle.

The Gist

Imagine you are standing by a calm, deep lake. Usually, we think of water as just water—wet, heavy, and obeying the laws of physics like gravity and friction. But what if I told you that if you look at a very specific, swirling whirlpool in that lake, it might start behaving exactly like a tiny subatomic particle, such as an electron?

That is the wild idea behind this paper. The author, D.M.F. Bischoff van Heemskerck, is essentially saying: "What if the weird rules of the quantum world (where particles act like waves) are actually just the rules of fluid dynamics (how water moves), but we just haven't looked at the right kind of whirlpool yet?"

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Setup: A Magical Whirlpool

In the real world, whirlpools usually spin down and disappear. But the author imagines a "perfect" whirlpool in a special kind of fluid (one that doesn't get sticky or change temperature).

To make this whirlpool act like a quantum particle, the author gives it three special "superpowers":

• It spins perfectly: It's an "irrotational" vortex, meaning the water flows smoothly around a center without churning up turbulence.
• It has a specific spin amount: The author forces the whirlpool to spin with an amount of "angular momentum" that exactly matches a fundamental constant of nature called Planck's constant (the "currency" of the quantum world).
• It has a "skin": The center of the whirlpool has a steep pressure change, creating a surface tension effect (called Korteweg stress). Think of this like the whirlpool having a tight, elastic skin that resists being squashed.

2. The Big Discovery: Water Math = Quantum Math

Once these three conditions are met, the author runs the math. And here is the magic trick: The equations that describe how this water whirlpool moves turn out to be mathematically identical to the equations that describe how electrons move.

• The Schrödinger Equation: This is the famous equation that tells us how a quantum particle (like an electron) behaves as a wave. The paper shows that if you take the equations for this special whirlpool and look at it from a distance (where the water is moving slowly), you get the Schrödinger equation.

  • Analogy: Imagine you are watching a dancer from far away. You can't see their individual steps, only the smooth flow of their dress. The paper says the "dress" of the whirlpool follows the exact same rules as the "dress" of an electron.

• The Born Rule: In quantum mechanics, we can't say exactly where a particle is; we can only say the probability of finding it there. The paper shows that if you have a bunch of these whirlpools, the density of the water (how much water is in a spot) maps perfectly to the probability of finding a particle there.

  • Analogy: If you drop a thousand marbles into a maze, they will spread out. The places where the marbles pile up the most are the "most likely" spots. The paper shows that for these whirlpools, the water density is that pile-up.

3. The Relativity Twist: The Fast Boat

The paper goes a step further. What happens if the fluid itself is moving fast, like a river flowing past the whirlpool?

• The Speed Limit: In our universe, nothing can go faster than light. In this fluid model, the "speed of sound" in the water acts as the speed limit.
• Time Travel (Sort of): When the whirlpool moves fast relative to the water, the math changes. The author shows that to describe the whirlpool correctly, you have to use Lorentz transformations. These are the same math tools Einstein used for Special Relativity to explain why time slows down and lengths shrink for fast-moving objects.
• The Klein-Gordon Equation: This is the "relativistic" version of the Schrödinger equation. The paper proves that when you account for the "delay" in how the wave travels through the moving fluid, the whirlpool's behavior matches this equation perfectly.
  • Analogy: Imagine a boat moving through water. If the boat goes slow, the waves look simple. If the boat goes near the speed of the water's ripples, the waves get squashed and distorted in a very specific way. The paper says this distortion follows the exact same rules as time slowing down for a spaceship moving near the speed of light.

4. The "Uncertainty Principle" Explained Simply

You've heard that you can't know both where a particle is and how fast it's going at the same time. The paper explains this using a simple wave concept.

• The Wave Packet: Imagine trying to pinpoint a specific ripple in a pond. If you make the ripple very small and sharp (to know exactly where it is), it becomes a mix of many different wave speeds (so you don't know how fast it's going). If you make it a long, smooth wave (to know the speed), you can't tell exactly where the "center" of the wave is.
• The Conclusion: The paper shows that this trade-off isn't a mysterious quantum magic trick; it's just a mathematical fact about how waves work. If you treat a particle as a whirlpool made of waves, the "Uncertainty Principle" pops out naturally.

5. The Catch: Is it Real?

This is the most important part. The author is very careful to say: "This is a mathematical analogy, not necessarily a physical truth."

• The "What If": The paper asks, "What if electrons are actually tiny whirlpools in a hidden, fundamental fluid?"
• The Reality Check: The author admits this is probably not true. If electrons were just whirlpools in a fluid, we would have to find that fluid. Also, the math gets very hard when you try to explain things like "spin" (a quantum property with no real-world fluid equivalent) or "entanglement" (where two particles are linked instantly across the universe).
• The Value: Even if electrons aren't actually whirlpools, this analogy is useful. It's like using a map of a city to understand traffic flow. Even if the map isn't the city itself, it helps engineers design better roads. This paper helps physicists understand the deep mathematical connections between the fluid world and the quantum world.

Summary

Think of this paper as a translator. It takes the complex, confusing language of quantum mechanics (electrons, probabilities, uncertainty) and translates it into the language of fluid mechanics (whirlpools, water density, waves).

It shows that if you build a whirlpool with very specific, magical properties, it will follow the exact same rules as a subatomic particle. It doesn't prove that the universe is made of water, but it does prove that the math of water and the math of quantum particles are two sides of the same coin.

Technical Summary

1. Problem Statement

The paper addresses the long-standing formal analogies between fluid mechanics, quantum mechanics, and relativity. While previous works (e.g., Madelung, Unruh, Visser) have established correspondences between fluid equations and quantum or acoustic metrics, this paper seeks to go beyond formal similarity. The specific problem is to determine whether a classical, inviscid, barotropic fluid model can be rigorously constructed such that its governing equations for a specific type of excitation (a vortex) are mathematically equivalent to the fundamental equations of quantum mechanics (Schrödinger) and special relativity (Klein–Gordon), rather than just analogous.

2. Methodology

The author employs a deductive approach, constructing a specific fluid dynamical model and applying a series of structural assumptions to derive wave equations.

• Fluid Model: The study assumes an inviscid, isothermal, and barotropic fluid where the speed of sound ($c_s$) is equal to the speed of light ($c$). The equation of state is defined as $P = \rho c^2$.
• Vortex Excitation: The fluid is assumed to sustain irrotational vortices. The velocity field is derived from a potential $\Phi$.
• Quantization Assumptions:
  • The circulation of the vortex is quantized, leading to a constant angular momentum $\ell$.
  • An effective mass $m$ is assigned to the vortex.
  • Crucial Assumption: The total angular momentum of the vortex is set equal to the reduced Planck constant ($J = m\ell = \hbar$).
• Korteweg Stress (Capillary Force): To account for the steep pressure/density gradients at the vortex core, the model incorporates Korteweg stress (a capillary force arising from free energy gradients). The capillary coefficient $K$ is specifically set to $K = \hbar^2 / (4m^2\rho)$. This term is mathematically identified as the Bohm quantum potential.
• Complex Wave Function: A complex wave function $\psi = \sqrt{\rho} e^{i\theta}$ is formulated, where $\rho$ is mass density and $\theta$ is the phase derived from the velocity potential.
• Frame Transformations: The model analyzes the system in two regimes:
  1. Low-Mach/Drift Limit: Introducing a uniform convection velocity $v_d$ relative to the fluid.
  2. Retarded Propagation: Accounting for the finite speed of signal propagation (sound speed $c$) from a moving source, necessitating a transformation of coordinates.

3. Key Contributions

The paper derives several fundamental physical relations and equations directly from classical fluid dynamics under the specified constraints:

• Derivation of the Schrödinger Equation: By neglecting the azimuthal flow field (far-field approximation) and assuming constant enthalpy, the coupled continuity and Euler-Korteweg equations combine to form the linear Schrödinger equation:
  $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V \right] \psi$$
  Here, the mass density $\rho$ plays the role of probability density.
• Derivation of the Born Rule: The paper constructs a statistical argument where an ensemble of vortices with uncertain initial positions yields a probability distribution $P$. It demonstrates that $P = |\psi|^2$, recovering the Born rule as a consequence of the continuity of mass density in the classical limit.
• Derivation of Relativistic Relations:
  • Einstein-Planck Relation: By analyzing the phase time derivative, the paper derives $E = \hbar \omega$ and $E = mc^2$.
  • De Broglie Wavelength: The drift velocity of the vortex leads to a plane wave solution with wavelength $\lambda = h / (mv_d)$.
  • Uncertainty Principle: Using Fourier analysis of a wave packet constructed from the fluid velocity field, the paper derives the Heisenberg uncertainty principle $\Delta x \Delta p \geq \hbar/2$ as a mathematical consequence of wave packet localization.
• Derivation of the Klein–Gordon Equation: By accounting for the retarded propagation of the wave field from a moving vortex, the author demonstrates that Galilean transformations fail. Instead, the system requires the Prandtl-Glauert-Lorentz transformation. Applying this transformation to the wave equation yields the Klein–Gordon equation:
  $$\frac{\partial^2 \psi}{\partial t^2} - c^2 \nabla^2 \psi = -\omega_c^2 \psi$$
  This establishes the Schrödinger equation as the low-Mach-number limit of the Klein–Gordon equation within this fluid model.

4. Results

• Mathematical Equivalence: The study successfully proves that under the specific assumptions of irrotationality, quantized angular momentum ($\hbar$), and Korteweg stress ($\hbar^2/4m^2$), the classical fluid equations are mathematically identical to the Schrödinger and Klein–Gordon equations.
• Emergent Relativity: The model naturally produces Lorentz transformations and relativistic energy-momentum relations ($E^2 = p^2c^2 + m^2c^4$) when the retardation of the wave field is considered.
• Limitations Identified: The derivation of the Schrödinger equation relies on suppressing the "rest" rotational velocity field of the vortex. Consequently, the Schrödinger equation in this model is a weak-field approximation valid only at low Mach numbers.

5. Significance and Discussion

• Theoretical Insight: The paper provides a rigorous mathematical bridge showing that quantum and relativistic wave equations can emerge from classical continuum mechanics without invoking quantum postulates a priori. It suggests that quantum behavior could, in principle, be an emergent property of a specific type of fluid excitation.
• Caveats on Physical Interpretation: The author explicitly cautions against interpreting this as a physical reconstruction of reality.
  • Preferred Frame: The model implies a preferred frame (the rest frame of the fluid), which conflicts with the principles of special relativity if the fluid itself is composed of these "particles."
  • Bell's Theorem: The model is local (changes propagate at the speed of sound), whereas quantum mechanics exhibits non-locality (entanglement). The paper notes that extending this fluid model to reproduce the Standard Model, spin, and non-local correlations (Bell inequalities) faces significant, likely insurmountable, hurdles.
• Pedagogical Value: Despite the likely impossibility of this being a "true" theory of quantum mechanics, the work offers a valuable pedagogical tool. It demonstrates how complex quantum and relativistic formalisms can be understood as hydrodynamic analogues, potentially aiding in the visualization of abstract concepts like wave-particle duality and relativistic time dilation.

Conclusion:
The paper establishes a "fluid-mechanical analogue" where the Schrödinger and Klein–Gordon equations are not fundamental laws but emergent descriptions of a capillary vortex in a specific classical fluid. While it does not claim to replace quantum mechanics, it successfully demonstrates a deep structural isomorphism between fluid dynamics and quantum field theory under specific constraints.