Unified Hydrodynamic Analogue of Aharonov-Bohm and Lense-Thirring Effects

Abstract: We show that surface waves in a draining-bathtub vortex provide a hydrodynamic realization of both Aharonov-Bohm phase shifts and Lense-Thirring frame dragging within a single system. A static time transformation maps the flat (2+1)-dimensional wave equation onto the convected shallow-water equation, yielding an effective vector potential set by the background flow. In this geometry, the circulation defines a global phase holonomy that controls wave structure. Traveling waves exhibit wavefront dislocations characteristic of Aharonov-Bohm scattering, while standing-wave superpositions produce nodal patterns that rotate at an angular velocity fixed by the circulation, providing a direct analogue of frame dragging. For noninteger circulation, the problem is naturally defined on the universal cover, ensuring single-valued partial-wave solutions. Experiments on a controlled vortex confirm these predictions and establish a laboratory platform in which topological phase and inertial effects, central to gauge and gravitational physics, emerge from a measurable velocity field.

• The paper demonstrates a unified hydrodynamic platform that exhibits analogues of Aharonov-Bohm phase shifts and Lense-Thirring frame-dragging effects.
• It maps the relativistic wave equation to the convected shallow-water equation, introducing an effective vector potential governed by vortex circulation.
• Experimental results confirm predicted phase singularities and rotating nodal lines, providing measurable parallels to quantum and gravitational phenomena.

Unified Hydrodynamic Analogue of Aharonov-Bohm and Lense-Thirring Effects

Introduction

This work develops and experimentally demonstrates a unified hydrodynamic platform that realizes classical analogues of both the Aharonov-Bohm (AB) phase shifts and the Lense-Thirring (LT) frame-dragging effect within a single system. In particular, surface waves propagating atop a draining-bathtub vortex are shown to exhibit signatures directly corresponding to global topological phenomena typically associated with quantum gauge fields (AB effect) and general relativity (LT effect). The unification is achieved via a mapping between a relativistic wave equation and the convected shallow-water equation, with the effective vector potential set by the circulation of the underlying flow.

Theoretical Framework and Mapping

The authors present a construction where the flat $(2+1)$-dimensional wave equation,

$$\square\Psi = \left(\nabla^2 - \frac{1}{c^2}\partial_t^2\right)\Psi = 0,$$

is mapped, via a static time transformation, onto the convected wave equation for surface deformation $\eta$ in shallow water. For sufficiently weak and slowly varying background flows $(|U|\ll c)$, this yields

$$\frac{1}{c^2}\left( \partial_t + U \cdot \nabla \right)^2 \eta = \nabla^2 \eta.$$

The mapping introduces an effective vector potential into the wave dynamics, crucially controlled by the circulation of the background flow $\Gamma$. The transformation formalizes the analogy between the electromagnetic vector potential in the AB effect and the velocity field in the hydrodynamic setting.

For globally potential flows, circulation enters as a geometric phase shift affecting traveling waves. However, with a vortex—rendering the potential multivalued—the phase shift becomes topologically nontrivial, encoding the holonomy that underlies both AB and LT phenomena.

Manifestation of Topological Effects

Aharonov-Bohm Analogues

In the presence of a vortex, traveling waves interacting with the circulation acquire an additional angular phase factor $e^{-i\alpha\varphi}$ with dimensionless circulation parameter $\alpha = \Gamma\nu/(2\pi c^2)$. For integer $\alpha$, the resulting field is single-valued, while for non-integer circulation, the field must be defined on the universal cover of the punctured plane, effectively capturing the winding history and phase monodromy encountered in quantum AB settings. The observable consequence is the formation of wavefront dislocations—nodal line singularities in the wave amplitude—that correspond to physical realizations of AB scattering patterns.

Lense-Thirring Analogues

By superposing counter-propagating traveling waves (standing wave configuration), interference in the presence of vortex-induced circulation yields rigidly rotating nodal lines. These lines rotate with angular velocity $\Omega = \nu/\alpha$, encapsulating the inertial frame dragging central to the LT effect. Unlike the local phase effects in traveling waves, the rotation of these nodal structures offers a direct, measurable signature of global topological constraints imposed by the circulation, closely paralleling the precession phenomena predicted in general-relativistic spacetimes near rotating masses.

Topology, Partial Waves, and Universal Cover

Non-integer circulation introduces a multivaluedness in the phase, requiring the use of the universal cover in angular coordinate space. The authors deploy a partial-wave expansion, which in this context reduces to the standard Jacobi-Anger expansion for integer circulation but realizes the genuine AB partial-wave (and holonomy) structure otherwise. This construction fundamentally reflects the nontrivial topology of the punctured plane and illustrates how the mathematical machinery typically reserved for quantum mechanical or gravitational holonomies finds a direct analogue in classical fluid systems.

Experimental Realization and Results

A controlled draining-bathtub vortex in a 2×1 m water tank is used to validate the theoretical predictions. Wavefronts are driven using acoustic transducers, and the vortex strength is precisely regulated via a recirculating flow. Vortex circulation is independently measured using particle image velocimetry (PIV), and surface deformations are imaged using a caustic-projection technique.

The experimental results robustly confirm the predicted phenomena:

• Traveling waves exhibit phase singularities and dislocation patterns in close agreement with analytic expectations for AB-type interference.
• Standing waves reveal system-spanning nodal lines that rotate rigidly with a circulation-dependent angular velocity, as predicted for LT analogues.
• Quantitative agreement is obtained between experiment and theory for both the number and velocity of nodal lines, with deviations at large radii attributed to viscous and capillary dissipation, consistent with the predicted capillary-drag scale cutoff.

Implications and Theoretical Significance

The unification of AB and LT effects within the same hydrodynamic system demonstrates that topologically induced phase phenomena are not unique to quantum or relativistic frameworks but emerge generically from global properties of the underlying field configuration—in this case, fluid circulation. The work places electromagnetic and gravitational holonomies on an equal theoretical footing, as both can be understood as geometric phases acquired in multiply connected spaces with nontrivial vector potentials.

Practically, the classical hydrodynamic system provides a directly measurable and fully controllable platform for exploring effects usually constrained to inaccessible quantum or astrophysical regimes. Direct measurement of the velocity field distinguishes this setup from traditional quantum or gravitational analogues, where underlying vector fields or spacetime metrics are not physically observable.

Theoretically, the correspondence supports the broader paradigm wherein gauge and topological structures manifest across diverse domains, from condensed matter to gravitational and high-energy physics. The present framework is directly extensible to more complex configurations, such as multi-vortex systems, facilitating experimental studies of holonomies, gauge structures, and wave scattering in spaces of arbitrary topology.

Future Directions

This unified hydrodynamic framework opens new avenues for analog gravity, topological physics, and the study of geometric phases in classical systems. Richer configurations (multi-vortex, rapidly rotating flows) will permit the exploration of non-Abelian holonomies and effective gauge structures. Coupling these platforms to nonlinear or dispersive effects could further model dynamical phenomena pertinent to black hole mergers, fast scramblers, or topologically nontrivial quantum fields. The experimental accessibility of all relevant observables ensures that such analogues can be probed well beyond the current limitations of quantum or gravitational settings.

Conclusion

The paper rigorously establishes a hydrodynamic system that brings together the physical mechanisms and topological structures underlying the Aharonov-Bohm and Lense-Thirring effects. The analytic and experimental demonstration that both phenomena are governed by a unified circulation-induced holonomy offers a concrete realization of deep theoretical connections between quantum mechanics, gauge theory, and gravitation, and provides a versatile, high-precision laboratory platform for future topological physics research (2604.21543).