The authors model a one-dimensional, dilute bosonic gas in a ring of length 20, with periodic boundaries and a circulating flow whose wave number is quantized as k_n = 2nπ/l. Their mean-field description is the Gross–Pitaevskii equation, which evolves the condensate wave function in one space and one time dimension. Before the interaction is switched on, the condensate is uniform, with a constant density ρ₀. At positive time, a spatially varying Gaussian coupling is introduced, causing the local sound speed to vary around the ring while the flow velocity remains tied to the circulation.

The variation creates regions where the local Mach number, defined as flow velocity divided by sound speed, crosses unity. In the n = 3 configuration, the flow becomes supersonic between two points: the first crossing is an analogue black-hole horizon, and the second, where the flow returns to subsonic speed, is an analogue white-hole horizon. Phonons generated in the supersonic region cannot propagate upstream against the flow, providing the acoustic analogue of a one-way event horizon. By comparison, the n = 2 case remains subsonic everywhere and contains no such horizon pair. Numerical evolution of the mean field indicates broadly similar density behavior in the two configurations over an initial interval from time zero to 3.9.

To measure what happens outside the coherent condensate, the study decomposes the bosonic field into a mean-field wave function and a fluctuation operator. The linearized fluctuation equation includes both the usual density perturbations and anomalous terms that pair creation and annihilation of excitations. A Bogoliubov transformation mixes particle annihilation and creation operators, allowing the quadratic fluctuation Hamiltonian to be expressed in terms of quasiparticles. In that transformed vacuum, a nonzero single-particle occupation equal to the squared magnitude of each Bogoliubov coefficient persists even at zero temperature. This is the origin of quantum depletion: interactions generate a population of particles outside the condensate mode even when there is no thermal excitation.

The same mode-mixing mathematics links the condensate calculation to the description of Hawking particle production, where outgoing modes are built from incoming positive- and negative-frequency modes. In both cases, a vacuum defined in one basis contains particles when measured in another. The authors emphasize that quantum depletion and Hawking radiation are not the same physical process. In the analogue system, the acoustic horizons supply a mechanism for Hawking-like phonon production, and the objective is to determine how that production contributes to depletion of the condensate as the analogue Hawking temperature changes.

For the evolving, spatially inhomogeneous background, the researchers solve the time-dependent Bogoliubov–de Gennes equations numerically. The resulting local depletion density is the sum of the squared Bogoliubov coefficients across modes, while the global depletion D is its spatial average over the ring. Early in the evolution, configurations with and without horizons deplete at similar rates because mean-field interaction dynamics act across the condensate immediately. A horizon-related difference emerges only after phonons have had time to propagate through the system, on a characteristic timescale of L/c_s, where L is the system size and c_s is the sound speed. The separation between these two effects makes the comparison clearest at later times.

To distinguish steady particle production from short-lived transients, the authors fit the depletion evolution with a constant offset, a linear growth term and a damped oscillatory component. The constant accounts for depletion already present immediately after the quench; the linear coefficient, Γ, is interpreted as the asymptotic depletion rate; and the damped term captures transient excitations and interference among finite-size modes. The numerical data were reproduced with R² greater than 0.995 for all investigated Hawking temperatures. The resulting rate rises monotonically with analogue Hawking temperature, indicating that hotter analogue horizons are associated with faster accumulation of particles outside the condensate mode.

The baseline analogue Hawking temperature is approximately 0.027 in the paper’s dimensionless energy units, expressed there as mc₀² = 1/ξ₀². To examine a broader temperature range while retaining the black-hole–white-hole pair, the authors vary the background density by as much as a factor of ten. At the upper end of the explored range, around T_H ≈ 0.1 mc₀², global depletion reaches approximately 25%. At that level, about 75% of particles remain in the condensate mode, so the state is still predominantly condensed. Nevertheless, the authors argue that the linearized treatment is becoming less reliable because higher-order correlations, beyond-mean-field physics and backreaction are no longer negligible.

The temperature increase creates a trade-off that is central to the proposed connection between theory and tabletop analogue gravity. A higher Hawking temperature produces a faster depletion signal, which may be easier to distinguish from noise or finite-temperature effects during an experiment. At the same time, it pushes the condensate toward a regime in which quantum excitations can influence the mean-field background and the Bogoliubov–de Gennes approximation eventually breaks down. The study therefore does not claim a sharp universal transition at 25%; rather, it uses that level as a practical benchmark within its modeled parameter range. Tuning the background density is also more controllable in the calculations than changing the spatial interaction profile, which brings the two horizons close together and makes local depletion harder to resolve.

Overall, the simulations suggest that the long-time depletion rate, rather than the accumulated depletion at an arbitrary observation time, is the clearest indicator of horizon-induced production in this model. The approach offers a way to study microscopic effects associated with analogue horizons while condensation remains robust, but it also identifies where the present framework must give way. The authors propose future work beyond Bogoliubov theory that explicitly incorporates strong backreaction, as well as a closer examination of the microscopic structure of depletion near the two horizons. Their results frame condensate depletion as a possible experimental probe of Hawking-related mode mixing, without equating the laboratory system with an astrophysical black hole or claiming that gravitational Hawking radiation has been observed.

Subject of Research: Numerical study of Hawking temperature and quantum depletion in a ring-shaped Bose–Einstein condensate with analogue horizons

Article Title: Probing Hawking temperature threshold via quantum depletion in Bose–Einstein condensate

Article References: Rana, A. (2026). Probing Hawking temperature threshold via quantum depletion in Bose–Einstein condensate. The European Physical Journal C, 86(10), Article 1193. https://doi.org/10.1140/epjc/s10052-026-16404-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16404-6

Keywords: Bose–Einstein condensate, Hawking temperature, quantum depletion, analogue gravity, Bogoliubov theory, black hole horizon, backreaction, condensate coherence, phonon production, Probing, Hawking, temperature

The framework at the center of the paper is known as extended Aharonov-Bohm electrodynamics. In this formulation, the Lagrangian of the electromagnetic field is supplemented with an additional term proportional to the square of the divergence of the four-potential. When the primary source current fails to satisfy a local continuity equation, the field equations generate a non-local secondary current whose divergence exactly compensates that of the primary current, so that the observable total source remains conserved. The authors emphasize that this construction, sometimes described as charge-conservation censorship, has been explored in earlier work on non-local quantum sources, radiative multipoles, gauge waves and couplings to scalar-tensor gravity. Crucially, they also stress that the theory is not an alternative explanation of the well-established magnetic Aharonov-Bohm interference effect; the name refers instead to the reduced-gauge line of development associated with Ohmura and with the later discussion by Aharonov and Bohm of the physical role of the electromagnetic potentials.

The central question the authors pose is deceptively simple. Can rotation, or more generally the shift-vector structure associated with a rotating frame or a stationary gravitomagnetic background, generate an effective source of the kind required by the scalar sector of extended electrodynamics, even when the underlying charged matter is entirely standard and its microscopic current remains conserved? Their answer comes in two parts. At the microscopic level, the answer is a firm no. Using what is essentially Noether’s theorem in curved spacetime, they prove that for any matter field minimally coupled to electromagnetism and gravity through a generally covariant, locally gauge-invariant action, the physical four-current is covariantly conserved on shell. Neither a gravitomagnetic field produced by rotating masses nor a mere change of coordinates can manufacture a genuine charge anomaly.

At the level of observer-adapted transport variables, however, something genuinely interesting happens. When the conserved four-current is decomposed using the standard three-plus-one splitting of spacetime into space and time slices, the continuity equation for the charge density and transport current measured by observers attached to the rotating frame acquires an exact additional term. This term, which the authors call the observer-split source and write as the spatial divergence of the charge density multiplied by the shift vector, reduces in the weak-field limit of rigid rotation to a remarkably compact expression: the angular velocity times the angular derivative of the charge density. The term is not a new law of physics, the authors insist, but rather the unavoidable bookkeeping that appears when an exact conservation law is rewritten in variables adapted to a rotating device.

A subtle but important part of the analysis concerns what physically selects the observer structure. The authors argue that the relevant foliation of spacetime is not an arbitrary coordinate choice. In a laboratory realization, the synchronization and readout protocol of the experiment selects the hypersurface normal, while the rotor’s phase and angular motion fix the evolution field relative to those hypersurfaces. The resulting source is therefore relational: it depends on the specified laboratory observer structure together with the rotating source motion, much as a medium four-velocity forms part of the physical specification in relativistic descriptions of matter in motion. A passive change of coordinates cannot alter any predicted observable, but changing the physically supplied foliation or evolution field genuinely changes the transport variables and hence the split source itself.

The phenomenological step of the paper is deliberately restricted. The authors postulate that the exact observer-split divergence can act as the source entering the scalar field equation of extended electrodynamics, which in curved spacetime reads as the covariant d’Alembert operator acting on the scalar field equaling the split source. They show that the split source is the four-divergence of a geometrically well-defined auxiliary current, constructed from the physical current plus a shift-vector contribution, and they introduce a minimal longitudinal surrogate primary current whose associated secondary current restores total conservation while leaving the ordinary Maxwell source untouched. In this minimal closure, the standard electromagnetic field remains exactly as Maxwell predicts; only the additional scalar and potential sector responds to the rotating-frame source.

What makes the proposal scientifically serious is that the effective source has sharply defined, testable properties. It is odd under reversal of the rotation direction, so flipping the sense of spin must flip the sign of any linearly related response. It vanishes for a spatially uniform or exactly axisymmetric charge distribution, since only angular gradients of the density contribute. It is controlled by spatial gradients rather than density offsets, and on a closed ring its positive and negative angular contributions cancel exactly for a single-valued periodic density. Because the term is a spatial divergence, its volume integral is governed by boundary fluxes, naturally producing source-sink or bipolar patterns. A genuine microscopic extra-current need not obey any of these restrictions, which is precisely why the model is falsifiable: an effect that survives rotation reversal or persists when axisymmetry is restored would not support the proposed closure.

The authors also spell out what would be needed for a true microscopic violation of charge conservation, going beyond their effective bridge. Three routes are identified: intrinsically non-local matter dynamics such as fractional wave equations, quantum-coherent macroscopic sectors that could exhibit an effective anomaly in the current operator, or a genuinely new coupling between the scalar sector and a gravitomagnetic invariant built from the vorticity of the rotating congruence. These possibilities lie beyond the present paper, but they frame the effective construction as a computable, bounded surrogate rather than a claim of fundamental charge non-conservation. The authors note that any sufficiently small freely falling frame removes the apparent non-conservation entirely, so the split source acquires meaning only after coarse-graining over the spatial and temporal scales set by the apparatus and its measurement protocol.

On the experimental side, the paper connects to a companion proposal involving a small rotating aluminum disc carrying a fiber-optic ring, of which only a localized angular sector is optically active. Illumination modulates an effective surface charge density in the metallic carrier, to be calibrated independently by electrometric or capacitive means synchronized with the optical modulation and rotor phase. The effective source then takes the form of the angular velocity times the angular gradient of the induced density, and its peak amplitude scales linearly with the rotation rate and inversely with the angular width of the illuminated sector. Four direct controls are proposed: optical on-off gating, clockwise versus counterclockwise reversal, comparison of a localized sector with nearly uniform illumination, and comparison of a conductive carrier with insulating dummy controls. The authors further show, through an exact Gaussian smoothing model and a general Fourier-mode theorem, that any linear suppression network with bounded gain can only reduce the effective source, giving a compact suppression law for low-pass filtering of transients.

Looking forward, the authors identify analogue-gravity systems as another promising arena, since acoustic or optical analogue spacetimes provide a material flow that naturally selects a preferred laboratory congruence while perturbations propagate on an effective curved metric. The overall message is one of disciplined modesty combined with genuine opportunity. Rotation does not violate charge conservation, and the authors prove it. But the exact observer-split term that emerges in rotating descriptions of conserved currents is computable, symmetry-constrained and experimentally addressable, and whether it triggers any real macroscopic scalar response in extended electrodynamics is now framed as a sharply posed experimental question rather than a matter of speculation. A null result would bound the proposed scalar response; a signal lacking the predicted reversal or axisymmetry dependence would refute the closure outright.

Subject of Research: Effective observer-split source terms arising from rotating frames and gravitomagnetic backgrounds as phenomenological sources in extended Aharonov-Bohm electrodynamics

Article Title: Effective observer-split source terms in rotating frames and gravitomagnetic backgrounds in extended Aharonov-Bohm electrodynamics

Article References: Iadicicco, A., Modanese, G., & Verolino, L. (2026). Effective observer-split source terms in rotating frames and gravitomagnetic backgrounds in extended Aharonov-Bohm electrodynamics. General Relativity and Gravitation, 58(9), Article 106. https://doi.org/10.1007/s10714-026-03606-2

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03606-2

Keywords: extended electrodynamics, Aharonov-Bohm electrodynamics, observer splitting, rotating frames, gravitomagnetism, charge conservation, ADM decomposition, shift vector, scalar field, falsifiability, analogue gravity, general relativity