Macroscopic Quantum Tunneling Between Acoustic Black and White Holes in the Michel Flow

The relativistic Michel flow possesses two degenerate stationary solutions: the accretion branch (υ < 0), which forms an acoustic black hole, and the wind branch (υ > 0), which forms an acoustic white hole. Both solutions are stationary points of a single energy functional; the barrier between them is of inertial origin and is due to the necessity of stopping and reversing the macroscopic flow. Within a one-parameter model reduction, an instanton trajectory in Euclidean time is constructed, connecting the two branches through the saddle point of the phase space. The instanton action is expressed through the effective moment of inertia of the flow and the height of the energy barrier; both factors are finite owing to the convergence of the integral with the υ² factor. Numerical estimates for a∞ ∈ \ 0.05, 0.1 \ show that the instanton action is controlled by the macroscopic scales of the acoustic horizon. The transition probability is exponentially suppressed and acquires physical meaning only for systems with a quantum microstructure (finite effective Planck constant eff > 0). The detailed balance between the forward and reverse transitions is broken owing to the classical instability of the acoustic white hole. The construction is consistent with the invariant quantities established in the companion studies. Principal results - The accretion and wind branches are identified as degenerate stationary points of a single energy functional; the barrier is shown to be of inertial-kinematic nature (Δ F = 0), not a thermodynamic phase barrier. - The instanton trajectory q(τ) = (τ/τ₀) is obtained in closed form, with the action Sᵢₙₛₜ = 4/3√2\,I\,Δ E finite and definite. - Numerical estimates yield Sᵢₙₛₜ = 30 ~ 590 for γ ∈ \ 4/3, 5/3 \ and a∞ ∈ \ 0.05, 0.1 \; the tunneling is more strongly suppressed for colder flows. - The structural analogy is established with the persistent-current reversal in a superconducting ring (SQUID) and with the vacuum black-hole/white-hole tunneling mechanism of Volovik, Ann. Phys. 449, 169219 (2023). - Coleman bubble nucleation is shown to be inapplicable (Δ E = 0; unstable wind seed); the reverse transition is identified as a fast classical decay over a time ~ κ⁻¹ rather than tunneling. Companion studies in this series Acoustic geometry and sound-ray dynamics in relativistic spherical accretion Perturbation spectrum of Michel flow: acoustic, vortical, and entropy modes Global acoustic geometry of the Michel flow in Painlevé-Gullstrand coordinates Files The upload contains the author's preprint in English and Russian: compiled PDF versions and LaTeX sources. Keywords Michel accretion; analogue gravity; acoustic black hole; acoustic white hole; macroscopic quantum tunneling; instanton; Painlevé-Gullstrand coordinates; surface gravity; Hawking radiation; Bose-Einstein condensate