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

Abstract: Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper examines under which conditions a classical fluid model may reproduce the most basic mathematical formalism of both theories. By assuming that the angular momentum of an irrotational vortex in an inviscid, barotropic, isothermal fluid with sound speed c is equal in magnitude to the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity equations of a Euler-Korteweg vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is equivalent to Schrödinger's equation. The model is shown to yield classical analogues to de Broglie wavelength, the Einstein-Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wavefield of a vortex in convection produces the Lorentz transformation and the Klein-Gordon equation, with Schrödinger's equation appearing as the low-Mach-number limit. These results demonstrate that, under explicit assumptions, a classical continuum can reproduce the mathematical formalism of quantum and relativistic theory in their simplest form, without assuming the postulates principal to those theories.