Quantum-Fluid Correspondence in Relativistic Fluids with Spin: From Madelung Form to Gravitational Coupling

Abstract: This paper explores the quantum-fluid correspondence in a charged relativistic fluid with intrinsic spin. We begin by examining the nonrelativistic case, showing that the inclusion of spin introduces a quantum correction to the classical fluid energy. This correction, coupled with Maxwell's equations, naturally leads to the Schr\"odinger equation in Madelung form. Building on this foundation, we extend the formalism to a relativistic perfect fluid, identifying the system's stress-energy-momentum tensor. Our analysis reveals that the trace of the quantum correction to this tensor corresponds to the energy density of an oscillator, with its frequency determined by the vorticity of the spin motion. We then use the stress-energy-momentum tensor to establish the relationship between the Ricci scalar curvature, as dictated by the Einstein field equations, and the fluid density in a static, spherically symmetric configuration. Lastly, we generalize the Madelung transformation to compressible Navier-Stokes flows with vorticity and viscosity by developing a tailored Clebsch representation of the velocity field. This theoretical framework offers potential applications for studying fluid-like systems with internal rotational degrees of freedom, commonly encountered in astrophysical settings.