Thermodynamic compatibility, holographic property and wave-function representation of generalized weakly nonlocal self-gravitating non-relativistic fluids

Abstract

Two pillars of physics---gravity and quantum mechanics---are usually formulated in entirely separate languages. Surprisingly, both can be expressed in the language of a fluid: a self-gravitating fluid for Newtonian gravity, and a quantum-pressure-carrying fluid for the Schrödinger equation. In this work we show that this resemblance can be understood within a common framework based on how the Second Law of Thermodynamics constrains the class of non-dissipative continua considered here. Using a transparent thermodynamic procedure, we derive the most general fluid model within a chosen constitutive state space involving density, velocity, energy, an internal scalar field, and their spatial gradients. The framework recovers, as special cases, the Euler equations, a restricted family of Korteweg fluids (including the hydrodynamic form of the Schrödinger and Gross--Pitaevskii equations), Newtonian self-gravitating fluids, and the Schrödinger--Newton system. We identify the conditions under which perfect fluids exhibit \emph{classical holography}: the pressure-tensor divergence can then be represented by the gradient of a scalar potential. Classical holography yields a vorticity transport law that preserves sufficiently regular, initially irrotational flows and, under additional assumptions, enables a single complex-valued Schrödinger-like representation. While gravitational and quantum-hydrodynamic theories were investigated separately in our previous works, the present study develops a substantially more general scalar-internal-variable construction, refines its perfect-fluid analysis, and systematically embeds the earlier models within a unified framework. The framework also identifies a thermodynamically admissible density--gravity coupling that connects classical and quantum self-gravitating fluids, although its stability, and observational consequences remain to be explored.

arXiv comment33 pages, 1 figure