Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit

TL;DR

This paper derives a hydrodynamic form of the Klein-Gordon-Einstein equations in the weak field limit using a generalized Madelung transformation, connecting relativistic scalar field models with nonrelativistic quantum hydrodynamics.

Contribution

It introduces a generalized Madelung transformation to obtain hydrodynamic equations from the Klein-Gordon-Einstein framework in the weak field limit, including self-interacting scalar fields.

Findings

Hydrodynamic equations reduce to Schrödinger-Poisson or Gross-Pitaevskii-Poisson forms in the nonrelativistic limit.

Comparison shows consistency between the derived relativistic hydrodynamics and simplified models with gravitational potential.

The approach generalizes previous models by incorporating arbitrary scalar field potentials.

Abstract

Using a generalization of the Madelung transformation, we derive the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. We consider a complex self-interacting scalar field with an arbitrary potential of the form $V(|\varphi|^2)$. We compare the results with simplified models in which the gravitational potential is introduced by hand in the Klein-Gordon equation, and assumed to satisfy a (generalized) Poisson equation. Nonrelativistic hydrodynamic equations based on the Schr\"odinger-Poisson equations or on the Gross-Pitaevskii-Poisson equations are recovered in the limit $c\rightarrow +\infty$.