Pressure-Mediated Gravity (PMG): A Speculative Toy Model for Emergent Gravity via Superfluid Vacuum Dynamics

Abstract "We propose the JERROW Sea as a speculative toy model in which spacetime is treated as a superfluid vacuum medium whose density perturbations generate emergent gravitational phenomena. The framework aims to reproduce Newtonian gravity in the high-acceleration regime via a depletion equation ∇²ρⱼ = +αρ_b (corrected sign), and predicts MOND-like scaling a ∼√(a₀a_N) in the deep-vacuum limit with characteristic acceleration a₀ = ℏcₛ³/(G mⱼ) derived from vacuum microphysics with quantum corrections. We demonstrate that the relativistic Gross-Pitaevskii equation for a logarithmic superfluid provides a quantum mechanical foundation for the depletion dynamics. An emergent spacetime metric in Painlevé-Gullstrand form reproduces aspects of curved spacetime geometry. Crucially, we emphasize that this work presents a non-relativistic toy model with significant unresolved challenges: no demonstration of full general relativity recovery (lensing, cosmology, gravitational waves), unaddressed Lorentz violation constraints requiring healing length ξ ≲10^{-20}ℓ_P, and physically implausible parameter values in the naive formulation. The framework should be viewed as a conceptual exploration of hydrodynamic gravity analogs—not as an established theory." Keywords: emergent gravity (speculative), superfluid vacuum analogs, MOND phenomenology, quantum gravity toy models