Conditions for Emergent Gravitational Light Bending from a Logarithmic Superfluid Vacuum

Purely scalar theories of gravitation notoriously fail to predict the correct light bending, yielding a Parameterized Post-Newtonian (PPN) parameter of γ = 0 instead of the general relativistic γ = 1. This manuscript traces this failure to a structural limitation of real scalar wave equations: the geometric-optics propagation speed is strictly fixed by the D'Alembertian operator, preventing any background-dependent refractive deflection. To escape this kinematic trap, this paper models the vacuum as a complex superfluid governed by a relativistic logarithmic nonlinearity. By applying the Madelung hydrodynamic decomposition, the framework introduces two critical degrees of freedom absent in standard scalar wave equations: a density-dependent sound speed and a macroscopic fluid flow. Using the relativistic acoustic-metric formalism, the sound speed is derived directly from the logarithmic equation of state as cs² = 2 ln (ρ/ρc) + 3⁻¹, demonstrating a strict dependence on local vacuum density that is entirely absent in the non-relativistic limit. The paper demonstrates that a purely static acoustic metric for this logarithmic fluid is mathematically pathological. Its equation of state exponent at the background density (α = -1) places the static PPN parameter exactly at a divergent pole, meaning no finite weak-field expansion exists. However, introducing a non-static acoustic metric with macroscopic vacuum flow regularizes this divergence. The manuscript proves that if the vacuum flow follows the free-fall Painlevé-Gullstrand profile, v (r) = √ (2Gₑff M / r), the acoustic metric successfully closes the self-consistency loop and yields γ = 1 exactly for any barotropic fluid. Finally, a self-consistent Bondi accretion calculation reveals that this required v ∝ r⁻¹/² velocity profile cannot emerge from the steady-state accretion of a single microscopic soliton, which decays far too rapidly as r⁻². The manuscript concludes by identifying the central open problem for superfluid vacuum gravity: proving that the macroscopic Painlevé-Gullstrand flow arises as a collective, many-body mean-field effect of microscopic topological defects (vortex-Gaussons) within the condensate.