Conditions for Emergent Gravitational Light Bending from a Logarithmic Superfluid Vacuum

Purely scalar theories of gravitation predict a PPN parameter γ = 0 — only half the observed light deflection. This paper traces that failure to a structural feature of real scalar wave equations: the propagation speed is fixed by the D'Alembertian and cannot depend on the background field, regardless of the self-interaction. Modeling the vacuum instead as a complex superfluid with a relativistic logarithmic nonlinearity, analyzed via the Madelung transformation, introduces two degrees of freedom absent from the wave-equation form — a density-dependent sound speed and a macroscopic flow velocity. Using the Barceló–Liberati–Visser acoustic-metric formalism, the sound speed c_s² = [2 ln(ρ̄/ρ_c) + 3]⁻¹ follows from the logarithmic equation of state, and the PPN parameter becomes γ = (1−α)/(1+α). At the background, α = −1 places the static acoustic metric exactly at the pole, where it is mathematically ill-defined. A non-static metric with macroscopic vacuum flow regularizes this and yields γ = 1 exactly — the deflection arising from advection of the signal by the flow, not from a refractive sound-speed gradient — provided the flow follows the Painlevé–Gullstrand profile v(r) = √(2G_eff M/r), the unique self-consistent irrotational weak-field solution. Light propagation is exactly achromatic (ω = ck to all orders), consistent with GW170817. A self-consistent Bondi accretion calculation shows the logarithmic equation of state does not naturally produce this profile — the far field decays as r⁻² rather than the required r⁻¹ᐟ². Deriving the macroscopic flow as a collective effect of the microscopic soliton dynamics is identified as the central open problem.