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 = c²/[2ln(ρ̄/ρ_c) + 3] follows from the logarithmic equation of state, and the static PPN parameter becomes γ = (α−1)/(α+1). This equals 1 for no value of α, so the static density channel cannot reproduce general-relativistic deflection under any barotropic equation of state. The logarithmic case is worse still: at the background α = −1 sits exactly at the pole, where the static metric is not merely wrong but mathematically ill-defined. The deflection must be carried by another channel. A non-static metric with macroscopic vacuum flow supplies it, yielding γ = 1 exactly, with the deflection arising from advection of the signal by the flow rather than from a refractive sound-speed gradient. This requires the Painlevé–Gullstrand profile v(r) = √(2G_eff M/r), the only self-consistent irrotational weak-field solution, since the PG acoustic metric is identically Schwarzschild for any barotropic equation of state. Because the bending is advective it acts identically at all frequencies: propagation is exactly achromatic (ω = ck to all orders in the continuum theory), consistent with GW170817, with any dispersion confined to Planck-scale granularity outside the present treatment. A self-consistent Bondi accretion calculation shows that 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.