Conditions for Emergent Gravitational Light Bending from a Logarithmic Superfluid Vacuum

Purely scalar theories of gravitation predict a Parameterized Post-Newtonian (PPN) parameter γ = 0, yielding only half the observed light deflection.We trace this failure to an inescapable kinematic constraint: in any real scalar wave equation, the characteristic propagation speed is fixed by the D'Alembertian operator and cannot depend on the background field, regardless of the self-interaction potential.We show that modeling the vacuum as a complex superfluid with a relativistic logarithmic nonlinearity, analyzed via the Madelung transformation, introduces two new degrees of freedom absent in the wave equation form: a density-dependent sound speed and a macroscopic flow velocity.We derive the relativistic sound speed c 2 s = [ln(ρ/ρ c ) + 2] -1 from the logarithmic equation of state, demonstrating its dependence on the local vacuum density-an effect absent in the non-relativistic limit.We compute the PPN parameter for the static acoustic metric, finding γ = (1 -α)/(1 + α) where α = d ln c s /d ln ρ, and show that no static barotropic fluid yields γ = 1 unless α = 0 (constant sound speed), which produces zero deflection.We then demonstrate that a nonstatic acoustic metric with macroscopic vacuum flow can achieve γ = 1 exactly, provided the flow velocity satisfies v(r) = 2G eff M/r-the Painlevé-Gullstrand profile.Using a self-consistent Bondi accretion calculation, we show that the logarithmic equation of state does not naturally produce this profile in the far field, where the flow decays as r -2 rather than the required r -1/2 .We further show that the Painlevé-Gullstrand flow is the only selfconsistent weak-field solution of the acoustic metric equations: the static metric fails because its inferred gravitational potential does not match the assumed one for general equations of state, whereas the flowing metric closes the self-consistency loop for any barotropic fluid.The remaining open problem is to derive this macroscopic flow as a collective effect of the microscopic soliton dynamics.