Conditions for Emergent Gravitational Light Bending from a Logarithmic Superfluid Vacuum

Purely scalar theories of gravitation predict a Parameterized Post-Newtonian (PPN) parameter \(\gamma = 0\), yielding only half the observed light deflection. We trace this failure to a structural feature of real scalar wave equations: 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. Using the standard relativistic acoustic-metric formalism of Barceló-Liberati-Visser, we derive the sound speed \(c_{s}^{2} = {\lbrack{{2{\ln{({\overline{\rho}/\rho_{c}})}}} + 3}\rbrack}^{- 1}\) 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 \(\gamma = {{({1 - \alpha})}/{({1 + \alpha})}}\) where \(\alpha = {d{\ln{{c_{s}/d}{\ln\rho}}}}\). The logarithmic equation of state at the background density gives \(\alpha = {- 1}\), placing \(\gamma_{static}\) exactly at the pole of this expression: the static acoustic metric is not merely numerically wrong but mathematically ill-defined. We then demonstrate that a non-static acoustic metric with macroscopic vacuum flow regularizes this pathology and yields \(\gamma = 1\) exactly, provided the flow velocity satisfies \({v{(r)}} = \sqrt{{2G_{eff}M}/r}\)—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_ self-consistent weak-field solution of the acoustic metric equations: the static metric is divergent, 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.