Abstract
<jats:p>In a companion paper, we showed that the Painlevé-Gullstrand (PG) flow \(v = \sqrt{{2GM}/r}\) regularizes the static-metric pathology of the logarithmic superfluid vacuum, yielding the PPN parameter \(\gamma = 1\) for any barotropic equation of state. Here we analyze this macroscopic flow as a collective effect of \(N\) microscopic vortex-Gausson solitons—the topological defects of the logarithmic condensate—and establish the conditions a self-consistent collective flow must satisfy. We show that the self-consistency requirement is met by the PG flow but not by the static configuration: the static solution is ill-defined (the spatial-to-temporal metric ratio diverges at the background equation-of-state exponent \(\alpha = {- 1}\)), while the PG flow closes the self-consistency loop. The PG profile is the unique _irrotational_ weak-field solution carrying a Newtonian potential; whether vortical or multi-fluid configurations admit others is left open. Postulating that each soliton sources the acoustic metric through a localized perturbation, we derive the emergent gravitational constant \(G_{eff} \sim {c^{2}/{({\xi^{2}\rho_{0}})}}\), where \(\xi\) is the healing length and \(\rho_{0}\) is the vacuum condensate mass density. This expression is universal—independent of the source mass \(M\) and the distance \(r\). Matching \(G_{eff}\) to Newton’s constant, combined with the natural condensate packing density \(n_{0} \sim {1/\xi^{3}}\), determines \(\xi \sim \ell_{P}\) (the Planck length) and \(\rho_{0} \sim {m_{P}/\ell_{P}^{3}}\) (the Planck density). An independent derivation from the Hamilton-Jacobi dynamics of the logarithmic fluid confirms the profile: at \(\alpha = {- 1}\) the potential is carried by the flow alone, so a Newtonian \(1/r\) potential requires exactly the PG velocity profile (\(v^{2} \propto {1/r}\)); the accompanying density depletion is real but gravitationally inert. The continuity constraint is discussed as an open problem.</jats:p>