Orbital Dynamics and Weak-Field Gravity From a Barotropic Space-Time Fluid

We model space–time as a compressible, barotropic, viscoelastic medium with state variables \left(\rho,p,\eta,c_s\right). In the static, weak-field limit a scalar enthalpy potential h obeys a Gauss/Poisson law, yielding an emergent inverse-square field h=-\mu/r and Kepler’s third law T=2\pi\sqrt{a^3/\mu} for test-body orbits—without assuming Newton’s law or the Einstein field equations. The same 1/r potential follows independently from pressure, density-response, and variational/free-energy routes, establishing uniqueness under linearity, locality, and isotropy. An EOS-controlled correction scales as \varepsilon\left(r\right)\sim\mu/\left(c_s^2r\right), implying ppm-level bounds from Solar-System orbits. We give measurement-mode reconstructions (JPL Horizons, epoch J2000 TDB) with ppm residuals and a realistic nonzero Moon residual at the {10}^{-3} level (expected from tides and oblateness). We relate the framework to thermodynamic/emergent-gravity programs and outline constraints from gravitational-wave propagation (GW170817). The model is falsifiable via \left(c_s,\eta\right) and reduces to the Newtonian/PPN limit in the Solar System.