Spacetime Exterior to a Star: Against Asymptotic Flatness

In many circumstances the perfect fluid conservation equations can be directly integrated to give a Geometric-Thermodynamic equation: typically that the lapse $N$ is the reciprocal of the enthalphy $h$, ($N=1/h$). This result is aesthetically appealing as it depends only on the fluid conservation equations and does not depend on specific field equations such as Einstein's. Here the form of the Geometric-Thermodynamic equation is derived subject to spherical symmetry and also for the shift-free ADM formalism. There at least three applications of the Geometric-Thermodynamic equation, the most important being to the notion of asympotic flatness and hence to spacetime exterior to a star. For asymptotic flatness one wants $h\to 0$ and $N\to 1$ simultaneously, but this is incompatible with the Geometric-Thermodynamic equation. Observational data and asymptotic flatness are discussed. It is argued that a version of Mach's principle does not allow asymptotic flatness.