0 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 enthalpy 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 geometricthermodynamic 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 asymptotic flatness and hence to spacetime exterior to a star. For asymptotic flatness one wants h → 0 and N → 1 simultaneously, but this is incompatible with the geometricthermodynamic equation. A first shot at modeling spacetime exterior to a star is to choose an idealized geometric configuration and then seek a vacuum-Einstein solution. Now the assumption of a vacuum is an approximation, in any physical case there will be both fields and fluids present. Here it is shown that the requirement that the star is isolated and the presence of a fluid are incompatible in most isentropic cases, the pressure free case being an exception. Thus there is the following dilemma: either an astrophysical system cannot be isolated or the exterior fluid must depend explicitly on the entropy. That a system cannot be isolated is another way of stating Mach’s principle. The absence of asymptotically flat solutions depends on: i)the equation of state, ii)the admissibility of vector fields, and iii)the requirement that the perfect fluid permeates the whole spacetime. The result is robust against different choices of geometry and field equations because it just depends on the fluid conservation equations and the ability to introduce a suitable preferred vector field. For example with spherical symmetry there is the preferred vector field tangent to the 3-sphere; furthermore for asymptotically flat spacetimes there is the preferred vector field tangent to the 3-sphere at infinity. The properties of spherically symmetric geodesics are studied both to examine whether a fluid spheres geodesics can explain solar system dynamics and also to model hypothetical galactic halo with spherically symmetric fluids so as to produce constant galactic rotation curves. The rate of decay of fields and fluids is discussed in particular whether there are any non-asymptotically flat decays and what gravitation modifies the Yukawa potential to be. The Tolman-Ehrenfest relation follows immediately from N = 1/h. The equations relating the enthalpy to the lapse have a consequence for the cosmic censorship hypothesis this is briefly discussed.