Spacetime Exterior to a Star

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 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 asympotic 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 Geometric-Thermodynamic equation. Consider the exterior to a star, or the exterior of any real or hypothetical astrophysical system. A first shot at modeling spacetime in this exterior region 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 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 be of a contrived nature. Specifically it is shown that many perfect fluids do not have asymptotically flat solutions. The result 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 for spherical symmetry there is the preferred vector field tangent to the 3-sphere; futhermore for asymptotically flat spacetimes there is the preferred vector field tangent to the 3-sphere at infinity. The Tolman-Ehrenfest relation follows immediately from N = 1/h. The equations relating the enthalphy to the lapse have consequences for the cosmic censorship hypothesis and for solar system dynamics and these are briefly mentioned. Pacs:97.90+j,04.20Me,98.90s; MR:83C55,83C57,83C75,83C10. 6-