A litsearch found this paper, and Meeker sent me a pdf of it: Urs M. Schaudt and Herbert Pfister: The Boundary Value Problem for the Stationary and Axisymmetric Einstein Equations is Generically Solvable, Phys. Rev. Lett. 77,16 (Oct 1996) 3284-3287 ABSTRACT We prove existence, uniqueness, and regularity for Dirichlet solutions of the stationary and axisymmetric Einstein equations in vacuum and in rigidly rotating ideal fluid matter for data (on a ball) whose absolute values are in a characteristic way limited by the "diameter" of the ball. These results have important consequences for the existence of exterior vacuum and interior matter solutions for rotating stars. The mathematical procedures and results have remarkable connections with a numerical solution technique for rotating stars. They also cite: Uwe Heilig: On the existence of rotating stars in general relativity, Commun Math'l Phys. 166,3 (1995) 457-493 http://projecteuclid.org/euclid.cmp/1104271700 ABSTRACT Abstract: The Newtonian equations of motion, and Newton's law of gravitation can be obtained by a limit L=1/c^2-->0 of Einstein's equations. For a sufficiently small constant A the existence of a set of solutions (0<L<A) of Einstein's equations of a stationary, axisymmetric star is proven. This existence is proven in weighted Sobolev spaces with the implicit function theorem. Since the value of the causality constant L depends only on the units used to measure the velocity, the existence of a solution for any small L is physically interesting. --- Here is my quick attempt to summarize these papers. If you postulate some equation-of-state for the matter inside a "star" and assume it is rigidly rotating blob of matter in hydrostatic equilibrium, then Einstein GR equations can be viewed as a Dirichlet problem -- given data on the boundary, can we solve it inside that boundary? Schaudt & Pfister set up an iteration which "improves" an approximate solution, and prove this is a "contraction" in a Banach space, and therefore an attractor exists, and therefore a solution exists. Hence, they get a theorem saying, "rigidly rotating stars exist" provided the mass is not too large and the spin is not too large -- to make their proof work they need masses and spins upper bounded by about 1% of what they presume would really be the maximum allowed. So, if the boundary data is chosen to agree with the Kerr exterior, I think this means that this proves that a matching "rigid rotating" interior always exists provided (1) mass and densities never too large (2) spin not too large (3) postulated equation of state for the matter (in the right class). But they do not actually FIND any such interior solution, they merely prove nonconstructively that at least one exists. It would be more desirable to have an explicit example that was not too hard to deal with. But it seems like the Banach-contaction proof should yield a numerical procedure for approximating such a solution to arbitrary accuracy. Heilig starts from a known exact solution in Newtonian gravity (rotating fluid ellipsoid, constant density fluid) and then tries to perturb it to work in Einstein gravity using L=1/c^2, where L=0 for Newton and L>0 for Einstein; he claims he proves such a perturbation exists if L is not too large. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)