On unbounded bodies with finite mass: asymptotic behaviour

There is introduced a class of barotropic equations of state (EOS) which become polytropic of index $n = 5$ at low pressure. One then studies asymptotically flat solutions of the static Einstein equations coupled to perfect fluids having such an EOS. It is shown that such solutions, in the same manner as the vacuum ones, are conformally smooth or analytic at infinity, when the EOS is smooth or analytic, respectively.Comment: 6 page

Celestial mechanics of elastic bodies

We construct time independent configurations of two gravitating elastic bodies. These configurations either correspond to the two bodies moving in a circular orbit around their center of mass or strictly static configurations.Comment: 16 pages, 2 figures, several typos removed, erratum appeared in MathZ.263:233,200

Helical Solutions in Scalar Gravity

We construct solutions, for small values of $G$ and angular frequency $\Omega$, of special relativistic scalar gravity coupled to ideally elastic matter which have helical but no stationary or axial symmetry. They correspond to a body without any symmetries in steady rotation around one of its axes of inertia, or two bodies moving on a circle around their center of gravity. Our construction is rigorous, but modulo an unproved conjecture on the differentiability of a certain functional.Comment: 11 page

Static axisymmetric space-times with prescribed multipole moments

In this article we develop a method of finding the static axisymmetric space-time corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static space-time for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernandez-Pastora and Martin concerning the monopole-quadropole solution.Comment: 14 page

Late time behaviour of the maximal slicing of the Schwarzschild black hole

A time-symmetric Cauchy slice of the extended Schwarzschild spacetime can be evolved into a foliation of the $r>3m/2$-region of the spacetime by maximal surfaces with the requirement that time runs equally fast at both spatial ends of the manifold. This paper studies the behaviour of these slices in the limit as proper time-at-infinity becomes arbitrarily large and gives an analytic expression for the collapse of the lapse.Comment: 18 pages, Latex, no figure

(In)finiteness of Spherically Symmetric Static Perfect Fluids

This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in General Relativity we improve earlier results

Vacuum Spacetimes with Future Trapped Surfaces

In this article we show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data is defined on a manifold with topology $R^3$ with a regular center and is asymptotically flat. Further, this initial data will contain an annular region which is foliated by two-surfaces of topology $S^2$. These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete.Comment: 19 page

The static spherically symmetric body in relativistic elasticity

In this paper is discussed a class of static spherically symmetric solutions of the general relativistic elasticity equations. The main point of discussion is the comparison of two matter models given in terms of their stored energy functionals, i.e., the rule which gives the amount of energy stored in the system when it is deformed. Both functionals mimic (and for small deformations approximate) the classical Kirchhoff-St.Venant materials but differ in the strain variable used. We discuss the behavior of the systems for large deformations.Comment: 19 pages, 8 figure