On "hard stars" in general relativity

We study spherically symmetric solutions to the Einstein-Euler equations which model an idealized relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets the speed of sound equal to the speed of light. We demonstrate the existence of a 1-parameter family of static solutions, or ''hard stars,'' and describe their stability properties: First, we show that small stars are a local minimum of the mass energy functional under variations which preserve the total number of particles. In particular, we prove that the second variation of the mass energy functional controls the ''mass aspect function.'' Second, we derive the linearisation of the Euler-Einstein system around small stars in ''comoving coordinates,'' and prove a uniform boundedness statement for an energy, which is exactly at the level of the variational argument. Finally, we exhibit the existence of time periodic solutions to the linearised system, which shows that energy boundedness is optimal for this problem.Comment: v1: 30 pages, 2 figures. v2: 41 pages, 2 figures; Section 4 now includes a linearisation of the Einstein-Euler equations, a new uniform boundedness result, and the construction of periodic solutions for the linearised system; abstract and Section 1.3 extended to reflect these additions; various remarks and references added; to appear in AH

Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.Comment: 33 pages, 1 figure. v2 matches final published versio

On the asymptotic behavior of static perfect fluids

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear and polytropic-type equations of state with index n>5. In order to capture the asymptotic behavior we introduce a notion of scaled quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In particular, these spacetimes are asymptotically simple.Comment: 32 pages; minor changes in v2, final versio

Static self-gravitating Newtonian elastic balls

The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.Comment: 30 pages, 2 figures. The order of presentation of the results and the notation have been changed considerably to improve the reading flow of the article. Some assumptions and theorems have been reformulated in a more clear way and several new remarks have been adde

Dynamical Boson Stars

The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called {\em geons}, but none were found. Instead, particle-like solutions were found in the late 1960s with the addition of a scalar field, and these were given the name {\em boson stars}. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.Comment: 79 pages, 25 figures, invited review for Living Reviews in Relativity; major revision in 201

A numerical stability analysis for the Einstein-Vlasov system

We investigate stability issues for steady states of the spherically symmetric Einstein-Vlasov system numerically in Schwarzschild, maximal areal, and Eddington-Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle from the viewpoint of our numerical results. The physical reliability of the latter is strengthened by obtaining consistent results in the three different coordinate systems and by the systematic use of dynamically accessible perturbations.Comment: 35 pages, 12 figure