Static, Self-Gravitating Elastic Bodies

There is proved an existence theorem, in the Newtonian theory, for static, self-gravitating bodies composed of elastic material. The theorem covers the case where these bodies are small, but allows them to have arbitrary shape.Comment: 8 page

Elastic deformations of compact stars

We prove existence of solutions for an elastic body interacting with itself through its Newtonian gravitational field. Our construction works for configurations near one given by a self-gravitating ball of perfect fluid. We use an implicit function argument. In so doing we have to revisit some classical work in the astrophysical literature concerning linear stability of perfect fluid stars. The results presented here extend previous work by the authors, which was restricted to the astrophysically insignificant situation of configurations near one of vanishing stress. In particular, "mountains on neutron stars", which are made possible by the presence of an elastic crust in neutron stars, can be treated using the techniques developed here.Comment: 29 page

Critical behavior in vacuum gravitational collapse in 4+1 dimensions

We show that the 4+1 dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the $(t,r)$ plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar Type II critical behavior at the threshold of black hole formation.Comment: 4 pages, 7 figure

How to bypass Birkhoff through extra dimensions (a simple framework for investigating the gravitational collapse in vacuum)

It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a 1+1 dimensional system of partial differential equations. Due to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay we describe a new cohomogeneity-two symmetry reduction of the vacuum Einstein equations in five and higher odd dimensions which evades Birkhoff's theorem and admits time dependent asymptotically flat solutions. We argue that this model provides an attractive 1+1 dimensional geometric setting for investigating the dynamics of gravitational collapse in vacuum.Comment: 7 pages, received "honorable mention" in 2006 Gravity Research Foundation essay contes

Helically symmetric N-particle solutions in scalar gravity

Within a scalar model theory of gravity, where the interaction between particles is given by the half-retarded + half-advanced solution of the scalar wave equation, we consider an N-body problem: we investigate configurations of N particles which form an equilateral N-angle and are in helical motion about their common center. We prove that there exists a unique equilibrium configuration and compute the equilibrium radius explicitly in a post-Newtonian expansion.Comment: 5 pages, 1 figure; minor corrections and changes; accepted for publication in Physical Review Letter

Codimension-two critical behavior in vacuum gravitational collapse

We consider the critical behavior at the threshold of black hole formation for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discretely self-similar solution with two unstable modes.Comment: 4 pages, 5 figures, typos correcte

Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains

We rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions

Helical symmetry in linear systems

We investigate properties of solutions of the scalar wave equation and Maxwell's equations on Minkowski space with helical symmetry. Existence of local and global solutions with this symmetry is demonstrated with and without sources. The asymptotic properties of the solutions are analyzed. We show that the Newman-Penrose retarded and advanced scalars exhibit specific symmetries and generalized peeling properties

Z-osteotomy in hallux valgus: clinical and radiological outcome after Scarf osteotomy

Correction osteotomies of the first metatarsal are common surgical approaches in treating hallux valgus deformities whereas the Scarf osteotomy has gained popularity. The purpose of this study was to analyze short- and mid-term results in hallux valgus patients who underwent a Scarf osteotomy. The subjective and radiological outcome of 131 Scarf osteotomies (106 hallux valgus patients, mean age: 57.5 years, range: 22–90 years) were retrospectively analyzed. Mean follow-up was 22.4 months (range: 6 months–5 years). Surgical indications were: intermetatarsal angle (IMA) of 12–23°; increased proximal articular angle (PAA>8°), and range of motion of the metatarsophalangeal joint in flexion and extension >40°. Exclusion criteria were severe osteoporosis and/or osteoarthritis. The mean subjective range of motion (ROM) of the great toe post-surgery was 0.8±1.73 points (0: full ROM, 10: total stiffness). The mean subjective cosmetic result was 2.7±2.7 points (0: excellent, 10: poor). The overall post-operative patient satisfaction with the result was high (2.1±2.5 points (0: excellent, 10: poor). The mean hallux valgus angle improvement was 16.6° (pre-operative mean value: 37.5°) which was statistically significant (p<0.01). The IMA improved by an average of 5.96° from a pre-operative mean value of 15.4° (p<0.01). Neither osteonecrosis of the distal fragment nor perioperative fractures were noted during the follow-up. In keeping with our follow-up results, the Scarf osteotomy approach shows potential in the therapy of hallux valgus