Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$\Delta u = -\chi_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss

The time evolution of marginally trapped surfaces

In previous work we have shown the existence of a dynamical horizon or marginally trapped tube (MOTT) containing a given strictly stable marginally outer trapped surface (MOTS). In this paper we show some results on the global behavior of MOTTs assuming the null energy condition. In particular we show that MOTSs persist in the sense that every Cauchy surface in the future of a given Cauchy surface containing a MOTS also must contain a MOTS. We describe a situation where the evolving outermost MOTS must jump during the coalescence of two seperate MOTSs. We furthermore characterize the behavior of MOTSs in the case that the principal eigenvalue vanishes under a genericity assumption. This leads to a regularity result for the tube of outermost MOTSs under the genericity assumption. This tube is then smooth up to finitely many jump times. Finally we discuss the relation of MOTSs to singularities of a space-time.Comment: 21 pages. This revision corrects some typos and contains more detailed proofs than the original versio

A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic Equations with Applications to Lorentzian and Riemannian Geometry

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.Comment: 37 pages, 1 figure, ams-latex using eepi

Seismology of adolescent neutron stars: Accounting for thermal effects and crust elasticity

We study the oscillations of relativistic stars, incorporating key physics associated with internal composition, thermal gradients and crust elasticity. Our aim is to develop a formalism which is able to account for the state-of-the-art understanding of the complex physics associated with these systems. As a first step, we build models using a modern equation of state including composition gradients and density discontinuities associated with internal phase-transitions (like the crust-core transition and the point where muons first appear in the core). In order to understand the nature of the oscillation spectrum, we carry out cooling simulations to provide realistic snapshots of the temperature distribution in the interior as the star evolves through adolescence. The associated thermal pressure is incorporated in the perturbation analysis, and we discuss the presence of $g$-modes arising as a result of thermal effects. We also consider interface modes due to phase-transitions and the gradual formation of the star's crust and the emergence of a set of shear modes.Comment: 27 pages, 14 figure

The Feynman-Wilson gas and the Lund model

We derive a partition function for the Lund fragmentation model and compare it with that of a classical gas. For a fixed rapidity ``volume'' this partition function corresponds to a multiplicity distribution which is very close to a binomial distribution. We compare our results with the multiplicity distributions obtained from the JETSET Monte Carlo for several scenarios. Firstly, for the fragmentation vertices of the Lund string. Secondly, for the final state particles both with and without decays.Comment: Latex, 21+1 pages, 11 figure

The Cosmological Time Function

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere.Comment: 19 pages, AEI preprint, latex2e with amsmath and amsth

Equilibrium spin pulsars unite neutron star populations

Many pulsars are formed with a binary companion from which they can accrete matter. Torque exerted by accreting matter can cause the pulsar spin to increase or decrease, and over long times, an equilibrium spin rate is achieved. Application of accretion theory to these systems provides a probe of the pulsar magnetic field. We compare the large number of recent torque measurements of accreting pulsars with a high-mass companion to the standard model for how accretion affects the pulsar spin period. We find that many long spin period (P > 100 s) pulsars must possess either extremely weak (B < 10^10 G) or extremely strong (B > 10^14 G) magnetic fields. We argue that the strong-field solution is more compelling, in which case these pulsars are near spin equilibrium. Our results provide evidence for a fundamental link between pulsars with the slowest spin periods and strong magnetic fields around high-mass companions and pulsars with the fastest spin periods and weak fields around low-mass companions. The strong magnetic fields also connect our pulsars to magnetars and strong-field isolated radio/X-ray pulsars. The strong field and old age of our sources suggests their magnetic field penetrates into the superconducting core of the neutron star.Comment: 6 pages, 4 figures; to appear in MNRA

R-mode oscillations and rocket effect in rotating superfluid neutron stars. I. Formalism

We derive the hydrodynamical equations of r-mode oscillations in neutron stars in presence of a novel damping mechanism related to particle number changing processes. The change in the number densities of the various species leads to new dissipative terms in the equations which are responsible of the {\it rocket effect}. We employ a two-fluid model, with one fluid consisting of the charged components, while the second fluid consists of superfluid neutrons. We consider two different kind of r-mode oscillations, one associated with comoving displacements, and the second one associated with countermoving, out of phase, displacements.Comment: 10 page

Non-radial oscillation modes as a probe of density discontinuities in neutron stars

A phase transition occurring in the inner core of a neutron star could be associated to a density discontinuity that would affect the frequency spectrum of the non-radial oscillation modes in two ways. Firstly, it would produce a softening of the equation of state, leading to more compact equilibrium configurations and changing the frequency of the fundamental and pressure modes of the neutron star. Secondly, a new non-zero frequency g-- mode would appear, associated to each discontinuity. These discontinuity g--modes have typical frequencies larger than those of g--modes previously studied in the literature (thermal, core g-- modes, or g--modes due to chemical inhomogeneities in the outer layers), and smaller than that of the fundamental mode; therefore they should be distinguishable from the other modes of non radial oscillation. In this paper we investigate how high density discontinuities change the frequency spectrum of the non-radial oscillations, in the framework of the general relativistic theory of stellar perturbations. Our purpose is to understand whether a gravitational signal, emitted at the frequencies of the quasi normal modes, may give some clear information on the equation of state of the neutron star and, in particular, on the parameters that characterize the density discontinuity. We discuss some astrophysical processes that may be associated to the excitation of these modes, and estimate how much gravitational energy should the modes convey to produce a signal detectable by high frequency gravitational detectors.Comment: submitted to MNRA