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

Bouncing Palatini cosmologies and their perturbations

Nonsingular cosmologies are investigated in the framework of f(R) gravity within the first order formalism. General conditions for bounces in isotropic and homogeneous cosmology are presented. It is shown that only a quadratic curvature correction is needed to predict a bounce in a flat or to describe cyclic evolution in a curved dust-filled universe. Formalism for perturbations in these models is set up. In the simplest cases, the perturbations diverge at the turnover. Conditions to obtain smooth evolution are derived.Comment: 7 pages, 1 figure. v2: added references

Comparison and Rigidity Theorems in Semi-Riemannian Geometry

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-dimensional rigidity results and using an inductive technique, a new class of gap-type rigidity theorems is proved for semi-Riemannian manifolds of arbitrary index, generalizing those first given by Gromov and Greene-Wu. As applications we prove rigidity results for semi-Riemannian manifolds with simply connected ends of constant curvature.Comment: 46 pages, amsart, to appear in Comm. Anal. Geo

A covariant action principle for dissipative fluid dynamics: From formalism to fundamental physics

We present a new variational framework for dissipative general relativistic fluid dynamics. The model extends the convective variational principle for multi-fluid systems to account for a range of dissipation channels. The key ingredients in the construction are i) the use of a lower dimensional matter space for each fluid component, and ii) an extended functional dependence for the associated volume forms. In an effort to make the concepts clear, the formalism is developed in steps with the model example of matter coupled to heat considered at each level. Thus we discuss a model for heat flow, derive the relativistic Navier-Stokes equations and discuss why the individual dissipative stress tensors need not be spacetime symmetric. We argue that the new formalism, which notably does not involve an expansion away from an assumed equilibrium state, provides a conceptual breakthrough in this area of research and provide an ambitious list of directions in which one may want to extend it in the future. This involves an exciting set of problems, relating to both applications and foundational issues.Comment: 21 pages RevTex, 3 pdf figures, matches the published version. arXiv admin note: text overlap with arXiv:1107.1005 by other author

The dynamics of neutron star crusts: Lagrangian perturbation theory for a relativistic superfluid-elastic system

The inner crust of a mature neutron star is composed of an elastic lattice of neutron-rich nuclei penetrated by free neutrons. These neutrons can flow relative to the crust once the star cools below the superfluid transition temperature. In order to model the dynamics of this system, which is relevant for a range of problems from pulsar glitches to magnetar seismology and continuous gravitational-wave emission from rotating deformed neutron stars, we need to understand general relativistic Lagrangian perturbation theory for elastic matter coupled to a superfluid component. This paper develops the relevant formalism to the level required for astrophysical applications.Comment: 31 pages, double spacing, minor typos fixe

Blowup of Jang's equation at outermost marginally trapped surfaces

The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence near a strictly stable MOTS.Comment: 15 pages. This revision corrects some typo

The nonlinear development of the relativistic two-stream instability

The two-stream instability has been mooted as an explanation for a range of astrophysical applications from GRBs and pulsar glitches to cosmology. Using the first nonlinear numerical simulations of relativistic multi-species hydrodynamics we show that the onset and initial growth of the instability is very well described by linear perturbation theory. In the later stages the linear and nonlinear description match only qualitatively, and the instability does not saturate even in the nonlinear case by purely ideal hydrodynamic effects.Comment: 15 pages, 9 figure