Related papers: Pathologies of hyperbolic gauges in general relati…
We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can…
An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime…
We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.
In hyperbolic reductions of the Einstein equations the evolution of gauge conditions or constraint quantities is controlled by subsidiary systems. We point out a class of non-linearities in these systems which may have the potential of…
With the aim of deriving symmetric hyperbolic free-evolution systems for GR that possess Hamiltonian structure and allow for the popular puncture gauge condition we analyze the hyperbolicity of Hamiltonian systems. We develop helpful tools…
Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations,…
After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of…
The majority of extensions to General Relativity display mathematical pathologies (higher derivatives, character change in equations that can be classified within PDE theory, and even unclassifiable ones) that cause severe difficulties to…
We consider gauge theories from the free evolution point of view, in which initial data satisfying constraints of a theory are given. Because the constraints are compatible with the field equations they remain so. We study a model…
The family of generalized-harmonic gauge conditions, which is currently used in Numerical Relativity for its singularity-avoidant behavior, is analyzed by looking for pathologies of the corresponding spacetime foliation. The appearance of…
The existence of gauge pathologies associated with the Bona-Masso family of generalized harmonic slicing conditions is proven for the case of simple 1+1 relativity. It is shown that these gauge pathologies are true shocks in the sense that…
We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.
We consider gauge vortices in symmetry breaking models with a non-canonical kinetic term. This work extends our previous study on global topological k-defects (hep-th/0608071), including a gauge field. The model consists of a scalar field…
By analogy to the theory of harmonic fields on the complex plane, we build the theory of wave-like fields on the plane of double variable. We construct the hyperbolic analogues of point vortices, sources, vortice-sources and their…
Testing General Relativity and exploring possible departures has received further input with the possibility to do so through gravitational waves emitted in strongly gravitating/highly dynamical scenarios and also through the availability…
This paper studies of a variation of the hyperbolic blow up scenario suggested by Hou and Luo's recent numerical simulation [12]. In particular, we propose a "hyperbolic" surface quasi-geostrophic equation characterized by a incompressible…
We study a class of two dimensional partially hyperbolic systems, not necessarily skew products, trying to establish the germ of a general theory. To illustrate the scope of the theory, we apply our results to the case of fast-slow…
The generalized harmonic representation of Einstein's equation is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity…
In this paper we shall analyse the structure of the Cauchy Problem (CP briefly) for General Relativity (GR briefly) by applying the theory of first order symmetric hyperbolic systems.
It is well-known that the $L^2$ metric on the moduli space of hyperbolic monopoles, defined using the Coulomb gauge-fixing condition, diverges. This article shows that an alternative gauge-fixing condition inspired by supersymmetry cures…