A Numerical Method for Singular Two Point Boundary Value Problems

The numerical solution of boundary value problems for linear systems of first order equations with a regular singular point at one endpoint is considered. The standard procedure of expanding about the singularity to get a nonsingular problem over a reduced interval is justified in some detail. Quite general boundary conditions are included which permit unbounded solutions. Error estimates are given and some numerical calculations are presented to check the theory

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in $\mathbb{R}^3$ exhibit singular behavior from the edges and corners. We present quasi-optimal $hp$-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an $hp$-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.Comment: 41 pages, 11 figure

Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a Physical Review Letter, and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulae for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest non-static solutions (which are of genus two) to which the rigidly rotating dust disk belongs.Comment: 32 pages, 2 figures, uses pstricks.sty, updated version (October 7, 1998), to appear in Phys. Rev.

Who's Afraid of Naked Singularities?

To probe naked spacetime singularities with waves rather than with particles we study the well-posedness of initial value problems for test scalar fields with finite energy so that the natural function space of initial data is the Sobolev space. In the case of static and conformally static spacetimes we examine the essential self-adjointness of the time translation operator in the wave equation defined in the Hilbert space. For some spacetimes the classical singularity becomes regular if probed with waves while stronger classical singularities remain singular. If the spacetime is regular when probed with waves we may say that the spacetime is `globally hyperbolic.'Comment: 25 pages, 3 figures, Accepted for publication in Phys.Rev.

Gasdynamic wave interaction in two spatial dimensions

We examine the interaction of shock waves by studying solutions of the two-dimensional Euler equations about a point. The problem is reduced to linear form by considering local solutions that are constant along each ray and thereby exhibit no length scale at the intersection point. Closed-form solutions are obtained in a unified manner for standard gasdynamics problems including oblique shock waves, Prandtl–Meyer flow and Mach reflection. These canonical gas dynamical problems are shown to reduce to a series of geometrical transformations involving anisotropic coordinate stretching and rotation operations. An entropy condition and a requirement for geometric regularity of the intersection of the incident waves are used to eliminate spurious solutions. Consideration of the downstream boundary conditions leads to a formal determination of the allowable downstream matching criteria. By retaining the time-dependent terms, an approach is suggested for future investigation of the open problem of the stability of shock wave interactions

A Method for Calculating the Structure of (Singular) Spacetimes in the Large

A formalism and its numerical implementation is presented which allows to calculate quantities determining the spacetime structure in the large directly. This is achieved by conformal techniques by which future null infinity (\Scri{}^+) and future timelike infinity ($i^+$) are mapped to grid points on the numerical grid. The determination of the causal structure of singularities, the localization of event horizons, the extraction of radiation, and the avoidance of unphysical reflections at the outer boundary of the grid, are demonstrated with calculations of spherically symmetric models with a scalar field as matter and radiation model.Comment: 29 pages, AGG2

Singularities of the Casimir Energy for Quantum Field Theories with Lifshitz Dimensions

We study the singularities that the Casimir energy of a scalar field in spacetimes with Lifshitz dimensions exhibits, and provide expressions of the energy in terms of multidimensional zeta functions for the massless case. Using the zeta-regularization method, we found that when the 4-dimensional spacetime has Lifshitz dimensions, then for specific values of the critical exponents, the Casimir energy is singular, in contrast to the non-Lifshitz case. Particularly we found that when the value of the critical exponent is $z=2$, the Casimir energy is singular, while for $z\geq 3$ the Casimir energy is regular. In addition, when flat extra dimensions are considered, the critical exponents of the Lifshitz dimensions affect drastically the Casimir energy, introducing singularities that are absent in the non-Lifshitz case. We also discuss the Casimir energy in the context of braneworld models and the perspective of Lifshitz dimensions in such framework.Comment: Major Revision, Similar to Journal Versio