Jörg Frauendiener

Triads and the Witten equation

It is shown that the Witten spinor, i.e. a solution to the Witten equation on a maximal spacelike hypersurface S of a four-dimensional spacetime M, is completely equivalent to a set of three mutually orthogonal divergence-free vector fields of equal length on S that satisfy an additional cyclic condition. Together with the surface normal these define a tetrad with respect to which the Sparling 3-form is timelike.

Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. 1. The Conformal field equations

This is the first in a series of articles on the numerical solution of Friedrichs conformal field equations for Einsteins theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to physical problems and why there is good hope that this might even be a good idea from the numerical point of view. We describe in detail the derivation of the conformal field equations in the spinor formalism which we use for the implementation of the equations, and present all the equations as a reference for future work. Finally, we discuss the implications of the assumptions of a continuous symmetry.

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP Solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev–Petviashvili and the Korteweg–de Vries equations. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.

Discrete differential forms in general relativity

A major obstacle in the numerical simulation of general relativistic spacetimes is the fact that coordinates have to be specified in order to obtain a well-defined numerical evolution. While the choice of coordinates has no impact on the geometry, it does influence the evolution in ways which are rather difficult to predict. For this reason it might be useful to devise numerical procedures which are manifestly coordinate invariant. In this paper we present an alternative way to obtain discrete versions of the Einstein equations. We formulate the Einstein equations as an exterior system in terms of differential forms. Such a formulation can be used to interpret the variables and equations in a discrete setting as discrete differential forms. We describe the basic equations and discuss the accuracy of the discrete equations.

Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. III. On the determination of radiation

We discuss the issue of radiation extraction in asymptotically flat spacetimes within the framework of conformal methods for numerical relativity. Our aim is to show that there exists a well defined and accurate extraction procedure which mimics the physical measurement process. It operates entirely intrinsically within + so that there is no further approximation necessary apart from the basic assumption that the arena should be an asymptotically flat spacetime. We define the notion of a detector at infinity by idealizing local observers in Minkowski space. A detailed discussion is presented for Maxwell fields and the generalization to linearized and full gravity is performed by way of the similar structure of the asymptotic fields.

A note on the relativistic Euler equations

We present an alternative way to write the Euler equations for an ideal isentropic fluid as a symmetric hyperbolic system of evolution equations for a timelike vector field. An equation of state has to be provided which relates the length of this vector to the sound velocity. The relation to the conventional formulation is established and some of the consequences are discussed.

Some Aspects of the Numerical Treatment of the Conformal Field Equations

This article discusses some of the numerical issues which become relevant in the numerical computation of initial data for the conformal field equations. In particular, the problem is addressed of how to obtain a solution of the Lichnerowicz-Yamabe-Equation which extends as smoothly as possible beyond the boundary. Another problem which is discussed is the division process by which some of the initial data have to be constructed.

Calculating initial data for the conformal Einstein equations by pseudo-spectral methods

Abstract We present a numerical scheme for determining hyperboloidal initial data sets for the conformal field equations by using pseudo-spectral methods. This problem is to split into two parts. The first step is the determination of a suitable conformal factor which transforms from an initial data set in physical space–time to a hyperboloidal hypersurface in the ambient conformal manifold. This is achieved by solving the Yamabe equation, a nonlinear second-order equation. The second step is a division by the conformal factor of certain fields which vanish on I , the zero set of the conformal factor. The challenge there is to numerically obtain a smooth quotient. Both parts are treated by pseudo-spectral methods. The nonlinear equation is solved iteratively while the division problem is treated by transforming the problem to the coefficient space, solving it there by the QR-factorisation of a suitable matrix, and then transforming back. These hyperboloidal initial data can be used to generate general relativistic space–times by evolution with the conformal field equations.

Discretizations of axisymmetric systems

In this paper we discuss stability properties of various discretizations for axisymmetric systems including the so called cartoon method which was proposed by Alcubierre, Brandt et.al. for the simulation of such systems on Cartesian grids. We show that within the context of the method of lines such discretizations tend to be unstable unless one takes care in the way individual singular terms are treated. Examples are given for the linear axisymmetric wave equation in flat space.

The kernel of the edth operators on higher-genus spacelike 2-surfaces

The dimension of the kernels of the edth and edth-prime operators on closed, orientable spacelike 2-surfaces with arbitrary genus is calculated, and some of its mathematical and physical consequences are discussed.