Hyperelliptic Theta-Functions and Spectral Methods

A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for solutions of the Ernst equation where the branch points of the Riemann surface are parameterized by the physical coordinates. An exploration of the whole parameter space of the solution is thus only possible with an efficient code. The use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision. The case of almost degenerate Riemann surfaces is addressed. Tests of the numerics using identities for periods on the Riemann surface and integral identities for the Ernst potential and its derivatives are performed. It is shown that an accuracy of the order of machine precision can be achieved. These accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary Einstein equations. The resulting solution agrees with the theta-functional solution to very high precision.Comment: 25 pages, 12 figure

Local twistors and the conformal field equations

This note establishes the connection between Friedrich's conformal field equations and the conformally invariant formalism of local twistors.Comment: LaTeX2e Minor corrections of misprints et

Numerical evolution of axisymmetric, isolated systems in General Relativity

We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended `conformal' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary.Comment: 10 pages, 8 figures, uses revtex4, replaced with revised versio

On spin-(3/2) systems in Ricci flat space-times

The Dirac formulation of massless spin-(3/2) fields is discussed. The existence and uniqueness for the solutions of the spin-(3/2) field equations in Dirac form is proven. It is shown that the system of equations can be split into a symmetric hyperbolic system of evolution equations and a set of constraint equations. The constraints are shown to propagate on a curved manifold if and only if it is an Einstein space. The gauge freedom present in the spin-(3/2) system is discussed and it is shown that the complete system ``solutions modulo gauge'' has a well posed Cauchy problem if and only if the Einstein equations hold

From Now to Timelike Infinity on a Finite Grid

We use the conformal approach to numerical relativity to evolve hyperboloidal gravitational wave data without any symmetry assumptions. Although our grid is finite in space and time, we cover the whole future of the initial data in our calculation, including future null and future timelike infinity.Comment: 15 pages, 14 figures, revtex

On Killing vectors in initial value problems for asymptotically flat space-times

The existence of symmetries in asymptotically flat space-times are studied from the point of view of initial value problems. General necessary and sufficient (implicit) conditions are given for the existence of Killing vector fields in the asymptotic characteristic and in the hyperboloidal initial value problem (both of them are formulated on the conformally compactified space-time manifold)

A Fully Pseudospectral Scheme for Solving Singular Hyperbolic Equations

With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving hyperbolic equations. The calculations are carried out within the framework of conformally compactified space-times. In our formulation, the equation becomes singular at null infinity and yields regular boundary conditions there. In this manner it becomes possible to avoid "artificial" conditions at some numerical outer boundary at a finite distance. We obtain highly accurate numerical solutions possessing exponential spectral convergence, a feature known from solving elliptic PDEs with spectral methods. Our investigations are meant as a first step towards the goal of treating time evolution problems in General Relativity with spectral methods in space and time.Comment: 24 pages, 12 figure