257 research outputs found
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
Interactive visualization of a thin disc around a Schwarzschild black hole
In the first course of general relativity, the Schwarzschild spacetime is the
most discussed analytic solution to Einstein's field equations. Unfortunately,
there is rarely enough time to study the optical consequences of the bending of
light for some advanced examples. In this paper, we present how the visual
appearance of a thin disc around a Schwarzschild black hole can be determined
interactively by means of an analytic solution to the geodesic equation
processed on current high performance graphical processing units. This approach
can, in principle, be customized for any other thin disc in a spacetime with
geodesics given in closed form. The interactive visualization discussed here
can be used either in a first course of general relativity for demonstration
purposes only or as a thesis for an enthusiastic student in an advanced course
with some basic knowledge of OpenGL and a programming language.Comment: 9 pages, 4 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
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