Adding initial data

Writing an Initial Data thorn for the Einstein Toolkit

This page explains how to write a new initial data thorn for the Einstein Toolkit. These initial data will set up ADMBase or HydroBase variables, so that they are usable for all other thorns that are based Einstein Toolkit.

Introduction

You need:

• a prescription or routine that actually generates the initial data,
• to decide whether this are initial data for the spacetime, for hydrodynamics, or both,
• these instructions.

Example

Let us assume, for the sake of simplicity, that we are going to set up initial data for the spacetime only. Let us pick a concrete example: the Kruskal coordinates. The Kruskal coordinates describe a single, static black hole (the Schwarzschild spacetime), and their advantage over many other coordinate systems is that they cover the whole, extended Schwarzschild spacetime (i.e. including both asymptotically flat ends, the worm hole, the black and white hole horizons), and that they do not introduce coordinate singularities. Of course there are curvature singularities inside the white and black holes, but there are no "artificial" singularities, e.g. at the horizons. The Krukal coordinates are beautifully described in Sean M. Carroll's "A No-Nonsense Introduction to General Relativity", 2001, at http://pancake.uchicago.edu/~carroll/notes/grtinypdf.pdf .

To make things a bit more interesting, we are also going to apply a coordinate transformation to the Kruskal coordinates, so that we can evaluate them on arbitrary slices. Of course that means that the actual coordinates are not the Kruskal coordinates. This means that the slicing can in principle be chosen arbitrarily within the extended Schwarzschild spacetime. Other coordinates (e.g. ingoing Eddington-Finkelstein coordinates) are regular only a in part of the spacetime. This will allow us to examine arbitrary slicings of Schwarzschild -- even slicings that are not spherically symmetric -- which is very interesting when one studies trapped surfaces or apparent horizons.

In particular, we want to implement the following slicings:

• Kruskal
• Schwarzschild
• the Wald-Iyer slicing introduced in Phys. Rev. D 44, R3719 (1991), "Trapped surfaces in the Schwarzschild geometry and cosmic censorship"

As described in this paper, this slicing has the property that it comes arbitrarily close to the (future) curvature singularity in the Schwarzschild spacetime, but does not contain any apparent horizons. We want to examine this numerically.