Principal Investigators: Jörg Frauendiener
The work within project A2 over the last years has been concerned to a large part with the investigation of various general fundamental issues for the discretisation of the Einstein equation motivated by the growing experience with the conformal field equations approach. In the application we have outlined five topics of research. The topics are a) the construction of initial data, b) the implementation of an evolution code, c) the investigation of compatible boundary conditions, d) the problem of the constraint propagation and e) the choice of gauge. It should be stressed that most of these topics are not specific to the conformal approach but appear in almost any formulation of the Einstein equations.
The numerical construction of initial data is based on work by Anderson, Chrusciel and Friedrich, who have described a method for solving the Lichnerowicz-Yamabe equation in the context of hyperboloidal initial hypersurfaces. This method is currently being implemented by Andreas King who will finish his PhD dissertation within this year. In a further development M. Pareja has started to consider the so called `thin-sandwich conjecture' which provides a different way to construct initial data for the Einstein equations. While this method is ill-posed on asymptotically euclidean hypersurfaces it turns out that it leads to well-posed equations on hyperboloidal hypersurfaces. Investigation of these equations in the context of spherically symmetric space-times has been started.
The implementation of the evolution module has been the second major task within this project which was carried out by Raffaele Rani. The work in this area has progressed to the stage where the conformal field equations in a frame formalism have been coded using the symbolic package KRANC. This was developed at the AEI. The need to deal with frame vectors turned out to be a major challenge for KRANC but also a good test for this software. The KRANC module for the conformal field equations is now available and can be used to generate a Cactus thorn.
The third issue concerns boundary conditions. It is a major problem in any evolution code that boundary conditions are not necessarily compatible with the time evolution scheme. While this is not such a big problem for the conformal approach because the boundary of the computational domain is in the unphysical region of the conformal manifold this is a problem for standard approaches which is largely ignored. In a diploma thesis and continuing with a PhD thesis M. Hary investigated the initial boundary value problem for the conformal field equations. Unfortunately, due to the lack of perspective within project A2 he did not finish the dissertation but tried to find a position outside academia.
In a diploma thesis Tilman Vogel analysed the Weyl subsystem to find mechanisms for constraint violation. The surprising answer is that constraint violation depends strongly on the gauge conditions. Using freezing of coefficients and Fourier transforms one can reformulate the so called subsidiary system which gives information about the behaviour of the constraint variables. The resulting system of ODEs can be analysed with well-established methods. The Routh-Hurwitz criterion for determination of the sign of the eigen-values of a matrix has been used to get information about the spectrum of the propagation matrix. It turns out that the geometry of the space-like slices is crucial for the stability of the constraint propagation. In particular, one finds that hyperboloidal slices are superior to asymptotically euclidean ones. Since the geometry of these slices is largely determined by the choice of the time coordinate, this result shows clearly, that the issue of gauge cannot be treated separately but is intimately connected to all the other issues in the numerical implementation of the Einstein equations. Having finished his Diploma thesis in Tübingen, T. Vogel went on to the AEI to work on related issues within project A1. In the meantime, he has developed a code to demonstrate his theoretical findings in a numerical evolution.
The experience gained within project A2 indicates that the numerical implementation of the Einstein equations should strive to benefit from the so called structure preserving discretisation methods. Motivated by modern aspects in computational electrodynamics and discrete geometry we developed an entirely new way to discretise the Einstein equation. The method relies on the formulation of the field equations as an exterior system and uses so called discrete differential forms. Here one tries to get rid of coordinates by discretising geometrically different object in a geometrically natural way, thus maintaining as much as possible the meaning of the various quantities. Meanwhile, this line of research has been diverted into project B9 of the SFB 382 at Tübingen.
Another line of thought motivated by the issue of constraint propagation is based on the fact that evolution and constraint equations arise from the variation of a singular Lagrangian density. So it is obvious, that the direct discretisation of the action should result in a system of compatible discrete evolution and constraint equations. We tried to develop model systems to investigate this idea further and to see whether one can make use of this in such complicated issues as the Einstein equations. Meanwhile, also this idea has found its home as project B5 of the SFB 382 at Tübingen.
It has been stressed above that the issues facing any approach to the numerical implementation of the vacuum Einstein equations are very similar. Therefore, the projects A1, A2 and B5 have collaborated on various occasions such as the common organisation of workshops, exchange of students and informal discussions. The above mentioned work of Vogel is a prime example for the cooperation between the projects A1 and A2 which has been very close. The connections to the B-projects in Tübingen has been maintained in common seminars and informal gatherings such as the annual Oberjoch seminars which have developed over the years from a work group gathering to a workshop for the research groups based at Tübingen, Golm and Jena. A new formulation for the Euler equations has been found and raised so much interest in the hydrodynamics community that the paper was elected as a research highlight 2004 by the IOP. It is hoped that this formulation is useful also for the projects concerned with source modeling.
|Joerg Frauendiener||Senior, PI 2003-2006|
|Andreas King||Student, 2003-2006|
|Raffaele Rani||Student, 2003-2006|
Discrete Differential Forms in General Relativity
J. Frauendiener, Class. Quantum Grav. 23, S369-S385 (2006)
Asymptotic structure and conformal infinity
J. Frauendiener, In M. Ruck (ed.), Encyclopedia in Mathematical Physics (Elsevier, 2006)
The computational aspects of General Relativity
J. Frauendiener, In Proceedings 2nd Russian-German Advanced Research Workshop on Computational Science and High Performance Computing (Springer Verlag, 2006)
Application of Discrete Differential Forms to Spherically Symmetric Systems in General Relativity
R. Richter, J. Frauendiener and M. Vogel, ePrint (2006)
Constant scalar curvature hypersurfaces in extended Schwarzschild space-time
M. Pareja and J. Frauendiener, Phys.Rev. D74 (2006) 044026
Algebraic stability analysis of constraint propagation
J. Frauendiener and T. Vogel, Class. Quantum Grav. 22, 1769-1793 (2005)
J. Frauendiener, Living Rev. Relativity 7 (2004)
Current issues in numerical relativity
J. Frauendiener, In I. Racz (ed.), Proceedings of the 6th Hungarian Relativity Meeting (2004)
Note on the relativistic Euler equations
J. Frauendiener, Class. Quantum Grav. 20(14), 193-196 (2003)
Zum Anfangsrandwertproblem der konformen Feldgleichungen
M. Hary, Diplomarbeit, Universität Tübingen, 2003
Diskrete Differentialformen in der Allgemeinen Relativitätstheorie am Beispiel der Schwarzschild-Raumzeit in Kruskal-Koordinaten
M. Vogel, Diplomarbeit, Universit"at T"ubingen, 2004
Kugelsymmetrische Einstein-Yang-Mills Systeme auf deSitter-artigen Mannigfaltigkeiten
R. Peter, Diplomarbeit, Universität Stuttgart, 2005