Spectral approach to axisymmetric evolution of Einstein's equations

Spanish Relativity Meeting: almost 100 years after Einstein's revolution

Schell, Christian^1 ; Rinne, Oliver^2

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Golm

14476, Germany^1

Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 26, Berlin

14195, Germany^2

关键词: Axis of symmetry;    Computational advantages;    Coordinate singularity;    Einstein's equations;    Harmonic expansion;    Spherical harmonics;    Spherical polar coordinates;    Vanishing twist;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/600/1/012060/pdf
DOI  :  10.1088/1742-6596/600/1/012060

来源: IOP

【 摘 要 】

We present a new formulation of Einstein's equations for an axisymmetric spacetime with vanishing twist in vacuum. We propose a fully constrained scheme and use spherical polar coordinates. A general problem for this choice is the occurrence of coordinate singularities on the axis of symmetry and at the origin. Spherical harmonics are manifestly regular on the axis and hence take care of that issue automatically. In addition a spectral approach has computational advantages when the equations are implemented. Therefore we spectrally decompose all the variables in the appropriate harmonics. A central point in the formulation is the gauge choice. One of our results is that the commonly used maximal- isothermal gauge turns out to be incompatible with tensor harmonic expansions, and we introduce a new gauge that is better suited. We also address the regularisation of the coordinate singularity at the origin.