Abstract
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.
Highlights
In this paper we consider the vacuum Einstein equations under the assumption of axisymmetry, i.e. there is an everywhere spacelike Killing vector field with closed orbits, ∂φ in spherical polar coordinates (t, r, θ, φ) adapted to the symmetry
Spherical harmonics are manifestly regular on the axis and take care of that issue automatically
One of our results is that the commonly used maximalisothermal gauge turns out to be incompatible with tensor harmonic expansions, and we introduce a new gauge that is better suited
Summary
In this paper we consider the vacuum Einstein equations under the assumption of axisymmetry, i.e. there is an everywhere spacelike Killing vector field with closed orbits, ∂φ in spherical polar coordinates (t, r, θ, φ) adapted to the symmetry. For the choice of spherical coordinates and spectral expansion in general relativity see for example [8, 9, 10]. In these proceedings we mainly focus on conceptual issues of the formulation, in particular the gauge choice. Because of the assumed twist-free axisymmetry, all quantities are φ-independent This implies in particular that the spherical harmonics reduce to the m = 0 harmonics. We obtain six evolution equations each for γij and Kij, a Hamiltonian constraint and three momentum constraints In this setting, twist-free axisymmetry means that all variables are φ-independent, γrφ = γθφ = 0 and βφ = 0. To fix the gauge we have to choose a slicing condition and two spatial gauge conditions
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
