Avenues for Analytic exploration in Axisymmetric Spacetimes. Foundations and the Triad Formalism
arXiv:1303.1919 · doi:10.1103/PhysRevD.88.044039
Abstract
Axially symmetric spacetimes are the only models for isolated systems with continuous symmetries that also include dynamics. For such systems, we review the reduction of the vacuum Einstein field equations to their most concise form by dimensionally reducing to the three-dimensional space of orbits of the Killing vector, followed by a conformal rescaling. The resulting field equations can be written as a problem in three-dimensional gravity with a complex scalar field as source. This scalar field, the Ernst potential is constructed from the norm and twist of the spacelike Killing field. In the case where the axial Killing vector is twist-free, we discuss the properties of the axis and simplify the field equations using a triad formalism. We study two physically motivated triad choices that further reduce the complexity of the equations and exhibit their hierarchical structure. The first choice is adapted to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a simplification of the three-dimensional Ricci tensor and the boundary conditions on the axis. We illustrate its properties by explicitly solving the field equations in the case of static axisymmetric spacetimes. The other choice of triad is based on geodesic null coordinates adapted to null infinity as in the Bondi formalism. We then explore the solution space of the twist-free axisymmetric vacuum field equations, identifying the known (unphysical) solutions together with the assumptions made in each case. This singles out the necessary conditions for obtaining physical solutions to the equations.
40 pages, 3 figures, reflects published version
References in corpus (11)
- LIGO: The Laser Interferometer Gravitational-Wave Observatory
- Phenomenological template family for black-hole coalescence waveforms
- Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-spinning, equal-mass black holes
- Accurate Effective-One-Body waveforms of inspiralling and coalescing black-hole binaries
- Understanding the "anti-kick" in the merger of binary black holes
- Nonlinear Gravitational Recoil from the Mergers of Precessing Black-Hole Binaries
- Hyperboloidal Einstein-matter evolution and tails for scalar and Yang-Mills fields
- Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction
- Head-on collisions of unequal mass black holes in D=5 dimensions
- On critical collapse of gravitational waves
- The inequality between mass and angular momentum for axially symmetric black holes