ES1 - Exact Solutions in Four and Higher Dimensions: Mathematical Aspects |
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Speaker |
Dobkowski-Rysko, Denis |
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Coauthors |
Dobkowski-Rylko, Denis; Kaminski, Wojciech; Pawlowski, Tomasz; Szereszewski, Adam |
| Talk Title |
Stationary isolated horizons of the Petrov type D |
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Abstract |
3-dimensional null surfaces that have local properties of Killing horizons are defined and called stationary to the second order isolated horizons (in short: stationary horizons). They are embedded in 4-dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Intrinsic geometry of stationary horizon consists of induced degenerate metric tensor and induced twist free connection. In the case of non-degenerate stationary horizon it determines its spacetime Weyl tensor. An assumption that the Weyl tensor is of the Petrov type D amounts to some complex equation on invariants of the intrinsic geometry. The equation is shown to be an integrability condition for the so called Near Horizon Geometry equation. The emergence of the Near Horizon Geometry in this context is equivalent to the hyper-suface orthogonality of the principal null direction of the Weyl tensor transversal to the horizon. In the case of bifurcated stationary horizon the type D equation implies the axial symmetry. In this way the axial symmetry is ensured without the rigidity theorem. For cross-sections of genus > 0 the Petrov type D equation is solved completely due to methods of the algebraic topology. For topologically spherical cross-sections the equation can be exactly solved in the axisymmetric case. The result is the subject to accompanying talk. |
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