Seminar

# Isolated horizons, near horizon geometries and the Petrov type D equation

#### General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein

#### 10 December 10:00 - 11:00

#### Jerzy Lewandowski - University of Warsaw

3-dimensional null surfaces that are Killing horizons to the second order are considered. They are embedded in 4-dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Internal geometry of 2-dimensional spacial sections of the horizons consists of induced metric tensor and a rotation 1-form potential. We impose on it the type D equation. The equation is interesting from both, mathematical and physical points of view. Mathematically it involves geometry, holomorphic structures and algebraic topology. Physically, the equation knows the secret of black holes: the only axisymmetric solutions on topological sphere correspond to the the Kerr / Kerr– de Sitter / Kerr–anti de Sitter non-extremal black holes or to the near horizon limit of the extremal ones. In case of the bifurcated horizons the type D equation implies another spacial symmetry. In this way the axial symmetry may be ensured without the rigidity theorem. The type D equation does not allow rotating horizons of topology different then that of the sphere (or its quotient). That completes a local no-hair theorem. The type D equation is also an integrability condition for the Near Horizon Geometry equation and leads to new results on the solution existence issue. The NHG equation is the equation of extremal isolated horizons (Killing horizons to the 1st order). We present also the second equation satisfied by the geometry of IHs if they admit an embedding as extremal Killing horizons. The extremal horizons equations are important for the black hole existence/uniqueness issues. The type D equation can be also considered for horizons that have the Hopf (or more general Dirac monopol) fibration structure. Our new results in that case lead to a new class of the Kerr - (A)dS - NUT spacetimes.

Organizers

Lars Andersson

Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Mattias Dahl

KTH Royal Institute of Technology

Philippe G. LeFloch

Sorbonne University

Richard Schoen

University of California, Irvine