# Jerzy Lewandowski: Isolated horizons, the Petrov type D equation and the Near Horizon Geometry equation

**Time:
**
Tue 2018-10-16 10.15

**Lecturer: **
Jerzy Lewandowski

**Location: **
Room 3418, Lindstedtsvägen 25, 4th floor, Department of mathematics, KTH

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

cross sections of the horizons consists of induced metric tensor

and a rotation 1-form potential. It is subject to the type D equation.

The equation is interesting from both the mathematical and physical

points of view. Mathematically it involves geometry, holomorphic

structures and algebraic topology. Physically, the equation knows the

secrete of black holes: the only axisymmetric solutions on topological

sphere correspond to the Kerr / Kerr-de Sitter /Kerr-anti-de-Sitter

non-extremal black holes or to the near horizon limit of the extremal

ones. In the case of 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 than that of

the sphere (or its quotient). That completes a new 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.