Geometric Properties of Stationary and Axisymmetric Killing Horizons

We study some geometric properties of Killing horizons in 4-dimensional stationary and axisymmetric space-times with electromagnetic field and cosmological constant. Using a $(1+1+2)$ space-time split, we construct relations between the space-time Riemann tensor components and components of the Riemann tensor corresponding to the horizon surface. The Einstein equations allow to derive the space-time scalar curvature invariants, Kretschmann, Chern-Pontryagin, and Euler, on the 2-dimensional spacelike horizon surface. The derived relations generalize the relations known for Killing horizons of static and axisymmetric 4-dimensional space-times. We also present the generalization of Hartle’s curvature formula.