Geometric Properties of Stationary and Axisymmetric Killing Horizons

TL;DR

This paper investigates the geometric properties of Killing horizons in 4D stationary, axisymmetric spacetimes with electromagnetic fields and cosmological constant, deriving relations between curvature invariants on the horizon surface.

Contribution

It generalizes known relations for static and axisymmetric horizons to more complex stationary, axisymmetric spacetimes with electromagnetic fields and cosmological constant.

Findings

Derived relations between spacetime and horizon surface Riemann tensor components.

Expressed scalar curvature invariants on the horizon surface.

Generalized Hartle's curvature formula for these spacetimes.

Abstract

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.