Universal properties of the near-horizon optical geometry

Abstract

Making use of the fact that the optical geometry near a static nondegenerate Killing horizon is asymptotically hyperbolic, we investigate some universal features of black-hole horizons. Applying the Gauss-Bonnet theorem allows us to establish some general properties of gravitational lensing, valid for all black holes. Hyperbolic geometry allows us to find rates for the loss of scalar, vector, and fermionic ``hair'' as objects fall quasistatically towards the horizon, extending previous results for Schwarzschild to all static Killing horizons. In the process we find the Li\'enard-Wiechert potential for hyperbolic space and calculate the force between electrons mediated by neutrinos, extending the flat space result of Feinberg and Sucher. We further demonstrate how these techniques allow us to derive the exact Copson-Linet potential due to a point charge in a Schwarzschild background in a simple fashion.