Surface Geometry of 5D Black Holes and Black Rings

TL;DR

This paper explores the geometric properties of five-dimensional black hole horizons, including black rings, by calculating invariants, embedding these surfaces in flat space, and relating them to four-dimensional black holes via Kaluza-Klein reduction.

Contribution

It provides new insights into the geometry of 5D black holes and black rings, including global embeddings and connections to 4D solutions through Kaluza-Klein reduction.

Findings

Calculated geometrical invariants of 5D horizons

Obtained global embeddings into flat space

Related 5D black rings to 4D black holes via Kaluza-Klein reduction

Abstract

We discuss geometrical properties of the horizon surface of five-dimensional rotating black holes and black rings. Geometrical invariants characterizing these 3D geometries are calculated. We obtain a global embedding of the 5D rotating black horizon surface into a flat space. We also describe the Kaluza-Klein reduction of the black ring solution (along the direction of its rotation) which relates this solution to the 4D metric of a static black hole distorted by the presence of external scalar (dilaton) and vector (`electromagnetic') field. The properties of the reduced black hole horizon and its embedding in $\E^3$ are briefly discussed.