Axisymmetric metrics in arbitrary dimensions

TL;DR

The paper develops a unified framework for axisymmetric static spacetimes in arbitrary dimensions (Weyl metrics), revealing a two-dimensional dilaton gravity reduction with a Liouville potential and uncovering a duality that connects vacuum internal-curvature solutions to Λ-backgrounds without internal curvature. It classifies solutions into three canonical classes (I–III) and provides explicit higher-dimensional generalizations of Weyl and C-metric-like spacetimes, plus novel braneworld black hole constructions via AdS slicing. The work demonstrates how lower-dimensional dilaton solutions map to higher-dimensional axisymmetric geometries, clarifies the role of internal curvature versus cosmological constant, and offers practical solution-generation tools, while also highlighting significant obstacles (notably horizon singularities) in extending the C-metric beyond four dimensions. These results advance the understanding of braneworld cosmology and AdS/CFT-inspired spacetimes, with implications for black holes localized on branes and higher-dimensional gravity phenomenology.

Abstract

We consider axially symmetric static metrics in arbitrary dimension, both with and without a cosmological constant. The most obvious such solutions have an SO(n) group of Killing vectors representing the axial symmetry, although one can also consider abelian groups which represent a flat `internal space'. We relate such metrics to lower dimensional dilatonic cosmological metrics with a Liouville potential. We also develop a duality relation between vacuum solutions with internal curvature and those with zero internal curvature but a cosmological constant. This duality relation gives a solution generating technique permitting the mapping of different spacetimes. We give a large class of solutions to the vacuum or cosmological constant spacetimes. We comment on the extension of the C-metric to higher dimensions and provide a novel solution for a braneworld black hole.