Birkhoff's Theorem for Three-Dimensional AdS Gravity

TL;DR

This paper classifies all three-dimensional, matter-free spacetimes with negative cosmological constant that exhibit cyclic symmetry, revealing that only the BTZ black hole and specific self-dual solutions fit these criteria.

Contribution

It provides a complete classification of cyclic symmetric solutions in 3D AdS gravity, extending Birkhoff's theorem to this setting.

Findings

Identified the BTZ black hole as the unique cyclic solution with SO(2) x R symmetry.

Discovered self-dual Coussaert-Henneaux spacetimes with specific isometry groups.

Established the uniqueness of these solutions under the given symmetry conditions.

Abstract

All three-dimensional matter-free spacetimes with negative cosmological constant, compatible with cyclic symmetry are identified. The only cyclic solutions are the 2+1 (BTZ) black hole with SO(2) x R isometry, and the self-dual Coussaert-Henneaux spacetimes, with isometry groups SO(2) x SO(2,1) or SO(2) x SO(2).