The BTZ black hole as a Lorentz-flat geometry

TL;DR

This paper demonstrates that 2+1 dimensional anti-de Sitter spacetimes, including the BTZ black hole, are Lorentz-flat, allowing their patches to have a locally pure gauge Lorentz connection, with implications for various 3D geometries.

Contribution

It establishes the Lorentz-flatness of 2+1D anti-de Sitter spacetimes and extends this property to broader classes of black holes and point particles in 3D.

Findings

Any simply-connected patch of BTZ black hole is Lorentz-flat.

The Lorentz-flatness property extends to other 3D black hole geometries.

The Lorentz connection can be locally expressed as a pure gauge.

Abstract

It is shown that 2+1 dimensional anti-de Sitter spacetimes are Lorentz-flat. This means, in particular, that any simply-connected patch of the BTZ black hole solution can be endowed with a Lorentz connection that is locally pure gauge. The result can be naturally extended to a wider class of black hole geometries and point particles in three-dimensional spacetime.