Central Singularity of Three-Dimensional Kerr-de Sitter Black Holes

TL;DR

This work identifies the distributional central sources that generate the three-dimensional Kerr-de Sitter geometry by recasting 3D gravity as a Chern-Simons theory with gauge group $SO(1,3)$ and employing the non-abelian Stokes theorem. The holonomy around the central region fixes delta-function curvature and torsion terms, yielding a symmetric, conserved energy-momentum tensor and a spin tensor, interpreted as a spinning point mass with mass $m$ and angular velocity $\omega$. The authors compute observer-measured charges, obtaining the energy $E_o$ and angular momentum $J_o$ for a co-moving observer, and show the relation $E_o - \omega_o J_o = m$, linking local observables to the central source. These results illuminate the observable algebra for a local observer in Kerr-de Sitter space and may inform microscopic holographic descriptions in de Sitter spacetimes.

Abstract

For three-dimensional Kerr-de Sitter space-time, we find the singular energy-momentum and spin tensor sources that generate the non-trivial geometry. The energy-momentum tensor is symmetric, conserved and compatible with a spinning massive point particle whose mass and angular velocity we determine. The calculation is based on the analysis of the holonomy for a closed loop around the singularity of the $SO(1,3)$ Chern-Simons gauge field appropriate for gravity in the presence of a positive cosmological constant. This holonomy is related, via the non-abelian Stokes theorem, to the singular source terms at the center. Our results may be helpful for a better understanding of the algebra of observables of a local observer in the Kerr-de Sitter space-time.