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

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.

💡 Research Summary

The paper investigates the microscopic source that underlies the three‑dimensional Kerr‑de Sitter (KdS) geometry, focusing on the singularity at the origin (r = 0). Although the KdS metric is a vacuum solution of Einstein’s equations with a positive cosmological constant, the non‑trivial topology implies that a distributional source must be present at the centre. The authors adopt the first‑order (vielbein‑spin‑connection) formulation of gravity, where the fundamental fields are the dreibein e⁽ᵃ⁾ and the Lorentz connection ω⁽ᵃᵇ⁾. From the Einstein–Hilbert action with Λ > 0 they derive the field equations for the curvature R⁽ᵃᵇ⁾ and torsion T⁽ᵃ⁾, together with the associated energy‑momentum tensor T^{ab} and spin tensor S^{cab}. In three dimensions the Weyl tensor vanishes, so the full Riemann tensor is algebraically fixed by the Ricci tensor; this simplification is repeatedly used throughout the analysis.

To expose the hidden source, the authors recast three‑dimensional de Sitter gravity as an SO(1,3) Chern‑Simons gauge theory. The gauge field is defined as
A = ½ ω^{ab} J_{ab} + ℓ^{‑1} e^{a} J_{a3},
with field strength
F = dA + A∧A = ½ E^{ab} J_{ab} + ℓ^{‑1} T^{a} J_{a3}.
In regions where the vacuum equations hold, F vanishes; thus any non‑zero contribution must be localized at singular points. The key tool is the non‑abelian Stokes theorem, which relates the Wilson loop U