Microstate Counting for rotating (type~II) isolated horizons

TL;DR

This paper proposes a novel method for counting black hole microstates in Loop Quantum Gravity for rotating horizons by decomposing the horizon into rings, enabling local Chern--Simons quantization and incorporating angular momentum.

Contribution

It introduces a local decomposition approach for rotating isolated horizons, allowing microstate counting with a consistent Chern--Simons framework in LQG.

Findings

Restores local CS description for rotating horizons

Includes angular momentum in microstate counting

Consistent with the first law of black hole mechanics

Abstract

We present a proposal for black hole microstate counting in Loop Quantum Gravity (LQG) for rotating (type~II) isolated horizons. The key obstacle in extending the standard nonrotating entropy derivation arises from the $\theta$-dependent rotation 1-form, which breaks the global Chern--Simons (CS) structure on the horizon. We propose a local decomposition of the horizon $S^2$ into narrow concentric rings, each approximated as a locally nonrotating patch with a constant effective CS level. Each ring is quantized independently using standard LQG techniques, and the total entropy is obtained by integrating over the entire horizon. This method restores a local CS description, includes the contribution of angular momentum, and is consistent with the first law of black hole mechanics.