Kerr Isolated Horizons in Ashtekar and Ashtekar-Barbero Connection Variables

TL;DR

This paper derives and analyzes the Ashtekar and Ashtekar-Barbero connection variables for Kerr isolated horizons, revealing differences from Schwarzschild horizons and proposing a method to modify the connection for compatibility with loop quantum gravity boundary conditions.

Contribution

It introduces a new method to construct a connection with curvature satisfying horizon boundary conditions for Kerr isolated horizons, extending previous formulations.

Findings

Derived Ashtekar and Ashtekar-Barbero connections for Kerr horizons.

Identified that the Ashtekar-Barbero curvature does not meet Schwarzschild boundary conditions.

Proposed a method to modify the connection to satisfy horizon boundary conditions in Kerr cases.

Abstract

The Ashtekar and Ashtekar-Barbero connection variable formulations of Kerr isolated horizons are derived. Using a regular Kinnersley tetrad in horizon-penetrating Kruskal-Szekeres-like coordinates, the spin coefficients of Kerr geometry are determined by solving the first Maurer-Cartan equation of structure. Isolated horizon conditions are imposed on the tetrad and the spin coefficients. A transformation into an orthonormal tetrad frame that is fixed in the time gauge is applied and explicit calculations of the spin connection, the Ashtekar and Ashtekar-Barbero connections, and the corresponding curvatures on the horizon 2-spheres are performed. Since the resulting Ashtekar-Barbero curvature does not comply with the simple form of the horizon boundary condition of Schwarzschild isolated horizons, i.e., on the horizon 2-spheres, the Ashtekar-Barbero curvature is not proportional to the Plebanski 2-form, which is required for an SU(2) Chern-Simons treatment of the gauge degrees of freedom in the horizon boundary in the context of loop quantum gravity, a general method to construct a new connection whose curvature satisfies such a relation for Kerr isolated horizons is introduced. For the purpose of illustration, this method is employed in the framework of slowly rotating Kerr isolated horizons.