Kerr isolated horizons in Ashtekar and Ashtekar–Barbero connection variables

Kerr isolated horizons in Ashtekar and Ashtekar–Barbero connection variables 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png General Relativity and Gravitation Springer Journals

Kerr isolated horizons in Ashtekar and Ashtekar–Barbero connection variables

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Physics; Theoretical, Mathematical and Computational Physics; Classical and Quantum Gravitation, Relativity Theory; Differential Geometry; Astronomy, Astrophysics and Cosmology; Quantum Physics
ISSN
0001-7701
eISSN
1572-9532
D.O.I.
10.1007/s10714-017-2251-6
Publisher site
See Article on Publisher Site

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.

Journal

General Relativity and GravitationSpringer Journals

Published: Aug 8, 2017

References

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