A Note on the Symmetry Reduction of SU(2) on Horizons of Various Topologies

Abstract

It is known that the SU(2) degrees of freedom manifest in the description of the gravitational field in loop quantum gravity are generally reduced to U(1) degrees of freedom on an $S^2$ isolated horizon. General relativity also allows black holes with planar, toroidal, or higher genus topology for their horizons. These solutions also meet the criteria for an isolated horizon, save for the topological criterion, which is not crucial. We discuss the relevant corresponding symmetry reduction for black holes of various topologies (genus 0 and $\ge 2$) here and discuss its ramifications to black hole entropy within the loop quantum gravity paradigm. Quantities relevant to the horizon theory are calculated explicitly using a generalized ansatz for the connection and densitized triad, as well as utilizing a general metric admitting hyperbolic sub-spaces. In all scenarios, the internal symmetry may be reduced to combinations of U(1).

13 pages, two figures. Version 2 has several references updated and added, as well as some minor changes to the text. Accepted for publication in Class. Quant. Grav

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• Detailed black hole state counting in loop quantum gravity
• Topological deformation of isolated horizons
• Quantum geometry and microscopic black hole entropy
• The combinatorics of the SU(2) black hole entropy in loop quantum gravity
• The Evolution of $\Lambda$ Black Holes in the Mini-Superspace Approximation of Loop Quantum Gravity
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• Anyonic statistics and large horizon diffeomorphisms for Loop Quantum Gravity Black Holes
• Loop quantum gravity without the Hamiltonian constraint
• Anisotropic Structures and Wormholes with Loop Quantum Gravity Holonomy Corrections