Note on SU(2) isolated horizon

Abstract

We point out that the symplectic structure, written in terms of the Sen-Ashtekar-Immirzi-Barbero variables, of a spacetime admitting an isolated horizon as the inner boundary (SS in short), involves a positive constant parameter, say σ, if γ = ±i, where γ is the Barbero-Immirzi parameter. The parameter σ represents the rescaling freedom that defines the equivalence class of null generators of the isolated horizon. The validity of the laws of mechanics and the value of surface gravity associated with the isolated horizon does not depend on the choice of σ. The SS diverges for σ 2 = (1 + γ 2 ) −1 , implying that the SS is not compatible with the full symmetry of the isolated horizon. We offer some remarks on black hole entropy calculation, with real γ, considering the restriction σ 2 = (1 + γ 2 ) −1 . * Electronic address: abhishek.majhi@gmail.com 1 See [1] for the historical reasons behind this nomenclature.