Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy

TL;DR

This work develops a manifestly SU(2) invariant treatment of isolated horizons, showing that non-static horizons break the diffeomorphism invariance of the pre-symplectic structure, and proposes an extended horizon phase space to restore it. For static isolated horizons, the boundary degrees of freedom admit a description in terms of two SU(2) Chern-Simons theories whose levels encode the distortion and link the bulk Immirzi parameter to the boundary theory. By quantizing the boundary and bulk together and counting horizon microstates under area constraints, the authors recover Hawking's area law, first under the usual paradigm (where the CS level scales with area and the Immirzi parameter is fixed), and then via a paradigm shift that treats the CS level as an independent input, yielding a family of area-independent boundary theories with a specified relation between $\beta$ and the CS level $k$. This reframed view eliminates the need to fix $\beta$ a priori and provides a broader, more universal connection between bulk and boundary degrees of freedom, potentially extending to distorted and rotating horizons. The results illuminate how horizon entropy may emerge from a boundary CS description compatible with loop quantum gravity, highlighting a flexible interdependence between bulk and boundary quantization data.

Abstract

We study the classical field theoretical formulation of static generic isolated horizons in a manifestly SU(2) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area A of the horizon is fixed macroscopically-states with fluctuations away from spherical symmetry are allowed-we show that it is possible to obtain agreement with the Hawking's area law---S = A/4 (in Planck Units)---without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.

Figures (2)

• Figure 1: The characteristic data for a (vacuum) spherically symmetric isolated horizon corresponds to Reissner-Nordstrom data on $\Delta$, and free radiation data on the transversal null surface with suitable fall-off conditions. For each mass, charge, and radiation data in the transverse null surface there is a unique solution of Einstein-Maxwell equations locally in a portion of the past domain of dependence of the null surfaces. This defines the phase space of Type I isolated horizons in Einstein-Maxwell theory. The picture shows two Cauchy surfaces $M_1$ and $M_2$ "meeting" at space-like infinity $i_0$. A portion of ${\mathfs {I}}^+$ and ${\mathfs {I}}^-$ are shown; however, no reference to future time-like infinity $i^+$ is made as the isolated horizon need not to coincide with the black hole event horizon.
• Figure 2: The value of the Immirzi parameter $\beta_{k}$ as a function of $k\in \mathbb{N}$ for the first few integers. The value $\beta_{1}=0.172217...$ is exact as well as the asymptotic value $\beta_{\infty}=0.343599...$. The other points have been computed using (\ref{['eq:lambda']}) which is only valid in the large $k$ limit.