Black hole entropy from the $SU(2)$-invariant formulation of type I isolated horizons

A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly $SU(2)$ invariant formulation where the (effective) horizon degrees of freedom are described by an $SU(2)$ Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy.

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 falloff 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 spacelike infinity $i_0$. Portions of ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$ are shown; however, no reference to future timelike infinity $i^{+}$ is made as the isolated horizon need not coincide with the black hole event horizon.