Black hole entropy from Quantum Geometry

TL;DR

This paper revisits black-hole entropy counting in the ABCK framework of Quantum Geometry, identifies a spurious constraint that underestimated horizon-state degeneracy, and introduces a corrected combinatorial formulation showing higher-spin labels contribute to the leading entropy. It derives entropy bounds $S(a)$ between $\frac{\ln 2}{4\pi \gamma l_{Pl}^2} a$ and $\frac{\ln 3}{4\pi \gamma l_{Pl}^2} a$, which imply $\gamma$ lies in $[\tfrac{\ln 2}{\pi}, \tfrac{\ln 3}{\pi}]$ and challenge the earlier ABCK value. A companion work meis provides the exact asymptotics $\ln N(a) = (\gamma_M/\gamma)\,a/(4 l_{Pl}^2) - \tfrac{1}{2}\ln a + O(1)$ with $\gamma_M$ defined by a defining equation, confirming area proportionality of the leading term and giving subleading corrections. The paper also derives the asymptotic spin distribution $P(m)=e^{-2\pi \gamma_M \sqrt{|m|(|m|+1)}}$ for large horizons and discusses implications for horizon microstructure and the relation to quasi-normal modes.

Abstract

Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.