Entropy of Non-Extremal Black Holes from Loop Gravity

TL;DR

The paper addresses deriving the Bekenstein-Hawking entropy from Loop Quantum Gravity for non-extremal black holes by introducing a quantum Rindler horizon. It defines horizon states via gamma-simple Lorentzian representations and identifies the boost Hamiltonian as the horizon energy, obtaining $E = (A a)/(8\pi G)$ and a canonical Unruh temperature $T = (\hbar a)/(2\pi)$. Entropy is derived from the Clausius relation, yielding $S = A/(4 G \hbar)$ with the Immirzi parameter $\\ ext{gamma}$ contributing only quantum corrections for small black holes; for large horizons the entropy is independent of $\\gamma$. The partition function links to the Spinfoam path integral and Euclidean Regge action, providing a covariant, horizon-local microscopic account of black hole thermodynamics within Loop Gravity.

Abstract

We compute the entropy of non-extremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the Bekenstein-Hawking expression S = A/4 with the one-fourth coefficient for all values of the Immirzi parameter. The near-horizon geometry of a non-extremal black hole - as seen by a stationary observer - is described by a Rindler horizon. We introduce the notion of a quantum Rindler horizon in the framework of Loop Gravity. The system is described by a quantum surface and the dynamics is generated by the boost Hamiltonion of Lorentzian Spinfoams. We show that the expectation value of the boost Hamiltonian reproduces the local horizon energy of Frodden, Ghosh and Perez. We study the coupling of the geometry of the quantum horizon to a two-level system and show that it thermalizes to the local Unruh temperature. The derived values of the energy and the temperature allow one to compute the thermodynamic entropy of the quantum horizon. The relation with the Spinfoam partition function is discussed.

Figures (1)

• Figure 1: The distribution $\varrho(\lambda)=|\langle\lambda|j\rangle|^2$ is plotted as a function of $\lambda$ for $j=1$ and $\gamma=1$. It represents the probability of finding an area eigenstate $|j\rangle$ in the state of energy $E=\hbar \lambda a$. The average is $\langle\lambda\rangle=\gamma j$, the dispersion $\Delta \lambda=\sqrt{1+\gamma^2}\sqrt{\frac{2j+1}{2j+3}}$. There is an exponential fall-off for large $\lambda$.