Entropy of Non-Extremal Black Holes from Loop Gravity

TL;DR

This paper calculates the entropy of non-extremal black holes within Loop Gravity, showing it aligns with the Bekenstein-Hawking formula and introduces a quantum Rindler horizon concept.

Contribution

It introduces a quantum Rindler horizon in Loop Gravity and demonstrates how its entropy reproduces the classical black hole entropy formula.

Findings

Entropy scales linearly with horizon area.

Quantum horizon energy matches known local horizon energy.

Quantum horizon thermalizes at Unruh temperature.

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.