Geometric temperature and entropy of quantum isolated horizons

TL;DR

This paper introduces a geometrical notion of temperature for quantum isolated horizons within loop quantum gravity, linking thermality, entanglement, and entropy to microscopic horizon states and their boundary conditions.

Contribution

It establishes a new framework connecting Lorentz invariance, thermality, and entropy in quantum isolated horizons using self-dual variables and boundary conditions.

Findings

Derived an exact temperature formula for quantum horizons.

Showed entanglement and Boltzmann entropies coincide and recover Bekenstein-Hawking entropy.

Linked horizon thermality to the analytic continuation of variables and the Connes-Rovelli thermal time.

Abstract

By reintroducing Lorentz invariance in canonical loop quantum gravity, we define a geometrical notion of temperature for quantum isolated horizons. This is done by demanding that the horizon state satisfying the boundary conditions be a Kubo-Martin-Schwinger state. The exact formula for the temperature can be derived by imposing the reality conditions in the form of the linear simplicity constraints for an imaginary Barbero-Immirzi parameter. Thus, our analysis reveals the connection between the analytic continuation to the Ashtekar self-dual variables and the thermality of the horizon. The horizon thermal equilibrium state can then be used to compute both the entanglement and the Boltzmann entropies. We show that the two provide the same finite answer, which allows us to recover the Bekenstein-Hawking formula in the semi-classical limit. In this way, we shed new light on the microscopic origin of black hole entropy by revealing the equivalence between the near-horizon degrees of freedom entanglement proposal and the state-counting interpretation. The connection with the Connes-Rovelli thermal time proposal for a general relativistic statistical mechanics is worked out.