Exact Renormalization Group, Entanglement Entropy, and Black Hole Entropy

TL;DR

This paper uses the exact renormalization group to analyze quantum contributions to black hole entropy, revealing how entanglement and quantum corrections influence the total entropy and resolving a previous mismatch issue.

Contribution

It provides a detailed RG flow analysis of black hole entropy, distinguishing entanglement and quantum correction contributions, and demonstrates the constancy of renormalized entropy.

Findings

Quantum fluctuations contribute to black hole entropy via entanglement and quantum corrections.

The renormalized black hole entropy remains constant throughout the RG flow.

No mismatch exists between coupling constant renormalization and entropy renormalization.

Abstract

The study of black hole physics revealed a fundamental connection between thermodynamics, quantum mechanics, and gravity. Today, it is known that black holes are thermodynamical objects with well-defined temperature and entropy. Although black hole radiance gives us the mechanism from which we can associate a well-defined temperature to the black hole, the origin of its entropy remains a mystery. Here we investigate how the quantum fluctuations from the fields that render the black hole its temperature contribute to its entropy. By using the exact renormalization group equation for a self-interacting real scalar field in a spacetime possessing a bifurcate Killing horizon, we find the renormalization group flow of the total gravitational entropy. We show that throughout the flow one can split the quantum field contribution to the entropy into a part coming from the entanglement between field degrees of freedom inside and outside the horizon and a part due to the quantum corrections to the Wald entropy coming from the Noether charge. The renormalized black hole entropy is shown to be constant throughout the flow while the balance between the effective black hole entropy at low energies and the infra-red entanglement entropy changes. A similar conclusion is valid for the Wald entropy part of the total entropy. Additionally, our calculations show that there is no mismatch between the renormalization of the coupling constants coming from the effective action or the gravitational entropy, solving an apparent "puzzle" that appeared to exist for interacting fields.