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Entanglement Interpretation of Black Hole Entropy in String Theory

Ram Brustein, Martin B. Einhorn, Amos Yarom

TL;DR

The paper argues that black hole entropy can be understood as entanglement entropy of the vacuum across horizons. It develops a path-integral framework to realize the Hartle-Hawking state as a thermofield double, and then shows that, under AdS/CFT duality, the dual field theory’s thermal (entanglement) entropy matches the Bekenstein-Hawking entropy. The authors illustrate this equivalence with a detailed AdS$_5$ black hole example, showing how near-horizon entanglement and holographic degrees of freedom reproduce the area law and its UV structure, while boundary CFT calculations corroborate the result up to scheme-dependent factors. This work connects quantum information concepts with gravitational thermodynamics, offering a unified entanglement-based perspective on black hole entropy and holography. It highlights how the UV structure and holographic degrees of freedom govern the entropy, and suggests broad applicability to spacetimes with bifurcating Killing horizons beyond black holes.

Abstract

We show that the entropy resulting from the counting of microstates of non extremal black holes using field theory duals of string theories can be interpreted as arising from entanglement. The conditions for making such an interpretation consistent are discussed. First, we interpret the entropy (and thermodynamics) of spacetimes with non degenerate, bifurcating Killing horizons as arising from entanglement. We use a path integral method to define the Hartle-Hawking vacuum state in such spacetimes and discuss explicitly its entangled nature and its relation to the geometry. If string theory on such spacetimes has a field theory dual, then, in the low-energy, weak coupling limit, the field theory state that is dual to the Hartle-Hawking state is a thermofield double state. This allows the comparison of the entanglement entropy with the entropy of the field theory dual, and thus, with the Bekenstein-Hawking entropy of the black hole. As an example, we discuss in detail the case of the five dimensional anti-de Sitter, black hole spacetime.

Entanglement Interpretation of Black Hole Entropy in String Theory

TL;DR

The paper argues that black hole entropy can be understood as entanglement entropy of the vacuum across horizons. It develops a path-integral framework to realize the Hartle-Hawking state as a thermofield double, and then shows that, under AdS/CFT duality, the dual field theory’s thermal (entanglement) entropy matches the Bekenstein-Hawking entropy. The authors illustrate this equivalence with a detailed AdS black hole example, showing how near-horizon entanglement and holographic degrees of freedom reproduce the area law and its UV structure, while boundary CFT calculations corroborate the result up to scheme-dependent factors. This work connects quantum information concepts with gravitational thermodynamics, offering a unified entanglement-based perspective on black hole entropy and holography. It highlights how the UV structure and holographic degrees of freedom govern the entropy, and suggests broad applicability to spacetimes with bifurcating Killing horizons beyond black holes.

Abstract

We show that the entropy resulting from the counting of microstates of non extremal black holes using field theory duals of string theories can be interpreted as arising from entanglement. The conditions for making such an interpretation consistent are discussed. First, we interpret the entropy (and thermodynamics) of spacetimes with non degenerate, bifurcating Killing horizons as arising from entanglement. We use a path integral method to define the Hartle-Hawking vacuum state in such spacetimes and discuss explicitly its entangled nature and its relation to the geometry. If string theory on such spacetimes has a field theory dual, then, in the low-energy, weak coupling limit, the field theory state that is dual to the Hartle-Hawking state is a thermofield double state. This allows the comparison of the entanglement entropy with the entropy of the field theory dual, and thus, with the Bekenstein-Hawking entropy of the black hole. As an example, we discuss in detail the case of the five dimensional anti-de Sitter, black hole spacetime.

Paper Structure

This paper contains 9 sections, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of entanglement entropy
  3. Entanglement in Spacetimes
  4. Non-degenerate Single Horizon
  5. Non-degenerate multiple killing horizons
  6. Extremal black holes
  7. Entanglement in field theory duals
  8. AdS black holes - an example
  9. Discussion

Figures (1)

  • Figure 1: Diagram of a line of constant radial coordinate $r$ in complex time. The circle is a line of constant $r$ in imaginary time where $\tau$ is an angle, while the hyperbolas are lines of constant $r$ in real time. The dashed lines represent, in real time, the bifurcation surface that splits the spacetime into the four wedges R, L, F, P.