Entanglement entropy of black holes (1104.3712v1)

Abstract: The entanglement entropy is a fundamental quantity which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff which regulates the short-distance correlations. The geometrical nature of the entanglement entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in 4 and 6 dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as 't Hooft's brick wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields which non-minimally couple to gravity is emphasized. The holographic description of the entanglement entropy of the black hole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.

• The paper demonstrates that entanglement entropy follows an area law analogous to Bekenstein-Hawking entropy, linking quantum measures to black hole surfaces.
• The analysis employs heat kernel methods and conical singularities to address ultraviolet divergences and systematically renormalize the entropy.
• Comparative studies with holographic models and diverse geometric settings suggest a unified quantum-gravitational description of black hole entropy.

Entanglement Entropy of Black Holes

The paper "Entanglement entropy of black holes" by Sergey Solodukhin offers a thorough examination of the interplay between quantum field theories, spacetime geometry, and black hole thermodynamics through the lens of entanglement entropy. The entanglement entropy is a quantum mechanical measure of the amount of information shared between two subsystems and is fundamentally linked to the area of the dividing surface between them, known as the entangling surface. This intrinsic property becomes particularly intriguing when applied to black holes, with their event horizons serving as natural boundaries.

Summary of Findings

Solodukhin begins by outlining the foundational principles of entanglement entropy and its measurement across a given surface within a larger quantum system. This measurement reveals a profound "area law" similarity with Bekenstein-Hawking entropy, which states that black hole entropy is proportional to the horizon's area. The proportional relationship hints at broader theoretical implications, potentially indicating a unifying description of gravitational entropy through quantum mechanical correlates.

The paper explores several essential mathematical tools and methodologies utilized in calculating the entanglement entropy of black holes. These include spaces with conical singularities, the heat kernel method, and the examination of ultraviolet (UV) divergences that naturally arise due to short-distance correlations in quantum field theories. Solodukhin gives special attention to the analysis of these divergences and the pursuit of their systematic renormalization, which remains pivotal in rendering entanglement entropy finite and measurable.

A key highlight of this work is its focus on the variance of entanglement entropy calculations across multiple geometrical settings and comparative methodologies, such as the 't Hooft's brick wall model and the Euclidean path integral approach. These comparisons reveal consistent but nuanced differences arising due to factors such as non-minimal coupling to gravity, dimensional embeddings, and boundary conditions. These variations directly inform on conformal anomalies and provide insights into the contributions from different fields of varying spin and boundary behaviors.

The paper concludes by exploring the possibility of interpreting the classical Bekenstein-Hawking entropy entirely as entanglement entropy by considering holographic descriptions, particularly through the AdS/CFT correspondence framework, which relates gravitational theories in AdS space to conformal field theories on its boundary. The holographic principle thus offers potential pathways to bridge classical and quantum descriptions of black hole entropy, aligning with concepts of quantum gravity.

Implications and Future Directions

The ramifications of Solodukhin's analysis extend into multiple theoretical domains. Practically, understanding how entanglement entropy relates to observable thermodynamic properties of black holes could illuminate the nature of spacetime at the quantum level and support the formulation of quantum gravity theories. This exploration challenges assumptions about the static nature of classical entropy constructs and invites reconsideration of the underlying information paradigms governing black holes.

The paper suggests various avenues for future research, particularly the need for deeper explorations into string theoretic and holographic models of entanglement entropy. These lines of inquiry may yield novel insights into longstanding puzzles like the non-minimal coupling anomaly and the reproduction of black hole entropy directly through string theory frameworks, potentially resolving inconsistencies that arise in classical interpretations.

Overall, Solodukhin's work serves as a comprehensive introductory guide and a call to innovation for understanding black hole entropy through the quantum lens of entanglement.