Generalized entropy of gravitational fluctuations (2501.08308v2)

Abstract: The corrections to holographic entanglement entropy from bulk quantum fields in a classical gravitational background are now well understood. They lead, in particular, to unitary Page curves for evaporating black holes. However, the correct treatment of quantum fluctuations of the metric, including graviton excitations, is a longstanding problem. We provide a gauge-invariant prescription for the generalized entropy of gravitons in anti-de Sitter space in terms of areas and bulk entanglement entropy, generalizing the quantum extremal surface prescription to accommodate fluctuations in the semiclassical spacetime geometry. This task requires a careful treatment of the area operator on the graviton Hilbert space and the definition of a "quantum extremal gauge" in which the extremal surface is unperturbed. It also requires us to determine the correct vacuum modular Hamiltonian for the graviton field, which we fix by requiring that it doesn't contain a boundary term in extremal gauge. We check our prescription with an explicit computation of the vacuum-subtracted generalized entropy of states containing a graviton in an AdS-Rindler background. Our results exactly match vacuum-subtracted von Neumann entropies for stress-tensor excited states in holographic conformal field theory with $d>2$ dimensions. We also use covariant phase space techniques to give a partial proof of our prescription when the entanglement wedge for the background spacetime has a bifurcate Killing horizon. Along the way, we identify a class of perturbative graviton states that have parametrically larger generalized entropy, in the small $G_N$ expansion, than any low-energy excitations of an ordinary quantum field.

• The paper introduces a gauge-invariant prescription to include quantum fluctuations of the metric (gravitons) in generalized entropy frameworks.
• It proposes a Quantum Extremal Gauge concept and uses phase space techniques to extend the quantum extremal surface prescription for gravitons in AdS space.
• The research shows consistency with holographic CFT predictions and provides a foundational framework for integrating gravitational fluctuations into quantum gravity entropy calculations.

Overview of Generalized Entropy of Gravitational Fluctuations

The paper `Generalized Entropy of Gravitational Fluctuations' addresses the ongoing challenge in quantum gravity of incorporating quantum fluctuations of the metric, specifically graviton excitations, into the existing frameworks of generalized entropy. The research investigates the broader implications of considering gravitational fluctuations, and extends the quantum extremal surface (QES) prescription to accommodate these quantum state fluctuations, while maintaining formalisms within the anti-de Sitter (AdS) space.

Context and Objectives

The primary objective of the paper is to define a gauge-invariant prescription for the generalized entropy of gravitons, which has remained a challenging task due to the complex nature of graviton field behaviors and the prevailing semiclassical treatments of spacetime geometries. The authors build upon existing knowledge on holographic entanglement entropy, known by its alignment with von Neumann entropy within the AdS/CFT correspondence, but now tackle the inclusion of the dynamic quantum metric contributions — a non-trivial advancement.

Methodology and Contributions

The authors propose several novel theoretical contributions:

• Quantum Extremal Gauge: This novel concept allows the extremal surface to remain unaltered while ensuring that the formulation remains gauge-invariant within perturbative quantum diffeomorphisms. The main task is to select a state within the space of gauge-equivalent states that minimizes the generalized entropy function.
• Vacuum Modular Hamiltonian: The research derives an appropriate Hamiltonian devoid of boundary terms in the extremal gauge. This formulation is in alignment with von Neumann entropies derived from holographic conformal field theories (CFT), hence preserving consistency and theoretical robustness.
• Phase Space Techniques: Through covariant phase space methodologies, a partial proof of the prescription's validity is provided, especially in the presence of bifurcate Killing horizons, confirming the theoretical framework's soundness under specially symmetric conditions.
• Perturbative Expansion Approach: The paper embeds the concept of perturbations order-by-order, acknowledging larger generalized entropy contributions from graviton fluctuations over conventional quantum fields.

Results and Implications

The work showcases detailed computational checks against states within an AdS-Rindler framework containing a singular graviton, attaining perfect consistency with holographic CFT predictions. The derived calculations establish a solid correlation between quantum fluctuations on the bulk side and corresponding entropic measures that align well with existing vignette models within holographic theory.

Future Directions

This paper lays a solid foundational framework for integrating gravitational fluctuations into quantum gravity's entropy landscape. However, it notably opens avenues for exploring higher-order calculations beyond initial perturbative approaches. Further investigations could enrich holographic tensor network models, specifically by trying to depict gravitational interactions more holistically and precisely. Moreover, this research could incite the incorporation of interactions and higher derivative corrections, thereby enriching our understanding of quantum gravity's theoretical tapestry in more complex scenarios.

Ultimately, the paper broadens the understanding of quantum extremality and sets the groundwork toward comprehensive quantum gravity formulations integrated within a fully quantized spacetime structure, anticipating both theoretical and empirical advancements in the discipline.