Generalized gravitational entropy (1304.4926v2)

Abstract: We consider classical Euclidean gravity solutions with a boundary. The boundary contains a non-contractible circle. These solutions can be interpreted as computing the trace of a density matrix in the full quantum gravity theory, in the classical approximation. When the circle is contractible in the bulk, we argue that the entropy of this density matrix is given by the area of a minimal surface. This is a generalization of the usual black hole entropy formula to euclidean solutions without a Killing vector. A particular example of this set up appears in the computation of the entanglement entropy of a subregion of a field theory with a gravity dual. In this context, the minimal area prescription was proposed by Ryu and Takayanagi. Our arguments explain their conjecture.

• The paper extends gravitational entropy beyond U(1) symmetry by adapting the replica trick to compute entanglement entropy in generalized Euclidean metrics.
• The paper employs analytical continuation and minimal surface methods to bridge gravitational actions with holographic entanglement entropy formulas.
• The paper validates its theoretical framework with concrete examples, confirming consistency with the Ryu-Takayanagi paradigm in diverse settings.

Overview of "Generalized Gravitational Entropy"

The paper "Generalized Gravitational Entropy" by Aitor Lewkowycz and Juan Maldacena contributes to the ongoing elaboration of entropy in the context of quantum gravity and holography. The central focus is extending the notion of gravitational entropy beyond traditional cases with a U(1) symmetry, specifically addressing Euclidean solutions that allow for more generalized metrics. This breakthrough is motivated by the need to gain insights into entanglement entropy calculations in field theories that possess gravitational duals.

The authors build on earlier work that derives gravitational entropy from the area of minimal surfaces, a well-known result in the context of black hole thermodynamics. This foundational formula is revisited here without the prescription of a Killing vector, effectively broadening the applicability of the concept. The extension of such calculations provides a tangible method of analytically continuing classical solutions, yielding entropies computed as the trace of density matrices reconstructed from the field's data.

Methodological Insights

One of the key technical achievements in this research is the effective use of the "replica trick," a method well known in quantum field theory contexts for computing entanglement entropy. The method is adeptly adapted herein for use in gravitational settings, resulting in the formulation of entropy from gravitational configurations. Specifically, the central focus is on the mathematical prescription $$S = -\lim_{n \to 1} n \frac{d}{dn}\left[ \log Z(n) - n\log Z(1)\right]$$, where $Z(n)$ represents the partition function of a configuration with a circular symmetry that is not necessarily contractible.

The absence of a U(1) symmetry brings up challenges that are addressed by hypothesizing a smooth subset of solutions in gravitational settings. Through precise calculations, the authors demonstrate that the entropy can still be largely accounted for by an invariant surface's area, adhering to a minimal area condition equivalent to the Ryu-Takayanagi conjecture for AdS/CFT correspondence. This feature underscores classical general relativity within the holographic principle.

Results and Computational Validations

Lewkowycz and Maldacena's assumptions about analytic solutions bear fruit in ensuring uniformity across diverse boundary conditions. The rigorous derivation aligns their theoretical predictions with the established Ryu-Takayanagi minimal surface paradigm, confirming its role in computing entanglement entropies. Specific examples, including cases of scalar fields interacting with the gravitational geometry, elucidate the authors' broader conjectures about the holographic nature of gravity.

The implementation of these specific cases confirmed conjectures regarding the contribution of fields to entropy, aligning them accurately with expected results based on traditional entropic formulations when symmetries are present. Calculations reproduce the established entropy formulas and validate that the conjectured principles appear feasible within specific constraints of holography theories.

Implications and Future Directions

The paper opens up new pathways in our understanding of entropy in quantum gravity, suggesting that deeper connections between gravitational actions and entropic measures are possible even in the absence of traditional symmetries. The findings promise multifaceted implications for theories grounded in AdS/CFT correspondence and may offer insight into more generalized settings beyond strictly defined U(1) symmetry cases.

Going forward, the authors propose further extensions to situations involving time-dependent dynamics which challenge the static assumptions presently in use. Additionally, they suggest broader exploration into theories that include higher derivative corrections and transcend traditional Einstein gravity, anxiously conjecturing that the Wald-Iyer entropy formula may have a role in these more complex systems.

In conclusion, "Generalized Gravitational Entropy" advances our conceptual understanding of gravitational actions concerning entropy, effectively generalizing and extending the horizon of research in the interplay between gravitational theory and quantum phenomena. The work lays a substantial foundation for future inquiries into the quantification of entropy in non-traditional spacetimes, driving progress in theoretical physics’ inquiry into the nature of space, time, and information.