Gravity Edges Modes and Hayward Term

TL;DR

This paper identifies the Hayward corner term as the essential encoding of gravity edge modes that give rise to gravitational area entropies such as $S_{BH}$ and holographic entanglement entropy. It demonstrates, via replica calculus, that the Hayward term yields the correct entropy $S_{\Sigma_A}=\frac{A(\Gamma)}{4G_N}$ and derives general pasting rules for gravitational spaces with wedges. It then offers two holographic interpretations—a generalized holography with Dirichlet boundaries and an AdS/BCFT framework with Neumann boundaries—showing the Hayward term accounts for edge degrees of freedom and conformal anomalies. Finally, it presents a string-theoretic perspective in which open strings anchored to a Rindler horizon reproduce the edge-mode entropy, supporting the view that gravity edge modes are fundamental and present in a complete quantum gravity description.

Abstract

We argue that corner contributions in gravity action (Hayward term) capture the essence of gravity edge modes, which lead to gravitational area entropies, such as the black hole entropy and holographic entanglement entropy. We explain how the Hayward term and the corresponding edge modes in gravity are explained by holography from two different viewpoints. One is an extension of AdS/CFT to general spacetimes and the other is the AdS/BCFT formulation. In the final part, we explore how gravity edge modes and its entropy show up in string theory by considering open strings stuck to a Rindler horizon.

Figures (12)

• Figure 1: A sketch of Euclidean space $M$ with a wedge. The wedge is situated along the co-dimension two surface $\Gamma$ with the angle $\theta$.
• Figure 2: A sketch of Lorentzian gravity setup for the canonical formalism.
• Figure 3: Sketches of Replica Calculations of Gravitational Entropy.
• Figure 4: Pasting Rules of Gravity Action.
• Figure 5: Pasting Two Wave Functions (Top) and its Doubled Version (Bottom).