The Area Operator in Gravity and Holography

The area operator plays a central role in our understanding of holography and black hole thermodynamics. In this talk I will discuss a formulation of the area operator at the level of classical gravity, the “kink transform”, which defines a notion of a half-sided boost of gravitational initial data. I will show that it satisfies the gravitational constraints only when done across an extremal surface. I will then consider the kink transform in holographic spacetimes, about the Ryu-Takayanagi surface of a boundary cut. For a flat boundary cut, I will demonstrate that the resulting Wheeler-DeWitt patch is the bulk dual to the boundary state obtained by a particular manifestation of modular flow: the Connes cocycle (CC) flow, which is a smooth notion of a half-sided boost of arbitrary states in quantum field theory. This is done by verifying that the kink transform is consistent with known properties of operator expectation values and subregion entropies under CC flow. Moreover, CC flow generates a stress tensor shock at the cut, controlled by a shape derivative of the entropy. I will show that the kink transform reproduces this shock holographically through a gravitational shock. I will also discuss the relationship to one-sided modular flow and JLMS.