Geometric Entropies and their Hamiltonian Flows

TL;DR

This paper investigates the flows generated by geometric entropy in higher derivative gravity theories, revealing new features and extending previous understanding from Einstein-Hilbert and JT gravity to more complex models.

Contribution

It introduces a covariant approach to analyze geometric entropy flows in higher derivative gravity, generalizing known results to broader theories with matter and complex interactions.

Findings

Flows retain geometric form on bifurcation surfaces of Killing horizons.

New non-geometric features emerge in flows for general spacetimes.

Poisson brackets can be used to derive these flows.

Abstract

In holographic theories, the Hubeny-Rangamani-Takayanagi (HRT) area operator plays a key role in our understanding of the emergence of semiclassical Einstein-Hilbert gravity. When higher derivative corrections are included, the role of the area is instead played by a more general functional known as the geometric entropy. It is thus of interest to understand the flow generated by the geometric entropy on the classical phase space. In particular, the fact that the associated flow in Einstein-Hilbert or Jackiw-Teitelboim (JT) gravity induces a relative boost between the left and right entanglement wedges is deeply related to the fact that gravitational dressing promotes the von Neumann algebra of local fields in each wedge to type II. This relative boost is known as a boundary-condition-preserving (BCP) kink-transformation. In a general theory of gravity (with arbitrary higher-derivative terms), it is straightforward to show that the flow continues to take the above geometric form when acting on a spacetime where the HRT surface is the bifurcation surface of a Killing horizon. However, the form of the flow on other spacetimes is less clear. In this paper, we use the manifestly-covariant Peierls bracket to explore such flows in two-dimensional theories of JT gravity coupled to matter fields with higher derivative interactions. The results no longer take a purely geometric form and, instead, demonstrate new features that should be expected of such flows in general higher derivative theories. We also show how to obtain the above flows using Poisson brackets.