A New Covariant Entropy Bound from Cauchy Slice Holography

TL;DR

This paper proposes a new covariant entropy bound derived from Cauchy slice holography, linking bulk gravity states to boundary theories, with explicit calculations in 3D gravity and implications for tensor networks.

Contribution

It introduces a novel covariant entropy bound based on Cauchy slice holography and demonstrates its properties and calculations in 3D gravity with T T̄ deformation.

Findings

The bound depends only on codimension-2 data on the surface.

In pure 3D gravity, the bound aligns with certain extremal surfaces.

The bound exceeds the area for trapped surfaces inside black holes.

Abstract

We begin an investigation of a new holographic covariant entropy bound (HCEB) in gravity. This bound arises from Cauchy slice holography, a recently proposed duality between the bulk gravity theory and a `boundary' theory that lives on Cauchy slices. The HCEB is the logarithm of the maximum number of states of this theory that can pass through a given cut $\sigma$ of a Cauchy slice $\Sigma$ ($\sigma$ is thus a codimension-2 surface in the bulk). We show that the bound depends only on the codimension-2 data on $\sigma$, and is thus independent of the choice of slice $\Sigma$. For classical states, the HCEB upper bounds the entanglement between two subregions of the boundary of $\Sigma$. We calculate the bound explicitly in pure three-dimensional GR with negative cosmological constant, where the Cauchy slice theory is the $T \overline{T}$-deformation of the dual CFT. We find that the imaginary energy eigenstates in the spectrum of the deformed theory play a crucial role for obtaining a valid bound in Lorentzian signature. Our bound agrees with the area of a surface at certain marginal and extremal surfaces, but differs elsewhere. In particular, it exceeds the area by an arbitrarily large amount for (anti)trapped surfaces, such as those that lie inside a black hole. Finally, we discuss how these results can be used to write down tensor networks corresponding to arbitrary Cauchy slices.