A New Covariant Entropy Bound from Cauchy Slice Holography

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.