Local conditions for the generalized covariant entropy bound

Abstract

A set of sufficient conditions for the generalized covariant entropy bound given by Strominger and Thompson is as follows: Suppose that the entropy of matter can be described by an entropy current ${s}^{a}$. Let ${k}^{a}$ be any null vector along $L$ and $s\ensuremath{\equiv}\ensuremath{-}{k}^{a}{s}_{a}$. Then the generalized bound can be derived from the following conditions: (i) ${s}^{\ensuremath{'}}\ensuremath{\le}2\ensuremath{\pi}{T}_{ab}{k}^{a}{k}^{b}$, where ${s}^{\ensuremath{'}}={k}^{a}{\ensuremath{\nabla}}_{a}s$ and ${T}_{ab}$ is the stress-energy tensor; (ii) on the initial 2-surface $B$, $s(0)\ensuremath{\le}\ensuremath{-}\frac{1}{4}\ensuremath{\theta}(0)$, where $\ensuremath{\theta}$ is the expansion of ${k}^{a}$. We prove that condition (ii) alone can be used to divide a spacetime into two regions: The generalized entropy bound holds for all light sheets residing in the region where $s<\ensuremath{-}\frac{1}{4}\ensuremath{\theta}$ and fails for those in the region where $s>\ensuremath{-}\frac{1}{4}\ensuremath{\theta}$. We check the validity of these conditions in FRW flat universe and a scalar field spacetime. Some apparent violations of the entropy bounds in the two spacetimes are discussed. These holographic bounds are important in the formulation of the holographic principle.