Local conditions for the generalized covariant entropy bound
  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 $s\equiv -k^a s_a$. Then the generalized bound can be derived from the following conditions: (i) $s'\leq 2\pi T_{ab}k^ak^b$, where $s'=k^a\grad_a s$ and $T_{ab}$ is the stress energy tensor; (ii) on the initial 2-surface $B$, $s(0)\leq -{1/4}\theta(0)$, where $\theta$ 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 $s<-{1/4}\theta$ and fails for those in the region where $s>-{1/4}\theta$. 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. 
Sijie Gao, Jose' P. S. Lemos