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≡-k^as_a. Then the generalized bound can be derived from the following conditions: (i) s^′≤2πT_abk^ak^b, where s^′=k^a∇_as and T_ab is the stress-energy tensor; (ii) on the initial 2-surface B, s(0)≤-1/4θ(0), where θ 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<-1/4θ and fails for those in the region where s>-1/4θ. 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.