Local conditions for the generalized covariant entropy bound

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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 sᵃ. Let kᵃ be any null vector along L and s≡ -kᵃ sₐ. Then the generalized bound can be derived from the following conditions: (i) s'≤ 2πTabkᵃkᵇ, where s'=kᵃₐ s and Tab is the stress energy tensor; (ii) on the initial 2-surface B, s(0)≤ -1/4θ(0), where θ is the expansion of kᵃ. 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.