Simple sufficient conditions for the generalized covariant entropy bound

The generalized covariant entropy bound is the conjecture that for any null hypersurface which is generated by geodesics with nonpositive expansion starting from a spacelike 2-surface B and ending in a spacelike 2-surface $B^{\prime },$ the matter entropy on that hypersurface will not exceed one quarter of the difference in areas, in Planck units, of the two spacelike 2-surfaces. We show that this bound can be derived from the following phenomenological assumptions: (i) matter entropy can be described in terms of an entropy current $s_a;$ (ii) the gradient of the entropy current is bounded by the energy density, in the sense that $|k^a{k}^b{\nabla }_a{s}_b|<~2\pi T_{\mathrm{ab}}{k}^a{k}^b/ħ$ for any null vector $k^a$ where $T_{\mathrm{ab}}$ is the stress energy tensor; and (iii) the entropy current $s_a$ vanishes on the initial 2-surface B. We also show that the generalized Bekenstein bound—the conjecture that the entropy of a weakly gravitating isolated matter system will not exceed a constant times the product of its mass and its width—can be derived from our assumptions. Though we note that any local description of entropy has intrinsic limitations, we argue that our assumptions apply in a wide regime. We closely follow the framework of an earlier derivation, but our assumptions take a simpler form, making their validity more transparent in some examples.