Proof of the entropy bound on dynamical horizons

The entropy bound conjecture concerning black hole dynamical horizons is proved. The conjecture states, if a dynamical horizon, $D_H$, is bounded by two surfaces with areas of $A_B$ and $\abp$ ($\abp>A_B$), then the entropy, $S_D$, that crosses $D_H$ must satisfy $S_D\leq {1/4}(\abp-A_B)$. We show that this conjecture is implied by the generalized Bousso bound. Consequently, the generalized second law holds for dynamical horizons. Finally, we show that the lightlike bousso bound and its spacelike counterpart can be unified as one bound.