Figure 1

(a) An eternal black hole spacetime is shown. The GSL says that the generalized entropy must increase from time slice 1 to time slice 2. However, all information outside of the horizon must either fall across the horizon $H$ or else reach future null infinity ${\mathcal{I}}^{+}$ (determinism). Hence one can “push forward” each of the two time slices to part of $H$ and all of ${\mathcal{I}}^{+}$ without losing any information. In addition to the Killing symmetry which acts on the horizon as a dilation, there is a translation symmetry of $H$ (shown as an arrow) which is not a symmetry of the whole spacetime. (b) A transverse view of $H$ in the same spacetime is shown. Vertical lines represent horizon generators. Each horizon generator can be independently translated and dilated (local Lorentz symmetry); this permits any two slices on $H$ to be translated into each other, and ensures that region above each slice on $H$ is thermal with respect to dilations about that slice. In order to prove the GSL this thermal property is needed for both slice 1 and slice 2.Reuse & Permissions