  The generalized second law can be used to prove a singularity theorem, by
generalizing the notion of a trapped surface to quantum situations. Like
Penrose's original singularity theorem, it implies that spacetime is null
geodesically incomplete inside black holes, and to the past of spatially
infinite Friedmann--Robertson--Walker cosmologies. If space is finite instead,
the generalized second law requires that there only be a finite amount of
entropy producing processes in the past, unless there is a reversal of the
arrow of time. In asymptotically flat spacetime, the generalized second law
also rules out traversable wormholes, negative masses, and other forms of
faster-than-light travel between asymptotic regions, as well as closed timelike
curves. Furthermore it is impossible to form baby universes which eventually
become independent of the mother universe, or to restart inflation. Since the
semiclassical approximation is used only in regions with low curvature, it is
argued that the results may hold in full quantum gravity. An introductory
section describes the second law and its time-reverse, in ordinary and
generalized thermodynamics, using either the fine-grained or the coarse-grained
entropy. (The fine-grained version is used in all results except those relating
to the arrow of time.) A proof of the coarse-grained ordinary second law is
given.
