Borde-Guth-Vilenkin theorem in extended de Sitter spaces

The Borde-Guth-Vilenkin theorem states that any spacetime with net positive expansion must be geodesically incomplete. We derive a new version of the theorem using the fluid flow formalism of general relativity. The theorem is purely kinematic, depending on the local expansion properties of geodesics, and makes no assumptions about energy conditions. We discuss the physical interpretation of this result in terms of coordinate patches on de Sitter space, and apply the theorem to Penrose's model of conformal cyclic cosmology. We argue that the conformal cyclic extension of an asymptotically de Sitter universe is geodesically incomplete.