The Borde-Guth-Vilenkin Theorem in extended de Sitter spaces

TL;DR

This paper presents a new, purely kinematic version of the Borde-Guth-Vilenkin theorem applicable to extended de Sitter spaces, with implications for cyclic cosmological models.

Contribution

It derives a novel, energy-condition-independent form of the BGV theorem using fluid flow formalism, applicable to extended de Sitter spacetimes.

Findings

The theorem is purely kinematic and depends only on local expansion.

It applies to Penrose's Conformal Cyclic Cosmology, showing geodesic incompleteness.

The extension of asymptotically de Sitter universes is geodesically incomplete.

Abstract

The Borde-Guth-Vilenkin (BGV) 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.