The BGV Theorem and the Null Convergence Condition

We examine the relationship between the Null Convergence Condition (NCC) and the Borde-Guth-Vilenkin (BGV) Theorem. We first show that, for an expanding spacetime foliated orthogonally to a timelike geodesic congruence with vanishing shear and vorticity, the BGV Theorem follows when the NCC holds and the spatial curvature is non-positive, ${}^3\mathcal{R} \leq 0$. The situation becomes more complex in the presence of shear, or non-geodesic threading of the spacetime. In these cases, the local expansion that enters the BGV construction depends not only on the local scalar expansion, but acquires terms given by the contraction of the shear and acceleration with locally defined spatial unit vectors. In the general case, null convergence and non-positive curvature are no longer sufficient to guarantee that the BGV Theorem holds. We discuss the result in the context of eternal inflation in the presence of comoving curvature perturbations and state a more general version of the BGV condition relevant to asymptotically past-de Sitter eternal inflation.