Geodesic focusing, energy conditions and singularities

Standard work on geodesic focusing assumes that a pointwise inequality, known as an energy condition, holds. This pointwise condition has been weakened by Tipler (1978) to the requirement that a certain integral comes out to be non-negative for large enough parameter intervals. In this paper, this integral condition is further weakened to cover cases where the integral of interest is only periodically non-negative. Thus, some situations where there are repeated violations of the energy conditions are shown to still lead to the focusing of geodesics, despite these violations. Focusing effects are discussed both for full as well as half geodesics. The existence of singularities in situations where the energy conditions are violated, as in inflationary cosmological models, is also discussed.