0 Theorems on gravitational time delay and related issues

Two theorems related to gravitational time delay are proven. Both theorems apply to spacetimes satisfying the null energy condition and the null generic condition. The first theorem states that if the spacetime is null geodesically complete, then given any compact set K, there exists another compact set K ′ such that for any p, q 6∈ K , if there exists a “fastest null geodesic”, γ, between p and q, then γ cannot enter K. As an application of this theorem, we show that if, in addition, the spacetime is globally hyperbolic with a compact Cauchy surface, then any observer at sufficiently late times cannot have a particle horizon. The second theorem states that if a timelike conformal boundary can be attached to the spacetime such that the spacetime with boundary satisfies strong causality as well as a compactness condition, then any “fastest null geodesic” connecting two points on the boundary must lie entirely within the boundary. It follows from this theorem that generic perturbations of anti-de Sitter spacetime always produce a time delay relative to anti-de Sitter spacetime itself.