Theorems ongravitational time delay and related issues

Abstract

Two theorems related to gravitational time delay are proven. Boththeorems apply to spacetimes satisfying the null energy condition and thenull generic condition. The first theorem states that if the spacetime isnull geodesically complete, then given any compact set K, there existsanother compact set K' such that for any p,q∉K', if there exists a `fastest null geodesic', γ, betweenp and q, then γ cannot enter K. As an application of thistheorem, we show that if, in addition, the spacetime is globally hyperbolicwith a compact Cauchy surface, then any observer at sufficiently latetimes cannot have a particle horizon. The second theorem states that if atimelike conformal boundary can be attached to the spacetime such thatthe spacetime with boundary satisfies strong causality as well as acompactness condition, then any `fastest null geodesic' connecting twopoints on the boundary must lie entirely within the boundary. It followsfrom this theorem that generic perturbations of anti-de Sitter spacetimealways produce a time delay relative to anti-de Sitter spacetime itself.