Theorems on gravitational time delay and related issues

Abstract

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 ∈ 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.

Causal structures and holography

“…The description of the bulk causal structure from the dual point of view has been the subject of a number of investigations in the AdS/CFT correspondence. Bulk causality in global AdS spacetime is consistent with boundary causality [2]; for example, propagation through the bulk is never faster than propagation along the boundary (this was extended to asymptotically AdS spacetimes satisfying the weak energy condition in [3]). When we restrict to Poincaré-invariant states, the light cone in the boundary directions will automatically agree in the bulk and boundary, and the focus is on understanding the causal structure in the bulk radial direction.…”

Closed string tachyon condensation and worldsheet inflation

“…At this point the geometry is on the cusp of having a singularity, and indeed there will be a big crunch (Milne singularity) if we add more matter energy, since the solutions will cross over into the hyperbolic class (9). Note that the addition of matter makes the de Sitter space 'taller', see figure 2; higher-dimensional gravity also exhibits this property, as shown in [69] and applied to holographic bounds in [70,71].…”

Bulk curves from boundary data in holography

“…Our proof has an immediate corollary. Suppose we attempted to compute the length of an open curve in AdS 3 using formula (39). The mistake we would have made is:…”