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Theorems on gravitational time delay and related issues

Sijie Gao, Robert M. Wald

TL;DR

The paper develops a background-independent treatment of gravitational time delay by proving two theorems under the null energy and null generic conditions. The first theorem shows that sufficiently long fastest null geodesics cannot exploit a compact region to achieve a time advance, yielding a no-particle-horizon corollary for globally hyperbolic spacetimes with compact Cauchy surfaces. The second theorem analyzes spacetimes with a timelike conformal boundary (notably asymptotically anti-de Sitter) and proves that fastest boundary-to-boundary null geodesics must lie on the boundary, implying a time delay relative to AdS for generic perturbations. Collectively, the results advance a geometric, background-free understanding of time delay with implications for cosmological horizons and AdS/CFT contexts.

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 $p,q \not\in K'$, 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.

Theorems on gravitational time delay and related issues

TL;DR

The paper develops a background-independent treatment of gravitational time delay by proving two theorems under the null energy and null generic conditions. The first theorem shows that sufficiently long fastest null geodesics cannot exploit a compact region to achieve a time advance, yielding a no-particle-horizon corollary for globally hyperbolic spacetimes with compact Cauchy surfaces. The second theorem analyzes spacetimes with a timelike conformal boundary (notably asymptotically anti-de Sitter) and proves that fastest boundary-to-boundary null geodesics must lie on the boundary, implying a time delay relative to AdS for generic perturbations. Collectively, the results advance a geometric, background-free understanding of time delay with implications for cosmological horizons and AdS/CFT contexts.

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 , there exists another compact set such that for any , if there exists a ``fastest null geodesic'', , between and , then cannot enter . 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.

Paper Structure

This paper contains 3 sections, 4 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Avoidance of Compact Sets by Sufficiently Long "Fastest Null Geodesics"
  3. Time Delay in Spacetimes with a Timelike Conformal Boundary

Key Result

Theorem 1

Let $(M, g_{ab})$ be a null geodesically complete spacetime satisfying the null energy condition (nec) and the null generic condition (see eq.(gc)). Then given any compact region $K \subset M$, there exists another compact region $K^\prime$ containing K such that if $q,p \notin K^\prime$ and $q \in

Figures (1)

  • Figure 1: In a null geodesically complete spacetime satisfying the null energy condition and the null generic condition, given a compact region $K$, there exists another compact region $K^\prime$ such that if $p,q \not\in K'$ and $q \in J^+(p)-I^+(p)$, then any causal curve connecting $p$ to $q$ cannot intersect $K$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Theorem 2