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On the uniqueness and global dynamics of AdS spacetimes

Michael T. Anderson

TL;DR

The paper proves a rigidity result for complete, non-singular asymptotically locally AdS spacetimes: if a solution to the vacuum Einstein equations with negative cosmological constant is weakly asymptotically stationary in both time directions and the linearized equations satisfy a suitable unique continuation at the conformal boundary, then the spacetime is globally stationary. The main tool is a fundamental identity (3.1) tying the holographic stress-energy tensor to variations of boundary data and the associated conserved holographic mass, together with a boundary-to-bulk extension argument and a unique continuation result to promote boundary symmetries into bulk isometries. This yields strong rigidity and uniqueness statements for AdS and soliton-type spacetimes and clarifies the role of conformal infinity in determining bulk dynamics within the AdS/CFT-inspired framework. The results delineate how boundary symmetry and mass conservation constrain global evolution, while highlighting limitations in the presence of horizons or singularities.

Abstract

We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationary. This gives certain rigidity or uniqueness results for exact AdS and related spacetimes.

On the uniqueness and global dynamics of AdS spacetimes

TL;DR

The paper proves a rigidity result for complete, non-singular asymptotically locally AdS spacetimes: if a solution to the vacuum Einstein equations with negative cosmological constant is weakly asymptotically stationary in both time directions and the linearized equations satisfy a suitable unique continuation at the conformal boundary, then the spacetime is globally stationary. The main tool is a fundamental identity (3.1) tying the holographic stress-energy tensor to variations of boundary data and the associated conserved holographic mass, together with a boundary-to-bulk extension argument and a unique continuation result to promote boundary symmetries into bulk isometries. This yields strong rigidity and uniqueness statements for AdS and soliton-type spacetimes and clarifies the role of conformal infinity in determining bulk dynamics within the AdS/CFT-inspired framework. The results delineate how boundary symmetry and mass conservation constrain global evolution, while highlighting limitations in the presence of horizons or singularities.

Abstract

We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationary. This gives certain rigidity or uniqueness results for exact AdS and related spacetimes.

Paper Structure

This paper contains 4 sections, 4 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Background Material.
  3. Proofs of the Results.
  4. Discussion

Key Result

Theorem 1.1

Suppose $g \in {\mathcal{E}}$ is weakly asymptotically stationary in the sense that there exist times $t_{i} \rightarrow \infty$ and $t_{i} \rightarrow -\infty$ such that $(g_{t_{i}}, K_{t_{i}})$ converges to a globally stationary solution of the vacuum constraint equations, as $i \rightarrow \infty

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 3.1