Killing boundary data for anti-de Sitter-like spacetimes

TL;DR

This paper investigates conditions on the conformal boundary of anti-de Sitter-like spacetimes that guarantee the existence of Killing vectors, using conformal wave equations and identifying an obstruction tensor related to boundary flatness.

Contribution

It introduces an analysis of boundary conditions ensuring Killing vectors in anti-de Sitter-like spacetimes, highlighting the role of an obstruction tensor and boundary conformal flatness.

Findings

Obstruction tensor's vanishing is necessary for Killing vectors.

Conformal boundary's flatness implies vanishing of the obstruction tensor.

Conditions on the boundary ensure spacetime symmetries.

Abstract

Given an initial-boundary value problem for an anti-de Sitter-like spacetime, we analyse conditions on the conformal boundary ensuring the existence of Killing vectors in the spacetime arising from this problem. This analysis makes use of a system of conformal wave equations describing the propagation of the Killing equation first considered by Paetz. We identify an obstruction tensor constructed from Killing vector candidate and the Cotton tensor of the conformal boundary whose vanishing is a necessary condition for the existence of Killing vectors in the spacetime. This obstruction tensor vanishes if the conformal boundary is conformally flat.