Revisiting the Aretakis constants and instability in two-dimensional Anti-de Sitter spacetimes

TL;DR

This paper analyzes the Aretakis constants and instability for massive Klein-Gordon fields in two-dimensional Anti-de Sitter spacetime, revealing their geometric origin and demonstrating their universal presence on all null hypersurfaces.

Contribution

It provides a geometric interpretation of Aretakis constants and instability in $AdS_2$, linking them to conformal Killing tensors and ladder operators, and shows their universality across null hypersurfaces.

Findings

Aretakis constants are related to conformal Killing tensors.

Instability occurs on all null hypersurfaces, not just the horizon.

Aretakis constants can be derived from conformal symmetry ladder operators.

Abstract

We discuss dynamics of massive Klein-Gordon fields in two-dimensional Anti-de Sitter spacetimes ($AdS_2$), in particular conserved quantities and non-modal instability on the future Poincar\'e horizon called, respectively, the Aretakis constants and the Aretakis instability. We find out the geometrical meaning of the Aretakis constants and instability in a parallel-transported frame along a null geodesic, i.e., some components of the higher-order covariant derivatives of the field in the parallel-transported frame are constant or unbounded at the late time, respectively. Because $AdS_2$ is maximally symmetric, any null hypersurfaces have the same geometrical properties. Thus, if we prepare parallel-transported frames along any null hypersurfaces, we can show that the same instability emerges not only on the future Poincar\'e horizon but also on any null hypersurfaces. This implies that the Aretakis instability in $AdS_2$ is the result of singular behaviors of the higher-order covariant derivatives of the fields on the whole $AdS$ infinity, rather than a blow-up on a specific null hypersurface. It is also discussed that the Aretakis constants and instability are related to the conformal Killing tensors. We further explicitly demonstrate that the Aretakis constants can be derived from ladder operators constructed from the spacetime conformal symmetry.