Classical Klein-Gordon solutions, symplectic structures and isometry actions on AdS spacetimes

TL;DR

This paper classifies Klein-Gordon solutions on AdS spacetimes across various regions, analyzes their symplectic structures, and examines invariance under AdS isometries, with comparisons to Minkowski spacetime.

Contribution

It provides a complete characterization of Klein-Gordon solutions and symplectic structures on AdS regions, including boundary data and isometry actions, which was not previously detailed.

Findings

Explicit solutions for Klein-Gordon equation on AdS regions.

Symplectic structures are invariant under AdS isometries.

One-to-one correspondence between initial data and solutions.

Abstract

We study classical, real Klein-Gordon theory on Lorentzian Anti de Sitter (AdS_{1,d}) spacetimes with spatial dimension d. We give a complete list of well defined and bounded Klein-Gordon solutions for three types of regions on AdS: slice (time interval times all of space), rod hypercylinder (all of time times solid ball in space), and tube hypercylinder (all of time times solid shell in space). Hypercylinder regions are of natural interest for AdS since the neighborhood of the AdS-boundary is a tube. For the solution spaces of our regions we find the actions induced by the AdS isometry group SO(2,d). For all three regions we find one-to-one correspondences between initial data and solutions on the regions. For rod and tube regions this initial data can also be given on the AdS boundary. We calculate symplectic structures associated to the solution spaces, and show their invariance under the isometry actions. We compare our results to the corresponding expressions for (3+1)-dimensional Minkowski spacetime, arising from AdS_{1,3} in the limit of large curvature radius.