On the Klein-Gordon equation near a De Sitter brane in an Anti-de Sitter bulk

TL;DR

This paper analyzes the Klein-Gordon equation near a De Sitter brane within an Anti-de Sitter bulk, demonstrating the stability of gravitational fluctuations and expressing solutions as a superposition of free fields.

Contribution

It provides a global solution framework for the Klein-Gordon equation in this setting and shows the De Sitter brane's linear stability through a Kaluza-Klein decomposition.

Findings

Solutions can be expressed as a superposition of free fields.

The leading gravitational fluctuation is a massless graviton.

The De Sitter brane is linearly stable.

Abstract

In this paper we investigate the Klein-Gordon equation in the past causal domain of a De Sitter brane imbedded in a Anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza-Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviours. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the De Sitter brane is linearly stable.