Mode solutions of the Klein-Gordon equation in warped spacetimes

TL;DR

This paper generalizes mode solutions of the Klein-Gordon equation in warped spacetimes, enabling variable separation and conservation laws, with potential extensions to curvature coupling in specific geometries.

Contribution

It introduces a new framework for mode solutions in warped spacetimes, extending known methods from Robertson-Walker universes to more general settings.

Findings

Mode solutions are well-defined in warped spacetimes.

A conserved Gordon current is established for these solutions.

Orthogonality of modes is demonstrated in Lorentzian factor-manifolds.

Abstract

In order to reduce the Klein-Gordon equation (with minimal coupling), we introduce a generalization of the so-called "mode solutions" that are well-known in the special case of a Robertson-Walker universe. After separation of the variables, we end up with a partial differential equation in lower dimension. A reduced version of the Gordon current arises and is conserved. When the first factor-manifold is Lorentzian, distinct modes appear as mutually orthogonal in the sense of the sesquilinear form obtained from the customary Gordon current. Moreover, a sesquilinear form is defined on the space of solutions to the reduced equation. Extension of this picture to curvature coupling is possible when the second factor-manifold has a constant scalar curvature.