Spacetimes with a separable Klein-Gordon equation in higher dimensions

TL;DR

This paper explores higher-dimensional spacetimes that allow the Klein-Gordon equation to be separated, deriving new solutions including warped geometries and extending known metrics like Kerr-NUT-(A)dS.

Contribution

It introduces a higher-dimensional metric ansatz inspired by four-dimensional work, finds solutions to the Klein-Gordon and Einstein equations, and discovers new warped geometry solutions.

Findings

Recovered Kerr-NUT-(A)dS spacetime as a solution.

Derived Einstein-Kähler metrics with Euclidean signature.

Found new solutions in warped geometries with separable Klein-Gordon equations.

Abstract

We study spacetimes that lead to a separable Klein-Gordon equation in a general dimension. We introduce an ansatz for the metric in higher dimensions motivated by analogical work by Carter in four dimensions and find solutions of the Klein-Gordon equation. For such a metric we solve the Einstein equations and regain the Kerr-NUT-(A)dS spacetime as one of our results. Other solutions lead to the Einstein-K\"ahler metric of a Euclidean signature. Next we investigate a warped geometry of two Klein-Gordon separable spaces with a properly chosen warped factor. We show that the resulting metric leads also to a separable Klein--Gordon equation and we find the corresponding solutions. Finally, we solve the Einstein equations for the warped geometry and obtain new solutions.