Spacetimes with a separable Klein-Gordon equation in higher dimensions

We study spacetimes that lead to a separable Klein-Gordon equation in a general number of dimensions. 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-Kähler 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.