Generating Generalized $G_{D-2}$ solutions

TL;DR

This paper presents a systematic method to generate higher-dimensional vacuum solutions to Einstein's equations with multiple symmetries, extending known solutions and analyzing their geometric structures.

Contribution

It introduces a new construction technique using 4D Einstein-scalar seed solutions to produce $G_{D-2}$ solutions in higher dimensions, including cases with timelike Killing vectors.

Findings

Constructed explicit higher-dimensional solutions with specified symmetries.

Generalized known stationary solutions to higher dimensions.

Analyzed rod structures and horizon features of the solutions.

Abstract

We show how one can systematically construct vacuum solutions to Einstein field equations with $D-2$ commuting Killing vectors in $D>4$ dimensions. The construction uses Einstein-scalar field seed solutions in 4 dimensions and is performed both for the case when all the Killing directions are spacelike, as well as when one of the Killing vectors is timelike. The later case corresponds to generalizations of stationary axially symmetric solutions to higher dimensions. Some examples representing generalizations of known higher dimensional stationary solutions are discussed in terms of their rod structure and horizon locations and deformations.