Rotating spacetimes with a free scalar field in four and five dimensions

TL;DR

This paper constructs explicit rotating solutions in four and five-dimensional Einstein gravity with a free scalar field, extending known solutions and providing new higher-dimensional geometries with scalar hair and multiple angular momenta.

Contribution

It introduces a method to derive rotating scalar field solutions in four and five dimensions, including new higher-dimensional geometries with multiple angular momenta and scalar hair.

Findings

Derived four-dimensional rotating scalar solutions.

Found five-dimensional solutions with two angular momenta.

Extended static solutions to include scalar multipolar hair.

Abstract

We construct explicit rotating solutions in Einstein's theory of relativity with a minimally coupled free scalar field rederiving and finding solutions in four or five spacetime dimensions. These spacetimes describe, in particular, the back-reaction of a free scalar field evolving in a Kerr spacetime. Adapting the general integrability result obtained many years ago from Eri\c{s}-G\"{u}rses to simpler spherical coordinates, we present a method for rederiving the four-dimensional Bogush-Gal'tsov solution. Furthermore, we find the five-dimensional spacetime featuring a free scalar with two distinct angular momenta. In the static limit, these five-dimensional geometries provide higher-dimensional extensions of the Zipoy-Voorhees spacetime. Last but not least, we obtain the four-dimensional version of a Kerr-Newman-NUT spacetime endowed with a free scalar, where the scalar field's radial profile is extended to incorporate dependence on the polar angular coordinate. Our results offer a comprehensive analysis of several recently proposed four-dimensional static solutions with scalar multipolar hair, representing a unified study of spacetimes with a free scalar field in both four and five dimensions under the general integrability result of Eri\c{s}-G\"{u}rses.