New construction of a vacuum doubly rotating black ring by the Ehlers transformation

Using the Ehlers transformation, we derive an exact solution for a doubly rotating black ring in five-dimensional vacuum Einstein theory. It is well-known that the vacuum Einstein theory with three commuting Killing vector fields can be reduced to a non-linear sigma model with $SL(3,{\mathbb R})$ target space symmetry. As shown previously by Giusto and Saxena, the $SO(2,1)$ subgroup in the $SL(3,{\mathbb R})$ can generate a rotating solution from a static solution while preserving asymptotic flatness. This so-called Ehlers transformation actually transforms the five-dimensional Schwarzschild black hole into the five-dimensional Myers-Perry black hole. However, unlike the case with the black hole, applying this method directly to the static black ring or the Emparan-Reall black ring, does not yield a regular rotating black ring due to the emergence of a Dirac-Misner string singularity. To solve this undesirable issue, we use a singular vacuum solution of a rotating black ring/lens that already possesses a Dirac-Misner string singularity as the seed solution for the Ehlers transformation. The resulting solution is regular, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that this solution obtained by the Ehlers transformation coincides precisely with the Pomeransky-Sen'kov solution. We expect that applying this method to other theories may lead to the finding of new exact solutions, such as solutions for black lenses and capped black holes, as well as black ring configurations.