Analytical expressions for Einstein–Scalar black ring solutions

Ascertain whether closed-form analytical solutions exist for the five-dimensional Einstein equations with a minimally coupled real massless scalar field that describe stationary and axisymmetric black rings (generalizing the vacuum black ring), and, if they exist, construct such solutions in a suitable coordinate system.

Background

The vacuum black ring is a five-dimensional stationary and axisymmetric solution, suggesting that generalizations with a minimally coupled scalar field might be accessible via extensions of the Eriş–Gürses framework used in the paper. However, known representations of the black ring typically employ prolate coordinates, complicating direct application of the method.

The authors explicitly state uncertainty about the feasibility of obtaining analytic closed-form expressions for such scalar-deformed black rings, highlighting coordinate-related obstacles. Resolving this would clarify whether an explicit Einstein–Scalar black ring can be written in closed form.