Charged five-dimensional geodesic polynomials

Determine whether the separated radial and polar equations for charged test-particle motion in the charged five-dimensional Kerr–(A)dS generalization in minimal gauged supergravity retain the form of Eq. (PQode), with both orbital polynomials having degree at most four.

Background

The paper establishes closed-form geodesic solutions for the uncharged five-dimensional Kerr–(A)dS black hole by reducing the radial and polar equations to polynomial equations whose degrees are at most four. It then considers the charged generalization known in minimal five-dimensional gauged supergravity and asks whether the same polynomial structure survives the inclusion of charge.

The question is motivated by the supersymmetric Gutowski–Reall solution, for which prior work reportedly finds a quartic radial polynomial and a quadratic polar polynomial. Whether this degree bound persists for the general charged family, including unequal rotations and non-supersymmetric solutions, is left unresolved in the paper.

References

Finally there is the charged case, where part of the answer is already in. A charged generalization of the present spacetime is known in minimal five-dimensional gauged supergravity, and the question raised here is whether eq:PQode survives the addition of charge with both polynomials still of degree at most four.

— Geodesics of the five-dimensional Kerr-(A)dS black hole in closed form  (2610.26242 - Sarıoğlu et al., 19 Oct 2026) in Section 8, Conclusions and outlook