Monodromy-Matrix Description of Doubly Rotating Black Rings

TL;DR

This work extends the Breitenlohner-Maison linear system to five-dimensional vacuum gravity with two independent angular momenta, framing the problem within an $SO(4,4)$ coset and introducing a monodromy-matrix approach that encodes black-hole data in meromorphic functions of the spectral parameter $w$. By constructing and factorizing explicitly the monodromy matrices for both the doubly rotating Myers-Perry black hole and the unbalanced Pomeranski-Sen'kov black ring, the authors demonstrate a direct read-off of spacetime geometry from $\mathcal{M}(w)$ via a BM-Riemann-Hilbert problem, thereby reconstructing the full metrics from purely algebraic data. The extremal MP case exhibits a nilpotent charge matrix and a compact exponential form for the monodromy, whereas the PS ring’s extremal limit does not rely on nilpotency, highlighting a structural difference between these two topologies. The paper further analyzes a network of limits, including balanced PS, ER, MIF, and MP, showing how pole-structure degenerations reproduce known solutions and elucidate relations among seemingly distinct five-dimensional black holes and rings. Overall, the BM framework provides a unified, algebraic pathway to generate and classify higher-dimensional rotating black holes and lays the groundwork for constructing more intricate configurations such as multi-ring or multi-center systems through carefully chosen monodromy data.

Abstract

Extending the single-angular-momentum case analyzed in our previous work, we investigate the solution-generating technique based on the Breitenlohner-Maison (BM) linear system for asymptotically flat, stationary, bi-axisymmetric black hole solutions with two angular momenta in five-dimensional vacuum Einstein theory. In particular, we construct the monodromy matrix associated with the BM linear system for the doubly rotating Myers-Perry black holes and the Pomeransky-Sen'kov black rings. Conversely, by solving the corresponding Riemann-Hilbert problem using the procedure developed by Katsimpouri et al., we demonstrate that the factorization of the monodromy matrix precisely reproduces these vacuum solutions, thereby reconstructing both geometries.

Figures (3)

• Figure 1: Rod diagram for 5D Myers-Perry solution. The positions are $w_1=-\frac{1}{2}\alpha\,, w_2=\frac{1}{2}\alpha$ with $\alpha>0$.
• Figure 2: Rod diagram for the 5D unbalanced Pomeransky-Sen’kov black ring. The intersection points $w_i$ of rods satisfy $w_1<w_2<w_3$.
• Figure 3: Degenerate limits of the unbalanced Pomeransky-Sen’kov black ring. Through various limiting procedures, the unbalanced Pomeransky-Sen’kov (PS) black ring reduces to the balanced PS black ring, extremal PS black ring, non-extremal and extremal Myers-Perry (MP) black holes, Emperan-Reall (ER) black ring, and Mishima-Iguchi-Figueras (MIF) black ring.