Description of Non-Spherical Black Holes in 5D Einstein Gravity via the Riemann-Hilbert Problem

Abstract

We investigate the solution-generating technique based on the Breitenlohner-Maison (BM) linear system, for asymptotically flat, stationary, bi-axisymmetric black hole solutions with various horizon topologies in five-dimensional vacuum Einstein theory. We construct the monodromy matrix associated with the BM linear system, which provides a unified framework for describing three distinct asymptotically flat, vacuum black hole solutions with a single angular momentum in five dimensions, each with a different horizon topology: (i) the singly rotating Myers-Perry black hole, (ii) the Emparan-Reall black ring, and (iii) the Chen-Teo rotating black lens. Conversely, by solving the corresponding Riemann-Hilbert problem using the procedure developed by Katsimpouri et al., we demonstrate that factorization of the monodromy matrix exactly reproduces these vacuum solutions, thereby reconstructing the three geometries. These constitute the first explicit examples in which the factorization procedure has been carried out for black holes with non-spherical horizon topologies. In addition, we discuss how the asymptotic behavior of five-dimensional vacuum solutions at spatial infinity is reflected in the asymptotic structure of the monodromy matrix in the spectral parameter space.

Figures (3)

• Figure 1: Rod diagram for 5D Myers-Perry solution. The positions of the intersection points of rod vectors are $w_1=-\frac{1}{2}\alpha\,, w_2=\frac{1}{2}\alpha$ with $\alpha>0$.
• Figure 2: The rod diagram for the 5D Emparan-Reall black ring. The corner points $w_i$ satisfy $w_1<w_2<w_3$.
• Figure 3: Rod diagram for 5D rotating black lens. The corner points satisfy $w_1<w_2<w_3$.