Orthogonal black di-ring solution

Abstract

We construct a five dimensional exact solution of the orthogonal black di-ring which has two black rings whose $S^1$-rotating planes are orthogonal. This solution has four free parameters which represent radii of and speeds of $S^1$-rotation of the black rings. We use the inverse scattering method. This method needs the seed metric. We also present a systematic method how to construct a seed metric. Using this method, we can probably construct other solutions having many black rings on the two orthogonal planes with or without a black hole at the center.

Figures (5)

• Figure 1: the rod structure of the orthogonal black di-ring solution we want.
• Figure 2: the rod structure of the seed metric of the orthogonal black di-ring the solid (dashed) lines means positive (negative) density rods.
• Figure 3: General rod structure after the transformation. Although this solution has the singularities at $z=a_1$ and $z=a_7$, we can remove these singularities if we choose the appropriate parameters.
• Figure 4: The rod structure of the orthogonal black di-ring, which is obtained by removing the singularities by fixing $c$ and $b$. The vectors accompanied with rods are such defined in Ref. Harmark.
• Figure 5: The expected rod structure of the seed metric of the many black rings solution.