Isolated black holes without $\mathbb{Z}_2$ isometry

Understand

A mechanism to construct asymptotically flat, isolated, stationary black hole (BH) spacetimes with no $\mathbb{Z}_2$ (No$\mathbb{Z}$) isometry is described.

• In particular, the horizon geometry of such No$\mathbb{Z}$ BHs does not have the usual north-south (reflection) symmetry.
• We discuss two explicit families of models wherein No$\mathbb{Z}$ BHs arise.
• In one of these families, we exhibit the intrinsic horizon geometry of an illustrative example by isometrically embedding it in Euclidean 3-space, resulting in an "egg-like" shaped horizon.