Two-body escape/merger threshold conjecture for prescribed ADM parameters

Let $(\mathbf{E}_1,\mathbf{P}_1,\mathbf{J}_1)$ and $(\mathbf{E}_2,\mathbf{P}_2,\mathbf{J}_2)$ lie in $\mathbb{R}_+\times\mathbb{R}^3\times\mathbb{R}^3$, with $\mathbf{E}_i>|\mathbf{P}_i|$, and set

$$m_i:=\sqrt{\mathbf{E}_i^2-|\mathbf{P}_i|^2}.$$

Assume $|\mathbf{J}_i|<m_i^2$ for $i=1,2$, the linear momenta are collinear and oppositely directed, and let $d_{12}>0$ be the Euclidean separation between the two centers in the constructed initial data $(\mathbb{R}^3,g,k)$. Two-body escape/merger threshold conjecture. There exists a universal constant $\kappa>0$ such that, for $d_{12}$ large compared to the gluing scales and short-pulse parameters: (i) if

$$d_{12}\geq\kappa\,\frac{m_1m_2}{(\mathbf{E}_1+\mathbf{E}_2)-(m_1+m_2)},$$

then the maximal future development contains two disjoint black hole regions, each asymptotic to a Kerr spacetime with ADM parameters close to $(\mathbf{E}_i,\mathbf{P}_i,\mathbf{J}_i)$; and (ii) if

$$d_{12}\leq\kappa^{-1}\,\frac{m_1m_2}{(\mathbf{E}_1+\mathbf{E}_2)-(m_1+m_2)},$$

then the future event horizon is connected. This conjecture predicts a sharp qualitative dichotomy between escape into two separate Kerr-like black holes and merger into a spacetime with connected event horizon, in a regime where the long-time evolution is not established by the initial-data construction.