Binary black hole merging condition

Question

Assuming two black holes with the same rest mass $m$ collid coming from infinity with velocity $v$ and impact parameter $b$. Lets ignore spin at first. For which values of $v$ and $b$ would these holes scatter or merge?

Using $G=c=m=1$ one unit of length would be half of the Schwarzschild radius from one of the black holes. Therefore for $b<4$ the holes would clearly merge because the event horizons will overlap. But what would happen for greater values of $b$ is there some simple merging condition?

Answer 1

The critical impact parameter $b_{\rm crit}(v)$ depends on the highly non-linear general relativistic interaction of the two black holes. The only way we currently have to address this question is with numerical relativity simulations on super computers. I.e. there is no simple expression for $b_{\rm crit}(v)$

Some simple things we can easily infer:

• $b_{\rm crit}(v)$ is a monotonically decreasing function of $v$
• $b_{\rm crit}(v)$ diverges as $1/v$ for small v.
• $b_{\rm crit}(1) > 3\sqrt{3}GM/c^2$, where $M=2m$.