# Binary black hole merging condition

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?

• FYI: The trajectories will not be as you drew them, with straight segments and a bend in the middle. Each BH will follow a smooth hyperbolic path with the common center of mass of the two BH system as one of the hyperbola's foci. (That means that your parameter, $b$ is not defined as simply as what your diagram shows.) Commented Jun 25 at 12:25
• Commented Jun 25 at 12:56
• If they are close enough to merge, they are close enough for general relativity and gravitational radiation to be important. The paths would be an inspiral if just close enough, and not be hyperbolic for a near miss. Commented Jun 25 at 13:24
• @SolomonSlow I just draw it for clarification the impact parameter b is the seperation of the BHs for t->-inf. Commented Jun 25 at 13:51
• @mmesser314 What if the starting velocity is high enough? Won't the paths be hyperbolic even though the holes nearly miss each other. Clearly the emitted radiation won't be as high as for mergers but it may be measurable in the future. Commented Jun 25 at 13:59

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 $$b_{\rm crit}(v)$$
• $$b_{\rm crit}(v)$$ 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 $$M=2m$$.