# Collapse of two large black holes in AdS

In $4d$ flat space, two black holes of mass $M$ can collapse to form another one of (roughly) mass $2M$. This process is spontaneous, as reflected by the fact that the black hole entropy $S=M^2$ satisfies $S(2M)>2S(M)$.

In $4d$ AdS, this continues to hold for small enough black holes. However, for black holes larger than the AdS radius, the entropy grows as $M^{2/3}$. As a result, $S(2M)$ is actually lower than $2 S(M)$ for large $M$.

On the other hand, I would expect that throwing two black holes of mass $M$ in AdS against each other would yield a black hole of mass $2M$, since the AdS “reflecting” boundary conditions mean all the energy sent to infinity during the collision eventually gets back to the black hole.

There seems to be a contradiction, so where is the faulty assumption(s) in the reasoning above? Is it somehow not possible to have two large AdS black holes far away from each other at the same time, so that entropy is approximately additive? Perhaps one cannot simply add up these masses, so that this two-black hole system actually has a mass lower than $2M$?

Thanks!