This paper presents a new, systematic way to compute how two black holes scatter under Einstein's theory of gravity. The authors derive a set of recursion relations that build higher-order pieces of the classical interaction from lower-order ones. As a demonstration, they solve these equations through the third post‑Minkowskian order — that is, through terms proportional to the third power of the gravitational coupling constant G — and recover the known result, including the effects of radiation on the motion.

A recursion relation is a rule that expresses complicated quantities in terms of simpler, previously computed ones. Here the idea is to treat the interaction perturbatively: expand the motion and fields in powers of G (the constant that sets the strength of gravity) and use the recursion to generate the next order from what is already known. “Post‑Minkowskian” means this expansion is in powers of G, not in powers of velocity; it is a way to probe gravitational effects without assuming slow motion.

Concretely, the authors write down equations that link successive orders in the expansion and then solve those equations up to third order in G. The solution at that order matches the correct scattering result reported in the literature. Importantly, their calculation includes back‑reaction from gravitational radiation — the small changes in the black holes’ trajectories caused by energy and momentum carried away by gravitational waves.

This work matters because it gives a compact, repeatable tool for doing perturbative calculations in classical gravity. Recursion relations can reduce the algebraic work needed at each order and make it easier to push calculations to higher precision. Such perturbative results are useful as checks on other methods and can inform the theory of how compact objects interact at distances where a series in G is meaningful.