The small mass-ratio expansion of the relativistic two body problem

To predict the gravitational waves emitted by a black hole binary, one needs to understand the dynamics of the binary in general relativity. No closed form solutions of this problem exist. Instead one must introduce some form of approximation. One such approximation, can be made if one of the components is much heavier than the other, suggesting a perturbative expansion in the mass-ratio. I will review this small mass-ratio (SMR) expansion of the dynamics, and the progress that has been made over the last two decades. In particular, I will discuss some recent results on the convergence of this series, suggesting that at relatively low orders this SMR expansion can be used to model even equal mass binaries.