The two body problem in general relativity is not exactly solvable. A viable strategy for weakly gravitating systems is to perform a (so-called post-Newtonian) perturbative expansion in terms of the small parameter G (m1+m2) / r. Nevertheless, going up in the perturbative series becomes encreasingly difficult, and the energy of the two-body system is presently known only at the third post-Newtonian order.
In this paper we reproduce precisely this quatity by using the effective field theory methods for the gravitationally bound two-body system proposed by Goldberger and Rothstein. This result has been obtained by automatizing the computation of Feynman amplitudes within a Mathematica algorithm, paving the way for higher-order computations not yet performed by traditional methods.