Gravity between two masses beyond Newton's law

Abstract

Newton's law of universal gravitation describes the attraction between two masses through an inverse-distance potential. In Einstein's theory gravity gravitates: each mass reshapes the spacetime geometry through which the other acts. More than a century later, extracting the two-body interaction from this coupled geometry remains a fundamental analytical challenge. Even without motion or radiation, the static sector of the interaction harbours an infinite hierarchy of field couplings: the purest laboratory for gravitational non-linearity. Here we determine the static interaction from ninth to twelfth order in Newton's constant for arbitrary mass ratios, and derive a closed expression for the first backreaction beyond the Schwarzschild test-body limit to all orders in the gravitational coupling. We achieve both advances by formulating the problem around the stationary interaction energy. An exact one-body background resums one source's self-interaction, stationarity reduces the fields required from the other, and integrated response identities reveal cancellations across the coupled modes. Complementary expansions then resolve the complete mass dependence at finite order. Together, these results bridge the gap between an infinitesimal probe and comparable masses, exposing hidden analytical simplicity within Einstein's theory and providing essential building blocks for modelling orbital dynamics and gravitational-wave generation.