For situations in which gravity is very weak, general relativity and Newton’s theory of gravity lead to very similar predictions for the motion of bodies (e.g. the planets in our solar system) and the propagation of light. Such situations can be described by starting out with the Newtonian description and then, step by step, adding correction terms that take into account the effects of general relativity. The post-Newtonian formalism is a method for performing those step-by-step corrections. As the correction terms are ordered in a systematic way (the largest effects are called “of first post-Newtonian order, 1pN”, the next smallest ones of second order, and so on), the progression of ever smaller corrections is also called the post-Newtonian expansion.
The post-Newtonian formalism is also crucial to describe relativistic effects in binary systems (binary neutron stars or black holes). It therefore plays a key role in the direct detection of gravitational waves: Based on the post-Newtonian formalism, various possible gravitational waveforms are modeled and used for searching the detector data for signals. The post-Newtonian formalism is a vital ingredient for all waveform models, but is particularly important for binary neutron stars. For those systems, most of the observed signal is in the regime where post-Newtonian terms are dominant. It was thanks to a post-Newtonian model that gravitational waves from the merger of two orbiting neutron stars were detected for the first time in 2017.