Historically, studying the motion of astrophysical object has lead to big advance, from the prediction of the existence of Neptune (predicted in 1843 and observed 1846), to the first success of GR (Mercury’s perihelion) and the prediction of a supermassive BH candidate in the center of galaxies.
While the two (or more) body problem in Newtonian gravity is tractable exactly, in GR, this is far from being the case. Only about a decade ago, it has been possible to simulate long time stable evolution for a black hole binary. This is part of the reason why approximate techniques such as the Post-Newtonian formalism have been developed. Here we give a brief overview of the principle of the PN formalism and of the Effective field theory approach to gravitational astrophysics.
Effective Field Theory and Post-Newtonian approximation – EFT  is a formalism commonly used in quantum field theory. It consists in giving an effective description of some parts of a full – fundamental – theory, by essentially truncating the theory in a mathematically consistent way. More precisely, the idea is typically to integrate out the ultra violet regime, i.e. the small scales out of the theory, leaving a low energy effective model, that can be studied on its own right. In some cases where the fundamental theory is too complex for this procedure to be carried out explicitly, an effective model is built from the symmetries of the theory, in this case, there are unknown couplings that need to be matched to the full model. In the astrophysical context, this is exactly the same principle: the compact objects, including their fine details like internal structure are effectively modeled by point particles . Finite size effects are incorporated in multipolar degrees of freedom propagating along the objects’ world-lines . There has been considerable amount of work relying on the EFT approach; for example the energy momentum up to the quadrupolar approximation is given in , inclusion of dissipative effects in the black hole dynamics has been considered in . As all EFTs, the approach has a limit of validity and breaks down once the system probes smaller scales, for example during the merger in a binary system.
Analytic or semi analytic treatments of the relativistic two body problem have been studied within the PN approximation , perturbatively or in the self-force context . These methods are complementary to numerical approaches, that are still computationally expensive. Here we will focus on the PN approach. The first work using the principle of the PN approximation was performed by Einstein in order to predict the perihelion advance of Mercury and formalized by Drost in 1918, where the equations of motion were derived up to 1 PN order (first order in 1/c²). In 1938 Einstein, Infeld and Hoffmann justified the point particle approximation and Chandrasekhar incorporated fluids in the sources in 1960. The rebirth of the PN formalism is due to Damour and collaborators in the early 80’s , and led to the 2.5 PN equations of motion (1/c5). Blanchet (from IAP) extended and systematized the construction of higher PN equations of motion in harmonic gauge and developed the formalism for gravitational waveform extraction . The PN framework provides a modeling of the evolution of binary systems, but also allows to reconstruct the metric, in turn allowing to extract the waveform and all geometrical quantities. This is very important for testing the theories against astrophysical observations.
Binary systems: a dance-floor for gravity – In a binary system, the main channel of interaction is gravity, through tidal interactions. Tidal forces have been well understood in Newton’s gravity for more than a century . The so-called Love numbers – named after Augustus Edward Hough Love – encode how much the body responds to a tidal external perturbation. In the relativistic case, they are better understood in the adiabatic limit, i.e. when the deformation is aligned with the perturbing source. Relativistic tidal interactions of compact objects were studied by Damour and Nagar , and Binnington and Poisson  in the effective action (resp. EFT) context and in the adiabatic limit. The perturbing field was chosen to be static, which means that the time scale of the response of the compact object to the perturbation is much shorter than the time scale of the perturbation itself. Among the results, it turns out that tidal coefficients (Love numbers) of stationary black holes vanish in 4 dimensions. This is in fact an accident, in arbitrary number of dimensions and depending on the angular momentum, the Love numbers can be divergent . They can be renormalized using standard regularization schemes, since all usual tools of statistical field theory apply to effective field theory. Note that relativistic tidal coefficients including rotation are presently unknown. Observations of binary systems involving compact objects provides invaluable informations, see the case of the historical Hulse-Taylor binary pulsar.