Guillaume Faye (Institut d'Astrophysique de Paris)
The post-Newtonian (PN) approximation, as a small-velocity perturbation scheme, is suitable to describe the inspiraling motion of two compact objects in general relativity. Indeed, at that stage of the binary evolution, the orbital velocity remains moderate and the separation of the bodies is still large compared with their radius, so that they may safely be regarded as point particles. Far enough from the system, the approximation beaks down due to the finite speed of propagation effects. However, a general multipolar post-linear expansion of the metric can be obtained by solving iteratively the Einstein field equations outside the source in harmonic gauge. Inside the source, or more precisely in the so-called near zone, the metric is constructed from the same equations by means of the PN iteration in the presence of matter. All originally unspecified functions entering the expressions of the gravitational fields are fixed by imposing that those two expressions should match in their common domain of validity.
Based on this method, the gravitational-wave signal produced by a compact binary has been determined up to the third PN order (3PN). Spin contributions have also been computed at a comparable level while only the first finite-size corrections have been included. Now, to build unbiased analytic waveforms and allow for more accurate comparisons with numerical simulations, the dynamics must be controlled at even higher orders. We shall thus report on our recent works about the 4PN equations of motions, showing how to construct a Fokker Lagrangian for two point masses by combining dimensional regularisation and renormalisation techniques. Special attention will be paid to the treatment of the tail potential, which is the near-zone manifestation, in the Fokker action, of the non-linear wave tails associated with the violation of the Huygens's principle in general relativity. The matching procedure plays a crucial role in our approach.