Conservative dynamics of two body systems at the fourth post Newtonian approximation of general relativity
The fourth post Newtonian (4PN) two body dynamics has been recently tackled by several different approaches: effective field theory, Arnowitt Deser Misner Hamiltonian, action angle Delaunay averaging, effective one body, gravitational self force, first law of dynamics, and Fokker action. We review the achievements of these approaches and discuss the complementarity of their results. Our main conclusions are: (i) the results of the first complete derivation of the 4PN dynamics [T.Damour, P.Jaranowski, and G.Sch\"afer, Phys. Rev. D 89, 064058 (2014) have been, piecewise, fully confirmed by several subsequent works; (ii) the results of the Delaunay averaging technique [T.Damour, P.Jaranowski, and G.Sch\"afer, Phys. Rev. D 91, 084024 (2015) have been confirmed by several independent works; and (iii) several claims in a recent Fokker action computation [L.Bernard et al. arXiv:1512.02876v2 [gr qc] are incorrect, but can be corrected by the addition of a couple of ambiguity parameters linked to subtleties in the regularization of infra red and ultra violet divergences.